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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>Casati, Paolo</author> <title>Mechanica, old version (815 pages)</title> <date>1684</date> <place>Lyon</place> <translator></translator> <lang>la</lang> <cvs_file>casat_mecha_017_la_1684.xml</cvs_file> <cvs_version></cvs_version> <locator>017.xml</locator> <echodir>/permanent/archimedes/casat_mecha_017_la_1684</echodir> </info> <text> <front> </front> <body> <chap> <pb xlink:href="017/01/001.jpg"></pb> <p type="head"> <s id="s.000001"><emph type="center"></emph>R. P. PAULI <lb></lb>CASATI <lb></lb>MECHANICA.<emph.end type="center"></emph.end></s> </p> <pb xlink:href="017/01/002.jpg"></pb> <pb xlink:href="017/01/003.jpg"></pb> <p type="head"> <s id="s.000002"><emph type="center"></emph>R. P. PAULI <lb></lb>CASATI <lb></lb>PLACENTINI <lb></lb>SOCIET. JESU <lb></lb>MECHANICORUM <lb></lb>LIBRI OCTO, <lb></lb>IN QUIBUS UNO EODEMQUE<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000003"><emph type="center"></emph>principio Vectis vires Phyſicè explicantur & Geometricè <lb></lb>demonſtrantur,<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000004"><emph type="center"></emph><emph type="italics"></emph>Atque Machinarum omnis generis componendarum methodus <lb></lb>proponitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <figure id="id.017.01.003.1.jpg" xlink:href="017/01/003/1.jpg"></figure> <p type="head"> <s id="s.000005"><emph type="center"></emph>LUGDUNI, <lb></lb>Apud ANISSONIOS, JOAN, POSUEL <lb></lb>& CLAUDIUM RIGAUD.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000006"><emph type="center"></emph><emph type="italics"></emph>M. </s> <s id="s.000007">D C. LXXXIV.<emph.end type="italics"></emph.end><lb></lb>CUM PRIVILEGIO REGIS.<emph.end type="center"></emph.end></s> </p> <pb xlink:href="017/01/004.jpg"></pb> <figure id="id.017.01.004.1.jpg" xlink:href="017/01/004/1.jpg"></figure> <pb xlink:href="017/01/005.jpg"></pb> <p type="head"> <s id="s.000008"><emph type="center"></emph>CHRISTIANISSIMO <lb></lb>GALLIARUM <lb></lb>ET NAVARRÆ REGI <lb></lb>LUDOVICO <lb></lb>MAGNO.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000009"><emph type="italics"></emph>AD Majeſtatis Tuæ pedes,<emph.end type="italics"></emph.end><lb></lb>REX INVICTISSIME, <lb></lb><emph type="italics"></emph>me, meámque hanc de rebus <lb></lb>Mechanicis lucubrationem, <lb></lb>ignotus homo, vix fortaſſe cre<lb></lb>dibili confidentiâ, ſiſto: Sed <lb></lb>quâ Regiâ comitate omnium <lb></lb>animos concilias, eâdem ſuſtentor, ne repulſam <lb></lb>timeam. </s> <s id="s.000010">In Te Orbis univerſi conjecti ſunt oculi, <lb></lb>quos Tuæ Gloriæ ſplendor allicit: à communi feli-<emph.end type="italics"></emph.end><pb xlink:href="017/01/006.jpg"></pb><emph type="italics"></emph>citate quid me paterer excludi? </s> <s id="s.000011">Amplißima <lb></lb>Tua in Societatem noſtram merita, quorum nullam <lb></lb>partem, ne cogitandâ quidem gratiâ, conſequi <lb></lb>poſſumus, hoc ſaltem officij ab univerſo Ordine re<lb></lb>petunt, ut ſinguli, quem cordi penitißimè impreſſum <lb></lb>geſtamus non ingrati LVDOVICVM, in <lb></lb>libris palàm inſcriptum velimus. </s> <s id="s.000012">Me verò Natu<lb></lb>ræ atque Artis mutuam ſocietatem coëuntium in <lb></lb>Machinis, ferè dixerim, miracula contemplari <lb></lb>aſſuetum rapuere ad mirabundum, quæ ipſe patraſti, <lb></lb>& bello, & pace, egregia atque præclara facinora <lb></lb>non modò mirabilia, ſed prodigiis ſimilia. </s> <s id="s.000013">Neque <lb></lb>illa quidem aut ex rerum magnitudine ac difficul<lb></lb>tate, aut ex multiplicato numero, aut ex diſsimi<lb></lb>lium varietate, aut ex ſerie non interruptâ, me<lb></lb>tienda duxi, quamquam & in his admirabilitatis <lb></lb>plurimum inſit: Verùm longè omnem admirationem <lb></lb>multúmque ſuperare mihi videtur, quòd paucis <lb></lb>luſtris vel ſæcula complexus, unus pluribus Regibus <lb></lb>par, tot, tantáque perficere valuiſti. </s> <s id="s.000014">Ingentis pon<lb></lb>deris gravitatem vincit adhibita Machina, ſed <lb></lb>diuturno impulſu agitanda, ut proficiat aliquid: At <lb></lb>plurima immenſis munita difficultatibus exiguo tem<lb></lb>poris ſpatio expugnare, atque ad optatum exitum <lb></lb>perducere, ita Tuum eſt, REX INVICTISSIME, <lb></lb>ut quemadmodum rerum geſtarum gloriâ, ac nomi<lb></lb>nis celebritate, nemini ſuperiorum Regum ſecundus <emph.end type="italics"></emph.end><pb xlink:href="017/01/007.jpg"></pb><emph type="italics"></emph>prædicaris, ſic Tibi ſecundum, qui Tuis planè in<lb></lb>ſiſtat veſtigiis, ventura ſæcula ſperare vix audeant. </s> <lb></lb> <s id="s.000015">Patere igitur pro ſummâ, quâ præditus es, huma<lb></lb>nitate, qualemcumque hanc rerum Mechanicarum <lb></lb>tractationem Regio inſigniri Nomine, ut, quos <lb></lb>meas haſce commentationes legere non piguerit, <lb></lb>vel hinc diſcant, aliud eſſe non imitabile genus <lb></lb>Facultatis, quâ ingentia citò perficiantur, ſi <lb></lb>LVDOVICI MAGNI mens acceſſerit. </s> <lb></lb> <s id="s.000016">Incolumem Te diu ſervet DEVS Catholicæ Fi<lb></lb>dei incremento, Regníque Tui felicitati; audiát<lb></lb>que bonorum omnium Largitor vota, quæ pro Ma<lb></lb>jeſtate Tuâ ſupplex nuncupat<emph.end type="italics"></emph.end></s> </p> <p type="head"> <s id="s.000017"><emph type="center"></emph><emph type="italics"></emph>MAJESTATIS Tuæ<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000018">Parmæ Kal, Maij 1683. </s> </p> <p type="main"> <s id="s.000019">Humillimus atque Obſequentiſſimus <lb></lb>Servus <lb></lb>PAULUS CASATUS è SOC. JESU. <pb xlink:href="017/01/008.jpg"></pb><emph type="center"></emph><emph type="italics"></emph>Facultas R. P. Provincialis Societatis Jeſu <lb></lb>in Provincia Veneta.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000020">EGo Octavius Rubeus Societatis Jeſu in Provincia Veneta <lb></lb>Præpoſitus Provincialis, poteſtate ad id mihi factâ ab <lb></lb>Adm. </s> <s id="s.000021">R. P. N. </s> <s id="s.000022">Præpoſito Generali Jo. </s> <s id="s.000023">Paulo Oliva, faculta<lb></lb>tem facio, ut Opus inſcriptum, <emph type="italics"></emph>Mechanichorum Libri octo, <lb></lb>Authore P. Paulo Caſato Societatis Noſtræ Sacerdote,<emph.end type="italics"></emph.end> ejuſdem <lb></lb>Societatis Doctorum hominum judicio approbatum, typis <lb></lb>mandetur, ſi ita iis, ad quos pertinet, videbitur. </s> <s id="s.000024">Cujus rei <lb></lb>gratiâ has litteras meâ manu ſubſcriptas, & ſigillo officij mei <lb></lb>munitas dedi. </s> <s id="s.000025">Parmæ 23. Februarij 1681. </s> </p> <p type="main"> <s id="s.000026">OCTAVIUS RUBEUS. <lb></lb></s> </p> <p type="main"> <s id="s.000027"><emph type="center"></emph><emph type="italics"></emph>Summa Privilegy à Chriſtianiſſimo Rege conceſſi.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000028">LUDOVICUS MAGNUS Galliarum & Navarræ Rex Chriſtianiſſimus, <lb></lb>Diplomate ſuo ſanxit, nequis per univerſos Regnorum ſuorum fines <lb></lb>intra decem proximos annos à die publicationis exemplarium computandos, <lb></lb>imprimat ſeu typis excudendum curet & venale habeat Opus quod inſcribi<lb></lb>tur, <emph type="italics"></emph>Mechanicorum Libri octo, Authore R. P. Paulo Caſato Soc. </s> <s id="s.000029">Ieſu<emph.end type="italics"></emph.end>; præter <lb></lb>Aniſſonios Bibliopolas Lugdunenſes, aut illos quibus ipſimet conceſſerint. </s> <lb></lb> <s id="s.000030">Prohibuit inſuper eadem auctoritate Regia omnibus ſuis ſubditis, idem <lb></lb>Opus extra Regni ſui limites imprimendum curare, & impreſſum divende<lb></lb>re, vel quempiam ubicumque fuerit ad id agendum impellere; ac inſtigare <lb></lb>ſine conſenſu dictorum ANISSONIORUM; Qui ſecus faxit, confiſca<lb></lb>tione librorum, aliaque gravi pœnâ multabitur, uti latius patet in diplo<lb></lb>mate regio. </s> <s id="s.000031">Dabatur Verſalis die vigeſima prima Januarij anno Dom. 1684. </s> </p> <p type="main"> <s id="s.000032"><emph type="italics"></emph>Ex mandato Regis.<emph.end type="italics"></emph.end></s> </p> <p type="head"> <s id="s.000033">JUNQUIERES. </s> </p> <p type="head"> <s id="s.000034">MECHA </s> </p> <pb xlink:href="017/01/009.jpg"></pb> <figure id="id.017.01.009.1.jpg" xlink:href="017/01/009/1.jpg"></figure> <p type="head"> <s id="s.000035"><emph type="center"></emph>AD LECTOREM.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000036">SERO in lucem prodit hæc Me<lb></lb>chanicorum tractatio, & vix fide <lb></lb>me abduco, quam dedi, cùm Diſ<lb></lb>ſertationes de <emph type="italics"></emph>Terrâ Machinis motâ <emph.end type="italics"></emph.end><lb></lb>quaſi Prodromum emiſi ante plures <lb></lb>annos: ſcilicet à ſtudiis tunc abſtra<lb></lb>ctus, utpote alieni juris, & ad mu<lb></lb>nera his non affinia tranſlatus, mul<lb></lb>tam ſalutem & Mathematicis diſciplinis & Phyſicis dicere <lb></lb>coactus ſum; adeò ut demum tot elapſis annis urgente jam <lb></lb>ſenio cogitationem omnem abjecerim de hujuſmodi com<lb></lb>mentationibus, diffidens me poſſe ad hanc ſcriptionem <lb></lb>ſatis temporis invenire, quin eam proxima mors interci<lb></lb>peret, & ſuſceptum alieniſſimo tempore laborem irritum <lb></lb>faceret. </s> <s id="s.000037">Adde quòd (pro meâ negligentiâ, quæ calamo <lb></lb>parcit) temporis diuturnitate deletæ ex animo pleræque <lb></lb>imagines vix tenue veſtigium reliquerant, cui novis indu<lb></lb>ctis coloribus eas redintegrari poſſe confiderem. </s> <s id="s.000038">Amico<lb></lb>rum tamen officioſis ſtimulis me urgeri paſſus ſum, ut ſub<lb></lb>ciſivis, quæ incurrebant, temporibus tentarem, an deſti<lb></lb>natam animo tractationem, cujus brevem Synopſim au<lb></lb>ditoribus meis in Romano Collegio, anno labentis ſæculi <lb></lb>decimi ſeptimi quinquageſimo quarto, tradideram, re<lb></lb>dordiri, & aliquâ ratione perficere liceret. </s> <s id="s.000039">Licuit autem, <lb></lb>præter ſpem, toties dimiſſum calamum reſumere, ut tan-<pb xlink:href="017/01/010.jpg"></pb>dem de ſingulis Mechanicis Facultatibus aliquid me ſcrip<lb></lb>ſiſſe invenerim, quod Mathematicarum diſciplinarum can<lb></lb>didatis profuturum amici cenſuerunt, ſi publici juris fieret. </s> <lb></lb> <s id="s.000040">Quapropter alienæ utilitati ſerviendum potiùs fuit, quàm <lb></lb>meæ voluntati. </s> </p> <p type="main"> <s id="s.000041">Verùm nete moveat, Amice Lector, quòd Mechanici <lb></lb>inſcribantur libri, cùm tamen aliqua ad Centrobaryca, ali<lb></lb>qua ad Statica pertineant. </s> <s id="s.000042">Cùm enim hæc ad pleniorem <lb></lb>eorum intelligentiam, quæ de Machinis diſputanda erant, <lb></lb>referantur, nomen à ſcopo deſumendum fuit: Nec decrat <lb></lb>ex Ariſtotele (ſi tamen ipſi tribuenda ſit illa tractatio) ſuf<lb></lb>fragium, qui Mechanicas Quæſtiones inſcripſit libellum, <lb></lb>in quo non de ſolis Mechanicis facultatibus agitur. </s> </p> <p type="main"> <s id="s.000043">Methodum ne culpes, quòd non in Theoremata & <lb></lb>Propoſitiones rem totam digeſſerim, ſed in Capita diſtri<lb></lb>buerim, & quidem aliquando longiuſcula: Brevitati nimi<lb></lb>rum ſtudens non amavi codicem titulis implere, ne fortè, <lb></lb>ad oſtendendam conſequentium cum præcedentibus con<lb></lb>nexionem, cogerer idem ſæpiùs inculcare. </s> <s id="s.000044">Facilius au<lb></lb>tem duxi ea, quæ conjuncta ſunt, uno eodemque ca<lb></lb>pite complecti, ut ex ipsâ verborum conſecutione re<lb></lb>rum cognatio innoteſcat. </s> <s id="s.000045">Præterquam quod, ſi formâ <lb></lb>illâ Mathematicis familiari uſus fuiſſem, animum fortaſſe <lb></lb>induxiſſes, me mihi ineptè blandiri, & quaſi Geometri<lb></lb>cas ratiocinationes obtrudere ea, quæ ſatis probabili con<lb></lb>jecturâ ſtabilire conatus ſum. </s> <s id="s.000046">Quamvis enim non pauca <lb></lb>attulerim, quæ Geometricas demonſtrationes recipiunt, <lb></lb>nec mihi videar pſeudographis ſyllogiſmis deceptus; quia <lb></lb>tamen & apud Phyſicos & apud Mathematicos agenda <lb></lb>erat cauſa, multa fuere ad Philoſophicas rationes revocan<lb></lb>da; & quidem, quoad ejus fieri potuit, à receptis in ſcho-<pb xlink:href="017/01/011.jpg"></pb>lis opinionibus mihi non erat hìc recedendum, ne quid <lb></lb>temerè ſine argumentis proferrem, aut ne longiùs ab in<lb></lb>ſtituto recederem, ſi quid novi, quæſitâ veri ſimilitudine, <lb></lb>molirer. </s> <s id="s.000047">Hoc videlicet mihi potiſſimum curæ fuit, ut Phy<lb></lb>ſicam admirandorum per Machinas motuum cauſam in<lb></lb>veſtigarem: in Phyſicis autem modum ſciendi Geome<lb></lb>tricum inquirens, ne ab Ariſtotele redarguerer, timerem. </s> <lb></lb> <s id="s.000048">Quare alia Geometricè, alia Phyſicè tractata æquo animo <lb></lb>patere. </s> </p> <p type="main"> <s id="s.000049">Stylum autem quid excuſem? </s> <s id="s.000050">Non eſt, fateor, con<lb></lb>ſtans & perpetuus, ſuíque ſimilis: tum quia non eadem <lb></lb>ſemper ſubjecta materia eſt, tum quia, prout tempus fe<lb></lb>rebat, animum inæqualiter affectum ad ſcribendum at<lb></lb>tuli; nec poterat æquabiliter fluere toties interciſa oratio. </s> </p> <p type="main"> <s id="s.000051">Unum eſt inter cætera, quod fortaſſe deſideres, nimi<lb></lb>rum illorum, qui de hoc eodem argumento ſcripſerunt, <lb></lb>ſententias explicari, & quæ à me dicuntur, eorum autho<lb></lb>ritate muniri. </s> <s id="s.000052">Plurimum ſanè mihi lucis affulſiſſet ex do<lb></lb>ctorum virorum Commentariis, neque contemnenda or<lb></lb>namenti acceſſio hujus meæ lucubrationis tenuitati fieret ex <lb></lb>diverſis Authorum opinionibus: Verùm ut nunc resſe ha<lb></lb>bet, opportunâ librorum ſupellectile deſtitutus authorum <lb></lb>mentionem facere plenam non potui, jejunam non debui, <lb></lb>ne quis per <expan abbr="contemptũ">contemptum</expan> prætermiſſus videretur. </s> <s id="s.000053">Mihi autem <lb></lb>non ea eſt memoriæ firmitas, quæ, quid aliquando lege<lb></lb>rim, aut ubi legerim, ſatis explicatâ recordatione ſuggerat. </s> <lb></lb> <s id="s.000054">Quòd ſi placuiſſet, corrogatis aliunde libris, magnificam <lb></lb>hanc eruditionis pompam meæ qualicumque commenta<lb></lb>tioni adhibere, non ſatis otii ad legendum ſuppetebat, & <lb></lb>nimium temporis poſtulaſſet ſcriptio, ſi exponendæ pri<lb></lb>mùm, dein confirmandæ aut refellendæ fuiſſent aliorum <pb xlink:href="017/01/012.jpg"></pb>ſententiæ: propterea ſatius duxi, quæ animo occurrebant, <lb></lb>pro meâ conſuetudine breviter ſimplicitérque ſcribere, <lb></lb>vix aliquando tactâ alicujus Authoris opinione, quam in <lb></lb>adverſariis jampridem notatam inveni. </s> </p> <p type="main"> <s id="s.000055">Nec te pluribus volo, Amice Lector. </s> <s id="s.000056">Multa habebis, <lb></lb>quæ pro tuâ humanitate mihi condones, plura quæ ama<lb></lb>nuenſi, plurima fortaſſe quæ Typographo, ubi præſertim <lb></lb>de Numeris, & de Majori aut Minori Ratione ſermo eſt; <lb></lb>facilis enim contingit oſcitanti hallucinatio, ut ab Auto<lb></lb>grapho aberret exemplar, & Numerus numero, verbum <lb></lb>verbo commutetur: Non ægrè tamen ex adjunctis peti <lb></lb>poterit correctio. </s> <s id="s.000057">In iis verò, in quibus à me per impru<lb></lb>dentiam peccatum fuerit, à tuâ Sapientiâ facilè patiar me <lb></lb>dedoceri. </s> <s id="s.000058">Vale. </s> </p> <figure id="id.017.01.012.1.jpg" xlink:href="017/01/012/1.jpg"></figure> <p type="head"> <s id="s.000059">ELENCHUS </s> </p> <pb xlink:href="017/01/013.jpg"></pb> <figure id="id.017.01.013.1.jpg" xlink:href="017/01/013/1.jpg"></figure> <p type="head"> <s id="s.000060"><emph type="center"></emph>ELENCHUS CAPITUM.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000061"><emph type="center"></emph>LIBER PRIMUS. </s> <s id="s.000062">De Centro Gravitatis.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>QVid ſit Centrum Gravium & Levium.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>An corpora prædita ſint gravitate & levitate.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>Quid ſit Centrum Gravitatis, & Linea Directionis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>An gravia centro vicina minùs gravitent.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>Qua ratione Centrum gravitatis corporum inveniatur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"></emph>Affertur ratio prædictarum praxeon.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"></emph>Quomodo gravia ſponte aſcendentia deſcendant.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"></emph>Cur gravium in plano inclinato deſcendentium alia repant, alia rotentur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"></emph>Cur turres inclinatæ non corruant.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"></emph>An plurium ſtructurarum capax ſit Mons, quàm ſubjecta planities.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"></emph>Quomodo animalium motus ordinentur ex centro gravitatis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"></emph>An tellus moveatur motu trepidationis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"></emph>Qua ratione minuatur gravitatio in plano inclinato.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XIV.</cell> <cell><emph type="italics"></emph>Qua ratione corpus gravitet in planum inclinatum.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XV.</cell> <cell><emph type="italics"></emph>Inquiruntur Rationes gravitationis corporum ſuſpenſorum.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XVI.</cell> <cell><emph type="italics"></emph>Tractiones ac elevationes obliquæ expenduntur.<emph.end type="italics"></emph.end></cell> </row> </table> <p type="head"> <s id="s.000063"><emph type="center"></emph>LIBER SECUNDUS. De Cauſis Motûs Machinalis.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>QVem ad finem Machinæ inſtruantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>Impetûs motum proximè efficientis natura explicatur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>Qua ratione ſemel conceptus impetus pereat.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>Qua ratione vis movendi cum impedimentis comparetur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>In quo Machinarum vires ſita ſint.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"></emph>Quid attendendum ſit in Machinæ collocatione, at que materiæ.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"></emph>Præſtetne Machinam augere? an componere?<emph.end type="italics"></emph.end></cell> </row> <pb xlink:href="017/01/014.jpg"></pb> <row> <cell>VIII.</cell> <cell><emph type="italics"></emph>Cur majores rotæ motum juvent præ minoribus.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"></emph>Quid cylindri & Scytalæ ad faciliorem ponderis motum præſtent.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"></emph>Circulorum Concentricorum motus explicatur.<emph.end type="italics"></emph.end></cell> </row> </table> <p type="head"> <s id="s.000064"><emph type="center"></emph>LIBER TERTIUS. De Libra.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>LIbræ forma & natura exponitur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>Libræ inæqualium brachiorum expenditur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>Quomodo Corporum æquilibria explicentur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>An, & cur libra ab æquilibrio dimota ad illud redeat.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>An fieri poſſit libra Curva.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"></emph>Quanam libræ ſint omnium exactißimæ.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"></emph>Libræ doloſæ vitia reteguntur,<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"></emph>Stateræ Natura & Forma explicatur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"></emph>Antiquorum Statera examinatur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"></emph>Libræ & Stateræuſus extenditur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"></emph>Fundamenta pramittuntur ad explicandum, Cur gravia ſuſpenſa modò præponderent, modò æquilibria ſint.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"></emph>Præponderatio & Æquilibritas gravium fune ſuſpenſorum conſideratur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"></emph>An aliqua ſit Libræ Obliquæ utilitas.<emph.end type="italics"></emph.end></cell> </row> </table> <p type="head"> <s id="s.000065"><emph type="center"></emph>LIBER QUARTUS. De Vecte.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>VEctis forma & vires explicantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>Quid in hypomochlij collocatione ſit obſervandum.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>Quaratione ſtatuendus ſit Ponderi locus in Vecte primi generis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>Momenta Ponderis in Vecte ſeaundi generis conſiderantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>Quæ ſit Ratio Vectis hypomochlium mobile habentis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"></emph>Quanam ſint momenta Vectis Pondus fune connexum tra-hentis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"></emph>Quid conferat Potentiæ moventis applicatio ad Vectens.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"></emph>Oneris ex Vecte pendentis momentum inquiritur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"></emph>An duo pondus geſtantes æqualiter premantur.<emph.end type="italics"></emph.end></cell> </row> <pb xlink:href="017/01/015.jpg"></pb> <row> <cell>X.</cell> <cell><emph type="italics"></emph>An vis Elaſtica ad aliquod Vectis genus pertineat.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"></emph>Cur longiora corpora faciliùs flectantur, difficiliùs ſuſtineantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"></emph>Vnde oriantur forcipum, & forficum vires.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"></emph>Cur Tollenones juxta puteos conſtituantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XIV.</cell> <cell><emph type="italics"></emph>Remoram vires in agenda navi expenduntur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XV.</cell> <cell><emph type="italics"></emph>Quomodo Naves à Gubernaculo moveantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XVI.</cell> <cell><emph type="italics"></emph>An Malus in motu navis habeat Rationem Vectis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XVII.</cell> <cell><emph type="italics"></emph>An ex Rationibus Vectis pendeat uſus Anchoræ.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XVIII.</cell> <cell><emph type="italics"></emph>Plures Vectis uſus exponuntur.<emph.end type="italics"></emph.end></cell> </row> </table> <p type="head"> <s id="s.000066"><emph type="center"></emph>LIBER QUINTUS. De Axe in Peritrochio.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>Axis in Peritrochio forma, & vires deſcribuntur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>Succulæ & Ergata uſus conſideratur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>Tympani à calcante circumacti vires expenduntur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>An Axis in Peritrochio inveniatur etiam ſinè tractione.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>Axium in ſuis Peritrochiis Compoſitione vires augentur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"></emph>Tympanorum dentatorum uſus. & vires exponuntur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"></emph>Moletrinarum artificium ex Axe in Peritrochio pendet.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"></emph>Axis cum Vecte compoſitus auget Potentiæ momenta.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"></emph>Multiplex Rotarum dentatarum uſus innuitur.<emph.end type="italics"></emph.end></cell> </row> </table> <p type="head"> <s id="s.000067"><emph type="center"></emph>LIBER SEXTUS. De Trochlea.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>TRochlearum forma & vires exponuntur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>An Trochlea ad Vectem revocanda ſit.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>An Orbiculi Magnitudo quicquam conferat.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>Qua Ratione Trochlearum vires augeantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>Trochlea Trochleis additæ plurimum augent momenta Po-tentiæ.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"></emph>Trochlearum ope moveri poteſt pondus velociter.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"></emph>Quàm validum eſſe oporteat Trochlearum retinaculum.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"></emph>Aliqui Trochlearum uſus indicantur.<emph.end type="italics"></emph.end></cell> </row> </table> <pb xlink:href="017/01/016.jpg"></pb> <p type="head"> <s id="s.000068"><emph type="center"></emph>LIBER SEPTIMUS. De Cuneo, & Percuſſionibus.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>CVnei farma & vires explicantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>Cunei inflexi vſus ad movendum.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>Cuneus Perpetuns circulo excentrico effingitur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>Ex Cylindro conſtrui poteſt Cuneus Perpetuus.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>Cuneum Perpetuum Circulus inclinatus imitatur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"></emph>Vnde oriatur vis Percuſſionis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"></emph>Quàm diſpares ex motûs velocitate ſint Percuſſiones.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"></emph>An validior ſit ictus Malles à Situ Verticali ad Horizonta-lem, an verò ab Horizontali ad Verticalem deſcendentis.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"></emph>Quomodo Percuſſiones ex Mele pendeant.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"></emph>Quid conferat reſiſtentia corporis percuſſi.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"></emph>Quomodo ex Percuſſionibus determinentar Reflexiones.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"></emph>Quomodo Impetus in Percuſſions communicetur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"></emph>Cunei uſus promovetur.<emph.end type="italics"></emph.end></cell> </row> </table> <p type="head"> <s id="s.000069"><emph type="center"></emph>LIBER OCTAVUS. De Cochlea.<emph.end type="center"></emph.end></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"></emph>COchleæ forma & virtus deſcribitur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"></emph>An utilis ſit Cochlea duplex contraria.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"></emph>Cochlea cum Vecte, atque cum Axe componitur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"></emph>Cochleæ Infinitæ vires explicantur.<emph.end type="italics"></emph.end></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"></emph>Cochlea uſus aliqui indicantur.<emph.end type="italics"></emph.end></cell> </row> </table> <pb n="1" xlink:href="017/01/017.jpg"></pb> <figure id="id.017.01.017.1.jpg" xlink:href="017/01/017/1.jpg"></figure> <p type="head"> <s id="s.000070"><emph type="center"></emph>MECHANICORUM<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000071"><emph type="center"></emph>LIBER PRIMUS.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000072"><emph type="center"></emph><emph type="italics"></emph>De Centro Gravitatis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000073">MACHINARUM vires, quibus innatæ corporum in <lb></lb>motum aut quietem propenſioni obſiſtimus, explo<lb></lb>raturus, præterire non poſſum gravitatem ipſam: <lb></lb>ne ſcilicet ignoretur, quid arte vincendum ſit. </s> <s id="s.000074">Ideò <lb></lb>primum hunc Librum Centro gravitatis tribuen<lb></lb>dum cenſui, cùm plura ex illo pendeant examinanda in poſte<lb></lb>rioribus. </s> <s id="s.000075">Neque tamen hîc ſubtiliſſimam illam ſtatices partem <lb></lb>perſequar, quæ in corporibus ſingulis gravitatis centrum in<lb></lb>veſtigat: id enim, & abundè ab aliis præſtitum, & mihi in hac <lb></lb>tractatione minimè neceſſarium; quippe cui ſatisfuerit cen<lb></lb>trum illud phyſicè perſpectum habere, quatenus præcaven<lb></lb>dum eſt, ne alienâ ponderis ad machinam applicatione longè <lb></lb>alia fiat momentorum ratio, quàm oporteat. </s> <s id="s.000076">Ut autem Centri <lb></lb>gravitatis notitia clarior habeatur, non inutile ducam quæſtio<lb></lb>nes aliquot ad illud enucleatiùs explicandum pertinentes ad<lb></lb>dere, ut ipſis etiam tyronibus fiat ſatis: quamquam enim illis <lb></lb>machinalis ſcientia carere poſſe alicui fortaſſe videatur, rem <lb></lb>tamen penitiùs introſpiciens eas extrà mechanicæ conſidera<lb></lb>tionis fines poſitas non eſſe cognoſcet.</s> </p> <p type="head"> <s id="s.000077"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000078"><emph type="center"></emph><emph type="italics"></emph>Quid ſit Centrum gravium, & levium.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000079">QUoniam hæc rerum univerſitas corpora diverſæ inter ſe <lb></lb>rationis complectitur, eorum ordo aliquis neceſſariu, fuit <pb n="2" xlink:href="017/01/018.jpg"></pb>ut ſuo unumquodque loco diſponeretur; atque adeò æquum <lb></lb>fuit, ut ſingulis à natura ea tribueretur facultas, quâ & ſe ſuo <lb></lb>in loco, hoc eſt, juxta inſitam propenſionem ſibi debito, con<lb></lb>ſervare poſſint, & ad illum ſe ipſa promovere, ſi fortè indè <lb></lb>dimota fuerint. </s> <s id="s.000080">Quia verò æqualia non niſi æqualiter, ſimili<lb></lb>que ratione diſponenda erant, nullum autem corpus præter <lb></lb>ſphæram habet perfectam in partium diſpoſitione æqualitatem, <lb></lb>debuerunt corpora omnia orbem unum conſtituere. </s> <s id="s.000081">At in <lb></lb>ſphæra punctum unum eſt, à quo æqualibus radiis extremæ <lb></lb>ſuperficiei partes removentur: igitur ex ordine ad punctum <lb></lb>hoc, quod Centrum dicitur, comparanda ſunt corpora; qua<lb></lb>tenus cùm naturâ impellente moventur, ut in loco ſibi debito, <lb></lb>à quo per vim ſejuncta fuere, demum conſiſtant, vel ad cen<lb></lb>trum hoc accedunt, vel ab eo recedunt. </s> </p> <p type="main"> <s id="s.000082">Et quidem ſi ad centrum accedant, gravitare dicuntur, ſi <lb></lb>verò recedant, levitare: & quæ propiora centro conſiſtunt, <lb></lb>graviora, quæ autem remotiora, leviora quoque cenſentur <lb></lb>ſecundùm ſpeciem gravitatis, & levitatis: quicquid ſit quod <lb></lb>æqualia eſſe poſſint ſecundùm gravitatem abſolutam, aut etiam <lb></lb>ſæpè contingat minus habere gravitatis abſolutæ id, quod eſt <lb></lb>gravius ſecundùm ſpeciem. </s> <s id="s.000083">Sic libra plumbi æqualis eſt libræ <lb></lb>aquæ, immò minor centum libris aquæ; quia tamen plum<lb></lb>bum infra aquam deſcendens fit centro vicinius, etiam gra<lb></lb>vius eſt ſecundùm ſpeciem. </s> <s id="s.000084">Quod ſi comparare velis duo cor<lb></lb>pora ſolida, quæ ſibi ſua duritie ita obſiſtunt, ut neutrum intra <lb></lb>alterum moveri poſſit tanquam in medio; illud eſſe ſecundùm <lb></lb>ſpeciem gravius affirmabis, quod datâ paritate molis cum alio <lb></lb>corpore, cum quo comparatur, ſtaterâ expenſum in eodem <lb></lb>medio, in quo utrumque gravitat puta in aëre, plus habere <lb></lb>ponderis deprehendes. </s> <s id="s.000085">Sic aurum eſt ferro gravius in ſpecie, <lb></lb>quia ex æqualibus molibus auri & ferri, aurea eſt ponderoſior. </s> </p> <p type="main"> <s id="s.000086">Generatim autem loquendo ea ſunt in ſpecie graviora, quæ <lb></lb>ſunt denſiora, ea verò in ſpecie leviora, quæ rariora: nam & <lb></lb>inflata veſica ob aërem conſtipatum gravior eſt, quàm flaccida; <lb></lb>& Æolipilam candentem, aëre intus vi caloris raro, leviorem <lb></lb>primùm, poſteà, ubi refrixerit, graviorem eſſe experimento <lb></lb>didicimus, aëre aſſumptam raritatem abjiciente. </s> <s id="s.000087">Cùm enim <lb></lb>radij à ſphæræ centro ad ſuperficiem ducti longiùs à ſe invi- <pb n="3" xlink:href="017/01/019.jpg"></pb>cem recedant, æquum fuit, ut quæ plus habent materiæ atque <lb></lb>ſubſtantiæ ſub minori mole, in anguſtiore ſpatio collocarentur; <lb></lb>ea verò, quæ ſub majoribus dimenſionibus continentur, am<lb></lb>pliora ſpatia occuparent, ubi radij magis diſtant: ut videlicet <lb></lb>hac ratione æqua ſubſtantiæ diſtributio fieret in totâ ſphærâ. </s> <lb></lb> <s id="s.000088">Hinc vides, cur idem corpus, eo ipſo quod rarum fit, aſcendat, <lb></lb>ut aqua in vaporem reſoluta (niſi aliunde ad deſcendendum <lb></lb>determinetur, ut aurum fulminans) quia materies eadem ſub <lb></lb>majoribus dimenſionibus petit longiùs abeſſe à centro, ibiquè <lb></lb>tantiſper conquieſcit, dum conſtipata, atque minorem in mo<lb></lb>lem redacta, iterum deſcendat. </s> </p> <p type="main"> <s id="s.000089">Quare centrum hoc, quod motus, vel quies corporum reſpi<lb></lb>cit, dicitur <emph type="italics"></emph>Centrum gravium, & levium<emph.end type="italics"></emph.end>; atque idem creditur <lb></lb>eſſe cum centro univerſi: vel ſaltem (ne parùm utili nos diſpu<lb></lb>tatione torqueamus) centrum eorum, quæ in hac ſphærâ ele<lb></lb>mentari gravia, aut levia dicuntur, idem eſt cum centro ter<lb></lb>raquei hujus globi, ut quotidiana docet experientia: quicquid <lb></lb>ſit, an pars lunaris globi, ſi à lunâ ſejungeretur, reditura eſſet <lb></lb>ad lunam, ut ad centrum ſui motus. </s> <s id="s.000090">Tam itaquè, quæ hujuſmo<lb></lb>di centro proxima ſunt, deorſum poſita dicuntur, ſurſum verò, <lb></lb>quæ ab eo longiùs collocata ſunt. </s> <s id="s.000091">Hinc telluris ſuperficiei in<lb></lb>ſiſtentes caput ſurſum, pedes deorſum habere dicimur. </s> <s id="s.000092">Ille <lb></lb>verò, quamvis rectus, & pedes, & caput ſurſum haberet, cu<lb></lb>jus umbilicus huic centro univerſi congrueret. </s> <s id="s.000093">Per quod pa<lb></lb>riter centrum ſi ſcala ducta intelligatur, duo poſſent ſibi non <lb></lb>occurrere invicem, licet alter aſcenderet, alter deſcenderet; <lb></lb>hic ſiquidem accederet ad centrum, ille inde recederet: per <lb></lb>eam verò poſſet uterque aſcendere, & tamen licet, æquali mo<lb></lb>tu moverentur, ſemper invicem diſtarent magis, quò à centro <lb></lb>ad oppoſitas partes recederent. <lb></lb></s> </p> <p type="head"> <s id="s.000094"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000095"><emph type="center"></emph><emph type="italics"></emph>An corpora prædita ſint gravitate, & levitate.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000096">INter ea, quæ planè homogenea ſunt, ordo eſſe non poteſt à <lb></lb>naturâ inſtitutus: hinc ſi nulla eſſet corporum diſſimilitudo, <pb n="4" xlink:href="017/01/020.jpg"></pb>ſed ex omninò ſimilibus ſubſtantiæ partibus totus hic orbis <lb></lb>conflaretur, nulla quoque eſſet aut gravitas, aut levitas. </s> <s id="s.000097">Quid <lb></lb>enim hæc potiùs pars, nulla naturæ conditione à cæteris diſcre<lb></lb>ta, petat abeſſe à centro, illa verò exigat in eo conquieſcere? </s> <s id="s.000098"><lb></lb>verùm quia multiplici corporum genere coagmentata rerum <lb></lb>univerſitas inconcinna eſſe non potuit, ſuum cuique locum na<lb></lb>tura tribuit, in quo ſe ſiſteret, ut infra hæc quidem deſcende<lb></lb>ret, ſuprà illa verò aſcenderet, ſi quando ſibi invicem con<lb></lb>tigua fierent ordine præpoſtero, nec ullus eſſet motui obex. </s> <lb></lb> <s id="s.000099">Cùm itaque corpora ſingula inſitam habeant propenſionem <lb></lb>(ab Ariſtotele dicitur <foreign lang="grc">ὁρμή</foreign>) qua petunt certum locum in uni<lb></lb>verſo; conſtat præter deſcendentium gravitatem dari etiam po<lb></lb>ſitivam levitatem, quâ corpus aliquod ſe ipſum promovet ad <lb></lb>ſuperiores partes univerſi à centro magis diſtantes, neque ſo<lb></lb>lùm admittendam levitatem negativam, quâ corpora minùs <lb></lb>gravia cenſentur levia, ſi eorum cum gravioribus fiat compa<lb></lb>ratio. </s> <s id="s.000100">Nam ſi ea, quæ levia dicuntur, eatenus dicas aſcendere, <lb></lb>quatenus à gravioribus in inferiorem locum deſcendentibus <lb></lb>propelluntur; mihi æquè liberum erit tollere omnem poſiti<lb></lb>vam gravitatem, ſolâ levitate admiſsâ; & omnia pariter ſol<lb></lb>vam dicendo ea gravia cenſeri, quæ minùs levia ſunt, atque <lb></lb>ideò tantùm deſcendere, quòd extrinſecùs à levioribus aſcen<lb></lb>dentibus loco pulſa detrudantur, non quòd ab internâ faculta<lb></lb>te deorſum impellantur. </s> <s id="s.000101">Quod ſi vel gravitas de medio tollen<lb></lb>da ſit, vel levitas, ſatius eſt levitatem relinquere; naturâ vi<lb></lb>delicet ad altiora ſemper, & perfectiora aſpirante, nec adeò <lb></lb>contendente de infimo loco. </s> <s id="s.000102">Quare cùm per gravitatem ſolam <lb></lb>æquè ac per ſolam levitatem motus iſti explicentur, cætero qui <lb></lb>autem ingenita ſit unicuique corpori ſui loci exigentia; utram<lb></lb>que admittere rationi maximè conſentaneum fuerit. </s> </p> <p type="main"> <s id="s.000103">Vitreum globum vacuum, qui in tubulum recurvum deſi<lb></lb>nat, quoad fieri poteſt, calefactum, ut incluſus aer rareſcat, <lb></lb>Hermeticè claude: tum adjiciatur congruens plumbi gravitas, <lb></lb>quâ infra aquam deprimatur. </s> <s id="s.000104">Sit autem globus, unà cum ad<lb></lb>jecto plumbo, connexus cum exquiſitæ libræ brachio, aut lan<lb></lb>ce, ejúſque gravitas intrà aquam exploretur: ubi gravitas in<lb></lb>notuerit, adhuc ſub aquâ retineatur globus, ſed longiore for<lb></lb>cipe extremum tubuli caput occluſum frangatur: & animad- <pb n="5" xlink:href="017/01/021.jpg"></pb>vertes globi vitrei cum appenſo plumbo gravitatem augeri; cu<lb></lb>jus incrementum indicabitur ab addito in oppoſitâ lance pon<lb></lb>dere ad conſtituendum æquilibrium. </s> <s id="s.000105">Cùm itaque idem maneat <lb></lb>vitrum, idémque plumbum, & nulla facta ſit alicujus gravita<lb></lb>tis acceſſio, illud unum ſupereſt, quòd aör rarus intrà globum <lb></lb>concluſus levior, quàm idem aör, aperto tubulo, ſibi reſtitu<lb></lb>tus, plus elidit gravitatis plumbi & vitri; atque moles compo<lb></lb>ſita ex plumbo, vitro, & aëre raro, ſecundùm ſpeciem levior <lb></lb>eſt, quàm moles ex plumbo, vitro, & aëre non raro. </s> <s id="s.000106">Aër igi<lb></lb>tur intra aquam ita levis eſt, ut aliquid gravitatis imminuat: <lb></lb>Nam ſi globum eundem ex aquâ extractum, omni aëre exclu<lb></lb>ſo, aquâ repleveris, & iterum eodem plumbo adjecto ejuſdem <lb></lb>gravitatem intrà aquam examinaveris, illam adhuc majorem <lb></lb>deprehendes; quia ſcilicet nulla levitas aöris adeſt, quæ ali<lb></lb>quam deterat gravitatem, ſed illa ſolùm perire videtur, quam <lb></lb>infert diſcrimen gravitatum ſecundùm ſpeciem, ut ex Hy<lb></lb>droſtaticis conſtat. </s> <s id="s.000107">Neque ſuſpiceris hæc gravitatum incre<lb></lb>menta oriri ex aquâ ſubeunte per apertum tubulum, cùm aër <lb></lb>aſſumptam ex calore raritatem abjicit, ſe in naturalem ſuam <lb></lb>molem reſtituens, ſivè, aëre prorſus excluſo, ex aquæ globum <lb></lb>implentis gravitate. </s> <s id="s.000108">Si enim vitrum aliud aut nullius, aut mo<lb></lb>diciſſimæ aquæ capax, ſed ejuſdem in aëre ponderis cum aſ<lb></lb>ſumpto globo, ſimiliter in aquâ expendas, eandem invenies <lb></lb>gravitatem, ſive multâ, ſive modicâ aquâ repletum fuerit. </s> <lb></lb> <s id="s.000109">Non igitur aqua intrà aquam gravitatem auget. </s> </p> <p type="main"> <s id="s.000110">Sed illud, ut reliqua ſileam, non leviter ſuadere poteſt cor<lb></lb>pora ſuis nutibus non deorſum tantùm, ſed etiam ſurſum co<lb></lb>nari, quod mihi haud ita pridem aliud inveſtiganti contigit <lb></lb>obſervare. </s> <s id="s.000111">Cum enim animadvertiſſem aliquando, quàm diſ<lb></lb>par eſſet gravitas aquæ dimidiam ſitulam implentis, ſi illa in ſu<lb></lb>perficie horizontali libraret ſeſe, ac quandò ſuppoſita ligneo <lb></lb>globo firmiter cum ſuperiore tigillo cohærenti altiùs ad latera <lb></lb>aſſurgebat locum globo concedens, quem tamen non ſuſtine<lb></lb>bat; ſubiit animum cupido tentandi, an bubula veſica inflata <lb></lb>tranſverſis virgulis infra vaſis labra depreſſa ita, ut eam aqua <lb></lb>circumplecteretur, vim haberet pariter augendi momenta gra<lb></lb>vitatis; aquam ſiquidem cogebat aſſurgere ad altitudinem ma<lb></lb>jorem perpendicularem, ac quandò, veſicâ liberè innatante, <pb n="6" xlink:href="017/01/022.jpg"></pb>ſubſidebat. </s> <s id="s.000112">Inveni tamen nullum planè obſervari poſſe in <lb></lb>gravitate diſcrimen, quamvis tam ampla eſſet veſica, ut facilè <lb></lb>dimidiam vaſis capacitatem impleret: in utroque enim caſu pon<lb></lb>dus fuit lib. 44 1/2. </s> <s id="s.000113">Id mihi, fateor, accidit præter opinionem: <lb></lb> <figure id="id.017.01.022.1.jpg" xlink:href="017/01/022/1.jpg"></figure><lb></lb>Nam ſi ex pariete extet tigillus, cui adnectatur <lb></lb>cylindrus P, aut veſica ritè firmata, ferè im<lb></lb>plens capacitatem vaſis AB, vaſque illi ſup<lb></lb>ponatur ita, ut aqua deinde infuſa poſſit libe<lb></lb>rè cylindro circumfundi; percipies onus lon<lb></lb>gè majus, quàm pro gravitate aquæ infuſæ, <lb></lb>ſi permitteretur ſubſidere: & ſi vas ex ſtaterâ <lb></lb>pendeat, adducto reductóve ſacomate appa<lb></lb>rebunt momenta gravitatis longè majora, quàm ſi tota illa <lb></lb>aqua fundum peteret, & cylindri pars, quæ priùs immerge<lb></lb>batur, abſciſſa, aut veſica innataret. </s> <s id="s.000114">Intelligebam id ex majori <lb></lb>altitudine perpendiculari aquæ ſupra eandem baſim oriri; nam <lb></lb>depreſſo vaſe ita, ut paulatim cylindus emergat, & aqua ſub<lb></lb>ſidat, ſemper minuitur pondus: idem futurum ſperabam, ſi <lb></lb>veſica intra aquam non ab extrinſeco obice detineretur, ſed à <lb></lb>virgulis cum vaſe ipſo connexis; quandoquidem aqua ad ean<lb></lb>dem pariter altitudinem aſſurgebat ſuper baſim eandem: at <lb></lb>ſpem fefellit eventus. </s> <s id="s.000115">Nec alia mihi ſe obtulit probabilior ra<lb></lb>tio, quàm ut exiſtimarem aquam altiorem vehementius qui<lb></lb>dem deorſum niti, veſicam tamen leviorem altiùs depreſſam, <lb></lb>conantem ſurſum, æqualiter contendere, ut emergeret; cùm <lb></lb>verò niſus iſte ſurſum oppoſitas virgulas, atque adeò vas cum <lb></lb>illis connexum urgeret, elidi adverſum impetum deorſum, <lb></lb>qui à majore altitudine perpendiculari aquæ oriebatur, & ſo<lb></lb>lum remanere conatum ex ipſorum corporum ſubſtantiâ pro<lb></lb>manantem, quæ ſicut eadem ſemper erat, ſivè innataret veſi<lb></lb>ca, ſivè per vim immergeretur, ita eadem obtinebat gravita<lb></lb>tis momenta. </s> <s id="s.000116">Quo experimento (quamquam non me lateat, <lb></lb>quid pro ſe afferre hîc poſſent aliter ſentientes) viſus mihi <lb></lb>ſum deprehendere non obſcurum poſitivæ levitatis veſti<lb></lb>gium. </s> </p> <p type="main"> <s id="s.000117">Ut autem levitatem corporibus adimendam aſſererent in<lb></lb>genioſi Academici, hoc potiſſimum ducti ſunt experimento. </s> <s id="s.000118"> <pb n="7" xlink:href="017/01/023.jpg"></pb>Ligneum <expan abbr="cylindrũ">cylindrum</expan> ABC <lb></lb> <figure id="id.017.01.023.1.jpg" xlink:href="017/01/023/1.jpg"></figure><lb></lb>plano horizontali D, E, <lb></lb>perpendicularem ſtatue<lb></lb>runt; & ut cylindri ba<lb></lb>ſis ſubjecto plano exactè <lb></lb>congrueret, laminas duas <lb></lb>accuratiſſimè lævigatas, <lb></lb>tùm cylindri baſi, tùm <lb></lb>ſubjecto plano firmiter <lb></lb>adnexas voluerunt. </s> <s id="s.000119">Tùm <lb></lb>ne aër facilè inter utrum<lb></lb>que ſubiret, erecto ſupra <lb></lb><expan abbr="planũ">planum</expan> in orbem ex cretâ, <lb></lb>aut cerâ aggerulo, <expan abbr="argen-tũ">argen<lb></lb>tum</expan> vivum infuderunt. </s> <s id="s.000120">Cylindrum extremo libræ jugo G, alligâ<lb></lb>runt, addito in oppoſitâ libræ extremitate H pondere L cylin<lb></lb>dri pondus adæquante; quod utique cylindrum elevare non po<lb></lb>teſt. </s> <s id="s.000121">Additum igitur eſt & aliud pondus M uſque eò, dum cy<lb></lb>lindrus à ſubjecto plano avelleretur, & fuit librarum circiter <lb></lb>trium: quam menſuram arguunt eſſe reſiſtentiæ cylindri con<lb></lb>tiguo plano adhærentis metu vacui. </s> <s id="s.000122">His peractis concavum <lb></lb>vas cylindricum NOP, æqualis aut majoris altitudinis parâ<lb></lb>runt, laminâ pariter perpolitâ vaſis fundo adnexâ, cui impo<lb></lb>ſitus fuit cylindrus, adeoque adhæſit, ut, pleno-mercurij <lb></lb>vaſe, omninò non avelleretur, ut innataret; ſed tunc demum <lb></lb>argento vivo innatavit, cùm per vim à vaſis fundo avulſus eſt <lb></lb>cylindrus: cui, ut iterum fundum peteret, & argento vivo <lb></lb>immergeretur, imponendum fuit pondus Q librarum circiter <lb></lb>quinque. </s> <s id="s.000123">Vis ergò levitatis ligni in mercurio (ſi qua levitas <lb></lb>eſſet) æſtimanda eſſet ut quinque, cùm vis adhæſionis metu <lb></lb>vacui ſolùm inventa ſit ut tria: debuiſſet igitur levitas ita præ<lb></lb>valere, ut adhæſionem vinceret, & cylindrus ſponte elevaretur. </s> <lb></lb> <s id="s.000124">Non eſt itaque levitas, quæ ligneum cylindrum innatare cogit, <lb></lb>ſed mercurij gravitas major ipſa eſt, quæ lignum elevat, cum <lb></lb>primùm locus patet, in quem deſcendat. </s> </p> <p type="main"> <s id="s.000125">Sed antequam experimentum hoc ad examen revocemus, <lb></lb>ut innoteſcat, quid hinc confici poſſit ad levitatem excluden<lb></lb>dam, haud ægrè permiſerim, cùm in abeuntis ſuâ ſponte cor- <pb n="8" xlink:href="017/01/024.jpg"></pb>poris locum corpus aliud ſuapte vi, & naturâ ſuccedit, ab hoc <lb></lb>illud urgeri poſſe, ut velociùs moveatur: duo ſcilicet corpora <lb></lb>diverſæ ſecundùm ſpeciem gravitatis ſi fuerint perturbatè diſ<lb></lb>poſita intrà medium, in quo utrumque gravitat, nil mirum, ſi <lb></lb>à graviore majori niſu conante extrudatur minùs grave: id <lb></lb>quod etiam de duobus levibus dicendum perturbatè diſpoſitis <lb></lb>in medio, ubi utrumque levitat: duobus enim ſimul currenti<lb></lb>bus, ab eo qui ponè ſubſequitur, ſi majoribus viribus polleat, <lb></lb>priorem urgeri atque impelli palam eſt, quamquam motus uni<lb></lb>verſus impulſioni tribuendus non ſit. </s> <s id="s.000126">Ita quoque aſcendentem <lb></lb>in mercurio ligneum cylindrum à deſcendente mercurio ſur<lb></lb>ſum urgeri aliquatenus poſſe non diffitebor, ſicut & mercu<lb></lb>rium ipſum repugnare, ne ſurſum propellatur, atque ab eodem <lb></lb>lignum innatans prohiberi, ne deſcendat: hinc tamen non <lb></lb>ſequitur ligni aſcendentis motum, aut innatantis quietem, <lb></lb>prægravis mercurij viribus omnino adſcribi jure debere, nam, <lb></lb>& ſua vis aſcendendi, atque conſiſtendi, ligno ipſi tribuen<lb></lb>da eſt. </s> </p> <p type="main"> <s id="s.000127">Quid quòd ipſæ innatantis cylindri portiones, altera quidem <lb></lb>mercurio immerſa, altera verò extans, levitatem ipſi ligno in<lb></lb>ſitam declarant? </s> <s id="s.000128">Quid enim partis immerſæ ad extantem (ſi <lb></lb>moles ſpectetur) ea ratio eſt, quæ ſpecificæ gravitatis ligni ad <lb></lb>differentiam gravitatum ligni, atque mercurij? </s> <s id="s.000129">niſi quia por<lb></lb>tionis mercurio immerſæ levitas, atque extantis in aëre gravi<lb></lb>tas, æquilibritatem conſtituent; quemadmodum in <emph type="italics"></emph>Terra ma<lb></lb>chinis mota differt.<emph.end type="italics"></emph.end> 5. <emph type="italics"></emph>n.<emph.end type="italics"></emph.end> 105. explicatum eſt. </s> <s id="s.000130">Hanc porrò æqua<lb></lb>litatem Algebricè ſic oſtendo. </s> <s id="s.000131">Ratio gravitatis ligni ad gravi<lb></lb>tatem mercurij ſit ut S. ad R; differentia eſt R—S. Ponatur <lb></lb>cylindri pars immerſa. A. </s> <s id="s.000132">Quia igitur ut ſpecifica gravitas <lb></lb>corporis innatantis ad differentiam gravitatum, hoc eſt ut <lb></lb>S ad R — S, ita pars cylindri immerſa A, ad extantem <lb></lb>(R in A—S in A/S); Si pars extans in aëre in ſuam gravitatem S du<lb></lb>catur, pars verò immerſa A in differentiam gravitatum R—S, <lb></lb>hoc eſt in — R + S, quia eſt deficiens, efficitur hinc quantitas <lb></lb>R in A — S in A, hinc verò — R in A + S in A, quæ ſe invi<lb></lb>cem elidunt. </s> <s id="s.000133">Æqualia igitur ſunt levitatis, & gravitatis mo<lb></lb>menta. </s> <s id="s.000134">Sit enim exempli causâ gravitas ligni ad gravitatem <pb n="9" xlink:href="017/01/025.jpg"></pb>mercurij, ut S. ad 13. differentia eſt 8. </s> <s id="s.000135">Eſt igitur cylindri <lb></lb>pars immerſa ejuſdem (5/13), extans verò (8/13): at portio immerſa de<lb></lb>ficit à gravitate mercurij ſecundùm ſpeciem ut 8; igitur (5/13) in - 8 <lb></lb>dant (40/13): item partis extantis gravitas in aëre eſt S; igitur (8/13) <lb></lb>in 5 dant (40/13): confligunt itaque inter ſe pari conatu levitas (-40/13) & <lb></lb>gravitas (40/13), adeóque fit conſiſtentia & innatat lignum. </s> </p> <p type="main"> <s id="s.000136">Sed jam ad propoſiti experimenti examen deſcendamus. </s> <s id="s.000137">Aio <lb></lb>cylindri reſiſtentiam ex adhæſione metu vacui non ſatis explo<lb></lb>ratam fuiſſe per libram; hæc enim dum ex pondere M deorſum <lb></lb>inclinatur, extremitas G ſurſum elevata arcum deſcribit, ac <lb></lb>proinde cylindri aſcendentis motus non eſt per lineam horizon<lb></lb>tali plano perpendiculariter inſiſtentem, ſed per inclinatam: <lb></lb>Quare cùm A. versùs I libræ centrum trahatur, cylindri baſis <lb></lb>non incipit elevari parallela horizonti, ſed cum inclinatione, ita <lb></lb>ut C priùs elevetur, quàm B: ea autem, quæ ſibi invicem adhæ<lb></lb>reſcunt, multò faciliùs divelli manifeſtum eſt, ſi id cum inclina<lb></lb>tione fiat, quàm ſi ſervandus ſit paralleliſmus. </s> <s id="s.000138">Adde in hac in<lb></lb>clinatione faciliùs adhuc divelli cylindrum à ſuppoſito plano, <lb></lb>quò longior cylindrus fuerit; habet ſcilicet rationem vectis, <lb></lb>cujus potentia eſt in A, hypomochlion in B, reſiſtentia vin<lb></lb>cenda in C. </s> <s id="s.000139">Quare pondus M non aptè metitur reſiſtentiam, <lb></lb>quæ oritur ex corporum adhæreſcentiâ, metu vacui, ſed hæc <lb></lb>multò major eſt, ſi ad perpendiculum motus fieri debeat; <lb></lb>quemadmodum & fieri oporteret, ſi in vaſe NOP mercurij <lb></lb>pleno cylindrus fundo adhærens rectâ aſcenderet. </s> <s id="s.000140">Quamvis <lb></lb>igitur pondus Q librarum quinque admitteretur menſura levi<lb></lb>tatis, non continuò argui poteſt hujus exceſſus ſupra reſiſten<lb></lb>tiam adhæſionis. </s> <s id="s.000141">Quin immo affirmare auſim, ſi libræ loco <lb></lb>adhibita fuiſſet amplior trochlea, & ex funiculo ejus orbitam <lb></lb><expan abbr="cõplectente">complectente</expan> hinc cylindrus A, hinc verò pondus M ad perpen<lb></lb>diculum pependiſſent, non ſatis futurum fuiſſe <expan abbr="põdus">pondus</expan> librarum <lb></lb>trium, ſed multò majus adhibendum fuiſſe, ut cylindri reſiſten<lb></lb>tiam ſuperaret; fuiſſet enim avellenda baſis ſervato paralleliſmo. </s> </p> <p type="main"> <s id="s.000142">Quantum autem virium, ferè ſupra fidem, habeat vacui <lb></lb>horror ad corpora retinenda, ſatis apertè declarant gravia, quæ <lb></lb>ſuſpenduntur. </s> <s id="s.000143">Ego ſanè vidi marmoreum mortarium commu<lb></lb>nis magnitudinis ſatis vulgari artificio ſuſpendi vitreo cyatho: <pb n="10" xlink:href="017/01/026.jpg"></pb>mortarij ſcilicet fundo exteriùs aptata fuerat maſſa ex farinâ <lb></lb>ad formandos panes recens macerata, & aquâ ita ſubacta, ut <lb></lb>illi tenaciter cohæreret: tum vitreo calici injecta ſtuppa admo<lb></lb>to igne exarſit, applicituſque calix maſſæ eam attraxit, ſicut & <lb></lb>medicorum cucurbitulæ carnem attrahunt: quare accepto ca<lb></lb>licis vitrei pede facile fuit mortarium elevare, & ſuſpendere. </s> <lb></lb> <s id="s.000144">Quod ſi marmoreum mortarium ex metu vacui in aëre pendu<lb></lb>lum hæſit, quid mirum ſi & ligneus cylindrus ſubjecto plano <lb></lb>adhæreſcens in mercurio ſtetit? </s> </p> <p type="main"> <s id="s.000145">Nondum itaque ex hoc experimento, aut ex ſimilibus, ubi <lb></lb>metu vacui ſuos motus moliri corpora non poſſunt, ſatis habe<lb></lb>mus argumenti, quo levitatem, ſolâ gravitate retentâ, expun<lb></lb>gamus. </s> <s id="s.000146">Hujuſmodi eſt illud, ubi in lignei vaſis fundo exca<lb></lb>vatur ſcaphium, cui exquiſitè congruat eburneus globus, qui <lb></lb>ſuperinfuſo hydrargyro non aſcendit. </s> <s id="s.000147">Neque enim ideò non <lb></lb>aſcendit, quia rima nulla patet argento vivo, per quam ſubiens <lb></lb>extrudat eburneum globum, ſed quia ita ſibi exquiſitè con<lb></lb>gruunt ebur, & lignum, ut vis ipſa aſcendendi vincere non va<lb></lb>leat vim adhæreſcentiæ. </s> <s id="s.000148">Nam & eadem vis in aere ſuſpendit <lb></lb>corpora gravia, ne deſcendant. </s> <s id="s.000149">Quamvis autem non totum <lb></lb>hemiſphærium globi eburnei, ſed ſolùm ejus maximus circu<lb></lb>lus congrueret excavato ligno, & cavitas ipſa aëre repleretur, <lb></lb>non propterea tollitur vis adhæreſcentiæ illius annularis; quia <lb></lb>ſcilicet vis aſcendendi in hydrargyro tanta non eſt, ut valeat <lb></lb>incluſum ibi aërem diſtrahere, ſicut opus eſſet ad incipiendum <lb></lb>motum citra periculum vacui, & præterea ſuperanda eſt re<lb></lb>ſiſtentia hydrargyri dividendi; corpora enim in motu divi<lb></lb>dunt medium, pro cujus craſſitudine reſiſtentiam experiuntur. </s> <lb></lb> <s id="s.000150">Adde hemiſphærium inferius in aëre tanquam in loco poſitum <lb></lb>gravitare non minùs, quàm hemiſphærium ſuperius levitet in <lb></lb>hydrargyro; proinde nil mirum, ſi globus non aſcendat. </s> <s id="s.000151">Quod <lb></lb>ſi aëre excluſo locum illum impleveris hydrargyro, & ebur<lb></lb>neum globum ita foramini aptaveris, ut illi exquiſitè congruat; <lb></lb>ſi in ſuperinfuſo hydrargyro globus non aſcendat, indicio eſt <lb></lb>ita globum eſſe foramini infixum, ut neque valeat elevari à ſub<lb></lb>jecto hydrargyro in ſcaphij formam per vim excavato: neque <lb></lb>enim facilè mihi perſuadebis ſpecificarum gravitatum diffe<lb></lb>rentiam exigere, ut hemiſphærium integrum præcisè extet: <pb n="11" xlink:href="017/01/027.jpg"></pb>præter quam quod ſi non valebat ſubjectum aërem diſtrahere, <lb></lb>multò minùs id in hydrargyro præſtare poteſt, ut vacuum <lb></lb>evitetur. </s> </p> <p type="main"> <s id="s.000152">At, inquis, fiſtulam quadricubitalem ſpiritu vini plenam <lb></lb>cum globulo innatante ſi clauſeris, & inverteris deorſum, <lb></lb>aſcendet globulus ſpatio 200 vibrationum perpendiculi; in eâ<lb></lb>dem verò fiſtulâ communis, & ſimplicis aquæ plenâ aſcendet <lb></lb>ſubduplo tempore 100 vibrationum. </s> <s id="s.000153">Cur hoc? </s> <s id="s.000154">niſi quia aqua <lb></lb>ut pote gravior validiùs extrudit globulum, quàm ſpiritus vini. </s> <lb></lb> <s id="s.000155">Nihilominus: ſi gravia in levibus magis gravitant, & velociùs <lb></lb>deſcendunt, quò major eſt ſpecificarum gravitatum differen<lb></lb>tia; viciſſim levia in gravibus magis levitant, & velociùs <lb></lb>aſcendunt, quò major eſt ſecundùm ſpeciem levitatis differen<lb></lb>tia: Atqui ſpiritus vini magis accedit ad ſpecificam levitatem <lb></lb>innatantis globuli, aqua autem magis differt; in aquâ igitur <lb></lb>globulus magis levitat, & velociùs aſcendit, ſicut lapis in aëre <lb></lb>velociùs deſcendit quàm in aqua, aut in melle. </s> </p> <p type="main"> <s id="s.000156">Addis iterum. </s> <s id="s.000157">Vitreo vaſculo, cui longior fiſtula adhæreat, <lb></lb>fomitem cum filo ſulphurato ope fili ferrei ingere, ut vitrum <lb></lb>tangat: totum imple hydrargyro, & converſo deorſum oſculo <lb></lb>deſcendit hydrargyrus; atque ſubſiſtit in altitudine cubiti, & <lb></lb>quadrantis: admotâ lucernâ vitrarij vitrum calefiat, ut fomes <lb></lb>cum filo ſulphurato accendatur: fumus deſcendit, nec niſi <lb></lb>aperto ſuperiore vaſis oſculo aſcendit, aëre videlicet ſubeunte, <lb></lb>à quo extrudatur ſurſum. </s> <s id="s.000158">Nego fumum ab aëre ſurſum extru<lb></lb>di, ſed qui gravior ſpiritu raro mercurij in illo deſcendebat, <lb></lb>ubi aërem tangit, ut pote levior in illo aſcendit. </s> </p> <p type="main"> <s id="s.000159">Non auſim tamen in lapide, qui gravitatem in aquâ & aëre, <lb></lb>levitatem in mercurio, aut plumbo liquente obtinet, duplicem <lb></lb>ſtatuere virtutem, quarum altera ſurſum, altera deorſum con<lb></lb>nitatur: Cum enim impetus motum efficiens (ut infrà conſta<lb></lb>bit) ejuſdem naturæ ſit, in quamcunque demum orbis plagam <lb></lb>dirigatur motus; ſatis video ab uno eodemque principio, pro <lb></lb>variâ contigui corporis conditione, aſcenſum, deſcensúmve <lb></lb>prodire poſſe. </s> <s id="s.000160">Quandoquidem motus, qui in eadem lineâ per<lb></lb>ficitur, ſimiles planè includit ubicationes ſucceſſivè acquiſi<lb></lb>tas, ſivè aſcenſus ſit, ſivè deſcenſus, ordine tantùm in earum <lb></lb>adeptione, commutato. </s> <s id="s.000161">Quare cum aſcenſus à deſcenſu hoc <pb n="12" xlink:href="017/01/028.jpg"></pb>uno differat, quòd quam ubicationem lapis demùm obtineret <lb></lb>poſt alias propè finem motûs, ſi fuiſſet centro propior quàm <lb></lb>mercurius, eam acquirat ſub initium motûs ante alias, ſi in <lb></lb>mercurij locum aër aut aqua ſurrogetur centro vicinior quàm <lb></lb>lapis: ad ordinem hunc permutandum non videtur neceſſaria <lb></lb>virtutis motricis diſſimilitudo; nihil quippe producitur diſſimi<lb></lb>le. </s> <s id="s.000162">Sed ſi quis ſufficere dicat conditionum varietatem, nihil <lb></lb>abſonum fortè loquatur: debuit enim una virtus activa in ſui <lb></lb>effectus productione non uni tantùm conditioni alligari, ſed pro <lb></lb>earum varietate modum quoque operandi mutare poſſe, modò <lb></lb>præſtitutos fines, quoad ſubſtantiam, non tranſiliret. </s> </p> <p type="main"> <s id="s.000163">Neque arbitror hoc tantùm ſenſu negatam ab aliquibus levi<lb></lb>tatem poſitivam; potuiſſent enim æquè negare gravitatem, ad<lb></lb>miſſa ſolùm potentia motrice. </s> <s id="s.000164">Sed ſi vis iſta ſe movendi deor<lb></lb>ſum gravitas poſitiva dicenda eſt, cùm eadem ſit virtus ſe mo<lb></lb>vendi ſursùm, cur levitas poſitiva non fuerit? </s> <s id="s.000165">Qui enim levita<lb></lb>tem à gravitate ſejunctam negat, non illicò levitatem expun<lb></lb>git: quemadmodum Angelos intelligentiâ aut voluntate dimi<lb></lb>nutos non aſſerunt ij, qui vitalium facultatum diſtinctionem <lb></lb>non agnoſcunt. </s> <s id="s.000166">Nullum igitur corpus ſimpliciter, & abſolutè <lb></lb>grave dicendum eſt, niſi quod cæteris omnibus ita petat ſubeſſe, <lb></lb>ut nequeat raritatem aſſumere, vi cujus evadat levius corpore <lb></lb>ſimili quidem ſecundùm naturam, diſſimilis tamen raritatis: <lb></lb>nullum ſimpliciter, & abſolutè leve, niſi quod ita exigat extre<lb></lb>mam orbis laciniam occupare, ut nunquam conſtipari poſſit, ac <lb></lb>fieri gravius proximo corpore rariore. </s> <s id="s.000167">Reliqua omnia non niſi <lb></lb>comparatè gravia, aut levia dici poſſunt: ſic plumbum grave eſt <lb></lb>in aëre, grave in aqua, at pariter leve in mercurio, leve ſi cum <lb></lb>auro conferatur. </s> </p> <p type="main"> <s id="s.000168">Hinc corpus in loco ſibi debito conſtitutum, sèque ibi con<lb></lb>ſervans (extra tamen ſphæræ centrum, nec in extimâ orbis ele<lb></lb>mentaris ſuperficie) ob idipſum, quia obſiſtit non tantùm, ne <lb></lb>infra ſubjectum corpus deprimatur, verùm etiam, ne in locum <lb></lb>ſuperioris attollatur, & levitare ſimul dicendum eſt, & gravi<lb></lb>tare. </s> <s id="s.000169">At ſi in alienum locum transferatur, quia in medio levio<lb></lb>re ita repugnat aſcenſui, ut petat deſcendere, ſolùm gravitat; <lb></lb>quia verò in graviore ita depreſſioni reluctatur, ut exigat ad <lb></lb>ſuperiora evadere, ſolùm levitat. </s> <s id="s.000170">Quod ſi corpora hujuſmodi <pb n="13" xlink:href="017/01/029.jpg"></pb>in actu ſecundo gravitare aut levitare tunc ſolùm dixeris, quan<lb></lb>do illa in locum non ſuum tranſlata aut deſcendere expetunt, <lb></lb>aut aſcendere, vel re etiam ipsâ deſcendunt, aut aſcendunt, <lb></lb>non admodum repugnabo; modò conatum illum, quo ſe ſuo <lb></lb>tutantur in loco, gravitationem, & levitationem ſaltem in actu <lb></lb>primo, aut pariter aſſeras, aut pariter neges. </s> </p> <p type="main"> <s id="s.000171">Porrò motus omnis gravium, & levium ſicut in vacuo exer<lb></lb>ceri non poteſt (ut in <emph type="italics"></emph>Vacuo Proſcripto cap.<emph.end type="italics"></emph.end>2. <emph type="italics"></emph>num.<emph.end type="italics"></emph.end>9. oſtendi) ita <lb></lb>in medio fit, vel tardiùs, vel citiùs, tùm pro majori vel minori <lb></lb>ipſius medij reſiſtentia ad ſciſſionem partium magis, vel minùs <lb></lb>connexarum, tùm comparatâ gravitate ſeu levitate mobilis <lb></lb>cum levitate ſeu gravitate medij. </s> <s id="s.000172">Hinc eſt gravibus minus <lb></lb>reſiſtere leviora, magis verò, quæ minùs levia, cæteris pari<lb></lb>bus: ſic aër minùs reſiſtit lapidi cadenti, quàm ſi idem lapis in<lb></lb>ciperet moveri in aquâ, quæ minùs levis eſt, quàm aër. </s> <lb></lb> <s id="s.000173">Ex oppoſito autem levibus graviora minùs reſiſtunt, quæ au<lb></lb>tem minùs gravia, magis reſiſtunt: ſic exhalatio ex fundo <lb></lb>aquæ, in vitreâ phialâ ad ignem expoſitâ, per aquam aſcendit <lb></lb>velociùs, quàm deinde extra aquam poſita aſcendat in aëre, <lb></lb>ubi fumeam naturam induerit. </s> <s id="s.000174">Unde patet non adeò ſolidum <lb></lb>ab aliquibus ex hoc experimento ſumi argumentum negandi <lb></lb>poſitivam levitatem. </s> <s id="s.000175">Quæ enim de gravibus ex comparatione <lb></lb>cum levibus dicuntur, ea de levibus, proportione ſervatâ, di<lb></lb>cenda ſunt, ſi cum gravibus conferantur. </s> <s id="s.000176">Cur autem gravibus <lb></lb>leviora, levibus graviora minùs reſiſtant, ratio eſt, quia mo<lb></lb>bile movetur in medio propter diſſimilitudinem; nam ſi corpus <lb></lb>contiguum eſſet, ſimile non moveretur; quando igitur major <lb></lb>eſt diſſimilitudo, debet velociùs moveri, ſegniùs autem, & len<lb></lb>tiùs, quò propiùs abeſt à ſimilitudine, donec in ſimili demum <lb></lb>quieſcat. </s> </p> <p type="main"> <s id="s.000177">Eſt itaque in corporibus gravitas, & levitas, vi cujus motus ali<lb></lb>quos juxta naturæ propenſionem perficiunt, ut certo denique in <lb></lb>loco conſiſtant, ejuſdemque vi reſiſtunt, ne oppoſitis motibus <lb></lb>cieantur, & à ſuæ quietis loco avellantur. </s> <s id="s.000178">Quamvis autem <expan abbr="eadẽ">eadem</expan> <lb></lb>maneat gravitas aut levitas, non <expan abbr="idẽ">idem</expan> tamen eſt ſemper <expan abbr="momentũ">momentum</expan> <lb></lb>(Græcis <foreign lang="grc"><gap></gap>πη</foreign>) hoc eſt actualis ad motum inclinatio, dum in actio<lb></lb>ne eſt; hæc enim, ut infra patebit, ut plurimum ex poſitione, & <lb></lb>ſitu mutatur, vel comparatè ad <expan abbr="mediũ">medium</expan>, in quo perficitur motus. <pb n="14" xlink:href="017/01/030.jpg"></pb> </s> </p> <p type="head"> <s id="s.000179"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000180"><emph type="center"></emph><emph type="italics"></emph>Quid ſit centrum gravitatis, & linea directionis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000181">QUamvis non minùs levitate, quàm gravitate prædita ſint <lb></lb>corpora, quia tamen frequentiùs gravitatem vincere co<lb></lb>namur, quàm levitatem; ideò illa potiſſimùm cadit ſub con<lb></lb>templationem ſcieǹtiæ Machinalis: vix enim aliquando con<lb></lb>tingere poterit, ut opus ſit infra aquam corpus aliquod leve <lb></lb>per vim deprimere. </s> <s id="s.000182">Hinc factum eſt, ut de ſolo gravitatis cen<lb></lb>tro ſermo communiter ſit, levitatis autem centrum ſilentio <lb></lb>obvolvatur: quia nimirùm quæ de gravitate deſcendente ex<lb></lb>plicantur, ea de levitate aſcendente, pro rata portione, dicta <lb></lb>facile intelliguntur. </s> </p> <p type="main"> <s id="s.000183">Ad centrum terræ (quod & centrum gravium ac levium <lb></lb>dicimus) properant corpora quæcumque gravia in medio le<lb></lb>viore conſtituta ſibi redduntur, ut motus ſuos perficiant. </s> <s id="s.000184">Quo<lb></lb>niam verò natura finem propoſitum per media, quæ poteſt, bre<lb></lb>viſſima proſequitur, ambages, & diverticula fugiens; mo<lb></lb>ventur per lineam rectam, ut pote breviſſimam, niſi externo <lb></lb>aliquo impedimento cogantur à rectitudine deflectere: Hæc <lb></lb>autem recta linea intelligi debet ex terræ centro ducta ad cor<lb></lb>pus ipſum, quod movetur; ac proinde tùm in ſphæricam ſu<lb></lb> <figure id="id.017.01.030.1.jpg" xlink:href="017/01/030/1.jpg"></figure><lb></lb>perficiem, tùm in planum Horizon<lb></lb>tis ad perpendiculum cadit. </s> <s id="s.000185">Sed quia <lb></lb>corpus, quod deorſum contendit, <lb></lb>plures habet partes, quibus conſtat, <lb></lb>ſingulas ſuâ gravitate præditas, lineæ <lb></lb>verò à ſingulis hiſce partibus exeun<lb></lb>tes in terræ centro concurrunt; fieri <lb></lb>non poteſt, ut ſervatâ corporis figu<lb></lb>râ, atque continuo partium nexu non <lb></lb>diſſoluto, per rectam ſuam lineam ad <lb></lb>centrum ductam unaquæque pars <lb></lb>deſcendat. </s> <s id="s.000186">Si enim parallelepipe<lb></lb>dum AB in aëre dimittatur, ut ſpon- <pb n="15" xlink:href="017/01/031.jpg"></pb>te ſua deſcendat, fieri non poteſt, ut A rectam AC percur<lb></lb>rat, quin oppoſitum extremum B à recta BC longiſſime rece<lb></lb>dat, & contra: utramque verò extremitatem ſimul A & B <lb></lb>rectâ in centrum C tendere non poſſe eſt manifeſtum: Quare <lb></lb>cum ſibi invicem obſiſtant æqualiter, ob gravitatis æqualita<lb></lb>tem, eas ex perpendicularibus AC, BC æqualiter ſecedere <lb></lb>oportet ad latera, atque parallelas BE, AF deſcendendo deſ<lb></lb>cribere. </s> <s id="s.000187">Eadem eſt ratio de cæteris partibus æquali intervallo <lb></lb>ſejunctis à medio D; omnes enim à ſuis perpendiculis rece<lb></lb>dunt, præter punctum medium D, cujus perpendicularis <lb></lb>DC parallela eſt lineis à reliquis partibus in motu deſcriptis. </s> <lb></lb> <s id="s.000188">Ex omnibus itaque particulis datum grave componentibus, eæ <lb></lb>ſolùm, quæ puncto D imminent, per rectam DC in centrum <lb></lb>moventur; quæ tàm plano horizontis in C, quàm ſuperficiei <lb></lb>ſphæricæ in H perpendicularis eſt; cæteræ verò parallelæ BE, <lb></lb>AF perpendiculares quidem in horizontem cadunt, ſed ſphæ<lb></lb>ricam ſuperficiem obliquè ſecant. </s> </p> <p type="main"> <s id="s.000189">Jam verò ſi ejuſdem parallelepipedi aliud planum AO hori<lb></lb>zonti parallelum moveri versùs C intelligas, erit in eo ſimiliter <lb></lb>aliud punctum unicum, quod rectam DC percurrat; & intra <lb></lb>corporis ſoliditatem unica linea puncto illi imminens viâ eâdem <lb></lb>in centrum perget non declinans à perpendiculo: cæteræ partes, <lb></lb>tam quæ ad <expan abbr="dextrã">dextram</expan>, quàm quæ ad <expan abbr="levã">levam</expan>, tam quæ antè, quàm quæ <lb></lb>ponè, ſibi mutuò adverſantes à recto in <expan abbr="centrũ">centrum</expan> itinere deflectent <lb></lb>æqualiter. </s> <s id="s.000190">Cum itaque, in priori poſitione, linea puncto D <lb></lb>imminens, eſſet in communi ſectione planorum, quorum alte<lb></lb>rum partes dextras à ſiniſtris, alterum anteriores à poſterioribus <lb></lb>æqualiter ſecernebat; in ſecundâ autem poſitione linea à per<lb></lb>pendiculo non recedens ſit quoquè in duorum planorum com<lb></lb>muni ſectione, quibus pariter corporis gravitas in æquas tribui<lb></lb>tur partes; unum verò ex planis ſecantibus ſit utrique poſitioni <lb></lb>commune; unicum eſt punctum tribus planis commune, in quo <lb></lb>binorum planorum ſectiones ſe invicem ſecant, & ſit ex. gr. <lb></lb>punctum I; quod unicum rectâ pergit in centrum C, quemcum<lb></lb>que tandem ſitum in motu obtineat corpus datum AB, ipſum <lb></lb>enim eſt duabus illis lineis commune, quæ in ſingulis poſitioni<lb></lb>bus ad ſui perpendiculi latera non recedunt: cætera illarum li<lb></lb>nearum puncta, mutatâ poſitione corporis, lineam quoque mo<lb></lb>tûs mutant. </s> </p> <pb n="16" xlink:href="017/01/032.jpg"></pb> <p type="main"> <s id="s.000191">Illud itaquè punctum in quocumque corpore gravi, quod <lb></lb>ſemper in motu deſcribit lineam rectà in terræ centrum <lb></lb>ductam, dicitur <emph type="italics"></emph>Centrum Gravitatis<emph.end type="italics"></emph.end>; & linea, quæ centrum <lb></lb>gravitatis conjungit cum terræ centro, <emph type="italics"></emph>Linea directionis<emph.end type="italics"></emph.end> dicitur; <lb></lb>ſecundùm quam videlicet dirigitur motus, & dimentienda eſt <lb></lb>corporis à centro terræ diſtantia, ſi quatenus grave conſidere<lb></lb>tur. </s> <s id="s.000192">Porrò punctum I centrum gravitatis dicitur, quia centri <lb></lb>nomen tribuitur puncto, quod eſt medium: & quemadmodum <lb></lb>magnitudinis alicujus centrum vocatur punctum illud, quod <lb></lb>æquales magnitudines circunſtant, ſi partes, quæ ex adverſo <lb></lb>ſunt, accipiantur; ita in gravibus centrum gravitatis dicitur, <lb></lb>quod æquales gravitates, vel æqualia gravitatum momenta cir<lb></lb>cunſtant. </s> <s id="s.000193">Quod ſi punctum I non haberet hinc, & hinc æqua<lb></lb>les gravitatum vires, ab alterutrâ parte præſtante viribus pro<lb></lb>pelleretur in latus extra lineam directionis, à quâ nunquam re<lb></lb>cedit, ſi liberè moveatur. </s> <s id="s.000194">Cave tamen, ne partium æqualita<lb></lb>tem dimetiaris linearum longitudine à céntro gravitatis exeun<lb></lb>tium, ita ut ſingulas lineas æqualiter dividendas putes; ſed to<lb></lb>tum corpus debet intelligi diviſum bifariam à plano per cen<lb></lb>trum gravitatis ipſius corporis, & per centrum gravium ac le<lb></lb>vium tranſeunte, ita ut ſi planum à dextrâ in ſiniſtram ductum <lb></lb>ſecernat partes anteriores à poſterioribus, æqualia ſint gravita<lb></lb>tum momenta antè, & ponè; ſi aliud planum per eandem di<lb></lb>rectionis lineam ductum partes dextras à ſiniſtris diſtinguat pa<lb></lb>ria ſimiliter hinc & hinc gravitatum momenta relinquat. </s> </p> <p type="main"> <s id="s.000195">Gravitatum, inquam, momenta, non gravitates; ne locus <lb></lb>pateat æquivocationi; neque enim quoties æqualia ſunt mo<lb></lb>menta, toties æquales ſunt gravitates hinc & hinc centrum gra<lb></lb>vitatis complectentes, ut patebit ex iis, quæ de æquilibrio dice<lb></lb>mus. </s> <s id="s.000196">Unde fit in iis tantùm corporibus, quæ partibus unius ejuſ<lb></lb>demque naturæ, ac ductu perpetuo ſimiliter conſtitutis, <expan abbr="conſtãt">conſtant</expan>, <lb></lb> <figure id="id.017.01.032.1.jpg" xlink:href="017/01/032/1.jpg"></figure><lb></lb>idem eſſe centrum gravitatis atque magni<lb></lb>tudinis; reliqua certis regulis non circum<lb></lb>ſcripta, aut ex variis naturis compoſita, in <lb></lb>alio puncto, molis centrum habere, in alio, <lb></lb>gravitatis. </s> <s id="s.000197">Si enim duo ſolida VT, cujus <lb></lb>centrum gravitatis, & magnitudinis R, <lb></lb>& MN, cujus centrum S, æqualia ſecun- <pb n="17" xlink:href="017/01/033.jpg"></pb>dùm gravitatem coagmententur, non erit centrum gravitatis <lb></lb>totius molis compoſitæ in I, ubi planum tranſiens per VN ſe<lb></lb>cat lineam RS jungentem centra ſingularum gravitatum æqua<lb></lb>lium, ſed erit in L, ubi recta RS bifariam dividitur: planum <lb></lb>autem per centrum terræ, & punctum L ductum non ita ſecat <lb></lb>hanc molem, ut ſint æquales hinc, & hinc gravitates, quamvis <lb></lb>æqualia ſint gravitatum inæqualium momenta, quæ ex figuræ <lb></lb>poſitione potiſſimùm pendent. </s> <s id="s.000198">Quod ſi corporis VT gravitas <lb></lb>ad corporis MN gravitatem, eam haberet rationem, quam SI <lb></lb>ad IR, eſſet I gravitatis centrum molis compoſitæ, quæ à plano <lb></lb>per terræ centrum, & punctum I ducto non in gravitates æqua<lb></lb>les, ſed in momenta æqualia divideretur; ut in loco inferiùs ex<lb></lb>plicabitur. </s> </p> <p type="main"> <s id="s.000199">Obſerva autem non ſemper centrum gravitatis eſſe in ipſo <lb></lb>corpore gravi, ut patet in corporibus annularibus, aut angulos <lb></lb>cavos habentibus, in quibus nullum eſt punctum per quod tran<lb></lb>ſeuntia plana quæcunque dividant in æquas partes momenta <lb></lb>gravitatum: ita tamen eſt extra corporis cavi ſoliditatem, ut ſit <lb></lb>intra ipſam cavitatem punctum, ex quo ſi intelligatur annulus, <lb></lb>vel fruſtum annulare ſuſpendi, manet poſitionem habens hori<lb></lb>zonti parallelam, cum habeat æqualia hinc, & hinc gravita<lb></lb>tum momenta. </s> <s id="s.000200">Quod ſi corpus in cavos angulos ſinuatum ha<lb></lb>beat particulam aliquam procurrentem, poteſt contingere, ut <lb></lb>in illius particulæ extremo ſit totius molis centrum gravitatis: <lb></lb>ſic brevioris alicujus bacilli extremitati alteri ſi duos cultros in<lb></lb>fixeris, ut ſinguli cum bacillo hinc, & hinc angulum acutum <lb></lb>ad eaſdem partes conſtituant, ita inclinari poſſunt, ut extremo <lb></lb>ungue ſuppoſito reliquæ bacilli extremitati tota illa moles ſuſti<lb></lb>neatur citrà periculum cadendi, cùm gravitatis centrum in illa <lb></lb>extremitate, intrà cavitatem, quam inclinati cultri faciunt, <lb></lb>æqualia habeat ex omni parte gravitatum momenta, ſi planum <lb></lb>ſecans per illud tranſeat. <pb n="18" xlink:href="017/01/034.jpg"></pb></s> </p> <p type="head"> <s id="s.000201"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000202"><emph type="center"></emph><emph type="italics"></emph>An gravia centro vicina minùs gravitent.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000203">COrpora non intelliguntur gravitare niſi in alieno loco; <lb></lb>quando ſcilicet corpus contiguum inter illa & centrum <lb></lb>terræ interjectum, quod medii rationem habere poteſt, levius <lb></lb>eſt; petit enim infra illud eſſe: niſum autem hunc deorsùm <lb></lb><emph type="italics"></emph>Gravitationem<emph.end type="italics"></emph.end> dicimus. </s> <s id="s.000204">Sed quoniam niſus iſte videtur idcircò <lb></lb>à naturâ inſtitutus, ut perturbatus corporum ordo reſtituatur; <lb></lb>ſi ex fine ratio petenda ſit, ſatis apparet corpora gravia ćentro <lb></lb>terræ vicina minùs gravitare. </s> <s id="s.000205">Quemadmodum enim quotieſ<lb></lb>cunque aliquis à propoſito fine magis diſtat, eò magis anxius <lb></lb>eſt, atque ſolicitus de mediis ad illum aſſequendum neceſſariis, <lb></lb>& animo æquiore toleratur modica, quàm multa violentia; ita <lb></lb>natura minorem ordinis debiti perturbationem ſentiens, ſi gra<lb></lb>ve parùm abſit, quàm ſi longè abeſſet, à loco, ubi juxta inge<lb></lb>nitam propenſionem exigit conſiſtere, minùs ſolicita eſſe debet <lb></lb>de illo reſtituendo, nec adeò vehementi conatu, hoc eſt gravi<lb></lb>tatione, illud urgere debet in locum ſuum. </s> </p> <p type="main"> <s id="s.000206">Ad hæc omnibus apertiſſimè liquet eò majore naturæ impe<lb></lb>tu corpora deorsùm niti, quò levius eſt corpus, in quo tan<lb></lb>quam in medio perficiendus eſt motus, ſi dimittantur. </s> <s id="s.000207">Sic à <lb></lb>ſaxo in aëre pendente manum deorsùm validiùs trahi ſenti<lb></lb>mus, quàm ab eodem aquæ immerſo trahatur, & multò lan<lb></lb>guidiùs conatur deorſum lapis in melle deſcendens, quàm in <lb></lb>aqua; quia videlicet aqua levior eſt melle, & aër levior aquâ. </s> <lb></lb> <s id="s.000208">Hinc eſt quod, ſi medij partes fuerint diversâ gravitate prædi<lb></lb>tæ, pars centro terræ propior etiam erit gravior; atque ideò <lb></lb>corpus in parte medij graviore minùs gravitabit propè centrum <lb></lb>terræ, quàm procul. </s> <s id="s.000209">Eſſe autem ejuſdem medij non commoti <lb></lb>partes graviores in imo, omnium ferè hominum ſenſus eſt: <lb></lb>quotus enim quiſque eſt, qui neſciat mellis optimam partem <lb></lb>eſſe, quæ in vaſis fundo, vini quæ in medio, olei quæ in ſum<lb></lb>mo? </s> <s id="s.000210">id autem verum non eſſet, niſi liquoris ejuſdem partes <lb></lb>eſſent diversâ gravitate delatæ in loca à terræ centro diſpari- <pb n="19" xlink:href="017/01/035.jpg"></pb>bus intervallis remota: Quia enim oleum eò perfectius eſt, <lb></lb>quò propiùs aëris levitatem ſpirituum ſubtilitate æmulatur, <lb></lb>ideò quod in ſummo vaſe innatat, optimum eſt: At vini ſua<lb></lb>vitas in exquiſitâ ſui tartari ſufficienti humore diluti cum ſpi<lb></lb>ritibus permiſtione conſiſtens medium locum in vaſe exigit, <lb></lb>ſicut media eſt illius gravitas inter vagantium ſpirituum levita<lb></lb>tem, & fæculenti tartari gravitatem: Mellis demùm dulcedo <lb></lb>ex ſui ſalis, ſeu ſacchari, copiâ proveniens iis partibus potiſſi<lb></lb>mum ineſt, quæ multo ſale refertæ graviores quoquè ſunt, & <lb></lb>in fundo ſubſidunt. </s> <s id="s.000211">Nec eſt iis abroganda fides, qui in altiſſi<lb></lb>mo mari adeò gravem aquam à ſe deprehenſam alicubi teſtan<lb></lb>tur, ut ſupta reliquum maris fundum ambulantes ad altiſſi<lb></lb>mam foſſam venerint, in quam penetrare ſæpiùs irrito conatu <lb></lb>tentârint: his enim non ægrè fidem habeo, qui aërem in imis <lb></lb>vallibus craſſiorem atquè graviorem, in ſummis verò montibus <lb></lb>puriorem atque leviorem ab omnibus admitti video. </s> <s id="s.000212">Cum ita<lb></lb>que (ſi ex notis ad minùs nota progredi philoſophando liceat) <lb></lb>propè centrum gravium ac levium medij partes graviores ſint, <lb></lb>quam procul ab illo; minor eſt gravitatio corporum, ſi centro <lb></lb>propiora fiant, ac quando longè ab illo remota detinebantur. </s> <lb></lb> <s id="s.000213">Hinc autem reſponderi poteſt quærentibus, cur in fodinis lon<lb></lb>gè faciliùs crudi metalli maſſa moveatur, quàm in ſuperficie <lb></lb>terræ: aër ſcilicet profundis illis cuniculis incluſus gravior mul<lb></lb>tò ac craſſior eſt aëre iſto, quem inſpiramus, atque adeò ibi <lb></lb>metallum minùs gravitat. </s> </p> <p type="main"> <s id="s.000214">Quòd ſi libeat minorem hanc gravitationem experimento <lb></lb>deprehendere, ſume vitream fiſtulam ſupernè clauſam longio<lb></lb>rem pedibus tribus Romanis, eam imple argento vivo, digito<lb></lb>que oſculum accuratè claudens inverte, ac argento vivo ſub<lb></lb>jecti vaſis immerge; tùm amoto digito deſcendet mercurius in <lb></lb>fiſtulâ, iterúmque aſcendet, & in certâ demum altitudine per<lb></lb>pendiculari quieſcet. </s> <s id="s.000215">Obſervatâ igitur altitudine perpendicu<lb></lb>lari, quam mercurius obtinet, ſi in imâ valle experimentum <lb></lb>inſtituatur, eâque comparatâ cum altitudine perpendiculari, <lb></lb>in qua conſiſtit, cùm in ſummo montis altiſſimi vertice expe<lb></lb>rimentum idem ſumitur, animadvertes altitudinem mercurij <lb></lb>per vim in fiſtulâ ſuſpenſi minorem eſſe in ſummo monte, quàm <pb n="20" xlink:href="017/01/036.jpg"></pb>in valle; Quia nimirum mercurius intra fiſtulam detentus tan<lb></lb>quàm in vaſe, eſt in aëre fiſtulam ambiente tanquam in loco; <lb></lb>in aëre autem leviori cùm magis gravitet, in minori etiam al<lb></lb>titudine perpendiculari conſiſtit. </s> <s id="s.000216">Experimentum hoc in valle, <lb></lb>& in monte ſumere mihi otium non fuit, quamvis in eo ſæ<lb></lb>piùs me exercuerim: ſed de illius veritate ambigere non ſinunt <lb></lb>teſtes in Galliâ luculentiſſimi, qui diſcrimen hoc in mercuri; <lb></lb>altitudine obſervârunt in altioribus montibus. </s> </p> <p type="main"> <s id="s.000217">Verùm, ex alio præteteà capite imminui debet gravitatio <lb></lb>corporum in minori à centro remotione, habitâ ſolùm ratione <lb></lb>ſitûs. </s> <s id="s.000218">Cùm enim totius corporis gravitatio conflata ſit ex ſin<lb></lb>gularum partium impetu, quo deorſum nituntur, manifeſtum <lb></lb>eſt ſingulis partibus languidiùs deorſum conantibus, totius cor<lb></lb>poris gravitationem eſſe pariter languidiorem. </s> <s id="s.000219">Quoniam verò <lb></lb>quicquid in motu cogitur à recto ſecundùm naturam tramite <lb></lb>deflectere, lentiùs atque remiſſiùs pergit ad præſtitutum mo<lb></lb>tûs terminum; particulæ autem corporis ſolidi gravis, propio<lb></lb>res centro factæ, magis à ſuo perpendiculo, ſibi invicem ad<lb></lb>verſantes, declinant; ſatis conſtat ſingulas fractis quodammo<lb></lb>do viribus languentes plurimum de conatu remittere. </s> <s id="s.000220">Si enim <lb></lb> <figure id="id.017.01.036.1.jpg" xlink:href="017/01/036/1.jpg"></figure><lb></lb>ſolidum AB fiat centro vicinius ita, <lb></lb>ut A ſit in K, & B in L, lineæ di<lb></lb>rectionis partium extremarum ſunt <lb></lb>KC, LC: at coguntur per lineas <lb></lb>KF, LE parallelas deſcendere, <lb></lb>fiuntque anguli CKF, CLE ex<lb></lb>terni majores internis CAK, CBL <lb></lb>per 16. l. 1. magis igitur in K & L <lb></lb>recedunt à perpendiculo, quàm re<lb></lb>cederent in A & B. </s> <s id="s.000221">Quia itaque <lb></lb>pars in K exiſtens magis impeditur <lb></lb>ab oppoſitâ extremitate, quæ in L, <lb></lb>ne per KC deſcendat (niſi enim <lb></lb>pars, quæ in L, urgeret oppoſitam <lb></lb>tentans per LC deſcendere, non cogeretur pars in K exiſtens <lb></lb>adeò recedere à ſuâ directionis lineâ) minori etiam impetu <lb></lb>deorſum fertur. </s> <s id="s.000222">Eſt autem eadem de reliquis partibus ratio, <pb n="21" xlink:href="017/01/037.jpg"></pb>præter eas, quæ in eâdem directionis lineâ ſunt cum centro <lb></lb>gravitatis; ſingulæ enim ad centrum terræ accedentes magis à <lb></lb>ſuo perpendiculo recedunt, minúſque deorſum gravitant. </s> <s id="s.000223">Quî <lb></lb>igitur fieri poſſit, ut debilitato ſingularum particularum cona<lb></lb>tu, atque impetu deorſum, non minuatur pariter totius cor<lb></lb>poris gravitatio, ſi fiat centro vicinius? </s> </p> <p type="main"> <s id="s.000224">Illud tamen non diffiteor, quod ſi medij levitates, aut angu<lb></lb>lorum CLE, CBL inclinationes eo tantùm diſcrimine ſecer<lb></lb>nantur, quod omnem ſenſum fugiat, vel ſaltem ex medij gra<lb></lb>vitate, & anguli magnitudine conjunctim ſumptis oriri non <lb></lb>poſſit varietas, quæ ſub ſenſum cadat; neque percipietur gra<lb></lb>vitationis differentia in majori vicinitate. </s> <s id="s.000225">Sed hoc non facit, <lb></lb>quin inter gravitationes diſcrimen intercedat; neque enim <lb></lb>continuò, ſi quid ſenſum latet, id omninò non eſſe dicendum <lb></lb>eſt: contingere ſi quidem poteſt motum aliquem ita ſenſim, & <lb></lb>ſine ſenſu fieri, ut non niſi elapſo temporis ſpatio demùm inno<lb></lb>teſcat. </s> <s id="s.000226">Sic ſi vinum, cujus gravitas vix minor ſit gravitate aquæ <lb></lb>arte ſatis notâ affuderis aquæ ita, ut innatet, & ſupremam va<lb></lb>ſis partem occupet, aliudque vas ſimili vino plenum, ſed paulò <lb></lb>altius, habeas, tum ex libra centrum motûs habente in cen<lb></lb>tro gravitatis jugi pendeant æqualia pondera intrà vinum <lb></lb>utriuſque vaſis; fiet utique ponderum æquilibrium, & con<lb></lb>ſiſtent eo in ſitu, quem illis dederis: at ſi alterum libræ extre<lb></lb>mum ita deprimas, ut pondus, quod ex eo pendet, ex vino ad <lb></lb>aquam vix graviorem tranſeat, reliquo pondere intra vinum <lb></lb>manente; initio quidem non apparebit motus libræ ſe reſti<lb></lb>tuentis, quia pondus in vino non excedit gravitationem pon<lb></lb>deris æqualis in aquâ niſi eo exceſſu, quo gravitas aquæ ſupe<lb></lb>rat gravitatem vini; hic autem exceſſus cum minimus ſit, mo<lb></lb>tum quoque efficiet, quem ægrè à quiete diſcernas, niſi ubi <lb></lb>poſt aliquod tempus deprehenderis pondus altius deſcendiſſe, <lb></lb>depreſſius autem aſcendiſſe. </s> <s id="s.000227">Haud ſecus philoſophandum eſt <lb></lb>de majore, aut minore corporum gravitatione, ſi diſparibus in<lb></lb>tervallis à terræ centro removeantur, diutiùs enim propè cen<lb></lb>trum incumbere poterunt ſuſtinenti, quàm procul: id quod <lb></lb>ſatis erit ad minorem gravitationem patefaciendam, quæ non <lb></lb>ſtatim innoteſcat. </s> </p> <pb n="22" xlink:href="017/01/038.jpg"></pb> <p type="main"> <s id="s.000228">Hæc autem non leviter confirmari videntur ex iis, quæ quo<lb></lb>tidiè ferè videmus; nam ſi circinus, quo circulos deſcribere <lb></lb>ſolemus, cadat, ſemper nodus prævertit cuſpides, & prior ter<lb></lb>ram ferit; niſi fortè nodus ad perpendiculum immineat cru<lb></lb>ribus: & omnia ferè corpora, quæ centrum gravitatis ex una <lb></lb>parte habent, ſi ex modicâ altitudine dimittantur, videntur <lb></lb>quidem cadere parallela; ſed ex majori altitudine ſi deſcen<lb></lb>dant, pars gravior prior terram attingit. </s> <s id="s.000229">Sit enim corpus ES, <lb></lb> <figure id="id.017.01.038.1.jpg" xlink:href="017/01/038/1.jpg"></figure><lb></lb>cujus gravitatis centrum H, linea <lb></lb>directionis HA; ſi horizonti paral<lb></lb>lelum deſcenderet, per rectas EI, <lb></lb>SR parallelas lineæ directionis mo<lb></lb>veretur; id quod in modicâ tantùm <lb></lb>altitudine contingere videtur, quia <lb></lb>nondum facta eſt ea gravitationis <lb></lb>imminutio in extremitate S, quæ <lb></lb>percipi poſſit. </s> <s id="s.000230">Si enim E per EI <lb></lb>deſcenderet, S verò per SR, an<lb></lb>gulus IEA æqualis alterno EAH <lb></lb>per 29. lib. 1. minor eſſet angulo <lb></lb>RSA, qui æqualis eſt alterno <lb></lb>HAS; nam ex hypotheſi minùs <lb></lb>diſtat E, quàm S, à centro gravi<lb></lb>tatis H, & eſt angulus EAH minor angulo HAS; pars igi<lb></lb>tur S magis deflecteret à ſuo perpendiculo SA, quàm E de<lb></lb>flecteret ab EA; cùm itaque S magis in latus propelleretur, <lb></lb>plus etiam de conatu deorſum remitteret, quàm E; atque adeò <lb></lb>non poſſet æqualiter deſcendere ac moveri, contra hypotheſim <lb></lb>paralleliſmi. </s> <s id="s.000231">Dicendum eſt igitur non per parallelas EI, SR <lb></lb>fieri motum, ſed intra illas paulatim partem E graviorem præ<lb></lb>currere: quia ſcilicet partes omnes extra lineam directionis <lb></lb>AH conſtitutæ dum removentur à ſuo perpendiculo, aliquid <lb></lb>amittunt de impetu, quo deorſum nituntur, propiores quidem <lb></lb>minus, remotiores autem plus; pars ſi quidem G in principio <lb></lb>motûs deſcendens parallela lineæ directionis per GM facit an<lb></lb>gulum AGM internum per 16.lib.1. minorem externo GMS, <lb></lb>qui per 29. 1. eſt æqualis alterno MSR. </s> <s id="s.000232">Quia ergo AGM <pb n="23" xlink:href="017/01/039.jpg"></pb>minor eſt angulo ASR, pars G minus de ſuo impetu deorſum <lb></lb>amittit, quàm pars S; & quamvis initio diſcrimen hoc non <lb></lb>percipiatur, demum fit, ut additis pluribus differentiis mani<lb></lb>feſtè appareat partem S minùs gravitare, quia tardiùs deor<lb></lb>ſum movetur; & tandem ipſa ſequitur partem E præcur<lb></lb>rentem, poſtquam minori illâ gravitatione permiſit parti E, <lb></lb>ut propiùs accederet ad lineam directionis, fieretquè quæ<lb></lb>dam virtualis converſio circa centrum gravitatis H, in qua <lb></lb>extremitas E occuparet infimum locum, S autem ſupre<lb></lb>mum. </s> <s id="s.000233">Quare cùm nos doceat experientia partem HS <lb></lb>æquiponderantem parti HE, ſi ſuſpendantur ex H, in mo<lb></lb>tu tamen minùs gravitare, quàm oppoſitam, ideóque fieri <lb></lb>illam converſionem, ut pars E fiat inferior; neque aptior <lb></lb>aſſignari poſſit ratio, quàm quæ petitur ex receſſu partium <lb></lb>majori à ſuo perpendiculo: ſatis liquet, quantum momenti <lb></lb>habeat hæc declinatio à perpendiculo ad minuendam gra<lb></lb>vitationem. </s> <s id="s.000234">Ex majori igitur declinatione à lineâ perpen<lb></lb>diculari, quæ conſequitur corpus conſtitutum non adeò <lb></lb>procul à centro terræ ut priùs, non ineptè arguitur minor <lb></lb>corporis gravitatio in eo ſitu, ſi cætera ſint paria: neque <lb></lb>enim comparo corpus, quod per motum deſcendit, perſe<lb></lb>verans in ſuo motu, cum corpore in loco altiori tranſeun<lb></lb>te à quiete ad motum; nam tunc ex impetu per motum <lb></lb>concepto major eſt gravitatio in loco inferiore, quàm in ſu<lb></lb>periore: ſed tantùm corpora invicem comparo, vel pariter <lb></lb>quieſcentia, vel æquali tempore mota, illudque, quod ter<lb></lb>ræ vicinius eſt, aſſero, vel minori niſu conari à quiete in <lb></lb>loco alieno tranſire ad motum, vel æquali tempore, quo præ<lb></lb>ceſſit motus, minus impetus acquiſiiſſe ac minoribus viribus <lb></lb>motum continuare. </s> </p> <p type="main"> <s id="s.000235">Ex his quæ de gravibus hactenus diſputata ſunt, aliquis <lb></lb>fortaſsè inferat levia à centro remotiora minùs levitare, ſi<lb></lb>cut gravia centro propiora minùs gravitant. </s> <s id="s.000236">Verùm res eſt <lb></lb>penſiculatiùs examinanda, nec ſimpliciter ex oppoſitis gra<lb></lb>vium, ac levium naturis definienda, quaſi ob id ipſum, <lb></lb>quia ſibi gravitas atque levitas adverſantur, contraria ha<lb></lb>berent omnia conſequentia. </s> <s id="s.000237">Et quidem quod ſpectat ad <pb n="24" xlink:href="017/01/040.jpg"></pb>ſolam corporis levioris poſitionem, non minuitur levitatio, <lb></lb>ſed potiùs augetur in majoribus à terræ centro intervallis; <lb></lb>ubi minùs à ſuo perpendiculo declinant partes centrum le<lb></lb>vitatis circunſtantes, & idcirco minùs de conatu remit<lb></lb>tunt, quò nituntur ad ſupe<lb></lb> <figure id="id.017.01.040.1.jpg" xlink:href="017/01/040/1.jpg"></figure><lb></lb>riora evadere. </s> <s id="s.000238">Sit namque <lb></lb>Globus HG, cujus centrum <lb></lb>levitatis M, & linea diſcretio<lb></lb>nis OMN; cui parallelæ <lb></lb>ſunt HD & GF, quas deſ<lb></lb>cribunt aſcendendo extremi<lb></lb>tates H & G, & motum eum<lb></lb>dem continuabunt, ſi globus <lb></lb>in N tranſlatus intelligatur. </s> <lb></lb> <s id="s.000239">Quando igitur globus eſt in M, <lb></lb>extremitas H recedit à per<lb></lb>pendiculo OI, & cum eo <lb></lb>facit angulum IHT; quan<lb></lb>do autem eſt in N, extremi<lb></lb>tas T aſcendens per TD fa<lb></lb>cit cum perpendiculo OR an<lb></lb>gulum RTD, qui per 15.lib.1. <lb></lb>æqualis eſt angulo HTO ad <lb></lb>verticem, hic autem, inter<lb></lb>nus cum ſit, per 16. 1. minor <lb></lb>eſt externo IHT. </s> <s id="s.000240">Eſt ergo <lb></lb>RTD minor angulo IHT, <lb></lb>atque ideò plus habet mo<lb></lb>menti ſurſum, ubi minus à <lb></lb>recto ſecundum naturam tra<lb></lb>mite deflectit. </s> </p> <p type="main"> <s id="s.000241">Diſcrimen hoc momentorum ab angulorum inæqualitate <lb></lb>proveniens optimè intelligit natura, quæ ita motum perfi<lb></lb>cit, ut, ſi duo inæqualiter levia coagmentata fuerint, le<lb></lb>vius præcurrat. </s> <s id="s.000242">Sic ſi A cortex ſuberis coagmentetur ligno <lb></lb>fagino B, & intra aquam mediocriter profundam horizon<lb></lb>taliter collocetur ſolidum DC, ita per lineam directio- <pb n="25" xlink:href="017/01/041.jpg"></pb>nis TO aſcendit centrum <lb></lb> <figure id="id.017.01.041.1.jpg" xlink:href="017/01/041/1.jpg"></figure><lb></lb>levitatis, ut demum A in <lb></lb>loco ſuperiore, B autem in <lb></lb>inferiore conſtituatur, ex<lb></lb>tremo D per rectam DO <lb></lb>aſcendente: Quo in motu <lb></lb>natura magnum invenit <lb></lb>compendium. </s> <s id="s.000243">Quia enim <lb></lb>partes centro levitatis vi<lb></lb>ciniores magis levitant, <lb></lb>quòd linea parallela lineæ <lb></lb>directionis faciat minorem <lb></lb>angulum cum earum per<lb></lb>pendiculo (ſic ſi linea di<lb></lb>rectionis ſit FL, eique pa<lb></lb>rallelæ NG, RX, angu<lb></lb>lus NGX internus per <lb></lb>29. 1. eſt æqualis externo <lb></lb>RXY, at PGX externus per 16. 1. major eſt interno GXF, <lb></lb>hoc eſt VXY ad verticem, ergo PGX major eſt angulo <lb></lb>VXY, & ſi uterque auferatur ex æqualibus NGX, RXY, <lb></lb>remanet NGP minor angulo RXV, ideoque G magis levi<lb></lb>tat, quam X) ex majore impedimento, quod initio motûs ha<lb></lb>betur ob anguli HDI magnitudinem, dum pars D minùs le<lb></lb>vitat, centrum levitatis per SO aſcendens inclinat corpus DC, <lb></lb>& extremitas D in recta DO conſtituitur, in qua longê ci<lb></lb>tiùs minuuntur impedimenta, quàm ſi per parallelam DI <lb></lb>aſcenderet: vix enim aſcendit in E, cum impedimenta ſunt <lb></lb>æquè diminuta, ac ſi aſcendiſſet in I; quandoquidem angu<lb></lb>lus KEI per 29. 1. eſt æqualis alterno EID, atque adeò <lb></lb>etiam angulo, quem in I faceret parallela DI cum perpendi<lb></lb>culo; eſt igitur angulus KEI minor quocunque alio angulo, <lb></lb>qui fieret in punctis intermediis lineæ DI; ſed quoniam cen<lb></lb>trum levitatis aſcendendo acquiſivit majorem impetum, quàm <lb></lb>extremitas in E exiſtens, per vim illam rapit extra paralle<lb></lb>lam EK, trahitque per lineam EO, & perpendiculum facit <lb></lb>angulum ſemper minorem cum lineâ directionis; unde fit <lb></lb>partem inferiorem ſemper faciliùs trahi, quo minùs in diverſa <pb n="26" xlink:href="017/01/042.jpg"></pb>abit ejus perpendiculum, cum quo ſemper minorem, & mi<lb></lb>norem angulum facit linea motûs DO; donec demùm to<lb></lb>tum ſolidum obtineat ſitum perpendicularem; quod initio erat <lb></lb>in æquilibrio. </s> </p> <p type="main"> <s id="s.000244">Cæterum, quamvis habitâ ratione ſitûs, levia altiora magis <lb></lb>levitent, ſivè parallela horizonti jaceant extrema, ſivè incli<lb></lb>nata, ratione tamen medij, quod in ſuperioribus eſt levius, <lb></lb>quàm in inferioribus, minùs levitant: experientia enim oſten<lb></lb>dit ea lentiùs aſcendere, quæ propiùs accedunt ad medij na<lb></lb>turam ſecundùm levitatem: nam ex tribus globulis ſphæricis, <lb></lb>quorum diameter unc. 2 1/5 pedis Romani, cereus erat ponderis <lb></lb>drachmarum 24, faginus drachm. 22, vitraëreus drachm. 7. <lb></lb>in aëre expenſi, ſed eorum motus in aquâ ad altitudinem pe<lb></lb>dum 14, valdè inæqualis fuit, numeratis vibrationibus ejuſ<lb></lb>dem perpendiculi; cereus ſiquidem aſcendit lentiſſimè vibra<lb></lb>tionibus 88, faginus vibrationibus 37, vitraëreus vibrationi<lb></lb>bus 33: unde patet cereum, qui minimùm ab aquâ differt in <lb></lb>pondere (aquæ etenim molis æqualis eſt drachm. 25 3/5) minùs <lb></lb>in eâ levitare. </s> <s id="s.000245">Sicut igitur diverſa levia in eodem medio inæ<lb></lb>qualiter levitant, ſic idem leve in medio diſſimili inæqualiter <lb></lb>levitabit pro majore aut minore levitatum diſſimilitudine. </s> <lb></lb> <s id="s.000246">Conveniunt itaque gravia, & levia, quod hæc procul à cen<lb></lb>tro offendentia medium levius minùs levitant, illa propè cen<lb></lb>trum habentia medium gravius minùs gravitant. </s> <s id="s.000247">Differunt au<lb></lb>tem ratione poſitionis, quia, in loco remotiore à centro, per<lb></lb>pendicula omnia concurrunt ad angulos magis acutos, minúſ<lb></lb>que differunt à lineâ rectâ, ideo quaſi collatis viribus magis <lb></lb>gravitant, & magis levitant; at prope centrum cum perpendi<lb></lb>cula magis in diverſa abeant, & levia minùs levitant, & gravia <lb></lb>minùs gravitant. </s> <s id="s.000248">Porrò hanc ſimilitudinem gravitationis gra<lb></lb>vium, & levitationis levium in eodem loco, à me vocari diſcri<lb></lb>men, & differentiam, quia habita ratione oppoſitorum videba<lb></lb>tur leve remotius debere minùs levitare, ſicut grave propius <lb></lb>minùs gravitat, ne te moveat; litem de verbo non faciam. <pb n="27" xlink:href="017/01/043.jpg"></pb></s> </p> <p type="head"> <s id="s.000249"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000250"><emph type="center"></emph><emph type="italics"></emph>Quâ ratione centrum gravitatis corporum <lb></lb>inveniatur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000251">OPus mechanicum plerunque non indiget puncto illo, <lb></lb>quod intra corporum ſoliditatem latet, ac centrum gra<lb></lb>vitatis definivimus; ſed ſatis eſt ſi in extimâ corporis ſuperfi<lb></lb>cie innoteſcat punctum, aut linea imminens ipſi gravitatis <lb></lb>centro, pro ratione ſitûs, in quo corpus grave conſiſtere cu<lb></lb>pimus. </s> <s id="s.000252">Ideo geometricum laborem inveniendi punctum illud <lb></lb>intimum Centrobarycæ relinquens, mechanica tantùm inqui<lb></lb>ſitione, & quaſi tentans, perveſtigo punctum illud, aut li<lb></lb>neam in corporis ſuperficie, cui reſpondet planum per lineam <lb></lb>directionis ductum, & ſecans corpus in certo ſitu conſtitu<lb></lb>tum. </s> <s id="s.000253">Et quidem ſi corpus ſphæricum fuerit ex partibus ejuſ<lb></lb>dem naturæ conflatum, aut ſaltem ex partibus heterogeneis <lb></lb>quidem, ſed circa ſphæræ centrum ſimiliter diſpoſitis ita, ut <lb></lb>intima ſphærula folliculis quibuſdam obvolvatur; quia idem <lb></lb>eſt molis atque gravitatis centrum, punctum quodcumque in <lb></lb>ſphærica ſuperficie aſſumatur, aptum erit; ſingula enim ſi<lb></lb>milem habent poſitionem. </s> <s id="s.000254">Sin autem aut ſphæræ ſegmentum, <lb></lb>aut ſphæra ex partibus heterogeneis inæqualiter diſpoſitis fue<lb></lb>rit; imponatur plano horizontali accuratè levi, & maximè æqua<lb></lb>bili; & quod punctum tangetur à ſuppoſito plano, ubi motus <lb></lb>omnis ceſſaverit, illud eſt, quod potiſſimùm quæritur, ac <lb></lb>punctum ſuperius, quod huic è regione eſt, erit pariter aptum <lb></lb>ad propoſitum finem. </s> </p> <p type="main"> <s id="s.000255">Quod ſi cylindricum fuerit oblatum corpus, aut priſma quod<lb></lb>cunque continuo, & ſimili ductu productum; ſecetur bifariam <lb></lb>longitudo, & punctum habebitur cylindri centro gravitatis <lb></lb>reſpondens: priſmatis autem ſingula plana parallelogramma ſi <lb></lb>dividantur in æquas tum longitudinis, tum latitudinis partes, <lb></lb>planum per inventa puncta ductum tranſibit per centrum <lb></lb>gravitatis priſmatis, dividet enim in partes æquales, & ſimi<lb></lb>liter poſitas, unde oritur momentorum gravitatis æqualitas. </s> <pb n="28" xlink:href="017/01/044.jpg"></pb> <figure id="id.017.01.044.1.jpg" xlink:href="017/01/044/1.jpg"></figure> <lb></lb> <s id="s.000256">Ut ſi parallelepipedi BC plana ita <lb></lb>dividantur, ut habeant puncta me<lb></lb>dia I, & O, & per ea agatur pla<lb></lb>num, conſtat æqualia eſſe momenta <lb></lb>gravitatis partium IB, & IC, cùm <lb></lb>nullo ex capite poſſit oriri momento<lb></lb>rum inæqualitas. </s> <s id="s.000257">At ſi non facies parallelogrammæ priſmatis <lb></lb>dividendæ ſint, ſed potius baſis, quæ ſæpè varia eſt, & irre<lb></lb>gularis, tunc inveniendum eſt in ea punctum, in quo ſibi oc<lb></lb>currunt ſectiones planorum ſecantium datum corpus in mo<lb></lb>menta æqualia, illudque reſpondet centro gravitatis intra ſo<lb></lb>liditatem exiſtenti. </s> </p> <figure id="id.017.01.044.2.jpg" xlink:href="017/01/044/2.jpg"></figure> <p type="main"> <s id="s.000258">Sit autem primò baſis priſmatis <lb></lb>trigona AHI; dividatur unum <lb></lb>ex lateribus ex. gr. HI bifariam <lb></lb>in G, planum enim tranſiens per <lb></lb>A & G, atque bifariam ſecans pa<lb></lb>rallelogrammum HV tranſibit per <lb></lb>centrum gravitatis priſmatis trigo<lb></lb>ni. </s> <s id="s.000259">Nam ſi datum priſma ſecetur <lb></lb>pluribus planis parallelis plano <lb></lb>HV facientibus ſectiones ML, <lb></lb>BO, NS, CE, & ex harum <lb></lb>ſectionum extremis exeant alia <lb></lb>plana ſecantia parallela plano AG; <lb></lb>abſcinduntur ex priſmate dato pa<lb></lb>rallelepipeda LF, OK &c. quæ à plano AG dividuntur in <lb></lb>partes GL, GM æquales ac ſimiliter poſitas; item DO, DB, &c. </s> <lb></lb> <s id="s.000260">Igitur ſingula in eodem plano AG habent gravitatis centrum, <lb></lb>ac proinde tota moles ex iis parallelepipedis compoſita in eo<lb></lb>dem plano habet centrum gravitatis. </s> <s id="s.000261">Quoniam verò, ſi adhuc <lb></lb>plana ſecantia frequentiora ſint, plura fiunt parallelepipeda, <lb></lb>quorum omnium moles compoſita adhuc minus differt à mole <lb></lb>totius priſmatis dati, ita ut toties multiplicari poſſit biſectio, <lb></lb>ut demum relinquatur differentia minor quacunque minimâ <lb></lb>mole excogitabili; hinc fit molem compoſitam ex parallelepi<lb></lb>pedis illis infinitis (ſic loqui liceat, quia non eſt certus eorum <lb></lb>numerus explicabilis) habere centrum gravitatis in plano AG; <pb n="29" xlink:href="017/01/045.jpg"></pb>ac proinde etiam priſma trigonum ex iis conflatum parallelepi<lb></lb>pedis habere in eodem plano AG centrum ſuæ gravitatis, <lb></lb>quandoquidem non differt ab illis niſi differentiâ minore qua<lb></lb>cumque minimâ excogitabili. </s> <s id="s.000262">Sunt igitur partium AGH, <lb></lb>AGI momenta æqualia; quia ſi inæqualia eſſent haberent <lb></lb>differentiam, qua poſſet dari minor (neque enim eſſet indivi<lb></lb>dua) hæc autem differentia ſi eſſet, alia non eſſet, quàm quæ <lb></lb>intercedit inter priſma datum, & omnia parallelepipeda, cu<lb></lb>jus differentiæ inæquales partes eſſent in AGH, & AGI: <lb></lb>igitur differentia partium AGH, AGI eſſet minor diffe<lb></lb>rentiâ priſmatis, & omnium parallelepipedorum; nam eſſe non <lb></lb>poteſt major, vel illi æqualis: ſed jam ex hypotheſi differentia <lb></lb>inter molem compoſitam ex omnibus parallelepipedis, & priſ<lb></lb>ma, eſt minor quacumque minimâ datâ, ergo ſi eſſent inæ<lb></lb>qualia momenta partium AGH, AGI haberent differen<lb></lb>tiam minorem, & non minorem eâdem differentiâ inter priſ<lb></lb>ma & omnia parallelepipeda. </s> <s id="s.000263">Non ſunt igitur inæqualia. </s> <s id="s.000264">Res <lb></lb>autem fortaſsè ſic breviùs explicabitur; ſi partes AGH, AGI <lb></lb>non ſunt æquales, ſit AGH minor quàm AGI, differentiâ Y. </s> <lb></lb> <s id="s.000265">Tot autem fiant biſectiones, ut parallelepipeda relinquant <lb></lb>differentiam minorem quàm Y. </s> <s id="s.000266">Quia ergo parallelepipeda <lb></lb>in AGI habent differentiam minorem quàm Y, à parte priſ<lb></lb>matis AGI, illa ſunt majora quàm pars priſmatis AGH, <lb></lb>quæ deficit à parte AGI differentiâ Y. </s> <s id="s.000267">Atqui parallelepepida <lb></lb>in AGH ſunt æqualia parallelepipedis in AGI, ergo etiam <lb></lb>parallelepipeda in AGH majora ſunt, quàm tota pars AGH, <lb></lb>quod eſt manifeſtè falſum. </s> <s id="s.000268">Non eſt igitur altera pars major, <lb></lb>altera minor. </s> <s id="s.000269">Porrò ex continua biſectione laterum AC, <lb></lb>& CN &c. relinqui ſemper minorem differentiam, hoc eſt ſe<lb></lb>miſſem præcedentis differentiæ, conſtat, quia ſi AC ſecetur <lb></lb>in P, & ducantur plana parallela planis AG, & HV, dividi<lb></lb>tur CT bifariam in Q, & eſt TP parallelepipedum ablatum <lb></lb>duplum priſmatis trigoni CPQ, cui æquale eſt priſma APX; <lb></lb>adeóque duobus hiſce priſmatis æquale eſt ablatum parallele<lb></lb>pipedum TP, quod eſt ſemiſſis differentiæ ATC, quæ priùs <lb></lb>relinquebatur: & eadem eſt de cæteris ratio. </s> <s id="s.000270">Quare ſi ex datâ <lb></lb>quantitate auferatur ſemiſſis, & iterum ſemiſſis reſidui, & ſic <lb></lb>in infinitum, neceſſe eſt aliquando eò devenire, ut reſidua <pb n="30" xlink:href="017/01/046.jpg"></pb>quantitas minor ſit quacunque datâ quantitate, ut colligitur <lb></lb>ex prop. 1. lib. 10. Eucl. </s> <s id="s.000271">Ideo fieri non poteſt, ut priſmate di<lb></lb>viſo à plano AG, altera pars excedat momenta alterius quan<lb></lb>titate Y, quia tot poſſunt abſcindi purallelepipeda, ut relin<lb></lb>quatur differentia illorum à priſmate minor, quàm ſit Y: pla<lb></lb>num autem AG æqualiter dividit momenta parallelepipedo<lb></lb>rum, igitur cum tota reſidua differentia minor ſit quam Y, <lb></lb>eſſe omnino non poteſt, ut altera pars habeat exceſſum quan<lb></lb>titati Y reſpondentem ſi enim quantitates illæ differrent, poſ<lb></lb>ſet dari quantitas minor illarum differentiâ; ſed non poteſt hu<lb></lb>juſmodi minor quantitas dari, nam quælibet data eſt major, <lb></lb>igitur non differunt, ſed ſunt æquales. </s> </p> <p type="main"> <s id="s.000272">His ita conſtitutis facilè definitur punctum centro gravitatis <lb></lb>imminens in baſi priſmatis: quia enim oſtenſum eſt planum <lb></lb>ab angulo per medium latus oppoſitum ductum tranſire per <lb></lb>centrum gravitatis, & dividere in momenta æqualia totum <lb></lb>priſma, centrum gravitatis erit non ſolùm in plano AG, ſed <lb></lb>etiam in plano IN propter eandem rationem. </s> <s id="s.000273">Punctum igi<lb></lb> <figure id="id.017.01.046.1.jpg" xlink:href="017/01/046/1.jpg"></figure><lb></lb>tur D, in quo occurrunt ſibi communes <lb></lb>ſectiones planorum ſecantium, & baſis, eſt <lb></lb>punctum, quod quæritur, imminens centro <lb></lb>gravitatis. </s> <s id="s.000274">Punctum D autem ſecare rectam <lb></lb>NI ita, ut ND ad DI ſit ut 1 ad 2, ſic <lb></lb>oſtenditur. </s> <s id="s.000275">Ducatur recta NG, quæ per 2. lib. 6. eſt paral<lb></lb>lela ipſi AI; ergo ut HG ad HI, ita NG ad AI per 4. lib. 6. <lb></lb>ergo NG ad AI eſt ut 1 ad 2: ergo triangula NGA, AGI <lb></lb>ſunt ut 1 ad 2, per 1. lib. 6. </s> <s id="s.000276">Cum autem ut ND ad DI, <lb></lb>ita NDA ad DIA, & NDG ad DIG per 1. 6. erit <lb></lb>etiam, ex 12. lib. 5. ut ND ad DI, ita NGA ad AGI, <lb></lb>hoc eſt 1 ad 2. </s> <s id="s.000277">Eadem ratione oſtenditur GD ad DA eſſe, <lb></lb>ut 1 ad 2. </s> <s id="s.000278">Vel etiam breviùs: Quia enim NG, AI ſunt pa<lb></lb>rallelæ, triangula NDG, ADI ſunt ſimilia propter angulo<lb></lb>rum æqualitatem; ergo ut NG ad AI, hoc eſt ut 1 ad 2, <lb></lb>ita GD ad DA, & ND ad DI. </s> <s id="s.000279">Quare ſatis erit latus unum <lb></lb>trianguli bifariam ſecare, & ab oppoſito angulo rectam duco<lb></lb>re; cujus tertia pars verſus baſim diviſam dabit centrum gravi<lb></lb>tatis trianguli. </s> </p> <p type="main"> <s id="s.000280">Jam verò ſi baſis priſmatis quadrangula fuerit parallelogram- <pb n="31" xlink:href="017/01/047.jpg"></pb>ma, ductis diametris apparebit quæſitum punctum, per quod <lb></lb>tranſeunt omnia plana dividentia æqualiter corporis dati mo<lb></lb>menta, cum ſint partes utrinque æquales, & ſimiliter poſitæ. </s> <lb></lb> <s id="s.000281">Et ob eandem rationem ſi baſis priſmatis fuerit aliqua ex figu<lb></lb>ris ordinatis, ſeu æquilateris; centrum figuræ eſt punctum im<lb></lb>minens centro gravitatis; planum ſi <expan abbr="quidẽ">quidem</expan> per illud tranſiens, & <lb></lb>per <expan abbr="unũ">unum</expan> angulorum, dividit <expan abbr="totũ">totum</expan> priſma in partes æquales ſimi<lb></lb>literque poſitas; atque adeò momenta hinc, & hinc ſunt æqualia. </s> </p> <p type="main"> <s id="s.000282">At ſi baſis trapezia fuerit, duc utramque <lb></lb> <figure id="id.017.01.047.1.jpg" xlink:href="017/01/047/1.jpg"></figure><lb></lb>diametrum EC, & BD: tum in baſi trigo<lb></lb>nâ BCD priſmatis partialis inveniatur <lb></lb>punctum centro gravitatis reſpondens (pun<lb></lb>ctum hoc deinceps, brevitatis gratiâ, dice<lb></lb>tur centrum gravitatis, quamvis per abuſionem) & ſit H; & in <lb></lb>oppoſita baſi trigona reliqui priſmatis BDE pariter invenia<lb></lb>tur punctum F; & per utrumque punctum tranſeat planum <lb></lb>FH; nam in hoc eodem plano eſt centrum gravitatis totius <lb></lb>priſmatis trapezij, quod dividitur in momenta æqualia: hoc ſi<lb></lb>quidem planum tranſiens per H gravitatis momenta æqualia <lb></lb>habet hinc, & hinc in priſmate trigono BDC; ſimiliter cum <lb></lb>tranſeat per F, habet hinc, & hinc momenta æqualia gravitatis <lb></lb>priſmatis trigoni BED: ſi igitur æqualia æqualibus jungantur, <lb></lb>planum idem æqualiter partitur momenta gravitatis priſmatis <lb></lb>trapezij EDCB, & in eo eſt centrum gravitatis illius. </s> <s id="s.000283">Eadem <lb></lb>ratione in baſi trigona EBC inveniatur punctum G, & in baſi <lb></lb>EDC punctum S, per quæ ſi agatur planum GS, in eo pariter <lb></lb>erit <expan abbr="centrũ">centrum</expan> gravitatis totius priſmatis trapezij. </s> </p> <p type="main"> <s id="s.000284">Eſt igitur centrum gravitatis in communi <lb></lb> <figure id="id.017.01.047.2.jpg" xlink:href="017/01/047/2.jpg"></figure><lb></lb>ſectione planorum FH, & GS; ac proinde <lb></lb>punctum I illud eſt, quod quæritur. </s> <s id="s.000285">Aliter <lb></lb>etiam, & facillimè in baſi trapezia ABCD <lb></lb>invenitur centrum gravitatis: ductis enim <lb></lb>diametris AC, BD, altera diameter ex. gr. <lb></lb>AC bifariam ſecetur in E, ducanturque rectæ <lb></lb>DE, BE; trianguli ADC centrum gravi<lb></lb>tatis eſt in recta DE, & quidem in F, ita ut EF <lb></lb>ſit tertia pars totius ED, ut conſtat ex paulò <lb></lb>ante demonſtratis. </s> <s id="s.000286">Ducatur igitur FG pa- <pb n="32" xlink:href="017/01/048.jpg"></pb>rallela alteri diametro BD, & erit ſimiliter G centrum gravita<lb></lb>tis trianguli ABC, quia per 2. lib. 6. ut EF ad FD, ita EG <lb></lb>ad GB; Quia ergo diameter AC ſecatur in H, ſumatur FO <lb></lb>æqualis ipſi GH, & eſt O centrum gravitatis trapezij, eſt enim <lb></lb>triangulum ABC ad triangulum ADC, ut FO ad OG, hoc <lb></lb>eſt ut HG ad HF. </s> <s id="s.000287">Eſt autem HG ad HF ut BI ad ID pro<lb></lb>pter paralleliſmum linearum GF, BD. </s> <s id="s.000288">Porrò conſtat triangu<lb></lb>lum ABC ad triangulum ADC eſſe ut BI ad ID, nam trian<lb></lb>gula ABI, ADI ſunt ut baſes BI, DI, item BCI, DCI ſunt <lb></lb>ut eædem baſes BI, DI per 1. lib. 6; igitur, & totum triangu<lb></lb>lum ABC ad totum ADC eſt ut BI ad DI: igitur, & trian<lb></lb>gulum ABC ad triangulum ADC eſt ut FO ad OG. </s> </p> <figure id="id.017.01.048.1.jpg" xlink:href="017/01/048/1.jpg"></figure> <p type="main"> <s id="s.000289">Hinc facilis patet via ad inveſti<lb></lb>gandum idem punctum in baſi priſ<lb></lb>matis pentagoni BDEAC. </s> <s id="s.000290">Pri<lb></lb>mùm enim ducto plano per BE, in<lb></lb>veniatur in baſi trigonâ BDE <lb></lb>punctum R, & in baſi BEAC qua<lb></lb>drangulâ punctum P; & ducto plano <lb></lb>per RP, in eo erit centrum gravi<lb></lb>tatis priſmatis pentagoni, cum in eo<lb></lb>dem ſint centra gravitatis partium. </s> <lb></lb> <s id="s.000291">Deinde ducto per D & A plano, inveniatur in baſi trigona <lb></lb>DEA punctum L centrum gravitatis, & in baſi quadrangu<lb></lb>lâ ACBD punctum M centrum gravitatis: in plano pariter <lb></lb>ducto per ML eſt centrum gravitatis totius priſmatis pentago<lb></lb>ni, quod proinde eſt in communi planorum per PR, & LM <lb></lb>ductorum ſectione; atque adeò punctum, quod quæritur, eſt O. </s> <lb></lb> <s id="s.000292">Eadem eſt methodus in priſmate hexagono; ducto enim plano <lb></lb>dividente in duo priſmata, quorum alterum eſt trigonum, al<lb></lb>terum pentagonum, inveniatur utriuſque centrum gravitatis, <lb></lb>& per inventa puncta agatur planum. </s> <s id="s.000293">Deinde iterum alio pla<lb></lb>no ſecetur in duo priſmata, quorum alterum pariter ſit trigo<lb></lb>num, alterum pentagonum, & per inventa ſingularia gravi<lb></lb>tatum centra agatur planum: duo ſiquidem plana ducta per <lb></lb>centra gravitatis partium, tranſeunt pariter per centrum gra<lb></lb>vitatis totius, quod eſt in communi eorum ſectione. </s> <s id="s.000294">Eademque <lb></lb>de reliquis priſmatis eſt ratio. </s> </p> <pb n="33" xlink:href="017/01/049.jpg"></pb> <p type="main"> <s id="s.000295">Sed hæc indicaſſe ſufficiat, quæ operi Mechanico ſatis eſſe <lb></lb>poſſunt in omnibus ferè priſmatis: Si enim baſis non fuerit <lb></lb>planè rectilinea, inſcripto polygono rectilineo, quod mini<lb></lb>mùm differat à plano baſis, quæres ejus centrum gravitatis, <lb></lb>methodo jam tradita; illoque uſurpato tanquam vero dati priſ<lb></lb>matis centro quæſito, minimùm aberrabis; aliquando tamen <lb></lb>aberrabis, aliquando continget, ut inventum cum quæſito <lb></lb>conveniat. </s> <s id="s.000296">Quod ſi accuratiori inveſtigatione opus fuerit: <lb></lb>quemadmodum in cæteris corporibus, quæ continuum ductum <lb></lb>non habent, ſed inæquali craſſitudine creſcunt, aut decreſ<lb></lb>cunt, ut in obeliſcis, aut pyramidibus truncatis, reliquiſquè <lb></lb>planè inordinatis molibus; tunc ad geometricam Centrobary<lb></lb>ces methodum confugiendum eſt; quam hic ego non perſe<lb></lb>quor. </s> <s id="s.000297">Praxes igitur aliquæ proponendæ ſunt, quibus centrum <lb></lb>gravitatis phyſicè perſpectum habere poſſimus in corporibus, <lb></lb>quorum frequentior, vulgariſque uſus eſſe poteſt. </s> </p> <p type="main"> <s id="s.000298">Prima praxis ſit ad inveniendum gra<lb></lb> <figure id="id.017.01.049.1.jpg" xlink:href="017/01/049/1.jpg"></figure><lb></lb>vitatis centrum in cingulis, quæ laminis <lb></lb>quoque communis eſſe poteſt. </s> <s id="s.000299">Sit datum <lb></lb>cingulum AH, quod primùm ſuſpenda<lb></lb>tur ex H, & inde pendens perpendicu<lb></lb>lum ſecet oppoſitum latus IA in C; note<lb></lb>tur igitur punctum C. </s> <s id="s.000300">Deinde iterum <lb></lb>ſuſpendatur ex R, & perpendiculum ca<lb></lb>dat in punctum F, quod notetur. </s> <s id="s.000301">His cognitis ducatur filum <lb></lb>ex R in F, ibique intentum alligetur; aliud filum ſimiliter <lb></lb>ex H in C ducatur, & ſecans in S filum RF, dabit punctum S <lb></lb>quæſitum centrum gravitatis: ex quo ſi ſuſpenderetur datum <lb></lb>cingulum, maneret horizonti parallelum. </s> <s id="s.000302">Quod ſi eſſet corpus <lb></lb>talis figuræ, ut ſpatium non clauderet, ſed haberet angulum <lb></lb>cavum, aut eſſet fruſtum annulare, eadem eſt methodus factâ <lb></lb>ſuſpenſione illius ex duobus punctis, ex quibus perpendiculum <lb></lb>cadere poſſit intrà corporis ſuperficiem; in qua ſi notentur <lb></lb>puncta, per quæ tranſit, & ducantur fila, ut priùs, eorum com<lb></lb>munis ſectio dabit quæſitum centrum gravitatis. </s> <s id="s.000303">Hinc ſi vel la<lb></lb>mina eſſet perforanda, ut axi infigeretur, vel cingulum eſſet <lb></lb>axi imponendum, in utrâque ſuperficie oppoſita quærere opor<lb></lb>teret punctum S, ut axis per centrum gravitatis tranſiret, eique <pb n="34" xlink:href="017/01/050.jpg"></pb>uterque polus reſponderet: in cingulis autem præterea haben<lb></lb>da eſſet ratio tranſverſariorum, per quæ axis infigendus eſſet, <lb></lb>ea enim poſſunt centrum gravitatis compoſitæ in alio puncto <lb></lb>conſtituere. </s> </p> <p type="main"> <s id="s.000304">Secunda praxis laminis potiſſimùm accommodata, in quibus <lb></lb>punctum medium ſatis accuratè inquiritur, ut ſi lamina metal<lb></lb>lica eſſet in calicem excavanda, hæc eſſe poteſt. </s> <s id="s.000305">Impone lami<lb></lb>nam acutæ cuſpidi cultri, aut ſtyli, eamque ultrò citróque <lb></lb>tantiſper move, dum conſiſtat citrà periculum cadendi: <lb></lb>punctum enim, quod à cultri aut ſtyli cuſpide notatur, cen<lb></lb>trum eſt quæſitum. </s> </p> <p type="main"> <s id="s.000306">Tertia praxis ſit iis corporibus conveniens, quæ præſtant <lb></lb>longitudine, qualia ſunt pſeudocylindrica, conica, pyrami<lb></lb>des &c. quæ ſi non prædita ſint multâ gravitate, imponantur <lb></lb>funiculo brevi horizontaliter extenſo, at ſi graviora fuerint, vel <lb></lb>cylindrulo vel aciei priſmatis trigoni imponantur, & uſque <lb></lb>dum in æquilibrio conſiſtant, promoveantur: ubi enim quie<lb></lb>verit corpus impoſitum, ex loco contactûs innoteſcet vel <lb></lb>punctum, ſi in puncto ſe contingant, vel linea, ſi in lineâ, per <lb></lb>quam ſi ducatur planum à centro terræ, diſtinguetur impoſi<lb></lb>tum corpus in momenta gravitatis æqualia. </s> <s id="s.000307">Inventâ autem hu<lb></lb>juſmodi lineâ facilè prodet ſe quæſitum punctum. </s> </p> <p type="main"> <s id="s.000308">Quarta praxis non multùm diſtat à ſuperiore: ſi nimirum <lb></lb>oblatum corpus impoſueris plano alicui horizontali, quod ta<lb></lb>men à pavimento abſit mediocri aliquo intervallo, habeat au<lb></lb>tem extremum marginem exactè rectum: extra ſuppoſiti pla<lb></lb>ni marginem illud paulatim promove, donec eò venerit, ut ſi <lb></lb>vel minimum ulteriùs promoveretur, ſponte caderet; ibíque <lb></lb>ſecundùm rectitudinem marginis plani duc ſtylo lineam in cor<lb></lb>pore impoſito. </s> <s id="s.000309">Deinde ſuperficie eâdem planum tangente, ſi <lb></lb>corpus, præter longitudinem, non modicam præterea habeat <lb></lb>latitudinem, convertatur aliquantulum, & ſimili methodo in<lb></lb>venietur linea alia ſecans priorem in puncto quæſito, quod ſci<lb></lb>licet reſpondet centro gravitatis intra corporis ſoliditatem de<lb></lb>liteſcenti. </s> </p> <p type="main"> <s id="s.000310">Hæc ſunt quæ Mechanices inſtituto ſufficere poſſint ad cen<lb></lb>trum gravitatis inveniendum; in molibus enim majoribus, quæ <lb></lb>plerumque vix differunt à priſmatis, non indigemus commu- <pb n="35" xlink:href="017/01/051.jpg"></pb>niter Geometricâ ſubtilitate. </s> <s id="s.000311">Illud reſtat, ut earum, quas at<lb></lb>tuli praxes, ratio, & cauſæ explicentur, ex quibus clarion ha<lb></lb>beatur notitia eorum, quæ ad centrum gravitatis pertinent. <lb></lb></s> </p> <p type="head"> <s id="s.000312"><emph type="center"></emph>CAPUT VI.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000313"><emph type="center"></emph><emph type="italics"></emph>Affertur ratio prædictarum praxeon.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000314">UT palam fiat praxibus capite ſuperiore allatis inveniri <lb></lb>punctum reſpondens centro gravitatis, quod inquiritur, <lb></lb>indicandi ſunt fontes, ex quibus illæ deducuntur. </s> <s id="s.000315">Earum ita<lb></lb>que ratio petenda eſt ex gravium naturâ, quæ extra locum ſibi <lb></lb>debitum conſtituta, in medio videlicet leviore, conantur de<lb></lb>orſum pro viribus, niſi impediantur: quod ſi interpellentur <lb></lb>quidem, non tamen prorſus deſcenſu prohibeantur, deſcen<lb></lb>dunt, prout fert obſtantium impedimentorum conditio. </s> <s id="s.000316">Sic <lb></lb>lapis ſphæricus in montis clivo poſitus cùm non valeat rectâ; <lb></lb>ſicut in aëre libero, deorſum ferri, per planum illud inclina<lb></lb>tum deſcendit: Sic plumbum, quod filo adnectitur laqueari, à <lb></lb>perpendiculo remotum deſcendit circulariter. </s> <s id="s.000317">Porrò quæ de <lb></lb>toto ipſo corpore vera eſſe intelligimus, ejus quoque partibus <lb></lb>ſingulis conveniunt; cùm enim ſingulæ ſuam habeant gravita<lb></lb>tem, niſi quid obſtet, deſcendunt. </s> <s id="s.000318">Jam verò ſi contingat ita <lb></lb>corpus grave oppoſito extrinſecùs obice impediri, ut cunctæ <lb></lb>ſimul partes, quaſi moles unà deſcendere nequeant; ſublato <lb></lb>partium nexu deſcendunt, quæcunque carent impedimento: <lb></lb>ut ſi ceream candelam, aut glaciem, quam manu ſuſtines, igni <lb></lb>admoveas; haud dubium, quin partes extremæ igni proximæ <lb></lb>liqueſcentes, ſolutâ unione cum cæteris, ſuis nutibus deorſum <lb></lb>latæ liberè deſcendant. </s> <s id="s.000319">At ſi partes omnes colligatæ invicem <lb></lb>permaneant, eandemque figuram ſervent; corpore illo ſuſpen<lb></lb>ſo aut ſuſtentato, fieri non poteſt, ut partes aliquæ deſcendant, <lb></lb>quin aliæ, ouæ è regione ſunt trans ſuſpenſionis, aut ſuſtenta<lb></lb>tionis punctum, aſcendant; id autem harum gravitati re<lb></lb>pugnat: non igitur aſcendere poſſunt, niſi deſcendentes op<lb></lb>poſitæ viribus ac momentis præſtent ita, ut harum gravitati <pb n="36" xlink:href="017/01/052.jpg"></pb>vim inferre valeant. </s> <s id="s.000320">Quare ſi fiat corporis ſuſpenſi, aut ſuſten<lb></lb>tati conſiſtentia, argumentum eſt æqualitatis momentorum <lb></lb>punctum ſuſpenſionis, aut ſuſtentationis hinc, & hinc uſque <lb></lb>quaque circunſtantium; ſi qua enim eſſet inæqualitas, alterutra <lb></lb>pars præponderaret, & ad motum incitaretur. </s> </p> <figure id="id.017.01.052.1.jpg" xlink:href="017/01/052/1.jpg"></figure> <p type="main"> <s id="s.000321">Sit corpus grave AB, cujus <lb></lb>centrum gravitatis H, linea di<lb></lb>rectionis HT in centrum uni<lb></lb>verſi producta. </s> <s id="s.000322">Si ſuſpendatur <lb></lb>ex puncto C, quod eſt in eadem <lb></lb>lineâ directionis, neceſſariò con<lb></lb>ſiſtit corpus horizonti paralle<lb></lb>lum, quia rectâ deſcendere non <lb></lb>poteſt per HT, cum in C reti<lb></lb>neatur; neque alterutra pars poteſt deſcendere, quia momen<lb></lb>ta partis HB, quibus deorſum nititur, æqualia ſunt momen<lb></lb>tis, quibus pars HA reſiſtit, no elevetur; & viciſſim viribus <lb></lb>gravitatis- AH cætero qui deſcenſuræ reluctatur gravitas HB <lb></lb>pari niſu repugnans, ne attollatur; totum ergo conſiſtit. </s> <s id="s.000323">At ſi <lb></lb>ex M puncto ſuſpendatur, non poteſt quidem per MT per<lb></lb>pendicularem deſcendere versùs terræ centrum, ſed neque <lb></lb>conſiſtet horizonti parallelum; quia ſi planum intelligatur ex <lb></lb>terræ centro per rectam MT ductum, non dividitur corpus in <lb></lb>momenta æqualia, cum non tranſeat per H centrum gravita<lb></lb>tis; igitur cum majora ſint momenta partis MB, quàm par<lb></lb>tis MA, illa præponderabit, atque deſcendens circa <lb></lb>punctum M permanens convertetur, donec centrum gra<lb></lb>vitatis H ſit in perpendiculari MT, cui congruat recta <lb></lb>MO: tunc autem demum conſiſtet, quia planum tranſiens <lb></lb>per MHO æqualiter diſpertit momenta gravitatis; neutrâ <lb></lb>autem parte præponderante, utraque quieſcit. </s> <s id="s.000324">Idem dicen<lb></lb>dum, ſi corpus ex I puncto ſuſpenderetur; tunc enim ſo<lb></lb>lùm fieret conſiſtentia, ubi in eadem directionis lineâ <lb></lb>eſſet punctum I atque H centrum gravitatis. </s> <s id="s.000325">Quod ſi du<lb></lb>plici funiculo ſuſpendatur pondus, & illi paralleli non ſint, <lb></lb>quia neque horizonti perpendiculares, illi ſi producantur, <lb></lb>concurrent in punctum aliquod lineæ directionis, ſivè ſupra <lb></lb>pondus, ſivè infra, pro ratione angulorum, quos conſtituunt. <pb n="37" xlink:href="017/01/053.jpg"></pb>Sit enim corpus AB, cujus cen<lb></lb> <figure id="id.017.01.053.1.jpg" xlink:href="017/01/053/1.jpg"></figure><lb></lb>trum gravitatis O, linea directio<lb></lb>nis IOC, ſi ex I ſuſpendatur <lb></lb>per O, in co ſitu manebit; ergo <lb></lb>etiam, ſi funiculi ſint IH, IL, <lb></lb>manebit: ergo etiam, ſi ſint PH, <lb></lb>SL, funiculorum enim longitudo <lb></lb>nihil facit; Idem etiam dicendum <lb></lb>cum funiculi ſunt DH, FL; pro<lb></lb>ducti enim concurrunt cum linea <lb></lb>directionis in C, ſemper ſcilicet <lb></lb>perinde ſe habet atque, ſi ex I ſuſpenderetur. </s> </p> <p type="main"> <s id="s.000326">Quæ verò de ſuſpenſione dicta ſunt, ea, analogiâ ſervatâ, de <lb></lb>ſuſtentatione quoque dicta intelligantur; tunc ſolùm videlicet <lb></lb>corpus conſiſtere, cùm ex centro gravitatis ducta directionis <lb></lb>linea tranſit per punctum ſuſtentationis, quia tunc ſolùm æqua<lb></lb>lia hinc, & hinc ſunt momenta virtutis ad deſcendendum, at<lb></lb>que reſiſtentiæ ad aſcendendum: ut quando corpus aliquod <lb></lb>imponitur cono, vel priſma ſphæræ, vel ſegmentum ſphæri<lb></lb>cum, plano, vel cylindrus aciei priſmatis trigoni in tranſver<lb></lb>ſum; cadet enim in alterutram partem impoſitum corpus, niſi <lb></lb>in eadem linea fuerint centrum terræ, punctum contactus, & <lb></lb>centrum gravitatis. </s> <s id="s.000327">Quod ſi corpus ſuſtentans, atque ſuſtenta<lb></lb>tum ſe tangant in linea, opus eſt lineam illam eſſe in plano per <lb></lb>lineam directionis ducto, ut fiat æqualium momentorum con<lb></lb>ſiſtentia. </s> <s id="s.000328">Quare ſi impoſitum corpus conſiſtat, certiſſimo ar<lb></lb>gumento conſtabit punctum, ſeu lineam, contactûs reſpon<lb></lb>dere centro gravitatis. </s> <s id="s.000329">Hinc patet ratio ſecundæ, & tertiæ <lb></lb>praxis. </s> </p> <p type="main"> <s id="s.000330">In prima praxi quia facies extima, ſupra quam perpendicu<lb></lb>lum liberè movetur, eſt in plano verticali, perpendiculum HC <lb></lb>eſt parallelum lineæ directionis corporis gravis, quæ tranſit <lb></lb>etiam per punctum ſuſpenſionis H: planum igitur tranſiens per <lb></lb>punctum ſuſpenſionis H, & per perpendiculum HC, tranſit <lb></lb>quoque per centrum gravitatis corporis. </s> <s id="s.000331">Cum verò idem pror<lb></lb>ſus dicendum ſit de plano tranſeunte per punctum ſuſpenſio<lb></lb>nis R, & perpendiculum RF, illud ſcilicet tranſire per cen<lb></lb>trum gravitatis corporis; apertum eſt centrum gravitatis eſſe in <pb n="38" xlink:href="017/01/054.jpg"></pb>communi illorum planorum ſectione, eique reſpondere <lb></lb>punctum S inventum. </s> </p> <p type="main"> <s id="s.000332">Quia demum, ſi corpus quod ſuſtinet, & id, quod ſuſtine<lb></lb>tur, in ſuperficie ſe tangant, corpus impoſitum in alterutram <lb></lb>partem cadere non poteſt (niſi fortè ſuppoſitum planum fuerit <lb></lb>inclinatum) quin planum per lineam directionis ductum ita ſit <lb></lb>extra ſuperficiem, in qua fit contactus, ut neque illam con<lb></lb>tingat; conſtat ratio quartæ praxis. </s> <s id="s.000333">Si namque planum ex ter<lb></lb> <figure id="id.017.01.054.1.jpg" xlink:href="017/01/054/1.jpg"></figure><lb></lb>ræ centro ductum per C cen<lb></lb>trum gravitatis dati corporis <lb></lb>OS, ſecet ſubjectum planum, <lb></lb>pars corporis extra marginem <lb></lb>FE in aëre extans minora ha<lb></lb>bet momenta gravitatis, quàm <lb></lb>reliqua pars; hæc igitur gra<lb></lb>vior non poteſt ab illa elevari: <lb></lb>ubi verò promotum corpus eò <lb></lb>venerit, ut planum per cen<lb></lb>trum gravitatis C ductum tangat extremum marginem ſub<lb></lb>jecti plani ita, ut in eodem plano, in quo eſt centrum gravi<lb></lb>tatis C, ſit etiam FE, æqualia ſunt gravitatis momenta par<lb></lb>tis CS in aëre extantis, ac CO partis plano incumbentis; & <lb></lb>ſi vel minimum ulteriùs promoveretur, pars extra planum ſub<lb></lb>jectum extans gravior eſſet, adeóque deſcenderet. </s> <s id="s.000334">Quare ſi in <lb></lb>corporis OS ſuperficie infimâ lineam deſcripſeris ſecundùm <lb></lb>marginem FE, ea erit in plano tranſeunte per centrum gravi<lb></lb>tatis. </s> <s id="s.000335">Quia verò idem contingit, ſi iiſdem ſuperficiebus ſe con<lb></lb>tingentibus alium ſitum corpori dederis, pariterque eò uſque <lb></lb>promoveris, ut citrà cadendi periculum promoveri ulteriùs <lb></lb>non poſſit; alia linea ſecundùm marginem FE ducta erit pari<lb></lb>ter in plano per gravitatis centrum tranſeunte, ſecabitque <lb></lb>priorem lineam, punctum mutuæ linearum ſectionis illud eſſe, <lb></lb>quod quæritur, ſatis liquet. </s> <s id="s.000336">Hæc eſt diſpar philoſophandi ra<lb></lb>tio, ſi pars CO adeò longa eſſet, ut etiam extaret extra an<lb></lb>guſtias ſubjecti plani; ſemper enim conſiſtit impoſitum corpus, <lb></lb>quandiù planum per lineam directionis tranſiens, aut tangit, <lb></lb>aut ſecat ſubjectum planum. </s> <s id="s.000337">Quandocunque enim linea di<lb></lb>rectionis non tranſit per punctum, vel lineam, vel ſuperficiem, <pb n="39" xlink:href="017/01/055.jpg"></pb>in quibus corpus grave tangitur à ſuſtentante (idem dic de <lb></lb>ſuſpenſione) ſemper in alterutram partem grave inclinatur, in <lb></lb>eam ſcilicet, in qua reperitur centrum gravitatis, cùm plura <lb></lb>ſint ex ea parte momenta gravitatis. <lb></lb></s> </p> <p type="head"> <s id="s.000338"><emph type="center"></emph>CAPUT VII.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000339"><emph type="center"></emph><emph type="italics"></emph>Quomodo gravia ſpontè aſcendentia deſcendant.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000340">EX his, quæ proximè dicta ſunt, grave ſuſtentatum in eam <lb></lb>partem inclinari, in qua eſt gravitatis centrum, oritur ali<lb></lb>quando aſcenſus gravium, qui rerum naturalium ignaros in <lb></lb>admirationem adducit non mediocrem, ſi maximè tunc cor<lb></lb>pus deſcendere intelligant, quando illud cernunt altiùs ab ho<lb></lb>rizonte aſcendere. </s> <s id="s.000341">Sit <lb></lb> <figure id="id.017.01.055.1.jpg" xlink:href="017/01/055/1.jpg"></figure><lb></lb>enim ſuper planum in<lb></lb>clinatum RN rota tantæ <lb></lb>latitudinis, ut poſſit in <lb></lb>plano verticali erecta <lb></lb>permanere, dum conver<lb></lb>titur; habeat autem ad <lb></lb>PO adnexam laminam <lb></lb>plumbeam craſſiorem, <lb></lb>adeò ut totius rotæ cen<lb></lb>trum gravitatis ſit S. </s> <s id="s.000342">Jam <lb></lb>verò ea ſit plani ſubjecti <lb></lb>inclinatio, ut rotâ illud <lb></lb>tangente puncto H, li<lb></lb>nea à terræ centro per H punctum contactûs tranſiens non <lb></lb>tranſeat per S centrum gravitatis (ſeu ut veriùs dicam, quia <lb></lb>extima ſuperficies rotæ cylindrica tangit planum in lineâ, pla<lb></lb>num ex centro terræ per lineam contactûs in H ductum non <lb></lb>tranſeat per S) ſed illud relinquat versùs ſuperiorem plani par<lb></lb>tem N; planum per rectam HO perpendicularem ductum <lb></lb>diſtinguit rotam in momenta gravitatis inæqualia: non poteſt <lb></lb>igitur rota in H conſiſtere, ſed convertitur, ita ut tangat pla<lb></lb>num in I primùm, deinde in E, demùm in P, ubi conſiſtet, <pb n="40" xlink:href="017/01/056.jpg"></pb>cùm linea directionis ex gravitatis centro S ducta in terræ cen<lb></lb>trum tranſibit per P locum contactús. </s> <s id="s.000343">In hac autem conver<lb></lb>ſione dum rotæ partes inter H & P deinceps aptantur ſubjecto <lb></lb>plano, centrum quidem molis aſcendit, ſed centrum gravita<lb></lb>tis S deſcendit. </s> <s id="s.000344">Lineam porrò SP minorem eſſe lineá SE, & <lb></lb>hanc minorem lineâ SI, & hanc lineâ SH, conſtat ex prop.7. <lb></lb>lib.3. Eucl. ſi nimirum per S, & C centrum agatur diameter. </s> <lb></lb> <s id="s.000345">Non eſt tamen cenſendum quamlibet ponderis additionem <lb></lb>in OP ſatis eſſe, ut in quolibet plano inclinato rota aſcendat; <lb></lb>ſi enim diſtantia centri gravitatis à centro rotæ minor fuerit, <lb></lb>quàm Sinus inclinationis plani, ſemper deſcendet; ſi eidem <lb></lb>Sinui æqualis, non aſcendet; ſi demum eo ſinu major, poterit <lb></lb>aſcendere. </s> </p> <figure id="id.017.01.056.1.jpg" xlink:href="017/01/056/1.jpg"></figure> <p type="main"> <s id="s.000346">Sit planum inclinatum <lb></lb>AB, quod in H contingat <lb></lb>circulum (hunc ſumo cir<lb></lb>culum, qui tranſeat per <lb></lb>centrum tum molis tum <lb></lb>gravitatis rotæ) cujus cen<lb></lb>trum C, & ducatur recta <lb></lb>CH, quæ cum perpendi<lb></lb>culari HO faciat angu<lb></lb>lum CHO. </s> <s id="s.000347">Quia enim <lb></lb>OH producta cadit in ho<lb></lb>rizontem AD perpendicularis, & angulus OHA per 32.lib.1. <lb></lb>æqualis eſt duobus internis HFA, FAH, eſt autem AHC ad <lb></lb>contingentem factus à ſemidiametro rectus per 18.lib. 3. ſicut <lb></lb>& HFA eſt rectus; reliquus CHO æqualis eſt angulo HAF <lb></lb>inclinationis plani. </s> <s id="s.000348">Certum eſt igitur, quòd in eam partem ro<lb></lb>ra convertetur, in qua fuerit centrum gravitatis. </s> <s id="s.000349">Quoniam <lb></lb>verò CI eſt Sinus anguli CHI, poſito radio CH, eſt au<lb></lb>tem CI minima omnium, quæ ex C puncto cadant in rectam <lb></lb>HO, manifeſtum eſt, quòd, ſi centrum gravitatis fuerit cen<lb></lb>tro rotæ vicinius, ut in R, rota ſemper deſcendet, quia cen<lb></lb>trum gravitatis reſpicit declivitatem plani: at, ſi fuerit <lb></lb>in I, aſcendere non poteſt, quia pars reſpiciens acclivita<lb></lb>tem plani non præponderat: ſi demum longiùs à centro <lb></lb>diſtiterit, ut in S, aſcendere poterit, uſque dum punctum S <pb n="41" xlink:href="017/01/057.jpg"></pb>fuerit in lineâ perpendiculari ad horizontem tranſeunte per <lb></lb>punctum contactûs. </s> </p> <p type="main"> <s id="s.000350">Ex his apertè conſtat futurum, ut rota deſcendat, ſi angulus, <lb></lb>quem in puncto contactûs faciunt lineæ ductæ ex centris mo<lb></lb>lis, & gravitatis (ſuppono molis centrum idem eſſe cum centro <lb></lb>rotæ, quâ rota eſt) minor fuerit angulo inclinationis plani, <lb></lb>tunc enim centrum gravitatis reſpicit declivitatem plani; fu<lb></lb>turum autem, ut rota aſcendat, ſi angulus ille major fuerit co<lb></lb>dem angulo inclinationis, quia centrum gravitatis reſpicit ac<lb></lb>clivitatem plani; futurum demùm, ut conſiſtat, ſi angulus il<lb></lb>le fuerit æqualis eidem angulo inclinationis plani, quia nimi<lb></lb>rum planum perpendiculare dividit æqualiter momenta gravi<lb></lb>tatis, cum tranſeat per centrum gravitatis exiſtens in lineâ <lb></lb>perpendiculari. </s> </p> <p type="main"> <s id="s.000351">Hinc patet ſemper deſcenſuram rotam, ſi habeat centrum <lb></lb>gravitatis R, quia ſemper facit angulum, de quo dictum eſt, <lb></lb>minorem angulo inclinationis, hoc eſt angulo CHI, nam ſi <lb></lb>ducatur ad CR perpendicularis RE, & ex centro ducatur <lb></lb>recta CE, angulus CER eſt maximus omnium, quos faciunt <lb></lb>lineæ ex punctis C, & R ductæ ad idem punctum circumfe<lb></lb>rentiæ, ut mox oſtendam; atqui CER minor eſt angulo CHI, <lb></lb>(quia ob lineas RE, IH parallelas, angulus IHC internus <lb></lb>per 29.lib.1. eſt æqualis externo RLC, & RLC externus per <lb></lb>16. lib. 1. major eſt interno CER, ac proinde IHC major <lb></lb>quàm CER) igitur quicunque angulus conſtitutus à rectis <lb></lb>exeuntibus ex C, & R minor eſt angulo inclinationis; atque <lb></lb>adeò ſemper deſcendet. </s> </p> <p type="main"> <s id="s.000352">At ſi centrum gravitatis fuerit S, ductâ ad CS perpendicu<lb></lb>lari SM, angulus omnium maximus eſt CMS: hic autem eſt <lb></lb>æqualis externo CKI, cum IK, & SM parallelæ ſint conſti<lb></lb>tutæ; angulus verò CKI externus major eſt interno CHI, <lb></lb>igitur angulus CMS major eſt angulo CHI, hoc eſt angulo <lb></lb>inclinationis. </s> <s id="s.000353">Aſcendere igitur poterit rota, quando angulus <lb></lb>ad contractum factus à lineis ex C, & S exeuntibus major eſt <lb></lb>angulo inclinationis; ſin autem contactus fiat in co puncto, ad <lb></lb>quod fit angulus æqualis, conſiſtet; ſi in iis punctis, ad quæ fit <lb></lb>angulus minor, deſcendet. </s> </p> <p type="main"> <s id="s.000354">Porrò quamvis iis, qui in Aſtronomicarum Proſtaphæreſeon <pb n="42" xlink:href="017/01/058.jpg"></pb>doctrinâ verſati ſunt, ſupervacaneum ſit oſtendere angulum <lb></lb>ad peripheriam factum à Radio circuli, & à linea perpendicu<lb></lb>lari in diametrum, eſſe maximum omnium, qui fieri poſſint à <lb></lb>Radio, & à lineâ ductâ ex eodem diametri puncto, in quod <lb></lb>cadebat perpendicularis; ut omnibus tamen fiat ſatis, non pi<lb></lb> <figure id="id.017.01.058.1.jpg" xlink:href="017/01/058/1.jpg"></figure><lb></lb>gebit hîc demonſtrare. </s> <s id="s.000355">Sit in diametro <lb></lb>circuli punctum R extra centrum C, & <lb></lb>ad CR ducatur perpendicularis HR, <lb></lb>quæ producta in G, bifariam dividitur <lb></lb>in R: & ductis ex centro rectis CH, <lb></lb>CG æqualibus, ſunt anguli CHR, <lb></lb>CGR æquales, per 5. vel 8. lib.1. </s> <s id="s.000356">Fiat <lb></lb>angulus CER, ductis ex C & R rectis <lb></lb>lineis ad idem punctum E peripheriæ. </s> <lb></lb> <s id="s.000357">Dico angulum CER minorem eſſe an<lb></lb>gulo CHR. </s> <s id="s.000358">Ducatur enim recta EG; & erunt in Iſoſcele <lb></lb>CEG æquales anguli CEG, CGE. </s> <s id="s.000359">Quia verò, per 7.lib.3. <lb></lb>RE major eſt quàm RG, angulus RGE major eſt angulo <lb></lb>REG, per 18.lib. 1. & ablatis æqualibus remanet REC mi<lb></lb>nor angulo RGC, hoc eſt RHC. </s> <s id="s.000360">Similiter oſtendetur angu<lb></lb>lum RIC minorem eſſe angulo RHC: ductâ enim IG, angu<lb></lb>li CIG, CGI ſunt æquales: & quoniam per 7.lib.3. RG ma<lb></lb>jor eſt quàm RI, angulus RIG major eſt angulo RGI, per <lb></lb>18.lib.1. ſi igitur ex æqualibus auferantur inæquales anguli, re<lb></lb>manet RIC minor, quàm RGC, hoc eſt quam RHC. </s> <s id="s.000361">Ea<lb></lb>dem erit methodus demonſtrandi angulos ad puncta periphe<lb></lb>riæ propiora puncto H eſſe majores angulo CER. </s> <s id="s.000362">Ductâ enim <lb></lb>RD æquali ipſi RE, ad punctum ſcilicet D æqualiter diſtans à <lb></lb>diametro, ac diſtet punctum E, & ducto radio CD, eſt angu<lb></lb>lus CDR æqualis angulo CER. </s> <s id="s.000363">Sit autem puncto H vicinior <lb></lb>angulus COR, quem dico eſſe majorem angulo CER per <lb></lb>7.lib.3. & 8.lib.1. </s> <s id="s.000364">Ducta lineâ OD, anguli COD, CDO <lb></lb>ſunt æquales, quia latera CO, CD æqualia ſunt: at per 7.lib.3. <lb></lb>RO minor eſt, quàm RE, hoc eſt RD, igitur angulus ROD <lb></lb>per 18.lib.1. major eſt angulo RDO, & ablatis æqualibus re<lb></lb>manet ROC major quam RDC, hoc eſt quàm REC. </s> <s id="s.000365">Angu<lb></lb>li itáque recedentes à puncto H ſemper fiunt minores, acce<lb></lb>dentes verò fiunt majores. </s> </p> <pb n="43" xlink:href="017/01/059.jpg"></pb> <p type="main"> <s id="s.000366">Hoc probato conſequens eſt illud, quod in rotæ peripheriâ <lb></lb>duo ſunt puncta, inter quæ quodlibet punctum contingat pla<lb></lb>num <expan abbr="inclinatũ">inclinatum</expan>, rota aſcendit, ſi angulus maximus factus à lineis <lb></lb>ductis ex centro rotæ, & ex centro gravitatis ſit major angulo <lb></lb>inclinationis; quia nimirum anguli à puncto H recedentes ad <lb></lb>utramque partem ſemper fiunt minores; ergo ad utramque eſt <lb></lb>angulus unus æqualis angulo inclinationis, & ſpatium inter <lb></lb>hujuſmodi angulos eſt quantitas peripheriæ, quæ aſcendens <lb></lb>poteſt coaptari plano inclinato: ac proinde ex horum puncto<lb></lb>rum diſtantia definietur ſpatium, quod poteſt rota aſcendens <lb></lb>percurrere. </s> </p> <p type="main"> <s id="s.000367">Sit igitur rota, cujus centrum C, & <lb></lb> <figure id="id.017.01.059.1.jpg" xlink:href="017/01/059/1.jpg"></figure><lb></lb>centrum gravitatis S: ſit autem CS par<lb></lb>tium 11, quarum CH Radius eſt 16: <lb></lb>eſt igitur CS æqualis Sinui gr. 43. 26′. <lb></lb>qui erit maximus angulus CIS ad peri<lb></lb>pheriam factus à Radio, & à lineâ IS <lb></lb>perpendiculari ad SC. </s> <s id="s.000368">Quare in quoli<lb></lb>bet plano habente minorem inclinatio<lb></lb>nem poterit aſcendere. </s> <s id="s.000369">Ponatur plani <lb></lb>inclinatio gr. 15, cui æqualis ſit angulus CHS. </s> <s id="s.000370">Fiat igitur <lb></lb>ut CS 11 ad CH 16, ita Sinus anguli CHS 25882 ad <lb></lb>37646 Sinum Anguli CSH gr. 22. 7′; eritque angulus <lb></lb>SCH gr. 142. 53′. </s> <s id="s.000371">Creſcet ergo ſupra angulum H angulus <lb></lb>ad peripheriam, ſi ultra punctum H fiat contactus rotæ <lb></lb>in alio puncto viciniore puncto I, ex quo ad SC perpendi<lb></lb>cularis cadit; & ex I decreſcit uſque dum in P fiat angu<lb></lb>lus SPC grad. 15 æqualis angulo inclinationis. </s> <s id="s.000372">In triangu<lb></lb>lo itaque SPC invenitur ex iiſdem datis angulus PSC <lb></lb>gr. 157. 53′. & angulus SCP gr. 7. 7′. qui ex angulo SCH <lb></lb>gr. 142. 53′ ablatus relinquit PCH gr. 135. 46′. quæ eſt quan<lb></lb>titas arcûs HIP, quæ plano coaptatur in aſcenſu. </s> <s id="s.000373">Quoniam <lb></lb>verò quarum partium CG Radius eſt 16, peripheria eſt 100 1/2 <lb></lb>earum parirer eſt arcus HP ferè 38, ſi Radius rotæ fuerit un<lb></lb>ciarum pedis 16, rota aſcendet in plano percurrens ſpatium <lb></lb>pedum 3, & eo ampliùs. </s> <s id="s.000374">Hinc poteris aut rotæ diametrum au<lb></lb>gere, aut plani inclinationem minuere, ſi volveris rotam lon<lb></lb>giore ſpatio moveri: auctâ enim rotæ diametro augetur peri- <pb n="44" xlink:href="017/01/060.jpg"></pb>pheria, ſervatâ ratione eadem diſtantiæ centri gravitatis. </s> <s id="s.000375">At ſi <lb></lb>data fuerit rota (oportet non ignorari diſtantiam centri gravi<lb></lb>tatis à centro rotæ, poterit autem primâ praxi cap.5. inveſtiga<lb></lb>ri) certum eſt illam non poſſe aſcendere niſi per ſpatium mi<lb></lb>nus longitudine ſemiperipheriæ; conſtituto autem ſpatio inve<lb></lb>nietur inclinatio plani neceſſaria, hac methodo. </s> <s id="s.000376">Data ſpatij <lb></lb>longitudo PH reducatur ad denominationem graduum, & erit <lb></lb>notus angulus PCH: & quoniam anguli ad H & ad P debent <lb></lb>eſſe æquales, anguli verò in R ad verticem ſunt æquales, erunt <lb></lb>pariter æquales PCH, & PSH, qui proinde notus eſt. </s> <s id="s.000377">Hujus <lb></lb>ſemiſſis auferatur ex recto CSI, & innoteſcet angulus CSH, <lb></lb>cum quo & duobus lateribus CS, CH invenietur per Trigo<lb></lb>nometriam angulus CHS æqualis angulo inclinationis plani <lb></lb>neceſſariæ. </s> <s id="s.000378">Quod autem angulus HSI ſit ſemiſſis totius HSP, <lb></lb>hoc eſt dati PCH, ſic oſtendo. </s> <s id="s.000379">Quia in duobus triangulis <lb></lb>CSP, CHS idem latus CS opponitur angulis æqualibus ad H, <lb></lb>& ad P, æqualia autem latera CH, & CP opponuntur angulis <lb></lb>quæſitis CSH, & CSP, conſtat horum duorum angulorum <lb></lb>eſſe unum eundemque ſinum; ergo ſimul ſumpti ſunt æquales <lb></lb>duobus rectis; auferatur ex eorum ſummâ unus rectus, rema<lb></lb>nebunt duo anguli ſimul CSH, ISP æquales uni recto, hoc <lb></lb>eſt angulo ISC: auferatur communis CSH, remanebit HSI <lb></lb>æqualis angulo ISP: id quod oportuit demonſtrare. </s> </p> <p type="main"> <s id="s.000380">Colligere poſſumus ex his, quæ hactenus explicata ſunt, fie<lb></lb>ri quidem poſſe, ut, ſi rota in plano inclinato primùm conſti<lb></lb>tuta exactè tangat in H, prorſus conſiſtat; id tamen vix poſſe <lb></lb>ſperari, quia ſi in alio puncto remotiore ab I tangat, cadet, ſi <lb></lb>in puncto viciniore, aſcendet. </s> <s id="s.000381">At ubi venerit in P, ſi ex con<lb></lb>cepto impetu pergat adhuc aliquantulum aſcendere; centro <lb></lb>gravitatis S tranſlato versùs plani declivitatem, & diminuto <lb></lb>angulo, deſcendet; & ubi tranſilierit punctum P, iterùm aucto <lb></lb>angulo aſcendet, donec omninò in P conſiſtat. </s> <s id="s.000382">Ubi licet <lb></lb>animadvertere non idem eſſe punctum contactus, in quo <lb></lb>quieſceret in plano horizontali, ac inclinato; in plano enim <lb></lb>horizontali quieſceret in O, ubi linea à centro rotæ C perpen<lb></lb>dicularis horizonti, ac tranſiens per S centrum gravitatis, ter<lb></lb>minatur: in eo autem puncto O conſiſtere non poſſe ſupra pla<lb></lb>num inclinatum ſatis patet ex dictis. </s> <s id="s.000383">Porrò hæc, quæ de rotâ <pb n="45" xlink:href="017/01/061.jpg"></pb>conſiſtente, aut cadente diſputata ſunt, dicenda eſſe de ſphæ<lb></lb>râ quieſcente in plano inclinato, clarius eſt, quàm ut oporteat <lb></lb>pluribus explicare. </s> </p> <p type="main"> <s id="s.000384">Unum ſupereſſe videtur oſtendendum, quî verum ſit cen<lb></lb>trum gravitatis deſcendere ita, ut fiat horizonti vicinius, dum <lb></lb>rota aſcendit, & fit remotior. </s> <s id="s.000385">Id ut manifeſtum fiat, primò in<lb></lb>veniatur HS: & ſit ut Sinus anguli CHS gr. 15. ad ſinum an<lb></lb>guli SCH gr. 14.2. 53′. hoc eſt ut 25882 ad 60344, ita CS <lb></lb>partium 11 ad HS 25 2/3: quæ eſt altitudo centri gravitatis ante <lb></lb>motum. </s> <s id="s.000386">Deinde inveniatur SP; & ſit ut Sinus SPC gr. 15 ad <lb></lb>Sinum SCP gr.7. 7′ hoc eſt, ut 25882 ad 12389, ita CS par<lb></lb>tium 11 ad SP 5 1/4, quæ in fine motus erit altitudo centri gravi<lb></lb>tatis ſupra planum inclinatum; huic autem addenda eſt altitu<lb></lb>do, quam ſupra horizontem habet punctum illud plani inclinati, <lb></lb>in quo tanget P. </s> <s id="s.000387">Quia ergo inclinatio plani eſt gr. 15, & HP <lb></lb>eſt partium 38, tantum eſt ſpatium, quod in plano percurritur <lb></lb>à rota aſcendente, fiat ut Radius 100000 ad 25882 Sinum an<lb></lb>guli inclinationis, ita 38 ad 9 4/5 altitudinem ſupra horizontem, <lb></lb>cui ſi addas SP 5 1/4, erit in fine motûs altitudo centri gravitatis <lb></lb>ſupra horizontem partium 15, cùm initio diſtaret partibus 25 2/3. </s> <lb></lb> <s id="s.000388">Centrum igitur gravitatis ſimpliciter, & abſolutè deſcendit, <lb></lb>dum rota in plano inclinato aſcendit. </s> </p> <p type="main"> <s id="s.000389">Poſſem hîc afferre aquam vi ſuæ gravitatis aſcendentem in <lb></lb>cochleâ Archimedis, dum cylindrus, quem cochlea ambit, <lb></lb>convertitur: abſtineo tamen, quia non vacat hîc examinare, <lb></lb>an motus ille compoſitus ſit ex converſione, quâ pulſu externo <lb></lb>agitata aqua attollatur, & ex naturali deſcenſu, quo per tubum <lb></lb>in ſpiras ſinuatum deſcendat; an verò quemadmodum ſuppoſi<lb></lb>to cuneo reluctans pondus elevatur, vel etiam cochleâ trahitur <lb></lb>in plano horizontali, ita dicendum ſit aquam vi ſuæ gravitatis <lb></lb>in imo perſiſtentem à cochleâ ſenſim ſubeunte elevari ſimul, & <lb></lb>trahi, quin illa ſponte ſua aſcendat: nam aquæ facilè tribuitur <lb></lb>aliquando motus, qui ſubjecto corpori, cui illa inſidet, conve<lb></lb>nit; ut liquet ſi ampliorem peluim ex fune ſuſpenderis, vel lu<lb></lb>brico in plano horizontali collocaveris, in qua ſit non multa <lb></lb>aqua in depreſſiore fundi parte quieſcens; vaſe ſiquidem ex <lb></lb>improviſo vehementiùs impulſo videtur aqua in oppoſitam par- <pb n="46" xlink:href="017/01/062.jpg"></pb>tem refluere, cum tamen vas ipſum potiùs infra aquam mo<lb></lb>veatur, quàm aqua in vaſe: quanquam ratione adhæſionis aquæ <lb></lb>ad peluim etiam ipſa motum concipiat. </s> <s id="s.000390">Quare in cenſu ſponte <lb></lb>aſcendentium numeranda non videtur aqua tubo ſpeciali cy<lb></lb>lindrum circumplexo elevata. </s> </p> <p type="main"> <s id="s.000391">Videatur fortaſſe aqua ſponte aſcenſura in tubo non æquabi<lb></lb>li ſed conico, in plano verticali rotæ ſpiraliter circumducto: <lb></lb>dum enim aqua æquilibrium ſuperficiei faciens in parte tubi <lb></lb>ampliore præponderat, convertitur rota, & illa iterum æqua<lb></lb>liter ſe librans totius molis compoſitæ centrum gravitatis trans<lb></lb>fert extra lineam perpendicularem: ſi tamen ea cautio adhi<lb></lb>beatur, ut tanta ſit aquæ quantitas, quæ non planam obtineat <lb></lb>ſuperficiem ſed tubi inflexione conformetur; neque ita ſit <lb></lb>ſpiræ aſcendentis ardua altitudo, ut aqua poſt ſuperficiei libra<lb></lb>tionem ex ea parte ob ſui paucitatem non præponderet; & præ<lb></lb>terea ejus figuræ ſit tubus, ut aqua in parte anguſtiore remo<lb></lb>tior à perpendiculari, non ita ratione ſitûs augeat momenta ſui <lb></lb>conatus deorſum, ut repugnare valeat aquæ ampliorem tubi <lb></lb>partem occupanti. </s> <s id="s.000392">Si hæc, inquam, obſerventur (an autem <lb></lb>ita facile ſit ea obſervare, ut quidam autumant, hic non de<lb></lb>finio) & centrum gravitatis transferatur extra perpendicula<lb></lb>rem versùs ampliorem tubi ſpiralis partem, futurum quidem <lb></lb>eſt, ut aqua aſcendat; id tamen non eſt opus centri gravitatis, <lb></lb>ſed potius virtutis illius, qua humor ſe æquabiliter librat. <lb></lb> </s> </p> <p type="head"> <s id="s.000393"><emph type="center"></emph>CAPUT VIII.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000394"><emph type="center"></emph><emph type="italics"></emph>Cur gravium in plano inclinato deſcendentium <lb></lb>alia repant, alia rotentur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000395">QUæ capite ſuperiori dixi de globi aut rotæ ſuper planum <lb></lb>inclinatum conſiſtentiâ in puncto, in quo linea à centro <lb></lb>globi, aut rotæ ducta cum eâ, quæ ex centro gravitatis duci<lb></lb>tur, facit angulum æqualem angulo inclinationis plani, non ita <lb></lb>intelligi velim, quaſi motus omnis deorſum adimatur rotæ aut <lb></lb>globo cujuſlibet gravitatis, & in quovis plano inclinato: ibi <lb></lb>enim conſiſtentiæ, aut quietis nomine ſolam converſionem <pb n="47" xlink:href="017/01/063.jpg"></pb>excipio, non lapſum nego. </s> <s id="s.000396">Fieri ſi quidem poteſt, ut adeò con<lb></lb>tinuo lævore lubricum ſit planum, exactéque rotundatus globus, <lb></lb>ut nullam ex eminulis particulis moram recipiens deorſum la<lb></lb>batur, volubilitate ipsâ motum nihil juvante, ſed ſolo pondere <lb></lb>urgente, cum in lineâ ad horizontem perpendiculari ſemper <lb></lb>maneat centrum gravitatis, & punctum contactûs. </s> </p> <p type="main"> <s id="s.000397">Neque eſſet diverſa ratio ſphæræ centrum gravitatis haben<lb></lb>tis extra centrum molis, ac cæterorum corporum non ſphæri<lb></lb>corum: Nam gravia quæcunque in plano inclinato conſtituta <lb></lb>tantum habent ad deſcendendum momenti, ut aſperitatis re<lb></lb>ſiſtentiam vincant, repunt quidem, ſi linea directionis ab eo<lb></lb>rum gravitatis centro in terræ centrum ducta tranſeat per con<lb></lb>tactum ſubjecti plani, & impoſiti gravis; rotantur verò, ſi di<lb></lb>rectionis linea in plani declivitatem cadat extra contactum: <lb></lb>ſivè demùm in puncto, ſivè in lineâ, ſivè in ſuperficie con<lb></lb>tactus fiat. </s> <s id="s.000398">Eſt autem animadvertendum non eſſe opus, ut una <lb></lb>continua ſuperficies ſit, aut linea, ſecundùm quam ſe tangant; <lb></lb>ſed pro ſuperficie aut linea contactûs accipitur totum illud ſpa<lb></lb>tium, quod inter extrema contingentia rectis lineis conjuncta <lb></lb>intercipitur. </s> </p> <p type="main"> <s id="s.000399">Sit planum inclinatum AB, <lb></lb> <figure id="id.017.01.063.1.jpg" xlink:href="017/01/063/1.jpg"></figure><lb></lb>cui globus C incumbit con<lb></lb>tingens in puncto D. </s> <s id="s.000400">Ex cen<lb></lb>tro gravitatis C, quod & cen<lb></lb>trum molis eſt ex hypotheſi, <lb></lb>cadat linea directionis CE <lb></lb>perpendicularis in horizon<lb></lb>tem FB; quæ neceſſariò ca<lb></lb>dit extra punctum contactûs <lb></lb>D; alioquin eadem linea CE <lb></lb>caderet ad angulos rectos ſu<lb></lb>pra planum inclinatum, & ſupra horizontale, id quod fieri <lb></lb>non poteſt, cum hujuſmodi plana non ſint invicem parallela. </s> <lb></lb> <s id="s.000401">Per D igitur punctum ſuſtentationis ductâ GH parallelâ lineæ <lb></lb>directionis, ſi per utramque plana parallela ducantur, planum <lb></lb>per GH ſecat ſphæram in partes inæqualiter graves; & idcir<lb></lb>co pars præponderans, in qua eſt centrum gravitatis globi, mo<lb></lb>vetur circa punctum ſuſtentationis D, atque adeò in gyrum <pb n="48" xlink:href="017/01/064.jpg"></pb>converſa circa centrum C deſcendit, ac rotatur. </s> <s id="s.000402">Quod ſi inæ<lb></lb>qualis fuerit ſphæræ ſubſtantia, & centrum gravitatis I in per<lb></lb>pendiculari GH, non deſcendet ſphæra in gyrum acta, ſed <lb></lb>tantùm repet, cum neutra pars præponderet. </s> </p> <p type="main"> <s id="s.000403">Simili ratione parallelepipedum KL, cujus centrum gravi<lb></lb>tatis M, non repit; quia, cùm linea directionis MN cadat ex<lb></lb>tra baſim KO, quæ contingit ſubjectum planum, ſi per extre<lb></lb>mam lineam KP tranſeat planum PQ horizonti perpendicu<lb></lb>lare, dividitur parallelepipedum in duo priſmata inæqualia, & <lb></lb>non æquiponderantia: cum verò priſma trapezium QLKP <lb></lb>præponderet priſmati trigono KOQ, quod ſuſtinetur à baſi, <lb></lb>illud neceſſariò deſcendit, & circa lineam KP convertitur. </s> <lb></lb> <s id="s.000404">Contrà autem quando intra baſim contactûs, ut in cubo PR, <lb></lb>cujus centrum S, cadit linea directionis ST, tunc repit, & non <lb></lb>rotatur cubus; quia ſcilicet ab extrema ſuſtentationis lineâ KP <lb></lb>ductum planum horizonti perpendiculare dividit cubum in <lb></lb>partes inæquales ita, ut pars illa, in qua eſt centrum gravitatis, <lb></lb>& quæ à ſubjecto plano tota ſuſtinetur, præponderet, nec poſ<lb></lb>ſit à reliquâ parte elevari, ut circa KP convertatur. </s> </p> <p type="main"> <s id="s.000405">Hinc apparet ad quantam altitudinem pertinere poſſit paral<lb></lb>lelepipedum, ut in dato plano inclinato non rotetur, ſed repat: <lb></lb>nam ab extremâ ſuſtentationis lineâ KP excitatum planum <lb></lb>horizonti perpendiculare PQ, quod bifariam in partes æqui<lb></lb>ponderantes dividit parallelepipedum KQ, determinat altitu<lb></lb>dinem maximam <expan abbr="Xq;">Xque</expan> in omni quippe majori altitudine non <lb></lb>repit, ſed rotatur, quia linea directionis cadit extra baſim <lb></lb>ſuſtentationis: in omni verò minori altitudine non rotatur, ſed <lb></lb>repit, quia linea directionis cadit intra baſim ſuſtentationis. </s> <lb></lb> <s id="s.000406">Hoc idem in corporibus cæteris, quamvis non parallelepipe<lb></lb>dis, obſervandum eſt, an ſcilicet linea directionis cadat extra <lb></lb>baſim ſuſtentationis, nec ne. </s> </p> <p type="main"> <s id="s.000407">Quæ tamen de cubo repente dicta ſunt, intelligi velim ſpecta<lb></lb>tâ per ſe gravium figurâ: quia per accidens fieri poteſt, ut cor<lb></lb>pus non repat, ſed rotetur, quamvis linea directionis cadat in<lb></lb>tra baſim, quæ planum inclinatum contingit. </s> <s id="s.000408">Nam ſi in motu <lb></lb>occurrat ſuper plano inclinato offendiculum aliquod, cui de<lb></lb>ſcendens corpus illidatur, fieri poteſt, ut impetus ex motu con<lb></lb>ceptus ita promoveat centrum gravitatis in anteriora, ut linea <pb n="49" xlink:href="017/01/065.jpg"></pb>directionis cadat extra baſim ultrà punctum illud, quod proxí<lb></lb>mum eſt offendiculo, ac proinde circa illud convertatur. </s> <s id="s.000409">Hæc <lb></lb>autem potiſſimùm eſt ratio, cur ex clivis deſcendentes lapides, <lb></lb>quamquam nec orbiculares, nec admodum alti, rotentur ta<lb></lb>men; quia ſcilicet multa offendicula in clivo occurrunt, & ab <lb></lb>impetu per motum concepto partes ſuperiores promoventur <lb></lb>ulteriùs, inferioribus retardatis. </s> <s id="s.000410">Sic ſæpè ceſpitantes cadimus, <lb></lb>quia ab offendiculo retinentur pedes, cum interim corpus re<lb></lb>liquum ex concepto impetu ulteriùs promoveatur, ita ut linea <lb></lb>directionis cadat extra baſim ſuſtentationis. <lb></lb></s> </p> <p type="head"> <s id="s.000411"><emph type="center"></emph>CAPUT IX.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000412"><emph type="center"></emph><emph type="italics"></emph>Cur turres inclinatæ non corruant.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000413">OBſervandum eſt, ait Vitruvius lib.6. cap. 11, uti omnes <lb></lb>ſtructuræ perpendiculo reſpondeant, neque habeant in <lb></lb>ulla parte proclinationes. </s> <s id="s.000414">Nemo eſt qui non intelligat præ<lb></lb>ceptum hoc ad ædificiorum conſiſtentiam pertinere; ſed neque <lb></lb>defuerunt, qui rem ſubtiliùs, quàm par ſit, perpendentes ina<lb></lb>ni timore ſe torquebant, ne fortè aliquando domus corrueret, <lb></lb>cujus parietes inter ſe paralleli fuerant conſtituti; cùm enim <lb></lb>perpendicula ſibi demum in terræ centro occurrant, fieri non <lb></lb>poſſe putabant, ut ſimul paralleli eſſent parietes. </s> <s id="s.000415">Id quod Geo<lb></lb>metricè quidem verum eſt; Phyſicè tamen paralleliſmus cum <lb></lb>perpendiculis conſentit: nam ſi funiculos duos longitudinis <lb></lb>ped. 100. clavo affixos ita extendas, ut extrema eorum palmi <lb></lb>intervallo diſtent, angulum facient acutiſſimum; & ſi lineas <lb></lb>duas bipedales duxeris eorum extremitatibus congruentes, vix <lb></lb>different à parallelis, cum intervalla jungentia utroſque linea<lb></lb>rum terminos differant inter ſe ſolum palmi parte quinquage<lb></lb>ſima. </s> <s id="s.000416">Longè autem majorem rationem terræ ſemidiameter ha<lb></lb>bet ad quamlibet ædificiorum altitudinem; ut proinde à paral<lb></lb>leliſmo multo minùs recedant parietes, etiamſi fuerint turrium <lb></lb>inſtar altiſſimi. </s> <s id="s.000417">Ponantur enim parietes duo, aut potiùs turres, <lb></lb>diſtare inter ſe paſſ.300; ſit autem parietum, vel turrium alti<lb></lb>tudo paſſ. 60, hoc eſt ped.300. </s> <s id="s.000418">Conſtat mihi, ut aliàs oſtendi, <lb></lb>terrenam ſemidiametrum non eſſe minorem paſſibus Rom. <pb n="50" xlink:href="017/01/066.jpg"></pb><expan abbr="antiq.">antique</expan> 4128635: quarè ſi fiat ut terræ ſemidiameter 4128635 <lb></lb>ad altitudinem 60, ita diſtantia parietum, aut turrium in imo <lb></lb>300, ad aliud, proveniet differentia, qua diſtantia turrium in <lb></lb>ſummo vertice ſuperat earum diſtantiam in imo pede, & erit <lb></lb>partium (4359/1000000) unius paſſus, quæ eſt minor quàm 2/5 digiti: quis <lb></lb>autem parallelas non dixerit turres, quæ vix uno aut altero <lb></lb>hordei grano diſtant à paralleliſmo? </s> <s id="s.000419">Quod ſi in tanta altitudine <lb></lb>atque diſtantiâ diſcrimen hoc adeò exiguum eſt, ſatis patet, <lb></lb>quid de columnarum paralleliſmo dicendum ſit. </s> <s id="s.000420">Conſtat autem <lb></lb>ex his ædificia in altiſſimis montibus conſtituta habere parie<lb></lb>tes minùs à paralleliſmo recedentes, ſi fuerint ad perpendicu<lb></lb>lum ædificati, quàm in locis depreſſioribus: atque adeò, ſi duæ <lb></lb>columnæ eandem inter ſe poſitionem ſervantes deſcenderent <lb></lb>cum ſubjecto plano, ita ut alterutra columnarum illarum ad <lb></lb>perpendiculum deſcenderet, reliqua demùm adeò inclinare<lb></lb>tur, ut caderet. </s> </p> <p type="main"> <s id="s.000421">Sed quàm inanem ſibi ſtruant ſolicitudinem, qui nimis exi<lb></lb>guè, & exiliter ad calculos revocant ſtructurarum perpendicu<lb></lb>la, ſatis indicant turres inclinatæ, quæ poſt aliquot ſecula con<lb></lb>ſiſtunt citrà ullum ruinæ periculum, quamvis illam timeant <lb></lb>imperiti. </s> <s id="s.000422">Duas habemus in Italiâ turres ob inſignem inclina<lb></lb>tionem conſpicuas; altera eſt Bononiæ quadrata opere lateri<lb></lb>tio, altera Piſis rotunda ex albo marmore affabrè expolito, & <lb></lb>columnis 284 rite diſpoſitis ornata. </s> <s id="s.000423">Ædificari cœpit anno <lb></lb>1173 Germano quodam architecto, quem ab aliis Guillel<lb></lb>mum, ab aliis Joannem OEnipontanum dici reperio. </s> <s id="s.000424">Rotunda <lb></lb>eſt forma duplici muro concludente ſcalas cochleæ in modum <lb></lb>ab imo ad ſummum ductas: parietis craſſities eſt cubitorum <lb></lb>6 1/3, turris altitudo cubitorum 78, ambitus in imo pede cubi<lb></lb>torum 80; unde colligitur diameter cubitorum ferè 25 1/2; incli<lb></lb>natio, ſeu intervallum inter baſim, & perpendiculum eſt cu<lb></lb>bitorum 7 1/3, ut ex literis ad me inde datis habeo; quamvis <lb></lb>apud aliquos legerim tantùm cubitos 7, apud alios 6 1/2. </s> <s id="s.000425">Factâ <lb></lb>ne fuerit illa inclinatio de induſtriâ, an verò ſubſidentibus fun<lb></lb>damentis, incertum eſt. </s> <s id="s.000426">Ego non facilè eo in illorum ſenten<lb></lb>tiam, qui id ſcribunt contigiſſe ex artificis imperitia, cui non <lb></lb>ſatis perſpecta eſſet ſoli natura; tum quia fundamenta altitudi- <pb n="51" xlink:href="017/01/067.jpg"></pb>nem habent, atque amplitudinem ingentem, quibus con<lb></lb>ſtruendis annus ſolidus ſatis non fuit; tum quia nullam unquam <lb></lb>egit rimam, id quod ſubſidente ſolo rariſſimum eſt; tum quia <lb></lb>potuit architectus excitari ad artis ſpecimen exhibendum à tur<lb></lb>ri Bononienſi Gariſendâ excitatâ anno 1110. </s> </p> <p type="main"> <s id="s.000427">Turris Bononienſis altitudinem habet pedum Bonon. 130; <lb></lb>exteriùs inclinatur ped. 9, interiùs verò ped. 1, & paulo am<lb></lb>plius: muri craſſities in parte infimâ eſt pedum 6 1/2, in ſupre<lb></lb>ma ped. 4; cava turris ped. 7. quare lateris longitudo eſt ped. <lb></lb>20, & ambitus, quoniam quadrata eſt, ped. 80. Ex his men<lb></lb>ſuris, quas in <emph type="italics"></emph>Bononïá Perluſtratâ<emph.end type="italics"></emph.end> anno 1650 typis evulgatâ at<lb></lb>tulit Antonius Pauli Maſini, turris ſpe<lb></lb> <figure id="id.017.01.067.1.jpg" xlink:href="017/01/067/1.jpg"></figure><lb></lb>ciem exhibeo, & eſt AB latus unum <lb></lb>ped. 20, BD inclinationis menſura <lb></lb>ped. 9. DC altitudo perpendicularis <lb></lb>ped.130; EB & AF ped. 6 1/2 craſſities <lb></lb>imi parietis, & CH ped. 4. craſſities <lb></lb>ejuſdem parietis EC exteriùs inclinati. </s> <lb></lb> <s id="s.000428">At quoniam inclinatio interior FI dici<lb></lb>tur eſſe ped.1, & paulo ampliùs, erit ID <lb></lb>paulo major ped.21; erecta autem ex I <lb></lb>perpendicularis dabit punctum G termi<lb></lb>num craſſitiei muri AG in parte ſupre<lb></lb>mâ, & erit CG major ped. 21, cum ſit <lb></lb>æqualis ipſi ID. </s> <s id="s.000429">Quare fieri non poteſt, <lb></lb>ut KG ſit ped. 4; quemadmodum HC; <lb></lb>alioquin eſſet CK ſaltem ped.25, cum <lb></lb>baſis AB ſit tantum ped.20. </s> <s id="s.000430">Hinc ſi li<lb></lb>ceat conjecturas perſequi (quandoqui<lb></lb>dem veritatem aſſequi non potui, cum <lb></lb>non careat periculo aſcenſus per ſcalas <lb></lb>ligneas à pluviis maximam partem cor<lb></lb>ruptas) exiſtimo AF majorem eſſe quàm <lb></lb>EB, hoc eſt majorem pedibus 6 1/2, KG <lb></lb>verò minorem quam HC, ut turri ſua <lb></lb>conſtet Eurithmia; id quod obtineretur, ſi ID uno, aut alte<lb></lb>ro pede minor eſſet quàm AB, differentia enim inter ID, <lb></lb>& AB eſſet craſſities KG. </s> <s id="s.000431">Et ſanè memini aliquando me au- <pb n="52" xlink:href="017/01/068.jpg"></pb>diviſſe ſupremam craſſitiem muri oppoſiti parti inclinatæ <lb></lb>non excedere integrum pedem. </s> <s id="s.000432">Id autem valde opportu<lb></lb>num accidebat, ut longè faciliùs paries AFGK ſuâ mole <lb></lb>ſtaret: neque enim caſu inclinatam fuiſſe turrim dicere po<lb></lb>teris, quam conſtat prope Aſinellam rectiſſimam ideò fuiſſe <lb></lb>conditam, ut multo clariùs appareret inclinatio: præterquam <lb></lb>quod inclinatio interior minor externâ ſatis oſtendit muros <lb></lb>nunquam fuiſſe parallolos. </s> </p> <p type="main"> <s id="s.000433">Porrò ut conſtet ex hujuſmodi inclinatione non magis <lb></lb>eſſe de ruinâ timendum, quàm ſi exactè perpendicularis eſ<lb></lb>ſet, examinemus, ſi placet, centrum gravitatis in turri Bo<lb></lb>nonienſi; hinc enim facilis erit conjectura de cæteris. </s> <s id="s.000434">Et <lb></lb> <figure id="id.017.01.068.1.jpg" xlink:href="017/01/068/1.jpg"></figure><lb></lb>primò parietis maximè inclinati ſectio <lb></lb>verticalis illum bifariam ſecans ac tran<lb></lb>ſiens per centrum gravitatis ſit HCBE: <lb></lb>cujus latera parallela HC, EB bifariam <lb></lb>ſecta in V & R jungantur rectâ VR, cu<lb></lb>jus longitudo inveſtiganda eſt, ut in eâ <lb></lb>definiatur punctum S centrum gravitatis, <lb></lb>ac innoteſcat utrum perpendicularis SX, <lb></lb>ſcilicet linea directionis cadat intra ba<lb></lb>ſim EB ſuſtentantem. </s> <s id="s.000435">Et ut à fractioni<lb></lb>bus minus incommodi ſubeamus, liceat <lb></lb>aſſumere pedem in partes centeſimas di<lb></lb>viſum. </s> <s id="s.000436">Cum autem EB ſit ped. 6 1/2, ſemiſ<lb></lb>ſis RB eſt ped. 3. 25″; & quia HC eſt <lb></lb>ped. 4, VC eſt ped. 200″. </s> <s id="s.000437">Et ducatur <lb></lb>recta BV. </s> </p> <p type="main"> <s id="s.000438">In triangulo BDC rectangulo datis BD, <lb></lb>inclinatione ped. 90′0′, & altitudine per<lb></lb>pendiculari CD ped. 130′0′, additis late<lb></lb>rum quadratis fit quadratum hypothenu<lb></lb>ſæ BC, quæ eſt ped. 13031″. </s> <s id="s.000439">Ex datis autem lateribus BD, <lb></lb>& DC invenitur angulus CBD gr. 88. 33′, cui æqualis eſt <lb></lb>inter parallelas VC, BD alternus VCB: angulus verò CBR <lb></lb>gr. 91. 27′. </s> </p> <p type="main"> <s id="s.000440">In triangulo VCB datis lateribus VC ped. 2.0′0′, CB <lb></lb>ped. 130. 31″, & angulo verticali VCB gr. 88. 33′, reperitur <pb n="53" xlink:href="017/01/069.jpg"></pb>CVB gr. 90. 34′. 14″, & VBC gr. 0. 52′. 46″.. </s> <s id="s.000441">Ex his autem <lb></lb>inveſtigatur VB ped. 130. 26″. </s> </p> <p type="main"> <s id="s.000442">Quoniam autem angulus CBR notus erat gr. 91. 27′, ſi de<lb></lb>matur ex illo angulus VBC gr. 0. 52′. 46″. remanet VBR <lb></lb>gr. 90. 34′, 14″, æqualis angulo CVB alterno inter parallelas; <lb></lb>& nota ſunt latera illum conſtituentia BR ped 3. 25″. & BV <lb></lb>ped. 130. 26″. </s> <s id="s.000443">Ex quibus datis invenitur angulus BRV gr. 88. <lb></lb>0′. 2″, BVR gr. 1. 25′. 44″ & baſis VR ped. 130. 326tʹ. </s> </p> <p type="main"> <s id="s.000444">Jam verò, ex prop. 15 lib.1. Æquipond. Archimedis, divi<lb></lb>datur VR in S eâ ratione, ut ſit VS ad SR, ut duplum EB <lb></lb>majoris parallelarum unâ cum minore HC, ad duplum HC <lb></lb>unâ cum majore EB, hoc eſt (quia EB eſt ped. 6 1/2) & HC <lb></lb>ped.4.) ut 17 ad 14 1/2. </s> <s id="s.000445">Igitur ut 31 1/2 ad 14 1/2, ita VR 130. 326tʹ, <lb></lb>ad SR ped. 59. 99″. </s> <s id="s.000446">Demum ex S ducta perpendiculari SX, <lb></lb>quia in triangulo RXS rectangulo datur angulus SRX gr.88. <lb></lb>0′.. 2″. atque adeò ejus complementum RSX gr.1. 59′. 58″. & <lb></lb>latus SR ped. 59. 99″. invenitur latus RX ped. 209″. </s> <s id="s.000447">Eſt igi<lb></lb>tur RX linea minor, quàm RB poſita ped. 3. 25″; & idcirco <lb></lb>perpendicularis linea directionis SX cadit intrà baſim parie<lb></lb>tis EBCH. </s> </p> <p type="main"> <s id="s.000448">Sed quia facturum me puto rem aliquibus gratam, ſi quas <lb></lb>inij rationes hîc exhibeam, calculi totius progreſſum per lo<lb></lb>garithmos hîc addo, ut illum poſſis, ſi placeat examinare. <lb></lb> </s> </p> <table> <row> <cell>In Triangulo BDC rectang</cell> <cell>In Triangulo VBR</cell> </row> <row> <cell>BD ped. 900′ —— r l</cell> <cell>7,04575,74906</cell> <cell>VB + BR ped. 13351 —— r l</cell> <cell>5,87448,62041</cell> </row> <row> <cell>DC ped.130.00″. — l.</cell> <cell>4.11394,33523</cell> <cell>VB - BR ped. 1270<gap></gap> —— l</cell> <cell>4,1038;,79160</cell> </row> <row> <cell>CBD gr.88.33. m</cell> <cell>1,15970,08429</cell> <cell>Semiſumma ang.</cell> <cell>gr.44.42′.53″,-m</cell> <cell>9,99567.51920</cell> </row> <row> <cell></cell> <cell></cell> <cell>differentia</cell> <cell>gr.43.17, 9 m</cell> <cell>9,97399,93121</cell> </row> </table> <pb n="54" xlink:href="017/01/070.jpg"></pb> <p type="main"> <s id="s.000449">Quod ſi paries exteriùs inclinatus etiam ſolitarius conſiſtere <lb></lb>poſſet, modò ea eſſet partium connexio, ut unum quid ſoli<lb></lb>dum conflarent, quia directionis linea intra baſim ſuſtentan<lb></lb>tem cadit, & planum per extremam baſis lineam, & terræ cen<lb></lb>trum tranſiens relinquit interiorem parietis partem præponde<lb></lb>rantem exteriori: quis poſſit de turris ruinâ dubitare, ſi eâdem <lb></lb>methodo deprehendat oppoſiti parietis AG centrum gravita<lb></lb>tis eſſe in O, ac proinde comparatis reliquorum duorum pa<lb></lb>rietum centris gravitatum, totius turris centrum gravitatis eſſe <lb></lb>in intimis turris partibus? </s> <s id="s.000450">Quò igitur firmiùs ſibi cohærebunt <lb></lb>partes turris, eò major erit inclinatio, quam obtinere poteſt ci<lb></lb>tra cadendi periculum. </s> <s id="s.000451">Id quod pueris ipſis notiſſimum eſt, <lb></lb>qui turriculas inclinatas architectantur ex buxeis orbiculis, <lb></lb>quibus in alveolo ludunt. </s> </p> <p type="main"> <s id="s.000452">Et ut res iſta planiſſimè oſtendatur, <lb></lb><figure id="id.017.01.070.1.jpg" xlink:href="017/01/070/1.jpg"></figure><lb></lb>ſit ſupra planum inclinatum AB, pa<lb></lb>rallelepipedum ligneum ID ita, ut <lb></lb>recta CE ad horizontem perpendicu<lb></lb>laris tranſeat per centrum gravitatis: <lb></lb>conſtat ex dictis cap. 8. futurum eſſe, <lb></lb>ut grave ID repat, non autem rote<lb></lb>tur, quia pars CED non præponderat parti CEI, ſiqui<lb></lb>dem poſſit deſcendere per planum inclinatum; quod ſi à lap<lb></lb>ſu impediatur, ſubſiſtet. </s> <s id="s.000453">Jam verò intellige per C planum <lb></lb>FH horizontale, & adnecti priſma trigonum CIK pa<lb></lb>rallelepipedo ID; utique pars CEK præponderat parti <lb></lb>CED, multóque minùs dubitandum erit de ſolidi KD rui<lb></lb>nâ verſus H. </s> <s id="s.000454">Quid autem aliud eſt ſolidum KD, quam tur<lb></lb>ris inclinata? </s> </p> <p type="main"> <s id="s.000455">Scripſeram hæc jam tum ab anno labentis ſæculi quinquage<lb></lb>ſimo ſexto; cum animum ſubiit ſuſpicari, an ſuperiùs allatæ ex <lb></lb>Maſino turris Bononienſis menſuræ omninò veritati reſponde<lb></lb>rent. </s> <s id="s.000456">Quare litteris ad P. Franciſcum Mariam Grimaldum da<lb></lb>tis rogavi, ut pro eâ, quam ad res omnes conferre ſolebat, di<lb></lb>ligentiâ, accuratè menſuras illas inquireret: hæc igitur ex ejus <lb></lb>reſponſione habui, quibus ſuperiùs dicta corrigenda ſunt; quæ <lb></lb>tamen expungere nolui, ut ſi lubeat, vulgarem opinionem ſe<lb></lb>qui valeas. </s> </p> <pb pagenum="55" xlink:href="017/01/071.jpg"></pb> <p type="main"> <s id="s.000457">Extimus turris ambitus tam in imâ, quam in ſupremâ parte <lb></lb>æqualis eſt, adeò ut oppoſitæ facies parallelæ excurrant: ſin<lb></lb>gulorum autem laterum ad baſim latitudo eſt ped. Bonon. 17. <lb></lb>unc. </s> <s id="s.000458">8. murorum craſſities in imo æqualis eſt; eo tantum diſ<lb></lb>crimine, quod murus, qua parte oſtium patet, craſſus eſt ped.5. <lb></lb>unc.11. qui verò Septentrionem ſpectat, propiùs accedit ad pe<lb></lb>des 6. Porrò in ſummâ turri murorum craſſities pariter æqualis <lb></lb>eſt, & vix deficit à pedibus 5, quantum quidem ex aſpectu à <lb></lb>ſuperiori proximæ turris Aſinellæ podio conjicere potuit ſingu<lb></lb>lorum murorum lateres numerans. </s> <s id="s.000459">Areæ demum vacuæ ad ba<lb></lb>ſim latus unum eſt ped. 6. alterum ped.6. unc.1. </s> </p> <p type="main"> <s id="s.000460">Cum autem pluviæ per hiantem, & patulum turris verticem <lb></lb>deciduæ ſcalas corruperint, nec eò veniri poſſit, ut demiſſo <lb></lb>perpendiculo altitudo turris inveſtigetur, ſubſidium peten<lb></lb>dum fuit ex Trigonometriâ, & ex proximâ turri Aſinellâ, cu<lb></lb>jus menſuræ multiplici obſervatione innotuerant. </s> <s id="s.000461">Sit itaque <lb></lb>turris inclinata DC, ſuperioris autem podij <lb></lb><figure id="id.017.01.071.1.jpg" xlink:href="017/01/071/1.jpg"></figure><lb></lb>Aſinellæ altitudo EB ped.234 1/2, unde obſer<lb></lb>vatus eſt angulus CEB gr. 18. 40′. </s> <s id="s.000462">Item in <lb></lb>eadem turri Aſinellâ patet feneſtra in F, adeò <lb></lb>ut diſtantia EF ſit ped.141: ibi pariter obſer<lb></lb>vatus eſt angulus EFC gr. 51. 51′. </s> <s id="s.000463">Quare in <lb></lb>triangulo CEF, notum eſt latus EF, & duo <lb></lb>anguli adjacentes, ex quibus datis colligi<lb></lb>tur EC diſtantia ped. (117 7/12). Jam verò intelli<lb></lb>gantur ex C cadere duæ perpendiculares, al<lb></lb>tera quidem CH in planum horizontale, alte<lb></lb>ra verò CG in turrim Aſinellam; erit enim al<lb></lb>titudo CH æqualis altitudini GB, nam CG <lb></lb>eſt parallela horizonti, cui turris EB perpen<lb></lb>dicularis inſiſtit. </s> <s id="s.000464">Ut igitur innoteſcat quæſi<lb></lb>ta altitudo, inveniatur in triangulo rectangu<lb></lb>lo CGE, ex datis latere CE ped. (117 7/12) & <lb></lb>angulo obſervato CEG, gr.18.40′, latus EG <lb></lb>ped. (111 5/12). Jam verò ſi EG ped.(111 5/12) dematur <lb></lb>ex EB ped. 234 1/2, remanet altitudo GB, hoc eſt CH, <lb></lb>ped. (123 1/12). </s> </p> <pb pagenum="56" xlink:href="017/01/072.jpg"></pb> <p type="main"> <s id="s.000465">Demum ad inveſtigandam turris inclinationem, applicito <lb></lb>ad punctum I perpendiculo obſervatus eſt angulus DIL <lb></lb>gr. 3. 10′.: cùm autem IL parallela ſit perpendiculari CH, erit <lb></lb>pariter angulus DCH gr.3.10′. </s> <s id="s.000466">Igitur in triangulo DCH <lb></lb>rectangulo ad H notum eſt latus CH ped.(123 1/12), & angulus <lb></lb>DCH gr.3.10′, ergo & innoteſcit latus DH ped.6. (10/12), quæ eſt <lb></lb>menſura inclinationis quæſitæ. </s> </p> <p type="main"> <s id="s.000467">Ex his accuratioribus menſuris indagemus, ſi placet, in <lb></lb>orientali pariete inclinato centrum gravitatis, & lineam di<lb></lb>rectionis methodo eâdem, qua ſuperiùs uſi ſumus; eademque <lb></lb>figura ſectionis verticalis reſumatur. </s> <s id="s.000468">Eſt igitur EB ped. 6. ac <lb></lb>propterea RB ped. 300″; & quia HC eſt ped. 5, VC eſt <lb></lb>ped.2. 50″. </s> <s id="s.000469">BD autem eſt ped. 6. unc.10, hoc eſt ped.(6 10/12). </s> </p> <figure id="id.017.01.072.1.jpg" xlink:href="017/01/072/1.jpg"></figure> <p type="main"> <s id="s.000470">In Triangulo BDC rectangulo datis BD <lb></lb>ped. 6. (10/12), & altitudine perpendiculari CD <lb></lb>ped. (123 1/12), additis laterum quadratis fit qua<lb></lb>dratum hypothenuſæ BC, quæ eſt ped.123.27″. </s> <lb></lb> <s id="s.000471">Fiat igitur ut CB ped. 123. 27″, ad BD <lb></lb>ped. 6. 83″. </s> <s id="s.000472">ita Radius ad ſinum anguli BCD <lb></lb>gr. 3. 10′ 34″. </s> <s id="s.000473">Quare angulus reliquus CBD <lb></lb>gr. 86. 49′. </s> <s id="s.000474">26″, cui æqualis eſt alternus VCB <lb></lb>inter parallelas VC, RD; angulus autem, <lb></lb>qui eſt deinceps, CBR gr. 93. 10′. </s> <s id="s.000475">34′. </s> <s id="s.000476">In <lb></lb>triangulo VCB datis lateribus VC ped.2-50″, <lb></lb>CB ped. 123. 27″, & angulo verticali VCB <lb></lb>gr. 86. 49′. </s> <s id="s.000477">26″, reperitur CVB gr. 92. 0′. </s> <s id="s.000478">36″, <lb></lb>& VBC. gr. 1. 9′, 58″. </s> <s id="s.000479">Ex his verò invenitur <lb></lb>VB ped. 122. 76″. </s> </p> <p type="main"> <s id="s.000480">Jam verò in Triangulo VBR, notus eſt <lb></lb>angulus RBV æqualis alterno CVB gr.92. <lb></lb>0′. </s> <s id="s.000481">36′. </s> <s id="s.000482">& nota ſunt latera RB ped. 300″, & <lb></lb>VB ped. 122. 76″. </s> <s id="s.000483">Quare invenitur angulus <lb></lb>VRB gr. 86. 35′ 43″. BVR gr. 1. 23′. </s> <s id="s.000484">41″, & baſis VR <lb></lb>ped. 123. 17″. </s> </p> <p type="main"> <s id="s.000485">Tum fiat ut 17 ad 16; hoc eſt duplum majoris EB cum mi<lb></lb>nore HC, ad duplum minoris HC cum majore EB, ita VS <lb></lb>ad SR, & erit SR ped.59.72″. </s> <s id="s.000486">Ductâ igitur ex S centro gra-<pb pagenum="57" xlink:href="017/01/073.jpg"></pb>vitatis perpendiculari lineâ directionis SX, ex datis latere SR <lb></lb>ped. 59. 72″, & angulo VRX gr. 86, 35′, 43″, innoteſcit RX <lb></lb>ped. 3. 54″. </s> <s id="s.000487">Quare RX major eſt quàm RB: & ſi paries ille <lb></lb>ſolitarius eſſet, non utique conſiſteret; ſed quoniam reliqui <lb></lb>tres parietes adjecti ſunt, conſtat ita totius molis centrum gra<lb></lb>vitatis eſſe in intima turris parte, ut linea directionis cadat in<lb></lb>trà turris baſim ſuſtentantem. </s> </p> <p type="main"> <s id="s.000488">Ex his diſcuties timorem eorum, qui ſoliciti ſunt de obeliſ<lb></lb>corum conſiſtentiâ, ex inclinatione aliquâ verticis ruinam <lb></lb>proximam præſagientes: cum enim in hujuſmodi molibus cen<lb></lb>trum gravitatis vicinius ſit baſi quàm vertici, ſi centrum incli<lb></lb>netur in alterutram partem ſpatio tantùm digitali, vertex in<lb></lb>ſignem acquiret inclinationem, conſiſtet tamen, quandiu linea <lb></lb>directionis tranſibit per baſim ſuſtentationis. </s> <s id="s.000489">Inclinatio enim <lb></lb>non eſt ſpatium illud, quod inter baſim, & perpendiculum à <lb></lb>turris, vel obeliſci vertice demiſſum intercipitur (quamvis hoc <lb></lb>vocabulo hactenus abuti placuerit, ne à vulgo diſcreparem) <lb></lb>ſed eſt angulus, quem turris facit cum plano; & manente ea<lb></lb>dem inclinatione, intervallum illud mutari poteſt pro majore, <lb></lb>aut minore turris longitudine. </s> <s id="s.000490">Quare quò longior eſt moles in<lb></lb>clinata, cæteris paribus, minùs eſt timendum, quia minor eſt <lb></lb>declinatio à perpendiculari: ſi enim KE ſit pedum 100, KC <lb></lb>verò ped.1. angulus KEC æqualis declinationi à perpendiculo <lb></lb>eſt gr. 0. 34. 22″. at ſi KE ſit ped. 50, & KC iterum ped. 1. <lb></lb>angulus KEC eſt grad. 11. 32′. </s> <s id="s.000491">13″. </s> </p> <p type="main"> <s id="s.000492">Hîc autem quaſi præteriens ſatisfaciam quærenti, cur lon<lb></lb>giores haſtas faciliùs, quàm breviores virgas digiti extremitate <lb></lb>ſuſtineamus, quin cadant. </s> <s id="s.000493">Quia nimirum minimus angulus <lb></lb>declinationis à perpendiculo ſtatim ſe prodit haſtæ vertice ad <lb></lb>partem unam ſecedente, cui ſtatim occurrimus haſtæ calcem <lb></lb>manu transferentes, ac ſub vertice collocantes: verùm quia fa<lb></lb>cilior haſtæ conſiſtentia innoteſcit etiam, quando à ſuppoſitâ <lb></lb>manu calx ejus non movetur (nam ſi militarem ſariſſam terræ <lb></lb>perpendiculariter inſiſtentem conſtitueris, potes te ſemel in gy<lb></lb>rum contorquere, & illam quaſi perpendicularem recipere, id <lb></lb>quod in breviore haſtâ non obtinebis) alia eſt ratio petenda <lb></lb>primùm ex dictis, quia ſcilicet longior haſta, cæteris paribus, <lb></lb>minùs declinat à perpendiculo, ideóque difficiliùs deſcendit; <pb pagenum="58" xlink:href="017/01/074.jpg"></pb>deinde quemadmodum longiorem haſtam ſi in aquá agitaveris <lb></lb>majorem percipies reſiſtentiam, quàm ſi breviorem virgam in<lb></lb>citares; ita aërem variis ſemper motibus turbatum plus etiam <lb></lb>impedire deſcenſum longioris haſtæ cenſendum eſt, præſertim <lb></lb>ſi in ſuperiore parte aër versùs unam, in inferiore autem versùs <lb></lb>aliam partem moveatur: id quod in breviore virgâ non accidit, <lb></lb>quam modicus aër contingit, nec poteſt aut adeò reſiſtere di<lb></lb>viſioni, aut adeò diverſis motibus cieri. </s> <s id="s.000494">Hinc aſta longior <lb></lb>tardiùs deſcenſum molitur, & faciliùs ſuſtinetur, quia major <lb></lb>aëris dividendi quantitas, ac motus varius, magis reſiſtit, & <lb></lb>datâ æqualitate motûs minùs declinat à perpendiculo. <lb></lb></s> </p> <p type="head"> <s id="s.000495"><emph type="center"></emph>CAPUT X.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000496"><emph type="center"></emph><emph type="italics"></emph>An plurium ſtructurarum capax ſit mons, quàm <lb></lb>ſubjecta planities.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000497">POteſt mons cum ſubjectâ planitie, cui inſiſtit, dupliciter <lb></lb>comparari; primùm conferendo ſolam planitiem in ver<lb></lb>tice montis exiſtentem cum parte ſubjecti plani ſibi reſ<lb></lb>pondente; deinde clivum montis comparando cum plano <lb></lb>horizontali. </s> <s id="s.000498">Et ſanè ſi planities in ſummo montis jugo con<lb></lb>ſideretur, certum eſt illam eſſe plurium ſtructurarum ca<lb></lb>pacem, quàm ſubjectum planum in ſuperficie globi ter<lb></lb>reſtris: Quemadmodum enim ſuperficies ſphæræ majoris <lb></lb>plura capit ædificia, quàm minor, ita etiam ſphærarum <lb></lb>inæqualium partes ſimiles inæqualis ſunt capacitatis: Conſtat <lb></lb>autem planitiem in ſummo monte pertinere ad ſphæram <lb></lb>majorem, quàm pertineat ſimilis planities illi ſubjecta; ac <lb></lb>proinde & amplior eſt, & magis capax. </s> <s id="s.000499">Harum verò pla<lb></lb>nitierum differentia ea erit, quæ eſt quadratorum diſtan<lb></lb>tiarum à centro terræ: quòd ſi quadratorum hujuſmodi <lb></lb>differentia exigua ſit & contemnenda, eo quod ad illam <lb></lb>quadratum ſemidiametri terræ habeat nimis magnam ratio<lb></lb>nem; planitierum pariter differentia fugiet omnem ſenſum. <pb pagenum="59" xlink:href="017/01/075.jpg"></pb>Sit terræ ſemidiameter CS, altitudo au<lb></lb><figure id="id.017.01.075.1.jpg" xlink:href="017/01/075/1.jpg"></figure><lb></lb>tem montis SR, in cujus vertice ſit pla<lb></lb>nities RH, cui ſimilis eſt in ſuperficie <lb></lb>globi terreni planities SO illi parallela: <lb></lb>hæ autem planities ſimiles habent, per <lb></lb>20. lib. 6. duplicatam Rationem laterum <lb></lb>RI, SL, hoc eſt, per 4. lib. 6. duplica<lb></lb>tam Rationis, quam habet CR ad CS. </s> <lb></lb> <s id="s.000500">Eſt igitur ut quadratum diſtantiæ CR. <lb></lb>ad quadratum diſtantiæ CS, ita plani<lb></lb>ties RH ad planitiem SO. </s> <s id="s.000501">Plura itaque <lb></lb>ædificia perpendiculariter inſiſtentia <lb></lb>poſſunt in planitie RH majori excitari <lb></lb>in montis vertice, quàm in ſubjectâ <lb></lb>plani tie. </s> </p> <p type="main"> <s id="s.000502">At ſi montis clivus RMOL comparetur cum ſubjectâ pla<lb></lb>nitie SO, certum eſt illum eſſe majorem, ſicuti latus RL op<lb></lb>poſitum angulo RSL, qui non eſt minor recto, majus eſt la<lb></lb>tere SL in triangulo RSL, & RM ad SF eſt ut RC ad SC: <lb></lb>ſuperficies igitur LM comprehenſa ſub majoribus lateribus, <lb></lb>& angulis non minoribus, quàm ſuperficies SO, major erit, <lb></lb>ſi illa per ſe conſideretur. </s> <s id="s.000503">Non tamen continuò major dicenda <lb></lb>eſt capacitas, quæ plura aut ampliora recipiat ædificia; niſi <lb></lb>mons ad ingentem altitudinem aſcendat; tunc enim perpendi<lb></lb>cula non ſunt inter ſe parallela, propter inſignem eorum <lb></lb>diſtantiam. </s> <s id="s.000504">Nam ſi ſuper clivo AB <lb></lb>ſit ſtructura AL, cujus parietes per<lb></lb><figure id="id.017.01.075.2.jpg" xlink:href="017/01/075/2.jpg"></figure><lb></lb>pendiculares, ſint etiam paralleli <lb></lb>LB, DA, illi non magis inter ſe <lb></lb>diſtant, quàm ſi ſuper plano hori<lb></lb>zontali NB fuiſſent excitati: quic<lb></lb>quid ſit, quod, ſicut linea AB ma<lb></lb>jor eſt quàm NB, ita planum incli<lb></lb>natum majus ſit plano horizontali. </s> <lb></lb> <s id="s.000505">Non igitur plures aut ampliores ſtructuras recipit clivus collis, <lb></lb>quàm ſubjectum planum horizontale. </s> <s id="s.000506">Quod verò de ſtructuris <lb></lb>dicitur, de cæteris quoque intelligendum eſt, quæ perpendi<lb></lb>cularia inſiſtunt, & ſpatium implent; at ſi ita ſe habeant, ut <pb pagenum="60" xlink:href="017/01/076.jpg"></pb>perpendicularia non inſiſtant, certum eſt plures aut longiores <lb></lb>homines jacere poſſe in clivo AB, quos non capit planum NB: <lb></lb>vel ſi in clivo ſe minùs invicem impediant, tunc plura hujuſ<lb></lb>modi corpora in colle eſſe poſſunt quàm in planitie: ſi enim ra<lb></lb>mi arboris inferioris reſpondeant trunco ſuperioris, certum eſt <lb></lb>quod multò viciniores eſſe poſſunt arbores, quàm in planitie, <lb></lb>ubi rami ſe viciſſim impedientes majorem poſtulant truncorum <lb></lb>diſtantiam; ac proinde etiam multo plures arbores intra eaſ<lb></lb>dem parallelas erunt. </s> <s id="s.000507">Sic plures homines eſſe poſſunt in gradi<lb></lb>bus amphitheatri, quàm in ſubjecto plano, quia graciliores, <lb></lb>partes ſuperiorum reſpondent craſſioribus inferiorum, & ſe <lb></lb>minùs invicem impedientes minus relinquunt ſpatij vacui: <lb></lb>quod ſi non homines, ſed parallelepipeda, ſtatueres in gradi<lb></lb>bus, non plura ſtatui in iis poſſent, quàm in planâ areâ gradi<lb></lb>bus ſubjectâ. </s> </p> <p type="main"> <s id="s.000508">Hæc autem ædificiorum æqualitas in clivo & in plani<lb></lb>tie, locum non habet niſi intra illud ſpatium, quod inter<lb></lb>cipitur à perpendiculis Phyſicè parallelis; ſtatim enim ac à <lb></lb>paralleliſmo recedunt perpendicula, ſi ea fuerit altitudo, ad <lb></lb>quam clivus aſcendens venit, ut planities parallela plano <lb></lb>horizontali in eâ altitudine major ſit, quàm ſimilis plani<lb></lb>ties depreſſior, etiam plura ædificia recipiet clivus, quàm <lb></lb>unica planities horizontalis ſubjecta. </s> <s id="s.000509">Ponamus enim per<lb></lb>pendicula GC, & OC jam non eſſe parallela, eamque eſſe <lb></lb>altitudinem KG, ut planum per G tranſiens horizonti <lb></lb>parallelum majus ſit plano per O intra eadem perpendicu<lb></lb>la intercepto, erit quidem capacitas plani inclinati GOLF <lb></lb>æqualis capacitati ſubjecti plani EKOL: at ulteriùs aſcen<lb></lb>dendo capacitas FGMR non erit æqualis capacitati plani <lb></lb>SK continuati cum priore plano EO, ſederit major, quip<lb></lb>pe quæ æqualis eſt capacitati plani VG; eſt autem pla<lb></lb>num VG ad planum ſimile SK, ut quadratum GC ad <lb></lb>quadratum KC: major igitur eſt totius clivi ML capacitas, <lb></lb>quàm planitiei SO. </s> </p> <p type="main"> <s id="s.000510">Et ut res apertius conſtet, quandoquidem clivi altiſ<lb></lb>ſimorum montium, ſi eandem ſervent inclinationem, non <lb></lb>ſunt ab imo pede ad ſummum jugum æquabili, & conti<lb></lb>nuo ductu extenſi, Sit terræ centrum H, & ſuperficies <pb pagenum="61" xlink:href="017/01/077.jpg"></pb>AD; cujus arcus dividatur in par<lb></lb><figure id="id.017.01.077.1.jpg" xlink:href="017/01/077/1.jpg"></figure><lb></lb>tes AB, BC, CD æquales, ita ut <lb></lb>ſinguli arcus pro rectâ lineâ, & ſu<lb></lb>perficies pro plano horizontali <lb></lb>Phyſicè uſurpari poſſint; & tunc <lb></lb>ſolùm intelligatur mutari horizon, <lb></lb>quando ex A jam venerit in B, <lb></lb>deinde in C &c. </s> <s id="s.000511">Si igitur ſit pla<lb></lb>num inclinatum AE, ubi venerit <lb></lb>in E punctum perpendiculi HB <lb></lb>producti, non poteſt rectâ progre<lb></lb>di, quin mutet inclinationem ſupra horizontem novum, ad <lb></lb>quem venit; quare ut ſervetur ſimilis inclinatio, deflectit in EF, <lb></lb>& eſt angulus HEF æqualis angulo HAE cui demum ubi ve<lb></lb>nerit in F, debet fieri æqualis angulus HEG. </s> <s id="s.000512">Centro autem H, <lb></lb>intervallis HE & HF deſcribantur arcus EI, & FK. </s> <s id="s.000513">Certum <lb></lb>eſt duarum linearum angulum conſtituentium partem aliquam <lb></lb>extremam eſſe, ſecundùm quam lineæ illæ non differunt, ſenſu <lb></lb>judice, à parallelis; at ſi major pars accipiatur, jam perit paral<lb></lb>leliſmus: Sic RA, & EB pro parallelis uſurpari ſi poſſint, non <lb></lb>poterunt ſimiliter pro parallelis accipi RA, & LB: Sic LE, & <lb></lb>FI ſumuntur tanquam parallelæ citrà errorem, at non item LB, <lb></lb>& MC. </s> <s id="s.000514">Quare perpendicula non ſolùm recedunt à paralleliſ<lb></lb>mo ſenſibili, quia majorem angulum in centro H conſtituunt, <lb></lb>ſed etiam quia major eorum pars aſſumitur, in qua jam apparet <lb></lb>convergentia, quæ in parte minore latebat. </s> </p> <p type="main"> <s id="s.000515">Cum itaque ſtructuræ perpendiculares in plano inclinato <lb></lb>occupent ſpatium eodem modo, ac ſi eſſent in plano horizon<lb></lb>tali intra eaſdem parallelas, jam conſtat clivi partem EF com<lb></lb>parandam eſſe cum plano EI, non autem cum plano BC; quia <lb></lb>in E, & I terminatur paralleliſmus linearum LE, FI. </s> <s id="s.000516">Eſt igi<lb></lb>tur capacitas clivi EF æqualis capacitati EI; at capacitas EI <lb></lb>major eſt quàm capacitas BC, ergo capacitas clivi AF major <lb></lb>eſt, quàm capacitas planitiei AC. </s> <s id="s.000517">Eademque eſto de cæteris <lb></lb>ratio. </s> <s id="s.000518">Hinc manifeſtum eſt non omninò in univerſum vera eſſe, <lb></lb>quæ paſſim dicuntur de æquali capacitate collium, & planitiei <lb></lb>ſubjectæ, niſi hæc certis limitibus circumſcribantur; videlicet <lb></lb>ſi ſermo ſit de iis quæ tantùm perpendiculariter inſiſtunt, & <pb pagenum="62" xlink:href="017/01/078.jpg"></pb>intrà illud ſpatium, ac in eá altitudine, ubi perpendiculorum <lb></lb>convergentia adeò exigua eſt, ut evaneſcat. </s> <s id="s.000519">Cæterùm ſatis <lb></lb>mihi videor oſtendiſſe fieri poſſe, ut clivus aliquis plures <lb></lb>ſtructuras recipere poſſit, quàm ſuperficies ſphærica globi illi <lb></lb>reſpondens. </s> <s id="s.000520">Si enim eadem eſt ſemper, ut ſupponitur, plani <lb></lb>inclinatio, etiam latera turrium, vel domorum parietes æquè <lb></lb>invicem remoti intercipient æquales partes plani inclinati: Si <lb></lb>ergo ſtructura intercipiens ſemiſſem plani AE transferatur in <lb></lb>EF, æqualem partem intercipiet; at hæc minor eſt ſemiſſe <lb></lb>ipſius EF, igitur duæ ſtructuræ occupantes totum planum AE, <lb></lb>tranſlatæ in EF æquale ſpatium occupabunt, & relinquent <lb></lb>adhuc partem ſpatij inanem. </s> <s id="s.000521">Eſſe autem EF lineam majorem <lb></lb>linea AE patet; quia triangula AHE, EHF æquiangula <lb></lb>ſunt, & latera habent proportionalia, adeóque ut AH ad HE, <lb></lb>ita AE ad EF; atqui HE excedit lineam HA; igitur & EF <lb></lb>major eſt quàm AE: ergo multo major erit ſuperficies ipſius <lb></lb>EF, quàm ſuperficies ſimilis ipſius AE. </s> <s id="s.000522">In ſpatio igitur, quo <lb></lb>ſuperficies EF excedit ſuperficiem AE, poterit alia præterea <lb></lb>ſtructura excitari. <lb></lb></s> </p> <p type="head"> <s id="s.000523"><emph type="center"></emph>CAPUT XI.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000524"><emph type="center"></emph><emph type="italics"></emph>Quomodo animalium motus ordinentur ex centro <lb></lb>gravitatis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000525">DEi ſapientiam nunquam ſatis admirari poſſumus, quæ in <lb></lb>ordinandis naturæ motibus elucet; animalia enim ſolo <lb></lb>naturæ ductu adeò accuratè ſe ipſa ſiſtunt in lineâ directionis, <lb></lb>ut nemo mathematicus Geometriæ apices perſcrutatus poſſit <lb></lb>tam ſubtiliter deprehendere, ac breviſſimo temporis momento, <lb></lb>centrum gravitatis. </s> <s id="s.000526">Quandoquidem ſive conſiſtentium quie<lb></lb>tem, ſivè gradientium motum, ſivè reclinantium ſe ſe inflexio<lb></lb>nem conſideres, miram naturæ artem intelliges, quâ præcavit, <lb></lb>ne corpus ingenitâ gravitate delatum præceps caderet. </s> <s id="s.000527">Id au<lb></lb>tem aſſecuta eſt motus ita diſponendo, ut linea directionis nun-<pb pagenum="63" xlink:href="017/01/079.jpg"></pb>quam caderet extrà baſim ſuſtentationis, niſi fortè in curſu, in <lb></lb>quo tamen ſatis conſultum eſt animalis incolumitati, dum ab <lb></lb>anteriore pede, ubi terram attigerit, retinetur, ne ulteriùs <lb></lb>deſcendat. </s> </p> <p type="main"> <s id="s.000528">Baſis autem ſuſtentationis non ſunt ſoli pedes, ſed totum <lb></lb>illud ſpatium interceptum à lineis pedum extremitates jun<lb></lb>gentibus; ſic in quadrupedibus linea directionis debet cadere <lb></lb>intrà ſpatium comprehenſum lineis, quæ jungunt extrema <lb></lb>pedum terram contingentium, ut poſſit animal conſiſtere. </s> <lb></lb> <s id="s.000529">Hinc equus in poſteriores pedes ſe erigens flexis poplitibus <lb></lb>reclinat ſe ſe in poſteriora, & tantiſper in eo ſitu conſiſtit, <lb></lb>dum centrum gravitatis imminet ſpatio, quod à pedibus oc<lb></lb>cupatur, & ab illis intercipitur; & ſi extra illud ſpatium ca<lb></lb>dat linea directionis, vel averſus cadit, vel iterum quatuor <lb></lb>pedibus inſiſtit. </s> <s id="s.000530">Ubi tamen obſervandum eſt ex equo & equi<lb></lb>te fieri unam molem compoſitam unum habentem commune <lb></lb>centrum gravitatis: unde fit equum magis defatigari, ſi eques <lb></lb>non rectus inſideat; ſed inclinatus in alterutram partem, cen<lb></lb>tro enim gravitatis tranſlato motûs facilitas mutatur; & equite <lb></lb>in anteriora inclinato ac premente caput equi in poſteriores <lb></lb>pedes erecti, centrum gravitatis in anteriora transfertur, & <lb></lb>occurritur periculo, ne equus averſus cadat. </s> </p> <p type="main"> <s id="s.000531">Porrò dum ſpatium à pedibus occupatum voco baſim ſuſten<lb></lb>tationis, non ſemper ſatis eſt lineam directionis cadere non <lb></lb>extrà pedes; quia ſi pedes ipſi ſolùm ex parte tangant ſub<lb></lb>jectum corpus, ut contingit in funambulis, debet linea di<lb></lb>rectionis cadere in funem, cui inſiſtunt pedes, & ſi extra il<lb></lb>lum cadat, certa eſt ruina, quia latitudo pedum non juvat. </s> <lb></lb> <s id="s.000532">Cum autem difficillimum ſit diutiùs conſiſtere ita, ut centrum <lb></lb>gravitatis ſemper immineat funi, ideò funambuli, vel haſtam <lb></lb>plumbeis laminis gravem in extremitatibus manu tenent, vel <lb></lb>brachiis expanſis ſe librant, ut haſtam vel brachia extenden<lb></lb>tes in partem oppoſitam ei, in quam gravitas inclinat, cen<lb></lb>trum gravitatis conſtituatur in puncto, quod immineat funi <lb></lb>ſui tentanti. </s> <s id="s.000533">Hinc oritur difficultas conſiſtendi, quam expe<lb></lb>riuntur grallatores; cum enim grallæ exiguâ ſui parte tangant <lb></lb>terram, eſt quaſi linea, in qua fit ſuſtentatio, extra quam fa<lb></lb>cilè cadit linea directionis: ideò tertium geſtant baculum, cui <pb pagenum="64" xlink:href="017/01/080.jpg"></pb>innitantur, quoties quieſcere voluerint, lineâ directionis ca<lb></lb>dente intrà ſpatium triangulare comprehenſum à grallis, & <lb></lb>baculo. </s> </p> <p type="main"> <s id="s.000534">Hîc autem maximè ſe prodit naturæ providentia in tam va<lb></lb>riâ pedum conformatione, ut ad ſuſtentandum idonei eſſent: <lb></lb>quadrupedibus ſiquidem non adeò amplos pedes tribuit, quia <lb></lb>ex eorum inter ſe diſtantiâ plurimum ſpatium intercipitur, cui <lb></lb>immineat centrum gravitatis: bipedibus verò latiores tribuit <lb></lb>pedes, quâ parte timeri potuit caſus: ſic quia ex duorum cru<lb></lb>rum modicâ divaricatione non facilè periculum erat cadendi <lb></lb>in alterutrum latus, ideò humanis pedibus minorem dedit la<lb></lb>titudinem, quàm longitudinem; hanc verò non in æquas <lb></lb>diſtribuit partes, ſed minimam calci (præterquam in Scauris, <lb></lb>quos pravis fultos male talis appellat Horatius, talis ſcilicet <lb></lb>extantioribus) maximam anteriori parti conceſſit, ne impetu <lb></lb>per motum concepto tranſlatum centrum gravitatis in anterio<lb></lb>ra tranſiliret baſim ſuſtentationis. </s> <s id="s.000535">Aliquam tamen mediocrem <lb></lb>latitudinem pedibus conceſſit, ut poſſet homo, ſi res ferret, uni <lb></lb>tantùm pedi inſiſtere, & eſſet aliqua ſpatij amplitudo, intrà <lb></lb>quam quodlibet punctum opportunum eſſet conſiſtentiæ cen<lb></lb>tri gravitatis. </s> <s id="s.000536">Sic aves illæ, quæ uni pedi inſiſtunt, cujuſmodi <lb></lb>ſunt grues, & ciconiæ, digitos habens longiores, quos valdè <lb></lb>explicant quaſi in gyrum, ut amplior ſit baſis ſuſtentationis; in<lb></lb>trà quam ut cadat linea directionis, altero pede elevato inclina<lb></lb>tur corpus in oppoſitam partem, ut centrum gravitatis immineat <lb></lb>pedi ſuſtentanti. </s> <s id="s.000537">Eandem ob cauſam anſeres, & anates, quæ <lb></lb>multâ carne abundant, & amplo ſunt pectore, alternâ qua<lb></lb>dam in dextrum, & ſiniſtrum latus inclinatione gradiuntur, <lb></lb>ideóque ampliores habent palmas, ut citrà cadendi periculum <lb></lb>centrum gravitatis faciliùs vel immineat pedi ſuſtentanti, vel <lb></lb>minimùm ab eo declinet, ne majore, quàm par ſit, impetu <lb></lb>deſcendens corpus & anteriori pedi incumbens, tibiæ muſcu<lb></lb>los, & tendines lædat. </s> <s id="s.000538">Aves verò, quæ ſubtilioribus ramuſcu<lb></lb>lis inſident non palmipedes ſunt, ſed digitatæ (palmæ enim <lb></lb>avibus amphibiis ad natandum potiſſimum datæ videntur) ut <lb></lb>ramis tenaciùs inhæreant; quæ præterquàm quod exiguæ ſunt <lb></lb>gravitatis, facilè ſe ſiſtunt in lineâ directionis, quæ cadat in <lb></lb>ramuſculum, cui inſiſtunt, majore, vel minore angulo, quem <pb pagenum="65" xlink:href="017/01/081.jpg"></pb>faciunt tibiæ cum coxâ; ideò ubi ramum arripuerint, ſubſul<lb></lb>tantes ſe librant, ramumque arctè apprehentes prohibent, ne <lb></lb>repentino caſu circumagantur à centro gravitatis nondum im<lb></lb>minente baſi ſuſtentationis. </s> </p> <p type="main"> <s id="s.000539">Verùm quoniam ad aves delapſus ſum, prætereundus non <lb></lb>eſt uſus centri gravitatis involatu; quia enim avis dum alis <lb></lb>aërem verberans in volatu ſe librat atque ſuſpendit, ita alas <lb></lb>debet extendere, ut centrum gravitatis exiſtat intra illud <lb></lb>alarum ſpatium, in quo exercetur ſuſtentatio; ideò ſi vo<lb></lb>luerit ad ſuperiora volatum dirigere, alas in anteriora ver<lb></lb>ſus caput extendit, ut centro gravitatis in poſterioribus re<lb></lb>licto, ac deorſum præponderante, caput ſurſum dirigatur: <lb></lb>contra verò, ut motum deorſum dirigat, alas retrahit, ut <lb></lb>caput præponderet, ac deorſum feratur. </s> <s id="s.000540">Hinc ſatis patet, <lb></lb>cur ubi Pavo caudæ pompam explicuerit, erecto pectore & <lb></lb>capite inſiſtat pedibus, quibus immineat centrum gravita<lb></lb>tis: at ſi caput ad anteriora inclinare voluerit, & pectus <lb></lb>inflectere, cogitur explicatam caudam demittere, ut ſyrma<lb></lb>te illo æquilibrium ſtatuat corpori, ne proruat, ut verè pro<lb></lb>cumberet, ſi pectore inclinato expanſa cauda retineretur in <lb></lb>poſitione eâdem. </s> </p> <p type="main"> <s id="s.000541">Infinitum eſſet ſingulos animalium motus perſequi, in qui<lb></lb>bus centri gravitatis ratio habetur; ſatis fuerit obſervaſſe nos <lb></lb>ex declivi loco deſcendentes non inſiſtere plantis pedum ad <lb></lb>angulos rectos; ſed paululum in poſteriora inclinari; contra <lb></lb>verò aſcendentes jugum acclive curvari in anteriora; ut nimi<lb></lb>rum linea directionis cadat intrà ſpatium, cui pedes inſiſtunt; <lb></lb>extra quod illa ſi caderet, nec alteri fulcro inniteremur, quod <lb></lb>unà cum pedibus includeret baſim ſuſtentationis, neceſſariò <lb></lb>nobis cadendum eſſet. </s> <s id="s.000542">Quòd ſi quis onus habens dorſo impo<lb></lb>ſitum in montosâ regione iter habeat, multò magis curvari de<lb></lb>bet, cum aſcendit, ut pedibus immineat centrum gravitatis <lb></lb>compoſitæ ex corpore, & ex onere: quare ſapientiſſimè ruſtici <lb></lb>aliqui in Alpibus, quæ Germaniam ab Italiá diſterminant, ar<lb></lb>culam ex levibus aſſerculis, & virgulis compactam habent, cui <lb></lb>onera immittunt, baſis autem arculæ, quæ geſtantis corpori <lb></lb>adhæret, imitatur Reſc Hebraicum, ita ut pars quidem dor<lb></lb>ſo, pars autem capiti incumbat: unde fit, ut centrum gravita-<pb pagenum="66" xlink:href="017/01/082.jpg"></pb>tis compoſitæ minùs recedat à medio humani corporis, adeó<lb></lb>que faciliùs etiam motus perficiatur, quin opus ſit tantâ corpo<lb></lb>ris inflexione. </s> <s id="s.000543">Simile quid experimur, ſi quis à ſede ſurgat; <lb></lb>caput enim cum thorace in anteriora reclinat; pedes verò in <lb></lb>poſteriora versùs ſedem retrahit, ut nimirum pedes ſupponan<lb></lb>tur centro gravitatis, quod primùm imminet parti digitis proxi<lb></lb>mæ, deinde corpore erecto linea directionis versùs talos rece<lb></lb>dit. </s> <s id="s.000544">Hinc etiam patet cur homo ſupinus jacens ſurgere non <lb></lb>poſſit, niſi retractis ſub ſe pedibus, & thorace in anteriora pro<lb></lb>pulſo per impetum ſibi impreſſum. </s> <s id="s.000545">Vidi tamen non ſemel ho<lb></lb>minem, qui cum ſupinus jaceret, non retractis ſub ſe pedibus <lb></lb>ſurgebat planè rectus ſicut ſtipes; ad caput autem appone<lb></lb>bat, vel globum tormentarium majorem, vel ſaxum non <lb></lb>modicæ gravitatis; quod manu utrâque apprehenſum attol<lb></lb>lebat, & velociter in anteriora movebat, ſibique impetum <lb></lb>imprimebat: impetus enim impreſſus promovens ad ante<lb></lb>riora ſaxum, & corpus ipſum vincebat gravitatem corpo<lb></lb>ris cæteroqui caſuri; ex brachiis autem extenſis ſaxum à <lb></lb>corpore remotum tenentibus oriebatur, ut centrum gravi<lb></lb>tatis molis compoſitæ longè citiùs immineret pedibus, à <lb></lb>quibus ſuſtentabatur, etiam antequam planta terram at<lb></lb>tingeret, ſed cum adhuc ſoli calci inniteretur. </s> <s id="s.000546">Quantum <lb></lb>verò impetus valeat ad vincendam oppoſitam gravitatem <lb></lb>corporis, patet in ceſpitantibus, qui naturæ ductu illico bra<lb></lb>chia extendunt, & in contrariam partem projiciunt, ut ſci<lb></lb>licet impetus in oppoſitam partem exæquet exceſſum gravita<lb></lb>tis, quæ ad eam partem reperitur, in quam ex ceſpitatione <lb></lb>facta eſt inclinatio. </s> </p> <p type="main"> <s id="s.000547">Ex his quid in ſingulis motibus dicendum ſit, intelli<lb></lb>ges; neque enim otium eſt ire per ſingula. </s> <s id="s.000548">Caput hoc <lb></lb>claudo explicatione quæſtionis, qua quæritur, quantò ma<lb></lb>jus ſpatium percurrat caput quàm pedes; certum ſiquidem <lb></lb>eſt hominem in lineâ directionis imminere ſemper terræ <lb></lb>centro; ac proinde ſi pedes ex B venerunt in C, caput ex <lb></lb>F in E tranſlatum eſt per arcum FE majorem arcu BC. </s> <lb></lb> <s id="s.000549">Cum enim uterque arcus BC, FE ſubtendatur eidem an<lb></lb>gulo ad centrum, ſunt ſimiles, & ut arcus BC ad totam <lb></lb>ſuam peripheriam, ita arcus FE ad ſuam peripheriam; ſunt <pb pagenum="67" xlink:href="017/01/083.jpg"></pb>autem peripheriæ inter ſe ut ſemi<lb></lb><figure id="id.017.01.083.1.jpg" xlink:href="017/01/083/1.jpg"></figure><lb></lb>diametri, igitur BC ad FE, ut TB, <lb></lb>ad TF; atqui TF major eſt quàm <lb></lb>TB, igitur & FE arcus major arcu <lb></lb>BC: abſcindatur FI, quæ ex hypo<lb></lb>theſi intelligatur æqualis ipſi BC; <lb></lb>eſt igitur ut TB ad TF, ita FI ad <lb></lb>FE, & dividendo ut TB ad BF <lb></lb>ita FI, hoc eſt BC, ad IE. </s> <s id="s.000550">Fiat ita<lb></lb>que ut TB ſemidiameter terræ mil<lb></lb>liar. </s> <s id="s.000551">Rom. ant.4128.paſſ.635. ad BF <lb></lb>altitudinem hominis ex. </s> <s id="s.000552">gr. ped. Rom. ant. </s> <s id="s.000553">6. ita BC iter pe<lb></lb>dum mill. 500, ad IE exceſſum itineris capitis qui eſt (726632/1000000) <lb></lb>unius pedis. </s> <s id="s.000554">Quòd ſi fiat ut terræ ſemidiameter ad hominis al<lb></lb>titudinem, ita circulus terræ maximus mill. 25941 ad exceſ<lb></lb>ſum itineris capitis ſupra iter pedum terræ ambitum percurren<lb></lb>tium, proveniet exceſſus ped. 37. unc.8. hoc eſt paſſ.7. & pau<lb></lb>lò ampliùs: Quare vides in ſingulis milliariis motum capitis non <lb></lb>habere exceſſum niſi partium (17429/1000000) unciæ pedis Romani anti<lb></lb>qui; quæ differentia ſenſum omnem fugit. </s> </p> <p type="main"> <s id="s.000555">Liceat hic ex morâ, quam in hoc Tractatu perficiendo duxi, <lb></lb>id utilitatis capere, quod poſſim pro me ipſe brevi Apologiâ <lb></lb>reſpondere, ne videar in Ageometriam lapſus, cui nulla niſi ex <lb></lb>oſcitantiâ ſuppeteret excuſatio (nam & quandoque bonus dor<lb></lb>mitat Homerus) & quidem tunc, cùm Mathematicas diſcipli<lb></lb>nas in Collegio Romano publicè proſitentem maximè ocula<lb></lb>tum fuiſſe oportuerat. </s> <s id="s.000556">Incidi in Magiam Naturalem P. Gaſparis <lb></lb>Schotti part.3.lib.1. pag. </s> <s id="s.000557">71, ubi mihi tribuit ſententiam maxi<lb></lb>mè abſurdam, quaſi in mechanicâ meâ manuſcriptâ (quam <lb></lb>ſcilicet anno 1653. Romæ auditoribus meis tradidi) docuerim <lb></lb>exceſſum motûs capitis ſupra motum pedum <emph type="italics"></emph>eſſe valde modi<lb></lb>cum, nimirum ſolum pedum ſex cum dimidio, adeò ut in milliaribus<emph.end type="italics"></emph.end><lb></lb>500 <emph type="italics"></emph>tantum reperiatur exceſſus<emph.end type="italics"></emph.end> (15/17) <emph type="italics"></emph>unius pedis, poſitá hominis altitu<lb></lb>dine pedum ſex, & terræ ambitu milliariorum<emph.end type="italics"></emph.end> 21600. Hæſi pri<lb></lb>mùm attonitus, meamque oſcitantiam admiratus illicò anti<lb></lb>quàs illas meas ſchedulas perſcrutari cœpi; & nihil minus in<lb></lb>veniens errorem Typographo, qui pro paſſibus pedes ſuppo-<pb pagenum="68" xlink:href="017/01/084.jpg"></pb>ſuerit, tribuendum cenſuiſſem, niſi Author ipſe modicum il<lb></lb>lum exceſſum pedum ſex cum dimidio redargueret. </s> <s id="s.000558">Quare <lb></lb>contingere facile potuit, ut ille, qui tunc Romæ degebat, ex <lb></lb>aliquo manuſcripto codice meam ſententiam reſcribens, ubi <lb></lb>menſuram hanc pedibus definiebam, brevitatis ergo ad paſ<lb></lb>ſus revocaverit, quam litera P notatam demùm pro pedibus ſit <lb></lb>interpretatus. </s> <s id="s.000559">Cæterùm prudens, & attentus lector me facilli<lb></lb>mè ab hoc errore vindicabit, ſi terræ ambitum mill. 21600. di<lb></lb>vidat per mill.500; & quotientem 43 multiplicet per (15/17) unius <lb></lb>pedis; deprehendet enim totum exceſſum pedum ferè 38, qui <lb></lb>excedunt paſſus ſeptem cum dimidio. </s> <s id="s.000560">Quod ſi ex diametro pe<lb></lb>dum 34400000, & ex diametro pedum 34400012, quas ibi Au<lb></lb>thor ponit congruentes peripheriæ juxta Rationem 7 ad 22 con<lb></lb>ſiderentur, erit differentia circulorum pedum 38 eadem plane <lb></lb>cum noſtrâ; ſed longiſſimè minor eâ, quam ille ibi ſtatuit. </s> </p> <p type="main"> <s id="s.000561">Cæterùm quantus ſit peripheriæ majoris exceſſus ſupra mi<lb></lb>norem, habebitur facillimè, ſi majoris Radij TF, exceſſum <lb></lb>BF, ſtatuas tanquam circuli Radium; hujus namque circuli <lb></lb>peripheria eſt æqualis exceſſui illi. </s> <s id="s.000562">Quia enim ut minor Ra<lb></lb>dius TB ad majorem Radium TF, ita minor peripheria ad <lb></lb>majorem peripheriam, etiam convertendo & dividendo, ut <lb></lb>TB ad BF, ita minor peripheria ad exceſſum peripheriæ ma<lb></lb>joris, & viciſſim permutando ut Radius TB minor ad ſuam <lb></lb>minorem peripheriam, ita BF exceſſus Radij majoris ad exceſ<lb></lb>ſum majoris peripheriæ. </s> <s id="s.000563">Atqui exceſſus hic BF aſſumptus ut <lb></lb>Radius circuli habet ad ſuam peripheriam eandem Rationem, <lb></lb>quam TB Radius minor ad ſuam peripheriam; igitur eſt ea<lb></lb>dem Ratio BF exceſsûs Radij, ad exceſſum peripheriæ majo<lb></lb>ris, quæ eſt ejuſdem BF ut Radij ad ſuam peripheriam: ergo <lb></lb>per 9. lib. 5. hæc peripheria æqualis eſt illi exceſſui periphe<lb></lb>riæ majoris. </s> <s id="s.000564">Cum itaque Ratio diametri ad peripheriam ſit ut <lb></lb>7 ad 22, ſeu ut 113 ad 355, fiat ut Radius 7 ad peripheriam <lb></lb>44, ſeu ut 113 ad 710, ita BF altitudo ped. 6. ad ped. 37. <lb></lb>unc. 8: qui numerus conſentit cùm ſuperiore. <pb pagenum="69" xlink:href="017/01/085.jpg"></pb></s> </p> <p type="head"> <s id="s.000565"><emph type="center"></emph>CAPUT XII.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000566"><emph type="center"></emph><emph type="italics"></emph>An tellus moveatur motu trepidationis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000567">QUoniam centrum gravitatis eſt in quolibet corpore <lb></lb>punctum illud, quod æquales gravitates circumſtant, <lb></lb>manifeſtum eſt non permanere idem gravitatis centrum, ſi <lb></lb>aliqua corpori additio fiat, aut detractio; neque enim manet <lb></lb>eadem momentorum gravitatis æqualitas circa illud punctum; <lb></lb>ſed aliud eſt punctum, per quod ducta plana dividunt totius <lb></lb>corporis gravitatem in momenta æqualia, & eſt novum cen<lb></lb>trum gravitatis. </s> <s id="s.000568">Hinc patet in telluris globo, qui plurimas <lb></lb>mutationes ſubit, corporibus gravibus ex alio in alium locum <lb></lb>tranſlatis, tolli æqualitatem partium ſaltem in actu primo gra<lb></lb>vitantium, cum hæc quidem, quæ oppoſitæ parti ante erat <lb></lb>æqualis, ſubtractione nunc fiat minor, illa verò, quæ pariter <lb></lb>ſibi oppoſitæ parti proximè fuit æqualis, additione evadat ma<lb></lb>jor. </s> <s id="s.000569">Ex quo neceſſariò colligitur mutatio centri gravitatis. </s> </p> <p type="main"> <s id="s.000570">Sed quia, ut tellus ſuis librata ponderibus in loco ſibi debi<lb></lb>to conſiſteret, debuit initio ejus centrum gravitatis congrue<lb></lb>re centro univerſi, circa quod gravia & levia diſponuntur; id<lb></lb>circò dubitari poteſt, utrùm mutato gravitatis centro terra mo<lb></lb>veri debeat, ut novum gravitatis centrum collocetur in centro <lb></lb>univerſi. </s> <s id="s.000571">Quoniam verò huc illuc paſſim tranſlatis corpori<lb></lb>bus, terra nunc in hanc, nunc in illam partem moveretur, ut <lb></lb>proinde quaſi trepidaret; hinc factus eſt quæſtioni locus, an <lb></lb>tellus moveatur motu trepidationis; quicquid ſit an motus iſte <lb></lb>ſub ſenſum cadat, nec ne. </s> </p> <p type="main"> <s id="s.000572">Terram univerſam & ſingulas ejus partes ſuâ gravitate re<lb></lb>pugnare, ne ſurſum moveantur, certum eſt; at univerſi cen<lb></lb>trum occupare, toti quidem elemento graviſſimo convenit, ſed <lb></lb>non partibus ſingulis: neque enim gravitas eſt appetitus ſub<lb></lb>ſiſtendi in centro, quem natura non ſatis aptè gravibus ſingu<lb></lb>lis indidiſſet; cui nimirùm fieri ſatis non poteſt, niſi corpora <lb></lb>ſe invicem penetrent; unum autem grave in centro exiſtens <pb pagenum="70" xlink:href="017/01/086.jpg"></pb>cætera omnia inde excludit. </s> <s id="s.000573">Reſtituunt ſe gravia in locum <lb></lb>ſuum versùs centrum pergendo, non ut ad centrum veniant; <lb></lb>ſed ut nihil levius infra ſe habeant; quemadmodum & levia <lb></lb>versùs cælum aſcendunt, non ut cælum petant, ibíque demum <lb></lb>quieſcant, ſed ne quid gravius ſupra ſe patiantur. </s> <s id="s.000574">Cæterùm <lb></lb>hoc ipſo, quòd natura, & vacuitatem omnem eliminavit, & <lb></lb>corporum penetrationem proſcripſit, & vim ſe ſuis locis diſpo<lb></lb>nendi corporibus indidit, ſatis univerſi conſiſtentiæ & ordini <lb></lb>conſultum eſt. </s> <s id="s.000575">Quare corpori nihil levius infra ſe habenti nul<lb></lb>lam præterea gravitationem tribuendam cenſeo, præter re<lb></lb>ſiſtentiam, ne ſurſum moveatur. </s> <s id="s.000576">Gravitas ſiquidem non niſi <lb></lb>comparatè dicitur, habitâ ratione proximi corporis, in quo <lb></lb>tanquam in loco exiſtit id, quod grave dicitur; nam ſi orbis <lb></lb>univerſus conſtaret unico corpore homogeneo, nihil eſſet aut <lb></lb>grave aut leve, cum nihil eſſet, quòd præ aliis expoſceret pro<lb></lb>piùs admoveri centro univerſi. </s> <s id="s.000577">Cum itaque terra ad hoc uni<lb></lb>verſi centrum perinde ſe habeat, atque ſi corporibus levioribus <lb></lb>non circumfunderetur, his namque ſublatis illa nec propiùs ad <lb></lb>univerſi centrum accederet, nec longiùs ab eo recederet; ideò <lb></lb>pars terræ quæcumque cum reliquis comparata (ponatur hîc <lb></lb>tellus tota homogenea) nec gravis eſt nec levis; ac proinde, <lb></lb>cùm nulla pars centro propior eſſe exigat, quàm alia, nulla <lb></lb>quoque eſt, quæ aliam urgeat, aut premat propriè, ſed omnes, <lb></lb>& ſingulæ tantummodò repugnant, ne ſurſum in medium leve <lb></lb>transferantur. </s> </p> <p type="main"> <s id="s.000578">Hinc eſt quod terræ conſiſtentiam in loco ſuo, non propriè <lb></lb>ex libræ rationibus explicandam cenſeo; quia in librâ utraque <lb></lb>lanx non repugnat ſolùm, ne attollatur, verùm etiam in aere <lb></lb>conſtituta deorſum nititur; terræ autem partes ſuperiores nil <lb></lb>infrà ſe levius habentes non conantur deorſum. </s> <s id="s.000579">Et quemad<lb></lb>modum ſi libræ lanx utraque ſubjecto plano incumberet, ea<lb></lb>rum conſiſtentia non eſſet æquilibrio tribuenda, quamvis <lb></lb>æquilibres ſint, ſed idcircò ſolùm conſiſterent, quia infrà ſe <lb></lb>haberent corpus, quod permeare vel non exigit, vel non po<lb></lb>teſt earum gravitas: ita terræ partes licèt adeò æqualiter ſint <lb></lb>diſpoſitæ circa ſuum commune gravitatis centrum (in quo vi<lb></lb>res ſuas exererent tellure totâ in aeris locum tranſlatâ) ut ex illo <lb></lb>ſuſpensâ tellure in æquilibrio conſiſterent; re tamen ipsâ non <pb pagenum="71" xlink:href="017/01/087.jpg"></pb>conſiſtunt propter æquilibrium; ſed quia nulla pars habet in<lb></lb>fra ſe aliquid, ſub quo petat exiſtere, atque adeò nulla eſt, <lb></lb>quæ deorſum nitatur. </s> <s id="s.000580">Quare Poëticè ſolùm, non verò Philo<lb></lb>ſophicè dictum eſt. <lb></lb><emph type="italics"></emph>Terra pilæ ſimilis, nullo fulcimine nixa, <lb></lb>Aëre ſubjecto tam grave pendet onus.<emph.end type="italics"></emph.end><lb></lb>Aer ſi quidem non eſt ſubjectus terræ, ſed circumfuſus; ea <lb></lb>namque ſubjecta ſunt, quæ inferiora; inferiora autem, quæ <lb></lb>centro propiora. </s> <s id="s.000581">Terræ itaque globus nihil habet, in quod <lb></lb>gravitatis vires exerceat deorſum conando. </s> </p> <p type="main"> <s id="s.000582">Quæ cum ita ſint, nulla unquam continget in terrâ mutatio <lb></lb>atque gravium tranſlatio, quæ efficiat motum trepidationis. </s> <lb></lb> <s id="s.000583">Sit enim terræ globus AB, cujus cen<lb></lb><figure id="id.017.01.087.1.jpg" xlink:href="017/01/087/1.jpg"></figure><lb></lb>trum C ſit pariter centrum gravitatis: <lb></lb>ducto per C plano IL, hemiſphærium <lb></lb>IAL eſt æquale hemiſphærio IBL; <lb></lb>ex quo abſciſſa intelligatur portio <lb></lb>ſphærica DEB, in cujus locum ſuc<lb></lb>cedat aër. </s> <s id="s.000584">Si qua igitur pars deberet <lb></lb>deorſum versùs C niti, non alia uti<lb></lb>que eſſet præter D & E, quæ longiùs <lb></lb>à centro abſunt, quàm contiguus aër <lb></lb>DE. </s> <s id="s.000585">At portio IDEL prævalere non <lb></lb>poteſt hemiſphærio IAL, quod deberet ſurſum propelli; ergo <lb></lb>non poteſt centrum C moveri versùs A, ut punctum aliquod <lb></lb>inter C & K congruat centro univerſi. </s> <s id="s.000586">Sed neque hemiſphæ<lb></lb>rium IAL debet deſcendere, quia nullum habet corpus leve <lb></lb>ſibi contiguum, quod univerſi centro vicinius ſit; non ergo <lb></lb>debet propellere oppoſitum ſegmentum IDEL; cujus omnes <lb></lb>partes non ſolùm reluctantur motui, quo recedant ab univerſi <lb></lb>centro C, ſed etiam illarum aliquæ ſe ipſæ urgent, & conan<lb></lb>tur versùs C. </s> <s id="s.000587">Nondum igitur terra movetur. </s> </p> <p type="main"> <s id="s.000588">Quare Segmentum Sphæricum DKEB transferatur in op<lb></lb>poſitam partem, & addatur hemiſphærio ſuperiori etiam mons <lb></lb>FHG æqualis abſciſſæ portioni ſphæricæ. </s> <s id="s.000589">Aio ne dum factam <lb></lb>eſſe mutationem, quæ ad motum telluri conciliandum ſufficiat. </s> <lb></lb> <s id="s.000590">Quamvis enim mons ille FHG, quippe quem ambit aër le-<pb pagenum="72" xlink:href="017/01/088.jpg"></pb>vior vicinior centro, conetur deorſum; certum eſt illum de<lb></lb>ſcendere non poſſe, quin totam reliquam terram impellat, ejuſ<lb></lb>que reſiſtentiam ſuperet; reſiſtit autem primò ſegmentum <lb></lb>IDEL, cujus omnes partes magis à centro removerentur; ni<lb></lb>ſi igitur mons FHG major ſit ſegmento ſphærico IDEL <lb></lb>(vel ſaltem non multò minor, ſi quidem ob majorem à centro <lb></lb>diſtantiam augerentur momenta gravitatis, ex dictis cap. 4.) <lb></lb>non poterit ſubjectam terram loco dimovere. </s> <s id="s.000591">Præterea etiam <lb></lb>hemiſphærium IAL repugnat deſcenſui montis FHG, quia <lb></lb>fieri non poteſt hic motus, niſi hemiſphærij partes tranſiliant <lb></lb>planum IL, atque magis à centro recedant. </s> <s id="s.000592">Quanta igitur <lb></lb>gravitate præditum eſſe montem oporteret, qui tantam re<lb></lb>ſiſtentiam ſuperare valeret? </s> <s id="s.000593">At nunquam fieri tantam partium <lb></lb>permutationem, ut id quod transfertur, ſit non minus ſemiſſe <lb></lb>hemiſphærij, ut ſaltem ratione habitâ diſtantiæ à centro poſ<lb></lb>ſit prævalere, ita omnibus eſt manifeſtum, ut probatione non <lb></lb>indigeat. </s> <s id="s.000594">Quare neque hanc gravium tranſlationem motus ul<lb></lb>lus conſequitur, quo tellus trepidare dicatur. </s> </p> <p type="main"> <s id="s.000595">At, inquis, ſi in utrâque libræ lance ſint unciæ 100, & al<lb></lb>terutri uncia una addatur, lanx illa deprimitur, & oppoſita <lb></lb>elevatur; ergo exiguum pondus vim habet movendi ingens <lb></lb>pondus; ergo pariter mons FHG producere poteſt impetum, <lb></lb>qui ad movendum ſegmentum IDEL, quantumvis gravius, <lb></lb>abundè ſufficiat. </s> <s id="s.000596">Ego vero nego conſequentiam; quia non ab <lb></lb>unciâ illâ additâ ſolâ elevatur oppoſitum pondus, ſed omnes <lb></lb>unciæ ſimul in medio leviore ſuſpenſæ collatis viribus deorſum <lb></lb>conantur, atque præponderantes oppoſitæ lancis pondus at<lb></lb>tollunt. </s> <s id="s.000597">Hoc autem nil in rem noſtram facit, ubi neque mons <lb></lb>FHG ſolitariè ſumptus poteſt ſursùm propellere molem <lb></lb>IDEL majorem ſe, neque juvari poteſt ab hemiſphærio IAL, <lb></lb>quod cum nihil infrà ſe habeat, quod & levius ſit, & inter <lb></lb>ipſum ac univerſi centrum intercipiatur, neque poteſt ſe ipſum <lb></lb>versùs centrum urgere ſecundùm aliquas ſui partes ab eo remo<lb></lb>tiores, cum maximè partes centro proximæ valde reluctentur, <lb></lb>ne ab illo removeantur. </s> <s id="s.000598">Id quod in libræ lance, cui uncia fue<lb></lb>rit addita, reperire non poteris; totum ſiquidem lancis pon<lb></lb>dus deorſum nititur. </s> </p> <p type="main"> <s id="s.000599">Quod ſi ex librâ ſimilitudinem ducere placeat, petenda po-<pb pagenum="73" xlink:href="017/01/089.jpg"></pb>tiùs eſt ex librâ, cujus lanx altera ſubjecto plano incumbat, al<lb></lb>tera in aëre libera pendeat; ſi enim utraque lanx plena æquali<lb></lb>bus ponderibus conſiſtat in æquilibrio, & incumbenti lanci ad<lb></lb>datur ponderis pars, quæ à pendulâ lance detrahatur, lances <lb></lb>non moventur, nec inter ſe mutuò confligunt ponderum gra<lb></lb>vitates, niſi quatenùs lanx gravior ſemper magis reſiſtit leviori, <lb></lb>ne ab illâ elevetur: cæterùm gravior lanx non movet leviorem, <lb></lb>niſi ubi demum tanto pondere prægravata fuerit, ut ſubjecti <lb></lb>plani reſiſtentiam vincens illud aut frangat, aut ſaltem depri<lb></lb>mat. </s> <s id="s.000600">Sic hemiſphærium IAL habet rationem lancis non tan<lb></lb>tùm ſubjecto plano incumbentis, ſed, quod potius eſt, ſuo in <lb></lb>loco quieſcentis; cui quò plus addideris ponderis, auges qui<lb></lb>dem reſiſtentiam ne ſursùm versùs H propellatur, ipſum verò <lb></lb>non conatur deorſum versùs C; ſed totus conatus impoſito & <lb></lb>adjecto monti tribuendus eſſet, vel (ut ſim maximè liberalis) <lb></lb>etiam exceſſui illi, quo hemiſphærium IAL ſuperat ſegmen<lb></lb>tum ſphæricum IDEL, qui exceſſus eſt æqualis ipſi monti, <lb></lb>hoc eſt ſegmento DEB. </s> <s id="s.000601">Quare ſi fuerit abſciſſa tertia pars <lb></lb>hemiſphærij unius, & addatur alteri hemiſphærio è regione ſe<lb></lb>cundùm diametrum, tunc ad ſummum æqualis erit pars terræ <lb></lb>deorſum nitens FMGH parti oppoſitæ repugnanti IDEL; & <lb></lb>ſi velis partem FMGH remotiorem à centro magis gravitare <lb></lb>ita, ut ratio hujus exceſsûs in gravitando poſſit vincere non ſo<lb></lb>lùm reſiſtentiam ſegmenti IDEL, ne ſurſum propellatur, ſed <lb></lb>etiam ſegmenti FILG, ne ſecundùm partes IL centro proxi<lb></lb>mas ab eo removeatur; non admodum repugnabo. </s> <s id="s.000602">Sed cum <lb></lb>nunquam milleſima, ne dum ſexta, pars terreni globi ex alio <lb></lb>in alium locum ex diametro oppoſitum transferatur, nulla un<lb></lb>quam fit gravium permutatio, vi cujus tellus trepidet. </s> </p> <p type="main"> <s id="s.000603">Sed unum adhuc ſupereſt, quod per diſſimulantiam præ<lb></lb>tereundum non videtur. </s> <s id="s.000604">Eſto inquis, nulla fiat in tellure gra<lb></lb>vium tranſlatio, quæ tanta ſit, ut novum gravitatis centrum in <lb></lb>univerſi centro conſtituere valeat, ac proinde nulla ſit centri <lb></lb>terræ trepidatio: circa centrum ſaltem nutabit tellus motu <lb></lb>converſionis, validâ ventorum vi ſummos montes impellente, <lb></lb>orbemque totum, pro variâ ipſorum incurſione, modò hanc, <lb></lb>modò illam partem verſante: unde fortaſſe ortam acû magne<lb></lb>ticæ eodem in loco poſt aliquot annos variationem ſuſpicari <pb pagenum="74" xlink:href="017/01/090.jpg"></pb>quis poſſit. </s> <s id="s.000605">Cum enim tellus æqualibus circà centrum nutibus <lb></lb>librata permaneat, multo faciliùs omnem in partem converti <lb></lb>poſſe videtur, quàm rota ingens ſuo in axe ſuſpenſa: Rota ſci<lb></lb>licet ſuo pondere axem premens illum, dum convertitur, te<lb></lb>rit; hancque affrictûs difficultatem vincat neceſſe eſt, quod <lb></lb>una ex parte additur pondus, vel quæ applicatur Potentia, ut <lb></lb>converſionem efficiat: tellus verò in orbem diffuſa nec cen<lb></lb>trum premit, nec axem, cum quo ullus fiat affrictus; ac <lb></lb>proptereà faciliorem præbet converſionis anſam Potentiæ unam <lb></lb>aliquam in partem urgenti. </s> <s id="s.000606">Hujuſmodi autem Potentia ventus <lb></lb>eſt, non ad perpendiculum in terram incidens, ſed obliquè in <lb></lb>præaltos ſaltem montes incurrens; cujus viribus nihil obſtare <lb></lb>videtur, quin telluris globum ſibi obſecundantem inclinet; <lb></lb>quemadmodum, & ingentes naves, vela implens, impellit. </s> </p> <p type="main"> <s id="s.000607">Huic difficultati ut me ſubducam, non me in abditos magne<lb></lb>tiſmi receſſus recipio, aſſerendo tellurem ita arcanis nodis cæ<lb></lb>lo connexam, ut à ſummo axium polorumque cæleſtium atque <lb></lb>terreſtrium conſenſu divelli ac diſtrahi prorsùs nequeat: ne<lb></lb>que enim hiſce magnetiſmi latebris me ſatis protectum exiſti<lb></lb>marem; demptâ quippe ſolis Auſtralibus atque Borealibus ven<lb></lb>tis hâc facultate tellurem convertendi, ne ſcilicet terreſtres <lb></lb>poli à cæleſtibus diſcrepent, quid prohibeat reliquos ad Orti<lb></lb>vum, aut Occiduum limitem pertinentes, quin ſuo flatu or<lb></lb>bem hunc volvant, adhuc ſupereſſet explicandum. </s> <s id="s.000608">Hoc qui<lb></lb>dem ſatis eſſe videretur ad ſubmovendam ſuſpicionem illam de <lb></lb>acûs magneticæ variatione ob telluris converſionem; manente <lb></lb>nimirum axe terreſtri ita, ut cum cæleſti conveniat, aut illi <lb></lb>ſaltem parallelus exiſtat, nihil eſt quod, etiam tellure circa <lb></lb>axem conversâ, magneticam declinationem commutare queat: <lb></lb>nam quod ad ſyderum aſpectus ſpectat, parum intereſt, tellus<lb></lb>ne? </s> <s id="s.000609">an cælum volvatur; ſi igitur diurna cæli converſio magne<lb></lb>tis declinationem non mutat, neque ad illam mutandam ſuffi<lb></lb>ceret telluris circa ſuum axem converſio, vi cujus alia atque <lb></lb>alia ſydera reſpiceret: Præterquam quod non id temporum lap<lb></lb>ſu accideret; ſed ubi ventorum impetus elanguiſſet, illicò va<lb></lb>riatio illa declinationis magneticæ deprehenderetur: id quod <lb></lb>ab omni experimento longè abeſt. </s> <s id="s.000610">Verùm adeò à noſtris ſen<lb></lb>ſibus ſejunctæ ſunt magneticorum ſymptomatum cauſæ, ut ad <pb pagenum="75" xlink:href="017/01/091.jpg"></pb>aliarum difficultatum ſolutionem non facilè advocandus ſit in <lb></lb>Philoſophicam ſcenam magnetiſmus. </s> </p> <p type="main"> <s id="s.000611">Illud potius hìc attendendum videtur, quod montis altitu<lb></lb>do, atque magnitudo ad totius telluris molem Rationem habet <lb></lb>ſatis exiguam. </s> <s id="s.000612">Cum enim terræ ambitus probabiliter ſtatuatur, <lb></lb>ut aliàs oſtendi, milliarium Rom. <expan abbr="antiq.">antique</expan> 30598, ejuſque <lb></lb>propterea diameter ſit proximè mill. (9738 4/51), tota ſuperficies <lb></lb>ſphærica (ut pote quadrupla maximi circuli ex demonſtratis <lb></lb>ab Archimede) eſt mill. quadratorum 297. 987800 proximè. </s> <lb></lb> <s id="s.000613">Mons ſtatuatur altitudinis perpendicularis milliarium quin<lb></lb>que; hæc eſt ad terreſtrem diametrum ut 1 ad 1947: baſis <lb></lb>montis occupet milliaria quadrata 500; hæc eſt ad ſphæricam <lb></lb>totius globi ſuperficiem, ut 1 ad 595975. Finge jam pro mon<lb></lb>te granum hordei, quod promineat ſecundùm ſuam latitudi<lb></lb>nem ex ſphærâ habente diametrum granorum 1947, hoc eſt <lb></lb>paſſuum geometricorum ſex, ſeu pedum Rom. <expan abbr="antiq.">antique</expan> 30. cir<lb></lb>culi maximi ambitus erit pedum 94 1/4: quare hujus ſphæræ ſu<lb></lb>perficies habet pedes quadratos 2827, hoc eſt quadratas lati<lb></lb>tudines grani hordei paulò plures quàm 11. 579000. Igitur <lb></lb>grani hordei jacentis altitudo ad hujus ſphæræ diametrum <lb></lb>eandem ex hypotheſi habet rationem, quam prædicti montis <lb></lb>altitudo ad telluris diametrum: & ſi decem grana ſibi invicem <lb></lb>attigua diſponantur, ut montis baſim æmulentur, eadem erit <lb></lb>ratio ad ſuperficiem. </s> <s id="s.000614">Quamvis itaque ſphæra illa intelligatur <lb></lb>planè inanis ac leviſſima ſolam habens ſuperficiem papyra<lb></lb>ceam, ex qua granum ordei agglutinatum promineat, an pu<lb></lb>tas à flatu quantumvis valido per fiſtulam emiſſo in granum il<lb></lb>lud hordei incurrente convertendum eſſe globum papyra<lb></lb>ceum? </s> <s id="s.000615">Id ſanè ex cæteris experimentis conjicere non licet; <lb></lb>perinde enim eſt atque ſi nihil promineret; neque vel mini<lb></lb>mùm obeſt Phyſicæ rotunditati. </s> <s id="s.000616">Quare neque montis altitu<lb></lb>do conſtituta quicquam detrahet orbicularis figuræ, quod ſub <lb></lb>Phyſicam conſiderationem cadat; ac proptereà nihil virium ad <lb></lb>tellurem convertendam obtinet ventus in montem incurrens. </s> </p> <p type="main"> <s id="s.000617">Et quidem converſionem hanc re ipsâ non fieri manifeſtum <lb></lb>eſt; ſi quidem cum nulla vincenda eſſet gravitas, quæ longiùs <lb></lb>à centro gravium recederet, vel quæ axem tereret, facillima <lb></lb>videretur eſſe globi totius converſio circa centrum, non ſolùm <pb pagenum="76" xlink:href="017/01/092.jpg"></pb>validioribus atque incitatioribus, ſed temperatis etiam atque <lb></lb>mediocribus ventis flantibus. </s> <s id="s.000618">Hi autem aliquando diuturni <lb></lb>ſunt; cujuſmodi potiſſimum ſunt Eteſiæ, quibus maritimi cur<lb></lb>ſus celeres, & certi diriguntur. </s> <s id="s.000619">Tot igitur dierum ſpatio, ven<lb></lb>to oppoſitos montes vehementiùs urgente, non modica fieret <lb></lb>terreni globi inclinatio; ac propterea non eadem demum per<lb></lb>maneret eodem in loco Poli ſuprà Horizontem altitudo, quo<lb></lb>ties ab alterutro cardine Auſtrali Boreali ve, aut à ſolſtitiali <lb></lb>Brumali-ve limite tam ortivo quàm occiduo ventus ſpiraret, at<lb></lb>que multarum ædium facies non eandem ampliùs reſpicerent <lb></lb>cæli plagam; quare & ſcietherica Horologia quantumvis ac<lb></lb>curatè ſemel deſcripta poſt non adeò multas temporum inclina<lb></lb>tiones toto ferè cælo diſcreparent; aliis enim, atque aliis ſub<lb></lb>inde flantibus ventis, varia oriretur orbis converſio, atque alia <lb></lb>planorum cum circulis horariis ſectio, quæ deſcriptis lineis non <lb></lb>congrueret. </s> <s id="s.000620">Hujus autem mutationis nullum in toto terra<lb></lb>rum orbe veſtigium apparet, niſi fortè fabulas liceat com<lb></lb>miniſci. </s> </p> <p type="main"> <s id="s.000621">Quòd ſi converſionem hanc non omninò circa centrum <lb></lb>quamcumque in partem fieri, ſed tantummodo circa axem, <lb></lb>dixeris, ut argumenti vim effugias; Quid illud eſt, quod ita <lb></lb>terreſtrem axem cum cæleſti colligatum velit, ut tamen ter<lb></lb>reſtres meridianos à primâ mundi molitione conſtitutos tem<lb></lb>poris lapſu cum cæleſtibus meridianis non convenire permit<lb></lb>tat? </s> <s id="s.000622">Sed & aliud profectò, nec illud quidem leve, incommo<lb></lb>dum ſubeas neceſſe eſt; dum enim converſionem adſtruis ab <lb></lb>ortu in occaſum, & viciſſim ab occaſu in ortum, fieri poterit, <lb></lb>ut poſt aliquot annos non planè ſpernenda converſio facta fue<lb></lb>rit, ac proinde temporum numeratio cælo non reſpondeat. </s> <lb></lb> <s id="s.000623">Nam ſi ab ortu in occaſum ex. </s> <s id="s.000624">gr. proceſſerit tellus, minus tem<lb></lb>poris numerabitur quàm pro ratione cæleſtium motuum; ut <lb></lb>contigiſſe fertur navi cui à Victoriâ nomen inditum eſt, in ex<lb></lb>peditione Magellanicâ; cum ſcilicet poſt totius orbis ambitum <lb></lb>redux in Hiſpalenſem portum, ex quo ante tres annos ſolve<lb></lb>rat, intraret, tunc primùm obſervarunt ſe à rectâ temporis nu<lb></lb>meratione defeciſſe die uno; quippe qui cum juxta diurnam <lb></lb>cæli converſionem ab ortu in occaſum iter inſtituiſſent, juſto <lb></lb>tardiùs ſemper ſol illis occiderat, exiguo quidem ſingulis die-<pb pagenum="77" xlink:href="017/01/093.jpg"></pb>bus, quibus procedebant, diſcrimine, ſed quod demùm modi<lb></lb>cis illis acceſſionibus in integrum diem excreverat. </s> <s id="s.000625">Contra ve<lb></lb>rò accideret, ſi ab occaſu in ortum ſemper navigaretur; juſto <lb></lb>enim breviores eſſent dies, ac propterea eorum numerus ac<lb></lb>creſceret. </s> <s id="s.000626">Hæc autem in temporum numeratione inconſtan<lb></lb>tia, ſi ventorum impetu tellus modò in ortum, modò in occa<lb></lb>ſum converteretur, quantam perturbationem inveheret in <lb></lb>Aſtronomiam? </s> <s id="s.000627">Neque tibi quicquam ſuffragari exiſtimes, ſi <lb></lb>ex varia ventorum oppoſitas in plagas ſivè ſimul, ſivè ſubinde, <lb></lb>ſpirantium commutatione converſiones illas compenſari dixe<lb></lb>ris: id enim ad incertum revocat omnes Aſtronomorum calcu<lb></lb>los, ubi meridianorum circulorum ſectiones ſtabiles non perma<lb></lb>neant; cum ad orbem totum inclinandum, ut tu quidem au<lb></lb>tumas, ſatis ſit, ſi unâ aliquâ in regione ventus montes impel<lb></lb>lat; quî verò certus ſim factam ab Argeſte telluris converſio<lb></lb>nem in ortum, æquatam demum fuiſſe à Vulturno, aut ab <lb></lb>Euro-Auſtro? </s> </p> <p type="main"> <s id="s.000628">Verùm quàm infirmæ ſint validiſſimorum ventorum vires ad <lb></lb>globum hunc terraqueum inclinandum, expendamus, etiamſi <lb></lb>montium perpendicula non quinque tantùm milliaribus defini<lb></lb>ta velis, ſed multò altiora. </s> <s id="s.000629">Statue in ingenti lacu compoſitam <lb></lb>ex trabibus aliquot ratem, quam in littore ſtans facilè funiculo <lb></lb>modereris: Tùm ratem aliam paris quidem latitudinis, ſed cen<lb></lb>tuplò longiorem, compone: Poteris-ne hanc funiculo eodem, <lb></lb>ac labore non majori, trahere perinde atque priorem? </s> <s id="s.000630">Negabis <lb></lb>utique, quamvis enim utraque lacui ſtagnanti innatet, nec <lb></lb>vincenda ſit alterutrius gravitas, ut à centro gravium magis re<lb></lb>cedat; licet utraque parem in motu ab aquâ dividendâ reſiſten<lb></lb>tiam inveniat (ejuſdem quippe ſunt latitudinis ſolâ diſcrepan<lb></lb>tes longitudine, & æqualis eſt utriuſque immerſio propter ean<lb></lb>dem ſingularum trabium molem, atque ſpecificam gravitatem) <lb></lb>quia tamen diſpar eſt ratium magnitudo, & impetu extrinſe<lb></lb>cùs accepto utraque eget, ut moveatur, palàm eſt majore im<lb></lb>petu opus eſſe, ut ratis major trahatur, ac propterea poſſe hanc <lb></lb>adeò augeri, ut impetus ad illam movendam neceſſarius exce<lb></lb>dat vires Potentiæ ratem minorem funiculo moderantis. </s> <s id="s.000631">Ita <lb></lb>planè eſt. </s> <s id="s.000632">Sed jam animum transfer ad inſtitutam diſputatio<lb></lb>nem, ut diſpicias, undè irrepſerit dubitatio hæc de telluris <pb pagenum="78" xlink:href="017/01/094.jpg"></pb>converſione ex ventorum impulſu, & quàm facilè fucum fece<lb></lb>rit rota ſuo in axe ſuſpenſa, quæ levi negotio, nec valido im<lb></lb>pulſu, volvitur. </s> <s id="s.000633">Rota ſiquidem tota deorſum gravitat, ac <lb></lb>proptereà axem premit; quia autem in axe ſuſpenditur, fieri <lb></lb>non poteſt, ut pars altera deſcendat, quin oppoſita aſcendat. </s> <lb></lb> <s id="s.000634">Quandiu conatus ad deſcendendum æqualis eſt reſiſtentiæ ad <lb></lb>aſcendendum, rota quieſcit; nec volvitur, niſi alterutri parti <lb></lb>fiat acceſſio Potentiæ, quæ pariter deſcenſum juvet, vel quia <lb></lb>ipſa quoquè deorſum conatur cum parte deſcendente, vel quia <lb></lb>ſurſum nitens partem alteram elevat, oppoſitamque deprimet <lb></lb>ſuapte naturâ deſcendentem. </s> <s id="s.000635">Non tamen hujuſmodi rotæ ſuſ<lb></lb>penſæ converſio tribuenda eſt ſoli Potentiæ; ſed pars rotæ de<lb></lb>ſcendens atque Potentia collatis viribus elevant partem rotæ <lb></lb>aſcendentem, eíque impetum imprimunt. </s> <s id="s.000636">At in telluris circa <lb></lb>ſuum centrum, vel axem, converſione nihil adeſſet, quod Po<lb></lb>tentiam juvaret; quia nulla eſt pars, quæ deorſum conetur, <lb></lb>aut ſurſum, ut poſſit oppoſitæ parti impetum aliquem impri<lb></lb>mere; nulla etenim pars in hujuſmodi converſione ad centrum <lb></lb>gravium accederet, aut ab illo recederet. </s> <s id="s.000637">Totus igitur impe<lb></lb>tus à vento imprimendus eſſet toti telluris globo, ut à ſuâ, quæ <lb></lb>ſecundùm naturam eſt, quiete dimoveretur. </s> <s id="s.000638">Atqui globi ter<lb></lb>raquei ea eſt moles, ut contineat milliaria cubica proximè <lb></lb>48670. 200000 (omnis nimirum ſphæra æqualis eſt cono, cu<lb></lb>jus altitudo par eſt Radio ſphæræ, baſis autem æqualis ſuperfi<lb></lb>ciei ſphæræ, ex dictis verò paulò ſuperiùs, & ſuperficies & Ra<lb></lb>dius globi hujus innoteſcit) nullus igitur adeò vehemens eſt <lb></lb>ventus, qui tantæ moli impetum imprimere valeat; nullus ſi<lb></lb>quidem excogitari poteſt ventus, qui globum marmoreum, aut <lb></lb>etiam ex argillâ, in planitie æquiſſimâ conſtitutum, ſi mille <lb></lb>paſſus Geometricos in diametro numeret, convolvere valeat. </s> <lb></lb> <s id="s.000639">Adde in telluris converſione, ſi illa fieret, quò vehementior <lb></lb>eſſet ventus in montem incurrens, validior eſſet reſiſtentia aëris <lb></lb>à reliquis montibus dividendi; ſed & multorum ingentium <lb></lb>fluminum contrariam in partem labentium impetus obſiſteret, <lb></lb>ne tellus vento flanti obſecundaret. </s> <s id="s.000640">Quod ſi hæc levis eſſe mo<lb></lb>menti dixeris ad obſiſtendum, levis pariter momenti eſſe ven<lb></lb>torum impetum, neceſſe eſt, fatearis: neque hic arduum eſſet <lb></lb>ventorum atque fluminum vires invicem conferre, aquarum-<pb pagenum="79" xlink:href="017/01/095.jpg"></pb>que impetum multò validiorem oſtendere; ſed ad alia prope<lb></lb>randum eſt: ſatisfuerit monuiſſe non mediocrem intercedere <lb></lb>analogiam inter aquarum guttas in rivulos primùm, deinde in <lb></lb>majores rivos, ac demum in torrentem concurrentes, atque <lb></lb>terræ expirationes in ventum congregatas, quæ multum vi<lb></lb>rium obtinent, ſi plurimæ in unum coëant, quemadmodum <lb></lb>& aquis contingit. <lb></lb></s> </p> <p type="head"> <s id="s.000641"><emph type="center"></emph>CAPUT XIII.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000642"><emph type="center"></emph><emph type="italics"></emph>Quâ ratione minuatur gravitatio in plano <lb></lb>inclinato.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000643">PLanum inclinatum dicitur planum quodcumque non tran<lb></lb>ſit per centrum gravium & levium, hoc eſt per centrum <lb></lb>univerſi; hujuſmodi ſiquidem planum non cadit ad angulos <lb></lb>æquales in ſphæricam terræ ſuperficiem. </s> <s id="s.000644">Hinc etiam planum <lb></lb>horizonti parallelum reipsâ eſt inclinatum, niſi adeò exiguum <lb></lb>ſit ac breve, ut puncti vicem obtineat, ſi cum terreni globi ſu<lb></lb><figure id="id.017.01.095.1.jpg" xlink:href="017/01/095/1.jpg"></figure><lb></lb>perficie conferatur. </s> <s id="s.000645">Sit univerſi <lb></lb>centrum A, plana BA, & CA ſunt <lb></lb>verticalia & perpendicularia, qui<lb></lb>bus ſi corpus aliquod grave appli<lb></lb>cueris, illud non impedietur, quin <lb></lb>per ſuam directionis lineam deſcen<lb></lb>dat. </s> <s id="s.000646">At verò tam planum BC, quam <lb></lb>planum CD inclinata ſunt, nec cor<lb></lb>pus grave illis impoſitum poteſt <lb></lb>rectâ ſecundùm directionis lineam <lb></lb>deſcendere, ſed ab illâ declinare co<lb></lb>gitur plano obſiſtente. </s> <s id="s.000647">Sunt autem anguli inclinationis ABC, <lb></lb>ACD. </s> <s id="s.000648">Quod ſi planum parallelum horizonti ita exiguum ſit, <lb></lb>ut à ſphæricâ ſuperficie, quam tangit, non recedat; tunc in <lb></lb>quacumque ejus parte conſtituatur corpus grave, perinde eſt, <lb></lb>atque ſi in puncto D collocatum concipiatur. </s> <s id="s.000649">Sin autem ita à <pb pagenum="80" xlink:href="017/01/096.jpg"></pb>puncto D diſtiterit, ut à ſphæricâ ſuperficie recedat, quemad<lb></lb>modum ſi eſſet planum DF, illud eſt inclinatum, & fit angulus <lb></lb>DFA inclinationis. </s> <s id="s.000650">Ubi obſervandum eſt non eandem eſſe <lb></lb>ſingularum plani partium inclinationem; angulus enim in<lb></lb>clinationis AEC major eſt inclinatione ABC, per 16. <lb></lb>lib. 1. & ſimiliter AFD maior eſt angulo ACD. </s> <s id="s.000651">Quare <lb></lb>ſtatim atque ea eſt puncti E à puncto B diſtantia, ut an<lb></lb>gulus à perpendiculis in centro A factus contemni non poſ<lb></lb>ſit, alia eſt etiam phyſicè inclinatio, & corporis ejuſdem <lb></lb>gravitatio mutatur. </s> </p> <p type="main"> <s id="s.000652">Quoniam verò corpus grave plano inclinato impoſitum ita <lb></lb>aëre circumfunditur, ut petat infrà illum deſcendere, & re<lb></lb>ſiſtat, ne ſurſum moveatur; ideò gravitare dicitur. </s> </p> <p type="main"> <s id="s.000653">Sed cavendum eſt, ne ex vocabulorum ſimilitudine er<lb></lb>ror ſubrepat: quandoquidem aliud eſt <emph type="italics"></emph>gravitare in plano <lb></lb>inclinato,<emph.end type="italics"></emph.end> aliud <emph type="italics"></emph>gravitare in planum inclinatum:<emph.end type="italics"></emph.end> nam intrà <lb></lb>aërem corpus grave, putà, lapis, gravitat in quocunque <lb></lb>plano etiam perpendiculari, non tamen gravitat in pla<lb></lb>num perpendiculare, nullaſque vires ſuæ gravitatis con<lb></lb>tra illud exercet, quamvis in eo exiſtens, & reſiſtat ſur<lb></lb>ſum trahenti, & conetur, ut vincat vires retinentis, ac <lb></lb>quicquid moram infert, & impedimentum motui. </s> <s id="s.000654">In pla<lb></lb>no itaque inclinato exiſtens corpus grave (ſubjectum pla<lb></lb>num ſupponitur optimè lævigatum, nec motui officiens <lb></lb>partium prominularum aſperitate) gravitat quidem, ſed mi<lb></lb>nùs quàm in plano perpendiculari, & pro variâ planorum <lb></lb>inclinatione, varia pariter eſt gravitatio, ut quotidiana nos <lb></lb>docet experientia. </s> <s id="s.000655">Quâ igitur ratione gravitatio minuatur, <lb></lb>hîc eſt examinandum; capite ſequenti gravitatio in Planum <lb></lb>inclinatum explicabitur. </s> </p> <p type="main"> <s id="s.000656">Cognoſcitur autem gravitatio ex reſiſtentiâ, quâ corpus <lb></lb>repugnat contra vires illud retinentis, ne deorſum feratur, <lb></lb>aut ſurſum trahentis; neque enim alio niſu gravia gravi<lb></lb>tant, quàm quo reſiſtunt impedienti motum gravitati <lb></lb>convenientem. </s> <s id="s.000657">Et quidem experimento aliquo poteſt gra<lb></lb>vitationis varietas inveſtigari; ſi nimirum planum BO ex <lb></lb>ligno, aut marmore accuratè lævigetur, & extremitati B <lb></lb>adnectatur orbiculus D facillimè circa axem verſatilis, pon-<pb pagenum="81" xlink:href="017/01/097.jpg"></pb>deri autem A ſubjiciantur <lb></lb><figure id="id.017.01.097.1.jpg" xlink:href="017/01/097/1.jpg"></figure><lb></lb>rotulæ, & adnectatur funi<lb></lb>culus per D tranſiens, ex <lb></lb>cujus extremo pendeat lanx <lb></lb>E, cui pondera immitti poſ<lb></lb>ſint: pro variâ enim plani <lb></lb>BO inclinatione etiam pon<lb></lb>dera in lance mutare opor<lb></lb>tebit, ut pondus A ſuſti<lb></lb>neatur, & plura erunt, quò magis ad perpendiculare accedet <lb></lb>planum BO. </s> <s id="s.000658">Verùm quia nunquam carere poteris ſuſpicione, <lb></lb>an corporum affrictus aliquid afferat impedimenti; ideò ſeclu<lb></lb>ſis omnibus, quæ extrinſecùs accidere poſſunt, reſiſtentiam ex <lb></lb>ſolâ gravitate ortam opus eſt conſiderare. </s> </p> <p type="main"> <s id="s.000659">Reſiſtentia verò omnis reſpondet violentiæ, quam patitur <lb></lb>id quod reſiſtit; minori etenim conatu minorem vim illatam <lb></lb>propulſare ſtudet natura, quæ validiùs obſiſtit majori violen<lb></lb>tiæ: id quod ita rationi eſt conſonum, & obviis experimentis <lb></lb>manifeſtum, ut in hoc demonſtrando ſupervacaneum ſit im<lb></lb>morari. </s> <s id="s.000660">Conſtituantur itaque duo <lb></lb><figure id="id.017.01.097.2.jpg" xlink:href="017/01/097/2.jpg"></figure><lb></lb>æqualis ponderis corpora in D & <lb></lb>in C; ſingulis alligetur funiculus, <lb></lb>qui per B tranſeat, & ſurſum tra<lb></lb>hantur ſimul ita, ut æqualiter mo<lb></lb>veantur. </s> <s id="s.000661">Abſolutâ motûs particu<lb></lb>lâ, corpus alterum ex D aſcendit <lb></lb>in H in plano perpendiculari; al<lb></lb>terum in plano inclinato ex C ve<lb></lb>nit in E, & CE linea æqualis eſt <lb></lb>lineæ motûs DH. </s> <s id="s.000662">Non eandem <lb></lb>tamen utrumque grave ſubiit vio<lb></lb>lentiam; nam motus DH fuit ſimpliciter, & abſolutè violen<lb></lb>tus; at motus CE eatenus ſolùm gravitati adverſatur, quate<lb></lb>nus aſcendit; aſcenſum autem metitur linea DG, quam ab<lb></lb>ſcindit EG horizonti parallela. </s> <s id="s.000663">Hîc ſcilicet planum DC in<lb></lb>tellige horizontale nihil à ſphæricá ſuperficie diſcrepans, ut <lb></lb>communiter contingit: quòd ſi non ita ſe haberet; ſed eſſet <lb></lb>ampliſſimum planum, menſura violentiæ illatæ ponderi in C <pb pagenum="82" xlink:href="017/01/098.jpg"></pb>conſtituto, in E elevato deſumenda eſſet ex differentiâ inter <lb></lb>KC & OE. </s> <s id="s.000664">Eſt itaque gravitatio in plano perpendiculari ad <lb></lb>gravitationem in plano inclinato, ut reſiſtentia ad aſcenden<lb></lb>dum in uno ad reſiſtentiam ad aſcendendum in alio; reſiſtentiæ <lb></lb>autem ſunt, ut violentia, quam corpora ſubeunt in motu; vio<lb></lb>lentia demum eſt ut HD ad GD, hoc eſt per 7. lib. 5. ut CE <lb></lb>ad DG. </s> <s id="s.000665">Sed ut CE ad DG, ita EB ad GB, per 2. lib. 6. & <lb></lb>ut BE, ad BG ita BC ad BD, per 4. lib. 6. igitur gravitatio <lb></lb>in perpendiculari ad gravitationem in inclinato eſt ut BC ad <lb></lb>BD, hoc eſt ut Secans anguli inclinationis ad Radium. </s> </p> <p type="main"> <s id="s.000666">Quæ autem de totis DH, & CE lineis dicta ſunt, de ſingu<lb></lb>lis earum particulis æqualibus dicta intelligantur; ductis quip<lb></lb>pe parallelis horizonti, eadem eſt omnium Ratio: hîc namque <lb></lb>ſupponimus planum BC non adeò magnum eſſe, ut ſingula <lb></lb>ejus puncta cum diverſis horizontibus comparanda ſint, omnes <lb></lb>ſiquidem perpendiculares lineæ directionis non quaſi conver<lb></lb>gentes, ſed phyſicè parallelæ accipiuntur. </s> <s id="s.000667">Quòd ſi tam lon<lb></lb>gum eſſet planum, ut phyſicè mutatus intelligeretur angulus <lb></lb>inclinationis, non eadem eſſet Ratio gravitationis in toto, ac in <lb></lb>partibus: ſed mutato angulo inclinationis mutaretur utique <lb></lb>ejus Secans; ac proinde inæqualium Secantium Ratio ad eum<lb></lb>dem Radium inæqualis, gravitationum pariter inæqualem ra<lb></lb>tionem oſtenderet. </s> </p> <p type="main"> <s id="s.000668">Quod ſi aſcendentium per vim extrinſecùs illatam corporum <lb></lb>reſiſtentiam atque gravitationem metimur ex violentiâ, quam <lb></lb>pro planorum varietate ſubeunt; eorum pariter in deſcendendo <lb></lb>efficacitatem ex ipſo deſcenſu argui æquum eſſet, datâ motûs <lb></lb>in diverſis planis æqualitate. </s> <s id="s.000669">Sed quia deſcenſus naturæ pro<lb></lb>penſioni congruit, fieri non poteſt, ut in alio atque alio plano <lb></lb>æquales ſint motus iſochroni; tardior enim eſt, qui in plano in<lb></lb>clinato perficitur, neque, ſi æqualis ponderis corpora deſcen<lb></lb>dant ex H & E, quando illud ad D pervenit, hoc poteſt attin<lb></lb>gere punctum C: ideò non ex deſcenſu gravitationem metiri <lb></lb>oportet, cum motus æquales non habeantur: niſi fortè eaſdem <lb></lb>movendi vires tribuas gravitati non impeditæ in perpendicula<lb></lb>ri, ac impeditæ in plano inclinato. </s> <s id="s.000670">Qua propter gravitationis <lb></lb>momenta ad deſcendendum non aliunde meliùs æſtimantur, <lb></lb>quàm ex repugnantiâ ad aſcendendum: ſic enim vulgari argu-<pb pagenum="83" xlink:href="017/01/099.jpg"></pb>mento ſingulorum corporum gravitates librâ expendimus, tan<lb></lb>tumque iis ad deſcendendum virium tribuimus, quantum re<lb></lb>ſiſtunt, ne ab oppoſitâ libræ lance deorſum conante eleventur. </s> <lb></lb> <s id="s.000671">Eadem igitur eſt gravitationis Ratio, ſeu propenſionis ad de<lb></lb>ſcendendum, quæ eſt reſiſtentiæ ad aſcendendum: Cum verò <lb></lb>reſiſtentiam in plano inclinato ad reſiſtentiam in perpendicu<lb></lb>lari oſtenſum ſit eſſe, ut Radius ad Secantem anguli inclinatio<lb></lb>nis, hoc eſt ut BD ad BC, erit pariter vis deſcendendi in <lb></lb>plano BC ad vim deſcendendi in plano BD, reciprocè ut BD <lb></lb>ad BC. </s> </p> <p type="main"> <s id="s.000672">Eadem ratione in plano CD ſuperficiem globi tangente, <lb></lb>gravitatio in CD ad gravitationem in perpendiculari CA eſt <lb></lb>ut CD ad CA; eſt enim CA Secans anguli inclinationis <lb></lb>DCA. </s> <s id="s.000673">Si enim ducatur KF Tangens, triangula CKF, <lb></lb>CDA ſunt ſimilia, angulus enim ad C communis eſt, & am<lb></lb>bo rectangula ad D & K; quare ut CK ad CF, ita CD ad <lb></lb>CA; ſed gravitatio in CF ad gravitationem in CK eſt reci<lb></lb>procè ut CK ad CF: igitur gravitatio in plano inclinato CD <lb></lb>globum tangente, ad gravitationem in perpendiculari CA, eſt <lb></lb>ut CD ad CA. </s> </p> <p type="main"> <s id="s.000674">Hinc eſt quod in planis horizontalibus, quæ ut plurimum <lb></lb>habemus, corpora non deſcendant, aut moveantur: quia ni<lb></lb>mirum à puncto, in quo grave ſtatuitur, ex. </s> <s id="s.000675">gr. F, ductæ li<lb></lb>neæ FA perpendicularis & FD Tangens faciunt angulum <lb></lb>DFA inclinationis adeò magnum, ut Radius ad ejus ſecan<lb></lb>tem penè infinitam non habeat ſenſu perceptibilem Rationem, <lb></lb>vel ſaltem non tantam, ut gravitatio, quæ ratione inclinatio<lb></lb>nis plani congruit corpori, non elidatur à reſiſtentiâ, quæ ori<lb></lb>tur ex corporum aſperitate. </s> <s id="s.000676">Quare ſublatâ, aut potiùs impeditâ, <lb></lb>gravitatione corpus quieſcit in plano horizontali. </s> </p> <p type="main"> <s id="s.000677">Et hæc eſt ratio, cur violentiam determinans, quam grave <lb></lb>aſcendens patitur, aſſumpſerim in perpendiculari BA par<lb></lb>tem GD, quam abſcindit parallela horizonti; hæc enim <lb></lb>menſura phyſicè non diſcrepat à verâ menſurâ, quæ aſſumen<lb></lb>da eſſet, ſi mente concipias rectam lineam DC tangere circu<lb></lb>lum, cujus ſemidiameter ſit millecuplo major. </s> <s id="s.000678">Menſura ſi qui<lb></lb>dem aſcensûs petenda eſt ex exceſſu, quo perpendicularis EA <lb></lb>ſuperat perpendicularem AC; illo enim intervallo, quo magis <lb></lb>receſſit à centro, aſcendit. </s> </p> <pb pagenum="84" xlink:href="017/01/100.jpg"></pb> <p type="main"> <s id="s.000679">Ex quo fit quod, ſi planum inclinatum BC cum perpendi<lb></lb>culari CA faceret angulum acutum ACB, corpus ex C uſque <lb></lb>in L (in quod punctum cadit perpendicularis AL) deſcende<lb></lb>ret, quia ſemper magis ad centrum accederet: ex L autem in E <lb></lb>aſcenderet, & aſcenſum metiretur exceſſus perpendiculi EA <lb></lb>ſuprà perpendiculum LA. </s> <s id="s.000680">Quare ut ex C aſcenderet, debe<lb></lb>ret eſſe planum inclinatum IC, quod cum CA faceret angu<lb></lb>lum ICA ſaltem rectum. </s> <s id="s.000681">Ubi ex occaſione licet obſervare <lb></lb>poſſe dari duos montes, qui cum valle intermediâ planitiem <lb></lb>unam conſtituant; ſi nimirum montium vertices eſſent E, & C, <lb></lb>ex quibus in imam vallem L deſcenderetur: & aqua per mon<lb></lb>tium venas deſcendens in L poſſet fontem aut lacum creare. </s> </p> <p type="main"> <s id="s.000682">Re autem ipsâ ſemper contingit angulum BCA eſſe obtuſum <lb></lb>vel non minorem recto. </s> <s id="s.000683">Ponatur enim terræ ſemidiameter DA <lb></lb>1000, & planum DC: (eſſet autem planum DC longius <lb></lb>milliar.4.) erit angulus DAC, gr. 0. 3′. </s> <s id="s.000684">26′; atque adeò DCA <lb></lb>gr. 89. 56′. </s> <s id="s.000685">34″. </s> <s id="s.000686">Jam verò ſit CD ad DB ut 100 ad 87; erit <lb></lb>angulus BCD gr.4.1. 1′. </s> <s id="s.000687">23″: quare totus BCA gr.130. 57′. </s> <s id="s.000688">57′. </s> <lb></lb> <s id="s.000689">Nunc ſi libeat comparare perpendiculum EA cum perpendi<lb></lb>culo GA, ſtatue GD ſemiſſem totius BD; eſt igitur & GE <lb></lb>ſemiſſis ipſius DC: Quare GE eſt partium 50, quarum GA eſt <lb></lb>100043 1/2: addantur quadrata GE 2500 & GA 10008701892 1/4, <lb></lb>& ſummæ radix quadrata (100043 102543/200086) major verâ eſt EA, quæ <lb></lb>non excedit perpendicularem GA 100043 1/2 niſi particulis (2500/400172). <lb></lb>Quoniam autem DAC angulus inventus eſt grad. 0. 3′. </s> <s id="s.000690">26′; <lb></lb>ejuſque Secans AC eſt partium (100000 5017/100000), quarum AD <lb></lb>poſita eſt 100000; diſcrimen inter AC, & AE ſuperiùs in<lb></lb>ventam, eſt partium (43 46227/100000), quæ eſt proximè eadem menſu<lb></lb>ra, ac DG poſita partium 43 1/2. Quod ſi in plani inclinati lon<lb></lb>gitudine <expan abbr="tantã">tantam</expan> Rationem habente ad terræ <expan abbr="ſemidiametrũ">ſemidiametrum</expan>, quan<lb></lb>ta conſtit ita eſt, poteſt citrà errorem aſſumi tanquam menſura <lb></lb>aſcensûs pars perpendiculi BA intecepta ab horizontali DC, <lb></lb>& parallelâ EG, ſatis patet id multò magis licere in planorum <lb></lb>longitudinibus minorem Rationem habentibus ad eandem ter<lb></lb>ræ ſemidiametrum. </s> <s id="s.000691">Manet itaque conſtituta regula gravitatio<lb></lb>nis, videlicet gravitationem in plano inclinato ad gravitationem <lb></lb>in perpendiculari eſſe, ut eſt Radius ad ſecantem anguli incli<lb></lb>nationis. </s> </p> <pb pagenum="85" xlink:href="017/01/101.jpg"></pb> <p type="main"> <s id="s.000692">Quamvis verò in partibus inferioribus plani inclinati ſit ſem<lb></lb>per major angulus inclinationis, quàm in ſuperioribus, & pro<lb></lb>inde minor ſit Ratio, quam habet Radius ad ſecantem anguli <lb></lb>majoris, ac ea, quam idem Radius habet ad ſecantem anguli <lb></lb>minoris: non tamen ea eſt gravitationis differentia, cujus ratio <lb></lb>habenda ſit; cum enim adeò exiguus ſit angulus BAC, ejus <lb></lb>quantitas diſtribuitur per omnes inclinationis angulos, qui <lb></lb>fiunt in punctis intermediis inter B & C; atque adeò contem<lb></lb>nendum eſt in praxi diſcrimen illud, quod oritur ex alio atque <lb></lb>alio inclinationis angulo in codem plano. </s> <s id="s.000693">Quod ſi inſignis eſſet <lb></lb>Rationum varietas, notabilis quoque eſſet gravitationis diver<lb></lb>ſitas idem enim contingeret, ac ſi non idem eſſet planum. </s> <s id="s.000694">Sed <lb></lb>hoc communiter non accidit. </s> </p> <p type="main"> <s id="s.000695">Ex his illud manifeſtâ conſecutione conficitur, quod ſi duo <lb></lb>plana inclinata inter ſe comparentur, ejuſdem corporis gravita<lb></lb>tiones in illis ſunt reciproce ut Secantes angulorum inclinatio<lb></lb>nis: hoc eſt, ſi fuerint duo plana inclinata BS, BC, gravitatio <lb></lb>in BS ad gravitationem in BC eſt ut BC ad BS. </s> <s id="s.000696">Quia enim <lb></lb>gravitatio in BC ad gravitationem in BD eſt ut BD ad BC; <lb></lb>& gravitatio in BD ad gravitationem in BS eſt ut BS ad BD, <lb></lb>igitur ex æqualitate, per 23. lib.5. gravitatio in BC ad gravi<lb></lb>tationem in BS eſt ut BS ad BC. </s> </p> <p type="main"> <s id="s.000697">Hinc prætereà fit, ut, ſi gravia in planis conſtituta habeant <lb></lb>Rationem eandem, quam ſecantes angulorum inclinationis ha<lb></lb>bent inter ſe vel ad Radium, eorum gravitatione, ſint æquales. </s> <lb></lb> <s id="s.000698">Sit ad horizontalem, SC per<lb></lb><figure id="id.017.01.101.1.jpg" xlink:href="017/01/101/1.jpg"></figure><lb></lb>pendicularis BD, & inclina<lb></lb>tæ BS, BC, per quas lineas <lb></lb>ducta intelligantur plana, & <lb></lb>in planis gravia diverſa, & ut <lb></lb>BD ad BC ita pondus O ad <lb></lb>pondus M, & ut BD ad BS <lb></lb>ita pondus O ad pondus N. </s> <lb></lb> <s id="s.000699">Dico ponderum M, O, N, gravitationes in ſuis planis eſſe <lb></lb>æquales. </s> <s id="s.000700"><expan abbr="Quoniã">Quoniam</expan> enim duorum gravium gravitationes in eadem <lb></lb>perpendiculari BD ſunt ut <expan abbr="ipſorũ">ipſorum</expan> pondera, gravitatio M in per<lb></lb>pendiculari BD, ad gravitationem O in eadem perpendiculari, <lb></lb>eſt ut M ad O, hoc eſt ut BC ad BD; ſed gravitatio M in per-<pb pagenum="86" xlink:href="017/01/102.jpg"></pb>pendiculari BD, ad gravitationem ejuſdem M in inclinatâ <lb></lb>BC, eſt pariter ut BC ad BD; igitur per 11. lib. 5. gravita<lb></lb>tio M in perpendiculari ad gravitationem O in perpendiculari <lb></lb>eſt, ut gravitatio M in perpendiculari BD ad gravitationem <lb></lb>M in inclinatâ BC; igitur per 14. lib. 5. gravitatio O in per<lb></lb>pendiculari BD æqualis eſt gravitationi M in inclinatâ BC. </s> <lb></lb> <s id="s.000701">Eâdem methodo oſtenditur æqualem eſſe gravitationem N in <lb></lb>inclinatâ BS, gravitationi O in perpendiculari BD. </s> <s id="s.000702">Quare <lb></lb>gravitationes M & N æquales inter ſe ſunt, cum æquales ſint <lb></lb>gravitationi O. </s> </p> <p type="main"> <s id="s.000703">Conſtat itaque iiſdem viribus retineri poſſe, aut ſurſum trahi, <lb></lb>majus pondus in plano inclinato, quàm in perpendiculari, ea<lb></lb>dem enim eſt illorum gravitatio, ut oſtendi; vires autem reti<lb></lb>nentis aut trahentis debent gravitationi corporis proportione <lb></lb>reſpondere. </s> <s id="s.000704">Quare datis viribus, quæ poſſint datum pondus O <lb></lb>ſuſtinere in perpendiculari BD, cognoſci poteſt gravitas pon<lb></lb>deris quod eædem vires ſuſtinere valebunt in dato plano BC in<lb></lb>clinato: ſi nimirùm fiat ut Radius ad ſecantem anguli datæ in<lb></lb>clinationis, ita datum pondus O ad pondus M quæſitum. </s> <s id="s.000705">De<lb></lb>tur O lib. 15. & angulus DBC gr. 36. Fiat ut radius 10000000 <lb></lb>ad ſecantem 12360680, ita lib. 15. ad lib. 18 1/2; quod eſt pon<lb></lb>dus M æquè gravitans in plano BC cum pondere O in per<lb></lb>pendiculari. </s> <s id="s.000706">Contra verò dato pondere M ſuſtinendo iiſdem <lb></lb>viribus, quibus ſuſtinetur O in perpendiculari, invenietur in<lb></lb>clinatio plani: ſi fiat ut pondus O lib. 15. ad pondus M datum <lb></lb>lib. 50, ita Radius 10000000 ad 333.33333.ſecantem anguli in<lb></lb>clinationis DBC gr. 72. 32′. </s> <s id="s.000707">32″. </s> <s id="s.000708">Demum dato pondere & pla<lb></lb>ni inclinatione nota fiet potentia, ſi ut Secans datæ inclinatio<lb></lb>nis ad Radium, ita fiat datum pondus ad aliud pondus, quod <lb></lb>potentia valet ſuſtinere in perpendiculari. </s> <s id="s.000709">Sit enim DBC <lb></lb>gr. 36, & M lib. 50. Erit ut Secans 12360680 ad Radium <lb></lb>10000000, ita M lib. 50 ad pondus O ferè lib.40 1/2, quod poſſit <lb></lb>à potentia in aere libero ſuſtineri. </s> <s id="s.000710">Quare potentia ſuſtinens <lb></lb>pondus in plano inclinato eſt ad pondus, ut Radius ad Secan<lb></lb>tem anguli inclinationis; & potentia potens movere cum ſit ma<lb></lb>jor potentiâ ſuſtinente, etiam majorem habet Rationem quàm <lb></lb>habeat Radius ad Secantem. </s> <s id="s.000711">Id quod intelligitur ex vi præcisè <lb></lb>gravitationis; quicquid inferat diſcriminis partium conflictus. <pb pagenum="87" xlink:href="017/01/103.jpg"></pb></s> </p> <p type="head"> <s id="s.000712"><emph type="center"></emph>CAPUT XIV.<emph.end type="center"></emph.end></s> </p> <p type="head"> <s id="s.000713"><emph type="center"></emph><emph type="italics"></emph>Quâ ratione corpus gravitet in planum inclinatum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000714">COnſtituta Ratione gravitationis in plano inclinato, deter<lb></lb>minatis ſcilicet momentis, quæ ad deſcendendum obtinet <lb></lb>corpus grave exiſtens in plano inclinato, ſupereſt explicanda <lb></lb>gravitatio, quam idem corpus exercet in planum inclinatum <lb></lb>illud urgendo, atque deorſum premendo. </s> <s id="s.000715">Certum eſt autem <lb></lb>planum verticale ſeu perpendiculare nullo pacto urgeri à cor<lb></lb>pore gravi, quod liberè deſcendere poteſt per ſuam directionis <lb></lb>lineam, quæ cum non occurrat plano verticali, nullum ab eo <lb></lb>recipit impedimentum. </s> <s id="s.000716">Quare corporis gravitas vires totas <lb></lb>exercet, aut deſcendendo, aut repugnando contra retinentem, <lb></lb>qui non plus adhibere debet conatûs in retinendo, etiam ſi pla<lb></lb>num verticale amoveatur: atque adeò nihil omninò gravitat in <lb></lb>planum verticale. </s> <s id="s.000717">Contra verò in planum horizontale, quam <lb></lb>maximè gravitant corpora; eò quod directionis lineâ in illud <lb></lb>incurrente ad angulos rectos, motus omnis impeditur, & <lb></lb>cunctas gravitatis vires deorſum contendentes ita ſubjectum <lb></lb>planum excipit, ut nihil reliquum ſit virium, quas vel minimo <lb></lb>motu exerceat. </s> <s id="s.000718">Hinc ſi corporis in plano horizontali jacentis <lb></lb>anſam teneas, nihil tibi prorſus eſt laborandum, nec quicquam <lb></lb>percipis ponderis; at ſubmoto plano lacertis omnibus eſt con<lb></lb>tendendum, ut illud retineas; tota enim gravitatio cum reti<lb></lb>nente luctatur, quæ planum ſuſtinens urgebat. </s> <s id="s.000719">In hoc itaque <lb></lb>planum verticale cum horizontali comparatur, quod cum ver<lb></lb>ticale nihil impediat motum, corpus in plano verticali omninò <lb></lb>gravitat, ſed in illud non gravitat: cum autem horizontale <lb></lb>prorſus impediat motum, corpus in plano horizontali nihil gra<lb></lb>vitat, ſed in illud totam ſuam gravitationem exercet. </s> <s id="s.000720">Eædem <lb></lb>igitur vires, quæ ad deſcendendum in plano verticali impen<lb></lb>derentur, in urgendo plano horizontali inſumuntur. </s> </p> <p type="main"> <s id="s.000721">Quæ cum ita ſint, ſatis conſtat corpora gravia ita in pla<lb></lb>no inclinato gravitare, & obtinere momenta ad deſcenden-<pb pagenum="88" xlink:href="017/01/104.jpg"></pb>dum, ut etiam in illud, à quo impediuntur, gravitent, il<lb></lb>ludque urgeant. </s> </p> <p type="main"> <s id="s.000722">Id verò fieri non poteſt niſi pro ratione impedimenti & mo<lb></lb>ræ, quam ſubjectum planum motui infert ſuſtinendo corpora <lb></lb>gravia; quæ proinde ſibi relicta à directionis lineâ declinant, <lb></lb>motúmque deflectunt. </s> <s id="s.000723">Porrò in plano inclinato quantum ſub<lb></lb>ſit impedimenti, ſtatim apparet, ac innoteſcit, quantum reli<lb></lb>quum ſit virium ad deſcendendum; vires enim, quæ reliquæ <lb></lb>ſunt, ſi adjiciantur viribus impeditis, totam virium omnium <lb></lb>ſummam conflare debent. </s> <s id="s.000724">Atqui ex ſuperiori capite notæ ſunt <lb></lb>vires, quibus corpus gravitat in plano inclinato; igitur quæ eſt <lb></lb>differentia gravitationis in plano inclinato, à gravitatione in <lb></lb>plano verticali, quod & perpendiculare, ea eſt menſura im<lb></lb>pedimenti, quod à ſubjecto plano infertur motui; atque <lb></lb><figure id="id.017.01.104.1.jpg" xlink:href="017/01/104/1.jpg"></figure><lb></lb>adeò gravitationis corporis in planum. </s> </p> <p type="main"> <s id="s.000725">Cum itaque oſtenſum fuerit <expan abbr="gravitationẽ">gravitationem</expan> <lb></lb>in plano BS ad gravitationem in plano BD <lb></lb>eſſe reciprocè ut BD ad BS, hoc eſt, ut Ra<lb></lb>dius ad ſecantem anguli inclinationis cum <lb></lb>verticali, hoc eſt ut BV ad BS, patet vires <lb></lb>non impeditas ad vires impeditas eſſe ut <lb></lb>BV ad VS, quandoquidem totas gravita<lb></lb>tis vires refert BS. </s> <s id="s.000726">In planum igitur inclinatum BS gravitatio <lb></lb>eſt ut VS, quæ in planum horizontale eſſet ſecundùm totas <lb></lb>vires ut BS. </s> <s id="s.000727">Quare gravitatio in planum horizontale ad gra<lb></lb>vitationem in planum inclinatum eſt ut Secans BS ad exceſ<lb></lb>ſum Secantis ſupra Radium, VS; ſeu, quod in idem recidit, ſi <lb></lb>gravitatio in plano inclinato ad gravitationem in verticali po<lb></lb>natur ut Sinus complementi anguli inclinationis ad Radium, <lb></lb>ita BR Radius ad DR Sinum verſum anguli inclinationis. </s> <s id="s.000728">Id <lb></lb>autem, quod de plano BS dictum eſt, de plano quoque BC, & <lb></lb>cæteris quibuſcunque dictum intelligatur; cum enim gravita<lb></lb>tio in plano inclinato BC ad gravitationem in perpendiculari <lb></lb>ſit ut BD, hoc eſt BX, ad BC, erit gravitatio in planum ho<lb></lb>rizontale ad gravitationem in inclinatum ut BC ad XC, hoc <lb></lb>eſt ut BT ad DT. </s> <s id="s.000729">Quare gravitatio in planum BS ad gravi<lb></lb>tationem in planum BC, eſt ut DR Sinus verſus inclinationis <lb></lb>DBS, ad DT Sinum verſum inclinationis DBC; aſſumptis <pb pagenum="89" xlink:href="017/01/105.jpg"></pb>ſcilicet numeris tabulatis ad eundem Radium relatis; nam ſi li<lb></lb>neæ ſpectentur, non eſt Ratio ut DR ad DT, ſed ut OT <lb></lb>ad DT; neque enim idem eſt Radius BS & BC; ac proinde <lb></lb>OT major eſt, quàm DR, ſicuti Radius BI major eſt Radio <lb></lb>BS; vel aſſumpto eodem Radio BD, Ratio eſt ut VS ad XC, <lb></lb>exceſſus ſecantium ſupra Radium. </s> </p> <p type="main"> <s id="s.000730">Id verò ex dictis ſub finem capitis ſuperioris videtur mani<lb></lb>feſtum: nam ſi in plano BC retinetur pondus lib. 50. iiſdem <lb></lb>viribus, quibus in perpendiculari ſuſpenderentur lib. 40 1/2, pa<lb></lb>tet à plano ſuſtineri lib.9 1/2; ac proinde grave, quod habet gra<lb></lb>vitatem totam ut 100, in plano BC gravitabit ut 81, & urge<lb></lb>bit ut 19 ſubjectum planum. </s> </p> <p type="main"> <s id="s.000731">Ex his fieri poteſt ſatis quæ<lb></lb><figure id="id.017.01.105.1.jpg" xlink:href="017/01/105/1.jpg"></figure><lb></lb>renti, cur ſuſtinens columnam <lb></lb>OR plus gravitatis percipiat, <lb></lb>quàm qui ſuſtinet columnam <lb></lb>SR: quia nimirum, qui ſuſtinet, <lb></lb>eſt pars plani inclinati, in quo ja<lb></lb>cens concipitur columna: quan<lb></lb>do igitur eſt pars plani habentis <lb></lb>inclinationem LOR, gravitas, <lb></lb>quæ ſuſtinetur à ſubjecto plano, ſe habet ad totam gravitatem <lb></lb>ut Sinus Verſus anguli LOR ad Radium; Quando autem eſt <lb></lb>pars plani habentis inclinationem VSR, gravitatio in ſub<lb></lb>jectum planum ſuſtinens eſt ad totam gravitationem ut Sinus <lb></lb>Verſus anguli VSR ad eumdem Radium. </s> <s id="s.000732">Atqui Sinus Verſus <lb></lb>anguli VSR minoris minor eſt Sinu Verſo anguli LOR ma<lb></lb>joris; igitur minor eſt gravitatio SR, quam OR. </s> <s id="s.000733">Verum qui<lb></lb>dem eſt illud, quod ſi in R aliquo obice prohibeatur, ne de<lb></lb>ſcendat; variatâ inclinatione, quo fit minor ſuſtinentis labor, <lb></lb>eò augetur magis conatus potentiæ in R detinentis columnam, <lb></lb>ne juxta plani inclinationem deſcendat. </s> <s id="s.000734">Hinc ſi duo ſint co<lb></lb>lumnam inclinatam deferentes, qui illam in R ſuſtinet, plus <lb></lb>ſubit laboris, quàm qui in O, aut S: quia præter gravitatio<lb></lb>nem, quam percipit tanquam pars plani inclinati SR aut OR, <lb></lb>debet præterea retinere columnam proclivem ad deſcenſum <lb></lb>propter plani inclinationem; ideò cùm ſcalas, aut montis cli<lb></lb>vum conſcendunt, qui in ſuperiore loco eſt, minimum ſubit <pb pagenum="90" xlink:href="017/01/106.jpg"></pb>laboris. </s> <s id="s.000735">Huc etiam revocari poſſe videtur ratio, ob quam in <lb></lb>elevando pontes illos verſatiles, qui arcium portis opponuntur, <lb></lb>initio major percipiatur difficultas, ſed demùm facillimè ele<lb></lb>ventur. </s> <s id="s.000736">Verùm id ex dicendis inferiùs clariùs conſtabit; neque <lb></lb>enim omnium gravium, quocunque ſe tandem modo habeant, <lb></lb>eadem eſt ratio; cum animum diligenter advertere oporteat, ut <lb></lb>innoteſcat planum inclinatum, in quo ſuam gravitationem <lb></lb>exercent, & habent vires ad deſcendendum. </s> </p> <p type="main"> <s id="s.000737">Non eſt autem per diſſimulantiam prætereunda difficultas, <lb></lb>quæ faceſſere poſſet aliquid negotij, & gravitationis Rationem <lb></lb>conſtitutam convellere videretur. </s> <s id="s.000738">Eſt ſiquidem certum apud <lb></lb>omnes mechanicos, tam ubi de libra, quàm ubi de vecte ſermo <lb></lb>eſt, aliam ſervari Rationem quàm Sinuum Verſorum in mo<lb></lb>mento potentiæ, aut ponderis determinando. </s> <s id="s.000739">Sit vectis, aut <lb></lb><figure id="id.017.01.106.1.jpg" xlink:href="017/01/106/1.jpg"></figure><lb></lb>libræ brachium EC, hypomochlion <lb></lb>ſeu centrum C; attollatur in H, aut <lb></lb>in D; omnes conſentiunt momentum <lb></lb>potentiæ aut ponderis in E ad mo<lb></lb>mentum in H, eſſe ut HC ad IC, <lb></lb>ad momentum verò in D eſſe ut DC <lb></lb>ad FC. </s> <s id="s.000740">Eſt igitur, inquis, gravitatio <lb></lb>in planum DC ad gravitationem in <lb></lb>planum horizontale EC, ut FC ad DC; in planum verò HC, <lb></lb>ut IC ad HC, hoc eſt ut Sinus Rectus anguli inclinationis ad <lb></lb>Radium. </s> </p> <p type="main"> <s id="s.000741">Priùs verò, quàm me ab hac difficultate expediam, oſtendo <lb></lb>non ſatis aptè gravitationem in planum inclinatum deſumi poſ<lb></lb>ſe ex Sinu Recto anguli inclinationis. </s> <s id="s.000742">Quandoquidem vis de<lb></lb>ſcendendi in plano DC ad <expan abbr="totã">totam</expan> corporis liberi <expan abbr="gravitationẽ">gravitationem</expan> eſt <lb></lb>ut DF ad DC, igitur ſi gravitatio in <expan abbr="planũ">planum</expan> DC ad totam <expan abbr="gravi-tationẽ">gravi<lb></lb>tationem</expan> eſt ut FC ad DC, tota virium ſumma eſt DF plus FC, <lb></lb>ac tota vis gravitandi, ubi nullum eſt impedimentum, eſt DC; <lb></lb>igitur DC, & DF plus FC, æquales ſunt, contra 20.lib.1.Eucl. </s> <lb></lb> <s id="s.000743">Neque hic liceat ad æqualitatem potentiarum confugere, ut <lb></lb>ſicut per 47. lib. 1. Eucl. linea DC poteſt quadrata linearum <lb></lb>DF, FC, ita vis totius gravitatis æqualis gravitationibus in <lb></lb>plano inclinato & in planum inclinatum eandem ſervet pro<lb></lb>portionem laterum trianguli DFC, adeò ut totam gravitatem <pb pagenum="91" xlink:href="017/01/107.jpg"></pb>Secans anguli inclinationis exprimat, gravitationem in plano <lb></lb>inclinato Radius, Tangens verò gravitationem in planum in<lb></lb>clinatum. </s> <s id="s.000744">Si enim Quadratum DC æquale eſt quadratis DF, <lb></lb>& FC ſimul ſumptis, non tamen linea DC æqualis eſt aggre<lb></lb>gato linearum DF & FC: neque eadem eſt inter lineas DF <lb></lb>& DC Ratio, quæ inter earum quadrata; ſed eſt ſub duplica<lb></lb>tâ quadratorum: Quare cum gravitatio in plano inclinato DC <lb></lb>ad gravitationem in perpendiculari, non ſit ut quadratum DF <lb></lb>ad quadratum DC; ſed ut linea DF ad lineam DC, fruſtrà ad <lb></lb>quadrata confugimus, quorum nulla hîc habetur ratio. </s> </p> <p type="main"> <s id="s.000745">In eo itaque æquivocatio conſiſtit, quod pondus in D conſti<lb></lb>tutum, & applicatum brachio DC concipitur eſſe in plano in<lb></lb>clinato DC, contra quàm res eſt: in eo ſiquidem plano intel<lb></lb>ligendum eſt, in quo ad motum determinatur; illud autem eſt <lb></lb>planum DG, quod tangit circulum ED; & ſic deinceps, pro <lb></lb>ut diverſa circuli puncta à diverſis planis contingi poſſunt. </s> <lb></lb> <s id="s.000746">Quare in D momentum ad deſcendendum per DG ad totam <lb></lb>gravitationem eſt ut DF ad DG, hoc eſt ut FC ad CD, per <lb></lb>8. lib.6. hoc eſt ut FC ad EC. </s> <s id="s.000747">Eſt igitur brachium libræ ſeu <lb></lb>vectis CD, ſuſtinens pondus ſeu potentiam D, quæ cum ha<lb></lb>beat vires univerſas ut EC, gravitationis autem momenta ha<lb></lb>beat ſolùm ut FC, impeditur à ſuſtinente ut FE; eſt autem <lb></lb>EF Sinus Verſus anguli FCD, hoc eſt anguli inclinationis <lb></lb>FDG. </s> <s id="s.000748">Quare gravitatio ponderis contrà ſubjectum corpus, <lb></lb>quod impedit motum perpendicularem, ad totam gravitatio<lb></lb>nem eſt, ut Sinus Verſus anguli inclinationis plani, per quod <lb></lb>fieri poteſt motus, ad Radium. </s> </p> <p type="main"> <s id="s.000749">Hinc vides valdè diſparem eſſe rationem gravitationis in <lb></lb>ſuſtinendo corpore inclinato, ſi illud liberè moveri poſſit, ac ſi <lb></lb>circa centrum perfici debeat motus. </s> <s id="s.000750">Nam ſi DC ſit columna, <lb></lb>aut pons verſatilis, retineaturque in C, jam punctum C vicem <lb></lb>obtinens ſubjecti plani, illiuſque munere fungens, ſuſtinet <lb></lb>ponderis partem EF, reliqua FC, quæ eſt menſura momenti <lb></lb>ad deſcendendum, debet ſuſtineri à potentia motum impe<lb></lb>diente per DG. </s> <s id="s.000751">Sin autem per DC planum columna moveri <lb></lb>poſſit rectâ & deſcendere, vis deſcendendi ad totam gravitatio<lb></lb>nem eſt ut DF ad DC, gravitatio autem contra ſuſtinentem <lb></lb>eſt ad totam gravitationem ut Sinus Verſus anguli inclinationis <pb pagenum="92" xlink:href="017/01/108.jpg"></pb>FDC ad Radium; qui enim ſuſtinet grave, dum deſcendit in<lb></lb>clinatum, habet rationem plani inclinati. </s> <s id="s.000752">Neque id mirum vi<lb></lb>deri debet, quandoquidem plurimum refert, an per planum <lb></lb>DG an verò per DC ſit determinatio ad motum, & quâ ra<lb></lb>tione ſuſtinens opponatur virtuti motivæ: quare cùm diversâ <lb></lb>ratione opponatur motui circa centrum C, ac motui per pla<lb></lb>num DC, etiam diſpar erit in ſuſtinendo difficultas. </s> </p> <p type="main"> <s id="s.000753">Ex his, quæ tùm hoc, tùm ſuperiori capite diſputata ſunt, <lb></lb>habes quid funambulis reſpondeas volatum mentiri meditanti<lb></lb>bus, cum pectore inſiſtentes intento funi, diductis cruribus & <lb></lb>extenſis brachiis, corpus æqualibus momentis librant, séque <lb></lb>ex editâ turri in depreſſiorem locum præcipites dant; ſi fortè, <lb></lb>ut noverint, quàm ſolidus eſſe debeat ac validus funis, quo iis <lb></lb>utendum eſt, quærant, quantis momentis corpus urgeat ſub<lb></lb><figure id="id.017.01.108.1.jpg" xlink:href="017/01/108/1.jpg"></figure><lb></lb>jectum funem. </s> <s id="s.000754">Datâ enim turris altitudi<lb></lb>ne BD, & depreſſioris loci, in quem de<lb></lb>ſcendendum eſt, diſtantiâ DC, collectíſ<lb></lb>que in ſummam harum quadratis, Radix <lb></lb>ſummæ dabit BC funis longitudinem; ex <lb></lb>quâ ſi auferatur BX turris altitudini BD <lb></lb>æqualis, erit BC diviſa in X juxtà Ratio<lb></lb>nem momentorum, quæ corporis gravitas <lb></lb>exercet in plano inclinato, & in planum <lb></lb>inclinatum. </s> <s id="s.000755">Sic poſitâ BD ped. 150, & DC ped. 200, BC eſt <lb></lb>ped. 250: ex quâ ſi auferatur BD, erit BX 150, & XC 100. <lb></lb>Statue autem totius gravitatis corporis funambuli momenta <lb></lb>220; hæc dividantur in duas partes, quarum major ſit ſeſqui<lb></lb>altera minoris, ſicut BX inventa eſt ipſius XC ſeſquialtera, & <lb></lb>erunt momenta quidem ad deſcendendum in plano inclinato <lb></lb>132, momenta verò gravitationis in planum inclinatum, hoc <lb></lb>eſt in ſubjectum funem, 88. Hæc tamen intelligenda ſunt eâ <lb></lb>factâ hypotheſi, quòd funis rectâ intentus permaneret: cæte<lb></lb>rùm cum & ſuopte pondere, & ſub impoſiti corporis mole ſub<lb></lb>ſidat, atque inflectatur, præſertim circà medium, ſatis apparet <lb></lb>adhuc majorem ſubjecti plani inclinationem æſtimandam eſſe, <lb></lb>quàm quæ ex altitudine DB & diſtantiâ DC inferatur, quin <lb></lb>& illam pro diversâ ab extremitatibus diſtantiâ ſubinde muta<lb></lb>ri, ac proinde validiori fune opus eſſe. <pb pagenum="93" xlink:href="017/01/109.jpg"></pb></s> </p> <p type="main"> <s id="s.000756"><emph type="center"></emph>CAPUT XV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000757"><emph type="center"></emph><emph type="italics"></emph>Inquiruntur Rationes gravitationis corporum <lb></lb>ſuspenſorum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000758">COnſideratâ corporum gravitatione tùm in plano inclinato, <lb></lb>tùm in planum inclinatum, conſequens eſt, ut ad eorum<lb></lb>dem gravitationem, ſi ex fune ſuſpendantur, gradum facia<lb></lb>mus; hæc enim illi valdè affinis eſt ſpeculatio: id quod facilè <lb></lb>intelligat, quiſquis animum advertere voluerit, remque totam <lb></lb>penitiùs introſpicere. </s> <s id="s.000759">Ex his ſi quidem, quæ hactenus diſputa<lb></lb>ta ſunt, lux, opinor, non modica ad hanc, quam examinan<lb></lb>dam ſuſcipimus quæſtionem, derivabitur. </s> </p> <p type="main"> <s id="s.000760">Pendeat ex clavo C ad perpen<lb></lb><figure id="id.017.01.109.1.jpg" xlink:href="017/01/109/1.jpg"></figure><lb></lb>diculum globus ferreus A, quem <lb></lb>ſuppoſitum planum horizontale <lb></lb>BD ita exactè contingat, ut nihil <lb></lb>de funiculi CA intentione remit<lb></lb>tatur. </s> <s id="s.000761">Satis apparet ſubjecto pla<lb></lb>no BD non incumbere globum A, <lb></lb>ſed omnia ſuæ gravitationis, qua <lb></lb>deorſum nititur, momenta exer<lb></lb>cere contrà clavum C, ex quo ſuſpenſus ad perpendiculum <lb></lb>pendet. </s> <s id="s.000762">Quod ſi aut clavus C, nemine funem retinente, revel<lb></lb>leretur, aut funis CA præcideretur, jam tota vis deſcendendi, <lb></lb>quæ corpori A ineſt, urgeret ſubjectum planum BD; nec ta<lb></lb>men in motum erumperet globus, quia planum BD; pari uſque<lb></lb>quaque ad perpendiculum inclinatione libratur, atque adeò <lb></lb>motui prorſus obſiſtit. </s> </p> <p type="main"> <s id="s.000763">Jam verò ſi globum A pariter ex perpendiculo CA penden<lb></lb>tem contingat planum aliud non quidem horizontale, ſed in<lb></lb>clinatum EF, manifeſtum eſt totam pariter gravitationem <lb></lb>exerceri contra clavum C retinentem, planumque contingens <pb pagenum="94" xlink:href="017/01/110.jpg"></pb>omninò non urgeri, niſi præciſo funiculo ſibi relinquatur glo<lb></lb>bus, ut in inclinato plano EF ad deſcenſum pronus contra ſub<lb></lb>jectum planum nitatur, à quo cogitur, ut in motu à recto, quod <lb></lb>ad univerſi centrum eſt, itinere deflectat. </s> </p> <p type="main"> <s id="s.000764">Quod ſi planum inclinatum EF ita ſuſpenſo globo A ſubji<lb></lb>ciatur, ut recta linea centrum gravitatis A, & punctum ſuſ<lb></lb>penſionis H conjungens parallela ſit lineæ EF, quam in plano <lb></lb>inclinato deſcendens globus percurreret; momenta quidem <lb></lb>gravitationis, quæ in eo plano obtineret globus ad deſcenden<lb></lb>dum, exercebit adversùs clavum retinentem in H, ſubjectum <lb></lb>verò planum EF perinde urgebitur, atque ſi nullo retinente li<lb></lb>bera eſſet globo deſcendendi facultas: vis enim, quâ prohibe<lb></lb>tur globus, ne moveatur ſecundùm rectam lineam, ut conſtat, <lb></lb>opponitur deſcenſui in plano inclinato; ejus autem directio <lb></lb>AH non opponitur nitenti in planum, cui parallela eſt. </s> </p> <p type="main"> <s id="s.000765">Contra verò ſi globus in plano inclinato conſtitutus retinea<lb></lb>tur ſecundùm rectam lineam, quæ ad perpendiculum cadit in <lb></lb>ſubjectum planum EF, nimirum ſecundùm lineam LO, im<lb></lb>peditur quidem, ne contra planum nitatur; ſed vis iſta ſic reti<lb></lb>nens nullâ ratione adverſatur motui in plano inclinato, quin <lb></lb>iiſdem gravitatis momentis deſcendat globus in eo plano; ſi <lb></lb>quidem retinentis directio LO maneat ſemper adversùs illud <lb></lb>planum perpendicularis. </s> <s id="s.000766">Nam ſi potentia retinens ſecundùm <lb></lb>eam directionem agat, ut neque congruat perpendiculari LO, <lb></lb>neque parallelæ HA, obſiſtet gravitationi corporis ſivè in pla<lb></lb>no inclinato, ſivè in planum inclinatum pro ratione anguli, <lb></lb>quem retinentis directio inter perpendicularem LO, & paral<lb></lb>lelam HA interjecta, conſtituet cum plano inclinato. </s> <s id="s.000767">Quæ <lb></lb>enim inter LO & CA fuerit, elidet omnem corporis conatum <lb></lb>adversùs planum, à quo illud avellit; non autem omnem eum, <lb></lb>qui in plano inclinato deorſum rapit. </s> <s id="s.000768">Quæ verò fuerit inter <lb></lb>CA & HA, tollet quidem deſcenſum in plano EF inclinato; <lb></lb>ſed non omninò prohibebit, quin ſubjectum planum, cui aliqua<lb></lb>tenus nititur, urgeat. </s> <s id="s.000769">Id quod facilè intelligas, ſi plana ſubjecta <lb></lb>BD horizontale, & EF inclinatum ex maximè flexili mate<lb></lb>ria, puta, papyro, concipias; in quâlibet enim ſuſpenſione <lb></lb>inter C, & L, planum BD horizontale flectetur ex pondere, <lb></lb>non autem inclinatum EF: contrà verò in omni ſuſpenſione <pb pagenum="95" xlink:href="017/01/111.jpg"></pb>inter C & H, planum inclinatum EF flectetur; at non item ho<lb></lb>rizontale BD, quia nimirum inclinatum EF prohibet, ne recta <lb></lb>HA ad perpendiculum accedens verticalis fiat. </s> </p> <p type="main"> <s id="s.000770">Unum hic præterea conſiderandum venit, quod ſuperiori <lb></lb>capite ſubindicatum fuit; ſi videlicet non ex flexili fune deor<lb></lb>ſum pendeat globus, ſed rigido bacillo circà axem inferiùs po<lb></lb>ſitum verſatili adnectatur ſupe<lb></lb><figure id="id.017.01.111.1.jpg" xlink:href="017/01/111/1.jpg"></figure><lb></lb>riùs. </s> <s id="s.000771">Sic rectus bacillus AB, cujus <lb></lb>extremitas altera adnexum ha<lb></lb>beat globum B, altera ſit circà <lb></lb>axem A verſatilis. </s> <s id="s.000772">Satis aperta <lb></lb>conjectura eſt bacillum AB vi<lb></lb>cem ſubire plam, cui innitatur <lb></lb>globus in B, qui proinde prohi<lb></lb>betur, tùm ne ad perpendiculum <lb></lb>cadat per BD, tùm ne per BA <lb></lb>delabatur: linea igitur plani, per quod moliri motum poterit <lb></lb>globus B, nulla alia congruentiùs aſſignari queat præter BC, <lb></lb>quæ cum bacillo BA rectum angulum conſtituit. </s> <s id="s.000773">Perindè igi<lb></lb>tur in motum incitabitur, atque ſi in plano eſſet, cujus inclina<lb></lb>tio angulum efficeret æqualem angulo elevationis bacilli ſuprà <lb></lb>planum horizontale GA. </s> <s id="s.000774">Cum enim recta BD producta ca<lb></lb>dens in planum horizontale, angulum BSA Rectum efficiat, <lb></lb>reliqui duo ſimul SAB, ABS, Recto ABC æquales ſunt; & <lb></lb>communi ABS dempto, ſupereſt SAB elevationis angulus <lb></lb>æqualis angulo SBC inclinationis plani. </s> <s id="s.000775">Quare ductâ Tan<lb></lb>gente DE, erit BE Secans anguli inclinationis, BD verò Ra<lb></lb>dius: ac proptereà ad deſcendendum in hujuſmodi plano BC <lb></lb>momenta, ad totam gravitatem in perpendiculo BD, erunt ut <lb></lb>Radius BD ad Secantem BE, juxta ea, quæ cap. 13. hujus lib. <lb></lb>demonſtravimus. </s> </p> <p type="main"> <s id="s.000776">Quia tamen in motu globus ex bacilli converſione circà <lb></lb>axem A non poteſt percurrere rectam BC, ſed ita retinetur à <lb></lb>bacillo, cui adnectitur, ut deſcendat in F, jam in alio plano <lb></lb>minorem inclinationem habente conſtitutus intelligitur, nimi<lb></lb>rùm in plano FG, quod cum perpendiculo FL efficit angulum <lb></lb>inclinationis GFL æqualem angulo LAF elevationis: id quod <lb></lb>eâdem planè methodo, ac ſuperiùs factum eſt, demonſtratur. <pb pagenum="96" xlink:href="017/01/112.jpg"></pb>Ex quo fit, quemadmodum in hujuſmodi converſione globus <lb></lb>in alio atque alio plano inclinato conſtituitur, ita alia atque alia <lb></lb>obtinere gravitatis momenta: in B ſiquidem gravitat ut BD ad <lb></lb>BE, in F verò ut HF ad FI. </s> <s id="s.000777">Cum igitur Radius utrobique ex <lb></lb>hypotheſi æqualis ſit, videlicet DB, & HF, major autem ſit <lb></lb>BE Secans majoris anguli DBE, quàm FI Secans minoris an<lb></lb>guli HFI, conſtat ex 8. lib. 5. majorem Rationem eſſe HF ad <lb></lb>FI minorem, quàm DB ad BE majorem, atque adeò globum <lb></lb>magis in F quàm in B gravitare, ut deorſum moveatur, atque <lb></lb>adeò minùs etiam conniti contrà planum, in quo eſt, videlicet <lb></lb>adversùs bacillum FA, magis verò adversùs bacillum BA. </s> </p> <p type="main"> <s id="s.000778">Ex his attentè perpenſis facilis eſt tranſitus ad ſuſpenſorum <lb></lb>corporum gravitationem inveſtigandam. </s> <s id="s.000779">Sit enim jam non in<lb></lb><figure id="id.017.01.112.1.jpg" xlink:href="017/01/112/1.jpg"></figure><lb></lb>feriùs, ſed ſuperiùs poſitus <lb></lb>Axis A, circa quem verſa<lb></lb>tilis eſt funiculus AB, cui <lb></lb>globus B adnectitur. </s> <s id="s.000780">Con<lb></lb>ſtat ſanè non ad perpendi<lb></lb>culum BD cadere poſſe <lb></lb>globum B; ſed à recto deor<lb></lb>ſum tramite deflectere, fu<lb></lb>niculo ſcilicet AB eum re<lb></lb>tinente, quemadmodum ri<lb></lb>gidus bacillus OB eum ali<lb></lb>quatenùs ſuſtineret. </s> <s id="s.000781">Quia autem bacillo OB ſuſtinente, vis <lb></lb>deſcendendi ea eſſet, quæ per planum inclinatum BC, eadem <lb></lb>pariter eſt funiculo retinente; videlicet per planum BC, in <lb></lb>quod recta AB ad rectos angulos incidit. </s> <s id="s.000782">Momenta igitur gra<lb></lb>vitatis in eo plano inclinato, ad gravitatis momenta ſi corpus <lb></lb>liberè deſcenderet, in eâ ſunt Ratione, quæ eſt DB ad BE; <lb></lb>hoc eſt DO ad OB per 8. lib.6. hoc eſt KB ad BA per 4.lib.6. <lb></lb>Haud diſpari methodo ratiocinantes oſtendemus globi in F <lb></lb>conſtituti momenta ad gravitandum eſſe perinde, atque ſi eſ<lb></lb>ſet in plano inclinato FI, in quod ad rectos angulos cadit fu<lb></lb>niculus AF; ac proinde gravitatio in F, ſi deſcendendi vis <lb></lb>præcisè ſpectetur, ad gravitationem globi liberi, eſt ut HF <lb></lb>ad FI, hoc eſt, ut GF ad FA. </s> </p> <p type="main"> <s id="s.000783">Ex quo apertiùs liquet, quàm ut in eo explicando diutiùs <pb pagenum="97" xlink:href="017/01/113.jpg"></pb>immorari oporteat, alia ſubinde atque alia eſſe momenta gra<lb></lb>vitatis corporis ſuſpenſi, pro ut major aut minor eſt angulus <lb></lb>declinationis à perpendiculo AG, haud aliter quàm ſi in aliis <lb></lb>atque aliis planis inclinatis conſtitueretur; quo enim minor eſt <lb></lb>declinationis angulus GAF, eò major eſt angulus inclinatio<lb></lb>nis plani, quippe qui eſt illius complementum. </s> <s id="s.000784">Conſtat ſi qui<lb></lb>dem angulos GAF, GFA ſimul, eſſe æquales tùm Recto <lb></lb>AFI, tùm Recto GFH; ac proinde dempto communi GFI, <lb></lb>remanet HFI angulus inclinationis plani æqualis angulo <lb></lb>GFA, qui eſt complementum anguli declinationis GAF. </s> <lb></lb> <s id="s.000785">Quare quò declinationis angulus major eſt, eò minus eſt <lb></lb>complementum, ac propterea eſt minor angulus inclinationis <lb></lb>plani: in plano autem minùs inclinato majora ſunt gravitatis <lb></lb>momenta. </s> <s id="s.000786">Quò igitur corpus ſuſpenſum magis à perpendiculo <lb></lb>removetur, eò majora percipiuntur gravitatis momenta, ma<lb></lb>jorque vis requiritur in eo, qui motum prohibere voluerit, ut <lb></lb>& ipſa experientia unicuique facilè demonſtrat, & ratio evin<lb></lb>cit; cum enim AB & AF æquales ſint, major eſt Ratio KB <lb></lb>ad BA, quàm GF ad FA per 8. lib. 5. eſt nimirum KB major, <lb></lb>& GF minor. </s> </p> <p type="main"> <s id="s.000787">Quoniam verò quò major eſt gravitatio in plano inclinato, <lb></lb>minor eſt in planum inclinatum; hoc ipſo, quod facto declina<lb></lb>tionis angulo GAB majore, quàm GAF, major eſt ad deſcen<lb></lb>dendum propenſio, minor eſt conatus adversùs axem A reti<lb></lb>nentem. </s> <s id="s.000788">Id quod manifeſto etiam experimento deprehen<lb></lb>des, ſi obſervaveris minùs intentum eſſe funiculum AB, <lb></lb>quàm AF. </s> </p> <p type="main"> <s id="s.000789">Hinc & illud ſatis dilucidè apparet, quod longitudinis <lb></lb>funiculi non exigua ratio habenda eſt; ex eâ ſcilicet pen<lb></lb>det, quod in plano magis aut minùs inclinato conſtitutum <lb></lb>cenſeatur corpus grave ſuſpenſum. </s> <s id="s.000790">Si enim globus F ex fu<lb></lb>niculo AF pendeat, declinationis angulus eſt GAF: at <lb></lb>verò ſi funiculus, quo ſuſpenditur, ſit MF, angulum de<lb></lb>clinationis facit GMF, qui cum externus ſit, major eſt <lb></lb>interno MAF per 16. lib. 1. ac propterea minor eſt incli<lb></lb>natio plani FN facientis cum rectâ MF angulum Rectum, <lb></lb>quàm ſit inclinatio plani FI, cui perpendicularis eſt recta <lb></lb>AF. </s> <s id="s.000791">Plus igitur momenti ad gravitandum habet glo-<pb pagenum="98" xlink:href="017/01/114.jpg"></pb>bus F, ſi ex breviore funiculo MF pendeat, quàm ſi ex <lb></lb>longiore AF. </s> </p> <p type="main"> <s id="s.000792">Quæ cum ita ſint, haud ſanè incongrua ſe nobis offert me<lb></lb>thodus pondus ex depreſſiore in altiorem locum transferendi; <lb></lb>ſi videlicet id curemus, ut ex ſatis valido & longiore fune ſuſ<lb></lb>pendatur; ſublato etenim partium attritu, qui fieret, ſi per pla<lb></lb>num raptaretur pondus, minore virium jacturâ trahi poteſt. </s> </p> <p type="main"> <s id="s.000793">Sit corpus grave ubi A, quod at<lb></lb><figure id="id.017.01.114.1.jpg" xlink:href="017/01/114/1.jpg"></figure><lb></lb>tollere oporteat, & in ſuperiorem <lb></lb>locum RS transferre. </s> <s id="s.000794">Si ex C brevio<lb></lb>ri fune ſuſpendatur, trahere illud po<lb></lb>terit uſque in R, quicumque facto de<lb></lb>clinationis angulo ACR poteſt illud <lb></lb>cum aliquo virium exceſſu retinere, <lb></lb>& obſiſtere gravitatis momentis, quæ <lb></lb>obtinet in R. </s> <s id="s.000795">At ſi ex longiore fune <lb></lb>DA pendeat, idem corpus A trahi <lb></lb>poterit, & retineri in S, ne deorſum labatur, & quidem mino<lb></lb>re conatu; facto enim declinationis angulo ADS minore, <lb></lb>quàm ACR, in S pariter minùs gravitat quàm in R. </s> <s id="s.000796">Angu<lb></lb>lum autem ADS minorem eſſe angulo ACR conſtat, ſi rectæ <lb></lb>AR, AS ducantur: nam CA, CR æqualia ſunt latera ex hy<lb></lb>potheſi, item DA, DS æqualia; eſt ſcilicet idem funiculus, <lb></lb>qui primum perpendicularis cadit, deinde à perpendiculo re<lb></lb>movetur: in Triangulo Iſoſcele CAR anguli ad baſim AR <lb></lb>æquales ſunt per 5. lib. 1. item in triangulo Iſoſcele DAS an<lb></lb>guli ad baſim AS æquales inter ſe ſunt. </s> <s id="s.000797">Porrò angulus DAS <lb></lb>major eſt angulo CAR; ergo & reliquus DSA major reliquo <lb></lb>CRA. </s> <s id="s.000798">Cum itaque tres anguli utriuſque trianguli ſint æquales <lb></lb>duobus Rectis per 32. lib. 1. ſi ex ſummâ duorum Rectorum au<lb></lb>ferantur duo majores anguli DAS, DSA, relinquitur ADS <lb></lb>minor, quàm ſi ex eâdem duorum Rectorum ſummâ auferan<lb></lb>tur duo minores CAR, CRA, hoc eſt minor quàm ACR. </s> <lb></lb> <s id="s.000799">Ut autem clariùs innoteſcat, quænam ſit gravitationum Ratio <lb></lb>pro funiculi longitudine, ſit corpus grave in R: & primùm <lb></lb>quidem ex C pendeat funiculo breviore CR, deinde ex D lon<lb></lb>giore funiculo DR: quiſquis retineat corpus in R conſtitu<lb></lb>tum, atque deſcenſu prohibeat, faciliùs retinebit, cum ex D, <pb pagenum="99" xlink:href="017/01/115.jpg"></pb>quàm cùm ex C, pendebit; quia declinationis angulus XCR <lb></lb>major eſt angulo XDR per 16. lib. 1. Verùm qua Ratione, in<lb></lb>quis, vires, quas in utroque caſu retinens exerit, diſcriminan<lb></lb>tur? </s> <s id="s.000800">utique ſecundùm Reciprocam funiculorum Rationem co<lb></lb>natur obſiſtens corporis propenſioni ad deſcenſum; quæ enim <lb></lb>Ratio gravitationum corporis, ea eſt virium gravitationibus <lb></lb>repugnantium: comparatà autem corporis in R conſtituti gra<lb></lb>vitatione, ſi ex C pendeat, cum ejuſdem ibidem poſiti gravita<lb></lb>tione, ſi pendeat ex D, eſt reciprocè ut DR ad CR; igitur <lb></lb>& vires retinentis corpus ex C pendens ſunt ut DR, retinen<lb></lb>tis verò idem corpus ex D pendens ſunt ut CR. </s> <s id="s.000801">Id quod hinc <lb></lb>conficitur, quia corpus in ſuſpenſione, poſitionem habens CR, <lb></lb>gravitat ut XR ad RC, poſitionem verò habens DR gravitat <lb></lb>ut XR ad RD; duæ autem Rationes XR ad RC, & XR ad <lb></lb>RD ſunt reciprocè ut RD ad RC. </s> <s id="s.000802">Quotieſcumque enim duæ <lb></lb>ſunt Rationes, quarum idem eſt Antecedens terminus, & di<lb></lb>verſus Conſequens, eæ ſunt reciprocè ut conſequentes. </s> </p> <p type="main"> <s id="s.000803">Quòd ſi quis Rationes inter ſe comparare non aſſuetus de <lb></lb>hoc ambigeret, an Rationes eumdem vel æqualem anteceden<lb></lb>tem terminum habentes ſint reciprocè ut Conſequentes, facilè <lb></lb>intelliget, ſi animadvertat Rationes eumdem Conſequentem <lb></lb>terminum habentes eſſe inter ſe directè, ut antecedentes. </s> <lb></lb> <s id="s.000804">Quemcumque enim interrogaveris, quæ ſit Ratio 2/7 ad 6/7 illicò <lb></lb>reſpondebit eſſe ſubtriplam, ſecunda ſcilicet ter continet pri<lb></lb>mam, ut conſtat ſi ter poſitam Rationem 2/7 in ſummam colligas; <lb></lb>neque enim <expan abbr="idẽ">idem</expan> eſt Rationem Rationis eſſe ſubtriplam, ac ſub<lb></lb>triplicatam; Ratio ſiquidem 2/7 eſt ſubtriplicata Rationis (8/343). Si <lb></lb>igitur pariter quæras, quænam ſit Ratio 7/2 ad 7/6 rectè reſponde<lb></lb>bit eam eſſe triplam, hoc eſt reciprocè ut 6 ad 2: id quod ma<lb></lb>nifeſtè apparebit, ſi illas ad denominationem eandem, hoc eſt <lb></lb>ad eumdem Conſequentem terminum reduxeris, ſunt nimirum <lb></lb>ut (42/12) ad (14/12), hoc eſt ut 6 ad 2. </s> </p> <p type="main"> <s id="s.000805">Ex quibus obiter patet methodus exponendi per lineas pro<lb></lb>portionem duarum Rationum etiam numeris non explicabi<lb></lb>lium; ſi videlicet fiat ut Antecedens ſecundæ Rationis ad ſuum <lb></lb>Conſequentem, ita Antecedens datus primæ Rationis ad alium <lb></lb>novum Conſequentem; erit enim prima Ratio data ad ſecun-<pb pagenum="100" xlink:href="017/01/116.jpg"></pb>dam rationem datam reciprocè ut novus Conſequens terminus <lb></lb>ad datum Conſequentem primæ Rationis: aut etiam ſi fiat ut <lb></lb>Conſequens ſecundæ Rationis ad ſuum Antecedentem, ita con<lb></lb>ſequens primæ Rationis ad alium novum Antecedentem; erit <lb></lb>enim prima ratio data ad ſecundam Rationem datam, directè <lb></lb>ut datus Antecedens primæ Rationis ad novum Antecedentem. </s> </p> <p type="main"> <s id="s.000806">Conſideratâ hactenus unicâ & ſimplici corporis gravis ſuſ<lb></lb>penſione, gradum facere oportet ad gravitationis rationes in<lb></lb>veſtigandas, ſi duplex fuerit ſuſpenſio. </s> <s id="s.000807">Sit enim globus A tùm <lb></lb><figure id="id.017.01.116.1.jpg" xlink:href="017/01/116/1.jpg"></figure><lb></lb>ex B, tùm ex C ſuſpenſus fu<lb></lb>niculis BA & CA. </s> <s id="s.000808">Haud du<lb></lb>bium quin tota corporis gravi<lb></lb>tas ex B & C pendeat; ſed quâ <lb></lb>Ratione ſingulæ vires eidem <lb></lb>gravitati obſiſtant, de hoc po<lb></lb>teſt ambigi. </s> <s id="s.000809">Verùm niſi mea <lb></lb>mihi nimiùm blanditur opi<lb></lb>nio, ex dictis facilis videtur <lb></lb>explicatio. </s> <s id="s.000810">Corpus ſiquidem <lb></lb>ex duplici fune ſuſpenſum ita <lb></lb>conſtitutum eſt, ut alterutro <lb></lb>fune præciſo ex reliquo pen<lb></lb>deat, & deſcendens moveatur <lb></lb>circà punctum, cui alligatur <lb></lb>funis. </s> <s id="s.000811">Quare unuſquiſque obſiſtit momentis, quibus ex altero <lb></lb>gravitat; nimirum funiculus CA retinens globum, ne deſcen<lb></lb>dat, repugnat momentis gravitatis, quibus globus A ſe ipſe <lb></lb>deorſum urget circa punctum B ex fune BA: Contrà verò fu<lb></lb>niculus BA eundem globum retinet, ne circa punctum C ex <lb></lb>funiculo CA moveatur deſcendens, atque adcò obſiſtit, mo<lb></lb>mentis gravitatis ad deſcendendum circà idem punctum C. </s> <s id="s.000812">At<lb></lb>qui momenta deſcendendi ex fune BA ad gravitatem in per<lb></lb>pendiculo ſunt ut DA ad AB, & ex fune CA ſunt ut EA ad <lb></lb>AC, ex his, quæ ſuperiùs diſputata ſunt. </s> <s id="s.000813">Sunt igitur duæ Ra<lb></lb>tiones DA ad AB, & EA ad AC. </s> </p> <p type="main"> <s id="s.000814">Quare fiat angulus DAF æqualis angulo EAC, & eſt trian<lb></lb>gulum DAF ob angulorum æqualitatem ſimile triangulo <lb></lb>EAC; ac propterea per 4. lib. 6. ut EA ad AC, ita DA ad <pb pagenum="101" xlink:href="017/01/117.jpg"></pb>AF. </s> <s id="s.000815">Ergo vis deſcendendi ex CA eſt ut DA ad AF, & vis <lb></lb>deſcendendi ex BA eſt ut DA ad AB: igitur duæ hæ Ratio<lb></lb>nes ſunt reciprocè ut BA ad AF; atque adeò B quidem reti<lb></lb>nens, ne deſcendat ex CA, exerit vires ut BA; C verò reti<lb></lb>nens, ne deſcendat ex BA, adhibet conatum ut FA; & quæ <lb></lb>componitur ex BA, AF, totum gravitatis momentum, quod <lb></lb>corpori ſuſpenſo ineſt, repræſentat. </s> <s id="s.000816">Momentum, inquam, <lb></lb>gravitatis potiùs, quàm gravitatem totam; totius ſi quidem <lb></lb>gravitatis nomine vires ipſas deſcendendi intelligimus, quas <lb></lb>corpus grave obtinet ſibi prorsùs relictum ſecluſo quolibet im<lb></lb>pedimento, à quo certam deſcendendi regulam accipiat: Mo<lb></lb>menti autem vocabulo ipſas deſcendendi vires ſignificamus <lb></lb>non per ſe & ſolitariè acceptas; ſed quatenus ex corporis poſi<lb></lb>tione, cæterorumque quæ circumſtant, ad majorem aut mino<lb></lb>rem motùs velocitatem determinatur. </s> <s id="s.000817">Conſiderato itaque niſu <lb></lb>corporis A ad deſcendendum & cùm perpendicularis eſt funi<lb></lb>culus BD, & cum declinat BA, Ratio momentorum eſt ut <lb></lb>BA ad AD. </s> <s id="s.000818">Similiter momentum ex perpendiculari CE ad <lb></lb>momentum ex declinante CA eſt ut CA ad AE, hoc eſt ut <lb></lb>FA ad AD: eſt igitur corporis A ex duplici funiculo BA, CA <lb></lb>pendentis totum gravitandi momentum, quod ex lineis BA, <lb></lb>AF componitur. </s> </p> <p type="main"> <s id="s.000819">Hic autem hæſitantem videre mihi videor non neminem ex <lb></lb>iis, quæ dicebantur, colligentem corpus A primùm ex decli<lb></lb>nante BA æquè ac ex perpendiculari BD gravitare; deinde <lb></lb>plus ad deſcendendum momenti obtinere, ſi ex duobus funi<lb></lb>culis, quàm ſi ex unico pendeat. </s> <s id="s.000820">Si enim angulus declinatio<lb></lb>nis DBA ſit gr. 22. 12′; eſt DA ſinus dati anguli ad radium <lb></lb>BA ut 37784 ad 100000: & ſi angulus declinationis ECA <lb></lb>ſit gr. 54. 35, eſt EA ſinus dati anguli ad Radium CA ut <lb></lb>81496 ad 100000. At ex conſtructione triangulum DAF ſi<lb></lb>mile eſt triangulo EAC; igitur DA ad AF eſt ut 81496 ad <lb></lb>100000. Eſt autem DA in particulis Radij BA partium 37784; <lb></lb>igitur ſi fiat ut 81496 ad 100000, ita 37784, ad aliud, erit AF <lb></lb>earumdem particularum 46363, quarum BA eſt 100000. Qua<lb></lb>re compoſita BA, AF momenta ſunt 146363, cum tamen <lb></lb>momentum in perpendiculari AD ſit tantum 100000. Cum <lb></lb>verò dictum ſit B clavum reſiſtere ponderi A ut BA, C autem <pb pagenum="102" xlink:href="017/01/118.jpg"></pb>ut FA, manifeſtum eſt B clavum retinere ut 100000 quando <lb></lb>declinat BA à perpendiculo: Atqui etiam in perpendiculo BD <lb></lb>retinet ut 100000, igitur idem eſt ponderis tùm ex BD, tùm <lb></lb>ex BA momentum; id quod eſt abſurdum. </s> </p> <p type="main"> <s id="s.000821">Sed & illud prætereà ex dictis conſequi videtur, quod ejuſ<lb></lb>dem corporis majus ſit momentum, ſi ex duobus funiculis, quàm <lb></lb>ſi ex unico pendeat. </s> <s id="s.000822">Fiat enim angulus DBH æqualis angulo <lb></lb>declinationis ECA, & aſſumptâ BH æquali ipſi BA, ducatur <lb></lb>ad BD perpendicularis HI: erit utique triangulum BHI ſimi<lb></lb>le triangulo CAE, ac propterea ut EA ad AC, ita IH ad <lb></lb>HB, hoc eſt ad AB. </s> <s id="s.000823">Sunt igitur duæ Rationes eundem Con<lb></lb>ſequentem terminum habentes, atque adeò inter ſe in ratione <lb></lb>Antecedentium, ac proinde cùm vis deſcendendi ex BA ſit ut <lb></lb>DA ad AB, & vis deſcendendi ex CA ſit ut IH ad AB, vires <lb></lb>deſcendendi invicem comparatæ ſunt ut DA ad IH, totum<lb></lb>que momentum componitur ex DA 37784, & IH 81496. <lb></lb>Quare momentum quod in perpendiculari, ſi unico funiculo <lb></lb>penderet ex BD, eſſet 100000, pendente corpore A ex duo<lb></lb>bus funiculis BA, CA, fit majus, ſcilicet 119280. ut quid igi<lb></lb>tur ex pluribus funiculis illud ſuſpendere oportuit? </s> </p> <p type="main"> <s id="s.000824">Quibus difficultatibus ut fiat ſatis, & id, quod inquirimus, <lb></lb>enucleatiùs explicetur, illud obſervo, quod funiculus BA ſi <lb></lb>præcisè ſpectetur, quatenus ex eo corpus grave pendet, retinet <lb></lb>globum A, ne rectâ deſcendat per lineam ipſi BD parallelam, <lb></lb>ſed cogit illum deflectere in motu: quare adversùs clavum B, <lb></lb>globus A exercet ea momenta, quæ exerceret in planum incli<lb></lb>natum, cui BA ad rectos angulos inſiſteret. </s> <s id="s.000825">At ſi globus ex alio <lb></lb>prætereà funiculo CA pendeat, idem funiculus BA reſiſtit <lb></lb>etiam momentis illis, quibus globus A deſcenderet in plano in<lb></lb>clinato, cui CA ad rectos angulos inſiſteret, quæ momenta (ut <lb></lb>ſummum) ſunt ad BA radium ut 81496. Momenta verò qui<lb></lb>bus urgeret planum inclinatum perpendiculare ad BA, ſunt, ex <lb></lb>dictis ſuperiori capite, ut Sinus Verſus anguli inclinationis pla<lb></lb>ni; inclinatio autem plani, ut paulò ſuperiùs hoc eodem capite <lb></lb>demonſtravimus, eſt complementum anguli declinationis <lb></lb>DBA. </s> <s id="s.000826">Quare differentia inter DA 37784 ſinum rectum an<lb></lb>guli declinationis, & radium BA 100000, cum ſit Sinus Ver<lb></lb>ſus anguli inclinationis plani, ſunt momenta 62216 addenda <pb pagenum="103" xlink:href="017/01/119.jpg"></pb>prioribus 81496; adeò ut ſumma ſit 143712 momentorum, qui<lb></lb>bus funiculus BA repugnat, ſi pondus pendeat etiam ex CA; <lb></lb>cum tamen ſi ex ipſo tantùm funiculo BA penderet, & aliquis <lb></lb>eſſet præcisè obluctans viribus ad deſcendendum, idem funicu<lb></lb>lus BA reſiſteret ſolùm momentis 62216. </s> </p> <p type="main"> <s id="s.000827">Eâdem methodo deprehendes funiculum CA, ſi ex eo ſolo <lb></lb>globus pendeat, retinere momenta 18504: at ſi etiam ex BA <lb></lb>globus pendeat, additis momentis 37784, tota momentorum <lb></lb>ſumma eſt 56288. Jam ſummam hanc priori 143712 adde, & <lb></lb>erit tota momentorum ſumma 200000: perinde atque ſi corpo<lb></lb>ris gravitas fuiſſet duplicata. </s> <s id="s.000828">Id quod deprehendes, quoſcum<lb></lb>que demùm declinationis angulos ſtatueris ſivè majores, ſivè <lb></lb>minores; ſemper enim eandem ſummam momentorum om<lb></lb>nium invenies 200000: & funiculus minoris declinationis plus <lb></lb>momentorum ſuſtinebit, tùm quia Sinus Verſus majoris incli<lb></lb>nationis plani major eſt, tum quia Sinus Rectus alterius anguli <lb></lb>declinationis majoris item major eſt. </s> </p> <p type="main"> <s id="s.000829">Hæc tamen ut veritati congruant, ita ſolùm accipienda ſunt, <lb></lb>ut momenta ſingula ex utrâque funiculorum declinatione orta <lb></lb>particulatim ſumantur: pondus ſcilicet ex utroque ſuſpenſum <lb></lb>perinde hactenus conſideratum eſt, ac ſi momenta ipſa deſcen<lb></lb>dendi in diverſas partes abeuntia momentum quoddam ex <lb></lb>utriſque temperatum non conſtituerent; re autem ipsa quod ex <lb></lb>iis componitur momentum, non ex ipſorum momentorum ad<lb></lb>ditione conflatur, ſed ex ipſis temperatur. </s> <s id="s.000830">Si enim mobile ſit <lb></lb>ubi A, impetum verò cum tali <lb></lb><figure id="id.017.01.119.1.jpg" xlink:href="017/01/119/1.jpg"></figure><lb></lb>directione habeat, quâ deferri <lb></lb>poſſit æquabiliter per rectam <lb></lb>AB, alio autem impetu feratur <lb></lb>æquabiliter directum in C, no<lb></lb>tum omnibus eſt motum, qui ex <lb></lb>AB & AC componitur, non fieri ex earum additione, ſed tem<lb></lb>perari in lineam AD, quæ dimetiens eſt parallelogrammi, quod <lb></lb>ex earumdem linearum AB, AC longitudine, ac mutuâ incli<lb></lb>natione formam deſumit. </s> <s id="s.000831">Quâ in re plurimum intereſt, quam <lb></lb>invicem habeant inclinationem directiones motuum in diverſa <lb></lb>abeuntium; quò enim acutiorem angulum conſtituunt, eò lon<lb></lb>giùs provehitur mobile, ut AB, AC in acutum angulum <pb pagenum="104" xlink:href="017/01/120.jpg"></pb>coëuntibus mobile ex A in D venit: quò verò obtuſior fuerit <lb></lb>angulus, eò etiam brevius eſt iter ipſius mobilis, ut contingit, <lb></lb>ſi ex B directum per rectas BA, BD ad obtuſum angulum <lb></lb>conſtitutas moveatur, ſiſtitur enim in C, & brevior eſt diame<lb></lb>ter BC quàm AD, ut ex 24. lib. 1. ſatis manifeſtum eſt geo<lb></lb>metris, & ipſa motuum natura poſtulat; qui nimirum ſibi in<lb></lb>vicem magis adverſantur, magiſque in diverſa abeunt, ſe ma<lb></lb>gis elidunt, id quod fit ex angulo obtuſo DBA; qui verò mi<lb></lb>nùs in diverſa abeunt, id quod fit ex angulo acuto CAB, ſe pa<lb></lb>riter minùs elidunt. </s> </p> <p type="main"> <s id="s.000832">Sint itaque, ut priùs, funiculi BA, CA, ex quibus A pon<lb></lb>dus ſuſpenditur: ducatur ad BA perpendicularis AR, & eſt <lb></lb>planum inclinatum, in quo deſcendendi momentum eſt ut <lb></lb>DA; ſimiliter ad CA perpendicularis AG ducatur referens <lb></lb>planum inclinatum, in quo deſcendendi momentum eſt AE. </s> <lb></lb> <s id="s.000833">Sumatur igitur AR quidem ipſi AD æqualis, AG verò ipſi <lb></lb>AE pariter æqualis, ſi funiculi BA, & CA æquales fuerint; <lb></lb>ſin autem inæquales ſint, fiat angulus DBH æqualis angulo <lb></lb>declinationis ECA, & ſumptâ BH æquali ipſi BA, duca<lb></lb>tur ad BD perpendicularis HI, eritque ut EA ad AC, <lb></lb>ita IH ad HB, hoc eſt ad AB; ac propterea ipſi IH, quæ <lb></lb>refert momentum AE, ſumatur AG æqualis. </s> <s id="s.000834">Ex quo fit cor<lb></lb>pus A ſuſpenſum hâc ratione momenta deſcendendi habe<lb></lb>re in diverſas partes abeuntia AR, AG: perfecto igitur paral<lb></lb>lelogrammo ARNG, ex duobus illis momentis temperatur <lb></lb>momentum AN. </s> </p> <p type="main"> <s id="s.000835">Ipſius autem AN longitudinem inveſtigare non eſt diffici<lb></lb>le; cum enim noti ſupponantur anguli declinationum DBA, <lb></lb>ECA, angulus RAG conflatur ex eorum complementis, <lb></lb>quippe qui æqualis eſt duobus angulis inclinationis planorum <lb></lb>AR, & AG. </s> <s id="s.000836">Porrò ex hypotheſi ſunt angulus DBA gr. 22. <lb></lb>12′, & angulus ECA gr. 54. 35′: jungantur ſimul, & eorum <lb></lb>ſumma gr. 76. 47′ auferatur ex gr. 180, ut reſiduum gr. 103. <lb></lb>13′ ſit angulus RAG, cui æqualis eſt oppoſitus RNG; ac <lb></lb>proinde notus eſt angulus G, qui eſt ſuo oppoſito R æqualis, <lb></lb>uterque ſcilicet gr. 76. 47′ quæ eſt ſumma angulorum decli<lb></lb>nationis. </s> <s id="s.000837">Sunt igitur in triangulo AGN nota latera AG, <lb></lb>GN (eſt enim ex 34. lib. 1. GN oppoſito lateri AR æquale) <pb pagenum="105" xlink:href="017/01/121.jpg"></pb>unâ cum angulo G comprehenſo, & ex Trigonometriâ inno<lb></lb>teſcit tertium latus AN. </s> <s id="s.000838">Quare cum latus AG ſit ex ſupe<lb></lb>riùs conſtitutis 81496, & GN, hoc eſt AR, 37784, fiat ut <lb></lb>laterum AG, GN ſumma 119280 ad eorumdem differen<lb></lb>tiam 43712, ita ſemiſummæ angulorum ad baſim, hoc eſt <lb></lb>gr. 51. 36 1/2 Tangens 126205 ad 46249 Tangentem gr. 24. 49′ 2/5 <lb></lb>differentiæ infra, vel ſupra eandem ſemiſummam. </s> <s id="s.000839">Eſt igitur <lb></lb>angulus GAN gr. 26. 47′ (3/10). In triangulo itaque AGN noti <lb></lb>ſunt duo anguli A, & G, ac latus GN angulo A oppoſi<lb></lb>tum; igitur ut anguli A gr. 26. 47′ (3/10) Sinus 45070 ad anguli G <lb></lb>gr. 76. 47′ Sinum 97351, ita latus GN 37784 ad latus AN <lb></lb>81613. </s> </p> <p type="main"> <s id="s.000840">Ex quibus apparet deſcendendi momentum, quod compo<lb></lb>nitur ex momentis in planis inclinatis, non eſſe 119280 ex eo<lb></lb>rum ſummâ, ſed ita temperari, ut longè minus ſit, videlicet ſo<lb></lb>lùm 81613. </s> </p> <p type="main"> <s id="s.000841">Methodo eâdem operantes deprehendemus ponderis in H <lb></lb>conſtituti, ac ex funiculis BH, CH ſuſpenſi momentum ita <lb></lb>componi ex momento HI bis ſumpto (ſi quidem anguli decli<lb></lb>nationum DBH, ECH & funiculi æquales ſint) ut in unum <lb></lb>ex utroque nimirum HI & HO temperatum HS coaleſcat. </s> <lb></lb> <s id="s.000842">Unde conſtabit quò majores fuérint declinationum anguli, eò <lb></lb>longiorem futuram lineam HS, atque adeò etiam majus mo<lb></lb>mentum deſcendendi; plana ſiquidem inclinata acutiorem <lb></lb>angulum conſtituunt. </s> <s id="s.000843">Quam momentorum varietatem pau<lb></lb>lò inferiùs manifeſto experimento comprobabimus: ubi conſta<lb></lb>bit pondus hâc ratione ſuſpenſum ex duobus funiculis plus ha<lb></lb>bere aliquando momenti ad deſcendendum, quàm in perpen<lb></lb>diculari ſuſpenſione. </s> </p> <p type="main"> <s id="s.000844">Quemadmodum verò de momentis deſcendendi in planis <lb></lb>inclinatis ratiocinati ſumus, ita pariter in unum coaleſcere di<lb></lb>cenda ſunt momenta, quibus funiculi pondus retinentes ipſum <lb></lb>quodammodo avellere conantur à plano inclinato, ne illud ur<lb></lb>geat; hæc enim pariter momenta in diverſa abeunt ſecun<lb></lb>dùm ipſam funiculorum directionem. </s> <s id="s.000845">Sunt autem momenta <lb></lb>illa Sinus Verſi angulorum inclinationis planorum; qui haben<lb></lb>tur, ſi Sinus Recti complementorum, hoc eſt angulorum de-<pb pagenum="106" xlink:href="017/01/122.jpg"></pb><figure id="id.017.01.122.1.jpg" xlink:href="017/01/122/1.jpg"></figure><lb></lb>clinationis funiculorum, de<lb></lb>mantur ex Radio. </s> <s id="s.000846">Itaque ex <lb></lb>BA auferatur BF ipſi DA <lb></lb>æqualis, & eſt FA Sinus Ver<lb></lb>ſus anguli inclinationis: poſita <lb></lb>eſt autem declinatio DBA <lb></lb>gr.22. 12′, igitur FA eſt parti<lb></lb>cularum 62216; & declinatio <lb></lb>ECA gr. 54. 35′; igitur factâ <lb></lb>CG æquali ipſi AE, remanet <lb></lb>GA particularum 18504, quarum CA eſt 100000. Quare ut <lb></lb>habeantur particulæ ejuſdem rationis cum particulis AF, fiat <lb></lb>ut CA ad AG, ita BA ad AH, & eſt AH particularum 18504 <lb></lb>homologarum particulis AF. </s> <s id="s.000847">Perficiatur parallelogrammum <lb></lb>AHIF; & quia funiculus CA retrahit à plano inclinato juxta <lb></lb>momentum ac directionem HA, funiculus verò BA retrahit à <lb></lb>plano inclinato ſecundùm momentum ac directionem FA, di<lb></lb>rectionibus in diverſa abeuntibus, temperatur ex his momentis <lb></lb>momentum AI diameter parallelogrammi. </s> </p> <p type="main"> <s id="s.000848">Porrò in diametri AI inveſtigatione methodus eſt eadem, <lb></lb>quâ paulò antè utebamur: Cum enim tres anguli BAD, BAC, <lb></lb>CAE ſint duobus Rectis æquales, anguli verò BAD, CAE <lb></lb>noti ſint, quippe complementa angulorum declinationis DBA, <lb></lb>ECA, innoteſcit reliquus FAH, qui æqualis eſt ſummæ an<lb></lb>gulorum declinationis. </s> <s id="s.000849">Eſt igitur FAH gr.76.47′, ac proinde <lb></lb>angulus AFI gr.103.13′ notus eſt, unâ cum lateribus FA 62216 <lb></lb>& FI 18504. Fiat igitur ut laterum ſumma 80720 ad eorum<lb></lb>dem differentiam 43712, ita angulorum ad baſim AI ſemiſum<lb></lb>mæ gr. 38. 23′1/2. Tangens 79235 ad 42907 Tangentem dif<lb></lb>ferentiæ infra vel ſupra eandem ſemiſummam, hoc eſt gr. 23. <lb></lb>13′.1/2 dempta igitur hæc differentia ex ſemiſſummâ gr.38.23′ 1/2, <lb></lb>reliquum facit angulum FAI gr.15.10′. </s> <s id="s.000850">Fiat demùm ut anguli <lb></lb>FAI gr.15.10′. </s> <s id="s.000851">Sinus 26163 ad anguli AFI gr. 103. 13′. </s> <s id="s.000852">hoc eſt <lb></lb>ad ſupplementi gr.76.47′. Sinum 97351, ita latus FI 18504 <lb></lb>ad baſim AI 68852. </s> </p> <p type="main"> <s id="s.000853">Inventa itaque momenta compoſita tùm in planis inclinatis, <lb></lb>tùm in plana inclinata, dividantur juxta Rationem momento-<pb pagenum="107" xlink:href="017/01/123.jpg"></pb>rum ſimplicium, ut innoteſcat, quid demum cuique funicolo <lb></lb>tribuendum ſit in pondere retinendo. </s> <s id="s.000854">Momentum deſcenden<lb></lb>di compoſitum inventum eſt ſuſperiùs 81613, ſimplicia ſunt <lb></lb>81496, & 37784. Fiat ut igitur ut ſimplicium momentorum <lb></lb>ſumma 119280 ad eorum alterutrum, puta ad 37784, ita mo<lb></lb>mentum compoſitum 81613 ad aliud, & provenit 25852 pars <lb></lb>illius momenti pertinens ad funiculum CA, qui retinet pon<lb></lb>dus; cujus vis deſcendendi eſt DA 37784. Reliqua autem mo<lb></lb>menti 81613 pars 55761 pertinet ad funiculum BA retinentem <lb></lb>pondus, cujus vis deſcendendi eſt EA 81496. Pari ratione fiat <lb></lb>ut Sinuum Verſorum angulorum inclinationis ſimplicium <lb></lb>62216, atque 18504 ſumma 80720 ad eorum alterutrum, pu<lb></lb>ta ad 18504, ita momentum compoſitum inventum 68852 ad <lb></lb>aliud, & provenit pro minori 15783, pro majori verò 53069. <lb></lb>Quare funiculus BA minorem habens declinationem, & plus <lb></lb>ſuſtinet in ſuo plano magis inclinato, cui perpendicularis eſt, <lb></lb>nimirum ut 53069, & plus retinet in plano reliquo minùs in<lb></lb>clinato, nimirum ut 55761: contra verò funiculus CA, & mi<lb></lb>nus ſuſtinet, ſcilicet ut 15783, & minus retinet ſcilicet ut <lb></lb>25852. Funiculus itaque BA exercet vires ut 108830, & fu<lb></lb>niculus CA ut 41635, & totum corporis ſuſpenſi momentum <lb></lb>eſt 150465. </s> </p> <p type="main"> <s id="s.000855">Non ſola autem momenta deſcendendi in planis inclinatis <lb></lb>conſiderari oportere, ſed & ea, quæ eſſent adversùs plana <lb></lb>ipſa inclinata, uti dictum eſt, ex eo apertè conficitur, quòd <lb></lb>ubi funiculi concurrerent ad acutiſſimum angulum, vix quic<lb></lb>quam virium in retinendo pondere exercere opus eſſet; te<lb></lb>nuiſſimum quippe, eſſet momentum, quod ex parvis mo<lb></lb>mentis per acutiſſimorum angulorum Sinus Rectos definitis <lb></lb>componeretur: ſi verò nihil præterea momenti addendum eſ<lb></lb>ſet; à magnâ gravitatione, quæ in perpendiculari eſt, ad ferè <lb></lb>nullam tranſitus eſſet, facta vel modicâ à perpendiculo decli<lb></lb>natione; atque adeò vix intenti eſſe deberent funiculi: id quod <lb></lb>manifeſto experimento adverſatur. </s> </p> <p type="main"> <s id="s.000856">Illud poſtremò hîc oſtendendum ſupereſt, plus ſcilicet in<lb></lb>eſſe poſſe momenti ad deſcendendum corpori ex duobus funi<lb></lb>culis invicem inclinatis ſuſpenſo, quàm ſi ex unico ad per<lb></lb>pendiculum pendeat. </s> <s id="s.000857">Orbiculo circà ſuum axem C verſatili, <pb pagenum="108" xlink:href="017/01/124.jpg"></pb><figure id="id.017.01.124.1.jpg" xlink:href="017/01/124/1.jpg"></figure><lb></lb>ac ſecundùm extremam <lb></lb>oram excavato, inſeratur <lb></lb>funiculus AFB, ex quo <lb></lb>æqualia hinc, & hinc <lb></lb>pondera A, & B pen<lb></lb>deant: nullus planè ſe<lb></lb>quitur motus, quia utrum<lb></lb>que ex perpendiculo pen<lb></lb>det, & quantâ vi alterum conatur deorſum, pari nuſu alterum <lb></lb>repugnat, ne elevetur. </s> <s id="s.000858">Quærenti igitur, quantum momenti <lb></lb>pondus B habeat ad deſcendendum, utique reſpondebis omni<lb></lb>nò par eſſe momento ponderis A. </s> <s id="s.000859">Jam verò ſit funiculus AFD, <lb></lb>qui in D religetur, & ponderi A ſumatur æquale pondus E, <lb></lb>vel potiùs ipſum B transferatur in E, & funiculo AFD ad<lb></lb>nectatur in H; ut ſint quaſi duo funiculi DH, FH. </s> <s id="s.000860">Quæro <lb></lb>quantum ad deſcendendum momenti habeat pondus E, hoc eſt <lb></lb>pondus B in H tranſlatum, quod eſt æquale ponderi A: ſi tan<lb></lb>tumdem habet momenti, quantum pondus A, planè manebit <lb></lb>immotum, intento funiculo FD; at ſi E deſcendens cogat <lb></lb>aſcendere pondus A, utique plus momenti habet quàm A, hoc <lb></lb>eſt, pluſquàm B perpendiculariter pendens. </s> <s id="s.000861">Id quod re ipsâ <lb></lb>contingit; & quidem tàm certo experimento, ut non ſolùm <lb></lb>pondus E prævaleat ponderi A, ſi ſit ei æquale, verùm etiam ſi <lb></lb>minus ſit eodem pondere A. </s> <s id="s.000862">Non igitur hoc abſurdum eſt, <lb></lb>quod conſtitutam à nobis momentorum hypotheſim conſequa<lb></lb>tur, ſed potiùs ipſi naturæ noſtra conſentit hypotheſis, cui ro<lb></lb>bur adjicit experientia; nec ex eo capite perperam philoſopha<lb></lb>ti videmur, quòd in perpendiculo minus momenti, quàm ex <lb></lb>duplici funiculo ſuſpenſum pondus habere dicendum ſit. </s> </p> <p type="main"> <s id="s.000863">Ex his, quæ de corpore ex binis funiculis ſuſpenſo hactenus <lb></lb>diſputata ſunt, non difficilis erit conjectura eorum, quæ dicen<lb></lb>da ſint, ſi ex tribus aut quatuor ſuſpendatur, ſivè illi immedia<lb></lb>tè adnectantur ipſi ponderi, ſivè funiculus unus demum in plu<lb></lb>ra capita dividatur, ex quibus fiat ſuſpenſio: neque enim his <lb></lb>diutiùs ad nauſeam immorandum cenſeo. <pb pagenum="109" xlink:href="017/01/125.jpg"></pb></s> </p> <p type="main"> <s id="s.000864"><emph type="center"></emph>CAPUT XVI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000865"><emph type="center"></emph><emph type="italics"></emph>Tractiones ac elevationes obliquæ expenduntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000866">PRoxima eſt iis, quæ hactenus diſputata ſunt, præſens in<lb></lb>veſtigatio gravitationis corporum, ſive nisûs, quo motui <lb></lb>reſiſtunt, cùm obliquè in plano aliquo trahuntur, aut elevan<lb></lb>tur: ſicut enim toto conatu repugnant elevanti ad perpendicu<lb></lb>lum, & abſtrahenti à plano, cui inſident, ita pro majori, aut <lb></lb>minori obliquitate tractionis aut elevationis magis etiam, aut <lb></lb>minùs, obſiſtere experimur. </s> <s id="s.000867">Et primùm quidem ſuper plano <lb></lb><figure id="id.017.01.125.1.jpg" xlink:href="017/01/125/1.jpg"></figure><lb></lb>inclinato AB duo pondera <lb></lb>prorſus æqualia, & ſimilia <lb></lb>intelligantur poſita in B <lb></lb>& C, atque linea CE ſit <lb></lb>horizonti BE perpendicu<lb></lb>laris, ac pondus C filo DC <lb></lb>ad perpendiculum ſuſpen<lb></lb>datur, ita tamen, ut con<lb></lb>tingat planum in C, & ſit <lb></lb>recta DE. </s> <s id="s.000868">Item ex D <lb></lb>puncto ducatur filum DB, <lb></lb>ut ſurſum trahatur B pon<lb></lb>dus incumbens plano in<lb></lb>clinato, dum pariter pon<lb></lb>dus C ſurſum rectâ trahi<lb></lb>tur, & à plano avellitur: horum autem funiculorum trahatur <lb></lb>ex D pars æqualis. </s> <s id="s.000869">Quando igitur C venerit in V, æquali men<lb></lb>ſurâ BP multatum intelligitur filum DB, & remanet longi<lb></lb>tudo DP, hoc eſt DO; pondus enim, cum filum in D trahe<lb></lb>retur, ex B venit in O. </s> <s id="s.000870">Ductâ itaque lineâ ON horizonti pa<lb></lb>rallelâ, erit EN altitudo perpendicularis, ad quam aſcendit <lb></lb>pondus B in plano inclinato interea, dum pondus C venit in V, <lb></lb>aut E venit in M, eſt enim EM aſſumpta ipſi CV æqualis. </s> <lb></lb> <s id="s.000871">Quare cum pondus B obliquè trahitur ſuper planum inclina-<pb pagenum="110" xlink:href="017/01/126.jpg"></pb>tum, minorem ſubit violentiam, quàm cum ab illo perpendi<lb></lb>culari elevatione avellitur. </s> </p> <p type="main"> <s id="s.000872">Hoc tamen ita intelligendum eſt, ut obſervetur alia eſſe <lb></lb>momenta, cùm tractionis linea parallela eſt ipſi plano inclina<lb></lb>to, ac cùm in planum inclinatum cadit obliqua, ut hîc li<lb></lb>nea DB. </s> <s id="s.000873">Si enim in plano inclinato ſumatur BR æqualis <lb></lb>perpendiculari EM, gravitatio per rectam BC, ſeu per li<lb></lb>neam eidem parallelam, ad gravitationem in perpendiculo <lb></lb>CE eſt reciprocè ut EC ad BC, ſeu ut ES ad BR aut EM, <lb></lb>ex ſuperiùs dictis cap.13. At verò cum tractio obliqua eſt, <lb></lb>gravitatio eſt ut EN ad EM, ſivè ut BO ad BX: punctum <lb></lb>autem O altius eſt puncto R, ac proptereà in hujuſmodi <lb></lb>obliquâ tractione plus violentiæ infertur ponderi, quàm in <lb></lb>tractione parallelâ, plus enim aſcendit. </s> <s id="s.000874">Porrò lineam BO <lb></lb>longiorem eſſe lineâ BR eſt manifeſtum; ſiquidem duo la<lb></lb>tera DO, OB per 20. lib.1. majora ſunt reliquo DB: eſt <lb></lb>autem ex hypotheſi DP ipſi DO æqualis, ergo reliqua <lb></lb>BP minor eſt, quàm BO: ſed & ipſi BP, hoc eſt ipſi <lb></lb>EM, æqualis aſſumpta eſt BR; igitur BR minor eſt quàm <lb></lb>BO. </s> <s id="s.000875">Id quod etiam hinc conſtat, quia in triangulo Iſo<lb></lb>ſcele DOP angulus OPB infra baſim major eſt recto, <lb></lb>cum ſit deinceps angulo DPO ad baſim acuto; ergo per <lb></lb>19.lib.1. latus BO majus eſt latere BP, hoc eſt BR; igi<lb></lb>tur etiam EN major eſt quàm ES, & plus difficultatis <lb></lb>percipitur in obliquâ hâc tractione, quàm in tractione pa<lb></lb>rallelà. </s> </p> <p type="main"> <s id="s.000876">Similiter intelligatur pondus C elevatum fuiſſe ex D <lb></lb>(quod punctum D concipiatur multò altius, quàm in præ<lb></lb>ſenti ſchemate) ad perpendiculum altitudine æquali ipſi ET, <lb></lb>pondus verò B æquali tractione funiculi veniſſe ex B in G, <lb></lb>demptâ ſcilicet longitudine BF ipſi ET æquali, atque <lb></lb>adeò DF, DG æquales ſunt: ipſi autem ET æqualis ſu<lb></lb>matur BI; quæ ſimili ratione demonſtratur brevior, quàm <lb></lb>BG: ex quo pariter ſit hîc etiam ad majorem altitudi<lb></lb>nem perpendicularem EH elevari, quàm ſi tractio pa<lb></lb>rallela fuiſſet plano inclinato, & elevatio ad altitudi<lb></lb>nem EL. </s> </p> <p type="main"> <s id="s.000877">Ex his manifeſtum eſt plus virium requiri ad trahendum <pb pagenum="111" xlink:href="017/01/127.jpg"></pb>pondus idem per lineam DB, aut DO, aut DG obli<lb></lb>quas, quàm per lineam plani inclinati BC, aut illi paral<lb></lb>lelam: dum enim per obliquas illas lineas fit tractio, pon<lb></lb>dus quidem non omninò abſtrahitur à plano, ſicut in tractio<lb></lb>ne perpendiculari, ſed nec omninò incumbit plano, ſi<lb></lb>cut in tractione parallelâ ipſi plano; ac propterea, quò ma<lb></lb>gis tractio ad perpendicularem accedit, eò majorem inve<lb></lb>nit in pondere reſiſtentiam. </s> <s id="s.000878">Patet autem altitudinum per<lb></lb>pendicularium EH, EL differentiam HL majorem eſſe, <lb></lb>quàm ſit altitudinum perpendicularium EN, ES differen<lb></lb>tia NS. </s> <s id="s.000879">Comparatis enim triangulis iſoſcelibus DPO, <lb></lb>DFG, anguli ad baſim PO majores ſunt angulis ad baſim <lb></lb>FG, quia angulus PDO minor eſt angulo FDG: ergo <lb></lb>angulus BPO, qui eſt infra baſim, minor eſt angulo <lb></lb>BFG infra baſim. </s> <s id="s.000880">Fiat igitur ipſi BPO æqualis angulus <lb></lb>BFK, ac proinde K cadit inter puncta I & G. </s> <s id="s.000881">Sunt ergo <lb></lb>triangula BPO, BFK habentia angulum ad B communem <lb></lb>æquiangula, & ſimilia, ac per 4. lib.6. ut PB, hoc eſt BR, <lb></lb>ad BO, ita FB, hoc eſt BI, ad BK; & invertendo, ac <lb></lb>dividendo, & iterùm invertendo ut BR ad RO, ita BI <lb></lb>ad IK. </s> <s id="s.000882">Atqui IG major eſt quam IK, ergo per 8.lib.5. <lb></lb>Ratio BI ad IG minor eſt Ratione BI ad IK, hoc eſt BR <lb></lb>ad RO. </s> <s id="s.000883">Cum itaque per 2. lib.6. ut BR ad RO, ita ES <lb></lb>ad SN; & ut BI ad IG, ita EL ad LH, major eſt Ra<lb></lb>tio ES ad SN, quàm EL ad LH, & permutando major <lb></lb>eſt Ratio ES ad EL, quàm SN ad LH; eſt autem ES <lb></lb>minor quàm EL, ergo etiam SN multò minor eſt quàm <lb></lb>LH; ac proinde quo magis à perpendiculari recedet obli<lb></lb>qua tractio, momentum ponderis magis accedit ad momen<lb></lb>tum ejuſdem in plano inclinato per tractionem parallelam, <lb></lb>hoc eſt, minore differentiâ hoc excedit. </s> <s id="s.000884">Momentum igitur <lb></lb>perpendicularis tractionis ad momentum obliquæ tractionis <lb></lb>minorem Rationem habet, quàm ad momentum tractionis pa<lb></lb>rallelæ plano inclinato. </s> </p> <p type="main"> <s id="s.000885">Ex his obſervare eſt aliquod paradoxum, pondus ſcilicet obli<lb></lb>quâ hâc elevatione tractum plus moveri, quàm potentiam tra<lb></lb>hentem; hæc enim movetur ſecundùm menſuram funiculi <lb></lb>tracti, hoc eſt BP ſeu BR illi æqualis, oſtenſum eſt autem <pb pagenum="112" xlink:href="017/01/128.jpg"></pb>BR minorem eſſe quàm BO. </s> <s id="s.000886">Id quod etiam manifeſtum eſt, <lb></lb>ſi tractio obliqua non abſtrahat pondus à plano, ſed quaſi il<lb></lb><figure id="id.017.01.128.1.jpg" xlink:href="017/01/128/1.jpg"></figure><lb></lb>lud adversùs planum trahat. </s> <lb></lb> <s id="s.000887">Sit enim planum AB, ſuper <lb></lb>quo globus C, & funiculus <lb></lb>obliquus DC; ex D autem <lb></lb>pendeat ad perpendiculum <lb></lb>æquale pondus E. </s> <s id="s.000888">Uterque fu<lb></lb>niculus pariter trahatur, & <lb></lb>cum E venerit in F, æqualis <lb></lb>pars CG decedit funiculo <lb></lb>DC; remanet autem longitu<lb></lb>do DG æqualis longitudini <lb></lb>DH, & centrum globi C ve<lb></lb>nit in H. </s> <s id="s.000889">Dico CH motum <lb></lb>globi majorem eſſe ſupra CG <lb></lb>motum potentiæ trahentis. </s> <lb></lb> <s id="s.000890">Ducatur enim recta GH; eſt <lb></lb>Iſoſceles DGH, ergo angulus HGC infra baſim major eſt <lb></lb>recto; ergo CH per 19.lib.1. major eſt quàm CG. </s> <s id="s.000891">Ipſi autem <lb></lb>CH æqualem eſſe diſtantiam contactuum RS manifeſtum <lb></lb>eſt, quia ex centris H & C rectæ cadunt in S & R ad angu<lb></lb>los rectos, atque adeò ſunt parallelæ: ſunt æquales CR & HS, <lb></lb>ut pote Radij ejuſdem globi; igitur per 33.lib.1. CH, & RS <lb></lb>æquales ſunt & parallelæ. </s> <s id="s.000892">Quare ſivè centrum ſpectetur, ſivè <lb></lb>puncta contactuum, perinde eſt; ſemper enim major eſt glo<lb></lb>bi motus motu potentiæ trahentis; & quia RS major eſt quàm <lb></lb>CG, hoc eſt quàm motus, qui fieret in ipſo plano inclinato <lb></lb>tractione parallelâ, hinc eſt quod hujuſmodi obliquâ tractio<lb></lb>ne ad majorem altitudinem perpendicularem pari tempore tra<lb></lb>hitur, majorémque proptereà violentiam ſubiens majoribus <lb></lb>indiget viribus, quàm ſi tractione parallelâ elevaretur. </s> </p> <p type="main"> <s id="s.000893">Sed jam trahatur iterum funiculus ita, ut ipſi CG primæ <lb></lb>tractioni æqualis ſit ſecunda tractio HL; & crit centrum globi <lb></lb>in M, & æquales DM, DL. </s> <s id="s.000894">Anguli MDH, HDC ſi di<lb></lb>cantur æquales, etiam per 3.lib.6. ut MD ad DC ita MH <lb></lb>ad HC: eſt igitur MH minor quàm HC, major tamen quàm <lb></lb>HL, quia ſubtenſa eſt angulo MLH obtuſo, ut pote infra ba-<pb pagenum="113" xlink:href="017/01/129.jpg"></pb>ſim Iſoſcelis MDL. </s> <s id="s.000895">Atqui ex hypotheſi anguli MDL, HDG <lb></lb>ſunt æquales; ergo Iſoſcelium anguli infra baſes, hoc eſt MLH, <lb></lb>HGC ſunt æquales: angulus autem externus MHL major eſt <lb></lb>interno HCD, hoc eſt HCG, per 16.lib.1. igitur reliquus <lb></lb>HML minor eſt reliquo CHG. </s> <s id="s.000896">Itaque in duobus triangulis, <lb></lb>angulis CGH, HLM ex hypotheſi oſtenſis æqualibus ſub<lb></lb>tenditur illi quidem majus latus CH, huic verò minus HM, <lb></lb>& angulis inæqualibus CHG majori, HML minori æquale <lb></lb>latus CG, HL: id quod omninò abſurdum eſſe conſtat ex <lb></lb>doctrinâ & Canone Sinuum; ſubtenſæ ſiquidem inæquales an<lb></lb>gulorum æqualium ſunt in circulis inæqualibus, major in majori <lb></lb>circulo, minor in minori, in quibus utique fieri non poteſt, ut <lb></lb>angulorum inæqualium ſubtenſæ ſint æquales. </s> <s id="s.000897">Non igitur fieri <lb></lb>poteſt ut factá ſecunda tractione HL æquali priori CG, angu<lb></lb>lus MDH æqualis ſit angulo HDC; alioquin triangulum <lb></lb>HLM (cujus baſis HM ex hypotheſi arguitur minor baſe <lb></lb>CH, quæ tamen ſunt angulis ad G & L æqualibus ſubtenſæ) <lb></lb>eſſet in circulo minore, quàm ſit circulus, in quo eſſet triangu<lb></lb>lum CGH; in circulo autem minore, angulo minori HML <lb></lb>ſubtenſa HL eſſet æqualis ipſi CG ſubtenſæ angulo majori <lb></lb>CHG in circulo majore. </s> </p> <p type="main"> <s id="s.000898">Quod ſi dicatur angulus MDH minor, quàm HDC, ergo <lb></lb>angulus MLH infra baſim minor eſt angulo HGC infra ba<lb></lb>ſim: atqui angulus MHL externus major eſt <expan abbr="interño">interno</expan> HCG; <lb></lb>igitur reliquus angulus LMH vel eſt æqualis angulo GHC, <lb></lb>vel illo minor, vel illo major. </s> <s id="s.000899">Sit æqualis: quoniam æqualibus <lb></lb>lineis CG, HL ſubtenduntur, ſunt in circulis æqualibus; ergo <lb></lb>cùm angulus MHL major ſit angulo HCG, etiam oppoſitum <lb></lb>latus ML majus eſt quàm HG: ergo Iſoſceles MDL habens <lb></lb>angulum minorem ſub brevioribu lateribus habet majorem <lb></lb>baſim, & Iſoſceles HDG habens angulum majorem ſub late<lb></lb>ribus <expan abbr="lõgioribus">longioribus</expan> habet <expan abbr="breviorẽ">breviorem</expan> baſim; id quod eſt manifeſtè <expan abbr="ab-ſurdũ">ab<lb></lb>ſurdum</expan>, ut patet ex 24. & 25.lib.1.Fieri igitur non poteſt, ut anguli <lb></lb>LMH, GHC ſint æquales, ſi MDH minor eſt quàm HDC. </s> </p> <p type="main"> <s id="s.000900">Quandoquidem igitur LMH, GHC non ſunt æquales, dica<lb></lb>tur angulus LMH minor quàm GHC, & quia æqualibus li<lb></lb>neis HL, CG ſubtenduntur, triangulum HLM eſt in circulo <lb></lb>majore, triangulum verò CHG in minore. </s> <s id="s.000901">Cum autem angu-<pb pagenum="114" xlink:href="017/01/130.jpg"></pb>lus MHL, ex ſæpiùs dictis, ſit major quàm HCG, etiam ſub<lb></lb>tenſa illius, ut potè in circulo majori, ſcilicet ML major eſt <lb></lb>quàm HG ſubtenſa anguli minoris in circulo minori: atque <lb></lb>hinc idem quod priùs, ſequitur abſurdum angulum verticalem <lb></lb>MDL, ex hypotheſi minorem, & brevioribus lateribus com<lb></lb>prehenſum baſim habere majorem, quàm ſit baſis anguli verti<lb></lb>calis HDG majoris ſub lateribus longioribus. </s> </p> <p type="main"> <s id="s.000902">Sed neque dici poteſt angulus HML major quàm CHG; <lb></lb>quia, ſi MDL minor eſt quàm HDG, angulus DML ad ba<lb></lb>ſim Iſoſcelis major eſt quàm DHG pariter ad baſim; ergo ſi <lb></lb>DML majori addatur major HML, & DHG minori adda<lb></lb>tur minor CHG, erit totus DMH major toto angulo DHC, <lb></lb>internus ſcilicet major externo, contra 16.lib.1. Si igitur an<lb></lb>gulus HML comparatus cum angulo CHG non poteſt eſſe <lb></lb>æqualis, neque minor, neque major, factâ hypotheſi anguli <lb></lb>MDL minoris quàm HDC, neceſſariâ conſecutione confici<lb></lb>tur angulum MDL non eſſe minorem angulo HDG. </s> </p> <p type="main"> <s id="s.000903">Cum itaque angulus MDL neque æqualis, neque minor ſit <lb></lb>angulo HDG, ſequitur quod ſit major: igitur & angulus in<lb></lb>fra baſim MLH major eſt angulo HGC; item angulus MHL <lb></lb>major eſt quàm HCG; ergo HML reliquus minor eſt reliquo <lb></lb>CHG: at iſtis æquales lineæ HL, CG ſubtenduntur, igitur <lb></lb>triangulum HML eſt in majore circulo, ac proinde angulo <lb></lb>MLH majori, quàm CGH, etiam majus latus ſubtenditur: <lb></lb>quapropter MH, hoc eſt SN, illi parallela & æqualis, major <lb></lb>eſt quàm CH, hoc eſt RS: atque adeò ad majorem altitudi<lb></lb>nem elevatur per SN, quàm per RS factâ æquali tractione, ſeu <lb></lb>æquali motu potentiæ trahentis. </s> <s id="s.000904">Ex quo & manifeſtum eſt pro <lb></lb>majori obliquitate & receſſu tractionis à paralleliſmo cum pla<lb></lb>no inclinato etiam trahenti difficultatem augeri. </s> </p> <p type="main"> <s id="s.000905">Facilè ex dictis colliges, quanto laboris compendio Romæ <lb></lb>altioribus rotis inſtruantur birota (antiquis Ciſia dicebantur) <lb></lb>adeò ut unicus equus temoni applicitus, illumque ſubjecto pla<lb></lb>no proximè parallelum ſervans, dum clivum aſcendit, ingentia <lb></lb>pondera trahat, quibus ſanè par non eſſet, ſi rotarum axis mi<lb></lb>nùs à ſubjecto plano diſtaret, & equitractio eſſet obliqua ſur<lb></lb>ſum: quamvis, ut aliàs ſuo loco explicabitur, ipſa rotarum am<lb></lb>plitudo plurimum conferat. </s> <s id="s.000906">Similiter in navium tractione, quæ <pb pagenum="115" xlink:href="017/01/131.jpg"></pb>adverſo flumine deducuntur fune abſidi mali conjuncto, ali<lb></lb>quid juvare funis longitudinem, ut ſcilicet minùs obliqua ſit <lb></lb>tractio, ex dictis confirmatur: quamvis enim tractiones in plano <lb></lb>inclinato conſideraverimus, ut gravium elevationem expende<lb></lb>remus, aliquid etiam facit obliquitas tractionis in plano horizon<lb></lb>tali, cujuſmodi eſt aqua, cui navis innatat; pars ſiquidem de<lb></lb>merſa obſtantem undam repellere debet; nec planè inutile eſt, <lb></lb>ſecundùm quam lineam dirigatur motus potentiæ trahentis, vi <lb></lb>cujus impedimentum ſuperandum eſt. </s> </p> <p type="main"> <s id="s.000907">Hactenus nobis de tractione ſermo fuit, quæ motum inferens <lb></lb>non niſi ſpatiis, per quæ motus eſt, determinari potuit. </s> <s id="s.000908">Quo<lb></lb>niam verò in obliquis tractionibus non eandem ſemper analo<lb></lb>giam ſervari, quæ in parallelâ tractione eadem perpetuò eſt, de<lb></lb>prehendimus, inquirendum ſupereſt, quæ demum Ratio mo<lb></lb>mentorum ſit pro ſingulis obliquitatibus, ut conſtet, quibus vi<lb></lb>ribus retineri poſſit, ne in proclive labatur pondus, etiamſi vires <lb></lb>ad illud ulteriùs elevandum non ſuppetant. </s> <s id="s.000909">Quamquam autem <lb></lb>pondera quaſi molis expertia unico puncto expreſſimus in plano <lb></lb>ipſo inclinato, ut in 1.fig.hujus cap. re tamen verâ centrum gra<lb></lb>vitatis attendendum eſt, ut in 2. ſchemate, quod utique diſtat à <lb></lb>plano, cui corpus grave incumbit: hujus verò diſtantiam nulla <lb></lb>certior menſura definit, quàm linea ex eo cadens in ſubjectum <lb></lb>planum ad angulos rectos, hæc quippe omnium breviſſima eſt. <lb></lb><figure id="id.017.01.131.1.jpg" xlink:href="017/01/131/1.jpg"></figure><lb></lb>Sit igitur planum inclinatum AB, <lb></lb>cui impoſitus globus centrum ha<lb></lb>bet gravitatis C, & contingit pla<lb></lb>num in D; ac propterea etiam, quæ <lb></lb>à centro ad contactum ducitur <lb></lb>recta CD, diſtantiam determinat, <lb></lb>cum ſit plano perpendicularis ex <lb></lb>18.lib.3. Jam recta CE parallela <lb></lb>plano ducatur, & ſit linea ſuſpen<lb></lb>ſionis, quam claritatis gratiâ paral<lb></lb>lelam vocemus: & per D punctum, <lb></lb>in quod cadit linea diſtantiæ cen<lb></lb>tri gravitatis tranſeat perpendicu<lb></lb>laris horizonti linea FD quæ in G <lb></lb>ſecat lineam CE. </s> <s id="s.000910">Conſtat trian-<pb pagenum="116" xlink:href="017/01/132.jpg"></pb>gulum DGC ſimile eſſe triangulo BAS: quia enim GD pa<lb></lb>rallela eſt lineæ AS pariter perpendiculari ad horizontem, an<lb></lb>guli SAB, ADG alterni æquales ſunt per 27.lib.1. Et quo<lb></lb>niam angulus CDA ex conſtructione eſt rectus, complemen<lb></lb>tum CDG æquale eſt angulo complementi ABS; anguli verò <lb></lb>DCG, BSA ſunt recti, hic quidem ex hypotheſi, ille autem <lb></lb>propter linearum CE, DA paralleliſmum: igitur reliquus <lb></lb>CGD reliquo BAS æqualis eſt; ac proptereà per 4. lib. 6. ut <lb></lb>BA ad AS, ita DG ad GC. </s> <s id="s.000911">Quoniam itaque, ſi pondus in <lb></lb>plano inclinato ad pondus in perpendiculari ſit ut inclinata BA <lb></lb>ad perpendicularem AS, eorum momenta æqualia ſunt, & <lb></lb>æquiponderant, etiam globus æqualia ad deſcendendum habet <lb></lb>momenta, ac potentia habeat vires ad retinendum in parallelâ <lb></lb>EC, ſi globi gravitas ad potentiam retinendum ſit ut DG ad <lb></lb>GC. </s> <s id="s.000912">Verum quidem eſt globum non per lineam FD, ſed per <lb></lb>CT à centro gravitatis perpendicularem horizonti deorſum ni<lb></lb>ti: Sed quia CT ipſi FD parallela eſt, triangulum CTD <lb></lb>triangulo DGC ſimile eſt & æquale; atque adeò parùm in<lb></lb>tereſt, utrùm lineis DG, GC, an verò lineis CT, TD eadem <lb></lb>Ratio exponatur. </s> </p> <p type="main"> <s id="s.000913">Sed jam retineatur globus per rectam CH; utique perinde ſe<lb></lb>cundùm eam directionem ſe habet, atque ſi eſſet planum HCK; <lb></lb>globus enim ſuſtinetur per lineam DC, & retinetur ex H, ac <lb></lb>proinde ſecundùm <expan abbr="rectã">rectam</expan> HCK conatur deorſum co ſitu: quam<lb></lb>quam ſubjecti plani inclinatio obſtaret, ne ſecundùm rectam <lb></lb>HCK procederet, ſi ſibi dimitteretur, & alia atque alia plana <lb></lb>conſtituerentur. </s> <s id="s.000914">Planum itaque illud HC declinat à perpen<lb></lb>diculari, cum quâ conſtituit angulum CID æqualem externo <lb></lb>KCT propter paralleliſmum perpendicularium FD, CT per <lb></lb>27. lib. 1. qui utique CID minor eſt externo CGD per 16. <lb></lb>lib. 1. & quidem differentia anguli ICG per 32.lib.1. Fiat <lb></lb>ergo angulus BAP æqualis angulo CIG; quia BAS oſtenſus <lb></lb>eſt æqualis ipſi CGD, remanet PAS æqualis angulo ICG. </s> <lb></lb> <s id="s.000915">Quare BPA externus æqualis eſt duobus internis, ſcilicet recto <lb></lb>PSA, & acuto SAP, per 32.lib.1. igitur idem angulus BPA <lb></lb>æqualis eſt toti angulo DCI. </s> <s id="s.000916">Sunt itaque æquiangula & ſimi<lb></lb>lia duo triangula BAP & DIC, atque per 4.lib.6. ut BA ad <lb></lb>AP, ita DI ad IC. </s> <s id="s.000917">Atqui pondera ſuper BA & AP, quæ ſint <pb pagenum="117" xlink:href="017/01/133.jpg"></pb>ut BA ad AP, æquiponderant ex dictis cap. 13. ergo etiam <lb></lb>æqualium momentorum eſt globus, & potentia retinens per <lb></lb>HC, ſi globus ad potentiam ſit ut DI ad IC, hoc eſt ut CN <lb></lb>ad ND, ſi ex D intelligatur exire DN parallela ipſi HC. </s> </p> <p type="main"> <s id="s.000918">Eâdem ratione ſi linea obliqua, per quam globus retinetur, <lb></lb>ſit infra parallelam CE, ut ſi ſit CX, oſtendetur globi gravita<lb></lb>tem ad potentiam retinentem eſſe ut DQ ad QC, eſt enim <lb></lb>quaſi planum inclinatum faciens cum perpendiculari angulum <lb></lb>DQC majorem interno DGC, hoc eſt majorem angulo BAS <lb></lb>illi æquali. </s> <s id="s.000919">Fiat igitur angulo DQC æqualis angulus BAY: <lb></lb>& quia ABY æqualis eſt angulo CDQ, ut ſuperiùs dictum <lb></lb>eſt, triangula BAY, DQC ſunt æquiangula & ſimilia, ac per <lb></lb>4.lib.6. ut BA ad AX, ita DQ ad QC: ergo quia pondera ſu<lb></lb>per BA, & AY, quæ ſint in Ratione BA ad AY, æquiponde<lb></lb>rant, etiam globi & potentiæ retinentis momenta æqualia ſunt, <lb></lb>ſi fuerint ut DQ ad QC. </s> </p> <p type="main"> <s id="s.000920">Hic autem tria obſervanda occurrunt. </s> <s id="s.000921">Primum eſt, quòd <lb></lb>Rationes prædictæ momentorum potentiæ retinentis compara<lb></lb>tæ ad pondus idem, quamvis pro diversâ obliquitate aliis atque <lb></lb>aliis lineis explicentur DQ ad QC, & DG ad GC, DI ad <lb></lb>IC, omnes tamen exponuntur comparatè ad eandem BA in <lb></lb>triangulo BAY; in quo ipſæ quoque inter ſe invicem compara<lb></lb>ri poſſunt. </s> <s id="s.000922">Secundum eſt, quòd ſi obliquitas tàm ſupra, quàm <lb></lb>infra parallelam CE æqualis ſit, hoc eſt angulus ICG æqualis <lb></lb>ſit angulo GCQ, momenta potentiæ retinentis in H & X <lb></lb>æqualia ſunt; inter ſe ſiquidem ſunt ut AP, & AY, quæ lineæ <lb></lb>æquales ſunt; nam anguli PAS, YAS æquales ſunt ex hypo<lb></lb>theſi, & conſtructione, anguli autem ad S ſunt recti & latus <lb></lb>AS eſt utrique triangulo commune; ergo etiam per 26.lib.1.la<lb></lb>tera AP & AY æqualia ſunt. </s> <s id="s.000923">Tertium eſt, quòd in lineá CE <lb></lb>parallelâ minus virium exigitur ad retinendum globum, quàm <lb></lb>in cæteris: nam & linea AS vires potentiæ repræſentans om<lb></lb>nium minima eſt, utpote perpendicularis. </s> </p> <p type="main"> <s id="s.000924">Ex his & illud colligitur, quod ſi linea, ſecundùm quam <lb></lb>pondus retinetur in plano inclinato, ſit parallela horizonti, <lb></lb>eadem eſt philoſophandi methodus. </s> <s id="s.000925">Si enim ſuper plano in<lb></lb>clinato AB ſit pondus tangens in C, cujus gravitatis centrum <lb></lb>ſit D, & linea retentionis DE horizonti parallela, ducatur <pb pagenum="118" xlink:href="017/01/134.jpg"></pb><figure id="id.017.01.134.1.jpg" xlink:href="017/01/134/1.jpg"></figure><lb></lb>CF perpendicularis horizonti; & Ratio <lb></lb>ponderis ad vires retinentes erunt ut CF <lb></lb>ad FD. </s> <s id="s.000926">Fiat enim angulus BAH æqua<lb></lb>lis angulo CFD, qui utique eſt rectus, <lb></lb>cum DE ex hypotheſi ſit horizonti pa<lb></lb>rallela, FC verò perpendicularis: ergo <lb></lb>ſuper AB, AH æquiponderant pondera, <lb></lb>quæ ſint ut AB ad AH; paria igitur ſunt <lb></lb>momenta, ſi pondus ad vires potentiæ re<lb></lb>tinentis in eâdem Ratione ſit ut AB ad AH, hoc eſt ut CF ad <lb></lb>FD. </s> <s id="s.000927">Quia enim BAH angulus eſt rectus per 8.lib.6. eſt ut <lb></lb>BA ad AH, ita BG ad GA; eſt autem BG ad GA ut CF ad <lb></lb>FD; quia nimirum FC perpendicularis horizonti eſt paralle<lb></lb>la ipſi AG, & anguli BAG, FCA alterni ſunt æquales per <lb></lb>27.lib.1. DCA verò eſt rectus ex hypotheſi; igitur & DCF <lb></lb>complementum recti æquale eſt angulo ABG: utrumque <lb></lb>triangulum eſt rectangulum; ergo ut BG ad GA, ita CF <lb></lb>ad FD. </s> </p> <p type="main"> <s id="s.000928">Hinc apparet fieri poſſe, ut ad retinendum pondus in tali ſi<lb></lb>tu aliquando plus virium requiratur, quàm ad ſuſtinendum il<lb></lb>lud in perpendiculari; quando videlicet ex inclinatione plani <lb></lb>AB conſequitur lineam CF minorem eſſe quàm FD: immò <lb></lb>creſcit retinendi difficultas, ſi adhuc retentio fiat per lineam <lb></lb>inferiorem horizontali DE, quæ cum perpendiculari CF con<lb></lb>ſtituat angulum DIC obtuſum; cum enim creſceret linea DI <lb></lb>ſupra DF, & IC decreſceret infra FC, eſſet minor Ratio pon<lb></lb>deris in perpendiculo ad potentiam obliquè retinentem, <lb></lb>quæ proinde major eſſe deberet, ut fieret momentorum æqua<lb></lb>litas. </s> </p> <p type="main"> <s id="s.000929">Concipe autem ſublatum triangulum totum BAH, & DC <lb></lb>eſſe columnam, quæ in eodem ſitu inclinata retineri debeat: <lb></lb>jam ſatis conſtat ex dictis, quâ ratione diſponi oporteat funes, <lb></lb>ut qui funium extremitates tenent, minus laboris impendant. </s> <lb></lb> <s id="s.000930">Non eſt tamen eadem funis retinentis, & fulcri ſuſtentantis <lb></lb>ratio: in ſupponendis enim fulcris illud potiſſimùm attenditur, <lb></lb>quòd fulcrum ipſum integrum permaneat, citrà ſciſſionis aut <lb></lb>fractionis periculum; id quod habetur, quò magis perpendicu<lb></lb>lari ad horizontem ſitui proximum collocatur; parùm ſcilicet <pb pagenum="119" xlink:href="017/01/135.jpg"></pb>intereſt, quanto conatu ſubjectam tellurem urgeat modò certi <lb></lb>ſimus de fulcri ipſius firmitate. </s> <s id="s.000931">Cæterùm ſi tu ipſe fuſtem <lb></lb>manu tenens cogaris inclinatam columnam ſuſtinere, punctum <lb></lb>autem ſuſtentationis, cui fulcrum applicatur, magis à ſub<lb></lb>jecto plano diſtet, vel ſaltem non minùs, quàm centrum gra<lb></lb>vitatis columnæ, experieris minori conatu opus eſſe, ſi ful<lb></lb>crum axi columnæ perpendiculare ſit, qui ſitus reſpondet re<lb></lb>tentioni parallelæ plano inclinato, majorem verò adhiben<lb></lb>dum eſſe conatum, ſi fulcrum cum eodem axe acutum aut ob<lb></lb>tuſum angulum conſtituat; id quod obliquis elevationibus <lb></lb>reſpondet. </s> </p> <p type="main"> <s id="s.000932">Quòd ſi infra centrum gravitatis applicetur fulcrum, jam <lb></lb>conſtat hoc ita eſſe collocandum, ut ei idem centrum im<lb></lb>mineat, alioquin aut columna corruet, aut multis viri<lb></lb>bus tibi contendendum erit, ut illam ſuſtentes à lapſu; ſi <lb></lb>tamen ea ſit complexio tùm inclinationis, tùm obicis co<lb></lb>lumnæ pedem retinentis, ne excurrat, aut elevetur, tùm po<lb></lb>ſitionis fulcri, ut aliquatenus ſuſtineri columna poſſit, ne pror<lb></lb>sùs ruat. </s> </p> <p type="main"> <s id="s.000933">Sed quoniam hîc columnæ mentio incidit, præſtat ele<lb></lb>vationes corporum, quæ non tota elevantur, ſed eorum <lb></lb>altera extremitas ſubjecto alicui fulcro aut plano innititur, <lb></lb>altera elevatur aut ſuſpenditur, conſiderare: neque enim hîc <lb></lb>reputanda ſunt momenta gravitatis perinde, ac ſi totum cor<lb></lb>pus elevaretur aut ſuſpenderetur, quemadmodum paulò an<lb></lb>te dicebatur; immò verè longè minora ſunt pro ratione <lb></lb>diſtantiæ à centro gravitatis, ut ex inferiùs dicendis, ubi de <lb></lb>æquilibrio, atque de vecte ſermo erit, conſtabit. </s> <s id="s.000934">Cavendum <lb></lb>autem plurimum eſt ab æquivocationibus, quæ obrepere <lb></lb>poſſunt, niſi animum advertas ad gravitatem, ſivè per totam <lb></lb>longitudinem, quæ movetur, aut ad motum incitari poteſt, <lb></lb>diffuſam, ſivè quaſi in unum punctum ibi collectam, ubi ele<lb></lb>vans applicatur, ut in vecte, aut librâ; hinc enim non mo<lb></lb>dica momentorum inæqualitas oritur. </s> <s id="s.000935">Nam ſi puncto appli<lb></lb>cationis reſpondeat centrum gravitatis, multò majores ad <lb></lb>elevandum, aut ſuſpendendum corpus requiruntur vires, <lb></lb>quàm ſi centrum gravitatis à puncto applicationis aliquo in<lb></lb>tervallo ſejungatur. </s> </p> <pb pagenum="120" xlink:href="017/01/136.jpg"></pb> <figure id="id.017.01.136.1.jpg" xlink:href="017/01/136/1.jpg"></figure> <p type="main"> <s id="s.000936">Hinc ſi ſit priſma AB ho<lb></lb>rizontaliter collocatum, ejuſ<lb></lb>que extremitas A innitatur <lb></lb>apici pyramidis, altera verò <lb></lb>extremitas B ſuſpendatur per<lb></lb>pendiculari funiculo CB, vel <lb></lb>ſuſtentetur ſuppoſito ad <expan abbr="per-pendiculũ">per<lb></lb>pendiculum</expan> fulcro DB, æqua<lb></lb>liter res ſe habet, & pares requiruntur vires tam in ſuſpenden<lb></lb>te CB, quàm in ſuſtentante DB: hæ tamen vires non pares <lb></lb>eſſe debent toti ponderi priſmatis; ſed quia centrum gravita<lb></lb>tis E ab utroque extremo æqualiter diſtare ſupponitur, ſe<lb></lb>miſſis tantùm gravitatis percipitur in B. </s> <s id="s.000937">Quod ſi in codem <lb></lb>horizontali ſitu retineatur priſma ſivè à ſuſpendente obliquo <lb></lb>IB, ſivè ab obliquo ſuſtentante OB, utique retinentis, aut <lb></lb>ſuſtentantis vires æquipollere debent viribus retinentis aut <lb></lb>ſuſtentantis ad perpendiculum CB aut DB. </s> <s id="s.000938">Quemadmo<lb></lb>dum igitur pondera illa ſuper BO & BD æquiponderant, <lb></lb>quæ ſunt ut BO ad BD, ita vires, quæ ſecundùm eaſdem <lb></lb>lineas ac directiones æqualem effectum præſtare debent; in <lb></lb>eâdem Ratione BO ad BD eſſe oportet: Vires ergo retinen<lb></lb>tis BI obliqui ad vires retinentis CB ad perpendiculum ſunt <lb></lb>ut BO ad BD, hoc eſt, ductâ parallelâ CI, ut IB ad CB, <lb></lb>propter triangulorum OBD, CBI ſimilitudinem. </s> </p> <p type="main"> <s id="s.000939">Ut autem non hîc perperam nos philoſophari innoteſcat, <lb></lb>finge ſublatam ex A pyramidem, & conſtitutam in G ita, <lb></lb>ut ex B ad perpendiculum dependeat pondus aliquod æqui<lb></lb>librium efficiens cum priſmate: quo perpendiculari pondere <lb></lb>ſublato, ut priſma horizontale permaneat, certum eſt ſuper <lb></lb>plano inclinato BO requiri pondus, quod ad pondus per<lb></lb>pendiculare ex BD ſit ut BO ad BD: igitur ſi loco pon<lb></lb>deris applicentur ſecundùm eandem rectam lineam BO vires <lb></lb>alicujus viventis, à quo retineatur priſma in eodem ſitu ho<lb></lb>rizontali, ſatis apparet conatum debere eſſe ut BO ad cona<lb></lb>tum, qui ſecundùm perpendicularem requireretur ut BD. </s> <lb></lb> <s id="s.000940">Sicut itaque conatus deorſum trahens, cum fulcrum eſt in <lb></lb>G citrà centrum gravitatis E, ex inclinatione lineæ, ſecun<lb></lb>dùm quam fit, deſumitur, ita etiam conatus ſuſpendens IB, <pb pagenum="121" xlink:href="017/01/137.jpg"></pb>aut ſurſum urgens OB, cum fulcrum eſt in A ultrà centrum <lb></lb>gravitatis E, deſumendus eſt pariter ex inclinatione lineæ, ſe<lb></lb>cundùm quam applicatur priſmati, comparatè ad conatum per<lb></lb>pendicularem CB, vel DB, habita ſemper ratione diſtantiæ <lb></lb>fulcri à centro gravitatis. </s> </p> <p type="main"> <s id="s.000941">Ne quid verò dubitationis <lb></lb><figure id="id.017.01.137.1.jpg" xlink:href="017/01/137/1.jpg"></figure><lb></lb>ſuperſit, utrum OB deorſum, <lb></lb>& IB ſurſum trahentium pa<lb></lb>res ſint vires ſecundùm ean<lb></lb>dem rectam lineam OI, ſint <lb></lb>rotulæ duæ H & F circa ſuum <lb></lb>axem verſatiles infixæ extre<lb></lb>mitatibus regulæ, aut tigilli, <lb></lb>& ex funiculo rotularum ca<lb></lb>vitatibus inſerto dependeant <lb></lb>æqualia pondera L & G. </s> <s id="s.000942">Hæc <lb></lb>pondera ſibi viciſſim æquipon<lb></lb>derare manifeſtum eſt, quem<lb></lb>cumque tandem ſitum ſivè <lb></lb>perpendicularem, ſivè incli<lb></lb>natum, habeat regula, aut ti<lb></lb>gillus, cui rotulæ infixæ ſunt. </s> <s id="s.000943">Sit libræ jugum AB æqualiter <lb></lb>in E diviſum, circa quod punctum ſtabile moveri queat, & <lb></lb>in A adnectatur funiculo HF: ex B autem dependeat pondus <lb></lb>D æquale ponderi G, ſed ita obliquè diſpoſitum, ut linea BO <lb></lb>parallela ſit lineæ AF. </s> <s id="s.000944">Submove pondus L, remanent G <lb></lb>& D, quorum neutrum prævalere poteſt; ſunt enim æqualia <lb></lb>inter ſe, & per lineas ſimiliter inclinatas AF, BO agunt. </s> <s id="s.000945">Re<lb></lb>pone pondus L, & amove pondus G, item removeatur pon<lb></lb>dus D, & ſurſum ponatur æquale C; aio libræ jugum AB <lb></lb>adhuc retinere eumdem ſitum; quia ſcilicet pondera C & D <lb></lb>viciſſim æquiponderabant, ſicut etiam G & L: igitur quantum <lb></lb>virium habebat pondus D ad æquiponderandum ipſi G, tan<lb></lb>tumdem virium habet pondus C ad æquiponderandum ponde<lb></lb>ri L, hoc eſt eidem ponderi G. </s> <s id="s.000946">Sivè igitur in ſuperiori ſche<lb></lb>mate conſiderentur vires deorſum trahentes aut ſuſtentantes <lb></lb>OB, ſive retinentes IB, perinde eſt, & æqualium momento<lb></lb>rum cenſendæ ſunt. </s> </p> <pb pagenum="122" xlink:href="017/01/138.jpg"></pb> <figure id="id.017.01.138.1.jpg" xlink:href="017/01/138/1.jpg"></figure> <p type="main"> <s id="s.000947">Non jam horizontale ſit <lb></lb>priſma AB, ſed inclinatum, <lb></lb>& puncto A ſtabili innixum: <lb></lb>momenta ad deſcendendum, <lb></lb>ac proinde repugnantia ad <lb></lb>aſcendendum, ut ſuperiùs in<lb></lb>nuimus cap.14; æſtimanda <lb></lb>ſunt in plano DC inclinato, <lb></lb>quod cum AB angulos facit <lb></lb>rectos, & cum horizonte AE <lb></lb>concurrit in puncto E. </s> <s id="s.000948">Ducatur per B perpendicularis ad ho<lb></lb>rizontem FH, & ex H ad BE perpendicularis HO. </s> <s id="s.000949">Momen<lb></lb>ta gravitatis priſmatis in perpendiculari ad momenta ejuſdem <lb></lb>in inclinatà ſunt reciprocè ut inclinata EB ad perpendicula<lb></lb>rem BH, hoc eſt per 8.lib.6. ut HB ad BO, ſive (ductâ ex D <lb></lb>ſuper DB inclinatam perpendiculari DG ſecante rectam HF <lb></lb>in F) ut BF ad BD, propter ſimilitudinem triangulorum OBH, <lb></lb>DBF. </s> <s id="s.000950">Vires ergo retinentes in D ad vires retinentes in F ſunt <lb></lb>ut DB ad BF. </s> </p> <p type="main"> <s id="s.000951">Retineatur priſma ſecundùm obliquam GB, quæ producta <lb></lb>uſque ad Horizontalem concurrat in L. </s> <s id="s.000952">Iterum ex L ad DE <lb></lb>cadat ad angulos rectos LC, quæ perpendicularem FH ſecabit <lb></lb>in I: eſt autem IC parallela ipſi HO; ac propterea per 4.lib.6. <lb></lb>ut HB ad BO, ita IB ad BC, & per 11.lib.5. ut IB ad BC, <lb></lb>ita BF ad BD. </s> <s id="s.000953">Ad retinendum igitur priſma in eodem ſitu in<lb></lb>clinationis BAE per obliquam GB, vires æquipollentes viri<lb></lb>bus retinentibus in perpendiculari FB eſſe oportet ut BL ad <lb></lb>BI, quemadmodum retinentes per rectam DB ſunt ut BC. </s> </p> <p type="main"> <s id="s.000954">Quare datâ corporis inclinatione, cujus gravitas retinenda eſt <lb></lb>in eodem ſitu, ſumatur ejuſdem axis tranſiens per gravitatis <lb></lb>centrum, & ad axis extremitatem mobilem ducatur ipſi axi per<lb></lb>pendicularis DB, in quâ aſſumpto quolibet puncto D, ducatur <lb></lb>prædicto axi parallela DG, quæ ſecans lineas quaſlibet obli<lb></lb>quas, & perpendicularem ad Horizontem, dabit omnium obli<lb></lb>quarum ſuſpenſionum Rationem: Sic recta DG ſecans perpen<lb></lb>dicularem FB & obliquam GB determinat Rationem virium in <lb></lb>utrâque ſuſpenſione, ut ſcilicet ſint in Ratione BF ad BG, & <lb></lb>ſic de reliquis. </s> </p> <pb pagenum="123" xlink:href="017/01/139.jpg"></pb> <p type="main"> <s id="s.000955">Quòd ſi in gradibus data ſit inclinatio priſmatis, & funiculi <lb></lb>oblique ſuſpendenti declinatio a perpendiculo, ſtatim ex tabu<lb></lb>lis Sinuum, aut etiam Secantium, apparebit Ratio quæſita li<lb></lb>nearum: angulus enim, quem perpendicularis ad axem facit <lb></lb>cum perpendiculari ad Horizontem, æqualis eſt angulo incli<lb></lb>nationis priſinatis; angulo ſiquidem BAE inclinationis priſma<lb></lb>tis, æqualis eſt angulus EBH per 8.lib.6. ac proptereà etiam <lb></lb>ex 15.lib.1. qui illi eſt ad verticem DBF. </s> <s id="s.000956">Hinc ſi inclinatio<lb></lb>nis angulus ſit gr. 36. DB ad BF erit ut Radius ad Secantem <lb></lb>gr. 36. vel ut Sinus gr.54. complementi gr.36. ad Radium. </s> <s id="s.000957">At <lb></lb>angulus, quem facit linea obliquæ ſuſpenſionis cum perpendi<lb></lb>culari ad horizontem tranſeunte per priſmatis punctum; in quo <lb></lb>ſuſpenditur, eſt æqualis angulo, quem eadem ſuſpenſionis li<lb></lb>nea facit cum perpendiculo tranſeunte per aliud extremum <lb></lb>ejuſdem lineæ ſuſpenſionis, cui applicatur potentia retinens: <lb></lb>duæ enim perpendiculares prædictæ ſunt inter ſe parallelæ, & <lb></lb>linea ſuſpenſionis in eas incidens alternos angulos facit æquales <lb></lb>per 27.lib.1. Si igitur GB à ſuo perpendiculo, quod ex G in <lb></lb>horizontem cadat, declinat gr.25. etiam FBG eſt gr.25. To<lb></lb>tus igitur angulus DBG eſt aggregatum anguli inclinationis <lb></lb>priſmatis, & anguli declinationis funiculi ſuſpendentis: igitur <lb></lb>DBG eſt gr.61, & poſitâ DB ut Radio, erit BG Secans gr.61. <lb></lb>Vel ſi comparanda ſit BG cum BF, qui angulus GFB ex<lb></lb>ternus per 32.lib.1. æqualis eſt duobus internis oppoſitis tran<lb></lb>guli DBF, erit GFB gr.126; at FBG eſt gr.25, igitur FGB <lb></lb>eſt gr.29. Quare BF ad BG eſt ut Sinus gr. 29. ad Sinum <lb></lb>gr.126, hoc eſt ſupplementi gr.54. Apparet ex his primò minimas vires exerceri, ſi linea reten<lb></lb>tionis cadat ad perpendiculum in axem corporis elevati cum in<lb></lb>clinatione; quia ſcilicet cum in D ſit angulus rectus, recta BD <lb></lb>eſt omnium linearum ex B puncto excuntium, & in rectam <lb></lb>DG cadentium minima: quò autem major fuerit obliquitas, <lb></lb>eò etiam majores vires requiri, quia longiores ſunt Secantes <lb></lb>angulorum majorum in B poſito Radio BD. </s> </p> <p type="main"> <s id="s.000958">Secundò fieri poteſt, ut pare: vires requirantur, ſi linea re<lb></lb>tentionis faciat cùm axe corporis elevati angulum acutum, ac <lb></lb>ſi faciat cùm eodem angulum obtuſum, ut ſi fuerit recta MB; <lb></lb>ipſa enim pariter opponitur angulo recto BDM, ac proinde <pb pagenum="124" xlink:href="017/01/140.jpg"></pb>eò major eſt quàm recta BD, quò fuerit major angulus MBD, <lb></lb>qui poteſt eſſe æqualis angulo DBF, vel DBG; quo caſu <lb></lb>etiam ipſa BM æqualis erit ipſi BF aut BG. </s> <s id="s.000959">Ex quo ulteriàs <lb></lb>ſequitur, ſi à retinente obliquè fiat tractio elevando magis ac <lb></lb>magis priſma ſic inclinatum, mutari ſubinde momenta: hoc ta<lb></lb>men intercedit diſcrimen, quod trahentis linea initio applicata, <lb></lb>ut angulum faciat acutum cum axe priſmatis, in ipsâ tractione <lb></lb>ſemper majorem facit cum ipſo axe angulum, donec veniat ad <lb></lb>angulum rectum conſtituendum, ut ſi MB traheretur, donec <lb></lb>coincidat cùm DB, quæ pariter moveri intelligatur: contrà <lb></lb>verò trahentis linea applicata, ut cum axe faciat angulum ob<lb></lb>tuſum, in ipsâ tractione magis adhuc obtuſum angulum conſti<lb></lb>tuit, donec tractionis linea (ſi tamen fieri id poſſit) in unam <lb></lb>rectam lineam cum axe priſmatis conveniat. </s> <s id="s.000960">Quare in primâ <lb></lb>illâ tractione minuitur conatus, in hac ſecunda augetur. <lb></lb><figure id="id.017.01.140.1.jpg" xlink:href="017/01/140/1.jpg"></figure></s> </p> <pb pagenum="125" xlink:href="017/01/141.jpg"></pb> <figure id="id.017.01.141.1.jpg" xlink:href="017/01/141/1.jpg"></figure> <p type="main"> <s id="s.000961"><emph type="center"></emph>MECHANICORUM<emph.end type="center"></emph.end><emph type="center"></emph>LIBER SECUNDUS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000962"><emph type="center"></emph><emph type="italics"></emph>De cauſis motus Machinalis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000963">INNOTUIT, opinor, quantum ad præſens inſtitu<lb></lb>tum ſatis eſſe poſſit, centrum gravitatis ex iis, quæ <lb></lb>libro ſuperiore dicta ſunt: nunc propiùs ad ipſam <lb></lb>machinalem ſcientiam accedendum, quam Mecha<lb></lb>nicam dicimus. </s> <s id="s.000964">Hæc Geometriæ ſubjicitur; neque <lb></lb>enim, ut illa, puram corporum quantitatem moliſque exten<lb></lb>ſionem abſtractè conſiderat, ſed quatenus gravitati illigatam <lb></lb>aut levitati; nihil tamen ſolicita de ipsâ corporum materie, au<lb></lb>reáne ſit, an lapidea. </s> <s id="s.000965">Quamvis autem ea quoque Statices pars, <lb></lb>quam Hydroſtaticen indigitamus, ſe pariter in corporum gra<lb></lb>vitate conſiderandâ exerceat, aliam tamen ſibi contemplatio<lb></lb>nem aſſumit; motum ſiquidem corporum ſingulorum naturæ <lb></lb>congruentem, pro humorum, in quos incurrunt, diverſitate, <lb></lb>potiſſimùm ſpeculatur: Mechanice verò eatenus ſolùm ingeni<lb></lb>tam corporibus propenſionem in motum aut quietem explorat, <lb></lb>ut earum facultati perſpectæ vim poſſit opportunâ inſtrumento<lb></lb>rum machinatione inferre. </s> <s id="s.000966">Quapropter ut certâ methodo ma<lb></lb>chinas oneribus movendis pares conſtruere valeamus, motus <lb></lb>machinalis cauſas antè cognitas habere neceſſe eſt, quàm ma<lb></lb>chinas ipſas aggrediamur. </s> <s id="s.000967">His porrò jactis fundamentis ope<lb></lb>roſum non erit inædificare, & machinarum ſingularum vires, <lb></lb>ſivè ſimplices illæ ſint, ſivè compoſitæ, exponere: adeò ut iis <lb></lb>ritè intellectis, quæ hoc ſecundo libro diſputabuntur, vix quie<lb></lb>quam in reliquo opere ſuperſit difficultatis. <pb pagenum="126" xlink:href="017/01/142.jpg"></pb></s> </p> <p type="main"> <s id="s.000968"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000969"><emph type="center"></emph><emph type="italics"></emph>Quem ad finem Machinæ inſtruantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.000970">FInis, quò demum unaquæque actio refertur, primus animo <lb></lb>concipitur, præſtituiturque, & idonea ad agendum ſubſi<lb></lb>dia, quæ deligenda ſunt, moderatur. </s> <s id="s.000971">Hinc ille primus nobis <lb></lb>in hâc contemplatione occurrit; quem ſcilicet ad finem ma<lb></lb>chinæ inſtituantur, inſtruantúrque, conſiderandum; ut ad <lb></lb>hanc quaſi regulam cæteræ cauſæ dirigantur, & formentur. </s> <lb></lb> <s id="s.000972">Fortè dixerit quiſpiam magnificè, eo conſilio machinas à no<lb></lb>bis excogitatas, ut naturam arte vincamus; quemadmodum <lb></lb>enim ſcribit Antipho Poeta apud Ariſtotelem in quæſt.Mechan. <lb></lb>ſub initium, <foreign lang="grc">τέχνῃ κρατοῦμεν, ὠ̄ν φύσ<gap></gap> νικώμεθα. </foreign></s> <s id="s.000973">Sed hic planiſ<lb></lb>ſimè philoſophandi locus eſt, non gloriandi inſolentiùs. </s> <s id="s.000974">Quare <lb></lb>fatendum eſt apertè, adhiberi machinas in ſubſidium infirmi<lb></lb>tatis; ut quod virium imbecillitas onus loco movere, aut omni<lb></lb>nò, aut niſi ægerrimè ſola nequiret, illud demum facilè, quò <lb></lb>libuerit, aut trahat, aut impellat, aut etiam expellat quantum<lb></lb>vis reluctans, ſi machina accedat. </s> </p> <p type="main"> <s id="s.000975">Dupliciter autem inſita corporibus gravitas obſiſtit moventi, <lb></lb>ſi ab alio in alium locum transferenda fuerit: diſparibus enim <lb></lb>momentis mora infertur motui, ſi hic fluido in corpore ac ſe<lb></lb>quaci, puta in aëre aut aquâ, perficiatur, ac ſi ſuprà ſolidam <lb></lb>conſiſtentemque planitiem raptetur moles, ſive Horizonti pa<lb></lb>rallela jaceat planities, ſive molli aut arduâ inclinatione eriga<lb></lb>tur in clivum. </s> <s id="s.000976">Et quidem ſi ſolidum in corpus non incumbat <lb></lb>onus, ſed in aëre ſuſpenſum pendeat, ac ſurſum trahere opor<lb></lb>teat, certos ad calculos revocari gravitatis momenta poterunt, <lb></lb>quibus machina proportione reſpondeat: nam quamvis aër aëri <lb></lb>præſtet tenuitate, non ea tamen eſt in levitatibus differentia, ut <lb></lb>hinc in gravium corporum momentis diſſimilitudo notabilis <lb></lb>oriatur. </s> <s id="s.000977">Quare ſicut laberetur turpiter, qui machinam ſaxo ab <lb></lb>imo mari ad ſummam ſuperficiem elevando parem inſtrueret, ſi <lb></lb>nullâ factâ virium acceſſione illud in aërem extrahi poſſe ſibi <pb pagenum="127" xlink:href="017/01/143.jpg"></pb>perſuaderet; ita nimis exiguè & exiliter ad calculos revocaret <lb></lb>aërem, qui pro diſpari ejus levitate modum machinæ ſtatueret; <lb></lb>in materiâ etenim, ex quâ machina componitur, nullus eſt <lb></lb>huic minutæ ſubtilitati locus, quæ aciem omnem fugit, niſi <lb></lb>cum veritas in diſputatione limatur. </s> <s id="s.000978">Id quod de eâ pariter <lb></lb>gravitationis inæqualitate dictum velim, quæ ex inæquali à cen<lb></lb>tro gravium diſtantiâ ortum habet, ut lib.1. cap. 4. diſputatum <lb></lb>eſt: Quia in tantulo Spatio, in quo nos labor noſter exercet, <lb></lb>illa momentorum exuperantia ſub ſenſum non cadit. </s> <s id="s.000979">Quo cir<lb></lb>ca ſatis ſupérque habemus, quòd moventis vires ac molis mo<lb></lb>vendæ pondus reputantes ita inter ſe conferamus, ut virium <lb></lb>imbecillitas adhibitâ machinâ convaleſcat, & repugnanti one<lb></lb>ris gravitati non reſiſtat modò, ſed & præſtare poſſit, nullâ aut <lb></lb>loci aut aëris habitâ ratione. </s> </p> <p type="main"> <s id="s.000980">Verùm quàm facile eſt corporis gravitatem cùm ex mate<lb></lb>riæ ſpecie, tùm ex molis magnitudine inveſtigare; tàm mul<lb></lb>tis difficultatibus impedita res eſt, ſi examinandum ſit, <lb></lb>quantùm ex mutuo corporum ſe contingentium tritu retardetur <lb></lb>motus: non enim quiſquis pendulum in aere majoris campanæ <lb></lb>malleum poteſt à perpendiculo dimovere, earum eſt virium, ut <lb></lb>illum pariter in terrâ jacentem propellere valeat: & decennis <lb></lb>puer arrepto fune illigatam cymbam, modicè fluctuante ſalo, <lb></lb>ad ſe trahit; quam vix, aut ne vix quidem, robuſtioris lacerti <lb></lb>vir dimoveat, ubi arenoſo vado inſederit: cum tamen eadem aut <lb></lb>ligneæ cymbæ aut ferreo malleo gravitas innata permaneat. </s> <lb></lb> <s id="s.000981">Eſt autem tùm ſubjecti corporis conſiſtentis, tùm impoſiti one<lb></lb>ris movendi ſuperficies ſpectanda, quatenus ſe contingunt: <lb></lb>Nam ſi lapideum globum pondo 100 in planitie conſtitutum <lb></lb>non rotare modo, ſed & rectâ urgere poſſis, non itidem cubum <lb></lb>pondere parem & materiâ ſimilem æquali facilitate urgebis; <lb></lb>quia ſcilicet globus tenuiſſimâ ſui parte ſuppoſitam planitiem <lb></lb>contingens minus invenit impedimenti ex proximè ſubjecti <lb></lb>corporis aſperitate, quæ prominulas impoſiti globi particulas re<lb></lb>moretur; at cubus longè pluribus ſui partibus plano adhæret, at<lb></lb>que adeò multiplicatá partium hujus in illius partes incurren<lb></lb>tium reſiſtentiâ, augeri quoque movendi <expan abbr="difficultatẽ">difficultatem</expan> neceſſe eſt. </s> </p> <p type="main"> <s id="s.000982">Quoniam verò obtineri nequit, ut corporum ſe contingen<lb></lb>tium ſuperficies ſint continuo lævore lubricæ, earum autem <pb pagenum="128" xlink:href="017/01/144.jpg"></pb>aſperitates anomalæ ſunt ac multiformes, reſiſtentia indè pro<lb></lb>veniens ſub certam legem non cadit; ſed quantum conjectura <lb></lb>aſſequi valemus, illa potius ex antiquis experimentis æſtimanda <lb></lb>videtur, quàm mathematicis ratiocinationibus indaganda. </s> <s id="s.000983">In <lb></lb>hoc uno nimirùm facem præferre poteſt Geometria, ut ſi reli<lb></lb>qua prorſus paria ſint, nec alia ſit quàm molis aut figuræ diſſi<lb></lb>militudo, quantum ex hoc capite movendi difficultas augea<lb></lb>tur, minuaturve, innnoteſcat: cæterùm plenè atque perfectè <lb></lb>explicare, quantum reſiſtentiæ ex aſperarum ſuperficierum <lb></lb>conflictione oriatur, quis niſi temerè conetur? </s> </p> <p type="main"> <s id="s.000984">Poſteriori huic malo, quod ſuperficierum aliqua aſperitas <lb></lb>creat, occurritur, ſi pingui ſequacíque materiâ oblitæ lubri<lb></lb>cæ fiant: Sic Automatis, rotarum ſe ſe mutuá collabellatione <lb></lb>mordentium converſione, horas indicantibus velocitas conci<lb></lb>liatur, ſi quis denticulos oleo leviter perungat: ſic plauſtrorum <lb></lb>tarditatem, equorumque laborem, ut imminuant aurigæ, axes <lb></lb>rotarúmque modiolos axungiâ illinunt; & cæmentarij majora <lb></lb>ſaxa attollentes, trochleæ orbiculis ſapone perfricatis, quærunt <lb></lb>laboris compendium. </s> <s id="s.000985">Hinc Amſterodami paſſim obſervatur <lb></lb>lubricas fieri trahas ceruiſiæ doliis, ſimilíve pondere, onuſtas; <lb></lb>cum enim equus non procul abeſt à ponte, in quem aſcenden<lb></lb>dum eſt, is, qui equum agit, centonem unguine delibutum <lb></lb>currenti trahæ ſubſternit, ut expreſſus ex centone pinguis hu<lb></lb>mor inficiat duo illa longiora tigna, quibus traha inſiſtit, ac <lb></lb>proinde lubrica machina faciliùs raptetur per vias lateribus <lb></lb>ſtratas. </s> <s id="s.000986">Sic Dio lib.50. de Auguſto loquens. <emph type="italics"></emph>Audivi eum trire<lb></lb>mes ex mari exteriore per murum in ſinum tranſtuliſſe, & loco Pa<lb></lb>langum, per quos ducerentur, tergoribus animalium recens cæſorum <lb></lb>loco inunctis uſum,<emph.end type="italics"></emph.end> Et Silius Ital. </s> <s id="s.000987">lib.13.v.444. <lb></lb><emph type="italics"></emph>Lubrica roboreis aderant ſubſtramina plauſtris, <lb></lb>Atque recens cæſi tergo prolapſa juvenci, <lb></lb>Æquorcem rota ducebat per gramina puppim.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.000988">Verùm nec frequens eſſe poteſt, nec commodum, remedium <lb></lb>hoc ex pingui liquore petitum; illud certius erit ad imminuen<lb></lb>dam moram ex tritu corporum ortam, quod ea ſe invicem <lb></lb>quàm minimùm contingant. </s> <s id="s.000989">Quoniam verò deducendi one<lb></lb>ris ſuperficiem amplam mutare ſæpè nequimus, aut illud rap<lb></lb>tandum trahæ imponimus, quæ non niſi tigillis duobus læviga-<pb pagenum="129" xlink:href="017/01/145.jpg"></pb>tis ſubjectam planitiem tangit; aut in plauſtrum injicimus, cu<lb></lb>jus rotæ ſolum calcantes dum convertuntur, axem tantum<lb></lb>modo terunt, compendio ſanè mirabili; nam dum rotæ modio<lb></lb>lus axem ſemel terit, pedes circiter viginti provehitur onus, aut <lb></lb>demum ſublato corporum mutuo tritu cylindros, vel ſcytalas <lb></lb>illi ſubjicimus, ut nihil noceat ſoli aſperitas, niſi quatenus hæc <lb></lb>cylindrorum vel ſcytalarum converſionem remoratur. </s> </p> <p type="main"> <s id="s.000990">Huc ſpectat id, quod non ſine voluptate obſervare aliouan<lb></lb>do contigit Bononiæ. </s> <s id="s.000991">Tres erant viri nec admodum robuſti, <lb></lb>qui ut aliquot ingentes ſaccos farinâ plenos in domum infer<lb></lb>rent, paratum habuerunt axem binis rotulis circiter ſeſquipal<lb></lb>maribus inſtructum; axi jungebatur craſſiuſculus temo ſacco<lb></lb>rum longitudinem vix ſuperans. </s> <s id="s.000992">Erecto ſacco machinulam ap<lb></lb>plicabant, tùm ſaccum pariter cum temone reclinabant, & ne <lb></lb>temoni incumbens juxtà longitudinem ſaccus in alterutram <lb></lb>partem inclinaretur, duo hinc & hinc retinebant pariter, ac <lb></lb>propellebant, ut tertium arrepto temone trahentem labore le<lb></lb>varent: Hâc ratione alium atque alium ſaccum tenuiſſimo la<lb></lb>bore in domum importarunt; erectoque iterum temone delap<lb></lb>ſus eſt ex machinulâ ſaccus, ſtetitque erectus. </s> </p> <p type="main"> <s id="s.000993">Ex his itaque conſtat in machinâ inſtruendâ non ſolùm in<lb></lb>genitæ corpori movendo gravitatis rationem habendam eſſe; <lb></lb>ſed & plani, ſuper quo illud deducendum eſt, jacens-ne ſit? </s> <lb></lb> <s id="s.000994">an erectum? </s> <s id="s.000995">læve, an aſperum? </s> <s id="s.000996">amplâ, an tenui ſuperficie <lb></lb>contingat? </s> <s id="s.000997">hinc ſi quidem varia reſiſtentiæ momenta exur<lb></lb>gunt. </s> <s id="s.000998">Illud tamen plerumque contingit, quod ſi attollendo ad <lb></lb>perpendiculum oneri par fuerit machina, illa pariter ſufficiat <lb></lb>ad onus idem ſuper plano horizontali, aut inclinato deducen<lb></lb>dum: vix enim fieri poteſt (niſi ſumma ſit ſuperficierum ſe <lb></lb>contingentium aſperitas) ut quantum reſiſtentiæ demitur à <lb></lb>plano ſuſtinente, tantumdem addatur ex mutuo prominentium <lb></lb>particularum conflictu. </s> </p> <p type="main"> <s id="s.000999">Quamquam & ipſa aſperitas facit aliquod laboris compen<lb></lb>dium: nam licèt continens ac perpetuus non ſit motus, ſed al<lb></lb>ternâ quiete interruptus ſuper arduo clivo, modico tamen co<lb></lb>natu prohibetur moles, ne prolapſa ſiſipheum crect laborem; <lb></lb>quia aſpera ſuperſicies motui obſiſtens efficit ne corporis gravi<lb></lb>tas deorſum conetur pro plani inclinatione. </s> <s id="s.001000">Satis igitur fuerit <pb pagenum="130" xlink:href="017/01/146.jpg"></pb>abſolutæ oneris gravitati machinam ita reſpondere, ut illi ad <lb></lb>perpendiculum ſuſtollendo cæteroqui impares vires ſufficiant: <lb></lb>qui enim valuerit, adhibitâ machinâ, molem attollere, poterit <lb></lb>illam pariter, ejuſdem machinæ ope, in plano quocunque tra<lb></lb>here aut propellere; ſi maximè cylindri aut rotæ ei ſubji<lb></lb>ciantur. </s> </p> <p type="main"> <s id="s.001001">Hîc autem fortè nec à præſenti inſtituto alienum, nec lecto<lb></lb>ri injucundum accidat, ſi quæ, aliquando comminiſci placuit, <lb></lb>ſubjiciam, cum narrantem quendam audirem de campaná in<lb></lb>gentis ponderis facillimè agitatâ ſubjectis æneis rotulis, quæ <lb></lb>demum longo ævo confectæ diſſipatæ fuere; ſed quonam artifi<lb></lb>cio, quóve ordine diſpoſitæ fuiſſent, enarrare omninò non <lb></lb>poterat. </s> <s id="s.001002">Quare mecum ipſe reputans, quî fieri id potuiſſet, in <lb></lb>eam incidi ſententiam, ut exiſtimarem graviſſimam campanam <lb></lb>potuiſſe facilè pulſari, imminutâ reſiſtentiâ, quæ oritur ex mu<lb></lb><figure id="id.017.01.146.1.jpg" xlink:href="017/01/146/1.jpg"></figure><lb></lb>tuo fulcri, & axis tritu. </s> <s id="s.001003">Sint <lb></lb>enim binæ rotulæ B & C ex <lb></lb>ære ſolido, quarum diameter <lb></lb>ſit in aliquâ Ratione multiplici <lb></lb>ad diametrum axis, cui cam<lb></lb>pana innititur. </s> <s id="s.001004">Axis autem ſe<lb></lb>midiameter ſit AE, rotulæ ve<lb></lb>rò BE in ratione duplâ; ergo <lb></lb>& periphæriæ ſunt in eâdem Ratione: dum igitur punctum I <lb></lb>in H perficit quadrantem, convertit pariter rotulam; cujus pe<lb></lb>ripheriæ ſemiquadranti coæquatur. </s> <s id="s.001005">Quare ſi rotula infixa eſſet <lb></lb>axi, cujus ſemidiameter BG eſſet æqualis ſemidiametro AE, <lb></lb>fieret affrictus cum octante peripheriæ axis rotulæ B; ſed quia <lb></lb>etiam in rotulâ C fieret æqualis affrictus cum ejuſdem axe, jam <lb></lb>nihil ferè emolumenti haberetur, quia totus affrictus æquè eſ<lb></lb>ſet, ac ſi quadrans EO in fulcro ſtabili & cavo converteretur: <lb></lb>& potiùs laboris in agitandâ campanâ compendium eſſet, ſi ro<lb></lb>tulæ fixæ hærerent, axis ſi quidem cylindricus cum ſit, ſubjectas <lb></lb>rotulas in lineâ tangeret modico ſcilicet tritu; rotularum autem <lb></lb>axes concavis earum partibus congruunt in ſuperficie, quæ te<lb></lb>ritur, dum rotulæ convertuntur: niſi fortè cylindrica axis <lb></lb>BG ſuperficies convexa paulò minor eſſet concavâ rotulæ <lb></lb>ſuperficie, eæque propterea ſecundùm lineam ſe continge-<pb pagenum="131" xlink:href="017/01/147.jpg"></pb>rent, ut ex 13. lib.3. facilè eſt demonſtrare; id quod nec rarò <lb></lb>contingit. </s> </p> <p type="main"> <s id="s.001006">Verum non eſt neceſſe rotulis B & C tàm ſolidos axes dare; <lb></lb>nam ſi axis AE toti campanæ oneri ferendo par eſt, bini æqua<lb></lb>les axes duplici ponderi reſiſtunt: ſatis igitur eſſet, ſi axes ſin<lb></lb>guli B & C, oneris ſemiſſem ſuſtinerent. </s> <s id="s.001007">Cum verò cylindro<lb></lb>rum reſiſtentiæ, ne frangantur, ſint in triplicatâ Ratione ſua<lb></lb>rum diametrorum, ſufficeret inter ſemidiametrum AE, & ejus <lb></lb>ſemiſſem duas medias proportione continuâ reperire, quæ enim <lb></lb>proxime minor eſſet ipsá AE, eſſet ſufficiens ſemidiameter cy<lb></lb>lindri ſubduplam habentis ſoliditatem ac reſiſtentiam. </s> <s id="s.001008">Sed <lb></lb>adhuc minor requiritur ſemidiameter, quia onus axes rotula<lb></lb>rum B & C obliquè premit; ex quo fit campanæ gravitationem <lb></lb>in axes illos eſſe ſecundùm lineas AB, AC, non autem juxtà <lb></lb>perpendiculum AD: igitur ut AD ad AB, ita reciprocè gra<lb></lb>vitatio ſuper AB ad gravitationem ſuper AD: atqui gravita<lb></lb>tio in alterutrum axium, ut ſummum ſubdupla eſt totius gra<lb></lb>vitationis; ergo gravitatio ſuper BA minor eſt ſubduplâ. </s> <s id="s.001009">Quâ <lb></lb>autem Ratione minor ſit conſtat. </s> <s id="s.001010">Cum enim detur tùm ſemi<lb></lb>diameter AE, tùm etiam BE, nota eſt tota BA, & BD, pari<lb></lb>ter, ipſi BE æqualis, nota eſt; igitur ex 47 lib. 1. etiam AD <lb></lb>innoteſcit, cujus ſcilicet quadratum habetur, ſi ex BA quadra<lb></lb>to dematur quadraturm BD. </s> </p> <p type="main"> <s id="s.001011">Cum itaque, ex hypotheſi, BA ſit 3, cujus quadratum 9, & <lb></lb>BD 2, cujus quadratum 4, remanet quadratum 5, ejuſque Ra. </s> <lb></lb> <s id="s.001012">dix 2. 23″. eſt recta DA: gravitatio igitur ſuper BA ad totam <lb></lb>campanæ ſuper utrumque axem B, & C, gravitationem eſt <lb></lb>223 ad 600′. </s> <s id="s.001013">Quoniam verò ſolidorum ſimilium reſiſtentia <lb></lb>eſt in triplicatâ Ratione laterum homologorum (in cylindris <lb></lb>autem diametrorum ratio habetur) quærantur duo medij pro<lb></lb>portionales numeri inter 600″ & 223″. </s> <s id="s.001014">Id quod aſſequeris, ſi <lb></lb>cujuſlibet extremi quadratum ducas in alium extremum, pro<lb></lb>ducti enim Radix cubica eſt terminus proximus illi numero, <lb></lb>cujus quadratum aſſumpſiſti. </s> <s id="s.001015">Primi igitur 600 quadratum <lb></lb>360000 duc in 223, & producti 80280000, Radix cubica eſt <lb></lb>431 1/3 proximè: alterius verò extremi 223 quadratum 49729 <lb></lb>ductum in 600 dat 29837400, cujus Radix cubica 310 proxi<lb></lb>mè eſt alter medius. </s> <s id="s.001016">Sunt igitur quatuor numeri 600. 431 1/3. <pb pagenum="132" xlink:href="017/01/148.jpg"></pb>310. </s> <s id="s.001017">223 continuè proportionales proximè, ſpretis fractiuncu<lb></lb>lis. </s> <s id="s.001018">Quare ſi ſiat ut 600′ ad 431′, ita ſemidiameter AE ad BN, <lb></lb>erit hæc ſemidiameter quæſita ſufficienter reſiſtens. </s> </p> <p type="main"> <s id="s.001019">Quoniam itaque BE dupla eſt ipſius AE, & AE ad BN <lb></lb>facta eſt ut 600 ad 431, erit BE ad BN ut 1200 ad 431; & ſe<lb></lb>cundùm hane eandem Rationem ſe habebunt ſemiquadrantes <lb></lb>ab illis deſeripti. </s> <s id="s.001020">Atqui octans peripheriæ ex Radio BE æqua<lb></lb>lis eſt quadranti ex Radio. </s> <s id="s.001021">AE; igitur quadrans EO ad ſemi<lb></lb>quadrantem ex Radio BN eſt pariter ut 1200 ad 431: Qui igi<lb></lb>tur affrictus axis campanæ cum fulcro ſtabili & cavo eſſet 1200, <lb></lb>rotulæ B cum ſuo axe eſt 431, cui æqualis eſt alterius rotulæ C <lb></lb>affictus cum ſuo axe; ac proinde ſubjectis rotulis, quarum dia<lb></lb>meter ſit tantum dupla diametri axis. </s> <s id="s.001022">campanæ, affrictus eſt ut <lb></lb>862, ad affrictum qui eſſet ut 1200. Si itaque rotularum dia<lb></lb>meter ad campanæ axem. </s> <s id="s.001023">non tantùm dupla, ſed vel tripla, <lb></lb>vel quadrupla ſit, multò minor erit affrictus, majorque in agi<lb></lb>tanda campanâ facilitas. </s> </p> <p type="main"> <s id="s.001024">Quamvis autem iſtâ conſimilivè diligentiâ induſtriâque plu<lb></lb>rimum imminui poſſit particularum conflictus, quæ ſe viciſſim <lb></lb>terentes moram atque impedimentum motui inferrent; non illa <lb></lb>tamen ex eo propriè veréque dicitur motio machinalis, quòd <lb></lb>inſtrumento atque apparatu aliquo perficiatur, niſi, ſpectatâ <lb></lb>dumtaxat oneris gravitate, potentia illi movendo cæteroqui im<lb></lb>par, ſubſidium ſibi comparet ex machinâ. </s> <s id="s.001025">Machina autem non <lb></lb>idem eſt, ſi plenè atque perfectè interpretari velis, ac inſtru<lb></lb>mentum; licet enim machina omnis inſtrumentum ſit, non ta<lb></lb>men inſtrumentum quodlibet machinæ vocabulum continuò <lb></lb>ſortitur, ſi motionem aliquatenùs juvet; ſed illud prætereà ef<lb></lb>ficiat neceſſe eſt, quod ejus ope naturalem ac inſitam vim cor<lb></lb>poris loco dimovendi ſuperet vis minor extrinſecùs adhibita. </s> <lb></lb> <s id="s.001026">Cum ergò onus hærere in ſalebrâ, non ex inſitâ vi, ſed ex proxi<lb></lb>mi etiam atque continentis corporis aſperitate proveniat, & <lb></lb>inſtrumenta, quibus hoc tantummodo impedimentum tollitur, <lb></lb>idem planè efficiant, quod pinguis humor lubricum parans iter; <lb></lb>neque hæc machinæ magis dici poſſunt, quàm centones ungui<lb></lb>ne delibuti, ſi ritè ſubſternantur, neque motus propterea inter <lb></lb>machinales numerandus videtur, quorum hîc cauſas yeſtigare <lb></lb>nobis propoſitum eſt. </s> <s id="s.001027">Quamquam negandum non ſit hæc pari-<pb pagenum="133" xlink:href="017/01/149.jpg"></pb>ter ad mechanicam contemplationem pertinere; quippe quæ <lb></lb>machinis, præcipuo nimirum mechanices ſcopo. </s> <s id="s.001028">affinia ſunt; <lb></lb>etiamſi ad illas non velut ſubjectæ partes ad genus revocentur: <lb></lb>& inſtrumentis hujuſmodi ſi machinæ appellationem tribuere <lb></lb>placuerit, non admodum de nomine diſputabo; res enim hîc <lb></lb>ſpectatur, non verba penduntur. </s> </p> <p type="main"> <s id="s.001029">Sed neque hîc diſputare velim, utrùm in motuum machina<lb></lb>lium cenſum irrepant, an verò iis ritè annumerandi ſint motus <lb></lb>illi, quos ſurſum deorſum, ultrò citróque perficiendos eatenus <lb></lb>expeditè, nec exiguo laboris compendio, molimur, quatenus <lb></lb>eos intervallis ita diſtinguimus, ut nos quidem corpus deprima<lb></lb>mus, ut adducamus, ab alio verò extollatur, aut reducatur: in <lb></lb>his ſiquidem ſæpè nihil eſt, quod noſtram imminuat operam, <lb></lb>ſi motiones ſingulæ attendantur; quamquam motui univerſo <lb></lb>adjumentum importat continens illa conatûs noſtri, alienique <lb></lb>ſubſidij, viciſſitudo. </s> <s id="s.001030">Hinc ſi quis <lb></lb><figure id="id.017.01.149.1.jpg" xlink:href="017/01/149/1.jpg"></figure><lb></lb>ad contundendam in æneo morta<lb></lb>rio A contumacem aliquam mate<lb></lb>riam graviore piſtillo ferreo opus <lb></lb>habeat, haud dubium quin ei mul<lb></lb>tâ lacertorum vi contendendum <lb></lb>ſit, ut illum extollat; cumque ope<lb></lb>roſius multo ſit inflexum corpus <lb></lb>erigere, quàm erectum inclinare, <lb></lb>multóque moleſtius brachia tanto <lb></lb>pondere pre gravata attollere, quàm <lb></lb>eorum gravitati obſecundando de<lb></lb>primere, ſatis conſtat, quantum ſi<lb></lb>bi laboris detractum eat, ſi ſuperio<lb></lb>re in loco tranſverſum tigillum <lb></lb>CD circa axem E verſatilem ſtatuat, paribúſque intervallis <lb></lb>hinc ex C pendeat fune ſuſpenſus piſtillus B, hinc verò in D <lb></lb>plumbea maſſa adnectatur, quâ ita piſtillus præponderetur, ut, <lb></lb>nemine hunc retinente aut deprimente, illa aliquanto gravior <lb></lb>in ſubjectum prodeuntis è pariete tigni caput G recidens ſpon<lb></lb>te ſubſidat. </s> <s id="s.001031">Omnis ſcilicet extollendi piſtilli labore ſublato, <lb></lb>vel ſolum brachiorum pondus piſtillo additum ſatis eſſe ali<lb></lb>quando poterit ad leviuſculè tundendam materiam, licebitque <pb pagenum="134" xlink:href="017/01/150.jpg"></pb>modò contento, modò remiſſo conatu opus urgere. </s> <s id="s.001032">Id quod <lb></lb>pariter continget, ſi operâ unâ opus duplex efficere placuerit; <lb></lb>nam ſi ex D plumbeæ maſſæ loco alius pendeat æque, ac plum<lb></lb>bum, gravis piſtillus, pondere præpollens elevabit piſtillum B, <lb></lb>aliámque viciſſim in altero ſubjecto mortario conteret mate<lb></lb>riam ſponte ſuâ cadens: cumque piſtillorum gravitates non ad<lb></lb>modum inter ſe diſpares ſint, neque multum laboris eum ſubi<lb></lb>re neceſſe erit, cui piſtillum B deprimendi munus incumbit. </s> </p> <p type="main"> <s id="s.001033">Quâ in re, ſi motus univerſus ita tribuatur in partes, ut tun<lb></lb>dentis quidem motiones ſingulæ ſeorſim ſpectentur, non ille <lb></lb>profectò ſe juvari ſentit, quippe quem, præter vires ad commi<lb></lb>nuendam materiam neceſſarias, conatum quoque adhibere <lb></lb>oportet ad vincendam præponderantis plumbi, aut piſtilli gra<lb></lb>vitatem. </s> <s id="s.001034">Cæterùm ſi totius motûs, qui Arſi pariter conſtat ac <lb></lb>Theſi, habeatur ratio, inſiciari nemo poterit, minus multo la<lb></lb>boris impendi, quàm ſi hæc omnia ſublata intelligantur. </s> <s id="s.001035">Qua<lb></lb>re nec incongruum prorſus videatur motûs machinalis voca<lb></lb>bulum, cum verſatilis tigillus CD ad libræ Rationes manifeſtò <lb></lb>revocetur, quam certè ex machinarum albo nemo expungit, ni<lb></lb>ſi qui ſolas quinque facultates, & quæ ex his componuntur, ma<lb></lb>chinas indigitare voluerit, & libram ad vectem referri poſſe <lb></lb>pernegarit. </s> </p> <p type="main"> <s id="s.001036">Nec diſſimilis ineunda videtur dicendi ratio, ſi quid alternis <lb></lb>ciendum motibus ſic diſponitur, ut, cum primùm quidem mo<lb></lb>vetur, corpus aliud vi flectatur, quod poſtmodum facultate <lb></lb>elaſticâ, ſe reſtituens illud viciſſim moveat; quemadmodum <lb></lb>paſſim in eorum officinis videre eſt, qui rudes arborum, aut <lb></lb>elephantini dentis particulas in toreumata elaborant: primùm <lb></lb>enim artifex pede ſubjectum vectem premens, toreuma in gy<lb></lb>rum ducit, haſtulámque ſuperiore in loco poſitam pariter in<lb></lb>flectit; quæ ſibi mox ſuam reparans rectitudinem, funiculum<lb></lb>que cylindrulo verſatili circumplicatum retrahens, illud iterum <lb></lb>ſua per veſtigia verſat, ut accuratè exquiſitéque tornetur. </s> <s id="s.001037">Sic <lb></lb>aliquid ſubtiliter ac delicatè ſecturus, ut ſerrulam rectâ addu<lb></lb>cas, reducáſque, operæ tantùm ſemiſſem tibi reſervans, arcum <lb></lb>intentum ex adverſo ſtatuito, ac medio nervo ſerrulam alliga<lb></lb>to; hac enim adductâ magis flectetur arcus, qui ſe ſe mox reſti<lb></lb>tuens illam viciſſim reducet. </s> </p> <pb pagenum="135" xlink:href="017/01/151.jpg"></pb> <p type="main"> <s id="s.001038">Hæc ſanè laboris in movendo compendia ex elaſmate, vel ex <lb></lb>antiſacomate petita, quemadmodum & ea, quæ mutuum cor<lb></lb>porum tritum atque conflictum minuunt, ut pote Mechanico <lb></lb>artificio conſtituta, eumdemque in finem ac machinæ, quibus <lb></lb>hoc nomen præcipuè tribuitur, videlicet in infirmæ potentiæ <lb></lb>ſubſidium excogitata, eſto illis primas deferant, non tamen <lb></lb>omninò rejicerem, ſi in machinarum cenſu prodirent, iiſque <lb></lb>ſe peterent adſcribi. </s> <s id="s.001039">Triplicem enim in ſpeciem tribui poſſe vi<lb></lb>detur univerſum machinarum genus: Prima eas complectitur <lb></lb>facultates, quarum ope motui facilitas conciliatur, quocum<lb></lb>que tandem ex capite ſivè tantummodo ex inſitâ in corporibus <lb></lb>gravitate, ſivè non ex eâ dumtaxat, ſed ex partium aſperitate <lb></lb>movendi difficultas conſurgat. </s> <s id="s.001040">Altera eſt, quæ mutuam qui<lb></lb>dem corporum ſe contingentium conflictionem minuit, ſed ad <lb></lb>vincendam oneris gravitatem ipſi potentiæ momenta non addit. </s> <lb></lb> <s id="s.001041">Tertia demùm eatenus per ſe, quia talis eſt, moventem juvat, <lb></lb>quatenus ejus operam alternam efficit, cum tamen neque gra<lb></lb>vitatem vincat, neque quod ex partium triru impedimentum <lb></lb>oritur, extenuet, niſi cum alterutra, aut utraque ſuperiori ſpe<lb></lb>cie, amico fœdere copuletur. </s> <s id="s.001042">Alternam autem operam appel<lb></lb>lo, cum in motu ex duplici motione compoſito alterutram effi<lb></lb>cit potentia, ſivè illæ ſibi invicem adverſantes ſuccedant, ut <lb></lb>Arſis ac Theſis, Adductio atque Reductio, ſivè in unam tem<lb></lb>perentur, ut cum premere ſimul oportet ac agitare: ſic plana <lb></lb>vitra expolientes in ſpecula, inter ipſa, & lacunar bacillum in<lb></lb>flectunt, qui ſe reſtituere tentans vi elaſticâ, ſpeculum validè, <lb></lb>quantum opus eſt, admovet atque applicat ad ſubjectum pla<lb></lb>num, adeò ut ad artificem à preſſu immunem nil aliud ſpectet, <lb></lb>quàm ſpeculum urgere, retrahere, contorquere. </s> <s id="s.001043">Verùm ta<lb></lb>metſi de his omnibus in hac tractione paſſim ſe offeret dicendi <lb></lb>locus, primus tamen diſputationis noſtræ ſcopus erit prima illa <lb></lb>ſpecies, ipſæ nimirum facultates, quarum potiſſimum momen<lb></lb>ta expendimus, cum motûs machinalis cauſas inquirimus. <pb pagenum="136" xlink:href="017/01/152.jpg"></pb></s> </p> <p type="main"> <s id="s.001044"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001045"><emph type="center"></emph><emph type="italics"></emph>Impetùs motum proximè efficientis natura <lb></lb>explicatur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001046">QUicquid movetur, qualecumque eſt, cauſam habeat mo<lb></lb>ventem neceſſe eſt, ut hoc quidem ſponte ſuâ, illud ve<lb></lb>rò alienâ vi ex alio in alium locum migret. </s> <s id="s.001047">Suopte ingenio mo<lb></lb>ventur tùm corpora gravia aut levia, ut ſi extrà præſcriptum <lb></lb>ſibi à naturâ locum conſtituta fuerint, ſuo quæque ordine diſ<lb></lb>ponantur; tùm rara aut denſa, ut ſi per vim hæc extenuata fue<lb></lb>rint, illa concreverint, naturæ ſtatum ſibi reparent; tùm ani<lb></lb>mantia, quibus cum à naturâ tributum ſit, ut ſe, vitam, cor<lb></lb>puſque tueantur, ſtimulos admovet appetitus, ut ea declinent, <lb></lb>quæ nocitura videantur, omniaque, quæ ſint ad vivendum ne<lb></lb>ceſſaria, acquirant, & parent. </s> <s id="s.001048">Vi extrinſecus impreſsâ locum <lb></lb>mutant, quæcumque in motu non ſerviunt naturæ, ſed alieno <lb></lb>reguntur arbitrio; ut iis contingit, quæ raptantur, pelluntur, in <lb></lb>gyrum ducuntur, projiciuntur, & hujus generis motibus <lb></lb>cientur. </s> </p> <p type="main"> <s id="s.001049">Quoniam verò gravium, & levium celeritatem naturâ ur<lb></lb>gente incitari, jaculorum autem, ac miſſilium, motum uſque <lb></lb>eò ſenſim langueſcere, ut planè deficiat, obſervamus; etiamſi <lb></lb>moventi naturæ, quæ ex Philoſophi decretis ſubſtantia eſt, mo<lb></lb>tûs originem ultimam tribuamus, jure tamen optimo aliquid <lb></lb>naturæ ipſi ac motui, interjectum agnoſcimus (Impetum no<lb></lb>minamus) cujus intentionem ac remiſſionem velocitas ac tar<lb></lb>ditas conſequatur. </s> <s id="s.001050">Cum enim eadem deſcendentis lapidis na<lb></lb>tura perſeveret, nec illa in ſuâ poteſtate ſit, aut optione delatâ, <lb></lb>ut eligat utrum velit, motum arbitrio ſuo incitare, aut remit<lb></lb>tere valeat; qui fieri poſſit, ut deſcendens velocitatem augeat, <lb></lb>niſi ei, quem primùm produxit, alium atque alium momentis <lb></lb>ſingulis impetum adjiciat? </s> <s id="s.001051">Illud certè extrà omnem controver<lb></lb>ſiam poſitum videtur, naturam gravem ſponte ſuâ non aſcen-<pb pagenum="137" xlink:href="017/01/153.jpg"></pb>dere: quid ergo illud eſt, quod eburneum globulum in ſub<lb></lb>jectam rupem delapſum reſilire cogit, aut ſibi relictum plum<lb></lb>bum ex fune ſuſpenſum ultrà perpendiculum, naturá repugnan<lb></lb>te, ſurſum provehit, & eò quidem altiùs, quò ex altiore loco <lb></lb>globulus aut plumbum deciderunt? </s> <s id="s.001052">niſi quia conceptus naturâ <lb></lb>procurante impetus pergit motum efficere, ipsâ etiam naturâ <lb></lb>quantum poteſt, obſiſtente. </s> <s id="s.001053">Quòd ſi corpus alienâ vi longiùs <lb></lb>emiſſum moveatur, extrinſecùs impetum imprimi neceſſe eſt: <lb></lb>quem ſanè non concipit, ubi primùm à projiciente ſejunctum <lb></lb>fuerit; nihil enim prodeſſet ad longiorem lapidis jactum fun<lb></lb>dam iterum ac tertiò circumducere, niſi alium atque alium im<lb></lb>petum lapis conciperet, quandiù funditori adhærens unâ cum <lb></lb>ipſo movetur. </s> </p> <p type="main"> <s id="s.001054">Quæcumque igitur moventur, impetum habent, quo ferun<lb></lb>tur; cui ſatis probabili conjectura, proxima vis motum efficien<lb></lb>di tribuenda videtur. </s> <s id="s.001055">Id quod in projectis quidem, iiſque om<lb></lb>nibus, quæ naturâ repugnante moventur, ita manifeſtum eſt, <lb></lb>ut id pluribus demonſtrare non oporteat; nulla ſiquidem adeſt <lb></lb>inſita motûs cauſa; ab impetu igitur illo extrinſecùs impreſſo <lb></lb>motum effici neceſſe eſt. </s> <s id="s.001056">At in cæteris, quibus ſe movendi <lb></lb>principium ineſt, neme jure negaverit aut in motu impetum <lb></lb>acquiri, aut velocitatis incrementum ex impetus acceſſione ori<lb></lb>ri: quî enim fieret, ut excurrentes objectam foſſam ampliore <lb></lb>ſaltu tranſilirent faciliùs, quàm nullo præcedente curſu, ſi in <lb></lb>curſu ipſo conceptus impetus non augeretur? </s> <s id="s.001057">Jam verò ſi ſe<lb></lb>cundo temporis momento incitatur magis motus, quàm primo, <lb></lb>urgente ſcilicet etiam impetu, quem corpus priore motu acqui<lb></lb>ſivit; hic utique impetus, quem nunc gignere non poteſt <lb></lb>prior motus, cum perierit, extitit pariter cum priore motu: <lb></lb>natura igitur movens priore momento & motum effecit & im<lb></lb>petum. </s> <s id="s.001058">Atqui impetum ex eorum ſaltem genere eſſe, quæ mo<lb></lb>tum efficiant, conſtat ex velociore motu poſterioribus momen<lb></lb>tis, naturâ prorſus immutatâ, factoque impetûs incremento: <lb></lb>contrà verò motu, quâ motus eſt, impetum non augeri ſatis <lb></lb>indicant miſſilia, quorum velocitas, dum moventur, ſenſim <lb></lb>elangueſcit. </s> <s id="s.001059">Igitur & priore illo temporis momento non mo<lb></lb>tus impetum; ſed impetus motum proximè effecit; impetum <lb></lb>autem procreavit innata movendi vis; cui id circo motio tri-<pb pagenum="138" xlink:href="017/01/154.jpg"></pb>buitur, quia id illa gignit, quod proximè motus conſequitur, <lb></lb>& ad motum efficiendum natura deſtinavit. </s> <s id="s.001060">Quid? </s> <s id="s.001061">quòd mo<lb></lb>tui per ſe, quia ex alio in alium locum continuata migratio eſt, <lb></lb>efficientiam ægrè tribuere poſſumus: quippe qui, cum in <lb></lb>fluxione conſiſtat, ita ut locus loco, ſeu potius, ut ſcholæ lo<lb></lb>quuntur, Ubicatio Ubicationi, priori ſcilicet pereunti ſuccedat <lb></lb>poſterior æquè fugax, inferioris notæ cenſendus eſt quàm im<lb></lb>petus naturâ ſuâ aliquandiù permanens: labentia enim ſtanti<lb></lb>bus deteriora eſſe, cæteris paribus, quis neget? </s> <s id="s.001062">effectum au<lb></lb>tem causâ præſtabiliorem eſſe non poſſe ipſa originis notio ſua<lb></lb>det, ne quid effectus habeat, quod non acceperit, aut aliquid <lb></lb>cauſa dederit, quo ipſa careret. </s> <s id="s.001063">Non igitur impetum motus, <lb></lb>ſed motum impetus efficit. </s> </p> <p type="main"> <s id="s.001064">Porrò cum definitas ad agendum vires unaquæque cauſa ob<lb></lb>tineat, certa eſt impetûs menſura, quæ cum innatâ movendi <lb></lb>facultate ita adæquatur, ut eo quaſi termino circumſcripta cen<lb></lb>ſenda ſit potentia movens, nec unquam validiore conatu poſſit <lb></lb>ſe ipſa urgere; ſi tamen omnem impetum antecedente motu aſ<lb></lb>ſumptum mente ſecernas. </s> <s id="s.001065">Et quidem omne animal (quippe <lb></lb>cui ineſt appetitio & declinatio naturalis ejus, quod naturæ ac<lb></lb>commodatum eſt, aut infenſum) non ſemper univerſam illam <lb></lb>impetûs menſuram exequitur, ſed ut vult, ita utitur motu ſui <lb></lb>corporis, quem aucto aut diminuto impetu modò intendit, mo<lb></lb>dò remittit, pro ut interiore motu, rerumque appetitu ſimula<lb></lb>tur. </s> <s id="s.001066">Contrà verò inanimum non ſuo arbitrio motûs intentio<lb></lb>nem moderatur, ſed naturæ juribus obſequens nihil prætermit<lb></lb>tit impetûs, & quantum eniti poteſt, opportunum in locum, ſi<lb></lb>bique à naturâ conſtitutum, contendit. </s> <s id="s.001067">Cave tamen exiſtimes <lb></lb>parem eſſe lapidis ejuſdem, & in aëre, & in aquâ deſcendentis <lb></lb>impetum: natura ſcilicet ex medio dividendo, in quo perficien<lb></lb>dus eſt motus, metitur impetûs modum. </s> </p> <p type="main"> <s id="s.001068">Sed quoniam non pauca ſunt, quæ motui ſæpè adverſantur, <lb></lb>hinc eſt non ſemper eandem eſſe corporis ſe moventis velocita<lb></lb>tem, quamvis pari impetu producto connitatur: deteritur nimi<lb></lb>rum tantum impetus, quantum ſatis eſt ad impedimentum ſub<lb></lb>movendum. </s> <s id="s.001069">Sivè enim objectum corpus propellendum ſit, ſivè <lb></lb>medij particulæ locum ægrè dantes divellendæ aut compri<lb></lb>mendæ ſint, ſivè connexam molem pariter rapi oporteat, ſivè <pb pagenum="139" xlink:href="017/01/155.jpg"></pb>quid aliud hujuſmodi adſit, cui niſi vis inferatur, ut ex alio <lb></lb>in alium locum migret præter naturam, irritus reddatur corpo<lb></lb>ris in motum propenſi conatus; ſatis conſtat illud motu agitan<lb></lb>dum eſſe exteriùs: atque adeò quantum impetus illi imprimi<lb></lb>tur oppoſitæ propenſioni æquale, motui tantumdem ſub<lb></lb>trahitur. </s> </p> <p type="main"> <s id="s.001070">In iis ſanè, quæ alienâ vi extrinſecùs moventur, quia infi<lb></lb>nitè progredi non licet, aliqua demum origo deprehenditur, <lb></lb>cui naturalis ſit motus: natura ſiquidem vis eſt ciens motus in <lb></lb>corporibus neceſſarios; ita tamen certis tenetur legibus uni<lb></lb>verſitatis rerum concinnitatem ſpectantibus, ut ne ab iis diſce<lb></lb>dat, ſingularibus corporibus vim aliquam inferri permittat, ubi <lb></lb>adverſis propenſionibus inter ſe confligentibus validior præſtat <lb></lb>imbecilliori. </s> <s id="s.001071">Sic quia nefas eſt aut corpora inanitatibus inter<lb></lb>jectis conciſa hiare, aut unum in proximi corporis locum, niſi <lb></lb>eo recedente, penetrare, aut diverticula flexioneſque in motu <lb></lb>ſponte quærere; ideò & liquor in longiore ſiphonis, aut ſpiri<lb></lb>talis diabetis, crure deſcendens continuum liquorem in brevio<lb></lb>re crure aſcendere cogit, totumque ex vaſe demum exhaurit; & <lb></lb>rapidè lapſus torrens ſaxa rapit, objectaſque moles disjicit; & <lb></lb>ad perpendiculum cadens lapis ſubjectum vitrum comminuit, <lb></lb>ſuique veſtigium in terrâ validiùs preſsâ relinquit. </s> <s id="s.001072">Verùm il<lb></lb>lud firmum ac perpetuum eſt, quòd ubi plus violentiæ opus eſt, <lb></lb>parem conatum languidior motus conſequitur. </s> <s id="s.001073">Id quod in <lb></lb><figure id="id.017.01.155.1.jpg" xlink:href="017/01/155/1.jpg"></figure><lb></lb>ſiphone ABC obſervare in promptu eſt, ex <lb></lb>cujus oſculo C inæqualis aquæ copia de<lb></lb>fluit paribus temporis intervallis: quò enim <lb></lb>magis aquæ ſuperficies in vaſe deprimitur, <lb></lb>eò lentiùs aqua ex ſiphone dilabitur: <lb></lb>quamvis ſcilicet aquæ crus BC implentis <lb></lb>pares ſint ſemper ad deſcendendum vires, ſi <lb></lb>nihil, aut ſaltem non inæqualiter, repugnet, <lb></lb>aquæ tamen crus BD brevius, & BI longius, & BA adhuc <lb></lb>longius implentis diſpar eſt in aſcenſu repugnantia; ac pro<lb></lb>pterea cum earumdem virium BC minor ſit Ratio ad majorem <lb></lb>reſiſtentiam BI, quàm ad minorem BD, languidior quoque <lb></lb>motus eſt deſcendentis aquæ ex BC, cùm graviorem aquam <lb></lb>BI, quàm cùm minùs gravem BD ſursùm trahere oportet. </s> <s id="s.001074">At <pb pagenum="140" xlink:href="017/01/156.jpg"></pb>ſi externum ſiphonis crus ità decurtatum ſit in E, ut oſculum E <lb></lb>& aquæ in vaſe ſuperficies I paribus abſint ab Horizonte inter<lb></lb>vallis, aquam ideò hærere, nec amplius ex E fluere conſtat, <lb></lb>quia aquæ BE ad deſcendendum propenſionem, par aquæ BI <lb></lb>repugnantia, ne aſcendat, elidit. </s> <s id="s.001075">Quòd ſi demum aquam in <lb></lb>vaſe imminuas, ut ejus ſuperficies paulò infra I, atque adeò <lb></lb>infra E oſculum deprimatur, non jam aqua hæret in E, ſed ſua <lb></lb>per veſtigia in EB remeare cogitur, præponderatâ nimirum <lb></lb>majore gravitate aquæ implentis crus paulo longiùs quàm BI, <lb></lb>atque adeò quàm BE, quod illi ex hypotheſi conſtituimus <lb></lb>æquale; tantóque velociùs ab aquâ interioris cruris raperetur <lb></lb>exterior, quantò depreſſior facta fuiſſet in vaſe aquæ ſuper<lb></lb>ficies. </s> </p> <p type="main"> <s id="s.001076">Hinc itaque fit, ut pro variâ corporis motui obſiſtentis re<lb></lb>pugnantiâ modò plus, modò minus impetûs reliquum ſit, quo <lb></lb>motû, celeritas aut tarditas perficiatur. </s> <s id="s.001077">Et ſi tanta ſit eorum <lb></lb>omnium, quæ motui moram inferunt, obſiſtentia, ut ad eam <lb></lb>vincendam plus impetûs neceſſe ſit, quàm pro potentiæ facul<lb></lb>tate, tunc nullus efficitur motus, quo corpus ex loco in locum <lb></lb>transferatur, ſed aliqua ex peregrino impetu fit partium com<lb></lb>preſſio, aut diſtractio; neque enim omnes corporis particulæ <lb></lb>homogeneæ ſunt, aut ita compactæ citrà omnes poros, ut nul<lb></lb>la tenuiorum particularum compreſſio aut diſtractio conſequi <lb></lb>poſſit. </s> <s id="s.001078">Quod ſi ea ſit corporis per vim movendi natura aut poſi<lb></lb>tio, ut nullum planè ſivè lationis, ſivè rotationis, ſivè vibratio<lb></lb>nis, ſivè conſtipationis, ſivè dilatationis motum concipere poſ<lb></lb>ſit, aut violento in ſtatu permanere languido illo impetu, quem <lb></lb>vis extrinſeca efficere valeret, nullum quoque impetum reci<lb></lb>pit; quippe qui idcircò imprimeretur, ut motum præter natu<lb></lb>ram efficeret, aut ut naturalem motum retunderet, aut etiam <lb></lb>prorſus impediret. </s> <s id="s.001079">Quemadmodum enim ſi corporis alicujus <lb></lb>ſpecificam gravitatem in aquâ mutari non poſſe conſtet, infer<lb></lb>re continuò licet, corpus idem neque raritatem neque denſita<lb></lb>tem in aquâ aſſùmere poſſe; ex his ſiquidem ſpecificæ gravita<lb></lb>tis mutatio oriretur: ita pariter ubi nihil haberi poteſt eorum, <lb></lb>quæ impetum extrinſecùs impreſſum neceſſariò conſequuntur, <lb></lb>impetum quoque abeſſe non immeritò conjectamus. </s> </p> <p type="main"> <s id="s.001080">Si quis tamen animum diligentiùs adverrat, manifeſtò de-<pb pagenum="141" xlink:href="017/01/157.jpg"></pb>prehendet corpus idem magis repugnare motui, ſi celeriùs mo<lb></lb>vendum ſit, minùs verò, ſi tardiùs: ſic ferreæ anſæ cubiculi <lb></lb>oſtio infixæ magnetem armatum applicui, & ſiquidem paulò <lb></lb>velociùs magnetem traherem, disjungebatur ab ansâ; at len<lb></lb>tiùs trahentem ſubſequebatur oſtium, magnetis ſcilicet vim <lb></lb>non ſuperans, ubi lentè res peragebatur. </s> </p> <p type="main"> <s id="s.001081">An non oneri, quod potentia præ ſui tenuitate propellere <lb></lb>non poſſe videtur, motus, qui momentis ſingulis ſenſum om<lb></lb>nem fugiat, conciliari poteſt, adeò ut, ſi illa quidem conſtan<lb></lb>ter urgeat, elapſo demùm longo temporis intervallo appareat? </s> <lb></lb> <s id="s.001082">Sic incumbentem glebam tenerrimus naſcentis frugis caulicu<lb></lb>lus tandem diſcutit; duriſſima marmora ſcindens caprificus lo<lb></lb>co movet; & ædificia ſubſediſſe, ac inæquabile ſolum preſſiſſe, <lb></lb>rimæ demùm loquuntur. </s> <s id="s.001083">Tota igitur corporis, quod præter <lb></lb>naturam movendum eſt, repugnantia metienda eſt, quâ ex <lb></lb>principio ipſo motum detrectante, quâ ex motûs celeritate, aut <lb></lb>tarditate: adeò ut pro variâ horum connexione diſpar movendi <lb></lb>difficultas oriatur. </s> </p> <p type="main"> <s id="s.001084">Ex quo fit impetu eodem moveri celeriùs poſſe corpus, quod <lb></lb>minorem ſubit violentiam, tardiùs verò, cui vis major infer<lb></lb>tur, &, ſi eadem ſit reciprocè Ratio tarditatis ad velocitatem, <lb></lb>quæ eſt minoris violentiæ ad majorem violentiam, parem fore <lb></lb>utrobique movendi difficultatem, cùm par ſit repugnantia, quæ <lb></lb>ex motûs tùm ſpecie, tùm intentione componitur. </s> <s id="s.001085">Si enim mo<lb></lb>les aliquâ tantâ vi raptetur, ut, quo tempore decies arteria pul<lb></lb>ſum edit, paſſum unum conficiat; quantum virium adhiberi <lb></lb>oporteat, ut paribus temporis momentis ad tres paſſus eadem <lb></lb>moles promoveatur? </s> <s id="s.001086">utique, ſi cætera omnia paria ſint, triplo <lb></lb>majorem conatum adhibendum concedes, intenſione exten<lb></lb>ſionem compenſante: nam quemadmodum iterùm ac tertiò re<lb></lb>petendus fuiſſet prior ille conatus ad æquale ſemper ſpatium pa<lb></lb>ri tarditate percurrendum; ita quamvis conatui conatus non <lb></lb>ſuccedat, triplici tamen conatu opus erit, ut tempore eodem <lb></lb>motus ille triplo major perficiatur. </s> <s id="s.001087">Nonnè & agricolæ terram <lb></lb>ſubigentes foſſione glebarum, tam multiplices adhibent operas, <lb></lb>quàm breviori tempore opus abſolvere meditantur? </s> <s id="s.001088">Eò igitur <lb></lb>magis reſiſtit corpus motui, quò celeriùs agitandum eſt; con<lb></lb>trà verò minùs repugnat, quò tardiùs. </s> </p> <pb pagenum="142" xlink:href="017/01/158.jpg"></pb> <p type="main"> <s id="s.001089">Quare ſi duo ſint corpora, quorum alterum alteri præſtet <lb></lb>triplo majori gravitate, atque hæc pari celeritate attollenda ſint, <lb></lb>diſparem exigunt conatum pro gravitatis Ratione: ſi par ſit eo<lb></lb>rum gravitas, motus autem alterius reliquo triplo velocior eſſe <lb></lb>debeat, inæqualem pariter exigunt conatum, ſed pro ratione <lb></lb>velocitatis: ſi demùm & diſpar ſit gravitas, & inæqualis velo<lb></lb>citas, eam eſſe conſtat repugnantiam, quæ tùm ex gravitate, <lb></lb>tùm ex velocitate componitur; atque adeò ſi corpus alterum <lb></lb>triplo gravius triplo etiam velociùs movendum eſſet, noncuplex <lb></lb>eſſet ejus repugnantia; ſin autem triplo levius triplo majori <lb></lb>velocitate quàm corpus triplo gravius, moveretur, par eſſet eo<lb></lb>rum obſiſtentia, paremque conatum exigerent. </s> </p> <p type="main"> <s id="s.001090">Hinc ſatis apertè conſtat, datâ tum reſiſtentiarum, tum velo<lb></lb>citatum Ratione, ſi gravitas altera nota ſit, reliquam facilè inno<lb></lb>teſcere: ſi nimirùm nota gravitas per ſuam velocitatem ducatur, <lb></lb>& in datâ Ratione reſiſtentiarum reperiatur huic producto ter<lb></lb>minus homologus; quo per ignotæ gravitatis velocitatem da<lb></lb>tam diviſo, prodibit Quotiens index quæſitæ gravitatis. </s> <s id="s.001091">Sint <lb></lb>duo corpora inæqualia, & ad ea movenda requiratur conatus <lb></lb>in Ratione ſeſquialterâ, motus autem eorum ſint ut 7 ad 8, & <lb></lb>illud quod minùs reſiſtit, moveturque velocitate ut 7, numeret <lb></lb>gravitatis libras 4. Reliqui corporis validiùs reſiſtentis, cujus <lb></lb>velocitas eſt ut 8, gravitas ſic invenietur. </s> </p> <p type="main"> <s id="s.001092">Libræ 4 ducantur per numerum ſuæ velocitatis 7, & fit 28. <lb></lb>Quia igitur reſiſtentiæ ſunt, ut 2 ad 3 ex hypotheſi, & unius <lb></lb>corporis reſiſtentiâ, quæ ex gravitate & motûs velocitate com<lb></lb>ponitur, eſt 28, fiat ut 2 ad 3, ita 28 ad aliud, & erit 42 re<lb></lb>ſiſtentia alterius corporis compoſita ex ejus velocitate & gravi<lb></lb>tate. </s> <s id="s.001093">Atqui velocitas nota eſt 8; igitur divisâ totâ reſiſtentiâ <lb></lb>42 per 8; prodibit quotiens 5 1/4 index quæſitæ gravitatis. </s> <s id="s.001094">Quare <lb></lb>ad movendas libras 5 1/4 velocitate ut 8, requiritur conatus ſeſ<lb></lb>quialter conatûs neceſſarij ad movendas libras 4 velocitate <lb></lb>ut 7. Eadem eſto de reliquis ac ſimilibus conjectura. </s> </p> <p type="main"> <s id="s.001095">Ex his præterea manifeſtum eſt corporis per vim dimovendi <lb></lb>reſiſtentiam ex ſolâ naturâ, & principio inſito, quod motui re<lb></lb>pugnat, abſolutè definiri non poſſe; motum ſi quidem ab omni <lb></lb>prorsùs celeritatis aut tarditatis menſurâ ſejungere non poſſu<lb></lb>mus; idcircò non niſi habitâ ratione celeritatis, aut tarditatis, <pb pagenum="143" xlink:href="017/01/159.jpg"></pb>ex quibus reſiſtentia componitur, reſiſtentia ipſa innoteſcere <lb></lb>poterit. </s> <s id="s.001096">Quare & impetus à facultate movendi principium ha<lb></lb>bente productus major ſit neceſſe eſt, quàm dimoti corporis <lb></lb>repugnantia; quæ varia prorsùs cùm ſit, nunc quidem majo<lb></lb>rem, nunc verò minorem impetum exigit, ut ab eo vincatur; <lb></lb>nam ſi pares confligerent vires, à neutrâ parte ſtaret victoria. </s> </p> <p type="main"> <s id="s.001097">Quod autem ad ipſam motûs originem ſpectat, ea, quæ vi<lb></lb>vunt, ab iis, quæ vitâ omnino carent, ſecernenda ſunt: hæc <lb></lb>enim (ſcilicet non viventia) propterea motum expetunt, ut <lb></lb>violentiam, quam ſubeunt, excutiant, nec unquam à loco, ſeu <lb></lb>ſtatu, ſecundùm naturam opportuno ſponte recedunt; quem<lb></lb>admodum eunti per ſingula conſtabit. </s> <s id="s.001098">Sic gravibus & levibus <lb></lb>ſuis in locis quietem natura indixit, non motum; nec deor<lb></lb>ſum conantur aut ſurſum, niſi alieno in loco, hoc eſt, in me<lb></lb>dio diſpari gravitate aut levitate prædito conſtitutâ: ſic quæ<lb></lb>cumque elaſticâ facultate pollent, motum non moliuntur, niſi <lb></lb>cum ſibi naturalem partium figuram, ſitumque reparare opor<lb></lb>tet. </s> <s id="s.001099">At motum, cujus origo vita eſt, natura perficit, etiamſi <lb></lb>nulla præceſſerit violentia: ſic ſtirpes dum augentur, & creſ<lb></lb>cunt, earum particulæ locum mutant; ſic vitali facultate in<lb></lb>fluentibus per nervos in <expan abbr="animaliũ">animalium</expan> muſculos ſpiritibus, quos ani<lb></lb>males vocant, intenduntur muſculi, motuſque membrorum con<lb></lb>ſequitur: quamvis ante motum nec ſtirpis particulæ, nec anima<lb></lb>lis membra vim <expan abbr="ullã">ullam</expan> ſubierint in loco minimè congruo retenta. </s> </p> <p type="main"> <s id="s.001100">Quæcunque igitur ob id ipſum in motum prona ſunt, quia <lb></lb>vim patiuntur, impetum illicò concipiunt, ac vis iis illata eſt, <lb></lb>quo naturalem locum, ſeu ſtatum, recipere valeant, licèt ſæpè <lb></lb>irrito conatu, niſi quatenùs adverſo hoc impetu illatam ab ob<lb></lb>ſiſtente violentiam retundunt, vim aliquam illi viciſſim infe<lb></lb>rentes. </s> <s id="s.001101">Sic onera bajulorum humeros, quibus ſuſtinentur, <lb></lb>premunt, aut penduli brachij; ex quo ſuſpenduntur, muſcu<lb></lb>los ac ligamenta fatigant: id quod pariter in corpore inanimo <lb></lb>cernere licet; quemadmodum enim ex diuturnâ prementis <lb></lb>deorſum ponderis, ac muſculorum ſursùm urgentium luctâ, <lb></lb>diſſipatis ſpiritibus, laſſitudo in animali oritur, ita pariter ſub<lb></lb>jectum aſſerem longâ temporis morâ pondus curvat, aut etiam <lb></lb>demùm frangit, & funem, ex quo pendet, non intendit ſolùm, ſed <lb></lb>etiam tandem aliquando corrupto particularum nexu disjicit. </s> </p> <pb pagenum="144" xlink:href="017/01/160.jpg"></pb> <p type="main"> <s id="s.001102">Quo id autem pacto contingat, explicare operoſum non fue<lb></lb>rit funiculi texturam conſideranti; ex tenuiſſimis ſcilicet linei <lb></lb>aut cannabini corticis longâ maceratione, & plurimâ tunſione <lb></lb>extenuati particulis in ſpiram contortis filum cohæret; ex filis <lb></lb>autem pluſculis in ſpiram pariter contortis funiculus, & pluri<lb></lb>bus funiculis craſſiores rudentes conflantur: quod ſi diſſolvatur <lb></lb>omnis ſpira, non cohærent funiculi aut fili partes. </s> <s id="s.001103">Spira diſ<lb></lb>ſolvitur factâ in contrarium revolutione; quò autem laxioribus <lb></lb>gyris flectitur, eò faciliùs villi ſinguli ex cæteris, quibus im<lb></lb>plicantur, extrahuntur; & uno ab aliorum communione ſe<lb></lb>juncto, amplitudo ſpatij faciliorem exitum proximis relinquit: <lb></lb>ex quo fit faciliùs ſemper ac faciliùs poſſe funiculum frangi; <lb></lb>filo enim uno rupto, aut extracto, facilior eſt in contrarium re<lb></lb>volutio, & ſpira fit amplior, ac reliqua fila faciliùs extrahun<lb></lb>tur. </s> <s id="s.001104">Obſervamus autem non rarò appenſum ex funiculo pon<lb></lb>dus aliquandiu in gyrum contorqueri; dum ſcilicet ſuâ gravi<lb></lb>tate deorſum connitens intendit funiculum, contorta fila in <lb></lb>contrarium revolvuntur. </s> <s id="s.001105">Sed &, quamvis nulla fieret in con<lb></lb>trarium revolutio, ſatis conſtat ex illâ intenſione funiculum <lb></lb>diſtrahi, ac produci; atque adeò ſpiram laxiorem fieri, paula<lb></lb>timque unum aut alterum villum educi, locumque fieri vapo<lb></lb>ribus, qui proximum villum corrumpentes faciliori ſciſſioni pa<lb></lb>rant, atque adeò, ſerpente lue, demùm non tot integri ſuper<lb></lb>ſunt villi, qui poſſint ponderis gravitati obſiſtere, quin dif<lb></lb>fringantur. </s> <s id="s.001106">Ex quo ſatis apparet ſuſpenſum pondus, licèt non <lb></lb>omninò deſcendat, impetum tamen concipere, quo retinenti <lb></lb>repugnat, & vim aliquam viciſſim infert. </s> </p> <p type="main"> <s id="s.001107">Nec abſimili ratione in reliquis vim patientibus contingere <lb></lb>obſervabimus, ea ſcilicet moliri illicò naturalis ſtatûs repara<lb></lb>tionem, aliquidque efficere, licèt tenuiſſimum, quod demum <lb></lb>appareat, ubi temporis morâ augmentum ceperit. </s> <s id="s.001108">Sic haſtam <lb></lb>per vim inflexam ſi continuò dimittas, illa ſeſe reſtituit, facul<lb></lb>tate elaſticâ; at ſi dies aliquot, aut etiam diutiù per vim ſi<lb></lb>nuata permanſerit, ſibi dimiſſa antiquam rectitudinem non re<lb></lb>parat; elanguit nimirùm facultas elaſtica, quæ ex violentâ par<lb></lb>ticularum compreſſione aut diſtractione oriebatur. </s> <s id="s.001109">Cùm enim <lb></lb>primùm haſta flectitur, particulæ concavam curvaturæ partem <lb></lb>reſpicientes comprimuntur, contra verò, quæ convexam reſpi-<pb pagenum="145" xlink:href="017/01/161.jpg"></pb>ciunt, diſtrahuntur; quare tùm quæ, raræ, tùm quæ denſæ factæ <lb></lb>ſunt, dum vim illicò prorsùs excutere conantur, conſpirant, ut <lb></lb>priſtinam haſtæ rectitudinem moliantur: Quod ſi id non li<lb></lb>cuerit, hæ quidem aliam ex anguſtiis evadendi, quâ facilior <lb></lb>patet via, rationem tentant, ita ut demùm ſubtiliſſimas in ru<lb></lb>gas criſpentur, illæ verò ſeſe ad anguſtiora ſpatia ſenſim reci<lb></lb>pientes mutuum nexum ſolvunt, tenuiſſimoſque poros relin<lb></lb>quunt, aut ſi qui priùs interjecti fuerint, ampliùs hiare per<lb></lb>mittunt. </s> <s id="s.001110">Id quod ubi jam contigerit, fruſtrà ſubmoves, quæ <lb></lb>admoveras impedimenta; & ſpontè curvaturam haſta ſervat, <lb></lb>niſi fortè particulis omnibus adhuc per tempus non licuerit <lb></lb>vim totam excutere; tunc enim ſe ſe languidiùs reſtituunt, pro <lb></lb>ratione reliquæ violentiæ. </s> <s id="s.001111">Hinc patet arcum, quò fuerit con<lb></lb>tentus atque adductus vehementiùs, remitti aliquando, & ma<lb></lb>nualium tormentorum rotas interdum laxari oportere, ne vis <lb></lb>elaſtica languidior facta minùs utilis fiat. </s> </p> <p type="main"> <s id="s.001112">Ex his igitur paulò enucleatiùs explicatis, in quibus longio<lb></lb>re temporis fluxu motum aliquem tardiſſimum contigiſſe, at<lb></lb>que adeò etiam impetum jam tum ab initio ſtatim fuiſſe pro<lb></lb>ductum conſtat, conjecturam in reliquis capio, & ab iis impe<lb></lb>tum concipi ſtatuo, quæ aut loco naturali dimota, aut incon<lb></lb>gruam partium poſitionem nacta id repetunt, quod natura exi<lb></lb>git. </s> <s id="s.001113">Motus autem non pro impetûs tantum, ſed & pro re<lb></lb>ſiſtentiæ modo conſequitur. <lb></lb></s> </p> <p type="main"> <s id="s.001114"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001115"><emph type="center"></emph><emph type="italics"></emph>Quâ ratione ſemel conceptus impetus pereat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001116">UT impetûs natura, quam inquirimus, explicatiùs atque <lb></lb>diſtinctiùs innoteſcat, ex quo pariter, quæ corpora, quâ<lb></lb>ve ratione, impetum reſpuant, intelligamus, hîc nobis eſt <lb></lb>veſtigandum, quâ ratione conceptum ſemel impetum abji<lb></lb>ciant: hinc nimirum in uberiorem ipſius reſiſtentiæ notitiam <lb></lb>venientes ad explicandam motûs machinalis cauſam propiùs <lb></lb>accedemus. </s> </p> <pb pagenum="146" xlink:href="017/01/162.jpg"></pb> <p type="main"> <s id="s.001117">Et ſanè conceptum impetum, naturâ ſuâ, nec flabilem ſem<lb></lb>per permanere, nec ad unicum temporis punctum durare, ſa<lb></lb>tis conſtat: ſivè enim ſpontè profluat ex naturâ debitum ſibi <lb></lb>locum quærente, ſivè alienâ vi impreſſus ſuo loco corpus ex<lb></lb>trudat, perpetuus eſſe nequit; omnis ſcilicet motus terminum <lb></lb>habeat neceſſe eſt; nam ſi violentus quidem eſt, perennis uti<lb></lb>que non eſt; ſin autem naturalis, quem violentus præceſſerit, <lb></lb>certis definitur terminis; à loco enim, in quo quietem natura <lb></lb>indixit, corpus infinito intervallo non abeſt, ac proinde ubi <lb></lb>eum attigerit, demùm conquieſcet, nec impetu perpetuo opus <lb></lb>erit, cùm motum ceſſare oporteat. </s> <s id="s.001118">Sed neque temporis mo<lb></lb>mento circumſcribi impetum ſivè in naturali motu acquiſitum, <lb></lb>ſive in violento impreſſum, plura ſunt, quæ palam faciunt: ut <lb></lb>enim reliqua ſileam nullæ eſſent funependulorum oſcillatio<lb></lb>nes, nullus emiſſæ ſagittæ motus, ſi conceptus impetus illicò <lb></lb>periret. </s> </p> <p type="main"> <s id="s.001119">In duo autem veluti genera tribuendus eſt Impetus ex natu<lb></lb>râ dimanans; alius Innatus, ſeu quaſi inſitus, alius Acquiſitus <lb></lb>dicitur, Innatum, ſeu quaſi inſitum, voco, non quem corpus <lb></lb>jugiter obtineat, ſive ſuo in loco, ſive in alieno quieſcat; ſed <lb></lb>eum, qui facultati ſe movendi præcisè reſpondet, nullo facto <lb></lb>per continuam adjectionem incremento: quandiù enim corpus <lb></lb>ita ſimili ſecundùm gravitatem corpore circumfunditur, ut na<lb></lb>turali in loco conſiſtere dicendum ſit, quare conetur motum: <lb></lb>conatum autem hîc ab impetu non diſtinguo: ſatis igitur citrà <lb></lb>quemlibet impetum ſuo ſe tutatur in loco per hoc, quod eá fa<lb></lb>cultate ſit præditum, quæ in contrariam partem conniti valeat <lb></lb>illicò, ac vis inferri cæperit. </s> <s id="s.001120">Hinc nullum aquæ impetum tri<lb></lb>buo intrà aquam conſiſtenti; ſed tunc ſolùm cùm ſitula plena <lb></lb>è lacu extrahitur, ea aquæ pars impetum habet, quæ ſuprà ſub<lb></lb>jectam lacûs ſuperficiem aere circumfuia motum expetit, quo <lb></lb>ſuum repetat locum repugnans ſuſtinenti. </s> <s id="s.001121">Impetum hunc, qui <lb></lb>naturali ſe movendi facultati reſpondet, & eſt ipſa gravitatio, <lb></lb>ſeu naturalis ad deſcenſum propenſio, Innatum voco, & is eſt, <lb></lb>cui extrinſeca cauſa repugnat motum impediens. </s> <s id="s.001122">Quòd ſi ſuſ<lb></lb>penſum corpus ſibi relinquatur, ita ſuum in locum contendit, <lb></lb>ut vis naturalis æquè ſemper ad agendum applicata, nec impe<lb></lb>dita, momentis ſingulis novum impetum acquirat, qui propterea <pb pagenum="147" xlink:href="017/01/163.jpg"></pb>Acquiſitus dicitur, & poſterior priori additus intenſionem ef<lb></lb>ficit: ſapienti ſanè naturæ inſtituto; nam ſi corpora per ſe ipſa <lb></lb>ac ſuâ ſponte mota non accelerarent; ſed naturalis motus pla<lb></lb>ne æquabilis eſſet, tardè nimis locum ſuum conſequerentur; <lb></lb>atque adeò augendus continuò fuit impetus, ut & motus in<lb></lb>crementum acciperet: at ſi innatus impetus valdè <expan abbr="intẽſus">intenſus</expan> eſſet, <lb></lb>corpora nonniſi ægerrimè aliò transferri, aut alieno in loco re<lb></lb>tineri pro animalium, & hominis utilitate poſſent; finge ſcili<lb></lb>cet animo tibiam tanto impetu innato repugnare, ne attollatur, <lb></lb>quanto impetu in aëre ex 200 paſſuum altitudine deſcenderet; <lb></lb>quanto id tibi eſſet incommodo? </s> <s id="s.001123">Quare peropportunum acci<lb></lb>dit, ut vehemens non eſſet ſingularum particularum impetus <lb></lb>innatus, qui tamen ubi motum efficeret, novâ acceſſione poſ<lb></lb>ſet augeri. </s> </p> <p type="main"> <s id="s.001124">Quod ad impetum Innatum ſpectat, quem à gravitatione <lb></lb>ipsá & proxima motus exigentia non ſejungo, utique fruſtrà <lb></lb>eſſet, ſi omni prorſus effectu careret; impetus autem motum <lb></lb>aut efficit, aut ſaltem exigit: propterea illum ſtatim perire au<lb></lb>tumo, ac fuerit corpus in loco ſuo: Id quod hoc deprehendes <lb></lb>experimento. </s> <s id="s.001125">Scrobem defoſsâ humo altè excavato; ſitulam <lb></lb>aquæ pienam, & noti ponderis, intrà illam ſuſpendito; tùm <lb></lb>aquam in ſcrobem tantâ copiâ derivato; ut ſitulan uſquequa<lb></lb>que circumplectatur: illicò evaneſcet totius aquæ priùs in ſitu<lb></lb>lâ gravitantis pondus, quin & ſitula ipſa pro gravitatum ſecun<lb></lb>dùm ſpeciem diſſimilitudine levior apparebit, ut ex Hydroſta<lb></lb>ticis conſtat. </s> <s id="s.001126">Periit ergo innatus impetus, quo aqua ſitulam <lb></lb>replens deſcenſum moliebatur. </s> </p> <p type="main"> <s id="s.001127">At impetum Acquiſitum non continuò perire, ac eò ventum <lb></lb>fuerit, ubi quieſcendum eſſet, hinc ſaltem diſces, quod <lb></lb>ligneum globum aquæ cæteroqui innataturum ſi in ſublime at<lb></lb>tollas, & ex illâ altitudine cadere permittas, infrà aquæ ſuper<lb></lb>ficiem deſcendere, ac penitùs immergi videbis; quamquam <lb></lb>poſtea emergat, & ubi aliquoties ſubſultaverit, demùm pro <lb></lb>gravitatum aquæ, & ligni diſparitate emerſus quieſcat. </s> <s id="s.001128">Quæ <lb></lb>ſanè immerſio, niſi Acquiſitus impetus adhuc duraret, omninò <lb></lb>non contingeret. </s> <s id="s.001129">Verùm nihil rem per ſe ſatis abſtruſam æquè <lb></lb>in lucem evocat, ac funependulorum motus; plumbum enim <lb></lb>ex filo ſuſpenſum, & à perpendiculo dimotum, ita deſcendens <pb pagenum="148" xlink:href="017/01/164.jpg"></pb>arcum deſcribit, ut ferè parem arcum, & vix (aut fortè ne vix <lb></lb>quidem) minori tempore aſcendens deſcribat. </s> <s id="s.001130">Cui autem, re<lb></lb>pugnante plumbi gravitate à naturá inſitâ, tribuatur aſcenſus, <lb></lb>niſi impetui acquiſito dum deſcenderet, adhuc poſt deſcenſum <lb></lb>duranti? </s> <s id="s.001131">Quemadmodum verò in deſcenſu poſteriores motûs <lb></lb>partes prioribus velociores ſunt, factâ nimirum novi impetûs <lb></lb>acceſſione, ita ex oppoſito aſcenſus ex celeritate in tarditatem <lb></lb>deſinit, factâ acquiſiti impetûs deceſſione continuâ, donec ita <lb></lb>elanguerit, ut gravitas ipſa ſuperet, & iterum deſcendens al<lb></lb>ternas vibrationes efficiat. </s> <s id="s.001132">Perit igitur Acquiſitus impetus non <lb></lb>totus ſimul; ſed ſenſim extenuatur; idque non aliâ ratione, <lb></lb>quàm quâ proportione impeditur motus, quocumque tandem <lb></lb>ex capite impedimenta oriantur. </s> <s id="s.001133">Cum enim impetus contra<lb></lb>rium impetum non habeat, ſi præciſa quidem impetûs natura <lb></lb>ſpectetur (quippe qui unus & idem contrariorum motuum ori<lb></lb>go eſt, ut ex funependulis ultrò citróque ſponte vibratis & ex <lb></lb>pilâ luſoriâ deorſum cadente, ac vi concepti impetûs ſurſum <lb></lb>reſiliente, conſtat) reliquum eſt, ut pereat pro ratione eorum, <lb></lb>quæ aut motui corporis obſiſtunt, aut illud aliò quoquomodo <lb></lb>dirigunt. </s> </p> <p type="main"> <s id="s.001134">Præſtat autem hîc funependuli <lb></lb><figure id="id.017.01.164.1.jpg" xlink:href="017/01/164/1.jpg"></figure><lb></lb>motum paulò attentiùs conſiderare. </s> <lb></lb> <s id="s.001135">Sit plumbeus globulus B filo AB <lb></lb>connexus clavo in A. </s> <s id="s.001136">Si globulo li<lb></lb>ceret, quâ impetus innatus urget viâ, <lb></lb>deſcendere, utique rectam BC per<lb></lb>curreret; ſed funiculo retinente co<lb></lb>gitur arcum BK deſcribere, adeò ut <lb></lb>ſemper in alio & alio plano inclinato <lb></lb>conſtitutus, alia, & alia habeat gra<lb></lb>vitatis momenta, ut lib. 1. cap. 15 explicatum eſt; hæc autem <lb></lb>ſunt pro Ratione Sinuum angulorum declinationis à perpendi<lb></lb>culo AK. </s> <s id="s.001137">Quare totum momentum, quod in B eſſet ut AB, <lb></lb>ſingulis momentis in deſcenſu libero per rectam BC paribus <lb></lb>ſaltem incrementis augeretur (Quicquid ſit an etiam pro Ra<lb></lb>tione duplicatâ temporum, de quo alias diſputabimus) ſed <lb></lb>cum à rectitudine deflectat, cum venerit in D, non additur <lb></lb>momentum ut EF, ſed ut ED; ſimiliter in G momentum non <pb pagenum="149" xlink:href="017/01/165.jpg"></pb>eſt ut HI, ſed ut HG. </s> <s id="s.001138">Augetur igitur impetus in deſcenſu <lb></lb>BK non omninò pro Ratione <expan abbr="momentorũ">momentorum</expan> temporis, quo motus <lb></lb>durat, ſed pro Ratione momentorum gravitatis, quæ ſubinde <lb></lb>obtinet minora & minora; pars <expan abbr="ſiquidẽ">ſiquidem</expan> impetûs ab inſitâ globuli <lb></lb>gravitate producti deteritur in intendendo filo, quo retinetur. </s> <lb></lb> <s id="s.001139">Quapropter ubi in K venerit per arcum BK, non tantum ha<lb></lb>bet impetûs, quantum ſi per lineam perpendicularem arcui <lb></lb>BK æqualem deſcendiſſet; in motu enim ad perpendiculum <lb></lb>cum nihil retineat aut impediat, totus impetus ad deſcenſum <lb></lb>urget velociùs, quàm ubi repugnat aliquid. </s> <s id="s.001140">Ex quo fit quod, <lb></lb>cùm arcus BK ad Radium AB, hoc eſt ad BC æqualem <lb></lb>proximè ut 11 ad 7, ex Cyclometricis, multò plus temporis in <lb></lb>percurrendo arcu BK, quàm in rectâ BC, inſumitur; tardiùs <lb></lb>ſcilicet movetur quàm in perpendiculari, quæ ad BC eſſet ut <lb></lb>11 ad 7. manente itaque, quamdiu corpus naturâ urgente mo-<lb></lb>vetur, impetu acquiſito, qui reſiſtentiam excedit, in fine <lb></lb>deſcensûs in K totus impetus eſt ut aggregatum omnium Si<lb></lb>nuum Quadrantis: at in perpendiculari BC in fine deſcensùs <lb></lb>in C eſſet ut aggregatum omnium parallelarum ipſi AB in <lb></lb>Quadrato AC; ac propterea (in re Phyſicâ ſi liceat cum geo<lb></lb>metrizantibus per Indiviſibilia ratiocinari) erit impetus per ar<lb></lb>cum BK acquiſitus ad impetum per rectam BC acquiſtum ut <lb></lb>Quadrans ABK ad Quadratum AC, hoc eſt ut 11 ad 14, ex <lb></lb>iis quæ in Cyclometriâ demonſtrantur. </s> </p> <p type="main"> <s id="s.001141">Quoniam verò ubi ad perpendiculum AK globulus deſcen<lb></lb>dens venerit, nihil objicitur, quod motum prorſùs impediar, <lb></lb>quin ad eaſdem partes pergat ferri ex præconcepti impetûs di<lb></lb>rectione, non ſiſtit in perpendiculo; ſed ulteriùs pergens aſcen<lb></lb>dit, nec niſi per arcum circà centrum A, funiculo ſcilicet reti<lb></lb>nente. </s> <s id="s.001142">Sed jam repugnat aſcenſui gravitas plumbi, non qui<lb></lb>dem quantum in perpendiculo KA, verùm pro ratione Sinuum <lb></lb>angulorum declinationis; qui cum ſemper aſcendendo creſ<lb></lb>cant, major eſt etiam momentorum gravitatis Ratio nitentium <lb></lb>contrà impetum deſcendendo acquiſitum. </s> <s id="s.001143">Quare tantum abeſt, <lb></lb>ut novus ſingulis temporis punctis impetus ſurſum directus pro<lb></lb>ducatur, ut potius ex eo tantumdem dematur, quanta eſt <lb></lb>aſcendentis plumbi repugnantia. </s> <s id="s.001144">Hinc eſt aſcenſum initio ve<lb></lb>lociorem eſſe, quia adhuc multus eſt impetus acquiſitus, & pro <pb pagenum="150" xlink:href="017/01/166.jpg"></pb>Sinuum declinationis brevitate, exigua illius pars deteritur, <lb></lb>atque adeò motus efficitur celerior: quia verò diminuto ſenſim <lb></lb>impetu, & auctis <expan abbr="cõtrariæ">contrariæ</expan> gravitatis <expan abbr="momẽtis">momentis</expan> pro Sinuum decli<lb></lb>nationis <expan abbr="incremẽto">incremento</expan>, minor fit ipſius impetûs ad <expan abbr="contrariũ">contrarium</expan> niſum <lb></lb>Ratio, tardior ſequitur motus, & plus acquiſiti impetûs perit, do<lb></lb>nec demùm prorſus evanuerit, & ſuperante gravitate globulus <lb></lb>iterum deſcendat. </s> <s id="s.001145">Quamvis autem ſi poſitio ſola ſpectetur, iiſ<lb></lb>dem Reciproce gradibus minui videatur impetus, quibus fuit <lb></lb>auctus, totidemque momentis temporis, ita ut quantum poſtre<lb></lb>mo temporis puncto acceſſit, tantumdem primo decedat, adhuc <lb></lb>tamen aliqua eſt obſiſtentiæ appendicula ex aëre dividendo, ac <lb></lb>propterea paulo ampliùs extenuatur impetus acquiſitus, quàm <lb></lb>pro Ratione incrementi Sinuum declinationis: quò autem ve<lb></lb>locior eſt motus, magis etiam aër dividendus comprimitur, <lb></lb>denſatúſque plus obſiſtit quàm rarus; quòd ſi medium non fue<lb></lb>rit compreſſionis capax, ſaltem æquali tempore plures medij <lb></lb>partes ſcinduntur, quàm in motu tardiori, ac propterea etiam <lb></lb>multiplex eſt medij reſiſtentia: Ex quo fit arcum aſcensûs pau<lb></lb>lò minorem ſemper eſſe arcu deſcensûs, &, cum viciſſim glo<lb></lb>bus remaneat ex humiliore loco ac priùs deſcendens, brevio<lb></lb>rem pariter ſecundi aſcensûs arcum perfici, atque ita deinceps, <lb></lb>ut ſervatâ eâ in motu ſemper minori reciprocando conſtantiâ <lb></lb>demum quieſcat in perpendiculo. </s> </p> <p type="main"> <s id="s.001146">At, inquis, dura magis obſiſtunt corpori, ejúſque motum <lb></lb>validiùs impediunt, quàm mollia, quæ dum ſe comprimi pa<lb></lb>tiuntur, & loco pauliſper cedunt, motui aliquantulùm & ex <lb></lb>parte obſecundant: ſi igitur pro Ratione impedimenti debili<lb></lb>tatur acquiſitus impetus, minus detrahitur impetûs corpori, <lb></lb>quod ex alto decidens à ſubſtratis paleis excipitur, quàm ſi ad <lb></lb>ſaxum allideretur; vehementiùs igitur à luto quàm à ſàxo re<lb></lb>flecteretur, contrà quàm docet experientia. </s> </p> <p type="main"> <s id="s.001147">Fateor eburneum globum ſegniùs reſilire delapſum in gle<lb></lb>bam humore perfuſam, quàm in marmor; non tamen his con<lb></lb>ſequens eſt, ut impetûs acquiſiti diminutioni alius ſtatuendus <lb></lb>ſit modus, quàm ex impedimento: ubi enim globus cadens ex<lb></lb>timam ſubjecti corporis ſuperficiem attigerit, non quieſcit, ſed <lb></lb>pergit moveri, aut deorſum comprimendo corpus molle, aut <lb></lb>illicò ſursùm reflexum à duro. </s> <s id="s.001148">Ita autem à corpore molli ex-<pb pagenum="151" xlink:href="017/01/167.jpg"></pb>cipitur, ut licèt hoc cedat, impediat tamen & remoretur mo<lb></lb>tum; ac proinde quò magis cedit ſubjectum corpus, eò diutiùs <lb></lb>movetur globus cum ipſo, vel intrà ipſum; atque interea plus <lb></lb>impetûs perit: quid igitur mirum, ſi languidiùs poſtea reſiliat, <lb></lb>cum exigua impetûs portio reliqua ſit? </s> <s id="s.001149">Quòd ſi <expan abbr="durũ">durum</expan> eſſet ſub<lb></lb>jectum corpus, impetu nondum debilitato reflecteretur vali<lb></lb>diùs. </s> <s id="s.001150">Hinc fieri poteſt adeò molle eſſe ſubjectum corpus, ut <lb></lb>dum illud penetrat decidens globus, tantum impetûs deper<lb></lb>dat, ut, quod reliquum fit, non ſatis ſit ad vincendam inſitam <lb></lb>globo gravitatem, qui propterea neque reſilire valeat. </s> <s id="s.001151">Quam<lb></lb>vis itaque corpus molle minùs obſiſtat quàm durum, diutiùs <lb></lb>tamen reſiſtit; & per aliquot momenta aliquoties diminutus <lb></lb>impetus minore menſurâ, eò decrementi venire poteſt, ut ma<lb></lb>gis imminutus demum fuerit, quàm ſi unico momento magis <lb></lb>obſtitiſſet corpus durum. </s> <s id="s.001152">Cæterùm paribus momentis plus pe<lb></lb>rit impetûs ex alliſione ad corpus durum, quàm ad molle, quip<lb></lb>pe quod magis opponitur motui. </s> <s id="s.001153">Porrò huic rei explicandæ <lb></lb>ſimilitudo aliqua peti poſſet ex luce, cui ſanè ſi contingat per <lb></lb>medium diaphanum quidem, ſed denſum, pergere, languidiùs <lb></lb>multò reflectitur à ſpeculo, in quod incurrit, ſi denſioris me<lb></lb>dij longior fuerit tractus, quàm ſi brevior, perinde atque eò <lb></lb>minùs reflectitur corpus, quò molliori magiſque ſubſidenti cor<lb></lb>pori occurrit. </s> <s id="s.001154">ſed quoniam quæ de luce dicenda eſſent, fortè ob<lb></lb>ſcuriora acciderent, ab hujuſmodi ſimilitudine <expan abbr="prudẽ">prudem</expan>, abſtineo. </s> </p> <p type="main"> <s id="s.001155">Sed ex illud eſt in durorum corporum colliſione obſervan<lb></lb>dum, quod aliqua particularum compreſſio aliquando contin<lb></lb>git ſivè in alterutro, ſivè in utróque, quæ ſe facultate elaſticâ <lb></lb>reſtituentes motum reflexum juvant: id autem manifeſto ex<lb></lb>perimento conſtat in pilâ ex gummi, ut vocant, Indico, quæ <lb></lb>ad terram eliſa frequentiſſimè ſubſultat; at ubi in corpus molle <lb></lb>incidit, neque hujus neque illius partes violentam compreſſio<lb></lb>nem ſubeunt, quam ſeſe reſtituentes excutere debeant. </s> <s id="s.001156">Sic & <lb></lb>pilá in ſphæriſterio ludentes ſatis nôrunt eam validiùs reflecti <lb></lb>objecto recticulo, quàm ligneo batillo; intenti ſcilicet nervi ex <lb></lb>contortis ſiccatiſque animalium inteſtinis reticulum conſtituen<lb></lb>tes cùm pilæ ictum excipiunt, flectuntur quidem aliquantu<lb></lb>lum; ſed illicò ſibi priſtinam rectitudinem reparantes pilam ex<lb></lb>cutiunt (id quod ligneo baſtillo non contingit) novoque hoc <pb pagenum="152" xlink:href="017/01/168.jpg"></pb>impetu auctus reliquus pilæ impetus motum quoquè efficit <lb></lb>majorem: quòd ſi in reticulo flaccidi, & remiſſi ſint nervi, lan<lb></lb>guidè pila reflectitur. </s> </p> <p type="main"> <s id="s.001157">Ad quandam autem reflexionis ſpeciem pertinere cenſenda <lb></lb>eſt concuſſio, ſive vibratio, aliquarum ſaltem corporis partium, <lb></lb>ubi totum ex reliquo impetu reſilire nequit: ſic corpus ita at<lb></lb>tollens, ut ſummis pedibus innitaris, poſtmodum recidens in <lb></lb>talos, eò validiorem partium concuſſionem percipies, quò ve<lb></lb>lociùs recides. </s> <s id="s.001158">Simile quid etiam in inanimis contingere ratio <lb></lb>ſuadet, neque enim ita ſemper ſolida aut prorſus homogenea <lb></lb>tota moles eſt, ut nullæ omninò partes concuti valeant: quin <lb></lb>etiam alliſi corporis partes, ſi non adeò tenaci vinculo inter ſe <lb></lb>cohæreant, ex reliquo impetu aliæ aliò diſtractæ deſiliunt. </s> </p> <p type="main"> <s id="s.001159">Hinc, docente naturâ, ex alto deſilientes ubi terram pedi<lb></lb>bus attigerint, genua antrorſum inflectunt, quaſi calcaneis in<lb></lb>ſeſſuri, ne conceptus ex ſaltu impetus ſuperiorem corporis par<lb></lb>tem deorſum validiùs urgens ſubjectas tibias, & genua ita pre<lb></lb>mat, ut inde diviſio aliqua membrorum, aut oſſium luxatio, aut <lb></lb>nervorum ſeu tendinum nimia diſtenſio dolorem gignat: hoc <lb></lb>autem valet illa genuum inflexio ad extenuandum impetum, <lb></lb>quod & flexili mollitiâ ſubſidens terra uliginoſa, ſi quando la<lb></lb>pis in eam ex alto deciderit. </s> <s id="s.001160">Sic Atlas Sinicus pag. </s> <s id="s.001161">123. in XI. </s> <lb></lb> <s id="s.001162">Provinciâ Fokion, ubi ſermo eſt de flumine Min, quod vio<lb></lb>lento curſu per ſaxa volvitur, ait naves, quibus ibi navigatur, <lb></lb>ex diverbio vocari <emph type="italics"></emph>Papyraceas, eo quòd tenuibus ac minime re<lb></lb>ſiſtentibus conſtent aſſeribus, imò ne clavis quidem compaginatis; <lb></lb>ſed vimine quodam lentiſſimo; unde tametſi in ſaxa impingat na<lb></lb>vis, ſapè tamen minimè rumpitur, quia vix reſiſtit.<emph.end type="italics"></emph.end></s> <s id="s.001163"> Et pag.127. <lb></lb>de catadupis aquarum in flumine per quod ad Jenping naviga<lb></lb>tur loquens ait. <emph type="italics"></emph>Cum naves tranſeunt, ne cum aquâ decidentes <lb></lb>f actionis incurrant periculum, ſcitè præmittunt nautæ aliquot ſtra<lb></lb>minis faſces, ad quos navis leviùs impingat, ac tranſeat.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001164">Jam verò ad impetum extrinſecùs impreſſum mentem ocu<lb></lb>loſque intendente, non illum ſemper momento perire animad<lb></lb>vertimus, aut illicò, ac externus agitator ceſſat. </s> </p> <p type="main"> <s id="s.001165">Unde enim tit, ut concitato navigio, cùm vela nautæ con<lb></lb>traxerunt, aut remiges inhibuerunt, retineat tamen ipſa navis <lb></lb>motum & curſum ſuum, intermiſſo ventorum incurſu, pulsúve <pb pagenum="153" xlink:href="017/01/169.jpg"></pb>remorum? </s> <s id="s.001166">niſi quia navis, etiam nullo impellente, vi impreſsâ <lb></lb>urgetur. </s> <s id="s.001167">Quid rhedam curſu procedente faciliùs quàm initiò <lb></lb>promovet, equis licet languidius connitentibus? </s> <s id="s.001168">curve onus <lb></lb>aliquod ingens protrudentes, aut trahentes hoc maximè ca<lb></lb>vent, ne contentionem illam quies interrumpat, experientiâ <lb></lb>ſatis edocti incitatum ſemel minori labore propelli, quàm com<lb></lb>moveri quieſcens? </s> <s id="s.001169">niſi quia reliquus ex priore motu impetus <lb></lb>adhuc perſeverans poſteriorem motum juvat. </s> <s id="s.001170">Hoc tamen tria <lb></lb>hæc differunt, quòd onus, ceſſantibus iis, qui protrudebant, <lb></lb>conſiſtit illicò (niſi fortè volubilitatem habens, aut ſubjectis <lb></lb>cylindris innixum, adhuc modicum quid volvi aut progredi <lb></lb>pergat) rheda currentes equo, ſubita funium abruptione dis<lb></lb>junctos ſequitur ad paſſus aliquot non adeò multos pro viæ <lb></lb>æquabilitate præcedentiſque velocitatis ratione; navigium verò <lb></lb>ſubmiſſis antennis, remiſque ceſſatione torpentibus aliquandiu, <lb></lb>intervallo non ſanè contemnendo, provehitur. </s> <s id="s.001171">Oneris ſcilicet <lb></lb>motui, cui volubilitatem neque ars, neque natura dederit, im<lb></lb>pedimento eſt ipſa extremitas aſpera ſubjectam planitiem ſale<lb></lb>bris quandóque non carentem contingens, gravitaſque ita va<lb></lb>lidè premens, ut major futurus eſſet partium tritus, quàm pro <lb></lb>impetûs modo, qui reliquus eſſet, ſuperari poſſet: Id quod cur<lb></lb>renti rhedæ idcircò non contingere planum eſt, quia licèt <lb></lb>nihilo levior ſit quàm onus protruſum, minùs tamen rotarum <lb></lb>modioli leniter cum axibus confligentes motum retardant. </s> <s id="s.001172">At <lb></lb>navis ſponte ſuâ innatans, ventorum incurſione, remorúmve <lb></lb>pulſu diutiùs acta, vix, aut fortè ne vix quidem, mole ſuâ re<lb></lb>luctatur, niſi quatenus diffindenda eſt aqua; nec ſinè multo fa<lb></lb>cilitatis compendio, prior ſiquidem unda, quam prora impel<lb></lb>lens excitat, aliam ante ſe urget ad eaſdem partes: propterea <lb></lb>impreſſus navi impetus modicum nactus impedimentum diù <lb></lb>durat, illámque promovet. </s> <s id="s.001173">Quare idem de impetu extrinſecùs <lb></lb>aſſumpto dicendum eſt, quod de acquiſito; nimirùm minui pro <lb></lb>Ratione eorum, quæ inſtituto motui obſiſtunt, aut etiam pror<lb></lb>sùs perire. </s> </p> <p type="main"> <s id="s.001174">Præter ea autem quæ utrique motui tùm naturali, tùm vio<lb></lb>lento æquè opponuntur, (cujuſmodi eſt medium dividendum, <lb></lb>objecti corporis occurſus, aut contingentis tritus atque con<lb></lb>flictus, retinaculum, quod certo limite motum definiat, & alia <pb pagenum="154" xlink:href="017/01/170.jpg"></pb>id genus) illa eſt externo impulſui peculiaris repugnantia, <lb></lb>quæ ex inhærente corpori gravitate oritur, ſive illi innatus im<lb></lb>petus, ſive acquiſitus modum ſtatuat. </s> <s id="s.001175">Neque id ſimpliciter <lb></lb>tantùm, ſed comparatè conſiderandum eſt, quam ſcilicet in <lb></lb>plagam impulſus motum dirigat, & quatenus gravitatis pro<lb></lb>penſioni opponatur. </s> <s id="s.001176">Quemadmodum enim qui in pilâ aroma<lb></lb>ta pinſunt, nihil repugnantem, quin & impulſui obſecundan<lb></lb>tem, experiuntur piſtilli gravitatem deprimentes; contrà verò <lb></lb>attollentes fatigat eadem gravitas directò deorſum urgens; me<lb></lb>dium autem quiddam tenet in obſiſtendo, ſi motio tranſverſa <lb></lb>contingat; ſicut experiri licet, ſi ex funiculo pendens idem <lb></lb>piſtillus à perpendiculo dimoveatur; minore enim conatu opus <lb></lb>eſt: ita quò minùs in oppoſitam gravitati plagam dirigitur im<lb></lb>pulſus, eò etiam diutiùs perſeverat minus habens impedimenti. </s> <lb></lb> <s id="s.001177">Hinc eſt quod gravitas æquabiliter toto corpore fuſa ſi aut ex <lb></lb>centro ſuſpendatur, aut coni apici inſiſtat, levi negotio, ac ſa<lb></lb>tis diù, in gyrum convertitur; innatum videlicet gravitatis im<lb></lb>petum vis ipſa ſuſpendens aut ſuſtentans elidit; nihil verò im<lb></lb>pulſum remoratur præter aut funiculi ſuſpendentis ſpiras paulò <lb></lb>ſpiſſiores, aut tritum cum ſubjecto cono, aëriſque dividendi <lb></lb>reſiſtentiam; quæ tamen ſi tollatur in corpore orbiculari circà <lb></lb>centrum commoto, etiam longior fit converſio. </s> <s id="s.001178">Sic ferream <lb></lb>ſagittam palmarem craſſiuſculam inſtar acûs magneticæ in <lb></lb>æquilibrio conſtitutam leviſſimo impulſu ac diutiſſimè in gy<lb></lb>rum agi obſervavi; vix enim acutiſſimum verticem, cui innite<lb></lb>batur, terebat, & aëris intrà eumdem gyrum circumducti mo<lb></lb>dica erat reſiſtentia. </s> <s id="s.001179">Id autem multo luculentiùs apparet in <lb></lb>verticillo, cujus axem perpolito alveolo inſiſtentem extremo <lb></lb>pollice ac indice leviter comprimens, ac paulò celeriùs vertens, <lb></lb>eò diuturniori vertigine contorqueri videbis, quò pauciores <lb></lb>minoreſque offenderit in ſubjectâ tabulâ aſperitates, ad quas al<lb></lb>liſus paululùm inclinetur, aut aliò reflectatur. </s> </p> <p type="main"> <s id="s.001180">Quòd ſi magnetis polo ritè armato chalybeum axiculum <lb></lb>congruo verticulo inſtructum admoveris, ut planè à magnete <lb></lb>ſuſpendatur, tùm ſummis digitis opportunè axem terentibus <lb></lb>vertiginem ei delicatè ac molliter conciliaveris, miraculi loco <lb></lb>tibi erit tàm diuturna converſio; quippe cui non ſubjectialveoli <lb></lb>aſperitates ſaltitare cogentes, non gravitas ipſa premens, tritum-<pb pagenum="155" xlink:href="017/01/171.jpg"></pb>que augens, non ſuſpendentis funiculi violenta contortio ob<lb></lb>ſiſtunt, motúmve aliquatenus impedientes impreſſum impe<lb></lb>tam imminuunt; ſed magnetico radio ſuſpenſus intra ſe perpe<lb></lb>tuò volvitur læviſſimum chalybem magnetis polo adhærentem <lb></lb>leniſſimè terens. </s> </p> <p type="main"> <s id="s.001181">Illud etiam in motu, qui ab extrinſeco provenit, conſide<lb></lb>randum eſt, quòd contingere poteſt duos adeſſe motores, qui <lb></lb>corporis motum in diverſas partes dirigant: quare alter alteri <lb></lb>obſiſtit, & motus ex duplici directione compoſitus is eſt, qui <lb></lb>non reſpondeat menſuræ duplicis illius impetûs, ſi ſinguli in<lb></lb>tegrè accipiantur. </s> <s id="s.001182">Conſtat enim, ſi æquabili & æquali cona<lb></lb>tu urgeant corpus, moveri aut per diametrum Quadrati, ſi di<lb></lb>rectiones ſint ad angulum rectum conſtitutæ; aut per Diago<lb></lb>nalem lineam Rhombi, ſi directiones obliquæ ſint: ſi verò <lb></lb>æquabiles quidem ſint, ſed inæquales conatus, per diametrum <lb></lb>Rectanguli aut Rhomboidis moveri, pro ut ad rectum aut obli<lb></lb>quum angulum directiones ſibi invicem reſpondent. </s> <s id="s.001183">Semper <lb></lb>autem minor eſt motus quàm pro duorum illorum impulſuum <lb></lb>ratione; diameter ſiquidem brevior eſt aggregato duorum <lb></lb>adjacentium laterum. </s> <s id="s.001184">Quòd ſi æquabiles non ſint impetus, <lb></lb>vel ſaltem alter æquabilis ſit, alter acceleratus aut retardatus, <lb></lb>linea curva deſcribitur; quæ pariter minor eſt duabus rectis, <lb></lb>quæ vi ſingulorum impetuum deſcriberentur; ab illis ſi qui<lb></lb>dem continetur. </s> </p> <p type="main"> <s id="s.001185">Hîc tamen advertendus animus eſt, & obſervare oportet <lb></lb>æquabilem impulſum (ſi continuus ſit, nec morulis inter<lb></lb>ruptus) eſſe non poſſe, niſi ab animali ſemper æqualiter conan<lb></lb>te efficiatur; quia gravium deſcenſus naturaliter acceleratur; <lb></lb>elaſmata verò dum ſe reſtituunt, ſemper languidiùs ſingulis <lb></lb>momentis conantur, ſi quidem virtus elaſtica conſideretur: <lb></lb>quamquàm poſteriore momento quod eſt reliquum prioris im<lb></lb>petûs, intenſionem efficit additum poſteriori licèt remiſſo. </s> <lb></lb> <s id="s.001186">Vix igitur contingere poteſt motum unum à duplici impetu <lb></lb>extrinſecùs impreſſo fieri per lineam rectam niſi corpus à du<lb></lb>plici motore æquabiliter urgeatur. </s> </p> <p type="main"> <s id="s.001187">Cum itaque impetus acquiſitus, aut aliundè impreſſus, ſit <lb></lb>qualitas propter motum inſtituta, quæ non niſi in motu pro<lb></lb>ducitur, ita pariter niſi in motu, & cum motu non conſerva-<pb pagenum="156" xlink:href="017/01/172.jpg"></pb>tur. </s> <s id="s.001188">Quare ſi corpus eò deveniat, ut nullo prorſus pacto agi<lb></lb>tari queat, aut interiore motu cieri, quo momento impeditur <lb></lb>motus, ne ſit, eo momento impetus perit, ceſſante videlicet <lb></lb>causa effectiva ab ejus conſervatione eo ipſo quod ceſſat finis, <lb></lb>propter quem impetus eſt. </s> <s id="s.001189">Quod ſi impedimentum occurrat <lb></lb>non prorsùs motum tollens (ut ſi globus in plano horizontali <lb></lb>rotatus veniat ad planum inclinatum, per quod ex concepto <lb></lb>impetu aſcendat) tunc pro ratione impedimenti extenuatur <lb></lb>impetus, donec tandem pereat. <lb></lb> </s> </p> <p type="main"> <s id="s.001190"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001191"><emph type="center"></emph><emph type="italics"></emph>Quâ ratione vis movendi cum impedimentis <lb></lb>comparetur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001192">MOtus omnis nec in oppoſitas, nec in diverſas plagas, ſed <lb></lb>per certam lineam dirigitur; unico quippe in loco, non <lb></lb>in pluribus, eodem temporis puncto eſſe poteſt corpus. </s> </p> <p type="main"> <s id="s.001193">Nihil igitur motui moram & impedimentum inferre poteſt, <lb></lb>niſi directò aut obliquè illi ſecundùm eam lineam, per quam <lb></lb>inſtituendus eſſet, antè, ponè, ad dextram, ad lævam, ſurſum, <lb></lb>deorſum opponatur. </s> <s id="s.001194">Si enim duo corpora eádem pergerent viâ, <lb></lb>& maximâ velocitatis, aut tarditatis conſpiratione conſentirent, <lb></lb>tunc neque poſterius ab eo quod antè eſt, traheretur, neque <lb></lb>prius à poſteriore urgeretur, neque alterum alteri impedimen<lb></lb>to eſſet. </s> <s id="s.001195">Hinc manifeſtum eſt non poſſe impedimentum ſupe<lb></lb>rari, quin ei vis aliqua inferatur. </s> </p> <p type="main"> <s id="s.001196">Rem porrò univerſam duas in partes tribuere poſſumus, ut <lb></lb>duplex Reſiſtentiæ genus ſtatuatur; Formalem alteram, alte<lb></lb>ram Activam ſcholæ vocarent. </s> <s id="s.001197">Corpus enim, quod obſtat, aut <lb></lb>retinet, ſi motum prorsùs nullum conetur inſtituto aut deſti<lb></lb>nato motui adverſantem, reſiſtit quidem, ſed Formaliter; nihil <lb></lb>ſcilicet efficit, quo repugnet, ſed ſuo tantùm ſe tutatur in loco: <lb></lb>Sin autem & contrà nitatur, aut retrahat, jam non obſiſtit ſo<lb></lb>lùm, ne loco per vim dimoveatur; ſed etiam impetum in con-<pb pagenum="157" xlink:href="017/01/173.jpg"></pb>trariam plagam directum efficit, cujus vi motum impedit, ac <lb></lb>proptereà Activè reſiſtit. </s> <s id="s.001198">Huic autem verbo, cum <emph type="italics"></emph>Reſiſtere<emph.end type="italics"></emph.end> di<lb></lb>cimus, ſubjecta notio eſt, in causá eſſe ne motus fiat, aut ſal<lb></lb>tem non ea velocitate, quæ virtuti movendi non impeditæ cæ<lb></lb>teroqui reſponderet. </s> <s id="s.001199">Sic paries, in quem incurris, tibi reſiſtit <lb></lb>Formaliter, ne procedas, & aqua ſtagnans, cui collo tenus im<lb></lb>mergeris, progredienti reſiſtit Formaliter, ne velociter, ſicut <lb></lb>intra aërem movearis pro ratione impetus, quo conaris progre<lb></lb>di: qui verò occurrens te repellit, ut ſi coneris contra ictum <lb></lb>fluvij, non Formaliter tantùm, ſed etiam Activè reſiſtit; non <lb></lb>ſolùm enim obſtat, quia ejus in locum ſuccedere non potes, <lb></lb>niſi cum loco dimoveas, ſed etiam tibi adverſum impetum im<lb></lb>primit, ut te loco extrudat. </s> </p> <p type="main"> <s id="s.001200">Cum itaque impedimenta motûs externo impetu ſubmoven<lb></lb>da ſint, virtus autem movendi certa ſit ac definita, conſtat vi<lb></lb>res omnes, quæ in corpore promovendo, ſi nihil obſtaret, exer<lb></lb>cerentur, duas in partes diſtrahi, ad movendum ſcilicet cor<lb></lb>pus, & ad tollenda impedimenta, Concipit igitur impetum, <lb></lb>qui motum efficiat, & obſtanti corpori impetum imprimit, ut <lb></lb>loco cedat. </s> <s id="s.001201">Quid igitur mirum, ſi diſtractis viribus languidior <lb></lb>ſequatur metus? </s> <s id="s.001202">Quia verò quò majori velocitate corpus <lb></lb>obſtans propellendum eſt, aut trahendum, majori quoque im<lb></lb>petu impreſſo opus habet, palàm eſt majorem quoque in pro<lb></lb>pellente, aut ſecum rapiente, impetum requiri, ut majorem re<lb></lb>ſiſtentiam vincens ſe ipſum pariter moveat. </s> </p> <p type="main"> <s id="s.001203">Hic autem quid monuiſſe oporteat vim reſiſtendi ſuperan<lb></lb>dam eſſe à virtute movendi? </s> <s id="s.001204">quis enim ambigat, an, ſi pares <lb></lb>illæ fuerint, nullus futurus ſit motus? </s> <s id="s.001205">Quòd ſi impedimentum <lb></lb>prorsùs immotum adversùs conantem perſtat, nullum pariter <lb></lb>recipit impetum; qui ſcilicet, etiam ſi priùs fuiſſet, motu ceſ<lb></lb>ſante periret. </s> <s id="s.001206">Hinc in animali defatigatio membrorum oritur, <lb></lb>quando prorsùs in irritum conatus cadit; impetus enim, quem <lb></lb>concipit, ut æqualem motum imprimeret impedimento, ſi hoc <lb></lb>ſuperari poſſet, in animali ipſo motum aliquem efficit, ſed quia <lb></lb>progredi vetatur ab oſtante aut retinente impedimento, impe<lb></lb>tus ille non totius animalis motum ulteriùs promovet; ſed mem<lb></lb>brorum partes alias comprimit, alias diſtendit, unde & dolor <lb></lb>aliquis, & laſſitudo provenit. </s> <s id="s.001207">At ſi corpus, cui motus debetur, <pb pagenum="158" xlink:href="017/01/174.jpg"></pb>cùm inanimum ſit, nequeat impetum, quemadmodum animan<lb></lb>tes, ex arbitrio temperare, & quia ſolidum eſt ac durum, nul<lb></lb>lam pati compreſſionem aut diſtentionem partium poſſit, ſicut <lb></lb>& corpus obſtans aut retinens compreſſionem omnem aut <lb></lb>diſtentionem reſpuit; tunc nullum concipit aut imprimit im<lb></lb>petum præter innatam gravitationem, aut levitationem, cùm <lb></lb>per vim in loco non debito detineatur. </s> <s id="s.001208">Ex hoc conjecturam ca<lb></lb>pere licet de eo, quod contingit, quando virtute movendi re<lb></lb>ſiſtentiam vincente impedimentum ſubmovetur; impediri vi<lb></lb>delicet, ne producatur motus, juxta reſiſtentiæ modum atque <lb></lb>menſuram; quæ ſicuti non quâlibet minimâ vi ſuperari poteſt, <lb></lb>ita majori cedit. </s> </p> <p type="main"> <s id="s.001209">Verùm quonam id pacto contingat, ut explicare conemur, <lb></lb>illud obſerva, quòd ſi corpus idem quadruplo velociùs moveri <lb></lb>debeat, ac moveretur priùs certâ impetûs menſurâ, utique qua<lb></lb>druplo majorem impetum exigit, ut pro impetûs intenſione <lb></lb>aut remiſſione velocior aut tardior ſequatur motus. </s> <s id="s.001210">At ſi cor<lb></lb>pus aliud movendum quadruplo gravius exhibeatur, in hoc im<lb></lb>petus ille quadruplex ſubquadruplam efficiet intenſionem, ac <lb></lb>propterea etiam motum habebit tardiorem, ſi cætera ſint paria, <lb></lb>pro impetûs intenſione. </s> <s id="s.001211">Si cætera, inquam, ſint paria; ſæpè <lb></lb>enim aër, aut aqua plus velociori motui reſiſtunt, quàm tardio<lb></lb>ri, & moles major efficit, ut non omninò velocitas intenſioni <lb></lb>impetûs reſpondeat. </s> <s id="s.001212">Hæc tamen nunc mente ſecernamus, per<lb></lb>inde atque ſi nihil officerent motui. </s> </p> <p type="main"> <s id="s.001213">Quoniam igitur motus ab omni velocitatis aut tarditatis men<lb></lb>ſurâ ſejungi nequit, finge corpus per vim movendum hujuſ<lb></lb>modi eſſe, ut ſpectatâ mole ſeu materiâ, ac ſpecificâ gravitate, <lb></lb>ad percurrendum ſpatium paſſuum 100 unius horæ quadrante, <lb></lb>indigeret impetu, cujus intenſio eſſet particularum 4 in ſingu<lb></lb>lis corporis movendi partibus: molem autem, exempli gratiâ, <lb></lb>diſtinctam concipe in particulas 100 minimas. </s> <s id="s.001214">Quare ſpectatâ <lb></lb>tùm extenſione tùm intenſione impetûs, neceſſe eſt illi à mo<lb></lb>tore imprimi impetûs particulas 400. Quòd ſi corporis per vim <lb></lb>movendi moles ac materia eſſet quadruplex alterius, ſi nimi<lb></lb>rum ratione materiæ extenſionis particulas haberet 400, jam <lb></lb>impetus idem ſubquadruplam efficeret intenſionem, & ſingulæ <lb></lb>impetûs particulæ ſingulis corporis particulis ineſſent; atque <pb pagenum="159" xlink:href="017/01/175.jpg"></pb>adeò etiam hujus velocitas eſſet ſubquadrupla prioris velocita<lb></lb>tis: partamen utrobique eſſet, illud quidem velociùs, hoc tar<lb></lb>diùs movendi difficultas, cum in utroque particulas 400 impe<lb></lb>tûs produci oporteret; utriuſque enim impetûs extenſiones & <lb></lb>intenſiones eſſent Reciprocè in eadem Ratione. </s> <s id="s.001215">In corpore <lb></lb>itaque, ex quo motus originem ducit, tanta vis movendi ineſſe <lb></lb>debet, ut & corpori impedienti, quod ſubmovetur, congruen<lb></lb>tem motui impetum imprimat, hoc eſt particulas 400, & ipſum <lb></lb>ſe pariter promoveat: nihil enim accepto extrinſecùs impetu <lb></lb>agitatur à motore prorsùs immoto, ut eunti per ſingula patebit. </s> </p> <p type="main"> <s id="s.001216">Jam verò quoniam idem corpus modò remiſſiùs, modò con<lb></lb>citatiùs moveri pro impetûs intenſione videmus, probabilis <lb></lb>conjectura eſt in iis, quæ non ſuo arbitrio, ſed naturæ reguntur <lb></lb>imperio, totum impetum produci, qui virtuti efficiendi reſpon<lb></lb>det: hæc autem in impedimento, cujus reſiſtentia vincitur, <lb></lb>impetum eâ intenſionis menſurâ imprimit, quæ illi motûs ve<lb></lb>locitatem conciliet ipſius corporis moventis velocitati con<lb></lb>gruentem, adeò ut movendi facultas totas ſuas vires exerat <lb></lb>partim impetum imprimens ſubmovendo impedimento, partim <lb></lb>motum efficiens in ipſo corpore: ex quo fit quod eò remiſſiorem <lb></lb>motum in ſe motor efficiat, quò major ſecundùm intenſionem <lb></lb>impetus impeditur ab impedimento. </s> <s id="s.001217">Sic plumbeus globus bili<lb></lb>bri, ſi, funiculo excavatæ volubilis orbiculi curvaturæ inſerto, <lb></lb>connectatur cum globulo ſubduplæ gravitatis, non eá veloci<lb></lb>tate deſcendit, qua deſcenderet ſibi relictus abſque ullâ appen<lb></lb>dice; velociùs tamen movetur, quàm ſi eſſet globuli adjuncti <lb></lb>tantùm ſeſquialter; quia ſcilicet ut ad æqualem velocitatem <lb></lb>temperentur motus tùm impedimenti ſurſum, tùm corporis mo<lb></lb>ventis deorſum, minor intenſivè impetus impediendus eſt à glo<lb></lb>bulo ſubduplo quàm à ſubſeſquialtero; ac propterea major eſt <lb></lb>ſecundùm intenſionem reliquus impetus motum efficiens con<lb></lb>citatiorem. </s> </p> <p type="main"> <s id="s.001218">Quòd autem à globo deſcendente imprimatur impetus glo<lb></lb>bulo, quem ſurſum trahit, hinc conſtat, quod ſi globulus ille <lb></lb>non ſit admodum gravis, tùm demum ſubſilit, ubi globus ve<lb></lb>lociter deſcendens ſubjectum planum attigerit: quid enim il<lb></lb>lum ſubſilire cogeret quieſcente jam globo, à quo trahebatur, <lb></lb>niſi adhuc aliquid impreſſi impetûs remaneret? </s> <s id="s.001219">At quòd im-<pb pagenum="160" xlink:href="017/01/176.jpg"></pb>preſſus hîc impetus non ab ipſo motore, ſed ab impetu, quem <lb></lb>ille concepit, proximè efficiatur, hinc ſibi ſuadent plures, quia <lb></lb>ex alterâ parte impetum ab impetu produci poſſe manifeſtum <lb></lb>videtur ex percuſſionibus projectorum, ut cùm globus pro<lb></lb>jectus in quieſcentem globum impactus illum trudit; ex alterâ <lb></lb>cauſam proximam effectui homogeneam congruenter naturæ <lb></lb>ſtatuimus; ſic enim & calorem in nobis à calore potiùs quàm à <lb></lb>ſubſtantiâ ignis proximè produci exiſtimamus. </s> <s id="s.001220">Sed quid de <lb></lb>percuſſionum impetu dicendum ſit, ſuo loco conſtabit inferiùs. </s> </p> <p type="main"> <s id="s.001221">Motoris demùm velocitatem intenſioni impetûs concepti <lb></lb>non reſpondere experimur, cum valdè conantes ut onus rapte<lb></lb>mus; parùm progredimur; at ſi funis ex improviſo abrumpa<lb></lb>tur, illicò corruimus, impetu ſcilicet concepto motum validiùs <lb></lb>efficiente, ubi deſierit impetum oneri, quod raptabatur, im<lb></lb>primere. </s> </p> <p type="main"> <s id="s.001222">Hinc fit quòd, ſi ea fuerit corporum diſpoſitio, ut impedi<lb></lb>mentum tardè ſubmovendum ſit, ac proinde remiſſiore impetu <lb></lb>opus habeat, qui ſibi imprimatur; corpus verò, cui motus <lb></lb>omnis tribuitur, non æquali tarditate cum impedimento ferri <lb></lb>neceſſe ſit, ſed velociùs præ illo moveri poſſit, hoc ſanè eò mi<lb></lb>nùs habet reſiſtentiæ, quò minorem in intentione impetûs men<lb></lb>ſuram impedimento eidem imprimere debet, ut illud ſubmo<lb></lb>veatur. </s> <s id="s.001223">Contrà verò ſi ita fuerint diſpoſita, ut impedimentum <lb></lb>velociùs præ ipſo motore moveri oporteat, multò magis reſiſtit, <lb></lb>quàm ſi pariter moverentur, plus enim impetûs imprimendum <lb></lb>eſt, ut motus conſequatur. </s> </p> <p type="main"> <s id="s.001224">Hactenùs reſiſtentiam potiſſimùm Formalem, impedimento <lb></lb>nihil in adverſum conante, contemplati ſumus; jam ad Acti<lb></lb>vam tranſeamus, cum ſcilicet duo corpora invicem aut omni<lb></lb>nò, aut ex parte repugnant, quia motum in diverſas aut oppo<lb></lb>ſitas plagas directum moliuntur. </s> <s id="s.001225">In medio vaſe aquâ pleno ſta<lb></lb>tuatur lignea tabella craſſiuſcula, eique lapis imponatur: dum <lb></lb>illa conatur aſcendere, hic deſcendere, ſe invicem urgent; ſed <lb></lb>cum ſe viciſſim permeare nequeant, ſi paribus quidem viribus <lb></lb>confligant, ſine motu conſiſtunt; ſin autem imparibus, aut <lb></lb>ambo aſcendunt, aut ambo deſcendunt, pro ut ſive tabellæ le<lb></lb>vitas, ſive lapidis gravitas oppoſitam vicerit. </s> <s id="s.001226">Quod ſi lapis ta<lb></lb>bellæ non impoſitus, ſed ſuppoſitus, arctè tamen connexus <pb pagenum="161" xlink:href="017/01/177.jpg"></pb>fuerit, adhue contrarios motus conantur, non ſe tamen invi<lb></lb>cem urgent, ſed viciſſim retrahunt, quandiù vinculum non <lb></lb>revellatur, aut rumpatur. </s> <s id="s.001227">Hic verò ſubdubitet quiſpiam, <lb></lb>utrùm corpora, quæ contrario niſu reluctantur, ſibi viciſ<lb></lb>ſim impetum imprimant, nec ne, aut æqualem, ſi pares fue<lb></lb>rint vires, aut, ſi impares, inæqualem: Quando enim ob vi<lb></lb>rium æqualitatem utrumque corpus conſiſtit, codem pacto <lb></lb>quies ſequitur, ſi unumquodque ſuam gravitationem aut levi<lb></lb>tationem ſervans nihil alteri imprimat, ac ſi lignea tabella levi<lb></lb>tans partem impetûs ſurſum directi conferat impoſito lapidi, à <lb></lb>quo gravitante viciſſim recipiat tantumdem impetús deorſum <lb></lb>directi; ex quo fiat, ut lapis habens concepti ac innati impe<lb></lb>tûs deorſum directi vires æquales viribus impetûs ſurſum di<lb></lb>recti conſiſtat, idemque in ligneâ tabellâ contingat. </s> <s id="s.001228">Cùm ve<lb></lb>rò inæquales fuerint vires, id quod validius eſt, eodem modo <lb></lb>ſuperat, ſive nihil contrarij impetûs ab infirmiore oppoſito re<lb></lb>cipiat, ſed minorem motum vi ſui impetûs producat pro ratio<lb></lb>ne virium, quibus ſuperat; ſivè partem impetûs contrarij reci<lb></lb>piat, quæ proprij impetûs vires attenuet. </s> </p> <p type="main"> <s id="s.001229">Quotidianum eſt hujus æqualitatis aut inæqualitatis experi<lb></lb>mentum in iis, quæ innatant humori; hæc enim humori im<lb></lb>poſita, quia in aëre gravitant, deſcendunt; pars verò immerſa <lb></lb>levitat in humore; prægravata tamen à reliquâ parte extante <lb></lb>deorſum adhuc urgetur, donec inter partem immerſam & ex<lb></lb>tantem fiat æquilibrium, & tantumdem pars immerſa levitet in <lb></lb>humore, ac extans gravitat in aëre. </s> <s id="s.001230">Sic maſſa plumbea argento <lb></lb>vivo impoſita deſcendit, donec molis plumbeæ pars (2/13) extet; eſt <lb></lb>enim ſpecifica plumbi gravitas ad ſpecificam mercurij gravita<lb></lb>tem ut 11 ad 13. levitat itaque plumbum in mercurio ut 2, gra<lb></lb>vitat in aëre ut 11; igitur plumbeæ maſſæ partes 11 levitantes <lb></lb>ſingulæ ut 2 parem habent conatum ſurſum, ac partes 2 gra<lb></lb>vitantes ſingulæ ut 11 conantur deorſum. </s> <s id="s.001231">Quòd ſi ita depri<lb></lb>meretur plumbum, ut ejus partes 12 immergerentur, & una <lb></lb>extaret; jam unica pars gravitans ut 11 vinceretur à partibus <lb></lb>12 levitantibus ſingulis ut 2, ac propterea adhuc pars una <lb></lb>emergeret: quemad modum ſi quatuor partes extarent, & no<lb></lb>vem immergerentur, harum levitas 18 ab illarum gravitate 44 <lb></lb>vinceretur, ideóque adhuc duæ immergerentur. </s> </p> <pb pagenum="162" xlink:href="017/01/178.jpg"></pb> <p type="main"> <s id="s.001232">Jam ſi dixeris à partis immerſæ levitantis momentis 18 impe<lb></lb>diri momenta 18 partis extantis gravitantis, adeò ut ſuperſint <lb></lb>tantùm vires juxtà exceſſum gravitatis, ſcilicet momentorum <lb></lb>26, juxta quem exceſſum impetum imprimat parti immerſæ, ut <lb></lb>deprimatur, tunc autem cum paria ſuerint levitatis atque gra<lb></lb>vitatis momenta, jam non invicem agere, ſed ſe viciſſim impe<lb></lb>dire, probabilior fortaſſe videatur alicui philoſophandi ratio <lb></lb>hîc, ubi directè ſibi invicem adverſantur directiones; alteruter <lb></lb>enim aut neuter impetus movet oppoſitum corpus. </s> <s id="s.001233">Verùm <lb></lb>quoniam ubi lineæ directionum motûs non ſunt in directum <lb></lb>poſitæ; ſed inclinationem habent, motus mixtus, qui ſequitur, <lb></lb>ex utroque impetu unum motum temperari indicat, in eam fe<lb></lb>ror ſententiam, ut exiſtimem duo corpora obliquè ſibi invicem <lb></lb>repugnantia viciſſim imprimere, & recipere impetum in diver<lb></lb>ſas plaga directum pro modo virtutis uniuſcujuſque, adeò ut <lb></lb>ſi paria ſint momenta, medius planè inter utramque directio<lb></lb>nem ſequatur motus, ſi diſparia, ſequatur pro modo exceſsûs. </s> </p> <p type="main"> <s id="s.001234">Fieri autem hane mutuam impetûs communicationem hinc <lb></lb>apparet, quòd ſi duo corpora, quorum virtus movendi ut AB <lb></lb><figure id="id.017.01.178.1.jpg" xlink:href="017/01/178/1.jpg"></figure><lb></lb>& AC, inloco, ubi A, conſti<lb></lb>tuta moveri cœperint, alterum <lb></lb>quidem, quod ad dexteram eſt, <lb></lb>cum directione AB, alterum <lb></lb>verò, quod ad ſiniſtram, cum di<lb></lb>rectione AC, ita ſe impediunt, <lb></lb>ut quod ad lævam eſt, urgeat reliquum, ne per rectam AB proce<lb></lb>dat; hoc verò quod ad <expan abbr="dexterã">dexteram</expan> eſt, illud impediat, ne per rectam <lb></lb>AC incedat; ſed propellat ita, ut ambo habeant directionem <lb></lb>mixtam AD. </s> <s id="s.001235">Hæc autem lineæ AD cum major ſit ſingulis <lb></lb>lateribus AB, AC in rectangulo, aut rhomboide, ut quadra<lb></lb>to, aut rhombo, cavè nè putes ſingulis corporibus ſupra pro<lb></lb>prium impetûs modum factam eſſe aliquam ab externo impetu <lb></lb>virium acceſſionem: quî enim fieri poſſit, ut corpus nullo re<lb></lb>pugnante poſſit certo tempore percurrere lineam AB, dimi<lb></lb>nutis verò impetûs viribus ex reſiſtentià, pari tempore longio<lb></lb>rem lineam AD percurrat? </s> <s id="s.001236">An quia recipiat à corpore re<lb></lb>pugnante impetum, cujus acceſſione augeatur proprius impe<lb></lb>tus, qui reliquus eſt? </s> <s id="s.001237">At ſi propter virium æqualitatem percur-<pb pagenum="163" xlink:href="017/01/179.jpg"></pb>rant Quadrati diametrum, utique tantumdem alterum ab alte<lb></lb>ro recipit impetús, quantum tribuit: igitur non eſt major vis <lb></lb>impetus, quàm ſi nihil repugnaret: ex quo fit neque motum ve<lb></lb>lociorem eſſe poſſe, ut pari tempore diametrum percurrant, <lb></lb>quo ſingula deſcriberent latus Quadrati. </s> </p> <p type="main"> <s id="s.001238">Non igitur ex illà mutuá impetus in diversâ directi commu<lb></lb>nicatione fit in ſingulis corporibus impetûs intenſio major (ſi <lb></lb>propriè loquendum ſit, habent enim impetus illi, conceptus <lb></lb>ſcilicet, & impreſſus, directionem diverſam) quàm ferat pro<lb></lb>pria ſingulorum virtus: id autem potiſſimùm conſtat, quando <lb></lb><expan abbr="ſingulorũ">ſingulorum</expan> directiones valdè obtuſum <expan abbr="angulũ">angulum</expan> conſtituunt; cor<lb></lb>pora enim in motu breviorem Rhombi aut Rhomboidis <expan abbr="diame-trũ">diame<lb></lb>trum</expan> deſcribunt, quæ linea aliquando minor eſt ſingulis lateribus. </s> </p> <p type="main"> <s id="s.001239">Finge itaque corpus, quod percurreret AB, nullo impedi<lb></lb>mento prohiberi, quin moveatur eádem velocitate per AD; <lb></lb>utique ſolùm æquale ſpatium AI decurreret, impediret tamen, <lb></lb>ne aliud corpus habens directionem AC, illique perpetuò <lb></lb>adhærens, decurreret juxta ſuam directionem ſpatium æquale <lb></lb>ipſi AC; ſed tantùm EI, hoc eſt Sinum anguli BAD loco <lb></lb>Tangentis ejuſdem anguli, poſito Radio AI. </s> </p> <p type="main"> <s id="s.001240">Finge iterum alterum corpus habens directionem AC eâ<lb></lb>dem velocitate moveri per AD; utique non niſi ſpatium AF, <lb></lb>ipſi AC æquale, motu dimetiretur, prohiberetque, ne reli<lb></lb>quum corpus habens directionem AB, illique perpetuò adhæ<lb></lb>rens, progrederetur niſi in F, hoc eſt ſpatio æquali ipſi BD; <lb></lb>ſed versùs B non procederet niſi juxta menſuram AG mino<lb></lb>rem ipsâ AC. </s> <s id="s.001241">Atqui utrumque ſuam habet directionem, & <lb></lb>non per AD, ſeque viciſſim impediunt; igitur dum ſimul mo<lb></lb>ventur, neque ſubſiſtunt in F, neque veniunt in I; ſed medio <lb></lb>loco conſiſtunt, puta in O. </s> </p> <p type="main"> <s id="s.001242">Dixeris fortaſſe AO æqualem ipſi AE ita, ut ſit ſicut DB <lb></lb>ad BA, ita IE ad EA, hoc eſt ad AO, aut AO eſſe medio <lb></lb>loco proportionalem inter AF & AI, hoc eſt inter AC & AB <lb></lb>menſuras virium impetûs ſingulorum corporum. </s> <s id="s.001243">Hoc tamen <lb></lb>ſecundo loco propoſitum non facilè admiſerim, quia ubi æqua<lb></lb>les ſunt virtutes movendi, medio loco proportionalis eſt æqua<lb></lb>lis ſingulis extremis, ac propterea utrumque corpus impeditum <lb></lb>æque velociter moveretur, ac non impeditum. </s> <s id="s.001244">Primum verò, <pb pagenum="164" xlink:href="017/01/180.jpg"></pb>quod ſcilicet AO æqualis ſit ipſi AE, gratis aſſeritur; neque <lb></lb>enim potior ulla apparet ratio, cur ad inſtituendam analogiam <lb></lb>aſſumatur potiùs IE, quàm quælibet alia minor linea cadens <lb></lb>inter G & E. </s> <s id="s.001245">Ego autem libentiùs profiteor me neſcire, quà <lb></lb>Ratione analogia hæc inſtituatur, quam aliquid certi divinan<lb></lb>do ſtatuere. </s> </p> <p type="main"> <s id="s.001246">Verùm quamvis non utrumque corpus velociùs moveatur <lb></lb>quàm pro ſuâ virtute, alterum tamen quod urgetur, ſeu rapitur <lb></lb>à validiori, poteſt, factâ impetûs acceſſione, plus ſpatij percur<lb></lb>rere, quàm pro ſuis viribus: impeditur ſiquidem motus non ab<lb></lb>ſolutè, ſed juxtà eam directionem. </s> <s id="s.001247">Hinc fit corpus habens di<lb></lb>rectionem & velocitatem AC minorem velocitate AB promo<lb></lb>veri ultrà punctum F in linea mixti motûs AD. </s> </p> <p type="main"> <s id="s.001248">At inquis: an ſi nautæ remis incumbant, veliſque obliquis <lb></lb>ventum excipiant, tardior erit motus, quàm ſi navis vel à ſolis <lb></lb>remigibus, vel à ſolo vento impelleretur? </s> <s id="s.001249">contrarium ſanè vi<lb></lb>detur experientia evincere. </s> <s id="s.001250">Verùm ſi rem attentiùs conſideres, <lb></lb>aliam planè eſſe rationem deprehendes, cum duo corpora ſe <lb></lb>moventia viciſſim ſe impediunt, aliam cùm unum à duplici ex<lb></lb>trinſeco impetu in diverſa directo impellitur: de illis hactenùs <lb></lb>ſermo fuit, neque ulla ratio ſuadere poteſt velocius à tardiore <lb></lb>incitari, quamquam tardius à velociore urgeatur, ut dictum eſt. </s> </p> <p type="main"> <s id="s.001251">At ſi unum corpus à duobus æqualis aut inæqualis virtutis <lb></lb>impetum recipiat, utique magis intenſus, vel ſi intenſionem <lb></lb>propriè dictam neges, certè major eſt impetus, quàm ſi ab al<lb></lb>terutro tantùm reciperet impetum: quare nil mirum, ſi ea mo<lb></lb>tûs velocitas conſequatur, quæ utrumque impetum ſingillatim <lb></lb>ſumptum vincat, quamvis utroque ſimul ſumpto minor ſit, quia <lb></lb>habent directiones oppoſitas, ut alibi explicabitur. </s> <s id="s.001252">Hinc eſt <lb></lb>navim velociùs agi velis remiſque, quàm ſi aut ſolâ ventorum <lb></lb>vi, aut ſolâ remigum ope propelleretur, & cymbam, dum ſe<lb></lb>cundo flumine rapitur, ſimulque remis ad alteram ripam im<lb></lb>pellitur, velociùs moveri, quàm aut in ſtagno eâdem remigum <lb></lb>operâ, aut à flumine ceſſantibus remis ageretur. </s> <s id="s.001253">Quemadmo<lb></lb>dum enim neque ventus remos impellit, neque ab his ventus <lb></lb>impellitur, ita neque ſe viciſſim immediatè impediunt, aut ſibi <lb></lb>mutuò repugnant; atque adeò non eſt hîc eadem philoſophan<lb></lb>di ratio, ac cum duo corpora ſibi invicem immediatè reſiſtunt, <pb pagenum="165" xlink:href="017/01/181.jpg"></pb>& alterum alterius vires extenuat impediens, ne juxtà propriæ <lb></lb>virtutis menſuram motum concipiat. </s> </p> <p type="main"> <s id="s.001254">Ex his quæ hactenùs dicta ſunt, illud ſatis conſtare videtur, <lb></lb>quòd animal eatenùs in motu difficultatem ac reſiſtentiam per<lb></lb>cipit, quatenùs multum impetûs concipere debet, ex quo muſ<lb></lb>culorum contentio oritur, neque tamen ea ſequitur motûs ve<lb></lb>locitas, quæ tanto impetui reſponderet, dum ſubmovendo im<lb></lb>pedimento maximam virium partem impendit impetum impri<lb></lb>mens: unde fit plurimum influentis ſpiritûs animalis abſumi in <lb></lb>tàm diuturnâ, vel tàm validâ muſculorum contentione, ac <lb></lb>proinde laſſitudinem ſequi, atque aliquando etiam contento<lb></lb>rum muſculorum dolorem, cum id non contingat ſine aliquâ <lb></lb>partium compreſſione aut diſtentione. </s> <s id="s.001255">Quò igitur velociùs <lb></lb>moveri poteſt animal pro ratione concepti impetûs, eò mino<lb></lb>rem percipit in ſubmovendo impedimento difficultatem; & <lb></lb>quidem maximè ſi alternâ contentionis ac remiſſionis muſcu<lb></lb>lorum viciſſitudine labor miteſcat. </s> </p> <p type="main"> <s id="s.001256">Curioſiùs autem inquirenti, quam Rationem habeat motoris <lb></lb>impetus ad impetum corpori, quod movetur, quatenus move<lb></lb>tur, impreſſum, ut aliquatenus ſatisfaciam, aſſero ut minimum <lb></lb>duplam eſſe, non quidem intenſivè, aut extenſivè ſed enti<lb></lb>tativè. </s> <s id="s.001257">Quatenùs, inquam, movetur, hoc eſt quatenus vinci<lb></lb>tur ejus reſiſtentia: cæterùm potentia movens in ſe producit, & <lb></lb>in mobili æqualem impetum; ſed quemadmodum ubi calor fri<lb></lb>gori permiſcetur illud vincens, non percipitur niſi quatenus <lb></lb>excedit vim frigoris, ita impetus oneri impreſſus eatenus mo<lb></lb>vet, quatenùs ejuſdem reſiſtentiam ſuperat: Hunc autem ex<lb></lb>ceſſum ſubduplum impetûs motoris ſatis probabili conjecturâ <lb></lb>affirmo. </s> <s id="s.001258">Illud enim hoc mihi ſuadet, quòd motoris virtutem <lb></lb>metitur exceſſus impetûs, quem ille habet ſuprà impedimenti <lb></lb>reſiſtentiam: reſiſtentiæ autem modus, ut ſæpiùs dictum eſt, <lb></lb>ex velocitate motûs, quæ concilianda eſt gravitati corporis ſub<lb></lb>movendi, deſumitur; hoc enim ideò reſiſtit partibus ex gr.100 <lb></lb>impetûs, quia ſi ſolùm fuerint 100 partes impetûs, fieri non po<lb></lb>teſt ut moveatur tantâ velocitate, ſed pluribus impetûs parti<lb></lb>bus indiget: exceſſus igitur virtutis motoris æqualis eſt ut mi<lb></lb>nimum reſiſtentiæ mobilis; atque adeò tota virtus motoris, hoc <lb></lb>eſt impetus ab eo conceptus, æquivalet tùm reſiſtentiæ mobi-<pb pagenum="166" xlink:href="017/01/182.jpg"></pb>lis juxta menſuram requiſitam ad motum, qui ſequitur, tùm <lb></lb>principio motûs ejuſdem mobilis: atqui motus hic æqualis eſt <lb></lb>motui, cui illud reſiſtit, totus igitur impetus motoris duplus eſt <lb></lb>impetùs, qui motum efficit in mobili, quatenus movetur. </s> </p> <p type="main"> <s id="s.001259">Hinc eſt eodem conatu motoris diſparem effici motum, ſi <lb></lb>potentia æqualiter moveatur cum mobili, ut conſtat: quia ni<lb></lb>mirum impetus mobili impreſſus inæqualem habet intenſio<lb></lb>nem, quamvis entitativè æqualis ſit. </s> <s id="s.001260">Si enim tota motoris vir<lb></lb>tus ſit 20, & decem impetûs particulas reſiſtentiam ſuperantes <lb></lb>mobili imprimat, in quo intenſio fiat ut 1, in mobili gravitatis <lb></lb>ſeſquialteræ, particulæ eædem decem impetûs intenſionem ef<lb></lb>ficiunt ut 2/3; quare & hujus motus erit ſubſeſqui alter, ac pro<lb></lb>inde motor, qui æqualiter cum mobili movetur, etiam tardio<lb></lb>rem habet motum, quàm cùm motum priori mobili conci<lb></lb>liabat. </s> </p> <p type="main"> <s id="s.001261">Patet igitur ex his nunquam fieri poſſe, ut corpus grave mi<lb></lb>noris aut æqualis virtutis alterum moveat ita, ut planè in velo<lb></lb>citate conſentiant; illud enim corpus minùs aut æquè grave <lb></lb>concipere non poteſt impetum, qui & ſibi ad motum ſufficiat, <lb></lb>& alteri impetum imprimat: finge ſcilicet animo fuiſſe impe<lb></lb>tum impreſſum corpori æquè vel magis gravi; hîc utique cum <lb></lb>non excedat reſiſtentiam mobilis, nullum efficere poteſt mo<lb></lb>tum; igitur neque impreſſus fuit impetus, ne ſit omninò inuti<lb></lb>lis. </s> <s id="s.001262">Quòd ſi eâ ratione diſponantur ut motor velociùs moveri <lb></lb>poſſit quàm mobile, jam fieri poteſt, ut à minore majus movea<lb></lb>tur: nam ſi motor certâ quâdam velocitate movere poſſit pon<lb></lb>dus unius libræ motu ſibi æquali, eodem conatu & eádem ve<lb></lb>locitate ſe movens movebit pondus centum librarum, ſi hoc ita <lb></lb>ſit diſpoſitum, ut centuplo tardiùs moveatur: quia nimirum <lb></lb>idem entitativè impetus in hoc pondere centuplo remiſſior, <lb></lb>quàm in pondere unius libræ, ſufficit ad motum centuplo tar<lb></lb>diorem. </s> <s id="s.001263">Motus ſiquidem centum librarum ſubcentuplus in ve<lb></lb>locitate, æqualis eſt motui unius libræ centuplo in velocitate; <lb></lb>ſi enim libra percurrit centum ſpatij digitos ſibi ſuccedentes in <lb></lb>longitudine, pari tempore centum libræ percurrunt quidem <lb></lb>unicum digitum longitudinis ſpatij, centum tamen ſpatia digi<lb></lb>talia percurrunt, ſingulæ ſcilicet libræ digitum. <pb pagenum="167" xlink:href="017/01/183.jpg"></pb> </s> </p> <p type="main"> <s id="s.001264"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001265"><emph type="center"></emph><emph type="italics"></emph>In quo Machinarum vires ſitæ ſint.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001266">POtentiam oneri movendo cæteroqui imparem præſtare poſ<lb></lb>ſe, ſi machina adhibeatur, quotidiano experimento diſci<lb></lb>mus; adeò ut ipſa unica pluribus potentiis machinâ deſtitutis <lb></lb>virtute æqualis ſit, & quæ pondus ſolitarium ac ſimplex loco <lb></lb>prorſus movere non poterat, ubi ſe ad machinam applicuerit, <lb></lb>jam non ponderi tantùm, ſed & machinæ motum conciliet. </s> <lb></lb> <s id="s.001267">Quid ergo illud ſit, ex quo hujuſmodi virium incrementum <lb></lb>oritur, hîc perveſtigandum eſt; & ad illud cauſæ genus revo<lb></lb>catur, quam Scholæ Formalem appellant; eſt ſcilicet ratio, per <lb></lb>quam fit, ut ſit, atque dicatur Machina: hoc autem incremen<lb></lb>tum virium, ut ex dicendis conſtabit, ex machinæ figurâ pen<lb></lb>det ſecundùm quam potentiæ, & ponderis motus ſibi invicem <lb></lb>pro ratâ portione reſpondent. </s> </p> <p type="main"> <s id="s.001268">A machinâ quâ machina eſt, potentiæ moventis vires non <lb></lb>augeri certum eſt; nihil enim illi interioris virtutis impertitur, <lb></lb>& quâ machina eſt, ab omni innatâ gravitate ſejuncta intelli<lb></lb>gitur: vectis ſiquidem, ferreus ſit, ſive ligneus, machinæ ra<lb></lb>tionem non immutat, ſi ſola intercedat materiæ gravioris aut <lb></lb>levioris diſparitas. </s> </p> <p type="main"> <s id="s.001269">Quòd ſi faciliùs ferreo vecte tricubitali deorſum premens at<lb></lb>tollas ſaxum, quàm ſi ligneo vecte pariter tricubitali utaris <lb></lb>(quia nimirum ferreus vectis habet ſibi adnexam ex gravi ma<lb></lb>teriâ, quâ conſtat, potentiam, quæ deorſum urgendo te juvat, <lb></lb>ut ſaxum attollatur,) id planè eſſe extra vectis naturam, quâ <lb></lb>vectis eſt, manifeſtum erit, ſi non deorſum, ſed ſurſum, aut à <lb></lb>lævâ in dextram connitendum ſit, ut duo connexa disjungas; <lb></lb>tunc enim ferrei vectis gravitas ſuſtentanda laborem potiùs <lb></lb>creabit, quàm ut præ ſimili ligneo vecte motum hunc facilio<lb></lb>rem reddat. </s> <s id="s.001270">Quare præter Mechanicæ facultatis inſtitutum <lb></lb>machinis accidit, ut gravitate ſuâ potentiæ moventis vires ad<lb></lb>augeant, non quidem illam immutando, facto interiore virtu-<pb pagenum="168" xlink:href="017/01/184.jpg"></pb>tis additamento; ſed aliam potentiam, quæ conjunctis cum illi <lb></lb>viribus agat, conſociando. </s> </p> <p type="main"> <s id="s.001271">Sed & illud animadvertendum eſt, vix unquam fieri poſſe, <lb></lb>ut potentia movens nihil prorſus impedimenti à machina reci<lb></lb>piat: ſivè enim machinæ ipſius pars aliqua gravis elevanda eſt; <lb></lb>ſivè membrorum, in quæ machina diſtribuitur, invicem con<lb></lb>fligentium, ſeque viciſſim terentium aſperitas obſiſtit; ſive mo<lb></lb>tus (ut machinæ ipſi, cui applicatur potentia, obſecundet) à <lb></lb>ſuâ directione inflectitur; ſivè quid hujuſmodi intercedit, quod <lb></lb>aliquid de motûs velocitate imminuat, quæ cæteroqui concep<lb></lb>tum potentiæ ab omni machinâ abſolutæ impetum conſequere<lb></lb>tur. </s> <s id="s.001272">Ex his tamen aliqua ſunt, quæ ita motui potentiæ offi<lb></lb>ciunt, ut ad retinendum onus juvent; hujus ſiquidem gravitas <lb></lb>minùs adversùs potentiam valet, ſi & ipſum, quia machinæ il<lb></lb>ligatum à recto in centrum gravium tramite deflectere, vel <lb></lb>mutuum partium ſe terentium conflictum vincere cogatur, ut <lb></lb>vim potentiæ inferat. </s> <s id="s.001273">Verùm hæc, quamvis, ubi res ad praxim <lb></lb>deducitur, per incuriam diſſimulanda non ſint, ſub ſtaticam <lb></lb>conſiderationem hîc non cadunt, ubi machinarum vires ex<lb></lb>penduntur; harum enim figura perindè attenditur, atque ſi <lb></lb>nihil adjumenti, nihil detrimenti ex materiâ accederet. </s> </p> <p type="main"> <s id="s.001274">Ad rem itaque propiùs accedentibus recolenda ſunt ea, quæ <lb></lb>in ſuperioribus hujus libri capitibus diſputata ſunt, proximam <lb></lb>videlicet motûs effectricem cauſam impetum eſſe ſive ab inte<lb></lb>riore virtute manantem in iis, quæ ſponte ſuâ moventur, ſivè <lb></lb>extrinſecùs aliunde impreſſum iis, quæ naturâ repugnante per <lb></lb>vim cientur: ex cujus impetûs intenſione, quatenùs omnem <lb></lb>reſiſtentiam ſuperat, motuum velocitas oritur: nunquam autem <lb></lb>à velocitate aut tarditate motum ſejungi poſſe certum eſt, quip<lb></lb>pe qui nec ſinè ſpatio per quod decurratur, nec ſinè partium <lb></lb>ſibi certâ lege ſuccedentium continuatione ac ſerie intelli<lb></lb>gi poteſt. </s> <s id="s.001275">Quare & reſiſtentiæ momenta tùm ex corporis <lb></lb>movendi gravitate, tùm ex velocitate componi ſæpiùs innui<lb></lb>mus, ut hinc innoteſcat fieri facilè poſſe, ut, ſicut ejuſdem <lb></lb>gravitatis reſiſtentia inæqualis eſt, ſi velocitate inæquali mo<lb></lb>venda ſit, & gravitatum inæqualium diſparia ſunt reſiſtentiæ <lb></lb>momenta, ſi Ratio, quæ ex gravitatum & velocitatum Ratio<lb></lb>nibus componitur, ſit Ratio Inæqualitatis, quia gravior velo-<pb pagenum="169" xlink:href="017/01/185.jpg"></pb>ciùs, minùs gravis tardiùs movetur; ita gravitatum inæqualium <lb></lb>par ſit reſiſtentia, ſi quæ inter gravitates intercedit Ratio, ea<lb></lb>dem reciprocè inter velocitates inveniatur. </s> <s id="s.001276">Quemadmodum <lb></lb>enim quæcumque calori adverſantur, vehementiorem quidem <lb></lb>validiſſimè reſpuunt, tenuiſſimum verò facillimè admittunt; <lb></lb>haud diſpari ratione pondera, ſi velociùs incitare velis, im<lb></lb>pensiùs reluctantur, minimo ac tardiſſimo motui leviſſimè ob<lb></lb>ſiſtunt. </s> </p> <p type="main"> <s id="s.001277">Quoniam igitur naturâ definitum eſt, quantam gravitatem, <lb></lb>quantáque velocitate, pro certâ impreſſi impetûs menſurâ, mo<lb></lb>vere poſſit Potentia concepto impetu, qui pro ratâ portione <lb></lb>reſpondeat impetui quem illa oneri imprimit, ut Potentia, & <lb></lb>onus æquali velocitate moveantur; ſatis conſtat eandem impe<lb></lb>tús menſuram parem eſſe movendo oneri graviori, ſi quá Ra<lb></lb>tione poſterior hæc gravitas priorem gravitatem vincit, eâdem <lb></lb>Reciprocè Ratione prioris velocitas poſterioris tarditatem ſu<lb></lb>peret; utrobique ſcilicet par eſt reſiſtentia, ac proinde ab eâ<lb></lb>dem potentiâ vinci poteſt. </s> <s id="s.001278">Cùm enim ea, quæ ſimul æqualiter <lb></lb>moventur, æquali impetu ferantur; ſi Potentia tàm tardè mo<lb></lb>veretur ac pondus per machinam, indigeret impetu ex. </s> <s id="s.001279">gr. ſub<lb></lb>quintuplo ejus quo illa movetur quintuplo velociùs ac ipſum <lb></lb>Pondus. </s> <s id="s.001280">Verùm impetus hîc ſubquintuplus ineptus eſſet ad <lb></lb>oneris reſiſtentiam quintuplo ferè majorem vincendam; ſed ſo<lb></lb>lum ſuperare poſſet ac movere 1/5 ponderis. </s> <s id="s.001281">Quinque igitur im<lb></lb>petus huic æquales poſſunt totam reſiſtentiam ſuperare. </s> <s id="s.001282">Cum <lb></lb>itaque in motu quintuplo velociori Potentiæ ſit verè impetus <lb></lb>quintuplus, poterit etiam elevare pondus, quod eſt quintuplo <lb></lb>majus, quàm ſit 1/5 ipſius. </s> <s id="s.001283">Verùm híc ubi de motûs velocitate <lb></lb>ſermo eſt, non is quidem abſolutè accipiendus eſt; ſed quâ <lb></lb>parte gravium naturæ repugnat: ſi enim plumbeus globus <lb></lb>A ex C dependeat funiculo CA, & circà verſatilem or<lb></lb>biculum B ſtabili axi infixum ducatur filum connectens <lb></lb>globos A & D, cottum quidem eſt globum A, ſi uſque ad <lb></lb>B perveniat, tantumdem ſpatij in arcu AB percurrere, <lb></lb>non tamen tantumdem aſcendere, quantum globus D ſe<lb></lb>cundùm rectam BD deſcendit; ſed aſcenſum metitur AE, <lb></lb>nimirum Sinus Verſus arcûs AB, qui minor eſt codem <lb></lb>arcu (arcus ſiquidem major eſt rectâ AB lineâ ipſum ſub-<pb pagenum="170" xlink:href="017/01/186.jpg"></pb><figure id="id.017.01.186.1.jpg" xlink:href="017/01/186/1.jpg"></figure><lb></lb>tendente, quæ oppoſita <lb></lb>recto angulo E major eſt <lb></lb>quàm trianguli baſis AE) <lb></lb>ac propterea reſiſtentiæ <lb></lb>momenta non ea ſunt, quæ <lb></lb>ex velocitate motûs AB, <lb></lb>ſed AE, & ipsâ globi A <lb></lb>gravitate componuntur. </s> <s id="s.001284">Ex <lb></lb>quo fit globum D quam<lb></lb>vis minorem poſſe globo A <lb></lb>graviori præſtare, ac illum <lb></lb>ad certam altitudinem ele<lb></lb>vare, ut cuilibet experiri <lb></lb>licet, cum tamen illi aſcen<lb></lb>ſum ſuo deſcenſui æqualem <lb></lb>nullatenùs conciliare poſſit. </s> <lb></lb> <s id="s.001285">Quòd ſi idem globus A ex breviore funiculo HA dependeat, <lb></lb>experimento conſtat opus eſſe globo D gravitatem addere, ut <lb></lb>valeat illum per arcum AF elevare ad eandem altitudinem <lb></lb>AE: magis quippè laborioſum eſt breviore motu AF, quàm <lb></lb>longiore motu AB ad eandem altitudinem aſcendere; atque <lb></lb>adeò plus virium in D requiritur, ut globo A majorem impetum <lb></lb>imprimat, ex cujus intenſione plus ſingulis temporis momentis <lb></lb>aſcendat in hoc poſteriore motu, quàm in priore. </s> <s id="s.001286">Ne tamen <lb></lb>motui globi D tribue menſuram arcûs AB ſed rectæ AB. </s> </p> <p type="main"> <s id="s.001287">Sicut autem ubi potentiæ & oneris æquales eſſe debent mo<lb></lb>tus, potentiæ vires gravitate oneris majores eſſe oportet, ut vim <lb></lb>illi inferant; ita pariter ubi potentia & onus in motuum velo<lb></lb>citate diſſentiunt, & illa quidem velociùs, hoc tardiùs move<lb></lb>tur, neceſſe eſt majorem eſſe Rationem Potentiæ ad Onus (licet <lb></lb>illa minor ſit onere) quàm ſit Ratio tarditatis hujus ad illius <lb></lb>velocitatem; ut ſcilicet ratio Potentiæ ad onus, quæ ex mo<lb></lb>tuum, & virium Rationibus componitur, ſit Ratio majoris inæ<lb></lb>qualitatis. </s> <s id="s.001288">Sit ex. </s> <s id="s.001289">gr. Ratio motûs Potentiæ ad motum Oneris <lb></lb>ut 3 ad 2; ſi Ratio virium potentiæ abſolutè ſumptæ ad gravi<lb></lb>tatem oneris ſit Reciprocè ut 2 ad 3, Ratio ex his Rationibus <lb></lb>compoſita eſt Æqualitatis, ſcilicet 1 ad 1, & motus nullus ſe<lb></lb>quitur; multò minùs ſi fuerit Ratio minor quàm 2 ad 3; prove-<pb pagenum="171" xlink:href="017/01/187.jpg"></pb>niret enim Ratio minoris Inæqualitatis: debet ergo eſſe major <lb></lb>Ratione 2 ad 3. Sit ex hypotheſi Ratio 4 ad 5; jam Ratio com<lb></lb>poſita ex Rationibus 3 ad 2, & 4 ad 5, eſt Ratio 6 ad 5 majoris <lb></lb>Inæqualitatis. </s> </p> <p type="main"> <s id="s.001290">Neque hoc ita dictum intelligas, quaſi motus ipſe Potentiæ, <lb></lb>ejuſque velocitas, efficiendi vim haberet; ſed ex ipsá majore <lb></lb>potentiæ velocitate innoteſcit impetum, qui radix eſt motûs, <lb></lb>minus invenire impedimenti ex onere, quod minùs reſiſtit, eo <lb></lb>quòd tardiùs movendum eſt, quàm ſi æqualem velocitatis gra<lb></lb>dum cum potentiâ ſortiri deberet. </s> <s id="s.001291">Quare licèt potentia minor <lb></lb>ſit, ac pauciores entitativè particulas impetús producere valeat, <lb></lb>quàm potentia major, ſatis in aperto eſt fieri poſſe, ut potentia <lb></lb>major majorem inveniens reſiſtentiam nequeat impetum im<lb></lb>primere, ac movere onus, quod movebitur à minore potentiâ, <lb></lb>ſi onus idem minùs reſiſtat, cum ſit tardiùs movendum: impe<lb></lb>tus enim à minore potentiâ oneri impreſſus ſatis eſt ad vincen<lb></lb>dam minorem hanc reſiſtentiam; cum tamen potentia major <lb></lb>non ſatis habeat virtutis, ut eam impetús menſuram oneri im<lb></lb>primat, quæ majorem illius reſiſtentiam ſuperaret. </s> </p> <p type="main"> <s id="s.001292">In eo igitur totum Mechanices artificium conſiſtit, ut ſua <lb></lb>inſtrumenta ita diſponat, lociſque congruis ita Potentiam, & <lb></lb>Onus collocet, ut Potentiæ motus velocior ſit præ motu Oneris: <lb></lb>tùm horum motuum Ratione attentè perſpectâ definies, quæ<lb></lb>nam Potentia datum Onus movere, vel quodnam Onus à datâ <lb></lb>Potentiâ moveri queat; ſi nimirum Potentiæ vires ad oneris <lb></lb>gravitatem majorem habeant Rationem, quàm ſit Ratio motùs <lb></lb>Oneris ad motum Potentiæ. </s> <s id="s.001293">Neque enim Machina aut Poten<lb></lb>tiæ vires auget, aut oneris gravitatem minuit, ſed Ponderis re<lb></lb>ſiſtentiam ad Potentiæ virtutem accommodat. </s> </p> <p type="main"> <s id="s.001294">Phyſica autem cauſa hæc eſt, quia impetus à Potentiâ pro<lb></lb>ductus, qui in onere minori movendo æque velociter cum po<lb></lb>tentiâ <expan abbr="majorẽ">majorem</expan> haberet intenſionem, in onere majore ſed tardiùs <lb></lb>movendo minorem quidem habet intenſionem, ſed quæ ſatis eſt <lb></lb>pro minore <expan abbr="reſiſtẽtia">reſiſtentia</expan>. </s> <s id="s.001295">Fac enim oneris particulas graves eſſe 20, <lb></lb>illique à <expan abbr="Potẽtiâ">Potentiâ</expan> <expan abbr="aliquãto">aliquanto</expan> graviore imprimi particulas 100 impe<lb></lb>tûs, quibus vincitur Oneris reſiſtentia: intenſio in ſingulis par<lb></lb>ticulis gravitatis eſt particularum impetûs 5, juxtà quam inten<lb></lb>ſionis menſuram ſequitur motus æque velox Potentiæ & oneris, <pb pagenum="172" xlink:href="017/01/188.jpg"></pb>hujus quidem per vim ſursùm; illius verò juxtà naturam deor<lb></lb>ſum. </s> <s id="s.001296">Sit adhuc eadem Potentia; ſed offeratur Onus, cujus <lb></lb>particulæ gravitatis ſint non jam 20; ſed 50: Potentiæ virtus eſt <lb></lb>eadem; quapropter non niſi reſiſtentiam vincere poteſt, cui <lb></lb>vincendæ ſufficiant particulæ 100 impetus; hæ autem in One<lb></lb>re graviore ut 50 efficerent ſolùm intenſionem ut 2: Non igitur <lb></lb>Potentia & onus æquè veloci motu, qui reſpondeat intenſioni <lb></lb>ut quinque, ſicuti priùs, moveri poterunt; ſed ut onus moveri <lb></lb>poſſit, impetúmque à potentiâ recipere, opus eſt ita illud col<lb></lb>locare, ut quò magis Ratione gravitati reſiſtit; eò minùs ra<lb></lb>tione tarditatis motûs reſiſtat, ſeque eâ ratione temperent duæ <lb></lb>hæ reſiſtentiæ, ut una confletur reſiſtentia non major illâ, quæ <lb></lb>oriebatur ex onere gravi ut 20 æqualiter movendo: id quod <lb></lb>fiet, ſi motus Potentiæ, quatenùs machinæ applicatur, ad mo<lb></lb>tum oneris ſit ut 5 ad 2 in Reciprocâ Ratione intenſionum im<lb></lb>petûs producti. </s> <s id="s.001297">Quare motus Potentiæ ad motum oneris eſt <lb></lb>duplus ſeſquialter, quemadmodum poſterior hæc oneris gravi<lb></lb>tas ut 50 eſt prioris gravitatis ut 20 dupla ſeſquialtera: atque <lb></lb>hinc manifeſtum eſt particulas gravitatis 50 reſiſtentes ut 2 ra<lb></lb>tione motûs comparati cum motu potentiæ, requirere particu<lb></lb>las 100 impetûs, quemadmodum particulæ gravitatis 20 re<lb></lb>ſiſtentes ut 5 ratione motûs comparati cum motu ejuſdem Po<lb></lb>tentiæ requirunt particulas 100 impetûs. </s> <s id="s.001298">Quid igitur mirum, ſi <lb></lb>potentia eadem eodem conatu movet onus ut 50 velocitate ut 2, <lb></lb>quo conatu movet onus ut 20 velocitate ut 5? </s> </p> <p type="main"> <s id="s.001299">Servatur itaque perpetua quædam juſtitia inter potentiæ vi<lb></lb>res, oneris gravitatem, ſpatia motuum, ac tempora; quò enim <lb></lb>decreſcunt potentiæ vires, aut oneris gravitas augetur, eò bre<lb></lb>viora ſunt ſpatia, & longiora tempora motuum ipſius oneris; <lb></lb>ſed ampliora ſpatia motuum potentiæ debilioris, quæ præ one<lb></lb>re velociùs movetur. </s> <s id="s.001300">Hinc dato onere graviori ſubmovendo, <lb></lb>aut potentiam augeri, aut, ſi illa immutata permaneat, oneris <lb></lb>motum imminui, ſeu potentiæ motum augeri neceſſe eſt: Te<lb></lb>nui enim potentiâ ingens pondus citò moveri non poteſt. </s> </p> <p type="main"> <s id="s.001301">Formalem igitur Machinæ Rationem, quâ Machina eſt, in eo <lb></lb>ſitam eſſe deprehendimus, quòd ea figura ſit, quæ potentiæ, <lb></lb>& oneris motibus legem ita ſtatuat, ut Potentia velociter, Pon<lb></lb>dus lentè moveatur; ſic enim fit, ut minor oneris reſiſtentia vir-<pb pagenum="173" xlink:href="017/01/189.jpg"></pb>tuti vim movendi, etiamſi minorem, habenti pro ratâ portio<lb></lb>ne reſpondeat. </s> <s id="s.001302">Satis igitur erit, ubi ſingularum machinarum <lb></lb>vires expendendæ erunt motuum inire rationes, qui ex machi<lb></lb>næ agitatione oriuntur: nam ſi Potentia præ Onere velociùs <lb></lb>moveatur, operæ pretium faciet Machinator; modò non adeò <lb></lb>tenuis ſit motuum Ratio, ut quicquid utilitatis ex machinæ fi<lb></lb>gurà accedit, deferatur ex partium ſe terentium conflictu; nam <lb></lb>perinde eſſet, ac ſi oneri gravitas adderetur. </s> </p> <p type="main"> <s id="s.001303">Ex his liquet à non paucis plus operæ laboriſque conſump<lb></lb>tum, quàm par eſſet, ut Ariſtoteli adhærerent in referendis <lb></lb>machinarum viribus in circuli naturam planè admirandam: <lb></lb><emph type="italics"></emph>Quapropter<emph.end type="italics"></emph.end> inquit initio <expan abbr="qq.">qque</expan> Mechan. <emph type="italics"></emph>non eſt inconveniens ipſum <lb></lb>miraculorum omnium eſſe principium. </s> <s id="s.001304">Ea igitur quæ circà libram fiunt, <lb></lb>ad circulum referuntur, quæ verò circa vectem, ad ipſam libram; <lb></lb>alia autem ferè omnia, quæ circa mechanicas ſunt motiones, ad <lb></lb>vectem.<emph.end type="italics"></emph.end></s> <s id="s.001305"> Niſi enim fucum veritati faciamus, quæ demum mi<lb></lb>racula ita circulum à reliquo figurarum vulgo ſecernunt, ut in <lb></lb>cum admiratio omnis corrivata confluat, nec niſi hinc in cæte<lb></lb>ras derivetur? </s> <s id="s.001306">An quòd linea eadem, quâ circuli ambitus de<lb></lb>finitur, omnis latitudinis expers, cava pariter atque convexa <lb></lb>amico fœdere copulat, quæ ſibi invicem repugnant? </s> <s id="s.001307">Cavum <lb></lb>ſi quidem à convexo, quæ recto interjecto diſcriminantur, per<lb></lb>inde diſſidere cenſemus, atque minus à majori, inter quæ ſibi <lb></lb>adverſantia id, quod æquale eſt, intercedit. </s> <s id="s.001308">At hæc ita vulga<lb></lb>ria ſunt, ut non Hyperbolæ ſolùm, ac Parabolæ, aut Nicome<lb></lb>dis Conchoidi, aut Archimedis Spiralibus, aut Dinoſtrati <lb></lb>Quadratici, cæteriſque omnibus extrà Geometricas leges cur<lb></lb>vis lineis communia ſint; verùm etiam in angulo quocumque <lb></lb>rectilineo facilè ab omnibus obſerventur; cum lineæ rectæ, qui<lb></lb>bus inclinatis angulus conſtituitur, hinc quidem ſibi mutuis <lb></lb>nutibus annuere, hinc verò abnuere videantur; quibus oppo<lb></lb>ſitis nutibus media pariter interjacet directa poſitio, omni in<lb></lb>clinatione ſubmotâ. </s> </p> <p type="main"> <s id="s.001309">An ipsâ naſcentis Circuli exordia admiratione non carent, <lb></lb>quòd æquè ex Radij ejuſdem in centro ſubſiſtentis quiete, ac <lb></lb>circumlati motu oriatur? </s> <s id="s.001310">Sed quid hæc in circulo potiùs ſuſ<lb></lb>piciamus, quàm in Helice, cui geneſis haud diſpar contingit? </s> <lb></lb> <s id="s.001311">Quòd ſi circulo primas ideò deferendas exiſtimemus, quòd <pb pagenum="174" xlink:href="017/01/190.jpg"></pb>in ſe recurrens peripheria ibi ſui motûs terminum inveniat, <lb></lb>unde ſumpſit exordium; & circumacta, quæ ex adverſo <lb></lb>ſunt, partes oppoſitis cieat motibus, ita ut progredientibus <lb></lb>ſupremis infimæ regrediantur, & in ima detrudantur ſi<lb></lb>niſtræ, dextris in altiora provectis: Quid Ellipſim præjudi<lb></lb>cio repellimus? </s> <s id="s.001312">cum & hæc unico limite cavo pariter atque <lb></lb>convexo in ſeſe redeunte circumſcripta in contrarias partes <lb></lb>incitetur; nec à rectâ tantummodo lineâ alternis auctà cre<lb></lb>mentis, imminutáque decrementis altero terminorum quieſ<lb></lb>cente, ſed etiam (quod verè miraculo proximum eſt) <lb></lb>utroque extremo flexilis lineæ in binis Ellipſeos umbilicis <lb></lb>defixo ab illâ in alios, atque alios angulos ſinuata deſ<lb></lb>cribatur. </s> </p> <p type="main"> <s id="s.001313">At, inquis, in circulo ſemidiametri partes codem im<lb></lb>pellente circà centrum agitatæ ita diſpari velocitate ferun<lb></lb>tur, ut earum tarditas aut concitatio intervallo, quo ſin<lb></lb>gulæ à centro abſunt, ſit analoga. </s> <s id="s.001314">Verùm & hoc Ellipſi, <lb></lb>ac plano Helicoidi aliquatenùs pro ſuo modulo commune <lb></lb>eſt; ſemidiametri enim circumactæ puncta à centro remo<lb></lb>tiora velociùs feruntur. </s> <s id="s.001315">Partes autem quieſcenti centro pro<lb></lb>piores cunctabundas moveri, naturæ pro viribus oppoſita <lb></lb>diſterminantis inſtituto conſentaneum eſſe nemo non videt, <lb></lb>qui tarditatem interjici videt quietem inter, ac motûs ve<lb></lb>locitatem. </s> <s id="s.001316">Quare ſapientiſſimo conſilio factum, ut eorum, <lb></lb>quæ firmo nexu invicem ſolidata ſubſiſtunt, vel particu<lb></lb>læ omnes æquis paſſibus moveantur, vel ſi qua moræ diſ<lb></lb>pendium ſubeat, finitimarum velocitas, ſervatâ aliquâ vi<lb></lb>cinitatis analogiâ minuatur: ne ſcilicet ſolutâ compage diſ<lb></lb>ſiliant. </s> </p> <p type="main"> <s id="s.001317">Quæ verò ad explicandum, cur ea, quæ centro propiora <lb></lb>ſunt, tardiùs in gyrum contorqueantur, Author illius libri <lb></lb>Quæſt. mechan. </s> <s id="s.001318">comminiſcitur de duplici motu, naturali vi<lb></lb>delicet, ac præter naturam, quibus feratur ea, quæ circu<lb></lb>lum deſcribit linea (quaſi breviorem lineam vis major à tra<lb></lb>hente centro illata magis à naturali motu, qui ſecundùm <lb></lb>Tangentem eſt, deflecteret) ea ſunt, quæ facillimè cor<lb></lb>ruant, & minimè cum Ariſtotelis doctriná cohæreant, qui <lb></lb>lib. 1. de Cælo. </s> <s id="s.001319">ſumma 4. circularem motum & ſimplicem, & <pb pagenum="175" xlink:href="017/01/191.jpg"></pb>naturalem, & priorem recto diſertiſſimè pronunciat; <emph type="italics"></emph>Perfectum <lb></lb>enim,<emph.end type="italics"></emph.end> inquit text. </s> <s id="s.001320">12; <emph type="italics"></emph>prius naturâ eſt imperſecto; circulus autem <lb></lb>perfectorum eſt, recta verò linea nulla.<emph.end type="italics"></emph.end></s> <s id="s.001321"> Quis ergo in circulo <lb></lb>motus præter naturam? <emph type="italics"></emph>neceſſarium eſt,<emph.end type="italics"></emph.end> ait text. </s> <s id="s.001322">8. <emph type="italics"></emph>eſſe ali<lb></lb>quod corpus ſimplex, quod natum eſt ferri circulari motu ſecun<lb></lb>dùm ſuam ipſius naturam.<emph.end type="italics"></emph.end></s> <s id="s.001323"> Ea certè quibus inſita eſt in mo<lb></lb>tum propenſio, in gyrum aguntur, ut ſydera; aut ſaltem mo<lb></lb>tu in ſe recurrente circulum æmulantur, ut ex cerebri & cor<lb></lb>dis ſyſtole ac diaſtole ſpirituum ac ſanguinis circuitio oritur; <lb></lb>aut plurium circularium motuum commixtione unum tempe<lb></lb>rant motum, ut animalia cum progrediuntur; oſſa ſiquidem, <lb></lb>quibus membra ſubſiſtunt, ita à muſculis commoventur, ut <lb></lb>unum quod que ſui motus centrum conſtituat in eâ finitimi oſſis <lb></lb>parte, cui ſivè <foreign lang="grc">Καθ´<gap></gap>ἐνάρσθρωσιν</foreign>, ſive <foreign lang="grc">κατά διἀρθρωσιν</foreign> flexili com<lb></lb>page inſeritur. </s> <s id="s.001324">At motu recto, ut potè breviſſimo, nihil fertur, <lb></lb>niſi cui ex naturæ inſtituto cedit quies certo in loco, à quo <lb></lb>abſtractum fuerit, eóque ſibi redditum ſpontè remigrat. </s> <s id="s.001325">Nihil <lb></lb>igitur præter naturam in circuli motu deprehendi poteſt, ex <lb></lb>quo diſpar illa intimarum atque extimarum partium velocitas <lb></lb>petenda ſit; cum vix alium natura per ſe expetat ſimplicem <lb></lb>motum præter circularem. </s> <s id="s.001326">Cur autem qui ſecundùm rectam <lb></lb>extremæ ſemidiametro ad perpendiculum inſiſtentem lineam <lb></lb>fit motus, naturalis cenſeatur? </s> <s id="s.001327">An quia gravia ſuis nutibus ad <lb></lb>terræ centrum rectâ feruntur? </s> <s id="s.001328">Semidiametro igitur, niſi in <lb></lb>verticali plano conſtituatur horizonti parallela, motus qui ſe<lb></lb>cundùm lineam circuli Tangentem eſt, præter naturam con<lb></lb>tinget, quippe qui à rectâ, quæ gravia in centrum dirigit, de<lb></lb>flectat: & in circulo horizonti parallelo circumacta ſemidiame<lb></lb>ter nullo naturali motu agitabitur; nulla enim recta linea cir<lb></lb>culi Tangens in eo plano eſt, quæ lineæ directionis gravium <lb></lb>congruat: & tamen quemcumque demum ſitum circulus ejuſ<lb></lb>que ſemidiameter obtineat, eandem ſemper motuum analo<lb></lb>giam ſervant partes pro ratione intervalli à centro, citrà ullam <lb></lb>motuum naturalis, & præter naturam, commiſtionem. </s> </p> <p type="main"> <s id="s.001329">Verùm mirifica ſit circuli natura; quid hæc ad explicandam <lb></lb>Mechanicarum motionum cauſam? </s> <s id="s.001330">an ut hanc ignotam fatea<lb></lb>mur, quia admirandam prædicamus? </s> <s id="s.001331">ſed unico argumento, <lb></lb>commenta hujuſmodi disjiciamus. </s> <s id="s.001332">Si minor potentia majori <pb pagenum="176" xlink:href="017/01/192.jpg"></pb>ponderi prævaleat, nullúſque intercedat circularis motus, <expan abbr="certũ">certum</expan> <lb></lb>eſt hoc virtutis <expan abbr="incremẽtum">incrementum</expan> neque in Vectem, neque in libram <lb></lb>neque in Circulum referri poſſe: adeóque principium aliud eſſe <lb></lb>magis latè patens, à circulo abſolutum: Atqui citrà omnem cir<lb></lb>cularem <expan abbr="motũ">motum</expan> minor potentia præpollet graviori ponderi: Mani<lb></lb>feſtum eſt igitur fruſtrà ex circulo peti Mechanicarum motio<lb></lb>num principium; ſed illud eſſe, quod à nobis indicatum eſt, <lb></lb>quippe quod, ubicumque reperitur, hoc efficit, ut minor po<lb></lb>tentia majori ponderi motum conciliet, nec is unquam ſine illo <lb></lb>contingit. </s> </p> <p type="main"> <s id="s.001333">Aſſumptionis veritas ut innoteſcat, ingenſque pondus tardè <lb></lb>movendum à tenui virtute ſine circulari motu propelli poſſe <lb></lb>confirmem, non ego te in ſuburbanum campum deducam, ut <lb></lb>tenerrimo germini ſuppullulanti incumbentes glebas demùm <lb></lb>loco ceſſiſſe obſerves, aut marmora Meſſalæ ſcindentem capri<lb></lb>ficum obtrudam, turreſque longâ annorum ſerie labefactatas <lb></lb>enatis fruticibus atque virgultis; ne mihi fortè herbeſcentes <lb></lb>cuneos obtrudas, quos ad vectem, & circulum revocare velis. </s> </p> <figure id="id.017.01.192.1.jpg" xlink:href="017/01/192/1.jpg"></figure> <p type="main"> <s id="s.001334">Sed age raptandus ſit in plano horizontali, aut inclinato, aut <lb></lb>etiam elevandus ſit ad perpendiculum cylindrus A. </s> <s id="s.001335">Experire <lb></lb>primùm quanto labore id præſtes illum trahens illigato fune <lb></lb>in C, & arreptâ extremitate funis B. </s> <s id="s.001336">Tùm in B infixo firmi<lb></lb>ter paxillo ductarius funis alligetur; hic porrò inſeratur annu<lb></lb>lo C optimè ferruminato, & quoad ejus fieri poterit exquiſitè <lb></lb>polito, arreptáque alterâ funis extremitate D iterum trahe cy<lb></lb>lindrum, & quantò minori labore id perficias, tu te ipſe doce<lb></lb>bis. </s> <s id="s.001337">At hîc nulla circuli vides miracula; hîc libra nulla; nullus <lb></lb>hîc vecti locus: motus enim tùm potentiæ trahentis, tùm cy<lb></lb>lindri, rectus eſt. </s> <s id="s.001338">Facilitatis autem diſcrimen non ex ullo cir<lb></lb>culari motu, qui nuſquam apparet, ſed ex eo oritur, quòd pri<lb></lb>mùm potentia & onus æqualiter moventur; poſteà verò cylin-<pb pagenum="177" xlink:href="017/01/193.jpg"></pb>dri velocitas ſubdupla eſt velocitatis potentiæ; quia cum ex C <lb></lb>cylindrus venit in B funis ultrà B extenditur juxtà longitudi<lb></lb>nem CB uſque in E; ac propterea motus potentiæ duplus eſt, <lb></lb>ſcilicet CE. </s> </p> <p type="main"> <s id="s.001339">Statue item in pariete puncta duo A & B (quo autem majo<lb></lb>re intervallo disjuncta fuerint, res meliùs ſuccedet) ibique <lb></lb>clavos rotundos nihil ha<lb></lb><figure id="id.017.01.193.1.jpg" xlink:href="017/01/193/1.jpg"></figure><lb></lb>bentes aſperitatis infige. </s> <lb></lb> <s id="s.001340">Tùm pondera duo H & <lb></lb>G æqualia aſſume, eáque <lb></lb>funiculo nullis nodis aſpe<lb></lb>ro, ſive ſerico crudo, ſive <lb></lb>crinibus equinis connexa <lb></lb>impone claviculis A & B, <lb></lb>ut liberè ex iis depen<lb></lb>deant: ſuâ autem gravitate <lb></lb>funiculum AB intentum Horizonti parallelum ſervabunt, & <lb></lb>neutro prævalente ob gravitatis æqualitatem prorsùs immota <lb></lb>conſiſtent. </s> <s id="s.001341">Elige jam pondus tertium I, quod alteri datorum <lb></lb>H & G æquale ſit, aut etiam ſingulis aliquantò minus; illud<lb></lb>que in E extento funiculo AB adnecte: ſtatim pondus I ſecun<lb></lb>dùm rectam EF deſcendens videbis; pondera autem H & G <lb></lb>per rectas HA, & GB aſcendentia, quâ menſurâ funiculi in<lb></lb>flexi partes AF, BF ſimul ſumptæ excedunt rectam AB. </s> <s id="s.001342">Nul<lb></lb>lus igitur motus circularis hîc eſt; ſed omnes recti ad perpendi<lb></lb>culum, & tamen potentia I minor commovet majus pondus, <lb></lb>quod ex H & G conflatur. </s> </p> <p type="main"> <s id="s.001343">Id autem ideò contingere, quia motus EF deſcendentis I <lb></lb>major eſt motu aſcendentium H & G, hinc manifeſtum eſt, <lb></lb>quòd pondus I uſque ad certum terminum deſcendit, ibique <lb></lb>ſubſiſtit: quòd ſi illud manu apprehenſum adhuc deorſum <lb></lb>trahens eleves pondera H & G, ubi manum indè abſtraxeris, <lb></lb>pondera H & G prævalent, ac deſcendentia elevant pondus I <lb></lb>ad certum illum terminum, ubi ſponte ſubſtiterat: quia nimi<lb></lb>rum ultrà illum terminum non jam major eſt Ratio ponderis I <lb></lb>ad pondera HG, quàm ſit Ratio motuum H & G ad motum I. </s> <lb></lb> <s id="s.001344">Hæc autem inferiùs, ubi de librâ & Æquilibrio ſermo erit, <lb></lb>paulò fuſiùs & dilucidiùs explicabuntur; nunc enim ſatis eſt <pb pagenum="178" xlink:href="017/01/194.jpg"></pb>pro inſtitutâ diſputatione oſtendiſſe minorem gravitatem præ<lb></lb>pollere citrà omnem motum circularem. </s> </p> <p type="main"> <s id="s.001345">Ratum itaque eſto ad nullum certum machinæ genus cætera <lb></lb>eſſe revocanda; ſed omnibus commune eſſe principium, ex quo <lb></lb>vires deſumunt; impetûs ſcilicet à potentiâ producti proportio <lb></lb>ad ponderis reſiſtentiam (quæ eò minor eſt, quò tardiùs mo<lb></lb>veri debet) ea eſt, quæ motûs facilitatem conciliat; nullus <lb></lb>quippe adeò tenuis impetus reperitur, cui lentiſſimus aliquis <lb></lb>motus non reſpondeat, ſi intereà à velociori motu potentia non <lb></lb>prohibeatur. </s> <s id="s.001346">Ubi autem de potentiæ velocitate ſermo eſt, non <lb></lb>ea intelligatur, quæ eſſet, ubi præter ſe nihil ipſa moveret, ab<lb></lb>ſoluta ab omni reſiſtentiâ; ſed eam velocitatem intellige, quæ <lb></lb>comparatè dicitur, ubi ejus motus cum oneris motu confertur. </s> <lb></lb> <s id="s.001347">Semper tamen impetus, qui in Potentiâ reperitur quatenùs ex<lb></lb>cedit reſiſtentiam ponderis, majorem in eâ intentionem ha<lb></lb>bet, quàm in pondere, quamvis pares entitativè ſint impetus <lb></lb>Potentiæ, & oneris. </s> <s id="s.001348">Hæc autem clariùs patebunt lib.4. cap.1. <lb></lb> </s> </p> <p type="main"> <s id="s.001349"><emph type="center"></emph>CAPUT VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001350"><emph type="center"></emph><emph type="italics"></emph>Quid attendendum ſit in Machinæ collocatione, <lb></lb>atque materie.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001351">QUamvis inſtructarum Machinarum vires ad calculos revo<lb></lb>centur inſpectâ earum figurâ, ut Potentiæ atque oneris <lb></lb>motus invicem comparentur; quo tamen loco & ſitu Machina <lb></lb>ipſa collocetur, diſpiciendum eſt, ut innoteſcat, quanta illi vis <lb></lb>inferatur tùm ab oneris gravitate, tùm à potentiæ conatu: ex <lb></lb>hoc ſiquidem decernendum erit, quàm ſolidam conſtrui opor<lb></lb>teat Machinam. </s> <s id="s.001352">Quotus enim quiſque eſt, qui ignoret longè <lb></lb>ſolidiorem requiri machinam, ſi ex illa dependeat, aut illi in<lb></lb>cumbat onus, quàm ſi non machinæ; ſed ſubjecto plano, inni<lb></lb>tatur idem pondus, aut aliunde dependeat? </s> <s id="s.001353">alia ſcilicet ſunt <lb></lb>gravitatis momenta contrà virtutem ſuſtinentem etiam citrà <lb></lb>motum, alia verò momenta, quatenus motui adverſatur. <pb pagenum="179" xlink:href="017/01/195.jpg"></pb>Hinc operæ pretium fuerit non contemnendum, ſi res ita à <lb></lb>Machinatore diſponantur, ut pondus, quàm minimum fieri <lb></lb>poſſit, à machinâ ſuſtineatur: hâc enim ratione fiet, ut lon<lb></lb>giùs avertatur periculum luxationis aut fractionis membrorum, <lb></lb>quibus machina diſtinguitur, etiamſi exilior illa fuerit; & ma<lb></lb>chinæ gravitas aliqua ſubtrahetur, dum moles ipſa minuitur, <lb></lb>atque proinde movendi oneris difficultas non augebitur ex ma<lb></lb>chiná; quæ etiam minore impendio parabitur. </s> </p> <p type="main"> <s id="s.001354">Sit exempli gratiâ pondus A, quod ſit trochleâ attollendum <lb></lb>in D. </s> <s id="s.001355">Poterit id duplici ratione fieri; primùm raptando illud in <lb></lb>plano Horizontali ita, ut ex B <lb></lb><figure id="id.017.01.195.1.jpg" xlink:href="017/01/195/1.jpg"></figure><lb></lb>veniat in C, tùm alligatâ tro<lb></lb>cleâ in I illud attollendo ad <lb></lb>perpendiculum uſque in D: <lb></lb>cum raptatur, totum incumbit <lb></lb>pondus ſubjecto plano; cum at<lb></lb>tollitur, totum ex trochleâ de<lb></lb>pendet. </s> <s id="s.001356">At ſi trochleâ utaris, <lb></lb>de cujus firmitate ſubdubites, <lb></lb>& loci diſpoſitio ferat, ut poſ<lb></lb>ſit ex E & H onus ſuſpendere, <lb></lb>res faciliùs perficietur. </s> <s id="s.001357">Ponde<lb></lb>ri enim A adnecte funem OE, <lb></lb>ex quo pendere poſſit in E, ac <lb></lb>prætereà tantumdem funis OS liberè vagantis; trochleam au<lb></lb>tem alliga in F: ubi verò ope trochleæ adduxeris pondus ex O <lb></lb>in G, tùm funem OS liberè vagantem eleva, ac benè inten<lb></lb>tum adnecte in H, ut jam pondus ex H dependeat ad perpen<lb></lb>diculum: Ex hoc fiet, ut reſoluto fune OE, liberéque vagan<lb></lb>te, ope trochleæ in F alligatæ adducas pondus ex G in D mul<lb></lb>tò minori labore, quàm ſi ex B in C illud raptâſſes, & ex C <lb></lb>in D ſuſtuliſſes. </s> <s id="s.001358">Conſtat autem pondus idem minùs conniti <lb></lb>adversùs lineas FG aut FD, quàm adversùs perpendiculares <lb></lb>HG aut ID, ex iis quæ diſputata ſunt lib. 1. cap. 15, ac <lb></lb>propterea etiam minùs dubitari poteſt de trochleæ firmitate. </s> </p> <p type="main"> <s id="s.001359">Hoc autem compendium elevandi pondera perinde, atque <lb></lb>ſi per planum inclinatum attollerentur, ea ſcilicet ſuſpendendo <lb></lb>atque obliquè trahendo, ubi in praxim ritè deduxeris, appa-<pb pagenum="180" xlink:href="017/01/196.jpg"></pb>rebit quanto labori, & quàm magnis ſumptibus parcatur: cum <lb></lb>neque vincendus ſit partium tritus atque conflictus inter pon<lb></lb>dus, ac ſubjectum planum, neque ſternendum ſit multo robo<lb></lb>re planum ipſum, quod oneri ſuſtinendo non impar ſit. </s> <s id="s.001360">At ubi <lb></lb>funem EO, quoad ejus fieri poterit, intenderis, aquá largiter <lb></lb>imbuito; hoc enim fiet, ut ſeſe contrahens etiam paulò inten<lb></lb>tior, atque ad deſtinatum opus evadat aptior. </s> </p> <p type="main"> <s id="s.001361">Quæ cum ita ſint, alia ſe offert methodus elevandi pondera <lb></lb>non levi laboris compendio, ſi nimirùm duplex adhibeatur <lb></lb><figure id="id.017.01.196.1.jpg" xlink:href="017/01/196/1.jpg"></figure><lb></lb>trochlea, altera quidem in A imminens pon<lb></lb>deri ad perpendiculum, altera verò in B. </s> <lb></lb> <s id="s.001362">Adhibita igitur trochlea B elevabit pondus <lb></lb>ex C in D, ibique totum ex B pendebit: <lb></lb>tùm viciſſim trochleâ A utere, & ex D in E <lb></lb>aſcendet pondus, quod ibi totum ex A pen<lb></lb>debit: iterum igitur adhibe trochleam B, ut <lb></lb>ex E in F aſcendat; atque viciſſim, adhibitâ <lb></lb>trochleâ A aſcendet ex F in G; & ſic de<lb></lb>inceps. </s> </p> <p type="main"> <s id="s.001363">Ubi vides motum ponderis aſcendentis per arcus CDEFG <lb></lb>majorem eſſe quàm ſi rectâ ad perpendiculum elevatum fuiſſet <lb></lb>ex C in G. </s> <s id="s.001364">Quia verò altitudines perpendiculares ſingulis ar<lb></lb>cubus reſpondentes ſubinde majores fiunt, propterea plus vi<lb></lb>rium à potentia movente adhibendum eſt in progreſſu. </s> <s id="s.001365">Quâ <lb></lb>autem Ratione altitudines illæ perpendiculares creſcant, faci<lb></lb>lè innoteſcet, ſi arcuum ſingulorum Sinus verſos ſuis Radiis <lb></lb>reſpondentes ad calculos revocaveris; arcus enim ſuperiores & <lb></lb>plurium eſſe graduum, & ex Radio minori, manifeſtum eſt: <lb></lb>diſtantia autem parallelarum AC, BD perpendicularium ea<lb></lb>dem ſemper eſt; quapropter & æquales lineæ ſunt Sinus Recti <lb></lb>arcuum inæqualium in circulis inæqualibus, videlicet arcuum <lb></lb>majorum in circulis minoribus. </s> <s id="s.001366">Quamquam nec omninò ne<lb></lb>ceſſe eſt ità ſingulis tractionibus pondus attollere, ut ad per<lb></lb>pendiculum dependeat, ſi maximè trochleæ invicem non mo<lb></lb>dicum diſtarent; ſed ſufficeret alternis operis trochleas agita<lb></lb>re, ut aſcendens pondus modò ad hoc, modò ad illud perpen<lb></lb>diculum accederet, ita tamen ut ultró citróque tranſgrediatur <lb></lb>perpendiculum, quod medium cadit inter extremas AC & BD; <pb pagenum="181" xlink:href="017/01/197.jpg"></pb>alioquin par non eſſet utriuſque trahentis labor. </s> <s id="s.001367">Cæterùm <lb></lb>ſatius eſt A & B parùm diſtare. </s> </p> <p type="main"> <s id="s.001368">Ut autem exemplo aliquo res manifeſta fiat, ſtatuamus alti<lb></lb>tudinem AC eſſe pedum 70, diſtantiam verò AB pedum 30, <lb></lb>cui æqualis eſt ea, quæ ex D cadit perpendicularis in AC, ſci<lb></lb>licet DS. </s> <s id="s.001369">Quare in triangulo ASD rectangulo nota eſt Hy<lb></lb>pothenuſa AD, quæ æqualis eſt ipſi AC, & nota eſt Baſis <lb></lb>SD. </s> <s id="s.001370">Atqui conſtat Perpendiculum AS eſſe medio loco pro<lb></lb>portionale inter ſummam atque differentiam Hypothenuſæ ac <lb></lb>baſis, ſcilicet inter 100 & 40; igitur ducta prima in tertiam, <lb></lb>videlicet ducta ſumma in differentiam dabit 4000 Quadratum <lb></lb>Mediæ (hoc eſt perpendiculi AS) cujus Radix ped. 63 1/4 ferè <lb></lb>eſt Perpendiculum AS. </s> <s id="s.001371">Igitur elevatio CS eſt ped. 6 3/4. </s> </p> <p type="main"> <s id="s.001372">Cum itaque BD æqualis ſit ipſi AS (jungunt enim paral<lb></lb>lelas æquales AB & SD) iterum in triangulo BVE rectangu<lb></lb>lo nota eſt Hypothenuſa BE ped. 63 1/4, & Baſis EV eſt ped. 30: <lb></lb>Quare inter ſummam ped. 93 1/4, ac differentiam ped. 33 1/4 media <lb></lb>proportionalis ped. 55. 67″. </s> <s id="s.001373">eſt Perpendiculum BV; atque <lb></lb>adeò elevatio DV eſt ped. 7. 58″. major quàm CS. </s> <s id="s.001374">Et ſic de <lb></lb>reliquis. </s> </p> <p type="main"> <s id="s.001375">At ſtatue diſtantiam AB ſolùm ped. 20: reperies perpendi<lb></lb>culum AS vix excedere ped. 67; quare elevatio CS erit ped. 3 <lb></lb>ferè; ac propterea etiam Perpendiculum BV erit paulò majus <lb></lb>ped. 63. 94″; & elevatio DV ped. 3. 06″; & ſic de cæteris. </s> </p> <p type="main"> <s id="s.001376">Potentiæ verò elevantis motum metitur differentia, quæ <lb></lb>inter lineas BC & BD intercedit: quando autem diſtantia <lb></lb>AB eſt ped. 30, linea BC eſt ped. 76. 15″; at cum eſt ped. 20, <lb></lb>BC eſt ped. 72 4/5. Cum igitur in primo caſu BD ſit ped. 63 1/4, <lb></lb>motus potentiæ eſt ped. (12 9/10); in ſecundo autem caſu cum BD <lb></lb>ſit ped. 67; linea autem BC ſit ped. 72 4/5, motus potentiæ eſt <lb></lb>ped. 5 4/5. Quare in primo Ratio motûs Potentiæ ad motum <lb></lb>ponderis eſt (12 9/10) ad 6 3/4, in ſecundo Ratio eſt 5 4/5 ad 3: & factâ <lb></lb>reductione ad alias denominationes, prima Ratio eſt 86 ad 45, <lb></lb>ſecunda Ratio eſt 29 ad 15, quæ ſi ad eumdem denominato<lb></lb>rem 45 reducatur, erit 87 ad 45. Conſtat autem majorem eſſe <lb></lb>Rationem 87 ad 45, quàm 86 ad 45. per 8. l. </s> <s id="s.001377">5. Majorem igi<lb></lb>tur Rationem habet motus Potentiæ ad motum ponderis, quan-<pb pagenum="182" xlink:href="017/01/198.jpg"></pb>do A & B minùs diſtant, quàm cum ſeparantur intervallo ma<lb></lb>jore; atque adeò major eſt etiam movendi facilitas. </s> </p> <p type="main"> <s id="s.001378">Quòd ſi rei hujus minimè dubium experimentum ſumere <lb></lb>placeat, ipſiſque oculis rem totam ſubjicere citrà omnem de<lb></lb><figure id="id.017.01.198.1.jpg" xlink:href="017/01/198/1.jpg"></figure><lb></lb>ludentis phantaſiæ ſuſpicio<lb></lb>nem, firmetur in A orbiculus <lb></lb>circà ſuum axem verſatilis, & <lb></lb>ex eo æqualia pondera D & E <lb></lb>funiculo connexa dependeant <lb></lb>ad perpendiculum; quæ prop<lb></lb>ter gravitatis æqualitatem im<lb></lb>mota permanent. </s> <s id="s.001379">Tùm in B <lb></lb>firmetur orbiculus circà ſuum <lb></lb>axem pariter verſatilis, & aſ<lb></lb>ſumatur pondus C ponderi E <lb></lb>æquale, cui adnectatur funi<lb></lb>culo EBC. </s> <s id="s.001380">Si manu retineas <lb></lb>pondus C, ne gravitet, per<lb></lb>ſiſtit pondus E in ſuo perpendiculo: jam manu retine <lb></lb>pondus D, ne prorſus moveatur, ac dimitte pondus C, vi<lb></lb>debis hoc quidem deſcendere, pondus verò E aſcendere, <lb></lb>donec ex B dependeat, & in æquilibrio cum pondere C <lb></lb>ſubſiſtat. </s> <s id="s.001381">Iterum retine pondus C, & dimitte pondus D, <lb></lb>pariterque pondus D deſcendens videbis, E verò adhuc <lb></lb>aſcendens; & ſic deinceps uſque eò, dum pondus E uni<lb></lb>cum ambobus D & C æquipolleat, ut ſuperiori capite in<lb></lb>dicatum eſt. </s> <s id="s.001382">Id igitur quod à ponderibus D & C præſtatur, <lb></lb>à quâlibet potentiâ æquali in D & C conſtitutâ præſtari poſſe <lb></lb>manifeſtum eſt. </s> <s id="s.001383">Si itaque ſimplicibus orbiculis fit, ut pondus <lb></lb>æquale poſſit prævalere, multò magis id fiet, ſi trochleæ adhi<lb></lb>beantur. </s> </p> <p type="main"> <s id="s.001384">Ex his apparet, quid & in cæteris machinarum generibus, <lb></lb>analogiâ ſervatâ, dicendum ſit, ex quarum opportunâ col<lb></lb>locatione facilitas movendi augentur. </s> <s id="s.001385">Si enim, exempli gra<lb></lb>tiâ, cubus A marmoreus elevandus fuerit vecte BC, mul<lb></lb>tò faciliùs id fiet, ſi ille ſupponatur cubo, quàm ſi ex I ad <lb></lb>perpendiculum elevaretur eodem vecte ſuſpenſum: ex I ſci<lb></lb>licet totus cubus à vecte ſuſtineretur; at ſubjectus vectis <pb pagenum="183" xlink:href="017/01/199.jpg"></pb>BC ita cubum ſuſtentat, ut <lb></lb><figure id="id.017.01.199.1.jpg" xlink:href="017/01/199/1.jpg"></figure><lb></lb>etiam reliquo latere cubus <lb></lb>idem ſubjecto plano incumbat. </s> </p> <p type="main"> <s id="s.001386">Quemadmodum autem non <lb></lb>quemlibet vectem cuilibet <lb></lb>oneri <expan abbr="elevãdo">elevando</expan> parem eſſe om<lb></lb>nes intelligunt; ſed habita ra<lb></lb>tione materiæ, ex quâ conſtat, <lb></lb>congrua ſoliditas ei tribuenda <lb></lb>eſt; ita pariter in cæteris omnibus, quæ hùc ſpectant (ſive <lb></lb>ſint machinarum membra, ſive paxilli ſint aut tigilli, quibus <lb></lb>machinæ adnectuntur) materiæ ſoliditatem attendendam eſſe <lb></lb>manifeſtum eſt, ne frangantur. </s> <s id="s.001387">Et quidem quod ad materiam <lb></lb>attinet, non omnium ſolidorum partes pari nexu cohærent, <lb></lb>ſed alia aliis fragiliora ſunt: ſic lignum quernum difficiliùs <lb></lb>frangitur, quàm fraxineum aut populeum: neque enim in <lb></lb>omni ligno æque operoſa ſimiliſque ſtaminum textura repe<lb></lb>ritur; cum etiam lignum idem quaqua verſum findi non poſ<lb></lb>ſit pari facilitate; permagni quippe intereſt, recta ne juxtà <lb></lb>venarum ductum? </s> <s id="s.001388">an obliquè? </s> <s id="s.001389">ſectio facienda ſit. </s> <s id="s.001390">Id quod <lb></lb>in ipſis quoque lapidibus, atque marmoribus obſervare quan<lb></lb>doque neceſſe eſt, ubi non æquè per omnes partes compacta <lb></lb>materia venas habet ſciſſioni maximè obnoxias. </s> <s id="s.001391">In metallis <lb></lb>pariter eorum natura conſideranda eſt, molliſne illa ſit, ac <lb></lb>flexibilis? </s> <s id="s.001392">an verò dura? </s> <s id="s.001393">ut eam, quam ſemel induit figu<lb></lb>ram, conſtanter retineat. </s> <s id="s.001394">Ex quo fit, ut pro materiæ diſſi<lb></lb>militudine diſpar etiam craſſities requiratur: quis enim neſciat, <lb></lb>quantum ligneum inter ac ferreum ejuſdem molis vectem in<lb></lb>terſit? </s> </p> <p type="main"> <s id="s.001395">Verùm illud potiùs conſiderandum videtur, quod ad ſoli<lb></lb>ditatem ipſam ſpectat, etiamſi materies diverſa non ſit; pro <lb></lb>variâ enim craſſitudine mutatur frangendi difficultas; & quia <lb></lb>in mole majori plures inſunt partes diviſioni reſiſtentes, fran<lb></lb>gendi pariter difficultas augetur pro Ratione multitudinis par<lb></lb>tium, ſi cætera paria ſint. </s> <s id="s.001396">Dubitare videlicet nemo poteſt à <lb></lb>duplici partium dividendarum numero duplicem oriri reſiſten<lb></lb>tiam. </s> <s id="s.001397">Si cætera, inquam, ſint paria; nam ſi filum ſericum ut <lb></lb>rumpatur, requirit vim ut unum, & decem fila ſerica paris <pb pagenum="184" xlink:href="017/01/200.jpg"></pb>craſſitiei ac longitudinis parallela ſimul poſita requirant vim <lb></lb>decuplam; ſi in unum funiculum decem illa fila ritè contor<lb></lb>queantur, multò majorem vim quàm decuplam requiri, ut fu<lb></lb>niculus frangatur, manifeſtum eſt: quemadmodum & ligneus <lb></lb>tigillus multo validiùs reſiſtit fractioni, quàm virgarum faſci<lb></lb>culus eidem tigillo æqualis; major eſt enim particularum unio, <lb></lb>ubi in unum corpus coaleſcant, quàm ubi plura minora corpo<lb></lb>ra conſtituantur. </s> </p> <p type="main"> <s id="s.001398">Hinc ſi fuerint duo parallelepipeda quadrata A & B, quorum <lb></lb>latera ſint in Ratione quadruplâ, altitudines verò AC, & BD <lb></lb><figure id="id.017.01.200.1.jpg" xlink:href="017/01/200/1.jpg"></figure><lb></lb>æquales; conſtat ex 32. <lb></lb>l. 11 ea eſſe inter ſe ut ba<lb></lb>ſes; baſes autem ſunt qua<lb></lb>drata laterum; igitur pa<lb></lb>rallelepipedum B eſt ſede<lb></lb>cuplum parallelepipedi A. </s> <lb></lb> <s id="s.001399">Finge ſexdecim parallele<lb></lb>pipeda ipſi A æqualia in <lb></lb>faſciculum colligata, & <lb></lb>ſciſſionem faciendam jux<lb></lb>ta lineam OS vi oneris in <lb></lb>O poſiti: certum eſt faci<lb></lb>liùs frangi poſſe ſexdecim <lb></lb>illa parallelepipeda, quàm <lb></lb>parallelepipedum B illis <lb></lb>omnibus æquale; ut enim ſcindatur, curvari oportet vi oneris <lb></lb>incumbentis; illa autem ſexdecim faciliùs curvantur quàm <lb></lb>ipſum B. </s> <s id="s.001400">Id quod manifeſtum fiat, ſi virgam ex ſalicto <lb></lb>decerpens, eamque leniter inflectens obſerves, quâ quidem <lb></lb>parte virga curvata eſt, tenerum corticem in rugas aſſurge<lb></lb>re atque criſpari, quâ verò parte convexa eſt, corticem <lb></lb>diſtrahi atque diſtendi. </s> <s id="s.001401">Ex quo facilè arguimus, quid durio<lb></lb>ribus corporibus contingat, quæ non adeò manifeſtè corru<lb></lb>gari poſſunt; flecti ſcilicet nequeunt, quin aliqua fiat inte<lb></lb>riorum partium compreſſio, & exteriorum diſtractio. </s> <s id="s.001402">Hinc <lb></lb>in parallelepipedo B, quod flecti intelligitur, ut ſcindatur, <lb></lb>partes, quæ circa O, comprimuntur; quæ verò circà S, <lb></lb>diſtrahuntur: huic autem motioni repugnant omnes particu-<pb pagenum="185" xlink:href="017/01/201.jpg"></pb>læ vi nexûs, quo unaquæque cum ſibi proximè cohærentibus <lb></lb>particulis colligatur. </s> <s id="s.001403">Cum autem ſexdecim illa parallelepipe<lb></lb>da minora non ſint invicem connexa, quemadmodum particu<lb></lb>læ omnes parallelepipedi B in unam molem coaluerunt, conſtat <lb></lb>pauciores nexus faciliùs, quàm plures, diſſolvi. </s> </p> <p type="main"> <s id="s.001404">Hoc verò ut pleniùs atque apertiùs explicetur, intellige ſo<lb></lb>lidum longiuſculum RS in plures tenues laminas plano RI <lb></lb>parallelas diviſum, ſibi<lb></lb><figure id="id.017.01.201.1.jpg" xlink:href="017/01/201/1.jpg"></figure><lb></lb>que ita viciſſim con<lb></lb>gruentes, ut earum ex<lb></lb>tremitates conſtituant <lb></lb>planum HI. </s> <s id="s.001405">Omnes <lb></lb>haſce laminas ſecun<lb></lb>dùm extremitates ful<lb></lb>cris impoſitas pondus <lb></lb>ſuper DC conſtitutum <lb></lb>adeò premat, ut cur<lb></lb>vari aliquantulum cogantur. </s> <s id="s.001406">Obſervabis illicò extremitates <lb></lb>illas non jam ampliùs in eandem planitiem HI exæquari; ſed <lb></lb>eas quidem laminas, quæ cavitatem ſpectant, magis curvari; <lb></lb>minùs verò eas, quæ convexitati reſpondent, ac proptereà ex<lb></lb>timæ laminæ extremitatem ab extremitate intimæ laminæ, quæ <lb></lb>ponderi impoſito cohæret, magis recedere, quàm interme<lb></lb>diarum extremitates. </s> <s id="s.001407">Conſtat itaque in hoc motu ſingula<lb></lb>rum laminarum particulas, dum curvantur, non iis reſpon<lb></lb>dere adhærentis laminæ particulis, quas priùs contingebant, <lb></lb>cùm omnis curvitatis expertes erant, atque faciliùs potuiſſe <lb></lb>ſingulas laminas moveri, quia nullo nexu invicem copulan<lb></lb>tur. </s> <s id="s.001408">Quòd ſi ex iis unum ſolidum RS planè integrum coa<lb></lb>leſcat, manifeſtum eſt planitiem HI permanere, ac propterea, <lb></lb>dum curvatur, neceſſe eſt, ut interiores particulæ invicem <lb></lb>connexæ diſtrahantur, cum nequeant aliæ ab aliis ſecedere, <lb></lb>quemadmodum in laminis contingere obſervavimus. </s> <s id="s.001409">Hinc <lb></lb>oritur major ſolidi, quàm laminarum, reſiſtentia, ne fran<lb></lb>gatur. </s> <s id="s.001410">Non negarim tamen aliquando ſatius eſſe duobus me<lb></lb>diocribus tigillis uti, quàm craſſiore tigno illis æquali; quia <lb></lb>nimirum alterutro labem patiente rimaſvè agente, alter faci<lb></lb>liùs integer perſeverat; in craſſiore autem tigno, ſi rimam du-<pb pagenum="186" xlink:href="017/01/202.jpg"></pb>cere occœperit, periculum eſt, ne malum ſerpat juxta vena<lb></lb>rum aut fibrarum ductum. </s> <s id="s.001411">Cæterum ſublato hujuſmodi peri<lb></lb>culo, ubi reliqua paria ſint, craſſiora corpora difficiliùs fran<lb></lb>guntur. </s> </p> <p type="main"> <s id="s.001412">Quare ſolidorum reſiſtentia, ne frangantur, major eſt <lb></lb>quam pro Ratione ſectionum; hæc ſiquidem Ratio ſectionum <lb></lb>ſervari quidem intelligitur, ſi limâ aut ſerrâ ſecari corpora <lb></lb>oporteat; illæ enim tantummodo particulæ reſiſtunt., quæ <lb></lb>ſectionem admittunt; at ubi de fractione agitur, quæ præter <lb></lb>motum particularum, quæ dividuntur, motum etiam aliquem <lb></lb>exigit aliarum, quas comprimi aut diſtrahi opus eſt, plus, <lb></lb>minùs, pro Ratione vicinitatis, longè alia eſt Ratio, pro ut <lb></lb>compreſſio illa atque diſtractio particularum faciliùs aut dif<lb></lb>ficiliùs perfici poterit. </s> <s id="s.001413">Hoc autem ex ipsâ figurâ potiſſimùm <lb></lb>pendet: Solidi enim RS ſectio CDE eadem quidem eſt, ſi<lb></lb>vè illud circà DE longiorem lineam, ſivè circa CD brevio<lb></lb>rem, curvari debeat, ut frangatur; ſed non eadem eſt in <lb></lb>fractione CD ac in fractione DE frangendi difficultas; nam <lb></lb>cum propiores ſint puncto D partes, quæ ad C, quàm quæ <lb></lb>ad E ſitæ ſunt, conſtat has quidem magis cum circà lineam <lb></lb>CD curvatur ſolidum, illas verò, cùm circà lineam DE <lb></lb>curvatur, minùs diſtrahi oportere, ut fractio ſequatur. </s> <s id="s.001414">Quò <lb></lb>autem magis diſtrahi debent particulæ, quæ ex D ver ûs E <lb></lb>recedunt, magis interim comprimi neceſſe eſt eas, quæ ad D <lb></lb>accedunt ſecundùm lineam RO in plano RI. </s> <s id="s.001415">Major igi<lb></lb>tur eſt difficultas, ſi circà breviorem lineam CD curve<lb></lb>tur, & fractio ſecundùm longiorem lineam DE ſequatur, <lb></lb>quàm ſi contrà curvetur circà longiorem DE, & fractio ſit <lb></lb>juxtà breviorem CD. </s> </p> <p type="main"> <s id="s.001416">Jam igitur ſi duo ſolida invicem comparentur, quæ ejuſ<lb></lb>dem ſint materiæ ejuſdemque longitudinis, & in pari ab ex<lb></lb>tremitatibus diſtantiâ frangi oporteat, ſtatuatur in utroque <lb></lb>ſolido punctum fractionis, per quod intelligatur planum ſe<lb></lb>cans ſimiliter inclinatum, facienſque in utroque ſolido ſuper<lb></lb>ficies, quas vocemus <emph type="italics"></emph>Baſes.<emph.end type="italics"></emph.end></s> <s id="s.001417"> Item planum per quod movetur <lb></lb>Potentia vim frangendi habens, ita productum intelligatur, ut <lb></lb>Baſibus prædictis ſimili inclinatione occurrens deſcribat ſectio<lb></lb>num lineas, quas vocemus Craſſities. </s> <s id="s.001418">Ut ſi fuerint duo ſoli-<pb pagenum="187" xlink:href="017/01/203.jpg"></pb>da CD & EF æqualis longitu<lb></lb><figure id="id.017.01.203.1.jpg" xlink:href="017/01/203/1.jpg"></figure><lb></lb>dinis, parieti infixa ſecundùm <lb></lb>æquales partes CI & EH, ut <lb></lb>in punctis I & H fiat fractio, <lb></lb>ex hypotheſi. </s> <s id="s.001419">Si per ea puncta <lb></lb>agantur plana ſimiliter inclina<lb></lb>ta, erunt ſuperficies IL & <lb></lb>HM, quas vocamus hîc <emph type="italics"></emph>Baſes.<emph.end type="italics"></emph.end><lb></lb>Jam in extremitatibus D & F <lb></lb>æquè remotis à punctis I & H <lb></lb>ſint Potentiæ vim frangendi habentes, & per lineam motûs <lb></lb>hujuſmodi Potentiarum intelligantur plana cum ſimili inclina<lb></lb>tione occurrentia baſibus IL & HM, ponamuſque communes <lb></lb>horum planorum ſectiones eſſe lineas parallelas, & æquales li<lb></lb>neis IN & HO; quas ſectiones vocamus <emph type="italics"></emph>Craſſities<emph.end type="italics"></emph.end> ſolidorum, <lb></lb>atque pro earum menſurâ uſurpamus lineas IN & HO. </s> <s id="s.001420">Cum <lb></lb>itaque frangendi difficultas oriatur tùm ex numero partium, <lb></lb>quæ ſeparandæ ſunt, has autem ipſæ Baſes IL & HM defi<lb></lb>niunt, tùm ex violento motu diſtractionis partium, qui ex ipsâ <lb></lb>ſolidorum craſſitie IN, & HO dignoſcitur; illud conſequens <lb></lb>eſt, quòd Reſiſtentiæ ſolidorum Ratio ea ſit, quæ ex Ratione <lb></lb>Baſium, & Ratione Craſſitierum componitur. </s> <s id="s.001421">Hinc eſt quòd <lb></lb>ſi Baſes fuerint ſimiles, & quæ eſt Ratio laterum homologo<lb></lb>rum, ea etiam ſit Craſſitierum Ratio, reſiſtentiæ ad fractionem <lb></lb>invicem comparatæ erunt in Ratione triplicatâ laterum homo<lb></lb>logorum; ac propterea cylindrorum reſiſtentia ad fractionem <lb></lb>erit in Ratione triplicatâ Diametrorum, ſeu Craſſitierum. </s> </p> <p type="main"> <s id="s.001422">Hanc, de quâ hactenus nobis ſermo fuit, <emph type="italics"></emph>Reſiſtentiam abſolu<lb></lb>tam<emph.end type="italics"></emph.end> dicimus, quam ſolidum habet, ne dividatur: quò enim <lb></lb>plures partes debent præter naturam comprimi, aut diſtrahi, <lb></lb>plures ſunt reſiſtentiæ; & quò magis hoc motu debent mo<lb></lb>mento eodem præter naturam moveri, eò etiam magis re<lb></lb>ſiſtunt: quâ igitur ratione plures ſunt reſiſtentes, & quâ Ra<lb></lb>tione magis reſiſtunt, tota reſiſtentiæ ratio componitur; quæ <lb></lb>ex ipsâ corporis ſoliditate pendet, nullâ habitâ ratione longi<lb></lb>tudinis ipſius ſolidi: Propterea <emph type="italics"></emph>Abſoluta<emph.end type="italics"></emph.end> dicitur. </s> <s id="s.001423">Nam ſi lon<lb></lb>gitudines frangendorum corporum comparemus, quæ ſuâ va<lb></lb>rietate mutant frangendi difficultatem, aut facilitatem, re-<pb pagenum="188" xlink:href="017/01/204.jpg"></pb>ſiſtentia hæc dicenda erit <emph type="italics"></emph>Reſpectiva<emph.end type="italics"></emph.end>; quæ aliquando ea eſſe <lb></lb>poteſt, ut corpus majore reſiſtentiâ abſolutâ præditum redda<lb></lb>tur magis obnoxium fractioni; longitudo ſiquidem auget fran<lb></lb>gendi facilitatem: ideo autem <emph type="italics"></emph>Reſpectivam<emph.end type="italics"></emph.end> dicimus, quia com<lb></lb>paratè ad momenta potentiæ ſumitur; hæc verò momenta ex <lb></lb>variâ longitudine, ſeu diſtantia à puncto fractionis pendere <lb></lb><figure id="id.017.01.204.1.jpg" xlink:href="017/01/204/1.jpg"></figure><lb></lb>manifeſtum eſt. </s> <s id="s.001424">Sit enim <lb></lb>ſolidum AB, quod ita <lb></lb>flectatur, ut fiat fractio <lb></lb>CD: Potentia movens in <lb></lb>B conſtituta dum perficit <lb></lb>ſpatium BE, diſtractio par<lb></lb>ticularum ſolidi fit ſolùm <lb></lb>per ſpatium CD (aut ve<lb></lb>riùs per CHD, nam etiam partes inter C & H diſtrahuntur; <lb></lb>Sed hîc claritatis gratiâ ſolùm extremæ CD conſiderantur) <lb></lb>quod eſt multo minus ſpatio BE ſecundùm Rationem HD ad <lb></lb>HE. </s> <s id="s.001425">At ſi ſolidum frangendum ſit AF, aut ſi ſit totum AB, <lb></lb>tamen Potentia movens ſit ſolùm applicata in F, Potentia perfi<lb></lb>ciens ſpatium FG (quod eſt minus quàm BE in Ratione HF <lb></lb>ad HB) major eſſe debet quàm Potentia in B ſecundùm Ratio<lb></lb>nem Reciprocam motuum BE & FG, ut ſequatur idem motus <lb></lb>diſtractionis partium CD; nam ex 8. l. 5. minor eſt Ratio FG <lb></lb>ad CD, quàm ſit Ratio BE ad eandem CD. </s> <s id="s.001426">Conſtat igitur <lb></lb>à longitudine augeri facilitatem frangendi, ac proinde Re<lb></lb>ſiſtentiam hanc Reſpectivam eſſe ſecundùm Reciprocam Ra<lb></lb>tionem longitudinum. </s> </p> <p type="main"> <s id="s.001427">Ex quo obiter apparet, cur ſolida Horizonti perpendicularia <lb></lb>magis reſiſtant fractioni, ſi potentiæ motus, ſeu conatus, ſit ad <lb></lb>perpendiculum Horizonti: quia videlicet in hujuſmodi motu <lb></lb>ad perpendiculum æqualiter moveri oportet Potentiam cum <lb></lb>ſolidi particulis, quæ diſtrahi aut comprimi debent: ut autem <lb></lb>Potentia ſuperet vim reſtitivam, aut major eſſe debet Ratio <lb></lb>motûs potentiæ ad motum corporis reſiſtentis, quàm ſit Ratio <lb></lb>virium reſiſtendi ad virtutem movendi, aut virtus movendi ab<lb></lb>ſolutè major eſſe debet vi reſiſtendi: Cum itaque in motu per<lb></lb>pendiculari intercedere non poſſit motuum inæqualitas, ne<lb></lb>ceſſe eſt virtutem movendi vehementer augeri, ut ſuperet vim, <pb pagenum="189" xlink:href="017/01/205.jpg"></pb>quâ particulæ ſolidi invicem connexæ repugnant, ne diſtra<lb></lb>hantur, aut comprimantur. </s> </p> <p type="main"> <s id="s.001428">Hinc ex haſtâ ad perpendiculum ſuſpensâ pendebit ingens <lb></lb>ſaxum, & tigillum perpendiculariter terræ inſiſtentem pre<lb></lb>met moles, penè dixerim, immenſa, citrà haſtæ aut ti<lb></lb>gilli fractionem: quia omnes haſtæ atque tigilli partes & <lb></lb>æqualiter cum onere ſuſpenſo aut incumbente moveri de<lb></lb>berent, & omnes æqualiter reſiſtunt diſtractioni aut com<lb></lb>preſſioni: At ſi ad horizontem inclinata aut parallela fue<lb></lb>rint hujuſmodi ſolida (haſta videlicet atque tigillus) non <lb></lb>eſt æqualis omnium partium diſtractio aut compreſſio, mi<lb></lb>nùs enim diſtrahuntur, quæ puncto H proximæ ſunt, quam <lb></lb>quæ ad D accedunt (concipe H in media craſſitie) con<lb></lb>trà verò illæ magis, hæ minùs comprimuntur; quemad<lb></lb>modum neque motui diſtractionis aut compreſſionis eſſet <lb></lb>æqualis motus oneris deorsùm urgentis in haſtæ, vel tigil<lb></lb>li non perpendicularium extremitate conſtituti, ſed multò <lb></lb>major eſſet hîc oneris motus. </s> <s id="s.001429">Quoniam verò rerum natu<lb></lb>ra magis repugnat corporum penetrationi, ad quam quodam<lb></lb>modo accedere videtur compreſſio, quàm corporum unito<lb></lb>rum diviſioni, ubi vacui metus abſit; hinc eſt majorem <lb></lb>molem faciliùs ſuſtineri à fulcro ad perpendiculum ſubjecto, <lb></lb>quàm ſuſpendi ex ſolido perpendiculari citrà fractionis pe<lb></lb>riculum. </s> <s id="s.001430">Quamvis negandum non ſit ad hujuſmodi facili<lb></lb>tatem, quam experimur in ſuſtinendo potiùs, quàm in re<lb></lb>tinendo onere, conferre plurimum, quòd tellus, cui ful<lb></lb>crum infigitur, demùm non ſubſidit; at laqueare ſeu for<lb></lb>nix ex quo ſolidum pendet onere prægravatum, tantam <lb></lb>gravitatem non ita facilè ferre poteſt. </s> <s id="s.001431">Quare ad tollenda <lb></lb>in ſuperiores ædificiorum partes ingentia ſaxa multo cau<lb></lb>tiùs atque tutiùs ij operantur, qui longam trabem, aut plu<lb></lb>ra tigna ritè connexa, quaſi navis malum rudentibus uſ<lb></lb>quequaque firmatum, ne à perpendiculo deflectat, ſta<lb></lb>tuunt, cui ſuperiorem trochleam adnectant; quàm qui tra<lb></lb>bem Horizonti parallelam parieti infigunt ad idem munus <lb></lb>præſtandum; hæc ſiquidem horizonti parallela magis fractio<lb></lb>ni obnoxia eſt, quàm perpendicularis; præterquam quod <lb></lb>parietem aliquatenus labefactare poteſt, cum habeat ratio-<pb pagenum="190" xlink:href="017/01/206.jpg"></pb>nem vectis in ſuperiora propellentis ſaxo deorſum urgente; <lb></lb>niſi huic periculo ex arte obviam eatur. </s> </p> <p type="main"> <s id="s.001432">Comparatis itaque invicem ſolidorum frangendorum lon<lb></lb>gitudinibus, hoc eſt intervallis inter fractionum puncta & <lb></lb>locum, ubi potentia vim frangendi habens conſtituta intel<lb></lb>ligitur, quò major eſt longitudo, eò minor eſt reſiſtentia <lb></lb>ſolidi, ne frangatur. </s> <s id="s.001433">Qua propter ubi duo data ſolida con<lb></lb>ferantur, quæcumque demùm illa ſint, non ſolùm eorum <lb></lb>Reſiſtentia Abſoluta, quæ ex Rationibus Baſium, & Craſ<lb></lb>ſitierum componitur, attendenda eſt, ſed etiam Reſiſtentia <lb></lb>Reſpectiva, quæ ex longitudinibus pendet: atque adeò <lb></lb>adæquata Ratio reſiſtentiæ, ne frangantur, ea eſt, quæ <lb></lb>componitur ex Rationibus Baſium & Craſſitierum atque ex <lb></lb>Ratione longitudinum Reciprocè ſumptarum: cùm enim <lb></lb>longitudini majori reſpondeat minor reſiſtentia, manifeſtum <lb></lb>eſt longitudinum Rationem eſſe Reciprocè ſumendam, ut <lb></lb>reſiſtentiæ, quæ ex illis oritur, Ratio habeatur. </s> <s id="s.001434">Hinc eſt <lb></lb>fieri aliquando poſſe, ut ſolidum craſſius minùs reſiſtat <lb></lb>fractioni, quàm ſubtilius, ſi hoc breve ſit, illud verò valdè <lb></lb>longum, ſi videlicet longitudo craſſioris ad longitudinem <lb></lb>ſubtilioris Rationem habeat majorem, quàm ſit ea, quæ ex <lb></lb>Rationibus Baſium, & Craſſitierum componitur. </s> <s id="s.001435">Sic ſi duo <lb></lb>fuerint cylindri, & alter triplo craſſior fuerit reliquo, ſed <lb></lb>etiam trigecuplo longior fuerit illo, minùs etiam fractioni <lb></lb>reſiſtet; quia reſiſtentia abſoluta majoris cylindri ad mino<lb></lb>rem eſt ut 27 ad 1, ſed reſiſtentia Reſpectiva ejuſdem ma<lb></lb>joris ad minoris reſiſtentiam pariter reſpectivam eſt ut 1 ad <lb></lb>30: Ratio ergo ex his Rationibus 27 ad 1, & 1 ad 30 <lb></lb>Compoſita, eſt Ratio 27 ad 30, hoc eſt 9 ad 10, ac propterea <lb></lb>major cylindrus reſiſtit fractioni ut 9, minor verò fractioni <lb></lb>reſiſtit ut 10. </s> </p> <p type="main"> <s id="s.001436">Deſine jam mirari, ſi quando paxillum maximis viribus <lb></lb>reſiſtere videris; quia nimirùm potentia, quæ motum co<lb></lb>natur, proximè applicata eſt parieti aut plano, cui paxil<lb></lb>lus infigitur: quòd ſi remotior illa fuerit, etiam minùs hic <lb></lb>reſiſtet. </s> <s id="s.001437">Sic defixo in terram paxillo AB, cui funis AC al<lb></lb>ligatur, experientia docet paxillum eò reſiſtere validiùs, quò <lb></lb>propiùs ad A alligatur funis, debiliùs autem reſiſtere, quò <pb pagenum="191" xlink:href="017/01/207.jpg"></pb>magis ad B accedit; <lb></lb><figure id="id.017.01.207.1.jpg" xlink:href="017/01/207/1.jpg"></figure><lb></lb>in A nimirùm motus <lb></lb>potentiæ trahentis vix <lb></lb>excederet motum pa<lb></lb>xilli, qui ibi flectere<lb></lb>tur ex hypotheſi; at <lb></lb>fune in B poſito, po<lb></lb>tentia ibi conſtituta, <lb></lb>& per funem applica<lb></lb>ta multò velociùs mo<lb></lb>veretur, quàm paxilli <lb></lb>partes propè A, quæ <lb></lb>ibi flecterentur. </s> </p> <p type="main"> <s id="s.001438">Quòd ſi loci conditio, aut ipſa oneris movendi conſtitutio <lb></lb>id exigat, ut funis propè B alligetur, & de paxilli AB firmi<lb></lb>tate dubitetur, paxillum alterum DE paulò remotiorem com<lb></lb>modo loco depange ita, ut funis primùm in D firmetur, de<lb></lb>inde circa B convolutus extendatur, pro ut operis faciendi ra<lb></lb>tio fieret. </s> </p> <p type="main"> <s id="s.001439">Eâdem ratione ſi tigillus, ex quo onus dependere debet, pa<lb></lb>rieti ſit infixus, & ſit GH, fractioni magis erit obnoxius, quò <lb></lb>propiùs accedet pondus ad H: <lb></lb><figure id="id.017.01.207.2.jpg" xlink:href="017/01/207/2.jpg"></figure><lb></lb>propterea aut ei ſubjicitur brevior <lb></lb>tigillus IR omninò contiguus, <lb></lb>aut ſupponitur fulcrum OS in<lb></lb>clinatum; quod fractionem eò va<lb></lb>lidiùs impediet, quò minùs diſta<lb></lb>bunt H & S, & quò acutior fue<lb></lb>rit angulus, quem fulcrum SO <lb></lb>cum pariete conſtituit, ſeu, quod <lb></lb>eôdem recidit, quò magis ad <lb></lb>recti anguli quantitatem acce<lb></lb>det angulus GSO. </s> <s id="s.001440">Quæ omnia <lb></lb>ita ex dictis aperta ſunt, ut ulte<lb></lb>riori explicatione non egeant. </s> </p> <p type="main"> <s id="s.001441">Sed & illud hîc, ubi de Reſiſtentiâ Reſpectivâ ſermo eſt, <lb></lb>adjiciendum videtur, quòd ex ſolâ majori longitudine hæc non <lb></lb>minuitur, niſi cùm longitudo ſolidi ad perpendiculum inſiſtit <pb pagenum="192" xlink:href="017/01/208.jpg"></pb>Horizonti; tunc enim gravitas ipſa ſolidi tota incumbit <lb></lb>ſubjecto plano; & tantùm Potentia oblique atque in tranſ<lb></lb>verſum trahens applicata extremitati longioris ſolidi plus ha<lb></lb>bet momenti, quàm applicata extremitate brevioris, quin <lb></lb>velociùs, & faciliùs movetur ſecundùm Rationem longitu<lb></lb>dinum illarum. </s> <s id="s.001442">At quando ſolida ſunt horizonti parallela, <lb></lb>aut ad illum ita inclinata, ut centrum gravitatis partis illius, <lb></lb>quæ erumpit ex corpore, cui ſolidum infigitur, non immi<lb></lb>neat baſi ſuſtentationis, non ſola longitudo attendenda eſt, <lb></lb>ſed & ipſa gravitas, quæ etiam nullo addito extrinſeco mo<lb></lb>tore ſua habet momenta, quibus deorſum connititur. </s> <s id="s.001443">Ex <lb></lb>quo fit pro majori gravitate etiam frangendi facilitatem au<lb></lb>geri, ipſa nimirum gravitas eſt potentia conjuncta, quæ au<lb></lb>getur pro ratione materiæ; materia autem augetur pro ra<lb></lb>tione longitudinis (cætera ſiquidem paria eſſe hîc claritatis <lb></lb>gratiâ, ponamus) ac propterea longius priſma comparatum <lb></lb>cum breviori priſmate, eo quòd majorem habeat gravita<lb></lb>tem, minùs reſiſtit fractioni ſecundùm Reciprocam Ratio<lb></lb>nem longitudinum. </s> <s id="s.001444">Atqui Ratio motûs hujuſmodi Potentiæ <lb></lb>conjunctæ eſt ſecundùm Rationem longitudinum, & ex <lb></lb>dictis Ratio Reſiſtentiæ in ordine ad hujuſmodi motum eſt <lb></lb>permutatim ac Reciprocè ſecundùm eandem longitudinum <lb></lb>Rationem: igitur Ratio duplicatur, & reſiſtentia longioris <lb></lb>ad reſiſtentiam brevioris eſt ſecundùm ſubduplicatam Ratio<lb></lb>nem longitudinum reciprocè ſumptarum. </s> <s id="s.001445">Id quod etiam <lb></lb>hinc conſtat, quia cùm ſingula illius longitudinis puncta <lb></lb>ſuam habeant gravitatem, ſua omnibus inſunt momenta pro <lb></lb>Ratione diſtantiæ à puncto quod eſt veluti centrum motûs; <lb></lb>ergo aggregata momentorum ſunt ut ſectores ab illis longi<lb></lb>tudinibus tanquam à Radiis deſcripti: ſunt autem ſimiles <lb></lb>ſectores in duplicatâ Ratione Radiorum. </s> <s id="s.001446">Quare ſi longitudi<lb></lb>nes ſint ut 3 ad 2, Reſiſtentia reſpectiva longioris ad reſiſten<lb></lb>tiam brevioris eſt ut 4 ad 9. Tota igitur ſolidorum reſiſten<lb></lb>tia, ne frangantur, componitur ex Rationibus Baſium, & <lb></lb>Craſſitierum, & ex ſubduplicatâ Ratione longitudinum per<lb></lb>mutatim ac reciprocè ſumptarum. </s> </p> <p type="main"> <s id="s.001447">Ex his itaque, quæ de ſolidorum reſiſtentiâ, ne frangan<lb></lb>tur, hactenùs diſputata ſunt, conjecturam facilè accipiet <pb pagenum="193" xlink:href="017/01/209.jpg"></pb>prudens machinator, quàm ſolida & craſſa ſtatui debeant <lb></lb>quæque machinarum membra, quóve loco collocanda ſint, <lb></lb>ut & materia & forma reſpondeant fini, in quem machinæ <lb></lb>deſtinantur: neque enim ſatis eſt concinno, & eleganti dia<lb></lb>grammate machinam oculis repræſentaſſe, ejuſque vires ad <lb></lb>calculos revocâſſe, quantum quidem ex machinæ figurâ col<lb></lb>ligitur, ſi demùm, inſtituto motu machina pondere prægra<lb></lb>vata luxetur. </s> </p> <p type="main"> <s id="s.001448">Illud tamen præterea Machinator animadvertat, oportet, <lb></lb>quod ſpectat ad momenta virium, quas potentia movens <lb></lb>exercet; neque enim ſola ponderis gravitas machinam, aut <lb></lb>corpus, cui machina alligatur, aut innititur, urget aut pre<lb></lb>mit, ſed & ipſa potentia, dum adversùs ipſum pondus co<lb></lb>natur machinam movens, aliquando auget gravitatem ex <lb></lb>oppoſitâ parte, adeò ut & huic & ponderi reſiſtere debeat <lb></lb>machina, aut id, quod machinam retinet. </s> <s id="s.001449">Si enim fuerit <lb></lb>vectis AB in<lb></lb><figure id="id.017.01.209.1.jpg" xlink:href="017/01/209/1.jpg"></figure><lb></lb>nixus ſuper ba<lb></lb>culum CD, ex <lb></lb>B pendeat glo<lb></lb>bus plumbeus <lb></lb>E, & extremi<lb></lb>tas A quieſcat <lb></lb>aliquo corpore <lb></lb>retinente, ut ſi <lb></lb>fuerit parieti in<lb></lb>fixa; ſolo globo E gravitante minus periculum ſubeſt fractio<lb></lb>nis tùm vectis, tùm baculi CD ſuſtentantis, quàm ſi in A <lb></lb>ſit potentia F; cujus conatus deorſum oppoſitus conatui de<lb></lb>orſum ponderis E faciliùs curvitatem, aut etiam demùm <lb></lb>fractionem vectis efficere poteſt in I, ut patet; immò & ba<lb></lb>culus CD ſuſtentans vectem, non ſolùm momenta ponderis E, <lb></lb>ſed & momenta Potentiæ F, quæ in I uniuntur, in ſe recipit; <lb></lb>atque adeò utriſque ferendis par eſſe debet. </s> </p> <p type="main"> <s id="s.001450">Simile quiddam obſervare eſt, ſi ex orbiculo O, in clavo <lb></lb>M ſuſpenſo, circà ſuum axem verſatili, dependeat pondus S, <lb></lb>& Potentia in R deorſum conata cogat pondus S aſcendere: <lb></lb>certum eſt enim ab axe orbiculi, & à clavo M ſuſtineri non <pb pagenum="194" xlink:href="017/01/210.jpg"></pb><figure id="id.017.01.210.1.jpg" xlink:href="017/01/210/1.jpg"></figure><lb></lb>ſolùm pondus S, ſed & Poten<lb></lb>tiam, quæ eſt in R. </s> <s id="s.001451">Contrà ve<lb></lb>rò ſi orbiculus V ſit adnexus pon<lb></lb>deri T, funis autem orbiculo in<lb></lb>ſertus alligetur clavo in N, & po<lb></lb>tentia P ſurſum trahat, conſtat ab <lb></lb>axe quidem orbiculi ſuſtineri ſo<lb></lb>lum pondus T; à clavo verò N <lb></lb>non totum pondus T ſuſtineri, <lb></lb>ſed ejus ſemiſſem, nam etiam Po<lb></lb>tentia P ſuſtinet pondus. </s> <s id="s.001452">Validior <lb></lb>igitur eſſe debet clavus M quàm <lb></lb>clavus N, hic enim ponderis ſe<lb></lb>miſſem fert, ille verò plus quàm <lb></lb>duplum. </s> <s id="s.001453">Potentia enim R major <lb></lb>eſt pondere S. </s> </p> <p type="main"> <s id="s.001454">Quòd ſi tàm pondera S & T, <lb></lb>quàm clavi M & N, atque Po<lb></lb>tentiæ R & P non in plano Ver<lb></lb>ticali, ſed in Horizontali conſtituantur, certum eſt pondera <lb></lb>S & T non ſuſpenſa ſed jacentia, nihil adversùs clavos M & <lb></lb>N; aut adversùs ſuorum orbiculorum O & V axes conari, im<lb></lb>mò neque adversùs Potentias R & P; quandoquidem toto niſu <lb></lb>plano ſubjecto incumbunt, nullámque exercent Activam Re<lb></lb>ſiſtentiam; ſed Formalem tantummodo, quâ repugnent Po<lb></lb>tentiis moventibus: quæ quidem reſiſtentia, tùm ex ip â pon<lb></lb>derum gravitate, tùm ex attritu ſubjecti plani componitur. </s> <lb></lb> <s id="s.001455">Clavorum igitur M & N ea ſit, oportet, ſoliditas atque firmi<lb></lb>tas, quæ potentiarum R & P conatibus reſpondeat; ne forte <lb></lb>clavi ipſi frangantur faciliùs, aut revellantur, quàm pondera <lb></lb>ſuo loco dimoveantur. </s> <s id="s.001456">Sed hæc innuiſſe ſat fuerit, ut ſingula <lb></lb>diligenter à machinatore circumſpicienda eſſe intelligatur; ne<lb></lb>que tamen in his ad nauſeam diutiùs immorandum. <pb pagenum="195" xlink:href="017/01/211.jpg"></pb> </s> </p> <p type="main"> <s id="s.001457"><emph type="center"></emph>CAPUT VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001458"><emph type="center"></emph><emph type="italics"></emph>Præſtet-ne Machinam augere? </s> <s id="s.001459">an componere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001460">EX iis, quæ de Machinarum viribus diſputata ſunt ſatis <lb></lb>liquet nullum dari finitum Pondus quod data Potentia mo<lb></lb>vere non poſſit ſi congruens machina adhibeatur: cum etenim <lb></lb>data ſit Ratio Ponderis ad Potentiam, eo artificio Machina <lb></lb>diſponatur, ut Ratione illâ datâ fiat major Ratio motûs Potentiæ <lb></lb>ad motum Ponderis; & Pondus cedet Potentiæ moventi. </s> <s id="s.001461">Sic <lb></lb>viciſſim ſi oblata fuerit machina, examinandus primùm eſt lo<lb></lb>cus, ubi Potentia applicanda eſt, ubi Pondus collocandum; <lb></lb>tùm utriuſque motûs rationes ineundæ: & pronunciabis majo<lb></lb>rem requiri rationem Potentiæ ad Pondus, quàm ſit Ratio mo<lb></lb>tûs Ponderis ad motum Potentiæ. </s> <s id="s.001462">Sit enim ex. </s> <s id="s.001463">gr. motuum hu<lb></lb>juſmodi Ratio, quæ eſt 3 ad 8; Potentia vim movendi habens <lb></lb>ut 3 non movebit Pondus, cujus vis reſiſtendi, & momentum, <lb></lb>ſit ut 8; ſed opus eſt, ut illa major ſit quàm 3. At neque Po<lb></lb>tentiam augere potes, ut oportet, neque Ponderi quicquam de<lb></lb>trahere: vide igitur utrum fieri poſſit, ut mutetur in machinâ <lb></lb>motuum Ratio, aut Potentiæ motum augendo, aut ponderis <lb></lb>motum minuendo. </s> </p> <p type="main"> <s id="s.001464">Hinc manifeſtum eſt machinam majorem non plus afferre <lb></lb>facilitatis præ minore, ſi illæ quidem omninò ſimiles fuerint <lb></lb>(modò utraque ſatis ſolida ſit, ne fractioni ſit obnoxia) mo<lb></lb>tuum enim Ratio eadem eſt in utráque. </s> <s id="s.001465">Sic Vectis 100 pal<lb></lb>morum ſi ita ab hypomochlio diſtinguatur in partes ut hinc <lb></lb>palmos 20, hinc 80 relinquat, non majorem movendi faci<lb></lb>litatem præbebit, quàm vectis palmorum quinque ita divi<lb></lb>ſus ab hypomochlio, ut hinc palmus unus, hinc verò quatuor <lb></lb>relinquantur. </s> <s id="s.001466">Ut igitur longior ille Vectis utilior accidat, ſi <lb></lb>hypomochlium quidem transferri queat, remove illud à Po<lb></lb>tentiâ, & admove Ponderi, motuumque Ratio augebitur; pa<lb></lb>tet ſcilicet majorem eſſe Rationem 85 ad 15, quam 80 ad 20: <lb></lb>Quod ſi verò hypomochlium ita fixum ſit ac vecti adnexum, <pb pagenum="196" xlink:href="017/01/212.jpg"></pb>ut mutari loco nequeat, abſcinde palmos (5 15/17), adeò ut hinc ſint <lb></lb>palmi 80 ut priùs, hinc autem ſint palmi (14 2/17), & eadem erít <lb></lb>Ratio, quæ eſt 85 ad 15. Quare breviore vecte plus ponderis <lb></lb>movebis, quàm longiore; vis enim, quæ longiore illo 100 pal<lb></lb>morum movebat pondus librarum 100, breviore hoc palmo<lb></lb>rum (94 2/17) movebit libras 141 2/3: Quia quamvis in utroque Vecte <lb></lb>hypomochlium habente poſt palmum octuageſimum, Potentia <lb></lb>eodem ſemper motu moveatur, non tamen idem eſt ponderis <lb></lb>motus, qui in minore vecte minor eſt, in majore major, ac <lb></lb>proinde motûs Potentiæ ad motum Ponderis Ratio major eſt in <lb></lb>minore, minor in majore vecte. </s> <s id="s.001467">Quod ſi demùm nec hypo<lb></lb>mochlium transferre, nec vecte mutilato uti liceat, licebit ſa<lb></lb>nè fuſtem, vel quid ſimile, firmiter ad alligatum Vecti adjun<lb></lb>gere, potentiamque ab hypomochlio longiùs removere: opor<lb></lb>teret autem additamentum hujuſmodi eſſe palmorum 33 1/3; nam <lb></lb>ut 15 ad 85, ita 20 ad 113 1/3; adeóque totus vectis eſſet pal<lb></lb>morum 133 1/3. </s> </p> <p type="main"> <s id="s.001468">Porrò hîc obſerva, quantò facilius ſit ponderis motum mi<lb></lb>nuere, quàm potentiæ motum augere: in allato ſiquidem <lb></lb>exemplo, manente eodem potentiæ motu, minuitur ponderis <lb></lb>motus decurtato vecte ac diminuto palmis (5 15/17); manente au<lb></lb>rem eodem ponderis motu augetur Potentiæ motus acuto vecte <lb></lb>palmis 33 1/3: Quia nimirum in Ratione majoris Inæqualitatis ſi <lb></lb>Conſequens terminus minor minuatur, aut Antecedens termi<lb></lb>nus major augeatur, fit adhuc major Inæqualitas; ut autem <lb></lb>eadem Ratio ſervetur aucto Antecedente ac diminuto Conſe<lb></lb>quente, manifeſtum eſt, quæ pars Conſequentis integri eſt <lb></lb>conſequens diminutus, eam debere eſſe partem Antecedentis <lb></lb>aucti Antecedentem datum: atqui Antecedens datus eſt major <lb></lb>dato Conſequente; igitur plus addendum eſt Antecedenti, <lb></lb>quàm dematur Conſequenti. </s> <s id="s.001469">Sic data ſit Ratio 8 ad 6: Con<lb></lb>ſequens bifariam ſecetur, ejuſque ſemiſſis fiat novus Conſe<lb></lb>quens; erit Ratio 8 ad 3 majoris adhuc inæqualitatis; hæc enim <lb></lb>eſt dupla ſuperbipartiens tertias, illa verò erat ſolùm ſeſqui<lb></lb>tertia. </s> <s id="s.001470">Ut igitur retento priori Conſequente 6 fit eadem Ratio <lb></lb>dupla ſuperbipartiens tertias, ſicut Conſequens fuit bifariam <lb></lb>diviſus, ita datus Antecedens 8 eſt duplicandus, ut ſit Ratio <pb pagenum="197" xlink:href="017/01/213.jpg"></pb>16 ad 6: plus autem eſt totus antecedens major qui additur, <lb></lb>quàm ſit ſemiſſis Conſequentis minoris qui demitur. </s> <s id="s.001471">In re au<lb></lb>tem noſtrâ ſemper Ratio motûs Potentiæ per machinam vali<lb></lb>dioris factæ ad motum dati ponderis eſt Ratio Majoris inæqua<lb></lb>litatis: Quapropter ſatius eſt Ponderis motum minuere, quam <lb></lb>potentiæ motum auctâ machinâ augere. </s> </p> <p type="main"> <s id="s.001472">Hæc quidem, quæ in vecte propoſita facilè ac in promptu <lb></lb>eſt perſpicere, in cæteris pariter mechanicis Facultatibus, ut <lb></lb>in Trochleis, Cochleâ, & reliquis intelligenda ſunt, ut ex iis, <lb></lb>quæ inferiùs dicentur, ſuo loco manifeſtum fiet. </s> <s id="s.001473">Sed quoniam <lb></lb>ad ponderis motum extenuandum certos quoſdam fines ipſa <lb></lb>machinarum materia præſcribit; neque enim quemadmodum <lb></lb>quantitatem omnem, & corporum molem in ſubtiliores, ac <lb></lb>ſubindè ſubtiliores partes mente concidimus, ita etiam id re <lb></lb>ipsâ perficere atque in praxim deducere poſſumus: propterea <lb></lb>ut plurimum cogimur Potentiæ velociorem motum conciliare, <lb></lb>ut majorem obtineat Rationem ad motum Ponderis. </s> <s id="s.001474">Quis ete<lb></lb>nim non incaſſum uti poſſit Vecte, cujus hypomochlium à <lb></lb>pondere ſatis gravi non ampliùs diſtet, quàm per digiti ſemiſ<lb></lb>ſem? </s> <s id="s.001475">aut Cochleam adhibere, cujus ſpiras intervallum capilla<lb></lb>ceum ſecernat? </s> </p> <p type="main"> <s id="s.001476">Verùm cum id duplici methodo præſtare poſſimus, videlicet <lb></lb>aut Machinam ipſam, ſpecie non mutatâ, augentes, aut illam <lb></lb>ex pluribus membris componentes, ſive ejuſdem generis ſint, <lb></lb>ſive diverſi; operæ pretium fuerit perpendere, majuſ-ne in <lb></lb>augmento? </s> <s id="s.001477">an verò in compoſitione? </s> <s id="s.001478">compendium inveniatur. <lb></lb><emph type="italics"></emph>Augmentum<emph.end type="italics"></emph.end> voco (ne ullus ſubſit æquivocandi locus) cum ejuſ<lb></lb>dem Facultatis ſpecies immutata permanet, factâ ſolum partis <lb></lb>alicujus acceſſione; ut ſi, quia Vectis juſto brevior eſt, Poten<lb></lb>tiæ ab hypomochlio diſtantiam longiorem facias; cum Tro<lb></lb>chleæ adhibeantur oneri movendo impares, amplificatis locu<lb></lb>lamentis orbiculorum numerum augeas; quia Cochlea ob ſpi<lb></lb>rarum raritatem minùs valida eſt quàm oporteat, lineam ipſam <lb></lb>ita inclines, ut ſpiſſioribus ſpiris circumducatur. </s> <s id="s.001479">At verò <emph type="italics"></emph>Com<lb></lb>poſita<emph.end type="italics"></emph.end> dicitur Machina, cum invalidæ Facultati membra alia <lb></lb>adjiciuntur, aut generis ejuſdem, ut cum Vectis Vecti, Co<lb></lb>chleæ Cochlea, Trochleis Throchleæ adjunguntur; aut diver<lb></lb>ſi generis, ut cum facultates ipſæ permiſcentur, vecti trochleas, <pb pagenum="198" xlink:href="017/01/214.jpg"></pb>Cochleæ vectem, Trochleis Cochleam, & deinceps, adjun<lb></lb>gendo. </s> <s id="s.001480">Prioris Compoſitionis intrà idem genus ſpecimen ali<lb></lb>quod exhibui in <emph type="italics"></emph>Terrâ Machinis motâ: Diſſertat.<emph.end type="italics"></emph.end> 1. & inferius <lb></lb>ſuis locis de eâ redibit ſermo: Poſterioris autem Compoſitionis <lb></lb>diverſarum Facultatum, ubi de ſingulis diſpurabimus, exem<lb></lb>pla aliqua ſubjiciemus, ut diſcat Tyro Machinarum vires ritè <lb></lb>ad calculos revocare, ſolertiamque machinandi acquirat. </s> </p> <p type="main"> <s id="s.001481">Quamvis autem quæſtio hæc multò dilucidiùs explicaretur, <lb></lb>ſi unamquamque Facultatem ſingillatim attingeremus, quàm <lb></lb>ſi unâ comprehenſione omnia complectamur; hîc tamen <lb></lb>doctrinæ ratio exigit, ut dimiſſis rivulis fontem ipſum aperia<lb></lb>mus, ex quo in Machinam Compoſitam vis major, quàm in <lb></lb>Amplificatam, majore compendio derivatur. </s> <s id="s.001482">Et quidem cum <lb></lb>res tota ex potentiæ atque Ponderis motuum Ratione pendeat, <lb></lb>quamdiu in ſimplici aliquâ facultate conſiſtimus, motus Po<lb></lb>tentiæ ad motum Ponderis ſimplicem habet Rationem; ſi verò <lb></lb>Facultas una cum aliâ quâpiam facultate conjungitur, atque <lb></lb>connectitur, jam Potentiæ motus ad motum ponderis eam ha<lb></lb>bet Rationem, quæ ex ſingularum facultatum rationibus com<lb></lb>ponitur. </s> <s id="s.001483">Voco autem <emph type="italics"></emph>ſingularum Facultatum Rationem<emph.end type="italics"></emph.end> eam, quæ <lb></lb>inter ipſos Potentiæ ac Ponderis motus intercederet, ſi facul<lb></lb>tas illa ſolitaria adhiberetur; Atqui Ratio hæc motuum in ſin<lb></lb>gulis Facultatibus modum recipit ex Facultatis ipſius partibus, <lb></lb>quarum altera ad Potentiam, àd Pondus altera ſpectare vide<lb></lb>tur; ut per ſingulas Facultates eunti conſtabit. </s> <s id="s.001484">In Vecte enim <lb></lb>Ponderis ab hypomochlio diſtantia pertinet ad Pondus, Poten<lb></lb>tiæ autem diſtantia ab eodem hypomochlio penes potentiam <lb></lb>eſt: In Trochleis ipſarum Trochlearum diſtantia Pondus reſpi<lb></lb>cit; funis autem explicatio Potentiam: In Axe in Peritrochio <lb></lb>craſſities Axis Ponderi, Peritrochij amplitudo Potentiæ tribui<lb></lb>tur: In Cuneo longitudo ad Potentiam ſpectat, craſſities ad <lb></lb>Pondus: In Cochleâ demùm ſpiræ circumductæ perimeter ad <lb></lb>Potentiam attinet, extremitatum ſpiralis lineæ intervallum, ad <lb></lb>Pondus. </s> <s id="s.001485">Manifeſtum eſt igitur, ubi ſimplex motuum Ratio in <lb></lb>ſingulis Facultatibus augenda fuerit, manente eâ parte, quæ <lb></lb>ad Pondus ſpectat, neceſſariò ita augendam eſſe partem reli<lb></lb>quam, quæ Potentiæ tribuitur, ut majori illi motuum Rationi <lb></lb>reſpondeat. </s> <s id="s.001486">Sic dato Vecte palmorum ſex, quo potentia mo-<pb pagenum="199" xlink:href="017/01/215.jpg"></pb>veatur in quintuplâ Ratione ad Pondus, ſi maneat eadem pon<lb></lb>deris ab hypomochlio diſtantia, & motuum Ratio eſſe debeat <lb></lb>vigecupla, ſatis conſtat totum vectem requiri palmorum 21, ut <lb></lb>unus Ponderi cedat, Potentiæ autem viginti. </s> </p> <p type="main"> <s id="s.001487">At verò ſi motuum Ratio ex Rationibus componenda ſit, ſa<lb></lb>tisfuerit datæ Facultati minorem Rationem continenti, quàm <lb></lb>oporteat, Facultatem aliam adjicere, cujus Ratio cum priori <lb></lb>Ratione compoſita quæſitam Rationem conſtituat. </s> <s id="s.001488">Sic dato <lb></lb>Vecti quintuplam rationem continenti adjunge aliam quamli<lb></lb>bet facultatem quadruplæ Rationis; ex quadruplâ enim Ratio<lb></lb>ne & quintuplâ componitur Ratio vigecupla quæſita. </s> <s id="s.001489">Ita au<lb></lb>tem ſecunda hæc Facultas priori Facultati adnectenda eſt, ut <lb></lb>quemadmodum duorum Magnetum oppoſiti poli junguntur, <lb></lb>Auſtralis videlicet unius Aquilonari alterius, ſic duarum Fa<lb></lb>cultatum oppoſitæ partes connectantur, ut ſcilicet quo loco ad <lb></lb>priorem Facultatem applicanda eſſet Potentia, eidem admo<lb></lb>veatur locus Ponderi in ſecundâ Facultate deſtinatus: proinde <lb></lb>ſiquidem ſe res habebit, atque ſi pondus diminutum pro Ra<lb></lb>tione prioris facultatis, videlicet ſub quintuplum, in ſecun<lb></lb>dam hanc Facultatem transferretur, in quâ ejus motus ad mo<lb></lb>tum Potentiæ Rationem haberet ſubquadruplam: re enim ve<lb></lb>râ duabus hiſce Facultatibus junctis, Potentiæ motus vigecu<lb></lb>plus eſt ad motum Ponderis; nam Pondus in vectis extremita<lb></lb>te alterâ conſtitutum quintuplo tardiùs movetur, quàm reli<lb></lb>qua vectis extremitas; hæc autem poſteriori Facultati loco <lb></lb>Ponderis adjuncta quadruplo tardiùs movetur quàm Poten<lb></lb>tia; igitur Ponderis motus vigecuplo tardior eſt motu Po<lb></lb>tentiæ. </s> </p> <p type="main"> <s id="s.001490">Statuamus exempli gratiâ ſecundam hanc Facultatem Vecti <lb></lb>adjunctam eſſe pariter Vectem ejuſdem generis quinque pal<lb></lb>morum ita ab hypomochlio diſtinctum in partes, ut hæ in qua<lb></lb>druplâ ſint Ratione: Ecce quanto compendio rem aſſequamur; <lb></lb>id enim quod ſimplici Vecte palmorum 21 præſtandum eſſet, <lb></lb>compoſitis vectibus duobus altero palmorum ſex, altero palm. </s> <lb></lb> <s id="s.001491">quinque perficimus, ſervatâ ſemper eâdem Ponderis ab hypo<lb></lb>mochlio diſtantiâ, nimirum palmi unius. </s> <s id="s.001492">Hæc tamen de duo<lb></lb>bus hiſce vectibus dicta ita intelliges velim, ut ad motum ſim<lb></lb>pliciter pertineant; non verò ad motûs quantitatem; ſatis enim <pb pagenum="200" xlink:href="017/01/216.jpg"></pb>ſcio non ad eam diſtantiam promoveri poſſe Pondus adhibito <lb></lb>ſecundo hoc vecte, ad quam promoveretur Vecte palmorum 21: <lb></lb>Verùm hîc ſola movendi facilitas conſideratur. </s> <s id="s.001493">Quòd ſi non <lb></lb>alterum Vectem adhibeas; ſed aliud facultatis genus, ut Tro<lb></lb>chleas binis orbiculis inſtructas, & Vecti in loco Potentiæ ad<lb></lb>nexas, multò adhuc faciliùs movebitur Pondus, cujus motus <lb></lb>erit ſubvigecuplus motûs Potentiæ funem Trochlearum tra<lb></lb>hentis, & tantus erit Ponderis motus, quantus eſſet, ſi extre<lb></lb>mitati Vectis palmorum ſex apponeretur Potentia quadrupla <lb></lb>datæ Potentiæ. </s> <s id="s.001494">Idem planè de cæteris dicendum Faculta<lb></lb>tibus. </s> </p> <p type="main"> <s id="s.001495">Hinc manifeſtum eſt compoſitis tribus, quatuorve, aut plu<lb></lb>ribus Facultatibus, Rationem Compoſitam motus potentiæ ad <lb></lb>motum Ponderis fieri multò majorem; cui ſi æqualem Ratio<lb></lb>nem habere velimus unicâ atque ſimplici Facultate, hujus <lb></lb>magnitudinem aliquando enormem fieri neceſſe eſſet; ut ſuis <lb></lb>locis infrà declarabitur. </s> </p> <p type="main"> <s id="s.001496">In eo igitur elucebit Machinatoris induſtria, ſi Facultates <lb></lb>ipſas aptè congruenterque diſponat, atque permiſceat, ſpecta<lb></lb>tâ materiæ ſoliditate, ſpatij amplitudine, Ponderis poſitione, <lb></lb>Potentiæ virtute, temporis ad movendum conceſſi opportuni<lb></lb>tate: hæc enim omnia attentiſſimè perpendenda ſunt; ne, dum <lb></lb>nimis ſollicitè laborem imminuere ſtudet, motum plus æquo <lb></lb>imminuens, tardioremque efficiens temporis jacturam faciat, <lb></lb>aut totum ſpatium machina implens in eas anguſtias Potentiam <lb></lb>moventem conjiciat, ut motum expeditè perficere nequeat. <lb></lb></s> </p> <p type="main"> <s id="s.001497"><emph type="center"></emph>CAPUT VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001498"><emph type="center"></emph><emph type="italics"></emph>Cur majores Rotæ motum juvent præ minoribus.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001499">ONera ſi ex alio in alium locum deportanda fuerint, gemi<lb></lb>no labore opus eſt, conatu videlicet, quo ſuſtineantur, <lb></lb>& impetu, quo transferantur: proptereà ſatius eſt ita res diſpo<lb></lb>nere, ut vires omnes ad transferendum exerceantur, citrà co<lb></lb>natum ſuſtinendi; ut eâ ratione vel gravius onus vel idem mul-<pb pagenum="201" xlink:href="017/01/217.jpg"></pb>tò faciliùs à potentia moveatur, quàm ſi ea illud ſuſtinere <lb></lb>pariter atque transferre cogeretur. </s> <s id="s.001500">Quoniam verò (cum one<lb></lb>ra ſubjecto plano impoſita illud premant, atque tùm onerum <lb></lb>tùm ſubjecti plani facies, quæ ſe invicem contingunt, non ita <lb></lb>læves ſint, ut partes omnes in rectum directæ nihil habeant <lb></lb>aſperitatis; quin immò ut plurimum, & ſalebris impedita via <lb></lb>ſit, & movendi corporis partes aliæ præ aliis extent atque emi<lb></lb>neant) ex mutuo prominentium particularum tritu atque con<lb></lb>flictu difficultas ad movendum oriretur; idcircò optimo conſi<lb></lb>lio factum eſt, ut oneribus ipſis ſubjiciantur Cylindri aut Rotæ, <lb></lb>quæ dum in gyrum aguntur, conflictum illum partium tollunt, <lb></lb>qui vitari non poſſet, ſi onera ſuper plano raptarentur. </s> <s id="s.001501">Hinc Ci<lb></lb>ſia, Sarraca, Vehes, Carri & genus omne plauſtrorum. </s> <s id="s.001502">Id quod <lb></lb>etiam homines ipſi, ut terreſtre iter commodiùs habeant, & <lb></lb>minori jumentorum labore illud perficiant, quàm ſi iis inſi<lb></lb>dentes veherentur, ſuos in uſus retulerunt: Hinc Belgæ ſua <lb></lb>eſſeda, Galli petorita & rhedas, Hiſpani pilenta, Itali carpen<lb></lb>ta; & pro ſuâ quiſque voluntate diverſa vehiculorum genera <lb></lb>excogitârunt, quæ ſubjectis rotis aguntur: dum enim Rota <lb></lb>convertitur, ejuſque curvaturæ partes aliis atque ſubinde aliis <lb></lb>ſubjectæ planitiei partibus aptantur, adeóque currus promove<lb></lb>tur, ſolus rotæ modiolus axis ambitum axungiâ lubricum terit; <lb></lb>ex quo tritu aut nulla aut levis mora motui infertur. </s> </p> <p type="main"> <s id="s.001503">Illud autem eſt omnibus exploratiſſimum, & quotidiano ex<lb></lb>perimento confirmatum, quo majoribus rotis inſtructi currus <lb></lb>(niſi diſcrimen aliquod in cæteris intercedat) multò faciliùs <lb></lb>trahuntur, paſſimque obſervatur Romæ in vulgaribus illis vehi<lb></lb>culis (ab antiquis Ciſiis aut parum aut nihil diſtant) quæ cum <lb></lb>ex celeberrimi Architecti Bonarotæ præſcripto duas ingentes <lb></lb>rotas habeant, tantis ponderibus onuſta cernuntur, ut miracu<lb></lb>lo proximum videatur ab unico equo tam ingentia onera trahi <lb></lb>poſſe: id quod alibi neutiquam fieri poteſt, ubi minoribus Rotis <lb></lb>vehicula hujuſmodi inſtructa longè minoribus oneribus defe<lb></lb>rendis paria ſunt, ſi unicus equus adhibeatur. </s> </p> <p type="main"> <s id="s.001504">Hujus rei cauſam indaganti acquieſcendum non eſt iis, qui <lb></lb>illam ex rationibus Vectis petendam eſſe exiſtimant, perinde <lb></lb>atque ſi rotæ majoris ſemidiameter eſſet longior Vectis, mino<lb></lb>ris verò brevior; ac proptereà majore rotâ faciliùs moveretur <pb pagenum="202" xlink:href="017/01/218.jpg"></pb>vehiculum onuſtum, quàm minore, quia & longiore vecte fa<lb></lb>ciliùs pondera moventur, quàm breviore. </s> <s id="s.001505">Hoc, inquam, 1/4 <lb></lb>veritate abeſſe palam fiet, ſi animadvertamus potentiam tra<lb></lb>hentem medio temone applicatam eſſe axi, cui pariter axi in<lb></lb>nititur onus; atque adeò tùm onus tùm Potentiam concipi <lb></lb>quaſi in Rotæ centro, cujus ſemidiametri altera extremitas hy<lb></lb>pomochlij punctum deſignaret. </s> <s id="s.001506">Atqui Vectis, in quo Potentia <lb></lb>& onus ab hypomochlio eandem aut æqualem diſtantiam ha<lb></lb>bent, parùm aut nihil habet utilitatis: immò in Vecte, quâ <lb></lb>vectis eſt, tria puncta diverſa tribuenda ſunt Potentiæ, oneri, <lb></lb>& Hypomochlio, ut infrà, ubi de Vecte diſputabitur: in Rotâ <lb></lb>autem duo tantummodo puncta conſiderantur, ſcilicet cen<lb></lb>trum & ſemidiametri extremitas. </s> <s id="s.001507">Igitur in Rotâ ratio Vectis <lb></lb>non invenitur, ideóque neque major Rota accipienda eſt qua<lb></lb>ſi longior Vectis. </s> <s id="s.001508">Aliundè itaque petendam eſſe cauſam, cur <lb></lb>majores rotæ præ minoribus motum juvent, manifeſtum eſt. </s> </p> <p type="main"> <s id="s.001509">Et primùm quidem, quod ad moram illam attinet, quæ ex <lb></lb>modioli Rotæ atque axis tritu oritur, eam minorem eſſe in ma<lb></lb>joribus Rotis, ſatis conſtàt, ſi attendamus axis craſſitiem, non <lb></lb>Rotæ magnitudini reſpondere, ſed oneris gravitati, quam opus <lb></lb>eſt ſuſtinere; quapropter axi ſatis valido pro ratione ponderis <lb></lb>ſuſtinendi parùm refert, utrùm Rota, cujus radij bipalmares <lb></lb>ſint, an verò tripalmares, infigatur: manente igitur codem axe <lb></lb>aut major, aut minor Rota vehiculo ſubjici poteſt. </s> <s id="s.001510">Sed quo<lb></lb>niam Rota major, cujus diameter ſeſquialtera eſt minoris, dum <lb></lb>converſionem unam perficit, ſpatium quoque ſeſquialterum <lb></lb>decurrit, eumdem tamen axem, quem minor Rota, terit, hinc <lb></lb>fit, per 8. lib. 5. eumdem axis ambitum ad majoris Rotæ peri<lb></lb>metrum (hoc eſt ad ejus motum) minorem habere rationem <lb></lb>quàm ad perimetrum minoris Rotæ (hoc eſt ad minorem mo<lb></lb>tum) atque adeò tritus ille modioli, & axis minùs impedit ma<lb></lb>jorem motum quàm minorem. </s> </p> <p type="main"> <s id="s.001511">Deinde, ut cap.16. lib.1. ſubindicatum eſt ſuperiùs, majo<lb></lb>res rotæ efficiunt, ut axis magis à terrâ diſtet; ac proinde te<lb></lb>mo, cui alligatus eſt equus, vel ſubjecto plano parallelus eſt, <lb></lb>vel minimùm à paralleliſmo recedit: ex quo fit tractionem aut <lb></lb>parallelam eſſe, aut ſaltem minùs obliquam, quam ſi Rota mi<lb></lb>nor eſſet, & axis depreſſior: quò autem minor eſt tractionis <pb pagenum="203" xlink:href="017/01/219.jpg"></pb>obliquitas, minorem quoque eſſe trahendi difficultatem loco <lb></lb>citato explicatum eſt. </s> </p> <p type="main"> <s id="s.001512">Ad hæc viarum aſperitatem impedimento eſſe nemo neſcit; <lb></lb>offendicula autem, in quæ vehiculorum Rotæ incurrunt, ma<lb></lb>gis obſiſtere minori Rotæ, quàm majori, facilè oſtenditur; hîc <lb></lb>enim pariter (id quod de magnitudinibus demonſtrat Eucli<lb></lb>des lib. 5. prop. 8.) idem majorem habet Rationem ad minus, <lb></lb>quàm ad majus. </s> <s id="s.001513">Nam ſi <lb></lb><figure id="id.017.01.219.1.jpg" xlink:href="017/01/219/1.jpg"></figure><lb></lb>Rotæ minoris ſemidiame<lb></lb>ter CB fuerit, majoris au<lb></lb>tem CD, & in planis pa<lb></lb>rallelis BA, DE volvantur, <lb></lb>ut impedimentum ſimile ſi<lb></lb>militerque poſitum inve<lb></lb>nient, multò majus eſſe <lb></lb>oportet illud, quod majori <lb></lb>Rotæ objicitur, quàm quod <lb></lb>minori. </s> <s id="s.001514">Sit enim minoris <lb></lb>offendiculum GI; ducatur <lb></lb>ex centro per I recta, quæ <lb></lb>ſit CIE ſecans majoris Rotæ peripheriam in H: erit igitur ar<lb></lb>cus IB ſimilis arcui HD, & ille quidem minor, hic verò ma<lb></lb>jor, ut manifeſtum eſt. </s> <s id="s.001515">Ducatur in planum perpendicularis <lb></lb>HF, & hoc erit impedimentum majoris Rotæ ſimile impedi<lb></lb>mento minoris IG, nam ſimilem arcum à converſione circà <lb></lb>centrum cum plani contactu impedit; neceſſe quippe eſt Ro<lb></lb>tam majorem converti circà punctum H, ſicut & minorem cir<lb></lb>cà punctum I, ut tranſgrediantur obſiſtens offendiculum. </s> <lb></lb> <s id="s.001516">Porrò lineam HF majorem eſſe quàm IG ſic oſtenditur. </s> <s id="s.001517">Quo<lb></lb>niam AB & ED parallelæ ſunt, triangula CBA, & CDE <lb></lb>ſimilia ſunt: ergo per 4. lib.6. ut CB ad CD, hoc eſt ut CI <lb></lb>ad CH, ita CA ad CE; & permutando ut CI ad CA, ita <lb></lb>CH ad CE; & dividendo ut CI ad IA, ita CH ad HE: at <lb></lb>CI minor eſt quàm CH; igitur per 14. lib.5. etiam IA minor <lb></lb>eſt quàm HE. </s> <s id="s.001518">Item quia AB & ED ex hypotheſi parallelæ <lb></lb>ſunt, recta IE in illas incidens facit angulos IAG & HEF <lb></lb>æquales per 29. lib. 1. ſunt autem triangula IGA & HFE <lb></lb>rectangula ad G & F ex conſtructione; ſunt igitur ſimilia, & <pb pagenum="204" xlink:href="017/01/220.jpg"></pb>per 4. lib. 6. ut. </s> <s id="s.001519">IA ad IG, ita HE ad HF: quare cum ex <lb></lb>dictis IA minor ſit quàm HE, erit per 14.lib.5. etiam IG mi<lb></lb>nor quàm HF. </s> </p> <p type="main"> <s id="s.001520">Cum itaque HF major ſit quàm IG (aſſumptâ DM æqua<lb></lb>li ipſi IG, & ductâ perpendiculari MS, donec occurrat peri<lb></lb>phæriæ in S) inter Tangentem ED & arcum circuli ſtatuatur <lb></lb>perpendicularis SL æqualis ipſi IG; & ex centro C ducatur <lb></lb>per S recta CO. </s> <s id="s.001521">In triangulo igitur CEO angulus internus <lb></lb>E, per 16. lib. 1; minor eſt externo SOL; igitur etiam angu<lb></lb>lus SOL major eſt quàm IAG: adde utrique angulum <lb></lb>rectum, ergo duo SLO, SOL ſimul majores ſunt duobus <lb></lb>IGA, IAG ſimul; ac propterea etiam externus LSC major <lb></lb>eſt externo GIC per 32.lib.1. Quapropter ſemidiameter CS <lb></lb>obliquior incidit in offendiculum SL, quàm ſemidiameter CI <lb></lb>incidat in æquale offendiculum IG: minùs igitur impeditur <lb></lb>Rotæ majoris converſio, quàm minoris, quippe cui minus di<lb></lb>rectè opponatur æquale offendiculum. </s> </p> <p type="main"> <s id="s.001522">Præterea cum trahendi difficultas hinc oriatur, quòd Rota <lb></lb>incurrens in obſtantem lapidem, aut quid ſimile, jam non cir<lb></lb>cà ſuum centrum convoluta aptatur ſubjecto plano, ſed, dum <lb></lb>Rota adhæret atque inſiſtit offendiculo; neceſſe eſt plauſtrum <lb></lb>cum impoſito onere elevari pro objecti impedimenti altitudi<lb></lb>ne; faciliùs ab eâdem Potentia elevatur plauſtrum onuſtum, ſi <lb></lb>major fuerit Rota, quàm ſi minor, quia videlicet motus Poten<lb></lb>tiæ ad eandem elevationem majorem habet Rationem in Ro<lb></lb>tâ majore quàm in minore, cum illâ enim plus movetur, <lb></lb>quàm cum iſtâ. </s> <s id="s.001523">Sit majoris Rotæ impedimentum LS pla<lb></lb>nè æquale impedimento GI minoris; producatur perpendicu<lb></lb>laris LS in T, & perpendicularis GI in V: tùm intervallo SC <lb></lb>deſcribatur arcus CT, & intervallo IC deſcribatur arcus CV. </s> <lb></lb> <s id="s.001524">Certum eſt in motu Rotæ majoris propter obicem LS manente <lb></lb>puncto S transferri centrum C in T, ita ut ST ſit Rotæ ſemi<lb></lb>diameter æqualis ſemidiametro CD, & ſimiliter in motu Rotæ <lb></lb>minoris propter offendiculum GI manente puncto I transferri <lb></lb>centrum C in V, ita ut IV æqualis ſit ſemidiametro CB. </s> <s id="s.001525">Quo<lb></lb>niam verò CD, VG, TL ad angulos rectos ſubjecto plano in<lb></lb>ſiſtunt, & parallelæ ſunt, anguli alterni VIC, ICB æquales <lb></lb>ſunt per 29. lib.1, eorumque menſuræ, arcus videlicet VC & <pb pagenum="205" xlink:href="017/01/221.jpg"></pb>IB, æquales ſunt; & ob eandem Rationem anguli alterni <lb></lb>TSC, SCD, eorumque menſuræ arcus TC & SD, ſunt <lb></lb>æquales. </s> <s id="s.001526">Atqui arcus SD major eſt quàm IB; igitur & arcus <lb></lb>TC major eſt quàm VC; hi autem arcus TC & VC reſpon<lb></lb>dent motui Potentiæ trahentis: longiore igitur ac majore mo<lb></lb>tu Potentiæ fit eadem elevatio, ac proinde faciliùs in Rotâ ma<lb></lb>jore quàm in minore. </s> <s id="s.001527">Porrò arcum SD majorem eſſe arcu IB, <lb></lb>magiſque diſtare punctum S à puncto D, quàm punctum I à <lb></lb>puncto B, illicò manifeſtum fiet, ſi duos circulos datis duobus, <lb></lb>æquales deſcripſeris ſe intùs contingentes, & ad contactüs <lb></lb>punctum lineam Tangentem duxeris, quocumque enim poſito <lb></lb>minoris circuli offendiculo inter Tangentem, & circulum mi<lb></lb>norem interjecto, illud idem offendiculum longiùs à con<lb></lb>tactûs puncto removendum videbis, ut inter Tangentem <lb></lb>eandem, & circulum majorem interjici poſſit: Id quod adeò <lb></lb>manifeſtum eſt, ut non ſit in eo explicando diutiùs immo<lb></lb>randum. </s> </p> <p type="main"> <s id="s.001528">Quòd ſi ad calculos rem hanc curiosiùs revocare libeat, ſic <lb></lb>ex gr. Rotæ minoris ſemidiameter CA pedum duorum, ſcilicet <lb></lb>digitorum 32, offendiculi verò DE <lb></lb><figure id="id.017.01.221.1.jpg" xlink:href="017/01/221/1.jpg"></figure><lb></lb>altitudo digitorum 4. Cum igitur <lb></lb>FD & CA parallelæ ſint, ſicut & <lb></lb>FC ac DA per 34. lib. 1. FD & <lb></lb>CA æquales ſunt, remanetque EF <lb></lb>digit.28, & eſt Sinus anguli FCE, <lb></lb>quo cognito innoteſcit complemen<lb></lb>tum, arcus ſcilicet quæſitus EA. </s> <lb></lb> <s id="s.001529">Fiat itaque ut CE ad EF, hoc eſt <lb></lb>ut 32 ad 28, ſeu ut 8 ad 7, ita <lb></lb>100000. Radius ad 87500 Sinum <lb></lb>arcûs gr.61. 2′ 42″; erit enim quæſi<lb></lb>tus arcus EA gr. 28. 57′ 18″. </s> <s id="s.001530">Jam verò poſitâ ſemidiametro <lb></lb>CA digitorum 32, fiat ut 113 ad 355, ita data ſemidiameter <lb></lb>digit. </s> <s id="s.001531">32 ad ſemiperipheriam circuli digitorum ferè 100 1/2, ſci<lb></lb>licet 100. 53″: ergo arcus EA eſt proximè digitorum 16. </s> </p> <p type="main"> <s id="s.001532">At Rotæ majoris ſemidiameter BA ſit ſeſquialtera (quic<lb></lb>quid ſit quòd figura ſolùm exprimat ſeſquiquartam) pedum <lb></lb>ſcilicet trium, hoc eſt digitorum 48, & offendiculum GH <pb pagenum="206" xlink:href="017/01/222.jpg"></pb>pariter digit. </s> <s id="s.001533">4. Quare HI eſt digit. </s> <s id="s.001534">44 Sinus anguli IBH, <lb></lb>ex quo innoteſcet arcus complementi HA. </s> <s id="s.001535">Fiat ut BH 48 ad <lb></lb>HI 44, ſeu ut 12 ad 11, ita Radius 100000 ad 91666 Sinum <lb></lb>arcûs gr. 66. 26′. </s> <s id="s.001536">33″; & eſt quæſitus arcus HA gr.23.33′.27″. </s> <lb></lb> <s id="s.001537">Jam ſit ut 113 ad 355, ita ſemidiameter 48 ad ſemiperiphe<lb></lb>riam digitorum 150 4/5 ferè: igitur arcus HA eſt proximè di<lb></lb>gitorum 20. Cum itaque dum onus elevatur ut 4, Potentia <lb></lb>in minore Rotâ moveatur ut 16, in majore autem ut 20 <lb></lb>(ut paulò ſuperiùs oſtenſum eſt motum centri æqualem eſſe <lb></lb>arcubus EA, & HA) facilitas movendi, quæ hinc oritur, erit <lb></lb>ut 5 ad 4. </s> </p> <p type="main"> <s id="s.001538">Ex his manifeſtum eſt, in vehiculis, quæ quatuor rotis <lb></lb>inſtruuntur, quarum binæ, priores minores ſunt, poſteriores <lb></lb>verò majores, faciliùs ſuperari impedimenta à poſterioribus <lb></lb>rotis quàm à prioribus, ac propterea minori labore currum ab <lb></lb>equis trahi, quàm ſi poſteriores prioribus eſſent æquales. </s> <s id="s.001539">Id <lb></lb>quod opportunè factum eſt, quia ut plurimum (quemadmo<lb></lb>dum in antiquioribus Rhedis viatoriis cernere eſt) in poſte, <lb></lb>riorem potiùs, quàm in anteriorem currus partem, onus reji<lb></lb>citur, atque adeò poſterior axis magis premitur: quæren<lb></lb>dum igitur fuit aliquod laboris compendium. </s> <s id="s.001540">Quamquam <lb></lb>non negarim alio prorſus conſilio primùm excogitatam hanc <lb></lb>Rotarum inæqualitatem; ut nimirum onus conſtitutum quaſi <lb></lb>in plano trahentem versùs inclinato, faciliùs quoque illum <lb></lb>ex impreſſo anterioris tractionis impetu ſequeretur, ſi in pla<lb></lb>nitie quidem tractio fieret; ubi verò ſuperandus eſſet clivus, <lb></lb>ut minùs adversùs trahentem repugnaret onus ſe ipſum in <lb></lb>proclive urgendo; nam ſi Rotæ æquales eſſent, longè faciliùs <lb></lb>vehiculum in poſteriora relaberetur, pro ipſius clivi inclina<lb></lb>tione, cui parallelum eſſet planum oneri ſubjectum inſiſtens <lb></lb>axibus æqualium Rotarum: at Rotis inæqualibus poſitis, & <lb></lb>poſterioribus quidem majoribus, planum, cui onus incumbe<lb></lb>re intelligitur à poſteriori axe ad anteriorem deductum minùs <lb></lb>inclinatur, quàm collis proclivitas ferat; ac propterea trahen<lb></lb>tibus equis minùs repugnat. </s> <s id="s.001541">Licèt autem non ſemper aſcen<lb></lb>dendum ſit in colles & clivos, quorum aſcenſus manifeſtè ar<lb></lb>duus eſt atque difficilis, rarò tamen, aut ferè nunquam, adeò <lb></lb>æquata eſt viarum planities, quin leviter ſaltem inflexæ modò <pb pagenum="207" xlink:href="017/01/223.jpg"></pb>aſcendere cogant, modò deſcendere: in quâ aſcenſuum atque <lb></lb>deſcenſuum viciſſitudine non modicè utilis eſt illa Rotarum <lb></lb>inæqualitas. </s> </p> <p type="main"> <s id="s.001542">Hinc manualia illa curricula (ſeu ruſticæ vehes) quæ binis <lb></lb>brachiis inſtructa unicam habent in anteriore parte rotam & <lb></lb>ſublevatis brachiis converſa Rotâ promoventur, faciliùs <lb></lb>conſtrui poſſent, ſi propè vectorem duæ eſſent Rotæ majores <lb></lb>illâ anteriore Rotâ, ita ut harum diameter triplex eſſet diame<lb></lb>tri illius: hunc enim unicus homo multò majus pondus trans<lb></lb>ferre poteſt vel impellendo, cùm in planitie eſt, aut clivum <lb></lb>aſcendit, vel trahendo, cùm ex declivi deſcendit; levatur ſi<lb></lb>quidem labore ſuſtinendi, & omnes vires exercet impellendo <lb></lb>aut trahendo; & illa Rotarum inæqualitas in causâ eſt, cur fa<lb></lb>ciliùs impellatur pondus versùs illam partem, in quam incli<lb></lb>natur. </s> </p> <p type="main"> <s id="s.001543">Et quoniam in Rotarum inæqualium mentionem incidi, il<lb></lb>lud hîc pariter obſervandum videtur, commodiùs currum mo<lb></lb>veri, cùm anteriores Rotæ à poſterioribus aliquantulùm diſtant, <lb></lb>quàm cùm valdè vicinæ ſunt (ubi tamen reliqua omnia paria <lb></lb>fuerint, neque aliud præter Rotarum diſtantiam, intercedat <lb></lb>diſcrimen) ſi in planitie quidem, & viâ minimum flexuosâ de<lb></lb>ducendus ſit. </s> <s id="s.001544">Quia nimirum quo propiores fuerint axes, pla<lb></lb>num, cui onus incumbit, magis inclinatur, ac propterea an<lb></lb>teriores Rotas premens adversùs ſubjectam tellurem minus <lb></lb>obliquè conatur, ideóque pondus illam validiùs urgens majo<lb></lb>rem creat movendi difficultatem: contrà verò ſi axes invicem <lb></lb>paulò remotiores fuerint, minùs inclinato plano, minor eſt <lb></lb>priorum rotarum preſſus in ſubjectam tellurem. </s> <s id="s.001545">Sic ſi Rotæ <lb></lb>fuerinc A & B, pla<lb></lb><figure id="id.017.01.223.1.jpg" xlink:href="017/01/223/1.jpg"></figure><lb></lb>num, cui onus inſi<lb></lb>det, eſt AB, at ſi Ro<lb></lb>tæ fuerint A & C, <lb></lb>planum eſt AC, quod <lb></lb>utique minùs incli<lb></lb>natum eſt, magiſque <lb></lb>accedit ad paralleliſmum cum Horizonte DE, atque adeò <lb></lb>Rota B magis terram premit, quàm Rota C. </s> <s id="s.001546">Si enim in utro<lb></lb>que plano pondus fuerit ſimiliter poſitum (puta circà me-<pb pagenum="208" xlink:href="017/01/224.jpg"></pb>dium) linea directionis à centro gravitatis ponderis ducta ca<lb></lb>det ad angulos magis inæquales in planum AB magis inclina<lb></lb>tum, quàm in AC minùs inclinatum, atque momentum gra<lb></lb>vitatis ponderis magis accedet ad B quàm ad C, ut infrà ſuo <lb></lb>loco explicabitur, & ſubindicatum eſt ſuperiùs lib.1. cap. 14. <lb></lb>§. <emph type="italics"></emph>Ex his fieri poteſt.<emph.end type="italics"></emph.end></s> <s id="s.001547"> Hinc Hamburgenſia plauſtra, quibus <lb></lb>merces Hamburgo Norimbergam devehuntur, longiora ſunt, <lb></lb>quia nec altiores clivi in itinere frequentes occurrunt, nec <lb></lb>anguſtæ ſunt viarum flexiones, ex quibus oriatur aut aſcen<lb></lb>dendi, aut plauſtrum inflectendi difficultas. </s> <s id="s.001548">Quare illis & <lb></lb>majora onera imponi poſſunt, & ſex equi non bini & bini, ſed <lb></lb>ſinguli recto ordine adjunguntur; quo fit ut non in diverſa <lb></lb>trahentes, omninò ſimili impetu currum deducant. </s> <s id="s.001549">Quòd ſi <lb></lb>viæ plus haberent difficultatis tùm ex clivis, tùm ex flexioni<lb></lb>bus, non expediret tàm longa plauſtra conſtruere, nec equos <lb></lb>tam longâ ſerie diſponere, ut cuique rem vel leviter conſide<lb></lb>ranti ſtatim patebit. <lb></lb> </s> </p> <p type="main"> <s id="s.001550"><emph type="center"></emph>CAPUT IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001551"><emph type="center"></emph><emph type="italics"></emph>Quid Cylindri & Scytalæ ad faciliorem ponderis <lb></lb>motum præſtent.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001552">ADeò ingentia aliquando pondera transferenda proponun<lb></lb>tur, ut ea carris imponere tranſvehenda aut nimis opero<lb></lb>ſum ſit, aut periculo non vacet, ne rotarum axes pondere præ<lb></lb>gravati diffringantur, aut propter ſoli mollitudinem rotæ de<lb></lb>vorentur: propterea rationem aliquam inire oportet, quâ voti <lb></lb>compotes ſimus, citrà hujuſmodi pericula. </s> <s id="s.001553">Et quidem ſi cor<lb></lb>pus teres ſit, nec viarum ſalebræ, aut anguſtiæ impedimento <lb></lb>ſint, ipſum verſari in gyrum poterit ſimili artificio, quo ad <lb></lb>deportandos Epheſum ex lapicidinis ſcapos columnarum cen<lb></lb>tum viginti ſeptem altitudine pedum ſexaginta uſus eſt Cteſi<lb></lb>phon Gnoſſius (ſic eum vocat Plinius lib. 7. cap. 37. cum Vi<lb></lb>truvio lib.10. cap. 6, quem tamen idem Plinius lib.36. cap.14. <pb pagenum="209" xlink:href="017/01/225.jpg"></pb>cum Strabone vocat Cherſiphronem) celeberrimo Dianæ <lb></lb>templo conſtruendo præfectus, & quidem felici eventu: ca<lb></lb>pitibus enim ſcaporum, ubi axis extremitates deſinebant, ſub<lb></lb>ſcudis in modum inſeruit, atque implumbavit ferreos axes: <lb></lb>tùm de materiâ trientali ſcapos (hoc eſt ligneos tigillos craſſi<lb></lb>tudinis unciarum quatuor pedis, ſeu pollicum quatuor) duos <lb></lb>longiores juxtà columnæ longitudinem, duoſque breviores <lb></lb>tranſverſarios ita compegit, ut parallelogrammum conſtituen<lb></lb>tes columnam poſſent complecti; mediiſque tranſverſariis <lb></lb>ferreas armillas inſeruit, quibus axes ferrei infigebantur, <lb></lb>adeò ut liberè verſari poſſent, cum boves traherent; quem<lb></lb>admodum & in gyrum volvuntur cylindri marmorei aut la<lb></lb>pidei, quorum uſus eſt in exæquandis ambulationibus. </s> <s id="s.001554">Eſt <lb></lb>autem maximè veriſimile, & probabile, ita firmiter <lb></lb>ligneum illud parallelogrammum fuiſſe compactum, ut non <lb></lb>ſolùm extremis tranſverſariorum capitibus anterioribus alli<lb></lb>gari poſſent boves; ſed etiam per totam anterioris ſcapi lon<lb></lb>gitudinem diſtribui, ut faciliùs columna transferretur. </s> </p> <p type="main"> <s id="s.001555">Proſperum exitum conſecuta ſcaporum vectura animum <lb></lb>adjecit Methageni Cteſiphontis filio, ut paternam in<lb></lb>duſtriam æmularetur in Epiſtyliis vehendis: cum enim ho<lb></lb>rum figura non ea eſſet, quæ perinde atque cylindrica vol<lb></lb>vi poſſet, duabus rotis pedum circiter duodenûm ſingula <lb></lb>epiſtylia firmiter incluſit; rotarumque centris ferreos axes <lb></lb>infixit, qui in armillis ſimilem haberent verſationem, ac <lb></lb>dictum eſt in ſcaporum vecturâ. </s> <s id="s.001556">Cum enim boves ligneo <lb></lb>parallelogrammo alligati traherent, Rotæ volvebantur, at<lb></lb>que cum illis pariter epiſtylia Rotis cohærentia in gyrum <lb></lb>verſabantur; quippe quæ in ſubjectum ſolum non incurre<lb></lb>bant, cum ſolæ Rotæ terram attingerent. </s> <s id="s.001557">Hâc methodo <lb></lb>corporibus, quæ non ſunt ad volubilitatem rotundata, faci<lb></lb>lem conyerſionem conciliare poſſumus; ex Rotis nimirum & <lb></lb>pondere moles una compingitur, cujus extremitatibus cylin<lb></lb>dricis tota innititur, nihilque refert, cujus demum figuræ ſit <lb></lb>pars media, ſcilicet pondus, modò hæc à ſolo aliquantulum <lb></lb>diſtans motum non impediat. </s> <s id="s.001558">Quâ autem ratione aut Rotæ <lb></lb>conſtruantur, aut illis onus includatur, artificis ſeu architecti <lb></lb>ſolertiæ relinquitur. </s> </p> <pb pagenum="210" xlink:href="017/01/226.jpg"></pb> <p type="main"> <s id="s.001559">Methagenis artificium imitatus Paconius, teſte Vitruvio <lb></lb>lib. 10. cap. 6. lapideam baſim longam pedes duodecim, la<lb></lb>tam pedes octo, & altam pedes ſex Apollinis coloſſo reſti<lb></lb>tuendam, duabus Rotis pedum circiter quindecim, ſimili<lb></lb>ter incluſit: ſed aliâ ratione ac Methagenes deducere ſtatuit. </s> <lb></lb> <s id="s.001560">A Rotâ ad Rotam circâ lapidem fuſos ſextantales, hoc eſt <lb></lb>craſſitudinis pollicum duorum, ad circinum compegit ita, ut <lb></lb>fuſus à fuſo non diſtaret pedem unum. </s> <s id="s.001561">Tùm circà fuſos fu<lb></lb>nem involvit, qui bobus trahentibus explicabatur, & con<lb></lb>vertebantur Rotæ. </s> <s id="s.001562">Verùm quia funis circumvoluti ſpiræ ad <lb></lb>unam, aut ad alteram partem ſpectabant, non poterat viâ <lb></lb>rectâ ad lineam deduci moles illa; ſed modò in hanc, mo<lb></lb>dò in illam partem deflectebat, ut opus eſſet retroducere, <lb></lb>adeò ut ducendo & reducendo pecuniam contriverit, & ope<lb></lb>ram luſerit Paconius. </s> <s id="s.001563">Potuiſſet tamen huic malo occurrere, <lb></lb>nec ſui inventi laude fraudari, ſi circà fuſos non unicum, <lb></lb>ſed duplicem funem ita involviſſet, ut funium ſpiris vel ab <lb></lb>extremitatibus fuſorum, vel à medio, incipientibus, funis <lb></lb>uterque paribus ſemper intervallis à ſibi proximâ Rotâ diſta<lb></lb>rent; ſic enim factum fuiſſet, ut boves æqualiter utrumque <lb></lb>funem trahentes, æqualiterque evolventes, molem illam rectâ <lb></lb>viâ deducerent. </s> </p> <p type="main"> <s id="s.001564">Quamquam autem ſuâ laude non careant hujuſmodi arti<lb></lb>ficum inventa, expeditiſſimè tamen, & citrà impendium, one<lb></lb>ra ingentia traducuntur ſubjectis cylindris, qui pondere preſſi, <lb></lb>cùm illud trahitur, convertuntur. </s> <s id="s.001565">Palangas peculiari voca<lb></lb>bulo Veterès dixere freſtes teretes, qui navibus ſubjiciuntur, <lb></lb>cùm attrahuntur ad pelagus, vel cùm ad littora ſubducuntur; <lb></lb>ut apud Nonium Marcellum legiſſe me memini. </s> <s id="s.001566">Neque aliud <lb></lb>quidpiam cenſendus eſt Cæſar intellexiſſe, ubi lib. 3. Belli <lb></lb>Civil. </s> <s id="s.001567">ſcribit <emph type="italics"></emph>Quatuor biremes ſubjectis ſcutulis<emph.end type="italics"></emph.end> (fortaſſe <emph type="italics"></emph>ſcuta<lb></lb>lis<emph.end type="italics"></emph.end>; hoc eſt <emph type="italics"></emph>ſcytalis,<emph.end type="italics"></emph.end> antiquis enim Romanis <emph type="italics"></emph>is<emph.end type="italics"></emph.end> literam uſupari <lb></lb>ſolitam. </s> <s id="s.001568">loco <emph type="italics"></emph>y<emph.end type="italics"></emph.end> literæ Græcæ notum eſt) <emph type="italics"></emph>impulſas vectibus in <lb></lb>interiorem partem tranſduxit.<emph.end type="italics"></emph.end></s> <s id="s.001569"> Sunt autem ſcytalæ ut apud Sui<lb></lb>dam, rotunda & polita ligna: aliquid tamen peculiare. </s> <s id="s.001570">ad<lb></lb>dit Ariſtoteles in Mechan. quæſt. </s> <s id="s.001571">11. quærens, <emph type="italics"></emph>cur ſuper ſcy<lb></lb>talas faciliùs portantur onera quàm ſuper currus, cum tamen ij <lb></lb>magnas habeant rotas, illæ verò puſillas<emph.end type="italics"></emph.end>? </s> <s id="s.001572">Scytalis nimirum pu-<pb pagenum="211" xlink:href="017/01/227.jpg"></pb>ſillas rotas adjectas intelligit, <lb></lb><figure id="id.017.01.227.1.jpg" xlink:href="017/01/227/1.jpg"></figure><lb></lb>non eas quidem circà axem, <lb></lb>ſed cum axe ipſo, cui adnectun<lb></lb>tur, verſatiles; cujuſmodi eſ<lb></lb>ſent in hoc ſchemate rotulæ A <lb></lb>& B cum ſuo axe connexæ. </s> </p> <p type="main"> <s id="s.001573">Porrò duplicem hujuſmodi ſcytalarum uſum conſidero: ſi <lb></lb>enim onus impoſitum incumbat Rotulis ipſis, vel quia plana <lb></lb>ſit ejus ſuperficies, vel quia tabulato fuerit ſuperpoſitum, <lb></lb>perinde res ſe habet, atque ſi cylindrus eſſet, cujus diameter <lb></lb>idem eſſet cum rotularum diametro: neque tunc admodum <lb></lb>refert, cujuſnam figuræ ſit axis, quem onus non tangit, ſi<lb></lb>ve rotundus ille ſit, ſive angulatus. </s> <s id="s.001574">At ſi onus ipſi axi in<lb></lb>cumbat, promineantque hinc & hinc rotulæ, omninò ne<lb></lb>ceſſe eſt axem rotundum eſſe, ut fieri poſſit rotularum con<lb></lb>verſio, atque ita longum, ut inter rotulas onus laxè interci<lb></lb>piatur; maximè quippe cavendum eſt, ne rotulæ onus con<lb></lb>tingant, alioquin ex mutuo conflictu mora non mediocris <lb></lb>motui crearetur. </s> <s id="s.001575">Ideò autem excogitatæ videntur hujuſmo<lb></lb>di ſcytalæ, ut minimâ ſui parte ſecundùm extremitates tan<lb></lb>gerent ſubjectum planum, atque adeò in pauciora incurre<lb></lb>rent offendicula, quàm cylindri totâ ſua longitudine incum<lb></lb>bentes plano. </s> <s id="s.001576">Sed illæ ab uſu artificum jam diù intermiſſæ <lb></lb>locum ſimplicibus cylindris conceſſere, quippe qui ob con<lb></lb>tinentem ſibique ſemper ſimilem figuram ſolidiores ſunt, & <lb></lb>periculo carent, cui obnoxiæ ſunt ſcytalæ, ne videlicet Ro<lb></lb>tulæ illæ labem aliquam faciant cum rotunditatis, atque adeò <lb></lb>etiam motûs, detrimento. </s> <s id="s.001577">Illud verò commodum, quod ex <lb></lb>offendiculorum evitatione oriebatur, obtinemus pariter, ſi <lb></lb>duplicem planorum tigillorum ſeriem ſubſternamus capitibus <lb></lb>cylindrorum; hinc enim fit, ut viarum ſalebræ evitentur, & <lb></lb>Cylindri modicâ ſui parte contingant ſubjectos tigillos, qui <lb></lb>viam planam & æquabilem conſtituentes moram nullam mo<lb></lb>tui injiciunt. </s> </p> <p type="main"> <s id="s.001578">Sed & in hoc cylindrorum uſu communiter cenſetur ali<lb></lb>quid ineſſe facilitatis majoris ad onera deducenda, quàm ſi <lb></lb>illa currui imponerentur; tùm quia currui ſua ineſt gravitas, <lb></lb>quæ unâ cum impoſitâ ſarcinâ majus onus conſtituit, ac <pb pagenum="212" xlink:href="017/01/228.jpg"></pb>propterea in utroque transferendo is, qui trahit, majorem <lb></lb>impendit laborem; at ſubjectis oneri cylindris, horum gra<lb></lb>vitas nihil officit trahenti: Tùm quia currûs Rotæ, cum ſint <lb></lb>circà ſuum axem, cui infiguntur, mobiles, aut hûc & illuc <lb></lb>nutant, ſi laxa ſint capita, nec clavo exquiſitè coërceantur, <lb></lb>aut ſi arctiùs axi cohæreant, axem quem complectuntur, & <lb></lb>clavum quo coërcentur, validiùs terunt; & ex utroque hoc <lb></lb>capite movendi difficultas oritur, cùm aliquid impreſſi im<lb></lb>petûs aut in illâ inconſtantiâ, aut in hoc conflictu contera<lb></lb>tur: nihil autem hujuſmodi cylindris contingit. </s> <s id="s.001579">Tùm etiam <lb></lb>quia Rotæ modiolus ab axe premitur, & deorſum pondere <lb></lb>urgente, & antrorſum impetu ad anteriora trahente; ex quo <lb></lb>quantum difficultatis in movendo oriatur, hinc manifeſtum <lb></lb>eſt, quod niſi axungiâ aut amurcâ illinantur curruum axes, <lb></lb>ægrè convertuntur rotæ, & denſo ſtridore, quantus ſit par<lb></lb>tium tritus atque conflictus, teſtatum faciunt. </s> <s id="s.001580">At Cylindri <lb></lb>quantumvis ab onere premantur, nullo pingui liquore obli<lb></lb>nendi ſunt, ut lubrici fiant; nulla enim impoſiti oneris aſpe<lb></lb>ritas cylindrorum converſionem impedire poteſt. </s> <s id="s.001581">Nam ſi fue<lb></lb>rit ingens lapis AB cylin<lb></lb><figure id="id.017.01.228.1.jpg" xlink:href="017/01/228/1.jpg"></figure><lb></lb>dris ſubjectis impoſitus, & <lb></lb>cylindri punctum C cen<lb></lb>gruat puncto A lapidis, dia<lb></lb>metri CD altera extremitas <lb></lb>D tangit ſubjectum planum; <lb></lb>cum verò ſaxum ex B ver<lb></lb>sùs A propellitur, ſeu tra<lb></lb>hitur ex A, ita cylindrus <lb></lb>convertitur, ut DF ar<lb></lb>cus ſenſim ad ſubjectum <lb></lb>planum, contrà verò arcus CE ad impoſitum ſaxum accom. </s> <lb></lb> <s id="s.001582">modetur, citrà omnem ſaxi & cylindri affrictum. </s> </p> <p type="main"> <s id="s.001583">Hinc tamen aliquid etiam incommodi cylindris adhæret, ſi <lb></lb>cum plauſtrorum rotis conferantur; hæ ſcilicet motum con<lb></lb>tinuant, cum ſine fine volvantur, quippe quæ axi infixæ, im<lb></lb>poſito oneri pariter, ut ita loquar, cohærent; illos verò, ni<lb></lb>mirum cylindros, onus dum promovetur, poſt ſe relinquit; ac <lb></lb>proinde aut cylindrorum copia non exigua ſuppetere debet, <pb pagenum="213" xlink:href="017/01/229.jpg"></pb>qui longâ ſerie diſpoſiti onus alij ex aliis excipiant, aut qui <lb></lb>relinquuntur, ſubinde transferendi ſunt, ut iterùm oneri <lb></lb>ſubjiciantur. </s> <s id="s.001584">Verùm hæc alterna cylindrorum tranſlatio non <lb></lb>adeò gravis eſt; quin plus habeat adjumenti, quàm incom<lb></lb>modi; cum enim plurimùm referat, utrùm qui ſubjicitur cy<lb></lb>lindrus, reliquis poſterioribus cylindris parallelus, an obli<lb></lb>quus ſtatuatur, ut onus ad lineam viâ rectâ deducatur, aut <lb></lb>motus ſui veſtigium inflectat; facillimum eſt opportunâ cylin<lb></lb>dri tranſlati collocatione parallelâ, aut obliqua, deſtinatum <lb></lb>oneris motum adminiſtrare. </s> </p> <p type="main"> <s id="s.001585">Illud autem non immeritò hîc examinandum occurrit, utrùm <lb></lb>majores cylindri minoribus potiores cenſendi ſint, & an præſtet <lb></lb>ſubjicere oneri cylindrum GI majorem, an verò minorem <lb></lb>GH. </s> <s id="s.001586">Et quidem ſi figuræ dumtaxat magnitudo atque parvi<lb></lb>tas ſpectetur, hoc unum diſcrimen invenio, quòd ad certam <lb></lb>motûs menſuram perficiendam crebriùs volvi oportet cylin<lb></lb>drum minorem, quàm majorem; onus verò à ſubjecto plano <lb></lb>diſtare majoris diametri GI intervallo potiùs, quàm minoris <lb></lb>GH, non video, quid conferat ad motûs facilitatem; tantum <lb></lb>enim promovetur onus, quantus eſt peripheriæ arcus, cui illud <lb></lb>in motu aptatur, eíque æqualis eſt arcus oppoſitus, qui plano <lb></lb>pariter in motu congruit: ac propterea parum refert, utrùm <lb></lb>eadem arcus menſura ſit majoris circuli pars minor, an minoris <lb></lb>circuli pars major. </s> </p> <p type="main"> <s id="s.001587">Verùm ſi qua inter motum occurrant offendicula, hæc <lb></lb>minùs officere majori cylindro, quàm minori, dicendum eſt, <lb></lb>quemadmodum & de rotis majoribus dictum eſt ſuperiori ca<lb></lb>pite; ſiquidem majoris cylindri diameter obliquior incidit in <lb></lb>idem offendiculum, quod minùs directè opponitur motui, & <lb></lb>longiore motu Potentiæ fit eadem ponderis elevatio, ut ibi ex<lb></lb>plicatum eſt. </s> </p> <p type="main"> <s id="s.001588">Aliud eſt præterea, nec ſanè nullius momenti, quod majo<lb></lb>ri cylindro incitatiorem dat volubilitatem; quòd videlicet <lb></lb>(quemadmodum & globo majori contingit) major cylindrus, <lb></lb>quamvis Geometricam Rotunditatem non aſſequatur, tamen <lb></lb>propiùs accedit ad figuram exquiſitè Rotundam, quàm mi<lb></lb>nor: ſi enim à circulo Geometricè perfecto æqualiter recedant <lb></lb>utriuſque cylindri majoris ac minoris baſes, non tamen æqua-<pb pagenum="214" xlink:href="017/01/230.jpg"></pb>liter angulata eſt utraque baſis, ſed in majori major eſt angu<lb></lb>lus, in minori minor, atque adeò ille magis, quàm hic, ad <lb></lb>rotunditatem accedit. </s> <s id="s.001589">In majori autem circulo angulum, qui <lb></lb>peripheriam complectitur, majorem eſſe palam eſt, quia idem <lb></lb>exceſſus majori Radio additus conſtituit ſecantem anguli mi<lb></lb>noris, quàm ſi minori Radio addatur; ac propterea angulus <lb></lb>Complementi major eſt in majori, quàm in minori. </s> <s id="s.001590">Id quod, <lb></lb>per ſe quidem ſatis clarum, dilucidiùs explicabitur, ſi ex mi<lb></lb><figure id="id.017.01.230.1.jpg" xlink:href="017/01/230/1.jpg"></figure><lb></lb>nore circulo extet particula, cu<lb></lb>jus altitudo ſit ON, ex majore <lb></lb>autem circulo æqualis altitudo <lb></lb>emineat IM. </s> <s id="s.001591">Ductis Tangen<lb></lb>tibus & Radiis, certum eſt Se<lb></lb>cantis exceſſum ON ſupra Ra<lb></lb>dium LO minorem, habere <lb></lb>majorem Rationem ad ſuum <lb></lb>Radium, quàm habeat æqualis <lb></lb>exceſſus IM ad ſuum Radium <lb></lb>LI majorem ex 8.lib.5. Eſt igi<lb></lb>tur MLP angulus minor angulo NLS, & Complementum <lb></lb>LMP majus eſt Complemento LNS quare totus angulus <lb></lb>VMP major eſt toto angulo TNS, ac proinde magis ad ro<lb></lb>tunditatem accedit. <lb></lb></s> </p> <p type="main"> <s id="s.001592"><emph type="center"></emph>CAPUT X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001593"><emph type="center"></emph><emph type="italics"></emph>Circulorum Concentricorum motus explicatur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001594">CIrculi motus, ob id ipſum quia circulus eſt, circa ſuum <lb></lb>centrum perficitut eâ ratione, ut ſuperiores partes pro<lb></lb>grediantur, inferiores retrocedant, anteriores deſcendant, <lb></lb>poſteriores aſcendant, ſervatâ ſemper pari oppoſitorum pro<lb></lb>greſsûs atque regreſsûs, deſcensûs atque aſcensûs menſurâ; <lb></lb>pro ut unicuique rem vel leviter conſideranti patet. </s> <s id="s.001595">Quare <lb></lb>dum in gyrum circulus agitur, centrum quidem manet, reli<lb></lb>quæ verò partes ita ſingulæ ex alio in alium locum ſibi invi-<pb pagenum="215" xlink:href="017/01/231.jpg"></pb>cem ſuccedentes commeant, ut circulus totus ſpatium, in quo <lb></lb>volvitur, omninò non mutet. </s> <s id="s.001596">Quemadmodum obſervare eſt <lb></lb>in Solis orbitâ, quam Eclipticam vocant; hæc enim diurnâ <lb></lb>converſione circa Mundi axem Solem ſecum rapiens à ſuo lo<lb></lb>co non recedit, Sole ab ortu in Occaſum commigrante: id <lb></lb>multò magis in ſingulorum circulorum circà ſua centra revo<lb></lb>lutione manifeſtum apparet. </s> <s id="s.001597">Quod ſi circulus aut horizonti <lb></lb>parallelus, aut illi ad perpendiculum inſiſtens, raptetur; mo<lb></lb>tus ille nihil habet circulari affine, cum circà centrum non <lb></lb>perficiatur, ſed ſingula circuli puncta ſolo motu recto unâ cum <lb></lb>centro moveantur. </s> </p> <p type="main"> <s id="s.001598">Sin autem axis circulo verſatili infixus trahatur, jam circu<lb></lb>lus & cum-axe pariter movetur, & circa axem volvitur: atque <lb></lb>adeò ſingularum circuli partium motus is eſt, qui ex recto cen<lb></lb>tri, & circulari ipſius orbitæ componitur. </s> <s id="s.001599">Hinc ſemicirculi <lb></lb>ſuperioris partes cum progrediantur versùs cumdem locum, ad <lb></lb>quem centrum tendit, ſuum motum motui centri addunt: <lb></lb>Contrà verò inferioris ſemicirculi partes retrocedentes ſuum <lb></lb>motum à centri motu detrahunt. </s> <s id="s.001600">Rotæ igitur puncta omnia, <lb></lb>dum currus trahitur, ſi non ſummatim tota revolutio, ſed par<lb></lb>ticulatim, accipiatur, non æquali velocitate moventur. </s> <s id="s.001601">Sit <lb></lb>explicandi gratiâ, <lb></lb><figure id="id.017.01.231.1.jpg" xlink:href="017/01/231/1.jpg"></figure><lb></lb>circulus BD AE, <lb></lb>cujus centrum C <lb></lb>moveatur verſus F, <lb></lb>& ſit tangens GA, <lb></lb>cui in motu appli<lb></lb>catur ipſius circu<lb></lb>li orbita; in quâ <lb></lb>accipiatur ſextans <lb></lb>hinc & hinc AD, <lb></lb>& AE. </s> <s id="s.001602">Igitur in <lb></lb>Converſione, dum <lb></lb>Centrum C trahitur ad F, punctum D venit in G, & arcus <lb></lb>DA æqualis eſt rectæ GA, cui in motu ſubinde per partes <lb></lb>congruit: atque adeò, quarum partium ſemidiameter CA <lb></lb>eſt 21, earum arcus AD, & recta AG eſt 22, & motus cen<lb></lb>tri illi æqualis CF eſt pariter 22. Quoniam verò in motu or-<pb pagenum="216" xlink:href="017/01/232.jpg"></pb>bitæ circa ſuum centrum, punctum A aſcendens in E retroce<lb></lb>dit juxta menſuram ſinûs SE (qui ad Radium CA 21 eſt ut 18) <lb></lb>hinc eſt poſt converſionem, in qua D eſt in G, punctum A <lb></lb>ita aſcendiſſe, ut ſit in lineâ HE parallelâ Tangenti GA, ſed <lb></lb>motui centri tantum detraxerit, quantus eſt ſinus SE. </s> <s id="s.001603">Quia <lb></lb>igitur Radius CD ubi congruit punctis FG, ſecat in H <lb></lb>rectam HE, ſumatur HI æqualis ſinui SE, & puncti A totus <lb></lb>progreſſus remanet SI partium 4, quarum SH, ſeu CF eſt 22. <lb></lb>Quare A eſt in I, quando D eſt in G. </s> </p> <p type="main"> <s id="s.001604">Contrà verò in ſuperiore ſemicirculo ſumatur item ex B <lb></lb>hinc, & hic ſextans BK & BL; atque in converſione ubi cen<lb></lb>trum C venerit in F, & punctum orbitæ D in G, erit K in O, <lb></lb>& diameter DK ſecabit parallelam KN in M. </s> <s id="s.001605">Igitur punctum <lb></lb>B ita deſcendit ad parallelam NK, ut motui centri CF, hoc <lb></lb>eſt BO ſeu RM, addiderit ſuum progreſſum juxta menſuram <lb></lb>RL Sinum Sextantis BL, hoc eſt 18. Venit igitur B in N; <lb></lb>atque additis RM 22, & MN 18, totus progreſſus puncti B <lb></lb>eſt RN 40. Comparatis itaque invicem curvis lineis AI & <lb></lb>BN, manifeſtum eſt puncta B & A non æque velociter mo<lb></lb>veri, cum eodem temporis ſpatio inæqualia loci ſpatia per<lb></lb>currant. </s> </p> <p type="main"> <s id="s.001606">Eadem erit methodus, ſi reliquorum orbitæ punctorum ve<lb></lb>locitates aut tarditates conſiderandæ ſint: ſi tamen adverteris <lb></lb>non eandem eſſe omnium circuli Quadrantum rationem in de<lb></lb>terminandâ menſura motûs addendi, aut demendi motui cen<lb></lb>tri. </s> <s id="s.001607">Nam in anteriori Quadrante ſuperioris ſemicirculi, & in <lb></lb>poſteriori Quadrante inferioris ſemicirculi, menſura progreſ<lb></lb>sûs addendi in illo, & regreſſus demendi in iſto, attendenda <lb></lb>eſt ex Sinu Recto arcûs, qui deſcribitur in motu circa cen<lb></lb>trum à puncto, cujus velocitas inquiritur, aut tarditas: Et <lb></lb>quidem integer Sinus Rectus accipitur, ſi punctum à ſummo <lb></lb>vertice deſcendens, vel ab infimo contactûs puncto aſcendens <lb></lb>movetur, ut ex B vel ex A: ſin autem punctum conſideretur, <lb></lb>quod intrà eoſdem Quadrantes diſtet ab extremitatibus diame<lb></lb>tri ſubjecto plano inſiſtentis, puta L aut E, quæ moventur in <lb></lb>V, aut in P, progreſsûs aut regreſsûs menſura deſumitur ex dif<lb></lb>ferentiâ Sinuum Rectorum, qui reſpondent arcubus BL & BV, <lb></lb>aut arcubus AE & AP. </s> <s id="s.001608">In poſteriori verò Quadrante ſupe-<pb pagenum="217" xlink:href="017/01/233.jpg"></pb>rioris ſemicirculi, & in anteriori Quadrante inferioris ſemicir<lb></lb>culi, progreſſus addendus, aut regreſſus demendus, motui <lb></lb>centri, menſuram deſumit ex Sinubus Verſis, aut ex eorum <lb></lb>differentiâ, pro ut puncti motus aſcendens aut deſcendens in<lb></lb>cipit ab extremitate Quadrantis, aut à loco medio, ut facilè <lb></lb>cuique conſtat: neque enim ſchema multiplici linearum deſ<lb></lb>criptione ad confuſionem implere operæ pretium eſt. </s> </p> <p type="main"> <s id="s.001609">Cum itaque in oppoſitis Quadrantibus ſimilem menſuram <lb></lb>recipiant incrementa atque decrementa ſive à ſinubus Rectis, <lb></lb>ſive à Verſis, addenda aut demenda motui centri, mani<lb></lb>feſtum eſt punctum quodlibet in integrâ converſione demùm <lb></lb>progreſſum fuiſſe pari menſurâ cum motu centri. </s> <s id="s.001610">Si enim Al<lb></lb>gebricè ſtatuatur motus Centri Z, incrementum in ſuperiore <lb></lb>ſemicirculo addendum +A, decrementum in inferiore ſemicir<lb></lb>culo tollendum — A; manifeſtum eſt totum motum, qui com<lb></lb>ponitur, Z +A — A non eſſe niſi Z. </s> </p> <p type="main"> <s id="s.001611">His ita conſtitutis, quæ ita clara ſunt, ut nihil habere vi<lb></lb>deantur dubitationis, nec in controverſiam vocari queant, jam <lb></lb>eximendus eſt ſcrupulus, quem philoſophantibus injecit Ari<lb></lb>ſtoteles Mechanic. quæſt. </s> <s id="s.001612">24. de circulorum concentricorum <lb></lb>motu, quando alter ad alterius motum promoto communi cen<lb></lb>tro movetur. </s> <s id="s.001613">Sit <lb></lb><figure id="id.017.01.233.1.jpg" xlink:href="017/01/233/1.jpg"></figure><lb></lb>enim major circu<lb></lb>lus, cujus Radius <lb></lb>CB, minor autem, <lb></lb>cujus Radius CS; <lb></lb>quos tangant pa<lb></lb>rallelæ BF & ST, <lb></lb>quibus item recta <lb></lb>per centrum ducta <lb></lb>parallela ſit CO, <lb></lb>quam videlicet per<lb></lb>currit centrum, <lb></lb>dum trahitur. </s> <s id="s.001614">Ne<lb></lb>gari non poteſt in <lb></lb>hâc circulorum tractione & converſione peripherias tùm ma<lb></lb>joris, tùm minoris Circuli ſuis Tangentibus ita coaptari, ut <lb></lb>factâ Quadrantis BD converſione, fiat pariter Quadrantis SI <pb pagenum="218" xlink:href="017/01/234.jpg"></pb>converſio, & ubi punctum D venerit in F, punctum I ſit in T, <lb></lb>& centrum C in O, atque adeò Radius CD matato ſitu factus <lb></lb>ſit OF. </s> <s id="s.001615">Major igitur Quadrans percurrit ſpatium BF, & mi<lb></lb>nor ſpatium ST. </s> <s id="s.001616">At quia æquales rectæ OF & CB perpen<lb></lb>diculares ſunt ad eandem rectam BF, ctiam ſunt parallelæ, <lb></lb>jungúntque parallelas ST & BF, quæ propterea etiam ſunt <lb></lb>æquales, ex 34. lib.1. Igitur arcus SI minor arcu BD, coap<lb></lb>tatur ſpatio æquali ipſi arcui Quadrantis BD, cui ſupponitur <lb></lb>æqualis recta BF. </s> <s id="s.001617">Quarum itaque partium 7 eſt Radius CB, <lb></lb>earum eſt Quadrans BD, hoc eſt recta BF 11, eſtque pariter <lb></lb>ST 11. At quarum partium 7 eſt Radius CB, earum ſit Ra<lb></lb>dius CS 4; igitur Quadrans SI eſt 6 3/7 multo minor quàm <lb></lb>recta ST, cui ipſe Quadrans SI in motu congruit. </s> </p> <p type="main"> <s id="s.001618">Id enim verò tantum præ ſe fert difficultatis, ut mirum ſit, <lb></lb>quot Ixiones rota hæc torqueat, & quàm varias in partes ſe alij <lb></lb>aliter verſent; quorum ſententias ſi examinare liberet, in lon<lb></lb>gum nimis ſermonem me vocaret iſta diſputatio, nec ſatis ſci<lb></lb>rem, utrùm plus aliquid lucis propoſitæ quæſtioni affunderetur. </s> <lb></lb> <s id="s.001619">Quid igitur probabilius dicendum videatur, paucis expono. </s> </p> <p type="main"> <s id="s.001620">Priùs tamen obſerva in dictâ Quadrantis revolutione, quan<lb></lb>do Centrum C venerit in O, & D in F, & in I in T, tunc <lb></lb>punctum B eſſe in E (eſt enim OE æqualis Radio CB) atque <lb></lb>punctum S in V (eſt ſcilicet OV æqualis Radio CS) ita <lb></lb>ut B aſcendat per curvam BE, punctum autem S aſcendat <lb></lb>per curvam SV, & ſimiliter punctum D deſcendat per cur<lb></lb>vam DF, punctum verò I deſcendat per curvam IT. </s> <s id="s.001621">Ex quo <lb></lb>patet punctum S minoris circuli plus promoveri, quàm <lb></lb>punctum B majoris circuli; hujus enim progreſſus eſt CE, il<lb></lb>lius autem eſt CV: & pari ratione conſtat magis ad anterio<lb></lb>ra promoveri punctum I minoris circuli, cujus progreſsûs men<lb></lb>ſura eſt IO, quàm punctum D majoris circuli, cujus progreſ<lb></lb>ſus eſt DO. </s> </p> <p type="main"> <s id="s.001622">Et hæc quidem, quando centri motus legem accipit à pe<lb></lb>ripheriâ majoris circuli; ad cujus motum minor circulus con<lb></lb>centricus movetur; eo quod major circulus inſiſtit ſubjecto pla<lb></lb>no, cui orbita ſubinde coaptatur rectam lineam ſibi æqualem <lb></lb>deſignans ex hypotheſi, dumque movetur, ſecum rapit interio<lb></lb>rem circulum. </s> </p> <pb pagenum="219" xlink:href="017/01/235.jpg"></pb> <p type="main"> <s id="s.001623">Quod ſi minor circulus inſiſtat ſubjecto ſibi plano, legem<lb></lb>que det motui centri; quia minor peripheria deſignat rectam <lb></lb>ſibi æqualem, res contrario modo procedit, quia dum ad mi<lb></lb>noris circuli motum circulus major movetur, hujus orbita de<lb></lb>ſignat in plano ſubjecto lineam minori peripheriæ æqualem. </s> <lb></lb> <s id="s.001624">Hinc ſi arcus SI deſignat rectam SG ſibi æqualem, ubi I ve<lb></lb>nerit in G, etiam D erit in H, atque totus Quadrans BD de<lb></lb>ſignabit ſolùm rectam BH æqualem rectæ SG. </s> <s id="s.001625">Erit igitur <lb></lb>recta SG æqualis Quadranti SI 6 2/7; cui pariter æqualis eſt <lb></lb>BH: Ex quo fit punctum B, quia diſtat à centro C partibus 7, <lb></lb>non ſolùm non procedere in revolutione Quadrantis; ſed re<lb></lb>trocedere per 5/7 interea, dum commune centrum C promove<lb></lb>tur per 6 2/7. </s> </p> <p type="main"> <s id="s.001626">Non abſimili ratione punctorum B, & S jam in E & V <lb></lb>tranſlatorum motus per conſequentes circuli Quadrantes, do<lb></lb>nec integra revolutio perficiatur, conſiderandus eſt: & quæ <lb></lb>de uno puncto cujuſque circuli deprehenduntur, de ſingulis <lb></lb>ejuſdem orbitæ punctis dicta faciliùs intelliguntur, quàm ut <lb></lb>uberiori explicatione opus ſit. </s> </p> <p type="main"> <s id="s.001627">Ex his apertè liquet eam lineam rectam in ſubjecto plano de<lb></lb>ſignari à peripheriâ tùm majoris, tùm minoris circuli, quæ <lb></lb>æqualis ſit motui centri, prout ille legem accipit à majore aut <lb></lb>à minore orbitâ, ad cujus motum altera movetur; ac proinde <lb></lb>modò longiori, modò breviori lineæ rectæ in motu coaptantur <lb></lb>ambæ peripheriæ; ut enim rectè loquitur Ariſtoteles loc. </s> <s id="s.001628">cit. <lb></lb><emph type="italics"></emph>Quando hic quidem movet, ille verò movetur ab iſto, quantum uti<lb></lb>que moverit alter, tantum alter movebitur.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001629">Cur igitur parem lineam rectam deſignat in plano utraque <lb></lb>orbita major & minor? </s> <s id="s.001630">conſtat ex dictis: quia nimirum cu<lb></lb>juſlibet circuli quodlibet punctum dum trahitur ſimul, & vol<lb></lb>vitur, promovetur non niſi pro ratione motûs centri: ſed con<lb></lb>centricorum circulorum unum & idem eſt centrum; ergo uni<lb></lb>cus eſt centri motus, & ſecundùm unam eandemque menſu<lb></lb>ram motûs centri, omnia puncta tùm majoris, tùm minoris or<lb></lb>bitæ, demum abſolutâ converſione, promota ſunt; ſingulorum <lb></lb>enim incrementa, dum ſuperiorem ſemiperipheriam motu <lb></lb>deſcribunt, ab oppoſitis decrementis eliſa in inferioris ſemipe-<pb pagenum="220" xlink:href="017/01/236.jpg"></pb>ripheriæ deſcriptione, ſolum centri motum relinquunt. </s> <s id="s.001631">Nil <lb></lb>itaque mirum, ſi tres lineæ, quarum primam centrum percur<lb></lb>rit, ſecundam orbita minor deſignat, tertiam orbita major, pla<lb></lb>nè æquales ſunt; pendent enim ab unico & communi motu <lb></lb>centri, cui nihil additur, aut demitur ex integrâ converſione <lb></lb>circa centrum, ſivè illa latiùs excurrat in majore circulo, ſivè <lb></lb>arctiùs in minore coërceatur. </s> </p> <p type="main"> <s id="s.001632">At, inquis, difficile eſt cogitatione aſſequi, & oratione ex<lb></lb>plicare, quî fieri poſſit, ut peripheriâ utráque ſubjectum ſibi <lb></lb>planum ſemper tangente, nullóque puncto manente ſine mo<lb></lb>tu, ita ut plana ſubjecta ab aliis ſubinde atque aliis punctis tan<lb></lb>gantur, pauciora puncta minoris peripheriæ totidem punctis <lb></lb>rectæ lineæ coaptentur, ac plura puncta majoris peripheriæ. </s> </p> <p type="main"> <s id="s.001633">Sunt qui difficultatem hanc declinant adſtruentes infinita <lb></lb>puncta tùm in circulorum peripheriis, tùm in lineis rectis, ne<lb></lb>gantéſque inter infinitas multitudines, quæ invicem compa<lb></lb>rentur, affirmari poſſe totidem in unâ infinitâ multitudine, ac <lb></lb>in aliâ pariter infinitâ unitates reperiri, nulla enim eſt infiniti <lb></lb>ad infinitum Ratio, ac proinde nulla fieri poteſt, perinde ac in <lb></lb>multitudinibus finitis, comparatio minoris, aut majoris, aut <lb></lb>propriè, &, ut aiunt, poſitivè æqualis. </s> <s id="s.001634">Hæc tamen (quamvis <lb></lb>quod ad infinita Ratione carentia ſpectat, à me ultrò admit<lb></lb>tantur, Rationem ſcilicet habere dicuntur inter ſe magnitudi<lb></lb>nes, idem & de multitudinibus dicendum, quæ poſſunt mul<lb></lb>tiplicatæ ſe mutuò ſuperare, ut definit Euclides lib.5. ubi au<lb></lb>tem nullus eſt terminus, ut in infinito, nullus pariter exceſſus <lb></lb>intercedere poteſt quavis factâ multiplicatione) non facient <lb></lb>ſatis comparanti omnia puncta unius lineæ cum omnibus <lb></lb>punctis alterius lineæ, non quâ infinitæ punctorum multitudi<lb></lb>nes ſunt, ſed quâ finitæ magnitudines ex punctis illis quan<lb></lb>tumvis infinitis conſtituuntur: finitas autem magnitudines <lb></lb>comparari invicem poſſe, ac Rationem interſe habere nemo <lb></lb>negaverit. </s> <s id="s.001635">Supereſt igitur explicandum, quomodo peripheria <lb></lb>minor coaptetur lineæ rectæ æquali illi eidem, cui commenſu<lb></lb>ratur peripheria major. </s> </p> <p type="main"> <s id="s.001636">Propterea, duce Galilæo Dialog.1. de motu, obſervant ſimi<lb></lb>lium polygonorum concentricorum motum ac converſionem, <lb></lb>in quâ polygonum, ex quo centri motus legem accipit, ſingu-<pb pagenum="221" xlink:href="017/01/237.jpg"></pb>la latera ita æqualibus lineæ rectæ partibus accommodat, ut in <lb></lb>integrâ converſione linea recta ſubjecti plani ſit æqualis peri<lb></lb>metro polygoni: at non item partes omnes lineæ, cui alterum <lb></lb>polygonum in motu coaptatur, ſi unica comprehenſione ſu<lb></lb>mantur, lineam æqualem polygoni majoris perimetro conſti<lb></lb>tuunt. </s> <s id="s.001637">Res, clarita<lb></lb><figure id="id.017.01.237.1.jpg" xlink:href="017/01/237/1.jpg"></figure><lb></lb>tis gratia, explicetur <lb></lb>in Hexagonis, quo<lb></lb>rum commune cen<lb></lb>trum ſit A, & latera <lb></lb>BC, DE incumbant <lb></lb>parallelis lineis BH, <lb></lb>DK. </s> <s id="s.001638">Det primùm le<lb></lb>gem motui centri po<lb></lb>lygonum exterius, & majus, fiatque converſio circa punctum <lb></lb>C, demùm latus CF congruet rectæ CH, & centrum A per <lb></lb>arcum AF erit tranſlatum in F; latus verò minoris polygoni <lb></lb>EG congruet parti IK, intactam relinquens partem EI, ita <lb></lb>tamen; ut tota EK æqualis ſit ipſi CH. </s> <s id="s.001639">Id quod eſt mani<lb></lb>feſtum, quia factâ tranſlatione centri in F, ſemidiameter, quæ <lb></lb>ex F pertingit ad H, eſt parallela ipſi AC, cum ad ſimiles an<lb></lb>gulos incidat in ſubjectam lineam; ſunt autem parallelæ etiam <lb></lb>AF, DK, & BH; igitur tres lineæ AF, EK, CH ſunt æqua<lb></lb>les, ex 34. lib.1. Atqui quod uni lateri contingit, etiam reli<lb></lb>quis lateribus commune eſt; igitur factá integrâ converſione <lb></lb>Hexagonum majus deſignabit lineam ſextuplicem ipſius CH <lb></lb>æqualem toti perimetro, & Hexagonum minus percurret li<lb></lb>neam ſimiliter ipſius EK ſextuplicem, quæ æqualis eſt perime<lb></lb>tro majoris Hexagoni, ſumendo tàm partes lineæ DK, quas <lb></lb>intactas relinquit, quàm quæ tangunrur. </s> <s id="s.001640">Cæterùm ſi eæ ſo<lb></lb>lùm, quæ ab Hexagono minore tanguntur, accipiantur, patet <lb></lb>illas ſimul ſumptas non eſſe majores perimetro ejuſdem mino<lb></lb>ris Hexagoni. </s> </p> <p type="main"> <s id="s.001641">Deinde polygonum interius & minus det legem motui cen<lb></lb>tri, & converſio fiat circa punctum E, poſtquam latus EG <lb></lb>congruit lineæ EI, & centrum eſt in G (in hoc enim exem<lb></lb>plo ad vitandam in Schemate confuſionem literarum aſſump-<pb pagenum="222" xlink:href="017/01/238.jpg"></pb>tum eſt Hexagonum minus ſubquadruplum majoris, latera ſci<lb></lb>licet minotis ſubdupla ſunt laterum majoris) cum interim <lb></lb>punctum C retroceſſerit in L, & demum latus CF congruat <lb></lb>lineæ LM. </s> <s id="s.001642">Igitur majus polygonum ſolùm deſignat in motu, <lb></lb>quo progreditur, lineam CM æqualem lateri minoris polygoni <lb></lb>EI; & factâ integrâ converſione, deſignata erit linea ſextuplex <lb></lb>ipſius CM & ipſius EI; atque adeò utrumque polygonum <lb></lb>æqualem lineam progrediendo deſignat. </s> </p> <p type="main"> <s id="s.001643">Hæc quæ de Hexagonis concentricis exempli gratiâ dicta <lb></lb>ſunt, de omnibus ſimilibus atque concentricis polygonis dicta <lb></lb>intelliguntur, quotcumque ſint laterum. </s> <s id="s.001644">Jam verò Authores <lb></lb>illi concipiunt circulos tanquam polygona infinitorum late<lb></lb>rum: & quemadmodum minus polygonum totidem ſpatia ſub<lb></lb>jectæ lineæ intacta relinquit, totidemque tangit, quot habet <lb></lb>latera; ita pariter in circuli minoris converſione, infinita ſpa<lb></lb>tia vacua non-quanta (ne ſcilicet ſi quanta eſſent, opus eſſet <lb></lb>lineâ infinitâ) intermiſta ſpatiis, quæ tanguntur, adſtruunt, <lb></lb>adeò ut demùm ex omnibus ſpatiis tactis ſimul & intactis coa<lb></lb>leſcat linea æqualis ei, quæ tangitur à majore peripheriâ ma<lb></lb>joris circuli. </s> </p> <p type="main"> <s id="s.001645">Mihi tamen arridere non poteſt illa loquendi formula, quæ <lb></lb>circulum polygonum infinitorum (& quidem infinitorum ſim<lb></lb>pliciter) laterum dicit. </s> <s id="s.001646">Polygonum enim utique regulare cir<lb></lb>culus eſſet; polygonum autem eſſe non poteſt illud, quod angu<lb></lb>lis caret; neque anguli eſſe poſſunt, ubi non eſt lineæ ad li<lb></lb>neam inclinatio; in peripheriâ verò circuli linea nulla eſſe po<lb></lb>teſt, eſſent ſiquidem infinitæ lineæ æquales invicem, quæ uti<lb></lb>que conſtituerent extenſionem ſimpliciter infinitam. </s> <s id="s.001647">Quod ſi <lb></lb>infinita dixeris puncta; non eſt puncti ad punctum inclinatio, <lb></lb>quæ poſſit angulum conſtituere, ac proinde circulus non eſt po<lb></lb>lygonum infinitorum laterum, niſi vocabulis ad opinandi li<lb></lb>centiam immoderatè abutamur. </s> <s id="s.001648">Adde quod omnia diametri <lb></lb>puncta ad omnia puncta peripheriæ eſſent in Ratione, quam <lb></lb>Archimedes <emph type="italics"></emph>lib.de dimenſione circuli<emph.end type="italics"></emph.end> definivit contineri inter Ra<lb></lb>tionem 7 ad 22, & Rationem 71 ad 223: non igitur infinita eſſe <lb></lb>poſſunt aut diametri, aut peripheriæ, aut utriuſque puncta; ab <lb></lb>infinitis enim Rationem omnem ablegant iidem Authores. </s> <s id="s.001649">Si <pb pagenum="223" xlink:href="017/01/239.jpg"></pb>itaque circulus polygonus non eſt, adhuc indiget explicatione, <lb></lb>quomodo ad circulos concentricos traducantur ea, quæ de po<lb></lb>lygonorum concentricorum converſione conſiderata ſunt. </s> </p> <p type="main"> <s id="s.001650">Quòd ſi circulum ita in polygonum convertamus, ut nec <lb></lb>illi fixum definitumque laterum numerum tribuamus, nec ſim<lb></lb>pliciter infinitum; ſed liceat minora ſemper atque minora late<lb></lb>ra concipere, ut laterum ipſorum numerus ſemper augeatur, <lb></lb>ita ut non ſimpliciter infinitus, ſed indefinitus dicatur, non <lb></lb>abnuo: propoſita enim difficultas ſatis commodè hâc ratione <lb></lb>explicabitur. </s> <s id="s.001651">Verùm in hac laterum extenuatione, ſi ad mini<lb></lb>mam extenſionem deveniamus, quæ à puncto phyſicè non dif<lb></lb>ferat; non infinitus eſt hujuſmodi punctorum numerus, ſed <lb></lb>certus eſt atque definitus: Necipſis punctis, ſeu minimis Phy<lb></lb>ſicis ſua figura detrahenda eſt, in majori enim peripheriâ mi<lb></lb>nùs curvantur interiùs, minúſque convexa ſunt exteriùs, pro<lb></lb>piúſque ad lineam rectam accedunt; in minori autem orbitâ <lb></lb>puncta hæc circularia curvantur magis, magiſque convexa ſunt <lb></lb>exteriùs, & à rectitudine magis deflectentia ita abſunt à ſub<lb></lb>jectâ rectâ lineâ, ut, dum converſio fit circuli, & trahitur, deſ<lb></lb>cribant in motu lineam curvam magis obſecundantem motui <lb></lb>centri, quàm quæ deſcribitur à punctis ſimiliter poſitis in ma<lb></lb>jore peripheriâ. </s> </p> <p type="main"> <s id="s.001652">Cærerùm cavendum eſt maximè ab eo, quod quia ſubeſt <lb></lb>æquivocationi, difficultatem in hâc quæſtione auget; illud au<lb></lb>tem eſt, quod punctum peripheriæ cum puncto lineæ Tangen<lb></lb>tis perperam comparatur, quaſi in contactu coæquarentur; id <lb></lb>quod à veritate longè abeſt; ſe enim contingunt circulus & li<lb></lb>nea incommenſurabiliter, ſi contactus præcisè ſpectetur: at ſi <lb></lb>contactus & motus componantur, jam quædam extenſio conci<lb></lb>pitur, quæ aliquâ ratione comparari poteſt cum ſpatio lineæ, <lb></lb>quæ tangitur, quatenùs huic aut illi parti lineæ in motu coapta<lb></lb>tur circulus, aut ejus pars. </s> <s id="s.001653">Quare circuli minoris, qui ad ma<lb></lb>joris circuli motum movetur, ſingula puncta non aptè compa<lb></lb>rantur cum ſingulis ſubjectæ rectæ lineæ punctis, quaſi circuli <lb></lb>punctum, quod eſt tertium à contactu, antequam incipiat mo<lb></lb>tus, in converſione tangat tertium rectæ lineæ punctum; ſed <lb></lb>tanget fortaſſe quintum aut ſextum pro ratione magnitudinis <pb pagenum="224" xlink:href="017/01/240.jpg"></pb>aut parvitatis ipſius circuli; pro ut in polygonis concentricis <lb></lb>obiervare eſt; quò enim majus eſt interius polygonum, eò <lb></lb>etiam minora ſunt intervalla, quæ intacta relinquuntur. </s> <s id="s.001654">Ex <lb></lb>quamvis in circuli contactu intervalla hujuſmodi intacta non <lb></lb>admittantur, non eſt tamen abs re puncto circuli, quod volui<lb></lb>tur ſimul & trahitur cum ipſo circulo, vim tribuere tangendi <lb></lb>plus quàm unum ſubjectæ rectæ lineæ punctum, quemadmo<lb></lb>dum majoris peripheriæ punctum in motu contingit ex punctis <lb></lb>ſubjectæ lineæ rectæ non communicantibus minus quàm unum, <lb></lb>ſi ad interioris circuli motum circulus exterior moveatur: nam <lb></lb>ad majoris, & exterioris motum minor, & interior promovetur; <lb></lb>ad minoris verò & interioris motum major & exterior circulus <lb></lb>retroagitur. </s> <s id="s.001655">Quapropter ſi interior circulus in primo caſu ve<lb></lb>lociùs, & exterior in ſecundo caſu tardiùs movetur comparatè <lb></lb>ad ſpatium collocatum cum eorum peripheriis, nil mirum in <lb></lb>motu perfici ab illius puncto Phyſico plus ſpatij, quàm ferat <lb></lb>ejus magnitudo, ab hujus autem puncto Phyſico minus ſpatij: <lb></lb>in continuâ enim quantitate partes minores ſubinde ac minores <lb></lb>vera, ut opinor, Philoſophia admittit. </s> <s id="s.001656">Sed quia hæc eſſet in<lb></lb>finita, concertationumque plena diſputatio, ſatis ea ſint, quæ <lb></lb>diximus, & ad utiliora gradum faciamus. <lb></lb><figure id="id.017.01.240.1.jpg" xlink:href="017/01/240/1.jpg"></figure></s> </p> <pb pagenum="225" xlink:href="017/01/241.jpg"></pb> <figure id="id.017.01.241.1.jpg" xlink:href="017/01/241/1.jpg"></figure> <p type="main"> <s id="s.001657"><emph type="center"></emph>MECHANICORUM<emph.end type="center"></emph.end><emph type="center"></emph>LIBER TERTIUS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001658"><emph type="center"></emph><emph type="italics"></emph>De Libra.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001659">EXPLICATIS ſuperiore Libro Cauſis motûs Ma<lb></lb>chinalis, ordinis ratio poſtularet, ut ad ipſas Ma<lb></lb>chinas, ſeu, ut ab Antiquioribus apud Pappum <lb></lb>lib.8. Collect. Mathem. prop.10. vocantur, Facul<lb></lb>tates, ad quas Machinamenta ab artificibus exco<lb></lb>gitata reducuntur, aut ex quibus hæc componuntur, exami<lb></lb>nandas & explicandas progrederemur: Et fortè alicui videatur <lb></lb>ab inſtituto noſtro alienum libram hîc conſiderare, quippe quæ <lb></lb>non ad motum oneribus conciliandum inventa eſt, ideóque <lb></lb>nec inter Facultates enumeratur, ſed uſum omnem habet in <lb></lb>motu prohibendo, ubi factum fuerit ponderibus æquilibrium. </s> <lb></lb> <s id="s.001660">Nec eo quidem conſilio libræ momenta hic expendo, ut indè <lb></lb>Vectis rationes explicentur (quemadmodum non paucis placet) <lb></lb>non enim Vectis vires ad libræ Rationes revocandas exiſtimo, <lb></lb>cum ſua cuique Facultati cauſa inſit, communis illa quidem, <lb></lb>ſed quæ perinde in Vecte reperitur, atque ſi nulla prorſus <lb></lb>exiſteret libra. </s> <s id="s.001661">Verùm eatenus libram Mechanicæ contem<lb></lb>plationi inſerendam cenſeo, quatenus non minoris artis eſt ea, <lb></lb>quæ in motum prona ſunt, cohibere & ſiſtere, quàm onera <lb></lb>quieſcentia per vim ſuo loco dimovere: Cum maximè ad libram <lb></lb>pertineat Statera, in qua modicum pondus multò majori pon<lb></lb>deri æquipollet, æquatis in diſpari gravitate gravitationum <pb pagenum="226" xlink:href="017/01/242.jpg"></pb>momentis, ut infra in loco oſtendetur. </s> <s id="s.001662">Præterquam quod <lb></lb>explicato æquilibrio, faciliùs declaratur in motu Machinali, <lb></lb>quid præſtet major illa Ratio momentorum agendi ad momen<lb></lb>ta reſiſtendi, quàm ſit reciproca Ratio gravitatum, ſeu vi<lb></lb>rium oppoſitarum, abſolutè ſumptarum extrà machinam; ex <lb></lb>qua majore Ratione momentorum, etiam Potentiæ moventis <lb></lb>virtus innoteſcit. </s> <s id="s.001663">Nihil autem officit libræ dignitati, quod <lb></lb>Cain authorem agnoſcere videatur, qui, ut Joſephus lib. 1. <lb></lb><expan abbr="Antiq.">Antique</expan> Jud. cap.2. loquitur, <emph type="italics"></emph>Simplicem hactenus vivendi rationem <lb></lb>excogitatis menſuris & ponderibus immutavit, priſlinamque ſinceri<lb></lb>tatem & generoſitatem ignaram talium artium, in novam quan<lb></lb>dam virſutiam depravavit.<emph.end type="italics"></emph.end></s> <s id="s.001664"> Quid enim ſi quis præclaro artifi<lb></lb>cio ex naturæ theſauris deprompto abutatur? </s> <s id="s.001665">Dolos & fallacias, <lb></lb>aut errores, quibus inſici poteſt libræ uſus, ideò retegemus: <lb></lb>ut nimirum quod Juſtitiæ commutativæ ſymbolum datur, om<lb></lb>ni injuſtitiæ ſuſpicione vacet. </s> <s id="s.001666">Cæterùm quæ nobis ineſt arbi<lb></lb>trij libertas, potiſſima naturæ rationis compotis prærogativa, <lb></lb>libræ, aut ſtateræ jure merito comparatur, quâ iniqui abuten<lb></lb>tes dicuntur Pſalm. 61. <emph type="italics"></emph>Mendaces filij hominum in ſtateris:<emph.end type="italics"></emph.end> ubi <lb></lb>S. Baſilius hom. </s> <s id="s.001667">in Pſalm. 61. ait <emph type="italics"></emph>Cuilibet noſtrûm intus ſtatera <lb></lb>quædam eſt à Conditore omnium apparata, per quam rerum naturam <lb></lb>poſſis probè dignoſcere.<emph.end type="italics"></emph.end> & infra: <emph type="italics"></emph>Tibi namque propria datur libra, <lb></lb>quæ ſufficiens diſcrimen boni, ac mali demonſtrat. </s> <s id="s.001668">Corporea enim <lb></lb>pondera in libræ lancibus probamus; quæ verò ad inſtituendam vi<lb></lb>tam eligenda veniunt, per liberum arbitrium diſcernimus: quod & <lb></lb>ſtateram nominavit, quòd momentum æquale ad utrumlibet poſſit <lb></lb>capere.<emph.end type="italics"></emph.end><lb></lb></s> </p> <p type="main"> <s id="s.001669"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001670"><emph type="center"></emph><emph type="italics"></emph>Libræ forma, & natura exponitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001671">EO conſilio inſtituta eſt libra, ut certis, ac notis ponderi<lb></lb>bus, ignotæ gravitatis quantitas indagetur, quæ demùm <lb></lb>innoteſcit, cum æquatis hinc & hinc ponderum libræ adnexo<lb></lb>rum momentis, neutro prævalente, libra conſiſtit. </s> <s id="s.001672">In hoc <pb pagenum="227" xlink:href="017/01/243.jpg"></pb>inſtrumento conſideratur pri<lb></lb><figure id="id.017.01.243.1.jpg" xlink:href="017/01/243/1.jpg"></figure><lb></lb>mùm <emph type="italics"></emph>Iugum<emph.end type="italics"></emph.end>, ſeu <emph type="italics"></emph>ſcapus,<emph.end type="italics"></emph.end> ſeu <lb></lb><emph type="italics"></emph>librile<emph.end type="italics"></emph.end> AB: hoc bifariam divi<lb></lb>ditur in C, quod, <emph type="italics"></emph>Centrum<emph.end type="italics"></emph.end> li<lb></lb>bræ dicitur, non quia ſit ne<lb></lb>ceſſariò Centrum gravitatis li<lb></lb>bræ, ſed quia eſt Centrum, <lb></lb>circa quod agitur, ſeu verſa<lb></lb>tur jugum, infixo nimirum in C axiculo, qui & <emph type="italics"></emph>Agina<emph.end type="italics"></emph.end> Latinis, <lb></lb>Græcis apud Ariſtorelem in quæſt. </s> <s id="s.001673">Mechan. <emph type="italics"></emph>Spartum<emph.end type="italics"></emph.end> dicitur. </s> <lb></lb> <s id="s.001674">Partes autem jugi videlicet CA, & CB. <emph type="italics"></emph>Brachia, Radij,<emph.end type="italics"></emph.end> aut <lb></lb>etiam ab aliquibus <emph type="italics"></emph>Librilia<emph.end type="italics"></emph.end> vocantur. </s> <s id="s.001675">Ex medio jugi ad per<lb></lb>pendiculum aſſurgit lingula CD, quæ inſeritur anſæ EF com<lb></lb>plectenti capita axiculi, adeò ut ſuſpensâ ex F an â, quæ ho<lb></lb>rizonti ad perpendiculum immineat, tùm demùm intelligatur <lb></lb>factum æquilibrium, cum lingula anſæ congruit, & jugum <lb></lb>conſiſtit horizonti parallelum. </s> <s id="s.001676">Utrùm autem <emph type="italics"></emph>Trutina<emph.end type="italics"></emph.end> dicenda <lb></lb>ſit ipſa lingula, an verò anſa, non conveniunt Authores: li<lb></lb>tem Grammaticis dirimendam relinquo. </s> </p> <p type="main"> <s id="s.001677">Extremis brachiorum punctis A & B adnectitur utrumque <lb></lb>pondus, tam notum, quod eſt alterius menſura, quàm igno<lb></lb>tum; cujus gravitas examinatur. </s> <s id="s.001678">Nihil autem refert, an pon<lb></lb>dera uncinis adnexa dependeant, an verò lancibus indè pen<lb></lb>dentibus imponantur; id quod vulgare eſt magiſque uſitatum, <lb></lb>& libræ fecit nomen <emph type="italics"></emph>Bilanci.<emph.end type="italics"></emph.end></s> <s id="s.001679"> Illud enim præcipuum eſt, ac <lb></lb>maximè attendendum, quòd omnia hinc & hinc æqualia ſint, <lb></lb>nimirum pondus unius lancis cum funiculis ſeu catenulis æqua<lb></lb>le ſit ponderi alterius lancis cum ſuis appendiculis (pondus, in<lb></lb>quam, ponderi æquale ſit; nil enim intereſt æquales ne? </s> <s id="s.001680">an <lb></lb>inæquales fuerint utriuſque lancis funiculi ſecundùm longitu<lb></lb>dinem, modò in æquali diſtantiâ à centro adnectantur) & bra<lb></lb>chium alterum majus non ſit reliquo brachio non ſolùm quoad <lb></lb>gravitatem, quæ materiæ jugi ineſt, ſed potiſſimùm quoad <lb></lb>ipſorum brachiorum longitudinem. </s> </p> <p type="main"> <s id="s.001681">Porrò hæc brachiorum longitudo non eſt deſumenda, ut ita <lb></lb>loquar, materialiter, à centro jugi ad extremitatem, ubi mate<lb></lb>ria deſinit, ex quâ conſtat, ſivè ferrum ſit, ſivè lignum, ſivè <lb></lb>aliud quidpiam: ſed brachiorum longitudinem definiunt <pb pagenum="228" xlink:href="017/01/244.jpg"></pb>puncta jugi; ex quibus pondera dependent: horum etenim <lb></lb>diſtantiam à centro omnino æqualem eſſe oportet. </s> <s id="s.001682">Hujuſmodi <lb></lb>autem puncta non alia ſunt, quàm puncta contactûs jugi & an<lb></lb>nulorum ſeu uncinorum illi infixorum, quibus deinde lances <lb></lb>aut pondera adnectuntur. </s> <s id="s.001683">Hoc illud eſt, in quo maxima arti<lb></lb>ficis induſtria, atque diligentia collocanda eſt, ut exactiſſimam <lb></lb>brachiorum æqualitatem aſſequatur. </s> </p> <p type="main"> <s id="s.001684">Data itaque hac, quam diximus, brachiorum æqualitate, ſi <lb></lb>æqualia pondera hinc & hinc addantur, manifeſtum eſt jugum <lb></lb>libræ ex aginâ ſuſpenſum ad neutram partem inclinari, ſed ma<lb></lb>nere horizonti parallelum; fieri namque non poteſt, ut extremi<lb></lb>tas altera deſcendat, quin oppoſita extremitas cum adnexo pon<lb></lb>dere aſcendat, & quidem æquali motu propter brachiorum <lb></lb>æqualitatem. </s> <s id="s.001685">Finge enim pondus B deſcendere in F, utique <lb></lb><figure id="id.017.01.244.1.jpg" xlink:href="017/01/244/1.jpg"></figure><lb></lb>pondus A aſcendet in E, at<lb></lb>que deſcribent arcus BF & <lb></lb>AE æquales, quippe qui <lb></lb>æqualibus angulis ad verti<lb></lb>cem in C ſubtenduntur, & <lb></lb>ab æqualibus radiis CB, CA <lb></lb>deſcribuntur. </s> <s id="s.001686">At æqualis eſt in B vis deſcendendi atque in A <lb></lb>repugnantia ad aſcendendum; illa igitur præpollere non poteſt. </s> <lb></lb> <s id="s.001687">Siquidem vis deſcendendi componitur ex ponderis gravitate, <lb></lb>& non impeditâ motûs naturalis velocitate; repugnantia verò <lb></lb>ad aſcendendum componitur & ex ponderis contranitentis gra<lb></lb>vitate, & ex velocitate motûs præter naturam: ſunt autem gra<lb></lb>vitates ex hypotheſi æquales, motus etiam per arcus BF & AE <lb></lb>eſſent æquales; ac proinde vis tendendi deorſum inveniens <lb></lb>æqualem oppoſitam repugnantiam ad motum ſurſum nequit illi <lb></lb>imprimere impetum, quo per vim moveatur: ut enim ſequa<lb></lb>tur motus, aut gravitates diſpares eſſe oportet, aut motuum Po<lb></lb>tentiæ moventis & Ponderis moti velocitates inæquales, ut ma<lb></lb>jor ſit Ratio hujuſmodi velocitatum, quàm ſit reciproca Ratio <lb></lb>gravitatum: alioquin nulla eſſet virium movendi & reſiſtentiæ <lb></lb>inæqualitas, ubi omnia eſfent æqualia. </s> <s id="s.001688">Cum itaque in librâ ſic <lb></lb>conſtitutâ intercedat omnimoda æqualitas & brachiorum, qui<lb></lb>bus definitur motus, & gravitatum, quæ ſibi invicem æquali<lb></lb>ter obſiſtunt, ac proinde eadem ſit reciproca Ratio gravitatum <pb pagenum="229" xlink:href="017/01/245.jpg"></pb>& motuum, jugum libræ horizonti parallelum conſiſtere ne<lb></lb>ceſſe eſt; & in alteram partem ſi inclinerur, manifeſtum eſt in <lb></lb>illâ lance plus ponderis fuiſſe impoſitum, quàm in reliquâ. </s> </p> <p type="main"> <s id="s.001689">Ut autem quàm exactiſſimè ponderum ignota gravitas exa<lb></lb>minari queat, opus eſt ut axiculus jugo infixus (ſaltem in ſupe<lb></lb>riore parte, cui ſcapus incumbit) exquiſitè cylindricam figu<lb></lb>ram obtineat; hinc enim fiet, ut cum rotundo foramine ſcapi <lb></lb>contactus fiat in lineâ, quamcumque tandem poſitionem ha<lb></lb>beat ipſe ſcapus: nam quemadmodum ex prop. 13. lib. 3. duo <lb></lb>circuli ſe intùs contingentes tangunt in puncto, ita duæ ſuper<lb></lb>ficies cylindricæ, cava altera, altera convexa, ſe tangunt in li<lb></lb>neà. </s> <s id="s.001690">Id ſi fiat facilè ab æquilibrio deflectet ſcapus, ſi vel modi<lb></lb>ca intercedat ponderum inæqualitas. </s> <s id="s.001691">At ſi angulatus fuerit axi<lb></lb>culus, vel ſuperior foraminis pars rotunditatem non fuerit aſſe<lb></lb>cuta, jam non in unâ lineâ, ſed in pluribus contactus fieret, at<lb></lb>que adeò iners eſſet ad motum ſeapus, etiamſi non omninò <lb></lb>æqualia eſſent pondera lancibus impoſita. </s> </p> <p type="main"> <s id="s.001692">Quare artifices illos non probo, qui axem ita efſormant, ut <lb></lb>ſuperior pars in aciem deſinat, illud ſibi perſuadentes, quod <lb></lb>minore partium conflictu ſe tangentes axis & ſcapus faciliorem <lb></lb>relinquant in alterutram partem motum libræ. </s> <s id="s.001693">Id quod ut ve<lb></lb>rum ſit, non tamen vacat periculo, ne, dum axis capita inſe<lb></lb>runtur anſæ, acies illa planè ſursùm non dirigatur, ſed modi<lb></lb>cum in alterutram partem vergat: quæ declinatio ſi contingat, <lb></lb>foramen autem exactè rotundum fuerit, miraculo proximum <lb></lb>cenſe, ſi libra vacua æquilibrium conſtituat, ita ut lingula ritè <lb></lb>collocata congruat anſæ; acies ſi quidem illa dividit inæquali<lb></lb>ter ſcapi longitudinem, & brachium alterum altero longius eſt, <lb></lb>atque præponderat. </s> <s id="s.001694">Hoc vitium ubi libra contraxerit, inepti <lb></lb>artifices nihil ſuſpicati ab axe malè conformato, aut perperam <lb></lb>diſpoſito, ortum duxiſſe, vel brachium extenuant, vel lancem <lb></lb>immutant, donec æquilibrium inveniant. </s> <s id="s.001695">Verùm libram hu<lb></lb>juſmodi doloſam eſſe inferiùs conſtabit propter brachiorum in<lb></lb>æqualitatem: quæ quidem levem infert ponderum differen<lb></lb>tiam in rebus exigui momenti contemnendam; ſed in iis, quæ <lb></lb>exquiſitam ponderis menſuram exigunt, non leve damnum <lb></lb>hinc poteſt emergere. </s> </p> <p type="main"> <s id="s.001696">Quod ſi axis non ſit anſæ, ſed ſcapo, firmiter infixus, volua-<pb pagenum="230" xlink:href="017/01/246.jpg"></pb>turautem in anſæ foraminibus (id quod artificibus non paucis <lb></lb>magis arridet) jam non ſuperior; ſed inferior axiculi pars at<lb></lb>tendenda eſt; quippe quæ inferiorem foraminum anſæ partem <lb></lb>contingit; & eadem, quæ de ſuperiore parte dicebantur, ob<lb></lb>ſervanda ſunt. </s> <s id="s.001697">Illud tamen præterea in anſæ foraminibus ob<lb></lb>ſervandum venit, quod eorum infima pars ita ſit conſtituta, ut <lb></lb>axis illis incumbens parallelus ſit horizonti, quando anſa ſuſ<lb></lb>penditur, ut liberè pendeat, vel ita collocatur, ut ad perpen<lb></lb>diculum horizonti immineat: alioquin axe inclinato, jugum <lb></lb>urgeret aiteram anſæ partem, ab alterâ recederet; ex quo jugi <lb></lb>cuman, conflictu aliqua motui difficultas crearetur. </s> </p> <p type="main"> <s id="s.001698">Jam verò quod ad pondera attinet, ſupervacaneum eſt mo<lb></lb>nere non omnia pondera omnibus libris convenire: quamvis <lb></lb>enim libra, quâ libra eſt, nuliam prorſus reſpuat ponderum gra<lb></lb>vitatem, ſed omnem quorumcumque ponderum æqualitatem <lb></lb>apta ſit indicare ſuo æquilibrio; quia tamen ex materiâ conſtat, <lb></lb>quæ definitam habet ſoliditatem atque partium firmitatem (ut <lb></lb>nihil dicam de certis atque definitis viribus retinentis anſam, <lb></lb>& cum ansâ libram, ac utrumque pondus) fieri poteſt, ut adeò <lb></lb>gravia lancibus imponantur onera, quæ brachiorum rectitudi<lb></lb>nem inflectant, & eorum æqualitatem corrumpant: Quare te<lb></lb>nuioribus libris parva pondera examinantur, craſſioribus ma<lb></lb>jora. </s> <s id="s.001699">Illud potiùs cavendum eſt, ne pondera, quibus tanquam <lb></lb>menſurâ utimur, fallacia ſint, quia falſa, aut excedendo legi<lb></lb>timam gravitatis quantitatem, aut ab illâ deficiendo. </s> </p> <p type="main"> <s id="s.001700">Quamvis autem tot pondera minimæ menſuræ adhibere poſ<lb></lb>ſemus, quot numerare oporteret ad explorandam propoſitæ <lb></lb>gravitatis ignotæ quantitatem, hoc tamen valde incommodum <lb></lb>eſſet: quid enim, ſi lanius carnem in macello vendens grana <lb></lb>numerare cogeretur, quæ æquilibrium cum carne conſtituunt? <lb></lb></s> <s id="s.001701">ſed & inutilis eſſet labor, nam multa ſunt, quorum quantitas <lb></lb>non eſt ad vivum reſecanda, & minutiſſimæ particulæ fruſtra <lb></lb>inveſtigantur. </s> <s id="s.001702">Subtilitas hæc relinquatur gemmariis, aurifici<lb></lb>bus, auríque monetalis cuſoribus, quibus damnum eſſet minu<lb></lb>tias contemnere. </s> <s id="s.001703">Quamquam nec iſtis author fuerim, ut ſin<lb></lb>gularibus granis uterentur, ſed potiùs ponderibus, quæ pluri<lb></lb>bus granis æquivalerent; ſi enim ſingula grana à legitimo pon<lb></lb>dere dificiunt per centeſimam grani partem, quæ facilè ſensûs <pb pagenum="231" xlink:href="017/01/247.jpg"></pb>aciem fugit, additis centum hujuſmodi granis error eſt inte<lb></lb>gri grani deficientis; & in uncia libræ Romanæ ponderalis ad <lb></lb>monetam pertinentis cum grana 576 contineantur, in uncia <lb></lb>auri error eſſet granorum ferè ſex deficientium, & in integrâ <lb></lb>librâ, quæ eſt granorum 6912, eſſet error granorum 69; qui <lb></lb>tamen error vix contingat, ſi aſſumatur integra uncia, aut li<lb></lb>bra: illud ſi quidem, quod ſolitarium præ ſua tenuitate in con<lb></lb>ſpectum non cadit, cum pluribus ſimilibus conjunctum evadit <lb></lb>demum notabile atque conſpicuum. </s> <s id="s.001704">Quare ad paranda pon<lb></lb>dera hujuſmodi ſubtiliora, aſſume laminam metallicam ponde<lb></lb>re unius libræ, ſed æquabiliter extenſam, ejuſque duodecimam <lb></lb>partem accipe; hæc erit Uncia, quam ſepones. </s> <s id="s.001705">Alterius Unciæ <lb></lb>octavam partem aſſumens habebis Draclimam. </s> <s id="s.001706">Drachmæ pars <lb></lb>tertia dabit ſcrupulum. </s> <s id="s.001707">Scrupuli ſemiſſis eſt obolus. </s> <s id="s.001708">Oboli <lb></lb>triens eſt ſiliqua. </s> <s id="s.001709">Demùm ſiliquæ quadrans eſt Granum. </s> <s id="s.001710">Ex <lb></lb>hac minutâ diviſione ſatis conſtat, quàm obnoxiæ errori ſint <lb></lb>minores particulæ præ majoribus; idemque error, qui in unciâ <lb></lb>fingularis eſſet, & ut nullus conſideraretur, toties repetitus, <lb></lb>quot grana in unciâ continentur, jam non eſſet contemnen<lb></lb>dus. </s> <s id="s.001711">Id autem dictum intelligatur etiam in majoribus ponde<lb></lb>ribus, ubi unciæ non reputantur, ſatius eſſe majora pondera <lb></lb>habere, quàm minimam menſuram ſæpiùs multiplicatam aſ<lb></lb>ſumere. </s> </p> <p type="main"> <s id="s.001712">Sed quoniam adhuc incommodum accideret tot habere <lb></lb>menſuras, quæ juxta ſeriem naturalem numerorum creſcerent, <lb></lb>ut propoſitæ paucitatis examinandæ quantitas indagetur, ob<lb></lb>ſervatum eſt non leve compendium, quod offert progreſſio <lb></lb>Geometrica ab unitate incipiens, & in Ratione dupla aut tri<lb></lb>plâ progrediens. </s> <s id="s.001713">Nam maximum terminum progreſſionis du<lb></lb>plæ ſibimet ipſi additum ſi mulctaveris unitate, & in progreſ<lb></lb>ſione triplâ maximo termino unitate mulctato ſi reſidui ſemiſ<lb></lb>ſem addideris, numerum habebis gravitatum omnium, quæ <lb></lb>paucis illis ponderibus examinari poſſunt. </s> <s id="s.001714">Sic dentur octo pon<lb></lb>dera in Ratione duplâ incipiendo ab uncia 1; octavum eſt <lb></lb>unc. </s> <s id="s.001715">128: hunc numerum duplica, & à 256 aufer unitatem, <lb></lb>reliquus numerus 255 indicat octo illis ponderibus poſſe in li<lb></lb>brâ examinari omnes gravitates ab uncia 1 ad uncias 255. Si<lb></lb>mili modo in Ratione triplâ dentur quatuor pondera 1. 3. 9. 27. <pb pagenum="232" xlink:href="017/01/248.jpg"></pb>aufer ab ultimo unitatem, remanet 26, cujus ſemiſſis 13 addi<lb></lb>tus numero 27 dat 40: cujus igitur gravitatis eſt primum pon<lb></lb>dus ut 1, tot gravitates uſque ad 40 examinari poſſunt illis ſolis <lb></lb>quatuor ponderibus. </s> <s id="s.001716">Præſtat autem uti ponderibus in Ratio<lb></lb>ne duplâ, quia licèt plura pondera requirantur, omnia tamen <lb></lb>ſeorſim in propriâ libræ lance collocantur: at ſi Ratio ponde<lb></lb>rum ſit tripla, aliquâ commutatione uti neceſſe eſt, ut in ad<lb></lb>jecta Tabella obſervabis, quæ uſque ad numerum 40. exten<lb></lb>ditur: Ubi etiam vides in Ratione triplâ ſufficere quatuor pon<lb></lb>dera 1. 3.9. 27, at in duplâ exigi ſex videlicet 1. 2. 4. 8. 16. 32. </s> </p> <p type="table"> <s id="s.001717">TABELLE WAR HIER</s> </p> <pb pagenum="233" xlink:href="017/01/249.jpg"></pb> <p type="table"> <s id="s.001718">TABELLE WAR HIER</s> </p> <p type="main"> <s id="s.001719">At contingere poteſt paratis hiſce ponderibus in Ratione <lb></lb>duplâ aut triplâ aliquid abundare, & maximum terminum cæ<lb></lb>teris additum excedere quæſitum numerum, (ut hic, ſi opus <lb></lb>eſſet provenire ſolum ad 40, maximus terminus 32 eſt abun<lb></lb>dans) proptereà retentâ cæterorum ſummâ adde aliud pondus, <lb></lb>ut quæſitum numerum compleat, & eſt illud, quo opus eſt; <lb></lb>ſic 1. 2. 4. 8. 16. conficiunt ſummam 31; aufer 31 ex 40, reſi<lb></lb>duum eſt 9; ſit igitur ſextum pondus 9, & ſatis erit uſque ad <lb></lb>40; quia cum habeantur reliquis ponderibus omnes numeri <lb></lb>infra 31, jam ex 23 & 9 fit 32, ex 24 & 9 fit 33, & ſic de re<lb></lb>liquis deinceps. </s> <s id="s.001720">Idem dic de aliâ qualibet ſummâ majore <lb></lb>quàm ferant data pondera, minore tamen quàm opus ſit, ſi <lb></lb>adhuc unum pondus in eâdem progreſſione adderetur; ſufficit <lb></lb>enim reſiduum. </s> <s id="s.001721">Exemplum habes in ſuperiore Tabella pon<lb></lb>derum in Ratione triplâ, ubi quatuor conficiunt 40, ſed ſi ad<lb></lb>deretur quintum in eadem Ratione 81, eſſet nimis magnum, <pb pagenum="234" xlink:href="017/01/250.jpg"></pb>ſi ſolùm habere velimus pondera infra 121: quæratur uſque ad <lb></lb>52, & quia inter 40 & 52 differentia eſt 12, quintum pondus <lb></lb>ut 12 ſufficiet. </s> <s id="s.001722">Hinc quia ad libram requiruntur ſolum 24 ſe<lb></lb>munciæ, ad unciam 24 ſcrupuli, ad ſcrupulum 24 grana, ſi <lb></lb>pondera ſint in Ratione triplâ, ſufficiunt tria ponderâ 1. 3.9. <lb></lb>quæ conficiunt 13, & quartum pondus ſit 11, ut compleatur <lb></lb>ſumma 24: & in Ratione duplâ ſufficiunt quatuor pondera <lb></lb>1. 2. 4. 8. quæ conficiunt 15, & quintum pondus 9 complens <lb></lb>ſummam 24. illud eſt, quod requiritur, ut ex adjectis Tabel<lb></lb>lis liquet. </s> </p> <p type="table"> <s id="s.001723">TABELLE WAR HIER</s> </p> <p type="table"> <s id="s.001724">TABELLE WAR HIER</s> </p> <p type="main"> <s id="s.001725">Unum hîc, ubi de Ponderibus ſermo eſt, obiter moneo, libræ <lb></lb>nomen apud Romanos æquivocum fuiſſe, alia enim erat libra <lb></lb>Ponderalis aridorum, alia Menſuralis liquidorum (& potiſſi<lb></lb>mum olei, quod cornu librali metiebantur) quam inciſis & in<lb></lb>ſculptis lineis in uncias 12 partiebantur, quemadmodum & li<lb></lb>bra pondo in uncias pariter 12 diſtinguebatur: ſed inter utram<lb></lb>que libram, ſi materia ipſa ad pondus revocabatur, non exi<lb></lb>guum erat diſcrimen; ut enim ex proprio experimento teſta-<pb pagenum="235" xlink:href="017/01/251.jpg"></pb>tur Galenus lib. 6. cap. 8. <emph type="italics"></emph>de compoſitione medicam. </s> <s id="s.001726">per genera. <emph.end type="italics"></emph.end> <lb></lb>Libra menſura ſolùm uncias decem continebat, quarum li<lb></lb>bra pondo erat duodecim: quapropter uncia menſuralis ad un<lb></lb>ciam ponderalem erat ut 5 ad 6 ſpectatâ gravitate & quantita<lb></lb>te materiæ. <lb></lb></s> </p> <p type="main"> <s id="s.001727"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001728"><emph type="center"></emph><emph type="italics"></emph>Libra inæqualium brachiorum expenditur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001729">USus libræ brachiorum inæqualium minùs neceſſarius eſt, <lb></lb>ac propterea neque communis aut vulgaris, niſi quatenus <lb></lb>ad ſtateram traductus eſt: illam tamen hîc conſiderare erit <lb></lb>operæ pretium, ut æquilibrij rationes magis innoteſcant. </s> <s id="s.001730">Sit <lb></lb>libra AB, cujus centro C <lb></lb><figure id="id.017.01.251.1.jpg" xlink:href="017/01/251/1.jpg"></figure><lb></lb>dividatur jugum in brachia <lb></lb>inæqualia CA & CB. </s> <lb></lb> <s id="s.001731">Certum eſt, etiam ſi nul<lb></lb>lum addatur pondus, ju<lb></lb>gum ex centro C ſuſpen<lb></lb>fum retinere non poſſe po<lb></lb>ſitionem AB horizonti pa<lb></lb>rallelam; quia licet punctum C ſit centrum motûs libræ, non <lb></lb>eſt tamen centrum gravitatis illius; hoc enim eſt in puncto ju<lb></lb>gum (quod hîc æquabiliter ductum ponitur) bifariam dividen<lb></lb>te, videlicet in I, quod æquales gravitates IA & IB cir<lb></lb>cumſtant. </s> <s id="s.001732">Verùm interim ex hypotheſi fingamus lineam AB <lb></lb>omni gravitate carentem; & in ipſis libræ extremitatibus ſta<lb></lb>tuamus pondera eam inter ſe reciprocè Rationem habentia, <lb></lb>quæ eſt Ratio brachiorum, & ut CA ad CB, ita ſit pondus B <lb></lb>ad pondus A. </s> <s id="s.001733">Pondera hæc, quæ in lancibus libræ vulgaris <lb></lb>æqualium brachiorum magnam momentorum inæqualitatem <lb></lb>haberent, quia inæqualiter gravia, hîc æquilibrium conſti<lb></lb>tuunt, quamvis inæquales ſint eorum gravitates abſolutæ, quia <lb></lb>libræ brachia reciprocè: ſecundùm eandem Rationem in<lb></lb>æqualia: quatenus enim alligantur pondera hæc extremita-<pb pagenum="236" xlink:href="017/01/252.jpg"></pb>tibus libræ, æqualia obtinent momenta, nec jugum AB <lb></lb>poteſt in alterutram partem inclinari, cum neutrum pon<lb></lb>dus poſſit ab altero aſſumere vim, qua ſursùm moveatur, <lb></lb>majorem oppoſitâ virtute innatá deſcendendi, qua repu<lb></lb>gnat, ne elevetur. </s> <s id="s.001734">Sit CA ad CB ut 1 ad 4, & viciſſim pon<lb></lb>dus B ut 1 ad pondus A ut 4. Si gravitates dumtaxat con<lb></lb>ſiderentur, virtus ponderis A eſt ut 4, virtus verò ponderis B <lb></lb>ut 1: ſed quia à centro motûs C retinentur, nec liberè rectâ viâ <lb></lb>moveri poſſunt, impedimentum recipiunt pro brachiorum lon<lb></lb>gitudine, minûſque impeditur deſcenſus aut aſcenſus rectus <lb></lb>ponderis, quod longiori brachio adjacet, magis, quod brevio<lb></lb>ri. </s> <s id="s.001735">Illud igitur pondus, quod majori brachio adnectitur, ſi <lb></lb>deſcendat, magis deſcendit, ſi aſcendat, magis aſcendit; quod <lb></lb>verò breviori, ſi aſcendat, minùs aſcendit, & ſi deſcendat, <lb></lb>minùs deſcendit: atque adeò ſi B deſcenderet in E, menſura <lb></lb>deſcensùs eſſet perpendicularis EG, aſſenſum autem ponderis <lb></lb>A in D metiretur perpendicularis DF: idem dic ſi A deſcen<lb></lb>deret, & B aſcenderet. </s> <s id="s.001736">Porrò DF & EG ſunt in Ratione <lb></lb>brachiorum CA & CB ut patet, quia triangula rectangula <lb></lb>CFD, & CGE, præter rectos angulos ad F & G æquales, ha<lb></lb>bent etiam æquales ad C angulos ad verticem, & per 32. lib. 1. <lb></lb>ſunt æquiangula; igitur per 4 lib. 6. ut CD ad CE, ita DF <lb></lb>ad EG; at CD æqualis eſt ipſi CA, & CE ipſi CB (eſt enim <lb></lb>eadem linea, quæ mutatâ poſitione AB venit in DE) igitur <lb></lb>ut CA ad CB ita DF ad EG. </s> <s id="s.001737">Quare ratione poſitionis pon<lb></lb>dus B vim habet deſcendendi, & reſiſtit aſcenſui, ut 4, pon<lb></lb>dus autem A vim habet deſcendendi, ac proinde etiam re<lb></lb>ſiſtendi, ne aſcendat, ſolùm ut 1. </s> </p> <p type="main"> <s id="s.001738">Cum itaque momentum deſcendendi (idem eſto judicium <lb></lb>de momento repugnantiæ, ne aſcendat) componatur tùm ex <lb></lb>gravitate ponderis, tùm ex propenſione ad motum, hoc eſt ex <lb></lb>motûs, qui conſequi poſſet, velocitate, manifeſtum eſt gravi<lb></lb>tatem ut 4, cujus motus eſſet ut 1, nec poſſe vincere gravitatem <lb></lb>ut 1, cujus motus eſſet ut 4, nec viciſſim poſſe ab illâ vinci; <lb></lb>eſt ſiquidem inter gravitatem quadruplum ſemel, & gravita<lb></lb>tem ſubquadruplam quater Ratio æqualitatis; victoria autem <lb></lb>obtineri non poteſt, niſi intercedat virium inæqualitas. </s> <s id="s.001739">Si <lb></lb>enim pondera eſſent æqualia, ponderis A reſiſtentia ratione <pb pagenum="237" xlink:href="017/01/253.jpg"></pb>motûs eſſet ſubquadrupla, ſed quadruplicatur ratione gravita<lb></lb>tis, ergo reſiſtentia eſt æqualis: item ſi longitudines eſſent <lb></lb>æquales, reſiſtentia ponderis B eſſet ſubquadrupla ratione <lb></lb>gravitatis, ſed quadruplicatur ratione diſtantiæ CB; ergo in B <lb></lb>eſt æqualis. </s> </p> <p type="main"> <s id="s.001740">Neutrum igitur pondus poteſt oppoſito ponderi impetum <lb></lb>imprimere, quo elevetur; quia nimirum unaquæque gravitas <lb></lb>majorem impetum alteri communicare non poteſt, quàm poſ<lb></lb>ſit ipſa concipere, ac propterea impetus gravitatis B, quæ eſt <lb></lb>ut CA, potens conari deorſum ut GE, ſi imprimeretur gravi<lb></lb>tati A, quæ eſt ut CB, deberet illam elevare ut FD: Atqui <lb></lb>gravitas ipſius A, quæ eſt ut CB, conatur deorsùm ut FD, & <lb></lb>ejus impetus ſi gravitati B, quæ eſt ut CA, imprimeretur, il<lb></lb>lam elevare deberet ut GE: igitur in unaquâque gravitate <lb></lb>æqualis eſſet ejuſdem conatus deorsùm & vis illata nitens ſur<lb></lb>sùm, nec plus præſtare poſſet impetus impreſſus, quàm innatus. </s> <lb></lb> <s id="s.001741">Utraque igitur conſiſtere debet, & neutra impetum acquirit, <lb></lb>aut ab alterâ impetum accipit, quia fruſtra eſſet impetus acqui<lb></lb>ſitus aut impreſſus, quem nullus conſequi poteſt motus. </s> <s id="s.001742">Quare <lb></lb>cum eadem ſit gravitatum Ratio ut CA ad CB, atque motuum <lb></lb>reciprocè ut FD ad GE, ex 16 lib. 6. rectangulum ſub extre<lb></lb>mis CA, hoc eſt pondere B, ut 1, & motu GE, ut 4, æquale <lb></lb>eſt rectangulo ſub mediis CB, hoc eſt pondere A ut 4, & mo<lb></lb>tu FD ut 1: ſunt igitur æqualia momenta, quæ componuntur <lb></lb>ex gravitate ut 1 & motu ut 4, atque ex gravitate ut 4 & <lb></lb>motu ut 1. </s> </p> <p type="main"> <s id="s.001743">Ex his apertiſſimè liquet, cur ſuperiori capite tantopere in<lb></lb>culcata ſit brachiorum æqualitas in libræ jugo, ut ex æquili<lb></lb>brio innoteſcat propoſiti ponderis ignota gravitas; hæc enim <lb></lb>æqualis cenſetur notæ gravitati, ubi cùm oblato pondere illa <lb></lb>æquâ lance libratur: quia ſcilicet, ſi inæqualia eſſent brachia, <lb></lb>inæquales eſſent propenſiones ad motum, ſeu motuum veloci<lb></lb>tates, quæ ad componendam momentorum Rationem concur<lb></lb>runt; adeóque fieri non poſſet, ut æquales eſſent gravitates in <lb></lb>lancibus; nam minor gravitas ex brachio longiore plus habet <lb></lb>momenti, quàm ex breviore, pro ratione inæqualitatis brachio<lb></lb>rum. </s> <s id="s.001744">Verum eſt libram hujuſmodi brachiorum inæqualium <lb></lb>vacuam poſſe priùs ad æquilibritatem reduci, deinde, illâ ſic <pb pagenum="238" xlink:href="017/01/254.jpg"></pb>æquilibri conſtitutâ poſſe lancibus imponi Reciprocè pondera <lb></lb>pro Ratione inæqualium brachiorum, & ex æquilibrio argui <lb></lb>ponderum illorum Rationem, non tamen æqualitatem: ſedar<lb></lb>tificium hoc, quod peritioribus nihil officeret, anſam non mo<lb></lb>dicam furacibus, & doloſis mercatoribus præberet decipiendi <lb></lb>imperitos; quamvis enim libræ hujuſmodi æquilibri impoſitis, <lb></lb>hinc & hinc ponderibus adhuc fieret æquilibrium, ſignum <lb></lb>quidem eſſet æqualibus momentis addita eſſe æqualia momen<lb></lb>ta gravitatis, non tamen verùm eſſet additas eſſe æquales gra<lb></lb>vitates, ut rudioribus fortaſſe videretur. </s> <s id="s.001745">Hinc eſt libram bra<lb></lb>chiorum inæqualium in uſu non eſſe, ne locus pateat dolis. </s> </p> <p type="main"> <s id="s.001746">Dixi autem expreſsè priùs ſtatuendam eſſe libræ vacuz <lb></lb>æquilibritatem, deinde ſumenda pondera reciprocè pro Ratio<lb></lb>ne longitudinis brachiorum: niſi etenim priùs æquilibritas illa <lb></lb>ſtatueretur, ſi pondera impoſita eſſent reciprocè in Ratione <lb></lb>longitudinis brachiorum, ſemper pondus minus additum bra<lb></lb>chio longiori præponderaret, quia etiam ipſa brachij longioris <lb></lb>gravitas ſua habet momenta, & quidem non modica, majora <lb></lb>momentis brachij brevioris, quæ omninò computanda ſunt: <lb></lb>nam ſi ponderum in ea Ratione reciprocè poſitorum momenta <lb></lb>ſint æqualia, illiſque adjiciantur inæqualia gravitatis bra<lb></lb>chiorum momenta, manifeſtum eſt momentorum ſummam, cui <lb></lb>plus additur, majorem eſſe reliquâ, cui additur minus. </s> </p> <p type="main"> <s id="s.001747">Sed quænam ſunt, & quanta utriuſque brachij momenta? </s> <lb></lb> <s id="s.001748">Ut hæc inveſtigemus, & certâ ratione definiamus, ponamus <lb></lb>jugum ipſum ſecundùm ſuas omnes partes uniuſmodi, & gravi<lb></lb>tatem æquabiliter fuſam per totam illius longitudinem. </s> <s id="s.001749">Sit igi<lb></lb><figure id="id.017.01.254.1.jpg" xlink:href="017/01/254/1.jpg"></figure><lb></lb>tur datum priſma AB, quod <lb></lb>in quinque partes æquales <lb></lb>dividatur, ſingulas pondoli<lb></lb>bram unam; & per ſingula <lb></lb>gravitatis centra ducatur <lb></lb>recta <emph type="italics"></emph>a u<emph.end type="italics"></emph.end>: fiatque ſecun<lb></lb>dùm rectam HI, à qua pars <lb></lb>una C abſcinditur à reliquis, totius priſmatis ſuſpenſio, ita ut <lb></lb>centrum motûs ſit in S. </s> <s id="s.001750">Proculdubio unaquæque pars à cæteris <lb></lb>ſejuncta ſi appenderetur ſecundùm longitudinem jugi <emph type="italics"></emph>a u,<emph.end type="italics"></emph.end><lb></lb>quod infigeretur per centra gravitatum <emph type="italics"></emph>a, e, i, o, u,<emph.end type="italics"></emph.end> obtineret <pb pagenum="239" xlink:href="017/01/255.jpg"></pb>fuum momentum juxtà diſtantiam centri ſuæ gravitatis à <lb></lb>centro motûs. </s> <s id="s.001751">Quid autem refert (quod quidem attinet ad <lb></lb>hanc momentorum Rationem) ſi in unum continuum corpus <lb></lb>unitæ illæ partes coagmententur, an verò diviſæ ſolo contactu <lb></lb>ſibi invicem adhæreant? </s> <s id="s.001752">eadem quippe eſt gravitas ſingulis in<lb></lb>ſita, eadem ſingularum à centro diſtantia. </s> <s id="s.001753">Cum itaque centra <lb></lb>gravitatum <emph type="italics"></emph>a<emph.end type="italics"></emph.end> & <emph type="italics"></emph>e<emph.end type="italics"></emph.end> æqualiter diſtent ab S centro motûs, partes <lb></lb>C & D æquiponderant: at diſtantia <emph type="italics"></emph>S i<emph.end type="italics"></emph.end> tripla eſt diſtantiæ <emph type="italics"></emph>S a<emph.end type="italics"></emph.end>; <lb></lb>ergo momentum partis E triplum eſt momenti partis C; ſimi<lb></lb>lique ratione pars F habet momentum quintuplum, & pars G <lb></lb>ſeptuplum. </s> <s id="s.001754">Igitur componendo, momentum totius aggregati <lb></lb>quatuor partium D, E, F, G, eſt ſedecuplum momenti partis <lb></lb>C; neque enim ſingulæ partes ex hoc quod cum cæteris pen<lb></lb>deant, illiſque cohæreant, ſuum amittunt momentum. </s> <s id="s.001755">Hinc <lb></lb>fit momenta brachiorum eſſe inter ſe ut Quadrata longitudi<lb></lb>num eorumdem brachiorum: ſiquidem oſtenditur ſingularum <lb></lb>partium momentum creſcere ſecundùm Rationem numero<lb></lb>rum imparium, prout ſecundùm eandem Rationem creſcunt <lb></lb>diſtantiæ centrorum gravitatis illarum. </s> <s id="s.001756">Sic brachiorum <lb></lb>longitudines ſi eſſent in Ratione 2 ad 7, illorum momenta <lb></lb>ratione ſuæ gravitatis innatæ & ratione poſitionis eſſent ut 4 <lb></lb>ad 49. </s> </p> <p type="main"> <s id="s.001757">Hæc Ratio momentorum in Ratione Quadratorum longi<lb></lb>tudinis, ſi res attentè perpendatur, omnibus eſt manifeſta: <lb></lb>Nam ſingulorum brachiorum gravitates juxta hypotheſim <lb></lb>æquabiliter fuſæ per totum libræ jugum Rationem inter ſe <lb></lb>habent, quam illorum longitudinis propenſiones ad motum, <lb></lb>ſeu, quod eòdem recidit, diſtantiæ à centro motûs eandem <lb></lb>pariter Rationem habent, quam brachiorum longitudines: <lb></lb>Quoniam igitur (ut ſæpiùs dictum eſt, ſæpiúſque iterùm <lb></lb>inculcandum) momenta componuntur ex gravitatibus ratio<lb></lb>ne materiæ, & ex propenſionibus ad motum ratione ſitûs ſeu <lb></lb>poſitionis, componuntur duæ Rationes longitudinum; atque <lb></lb>adeó momentum unius brachij ad momentum alterius bra<lb></lb>chij eſt in duplicata Ratione ſuarum longitudinum, hoc <lb></lb>eſt, ut ipſarum longitudinum Quadrata. </s> <s id="s.001758">Id quod adhuc ul<lb></lb>teriùs ſic explicari poſſe videtur. </s> <s id="s.001759">Sit libræ jugum M. N, & <lb></lb>motûs centrum O: intelligatur moveri, ut obtineat poſitio-<pb pagenum="240" xlink:href="017/01/256.jpg"></pb>nem PR. </s> <s id="s.001760">Momentum brachij minoris OM referre videtur <lb></lb>ſector MOP, momentum verò brachij majoris ON referre <lb></lb><figure id="id.017.01.256.1.jpg" xlink:href="017/01/256/1.jpg"></figure><lb></lb>videtur ſector NOR; ſingularum <lb></lb>quippe partium motus ab arcu <lb></lb>deſcriptus illarum momentum ob <lb></lb>oculos ponit, & totius brachij mo<lb></lb>mentum illius motus, ſcilicet ſector <lb></lb>in motu deſcriptus. </s> <s id="s.001761">At ob æquali<lb></lb>tatem angulorum ad verticem in <lb></lb>O, ſectores MOP, NOR ſunt ſi<lb></lb>miles, &, quia uterque ſector eſt <lb></lb>ſimilis pars ſui circuli, eam inter ſe habent ſectores Rationem, <lb></lb>quæ eſt circulorum, per 15.lib.5. circuli autem ſunt in dupli<lb></lb>catâ Ratione diametrorum, ex 2.lib.12. ſeu Radiorum OM <lb></lb>& ON; igitur & ſectores ſunt in duplicatâ Ratione OM ad <lb></lb>ON, hoc eſt quadrati OM ad quadratum ON. </s> </p> <p type="main"> <s id="s.001762">At quæris. </s> <s id="s.001763">In propoſito priſmate AB, momentum brachij <lb></lb>SA ad momentum brachij SB eſt ut 1 ad 16: An, ut ha<lb></lb>beatur æquilibrium in S, addendum erit in A pondus libra<lb></lb>rum 15? quandoquidem pars C eſt libræ unius, reliquum au<lb></lb>tem brachium lib. 4, & longitudo SB eſt quadrupla longitu<lb></lb>dinis SA. </s> </p> <p type="main"> <s id="s.001764">Hoc ſanè non eſt iis, quæ dicta ſunt, conſequens, necex illis <lb></lb>efficitur: aliud quippe eſt momenta brachiorum eſſe ut 1 ad 16, <lb></lb>aliud verò perinde ſe habere, atque ſi ex brachiorum gravita<lb></lb>te carentium extremitatibus penderent libræ 1 & 16, ut ad <lb></lb>æquilibrium conſtituendum opus ſit breviori brachio addere <lb></lb>libras 15. Primum illud verum eſt, etiam ſi extremitatibus ad<lb></lb>necti intelligamus hinc quidem libræ ſemiſſem; hinc verò li<lb></lb>bras octo, mane ſcilicet eadem Ratio 1 ad 16. Alterum à for<lb></lb>mà veritatis prorsùs alienum videtur, nam licet libræ 4 in ex<lb></lb>tremitate B poſitæ æquivaleant libræ unciæ ſimul cum pondere <lb></lb>lib.15. in extremitate A; non eſt tamen eadem ratio librarum 4 <lb></lb>ſecundùm longitudinem brachij SB diſtributarum; quo enim <lb></lb>propiores ſunt partes centro motûs, eò minus habent mo<lb></lb>menti: non igitur libræ 4 ſic diſtributæ æquivalent libris <lb></lb>16, nec addendum erit pondus librarum 15 in oppoſitâ extre<lb></lb>mitate ad æquilibrium conſtituendum, quandoquidem nec ipſa <pb pagenum="241" xlink:href="017/01/257.jpg"></pb>unica libra partis C tantumdem habet momenti, quantum ha<lb></lb>beret ſi totâ ex A penderet. </s> </p> <p type="main"> <s id="s.001765">Equidem ex his, quæ paulò ante dicebam de ſectoribus re<lb></lb>ferentibus momenta brachiorum, aliquando eò deveni, ut ſuſ<lb></lb>picarer totam gravitatem brachij ON (idem dic de reliquo <lb></lb>OM) intelligendam eſſe ibi exercere totum momentum, ubi <lb></lb>eſt quaſi centrum omnium ſuorum momentorum, hoc eſt, ubi <lb></lb>momenta bifariam dividuntur. </s> <s id="s.001766">Si autem ſector NOR refert <lb></lb>totum momentum brachij ON; non eſt intelligendum cen<lb></lb>trum hoc momentorum eſſe punctum L, ubi eſt ſemiſſis bra<lb></lb>chij ON; quia Sector LOQ ad Sectorem NOR eſt in Ra<lb></lb>tione Quadrati OL ad Quadratum ON, quod eſt illius qua<lb></lb>druplum. </s> <s id="s.001767">Quod ſi inter OL & ON ſumatur media propor<lb></lb>tionalis OV, jam ſector VOT eſt ad Sectorem NOR in du<lb></lb>plicatâ Ratione Radiorum OV, & ON, hoc eſt ut OL ad <lb></lb>ON, hoc eſt ut 1 ad 2; ac propterea Sector VOT æqualis eſt <lb></lb>Trapezio NVTR; proinde in V videbantur diviſa æqualiter <lb></lb>momenta, Hinc arguebam vel totam brachij gravitatem cen<lb></lb>ſendam eſſe ſua exercere momenta in puncto diſtantiæ à centro <lb></lb>motûs mediæ proportionalis inter ſemiſſem brachij & totam <lb></lb>brachij longitudinem, vel in extremitate brachij cenſen<lb></lb>dam eſſe pendere gravitatem, quæ medio loco proportiona<lb></lb>lis ſit inter totam brachij ejuſdem gravitatem & ejus ſe<lb></lb>miſſem. </s> </p> <p type="main"> <s id="s.001768">Verùm, ut quod res eſt ſincerè eloquar, quamvis in Secto<lb></lb>ribus illis, quos paulò ante commemorabam, imaginem <lb></lb>quandam momentorum gravitatis ſecundùm brachiorum <lb></lb>longitudinem diſtributæ agnoſcerem, non tamen in re <lb></lb>Phyſicâ ſatis fidebam Geometricæ illi commentationi: quip<lb></lb>pe qui obſervabam à Sectoribus quidem poni ob oculos Ra<lb></lb>tionem momentorum ſingulorum brachiorum ex motu, qui <lb></lb>idem eſt, ſivè multa, ſivè modica ſit gravitas, ſivè in uno, <lb></lb>ſivè in alio puncto conſtituta intelligatur, non tamen defi<lb></lb>niri ipſius gravitatis momenta. </s> <s id="s.001769">Quare ſatius duxi ad experi<lb></lb>menta potiùs confugere, ut hinc lux aliqua ſuboriretur, qua <lb></lb>gravitatis quæſita momenta innoteſcerent. </s> </p> <p type="main"> <s id="s.001770">Primùm igitur aſſumptus eſt ligneus cylindrus, cujus dia<lb></lb>meter CE unc. </s> <s id="s.001771">1. 06″ pedis Romani antiqui, & addito in A <pb pagenum="242" xlink:href="017/01/258.jpg"></pb>pondere D unciarum 40 1/2 collocatus eſt in æquilibrio, quod <lb></lb>factum eſt in B puncto. </s> <s id="s.001772">Fuit autem longitudo BA unciarum <lb></lb><figure id="id.017.01.258.1.jpg" xlink:href="017/01/258/1.jpg"></figure><lb></lb>pedis Romani 7 2/5 BC ve<lb></lb>rò unc.(42 17/50). Reſecto de<lb></lb>mùm ſubtiliſſimè cylindro, <lb></lb>repertum eſt pondus AB <lb></lb>unciarum 2 1/8, pondus an<lb></lb>tem BC unc. </s> <s id="s.001773">13 1/2. Hisob<lb></lb>ſervatis cum nullus dubitarem, quin momenta brachiorum <lb></lb>eſſent ut quadrata longitudinum, ipſas longitudines AB <lb></lb>unc. </s> <s id="s.001774">7 2/5, & BC unc.(42 17/50) ad unicam <expan abbr="denominationẽ">denominationem</expan> reduxi, vi<lb></lb>delicet (370/50) & (2117/50): & aſſumptis numeratorum Quadratis 136900 <lb></lb>atque 4481689 hanc poſui Rationem momentorum. </s> <s id="s.001775">Tùm ſic <lb></lb>ratiocinatus ſum Algebricè; ut 136900 ad 4481689, ita mo<lb></lb>mentum BA 1 ℞ ad 32.73″ ℞ momentum BC. </s> <s id="s.001776">Cum igitur <lb></lb>æqualitas eſſet inter momentum brachij BC, & momentum <lb></lb>brachij BA plus ipſo pondere D; hæc enim conſtituebant <lb></lb>æquilibrium, æquatio Algebricè eſt inter momentum BC <lb></lb>32. 73″ ℞ & BA + D, hoc eſt 1 ℞ + unc. </s> <s id="s.001777">40 1/2: & per An<lb></lb>titheſim demptâ utrinque 1 ℞, æquatio eſt inter 37. 73″ ℞ & <lb></lb>unc. </s> <s id="s.001778">40 1/2. Factâ itaque numeri abſoluti 40 1/2 diviſione per nu<lb></lb>merum Radicum prodit pretium 1 ℞ pondo unc.1.27″, quod eſt <lb></lb>momentum brachij BA; ac proinde momentum brachij BC: <lb></lb>eſt pondo unc.41. 57″. </s> <s id="s.001779">Quare perinde eſt atque ſi gravitas <lb></lb>unc. </s> <s id="s.001780">1. 27″ poneretur in extremitate Alineæ Mathematicæ, ac <lb></lb>in extremitate C poneretur gravitas unc. </s> <s id="s.001781">41. 57″. </s> <s id="s.001782">At in A fuit <lb></lb>additum pondus unc. </s> <s id="s.001783">40 1/2: ergo momentum brachij BC æqui<lb></lb>valet ponderi D, & præterea unc.1.07″, qui eſt ſemiſſis gravitatis <lb></lb>brachij AB obſervatæ unc. </s> <s id="s.001784">2 1/8, hoc eſt in centeſimis paulò ul<lb></lb>tra 2. 12″. </s> <s id="s.001785">Si verò momentis brachij BA pondo unc. </s> <s id="s.001786">1.27″ ad<lb></lb>datur gravitas D pondo unc. </s> <s id="s.001787">40. 50″, fit aggregatum 41.77″, <lb></lb>quod excedit inventum momentum brachij BC unc.41.57″. <lb></lb>exceſſu (20/100) unciæ: quæ diſcrepantia facillimè potuit oriri ex <lb></lb>aliquâ exili, ac minime notabili differentiâ vel in dimetiendis <lb></lb>brachiorum longitudinibus, vel in ponderandis eorum gravi<lb></lb>tatibus; cum maximè reſegmina illa, & ſcobs, non computa<lb></lb>rentur in gravitate. </s> <s id="s.001788">Quod ſi fiat ut longitudo BC 2117 ad <pb pagenum="243" xlink:href="017/01/259.jpg"></pb>longitudinem AB 370, ita pondus in A unc.41.77″ ad pon<lb></lb>dus in B unc. </s> <s id="s.001789">7. 30″, conſtat eſſe ferè ſemiſſem gravitatis <lb></lb>unc. </s> <s id="s.001790">13 1/2: ſed eſt exceſſus ſemunciæ ob minùs accuratam ob<lb></lb>ſervationem. </s> </p> <p type="main"> <s id="s.001791">Qua propter aliud experimentum quàm accuratiſſimè inſti<lb></lb>tui ligneo parallelepipedo, cujus longitudo palmorum Roma<lb></lb>norum 7. unc.6. 566tʹ, ejus verò pondus lib. 1. unc.1 1/4. Alte<lb></lb>ri extremitati additus eſt <lb></lb><figure id="id.017.01.259.1.jpg" xlink:href="017/01/259/1.jpg"></figure><lb></lb>plumbeus cylindrus ad per<lb></lb>pendiculum pendens, cujus <lb></lb>pondus unc. </s> <s id="s.001792">20. Impoſitum <lb></lb>eſt parallelepipedum rotun<lb></lb>do claviculo ferreo, qui horizonti parallelus erat, & factum <lb></lb>eſt æquilibrium in puncto, ubi tota longitudo in duas partes <lb></lb>dividebatur, quarum minor ponderi adhærens fuit menſurâ <lb></lb>unc. </s> <s id="s.001793">18 1/6, partes verò major fuit menſurâ palm. </s> <s id="s.001794">6. unc.2/5. Cum <lb></lb>itaque longitudo CB obſervata fuerit unciarum menſuralium <lb></lb>72. 40″, & AC unciarum menſuralium 18. 16″, in eadem <lb></lb>pariter Ratione ponuntur brachiorum gravitates abſolutæ. </s> <lb></lb> <s id="s.001795">Quare CB pondo unc. </s> <s id="s.001796">1059, AC verò pondo unc. </s> <s id="s.001797">2. 66″. </s> <lb></lb> <s id="s.001798">Igitur ut longitudinis BC quadratum 52417600 ad longitudi<lb></lb>nis AC quadratum 3297856, ita momentum BC 1 ℞ ad <lb></lb>(3297856/52417600) ℞ momentum brachij AC: cui additur cylindrus D <lb></lb>unc.20: Eſt ergo æquatio inter AC + D, hoc eſt (3297856/52417600) ℞ + <lb></lb>unc. </s> <s id="s.001799">20.00″ & 1 ℞; & factâ Antitheſi eſt æquatio inter <lb></lb>unc. </s> <s id="s.001800">20.00″ & (49119744/52417600) ℞: demum inſtitutâ diviſione conſurgit <lb></lb>pretium 1 ℞, hoc eſt momentum BC, unc. </s> <s id="s.001801">21. 342tʹ & paulo <lb></lb>amplius: atque momentum brachij AC eſt pondo unc.1.343tʹ, <lb></lb>cui additâ gravitate cylindri fit ſumma unc. </s> <s id="s.001802">21. 343tʹ planè <lb></lb>æqualis momento brachij BC. </s> </p> <p type="main"> <s id="s.001803">Et ut hanc operandi methodum confirmarem, iterum inſti<lb></lb>tui argumentationem aſſumendo quadrata gravitatum utriuſ<lb></lb>que brachij, ſunt enim ex hypotheſi gravitates in Ratione lon<lb></lb>gitudinum. </s> <s id="s.001804">Cum igitur ſit CB pondo unc. </s> <s id="s.001805">10. 50″; & AC <lb></lb>pondo unc. </s> <s id="s.001806">2. 66.″ fiat ut quadratum CB 1121481 ad quadra<lb></lb>tum AC 70756, ita ipſius CB momentum 1 ℞ ad (70756/1121481) ℞ <lb></lb><expan abbr="momentũ">momentum</expan> ipſius AC. </s> <s id="s.001807">Quoniam verò AC + D hoc eſt (70756/1121481) ℞ <pb pagenum="244" xlink:href="017/01/260.jpg"></pb>+ unc. </s> <s id="s.001808">20.00″ æquatur momento BC hoc eſt 1 ℞, factâ per <lb></lb>Antitheſin communi ſubtractione (70756/1121481) ℞, remanet æquatio <lb></lb>inter pondus unc. </s> <s id="s.001809">20.00″ & (1050725/1121481) ℞, & factâ diviſione emer<lb></lb>git pretium 1 ℞, hoc eſt momentum BC pondo unc. </s> <s id="s.001810">21. 347tʹ. </s> <lb></lb> <s id="s.001811">atque adeò momentum ipſius AC eſt pondo unc. </s> <s id="s.001812">1. 347″; cui <lb></lb>ſi addatur cylindri D gravitas unc. </s> <s id="s.001813">20, totum momentum in A <lb></lb>eſt unc. </s> <s id="s.001814">21. 347tʹ, omnino æquale momento ipſius B: id quod <lb></lb>ab initio vix ſperare audebam, cum hæc operatio à ſuperiore <lb></lb>differat ſolùm per (1/1000). Hîc pariter brachij AC gravitas abſo<lb></lb>luta pondo unc. </s> <s id="s.001815">2. 66″. habet momentum unc. </s> <s id="s.001816">1. 347tʹ, cum <lb></lb>ejus ſemiſſis ſit unc. </s> <s id="s.001817">1. 330tʹ, quæ eſt minima atque prorsùs <lb></lb>contemnenda differentia: quî enim fieri potuit, ut, quantali<lb></lb>bet adhiberetur diligentia in metiendo, & ponderando, ne <lb></lb>pilum quidem à verò aberrarem? </s> <s id="s.001818">aut quis omninò certus ſit <lb></lb>omnes parallelepipedi partes æquali prorsùs fuiſſe præditas gra<lb></lb>vitate, itaut quæ pars ad arboris radicem vergebat, non fuerit <lb></lb>paulò denſior, aut interiùs nodulum aliquem latentem habue<lb></lb>rit, quo factum fuerit, ut vera gravitas inſtituto calculo non <lb></lb>exactiſſimè reſponderet? </s> <s id="s.001819">ſimili ratione ſemiſſis gravitatis bra<lb></lb>chij BC intelligitur in extremitate B: nam fiat ut longitudo <lb></lb>BC 72. 40″ ad longitudinem AC 18.16″, ita reciprocè pon<lb></lb>dus in A unc. </s> <s id="s.001820">21. 347tʹ ad pondus in B unc. </s> <s id="s.001821">5. 354tʹ: erat au<lb></lb>tem brachij BC gravitas abſoluta unc. </s> <s id="s.001822">10. 59″ cujus, ſemiſſis <lb></lb>5. 295tʹ. </s> <s id="s.001823">differt ab invento pondere ſolùm per (50/1000) unciæ, hoc <lb></lb>eſt ferè ſeſquiſcrupulum, ſeu grana 34. </s> </p> <p type="main"> <s id="s.001824">Ex his quidem ſatis apparebat brachij gravitatem in libræ <lb></lb>jugo intelligendam eſſe, quaſi ejus ſemiſſis in ipsâ extremitate <lb></lb>conſtitueretur, ſeu, quod idem eſt, tota gravitas brachij ad <lb></lb>mediam longitudinem applicaretur (eadem ſiquidem eſſe mo<lb></lb>menta totius gravitatis in dimidiatâ diſtantiâ, ac dimidiæ gra<lb></lb>vitatis in totâ diſtantiâ, ex ſæpiùs dictis eſt manifeſtum) mihi <lb></lb>tamen ſatisfactum non exiſtimabam, niſi ulteriore experimento <lb></lb>veritatis veſtigia perſequerer. </s> <s id="s.001825">Quare eundem plumbeum cy<lb></lb>lindrum, cujus longitudo erat palmi 1. unc. </s> <s id="s.001826">1. (9/10), ita in extre<lb></lb>mitate A collocavi, ut ſuper AI jaceret, & factum eſt æquili<lb></lb>brium in E, eratque EA longitudo unc. (22 4/10). Tùm diviſo bi<lb></lb>fariam in O ſpatio AI, quod cylindrus jacens occupabat, ex <pb pagenum="245" xlink:href="017/01/261.jpg"></pb>puncto O ſuſpendi cylindrum, & factum eſt pariter æquili<lb></lb>brium exactiſſimè in E, ſicut priùs, cum jacebat ſuper AI. </s> <lb></lb> <s id="s.001827">Deinde cylindrum eumdem iterum parallelepipedo impoſui ja<lb></lb>centem, ſed ea ratione illum ultrò citróque promovebam, ut <lb></lb>omnino propè fulcrum conſiſteret, donec demùm factum eſt <lb></lb>æquilibrium in H, & fuit HA palm.2. unc.(10 7/10): Factâ verò <lb></lb>ſuſpenſione cylindri ex L, ita ut HL eſſet dimidiata cylin<lb></lb>dri jacentis longitudo, æquilibrium pariter in H factum eſt. </s> </p> <p type="main"> <s id="s.001828">Relictâ igitur illâ ſectorum analogiâ, deprehendi per illas <lb></lb>quidem ob oculos poni motum, non verò momentum, ſeu pro<lb></lb>penſionem ad motum, quæ ex diſtantiâ à centro motûs in ipsâ <lb></lb>longitudine definienda eſt: & quod ad gravitatem attinet, nul<lb></lb>lus mihi relictus eſt dubitandi locus ita computandam eſſe to<lb></lb>tius brachij gravitatem per ipſum æquabiliter diffuſam, quaſi <lb></lb>tota in dimidiatâ diſtantiâ à centro motûs collocaretur: quam<lb></lb>vis enim particularum gravium, quæ ultrâ ſemiſſem longitudi<lb></lb>nis magis à centro removentur, momentum creſcat pro Ratio<lb></lb>ne diſtantiæ, reliquarum tamen numero æqualium citrà longi<lb></lb>tudinis ſemiſſem centro propiorum momentum ſimiliter pro <lb></lb>Ratione minoris diſtantiæ minuitur; ac proptereà tantùm iſta <lb></lb>momenta ſimul ſumpta decreſcunt, quantum illa ſimul ſumpta <lb></lb>augentur. </s> <s id="s.001829">Ex quo oritur quædam quaſi æqualitas, perinde at<lb></lb>que ſi momenta omnia majora & minora in illam particulam <lb></lb>confluerent, quæ media eſt Arithmeticè inter extrema (mo<lb></lb>menta ſi quidem ratione diſtantiæ Arithmeticè creſcunt, prout <lb></lb>Arithmeticè ipſa diſtantia creſcit) hæc autem eſt in ſemiſſe <lb></lb>longitudinis brachij. </s> <s id="s.001830">Ex quo iterum confirmatur momenta <lb></lb>brachiorum eſſe ut quadrata longitudinum; ſunt enim in du<lb></lb>plicatâ Ratione illarum; ſemiſſes quippè ſunt in Ratione inte<lb></lb>grarum longitudinum, gravitates ſunt in Ratione earumdem <lb></lb>longitudinum, ergo Ratio compoſita eſt duplicata ejuſdem Ra<lb></lb>tionis longitudinum. </s> </p> <p type="main"> <s id="s.001831">Hinc datâ jugi æquabilis, & uniformis gravitate abſolutâ, <lb></lb>& datâ Ratione longitudinum brachiorum inæqualium libræ, <lb></lb>dividatur data gravitas ſecundùm datam Rationem brachio<lb></lb>rum: tùm fiat ut longitudo minor ad longitudinem majorem, <lb></lb>ita dimidia gravitas majoris brachij ad aliud, ex quo quarto ter-<pb pagenum="246" xlink:href="017/01/262.jpg"></pb>mino invento ſi auferatur dimidia gravitas brachij minoris, re<lb></lb>ſiduum indicabit pondus addendum extremitati brachij mino<lb></lb>ris, ut fiat æquilibrium cum ſolâ gravitate brachij longioris. </s> <lb></lb> <s id="s.001832">Vel potiùs fiat ut quadratum longitudinis brachij minoris ad <lb></lb>differentiam inter quadrata brachiorum, ita ſemiſſis gravitatis <lb></lb>brachij minoris ad pondus ipſi addendum. <lb></lb> </s> </p> <p type="main"> <s id="s.001833"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001834"><emph type="center"></emph><emph type="italics"></emph>Quomodò corporum æquilibria explicentur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001835">QUamvis libro primo plura de Gravitatis centro, prout hu<lb></lb>jus operis inſtituto congruebat, diſputata ſint, eorum ta<lb></lb>men plenior explicatio ex his, quæ duobus præcedentibus ca<lb></lb>pitibus dicta ſunt, petenda eſt, ſi quidem Phyſicam æquilibrij <lb></lb>cauſam noſſe velimus. </s> <s id="s.001836">Neque enim Gravitatis centrum illud <lb></lb>eſt, quod æquales gravitates, ſed quod æquales gravitationes, <lb></lb>aut æqualia gravitatis momenta, hoc eſt æquales ad deſcen<lb></lb>dendum propenſiones ac vires circumſtant. </s> <s id="s.001837">Nam gravitas câ <lb></lb>Ratione per univerſum corpus grave diſtribuitur, quâ Ratio<lb></lb>ne materia ipſa, cui illa ineſt, diffuſa intelligitur; quæ ſi uniuſ<lb></lb>modi ſit & homogenea, ibi centrum habet, ubi eſt molis ipſius <lb></lb>centrum; ubi ſiquidem bifariam moles & materia, ibi pariter <lb></lb>gravitas illi inſita bifariam dividitur. </s> <s id="s.001838">Quoniam verò fieri po<lb></lb>teſt, ac ſæpiùs contingit, materiam quidem corporis & molem <lb></lb>invariatam permanere, figuram autem mutari; ex quo nunc in <lb></lb>hanc, nunc in illam partem migrat gravitatis centrum, quia <lb></lb>alia atque alia fiunt gravitatis momenta pro variâ corporis ſe<lb></lb>cundùm ſuas partes poſitiones; proptereà hujuſmodi momento; <lb></lb>rum æqualitas ex libræ Rationibus deſumenda eſt, ſivè æqua<lb></lb>lium, ſivè inæqualium brachiorum libra intelligatur, prout va<lb></lb>ria corporis gravis ſuſpenſio aut ſuſtentatio contingit. </s> </p> <p type="main"> <s id="s.001839">Sed quia in communi uſu non adeò frequens eſt illa ſuſpen<lb></lb>ſio, qua corpus pendeat quaſi ex puncto lineæ directionis tran<lb></lb>ſeuntis per centrum gravitatis, & ad univerſi centrum de<lb></lb>ductæ, aut illa ſuſtentatio, qua corpus grave acutiſſimo apici <pb pagenum="247" xlink:href="017/01/263.jpg"></pb>incumbat, cui immineat idem gravitatis centrum; quinimmò <lb></lb>ita plerumque ſuſpenditur, aut ſuſtinetur corpus, ut ductâ per <lb></lb>Gravitatis centrum lineâ, aut ex hujus extremitatibus tan<lb></lb>quam polis illud ſuſpendatur, aut ſubjecto fulcro lineæ huic <lb></lb>parallelo illud ſuſtineatur; ideò hujuſmodi lineam per centrum <lb></lb>gravitatis ductam liceat appellare <emph type="italics"></emph>Diametrum Gravitatis<emph.end type="italics"></emph.end>; quæ <lb></lb>diameter quaſi in librâ locum Axis ſeu Aginæ obtinet, corporis <lb></lb>verò partes hinc & hinc poſitæ rationem habent brachiorum <lb></lb>libræ, atque pro diſtantiarum ſeu longitudinum Ratione ſua <lb></lb>habent momenta. </s> <s id="s.001840">Sit propoſitum Trapezium, cujus gravita<lb></lb>tis centrum C puncto reſpondeat, & <lb></lb><figure id="id.017.01.263.1.jpg" xlink:href="017/01/263/1.jpg"></figure><lb></lb>ſuſtineatur ſecundùm rectam lineam <lb></lb>ACN (ſimilis eſſet philoſophandi <lb></lb>ratio, ſi aſſumeretur recta RCS) quæ <lb></lb>proptereà <emph type="italics"></emph>Diameter Gravitatis<emph.end type="italics"></emph.end> à me <lb></lb>dicitur, quia ſicut circuli diameter <lb></lb>per centrum ducta illum in ſemicircu<lb></lb>los æquales diſtinguit, ita hæc per <lb></lb>gravitatis centrum tranſiens dividit <lb></lb>Trapezium in momenta æqualia, itaut in neutram partem in<lb></lb>clinetur, juxta dicta de centro Gravitatis. </s> <s id="s.001841">Sed cur fiat æquili<lb></lb>brium intelliges ex Rationibus libræ Brachiorum inæqualium: <lb></lb>ducatur enim ad rectam AN per C perpendicularis DCE, & <lb></lb>fiunt brachia CD, CE inæqualia; ſunt igitur momenta CE <lb></lb>longioris majora momentis CD brevioris. </s> <s id="s.001842">Ductis verò ipſi <lb></lb>DE parallelis BF & ML, ſecatur diameter gravitatis AN in <lb></lb>punctis H & I: quare inæqualia ſunt brachia HB longius, & <lb></lb>HF brevius, & viciſſim IM eſt brevius, & IL longius: Ex quo <lb></lb>fit momenta in L & E majora eſſe momentis in M & D, at mo<lb></lb>mentum in F minus eſſe momento in B; atque adeò compo<lb></lb>nendo majora cum minoribus ex eâdem parte, fieri compoſi<lb></lb>tum momentum unius partis æquale toti momento oppoſitæ <lb></lb>partis. </s> <s id="s.001843">Vel ſi non placeat particulatim Trapezium diſtinguere <lb></lb>quaſi in tot libras, quot ductæ intelliguntur parallelæ, dic to<lb></lb>tius gravitatis ADN ſemiſſem intelligi in D, & totius gravi<lb></lb>tatis AEN ſemiſſem intelligi in E; & quamvis pars ADN ab<lb></lb>ſolutè & ſeorſim accepta major ſit & gravior parte AEN abſo<lb></lb>lutè ſumptâ, quia tamen ſunt reciprocè in Ratione diſtantia-<pb pagenum="248" xlink:href="017/01/264.jpg"></pb>rum CE & CD, propterea æquilibrium conſtituere; pars enim <lb></lb>minùs gravis ex poſitione majorem habet propenſionem ad mo<lb></lb>tum, qui eſſet velocior; partis verò gravioris minor eſt propen<lb></lb>ſio ad motum, qui eſſet tardior; atque adeò hæc minùs reſiſtit <lb></lb>ratione motûs, magis autem ratione gravitatis; at illa ex adver<lb></lb>ſo magis reſiſtit ratione motûs, ſed minùs ratione gravitatis, <lb></lb>ſervatâ reciprocè eâdem Ratione inter gravitates & motus. </s> <s id="s.001844">Nil <lb></lb>igitur mirum ſi æquatis hinc & hinc viribus agendi, & reſiſten<lb></lb>di ſequatur conſiſtentia. </s> </p> <p type="main"> <s id="s.001845">Hinc manifeſtum eſt, cur mutatâ figurâ centrum gravitatis <lb></lb>ad eam partem transferatur, quæ longiùs à ſuſtentationis vel <lb></lb>ſuſpenſionis loco recedit; quia nimirum creſcunt ex illâ parte <lb></lb>comparatè ad oppoſitam momenta ratione diſtantiæ majoris, ac <lb></lb>proinde, ut fiat momentorum æqualitas, centrum ad illam par<lb></lb>tem ſecedit. </s> <s id="s.001846">Sic ceſpitantes à naturâ docentur in partem op<lb></lb>poſitam illi, in quam inclinantur, brachium illicò extendere, <lb></lb>ut brachij gravitas longiùs à corpore tranſlata plus habeat mo<lb></lb>menti, quàm cùm reliquo corpori adhæret, atque hinc ſequa<lb></lb>tur centri gravitatis in illam partem tranſlatio. </s> <s id="s.001847">Veritas hæc ſa<lb></lb>tis nota eſt ipſis funambulis, cùm corpus univerſum ſuper ex<lb></lb>tento fune librant; neque enim temerè crura & brachia exten <lb></lb>dunt aut contrahunt, ſed certâ lege, ut centrum momento<lb></lb>rum gravitatis totius corporis hac vel illâ ratione diſpoſiti im<lb></lb>mineat, & incumbat funi. </s> <s id="s.001848">Sic plumbeæ virgæ rectæ ex medio <lb></lb>ſuſpenſæ, & in æquilibrio manentis, ſi brachium alterum in<lb></lb>flexeris, fieri non poteſt, ut reliquum brachium rectum ſervet <lb></lb>poſitionem horizonti parallelam, ſed deorſum inclinabitur, qun <lb></lb>cum longius ſit brachio inflexo, majora habet momenta ac <lb></lb>prævalet. </s> <s id="s.001849">Quod ſi ob inæqualem virgæ craſſitiem non planè <lb></lb>ad mediam illius longitudinem facta ſit ſuſpenſio, ſed æquili<lb></lb>britas contingat in puncto, quod propius eſt craſſiori extremi<lb></lb>tati virgæ, factâ alterutrius brachij inflexione tollitur æquili<lb></lb>brium, quia non jam ampliùs eadem eſt reciprocè Ratio longi<lb></lb>tudinum, quæ & gravitatum. </s> </p> <p type="main"> <s id="s.001850">Ex his pariter conſequens eſt aliquando minimam virtutem <lb></lb>ſatis eſſe ad dimovenda ab æquilibrio ingentia corpora, ſi ita <lb></lb>ſuſtineantur, ut fulcrum vel in puncto, vel in lineâ contingant: <lb></lb>quoniam ſi corpus grave inſiſtat apici coni, aut pyramidis, aut <pb pagenum="249" xlink:href="017/01/265.jpg"></pb>angulo ſolido, aut portioni ſphæricæ, quam contingat idem <lb></lb>corpus ſive planâ, ſive ſphæricè cavâ, ſive ſphæricam æmulan<lb></lb>te ſuperficie, contactus in puncto efficitur, ac propterea qua<lb></lb>cunque in extremitate corporis addatur vis movendi, æquili<lb></lb>brium tollitur, & quidem eò faciliùs, quo magis à puncto con<lb></lb>tactûs extremitas illa removetur; in illâ quippe diſtantiâ vis mo<lb></lb>vendi apta velociorem motum efficere, quàm ſi propior eſſet, <lb></lb>plus habet momenti: Id quod adhuc faciliùs accidit, ſi ab ex<lb></lb>tremitate, ubi vis movendi applicatur, ductâ per contingentis <lb></lb>fulcri punctum rectâ lineâ ad oppoſitam extremitatem, inæqua<lb></lb>liter diviſa ſit in puncto contactûs, & vis ipſa movendi in magis <lb></lb>diſtante extremitate conſtituta fuerit; tunc enim non ſua tan<lb></lb>tùm momenta addit, ſed illa multiplicat pro Ratione exceſsûs <lb></lb>ſuæ diſtantiæ; quemadmodum de inæqualibus libræ brachiis <lb></lb>dictum eſt. </s> <s id="s.001851">Sin autem fulcrum ſuſtinens, quod horizonti paral<lb></lb>lelum ponitur, ſit acies priſmatis, aut latus pyramidis jacentis, <lb></lb>aut portio cylindrica ſeu conica jacens; tunc in lineâ fit con<lb></lb>tactus, ſi vel plana ſit, vel circulariter concava corporis in<lb></lb>ſiſtentis ſuperficies: ſed ſi vis movendi, quantacumque ſit, ad<lb></lb>datur ſecundùm rectam lineam, quæ efficit Gravitatis diame<lb></lb>trum, puta in A vel N, non mutat æquilibritatem, ſi fulcrum <lb></lb>congruit toti diametro AN: ſi verò fulcrum brevius eſt quàm <lb></lb>AN, & ex. </s> <s id="s.001852">gr. congruit ſolùm ipſi AI, jam centrum motûs eſt <lb></lb>I, & oportet vim movendi tantam eſſe in N, ut aggregatum ex <lb></lb>parte MLN ac virtute additâ in N habeat ad partem MAL re<lb></lb>liquam majorem Rationem, quàm ſit Ratio diſtantiæ IA ad <lb></lb>diſtantiam IN. </s> <s id="s.001853">Quare in hujuſmodi contactu lineari vis mo<lb></lb>vendi, æquilibrium facilè tollens, eſſe debet ad latus diametri <lb></lb>gravitatis, & pro ratione diſtantiæ majus erit momentum; ma<lb></lb>ximum autem erit momentum in E diſtantiâ maximâ. </s> </p> <p type="main"> <s id="s.001854">Non igitur facilè inter fabulas rejicienda ſunt, quæ Atlas <lb></lb>Sinicus pag.32. de Montibus circa urbem Peking loquens ait, <lb></lb><emph type="italics"></emph>Púon mons altiſſimus ac præruptus varios attollens vertices, in cujus <lb></lb>ſummitate ingens eſt lapis, qui minimo contactu movetur ac titubat:<emph.end type="italics"></emph.end><lb></lb>fieri ſiquidem potuit, ut lapis ille in infimâ parte excavatus in<lb></lb>nitatur ſubjecto ſaxo, à quo vel in puncto, vel in lineâ tanga<lb></lb>tur, ſicuti dictum eſt; & cum ſit perfectè libratus, modico im<lb></lb>pulſu tangentis, quâ ſaltem parte ad illum patet acceſſus, po-<pb pagenum="250" xlink:href="017/01/266.jpg"></pb>teſt ab æquilibrio dimoveri: quòd ſi uſquequaque circum<lb></lb>obeundo lapidem quâcumque in parte tangatur, ſequitur illius <lb></lb>trepidatio, ſignum eſt contactum ſubjecti fulcri eſſe in puncto. </s> <lb></lb> <s id="s.001855">Simili ratione explicanda ſunt, quæ idem Atlas Sinicus in XI <lb></lb>Provincia Fokien habet pag. </s> <s id="s.001856">125, ubi ait, <emph type="italics"></emph>Versùs Vrbis <lb></lb>Changcheu Orientalem partem mons eſt Cio dictus, in quo lapida, <lb></lb>eſſe ſcribunt altum perticas quinque, craſſum decem & octo, qui quo<lb></lb>ties tempeſtas imminet, titubat omninò, ac movetur:<emph.end type="italics"></emph.end> hic enim la<lb></lb>pis in perfecto æquilibrio conſtitutus ſuprà fulcrum, à quo in <lb></lb>puncto, vel in lineâ tangatur, & fortaſſe etiam ab eodem fulcro <lb></lb>diſtinctus in longitudines inæquales, violento impulſu hali<lb></lb>tuum aut infernè ſubeuntium, aut ex ſuperiore nubium parte <lb></lb>obliquè reflexorum, facilè moveri poteſt ac titubare, ſi extre<lb></lb>mitas à fulcro remotior impellatur. </s> </p> <p type="main"> <s id="s.001857">Et quoniam de Sinenſibus mentio incidit, non injucundum <lb></lb>fuerit hîc aliud addere pertinens ad eorum induſtriam in ſer<lb></lb>vando æquilibrio. </s> <s id="s.001858">Idem Atlas Sinicus, cum ſermo eſt de Pro<lb></lb>vincia Peking, ubi ſolum eſſe arenoſum atque planiſſimum <lb></lb>teſtatur, hæc habet pag.28. <emph type="italics"></emph>Modus itineris faciendi hiſce locis <lb></lb>non infrequens, nec incommodus eſt. </s> <s id="s.001859">Plauſtrum adhibent cum unâ <lb></lb>rotâ ita conſtitutum, ut uni illius medium occupandi, & quaſi equo <lb></lb>inſidendi ſit locus, aliis duobus ab utroque latere adſidentibus; auri<lb></lb>ga plauſtrum retro ligneis vectibus urget ac promovet non ſecurè mi<lb></lb>nùs, quàm velociter.<emph.end type="italics"></emph.end></s> <s id="s.001860"> Si rem conjecturis indagare liceat, ego ro<lb></lb>tam concipio ita incluſam ligneo loculamento majoris ſegmen<lb></lb>ti circuli figuram habente, ut huic inſitus ſit rotæ axis, ad dex<lb></lb>tram autem & ad lævam extantia tabulata tantæ latitudinis, <lb></lb>ut quis modò propè rotam, modò longiùs adſidere queat ad <lb></lb>æquilibrium conſtituendum inter duos viatores inæqualiter <lb></lb>graves: Aurigæ locus eſt in ſuprema parte loculamenti, cui <lb></lb>quaſi equitans inſidet, binoſque contos, ſeu vectes concinnè <lb></lb>locatos, ut manubrium ante ſe habeat, extremitas altera (for<lb></lb>taſsè in acumen deſinens, ut leviter ſolo infigatur) poſt ſe ter<lb></lb>ram reſpiciat, utrâque manu apprehendens ſolum obliquè pre<lb></lb>mit, & currum in anteriora velociter promovet. </s> <s id="s.001861">Id quod nemi<lb></lb>ni difficile videatur, qui ſæpiùs obſervaverit à puero fabri <lb></lb>lignarij aut ferrarij rotam curulem identidem impulſam per <lb></lb>urbis vias velociter deduci; quæ dum impreſſo impetu veloci-<pb pagenum="251" xlink:href="017/01/267.jpg"></pb>ter converſa in anteriora promovetur, licet huc atque illuc <lb></lb>nutabunda inclinetur, ob velocem converſionem immunis eſt à <lb></lb>caſu: quemadmodum etiam ſtanneum aut argenteum orbem <lb></lb>apici cultri impoſitum, ſi in gyrum velociter agatur, à caſu im<lb></lb>munem videmus, etiamſi punctum ſuſtentationis non exactiſſi<lb></lb>mè centro reſpondeat. </s> <s id="s.001862">Sic aliquis ſuppoſitam ſphærulam altero <lb></lb>pede, etiam ſummis digitis premens, celeriter in gyrum totum <lb></lb>corpus contorquet, qui non ita facilè citrà cadendi periculum <lb></lb>eidem ſphærulæ inſiſtens quietus conſiſteret; ipsâ nimirum <lb></lb>converſionis celeritate gravitatis propenſionem eludente. </s> <s id="s.001863">Non <lb></lb>abſimili igitur ratione in hujuſmodi rotæ Sinici plauſtri conver<lb></lb>ſione veloci deteritur, quicquid in alterutram partem inclinatio<lb></lb>nis oriretur vel ex modicâ viæ inæqualitate, vel ex æquilibrio <lb></lb>non adeò exactè ſervato, ut etiam conſiſtente plauſtro inſiden<lb></lb>tes viatores conſiſterent æqualiter librati abſque alicujus artifi<lb></lb>cij ſubſidio: Quod artificium in promptu eſſe non dubito; ne<lb></lb>que enim Sinenſes ita ſibi præfidentes exiſtimo, ut aliquâ ratio<lb></lb>ne ſibi non præcaveant à periculo casûs, ſi fortè rotun in obicem <lb></lb>incurrente plauſtrum ſeu loculamentum in anteriorem, aut in <lb></lb>poſteriorem partem improvisâ inclinatione convertatur. </s> <s id="s.001864">Sed <lb></lb>ſingula perſequi nec otium eſt, nec operæ pretium: quapropter <lb></lb>generatim dicendum corporis æquilibrium ibi fieri, ubi in duas <lb></lb>partes ita diſtinguitur, ut illarum gravitates ſint reciprocè in <lb></lb>Ratione longitudinum ſeu diſtantiarum à puncto ſuſpenſionis <lb></lb>ſeu ſuſtentationis, quemadmodum in librâ dictum eſt. </s> <s id="s.001865">Quare ſi <lb></lb>tota moles propoſita eâdem gravitatis ſpecie prædita fuerit, nec <lb></lb>facile ſit in illâ centrum gravitatis invenire, quia nimis irregu<lb></lb>laris eſt, diſtingue illam in duas partes, & ſingularum inventa <lb></lb>centra gravitatis junge rectâ lineâ, quæ quaſi libræ jugum divi<lb></lb>datur in reciprocâ Ratione illarum partium; eſt enim punctum <lb></lb>illud, in quod cadit diviſio, punctum æquilibrij, & centrum gra<lb></lb>vitatis totius. </s> <s id="s.001866">Sic Trapezij, NPMQ in<lb></lb><figure id="id.017.01.267.1.jpg" xlink:href="017/01/267/1.jpg"></figure><lb></lb>venies punctum æquilibrij, ſi duorum <lb></lb>triangulorum NQM, NPM, in quæ di<lb></lb>viditur, ſingularia centra gravitatis inve<lb></lb>nias O & B: hæc jungantur rectâ OB; <lb></lb>tum fiat ut triangulum NQM ad trian<lb></lb>gulum NPM, ita reciprocè BD ad DO, <pb pagenum="252" xlink:href="017/01/268.jpg"></pb>& eſt D punctum æquilibrij, ſeu centrum gravitatis Trapezij <lb></lb>quæſitum. </s> <s id="s.001867">At ſi Trapezio addatur triangulum NLP ejuſdem <lb></lb>ſpecificæ gravitatis, emergit Pentagonum irregulare LPMQN: <lb></lb>inveniatur additi trianguli centrum ſingulare gravitatis A, & <lb></lb>jungatur recta AD; tùm fiat ut Trapezium ad triangulum ad<lb></lb>ditum, ita reciprocè AS ad SD, & eſt punctum S centrum <lb></lb>commune gravitatis totius Pentagoni, in quo fit æquilibrium; <lb></lb>perinde enim eſt ac ſi in jugo libræ AD inæqualiter diſtributæ <lb></lb>appenderetur ex A quidem triangulum NLP; ex D verò Tra<lb></lb>pezium NQMP, quæ in illis diſtantiis à centro motûs æqualia <lb></lb>haberent momenta. </s> </p> <p type="main"> <s id="s.001868">Quòd ſi tota moles propoſita conſtet partibus non ejuſdem <lb></lb>ſpecificæ gravitatis, non jam ſatis eſt inveniſſe ſingularia cen<lb></lb>tra, ut ducatur jugum libræ illa connectens, & notam eſſe Ra<lb></lb>tionem molis ad molem; ſed prætereà opus eſt notam habere <lb></lb>Rationem gravitatis ſpecificæ ad gravitatem ſpecificam; quiz <lb></lb>Ratio gravitatum abſolutarum componitur ex Rationibus <lb></lb>quantitatum, & gravitatum ſecundùm ſpeciem. </s> <s id="s.001869">Quamobrem <lb></lb>ſi additum triangulum habeat ſpecificam gravitatem majorem <lb></lb>gravitate ſpecificâ Trapezij, quia hoc ligneum eſt, illud fer<lb></lb>reum, non cadet in S punctum æquilibrij, ſed accedet ad <lb></lb>punctum A, quia factâ hujuſmodi Rationum compoſitione, <lb></lb>minor eſt inæqualitas gravitatum abſolutarum; ſi enim Trape<lb></lb>zium excedit mole Triangulum, cedit illi ſpecificâ gravitate. </s> <lb></lb> <s id="s.001870">Ponamus namque Rationem molis Trapezij ad molem Trian<lb></lb>guli eſſe ut & ad 2; ſpecificæ verò gravitatis Rationem ut 5 ad <lb></lb>42, gravitas abſoluta Trapezij lignei eſt ut 35, gravitas Trian<lb></lb>guli ferrei ut 84: ſunt igitur gravitates in Ratione 5 ad 12: di<lb></lb>vidatur itaque jugum AD in I reciprocè, ut ſit AI 5, ID 12, <lb></lb>& erit I centrum gravitatis compoſitæ, ac punctum æquilibrij, <lb></lb>quia ab illo inæquales gravitates habent ſuas diſtantias in Ra<lb></lb>tione reciprocâ ipſarum gravitatum. </s> <s id="s.001871">Eadem eſt in corporibus <lb></lb>omnibus Ratio, & methodus deprehendendi punctum æqui<lb></lb>librij, ſeu centrum gravitatis, per quod deinde duci poteſt dia<lb></lb>meter gravitatis, ut fiat opportuna ſuſpenſio. </s> </p> <p type="main"> <s id="s.001872">Quia tamen aliquando evenit ſuſpenſum corpus aut ſuſten<lb></lb>tatum, dum poſitionem horizonti parallelam ſervare contendit, <lb></lb>aliquod incommodum ſubire in motu corporis, cui innititur; <pb pagenum="253" xlink:href="017/01/269.jpg"></pb>proptereà huic occurrendum eſt artificio, quo ſitum eumdem <lb></lb>perpetuò ſervet. </s> <s id="s.001873">Rem exemplo declaro. </s> <s id="s.001874">In pyxide nauticâ in<lb></lb>ſiſtit cuſpidi acus magnetica æqualibus momentis librata, ut <lb></lb>horizonti parallela jaceat, quamcumque in partem dirigatur. </s> <lb></lb> <s id="s.001875">Si alicui navis plano pyxis ipſa adhæreret ita, ut infimâ ſui par<lb></lb>te illi congrueret, quamcumque in partem navis inclinaretur, <lb></lb>ipſum pariter pyxidis fundum inclinari manifeſtum eſt, & alte<lb></lb>ri acûs magneticæ poſitionem horizonti parallelam ſervantis <lb></lb>extremitati occurrens illius motum impediret, aut ſaltem retar<lb></lb>daret. </s> <s id="s.001876">Ut igitur ſemper pyxis tùm acui magneticæ, tùm hori<lb></lb>zonti parallela conſiſtat, ſuſpendenda fuit, non quidem funi<lb></lb>culo, ne incertis motibus jactaretur, ſed duobus polis, ſuper <lb></lb>quibus opportunè verſaretur æqualiter librata. </s> <s id="s.001877">Verùm duobus <lb></lb>hiſce polis non tollitur omne incommodum; ſi etenim poli <lb></lb>reſpiciant navis latera, elevatâ aut depreſsâ prorâ juvant, ſed <lb></lb>navi in dextrum aut in ſiniſtrum latus inclinatâ, alter deprime<lb></lb>retur, alter elevaretur, niſi & ipſi infigerentur circulo ſuper <lb></lb>alios polos proram & puppim reſpicientes verſatili. </s> <s id="s.001878">Sit pyxis <lb></lb>ipſa ABCD, in qua venti deſ<lb></lb><figure id="id.017.01.269.1.jpg" xlink:href="017/01/269/1.jpg"></figure><lb></lb>cripti ſint, & in centro O acus <lb></lb>magnetica volubilis inſiſtat: py<lb></lb>xidem circulus EIFH com<lb></lb>plectatur, cui poli D & B facilè <lb></lb>verſatiles infigantur, ut inclinatâ <lb></lb>navi in A vel in C pyxis horizon<lb></lb>ti parallela maneat; & ut eumdem <lb></lb>paralle iſmum ſervet, etiam ſi na<lb></lb>vis in B aut D inclinetur, circu<lb></lb>lus ille EIFH duos pariter polos <lb></lb>facilè verſatiles habeat in E & F <lb></lb>externæ pyxidi immobili infixos: <lb></lb>hac enim ratione fiet, ut in quacumque navis inclinatione <lb></lb>pyxis nautica à ſuo paralleliſmo & æquilibrio non recedat. </s> </p> <p type="main"> <s id="s.001879">Hoc eodem artificio conſtruitur luceina ferreo aut æneo <lb></lb>globo incluſa multipliciter perforato, ut fumo exitus pateat, <lb></lb>quæ citrà effuſionem olci in ſolo rotata non extinguitur; eſt ſi<lb></lb>quidem vaſculum plumbeum, ut ſua gravitate ſecuriùs deor<lb></lb>ſum vergat, polis verſatilibus ſuſpenſum in circulo, qui pariter <pb pagenum="254" xlink:href="017/01/270.jpg"></pb>polos inſerit ſecundo circulo, ſecundus ſimiliter tertio, tertius <lb></lb>demum ſcaphio, ſeu inferiori hemiſphærio globi, cui includi<lb></lb>tur, eâ diſpoſitione, ut quemadmodum pyxidis nauticæ hic <lb></lb>deſcriptæ ambitus in quatuor partes diſtinguitur à polis, ita lu <lb></lb>cernæ hujus ambitus in octo partes à polis diſtribuatur, atque <lb></lb>proinde facilior ſit globi in omnem partem volutatio citrà peri<lb></lb>culum inclinationis vaſculi oleum cum ellychnio continentis. </s> </p> <p type="main"> <s id="s.001880">Nec pluribus opus eſt hîc explicare, quàm proclive ſit arti<lb></lb>ficium hoc ad plura traducere, quorum uſus eſt in plano hori<lb></lb>zontali, ne libellâ ſemper & normâ indigeamus, ut illa ritè <lb></lb>collocentur: ut ſi horologium horizontale ſtatuendum ſit quo<lb></lb>cumque in plano, ſit illud pyxidi incluſum cum circulo, quem<lb></lb>admodum de pyxide nauticâ dictum eſt: ſi lectulum viatorium <lb></lb>in rhedâ ſternere oporteat, in quo citrà jactationem, etiam viâ <lb></lb>ſalebrosâ, quieſcere liceat, ferreo parallelogrammo complecte<lb></lb>re lectulum ex polis ſuſpenſum circâ medium eo loco, ut cor<lb></lb>pus in lectulo jacens ſit horizonti parallelum, ipſum verò paral<lb></lb>lelogrammum polis rhedæ infixis & verſatilibus ad caput & ad <lb></lb>pedes ſuſpendatur: & alia hujuſmodi, quæ facilè pro rerum <lb></lb>opportunitate excogitari poſſunt. </s> </p> <p type="main"> <s id="s.001881">Verùm quàm facilè eſt ſuper polos in æquilibrio conſtituere <lb></lb>corpora gravitatis centrum habentia vel in ipsâ ſuſtentationis <lb></lb>lineâ, vel infrà illam, tam multis difficultatibus implicitum <lb></lb>opus eſt in æquilibrio ſtatuere corpus, cujus gravitatis cen<lb></lb>trum in parte ſuperiori reperitur, & quidem maximè ſi mul<lb></lb>tùm inde removeatur; tunc enim ſuſſicit vel minima inclinatio, <lb></lb>ut totum corpus revolvatur, cum ex alterâ parte ſint plura gra<lb></lb>vitatis momenta, quàm in oppoſitâ. </s> </p> <p type="main"> <s id="s.001882">Nam ſi corpus BC, cujus centrum gravitatis ſit A, ſuſpen<lb></lb>datur ſuper polis in I, quando axi ſuſtentanti ad perpendiculum <lb></lb><figure id="id.017.01.270.1.jpg" xlink:href="017/01/270/1.jpg"></figure><lb></lb>reſpondet centrum gravitatis A, ma<lb></lb>net æquilibrium, ſed factâ corporis <lb></lb>inclinatione, ut A recedat à perpen<lb></lb>diculo, jam versùs C plures ſunt <lb></lb>partes gravitatis deſcendentes, quàm <lb></lb>versùs B ſint partes aſcendentes, & <lb></lb>illæ velociùs moventur deorſum, <lb></lb>quàm hæ ſurſum; quapropter illæ <pb pagenum="255" xlink:href="017/01/271.jpg"></pb>majora habent momenta, quibus deorium urgentibus corpus <lb></lb>revolvitur. </s> <s id="s.001883">Id quod multò magis contingit in Acrobarycis, quæ <lb></lb>nimirum gravitatem in ſummitate habent, ut ſi corpori BC in <lb></lb>ſuperiori parte adnexa eſſet pyramis D; cum enim totius com<lb></lb>poſitæ molis ex ſolido BC, & pyramide D, centrum commu<lb></lb>ne gravitatis non eſſet in A, ſed adhuc ſuperius procul à polo <lb></lb>I, qui eſt centrum motûs, factâ levi inclinatione multo plus <lb></lb>gravitatis eſſet ex parte C, quàm ex oppoſità B, ut conſtat: <lb></lb>nam quò altius & remotius eſt centrum gravitatis, eò faciliùs <lb></lb>linea directionis cadit extra punctum vel lineam ſuſtentationis, <lb></lb>facta pari inclinatione. </s> </p> <p type="main"> <s id="s.001884">Liceat autem hîc obiter, quaſi cerollarij loco, attingere <lb></lb>æquilibria corporum humido inſidentium, & Acrobary corum <lb></lb>fluitantium, in quibus pariter Rationes libræ agnoſcentur, ſi <lb></lb>rectè perpendatur, ubi fiat ſuſtentatio. </s> <s id="s.001885">In omni igitur corpo<lb></lb>re fluitante duplex pars conſideranda eſt, & quæ intrá humi<lb></lb>dum mergitur, & quæ in aëre extat: illa quidem utpote ſecun<lb></lb>dùm ſpeciem minùs gravis, quàm humor, levitat, hæc verò <lb></lb>aëre gravior gravitat: Quare & illa ſuum habet centrum levi<lb></lb>tatis, & hæc centrum gravitatis; nec poſſet corpus datam poſi<lb></lb>tionem ſervare, niſi in eâdem lineâ perpendiculari ad univerſi <lb></lb>centrum tendente eſſet utrumque centrum & levitatis & gra<lb></lb>vitatis; cumque par ſit virtus aſcendendi virtuti deſcendendi, <lb></lb>neutrâ prævalente, & ſibi viciſſim utrâque obſiſtente, conſiſtit <lb></lb>corpus. </s> <s id="s.001886">Quòd ſi non in eodem perpendiculo ſit utrumque <lb></lb>centrum, utrumque ſuâ viâ pergere poteſt, illud aſcendendo, <lb></lb>hoc deſcendendo. </s> <s id="s.001887">Sic baculum rectum in aquam immittens, <lb></lb>manúque retinens, ne in alterutram partem inclinetur, mergi <lb></lb>quidem illum videbis pro Ratione ſpecificæ ſuæ gravitatis, quæ <lb></lb>minor eſt ſpecificâ gravitate aquæ, ſed erectus non manebit, <lb></lb>niſi quandiù retinueris; nam ubi illum dimiſeris, ſtatim cen<lb></lb>trum gravitatis deſcendet, & levitatis centrum aſcendet, quia <lb></lb>vel exiguus aquæ motus partem immerſam inclinans ſatis eſt, <lb></lb>ut centra illa non eidem perpendiculo reſpondeant; ac prop<lb></lb>terea demùm baculus jacens innatabit. </s> </p> <p type="main"> <s id="s.001888">Quieſcente igitur corpore in humoris ſuperficie, mani<lb></lb>feſtum eſt centrum gravitatis partis extantis in eodem perpen<lb></lb>diculo eſſe cum centro levitatis partis demerſæ. </s> <s id="s.001889">Quare ſi <pb pagenum="256" xlink:href="017/01/272.jpg"></pb>ligneum priſina AC aquæ imponatur, & immergatur ita, ut <lb></lb>pars demerſa & levitans ſit EC, pars verò extans in aëre & <lb></lb><figure id="id.017.01.272.1.jpg" xlink:href="017/01/272/1.jpg"></figure><lb></lb>gravitans ſit AF, centrum gravi<lb></lb>tatis eſt G, centrum levitatis eſt <lb></lb>H, quæ ſibi directè adverſantia <lb></lb>in oppoſitas partes conantur <lb></lb>æqualibus viribus, atque prop<lb></lb>terea nullus ſequitur motus. </s> <lb></lb> <s id="s.001890">Quòd ſi aut H recederet versùs <lb></lb>D, aut G versùs B, & hoc poſſet <lb></lb>deſcendere, & illud aſcendere <lb></lb>neutro contranitente. </s> </p> <p type="main"> <s id="s.001891">Jam verò quieſcenti priſmati imponatur aliquod pon<lb></lb>dus, certum eſt partem in aëre extantem, conflatam ex <lb></lb>parte priſmatis & ex addito pondere, graviorem eſſe, ac <lb></lb>proinde prævalere viribus partis in aquâ levitantis, illam<lb></lb>que deprimere, quoaduſque fiat æqualitas inter levitatem <lb></lb>& gravitatem. </s> <s id="s.001892">Sed multùm intereſt, utrùm additi pon<lb></lb>deris centrum gravitatis in eodem perpendiculo ſit cum cen<lb></lb>tro gravitatis G, ut rectâ deprimatur priſma infrà ſuperfi<lb></lb>ciem aquæ; an verò ſit extrà illud perpendiculum; id <lb></lb>quod ſi accidat, commune centrum gravitatis transfertur ver<lb></lb>ſus A, aut B. </s> <s id="s.001893">Sit ex. </s> <s id="s.001894">gr. ad partes A propè S; cumque non <lb></lb>immineat puncto H centro levitatis, deſcendit priſma ad partes <lb></lb>A, & oppoſita pars aſcendit, ita ut E deprimatur infrà ſuperfi<lb></lb>ciem aquæ, F veró emergat. </s> <s id="s.001895">Sed dum ad partes CF priſma <lb></lb>emergit ex aquâ, ad partes autem DE deprimitur, centrum levi<lb></lb>tatis non manet in H, ſed ad majorem partem depreſſam ſecedit, <lb></lb>donec fiat V, atque in eodem <expan abbr="perpẽdiculo">perpendiculo</expan> ſit cum centro gravi<lb></lb>tatis S; & tunc quieſcit priſma, nec amplius demergitur in E, <lb></lb>aut emergit ex F. </s> <s id="s.001896">Suſtinetur itaque centrum gravitatis S à cen<lb></lb>tro levitatis V, & viciſſim centrum levitatis V retinetur à cen<lb></lb>tro gravitatis S; & fit tùm inter gravitates, tùm inter levitates <lb></lb>æquilibrium, quia gravitas in A major minùs diſtat à puncto, <lb></lb>vel potiusà lineâ ſuſtentationis factâ à plano tranſeunte per V, <lb></lb>& gravitas in B minor magis diſtat; ideóque neutra prævalet: <lb></lb>& ſimiliter ievitas in DE major minùs diſtat à lineâ detentio<lb></lb>nis facta à plano tranſeunte per S, ac levitas minor in C magis <pb pagenum="257" xlink:href="017/01/273.jpg"></pb>diſtat; quare vis tardiùs aſcendendi major prævalere non po<lb></lb>reſt minori virtuti repugnanti ad deſcendendum velociùs. </s> </p> <p type="main"> <s id="s.001897">Quemadmodum verò ſi tantum ponderis adderetur in A, ut <lb></lb>centrum commune gravitatis non poſſet imminere centro levi<lb></lb>tatis partis demerſæ, nemo non intelligit futuram omnimodam <lb></lb>depreſſionem partis A infrà ſuperficiem aquæ, & omnimodam <lb></lb>emerſionem oppoſitæ partis C; ita in Acrobarycis fluitantibus <lb></lb>manifeſtum eſt, quò altiùs attollitur gravitas, eò faciliùs factâ <lb></lb>inclinatione transferri commune centrum gravitatis ultrà per<lb></lb>pendiculum, in quo eſt centrum levitatis partis demerſæ. </s> <s id="s.001898">Sic <lb></lb>ſi juſto longior ſit in navi malus, factâ ex fluctibus inclinatione <lb></lb>in latus, aut ſaltem impulſu venti ſuprema carbaſa implentis, <lb></lb>facilis erit navis ſubmerſio, quia plus momentorum gravitatis <lb></lb>eſt ex alterâ parte, quàm ex oppoſitâ, tranſlato in navis latus, <lb></lb>aut ultra illud, centro gravitatis totius partis extantis in aëre. </s> <lb></lb> <s id="s.001899">Sed de his, Deo dante, pleniùs in Hydroſtaticis diſſerendum <lb></lb>erit, ubi oſtendetur ad navium ſtabilitatem neceſſariam eſſe <lb></lb>eam centrorum diſpoſitionem, ut centrum gravitatis totius na<lb></lb>vis cum omnibus impoſitis ſit infrà centrum levitatis partis de<lb></lb>merſæ in eodem perpendiculo, in quo pariter erit centrum gra<lb></lb>vitatis partis extantis. <lb></lb> </s> </p> <p type="main"> <s id="s.001900"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001901"><emph type="center"></emph><emph type="italics"></emph>An, & cur libra ab æquilibrio dimota ad illud <lb></lb>redeat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001902">NEmini dubium eſſe poteſt æquilibrium tolli ob momento<lb></lb>rum gravitatis inæqualitatem, vel quia in una libræ æqui<lb></lb>libris lance additum eſt pondus, vel quia altera jugi extremi<lb></lb>tas, alicujus elevantis aut deprimentis vi, recedit à poſitione <lb></lb>horizonti parallelâ. </s> <s id="s.001903">Illud in quæſtionem revocati poteſt, an <lb></lb>ſublato ponderis exceſſu, aut ceſſante impulſu extrinſeco, li<lb></lb>bra redeat ad æquilibrium, & poſitionem horizonti parallelam <lb></lb>ſibi ipſa reſtituat. </s> <s id="s.001904">Certè Keplerus in Aſtronomiâ Opticâ cap.1. <pb pagenum="258" xlink:href="017/01/274.jpg"></pb>prop. 20. aſſerit eum, qui negat libram brachiorum æqualium <lb></lb>ad horizontis æquilibrium redituram, <emph type="italics"></emph>non antiquitati tantum, <lb></lb>ſed rerum naturæ, ſed utilitati generis humani bellum indicere.<emph.end type="italics"></emph.end></s> <s id="s.001905"> At <lb></lb>ex adverſo Authores ferè omnes, qui de his accuratiùs ſcripſe<lb></lb>runt, triplicem libræ ſpeciem diſtinguentes unam tantummo<lb></lb>do agnoſcunt, quæ ſe reſtituat horizonti parallelam. </s> <s id="s.001906">Hoc ſi<lb></lb>quidem tanquam certum aſſumunt, corpus quodcumque gra<lb></lb>ve, quod ſuſpenſum, aut ſuſtentatum liberè in aëre pendeat, <lb></lb>in cò tantum ſitu quieſcere, in quo gravitatis centrum cum ſuſ<lb></lb>penſionis aut ſuſtentationis puncto in eâdem directionis lineâ <lb></lb>reperiatur; deſcendit enim quantum poteſt, neque ei opponi<lb></lb>tur punctum ſuſpenſionis aut ſuſtentationis, niſi in eodem per<lb></lb>pendiculo ad univerſi centrum ducto utrumque ſit. </s> <s id="s.001907">Cumitaque <lb></lb>libra ſit corpus grave ſuſpenſum, & ſuum habeat centrum gra<lb></lb>vitatis, tunc demùm quieſcet, ubi eam poſitionem obtinuerit, <lb></lb>in quâ ſuſpenſionis punctum, & gravitatis centrum in eâdem <lb></lb>ſint directionis lineâ. </s> <s id="s.001908">Punctum verò ſupenſionis libræ non il<lb></lb>lud hîc intelligitur, ex quo pendet anſa, cui libra inſeritur, ſed <lb></lb>ipſa Agina, ſeu ſpartum, ut Ariſtotelico vocabulo utar, eſt ſuſ<lb></lb>penſionis punctum; ex illo enim proximè libra ſuſpenditur. </s> </p> <p type="main"> <s id="s.001909">Hinc oritur triplex libræ ſpecies, quia tripliciter componi <lb></lb>poſſunt centrum motûs, & centrum gravitatis; primò ſcilicet <lb></lb>poſſunt in uno eodemque puncto convenire, deinde centrum <lb></lb>motûs poteſt eſſe ſuperius, demum inferius centro gravitatis. </s> </p> <p type="main"> <s id="s.001910">Et quidem ſi unum idemque punctum ſit motûs & gravita<lb></lb>tis centrum A, & æqualibus brachiis AB, AC æqualia ſint <lb></lb><figure id="id.017.01.274.1.jpg" xlink:href="017/01/274/1.jpg"></figure><lb></lb>adnexa pondera B & C, uti<lb></lb>que æquilibrium horizonta<lb></lb>le manet, propter momento<lb></lb>rum æqualitatem tùm ratio<lb></lb>ne gravitatum æqualium, <lb></lb>tùm ratione æqualium pro<lb></lb>penſionum ad motum. </s> <s id="s.001911">Si <lb></lb>igitur applicatâ manu in B <lb></lb>deprimatur libra, ut ſit DE; <lb></lb>amotâ manu, cur redeat libra ad priorem poſitionem BC? </s> <s id="s.001912"><lb></lb>adhuc enim momenta utrinque ſunt æqualia, & tantumdem <lb></lb>aſcendere deberet D, quantum deſcenderet E: par igitur eſt <pb pagenum="259" xlink:href="017/01/275.jpg"></pb>reſiſtentia ipſius D propenſioni ad motum ipſius E: neutro ita<lb></lb>que prævalente fiet in eo ſitu DE conſiſtentia. </s> </p> <p type="main"> <s id="s.001913">Attamen huic argumentationi, quamvis legitimæ, non ac<lb></lb>quieſcunt nonnulli, qui libram hujuſmodi in quácumque poſi<lb></lb>tione quieſcentem ſe viſuros deſperant, quia nunquam vide<lb></lb>runt: quare potiùs cauſam inquirunt, cur ad æquilibrium re<lb></lb>deat libra æqualium brachiorum, quamvis ex medio jugo ſuſ<lb></lb>pendatur. </s> <s id="s.001914">Exiſtimant aliqui poſſe vim argumenti eludi, ſi con<lb></lb>cedant quidem in uno eodemque puncto convenire centrum <lb></lb>motûs & centrum gravitatis jugi, non tamen libræ: nam ſi <lb></lb>præter jugum aſſumantur etiam uncini aut lances, quibus ad<lb></lb>nectuntur aut imponuntur pondera, multò magis ſi eadem pon<lb></lb>dera aſſumantur, centrum gravitatis hujuſce molis compoſitæ <lb></lb>reperiri aſſerunt infrà ipſum jugum, ac propterea nullam eſſe <lb></lb>hujuſmodi primam ſpeciem libræ. </s> </p> <p type="main"> <s id="s.001915">Sit libræ jugum AB; centrum motûs & gravitatis jugi ſit C: <lb></lb>pendeant lances D & E, ſingularúmque cum ſuis appendiculis <lb></lb>gravitas ſit æqualis gra<lb></lb><figure id="id.017.01.275.1.jpg" xlink:href="017/01/275/1.jpg"></figure><lb></lb>vitati jugi, ut facere con<lb></lb>ſueverunt accuratiores <lb></lb>monetarij. </s> <s id="s.001916">Lancium igi<lb></lb>tur ſimul ſumptarum <lb></lb>commune gravitatis cen<lb></lb>trum eſt in F: jungantur <lb></lb>centra gravitatum C & <lb></lb>F; & erit demum totius <lb></lb>libræ vacuæ DABE <lb></lb>commune gravitatis cen<lb></lb>trum in G. </s> <s id="s.001917">Quod ſi lan<lb></lb>cibus D & E imponan<lb></lb>tur æqualia pondera, <lb></lb>commune centrum gravitatis erit inter G & F, atque quò gra<lb></lb>viora erunt pondera, eò propiùs accedet ad F. </s> <s id="s.001918">Eſt igitur ma<lb></lb>nifeſtum centra motûs & gravitatis totius libræ non in eodem <lb></lb>puncto convenire, ſed gravitatis centrum eſſe infrà centrum <lb></lb>motûs, ſeu ſpartum C. </s> </p> <p type="main"> <s id="s.001919">Verum effugium hoc nullum eſſe cenſeo: inclinetur enim <lb></lb>libra, & acquirat poſitionem HI, jam HM & IN lineæ di-<pb pagenum="260" xlink:href="017/01/276.jpg"></pb>rectionis lancium ſunt æquales, quia cædem cum AD & BE, <lb></lb>& ſunt parallelæ, quia ambæ perpendiculares ad horizontem; <lb></lb>ac propterea ex 33. lib.1. æquales ſunt ac parallelæ HI & MN. </s> <lb></lb> <s id="s.001920">Cumque CF linea directionis centri gravitatis jugi ſit iiſdem <lb></lb>HM & IN parallela, & exeat ex C medio rectæ HI, cadet <lb></lb>pariter in medium rectæ MN ex 34 lib.1. & idem punctum F <lb></lb>eſt commune centrum gravitatum M & N; atque proinde li<lb></lb>bræ MHIN commune centrum gravitatis erit in eadem rectâ <lb></lb>lineá CF. </s> <s id="s.001921">Si itaque quieſcit corpus grave ſuſpenſum, quando <lb></lb>in eâdem directionis linea eſt punctum ſuſpenſionis, & gravi<lb></lb>tati, centrum, etiam in poſitione HI deberet libra quieſcere, <lb></lb>eſto in C non conveniant contra motûs & gravitatis totius <lb></lb>libræ. </s> </p> <p type="main"> <s id="s.001922">Nicolaus Tartalea lib. 8. quæſito 32. ideo libram ad paralle<lb></lb>liſmum horizontis redire exiſtimat, quia in inclinatione jugi <lb></lb>putat majora eſſe momenta brachij elevati, quàm depreſſi. </s> <lb></lb> <s id="s.001923">Id quod hâc methodo conatur oſtendere. </s> <s id="s.001924">Si ex C æqualiter <lb></lb><figure id="id.017.01.276.1.jpg" xlink:href="017/01/276/1.jpg"></figure><lb></lb>diſtent pondera æqualia A & B, <lb></lb>fuerintque ab æquilibrio remota, <lb></lb>deſcribunt circulum, in quo <lb></lb>ſumptis partibus æqualibus, dum <lb></lb>A deſcendit ex F in A, vis de<lb></lb>ſcendendi eſt NO, at ex A in G <lb></lb>vis deſcendendi eſt OP major, <lb></lb>quàm NO, ut conſtat ex doctri<lb></lb>nâ Sinuum. </s> <s id="s.001925">Similiter vis deſcen<lb></lb>dendi ipſius B ex I in B eſt KL <lb></lb>major, quàm LM vis deſcenden<lb></lb>di ex B in H. </s> <s id="s.001926">Eſt autem KL ipſi OP, & LM ipſi ON <lb></lb>æqualis; igitur OP eſt etiam major, quàm LM. </s> <s id="s.001927">Cum itaque <lb></lb>in ſitu ACB pondus B gravitet ſolùm ut LM, & pondus A <lb></lb>gravitet ut OP, major eſt potentia ipſius A, quàm ipſius B: <lb></lb>igitur ad æquilibrium deſcendere oportet pondus A. </s> </p> <p type="main"> <s id="s.001928">Sed peccat hæc Tartaleæ argumentatio, quia in pondere B <lb></lb>non eſt conſideranda vis deſcendendi in H, ſed repugnantia <lb></lb>ad aſcendendum in I, ſecundùm quam obſiſtit oppoſito pon<lb></lb>deri A; hujus autem reſiſtentiæ menſura eſt LK æqualis ipſi <lb></lb>OP potentiæ ſeu propenſioni ipſius A ad deſcendendum: <pb pagenum="261" xlink:href="017/01/277.jpg"></pb>æquatur ergo potentia reſiſtentiæ, nec ullus fieri poteſt motus, <lb></lb>quamdiu hæc æqualitas permanet. </s> </p> <p type="main"> <s id="s.001929">Joannes Keplerus Aſtronomiæ Opticæ loco citato, cur libræ <lb></lb>brachia revolvantur ad æquilibrium, infert ex eo, quòd altero <lb></lb>brachiorum prægravato additione ponderis, ita jugum libræ <lb></lb>conſiſtit, ut quod eſt gravius non planè imum locum petat, <lb></lb>& quod eſt levius, non planè in apicem attollatur. </s> <s id="s.001930">Cujus rei <lb></lb>cauſam inquirens ſtatuit libræ jugum <lb></lb><figure id="id.017.01.277.1.jpg" xlink:href="017/01/277/1.jpg"></figure><lb></lb>CD bifariam in A diviſum; & centro <lb></lb>A deſcripto circulo ducit perpendicu<lb></lb>lum BAF: ex quo manifeſtum eſt <lb></lb>neutrum pondus poſſe deprimi infra F, <lb></lb>aut attolli ſupra B. </s> <s id="s.001931">Sed quia pondus D <lb></lb>ponitur gravius, quàm pondus C, & <lb></lb>utrumque naturâ ſuâ ad imum tendit, <lb></lb>contenduntque invicem, partiuntur <lb></lb>inter ſe deſcenſum BF in proportione, <lb></lb>quâ ipſa ſunt: adeò ut BH deſcenſus <lb></lb>ponderis C ſit ad BG deſcenſum ponderis D, ut pondus C ad <lb></lb>pondus D. </s> <s id="s.001932">Eſt autem FG linea æqualis lineæ BH, quia ex <lb></lb>æqualibus AB & AF auferuntur æqualia latera AH & AG, <lb></lb>cum enim triangula CHA, DGA rectangula ſint, & angu<lb></lb>los ad verticem A æquales habeant, & latera AC, AD æqua<lb></lb>lia; etiam per 26. lib.1. latus AH eſt æquale lateri AG. </s> <s id="s.001933">Igitur <lb></lb>ut pondus C ad pondus D, ita FG ad GB. </s> </p> <p type="main"> <s id="s.001934">Ducatur ex F ad AD perpendicularis FK: ſimiliter triangula <lb></lb>AGD, AKF rectangula, & <expan abbr="cõmunem">communem</expan> angulum in A habentia, <lb></lb>cum latere AF æquali lateri AD, per eandem 26.lib.1. <expan abbr="habẽtla-tera">habent la<lb></lb>tera</expan> AG & AK æqualia: ergo & reſidua FG, DR æqualia ſunt. </s> <lb></lb> <s id="s.001935">Igitur propter <expan abbr="æqualitatẽ">æqualitatem</expan> <expan abbr="diametrorũ">diametrorum</expan> FB & DC, erit etiam GB <lb></lb>linea æqualis lineæ KC. </s> <s id="s.001936">Quare ut <expan abbr="põdus">pondus</expan> D ad pondus C, ita GB <lb></lb>ad GF, hoc eſt ita KC ad KD: ac propterea factâ jugi ſuſpen<lb></lb>ſione in K pondera C & D inæqualia ſecundùm Rationem bra<lb></lb>chiorum reciprocè poſita æquiponderabunt & conſiſtent. </s> <s id="s.001937">Cum <lb></lb>igitur in hac eâdem Ratione ſit deſcenſus BH & BG, ut eſt <lb></lb>pondus C ad pondus D, fiet conſiſtentia in ſitu CAD. <emph type="italics"></emph>Ergo <lb></lb>per ſubſumptionem patet,<emph.end type="italics"></emph.end> ſubdit Keplerus, cujus ſuperiorem <lb></lb>doctrinam conatus ſum paulo clariùs exponere, <emph type="italics"></emph>cur libræ brachia <emph.end type="italics"></emph.end><pb pagenum="262" xlink:href="017/01/278.jpg"></pb><emph type="italics"></emph>revolvuntur ad æquilibrium; cum enim æque ponderent, æquales etiam <lb></lb>in circulo fieri deſcenſus par eſt.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.001938">Meam hebetudinem diſſimulare non poſſum, qui hujuſce <lb></lb>Keplerianæ argumentationis vim ſatis aſſequi non valeo: quid <lb></lb>enim, ſi fieret æquilibrium horizontale ponderum, facta in K <lb></lb>ſuſpenſione? </s> <s id="s.001939">an propterea conſequens eſt fieri æquilibrium <lb></lb>etiam in ſitu CAD, niſi aliunde probetur? </s> <s id="s.001940">ſed quod ad rem <lb></lb>noſtram attinet, pondera alligata, & adnexa libræ non ita con<lb></lb>ſideranda ſunt, ut ambo deſcendant, ſi comparatè ſumantur, <lb></lb>ſed alterius propenſio ad motum deorſum comparanda eſt cum <lb></lb>alterius repugnantiâ ad motum ſurſum, & viciſſim hujus pro<lb></lb>penſio ad deſcendendum cum illius reſiſtentiâ, ne aſcendat. </s> <lb></lb> <s id="s.001941">Quapropter ſi ex D pondere majore auferatur exceſſus ſupra <lb></lb>pondus C, & fiant æqualia pondera, non poſſunt ad æquili<lb></lb>brium horizontale redire, niſi C deſcendat, D verò aſcendat: <lb></lb>Cum autem hujus aſcenſus GA ſit æqualis deſcenſui HA, nul<lb></lb>la eſt ratio, cur propenſio ponderis C vincere debeat æqualem <lb></lb>ponderis D reſiſtentiam. </s> </p> <p type="main"> <s id="s.001942">Deinde quid intelligendum eſt, cum dicitur ipſius C deſcen<lb></lb>ſus eſſe BH, ipſius verò D deſcenſus eſſe BG? </s> <s id="s.001943">ex B enim non <lb></lb>utrumque deſcendit, ſed alterutrum: & ſi pondus D deſcendiſ<lb></lb>ſet ex B, ex adverſo pondus C aſcendiſſet ex F; cúmque illius <lb></lb>deſcenſus eſſet BG, hujus aſcenſus eſſet FH; ſunt autem BG <lb></lb>& FH æquales. </s> <s id="s.001944">Quòd ſi non motus præcedens, ſed ſola pro<lb></lb>penſio ad deſcendendum & repugnantia ad aſcendendum con<lb></lb>ſideretur pro ratione poſitionis, pondus D habet menſuram <lb></lb>propenſionis ad deſcendendum, non motum (qui fortaſſe tran<lb></lb>ſiit) ex B in D, ſed quem in eo ſitu poſſet perficere ex D in F: <lb></lb>atque adeò ipſius D deſcenſus eſt GF, ejuſque reſiſtentia, ne <lb></lb>aſcendat uſque ad ſummum eſt GB, & viciſſim ponderis C pro<lb></lb>penſio ad deſcendendum non eſt ex B in C, ſed ex C in F, ſi <lb></lb>uſque ad imum deſcendat, habens menſuram HF, ejus verò <lb></lb>repugnantiam ad aſcendendum metitur HB. </s> <s id="s.001945">Eſt igitur mani<lb></lb>feſtum uniuſcujuſque ponderis propenſionem habere oppoſi<lb></lb>tam reſiſtentiam æqualem (eſt enim propenſio GF æqualis re<lb></lb>ſiſtentiæ HB, & propenſioni HF æquali eſt reſiſtentia GB) <lb></lb>ac proinde nullum ſequi poſſe motum ponderum æqualium à <lb></lb>centro A æqualiter diſtantium. </s> <s id="s.001946">At, inquis, quid cauſæ eſt, <pb pagenum="263" xlink:href="017/01/279.jpg"></pb>cur ſimilem libram in quácumque poſitione quieſcentem non <lb></lb>habemus? </s> <s id="s.001947">ſed omnis libra ea eſt, ut vel ad æquilibrium redeat, <lb></lb>vel omninò quantum poteſt deſcendat, qua parte habet bra<lb></lb>chium inclinatum Reſponſio in promptu eſt; quia ſcilicet dif<lb></lb>ficillimum eſt duo illa puncta exquiſitè convenire, hoc eſt cen<lb></lb>trum motus & centrum gravitatis, nimirùm punctum illud, <lb></lb>quod brachiorum longitudinem diſcriminat. </s> <s id="s.001948">Quod ſi vel mi<lb></lb>nimum duo illa centra diſcrepent, natura omnes ſui juris api<lb></lb>ces exactiſſimè perſequitur, & eſt ſpartum non in medio, ſed <lb></lb>aut in ſuperiore, aut in inferiore parte jugi (ſi quidem brachia <lb></lb>ſint æqualia; nam ſi ad latus eſſet in eadem recta linea, libra eſ<lb></lb>ſet inæqualium brachiorum, & tunc non adnexorum ponderum <lb></lb>æqualitas eſſet conſideranda, ſed eorum Ratio, ſumpta recipro<lb></lb>cè brachiorum Ratione) ex quo ſequitur aut reditus ad æquili<lb></lb>brium, aut ulterior deſcenſus brachij inclinati. </s> </p> <p type="main"> <s id="s.001949">Hinc eſt de illâ duplici tantummodo libræ ſpecie locutum <lb></lb>fuiſſe Ariſtotelem in Mechan. <expan abbr="q.">que</expan> 2. omiſsá priore hac, quæ vi<lb></lb>detur ſpeculantis intellectûs terminis coërceri, nunquam in <lb></lb>praxim niſi fortuito deducenda. </s> <s id="s.001950">Non enim ſatis eſt accuratiſ<lb></lb>ſimè inquirere centrum gravitatis jugi, ut illud ſit pariter cen<lb></lb>trum motûs, ſed neceſſe eſt punctum hoc in eádem rectá lineâ <lb></lb>eſſe, quæ jungit puncta contactuum jugi & annulorum, ex <lb></lb>quibus lances dependent: nam niſi hoc contingat, centrum il<lb></lb>lud gravitatis aſſumptum non eſt punctum, à quo brachiorum <lb></lb>longitudines diſcriminantur, ut inferiùs conſtabit dilucidiùs <lb></lb>ex iis, quæ de librâ curvâ dicentur. </s> </p> <p type="main"> <s id="s.001951">Quærendum eſt itaque, cur libra aginam habens in ſupe<lb></lb>riore loco, ſi ab æquilibrio horizontali dimoveatur, ad illud re<lb></lb>deat. </s> <s id="s.001952">Et ne locus æquivocationi pateat, dum ad hoc de<lb></lb>monſtrandum aſſumuntur puncta notabili intervallo inter ſe <lb></lb>diſtantia (ne videlicet linearum brevitas confuſionem aut ob<lb></lb>ſcuritatem pariat) obſerva lingulæ nomine non eam ſolùm par<lb></lb>tem intelligi, quæ ſupra libræ jugum intrà anſam excurrens <lb></lb>extat; ſed lingulæ, ſeu, ut aliis placet, trutinæ pars eſt etiam <lb></lb>linea, quæ in ipſa jugi craſſitie deſcripta intelligitur perpendi<lb></lb>cularis ad lineam longitudinis brachiorum, & tranſiens per <lb></lb>centrum motûs. </s> <s id="s.001953">Quare hujus lineæ pars intercepta inter cen<lb></lb>trum motûs, & lineam longitudinis brachiorum, ſivè exigua <pb pagenum="264" xlink:href="017/01/280.jpg"></pb>ſit, ſivè valde notabilis (quod quidem ad præſentem conſide<lb></lb>rationem attinet) nihil intereſt, nam eadem planè ſemper eſt <lb></lb>ratio, atque demonſtratio. </s> <s id="s.001954">Sit libra æqualium brachiorum <lb></lb><figure id="id.017.01.280.1.jpg" xlink:href="017/01/280/1.jpg"></figure><lb></lb>AB, cujus puncto medio C in<lb></lb>ſiſtat perpendicularis CD, & ſit <lb></lb>in ipsâ jugi craſſitie centrum mo<lb></lb>tûs punctum D, impoſitiſque <lb></lb>æqualibus ponderibus in A & B, <lb></lb>maneat in æquilibrio horizonta<lb></lb>li AB. </s> <s id="s.001955">Deprimatur extremitas A, <lb></lb>ut veniat in E, reliqua extremitas <lb></lb>B aſcendit in F, & C venit in G. </s> </p> <p type="main"> <s id="s.001956">Non poteſt igitur manere libra in poſitione EF ſublato de<lb></lb>primente in E, ſed manentibus æqualibus ponderibus redit ad <lb></lb>æquilibrium, séque reſtituit in AB; tùm quia centrum gravi<lb></lb>tatis non eſt in lineâ directionis tranſeunte per D punctum <lb></lb>ſuſpenſionis, tùm potiſſimum quia momenta ipſius F majora <lb></lb>ſunt momentis ipſius E ratione poſitionis & propenſionis ad <lb></lb>motum; poteſt enim F deſcendere juxta menſuram FH, dum <lb></lb>E aſcendit juxta menſuram EI; eſt autem major Ratio motûs <lb></lb>FH ad motum EI, quam ſit Ratio ponderum, quæ eſt Ratio <lb></lb>æqualitatis, nimirum ut FG ad GE. </s> <s id="s.001957">Nam per 8 lib.5. FO ad <lb></lb>GE majorem habet Rationem quàm FG ad GE, & FO ad <lb></lb>OE majorem habet Rationem quàm FO ad GE; ergo multo <lb></lb>major eſt Ratio FO ad OE, quàm FG ad GE. </s> <s id="s.001958">At ſimilia <lb></lb>ſunt triangula FHO, EIO, quia æquiangula (nam propter <lb></lb>paralleliſmum linearum directionis FH & IE, alterni E & F, <lb></lb>& alterni I & H, qui etiam recti ponuntur, & qui ad verticem <lb></lb>O, æquales ſunt) igitur per 4.lib. 6. ut FO ad OE, ita FH <lb></lb>ad EI. </s> <s id="s.001959">Eſt igitur major Ratio deſcensûs FH ad aſcenſum EI, <lb></lb>quàm ſit Ratio ponderum, quæ eſt ut FG ad GE. </s> </p> <p type="main"> <s id="s.001960">Hinc patet clara ſolutio quæſtionis à Keplero propoſitæ: <lb></lb>quia ſi pondus E majus ſit pondere F, illud non ad imum lo<lb></lb>cum deſcendet, ſed ibi libra obliquè ſubſiſtet, ubi pondera <lb></lb>erunt in Ratione reciprocâ motuum; quando ſcilicet ratione <lb></lb>poſitionis ita propenſio ad deſcendendum ponderis F erit ad <lb></lb>reſiſtentiam ponderis E, ne aſcendat, ut eſt viciſſim pondus E <lb></lb>ad pondus I: & tunc perpendicularis linea directionis ex D <pb pagenum="265" xlink:href="017/01/281.jpg"></pb>pancto ſuſpenſionis demiſſa cadet in centrum gravitatis compo<lb></lb>ſitæ libræ & ponderum. </s> <s id="s.001961">Cujus rei argumentum eſt mani<lb></lb>feſtum, quod libra quieſcens in poſitione EF ſi moveatur ab <lb></lb>aliquo deprimente ulteriùs aut elevante, ſibi relicta non minùs <lb></lb>redit ad eumdem ſitum obliquum, quam redeat ad æquilibrium <lb></lb>horizontale, ſi pondera ſint æqualia. </s> <s id="s.001962">Quæ omnia ex dictis pla<lb></lb>na ſunt & aperta; ſed an hoc idem rite probaverit Keplerus, <lb></lb>viderint alij. </s> </p> <p type="main"> <s id="s.001963">Eadem philoſophandi ratio erit in librâ brachiorum inæqua<lb></lb>lium LM, in qua ſint pondera L & M (computatis ipſorum <lb></lb>brachiorum gravitatibus juxta <lb></lb><figure id="id.017.01.281.1.jpg" xlink:href="017/01/281/1.jpg"></figure><lb></lb>momenta, quæ habent in illâ eâ<lb></lb>dem longitudine, ut dictum cap.2. <lb></lb>hujus libri) reciprocè in Ratione <lb></lb>brachiorum NM & NL. </s> <s id="s.001964">Depri<lb></lb>matur L in P, & elevabitur M in <lb></lb>Q, & N in V. </s> </p> <p type="main"> <s id="s.001965">Dico libram ſummoto deprimen<lb></lb>te, ad æquilibrium LM redituram. </s> <lb></lb> <s id="s.001966">Ducantur perpendiculares PT & QR, productâ LM horizon<lb></lb>tali, ſi opus fuerit. </s> <s id="s.001967">Triangula SQR, SPT ſunt ſimilia; igitur <lb></lb>per 4 lib.6. ut QS ad SP, ita ponderis Q propenſio ad deſcen<lb></lb>dendum QR, ad ponderis P reſiſtentiam, ne aſcendat, PT. </s> <lb></lb> <s id="s.001968">Eſt autem major Ratio QR ad PT, quàm ſit ponderis P ad <lb></lb>pondus <expan abbr="q;">que</expan> igitur pondus Q prævalebit. </s> <s id="s.001969">Majorem autem eſſe <lb></lb>Rationem ſic oſtenditur. </s> <s id="s.001970">Pondus P ad pondus Q eſt ut NM <lb></lb>ad NL ex hypotheſi, hoc eſt ut QV ad VP: ſed per 8. lib. 5. <lb></lb>major eſt Ratio QS ad VP, quàm QV ad VP, & major Ra<lb></lb>tio QS ad SP, quàm QS ad VP: igitur major eſt Ratio QS <lb></lb>ad SP, quàm QV ad VP, hoc eſt quàm pondus P ad pon<lb></lb>dus <expan abbr="q.">que</expan> Eſt autem demonſtratum ita eſſe QS ad SP, ut QR <lb></lb>ad PT; igitur major eſt Ratio deſcensûs QR ad aſcenſum PT, <lb></lb>quàm ſit Ratio ponderis P ad pondus Q: Ergo vis deſcendendi <lb></lb>major eſt; quàm oppoſita reſiſtentia, ac proptereà reſtituet ſe <lb></lb>libra in æquilibrio horizontali. </s> </p> <p type="main"> <s id="s.001971">Ex his manifeſtum eſt rem contrario modo ſe habere, quan<lb></lb>do ſpartum eſt in craſſitie jugi ira collocatum, ut ſit infra li<lb></lb>neam, quæ conſtituit longitudinem brachiorum; tunc enim al-<pb pagenum="266" xlink:href="017/01/282.jpg"></pb>tero brachiorum inclinato, tantum abeſt, ut libra revertatur ad <lb></lb>priorem paralleliſmum cum horizonte, ut potiùs, nullo ulteriùs <lb></lb>deprimente, brachium inclinatum deſcendat omninò, donec <lb></lb>impediatur ab ansá, in quam incurrit alterum brachium eleva<lb></lb>tum: quod ſi ſuperiori aut inferiori brachio nullum occurreret <lb></lb>impedimentum, ita fieret totius libræ converſio & revolutio, <lb></lb>ut ſpartum eſſet in loco ſuperiore, & tunc demùm in æquili<lb></lb>brio horizontali jugum quieſceret. </s> <s id="s.001972">Quæ omnia licet perſpicua <lb></lb>ſint, ſi ſuperiores duæ figuræ invertantur, clarioris tamen ex<lb></lb><figure id="id.017.01.282.1.jpg" xlink:href="017/01/282/1.jpg"></figure><lb></lb>plicationis gratiâ, ſit iterum jugum AB <lb></lb>æqualiter diviſum in C, & in perpen<lb></lb>diculari CD ſit axis, & centrum mo<lb></lb>tûs inferiùs in D: poſitis æqualibus <lb></lb>ponderibus A & B ſit æquilibrium ho<lb></lb>rizontale: & quoniam æqualia ſunt <lb></lb>pondera, atque æquales ad motum pro<lb></lb>penſiones, centrumque gravitatis eſt <lb></lb>in eâdem perpendiculari lineâ di<lb></lb>rectionis cum puncto ſuſtentationis D, manent in æquilibrio. </s> <lb></lb> <s id="s.001973">Deprimatur A in E, elevatur pariter B in F, & C deprimitur <lb></lb>in G. </s> <s id="s.001974">Dico libram, ſi ſibi ipſa dimittatur, non redituram ad po<lb></lb>ſitionem AB ſupra punctum D; ſed pondus E ulteriùs deſcen<lb></lb>ſurum. </s> <s id="s.001975">Ductis enim perpendicularibus EI & FH, propenſio <lb></lb>ponderis F ad motum deorſum, ut ſe reſtituat in priore æqui<lb></lb>librio, eſt FH, reſiſtentia ponderis E ad motum ſurſum eſt <lb></lb>EI. </s> <s id="s.001976">Eſt autem major Ratio reſiſtentiæ EI ad propenſionem <lb></lb>deorſum FH, quàm ſit Ratio ponderis F ad pondus E, aut vi<lb></lb>ciſſim; hæc enim æqualia ſunt ex hypotheſi, & eſt eorum Ra<lb></lb>tio ut AC ad CB, hoc eſt ut EG ad GF: Non igitur poteſt à <lb></lb>pondere F, cujus momenta minora ſunt elevari pondus E, cu<lb></lb>jus momenta ſunt majora ex diſpoſitione ad motum. </s> <s id="s.001977">Conſtat <lb></lb>verò major Ratio reſiſtentiæ EI ad propenſionem FH, quàm <lb></lb>ponderis F ad pondus E, quia in triangulis OIE, & OHF ſi<lb></lb>milibus eâdem eſt Ratio EI ad FH, quæ eſt EO ad OF; ſed <lb></lb>ex 8 lib.5. EO ad OF majorem habet Rationem quam EG ad <lb></lb>GF: igitur major eſt Ratio EI ad FH, quam EG ad GF, hoc <lb></lb>eſt ponderis ad pondus. </s> <s id="s.001978">Deſcendet itaque E, & nullo occur<lb></lb>rente obice ea fiet totius libræ revolutio circà centrum D, ut <pb pagenum="267" xlink:href="017/01/283.jpg"></pb>demum jugum EF ſit infrà punctum D, & quod inito fuit <lb></lb>punctum ſuſtentationis, fiat punctum ſuſpenſionis libræ. </s> <s id="s.001979">Ea<lb></lb>dem dicta intelligantur de librâ brachiorum inæqualium, quæ <lb></lb>ſupervacaneum eſt iterum inculcare. </s> </p> <p type="main"> <s id="s.001980">Oblata itaque librâ facilè dignoſces, cujus ſpeciei illa ſit, <lb></lb>quamvis ob punctorum propinquitatem, ſcilicet centri mo<lb></lb>tûs, & puncti brachiorum longitudinem diſcriminantis, non <lb></lb>valeat oculus dijudicare: impoſitis enim æqualibus ponderi<lb></lb>bus, ut habeat æquilibrium horizontale, aliquantulum depri<lb></lb>me alterutrum brachiorum, & ſublato deprimente, ſi quidem <lb></lb>manſerit obliqua (id quod rariſſimè continget) pronunciabis <lb></lb>centrum motûs convenire cum puncto brachiorum longitudi<lb></lb>nem diſcriminante: ſin autem ad æquilibrium redierit, cen<lb></lb>trum motûs erit in ſuperiore loco; ſi ulteriùs deſcenderit, cen<lb></lb>trum motûs erit infra lineam longitudinis brachiorum. </s> <s id="s.001981">Vel <lb></lb>etiam facto æquilibrio horizontali, adde pondus alteri lanci; <lb></lb>ſi deſcendat ita, ut jugum oblique conſiſtat aut magis aut mi<lb></lb>nùs, prout major aut minor factus eſt exceſſus ponderis, pro<lb></lb>nunciabis centrum motûs eſſe in ſuperiore loco: at ſi factâ <lb></lb>ponderum inæqualitate lanx gravior uſque ad imum deprima<lb></lb>tur, quantùm poteſt, indicabit centrum motûs eſſe in inferio<lb></lb>re loco, aut convenire cum puncto brachia diſcriminante: ſed <lb></lb>hoc ultimum temerè non affirmabis, niſi reſtitutâ ponderum <lb></lb>æqualitate, ſequatur quies in quacumque poſitione, aut con<lb></lb>versâ deorſum ansâ non contingat obliqua jugi conſiſtentia: <lb></lb>ſi enim factâ anſæ ſuſpenſione centrum illud fuiſſet in inferio<lb></lb>re loco, factâ converſione eſſet in ſuperiore loco, & continge<lb></lb>ret æquilibrium in poſitione obliquâ. <lb></lb></s> </p> <p type="main"> <s id="s.001982"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001983"><emph type="center"></emph><emph type="italics"></emph>An fieri poſsit libra Curva.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.001984">QUamvis ad ponderum examen inſtituendum rarò contin<lb></lb>gere poſſit, ut librâ Curvâ uti cogamur, quia tamen in <lb></lb>machinamentis aliquibus ita aut loci anguſtiæ, aut opportuna <pb pagenum="268" xlink:href="017/01/284.jpg"></pb>corporum movendorum diſpoſitio, exigunt collocari ponde<lb></lb>ra, ut & libræ Rationes ſerventur, & tamen jugi rectitudo nul<lb></lb>la appareat; non erit hî inutile libram curvam examinare, ut, <lb></lb>ſi quando eâ uti contigerit, innoteſcat, quænam ſint brachio<lb></lb>rum, & motuum Rationes. </s> <s id="s.001985">Libram autem curvam voco, quæ <lb></lb>a communi formâ deflectens latera habet non in directum poſi<lb></lb>ta, ſed in angulum concurrentia, aut in arcum ſinuata, quo<lb></lb>rum extremitates ſivè ſurſum, ſivè deorſum reſpiciunt: factâ <lb></lb>enim ſuſpenſione ſive ubi angulum latera conſtituunt, ſivè in <lb></lb>aliquo arcûs puncto, ea fieri poteſt hinc & hinc ponderum ad<lb></lb>ditio, quam horizontale æquilibrium conſequatur. </s> <s id="s.001986">Sed quia <lb></lb>imperitis fucum facere poſſet apparens hæc laterum longitudo, <lb></lb>caveant, ne ex illis jugum libræ deductum intelligant: contin<lb></lb>gere ſcilicet poteſt, ut planè varia ſit hujuſmodi libræ forma, <lb></lb>& magnitudo, idem tamen ſit ſemper libræ jugum, in quo <lb></lb>brachia deſumenda ſunt. </s> </p> <p type="main"> <s id="s.001987">Sint enim in angulum compacta duo latera recta AB & <lb></lb>AC; non eſt tota jugi magnitudo computanda ex horum late<lb></lb><figure id="id.017.01.284.1.jpg" xlink:href="017/01/284/1.jpg"></figure><lb></lb>rum longitudinibus; ſed ex ipsâ extre<lb></lb>mitatum B & C diſtantiâ BC; quæ ſem<lb></lb>per eadem eſt, ſivè ſit arcus BEFC, <lb></lb>ſivè alia ſint latera DB & DC, aut <lb></lb>GB & GC, atque ſuſpenſio fiat ſivè <lb></lb>in A, ſivè in D, ſivè in G, ſivè in quo<lb></lb>cumque alio puncto, quod ſit intra ſpa<lb></lb>tium à lineis AB, AC, BC comprehenſum. </s> <s id="s.001988">Eſt igitur idem <lb></lb>jugum BC, quia in B & C adnexa intelliguntur pondera, eo<lb></lb>rúmque diſtantia, prout libræ adnectuntur, ca eſt, quæ jugi <lb></lb>longitudinem determinat. </s> <s id="s.001989">Verùm an libra æqualium ſit po<lb></lb>tiùs, quàm inæqualium brachiorum, definiendum eſt ex <lb></lb>puncto ſuſpenſionis, à quo ad extremitates B & C deducen<lb></lb>dæ ſunt rectæ lineæ; quæ ſi æquales fuerint, libra eſt æqualium <lb></lb>brachiorum; ſin autem inæquales, inæqualium. </s> <s id="s.001990">Hinc ſi late<lb></lb>ra AB & AC jungantur tranſverſario HI, in eoque ſumatur <lb></lb>punctum ſuſpenſionis D, nil refert æqualia-ne, an inæqualia <lb></lb>ſint latera AB & AC? </s> <s id="s.001991">ſed attendenda eſt æqualitas aut in<lb></lb>æqualitas linearum ex D ductarum ad extremitates B & C. </s> </p> <p type="main"> <s id="s.001992">Neque me arguas, quòd dixerim jugum eſſe BC, & attenden-<pb pagenum="269" xlink:href="017/01/285.jpg"></pb>dam æqualitatem aut <expan abbr="inæqualitatẽ">inæqualitatem</expan> <expan abbr="linearũ">linearum</expan> ex puncto ſuſpenſio<lb></lb>nis ductarum, puta DB & DC; brachia ſiquidem in ipſo jugo <lb></lb>conſideranda ſunt; illæ <expan abbr="autẽ">autem</expan> lineæ nihil habent cum jugo com<lb></lb>mune præter puncta extrema B & C. </s> <s id="s.001993">Quamvis enim lineæ hu<lb></lb>juſmodi brachia libræ non ſint, ſi res proprie conſideretur, inſe<lb></lb>runt tamen æqualitatem aut inæqualitatem brachiorum, qua<lb></lb>tenus ex puncto ſuſpenſionis D ducta intelligitur ad BC jugum <lb></lb>perpendicularis DM, quæ jugum dividit in partes BM & CM <lb></lb>æquales aut inæquales. </s> <s id="s.001994">Nam quia triangula BMD & CMD <lb></lb>ſunt rectangula, quadrato BD, ex 47. lib.1. æqualia ſunt duo <lb></lb>quadrata DM & MB, & quadrato DC æqualia ſunt duo qua<lb></lb>drata DM & MC. </s> <s id="s.001995">Si igitur lineæ DB & DC æquales ſunt, <lb></lb>carum pariter quadrata ſunt æqualia; ex quibus dempto com<lb></lb>muni quadrato DM, remanent quadrata BM & CM æqualia, <lb></lb>ac proinde lineæ MB & MC æquales. </s> <s id="s.001996">Si verò lineæ BD & <lb></lb>CD ſunt inæquales, quadrata carum ſunt inæqualia; ex qui<lb></lb>bus dempto communi quadrato DM, reſidua ſunt quadrata <lb></lb>BM & CM inæqualia, eorumque latera (ſcilicet lineæ MB & <lb></lb>MC) inæqualia erunt pronuncianda. </s> </p> <p type="main"> <s id="s.001997">Brachia itaque hujus libræ curvæ propriè ſumpta non illa <lb></lb>ſunt, quæ apparent, & quia ex illis libræ curvæ moles conſtat, <lb></lb>vulgariter hoc vocabulo donantur; ſed ſunt ſegmenta lineæ <lb></lb>jungentis extremitates, quibus pondera adnectuntur; in quæ <lb></lb>ſegmenta dividitur à perpendiculo, quod ad illam ducitur ex <lb></lb>puncto, quod eſt motûs centrum. </s> <s id="s.001998">Cum igitur punctum hoc, <lb></lb>quod tanquam centrum legem dat motui, ſit extrà lineam ex<lb></lb>tremitates illas jungentem, aut in ſuperiore, aut in inferiore <lb></lb>loco crit; ac proptereà altera erit ex duabus illis ſpeciebus li<lb></lb>bræ, de quibus capite ſuperiore ſermo fuit, habentibus ſpar<lb></lb>tum aut ſuprà, aut infrà; & huic curvæ ea omnia convenient, <lb></lb>quæ ibi dicta ſunt, ut fiat æquilibrium horizontale, aut obli<lb></lb>quum. </s> <s id="s.001999">Si enim ſit libræ ſca<lb></lb><figure id="id.017.01.285.1.jpg" xlink:href="017/01/285/1.jpg"></figure><lb></lb>pus rectus AB bifariam divi<lb></lb>ſus, centrum motûs habens <lb></lb>in C & pondera adnexa in D <lb></lb>& E æqualia, habet æquilibrium horizontale, ad quod redit, ſi <lb></lb>ab illo dimoveatur; & ſi pondera D & E ſint inæqualia, ha<lb></lb>bet æquilibrium obliquum pro Ratione diſcriminis ponderum, <pb pagenum="270" xlink:href="017/01/286.jpg"></pb>quia ſcilicet centrum motûs C eſt ſupra lineam DE jungentem <lb></lb>puncta contactuum, quibus pondera adnectuntur. </s> <s id="s.002000">Facta au<lb></lb>tem figuræ converſione, ut C ſit in inferiore loco, & linea DE <lb></lb>in ſuperiore, in ſolo æquilibrio horizontali manet, à quo ſi re<lb></lb>moveatur, ad illud non redit, neque ullum habet æquilibrium <lb></lb>in poſitione obliquâ, ut dictum eſt. </s> <s id="s.002001">Jam ex jugo AB omnia <lb></lb>ſuperflua reſecentur, & remaneant virgulæ CD & CE con<lb></lb>nexæ in C centro motûs: manifeſtum eſt non eſſe immutata <lb></lb>ponderum momenta, & eundem eſſe motum libræ curvæ DCE <lb></lb>ac rectæ AB; ſivè C intelligatur in parte ſuperiori, ſivè in in<lb></lb>feriori. </s> <s id="s.002002">Quare & de hac curvâ, quod ad æquilibrium ſpectat, <lb></lb>eadem dicenda ſunt, quæ de librâ ſpartum ſuperiùs aut inferiùs <lb></lb>habente ſunt dicta. </s> </p> <p type="main"> <s id="s.002003">Et quidem ſi latera illa, quibus libra curva conſtat, ſecun<lb></lb>dùm longitudinem æqualia ſint, & paris gravitatis, additis <lb></lb>hinc & hinc æqualibus ponderibus fiet æquilibrium horizonta<lb></lb>le; quia vera linea jugi in ſegmenta æqualia dividitur, ſunt au<lb></lb>tem omnes Rationes Æqualitatis, omninò ſimiles. </s> <s id="s.002004">At ſi late<lb></lb>ra illa ſint inæqualia, non erunt addenda reciprocè pondera <lb></lb>(etiam computatâ ipſorum laterum gravitate) in Ratione illa<lb></lb>rum longitudinum; ſed in Ratione ſegmentorum jugi, ut fiat <lb></lb>æquilibrium: quia ex laterum illorum inæqualitate ſtatim qui<lb></lb>dem infertur etiam veram lineam jugi dividi in ſegmenta in<lb></lb>æqualia; ſed non illico conſequens eſt ſimilem eſſe Rationem <lb></lb>Inæqualitatis: Immò ſi inæqualia ſint illa latera, fieri omnino <lb></lb>non poteſt, ut ſegmenta, quæ fiunt à perpendiculari cadente <lb></lb>in baſim, videlicet in lineam jugi, ſint in eâdem Ratione; alio<lb></lb>quin ſi baſis ſegmenta eſſent in Ratione laterum adjacentium, <lb></lb>angulus, ex quo perpendicularis demittitur, eſſet bifariam <lb></lb>ſectus, per 3 lib.6. atque adeò duo triangula haberent duos an<lb></lb>gulos duobus angulis æquales, nimirum rectum & acutum, at<lb></lb>que latus haberent commune; ergo per 26.lib.1. & reliqua late<lb></lb><figure id="id.017.01.286.1.jpg" xlink:href="017/01/286/1.jpg"></figure><lb></lb>ra eſſent æqualia, contra hy<lb></lb>potheſim. </s> <s id="s.002005">Sit enim libra cur<lb></lb>va laterum inæqualium BAC, <lb></lb>linea recta BC eſt vera linea <lb></lb>jugi, in quam cadens perpen<lb></lb>diculum AD definit brachio-<pb pagenum="271" xlink:href="017/01/287.jpg"></pb>rum DB & DC longitudinem. </s> <s id="s.002006">Non eſt autem DB ad DC <lb></lb>ut BA ad AC, alioquin angulus BAC eſſet bifariam ſectus, <lb></lb>& duo triangula DAB, DAC haberent præter rectos ad D, <lb></lb>ctiam acutos ad A æquales, atque latus AD commune, ac <lb></lb>proinde eſſent etiam latera BA & AC æqualia contra hypo<lb></lb>theſim. </s> </p> <p type="main"> <s id="s.002007">Sunt igitur anguli ad A inæquales, & minor eſt, qui adja<lb></lb>cet minori lateri AC, quàm qui adjacet majori lateri AB: quia <lb></lb>in triangulo BAC major eſt angulus C oppoſitus majori lateri <lb></lb>BA, quàm angulus B oppoſitus minori lateri AC, ex 18.lib.1. <lb></lb>igitur in triangulis BDA, CDA rectangulis ad D, comple<lb></lb>mentum CAD minus eſt complemento BAD. </s> <s id="s.002008">Qua propter <lb></lb>ſi angulus BAC ſit bifariam dividendus, recta AE auferet ali<lb></lb>quid ex majore angulo BAD, & conſtituens angulum BAE <lb></lb>cadet in baſim inter B & D. </s> <s id="s.002009">Eſt itaque, per 3.lib.6. ut BA ad <lb></lb>AC, ita BE ad EC: ſed minor eſt Ratio BE ad EC quàm BD <lb></lb>ad EC, & multo minor quàm BD ad DC. per 8.lib.5. igitur <lb></lb>minor eſt Ratio BA ad AC, quàm ſit Ratio brachij BD ad <lb></lb>brachium DC. </s> <s id="s.002010">Si igitur pondera in C & B eſſent reciprocè ut <lb></lb>BA ad AC, haberent minorem Rationem, quàm BD ad DC, <lb></lb>ac propterea non eſſent apta ad conſtituendum æquilibrium <lb></lb>horizontale. </s> <s id="s.002011">Retento igitur pondere B, augendum eſſet pon<lb></lb>dus C, vel retento pondere C, minuendum eſſet pondus B, ut <lb></lb>eſſent in reciprocâ Ratione brachiorum BD & DC. </s> </p> <p type="main"> <s id="s.002012">Hinc etiam conſtat retentis eodem latere AB eadémque li<lb></lb>neâ horizontali BC cum eodem angulo B, ſi velis uti minori <lb></lb>pondere, quod cum pondere B faciat æquilibrium, addendum <lb></lb>eſſe in A latus majus latere AC, puta latus AF, itaut tota BF <lb></lb>ſit jugi longitudo, & brachia ſint BD & DF. </s> <s id="s.002013">Manifeſtum eſt <lb></lb>autem ex 8.lib.5. majorem Rationem eſſe ejuſdem BD ad DC <lb></lb>minorem, quàm ad DF majorem; ad pondera debent eſſe in F <lb></lb>& B ut BD ad DF; igitur minus pondus in F æquivalet eidem <lb></lb>ponderi B, cui in C æquivalet pondus majus. </s> <s id="s.002014">Porrò nemini <lb></lb>dubium eſſe poteſt, an latus AF majus ſit latere AC, quippe <lb></lb>quod in triangulo CAF opponitur angulo obtuſo ACF, per <lb></lb>19.lib.1. </s> </p> <p type="main"> <s id="s.002015">Sed ſi res fuerit in praxim deducenda, indicare oportet, quâ <lb></lb>methodo utendum ſit, ut quæſitam ponderum Rationem, hoc <pb pagenum="272" xlink:href="017/01/288.jpg"></pb>eſt ipſa jugi ſegmenta inveniamus, quippe quod ſolá mente <lb></lb>concipitur ad laterum extremitates jungedas deductum. </s> <s id="s.002016">Hæc <lb></lb>autem eſſe poterit praxis. </s> <s id="s.002017">Laterum AB & AC longitudine <lb></lb>metire, tùm ex B ad C extentum funiculum ad ſimilem men<lb></lb>ſuram revoca. </s> <s id="s.002018">His paratis certum eſt hanc jugi longitudinem <lb></lb>communiter majorem eſſe longitudine ſingulorum laterum, <lb></lb>ſemper tamen ſaltem alterius, tanto exceſſu, ut poſſit ab ea au<lb></lb>ferri pars, de quâ mox dicetur; debet ſcilicet excedere me<lb></lb>diam proportionalem inter aggregatum laterum, & eorum dif<lb></lb>ferentiam. </s> <s id="s.002019">Cum enim linea jugi à perpendiculo cadente ex <lb></lb>angulo verticali dividenda ſit, utrumque latus cum jugo facit <lb></lb>angulos acutos; alioquin ſi alteruter angulorum rectus eſſet, <lb></lb>aut linea jugi non eſſet parallela horizonti, aut latus eſſet idem <lb></lb>perpendiculum; & ſi obtuſus eſſet, perpendiculum caderet ex<lb></lb>tra lineam extremitates jungentem. </s> <s id="s.002020">Debet igitur tanta eſſe <lb></lb>jugi longitudo, ut differentia partium, in quas dividitur ad <lb></lb>differentiam laterum ſit ut ſumma laterum ad totum jugum. </s> </p> <p type="main"> <s id="s.002021">Quare fiat ut jugi longitudo funiculo deprehenſa ad laterum <lb></lb>ſummam, ita laterum differentia ad partem auferendam ex <lb></lb>longitudine jugi; cujus reſiduum bifariam diviſum dabit mi<lb></lb>noris brachij longitudinem. </s> <s id="s.002022">Hujus operationis ratio manifeſta <lb></lb>eſt ex corollario primo prop. 36.lib.3, & ex 3. ejuſdem lib.3. Sit <lb></lb>exempli gratia latus AB partium 20, latus AC partium 9, <lb></lb>diſtantia BC partium 23. Fiat ut 23 ad 29 ſummam laterum, <lb></lb>ita laterum differentia 11 ad (13 20/23) partem auferendam ex jugi <lb></lb>longitudine 23: Reſiduum partium (9 3/23) bifariam dividatur, & <lb></lb>ejus ſemiſſis (4 13/23) eſt longitudo brachij minoris DC; quod reli<lb></lb>quum eſt jugi partium (18 10/23) dat longitudinem alterius brachij <lb></lb>majoris BD. </s> <s id="s.002023">Eſt igitur brachiorum (atque adeò etiam ponde<lb></lb>rum reciprocè) Ratio ut 424 ad 105. </s> </p> <p type="main"> <s id="s.002024">Quod ſi his cognitis inveſtigare oporteat, quanta ſit hujus <lb></lb>lineæ horizontalis BC diſtantia à puncto ſuſpenſionis A, ni<lb></lb>mirum quanta ſit perpendicularis AD, ſtatim ex 47. lib.1. in<lb></lb>noteſcet, ſi ex quadrato lateris AC 81 auferas brachij DC <lb></lb>quadratum (20 445/529); nam reſiduum (60 84/529) eſt quadratum perpen<lb></lb>diculi AD, quod proinde eſt partium (7 17/23) proximè. </s> </p> <p type="main"> <s id="s.002025">At ſi pro ratione tui inſtituti nimia ſit hujus perpendiculi <pb pagenum="273" xlink:href="017/01/289.jpg"></pb>longitudo, & opportuniùs accidat jugum BC horizontale mi<lb></lb>nus diſtare à puncto ſuſpenſionis A, jam conſtat latera AB <lb></lb>& AC explicanda in majorem angulum; quapropter etiam <lb></lb>major erit jugi longitudo, ex 24.lib.1. Sit ergo definita per<lb></lb>pendiculi AD altitudo partium 4: hujus quadratum 16 aufer <lb></lb>ex 81 quadrato lateris AC, & reſiduum 65 eſt quadratum bra<lb></lb>chij minoris DC, quod idcircò eſt partium (8 1/16) ſerè. </s> <s id="s.002026">Simili<lb></lb>ter ipſius AD quadratum 16 aufer ex 400 quadrato lateris AB, <lb></lb>& reſiduum 384 eſt quadratum brachij majoris BD, quod eſt <lb></lb>partium (19 23/39) proximè; & totum jugum BC eſt partium (27 25/39). <lb></lb>Quare brachi BD ad brachium DC Ratio eſſet ut 764 ad 314, <lb></lb>quæ reciprocè eſſet & ponderum. </s> </p> <p type="main"> <s id="s.002027">Ex quibus perſpicuum eſt, poſitis iiſdem libræ curvæ late<lb></lb>ribus, diſparem eſſe ponderum Rationem: in priore enim poſi<lb></lb>tione Ratio eſt 424 ad 105, hoc eſt proxime ut 4 ad 1. in poſte<lb></lb>riore poſitione, ubi in majorem angulum latera explicantur, <lb></lb>Ratio eſt 764 ad 314, hoc eſt ut 2. 43 ad 1; quæ minor eſt <lb></lb>Ratio, quàm prior ut 4 ad 1. Si autem latera eadem eſſent in <lb></lb>directum conſtituta, eſſet ponderum Ratio ut 20 ad 9, hoc eſt <lb></lb>ut 2. 22′ ad 1; quæ eſt minima Ratio omnium, quæ intercede<lb></lb>re poſſunt inter pondera æquilibrium horizontale conſtituen<lb></lb>tia ex illorum laterum extremitatibus: quæ extremitates quo<lb></lb>minus diſtabunt, inflexis ſubinde lateribus, eo majus pondus <lb></lb>requiretur in extremitate lateris brevioris, ut æquè ponderet <lb></lb>cum uno eodemque pondere collocato in extremitate lateris <lb></lb>longioris. </s> </p> <p type="main"> <s id="s.002028">Porrò ubi de ponderum Ratione ſermo eſt, cave ne ipſorum <lb></lb>laterum inæqualium libræ curvæ gravitatem contemnas; ſi <lb></lb>enim æqualia illa eſſent, æqualia quoque eſſent eorum mo<lb></lb>menta tùm ratione gravitatis, tum ratione poſitionis, nam per<lb></lb>pendiculum caderet in medium jugum, & latera eſſent ſimi<lb></lb>liter inclinata, ac proinde ſola ponderum æqualitas ſpectaretur: <lb></lb>at laterum hujuſmodi inæqualium momenta ſunt ex utroque <lb></lb>capite inæqualia, videlicet & ratione gravitatis inſitæ, quæ ex <lb></lb>hypotheſi ſingulis lateribus ineſt pro Ratione molis inæqualis, <lb></lb>& ratione poſitionis, quæ valde diverſa eſt, cùm non ſint late<lb></lb>ra illa ſimili angulo ad perpendiculum inclinata; ſed magis in-<pb pagenum="274" xlink:href="017/01/290.jpg"></pb>clinatur latus longius faciens cum perpendiculo majorem an<lb></lb>gulum: pro variâ autem inclinatione ipſam ejuſdem lateris gra<lb></lb>vitatem varia obtinere momenta manifeſtum videtur. </s> <s id="s.002029">Pona<lb></lb><figure id="id.017.01.290.1.jpg" xlink:href="017/01/290/1.jpg"></figure><lb></lb>mus laminam metallicam AB clavo <lb></lb>infixam in A, circa quem quaſi cen<lb></lb>trum deſcribat ſemicirculum BDC. </s> <lb></lb> <s id="s.002030">Si obtineat perpendicularem poſitio<lb></lb>nem AB, tota gravitas innititur clavo <lb></lb>A ſuſtinenti, & nullam vim habet de<lb></lb>ſcendendi; ſimiliter in perpendiculari <lb></lb>poſitione AC tota gravitas retinetur à <lb></lb>clavo A, nec poteſt deſcendere. </s> <s id="s.002031">At ſi <lb></lb>poſitionem habeat AD horizonti pa<lb></lb>rallelam, omnino nec ſuſtinetur, nec <lb></lb>retinetur à clavo, ſed toto conatu ſuas <lb></lb>deſcendendi vires exerit. </s> <s id="s.002032">In locis igi<lb></lb>tur intermediis partim ſuſtinetur aut <lb></lb>retinetur à clavo A, partim conatum <lb></lb>deorſum exercet: ſic ex B veniens in E ſuſtinetur juxta men<lb></lb>ſuram FE, & deorſum tendit juxta menſuram GE; at ex B ve<lb></lb>niens in H ſuſtinetur juxta menſuram IH, & deorſum tendit <lb></lb>juxta menſuram KH. </s> <s id="s.002033">Simili modo contingit in quadrante in<lb></lb>feriore; nam in poſitione AL retinetur juxta menſuram IL, <lb></lb>nec deſcenſum poteſt habere niſi ut LM; atque in O impedi<lb></lb>mentum à retinente eſt ut FO, conatum deorſum metitur ON. </s> <lb></lb> <s id="s.002034">Quia ſcilicet ſi ab aliquo ſuſtineatur in L, perinde ſe habet ac <lb></lb>ſi eſſet in plano habente inclinationis angulum CAL; in quo <lb></lb>plano gravitatio eſt ad gravitationem in perpendiculo ut Ra<lb></lb>dius ad ſecantem, ſeu ut Sinus Complementi ad Radium, hoc <lb></lb>eſt ut IL ad AL: ac propterea vires clavi retinentis in eâ in<lb></lb>clinatione ad vires retinentis in perpendiculo debent eſſe ut IL <lb></lb>ad AC, hoc eſt ad AL: At gravitatio, quâ urgetur planum <lb></lb>inclinatum, eſt ut PC Sinus Verſus anguli inclinationis, qui <lb></lb>planè æqualis eſt ipſi LM. </s> <s id="s.002035">Cùm autem hîc nullum habeatur <lb></lb>ſubjectum planum, quod prematur à gravitante laminâ metal<lb></lb>licâ, exerit hunc conatum deorſum adversùs aliud oppoſitum <lb></lb>pondus, quod elevare conatur, vel cui conanti reſiſtit, ne ab <lb></lb>eo elevetur. </s> <s id="s.002036">Si igitur in lineá AC perpendiculari lamina AC <pb pagenum="275" xlink:href="017/01/291.jpg"></pb>contra clavum A exercet momenta totius gravitatis deorſum <lb></lb>nitentis, & in AL impeditur, ac retinetur ſecundum menſu<lb></lb>ram IL, fiat ut AC ad IL, ita tota gravitas laminæ ad aliud, <lb></lb>& prodibit quantitas gravitationis contra retinentem, reſi<lb></lb>duumque LM erit illa gravitatio, quæ conſideranda eſt in eâ <lb></lb>poſitione inclinata AL. </s> </p> <p type="main"> <s id="s.002037">Sed quoniam AL à centro motûs A diſtantiam habet AI, <lb></lb>comparanda erit hæc diſtantia cum diſtantia oppoſiti lateris li<lb></lb>bræ, ut habeantur momenta invicem comparata. </s> <s id="s.002038">Obſervan<lb></lb>dum tamen eſt non rem perinde ſe habere, ac ſi tota gravita<lb></lb>tio laminæ inclinatæ AL poſita eſſet in L, atque adeò in diſtan<lb></lb>tià AI; ſed quia diſtribuitur ſecundùm totam ipſam longitudi<lb></lb>nem AL, & partes remotiores plus habent momenti, quàm <lb></lb>propiores centro, juxtà Rationem diſtantiarum, proptereà vel <lb></lb>tota gravitas lateris AL, quæ eſt LM, intelligenda eſt in me<lb></lb>dia diſtantiâ inter A & I, vel ſemiſſis gravitationis AL, hoc eſt <lb></lb>ſemiſſis ipſius LM, intelligendus eſt in I, quemadmodum hu<lb></lb>jus libri 3. cap. 2. dictum eſt totam gravitatem AD intelligen<lb></lb>dam in mediâ diſtantiâ inter A & D, aut ejus ſemiſſem in ex<lb></lb>tremitate D. </s> <s id="s.002039">Quamvis autem ex inclinatione CAL oriatur <lb></lb>diſtantia AI, hæc tamen venire pariter in computationem <lb></lb>debet, quia comparari debent hæc momenta cum momentis <lb></lb>diſtantiæ oppoſitæ, quæ momenta orta ex Ratione diſtantiarum <lb></lb>eadem ſunt, ſive AL ſit lamina, ſive trabs; quamquam valde <lb></lb>diſpares ſint gravitates, quæ aſſumendæ ſunt ex eâdem inclina<lb></lb>tione; ac propterea & LM indicans gravitationem comparatè <lb></lb>ad totam gravitatem abſolutam, & AI definiens momentum <lb></lb>ex diſtantiâ, conſiderari debent. </s> <s id="s.002040">Hoc pacto habetur totum <lb></lb>momentum lateris AL; ſimiliterque habebitur momentum la<lb></lb>teris oppoſiti. </s> <s id="s.002041">Ex quo patet laterum inclinatorum in librâ cur<lb></lb>vâ momenta componi & ex Ratione diſtantiarum, & ex Ratio<lb></lb>ne momenti, quod habent ſingula latera ex inclinatione ad <lb></lb>perpendiculum. </s> </p> <p type="main"> <s id="s.002042">At ſubdubitas, utrùm iſta, quæ hîc dicuntur, cum iis aptè <lb></lb>cohæreant, quæ lib.1. cap.15. dicta ſunt, ubi ponderis in L <lb></lb>conſtituti vires ad deſcendendum definiri diximus à Sinu an<lb></lb>guli declinationis à perpendiculo CAL, qui æqualis eſt ipſi <lb></lb>AI: hîc verò laminæ AL gravitationem conſtituimus ex <pb pagenum="276" xlink:href="017/01/292.jpg"></pb>Sinu complementi ejuſdem anguli CAL, nimirum ex li<lb></lb>neâ IL. </s> </p> <p type="main"> <s id="s.002043">Quapropter obſerva non eandém eſſe rationem gravitationis <lb></lb>lateris AL libræ, atque ponderis adnexi in extremitate L; hu<lb></lb>jus enim momenta perinde computantur, ac ſi eſſet in I; quia <lb></lb>ſcilicet AI æqualis eſt brachio libræ PL, & planum inclina<lb></lb>tum, in quo pondus L conſtitutum intelligitur, non eſt AL, <lb></lb>ſed Tangens in L ad angulos rectos, ut loco citato explicatum <lb></lb>eſt. </s> <s id="s.002044">At libræ latus AL ſuam habens gravitatem aliter ſe habet: <lb></lb>nam quemadmodum ſi inniteretur clavo in A, non tamen illi <lb></lb>infigeretur, atque ab aliquo ſuſtineretur in puncto L, certum <lb></lb>eſt planum inclinatum, in quo moveretur, eſſe AL, contra <lb></lb>quæ momenta deſcendendi in plano inclinato reluctatur clavus <lb></lb>in A poſitus, & retinens; ita ſublato ſuſtinente in L, & poſito <lb></lb>contranitente reliquo latere libræ, non tollitur munus clavi A <lb></lb>retinentis, ſed ſubſtituitur latus illud oppoſitum loco ſuſtinen<lb></lb>tis in L: igitur contra illud latus hoc latus AL exercet eadem <lb></lb>momenta gravitationis, quæ exerceret adversùs ſuſtinentem <lb></lb>in L, hoc eſt in planum inclinatum; quæ momenta ea ſunt, <lb></lb>quæ remanent demptis IL momentis gravitationis in plano in<lb></lb>clinato, nimirum reſiduum LM. </s> <s id="s.002045">Quia verò qui ſuſtineret la<lb></lb>tus AL in L, non eſſet unicum ſuſtinens, ſed planum inclina<lb></lb>tum eſt AL, & ita latus retinetur in clavo A, ut etiam ab eo <lb></lb>aliquatenus ſuſtineatur, atque adeò lamina inclinata ſuſtinea<lb></lb>tur à duobus in A & L, retineaturque ſolùm ab A; propterea <lb></lb>non totum momentum LM, ſed ejus ſemiſſem accipiendum <lb></lb>diximus, ut habeantur momenta, quibus contranititur oppo<lb></lb>ſitum latus, ſi addantur momenta, quæ oriuntur ex diſtantiâ à <lb></lb>centro motûs, ut dictum eſt. </s> </p> <p type="main"> <s id="s.002046">Hæc autem ut exemplo clariora fiant, ſint eadem, quæ priùs <lb></lb>in præcedente figurâ poſita ſunt, latera libræ curvæ BAC, lon<lb></lb>gius BA partium 20, brevius CA partium 9, & quidem in eâ <lb></lb>poſitione, ut perpendiculum AD cadens in jugum ſit partium <lb></lb>(7 17/23), & brachium jugi DC adjacens minori lateri ſit partium <lb></lb>(4 13/23), reliquum verò jugi brachium DB partium (18 10/23). Primùm <lb></lb>quære momenta laterum ex eorum inclinatione: Cumque per<lb></lb>pendiculum AD ſit æquale Sinui Complementi anguli incli-<pb pagenum="277" xlink:href="017/01/293.jpg"></pb>nationis DAC, poſito Radio AC, notus eſt Sinus Verſus ejuſ<lb></lb>dem anguli inclinationis, ſcilicet differentia inter AD & AC, <lb></lb>quæ eſt partium (1 6/23): & ſimili methodo Sinus Verſus anguli in<lb></lb>clinationis DAB eſt partium (12 6/23). Ratio igitur gravitationis <lb></lb>lateris AB ad gravitationem lateris AC ex inclinatione eſt ut <lb></lb>282 ad 29; Ratio momentorum ex diſtantiâ à centro, ut ſupra <lb></lb>diximus, eſt ut 424 ad 105. Compoſitis igitur duabus hiſce <lb></lb>Rationibus, eſt totius momenti lateris AB ad totum momen<lb></lb>tum lateris AC Ratio ut 119568 ad 3045, hoc eſt in minimis <lb></lb>terminis ut 39. 267″ ad 1. Sit igitur gravitas abſoluta lateris <lb></lb>AB unciarum 20; gravitatio reſpondens ſemiſſi Sinus Verſi an<lb></lb>guli inclinationis eſt unciarum (6 3/23). Item gravitas abſoluta la<lb></lb>teris AC ſit unc. </s> <s id="s.002047">9: gravitatio reſpondens ſemiſſi Sinus Verſi <lb></lb>anguli inclinationis eſt unc. (29/46). Hæc gravitatio (29/46) ducatur in <lb></lb>diſtantiam à perpendiculo partium (4 13/23), & eſt momentum <lb></lb>2.878tʹ. </s> <s id="s.002048">Similiter gravitatio unc. (6 3/23) ducatur in diſtantiam à <lb></lb>perpendiculo partium (18 10/23), & eſt momentum 113.013tʹ. </s> <s id="s.002049">Di<lb></lb>viſo itaque majore numero 113013 per minorem 2878, in mi<lb></lb>nimis terminis Ratio eſt ut 39.268″ ad 1: quæ minimùm differt <lb></lb>à priore illa Ratione propter neglectas fractiunculas in divi<lb></lb>ſionibus. </s> </p> <p type="main"> <s id="s.002050">Nunc inquiramus, quantum ponderis addendum ſit lateri <lb></lb>minori, ut fiat æquilibrium cum ſolâ majoris lateris gravitate. </s> <lb></lb> <s id="s.002051">Statuatur pondus addendum Algebricè 1 ℞, cujus diſtantia à <lb></lb>perpendiculo cum ſit partium (4 13/23), ponderis additi momentum <lb></lb>eſt (105/23) ℞ addendum momento lateris minoris invento. </s> <s id="s.002052">Quare <lb></lb>2.878tʹ + (105/23) ℞ æquantur momento 113.013tʹ lateris majoris: <lb></lb>& utrinque demptis 2.878tʹ, remanet æquatio inter (105/23) ℞ & <lb></lb>110.135tʹ. </s> <s id="s.002053">Demum inſtitutâ diviſione prodit <expan abbr="pretiũ">pretium</expan> 1 ℞, hoc eſt <lb></lb>ponderis addendi, unciarum 24 1/8. Huic itaque ponderi additâ <lb></lb>gravitatione lateris minoris AC unc. (29/46) hoc eſt in milleſimis <lb></lb>630tʹ, erit in C totum pondus unc. </s> <s id="s.002054">24.755tʹ; & in B intelli<lb></lb>gitur gravitas unc. (6 3/23), hoc eſt in milleſimis unc. </s> <s id="s.002055">6.130tʹ ferè. </s> <lb></lb> <s id="s.002056">Vides igitur hæc pondera eſſe reciprocè poſita in Ratione <lb></lb>diſtantiarum DB & DC: & quamvis demum in his Ratio<lb></lb>nibus non ſibi exactiſſimè reſpondeant numeri, ſatis pa-<pb pagenum="278" xlink:href="017/01/294.jpg"></pb>tet exiguum hoc diſcrimen oriri ex neglectis fractiun<lb></lb>culis. </s> </p> <p type="main"> <s id="s.002057">Cæterùm hæc tam minutè perſequi in librâ curvâ, cujus <lb></lb>latera non adeò notabili gravitate ſunt prædita, labor quidem <lb></lb>videtur inutilis: ſed quoniam hujuſmodi libræ præcipuus uſus <lb></lb>eſſe poteſt in machinationibus, ubi latera libræ ſunt tigilli craſ<lb></lb>ſiores non mediocris gravitatis, operæ pretium fuit indicare, <lb></lb>quâ methodo ipſorum laterum gravitates & momenta compu<lb></lb>tari oporteat, ut non caſu, ſed ex certâ ratione pondera collo<lb></lb>centur, & æquipondia ſtatuantur. <lb></lb> </s> </p> <p type="main"> <s id="s.002058"><emph type="center"></emph>CAPUT VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002059"><emph type="center"></emph><emph type="italics"></emph>Quænam libræ ſint omnium exactiſsimæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002060">INſtrumenti cujuſque bonitas æſtimatur ex fine, ad quem fuit <lb></lb>inſtitutum, prout ad illum aſſequendum aptum fuerit, aut <lb></lb>ineptum, eóque melius cenſetur inſtrumentum, quò certiùs <lb></lb>per illud propoſitus finis obtinetur; quemadmodum per ſingu<lb></lb>la eunti facilè conſtabit. </s> <s id="s.002061">Ut igitur exactiſſimum libræ genus <lb></lb>innoteſcat, ſatis patet inquirendum eſſe, quænam libra facilli<lb></lb>mè ab æquilibrio recedat; quo receſſu indicans vel minimam <lb></lb>ponderum inæqualitatem, etiam ſuo æquilibrio exquiſitam <lb></lb>ponderum æqualitatem oſtendit; id quod per libram veſtiga<lb></lb>mus. </s> <s id="s.002062">Hîc autem de librâ æqualium brachiorum ſermo eſt, quâ <lb></lb>communiter uti ſolemus: quamquam aliqua etiam ad libram <lb></lb>inæqualium brachiorum proportione traduci queant. </s> <s id="s.002063">Ex du<lb></lb>plici capite libram, quà libra eſt, ponderum gravitates præ aliis <lb></lb>libris exquiſitè examinare contingit, videlicet aut ex brachio<lb></lb>rum longitudine, aut ex ſparti, ſeu centri motûs, poſitione; <lb></lb>reliqua enim impedimenta, aut adjumenta materiam potiùs ſe<lb></lb>quuntur, quàm libræ formam. </s> </p> <p type="main"> <s id="s.002064">Et quidem quod ad brachiorum longitudinem ſpectat, adeò <lb></lb>certum Ariſtoteli videtur majoribus libris, majori ſcilicet bra<lb></lb>chiorum longitudine præditis, accuratiùs examinari ponde<lb></lb>rum æqualitatem, ut in Mechanicis quæſtionibus hoc primum <pb pagenum="279" xlink:href="017/01/295.jpg"></pb>ab eo quæratur, <emph type="italics"></emph>Cur majores libræ exactiores ſunt minoribus?<emph.end type="italics"></emph.end> Cau<lb></lb>ſam autem ex eo deſumendam putat, quòd ſpartum ſit cen<lb></lb>trum, brachia verò quaſi lineæ à centro exeuntes; & quia Ra<lb></lb>dij longiores ab eodem centro cum brevioribus exeuntes ſi pa<lb></lb>riter moveantur, majorem arcum deſcribunt, propterea etiam <lb></lb>citius moveri neceſſe eſt extremitatem libræ, quò plus à ſparto <lb></lb>diſceſſerit. </s> <s id="s.002065">Hinc eſt in minore librâ poſſe aliquando ex tenui <lb></lb>inæqualitate ponderum fieri motum non conſpicuum, atque <lb></lb>adeò illam occultè diſcedere ab æquilibrio; id quod in majore <lb></lb>librâ contingere non poteſt, quia longioris brachij extremitas <lb></lb>notabili motu inclinatur. </s> <s id="s.002066">Sit enim li<lb></lb><figure id="id.017.01.295.1.jpg" xlink:href="017/01/295/1.jpg"></figure><lb></lb>bra longior AB, cujus ſpartum ſit C; <lb></lb>moveatur, & deſcribat arcus BG, & <lb></lb>AF, qui ſunt multò magis conſpicui <lb></lb>& majores, quàm qui à librâ minore <lb></lb>DE habente idem motûs centrum C, <lb></lb>deſcribantur arcus EI & DH. </s> <s id="s.002067">Con<lb></lb>ſtat igitur motum puncti E prorſus fugere omnem oculorum <lb></lb>aciem, ſi motus extremitatis B vix ſit conſpicuus. </s> <s id="s.002068">Ex quo il<lb></lb>lud etiam conſequens eſt, quod major libra clariùs indicat <lb></lb>æquilibrium. </s> </p> <p type="main"> <s id="s.002069">Verùm ſi hæc ita accipiantur, prout communi huic inter<lb></lb>pretationi ſubeſt Ariſtoteles, vix aliquid habent momenti: <lb></lb>quis enim pondera vix inæqualia bilance ſubtiliter examinans <lb></lb>jugi extremitates reſpicit, ut videat, an lineæ horizonti paral<lb></lb>lelæ congruat jugum? </s> <s id="s.002070">& non potiùs lingulam CO conſiderat, <lb></lb>an cum ansâ perpendiculari illa conveniat? </s> <s id="s.002071">Quod ſi lingula at<lb></lb>tendatur, idem eſt ejus motus ſive longior ſit libra AB, ſive <lb></lb>brevior DE; factâ enim inclinatione aut majore motu BG, aut <lb></lb>minore motu EI, eadem eſt lingulæ poſitio CS. </s> <s id="s.002072">Hoc tantùm <lb></lb>habent emolumenti brachia longiora, quod faciliùs dividuntur <lb></lb>bifariam æqualiter quàm breviora: & ſi minimum aliquod diſ<lb></lb>crimen intercedat, hoc minorem habet Rationem ad bra<lb></lb>chium longiùs, quàm ad brevius. </s> <s id="s.002073">Quare aliâ ratione acci<lb></lb>pienda eſt libra: nam ſi in uno eodemque puncto C conveniant <lb></lb>ſpartum & jugi diviſio, aut ſpartum ſit inferius, ſive longiora, <lb></lb>ſive breviora ſint brachia, ponderum inæqualitas illicò inno<lb></lb>teſcit, quia extremitas præponderans, ad imum locum, quan-<pb pagenum="280" xlink:href="017/01/296.jpg"></pb>tum poteſt, deſcendit. </s> <s id="s.002074">Locutus igitur videtur Ariſtoteles de <lb></lb>librâ ſpartum habente in ſuperiore jugi loco extrà lineam, quæ <lb></lb>jugi longitudinem definit. </s> </p> <p type="main"> <s id="s.002075">Sit iterum libra longior AB, & brevior DE, utraque bifa<lb></lb>riam diviſa in C; & ſit linea lingulæ perpendicularis CK, in <lb></lb><figure id="id.017.01.296.1.jpg" xlink:href="017/01/296/1.jpg"></figure><lb></lb>quâ ſumatur ſpartum, ſeu motús <lb></lb>centrum O, & reſiduum OK ſit <lb></lb>lingula, ex cujus declinatione à <lb></lb>perpendiculo anſæ, dignoſcitur <lb></lb>ſublatum æquilibrium. </s> <s id="s.002076">Sit pondus <lb></lb>A ad pondus B ut 5 ad 3: centrum <lb></lb>gravitatis jugi & ponderum commune non poteſt eſſe C, quod <lb></lb>brachia CA & CB æqualia conſtituit; ſed erit ut pondus A ad <lb></lb>pondus B, ita reciprocè longitudo BG ad longitudinem GA, <lb></lb>eritque punctum G centrum gravitatis, nec libra conſiſtet, ni<lb></lb>ſi recta GOH fiat perpendicularis horizonti: lingula igitur <lb></lb>OK declinabit à perpendiculo anſæ juxta angulum HOK. </s> <lb></lb> <s id="s.002077">Eadem pondera transferantur in minorem libram DE; & ſi <lb></lb>fiat ut pondus D 5 ad pondus E 3, ita EF ad FD, erit F cen<lb></lb>trum gravitatis libræ DE & ponderum: quare libra non con<lb></lb>ſiſtet, niſi recta FOI ſit horizonti perpendicularis, & tunc à <lb></lb>perpendiculo declinabit lingula OK juxta angulum IOK. </s> <lb></lb> <s id="s.002078">Quoniam verò eſt ut 4 ad 1, ita AC ad CG, ita DC ad CF, <lb></lb>& AC major eſt quàm DC, erit etiam ex 14 lib.5. GC major <lb></lb>quàm FC; igitur angulus COF minor eſt angulo COG, pars <lb></lb>minor toto; ac proinde ad verticem angulus KOI minor eſt <lb></lb>angulo KOH. </s> <s id="s.002079">Poſitis igitur ponderibus iiſdem in libræ lon<lb></lb>gioris AB extremitatibus, declinabit lingula à perpendiculo, <lb></lb>cum eo conſtituens angulum majorem, quàm ſit angulus ab <lb></lb>eadem lingulâ conſtitutus cum perpendiculo, quando ponde<lb></lb>ra illa inæqualia adnectuntur libræ breviori DE. </s> <s id="s.002080">Hinc eſt <lb></lb>quòd ſi inæqualitas ponderum exigua ſit, centrum gravitatis <lb></lb>in utrâque librâ non multùm recedat à puncto C, parùm in ma<lb></lb>jore, minimùm in minore, ac proinde lingulæ deflexio fortaſſe <lb></lb>inobſervabilis erit in minore librâ, quæ in majore evadet nota<lb></lb>bilis atque conſpicua. </s> <s id="s.002081">Hinc etiam patet, cur extremitas A <lb></lb>deſcendens magis moveatur, quàm extremitas D minoris li<lb></lb>bræ; quia ſcilicet angulus OGA, per 16. lib.1. major eſt quàm <pb pagenum="281" xlink:href="017/01/297.jpg"></pb>angulus OFD, ac propterea ubi OG facta ſit perpendicularis, <lb></lb>linea AG cum illà faciens obtuſiorem angulum, magis depri<lb></lb>metur infrà lineam AB horizontalem. </s> </p> <p type="main"> <s id="s.002082">Sed jam inquirendum eſt, utrùm expediat centrum motûs <lb></lb>magis diſtare à lineâ jugi, an verò illi propiùs admoveri, ut <lb></lb>clariùs innoteſcat receſſus jugi ab æquilibrio horizontali: illa <lb></lb>quippe ſparti poſitio eligenda eſt, quæ etiam minimum mo<lb></lb>tum indicet notabili lingulæ declinatione. </s> <s id="s.002083">Dico itaque ſpar<lb></lb>tum lineæ jugi proximum utilius eſſe, quàm remotum. </s> <s id="s.002084">Sit <lb></lb>enim libra AB bifariam in C di<lb></lb><figure id="id.017.01.297.1.jpg" xlink:href="017/01/297/1.jpg"></figure><lb></lb>viſa, & ex hoc puncto exeat per<lb></lb>pendicularis CI; in quâ pro cen<lb></lb>tro motûs eligatur punctum S; <lb></lb>ponantur verò pondera A & B ita <lb></lb>eſſe inæqualia, ut centrum gravi<lb></lb>tatis commune ſit D. </s> <s id="s.002085">Igitur DSR <lb></lb>eſt linea, quæ facta perpendicularis conſtituit cum lingulâ SI <lb></lb>angulum ISR. </s> <s id="s.002086">Deinde reliquis omnibus manentibus, ſit cen<lb></lb>trum motûs O remotius à lineâ jugi, & linea DOV facta per<lb></lb>pendicularis declinabit à lingulâ OI juxta angulum IOV, <lb></lb>quem conſtat eſſe minorem angulo ISR; nam angulus DSC <lb></lb>externus major eſt interno DOS, per 16. lib. 1. eſt autem huic <lb></lb>ad verticem IOV, & illi ad verticem ISR; igitur ISR angu<lb></lb>lus eſt major angulo IOV. </s> </p> <p type="main"> <s id="s.002087">Quòd ſi centrum motûs adhuc propiùs admoveatur medio <lb></lb>jugi puncto C, adhuc majorem angulum conſtituet cum lin<lb></lb>gulâ, ac proptereà adhuc multò notabilior erit deflexio lingu<lb></lb>læ à perpendiculo, etiam ſi exiguus ſit motus ex eo, quod cen<lb></lb>trum gravitatis D proximè accedat ad punctum C: eſt ſiqui<lb></lb>dem extrà controverſiam, quò minor eſt ponderum inæquali<lb></lb>tas, eò etiam minorem eſſe puncti D à puncto C diſtantiam. </s> <lb></lb> <s id="s.002088">Ex quo manifeſtum evadit exiguam ponderum differentiam <lb></lb>non dignoſci, ſi ſpartum notabili intervallo receſſerit à lineâ ju<lb></lb>gi; hæc enim ſparti diſtantia habet rationem Radij, diſtantia <lb></lb>centri gravitatis à medio jugi locum obtinet Tangentis; igitur <lb></lb>ſi fiat major ſparti diſtantia, eadem Tangens ad majorem Ra<lb></lb>dium minorem Rationem habebit, atque adeò ſubtendet mul<lb></lb>tò acutiorem angulum, qui proptereà minùs obſervari poterit. <pb pagenum="282" xlink:href="017/01/298.jpg"></pb>Quare pro eâdem ponderum inæqualitate dignoſcendâ, ſi con<lb></lb>currant minima ſparti à jugo diſtantia, & ob longitudinem ma<lb></lb>jorem brachiorum libræ major centri gravitatis diſtantia à me<lb></lb>dio jugi puncto, patet multò faciliùs dignoſci inæqualia eſſe <lb></lb>pondera, quia majore angulo linea deflectit à perpendiculo; & <lb></lb>poſito minimo Radio Tangens major angulo majori opponitur. </s> </p> <p type="main"> <s id="s.002089">Hæc quidem de libra ſpartum habente ſuprà lineam jugi <lb></lb>dicta accommodari poſſunt libræ ſpartum habenti infrà jugi li<lb></lb>neam, ſi eadem ſchemata inverſo ſitu poſita intelligantur: quò <lb></lb>enim maiore angulo deflectit à perpendiculo linea jungens gra<lb></lb>vitatis centrum, & centrum motus, eò faciliùs brachium, in <lb></lb>quo eſt gravitatis centrum, inclinatur. </s> <s id="s.002090">Verùm ſi duplex hæc <lb></lb>libræ ſpecies, quæ ſuprà, & quæ infra jugi lineam ſpartum ha<lb></lb>bet, invicem comparetur, ſatis apertum eſt multò faciliùs à <lb></lb>poſteriore hˊc ſpecie indicari ponderum inæqualitatem; quia <lb></lb>videlicet ſi centrum gravitatis in alterutram partem vel mini<lb></lb>mùm recedat à medio jugi, non ampliùs imminet ſparto in eo<lb></lb>dem perpendiculo, neque poteſt ſuſtineri, ſed illicò, quantùm <lb></lb>poteſt ad imum locum deſcendit. </s> <s id="s.002091">At in priore illa ſpecie libræ <lb></lb>ſpartum in ſuperiore loco habentis, recedente in alterutram <lb></lb>partem centro gravitatis, deſcendit illud quidem; ſed non niſi <lb></lb>pro ratione exceſsûs ponderis; qui deſcenſus inobſervabilis erit, <lb></lb>ſi exigua ſit ponderum differentia. </s> <s id="s.002092">Hinc non ſemel animadver<lb></lb>ti accuratiſſimas bilances, quibus aurearum monetarum ponde<lb></lb>ra examinantur, eas eſſe, quæ ſpartum in inferiore loco habent; <lb></lb>lanx enim, quæ pondere prægravatur, ad imum, quantùm po<lb></lb>teſt deſcendit: factâ autem libræ converſione ita, ut anſa infe<lb></lb>riùs ſuſtentata libram ſuſtineat, iiſdemque ponderibus impoſi<lb></lb>tis, lanx prægravata non deſcendit ad imum locum; ſed manet <lb></lb>libra in obliquâ poſitione, quæ ponderum inæqualitati congruè <lb></lb>reſpondet; &, ſi ea ſit ponderum inæqualitas, quæ omnem ob<lb></lb>ſervantis ſubtilitatem effugiat, videtur libra in æquilibrio hori<lb></lb>zontali poſita, cum tamen in priore ſitu, antequam libra inver<lb></lb>teretur, non poſſet in ullo æquilibrio conſiſtere. </s> </p> <p type="main"> <s id="s.002093">Non ita tamen hæc dicta intelligi velim, ut nulla ſit habenda <lb></lb>ratio materiæ, ex qua libra conſtat; hæc ſiquidem tantæ gravi<lb></lb>tatis eſſe poteſt, ut axem vehementiùs premens motum aliqua<lb></lb>tenus impediat, ac propterea levis illa virtus effectiva motus, <pb pagenum="283" xlink:href="017/01/299.jpg"></pb>qui ponderum adnexorum inæqualitatem cæteroqui conſeque<lb></lb>retur, ex hâc preſſione, & prominularum particularum ſe vi<lb></lb>ciſſim contingentium conflictu elidatur, atque jugi æquili<lb></lb>brium horizontale permaneat. </s> <s id="s.002094">Gravitatem autem motui im<lb></lb>pedimento eſſe ex eo conſtat, <emph type="italics"></emph>quòd faciliùs quando ſine pondere <lb></lb>eſt, movetur libra, quàm cum pondus habet,<emph.end type="italics"></emph.end> ut obſervavit <lb></lb>Ariſtoteles 9. 10. Mechan. Cui tamen in aſſignandâ hujus <lb></lb>difficultatis causa non aquieſco, licet ultrò concedam <emph type="italics"></emph>in con<lb></lb>trarium ei, ad quod vergit onus, movere difficile eſſe<emph.end type="italics"></emph.end>; ſi enim libræ <lb></lb>vacuæ lances minùs graves ſunt, impoſito autem pondere fiunt <lb></lb>graviores, & proptereà lanx elevanda facta gravior difficiliùs <lb></lb>movetur contra inſitam gravitati propenſionem, etiam viciſſim <lb></lb>lanx deprimenda facta gravior ex adnexo pondere faciliùs ob<lb></lb>ſecundat naturali gravium propenſioni, atque adeò augere de<lb></lb>beret movendi facilitatem, vel ſaltem hanc imminui non per<lb></lb>mitteret. </s> <s id="s.002095">Non aliunde igitur ortum ducere videtur hujuſmo<lb></lb>di difficultas movendi libram onuſtam, quàm ex majore pre<lb></lb>mentis gravitatis conatu: preſſione autem motum impediri quis <lb></lb>neget, ſi ſuper planam ſuperficiem continuo lævore lubricam <lb></lb>ducat regulam metallicam exquiſite politam, quam nunc te<lb></lb>nui, nunc validiori conatu premat? </s> <s id="s.002096">utique percipiet pro vario <lb></lb>prementis conatu aliam atque aliam eſſe trahendæ regulæ me<lb></lb>tallicæ difficultatem. </s> </p> <p type="main"> <s id="s.002097">Adde graviori libræ craſſiorem axem, ut ei proportione <lb></lb>reſpondeat, neceſſariò adjungi; hic autem ſi non ſit exquiſitè <lb></lb>cylindricus, quâ parte fit contactus, ſed aliquatenùs angulatus <lb></lb>duobus in locis contingat, ſatis manifeſtè apparet magis impe<lb></lb>diri motum libræ, quàm ſi axis tenuior eſſet, atque ſubtilior; <lb></lb>licet enim hic pariter ſimilique ratione angulatus eſſet, quia <lb></lb>tamen anguli minùs diſtarent invicem, quàm in axe craſſiore, <lb></lb>minùs etiam libræ converſionem impedirent. </s> <s id="s.002098">Idem accidit, ſi <lb></lb>axis quidem cylindricus, foramen autem, cui axis inſeritur, <lb></lb>non exquiſitè rotundum ſed angulatum fuerit. </s> <s id="s.002099">Cur autem libræ <lb></lb>converſio impediatur, ſi fiat contactus in duobus punctis, pa<lb></lb>làm eſt; quia nimirum quamdiu centrum gravitatis compoſitæ <lb></lb>interjicitur inter duos illos contactus (vel ſaltem linea directio<lb></lb>nis per illud centrum ducta tranſit per intervallum illud duo<lb></lb>rum contactuum) non poteſt fieri libræ in alterutram partem <pb pagenum="284" xlink:href="017/01/300.jpg"></pb>converſio; quæ proinde ut convertatur, tantum ponderis alte<lb></lb>ri lanci addi neceſſe eſt, ut centrum gravitatis omninò cadat <lb></lb>extrà illud ſpatium, quod à contactibus comprehenditur. </s> </p> <p type="main"> <s id="s.002100">Hinc patet, cur libræ craſſiores, & majores ingentibus ſar<lb></lb>cinis onuſtæ inertes fiant ad motum, etiam ſi adnexis ponderi<lb></lb>bus inſit aliquot unciarum, aliquando fortaſſe etiam librarum, <lb></lb>diſparitas. </s> <s id="s.002101">Contrà verò aurificibus, & gemmariis, quibus mi<lb></lb>nutias contemnere damno eſſet, valdè exiguæ libræ in uſu <lb></lb>ſunt; quippè quæ ſubtiliſſimo axe contentæ ſunt, & levi jugo <lb></lb>conſtant, cujus gravitati æqualis eſt ſingularum lancium gra<lb></lb>vitas: quare cum nec vehemens preſſio contingat, nec axis <lb></lb>adeò tenuis facilè angulos admittat, exilioribus hujuſmodi li<lb></lb>bris etiam minima ponderum inæqualitas exploratur, ſi cæte<lb></lb>róqui fuerint ritè conſtructæ. </s> </p> <p type="main"> <s id="s.002102">At quærat hîc quiſpiam. </s> <s id="s.002103">Proponitur libra, quæ vacua æqui<lb></lb>librium oſtendit, nec ita gravis eſt, ut de validiore axis preſ<lb></lb>ſione dubitetur: ut inquiratur, quàm facilè mobilis illa ſit, alte<lb></lb>ri lanci ſingula ſubinde grana delicatè imponuntur, quot ſatis <lb></lb>ſint ad primò tollendum æquilibrium, tùm aliâ librâ tenuiori <lb></lb>examinatum granorum omnium pondus (rejecto ultimo grano, <lb></lb>cujus additione primò facta eſt libræ inclinatio) deprehendi<lb></lb>tur unciæ unius, exempli gratiâ. </s> <s id="s.002104">Quæritur, an, ſi eidem lanci <lb></lb>imponantur merces, & oppoſitæ lanci legitima pondera, ſit <lb></lb>ſemper numeranda uncia una amplius, ut verum mercis pon<lb></lb>dus habeatur; quandoquidem deprehenſum eſt non mutari <lb></lb>æquilibrium, niſi uncia addatur. </s> </p> <p type="main"> <s id="s.002105">Ut quæſtioni ſatisfaciam, tanquam certum ſtatuamus hanc <lb></lb>libræ inertiam non oriri ex multâ jugi & lancium gravitate <lb></lb>axem premente; ſi enim ex hujuſmodi preſſione oriretur, ad<lb></lb>ditis hinc & hinc ponderibus multò major fieret preſſio, ex <lb></lb>quâ movendi difficultas major crearetur; & ſi minorem preſſio<lb></lb>nem vix unius unciæ exceſſus vincit, utique majorem preſſio<lb></lb>nem non niſi plurium unciarum exceſſus vincere poterit. </s> <s id="s.002106">De<lb></lb>finire autem hujuſmodi preſſionum vires motum libræ retar<lb></lb>dantes, meæ tenuitatis non eſt; quippè qui nec divinare au<lb></lb>deo, nec certam rationem preſſiones illas dimetiendi invenio. </s> <lb></lb> <s id="s.002107">Illud igitur reliquum eſt, ſeclusâ preſſione, quòd axis con<lb></lb>tactus non omninò in unico puncto, ſed in pluribus fiat, ac <pb pagenum="285" xlink:href="017/01/301.jpg"></pb>propterea alterutri vacuæ libræ lanci imponendam unciam, ut <lb></lb>primò diſpoſita ſit libra ad recedendum ab æquilibrio. </s> <s id="s.002108">Hoc au<lb></lb>tem indicat, libræ prorſus vacuæ centrum gravitatis eſſe inter <lb></lb>extrema puncta contactûs axis; ſed additâ unciâ compoſitæ gra<lb></lb>vitatis centrum convenire cum extremo puncto contactûs <lb></lb>axis. </s> </p> <p type="main"> <s id="s.002109">Quærendum eſt igitur, quo intervallo extremum hoc <lb></lb>punctum, quod etiam eſt gravitatis centrum, diſtet à medio <lb></lb>jugi puncto. </s> <s id="s.002110">Id quod ut innoteſcat, obſervetur jugi & lan<lb></lb>cium gravitas; tùm in extremitatibus jugi intelligatur ſemiſſis <lb></lb>ſingulorum brachiorum, & addatur ſingularum lancium gra<lb></lb>vitas: ſint autem hinc & hinc ex. </s> <s id="s.002111">gr. unciæ duodecim tota gra<lb></lb>vitas: alteri addatur uncia, & erunt hinc quidem unciæ 12; <lb></lb>hinc verò unciæ 13. Quare jugum reciprocè diſtinguatur in <lb></lb>duas partes, quarum altera ſit 13, altera 12: igitur punctum <lb></lb>hoc diviſionis jugi diſtat à medio jugi puncto parte unâ quin<lb></lb>quageſimá totius longitudinis ejuſdem jugi: hæc ſiquidem lon<lb></lb>gitudo diſtincta intelligitur in partes 25 æquales; punctum <lb></lb>medium ab extremitate diſtat partibus 12 1/2, centrum gravita<lb></lb>tis compoſitæ diſtat partibus 12; igitur punctorum iſtorum in<lb></lb>tervallum eſt (1/50). </s> </p> <p type="main"> <s id="s.002112">Jam imponatur alteri lanci merx, quæ cum pondere le<lb></lb>gitimo lib. 2. faciat æquilibrium: aio non poſſe pronuncia<lb></lb>ri mercem eſſe unc. </s> <s id="s.002113">25: nam ſi ponatur merx unc. </s> <s id="s.002114">25: ad<lb></lb>ditâ gravitate lancis & brachij unc. </s> <s id="s.002115">12 ex hypotheſi, hinc <lb></lb>quidem eſſent unciæ 37, hinc verò unciæ 36; igitur divi<lb></lb>ſo jugo in partes 73, centrum gravitatis diſtaret à medio jugi <lb></lb>puncto parte (1/146). At punctum extremum contactûs axis & jugi <lb></lb>diſtat parte (1/50), igitur multo majus pondus ſupra unciam adden<lb></lb>dum eſt merci, ut æquilibrium exquiſitè faciat cum pondere <lb></lb>legitimo lib. 2. Nimirum inſtituenda eſt analogia ut 12 ad 13, <lb></lb>ita unciæ 36 ad uncias 39; dempto igitur pondere lancis & bra<lb></lb>chij libræ, quantitas mercis eſt unc. </s> <s id="s.002116">27. Ex quo liquet, quò <lb></lb>majora pondera lancibus imponuntur, eò majorem eſſe diffe<lb></lb>rentiam à pondere legitimo. </s> <s id="s.002117">Hinc ulteriùs patet hujuſmodi <lb></lb>librâ ſatius eſſe multam mercem ſimul ponderare, quàm per <lb></lb>partes: pone enim eſſe uncias 12 legitimi ponderis, cum quo <pb pagenum="286" xlink:href="017/01/302.jpg"></pb>æquilibrium conſtituitur, merx erit unicarum 14, quia ut 12 <lb></lb>ad 13, ita unc. </s> <s id="s.002118">24 ad 26, & demptis unciis 12 ad brachium & <lb></lb>lancem ſpectantibus, remanent mercis unciæ 14: quare bis <lb></lb>facta ponderatione erit differentia unc. </s> <s id="s.002119">4; unica autem ponde<lb></lb>ratio dabat tantum uncias 3: quia videlicet ſingulis vicibus ad<lb></lb>ditui id, quod reſpondet gravitati lancis oppoſitæ; atque adeò <lb></lb>differentia ſæpiùs repetita major eſt, quàm ſimplex: ſic qua<lb></lb>tuor libris ponderis legitimi reſponderent in altera lance mer<lb></lb>cis lib.4.unc. </s> <s id="s.002120">5; quòd ſi quatuor vicibus operando ſingulas libras <lb></lb>expendiſſes, differentia demùm eſſet unciarum 8. </s> </p> <p type="main"> <s id="s.002121">Unum ad huc ſupereſſe videtur hîc obſervandum, quoniam <lb></lb>longioribus brachiis exquiſitiùs indicari æquilibrium diximus: <lb></lb>cavendum ſcilicet, ne in aliud incommodum incidamus, quo <lb></lb>illud idem pereat, quod perſequimur. </s> <s id="s.002122">Si enim longiora fiant <lb></lb>brachia, additur gravitas, quæ magis axem premens motui ali<lb></lb>quam difficultatem creat: quod ſi retentâ eâdem brachiorum <lb></lb>gravitate illorum craſſities extenuetur, & in longitudinem ex<lb></lb>tendantur, vide ne nimis exilia evadant ita, ut flexioni obnoxia <lb></lb>ſint, vel ſuâ ipſorum, vel expendendorum ponderum gravita<lb></lb>te. </s> <s id="s.002123">Præterquam quod longiora brachia plus habere videntur <lb></lb>momenti ad premendum axem, etiam ſi par ſit longiorum at<lb></lb>que breviorum libræ brachiorum gravitas abſoluta; cujus ſe<lb></lb>miſſis in extremitate brachij longioris plùs habet momenti ad <lb></lb>deſcendendum, quàm in extremitate brevioris. </s> <s id="s.002124">Et ſi longior <lb></lb>haſta ex medio ſuſpenſa faciliùs ſponte ſuâ flectitur circa me<lb></lb>dium (id quod breviori non accidit) indicio eſt obicem reti<lb></lb>nentem magis premi; idem igitur & axi libræ contingere po<lb></lb>teſt, cujus preſſio major eſſe videtur ex longioribus brachiis, <lb></lb>etiamſi in cæteris nullum intercedat diſcrimen. </s> <s id="s.002125">Sic Ariſtote<lb></lb>les quærit quæſt. </s> <s id="s.002126">27. Mechan. <emph type="italics"></emph>Cur ſi valde procerum fuerit idem <lb></lb>pondus, difficiliùs ſuper humeros geſſatur, etiam ſi medium quiſpiam <lb></lb>illud ferat, quam ſi brevius ſit?<emph.end type="italics"></emph.end> cujus difficultatis cauſam ille tri<lb></lb>buit validiori vibrationi extremitatum magis diſtantium ab hu<lb></lb>mero ſuſtinente: ſed hoc non niſi in motu contingit, & cùm <lb></lb>flexile eſt pondus, cujuſmodi eſſet longior haſta aut bractea <lb></lb>ferrea mediocris craſſitiei. </s> <s id="s.002127">Certè longiori columnæ marmoreæ <lb></lb>jacenti, cujus medio recens fulcrum ſubjectum fuit, jam pu<lb></lb>treſcentibus extremis fulcris, ſua longitudo obfuit, ut frange-<pb pagenum="287" xlink:href="017/01/303.jpg"></pb>retur: id quod æqualis ponderis columnæ breviori ex graviore <lb></lb>ſecundum ſpeciem marmore non ita facilè accidiſſet: non niſi <lb></lb>quia gravitas magis à fulcro diſtans plùs habet momenti, etiam<lb></lb>ſi non contingat vibratio corporis, quemadmodum in motu. </s> </p> <p type="main"> <s id="s.002128">Illud poſtremò non omittendum, quod ad lingulam perti<lb></lb>net, hanc enim longiuſculam eſſe præſtat, quàm brevem, ut <lb></lb>vel levi inclinatione libræ, apex lingulæ magis conſpicuo mo<lb></lb>tu extra anſam ad latus ſecedat, & ſublatum æquilibrium indi<lb></lb>cet. </s> <s id="s.002129">Dum tamen lingulæ longitudinem affectas, cavendum, <lb></lb>ne illa momentum addat ſuâ gravitate brachio, quod inclina<lb></lb>tur; quamvis enim hoc nihil referat, ubi ſublatum horizontale <lb></lb>æquilibrium indicatur; in librâ tamen, quæ in æquilibrio obli<lb></lb>quo poteſt conſiſtere, videretur indicare majorem ponderum <lb></lb>inæqualitatem, quàm revera ſit. </s> <s id="s.002130">Cæterùm communiter libræ <lb></lb>hoc periculo vacant; ſola enim ponderum æqualitas horizonta<lb></lb>li æquilibrio inquiritur, non ponderum Ratio obliquo æquili<lb></lb>brio inveſtiganda proponitur: quare communiter nil de lingu<lb></lb>læ gravitate timendum eſt, quod nos ſolicitos habeat. </s> </p> <p type="main"> <s id="s.002131">Quare præter exquiſitam brachiorum æqualitatem, & accu<lb></lb>ratam lingulæ cum ipſo jugo poſitionem ad angulos rectos, ad <lb></lb>libram exactiſſimam conſtituendam concurrunt brachiorum & <lb></lb>lingulæ longitudo, jugi & lancium modica gravitas, axis ſub<lb></lb>tilitas, ſparti & jugi quàm maxima propinquitas, & ipſius <lb></lb>ſparti infrà jugi lineam poſitio. </s> <s id="s.002132">Quæ tamen omnia cum rectâ <lb></lb>ratione ſunt adminiſtranda, ut ponderibus examinandis pro<lb></lb>portione reſpondeant libræ partes; majoribus enim ſarcinis va<lb></lb>lidior axis, & craſſiora libræ brachia conveniunt; & ſic de <lb></lb>reliquis. <lb></lb></s> </p> <p type="main"> <s id="s.002133"><emph type="center"></emph>CAPUT VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002134"><emph type="center"></emph><emph type="italics"></emph>Libræ doloſæ vitia reteguntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002135">LIbram doloſam voco, quæ ſolitariè accepta ſinè ponderi<lb></lb>bus juſta apparet, & æquilibrium oſtentat, re tamen verâ <lb></lb>injuſta eſt, quia adnexis ponderibus ſuo æquilibrio non tribuit <pb pagenum="288" xlink:href="017/01/304.jpg"></pb>æqualitatem, vel quia ponderum æqualitatem non indicat ve<lb></lb>to æquilibrio. </s> <s id="s.002136">Quare nullus mihi ſermo de iniquorum vendi<lb></lb>torum ſycophantiis, quibus, juſtam licèt libram adhibentes, <lb></lb>rudem ac ſimplicem emptorem circumveniunt, aut imprimen<lb></lb>do impetum ſurſum brachio, cui legitimum pondus adnectitur, <lb></lb>ut merx præponderare videatur, aut ponderibus iniquis & juſto <lb></lb>minoribus utendo, aut ſubjectam menſam, cui lanx mercis in<lb></lb>cumbit, materiâ aliquatenus tenaci illinendo, ut ſublatâ in <lb></lb>aërem librâ priùs attollatur lanx ponderis quàm mercis, quæ <lb></lb>omninò præponderans apparet, ſi libra ſpartum habeat infra <lb></lb>jugum, aut ſimiles impoſturas excogitando: ſed de illis tantum <lb></lb>deceptionibus agendum, quæ ex ipſius libræ conſtructione, <lb></lb>aut poſitione ortum habere poſſunt. </s> </p> <p type="main"> <s id="s.002137">Et primò quidem ſe offert dolus, cujus meminit Ariſtoteles <lb></lb>quæſt.1.Mechan. familiaris eo tempore vendentibus purpuram, <lb></lb>& ea, quorum modica quantitas pretium exigebat non contem<lb></lb>nendum: hi enim librâ utebantur, quæ brachiis non omninò <lb></lb>paribus conſtabat, ita tamen, ut hæc inæqualitas non ſe oculis <lb></lb>ſtatim proderet. </s> <s id="s.002138">Ut autem lateret dolus, ſcapum ſeu jugum <lb></lb>libræ ex ligno conſtruebant, cujus partes omnes non eandem <lb></lb>ſpecificam gravitatem obtinerent, quamvis nulla ſecundùm <lb></lb>molem diverſitas intuenti occurreret: quia enim nodi, & partes <lb></lb>radici propiores, ut potè magis denſæ, graviores ſunt, quàm <lb></lb>reliquæ partes à radice remotiores & nodis carentes, partem il<lb></lb>lam graviorem breviori brachio tribuebant, vel ſi materia pla<lb></lb>nè uniuſmodi eſſet, & æquabili gravitate prædita, breviori <lb></lb>brachio aliquid plumbi infundebant, ut materiæ gravitate mo<lb></lb>mentum, quod ratione poſitionis deerat, ſupplente, appareret <lb></lb>æquilibrium lancium in vacuâ librâ. </s> <s id="s.002139">Sed ubi demum merx <lb></lb>lanci longioris brachij imponebatur, hæc erat juſto minor, <lb></lb>quamvis cum oppoſito pondere eſſet æquilibris; non enim erat <lb></lb>illi æqualis, ſed in Ratione reciprocâ longitudinis brachij mi<lb></lb>noris ad longitudinem majoris. </s> <s id="s.002140">Hûc ſpectat inæqualitas bra<lb></lb>chiorum orta ex eo, quòd jugi ferrei pars altera ex validiore, & <lb></lb>diuturniore percuſſione mallei facta denſior, etiam gravior eſt; <lb></lb>nam puncto longitudinem jugi bifariam dividenti non reſpon<lb></lb>det centrum gravitatis; ſed recedit à medio versùs extremita<lb></lb>tem denſiorem, atque graviorem; ac proptereà, ut æquili-<pb pagenum="289" xlink:href="017/01/305.jpg"></pb>brium appareat, centrum motûs inæqualiter dividit longitudi<lb></lb>nem jugi. </s> <s id="s.002141">Similiter ſi jugi quidem materia æquabiliter ſit gra<lb></lb>vis, ſed brachiorum inæqualitatem ſuppleat lancium gravitas <lb></lb>reciprocè inæqualis; æquilibris erit libra vacua; ſed damno <lb></lb>emptoris merx longiori brachij adnectitur. </s> <s id="s.002142">Quare ut pateat <lb></lb>dolus, facto æquilibrio inter mercem ac pondus, ſtatim com<lb></lb>muta lances, & pondus majus ex longiore brachio multò plus <lb></lb>habebit momenti, quàm merx ex brachio breviore: idcircò, <lb></lb>ſi ex pondere dematur, quantùm ſatis ſit ad æquilibrium cum <lb></lb>merce iterum ſtatuendum, plus mercis habebit emptor, quàm <lb></lb>pro oppoſiti ponderis menſurâ. </s> </p> <p type="main"> <s id="s.002143">Secundò ſit jugi materia planè æquabilis, & ab axe jugum <lb></lb>dividatur omnino bifariam: ſed puncta contactuum annulo<lb></lb>rum, ex quibus pendent lances, non æqualiter diſtent à me<lb></lb>dio: etiamſi lancis propioris gravitas ſuppleat momentum, quod <lb></lb>deeſt ratione ſitûs, & æquilibris appareat libra vacua, non ta<lb></lb>men æqualia pondera lancibus impoſita conſtituent æquili<lb></lb>brium, ſed illud gravius apparebit, quod ex diſtantia majore <lb></lb>appendetur: & ſi pondera æquilibrium faciant, inæqualia <lb></lb>erunt reciprocè juxtà Rationem inæqualitatis diſtantiarum à <lb></lb>medio. </s> <s id="s.002144">Similiter igitur facto ponderum æquilibrio, lances <lb></lb>commuta, & quidem ſi poſt commutationem iterum æquili<lb></lb>brium fiat, juſta eſt libra, ſecùs verò ſi alterum gravius appa<lb></lb>reat, quod priùs æquale videbatur. </s> </p> <p type="main"> <s id="s.002145">At quæris, quá methodo poſſis deprehendere, quanta ſit bra<lb></lb>chiorum inæqualitas, quando quidem non habetur æquili<lb></lb>brium poſt factam lancium commutationem, & planè ignora<lb></lb>tur, quanta ſit mercis gravitas. </s> <s id="s.002146">Ut quæſtioni ſatisfaciam, acci<lb></lb>pio legitima pondera, & primùm facto æquilibrio obſervo legi<lb></lb>timi ponderis quantitatem: Commuto deinde lances, & cum <lb></lb>non fiat æquilibrium cum eâdem merce, tantum accipio legi<lb></lb>timi ponderis, quantum requiritur ad æquilibrium. </s> <s id="s.002147">Demum <lb></lb>inter hæc duo pondera legitima invenio terminum medio loco <lb></lb>proportionalem, & hoc eſt mercis pondus, quod collatum cum <lb></lb>alterutro ex legitimis ponderibus dat reciprocè longitudinis <lb></lb>brachiorum Rationem. </s> <s id="s.002148">Hanc methodum eſſe certam patet, <lb></lb>quia cum bis fiat æquilibrium, bis inter pondera eſt eadem Ra<lb></lb>tio reciproca brachiorum. </s> <s id="s.002149">Sint brachia, quæ brevitatis gratia <pb pagenum="290" xlink:href="017/01/306.jpg"></pb>vocemus R & S; igitur ut R ad S ita primum pondus legiti<lb></lb>mum in S ad mercem in R: & factâ commutatione ponitur <lb></lb>merx in S, & iterum fit ut R ad S, ita reciprocè merx eadem <lb></lb>in S ad ſecundum pondus legitimum in R: igitur, per 11.lib.5. <lb></lb>ut primum pondus ad mercem, ita merx ad ſecundum pondus: <lb></lb>ſunt autem nota duo pondera legitima; igitur & innoteſcit mer<lb></lb>cis gravitas: quæ ſi comparetur ut conſequens terminus cum <lb></lb>primo pondere, aut ut Antecedens cum ſecundo pondere, ha<lb></lb>bebitur Ratio R ad S. </s> <s id="s.002150">Sit itaque ex. </s> <s id="s.002151">gr. in primo æquilibrio <lb></lb>primum pondus legitimum unc. </s> <s id="s.002152">72, in ſecundo æquilibrio ſe<lb></lb>cundum pondus legitimum ſit unc. (69 18/100). Eſt ergo merx me<lb></lb>dio loco proportionalis unc. (70 576/1000); ac propterea R ad S eſt <lb></lb>ut 72 ad (70 1576/1000), aut ut (70 576/1000) ad (69 18/100), hoc eſt ut 4500 ad <lb></lb>4411. Sit demum totius jugi longitudo diſtincta in partes 200: <lb></lb>addantur termini Rationis inventæ, & fiat ut 8911 ad 4411 <lb></lb>ita 200 ad 99, & hæc eſt longitudo brachij brevioris, erit au<lb></lb>tem longioris brachij longitudo partium 101: diſtat ergo ſpar<lb></lb>tum à puncto medio per unam ducenteſimam partem totius ju<lb></lb>gi. </s> <s id="s.002153">Quòd ſi res ſubtiliſſimè ad calculos revocanda eſſet, hujus <lb></lb>ducenteſimæ partis gravitas, quæ eſt ſemiſſis gravitatis diffe<lb></lb>rentiæ brachiorum eſſet computanda, atque ſubducenda, vel <lb></lb>addenda, ut mercis pondus exquiſitè innoteſcat. </s> </p> <p type="main"> <s id="s.002154">Tertiò. </s> <s id="s.002155">Accidere poteſt lingulam ex medio libræ ſcapo aſ<lb></lb>ſurgere ad angulos rectos, lineamque lingulæ tranſeuntem per <lb></lb>centrum motûs ita occurrere lineæ jungenti puncta, ex quibus <lb></lb>lances pendent, ut eam bifariam æqualiter dividat, in eam ta<lb></lb>men ad angulos inæquales cadat. </s> <s id="s.002156">Aio nec brachia eſſe verè <lb></lb>æqualia, nec lingulam, quamvis anſæ congruens videatur, in<lb></lb>dicare æquilibrium horizontale, eſſe veram lingulam, etiamſi <lb></lb>pondera in eo æquilibrio conſiſtentia ſint æqualia, & non in <lb></lb>Ratione brachiorum. </s> </p> <p type="main"> <s id="s.002157">Sit ſcapus libræ AB, ex quo perpendicularis aſſurgat lingula <lb></lb><figure id="id.017.01.306.1.jpg" xlink:href="017/01/306/1.jpg"></figure><lb></lb>CD, & ex D per O centrum mo<lb></lb>tûs ducta recta linea occurrat li<lb></lb>neæ SV tangenti extrema puncta, <lb></lb>ex quibus lances pendent, eam<lb></lb>que bifariam dividat in I: ſed quo<lb></lb>niam punctum S eſt paulò altiùs <pb pagenum="291" xlink:href="017/01/307.jpg"></pb>quàm punctum V, fiat angulus SIO minor, & VIO major. </s> <lb></lb> <s id="s.002158">Dico lineam SV eſſe quidem jugum, ſed brachia non eſſe æqua<lb></lb>lia, non enim ſunt IS & IV: quandoquidem ductis rectis OS <lb></lb>& OV, eſt libra curva SOV latera habens inæqualia, SO <lb></lb>minus, & VO majus. </s> <s id="s.002159">Nam in triangulis SIO, VIO latus <lb></lb>IS ex hypotheſi eſt æquale lateri IV, latus IO commune eſt, <lb></lb>angulus SIO eſt ex hypotheſi minor, quàm angulus VIO; <lb></lb>ergo per 24.lib.1. baſis SO minor eſt baſi VO. </s> <s id="s.002160">Igitur ex O <lb></lb>perpendicularis linea cadens in jugum SV dividit illud in bra<lb></lb>chia inæqualia, & perpendiculum ex O cadit inter S & I, pu<lb></lb>ta in H, quia ex hypotheſi angulus SIO eſt acutus. </s> <s id="s.002161">Vera <lb></lb>igitur lingula non eſt ID, ſed linea, quæ ad angulos rectos <lb></lb>inſiſtens jugo SV ex H per O ducitur. </s> <s id="s.002162">Quare ſi CD con<lb></lb>gruit anſæ perpendicularis horizonti, jugum SV non eſt ho<lb></lb>rizonti parallelum, non eſt igitur æquilibrium horizontale, ſed <lb></lb>obliquum: quia tamen eſt I centrum commune gravitatis pon<lb></lb>derum æqualium in S & V, ac per illud tranſit perpendicu<lb></lb>lum ex O cadens in horizontem, proptereà poſſunt eſſe ponde<lb></lb>ra æqualia, & æquilibrium oſtendere, quod modicá obliquita<lb></lb>te inclinatum mentiatur æquilibrium horizontale. </s> <s id="s.002163">At ſi alia <lb></lb>fieret hypotheſis, ſcilicet lineam jugi SV non dividi æqualiter, <lb></lb>pondera non eſſent æqualia, ſed eſſent reciprocè in Ratione <lb></lb>motuum, quos perficere poſſent extremitates S & V, juxta ſu<lb></lb>periùs dicta cap. 4. hujus lib. 3. </s> </p> <p type="main"> <s id="s.002164">Vitium igitur hujus libræ non in eo conſiſtit, quòd ponde<lb></lb>ra non ſint æqualia, ſed quòd indicet æquilibrium horizontale, <lb></lb>cum ſit obliquum, & pondera æqualia nunquam poſſint ad <lb></lb>æquilibrium horizontale devenire; ut enim hoc fieret, ponde<lb></lb>ra eſſe oporteret inæqualia reciprocè in Ratione brachiorum <lb></lb>SH & HV. </s> <s id="s.002165">Quòd ſi contingat punctum O centrum motûs, <lb></lb>eſſe idem cum puncto I, pondera æqualia verè habebunt æqui<lb></lb>librium horizontale; ſed lingula CD declinabit ab ansâ, quaſi <lb></lb>æquilibrium non eſſet. </s> <s id="s.002166">Libræ hujuſmodi vitium deprehendi <lb></lb>non poteſt ponderum commutatione in lancibus; quia cùm <lb></lb>æqualia ex hypotheſi ſint pondera, eadem utrobique habent <lb></lb>momenta, ſervant quippè eamdem diſtantiam, & æqualiter <lb></lb>ſunt ad motum diſpoſita. </s> <s id="s.002167">Rarò tamen continget jugum SV <lb></lb>planè æqualiter dividi à lineâ lingulæ ad angulos obliquos in-<pb pagenum="292" xlink:href="017/01/308.jpg"></pb>cidente, quæ tamen ad ſcapum perpendicularis appareat: <lb></lb>proptereà facta ponderum in lancibus commutatione prodet ſe <lb></lb>momentorum inæqualitas. </s> </p> <p type="main"> <s id="s.002168">Quartò. </s> <s id="s.002169">Libra, quam diutiſſimè juſtam expertus es, poteſt <lb></lb>momento à ſua juſtitiâ deficere, ſi vel modicum inflectatur al<lb></lb>terutrum brachiorum, vel ſi utrumque non æqualiter flectatur; <lb></lb>hinc enim oritur brachiorum inæqualitas; quam deprehendes <lb></lb>commutatis ponderibus in utrâque lance; quæ ſcilicet æquili<lb></lb>brium conſtituebant propter reciprocam Rationem brachio<lb></lb>rum, quibus adnectebantur, non ampliùs eandem ſervant in <lb></lb>aliâ poſitione Rationem. </s> </p> <p type="main"> <s id="s.002170">Quintò. </s> <s id="s.002171">Axis, qui duobus in punctis contingat (ſcio con<lb></lb>tactum fieri in linea; ſed puncta aſſumo in ipſis lineis, per quæ <lb></lb>tranſit planum perpendiculare ad horizontem, in quo eſt linea <lb></lb>jugi) vel quia ipſe eſt angulatus, vel quia foramen, cui inſeri<lb></lb>tur, non exquiſitè rotundum, quâ ſaltem parte fit contactus, <lb></lb>libram conſtituit doloſam: quia videlicet duo illa puncta axis <lb></lb>perinde ſe habent, ac ſi duo eſſent centra motûs. </s> <s id="s.002172">Manifeſtum <lb></lb>eſt autem eandem jugi lineam non poſſe in duobus punctis <lb></lb>æqualiter dividi. </s> <s id="s.002173">Tripliciter poteſt hoc fieri. </s> <s id="s.002174">Primò unum ex <lb></lb>his punctis poteſt exactè reſpondere medio jugi; ſecundò po<lb></lb>teſt utrumque hoc punctum æqualiter à medio jugi diſtare; <lb></lb>Tertiò poſſunt ab eodem medio hinc & hinc inæqualiter <lb></lb>diſtare. </s> </p> <p type="main"> <s id="s.002175">Sit linea jugi AB, cujus medium C: puncta contactuum <lb></lb>axis, ex quibus ad jugum ducitur perpendicularis, ea ſint pri<lb></lb>mò, ut reſpondeant in jugo punctis <lb></lb><figure id="id.017.01.308.1.jpg" xlink:href="017/01/308/1.jpg"></figure><lb></lb>C & D. </s> <s id="s.002176">Si lanci in B imponatur le<lb></lb>gitimum pondus, tùm in A ponatur <lb></lb>merx uſque ad æquilibrium, à quo <lb></lb>proximè recederet, ſi aliquid am<lb></lb>plius mercis adderetur, fiet æqualitas, quia ex C puncto æqua<lb></lb>liter ab extremitatibus diſtante fit ſuſpenſio libræ. </s> <s id="s.002177">At ſi poſitâ <lb></lb>primùm merce in A, deinde legitima pondera addantur in B, <lb></lb>utique plura pondera, quàm par ſit, addentur: quia videlicet <lb></lb>non inclinabitur libra infrà B, niſi ponderum ad mercem Ra<lb></lb>tio excedat Rationem reciprocam brachiorum AD ad DB; eſt <lb></lb>enim D quaſi centrum motûs. </s> </p> <pb pagenum="293" xlink:href="017/01/309.jpg"></pb> <p type="main"> <s id="s.002178">Deinde puncta illa contactuum axis poſſunt reſpondere jugi <lb></lb>punctis E & D æqualiter à medio C diſtantibus: & tunc, ut <lb></lb>tollatur æquilibrium, neceſſe eſt tantum ponderis uni lanci ad<lb></lb>dere, ut pondera ſint in majori Ratione, quàm ſit Ratio reci<lb></lb>proca brachiorum; erit ſi quidem extremitas A proxime diſpo<lb></lb>ſita, ut facto additamento gravitatis inclinetur, ſi fuerit ut BE <lb></lb>ad EA, ita pondus in A ad pondus in B; & viciſſim extremitas <lb></lb>B erit proximè diſpoſita, ut auctà gravitate inclinetur, ſi ut AD <lb></lb>ad DB ita pondus in B ad pondus in A. </s> <s id="s.002179">Quia autem ex hypo<lb></lb>theſi DC & EC æquales ſunt, etiam reſidua EA & DB æqua<lb></lb>lia ſunt, item AD & BE: quapropter ut AD ad DB, ita BE <lb></lb>ad EA; ex quo conſequens eſt ex ſolâ lancium commutatione <lb></lb>(ſi centrum motûs modò ſit D, modò ſit E) non poſſe dignoſci <lb></lb>hoc libræ vitium, ſicut dignoſceretur in primo caſu, ſi ut AD <lb></lb>ad DB, ita pondus in B ad pondus in A; factâ enim lancium <lb></lb>commutatione, pondus ex B in A tranſlatum præponderaret <lb></lb>ex centro motûs C, cum tamen in priori poſitione circa cen<lb></lb>trum motûs D non tolleret æquilibrium. </s> </p> <p type="main"> <s id="s.002180">Similiter in tertio caſu, quando puncta contactuum axis eſ<lb></lb>ſent F & D à medio C inæqualiter diſtantia, & ut AF ad FB, <lb></lb>ita pondus in B ad pondus in A daret æquilibrium; factá pon<lb></lb>derum in lancibus commutatione non maneret æquilibrium, <lb></lb>quia pondus tranſlatum in B ad pondus tranſlatum in A poſt <lb></lb>hanc commutationem adhuc eſſet ut BF ad FA; ſed ad æqui<lb></lb>librium circa D centrum motûs deberet eſſe ut AD ad DB, <lb></lb>eſt autem BF prima major, quàm AD tertia, & FA ſecunda <lb></lb>minor eſt, quàm DB quarta; igitur eſt major Ratio BF ad FA, <lb></lb>quàm AD ad DB: igitur pondus, quod priùs erat in B, tranſla<lb></lb>tum in A impar eſt ad æquilibrium conſtituendum. </s> </p> <p type="main"> <s id="s.002181">Ad dignoſcendum, an libra hoc vitio laboret, uti poteris hac <lb></lb>methodo. </s> <s id="s.002182">Lancibus impone pondera, ut fiat æquilibrium: tùm <lb></lb>lances commuta; & ſiquidem iterum fiat æquilibrium, adde <lb></lb>alteri lanci aliquid ponderis, à quo ſi libra inclinetur, aufer ad<lb></lb>ditum pondus, & oppoſitæ lanci impone; quæ ſi perſiſtat non <lb></lb>inclinata, adde adhuc pondus, quantum ferre poteſt citrà in<lb></lb>clinationem: iterum commutatis lancibus, nullo pacto manere <lb></lb>æquilibrium videbis, & indicio erit contactum axis fieri in <lb></lb>duobus punctis, quorum alterum reſpondet medio jugi ſiqui-<pb pagenum="294" xlink:href="017/01/310.jpg"></pb>dem in primâ lancium commutatione manſit æquilibrium; & <lb></lb>eſt primus caſus. </s> <s id="s.002183">Quòd ſi facto æquilibrio, alterutri lancium <lb></lb>addas pondus, & æquilibrium maneat, adde quantum ſatis eſt, <lb></lb>ut libra ſit proximè inclinanda in eam partem, ſi adhuc pondus <lb></lb>adderetur, tùm oppoſitæ lanci ſimiliter additum pondus ſi non <lb></lb>tollat æquilibrium, indicat inter puncta contactuum axis eſſe <lb></lb>medium punctum C, quod bifariam dividit jugum: & videbis <lb></lb>poſſe ſine ſine alternis additamentis augeri pondera ſingularum <lb></lb>lancium, quia commune centrum gravitatis modò migrat ad <lb></lb>unum punctum contactûs, modò ad aliud extremum. </s> <s id="s.002184">Sed ad <lb></lb>internoſcendum, utrùm puncta hæc æqualiter, an inæqualiter <lb></lb>à puncto C medio diſtent, obſerva additamenta illa, æqualia ne <lb></lb>ſint? </s> <s id="s.002185">an inæqualia? </s> <s id="s.002186">Nam ut centrum gravitatis migret ex D in <lb></lb>E, & iterum ex E in D, æqualia addenda ſunt primùm in B, <lb></lb>deinde in A, pondera. </s> <s id="s.002187">At ut migret gravitatis centrum ex D <lb></lb>in F, plus addendum eſt ponderis in A, quàm addatur in B, ut <lb></lb>migret ex F in D; quia ſcilicet B magis diſtat à D centro mo<lb></lb>tús, quàm A diſtet ab F centro motûs: igitur plus ponderis ad<lb></lb>dendum eſt in A, ut habeat momentum æquale momento pon<lb></lb>deris additi in B. </s> <s id="s.002188">Hoc vitium minoribus libris, quarum exilis <lb></lb>eſt axis, non facilè inerit; majores libræ, quæ craſſiori axe in<lb></lb>digent, illi obnoxiæ eſſe poſſunt, niſi artificis induſtria in eo ex <lb></lb>poliendo ſolicita fuerit. </s> <s id="s.002189">Sed quid ſi axis, quâ parte contingit, <lb></lb>in angulum ſimplicem deſinat, non tamen in eum cadat per<lb></lb>pendicularis linea lingulæ, quæ jugum bifariam dividit? </s> <s id="s.002190">Jam <lb></lb>conſtat à centro motûs dividi jugum in brachia inæqualia, ac <lb></lb>proptereà æquilibrium horizontale eſſe non poſſe, inter pon<lb></lb>dera verè æqualia. </s> </p> <p type="main"> <s id="s.002191">Sextò. </s> <s id="s.002192">Si libra exactiſſimè habens brachia æqualia, & lin<lb></lb>gulam perpendicularem, & lances æquales, & funiculorum <lb></lb>pondera æqualia, habeat tamen funiculum alterum altero lon<lb></lb>giorem, incumbátque plano horizontali, impoſitis æqualibus <lb></lb>ponderibus non apparebit æquilibrium, ſi centrum motûs fue<lb></lb>rit in medio jugi puncto, vel infrà illud; ſed ad illam partem <lb></lb>inclinabitur, quæ breviorem funiculum habuerit. </s> <s id="s.002193">Hoc ideò <lb></lb>accidit, quia libram attollens extendit breviorem funiculum <lb></lb>longiori adhuc langueſcente, ac proinde pondus huic lanci im<lb></lb>poſitum non reſiſtit ſurſum trahenti, niſi cum funiculus iſte <pb pagenum="295" xlink:href="017/01/311.jpg"></pb>fuerit extentus: quare libræ jugum ex hâc parte aſcendit ſine <lb></lb>reſiſtentiâ, dum ex alterâ, quæ funiculum habet breviorem, <lb></lb>invenit reſiſtentiam; atque alterâ extremitate manente, alterâ <lb></lb>aſcendente, jugum inclinatur, extento demùm utroque funi<lb></lb>culo lanx utraque attollitur. </s> <s id="s.002194">Sed quia ex hypotheſi omnia ſunt <lb></lb>æqualia, vel remanet jugum in eâdem poſitione inclinatum, <lb></lb>ſi punctum libræ brachia diſterminans congruat centro motûs, <lb></lb>vel pars inclinata ulteriùs deſcendit, ſi ſpartum ſit inferiùs po<lb></lb>ſitum. </s> </p> <p type="main"> <s id="s.002195">Hinc pondera apparent inæqualia, quamvis verè æqualia <lb></lb>ſint; & non rarò accidit monetas aliquas aureas tanquam le<lb></lb>ves rejici, quamvis reverâ ſint juſti & legitimi ponderis; quia <lb></lb>lancis, cui imponuntur, funiculus longior eſt, & libra ad hanc <lb></lb>partem, in quâ eſt pondus, inclinatur; ideóque tribuitur mo<lb></lb>netæ levitas, quia libra vacua in aëre ſuſpenſa juſtiſſima appa<lb></lb>ret. </s> <s id="s.002196">Viciſſim igitur poteſt fieri, ut moneta levis appareat præ<lb></lb>ponderans, in librâ ſpartum inferiùs habentè, ſi moneta levis <lb></lb>fuerit impoſita lanci, cujus funiculus brevior eſt; factâ ſcilicet <lb></lb>jam jugi ad hanc partem inclinatione, cum poſtea lanx utra<lb></lb>que à plano ſeparatur, legitimum pondus, quod gravius qui<lb></lb>dem eſt, non poteſt deſcendere, niſi attollat oppoſitam lan<lb></lb>cem, cujus aſcendentis motus major eſſe deberet motu legitimi <lb></lb>ponderis deſcendentis; ac proptereà niſi ſit major Ratio pon<lb></lb>deris ad monetam, quàm motûs monetæ aſcendentis ad motum <lb></lb>ponderis deſcendentis, moneta videbitur præponderans: & <lb></lb>tantiſper latebit dolus, dum facta fuerit in lancibus ponderis, <lb></lb>& monetæ commutatio: apparebit ſiquidem levius id, quod <lb></lb>in lance pendet ex funiculo longiore. </s> <s id="s.002197">Quòd ſi libra hujuſmodi <lb></lb>funiculis inæqualibus inſtructa ſpartum haberet in loco ſupe<lb></lb>riore, initio quidem impoſita æqualia pondera apparerent in<lb></lb>æqualia, quia non viderentur æquilibria, ſed demùm ſe libra in <lb></lb>æquilibrio conſtitueret, ſi verè omnia æqualia ſint, ut fert hy<lb></lb>potheſis. </s> <s id="s.002198">At ſi, ut non paucis venditoribus vulgare eſt, ita li<lb></lb>bra ſit conſtituta, ut lanx altera, cui legitimum pondus impo<lb></lb>nitur juxtà quæſitam mercis quantitatem, ſubjecto piano in<lb></lb>ſiſtat, altera merci deſtinata in aëre pendeat, lingulâ anſæ <lb></lb>congruente, quæ æquilibrium oſtendit; ſit verò funiculus lan<lb></lb>cis plano incumbentis fortaſsè non ſatis extentus (quia ita con-<pb pagenum="296" xlink:href="017/01/312.jpg"></pb>textus, ut majore vi extendatur, quâ ceſſante ſe iterum con<lb></lb>trahat) merx videbitur præponderans, etiamſi non ſit major <lb></lb>legitimo pondere; quia deorſum ſuá gravitate connitens, dum <lb></lb>pondus ex alterâ parte reſiſtit, inclinat lingulam, & oppoſitæ <lb></lb>lancis funiculum extendit. </s> </p> <p type="main"> <s id="s.002199">Septimò. </s> <s id="s.002200">Ex ipſo plano, cui libra incumbit, antequam at<lb></lb>tollatur, oriri poteſt fallacia æqualibus ponderibus inæqualita<lb></lb>tem tribuens, etiamſi nullum libræ inſit vitium aut ratione in<lb></lb>æqualitatis brachiorum, aut ratione lingulæ perperam inclina<lb></lb>tæ ad jugum, aut ratione axis angulati, aut ratione funiculo<lb></lb>rum inæqualium. </s> <s id="s.002201">Nam ſi planum ab horizonte deflectat, & ad <lb></lb>illum inclinetur; cùm ad perpendiculum anſa attollitur, funi<lb></lb>culi pariter horizonti perpendiculares intelliguntur, & quia <lb></lb>æquales ſunt, jugum libræ eſt parallelum plano, ac proptereà <lb></lb>perpendiculum anſæ ad angulos inæquales incidit tùm in ju<lb></lb>gum libræ, tùm in planum inclinatum; lingula igitur, quæ ju<lb></lb>go inſiſtit ad angulos rectos, declinat ab ansâ, & ſublatâ in <lb></lb>aërem librâ, inclinatur lingula ad depreſſiorem plani partem, <lb></lb>manetque inclinata, quamvis pondera æqualia ſint, ſi centrum <lb></lb>motûs & punctum brachia diſterminans in codem puncto con<lb></lb>veniant; ſi verò ſpartum inferius ſit, adhuc magis inclinatur, <lb></lb>videturque lanx illa omninò præponderans: at ſi ſpartum in ſu<lb></lb>periore loco fuerit, libra primùm inclinata, demùm in aëre ſuſ<lb></lb>penſa ad æquilibrium horizontale veniet. </s> </p> <p type="main"> <s id="s.002202">Octavò. </s> <s id="s.002203">Si contingat ita pondus in lance collocari, ut ipſius <lb></lb>ponderis ſingulare centrum gravitatis non omninò in eodem <lb></lb>perpendiculo ſit cum puncto jugi, ex quo lanx illa dependet, <lb></lb>æquilibrium non indicabit æqualitatem ponderum in utráque <lb></lb>lance poſitorum: Nam ſi linea directionis per hujuſmodi cen<lb></lb>trum gravitatis tranſiens incurrat in jugi punctum, quod ſit <lb></lb>centro motûs vicinius, quàm punctum extremum brachij, op<lb></lb>poſitæ lancis pondus erit minus; ſin autem occurrat lineæ jugi <lb></lb>(quæ producta intelligitur) remotiùs à centro motûs, oppoſitæ <lb></lb>lancis pondus erit majus; quia ſcilicet hæc centri gravitatis <lb></lb>ponderis collocatio perinde ſe habet, atque ſi brachium illud <lb></lb>aut imminutum ſit, aut auctum: quapropter etiam pondera <lb></lb>æquilibria ſunt in Ratione reciprocâ brachiorum, ut ex ſæpius <lb></lb>dictis liquet. </s> <s id="s.002204">Hinc ſi pondus præter opinionem gravius aut le-<pb pagenum="297" xlink:href="017/01/313.jpg"></pb>vius appareat, ejuſque pars maxima extrà lancem extet, illud <lb></lb>aliter in lance diſpone, ut centro gravitatis ponderis facilè im<lb></lb>mineat punctum jugi, ex quo lanx illa ſuſpenditur; & tunc <lb></lb>certior fies, an verè gravitas illa ponderi inſit, an verò irrep<lb></lb>ſerit fallacia ex ineptâ ipſius ponderis poſitione priori. </s> <s id="s.002205">Hoc <lb></lb>tamen intellige, quando ex hujuſmodi poſitione ſequeretur in<lb></lb>æqualis velocitas motuum oppoſitorum ponderum. <lb></lb></s> </p> <p type="main"> <s id="s.002206"><emph type="center"></emph>CAPUT VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002207"><emph type="center"></emph><emph type="italics"></emph>Stateræ natura & forma explicatur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002208">HActenùs de librâ ſermo fuit, in quâ, cum brachia æqua<lb></lb>lia ſint, legitimum pondus eſt æquale gravitati rei, cujus <lb></lb>quantitatem ex gravitate inveſtigamus: & quidem quando exi<lb></lb>gua, vel etiam mediocria ſunt pondera, res commodè hujuſ<lb></lb>modi bilance perficitur; at ubi ingentium ſarcinarum quanti<lb></lb>tas examinanda eſt, prorsùs incommodum eſſet opportunas bi<lb></lb>lances aut habere, aut adhibere: quot enim & quanta pondera <lb></lb>parare oporteret, ut centenas aliquot fæni libras, ſeu mercato<lb></lb>rios faſces, ſeu ſaccos farinæ plenos expenderemus? </s> <s id="s.002209">& ex alio <lb></lb>in alium locum ſi transferenda eſſet libra cum legitimis ponde<lb></lb>ribus tantæ gravitatis, nonne opus eſſet plauſtro, ut tàm in<lb></lb>gens onus in deſtinatum locum tranſveheretur? </s> <s id="s.002210">Quare Statera <lb></lb>excogitata eſt tanquam libra brachiorum inæqualium, in quâ <lb></lb>pondus minus longiori brachio adnexum æqualia habet mo<lb></lb>menta cum majori pondere, quod ex breviore brachio ſuſpen<lb></lb>ditur. </s> <s id="s.002211">Sed ne varia pondera in promptu habere cogeremur, <lb></lb>quæ longioris brachij extremitati adnecterentur, pro variâ <lb></lb>oneris gravitate explorandâ, ſapientiſſimè à majoribus ſta<lb></lb>tera conſtructa eſt quæ eodem æquipondio modò in majo<lb></lb>re, modò in minore diſtantiâ à centro motûs, æquilibrium <lb></lb>conſtitueret. </s> <s id="s.002212">Ex quo fit ſtateram eandem vires ſubire plu<lb></lb>rium librarum, prout plura longioris brachij puncta percur<lb></lb>rit æquipondium; mutantur ſiquidem Rationes diſtantiarum <lb></lb>ponderum, manente eâdem mercium à ſparto diſtantiâ, ac <pb pagenum="298" xlink:href="017/01/314.jpg"></pb>proinde etiam idem æquipondium variam habet Rationem ad <lb></lb>merces inæquales. </s> </p> <p type="main"> <s id="s.002213">Sunt autem ſtateræ partes Jugum, Anſa, Uncus aut lanx, <lb></lb>Æquipondium, quod aliis Sacoma, aliis Curſorium dicitur. </s> <lb></lb> <s id="s.002214">Jugum eſt, quod in partes inæquales diviſum ab axe, qui An<lb></lb>ſæ inſeritur, definit Rationem ponderum, quæ momentis <lb></lb>æqualibus librantur. </s> <s id="s.002215">Anſa eſt, ex quâ ſuſpenditur ſtatera, ut <lb></lb>liberè utramque in partem verſetur. </s> <s id="s.002216">Uncus, aut lanx, oneri <lb></lb>ſuſtinendo deſtinatur; quæ enim facilè molem unam efficiunt, <lb></lb>poſſunt ex Unco ſuſpendi; ſed quæ ex pluribus non facilè in <lb></lb>unam molem coëuntibus conſtant, lance ſubjectá recipi oportet. <lb></lb></s> <s id="s.002217">Æquipondium eſt certæ gravitatis pondus, ex quo oppoſitæ <lb></lb>gravitatis Ratio innoteſcit. </s> </p> <p type="main"> <s id="s.002218">Sit AB jugum ab axe inæqualiter in C diviſum, ſitque CA <lb></lb>brachium minùs, cujus extremitati A catena aut funis adnecti<lb></lb><figure id="id.017.01.314.1.jpg" xlink:href="017/01/314/1.jpg"></figure><lb></lb>tur cum unco aut lance E, & CB <lb></lb>brachium majus, cujus longitu<lb></lb>dinem pro opportunitate percurrit <lb></lb>æquipondium F. </s> <s id="s.002219">Anſa reſpondens <lb></lb>lingulæ CD, ipſius axis extremi<lb></lb>tates recipit, ut facilè convolvi <lb></lb>poſſit. </s> <s id="s.002220">In minoribus & mediocri<lb></lb>bus ſtateris lingula craſſiuſcula ad<lb></lb>ditur, quæ anſæ intercapedinem ita impleat, eíque congruat, <lb></lb>ut tamen nullo partium conflictu impediatur motus; in majori<lb></lb>bus & longioribus ſtateris aliquando lingula omittitur, vel quia <lb></lb>ſpartum eſt infrà rectam lineam jugi, quod non niſi horizonta<lb></lb>liter conſiſtit, vel quia ſi ſpartum eſt in ſuperiore loco, non <lb></lb>multùm à vero pondere aberrare permittit ipſa brachij longitu<lb></lb>do, quæ facilè prodit paralleliſmum aut inclinationem ad ho<lb></lb>rizontem; mediocris autem error in mercibus, quæ hujuſmodi <lb></lb>magnis ſtateris expenduntur, neque emptori, neque venditori <lb></lb>incommodo eſt; quapropter in iis ſubtilitatem ſcrupulosè per<lb></lb>ſequi inutile eſt, & ineptum. </s> <s id="s.002221">Quæ in librâ circà Axem, lin<lb></lb>gulam, Anſam obſervanda monuimus, ſtateræ pariter commu<lb></lb>nia ſunt, neque hîc iterum inculcanda. </s> </p> <p type="main"> <s id="s.002222">Potiſſimum, quod in ſtaterâ obſervandum eſt, pertinet ad <lb></lb>diviſionem longioris brachij in minutiores particulas, ut exqui-<pb pagenum="299" xlink:href="017/01/315.jpg"></pb>ſitiùs innoteſcat Ratio mercis ad æquipondium, quæ denota<lb></lb>tur ab inciſis in brachio notis indicantibus Rationem brachij <lb></lb>longioris ad brevius; eſt ſcilicet minoris brachij longitudo <lb></lb>transferenda in alterum brachium, quoties fieri poteſt; & quia <lb></lb>hoc longius produci poteſt infinitè, proptereà ſtatera vocari <lb></lb>poteſt libra quaſi infinita brachiorum inæqualium. </s> <s id="s.002223">Sic diſtan<lb></lb>tia AC tranſlata in brachium CB ex. </s> <s id="s.002224">gr. quater, facit ut pon<lb></lb>dus in E poſſit eſſe quadruplum æquipondij F, ſi æquipondium <lb></lb>ſit in extremitate B: quia, ut dictum eſt de librâ brachiorum <lb></lb>inæqualium, ut AC ad CB, ita pondus in B ad pondus in A: <lb></lb>& ſ æquilibrium contingat ſacomate exiſtente in G, erit ut <lb></lb>AC ad CG ita Sacoma in G ad pondus in E. </s> </p> <p type="main"> <s id="s.002225">Hîc animadvertendum eſt diſtantiam AC, ſi ſit valdè nota<lb></lb>bilis, capacem eſſe multiplicis diviſionis, ac proptereà æqua<lb></lb>lem partem HG poſſe ſubtiliùs dividi, ut non ſolùm uncias, <lb></lb>ſed & unciæ quadrantes, aut etiam drachmas oſtendat, ſi tran<lb></lb>ſitus ex H in G ſit nota unius libræ. </s> <s id="s.002226">Verum eſt in brachio CB <lb></lb>hujuſmodi majores partes minori brachio æquales non multas <lb></lb>eſſe poſſe: ſed huic malo occurritur in adversâ parte jugi; con<lb></lb>verſa enim ſtatera aliam habet anſam, puta SV, quæ minùs <lb></lb>diſtat ab extremitate A; hæc autem diſtantia ſæpiùs iterata plu<lb></lb>res exhibet partes, & factâ ſuſpenſione VS, æquipondium in <lb></lb>extremitate B poſitum æquilibratur cum majori pondere, quàm <lb></lb>cùm ex DC ſtatera ſuſpenditur; eſt ſcilicet major Ratio BS ad <lb></lb>SA, quàm BC ad CA; nam ad eandem CA, majorem Ratio<lb></lb>nem habet BS major, quàm BC minor, & eadem BS majo<lb></lb>rem Rationem habet ad SA minorem, quàm ad CA majorem <lb></lb>ex 8 lib. 5. manifeſtum eſt igitur majorem eſſe Rationem BS <lb></lb>ad SA, quàm BC ad CA. </s> <s id="s.002227">Si igitur pondera ſunt reciprocè ut <lb></lb>brachiorum longitudines, idem æquipondium in extremitate B <lb></lb>poſitum minorem habet Rationem ad pondus in A, quando <lb></lb>brachia ſunt BS & SA, quàm cùm brachia ſunt BC & CA: <lb></lb>ac propterea tunc pondus in A eſt majus. </s> </p> <p type="main"> <s id="s.002228">Verùm hactenùs de ſtaterâ perinde locutus ſum, ac ſi nulla <lb></lb>illi ineſſet gravitas; quæ tamen omninò contemnenda non eſt, <lb></lb>quantumvis minuta ſit ipſa ſtatera atque exilis, hac enim mi<lb></lb>norum ponderum gravitatem ſcrupuloſiùs exploramus: ideò <lb></lb>autem gravitatem à materiâ mente præcidere ſatius duxi, ut <pb pagenum="300" xlink:href="017/01/316.jpg"></pb>ſtatim appareat vis momentorum, quæ pro variâ diſtantiâ obti<lb></lb>net æquipondium; prout ad majorem, aut ad minorem motum <lb></lb>comparatè cum motu ponderis in A, eſt diſpoſitum. </s> <s id="s.002229">Cæterùm <lb></lb>pondus in A, quod æquilibrium facit cum ſacomate F, majus <lb></lb>eſt quàm pro Ratione diſtantiarum reciprocè ſumptâ; quia vi<lb></lb>delicet ipſius brachij longioris gravitas ſua habet momenta ma<lb></lb>jora momentis brachij brevioris, ac propterea præter pondus, <lb></lb>quod Sacomati reſpondet, addendum eſt etiam pondus, quod <lb></lb>reſpondeat exceſſui momentorum brachij majoris ſuprà mo<lb></lb>menta brachij minoris. </s> <s id="s.002230">Cùm itaque ex dictis cap.2. hujus lib. <lb></lb>momenta brachiorum ſingulorum perinde ſe habeant, atque <lb></lb>ſi ſemiſſis gravitatis ſingulorum eſſet in extremitatibus, poſito <lb></lb>jugo æquabilis craſſitiei, ſi nota ſit totius jugi gravitas, & bra<lb></lb>chiorum Ratio, ſingulorum quoque gravitas innoteſcit; cujus <lb></lb>ſemiſſis per ſibi congruum terminum Rationis ductus exhibet <lb></lb>ſingulorum momenta. </s> <s id="s.002231">Sit AB jugum lib.5. unc.10, hoc eſt <lb></lb>omninò unc.70: Ratio AC ad CB ſit ut 2 ad 5; igitur gravi<lb></lb>tas AC eſt unc. </s> <s id="s.002232">20, & CB unc.50: ſemiſſis AC unc.10 ductus <lb></lb>per 2 (qui eſt terminus Rationis illi congruens) dat momen<lb></lb>tum 20: ſemiſſis CB unc. </s> <s id="s.002233">25 ductus per 5, dat momentum 125: <lb></lb>differentia momentorum eſt 105 dividenda per terminum Ra<lb></lb>tionis congruum diſtantiæ AC, videlicet per 2: Quare ut fiat <lb></lb>æquilibrium cum ſolâ gravitate brachij longioris, addendæ <lb></lb>ſunt extremitati A unciæ 52 1/2: igitur addito ſemiſſe gravita<lb></lb>tis AC, intelliguntur in A unciæ 62 1/2; & in B unciæ 25: ſunt <lb></lb>autem 62 1/2 ad 25, ut 5 ad 2, quæ eſt Ratio reciproca brachio<lb></lb>rum. </s> <s id="s.002234">Quare ſi jugum AB æquabile ſit, ut fert hypotheſis, & <lb></lb>in extremitate B ſit Sacoma lib.2, pondus in A (computatâ <lb></lb>etiam gravitate catenæ & unci AE) non erit ſolùm lib.5. ut <lb></lb>exigit Ratio longitudinis brachiorum, ſed prætereà unc.52 1/2, <lb></lb>hoc eſt omnino lib.9. unc.4 1/2. </s> </p> <p type="main"> <s id="s.002235">Quia verò aliquando accidit properatâ ad ſubitum uſum ſta<lb></lb>terâ uti, videlicet craſſiore tigillo, cujus gravitas non eſt planè <lb></lb>contemnenda, ſed valdè notabilis; proptereà hîc brevem <lb></lb>praxim adjicere placet, quæ etiam minùs peritis uſui eſſe poſſit, <lb></lb>ut ſtatim inveniant gravitatis quantitatem, quæ ſoli gravitati <lb></lb>brachij longioris reſpondet. </s> <s id="s.002236">Sit tigillus AB, in quo intelliga-<pb pagenum="301" xlink:href="017/01/317.jpg"></pb>tur ipſi AC brachio minori æqualis pars CH; eſt igitur bra<lb></lb>chiorum differentia HB. </s> <s id="s.002237">Ponamus totam jugi longitudinem <lb></lb>eſſe diſtinctam in partes 22, quarum AC ſit 4, CB 18, ac dif<lb></lb>ferentia HB 14. Sit verò tigilli pondus lib.84, cujus ſemiſſem <lb></lb>lib.42 accipio. </s> <s id="s.002238">Tum fiat ut longitudo brachij minoris 4 ad dif<lb></lb>ferentiam brachiorum 14, ita ſemiſſis gravitatis jugi lib.42 ad <lb></lb>aliud, & provenient lib.147 addendæ brachio minori, ut fiat <lb></lb>æquilibrium cum ſolâ gravitate longioris. </s> <s id="s.002239">Sic in ſuperiore <lb></lb>exemplo, ubi brachia erant ut 2 ad 5, differentia 3, pondus ju<lb></lb>gi unc.70, cujus ſemiſſis unc.35; fiat ut 2 ad 3, ita unc.35 ad <lb></lb>uncias 52 1/2, quod eſt pondus ibi inventum pluribus calculis. </s> <lb></lb> <s id="s.002240">Ex his infertur jugum æquabilis craſſitiei ſi ſuſpendatur ex <lb></lb>quartâ parte ſuæ longitudinis, ſuſtinere ſinè æquipondio pon<lb></lb>dus additum minori brachio, cujus gravitas æqualis ſit gravita<lb></lb>ti totius jugi. </s> <s id="s.002241">Si ex ſextâ parte ſuſpendatur, ſuſtinet pondus <lb></lb>duplex gravitatis ipſius jugi: ſi ex octavâ parte, ſuſtinet pon<lb></lb>dus triplex gravitatis jugi; ſi ex decima parte, ſuſtinet pondus <lb></lb>quadruplex; ſi ex duodecimâ, ſuſtinet pondus quintuplex, & <lb></lb>ſic deinceps. </s> </p> <p type="main"> <s id="s.002242">Ut igitur ex ratione & certâ methodo conſtrueretur ſtatera <lb></lb>exquiſitè diſtincta in ſuas particulas, oporteret brachium mi<lb></lb>nus cum adnexis appendiculis, catenâ, unco, ſeu lance, tantæ <lb></lb>gravitatis eſſe, ut cum ſolâ longioris brachij gravitate æquili<lb></lb>brium conſtitueretur: tùm diſtantia inter punctum, ex quo <lb></lb>onus ſuſpenditur, & centrum motûs transferenda eſſet ex eo<lb></lb>dem centro motûs in brachium longius, quoties fieri poſſet, & <lb></lb>ſingula intervalla in certas partes minores dividenda, vel pro <lb></lb>libito vel (quod magis rationi congruum eſt) in partes pro<lb></lb>prias menſuræ, quæ adhibetur, ut ſi libra ſit in uncias, ſi un<lb></lb>cia, in drachmas. </s> <s id="s.002243">Hoc autem pendet ex gravitate ſacomatis, <lb></lb>quod eligitur: nam ſi libram unam pendat unà cum ſuo annu<lb></lb>lo æquipondium, tot erunt ponderis libræ, quot partes minori <lb></lb>brachio æquales intercipiuntur inter ſpartum & ipſum æqui<lb></lb>pondium: at ſi bilibre ſit ſacoma, jam partes illæ aſſumptæ <lb></lb>æquales minori brachio ſunt bifariam dividendæ, ut ſingula<lb></lb>rum librarum notæ in jugo habeantur. </s> <s id="s.002244">Quod ſi conſtructá jam <lb></lb>hoc modo ſtaterâ, & majoribus partibus diſtinctis in particulas <lb></lb>ex libito aſſumptas, velis apponere æquipondium majus, quàm <pb pagenum="302" xlink:href="017/01/318.jpg"></pb>fortè ab artifice deſtinaretur, licebit; modò memineris reci<lb></lb>procam eſſe diſtantiarum Rationem & ponderum, quæ in æqui<lb></lb>librio ſunt. </s> </p> <p type="main"> <s id="s.002245">At ſi contigerit ea omnia, quæ breviori brachio adhærent, <lb></lb>non conſtituere æquilibrium cum brachio longiore ſeorſim <lb></lb>ſumpto abſque ſacomate, vel quia graviora ſunt, vel quia mi<lb></lb>nùs gravia; ſatis apparet æquipondium in diſtantia à ſparto du<lb></lb>plà brachij minoris non habere duplum momentum, ſed inve<lb></lb>niendum eſſe aliud punctum, à quo diſtantiæ menſura deſu<lb></lb>matur. </s> </p> <p type="main"> <s id="s.002246">Sit ſtatera ACB, quæ in C ſuſpendatur: gravitas brachio<lb></lb>rum ita ſe habet, ac ſi illius ſemiſſis in ſua cujuſque brachij <lb></lb><figure id="id.017.01.318.1.jpg" xlink:href="017/01/318/1.jpg"></figure><lb></lb>extremitate poneretur. </s> <s id="s.002247">Hujuſmodi ſe<lb></lb>miſſes gravitatum repræſententur à li<lb></lb>neis BD & AE, quæ ſunt utique invi<lb></lb>cem in Ratione brachiorum (quoniam ju<lb></lb>gum æquabile & uniforme ponitur) & ut <lb></lb>AC ad CB, ita AE ad BD. </s> <s id="s.002248">Sed ut fiat <lb></lb>æquilibrium debet eſſe viciſſim ut AC <lb></lb>ad CB, ita BD gravitas in B ad AF gra<lb></lb>vitatem in A: Eſt igitur AE ad AF in <lb></lb>duplicatâ Ratione brachiorum AC ad <lb></lb>CB, hoc eſt ut Quadratum AC ad Qua<lb></lb>dratum CB: Ergo etiam dividendo, per 17. lib.5. ut Quadra<lb></lb>tum CB minus Quadrato AC ad Quadratum AC, ita AF <lb></lb>minùs AE ad AE; hoc eſt ut, differentia Quadratorum utriuſ<lb></lb>que brachij ad Quadratum brachij minoris, ita FE pondus ad<lb></lb>dendum, ad AE ſemiſſem gravitatis brachij minoris, ut fiat <lb></lb>æquilibrium cum ſemiſſe gravitatis, & momento brachij CB <lb></lb>longioris. </s> <s id="s.002249">Id ſi factum fuerit, aſſumantur in CB, incipiendo à <lb></lb>puncto C, partes æquales ipſi CA, & tunc ad mercem addi<lb></lb>tam in F habebit gravitas ſacomatis H eam Rationem, quam <lb></lb>habuerit AC ad diſtantiam ejuſdem ſacomatis à puncto C, ut <lb></lb>ſuperiùs dicebatur. </s> </p> <p type="main"> <s id="s.002250">Verùm ſi præter AE gravitatem reſpondentem minori bra<lb></lb>chio AC, pendere intelligatur ex A non ſolùm gravitas EF, <lb></lb>quæ ſufficiat ad æquilibrium cum longiore brachio CB, ſed <lb></lb>præterea ſit etiam gravitas FG, ita ut tota gravitas addita ſit <pb pagenum="303" xlink:href="017/01/319.jpg"></pb>EG; tunc aſſumpto æquipondio H notæ gravitatis, debet fieri <lb></lb>ut pondus H ad pondus FG exceſſum ſuprà id, quod requiri<lb></lb>tur ad æquilibrium, ita diſtantia AC ad aliud ex. </s> <s id="s.002251">gr. CI: & <lb></lb>ex I initium ſumere debet diviſio transferendo in longius bra<lb></lb>chium, & iterando diſtantiam CA ita, ut AC æqualis ſit ipſi <lb></lb>IN: ſi enim in G addatur tantum mercis, cujus gravitas GM <lb></lb>ſit ad æquipondium H, ut IN ad AC, fiet in N æquilibrium. </s> <lb></lb> <s id="s.002252">Quia ſcilicet ut FG gravitas ad gravitatem H, ita IC diſtan<lb></lb>tia ad diſtantiam CA ex conſtructione; & ut gravitas H ad <lb></lb>gravitatem GM, ita CA diſtantia ad diſtantiam IN; erit ex <lb></lb>æqualitate per 22. lib.5. ut gravitas FG ad gravitatem GM, <lb></lb>ita diſtantia CI ad diſtantiam IN; Ergo componendo, per 18. <lb></lb>lib.5. ut FM ad GM, ita CN ad IN; ſed ut GM ad H, ita <lb></lb>IN ad CA ex hypotheſi; igitur ex æqualitate ut FM gravitas <lb></lb>ad gravitatem H, ita CN diſtantia ad diſtantiam CA. </s> <s id="s.002253">Cùm <lb></lb>itaque pondera addita ultrà æquilibrium, quod addità gravita<lb></lb>te EF fit in C puncto ſuſpenſionis, ſint in Ratione reciprocâ <lb></lb>diſtantiarum à ſparto C, neceſſariò ſequitur æquilibrium in N. </s> <lb></lb> <s id="s.002254">Idem dicendum de cæteris deinceps punctis iterando diſtan<lb></lb>tiam IN, prout brachij longitudo ferre poteſt, nam duplicatâ <lb></lb>diſtantiâ IN, poterit in G addi gravitas dupla gravitatis æqui<lb></lb>pondij H. </s> </p> <p type="main"> <s id="s.002255">Quod ſi demùm partes minori brachio CA adjacentes non <lb></lb>eſſent tantæ gravitatis, ut fieret cum longiore brachio CB <lb></lb>æquilibrium, quemadmodum ſi eſſent ut OE ad EA ſemiſſem <lb></lb>gravitatis brachij minoris; primò obſerva, quantum deſit gra<lb></lb>vitatis, ut fiat æquilibrium, ſcilicet ſit quantitas OF, quæ po<lb></lb>natur minor gravitate æquipondij H: intelligatur itaque gravi<lb></lb>tas æqualis gravitati æquipondij H, & ſit exceſſus FG. </s> <s id="s.002256">Quare <lb></lb>ſicuti paulò antè dicebatur, fiat ut pondus H ad gravitatem <lb></lb>FG, ita AC ad CI, & erit I punctum à quo incipienda eſt di<lb></lb>viſio jugi, ita tamen ut facto æquilibrio in I intelligatur addita <lb></lb>merx æqualis gravitatis cum æquipondio H, & erit ex. </s> <s id="s.002257">gr. pri<lb></lb>ma libra. </s> <s id="s.002258">At verò ſi OE tam modica gravitas eſſet, ut etiam <lb></lb>addita gravitas æqualis gravitati ſacomatis H, nondum adæ<lb></lb>quaret gravitatem EF, addatur duplex, triplex, quadruplex <lb></lb>gravitas ſacomatis H ita, ut demum excedat gravitatem EF <lb></lb>neceſſariam ad æquilibrium cum ſolo brachio longiore; tum fiat <pb pagenum="304" xlink:href="017/01/320.jpg"></pb>ſicuti priùs, ut pondus H ad exceſſum illum, ſcilicet ad FG, <lb></lb>ita AC ad CI, & eſt I punctum quæſitum, ex quo incipit divi<lb></lb>ſio, & in quo ſi fiat æquilibrium mercis cum ſacomate, indicat <lb></lb>mercis gravitatem eſſe duplam, triplam, quadruplam gravita<lb></lb>tis ſacomatis H, prout hanc duplicare oportuit, aut triplicare. </s> </p> <p type="main"> <s id="s.002259">Sed quas habemus communes ſtateras ab hác ſedulitate pro<lb></lb>cul remotas eſſe omnibus conſtabit, ſi obſervaverint amplitu<lb></lb>dines priorum diviſionum non omninò reſpondere brachij mi<lb></lb>noris longitudini, hoc eſt, intervallo, quo pondus diſtat à ſpar<lb></lb>to; neque id ſolùm, quia artifices tantam adhibere diligentiam <lb></lb>recuſant pro tenui mercede; verùm etiam ne adeò graves <lb></lb>exiſtant majores ſtateræ, ſi minori brachio tanta eſſet addita <lb></lb>gravitas, quæ longioris brachij momenta æquaret. </s> <s id="s.002260">Propterea <lb></lb>jugum conſtruunt, uncum ſeu lancem cum ſuis catenulis ad<lb></lb>nectunt, ex ansâ ſuſpendunt, ſacoma non certi ponderis ſed ex <lb></lb>arbitrio eligunt, quod tamen additæ lanci, aut unco aliquate<lb></lb>nus reſpondeat juxta minoris brachij longitudinem; nam ſi hoc <lb></lb>valde breve ſit, augent lancis pondus, & minuunt æquipon<lb></lb>dium; & ex adverſo, ſi illud longiuſculum ſit, minuunt lancem, <lb></lb>augent ſacoma; quia nimirum in illâ brevitate brachij minoris <lb></lb>majora ſunt momenta brachij longioris, & minus æquipon<lb></lb>dium plus habet momenti; contrà verò auctâ minoris brachij <lb></lb>longitudine decreſcunt momenta tùm longioris brachij tùm <lb></lb>æquipondij. </s> </p> <p type="main"> <s id="s.002261">His paratis ſtatuunt in lance legitimum aliquod pondus jux<lb></lb>tà denominationem menſuræ, quam aſſumunt tribuendam ſta<lb></lb>teræ, puta libram (idem dic de majoribus ponderibus in aversâ <lb></lb>ſtateræ parte inſcribendis, ut lib.25 aut 100 juxtà regionis mo<lb></lb>rem) deinde tantiſper ſacoma adducunt vel reducunt, dum fiat <lb></lb>exquiſitè æquilibrium; & punctum adnotant, in quo ſacoma <lb></lb>quieſcit. </s> <s id="s.002262">Tùm aliam adhuc libram, aut, primâ ſublatâ, bilibre <lb></lb>pondus, lanci imponunt, & ſacoma retrahunt, ut magis à mo<lb></lb>tûs centro diſtet; iterumque facto æquilibrio punctum notant. </s> <lb></lb> <s id="s.002263">Demum intervallum inter hæc duo notata puncta in jugo ite<lb></lb>rant, quoties poſſunt; & ut uncias habeant, ſingula intervalla <lb></lb>in duodecim æquales particulas diſtinguunt, quæ in minuſcu<lb></lb>lis ſtateris ad huc minores diviſiones recipiunt. </s> </p> <p type="main"> <s id="s.002264">Quod ſi adhuc pondera infrà libram unam, hoc eſt infra un-<pb pagenum="305" xlink:href="017/01/321.jpg"></pb>cias 12, hac ſtaterâ examinare libeat, inter punctum primò no<lb></lb>tatum atque ſpartum minuſculas illas diviſiones transferunt, <lb></lb>incipiendo ab illo puncto. </s> </p> <p type="main"> <s id="s.002265">Quid autem hîc meminerim puncta hujuſmodi omnia in ju<lb></lb>gi acie, ſeu angulo ſolido ſuperiore notari, majores autem di<lb></lb>viſiones certis lineis ad latus ductis ſignificari? </s> <s id="s.002266">hæc enim vul<lb></lb>garia ſunt. </s> <s id="s.002267">Illud potius notandum eſt, quod in unâ eâdemque <lb></lb>ſtaterâ trium regionum ſtateras habere poſſumus: quia enim <lb></lb>ſtateræ ſcapus communiter quadrangularis eſt, & in ſuperiore <lb></lb>angulo libras hujus regionis inſculpſit artifex, in duobus angu<lb></lb>lis hinc, & hinc libras duabus regionibus, cum quibus com<lb></lb>mercia miſcentur, peculiares inſcribere licebit (nam pondera <lb></lb>ſimili nomine in pluribus regionibus donata, non eſſe inter ſe <lb></lb>æqualia docemur experientiâ, quæ libras Pariſienſem, Ro<lb></lb>manam, Venetam inæquales eſſe oſtendit) & æquipondij an<lb></lb>nulus unâ eâdemque operâ in tribus angulis diverſarum regio<lb></lb>num pondus ejuſdem mercis indicabit. </s> </p> <p type="main"> <s id="s.002268">Hîc verò curiosiùs inquirenti, præſtantiorne dicenda ſit ſta<lb></lb>tera? </s> <s id="s.002269">an libra? </s> <s id="s.002270">vix poterit quiſquam abſolutè reſpondere: nam <lb></lb>minoribus ponderibus, ut gemmis, aureis monetis, & ſimili<lb></lb>bus examinandis parùm opportuna eſt ſtatera; at ingentibus <lb></lb>oneribus hæc aptiſſima eſt, libra autem incommoda. </s> <s id="s.002271">Compen<lb></lb>dium habet ſtatera unico ſacomate contenta; pluribus ponderi<lb></lb>bus eget libra. </s> <s id="s.002272">Viciſſim in librâ ſecuriùs artifices laborem im<lb></lb>pendunt, quia faciliùs æqualitatem aſſequuntur brachiorum, <lb></lb>quàm proportionem juſto æquilibrio neceſſariam; & in librâ <lb></lb>quidem ſi æqualitatem perfectam ſemel ſtatuant, nil eſt quæ<lb></lb>rendum ampliùs; ſed in ſtaterâ ſingula diviſionum puncta ſuam <lb></lb>habent Rationem, ſuamque expoſcunt diligentiam; in pluribus <lb></lb>verò aliquando peccare proclivius eſt, quàm in uno. </s> <s id="s.002273">Quòd ſi <lb></lb>libræ perfecta æqualitas deſit, ſaltem lancium & ponderum <lb></lb>commutatione, ut ſuperiùs monuimus, deprehenditur error; <lb></lb>at ſi falſa ſit ſtatera, non aliter innoteſcet, quàm ſi pondus idem <lb></lb>iterùm librâ examinemus, ut appareat, an ſibi conſtet eadem <lb></lb>gravitas: quis enim aliter iniqui venditoris impoſturam rete<lb></lb>gat, qui, ut major appareat mercis gravitas, ex æquipondio, <lb></lb>aut ex capite longioris brachij, quaſi nitidiùs illa expoliens, <lb></lb>notabilem aliquam gravitatis particulam limâ abraſit? </s> <s id="s.002274">cum ta-<pb pagenum="306" xlink:href="017/01/322.jpg"></pb>men à minore brachio expoliendo manum abſtinuerit; quippe <lb></lb>qui ſatis notat id fieri non poſſe citrà ipſius venditoris damnum: <lb></lb>conſtitutâ ſiquidem ſtaterâ, nihil ex hac aut ex illâ parte de<lb></lb>mendum, nihil addendum, ne mutetur Ratio, quæ intercedit <lb></lb>inter ipſorum brachiorum momenta, aut ne æquipondium di<lb></lb>minutis momentis magis removendum ſit à ſparto, quàm pro <lb></lb>gravitate mercis. </s> <s id="s.002275">Siverò hoc acciderit, occultum manet ſtate<lb></lb>ræ vitium, nec ipſa ſe prodit. </s> </p> <p type="main"> <s id="s.002276">Et quoniam de ſtateræ vitio ſermo incidit, cavendum vendi<lb></lb>tori eſt, ne illâ utatur, ſi facta fuerit curva; cùm enim recta <lb></lb>fuerit ab artifice ſuas in partes ritè diſtincta, & quidem juxta <lb></lb>Rationem brachiorum, curva non eandem ſervat Rationem, <lb></lb>ut oſtenſum eſt hîc cap.5. & venditoris damno plus mercis ad<lb></lb>dendum eſſet lanci, ut haberetur æquilibrium; ut ex ibi dictis <lb></lb>conſtat. <lb></lb></s> </p> <p type="main"> <s id="s.002277"><emph type="center"></emph>CAPUT IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002278"><emph type="center"></emph><emph type="italics"></emph>Antiquorum Statera examinatur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002279">DUbitatur à non paucis, utrùm noſtræ, quâ nunc utimur, <lb></lb>ſtateræ ſimilis eſſet Antiquorum, ſaltem Græcorum, ſta<lb></lb>tera. </s> <s id="s.002280">Dubitationi locum fecit Ariſtoteles in quæſt. </s> <s id="s.002281">20. Mechan. <lb></lb>quærens, <emph type="italics"></emph>Cur ſtatera, quâ carnes ponderantur, pauco appendiculo <lb></lb>magna ponderat onera?<emph.end type="italics"></emph.end> quæſtioni autem ſatisfaciens plurium <lb></lb>ſpartorum mentionem fecit. <emph type="italics"></emph>Quemadmodum autem ſi una li<lb></lb>bra multa ſint libræ; ſic talia inſunt ſparta multa in ejuſmodi li<lb></lb>brâ; quorum uniuſcujuſque quod intrinſecùs eſt ad appendicu<lb></lb>lum, ſtateræ eſt dimidium.<emph.end type="italics"></emph.end> & poſt pauca. <emph type="italics"></emph>Hujuſmodi autem <lb></lb>exiſtens multæ ſunt libræ, totque, quot fuerint ſparta. </s> <s id="s.002282">Semper au<lb></lb>tem quod lanci propinquius eſt ſpartum appenſoque oneri, majus <lb></lb>trahit pondus.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002283">Plura hæc ſparta, quorum Ariſtoteles meminit, Blancano in <lb></lb>locis Mathem. Ariſt. occaſionem præbuerunt ſtateram quan<lb></lb>dam comminiſcendi, quaſi illa fuerit Antiquorum ſtatera: cu<lb></lb>jus ſententiam probare non potui, cum Mechanicam doctri-<pb pagenum="307" xlink:href="017/01/323.jpg"></pb>nam anno labentis ſæculi 54 in Collegio Romano explicans, <lb></lb>publici juris facerem hæc eadem, quæ nunc poſt annos vigin<lb></lb>ti ſcribo. </s> <s id="s.002284">Quoniam verò quæ tunc Blancano oppoſui, video <lb></lb>placuiſſe Authori Magiæ Naturalis P. Gaſpari Schoto tunc ibi <lb></lb>degenti (eaque cum aliis quibuſdam in ſuam Magiam ſtaticam <lb></lb>tranſtulit, me identidem ſuprà meritum, pro ſuâ humanitate, <lb></lb>laudato) hîc iterum proferre non gravabor, ut meliùs ſtateræ <lb></lb>natura innoteſcat. </s> </p> <p type="main"> <s id="s.002285">Statuit itaque Blancanus ſtateram illam ſuiſſe haſtam oblon<lb></lb>gam AB in certas partes diſtributam inter ſe æquales, puta 12, <lb></lb>ex quibus exirent trutinæ diverſæ, ut modò ex hâc, modò ex <lb></lb>illâ ſuſpenderetur ſtatera, prout carnis vendendæ quantitas <lb></lb>poſtulabat, ſinguliſque trutinis inſculptam fuiſſe <expan abbr="notã">notam</expan> ponderis <lb></lb>mercis. </s> <s id="s.002286">In extremitate A <lb></lb><figure id="id.017.01.323.1.jpg" xlink:href="017/01/323/1.jpg"></figure><lb></lb><expan abbr="pẽdebat">pendebat</expan> lanx capax mer<lb></lb>cis, in oppoſitâ extremita<lb></lb>te B æquipondium, <emph type="italics"></emph>quod <emph.end type="italics"></emph.end><lb></lb>ut ille ait, <emph type="italics"></emph>debet habere <lb></lb>tantum pondus, quantum <lb></lb>eſt in lance nudâ, ut ſic tota <lb></lb>ſtatera ſit per ſe ſolam <lb></lb>æquilibralis; & præterea debet habere pondus ſtatum ac legitimum, <lb></lb>ex. </s> <s id="s.002287">gr. unius libræ, aut duarum, aut trium, prout magis trutinandæ <lb></lb>merci idoneum erit, & hoc erit proprium æquipondij pondus. </s> <s id="s.002288">Pona<lb></lb>mus æquipondium eſſe librarum<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>Dico quod trutina C dabit in <lb></lb>lance pondus mercis<emph.end type="italics"></emph.end> 12 <emph type="italics"></emph>lib. ſi ex eâ fiat æquilibrium; eſt enim ut AC <lb></lb>ad CB, it a permutatim æquipondium<emph.end type="italics"></emph.end> 12 <emph type="italics"></emph>ad mercem; ſed AC ipſi <lb></lb>CB eſt æqualis; ergo etiam æquipondium<emph.end type="italics"></emph.end> 12 <emph type="italics"></emph>erit merci æquale, hoc <lb></lb>eſt utrinque erit<emph.end type="italics"></emph.end> 12 <emph type="italics"></emph>lib. </s> <s id="s.002289">Similiter ſi fieret æquilibrium ex trutinâ D, <lb></lb>eſſet ut AD<emph.end type="italics"></emph.end> 3 <emph type="italics"></emph>ad DB<emph.end type="italics"></emph.end> 9, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 12 <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 36. <emph type="italics"></emph>Tandem trutinâ E æquilibrante, <lb></lb>eſſet ut AE<emph.end type="italics"></emph.end> 9 <emph type="italics"></emph>ad EB<emph.end type="italics"></emph.end> 3, <emph type="italics"></emph>ita<emph.end type="italics"></emph.end> 12 <emph type="italics"></emph>ad<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>Si igitur trutina C notetur<emph.end type="italics"></emph.end> 12 <lb></lb><emph type="italics"></emph>numero, trutina D numero,<emph.end type="italics"></emph.end> 36, <emph type="italics"></emph>trutina E numero<emph.end type="italics"></emph.end> 4, <emph type="italics"></emph>& idem de cæteris, <lb></lb>ſtatim facile erit quodlibet pondus per hujuſmodi ſtateram exhibere. </s> <lb></lb> <s id="s.002290">Vnde videas contrario ab illis modo in noſtris ſtateris æquipondium <lb></lb>totam haſtam percurrere, in illis verò manente æquipondio trutinam <lb></lb>quodammodo per haſtam moveri.<emph.end type="italics"></emph.end> Hæc ille. </s> </p> <p type="main"> <s id="s.002291">Plures haſce trutinas ſic expoſitas, quaſi ſolidas anſas haſtæ <lb></lb>infixas, quæ pro opportunitate apprehenderentur, nunquam <pb pagenum="308" xlink:href="017/01/324.jpg"></pb>potui in animum inducere, ut mihi perſuaderem fuiſſe anti<lb></lb>quis in uſu; cùm enim non poſſent ſummis digitis ſuſpendi ob <lb></lb>nimiam mercis gravitatem, puta lib.36 (& multò plurium, ſi <lb></lb>ex F ſtatera penderet) manu fuiſſent validè apprehendendæ; <lb></lb>quis autem non videt, quibus dolis obnoxia fuiſſet ſtatera ex <lb></lb>leviſſimâ manûs inclinatione æquilibrium mentiente? </s> <s id="s.002292">Neque <lb></lb>plicatiles fuiſſe hujuſmodi trutinas, videlicet funiculos forami<lb></lb>nibus inſitos in diviſionum locis, exiſtimo, quia vel nimis fre<lb></lb>quentes eſſe debuiſſent, vel, niſi æquipondium fuiſſet leviſſi<lb></lb>mum, non potuiſſent, citrà venditoris, aut emptorum incom<lb></lb>modum non leve, exhibere quæſitum pondus. </s> <s id="s.002293">Si enim (ut in<lb></lb>ſiſtam ratiocinantis Blancani veſtigiis) in D exhibentur libræ <lb></lb>36 mercis, in G exhiberentur libræ 60, quia ut AG 2 ad <lb></lb>GB 10, ita æquipondium 12 ad mercem 60: quâ igitur ratio<lb></lb>ne innoteſcere poterat pondus mercis, ſi deprehendebatur eſſe <lb></lb>majus quidem libris 36, ſed minus libris 60? </s> <s id="s.002294">Et ſi æquilibrium <lb></lb>fuiſſet inter F & G, pondus fuiſſet majus libris 60, minus li<lb></lb>bris 132: quàm latè igitur patuiſſet campus erroribus in tantâ <lb></lb>ponderum differentiâ? </s> </p> <p type="main"> <s id="s.002295">Quare ſi hoc ſtateræ genere utendum eſſet, in quâ manen<lb></lb>te æquipondio ſpartum percurreret jugi longitudinem, inſe<lb></lb>renda potius eſſet haſta annulo ſolidè firmato, intrà quem haſta <lb></lb>ipſa ultrò citróque promoveretur, donec haberetur æquili<lb></lb>brium; eâ enim ratione in minutiores particulas poſſet haſta <lb></lb>diſtingui; & plurima eſſent ſparta, ſeu centra motûs. </s> <s id="s.002296">Aut <lb></lb><figure id="id.017.01.324.1.jpg" xlink:href="017/01/324/1.jpg"></figure><lb></lb>etiam jugum parari <lb></lb>poſſet craſſioris lami<lb></lb>næ in ſpeciem, cujuſ<lb></lb>modi eſſet MO, per <lb></lb>cujus longitudinem <lb></lb>ductâ inciſurâ ſeu cre<lb></lb>nâ SI excurrere poſſet <lb></lb>axis exquiſitè cylin<lb></lb>dricus infixus anſæ <lb></lb>DE cujus anſæ extremitas in apicem E deſinens indicaret par<lb></lb>ticulas in lineâ MO notatas. </s> <s id="s.002297">Verùm quia adversùs haſce ſtate<lb></lb>ras faciunt pleræque rationes mox contrà Blancani ſtateram <lb></lb>afferendæ, proptereà illas ut parùm aptas rejicio. </s> </p> <pb pagenum="309" xlink:href="017/01/325.jpg"></pb> <p type="main"> <s id="s.002298">Et primùm quidem difficile videatur, quâ ratione fieri poſ<lb></lb>ſet, ut in C puncto medio indicetur mercis pondus lib.12, ſi ex <lb></lb>illo ſtatera ipſa eſt per ſe ſolam æquilibralis, ut Blancanus loqui<lb></lb>tur, poſitâ lance æqualis gravitatis cum æquipondio: Aſſumen<lb></lb>da fuiſſet trutina quarta H, quia ut AH 4 ad HB 8, ita 12 ad <lb></lb>24, & ſubductâ gravitate lancis 12, reliquæ fuiſſent lib.12 <lb></lb>mercis. </s> <s id="s.002299">Hinc patet neque in D indicari pondus mercis lib.36; <lb></lb>hoc enim eſt pondus mercis & lancis ſimul ſumptarum; quare <lb></lb>merx ſolum eſſet lib.24; & ut haberentur mercis lib.36, opor<lb></lb>teret ſpartum accipere, quod haſtam divideret in partes, qua<lb></lb>rum proxima lanci eſſet 1, reliqua 4, quia ut 1 ad 4, ita 12 ad <lb></lb>48, & demptâ lancis gravitate lib.12 remanerent mercis lib.36. <lb></lb>Sed illud à veritate longiſſimè abeſt, quod à Blancano additur, <lb></lb>ex trutinâ E indicari mercem lib.4. Immò addo nullum po<lb></lb>tuiſſe ibi fieri æquilibrium, & maximam partem illarum truti<lb></lb>narum futuram fuiſſe prorſus inutilem; nam ſi lanx A æquè <lb></lb>gravis eſt ac æquipondium B, lanx cum merce gravior eſt æqui<lb></lb>pondio; igitur lanx cum merce in diſtantiâ majore, quàm ſit <lb></lb>æquipondij diſtantia majora habet momenta quàm æquipon<lb></lb>dium, cum quo nunquam poterit æquilibrium conſtituere. </s> <lb></lb> <s id="s.002300">Quare omnes trutinæ inter B & C, & ipſa trutina C inutiles <lb></lb>ſunt, ſi lanx æqualis gravitatis ſit cum æquipondio B: proptereà <lb></lb>lancem multò leviorem eſſe oporteret, ut cum impoſitâ merce <lb></lb>poſſet habere ad æquipondium Rationem reciprocam diſtantia<lb></lb>rum à ſparto. </s> <s id="s.002301">Sed ſi lanx levior ſit æquipondio, ut inter C & B <lb></lb>haberi poſſit æquilibrium; jam non omnes quidem; ſed aliquæ <lb></lb>tantum trutinæ inter B & C inutiles evadent; ubi enim haſta <lb></lb>dividitur reciprocè in Ratione <expan abbr="gravitatũ">gravitatum</expan> lancis, & æquipondij, <lb></lb>ibi eſſet ſtatera per ſe ſolam æquilibralis, juxtà Blancani ratio<lb></lb>cinium: igitur nulla trutina inter illud punctum, & B eſſet uti<lb></lb>lis; quia diminutâ æquipondij à ſparto diſtantiâ, ejus momenta <lb></lb>decreſcunt, & auctâ lancis ab eodem ſparto diſtantiâ, ipſius lan<lb></lb>cis momenta augentur; igitur multò magis augentur facto pon<lb></lb>deris in lance additamento; ac proinde fieri non poterit æqui<lb></lb>librium. </s> </p> <p type="main"> <s id="s.002302">Verùm fortaſſe Author ille, cùm ſtateram dixit per ſe ſolam <lb></lb>æquilibralem ex lancis, & æquipondij gravitatibus æqualibus, <lb></lb>hoc tantùmmodo voluit (& ex ejuſdem verbis inferendum vi-<pb pagenum="310" xlink:href="017/01/326.jpg"></pb>detur) ut æquipondium ultrà libras 12 ſibi peculiares, tantam <lb></lb>prætereà haberet gravitatem, quæ ſi ſolitariè aſſumeretur, poſ<lb></lb>ſet cum lance vacuâ æquilibrium facere in C: quo pacto lanx <lb></lb>non eſſet lib.12; ſed levior. </s> <s id="s.002303">Per hæc tamen non omne incom<lb></lb>modum ſublatum eſſet, neque Blancani dicta conſiſterent; quia <lb></lb>ſit lanx unius libræ, & item æquipondium ultrà libras 12 habeat <lb></lb>libram unam; in C quidem eſſet æquilibrium cum merce <lb></lb>lib.12; quia merx cum lance, item æquipondium totum ſunt <lb></lb>lib.13. At facto æquilibrio in D, diſtantiæ eſſent ut 3 ad 9, igi<lb></lb>tur æquipondium ad mercem cum lance ut 13 ad 39; & ſub<lb></lb>ductâ lancis gravitate lib.1, eſſet merx lib.38, non verò 36. Sic <lb></lb>in E facto æquilibrio, diſtantiæ eſſent ut 9 ad 3, igitur æquipon<lb></lb>dium ad mercem cum lance ut 13 ad 4 1/3, & lancis gravitate <lb></lb>lib.1. demptâ, eſſet merx lib.3 1/3 non autem lib.4. Et in ultima <lb></lb>trutinâ prope B eſſet ut 11 ad 1, ita 13 ad (1 2/11), & lance ſublatâ <lb></lb>lib.1, eſſet merx lib. (2/11), cum juxta Blancani ratiocinium debe<lb></lb>ret eſſe ſolum lib. (1/11). </s> </p> <p type="main"> <s id="s.002304">Deinde jugi brachia ſua habent gravitatis momenta, quæ pro <lb></lb>variâ longitudine inæqualitatem ſubirent; & hæc in hujuſmo<lb></lb>di ſtaterâ modò majora, modò minora eſſent, aliquando adden<lb></lb>da lanci, aliquando æquipondio. </s> <s id="s.002305">Nam ſi ſpartum ſit in D, ab<lb></lb>ſcindens quartam jugi partem, ſola brachij DB gravitas ſuſti<lb></lb>net in A pondus æquale gravitati totius jugi; ac proinde facto <lb></lb>in D æquilibrio, pondus totum additum in A eſt non ſolùm tri<lb></lb>plum æquipondij, ut fert reciproca diſtantiarum Ratio; ſed eſt <lb></lb>præterea æquale gravitati jugi. </s> <s id="s.002306">At ſi ſpartum in F abſcindat ju<lb></lb>gi partem duodecimam, non ſolùm pondus unâ cum lance eſt <lb></lb>æquipondij undecuplum, ſed etiam quintuplum gravitatis jugi: <lb></lb>& ſic de cæteris. </s> <s id="s.002307">Contra verò ſi quando æquilibrium fieret in<lb></lb>ter C & B, ex æquipondio demenda eſſet gravitas reſpondens <lb></lb>momento brachij oppoſiti; tum ex reſiduo colligeretur gravitas <lb></lb>lancis cum merce, & ſubductâ demùm lance, gravitas mercis <lb></lb>innoteſceret. </s> <s id="s.002308">Sic in E facto æquilibrio, quia EB eſt quarta pars <lb></lb>jugi, ex æquipondio B lib.12 auferenda eſt gravitas jugi ex.gr. <lb></lb>lib.4, remanent lib. 8: igitur ut AE 3 ad EB 1, ita lib. 8 ad <lb></lb>lib. 2 2/3: ſi demas pondus lancis, quæ utique valde levis eſſe de<lb></lb>bet, vide quanta gravitas ſit demùm tribuenda merci. </s> <s id="s.002309">At ſi lanx <pb pagenum="311" xlink:href="017/01/327.jpg"></pb>adeò levis ſit, manifeſtum eſt, quantò plus mercis apponen<lb></lb>dum ſit, quando ſpartum à medio ſecedit verſus lancem A. </s> </p> <p type="main"> <s id="s.002310">Quare patet genus hoc ſtateræ, ut pote parùm utile, reji<lb></lb>ciendum, nec potuiſſe Antiquis uſitatum eſſe, quin facilè de<lb></lb>prehenderetur erroribus non levibus obnoxium; cum præſer<lb></lb>tim oblongam fuiſſe haſtam (non utique leviſſimam) commi<lb></lb>niſcatur Blancanus, & qui eum ducem ſequuti ſunt. </s> <s id="s.002311">Non ne<lb></lb>gârim quidem poſſe à perito mathematico ita iniri rationes, ut <lb></lb>certis mercium ponderibus ſua puncta in jugo inſcriberentur, in <lb></lb>quibus æquilibrium fieret cum æquipondio manente in extre<lb></lb>mitate jugi: ſed hunc laborem ſubiiſſe antiquos Mathematicos, <lb></lb>ut ſtateras carnem in macello vendentibus pararent, ſuaderi <lb></lb>non poteſt; artificibus autem tantum fuiſſe induſtriæ, omnem <lb></lb>fidem ſuperat. </s> <s id="s.002312">Ex his mihi certiſſimum videtur aliam prorsùs <lb></lb>adhibendam eſſe Ariſtotelicis verbis interpretationem: Nam <lb></lb>ponamus ſtateram illam, de quâ Ariſtoteles loquitur, planè ſi<lb></lb>milem fuiſſe noſtræ ſtateræ, quis neget unam libram brachio<lb></lb>rum inæqualium eſſe multas libras, hoc ipſo quod æquipon<lb></lb>dium in multis diſtantiis ab eodem puncto varias brachiorum <lb></lb>Rationes conſtituit? </s> <s id="s.002313">ſunt autem plura ſparta, quia punctum <lb></lb>idem diſterminans brachia varias Rationes habentia æquivalet <lb></lb>multis, & quàm multas Rationes brachiorum definire poteſt, <lb></lb>tàm multas conſtituit libras. </s> <s id="s.002314">Demùm quamvis lancis à ſparto <lb></lb>eadem materialiter ſit diſtantia, non eſt tamen eadem formali<lb></lb>ter, neque enim ſolitariè accipienda eſt, ſed comparatè cum <lb></lb>diſtantiâ æquipondij à ſparto; ac propterea cum major æqui<lb></lb>pondij diſtantia ad eandem lancis & oneris diſtantiam majo<lb></lb>rem habeat Rationem, poteſt etiam dici tunc ſpartum eſſe lan<lb></lb>ci & oneri propinquius; nam ſi in unâ æquipondij diſtantiâ bra<lb></lb>chia ſint, ut 2 ad 5, & remoto æquipondio Ratio diſtantiarum <lb></lb>ſit ut 2 ad 6, patet comparatè ad æquipondij diſtantiam, eſſe <lb></lb>minorem priore poſteriorem hanc lancis à ſparto diſtantiam. </s> <lb></lb> <s id="s.002315">Cùm itaque nulla hîc intercedat violenta interpretatio, nil pro<lb></lb>hibet exiſtimare Ariſtotelem de ſtaterâ noſtris non diſſimili lo<lb></lb>cutum fuiſſe. <pb pagenum="312" xlink:href="017/01/328.jpg"></pb> </s> </p> <p type="main"> <s id="s.002316"><emph type="center"></emph>CAPUT X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002317"><emph type="center"></emph><emph type="italics"></emph>Libræ & ſtateræ uſus extenditur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002318">QUæ ſemel aliquem in finem excogitata ſunt, non ea ſunt, <lb></lb>ut illis tantùm terminis coërceantur, ſed ad plura extendi <lb></lb>poſſunt; & fundamentis poſitis alia ſuperſtrui licet, modò non <lb></lb>deſit artificis induſtria atque ſolertia. </s> <s id="s.002319">Quos in uſus libra & ſta<lb></lb>tera à vulgo deſtinentur, omnes nôrunt; ſed ad quos alios tra<lb></lb>duci poſſint, iis manifeſtum eſt, qui illarum naturam diligen<lb></lb>tiùs ſcrutati ſunt. </s> <s id="s.002320">Qua propter ut aliquâ ratione induſtriis arti<lb></lb>ficibus præeam, qui ſimilia, & multò meliora comminiſci po<lb></lb>terunt, pauca quædam hoc capite innuam, quibus libræ & ſta<lb></lb>teræ uſus extenditur. </s> </p> <p type="main"> <s id="s.002321">Diſtinctionis autem atque claritatis gratiâ, in plures propo<lb></lb>ſitiones caput hoc tribuere commodum accidet. </s> </p> <p type="main"> <s id="s.002322"><emph type="center"></emph>PROPOSITIO I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002323"><emph type="italics"></emph>Libram conſtruere, quâ innatantium ſolidorum in humido ſpeci<lb></lb>ficam levitatem, & ipſorum humidorum ſpecificam gravita<lb></lb>tem inveſtigare poſſumus.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002324">ERigatur tigillus AB fulcro ritè inſtructus in B, ut firmiter <lb></lb>conſtitui poſſit horizonti perpendicularis: tranſverſa juga <lb></lb><figure id="id.017.01.328.1.jpg" xlink:href="017/01/328/1.jpg"></figure><lb></lb>duo CD, & EF bifariam æqua<lb></lb>liter diviſa, & circà ſuos axes <lb></lb>verſatilia inſerantur tigillo, pro<lb></lb>ut opportunius fuerit, ita tamen, <lb></lb>ut in eâdem perpendiculari li<lb></lb>neâ VS ſint axes, & inferiori <lb></lb>jugo addatur exteriùs axis capi<lb></lb>ti inſertus index GI, qui ubi <lb></lb>convenerit cum perpendiculari <lb></lb>lineâ VS in facie tigilli deſcrip-<pb pagenum="313" xlink:href="017/01/329.jpg"></pb>tâ, æquilibrium horizontale jugorum CD & EF indicet. </s> <lb></lb> <s id="s.002325">Tum extremitates C & E vel ſolido, vel plicatili vinculo CE <lb></lb>connectantur, & in D quidem addatur lanx; in C verò mo<lb></lb>mentum plumbi, ut æquilibrium ſuâ gravitate conſtituant. </s> <lb></lb> <s id="s.002326">Poſtquam in F adnexus fuerit ſtylus in triplicem cuſpidem de<lb></lb>ſinens, ut faciliùs deprimatur corpus ſolidum H infrà humo<lb></lb>rem, in quo levitat, addatur pariter in E aliquid plumbi, ut <lb></lb>jugum EF in æquilibrio maneat; niſi fortè tanta ſit ipſius vin<lb></lb>culi CE gravitas, ut plumbum addere non ſit opus. </s> <s id="s.002327">Demum <lb></lb>habeatur vas humore implendum, quod ſubjici poſſit extremi<lb></lb>tati F, unà cum ſolido H innatante. </s> </p> <p type="main"> <s id="s.002328">Primò quæritur levitas ſolidi H in aquâ. </s> <s id="s.002329">Expendatur ſoli<lb></lb>dum H exactè in aëre librâ communi & conſucta; ejuſque pon<lb></lb>dus adnotetur: deinde imponatur vaſi aquæ pleno, ita ut ſoli<lb></lb>dum totum immergatur; id quod tunc ſolùm fiet, cùm lanci in <lb></lb>D fuerit impoſitum pondus congruum, nam deſcenden<lb></lb>te D, aſcendit C, & ſecum trahit E ſurſum, ac proin<lb></lb>de F deprimit ſolidum H infrà aquam. </s> <s id="s.002330">Ubi lingula GI in<lb></lb>dicaverit æquilibrium ſolido H aquæ prorſus immerſo, obſer<lb></lb>va pondus lanci D impoſitum: hoc adde ponderi priùs in<lb></lb>vento ejuſdem ſolidi H in aëre; & pronunciabis, ut hæc ſum<lb></lb>ma ponderum ad pondus ſolidi in aëre, ita eſſe gravitatem <lb></lb>ſpecificam aquæ ad gravitatem ſpecificam propoſiti ſolidi. </s> <lb></lb> <s id="s.002331">Fuerit pondus in aëre unc. </s> <s id="s.002332">20; additæ ſint in lance D unciæ 5; <lb></lb>igitur ut 25 ad 20, hoc eſt ut 5 ad 4, ita gravitas ſpecifica <lb></lb>aquæ ad gravitatem ſpecificam ſolidi. </s> </p> <p type="main"> <s id="s.002333">Veritas oſtenditur ex iis, quæ in Hydroſtaticis certa ſunt. </s> <lb></lb> <s id="s.002334">Si enim ponamus aquæ gravitatem ad ſolidi H gravitatem ſe<lb></lb>cundùm ſpeciem eſſe ut 5 ad 4, emergit ex aquâ pars quinta <lb></lb>ſolidi gravitans ut 4; reliquæ quatuor infrà aquam levitant ſin<lb></lb>gulæ ut 1, quæ eſt differentia ſpecificarum gravitatum: igitur <lb></lb>pars quinta ſolidi extans eſt unc. </s> <s id="s.002335">4, quia totum in aëre eſt <lb></lb>unc. </s> <s id="s.002336">20; & pars immerſa levitat tanto niſu, ut æqualis ſit <lb></lb>contrario conatui unc. </s> <s id="s.002337">4. Igitur ſi quinque partes demergan<lb></lb>tur, reſiſtent unciis quinque, quæ ſolido ſuperimponeren<lb></lb>tur; idem autem eſt, ſi unciæ quinque imponantur lanci D; <lb></lb>eandem enim deprimendi vim habent. </s> <s id="s.002338">Si igitur ſolidum gra<lb></lb>ve in aëre ut 20, levitat in aquâ ut 5, aquæ moles æqualis <pb pagenum="314" xlink:href="017/01/330.jpg"></pb>eſt 25, atque adeò aqua ad ſolidum eſt ut 25 ad 20 ſecundùm <lb></lb>gravitatis ſpeciem. </s> </p> <p type="main"> <s id="s.002339">Secundò comparandi ſint humores, uter gravior ſit. </s> <lb></lb> <s id="s.002340">Idem ſolidum H notæ gravitatis in aëre unc. </s> <s id="s.002341">20, quod <lb></lb>priori aquæ immerſum requirebat in lance D uncias 5, im<lb></lb>mergatur eodem modo alteri aquæ, ita, ut in lance ſint <lb></lb>unc. </s> <s id="s.002342">4. drachmæ 5: igitur ſolidi gravitati in aere unc. </s> <s id="s.002343">20. <lb></lb>addantur unc. </s> <s id="s.002344">4. drach. </s> <s id="s.002345">5. & erit aquæ ſecundùm molem <lb></lb>æqualis ſpecifica gravitas unc. </s> <s id="s.002346">24 5/8; hæc ergo poſterior aqua <lb></lb>ad priorem aquam eſt ut 197 ad 200. </s> </p> <p type="main"> <s id="s.002347">Tertiò. </s> <s id="s.002348">Notâ ſolidi ſecundùm ſpeciem gravitate com<lb></lb>paratâ cum gravitate ſpecificâ humoris, cognoſcere poſſumus <lb></lb>alterius molis ejuſdem ſpeciei gravitatem in aëre. </s> <s id="s.002349">Sit cogni<lb></lb>ta Ratio gravitatum ſecundùm ſpeciem ut 4 ad 5. Requi<lb></lb>ratur in lance D pondus unc. </s> <s id="s.002350">8, ut infrà aquam deprima<lb></lb>tur ſolidum. </s> <s id="s.002351">Fiat ut differentia ſpecificarum gravitatum 1, <lb></lb>ad ſpecificam gravitatem ſolidi 4, ita unciæ 8, ad unc. </s> <s id="s.002352">32: <lb></lb>Eſt ergo ſolidum in aëre unciarum 32, & aquæ moles æqualis <lb></lb>unc. </s> <s id="s.002353">40. </s> </p> <p type="main"> <s id="s.002354">Placeat fortaſſe alicui rem hanc aliter perficere. </s> <s id="s.002355">Libræ ju<lb></lb>gum EF ita firmetur in G, ut alteri extremitati E ad<lb></lb><figure id="id.017.01.330.1.jpg" xlink:href="017/01/330/1.jpg"></figure><lb></lb>nexus funiculus aſcendat <lb></lb>orbiculo X circumvolu<lb></lb>tus, & appoſitâ lance D, <lb></lb>atque in F ſtylo tricuſpide, <lb></lb>omnia ſint æquilibrata, ad<lb></lb>dito, ſi opus fuerit, in F <lb></lb>plumbi momento: Pondus <lb></lb>etiam lanci impoſitum ſur<lb></lb>ſum trahens E deprimit F, <lb></lb>& pariter ſolidum ſubjecto <lb></lb>humori innatans à ſtylo deprimitur, & immergitur. </s> </p> <pb pagenum="315" xlink:href="017/01/331.jpg"></pb> <p type="main"> <s id="s.002356"><emph type="center"></emph>PROPOSITIO II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002357"><emph type="center"></emph><emph type="italics"></emph>Horologium arenarium ex librâ conſtruere, quod horæ minu<lb></lb>ta indicet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002358">JUgum libræ æqualium brachiorum AB paretur, ſpartum O <lb></lb>in ſuperiore loco habens: huic enim tantummodo libræ ſpe<lb></lb><figure id="id.017.01.331.1.jpg" xlink:href="017/01/331/1.jpg"></figure><lb></lb>ciei convenire poteſt æqui<lb></lb>librium obliquum. </s> <s id="s.002359">Lingu<lb></lb>lam OI habeat longiuſcu<lb></lb>lam, quæ indicis munere <lb></lb>fungi poſſit, & quam leviſ<lb></lb>ſima ſit. </s> <s id="s.002360">Tum aſſumpta <lb></lb>lanx, quæ figuram conicam <lb></lb>æmuletur, in imâ parte, quâ <lb></lb>apex deſinit, foramen ha<lb></lb>beat exiguum, ex quo poſ<lb></lb>ſit ſenſim arena fluere; cu<lb></lb>juſmodi ea eſt, quâ in vul<lb></lb>garibus horologiis arenariis <lb></lb>utimur. </s> <s id="s.002361">Suſpendatur lanx <lb></lb>ſeorſim à jugo, & impleatur <lb></lb>arenâ, quæ in ſubjectum vas defluat ſpatio horæ unius: horâ <lb></lb>elapsâ ſervetur arena, quam vas excepit, reliqua, quæ in lan<lb></lb>ce, rejiciatur. </s> </p> <p type="main"> <s id="s.002362">Sed quoniam ubi multum erat arenæ in lance, plus defluxit, <lb></lb>quàm par eſt, iterum arena hæc vaſis ſubjecti in lancem infun<lb></lb>datur, & toties experimentum repetatur rejiciendo reliquam, <lb></lb>quoties opus fuerit, ut certi ſimus arenæ defluxum exquiſitè <lb></lb>metiri unius horæ longitudinem. </s> </p> <p type="main"> <s id="s.002363">Habitâ jam congruâ arenæ quantitas diligenter ſervetur, ne <lb></lb>pereat aliquid illius, & novum laborem ſubire cogamur. </s> <s id="s.002364">Hujus <lb></lb>arenæ gravitas examinetur librâ exactiſſimâ: item lancis cum <lb></lb>ſuis appendiculis pondus inquiratur: quibus cognitis inter gra<lb></lb>vitatem ſolius lancis C vacuæ, & gravitatem lancis congruâ <lb></lb>arenâ plenæ inveniatur terminus medio loco proportionalis, qui <lb></lb>dabit gravitatem ponderis D ex oppoſito libræ brachio appen<lb></lb>dendi. </s> </p> <pb pagenum="316" xlink:href="017/01/332.jpg"></pb> <p type="main"> <s id="s.002365">Demùm intervallo OI longitudinis lingulæ, quæ ſcilicet à <lb></lb>ſparto incipit, deſcribatur vel in lamellâ, vel in craſſiore papy<lb></lb>ro ſextans circularis limbi EIF, qui diviſus in partes 60 ita ap<lb></lb>tandus eſt, ut lingula ſuo apice notata puncta percurrens me<lb></lb>dio puncto I congruat, ubi libræ jugum AB horizontale fuerit. </s> <lb></lb> <s id="s.002366">Quare cum lingulæ apex I erit in E, declinabit lingula à per<lb></lb>pendiculo angulo gr. 30: id quod pariter in oppoſitâ parte con<lb></lb>tinget, quando lingulæ apex venerit in F. </s> </p> <p type="main"> <s id="s.002367">Cum igitur jugum ſimiliter inclinari debeat, ut æquilibrium <lb></lb>ſimiliter obliquum fiat hinc lancis C arenâ plenæ depreſſæ cum <lb></lb>pondere D elevato, hinc ponderis D depreſſi cum lance vacuâ <lb></lb>elevatâ; conſtat eandem eſſe oportere Rationem gravitatis lan<lb></lb>cis C arenâ plenæ ad pondus D, quæ eſt ponderis D ad gravi<lb></lb>tatem lancis vacuæ: Eſt igitur ponderis D gravitas medio loco <lb></lb>proportionalis inter gravitates lancis vacuæ, & lancis plenæ. </s> <lb></lb> <s id="s.002368">Sit deprehenſa gravitas lancis vacuæ pondo unc. </s> <s id="s.002369">5 5/9, lancis au<lb></lb>tem cum arenâ unc. </s> <s id="s.002370">18: igitur pondus D requiritur unc. </s> <s id="s.002371">10. </s> </p> <p type="main"> <s id="s.002372">Sed quærendum eſt, quantum diſtare oporteat ſpartum à li<lb></lb>neâ jugi, ut fiat hujuſmodi æquilibrium obliquum gr. 30. Sit <lb></lb><figure id="id.017.01.332.1.jpg" xlink:href="017/01/332/1.jpg"></figure><lb></lb>CD libra, & in C pondus <lb></lb>unc. </s> <s id="s.002373">18. in D unc. </s> <s id="s.002374">10; & <lb></lb>fiat æquilibrium ita, ut OI <lb></lb>lingula faciat cum perpen<lb></lb>diculo HS angulum HOI <lb></lb>gr. 30. Ergo in S eſt cen<lb></lb>trum gravitatis, & eſt reci<lb></lb>procè ut pondus C ad pon<lb></lb>dus D, ita longitudo DS. <lb></lb>ad longitudinem SC: igi<lb></lb>tur quarum partium tota <lb></lb>CD eſt 28, & CG 14, <lb></lb>earum partium eſt GS 4. In triangulo igitur OGS rectan<lb></lb>gulo, GS eſt Sinus gr. 30, & GO eſt Sinus gr. 60; ac propterea. <lb></lb></s> <s id="s.002375">ſi GS eſt 4, GO eſt 6. 928tʹ: tanta itaque debet eſſe diſtantia <lb></lb>ſparti O à lineâ jugi. </s> </p> <p type="main"> <s id="s.002376">Hîc autem obſervabis lineam jugi inclinatam, cum lineâ ho<lb></lb>rizontali, quam ſecat, conſtituere angulum æqualem angulo <lb></lb>declinationis lingulæ à perpendiculo; nam angulo lingulæ cum <pb pagenum="317" xlink:href="017/01/333.jpg"></pb>perpendiculari HOI æqualis eſt ad verticem angulus SOG: <lb></lb>& quia horizontalis VR ſecat perpendiculum HS ad angulos <lb></lb>rectos in B, duo triangula OGS, & EBS rectangula, & com<lb></lb>munem angulum ad S habentia, ſunt æquiangula, atque adeò <lb></lb>angulò SOG, æqualis eſt angulus SEB, cui ad verticem <lb></lb>æqualis eſt angulus DER, qui proptereà æqualis eſt ipſi <lb></lb>HOI. </s> </p> <p type="main"> <s id="s.002377">Sed quoniam GS eſt 4, & GO eſt 6. 928tʹ, per 47. lib.1. <lb></lb>innoteſcit OS partium 7. 999tʹ ex quâ aufertur OB æqualis <lb></lb>ipſi GO (eſt enim diſtantia ſparti ab horizontali æqualis <lb></lb>diſtantiæ ejuſdem ſparti à jugo) remanet BS partium 1. 071tʹ. </s> <lb></lb> <s id="s.002378">In triangulis igitur SGO, SBE ſimilibus ut GS 4 ad SO <lb></lb>7.999tʹ, ita BS 1. 071tʹ ad SE partium 2. 142″: remanet <lb></lb>igitur EG partium 1. 858tʹ. </s> <s id="s.002379">Quare tota DE eſt partium 15. <lb></lb>858″, angulus E in triangulo EMD rectangulo eſt gr.30, ut <lb></lb>oſtenſum eſt; igitur DM altitudo, ad quam elevatur pondus <lb></lb>eſt partium 7. 929tʹ. </s> <s id="s.002380">Et ſimiliter quia EC eſt partium 12.142tʹ, <lb></lb>depreſſio NC eſt partium 6. 071tʹ. </s> <s id="s.002381">Ex quo habetur ſub<lb></lb>jectum vas, quod cadentem arenam excipit, hoc ſaltem inter<lb></lb>vallo depreſſum eſſe infrà lancem pendentem ex jugo horizon<lb></lb>tali poſito. </s> </p> <p type="main"> <s id="s.002382">Et ut ſubjecti vaſis longitudinem invenias, quâ poſſit caden<lb></lb>tem arenam excipere, invenienda eſt diſtantia lancis à per<lb></lb>pendiculo HS, & cùm in ſummâ depreſſione eſt, & cùm eſt <lb></lb>maximè elevata: Cùm depreſſa eſt, diſtat intervallo BN, cùm <lb></lb>horizontalis eſt, diſtat intervallo BV, cùm demùm eſt elevata, <lb></lb>diſtat intervallo æquali ipſi BM. </s> <s id="s.002383">Sunt inveſtigandæ diſtantiæ <lb></lb>BN & BM: Et quia in triangulo EMD rectangulo angulus <lb></lb>eſt gr. 30, & Radius ED eſt partium 15. 858tʹ; Sinus Com<lb></lb>plementi EM eſt partium 13. 733tʹ. </s> <s id="s.002384">Et in ſimili triangulo <lb></lb>ENC, quia EC Radius eſt partium 12. 142tʹ, Sinus Comple<lb></lb>menti EN eſt partium 10. 515tʹ. </s> <s id="s.002385">Et iterum in ſimili triangu<lb></lb>lo EBS, quia ES Radius inventus eſt partium 2. 142tʹ, Sinus <lb></lb>Complementi EB eſt partium 1. 855″. </s> <s id="s.002386">Itaque ex EN aufer <lb></lb>EB, remanet BN 8. 660tʹ, ipſi verò EM adde EB, eſt BM <lb></lb>partium 15. 588tʹ. </s> <s id="s.002387">Demum ex BM aufer BN, & reſiduum <lb></lb>partium 6. 928tʹ eſt longitudo, quam percurrit lanx aſcenden<lb></lb>do, & eſt æqualis diſtantiæ ſparti à lineâ jugi; ac propterea vas <pb pagenum="318" xlink:href="017/01/334.jpg"></pb>excipiendæ arenæ deſtinatum longitudinem habeat neceſſe eſt, <lb></lb>quæ ſaltem ſit quarta pars longitudinis totius jugi, quæ ex da<lb></lb>tis eſt partium 28. </s> </p> <p type="main"> <s id="s.002388">Hæc quæ hactenus dicta ſunt, eo conſilio attuli, ut ſi quis <lb></lb>velit rem ex certâ ratione peragere, intelligat, quâ ſit illi uten<lb></lb>dum methodo: Cæterùm nemini author fuerim, ut hæc omnia <lb></lb>calculis indagare eligat, cùm poſſit citrà laborem citiſſimè aſ<lb></lb>ſequi propoſitum finem Statutis enim ponderibus, ſcilicet <lb></lb>lance, arenâ, & æquipondio (quod, ut dixi, medio loco pro<lb></lb>portionale eſſe oportet inter vacuam lancem, & lancem ean<lb></lb>dem cum arenâ) aſſumatur libræ jugum quodcumque, modò <lb></lb>ſit æqualium brachiorum, & ſpartum in ſuperiore loco habeat, <lb></lb>tùm adnexis hinc lance cum arenâ, hinc æquipondio, libra <lb></lb>conſiſtat obliqua; & in plano Verticali libræ proximo notetur <lb></lb>punctum, cui lingulæ apex congruit: deinde extractâ arenâ <lb></lb>vacuam lancem relinquat, & librâ conſiſtente notetur pariter <lb></lb>punctum in plano, quod apici lingulæ reſpondet; & hæc ſunt <lb></lb>extrema puncta arcûs, qui à circumductâ lingulâ deſcribi po<lb></lb>teſt in eodem plano verticali, & dividi in quæſitas partes 60, ut <lb></lb>horæ minuta indicentur. </s> <s id="s.002389">Quò autem propius ad jugi lineam <lb></lb>accedet ſpartum, & longior fuerit lingula, major quoque erit <lb></lb>hujuſmodi arcus, & faciliùs in partes 60 dividetur. </s> <s id="s.002390">Vaſis de<lb></lb>mum longitudinem ipſa libræ poſitio duplex & cum arenâ, & <lb></lb>ſine arenâ ſtatim oſtendet. </s> <s id="s.002391">Hîc verò ubi de arcûs diviſione in <lb></lb>partes 60 ſermo eſt, liceat mihi diſſimulare partes illas, ſi res <lb></lb>ſubtiliſſimè examinetur, non eſſe omninò inter ſe æquales; ſed <lb></lb>in re Phyſicâ ſubtilitatem hanc perſequi inutile eſt. </s> </p> <p type="main"> <s id="s.002392"><emph type="center"></emph>PROPOSITIO III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002393"><emph type="center"></emph><emph type="italics"></emph>Ex Libræ Rationibus aliquod Motûs perpetui <lb></lb>rudimentum proponere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002394">HOc ſaxum jamdiu multi verſant; ſua cuique cogitata pla<lb></lb>cent; quem corporibus tribuere nondum potuerunt arti<lb></lb>fices perpetuum motum, hunc ſibi vendicant Philoſophorum <lb></lb>mentes inquietâ vertigine illius veſtigiis inſiſtentes; ſed nimis <pb pagenum="319" xlink:href="017/01/335.jpg"></pb>fugacem nunquam aſſequentes. </s> <s id="s.002395">Liceat & mihi hîc aliquid <lb></lb>proponere quaſi rudimentum naturæ motum perpetuum effice<lb></lb>re condiſcentis. </s> <s id="s.002396">Videtur autem omninò certum, ut motus ſe<lb></lb>mel inſtitutus ſine fine perſeveret (ſeclusâ materiæ corruptio<lb></lb>ne, quæ ævo confecta tabeſcit) opus eſſe alterno quodam vi<lb></lb>rium incremento atque decremento, ut idem viribus auctis <lb></lb>prævaleat, viribus diminutis minùs reſiſtat: propterea ſimpli<lb></lb>ciſſimam machinulam, quaſi duplicem libram æqualium bra<lb></lb>chiorum ad angulos rectos compactam aliquando excogitavi, <lb></lb>in quâ alterna hæc viciſſitudo contingere poſſe videtur. </s> </p> <p type="main"> <s id="s.002397">Scapi duo AB & DE ad angulos rectos in C compingantur, <lb></lb>& ſit in C axis, circa quem facilè verſari poſſint: quia verò <lb></lb>oppoſita brachia. </s> <s id="s.002398">ex hypotheſi <lb></lb><figure id="id.017.01.335.1.jpg" xlink:href="017/01/335/1.jpg"></figure><lb></lb>æqualia ſunt, & centrum motûs <lb></lb>planè in medio congruens cen<lb></lb>tro gravitatis ponitur, in quâ<lb></lb>cumque poſitione æqualibus mo<lb></lb>mentis librata quieſcunt. </s> <s id="s.002399">Sint <lb></lb>autem ſingula brachia tubi in <lb></lb>morem excavata ab extremitate <lb></lb>uſque ad decuſſationis locum <lb></lb>æqualiter, ita tamen, ut ex uno <lb></lb>brachio in aliud brachium ſivè <lb></lb>oppoſitum, ſivè proximum nul<lb></lb>lus pateat exitus: extremum autem tubi oſculum congruâ co<lb></lb>chleâ poſſit exquiſitè claudi. </s> <s id="s.002400">Hæc, inquam, omnia ea ſint, quæ <lb></lb>æquilibrium in quâcumque poſitione conſtituant: id quod im<lb></lb>probus labor accurati artificis aſſequi ſe poſſe non deſperat. </s> </p> <p type="main"> <s id="s.002401">Duplici hac librâ ſic paratâ, ſingulis brachiis certa & om<lb></lb>ninò æqualis quantitas Argenti Vivi infundatur, aut major aut <lb></lb>minor pro ratione magnitudinis & gravitatis tuborum, ita ta<lb></lb>men ut non ſit immodica quantitas. </s> <s id="s.002402">Occluſis diligentiſſimè tu<lb></lb>borum oſculis, erigatur DE ad perpendiculum. </s> </p> <p type="main"> <s id="s.002403">Utique hydrargyrus in ſuperiore brachio DC totus quieſcit <lb></lb>propè C, in inferiore brachio CE totus eſt in extremitate E: <lb></lb>in brachiis autem CA & CB horizonti parallelis ſe æqualiter <lb></lb>librat juxtà brachiorum longitudinem; quare libra tota manet <lb></lb>immota, cum ſint hinc & hinc æqualia momenta, tùm ratione <pb pagenum="320" xlink:href="017/01/336.jpg"></pb>brachiorum æqualium, tùm ratione argenti vivi æqualis, & <lb></lb>æqualiter ad motum diſpoſiti: illud verò quod eſt propè C, & <lb></lb>propè E, non poteſt mutare æquilibrium, ut patet. </s> <s id="s.002404">Incline<lb></lb>tur extremitas B aliquantulum deorſum; illicò totus hydrargy<lb></lb>rus brachij CB confluit ad extremitatem B, contrà verò qui <lb></lb>eſt in brachio CA, totus confluit propè centrum C: Facta eſt <lb></lb>igitur libra inæqualium brachiorum, & æqualia argenti vivi <lb></lb>pondera inæqualiter diſtant à centro motûs; ac proinde juxtà <lb></lb>naturam libræ ſpartum in ipsâ jugi lineâ habentis extremitas B <lb></lb>deſcendit quantum poteſt. </s> <s id="s.002405">Cum autem grave quodcumque <lb></lb>ſponte ſua deſcendens acquirat impetum non ſtatim pereun<lb></lb>tem, ſed qui adhuc juxta priorem directionem ad eaſdem par<lb></lb>tes ferat corpus grave etiam contra naturæ propenſionem, ut in <lb></lb>perpendiculo aſcendente eſt manifeſtum, quid prohibeat ex<lb></lb>tremitatem B hydrargyro prægravatam, ex concepto impetu <lb></lb>dum deſcendit, vel, modicum quid tranſilire perpendicularem <lb></lb>poſitionem ultrà punctum E? </s> <s id="s.002406">Id quod ſi accidat, extremitas D, <lb></lb>dum tota libra convertitur, inclinata infrà horizontalem AB <lb></lb>totum hydrargyrum habet non jam in C; ſed in D, quare & <lb></lb>illa ſimili modo deſcendit, nam hydrargyrus, qui erat in E, <lb></lb>elevato brachio CE ſupra horizontalem AB, totus confluit <lb></lb>prope C: neque difficilis eſt deſcenſus; quia B, ubi tranſilierit <lb></lb>perpendiculum DE, ulteriùs ex concepto impetu ſponte aſcen<lb></lb>deret; ſed multò magis aſcendit ex impetu impreſſo brachij <lb></lb>deſcendentis, à quo urgetur. </s> </p> <p type="main"> <s id="s.002407">Fateor equidem in primâ converſione poſt quietem, hydrar<lb></lb>gyrum E reluctari, nec juvare quicquam ad motum; quia ſci<lb></lb>licet, cùm debeat aſcendere ex ſolo impetu impreſſo brachij <lb></lb>CB deſcendentis, nihil confert ad motum, niſi quatenus E <lb></lb>initio ſui aſcensûs modicum aſcendit, B verò initio ſui deſcen<lb></lb>sûs multum deſcendit, ac propterea plus imprimi poteſt impe<lb></lb>tûs, ratione cujus, creſcente quamvis aſcenſuum menſurâ, ha<lb></lb>betur aliquid facilitatis ex prævio impulſu. </s> <s id="s.002408">Hinc eſt in primis <lb></lb>converſionibus opus eſſe manûs adjumento, quæ ſurſum pellat <lb></lb>infimum brachium CE: concepto autem jam impetu, nondum <lb></lb>video, cur motus ceſſaturus ſit. </s> <s id="s.002409">Nam ſi nullâ factâ ponderum <lb></lb>alternâ tranſlatione (quæ ſemper novum motûs principium af<lb></lb>fert) ſed ponderibus ſemper in extremitate brachiorum manen-<pb pagenum="321" xlink:href="017/01/337.jpg"></pb>tibus, poſt aliquot converſiones externo impulſu factas ſponte <lb></lb>ſua diu convertitur rota, aut etiam ſimplex ſcapus, non niſi ex <lb></lb>impreſſo impetu tamdiu permanente, quidni perſeveret in mo<lb></lb>tu, ſi in ſingulis converſionibus novum impetum concipiat? </s> <s id="s.002410">Sed <lb></lb>hæc indicaſſe ſufficiat, ut ſaltem longiorem motum, ſi non per<lb></lb>petuum, quis aſſequi poſſit ſuo inſtituto atque propoſito op<lb></lb>portunum: mihi enim ſatis eſt rationes libræ hujuſmodi com<lb></lb>mentatione aliquantò uberiùs explicare. </s> <s id="s.002411">Unum tamen hîc ad<lb></lb>dere fuerit operæ pretium, videlicet, ſi non placuerit ſcapos <lb></lb>AB & DE invicem ad angulum rectum compactos excavare, <lb></lb>ſed ſolidos retinere volueris, poſſe ſingulis brachiis æquales <lb></lb>tubulos hydrargyri quantitate æquali impletos adalligari, ita ta<lb></lb>men, ut ſimilem brachij faciem contingant, ex quo fiet, ut ſint <lb></lb>ipſi tubuli alternatim diſpoſiti, qui ſibi ex adverſo reſpondent, <lb></lb>nimirum alter ſuperior, alter inferior, alter ad dexteram, alter <lb></lb>ad ſiniſtram. </s> </p> <p type="main"> <s id="s.002412"><emph type="center"></emph>PROPOSITIO IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002413"><emph type="center"></emph><emph type="italics"></emph>Dato unico pondere legitimo examinare bilance <lb></lb>gravitatem multiplicem materiæ dividuæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002414">REs eſt facilis, non tamen omittenda, ne fortè quis ſibi per<lb></lb>ſuadeat non niſi longiſſimâ operâ id perfici poſſe. </s> <s id="s.002415">Datum <lb></lb>ſit unicum pondus legitimum, ex. </s> <s id="s.002416">gr. uncia, & oblata ſit ma<lb></lb>teria dividua, quæ particulatim examinari poſſit, ut ſal, & cæ<lb></lb>tera minuta. </s> <s id="s.002417">Non ſunt ſingulæ unciæ ponderandæ; ſed pri<lb></lb>mò quidem fiat cum unciâ æquilibrium ſalis; deinde in lancem <lb></lb>eandem cum pondere legitimo transferatur ſal; iterum cum <lb></lb>alio ſale fiat æquilibrium, & hic in lancem ponderis refundatur, <lb></lb>totiéſque ſimili methodo repetatur ponderatio, donec oblatæ <lb></lb>materiæ plus quàm ſemiſſem exhauſeris; & adnota, quoties <lb></lb>operam illam repetieris; tot enim termini in Ratione duplâ in<lb></lb>cipiendo ab unitate aſſumpti, & in ſummam redacti, dabunt <lb></lb>gravitatem ſalis jam examinati. </s> <s id="s.002418">Sint ex. </s> <s id="s.002419">gr. decem termini; <lb></lb>poſtremus eſt 512, cujus duplum demptâ unitate eſt ſumma <lb></lb>omnium; ſunt igitur unciæ 1023, hoc eſt libræ 85 1/4. Quod re-<pb pagenum="322" xlink:href="017/01/338.jpg"></pb>ſiduum eſt ſalis, iterum ſimili ratione examinetur, donec <lb></lb>habeas plus quàm ſemiſſem illius reſidui, acceptíſque ite<lb></lb>rum tot terminis progreſſionis duplæ habebis ejus quantita<lb></lb>tem: & ſic deinceps, donec totius propoſitæ molis pondus <lb></lb>innoteſcat. </s> </p> <p type="main"> <s id="s.002420">Quòd ſi certam ſalis menſuram extrahere ex totâ illa mole <lb></lb>deſideras, ex. </s> <s id="s.002421">gr. libras tres, hoc eſt uncias 36, obſerva quot <lb></lb>terminis progreſſionis duplæ proximè accedas ad propoſitam <lb></lb>quantitatem, & erunt quinque termini, quorum poſtremus <lb></lb>eſt 16, & tota ſumma 31. Quare operatio, ut ſuprà, quinquies <lb></lb>repetenda eſt, & habentur unciæ 31: quibus ſepoſitis inqui<lb></lb>rantur unciæ 5 addendæ, nam duplici operatione ſingulas un<lb></lb>cias accipiens in eandem lancem cum unciâ legitimâ repones, <lb></lb>& facto demum æquilibrio reliquas tres uncias habebis, ut <lb></lb>ſumma conficiatur 36. unc. </s> </p> <p type="main"> <s id="s.002422"><emph type="center"></emph>PROPOSITIO V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002423"><emph type="center"></emph><emph type="italics"></emph>Libram æqualium brachiorum conſtruere ad plura <lb></lb>pondera tùm multiplicia tùm ſubmultiplicia <lb></lb>ejuſdem æquipondij examinanda.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002424">ILlud, in quo præſtat libræ ſtatera, eſt, quòd uno eo<lb></lb>demque ſtateræ æquipondio plura pondera examinamus. </s> <lb></lb> <s id="s.002425">Non diſſimile compendium invenire poſſumus in libra æqua<lb></lb>lium brachiorum, quæ tamen ſpartum in ſuperiore loco <lb></lb>habeat; hæc enim pro diversâ ponderum inæqualitate va<lb></lb>riam habet inclinationem, in quâ quieſcat obliquè poſita. </s> <lb></lb> <s id="s.002426">Expedit autem ſpartum à lineâ jugi aliquanto intervallo <lb></lb>diſtare. </s> <s id="s.002427">Sit ſcapus planus, in quo linea jugi recta AB <lb></lb>bifariam dividatur in C; ex quo ad angulos rectos aſſur<lb></lb>gat firmiter adnexa quaſi lingula CD, ita tamen ut in <lb></lb>D ſtatuatur Axis anſæ inſertus, circà quem verſanda eſt <lb></lb>libra; & ex axe pendeat perpendiculum DE, cujus lon<lb></lb>gitudo tanta eſſe debet, ut non ſit minor intervallo DA aut <lb></lb>DB. </s> <s id="s.002428">Tum ex A ſumatur totius lineæ jugi AB tertia pars <pb pagenum="323" xlink:href="017/01/339.jpg"></pb>A 2, quarta A 3, quinta A 4, ſexta A 5, & ſic dein<lb></lb>ceps, quate<lb></lb><figure id="id.017.01.339.1.jpg" xlink:href="017/01/339/1.jpg"></figure><lb></lb>nus commo<lb></lb>dè fieri po<lb></lb>terit: quæ <lb></lb>eædem par<lb></lb>tes ex B in <lb></lb>alterum bra<lb></lb>chium <expan abbr="tranſ-ferãtur">tranſ<lb></lb>ferantur</expan> quàm <lb></lb>accuratiſſi<lb></lb>mè. </s> <s id="s.002429">Demum <lb></lb>ex A & B <lb></lb>æquales lan<lb></lb>ces pendeant, quæ æquilibrium conſtituant. </s> </p> <p type="main"> <s id="s.002430">Hujus libræ uſus eſt ad multiplicia vel ſubmultiplicia pon<lb></lb>dera cum uno eodemque æquipondio comparata invenienda: <lb></lb>Nam ubi æquipondium legitimum ſtatueris in lance B, mer<lb></lb>cem verò in lance A, ſi æqualitas intercedat, ita jugum ma<lb></lb>net, ut perpendiculum DE congruat puncto C: ſi merx ma<lb></lb>jor ſit æquipondio, inclinatur deorſum lanx A, & perpendicu<lb></lb>lum DE ad angulos inæquales ſecans lineam jugi congruit ali<lb></lb>cui ex punctis notatis inter C & A, ſcilicet in 2. ſi fuerit dupla, <lb></lb>in 3 ſi tripla, & ſic de reliquis: ſi demum merx fuerit minor <lb></lb>æquipondio, lanx B inclinabitur, & perpendiculum DE con<lb></lb>gruet alicui ex punctis inter C & B notatis, indicabitque mer<lb></lb>cem eſſe æquipondij aut ſemiſſem, aut trientem, aut quadran<lb></lb>tem, &c. </s> <s id="s.002431">Hinc ſi volueris plures uncias, aut unciæ partem ali<lb></lb>quotam habere, ſtatue in B legitimum unciæ pondus; ſi verò <lb></lb>plures libras, aut libræ partem aliquotam quæſieris, ſtatue in B <lb></lb>libram legitimam. </s> <s id="s.002432">Verùm potiſſima hujus libræ utilitas ſe pro<lb></lb>det, ubi dati ponderis, cujus gravitas ſecundùm legitimas men<lb></lb>ſuras ignota eſt, quæritur pars aliquota, aut illius multiplex <lb></lb>pondus. </s> <s id="s.002433">Hujus autem libræ conſtructio innititur ſuperiùs dictis, <lb></lb>& manifeſta eſt ratio, quia ex ponderum inæqualitate centrum <lb></lb>commune gravitatis reſpondet jugi puncto, quod congruit <lb></lb>perpendiculo pendenti ex eodem puncto ſuſpenſionis libræ. </s> </p> <pb pagenum="324" xlink:href="017/01/340.jpg"></pb> <p type="main"> <s id="s.002434"><emph type="center"></emph>PROPOSITIO VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002435"><emph type="center"></emph><emph type="italics"></emph>Staterâ examinare pondus majus, quàm ipſa communiter ferat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002436">CErtum eſſe pondus, quod unaquæque ſtatera ferat pro ra<lb></lb>tione ſuæ magnitudinis, & gravitatis æquipondij, omni<lb></lb>bus maniſeſtum eſt: & quidem ſi oblatum pondus dividuum <lb></lb>ſit, explorari poteſt per partes ejus gravitas, ut tota demùm in<lb></lb>noteſcat; ſed ſi moles quædam ſolida ſit, quæ ſe dividi non pa<lb></lb>tiatur, ſtatera autem ſit impar tanto oneri, artificium aliquod <lb></lb>adhiberi poteſt, quo gravitatem illam majorem hâc eâdem ſta<lb></lb>terâ inveſtigemus. </s> <s id="s.002437">Et primò quidem ponamus ſtateram ita <lb></lb>fuiſſe conſtructam, ut lancis gravitas ſuis momentis æquet mo<lb></lb>menta brachij longioris, adeò ut, dempto æquipondio ſtateræ <lb></lb>jugum conſiſtat in æquilibrio horizontaliter. </s> <s id="s.002438">Tunc certum <lb></lb>eſt æquipondium ad onus eſſe reciprocè in Ratione diſtantia<lb></lb>rum oneris, & æquipondij à centro motûs. </s> <s id="s.002439">Quare eadem ſta<lb></lb>tera poterit quodammodò multiplex fieri, ſi nimirum æquipon<lb></lb>dium duplicetur, aut triplicetur; poterit enim duplex aut tri<lb></lb>plex pondus ſtaterâ examinari; ut, ſi proprium ſtateræ æqui<lb></lb>pondium ſit unius libræ, & brachium longius ſit brevioris bra<lb></lb>chij quindecuplex, examinari poterit pondus ut ſummum li<lb></lb>brarum quindecim; aſſumptum verò æquipondium novum bi<lb></lb>libre habebit momentum æquale libris 30; ſi trilibre ſit no<lb></lb>vum æquipondium, momentum erit æquale libris 45; & ſic de <lb></lb>reliquis, etiam ſi æquipondium hoc novum non eſſet ad anti<lb></lb>quum omninò in Ratione multiplici; ſed in quâcumque alia <lb></lb>Ratione etiam ſuper particulari, aut ſuperpartiente; ducto <lb></lb>enim pondere novi æquipondij per numerum notatum in ſta<lb></lb>teræ brachio, habebitur quantitas ponderis, quod poteſt exa<lb></lb>minari; ſic ſi æquipondium novum ſit ad antiquum ut unc. </s> <s id="s.002440">20. <lb></lb>ad unc. </s> <s id="s.002441">12. ducto 20 per 15, fit pondus unc. </s> <s id="s.002442">300, hoc eſt lib.25, <lb></lb>quibus novum æquipondium in extremitate ſtateræ poſitum <lb></lb>æquivalet. </s> </p> <p type="main"> <s id="s.002443">Verùm illud eſt incommodum, quòd hujuſmodi æquipon<lb></lb>dio majori non poſſumus exploratam habere gravitatem pon<lb></lb>deris, ſi fortè gravitas æquipondij non ſit illius pars aliquotae <pb pagenum="325" xlink:href="017/01/341.jpg"></pb>nam ſi novum æquipondium ſit bilibre, non indicabit nume<lb></lb>rum diſparem librarum ponderis in punctis libras denotantibus <lb></lb>(ſed ſolummodo in punctis ſelibrarum) vel ſaltem ſingulas un<lb></lb>cias non indicabit, quia omnes numeri in ſtaterâ notati dupli<lb></lb>candi eſſent: ſimiliter dicendum de æquipondio triplici, quo <lb></lb>adhibito omnes numeri triplicandi eſſent. </s> </p> <p type="main"> <s id="s.002444">Propterea, ut huic incommodo occurratur, retineatur anti<lb></lb>quum æquipondium in jugo ſtateræ, ſed ſimul novum æqui<lb></lb>pondium in jugi extremitate apponatur duplum, vel triplum, <lb></lb>vel quadruplum antiqui æquipondij, prout proximè requiri<lb></lb>tur ad explorandam dati oneris gravitatem; tùm antiquum <lb></lb>æquipondium in jugo ſtateræ admoveatur vel removeatur, <lb></lb>quatenus opus fuerit ad æquilibrium conſtituendum. </s> <s id="s.002445">Nam ſi <lb></lb>numerus librarum novi æquipondij ducatur per numerum om<lb></lb>nium librarum, quas ferre poteſt ſtatera, huicque addatur nu<lb></lb>merus ab antiquo æquipondio indicatus, habebitur ipſa pon<lb></lb>deris gravitas, quæ inquiritur. </s> <s id="s.002446">Proponatur pondus aliquod <lb></lb>gravitatis ignotæ, quod ſtateræ lanci imponatur, & æquipon<lb></lb>dium antiquum ac proprium ſtateræ in extremitate poſitum <lb></lb>non valeat pondus elevare ad æquilibrium; addatur æquipon<lb></lb>dium duplum, hoc adhuc impar eſt; addatur triplum, neque <lb></lb>hoc ſatis eſt; addatur quadruplum, & hoc unà cum antiquo <lb></lb>æquipondio in extremitate brachij poſito præponderans illud <lb></lb>eſt, quod requiritur; manente enim novo hoc æquipondio qua<lb></lb>druplo in extremitate, antiquum æquipondium admoveatur <lb></lb>versùs ſpartum, & fiat æquilibrium in puncto lib. 7. unc. </s> <s id="s.002447">9: <lb></lb>quia ſtateræ numerus extremus eſt ex hypotheſi lib.15, & æqui<lb></lb>pondium novum eſt lib.4, jam ſunt lib.60; adde lib.7. unc. </s> <s id="s.002448">9. <lb></lb>tota gravitas ponderis quæſita eſt lib.67. unc.9. </s> </p> <p type="main"> <s id="s.002449">At quæris, an eodem hoc artificio uti liceat in communibus <lb></lb>ſtateris, quas noſtratibus artificibus conſtruere ſolemne eſt; in <lb></lb>quibus nec ſtatera eſt per ſe ſolam æquilibris, nec æquipondij <lb></lb>ſtateræ jugo ita innexi, ut inde pro libito auferri nequeat, gra<lb></lb>vitatem indagare poſſumus, ut æquipondium illius multiplex <lb></lb>eligamus. </s> <s id="s.002450">Opportunè utique dubitas; nam pondus & æquipon<lb></lb>dium in vulgaribus ſtateris non ſunt omninò in reciprocâ Ra<lb></lb>tione diſtantiarum à ſparto, ut ſuperiùs ſuo loco dictum eſt. </s> <lb></lb> <s id="s.002451">Propterea uti quidem poſſumus eodem artificio, ſed certâ ratio-<pb pagenum="326" xlink:href="017/01/342.jpg"></pb>ne: quia enim antiquum æquipondium cum ſtateræ notis lon<lb></lb>gè aliter ſe habet, ac in ſtaterá ſuperiùs aſſumpta, hoc retinea<lb></lb>tur, quod antiquum æquipondium indicabit gravitatem ponde<lb></lb>ris juxta notas, ſtateræ impreſſas; ſed æquipondium novum aſ<lb></lb>ſumatur certæ ac notæ gravitatis proxime tàm ſubmultiplicis <lb></lb>ponderis examinandi, quàm ſubmultiplex brachij longioris eſt <lb></lb>brachium minus ſtateræ; & hoc æquipondium adnectatur non <lb></lb>planè in ſtateræ extremitate, ſed in puncto, in quod cadit lon<lb></lb>gitudo multiplex brachij minoris. </s> <s id="s.002452">Sit ex. gr. ſtatera communis, <lb></lb>quæ elevet pondus lib.15; ſed comparato breviore brachio cum <lb></lb>longiore, hoc non eſt illius omnino quindecuplum; aſſumo, <lb></lb>quoties aſſumi poteſt brachium minus, ex. </s> <s id="s.002454">gr. quaterdecies; & <lb></lb>in illo puncto ſtatuendum erit novum æquipondium notæ gra<lb></lb>vitatis; & quoniam ſuſpicor propoſitam gravitatem non mul<lb></lb>tum abeſſe à lib.50, aſſumo æquipondium lib.3. quæ in notato <lb></lb>puncto æquivaleat libris 42 (nam ter 14 dant 42) & promoto <lb></lb>versùs ſpartum antiquo æquipondio, fit æquilibrium in puncto <lb></lb>lib.5. unc.3: erit igitur propoſita gravitas lib.47. unc.3. Id quod <lb></lb>eſt manifeſtum, quia antiquum æquipondium cum notis ſtate<lb></lb>ræ impreſſis indicat gravitatem ponderis habitâ ratione mo<lb></lb>mentorum brachij ſtateræ & cæterarum illius partium, quas <lb></lb>ſemel attendere opus eſt; reliquæ gravitatis momenta non niſi <lb></lb>ratione diſtantiarum conſideranda ſunt. </s> </p> <p type="main"> <s id="s.002455">Quòd ſi plurium æquipondiorum ſupellectile careas, & ur<lb></lb>geat neceſſitas ſtatim explorandi gravitatem illam majorem, ob<lb></lb>vium aliquod pondus, puta lapidem, vel quid ejuſmodi, ſtate<lb></lb>râ tuâ expende, ut ejus gravitas innoteſcat: hoc ſuſpende ex <lb></lb>opportuno ſtateræ puncto, de quo dictum eſt, & ejus gravita<lb></lb>tem duc per 14 (vel alium quemlibet numerum minorem aut <lb></lb>majorem, prout opportuna ejus ſuſpenſio, aut ſtateræ longitu<lb></lb>do feret) ut habeas gravitatem huic novo æquipondio reſpon<lb></lb>dentem: Cætera ut priùs abſolve. </s> <s id="s.002456">Non videtur autem neceſſa<lb></lb>riò monendus hîc lector poſſe plura nova æquipondia vel di<lb></lb>verſæ, vel paris gravitatis, addi in diverſis diſtantiis à ſparto; <lb></lb>ut ſi æquipondium lib.3. in diſtantia 14, & aliud lib.2 in diſtan<lb></lb>tia 11 ſimul apponantur, æquivalebunt lib.42 & 22, hoc eſt li<lb></lb>bris 64; hæc enim clariora ſunt, quàm indigeant uberiori ex<lb></lb>plicatione. </s> </p> <pb pagenum="327" xlink:href="017/01/343.jpg"></pb> <p type="main"> <s id="s.002457"><emph type="center"></emph>PROPOSITIO VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002458"><emph type="center"></emph><emph type="italics"></emph>Stateram parare ad minuſculas gravitates expendendas.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002459">STateræ hujus jugum non differt à vulgaribus; ſed æquipon<lb></lb>dij & ponderis eſt contraria poſitio; pondus enim longiori <lb></lb>brachio, breviori æquipondium adnectitur, & quò levius fue<lb></lb>rit pondus, eò magis à ſparto removetur. </s> <s id="s.002460">Paretur jugum cum <lb></lb>lance adnexâ, quæ ſuâ gravitate æquet momenta brachij lon<lb></lb>gioris, & in perfecto æquilibrio conſiſtat. </s> <s id="s.002461">Tum brevioris bra<lb></lb>chij longitudo accuratè transferatur in brachium majus, quod <lb></lb>minoris ſaltem decuplum vellem, & ſingulas partes iterum in <lb></lb>decem minores particulas tribuerem, ut totum longius bra<lb></lb>chium in centum particulas diſtingueretur. </s> <s id="s.002462">Sit ſtateræ jugum <lb></lb>AB ita in C à <lb></lb><figure id="id.017.01.343.1.jpg" xlink:href="017/01/343/1.jpg"></figure><lb></lb>ſparto diviſum, <lb></lb>ut CB ſit de<lb></lb>cuplex ipſius <lb></lb>CA: ex A au<lb></lb>tem pendeat lanx D ſuâ gravitate æquè librans momenta bra<lb></lb>chij CB longioris; quod diſtinctum in longitudines decem <lb></lb>æquales brachio minori CA, in ſingulis diviſionibus indicabit <lb></lb>Rationem ponderis ad æquipondium. </s> <s id="s.002463">Collocetur enim æqui<lb></lb>pondium in lance D, pondus examinandum ſi leviuſculum ſit <lb></lb>ita, ut ſerico crudo ſuſpendi poſſit, jugo CB imponatur, & à <lb></lb>ſparto removeatur, donec fiat æquilibrium: nam ſi in primo <lb></lb>puncto diviſionis conſiſtat, erit æqualis gravitatis cum æqui<lb></lb>pondio; ſi in ſecundo puncto, erit ſemiſſis gravitatis æquipon<lb></lb>dij; ſi in tertio, erit triens, ſi in quarto, quadrans, & ſic de cæ<lb></lb>teris. </s> <s id="s.002464">At ſingulis diviſionibus minori brachio æqualibus ite<lb></lb>rum in decem particulas diſtinctis, indicabitur gravitas à <lb></lb>fractione, cujus numerator eſt 10, denominator eſt numerus <lb></lb>particularum omnium, quæ inter ſpartum C & locum ponde<lb></lb>ris æquèlibrati intercipiuntur: ut ſi ex. </s> <s id="s.002465">gr. æquilibrium fiat in F, <lb></lb>hoc eſt in tertiâ particulâ poſt duas integras diviſiones priores, <lb></lb>jam ſunt particulæ 23; igitur pondus eſt (10/23); ipſius æquipondij <lb></lb>in D poſiti; ut conſtat ex co, quòd ut diſtantia CF 23 ad <pb pagenum="328" xlink:href="017/01/344.jpg"></pb>diſtantiam CA 10, ita æquipondium in D ad pondus in <lb></lb>F (10/23). </s> </p> <p type="main"> <s id="s.002466">Hujus ſtateræ utilitas ſatis latè patet, quia non alligatur cer<lb></lb>to æquipondio, ſed in lance D ſtatui poteſt ſivè drachma, ſi<lb></lb>vè uncia, ſivè libra, & ponderis minoris gravitas examinabitur; <lb></lb>quæ quidem habebitur ſecundùm Rationem partis ad aſſem, <lb></lb>ſed deinde ad certam ponderis menſuram, ſivè ſcrupula, ſivè <lb></lb>grana revocabitur. </s> </p> <p type="main"> <s id="s.002467">Quòd ſi pondus examinandum non facilè ſuſpendi poſſit ſe<lb></lb>rico crudo, ut dictum eſt, paratam habeto lancem minuſculam, <lb></lb>cui imponi poſſit pondus; & demùm facto æquilibrio, gravita<lb></lb>te ponderis inventâ, atque ad homogeneam cum æquipondio <lb></lb>menſuram redactâ, ſubducenda eſt hujus lancis cum ſuo funi<lb></lb>culo gravitas, ut ſola ponderis impoſiti gravitas habeatur. </s> <s id="s.002468">Ex <lb></lb>quo patet adhibitâ hujuſmodi lance, quæ percurrat ſtateræ ju<lb></lb>gum, poſſe expendi gravitatem multò minorem: propterea lan<lb></lb>cis hujus gravitas minor eſſe deberet, quàm ſubdecupla gravi<lb></lb>tatis æquipondij impoſiti lanci D, ut in extremo ſtateræ puncto <lb></lb>B fieri poſſet æquilibrium: verùm ſi æquipondium in D ſit un<lb></lb>cia, aut aliquid unciâ minus, majus tamen decimâ ejus parte, <lb></lb>ſatius fuerit lancem illam excipiendo ponderi deſtinatam eſſe <lb></lb>decimam unciæ partem. </s> </p> <p type="main"> <s id="s.002469">Ponatur enim æquipondium uncia, lanx ponderis curſo<lb></lb>ria (1/10) unciæ: impoſitum pondus faciat æquilibrium in F puncto <lb></lb>particulæ 23: eſt igitur pondus cum ſuâ lance (10/23) unciæ, aufer <lb></lb>ratione lancis (1/10) unciæ, reſiduum (77/230) unciæ eſt gravitas ponde<lb></lb>ris; hoc eſt ſcrupulorum 8. Similiter fiat æquilibrium in <lb></lb>puncto 99; ergo pondus cum lance eſt (10/99) unciæ; aufer (1/10), reſi<lb></lb>duum eſt (1/990) unciæ, quod eſt leviſſimum pondus paulò majus <lb></lb>ſemiſſe grani. </s> <s id="s.002470">Si in parte 98, pondus erit (1/490) unciæ, hoc eſt <lb></lb>grani 1 1/6, ſi in puncto 97, pondus erit (3/970) hoc eſt ferè grani <lb></lb>1 4/5: & ſic de cæteris. </s> </p> <pb pagenum="329" xlink:href="017/01/345.jpg"></pb> <p type="main"> <s id="s.002471"><emph type="center"></emph>PROPOSITIO VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002472"><emph type="center"></emph><emph type="italics"></emph>Ad ingentia onera examinanda ſtateras communes <lb></lb>componere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002473">SI opportunas ſtateras parare oporteret ingentibus oneribus <lb></lb>examinandis pares, cujuſmodi eſſet æs campanum, aut <lb></lb>bellicum tormentum majus, eas eſſe debere aut longiſſimas, <lb></lb>aut immani æquipondio inſtructas, manifeſtum eſt. </s> <s id="s.002474">Fac <lb></lb>enim tormentum eſſe lib. 17000 circiter, & ſtateram habe<lb></lb>re uncialem diſtantiam ſparti ab extremitate, cui pondus <lb></lb>adnectitur, æquipondium verò eſſe lib. 25; utique ut 25 <lb></lb>ad 17000, ita uncia pedis ad uncias 680, hoc eſt pedes 56. <lb></lb>tinc. </s> <s id="s.002475">8: atque adeò tota ſtatera eſſet ped. 56 3/4 ut minimum: <lb></lb>cui longitudini ſi congrua craſſities reſpondeat, an non ma<lb></lb>chinâ opus eſt, ut ſola ſtatera transferatur? </s> <s id="s.002476">præterquam <lb></lb>quod ipſa longioris brachij gravitas momenta non exigua <lb></lb>haberet. </s> <s id="s.002477">Quòd ſi, ut non paucis ſolemne eſt, ita trabem <lb></lb>ex mediâ longitudine ſuſpendas, ut æquilibris maneat, cùm <lb></lb>alteri extremitati propoſitum onus adnectas, oppoſitæ au<lb></lb>tem extremitati plura minora pondera adjicias, donec æqui<lb></lb>librium fiat, quorum ſingulæ gravitates in ſummam redactæ <lb></lb>propoſiti oneris gravitatem manifeſtam reddant, non ſolùm <lb></lb>methodus hæc artificio caret, ſed & falſitatis periculo non <lb></lb>vacat, incertum quippe eſt an trabis centrum gravitatis planè <lb></lb>in mediâ longitudine ſit, cùm pars radici proxima gravior ſit <lb></lb>reliquâ, ac proinde libra ſit inæqualium brachiorum, quæ <lb></lb>cenſetur æqualium. </s> </p> <p type="main"> <s id="s.002478">Satius igitur fuerit ſtateras plures minores componere, <lb></lb>ut indicatum eſt lib. 2. cap. 7, quàm ingentem ſtateram <lb></lb>conſtruere. </s> <s id="s.002479">Aſſumantur tres ſtateræ AB, DE, GH, qua<lb></lb>rum brachium minus ſit majoris ſubdecuplum, & ita om<lb></lb>nes ex ſuperiore loco ſuſpendantur, ut orbiculi M & N <lb></lb>facilè verſatiles inferiùs firmati excipere poſſint funiculos <lb></lb>BMD, & ENG, quibus extremitates junguntur: ex quo <lb></lb>fiet, ut dum H vi æquipondij deprimitur, extremitas E, at<lb></lb>que extremitas B pariter deprimantur, pondus verò in A ad-<pb pagenum="330" xlink:href="017/01/346.jpg"></pb>nexum elevetur. </s> <s id="s.002480">Motus autem ſtaterarum non ſunt æquales: <lb></lb>nam ſicut depreſſio ipſius H eſt decupla elevationis ipſius G, <lb></lb><figure id="id.017.01.346.1.jpg" xlink:href="017/01/346/1.jpg"></figure><lb></lb>cui elevationi æqualis eſt depreſſio extremitatis E, ita hæc ejuſ<lb></lb>dem E depreſſio decupla eſt elevationis ipſius D: quare depreſ<lb></lb>ſio H eſt centupla elevationis D; ac propterea quia depreſſio <lb></lb>B æqualis elevationi D eſt decupla elevationis A, depreſſio <lb></lb>æquipondij in H eſt millecupla elevationis ponderis in A <lb></lb>conſtituti. </s> <s id="s.002481">Ex quo ſequitur æquipondium in H æquivalere <lb></lb>ponderi millecuplo, quod in A appendatur. </s> <s id="s.002482">Igitur æquipon<lb></lb>dium lib.17 æquivalebit ponderi lib.17000. </s> </p> <p type="main"> <s id="s.002483">Quod autem hactenus de ſtateris æqualibus dictum eſt, etiam <lb></lb>de inæqualibus dictum intelligatur, componendo Rationes, <lb></lb>quas ſingularum ſtaterarum brachia habent. </s> <s id="s.002484">Hinc ſi Ratio <lb></lb>AC ad CB ſit 1 ad 10, Ratio DF ad FE ſit 1 ad 8, Ratio GI <lb></lb>ad IH ſit 1 ad 12, Ratio compoſita eſt 1 ad 960, quæ poteſt <lb></lb>intercedere inter æquipondium & onus. </s> <s id="s.002485">Hinc manifeſtum eſt <lb></lb>plures addi poſſe ſtateras, quot opus fuerit, quocumque tan<lb></lb>dem ordine collocentur, ſive ſecundùm rectam lineam, ſive <lb></lb>invicem parallelæ, prout commodius accidet, & loci oppor<lb></lb>tunitas feret. </s> </p> <p type="main"> <s id="s.002486">Si ſtateræ iſtæ fuerint ita conſtructæ, ut jugum dempto <lb></lb>æquipondio æquilibre ſit, quia extremitas brachij minoris gra<lb></lb>vitate tantâ prædita eſt, ut gravitati longioris brachij æquipol<lb></lb>leat, res planiſſima eſt, quia ſola brachiorum longitudinis Ra<lb></lb>tio attendenda eſt; & præterea æquipondium in H augeri poſ<lb></lb>ſet, aut minui. </s> <s id="s.002487">Immò hîc etiam adhiberi poſſet artificium, <lb></lb>de quo prop. 6. dicebatur, addendo novum æquipondium cer<lb></lb>tæ gravitatis, ut ſi præter æquipondium H etiam eſſet L lib. 2; <lb></lb>quod in puncto jugi ſeptimo æquivaleret ponderi ſeptingenties <lb></lb>majori, hoc eſt lib. 1400. Quâ methodo addi poſſunt etiam <lb></lb>plura æquipondia in punctis jugi diverſis: quod ſanè eſſet egre-<pb pagenum="331" xlink:href="017/01/347.jpg"></pb>gium compendium, & ut plurimum duabus ſtateris pondera<lb></lb>tio ipſa perficeretur. </s> </p> <p type="main"> <s id="s.002488">At ſi ſtateræ cujuſque jugum non fuerit æquilibre, contem<lb></lb>nenda non eſt brachij longioris gravitas, ut dati ponderis gra<lb></lb>vitas ritè examinetur: Nam ſemiſſis gravitatis brachij IH in <lb></lb>extremitate H conſtitutus æquivalet ponderi decuplo in G mi<lb></lb>nùs gravitate ſemiſſis brachij IG. </s> <s id="s.002489">Igitur perinde eſt, atque ſi <lb></lb>hujuſmodi pondus additum fuiſſet in E, ubi habet momentum <lb></lb>decuplum æqualis ponderis in D, & centuplum æqualis pon<lb></lb>deris in A. </s> <s id="s.002490">Quare momentum brachij IH eſt ut 50, & mo<lb></lb>mentum IG ut 1/2, atque adeò momentum ut 49 1/2 intelligitur <lb></lb>additum in E, quod propterea comparatum cum extremitate <lb></lb>A habet momentum ut 4950. Sic momentum gravitatis FE <lb></lb>comparatum cum extremitate A eſt ut 495; & momentum bra<lb></lb>chij CB eſt ut 49 1/2. Tota igitur momentorum, quæ ex bra<lb></lb>chiorum gravitate oriuntur (ſi illa æqualiter ducta intelligan<lb></lb>tur) ſumma eſt 5494 1/2, ſive ſint unciæ, ſive libræ, prout ſta<lb></lb>terarum moles requirit. </s> <s id="s.002491">Id quod quia ægrè innoteſcit, ſi ju<lb></lb>gum non fuerit æquabiliter ductum, idcircò expeditius fuerit <lb></lb>ſtateris ritè diſpoſitis, ac dempto æquipondio, addere in A tan<lb></lb>tum gravitatis, ut juga ſint horizonti parallela (cujus paralle<lb></lb>liſmi indicium potiſſimum dabit extremæ ſtateræ lingula, quæ <lb></lb>plus cæteris movetur) quæ gravitas ubi innotuerit, addenda <lb></lb>erit gravitati, quam deinde æquipondium indicabit, cùm onus <lb></lb>ipſum in A additum fuerit. </s> <s id="s.002492">Sic pone momenta illa 5494 1/2 eſſe <lb></lb>uncias, hoc eſt lib. 457. unc. </s> <s id="s.002493">10 1/2, & expendendo onus addi<lb></lb>tum in A, æquipondium librale H indicet æquilibrium in <lb></lb>puncto ſeptimo, hoc eſt lib.700, addantur lib.457. unc. </s> <s id="s.002494">10 1/2, <lb></lb>erit tota oneris gravitas lib.1157. unc. </s> <s id="s.002495">10 1/2. </s> </p> <p type="main"> <s id="s.002496">Verùm quia vulgares ſtateræ, quibus communiter utimur <lb></lb>ad majorum ponderum gravitatem examinandam, non ita ſunt <lb></lb>fabrefactæ, ut brachium longius in partes aliquotas minori bra<lb></lb>chio æquales diſtinguatur, propterea minoris brachij longitu<lb></lb>do, quoties fieri id poterit, transferatur in brachium longius, <lb></lb>ut inveniatur punctum, cui adnectendus eſt funiculus, quo <lb></lb>cum alterius proximæ ſtateræ extremitate connectitur. </s> <s id="s.002497">Sed an<lb></lb>tequam opus aggrediaris, amoto æquipondio ſecundæ & ter-<pb pagenum="332" xlink:href="017/01/348.jpg"></pb>tiæ ſtateræ, vide quantum ponderis ſingulæ, quantum con<lb></lb>nexæ requirant ad æquilibrium cum longiore brachio, ut inno<lb></lb>teſcat, quantum adhuc gravitati, oneri tribuendum ſit, præter <lb></lb>illam, quæ ab æquipondio indicatur. </s> </p> <figure id="id.017.01.348.1.jpg" xlink:href="017/01/348/1.jpg"></figure> <p type="main"> <s id="s.002498">Sit ex. </s> <s id="s.002499">gr. ſecunda ſtatera CD, cujus brachium longius <lb></lb>SD æquivaleat lib. 42, & tertiæ ſtateræ FG longius brachium <lb></lb>VG æquivaleat libris 37. Ponamus tertiæ ſtateræ (cui onus <lb></lb>erit adnectendum) brachium minus FV duodecies contineri <lb></lb>in longiore brachio uſque ad I, ubi funiculus connectit illud <lb></lb>cum extremitate C ſecundæ ſtateræ. </s> <s id="s.002500">Item ſecundæ ſtateræ <lb></lb>CD brachium minus CS tredecies ſumi poſſit in brachio lon<lb></lb>giore uſque ad punctum E, ubi illam funiculus connectit cum <lb></lb>primæ ſtateræ extremitate A. </s> <s id="s.002501">Igitur quia momentum brachij <lb></lb>SD æquivalet libris 42 ex hypotheſi, & intelligitur tranſlatum <lb></lb>in I, ubi duodecuplo velociùs movetur quàm punctum F, du<lb></lb>cantur lib. 42 per 12, & æquivalet libris 504, quibus addenda <lb></lb>ſunt momenta brachij VG lib.37, & ponderi invento ex æqui<lb></lb>pondio demum addendæ erunt lib. 541. Jam ſtatuamus æqui<lb></lb>pondium H primæ ſtateræ AB conſtituere æquilibrium in <lb></lb>puncto indicante libras 14: perinde igitur eſt, atque ſi libræ 14 <lb></lb>ponerentur in E; & quia ES ad SC eſt ut 13 ad 1, libræ 14 in <lb></lb>E æquivalent ponderi in C librarum 182, quæ in I poſitæ (quia <lb></lb>IV ad VF eſt ut 12 ad 1) æquivalent ponderi in F librarum <lb></lb>2184. Quod ſi punctum illud, in quo æquipondium H con<lb></lb>ſiſtit, non eſſet nota librarum ſimplicium 14, ſed ponderum, <lb></lb>quæ ſingula libras 25 continent (ut nobis Italis præſertim in <lb></lb>Galliâ Ciſalpinâ ſolemne eſt) utique onus in F adnexum eſſet <lb></lb>lib. 54600, quibus adhuc addendæ eſſent libræ 541, propter <lb></lb>momenta brachiorum ſecundæ & tertiæ ſtateræ, & eſſet tota <lb></lb>gravitas lib. 55141. </s> </p> <p type="main"> <s id="s.002502">At ſi ſtateras communes habeas, nec poſſis æquipondia jugo <lb></lb>inſerta amovere, ut inquirere poſſis momenta gravitatis brachij <pb pagenum="333" xlink:href="017/01/349.jpg"></pb>longioris, hoc unum in ſecundâ, & in tertiá ſtaterâ, aut etiam <lb></lb>pluribus, ſi opus fuerit, obſerva, quoties nimirum brachium <lb></lb>minus in longiore contineatur, ut punctum I & E innoteſcat, <lb></lb>quod cum proximæ ſtateræ extremitate C & A connectendum <lb></lb>eſt: in ſingulis autem ſtateris ſua æquipondia admoveantur, <lb></lb>vel removeantur, donec fiat æquilibrium. </s> <s id="s.002503">Non eſt autem ne<lb></lb>ceſſe ſingulas ſtateras inſculptas eſſe notis homogeneis gravi<lb></lb>tatum; prima enim AB poteſt habere notas indicantes quar<lb></lb>tam partem Centenarij, hoc eſt lib. 25, ſecunda verò & ter<lb></lb>tia poſſunt indicare tantum ſingulas libras cum ſuis unciis. </s> <s id="s.002504">Fac <lb></lb>enim conſtituto æquilibrio, æquipondium H eſſe in puncto <lb></lb>pond. </s> <s id="s.002505">9. lib.7. duc 9 per 25, & ſunt lib. 225, & additis lib.7, <lb></lb>ſunt lib. 232; quæ ducuntur primò per Rationem ſecundæ ſta<lb></lb>teræ 13 ad 1, & fiunt 3016, quæ ductæ per Rationem tertiæ <lb></lb>ſtateræ 12 ad 1, dant demum lib. 36192. Deinde æquipon<lb></lb>dium ſecundæ ſtateræ CD ſit in puncto lib. 7. unc. </s> <s id="s.002506">8: hæ du<lb></lb>cendæ ſunt per Rationem tertiæ ſtateræ 12 ad 1, & fiunt <lb></lb>lib. 92. Demum æquipondium tertiæ ſtateræ indicet lib. 5. <lb></lb>unc. </s> <s id="s.002507">6, addantur hi tres numeri 36192, 92, & 5. unc. </s> <s id="s.002508">6; tota <lb></lb>gravitas oneris in F adnexi erit lib.36289. unc. </s> <s id="s.002509">6. </s> </p> <p type="main"> <s id="s.002510">Ideò autem inquirenda dixi puncta I & E, ut longitudines <lb></lb>VI & SE ſint multiplices longitudinum brachiorum minorum <lb></lb>FV & CE, atque fractionum moleſtia evitetur. </s> <s id="s.002511">Cæterùm ſi <lb></lb>volueris extremitates ipſas G & D cum extremitatibus C & A <lb></lb>connectere, omnino licebit, ubi innotuerit, quota pars brachij <lb></lb>minoris ſit IG & ED. </s> <s id="s.002512">Nam ſi Ratio DS ad SC deprehen<lb></lb>datur ut 13 2/5 ad 1, Ratio autem GV ad VF ut 12 1/4 ad 1, gra<lb></lb>vitas indicata ab æquipondio H ducenda primùm erit per 13 2/5, <lb></lb>deinde numerus productus per 12 3/4 ductus dabit quæſitam <lb></lb>oneris gravitatem reſpondentem æquipondio H, quod, ex hy<lb></lb>potheſi ſuperiùs conſtitutá, indicans pond. </s> <s id="s.002513">9. lib.7, hoc eſt <lb></lb>lib.232, monet ducendas libras 232 per 13 2/5, & fit 3108 4/5, qui <lb></lb>numerus ducatur per 12 3/4, & fiunt lib.39637 1/5. Deinde æqui<lb></lb>pondium ſecundæ ſtateræ poſitum in puncto lib.7. unc. </s> <s id="s.002514">8 indi<lb></lb>cat has ducendas per 12 3/4, & erunt lib. 97 3/4: quibus ſi adda<lb></lb>tur numerus primæ ſtateræ, & numerus quem dat tertia ſtatera <lb></lb>lib.5. unc. </s> <s id="s.002515">6, ſumma erit omnino lib.39740. unc. </s> <s id="s.002516">5. </s> </p> <pb pagenum="334" xlink:href="017/01/350.jpg"></pb> <p type="main"> <s id="s.002517"><emph type="center"></emph>PROPOSITIO IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002518"><emph type="center"></emph><emph type="italics"></emph>In librâ brachiorum æqualium poſſe non æqualia eſſe ponderum <lb></lb>æqualium momenta.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002519">SIt libra AB, cujus centrum C, prorſus in medio, jugum in <lb></lb>brachia dividat æqualia: ſint autem in brachiorum extre<lb></lb><figure id="id.017.01.350.1.jpg" xlink:href="017/01/350/1.jpg"></figure><lb></lb>mitatibus annuli <lb></lb>vel unci, quibus <lb></lb>adnectenda ſunt <lb></lb>pondera, quæ aſ<lb></lb>ſumantur gravi<lb></lb>tatis exquiſitè æ<lb></lb>qualis, computatâ <lb></lb>etiam funiculo<lb></lb>rum gravitate. </s> <s id="s.002520">Sed <lb></lb>alterum quidem <lb></lb>pondus D unco <lb></lb>adnectatur unà <lb></lb>cum ſuo funicu<lb></lb>lo; alterius verò ponderis E funiculus ſuâ extremitate inferiùs <lb></lb>in F paxillo alligetur, & tranſiens per annulum, vel uncum <lb></lb>ſuſpendat connexum pondus E. </s> <s id="s.002521">Experimento diſces pondus E <lb></lb>ſemper prævalere æquali ponderi D, ſi per annulum vel uncum <lb></lb>funiculus liberè valeat excurrere, deſcendente ipſo pon<lb></lb>dere E. </s> </p> <p type="main"> <s id="s.002522">Sed rei primâ facie admiratione dignæ cauſam inquirenti illa <lb></lb>ſe ſtatim offert, quæ Machinalium motionum cauſa à nobis af<lb></lb>fertur; quia videlicet pondus E deſcendens duplo velociùs de<lb></lb>ſcendit, quàm pondus D aſcendat; ubi enim pondus E vene<lb></lb>rit in F, extremitas libræ B ibi conſiſtet, ubi duplicatus eſt fu<lb></lb>niculus, mediâ nimirum viâ; atqui extremitas A non niſi tan<lb></lb>tumdem aſcendit, & cum eâ pondus D; igitur pondus E velo<lb></lb>ciùs deſcendens potiora habet momenta, nec erit æquilibrium, <lb></lb>niſi pondus E ſit ponderis D ſubduplum. </s> </p> <p type="main"> <s id="s.002523">Cave tamen exiſtimes ſemper eſſe motuum Rationem du<lb></lb>plam; id enim tunc ſolùm accidit, cum funiculus extentus eſt <pb pagenum="335" xlink:href="017/01/351.jpg"></pb>horizonti perpendicularis, cujuſmodi eſt FE: at ſi fuerit in<lb></lb>clinatus, non eſt eadem motuum Ratio, ſed ut duplex funicu<lb></lb>li GE longitudo ad altitudinem perpendicularem EF, ita ſe <lb></lb>habet motus ponderis E ad differentiam, quâ excedit motum <lb></lb>ponderis D, ſeu depreſſionis libræ B. </s> <s id="s.002524">Sit funiculus GE, alti<lb></lb>tudo perpendicularis, per quam deſcendit pondus E, ſit EF; <lb></lb>diſtantia GF: deſcendente pondere E, ubi hoc attigerit pla<lb></lb>num horizontale in F, funiculus, qui erat GE, factus eſt GIF; <lb></lb>igitur libra deprimitur uſque in I, & eſt IF differentia motuum <lb></lb>EF & EI. </s> </p> <p type="main"> <s id="s.002525">Quare cùm GI ſit GE minùs IF, quadratum GI æquale <lb></lb>eſt quadrato GE plus quadrato IF, minùs rectangulo ſub GE <lb></lb>& IF bis comprehenſo. </s> <s id="s.002526">At eidem quadrato GI æqualia ſunt <lb></lb>quadrata IF & GF ſimul ſumpta ex 47. lib. 1: propterea aufe<lb></lb>ratur utrinque quadratum IF, & remanet quadratum GE, mi<lb></lb>nùs rectangulo bis ſub GE & IF comprehenſo æquale quadra<lb></lb>to GF: Addatur utrinque rectangulum ſub GE & IF bis, & <lb></lb>utrinque dematur quadratum GF, & eſt quadratum GE mi<lb></lb>nus quadrato GF (hoc eſt quadratum EF ex 47. lib.1.) æqua<lb></lb>le rectangulo bis ſub GE & IF. </s> <s id="s.002527">Igitur ex 17 lib 6. ut bis GE <lb></lb>ad EF, ita EF ad IF. </s> <s id="s.002528">Ponderis itaque motus deorſum EF <lb></lb>comparatus cum aſcenſu ponderis D, eſt ad differentiam mo<lb></lb>tuum IF, ut duplex longitudo funiculi GE ad altitudinem <lb></lb>perpendicularem EF, per quam deſcendit pondus E. </s> </p> <p type="main"> <s id="s.002529">Ex quo ulteriùs colligitur, quò obliquior eſt funiculus, eò <lb></lb>minorem eſſe differentiam IF, ac propterea minorem eſſe Ra<lb></lb>tionem deſcensûs EF ad aſcenſum ponderis oppoſiti, ideóque <lb></lb>etiam minus habere virium ad prævalendum. </s> <s id="s.002530">Hinc ex diver<lb></lb>sâ funiculi longitudine & obliquitate, ſi æquilibrium fiat, lice<lb></lb>bit arguere ipſam ponderum inæqualitatem, ratione habitâ mo<lb></lb>tuum reciprocè ſumptorum; qui motus cum habere non poſſint <lb></lb>Rationem multiplicem majorem duplâ, ut conſtat funiculi ipſius <lb></lb>flexionem conſideranti, neque pondus D poteſt eſſe minus pon<lb></lb>dere E, neque eodem majus quàm duplum, ſi fiat æquilibrium; <lb></lb>minus autem erit quàm duplum, ſi funiculus ſit obliquus, & ex <lb></lb>motuum differentiâ, quæ ſingulas funiculi obliquitates conſe<lb></lb>queretur, etiam ipſa ponderum inæqualium differentia in<lb></lb>fertur, </s> </p> <pb pagenum="336" xlink:href="017/01/352.jpg"></pb> <p type="main"> <s id="s.002531"><emph type="center"></emph>PROPOSITIO X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002532"><emph type="center"></emph><emph type="italics"></emph>Æqualia pondera ſimilis figuræ, ſed diverſæ ſubſtantiæ, ſimili<lb></lb>bus & æqualibus pyxidibus incluſa diſcernere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002533">SInt duo globi, alter ferreus H, alter argenteus S, incluſi <lb></lb>æqualibus & ſimilibus pyxidibus AB & CD ita æqualis <lb></lb><figure id="id.017.01.352.1.jpg" xlink:href="017/01/352/1.jpg"></figure><lb></lb>ponderis, ut pyxides vacuæ librà <lb></lb>examinatæ æquiponderent, & ad<lb></lb>jecti globi pariter ſint æquales ra<lb></lb>tione ponderis, quamvis moles <lb></lb>inæquales ſint, major enim eſt <lb></lb>ferreus, minor argenteus. </s> <s id="s.002534">Opor<lb></lb>teat igitur diſcernere, utra pyxis <lb></lb>argenteum globum contineat. </s> <lb></lb> <s id="s.002535">Singularum pyxidum longitudo <lb></lb>bifariam dividatur in F & E, ex <lb></lb>quibus punctis fiat ſuſpenſio; quâ factâ utique deſcendent ex<lb></lb>tremitates B & D. </s> <s id="s.002536">Addantur tum in A, tum in C pondera, <lb></lb>ut fiat æquilibrium. </s> <s id="s.002537">Pondus majus indicabit ibi eſſe globum <lb></lb>argenteum. </s> <s id="s.002538">Vel ſi unico æquipondio uti placeat, invento <lb></lb>æquilibrio unius pyxidis, idem æquipondium ad alteram pyxi<lb></lb>dem transferatur: ſi enim appoſita extremitas præponderet, ibi <lb></lb>eſt argentum, ſi ſurſum attollatur, ibi eſt ferrum. </s> <s id="s.002539">Manifeſta au<lb></lb>tem eſt ratio, quia majoris globi centrum gravitatis propius eſt <lb></lb>medio pyxidis, ex quo ſit ſuſpenſio, ac propterea minus habet <lb></lb>momenti, quàm minor globus, cujus centrum magis diſtat. </s> </p> <p type="main"> <s id="s.002540">Quamvis verò ſuſpenſio facta fuerit ex medio, nihil refert, <lb></lb>etiamſi ad alterutram extremitatem accedat ut in K, dummodo <lb></lb>æqualis aſſumatur diſtantia in L; eadem enim ſemper ratio pro <lb></lb>inæqualitate momentorum militat, inæqualis ſcilicet diſtantia <lb></lb>centrorum gravitatis. </s> </p> <p type="main"> <s id="s.002541">At ſi non ea eſſet pyxidum longitudo, ut extremitatibus A <lb></lb>& C facilè adnectatur æquipondium, aſſume regulam BZ lon<lb></lb>giorem ipsâ pyxide, eamque alliga funiculo per K tranſeunte, <lb></lb>& in Z æquipondium ſtatuatur: deinde regulam eandem ſimi<lb></lb>liter alliga alteri pyxidi, ut ſit DX, & funiculus per L tranſeat: <pb pagenum="337" xlink:href="017/01/353.jpg"></pb>nam idem æquipondium in X ſi nimis leve ſit, indicat ibi ar<lb></lb>gentum eſſe; id quod pariter indicabit æquipondium majus <lb></lb>faciens æquilibrium. </s> </p> <p type="main"> <s id="s.002542">Quòd ſi pondus idem utrobique faceret æquilibrium, indicio <lb></lb>eſſet aut incluſa corpora non eſſe ſecundùm molem ſimilia, aut <lb></lb>ſi ſimilia fuerint non eſſe in pyxidibus ſimiliter poſita in extre<lb></lb>mitate, contrà hypotheſim. </s> <s id="s.002543">Id quod ut deprehendas, ita pyxi<lb></lb>des converte, ut ad latus conſtituatur pars, quæ priùs erat infi<lb></lb>ma; tunc enim ponderis aliqua diverſitas apparebit. </s> <s id="s.002544">Si autem <lb></lb>adhuc æquilibrium conſtituatur, minorem molem ita ex arte <lb></lb>collocatam fuiſſe, ut centrum gravitatis æqualem diſtantiam <lb></lb>habeat à puncto ſuſpenſionis, ac moles major in alterâ pyxide, <lb></lb>manifeſtum eſt. </s> <s id="s.002545">Tunc igitur utraque pyxis intrà aquam pon<lb></lb>deranda eſt; quæ enim minùs gravis apparebit, continet ar<lb></lb>gentum; hoc quippe minus ſpatij occupans quàm ferrum, ma<lb></lb>jori aëris moli in pyxide locum relinquit: major autem aëris <lb></lb>moles plus deterit ponderis pyxidi intra aquam: pyxidum ſci<lb></lb>licet moles ponuntur æquales. <lb></lb></s> </p> <p type="main"> <s id="s.002546"><emph type="center"></emph>CAPUT XI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002547"><emph type="center"></emph><emph type="italics"></emph>Fundamenta præmittuntur ad explicandum, cur <lb></lb>gravia ſuſpenſa modò præponderent, modò <lb></lb>æquilibria ſint.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002548">LOcus hic eſt obſtrictam non ſemel in ſuperioribus fidem <lb></lb>liberandi, cùm me oſtenſurum ſuſcepi in corporibus ſuſ<lb></lb>penſis aliquando minùs gravia gravioribus prævalere, nec ta<lb></lb>men ullum libræ aut Vectis veſtigium deprehendi, neque mo<lb></lb>tum propriè circularem tribui poſſe potentiæ moventi, quæ vi <lb></lb>ſuæ gravitatis juxtà directionis lineam deorſum conatur, atque <lb></lb>movetur motu recto, ſurſum aſcendente rectà corpore gravio<lb></lb>re, quod per vim elevatur. </s> <s id="s.002549">Sed ut res tota capite ſequenti cla<lb></lb>riùs & breviùs explicari valeat, propoſitiones aliquot hîc lem<lb></lb>matum loco præmittendæ videntur, & problemata, quibus cer-<pb pagenum="338" xlink:href="017/01/354.jpg"></pb>ta methodus præſcribatur, ut pro inſtituto corpora ipſa gravia <lb></lb>eligantur, atque ſuis quæque locis diſponantur. </s> </p> <p type="main"> <s id="s.002550"><emph type="center"></emph>PROPOSITIO I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002551"><emph type="center"></emph><emph type="italics"></emph>Exceſſus ſecantis cujuſcumque anguli ſupra Radium, minor eſt <lb></lb>Tangente ejuſdem anguli.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002552">SIt datus angulus quilibet DBC, ejus Tangens DC, ſecans <lb></lb>BD, & exceſſus ſecantis ſupra Radium DE. </s> <s id="s.002553">Dico DE <lb></lb><figure id="id.017.01.354.1.jpg" xlink:href="017/01/354/1.jpg"></figure><lb></lb>minorem eſſe Tangente DC. </s> <s id="s.002554">Ducatur <lb></lb>recta CE dato angulo ſubtenſa faciens <lb></lb>angulos ad baſim æquales ex 5. lib. 1. ac <lb></lb>proinde acutos: igitur angulus DEC <lb></lb>complementum ad duos rectos eſt ob<lb></lb>tuſus, & maximus in triangulo DEC, <lb></lb>ac propterea ex 19. lib. 1. maximum la<lb></lb>tus eſt, quod illi opponitur, nimirum <lb></lb>Tangens DC. </s> </p> <p type="main"> <s id="s.002555"><emph type="center"></emph>PROPOSITIO II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002556"><emph type="center"></emph><emph type="italics"></emph>Cujuſlibet anguli Tangens eſt media proportionalis inter exceſſum <lb></lb>ſecantis ſupra Radium, & aggregatum ex Radio & <lb></lb>ſecante ejuſdem anguli.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002557">DAtus ſit idem angulus DBC, Tangens DC, exceſſus <lb></lb>ſecantis DE: producatur recta DB uſque in A, & eſt <lb></lb>recta DA aggregatum ex Radio BA & ſecante BD. </s> <s id="s.002558">Dico <lb></lb>Tangentem DC eſſe mediam proportionalem inter ED & <lb></lb>DA. </s> <s id="s.002559">Cùm enim ex 36. lib. 3. rectangulum ſub ED & DA <lb></lb>æquale ſit quadrato, quod à Tangente CD deſcribitur, per 17. <lb></lb>lib. 6. ſunt tres continuè proportionales ED, DC, DA. </s> </p> <p type="main"> <s id="s.002560">Hinc ſequitur exceſſum ſecantis ſupra Radium ad aggrega<lb></lb>tum ex Radio & ſecante habere Rationem duplicatam Ratio-<pb pagenum="339" xlink:href="017/01/355.jpg"></pb>nis, quam idem exceſſus habet ad Tangentem, hoc eſt, ſe ha<lb></lb>bere ut quadratum ED ad quadratum DC, ita ED ad DA, <lb></lb>igitar & dividendo ut quadratum ED ad differentiam quadra<lb></lb>torum ED & DC, ita exceſſus ED ad Radij duplum EA, dif<lb></lb>ferentiam inter ED & DA. </s> </p> <p type="main"> <s id="s.002561"><emph type="center"></emph>PROPOSITIO III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002562"><emph type="italics"></emph>Dato angulo, ad cujus ſecantis exceſſum ſupra Radium ſua <lb></lb>Tangens habet Rationem datam, cujuſcumque anguli mino<lb></lb>ris Tangens ad exceſſum ſuæ ſecantis habet Rationem majo<lb></lb>rem datâ; cujuſcumque autem anguli majoris Tangens ad <lb></lb>exceſſum ſuæ ſecantis habet Rationem minorem datâ Ratione.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002563">ANgulus DBC ſit datus, & illius Tangens DC ad DE ex<lb></lb>ceſſum ſuæ ſecantis habeat datam aliquam Rationem. </s> <s id="s.002564">Pri<lb></lb>mò ſit minor angulus FBC. </s> <s id="s.002565">Dico ejus Tangentem FC ad ſuæ <lb></lb>ſecantis exceſſum FZ habere majorem Rationem quàm DC ad <lb></lb>DE. </s> <s id="s.002566">Quia angulus CFB exterior major eſt interno CDB ex <lb></lb>16. lib.1. fiat huic æqualis angulus CFG, eruntque ex 28.lib.1. <lb></lb>parallelæ lineæ DB & FG, & ex 29.lib.1. DBF & GFH al<lb></lb>terni æquales: ſunt autem BHE & FHG æquales per 15. <lb></lb>lib.1. ut pote ad verticem; ergo & reliquus angulus BEH eſt <lb></lb>reliquo angulo FGH æqualis. </s> <s id="s.002567">Similia itaque ſunt triangula, <lb></lb>& per 4. lib. 6. ut EB ad BH, ita GF ad FH: eſt autem EB <lb></lb>major quàm BH (nam BH minor eſt Radio BZ, cui æqualis <lb></lb>eſt Radius BE) igitur & GF major eſt quàm FH; ergo & mul<lb></lb>tò major quàm FZ. </s> <s id="s.002568">Sed quoniam GF & ED ſunt parallelæ, & <lb></lb>triangula CFG, CDE ſunt æquiangula, ex 4. lib.6. eadem eſt <lb></lb>Ratio CF ad FG, quæ eſt CD ad DE: CF autem ad FG <lb></lb>majorem ex 8. lib. 5. habet minorem Rationem quàm ad FZ <lb></lb>minorem; ergo CF Tangens anguli minoris habet ad FZ <lb></lb>exceſſum ſuæ ſecantis ſupra Radium, Rationem majorem quàm <lb></lb>CD ad DE. </s> </p> <p type="main"> <s id="s.002569">Secundò ſit angulus IBC major dato angulo DBC: Dico <lb></lb>illius Tangentem CI ad ſuæ ſecantis exceſſum KI habere mi-<pb pagenum="340" xlink:href="017/01/356.jpg"></pb>norem Rationem, quàm CD ad DE. </s> <s id="s.002570">Quoniam externus an<lb></lb>gulus CDB major eſt interno CIB, fiat illi æqualis angulus <lb></lb>CIO, & lineæ CE productæ occurrat in O, linea IO, quæ <lb></lb>parallela eſt lineæ BD; & ſunt anguli OIL & EBL alterni <lb></lb>æquales, quemadmodum & anguli ad verticem in L æquales <lb></lb>ſunt. </s> <s id="s.002571">Quapropter in triangulis IOL & EBL æquiangulis per <lb></lb>4. lib.6. ut LB ad BE, ita LI ad IO: eſt autem LB major <lb></lb>quàm BE (nam LB major eſt Radio BK) ergo etiam LI, & <lb></lb>multo magis KI major eſt quàm IO. </s> <s id="s.002572">Sed ut CD ad DE ita <lb></lb>CI ad IO; ergo minor eſt Ratio CI ad IK majorem, quàm <lb></lb>ſit CI ad IO minorem; ergo eſt minor Ratio Tangentis CI ad <lb></lb>exceſſum ſuæ ſecantis KI, quàm ſit Ratio CD ad DE. </s> </p> <p type="main"> <s id="s.002573"><emph type="center"></emph>PROPOSITIO IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002574"><emph type="center"></emph><emph type="italics"></emph>Differentia inter Tangentes duorum quorumlibet angulorum <lb></lb>major eſt, quàm differentia inter eorum ſecantes.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002575">SInt anguli BAC, BAD, eorum Tangentes BC & BD, <lb></lb>quarum differentia CD: angulorum ſecantes AC & AD, <lb></lb><figure id="id.017.01.356.1.jpg" xlink:href="017/01/356/1.jpg"></figure><lb></lb>ſecantium differentia (aſſumptâ AG <lb></lb>æquali ipſi AC) eſt DG. </s> <s id="s.002576">Dico CD <lb></lb>eſſe majorem quàm DG. </s> <s id="s.002577">Ducatur <lb></lb>recta CG, & eſt triangulum CAG <lb></lb>iſoſceles, ideóque angulus CGA <lb></lb>acutus, & qui eſt illi deinceps, CGD <lb></lb>obtuſus, & maximus in triangulo <lb></lb>CGD: quare per 18. lib. 1. major eſt <lb></lb>CD Tangentium differentia quàm <lb></lb>DG ſecantium differentia. </s> </p> <p type="main"> <s id="s.002578"><emph type="center"></emph>PROPOSITIO V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002579"><emph type="center"></emph><emph type="italics"></emph>Ratio differentiæ Tangentium ad differentiam ſecantium fit <lb></lb>ſemper minor.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002580">ESto anguli BAC Tangens BC, anguli BAK Tangens <lb></lb>BK; deſcripto arcu COG, differentia ſecantium eſt KO, <pb pagenum="341" xlink:href="017/01/357.jpg"></pb>& Tangentium differentia eſt CK. </s> <s id="s.002581">Item anguli BAD Tan<lb></lb>gens BD, & deſcripto arcu KH, differentia Tangentium <lb></lb>BK & BD eſt KD, atque ſecantium AK & AD differentia <lb></lb>eſt HD. </s> <s id="s.002582">Dico majorem Rationem eſſe CK ad KO, quàm <lb></lb>KD ad DH. </s> <s id="s.002583">Ducantur rectæ CG & KH. </s> <s id="s.002584">In triangulis iſo<lb></lb>ſcelibus CAG & KAH, anguli ad baſim CG minores ſunt <lb></lb>angulis ad baſim KH, quia angulus CAG major eſt angulo <lb></lb>KAH: quapropter angulo CGA fiat æqualis angulus IHA. <lb></lb>Cum itaque ex 28.lib.I. IH & CG ſint parallelæ, per 2.lib.6. <lb></lb>ut CI ad HG, hoc eſt ad KO, ita ID ad DH: atqui CK ma<lb></lb>jor eſt quàm CI; ergo major eſt Ratio CK ad KO quàm CI <lb></lb>ad KO ex 8.lib.5. hoc eſt quàm ID ad DH. </s> <s id="s.002585">Sed ID eſt ma<lb></lb>jor quàm KD; ergo per 8.lib. 5. major eſt Ratio ID ad DH, <lb></lb>quàm KD ad DH; ergo multò major eſt Ratio CK ad KO, <lb></lb>quàm KD ad DH. </s> <s id="s.002586">Idem de cæteris conſequentibus angulis <lb></lb>nec diſſimili methodo demonſtrari poterit, minorem ſcilicet <lb></lb>fieri Rationem differentiæ Tangentium ad differentiam ſe<lb></lb>cantium. </s> </p> <p type="main"> <s id="s.002587"><emph type="center"></emph>PROPOSITIO VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002588"><emph type="center"></emph><emph type="italics"></emph>Dato Radio, & datâ Ratione Tangentis ad exceſſum ſecantis, <lb></lb>invenire Tangentem & ſecantem, earúmque angulum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002589">DAtus Radius ſit B, data Ratio Tangentis ad exceſſum ſe<lb></lb>cantis ſuprà Radium ſit R ad S. </s> <s id="s.002590">Oportet Tangentem <lb></lb>ipſam atque ſecantem inve<lb></lb><figure id="id.017.01.357.1.jpg" xlink:href="017/01/357/1.jpg"></figure><lb></lb>nire. </s> <s id="s.002591">Tangens eſto A: ut R <lb></lb>ad S ita A ad (A in S/R) exceſſum <lb></lb>ſecantis ſupra Radium; igitur <lb></lb>ſecans integra eſt B + (A in S/R); <lb></lb>hujus quadratum eſt B quad. <lb></lb>+ (2 B in A in S/R) + (A quad. </s> <s id="s.002592">in S quad./R quadr.)quod <lb></lb>ex 47. lib. 1. æquale eſt qua<lb></lb>dratis Radij & Tangentis ſi<lb></lb>mul, hoc eſt B quad. + A quad. </s> <s id="s.002593">Utrinque dempto B quad. </s> <lb></lb> <s id="s.002594">tum omnibus per A diviſis, deinde omnibus ductis per R quad. <pb pagenum="342" xlink:href="017/01/358.jpg"></pb>demum factâ Antitheſi A in R quad. </s> <s id="s.002595">—— A in S quad. </s> <s id="s.002596">æqua<lb></lb>tur 2 S in B in R. </s> <s id="s.002597">Quare revocatâ ad Analogiam æquatione, <lb></lb>eſt ut R quad. </s> <s id="s.002598">—— S quad. </s> <s id="s.002599">ad 2 S in R, ita B Radius ad A <lb></lb>Tangentem quæſitam. </s> <s id="s.002600">Tum fiat ut R ad S ita A inventa ad <lb></lb>aliud, & erit exceſſus ſecantis, qui additus Radio B dabit quæ<lb></lb>ſitam ſecantem. </s> </p> <p type="main"> <s id="s.002601">Sit R 3, S 2: horum quadratorum 9 & 4 differentia eſt 5; <lb></lb>duplum rectangulum ſub R & S eſt 12. Igitur ut 5 ad 12, ita B <lb></lb>Radius 100000 ad 240000 Tangentem gr. 67. 22′.48′. </s> <s id="s.002602">Iterum <lb></lb>ut 3 ad 2 ita 240000 ad 160000 exceſſum ſecantis; Igitur ad<lb></lb>dito Radio, Secans quæſita eſt 260000; quæ etiam in Canone <lb></lb>reſpondet eidem angulo. </s> </p> <p type="main"> <s id="s.002603">Itaque generatim loquendo, fiat ut differentia inter quadra<lb></lb>ta terminorum datæ Rationis ad rectangulum bis ſub iiſdem <lb></lb>terminis comprehenſum, ita datus Radius ad aliud, & prove<lb></lb>niet Tangens quæſita; quæ habita facile dabit ſecantis exceſ<lb></lb>ſum in Ratione datâ. </s> </p> <p type="main"> <s id="s.002604">Quòd ſi rem Geometricè perficere velis, circà majorem Ra<lb></lb>tionis datæ terminum R deſcribe ſemicirculum, & in eo ac<lb></lb>commoda minorem Rationis terminum S; nam linea T dabit <lb></lb>quadratum, quod eſt differentia quadratorum ex R & ex S, ut <lb></lb>eſt manifeſtum ex eo, quod angulus in ſemicirculo eſt rectus <lb></lb>per 31.lib.3.& ex 47 lib.1. quadratum unius lateris circa rectum <lb></lb>eſt differentia quadratorum hypothenuſæ & reliqui lateris. </s> <lb></lb> <s id="s.002605">Deinde inter alterutrum terminorum duplicatum, & reliquum <lb></lb>terminum quære mediam proportionalem, & ſit V potens qua<lb></lb>dratum æquale duplo rectangulo ſub terminis datis. </s> <s id="s.002606">Quoniam <lb></lb>verò ex 20.lib. 6. quadrata ſunt in duplicatâ Ratione laterum, <lb></lb>& T quadratum ad V quadratum eſt in duplicatâ Ratione T <lb></lb>ad V; inveniatur tertia proportionalis X. </s> <s id="s.002607">Demum ut T ad X <lb></lb>ita fiat B ad Z, quæ eſt quæſita Tangens, & ad angulum rectum <lb></lb>conſtituta cum Radio B dabit hypothenuſam ſecantem quæſi<lb></lb>tam, quæ cum Radio conſtituet quæſitum angulum. </s> </p> <p type="main"> <s id="s.002608">Vel etiam ex corollario prop.2. fiat ut differentia quadrato<lb></lb>rum ex R & ex S ad S quadratum, ita duplum Radij B ad exceſ<lb></lb>ſum ſecantis: deinde hic exceſſus inventus ad Tangentem <lb></lb>quæſitam fiat ut S ad R; & ſumma ex dato Radio atque exceſ<lb></lb>ſu invento dabit quæſitam ſecantem. </s> </p> <pb pagenum="343" xlink:href="017/01/359.jpg"></pb> <p type="main"> <s id="s.002609"><emph type="center"></emph>PROPOSITIO VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002610"><emph type="center"></emph><emph type="italics"></emph>Datá Tangente communi duorum circulorum inæqualium, & datis <lb></lb>Rationibus exceſſum Secantium ad eandem Tangentem, <lb></lb>invenire Circulorum Radios.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002611">SIt ſuper lineam CD indefinitam erecta ad perpendiculum <lb></lb>recta AB, quam in B oporteat tangere duos circulos inæ<lb></lb>quales, ita ut ſit Tangens <lb></lb><figure id="id.017.01.359.1.jpg" xlink:href="017/01/359/1.jpg"></figure><lb></lb>duorum angulorum inæ<lb></lb>qualium, exceſſus autem <lb></lb>ſecantis unius ſit ad da<lb></lb>tam Tangentem ut E ad <lb></lb>G, alterius verò ſecantis <lb></lb>exceſſus ſit ad eandem ut <lb></lb>F ad G: & hujuſmodi <lb></lb>circulorum ſemidiame<lb></lb>tros invenire oporteat. </s> </p> <p type="main"> <s id="s.002612">Fiat ut G ad E ita AB <lb></lb>data ad H; & ut H ad <lb></lb>AB ita AB ad MS, ex <lb></lb>quâ dematur MO ipſi H <lb></lb>æqualis, reliquæ OS ſe<lb></lb>miſſi RS æqualis ſuma<lb></lb>tur BD pro Radio circuli BL. </s> <s id="s.002613">Item fiat ut G ad F ita AB <lb></lb>data ad I; & ut I ad AB ita AB ad NT, ex quâ dematur <lb></lb>NP æqualis ipſi I, & reliquæ PT ſemiſſi VT æqualis ſtatua<lb></lb>tur BC ſemidiameter circuli BK. </s> <s id="s.002614">Junctis CA, & DA erunt <lb></lb>exceſſus ſecantium ſuprà ſuos Radios ad Tangentem, videlicet <lb></lb>KA & LA ad AB in datis Rationibus. </s> </p> <p type="main"> <s id="s.002615">Quia enim recta TP ſecta eſt bifariam in V, & adjecta eſt <lb></lb>illi PN, per 6. lib.2. quadratum NV eſt æquale quadrato VT <lb></lb>(hoc eſt quadrato CB) unà cum rectangulo TNP: huic au<lb></lb>tem rectangulo, ex 17. lib. 6. æquale eſt quadratum AB, quæ <lb></lb>ex conſtructione eſt media proportionalis inter PN, hoc eſt I, <lb></lb>& NT. </s> <s id="s.002616">At iiſdem quadratis CB & BA ſimul ſumptis æquale <pb pagenum="344" xlink:href="017/01/360.jpg"></pb>eſt quadratum CA ex 47. lib.1. igitur quadratum CA æquatur <lb></lb>quadrato NV, & linea CA æqualis eſt lineæ NV. </s> <s id="s.002617">Sunt au<lb></lb>tem VP & CK æquales (nam & æquales ſunt lineis VT & <lb></lb>CB) ergo etiam KA reliqua æqualis eſt reliquæ PN, hoc <lb></lb>eſt I. </s> <s id="s.002618">Cum itaque I ad AB ſit ut F ad G ex conſtructione, <lb></lb>etiam KA ad AB eſt in eâdem datâ Ratione F ad G. </s> </p> <p type="main"> <s id="s.002619">Nec diſſimili methodo utendum erit ad oſtendendum LA <lb></lb>ad AB eſſe in datâ Ratione E ad G: id quod indicaſſe ſuffi<lb></lb>ciat, nec pluribus eſt opus. </s> <s id="s.002620">Quare CB & DB ſunt quæſito<lb></lb>rum circulorum ſemidiametri. </s> </p> <p type="main"> <s id="s.002621"><emph type="center"></emph>PROPOSITIO VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002622"><emph type="italics"></emph>Datis duobus inæqualibus circulis ſe contingentibus in B, da<lb></lb>tiſque eorum Radiis CB & DB, invenire Tangentem com<lb></lb>munem BA, ad quam ſecantium exceſſus habeant datas Ra<lb></lb>tiones E ad G, & F ad G.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002623">OPortet ſecantis exceſſum, qui ad Tangentem habet majo<lb></lb>rem Rationem, quàm alter exceſſus; pertinere ad mino<lb></lb>rem circulum; qui verò minorem Rationem habet, pertinere <lb></lb>ad majorem circulum. </s> <s id="s.002624">Cum enim rectangula ſub exceſſibus <lb></lb>& aggregatis ſuarum ſecantium ſuorúmque Radiorum ſint in<lb></lb>ter ſe æqualia, ut pote ex 36. lib.3. eidem Tangentis quadrato <lb></lb>æqualia, erit per 16.lib. 6. ut exceſſus ſecantis majoris circuli <lb></lb>ad exceſſum minoris, ita aggregatum ex ſecante & Radio mi<lb></lb>noris ad aggregatum ex ſecante & Radio majoris. </s> <s id="s.002625">Sicut ergo <lb></lb>eadem Tangens habet majorem Rationem ad Radium minoris <lb></lb>circuli quàm ad Radium majoris, ſubtendítque majorem angu<lb></lb>lum in circulo minori quàm in majori; ita ſuæ ſecantis exceſſus <lb></lb>habet majorem Rationem ad eandem Tangentem, quàm ex<lb></lb>ceſſus ſecantis minoris anguli in circulo majori. </s> </p> <p type="main"> <s id="s.002626">Sit itaque major Ratio F ad G quàm E ad G, & pertinebit <lb></lb>ad circulum minorem. </s> <s id="s.002627">Fiat ut F ad G ita G ad QX, ex quâ <lb></lb>dematur QZ æqualis ipſi F. </s> <s id="s.002628">Tum fiat ut XZ ad ZQ, ita mi<lb></lb>noris Radij duplum TP ad PN: & inter PN & NT invenia<lb></lb>tur media proportionalis BA, quam ex B ad perpendiculum <pb pagenum="345" xlink:href="017/01/361.jpg"></pb>erectam jungat cum centro C rectâ CA: nam KA ad Tan<lb></lb>gentem AB habet datam Rationem F ad G. </s> <s id="s.002629">Cùm enim ea<lb></lb>dem AB, quæ ex conſtructione eſt media inter PN & NT, ſit <lb></lb>etiam ex 36. lib.3. & 17. lib.6. Media inter KA & ACB, & <lb></lb>extremarum NT & ACB exceſſus ſupra ſibi reſpondentes <lb></lb>extremas PN & KA ſint ex conſtructione æquales (ſunt ſcili<lb></lb>cet PT & KCB duplum Radij CB) etiam ipſæ extremæ ſunt <lb></lb>æquales, nimirum NT æqualis ipſi ACB, & PN, æqualis <lb></lb>KA. </s> <s id="s.002630">Atqui ut XZ ad ZQ, ita ex conſtructione TP ad PN, <lb></lb>& componendo atque convertendo ut ZQ ad QX ita PN ad <lb></lb>NT; ergo etiam ut ZQ ad QX ita KA ad ACB. </s> <s id="s.002631">Quare ſi<lb></lb>cuti ZQ ad QX eſt duplicata Rationis F ad G ex conſtructio<lb></lb>ne, etiam KA ad ACB eſt ejuſdem Rationis F ad G duplica<lb></lb>ta; ergo KA ad mediam AB, hoc eſt Exceſſus ſecantis ad Tan<lb></lb>gentem, eſt ut F ad G. </s> </p> <p type="main"> <s id="s.002632">Eádem methodo fiat ut E ad G ita G ad Y <emph type="italics"></emph>a,<emph.end type="italics"></emph.end> ex quâ dema<lb></lb>tur Y <emph type="italics"></emph>b<emph.end type="italics"></emph.end> æqualis ipſi E: & fiat ut <emph type="italics"></emph>ab<emph.end type="italics"></emph.end> ad <emph type="italics"></emph>b<emph.end type="italics"></emph.end> Y, ita Radij majoris <lb></lb>BD duplum SO ad OM; atque inter OM & MS erit media <lb></lb>proportionalis eadem AB: ſimilique ratiocinatione oſtendetur <lb></lb>exceſſum LA ad Tangentem AB eſſe in datâ Ratione E ad G. </s> </p> <p type="main"> <s id="s.002633">Ut in praxim res faciliùs deduci queat, exemplo illuſtretur. </s> <lb></lb> <s id="s.002634">Sit Radius minor CD 12, F ad G ut 16 ad 35: inveniatur his <lb></lb>tertia proportionalis QX (76 9/16). Dematur F 16, remanet XZ <lb></lb>(60 9/16). Fiat ut (60 9/16) ad 16, ita Radij duplum TP 24 ad <lb></lb>PN 6 2/5 proximè. </s> <s id="s.002635">Eſt ergo NT 30 2/5. Inter 6 2/5 & 30 2/5 me<lb></lb>dia eſt 14. </s> </p> <p type="main"> <s id="s.002636">Item ſit Radius major BD 18, E ad G ut 12 ad 35: inve<lb></lb>niatur his tertia proportionalis Y <emph type="italics"></emph>a<emph.end type="italics"></emph.end> 102 1/2, & auferatur E 12, <lb></lb>remanet <emph type="italics"></emph>a b<emph.end type="italics"></emph.end> 90 1/2. Fiat ut 90 1/2 ad 12, ita Radij duplum SO 36 <lb></lb>ad OM 4 7/9. Eſt ergo MS 40 7/9. Inter 4 7/9 & 40 7/9 eſt media <lb></lb>proportionalis 14: in his autem exemplis neglectæ ſunt <lb></lb>fractiunculæ. </s> </p> <pb pagenum="346" xlink:href="017/01/362.jpg"></pb> <p type="main"> <s id="s.002637"><emph type="center"></emph>PROPOSITIO IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002638"><emph type="italics"></emph>Si duorum circulorum ſe exteriùs contingentium centra jungat <lb></lb>recta linea, & ab unius centro ad alterius convexam peri<lb></lb>pheriam rectæ ducantur, ſubtenſa arcus abſciſſi major eſt <lb></lb>quàm differentia linearum angulum in illo centro conſti<lb></lb>tuentium.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002639">DUorum circulorum centra ſint A & B, qui ſe tangant in C, <lb></lb>& jungat centra recta AB. </s> <s id="s.002640">Ex centro A in alterius con<lb></lb><figure id="id.017.01.362.1.jpg" xlink:href="017/01/362/1.jpg"></figure><lb></lb>vexam peripheriam ducatur <lb></lb>recta AD abſcindens arcum <lb></lb>CD. </s> <s id="s.002641">Dico linearum AD & <lb></lb>AC angulum in centro A <lb></lb>conſtituentium differentiam <lb></lb>ED minorem eſſe ſubtensâ <lb></lb>CD. </s> <s id="s.002642">Quia ex 20. lib. 1. duæ <lb></lb>lineæ AC & CD ſimul majores ſunt rectâ AD; auferantur <lb></lb>AC & AE æquales, remanet CD major quàm ED. </s> <s id="s.002643">Simili <lb></lb>ratione CI major eſt quam IF. & ſi ſumatur angulus IAD, <lb></lb>etiam ID major eſt quàm DH differentia inter AI & AD, <lb></lb>quia in triangulo AID duo latera AI & ID majora ſunt reli<lb></lb>quo DA, demptíſque æqualibus AI & AH remanet ID ma<lb></lb>jor quàm DH. </s> </p> <p type="main"> <s id="s.002644"><emph type="center"></emph>PROPOSITIO X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002645"><emph type="italics"></emph>Si duo circuli ſe exteriùs contingant, & in uno æquales arcus <lb></lb>ſumantur, ad quorum extremitates ducantur rectæ à centro <lb></lb>alterius circuli; differentia ſinuum arcûs ſimpli & dupli ad <lb></lb>differentiam Exceſſuum harum rectarum ſupra ſuum Radium <lb></lb>habet minorem Rationem, quàm ſinus arcûs ſimpli ad Exceſ<lb></lb>ſum lineæ ad ipſum ductæ.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.002646">SInt duo circuli, quorum centra A & B, ſe contingentes in C, <lb></lb>ſumantur æquales arcus CI & ID, ad quos ex centro B <pb pagenum="347" xlink:href="017/01/363.jpg"></pb>ducantur rectæ BI & BD ſecantes circulum CE in F & E: <lb></lb>Radium B excedunt exceſſibus FI & ED, qui ex 8. lib. 3. in<lb></lb>æquales ſunt, & ma<lb></lb><figure id="id.017.01.363.1.jpg" xlink:href="017/01/363/1.jpg"></figure><lb></lb>jor eſt ED quàm FI <lb></lb>differentiâ KD. </s> <s id="s.002647">Ar<lb></lb>cuum ſubtenſæ CI <lb></lb>& ID æquales ſunt, <lb></lb>ſinuum IH & DG <lb></lb>differentia eſt LD. </s> <lb></lb> <s id="s.002648">Dico majorem Ra<lb></lb>tionem eſſe HI ad IF, quàm LD ad DK. </s> </p> <p type="main"> <s id="s.002649">Primò ducantur rectæ EF, KI: EF autem producatur ita, <lb></lb>ut occurrat rectæ DI productæ in O. </s> <s id="s.002650">Quia triangula BFE & <lb></lb>BIK ſunt iſoſcelia, & angulus BEF æqualis eſt angulo BKI, <lb></lb>rectæ EO & KI ex 28.lib.1. ſunt parallelæ: igitur ex 2 lib. 6. <lb></lb>in triangulo DOE ut DI ad IO, ita DK ad KE: Atqui DI <lb></lb>major eſt quàm IO, ergo etiam DK major quàm KE. </s> <s id="s.002651">Proba<lb></lb>tur autem DI majorem eſſe quàm IO; quia DI æqualis eſt <lb></lb>ipſi CI ex hypotheſi; punctum verò O eſt extra circulum CE, <lb></lb>quem linea EFO ſecat: ergo linea EF producta occurrit li<lb></lb>neæ IC citrà punctum C in S. </s> <s id="s.002652">Sed quoniam angulus BEF eſt <lb></lb>acutus, qui eſt illi deinceps DEO eſt obtuſus; ergo per 16. <lb></lb>lib.1. externus DOS multo magis eſt obtuſus: ergo per 19 <lb></lb>lib.1. major eſt IS quàm IO, ergo multò major eſt IC quàm <lb></lb>IO, hoc eſt ID major eſt quàm IO. </s> </p> <p type="main"> <s id="s.002653">Deinde angulus MCI major eſt angulo NID, majori enim <lb></lb>arcui MDI ille inſiſtit, hic autem minori ND ex 33. lib. 6: <lb></lb>triangula verò HIC & LDI rectangula æquales habent hy<lb></lb>pothenuſas, hoc eſt Radios CI & ID, ergo majoris anguli <lb></lb>HCI major eſt ſinus HI; minoris verò anguli LID minor eſt <lb></lb>ſinus LD. </s> <s id="s.002654">Igitur ex 8. lib.5. HI major ad KE, hoc eſt ad IF, <lb></lb>habet majorem Rationem quàm ad eandem KE habeat LD <lb></lb>minor: & eadem LD habet minorem Rationem ad DK ma<lb></lb>jorem quàm ad KE minorem: Ergo HI ad IF majorem habet <lb></lb>Rationem, quàm LD ad DK. <pb pagenum="348" xlink:href="017/01/364.jpg"></pb></s> </p> <p type="main"> <s id="s.002655"><emph type="center"></emph>CAPUT XII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002656"><emph type="center"></emph><emph type="italics"></emph>Præponderatio & Æquilibritas gravium fune <lb></lb>ſuſpenſorum conſideratur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002657">PRopoſitum eſt lib.2. capit. </s> <s id="s.002658">5. Experimentum, cujus hîc <lb></lb>ſymptomata explicanda, cauſam afferendo omninò conſo<lb></lb>nam iis, quæ ſæpiùs inculcata ſunt. </s> <s id="s.002659">Funiculi extremitatibus al<lb></lb>ligantur pondera prorsùs æqualia; tùm claviculis duobus à ſe <lb></lb>invicem aliquo intervallo disjunctis, ſed in eádem horizontali <lb></lb>lineâ conſtitutis (exquiſitè tamen, quoad ejus fieri poterit ro<lb></lb>tundis atque politis, ne ſuâ aſperitate motui impedimento ſint) <lb></lb>funiculus imponitur. </s> <s id="s.002660">Deinde tertium pondus aſſumitur duo<lb></lb>bus illis ſimul acceptis levius, aut ſingulis illis æquale, aut etiam <lb></lb>illis minus, & funiculo inter utrumque claviculum adnectitur: <lb></lb>hoc ſibi dimiſſum ita duobus illis ponderibus, quæ ob gravita<lb></lb>tis æqualitatem ſibi mutuo niſu obſiſtebant, ne moverentur, <lb></lb>prævalet, ut ipſum deſcendens vi ſuæ gravitatis cogat utrum<lb></lb>que illud aſcendere. </s> <s id="s.002661">Id quod admiratione carere non poteſt, <lb></lb>cum duo majora pondera, ſuum æqualem conatum ſingula vi<lb></lb>ciſſim elidentia, conjunctis viribus minori gravitati præſtare <lb></lb>non valeant. </s> </p> <p type="main"> <s id="s.002662">Funiculo CABD jungantur æquales gravitates C & D ex <lb></lb>claviculis A & B pendentes, quæ æqualiter deorſum conniten<lb></lb><figure id="id.017.01.364.1.jpg" xlink:href="017/01/364/1.jpg"></figure><lb></lb>tes, ſibique æqualiter repugnantes <lb></lb>ne aſcendant, quieſcunt. </s> <s id="s.002663">Adnecta<lb></lb>tur in E pondus: huic etiamſi mi<lb></lb>nori illæ gravitates C & D omnino <lb></lb>obſiſtere non poſſunt, quin ex E <lb></lb>deſcendat in F ex.gr. & funicu<lb></lb>lum trahens cogat illas aſcendere <lb></lb>C quidem in I, D verò in K. </s> <s id="s.002664">Qua<lb></lb>propter funiculo EBD æqualis eſt funiculus FBK, & funicu<lb></lb>lo EAC æqualis eſt funiculus FAI: cum autem rectæ BE & <lb></lb>BG æquales ſint (nam centro B, intervallo BE deſcriptus eſt <pb pagenum="349" xlink:href="017/01/365.jpg"></pb>arcus) his ablatis, BD æquatur ipſi FG plus BK; & demptâ <lb></lb>communi BK, remanet GF æqualis ipſi DK. </s> <s id="s.002665">Eadem ratione <lb></lb>HF oſtenditur æqualis ipſi CI. </s> <s id="s.002666">Eſt igitur menſura motús pon<lb></lb>derum C & D aſcendentium HFG, ponderis verò intermedij <lb></lb>deſcendentis EF. </s> <s id="s.002667">At ex prop.1. capitis ſuperioris Tangens EF <lb></lb>major eſt ſecantis BF exceſſu GF, item ſecantis AF exceſſu <lb></lb>HF: contingit autem aliquam Tangentem majorem eſſe utro<lb></lb>que exceſſu ſimul ſumpto: poteſt igitur gravitas minor velociùs <lb></lb>deſcendens præſtare utrique ponderi tardiùs aſcendenti. </s> </p> <p type="main"> <s id="s.002668">Quamdiu itaque ſpatium deſcendentis per Tangentem ma<lb></lb>jus eſt ſpatio aſcendentium, quod metitur exceſſus ſecantium, <lb></lb>ita ut Ratio motûs deſcendentis ad motum aſcendentium major <lb></lb>eſſe poſſit Ratione, quam habent pondera extrema ad pondus <lb></lb>intermedium; hoc minore illa majora præponderantur. </s> <s id="s.002669">Ubi ve<lb></lb>rò eò ventum ſit, ut jam neutra Ratio alteri præſtet, tunc pon<lb></lb>dera ſubſiſtunt, & quies eſt. </s> <s id="s.002670">Si demùm ponderi intermedio <lb></lb>pondus addatur, vel vis aliqua inferatur ponderis vicem ſubiens, <lb></lb>utique adhuc deſcendit, quia Ratio ponderum extremorum ad <lb></lb>pondus intermedium auctum facta eſt minor; ſed ſublato hoc <lb></lb>ponderis additamento, illa extrema majorem habent Rationem <lb></lb>ad pondus intermedium, quàm poſſit eſſe motuum reciprocè <lb></lb>ſumptorum Ratio; ac proinde illa deſcendentia hoc tantiſper <lb></lb>elevant, dum fiat Rationum æqualitas. </s> </p> <p type="main"> <s id="s.002671">Non eſt autem hîc opus ea, quæ uberiùs ſuperiore libro ex<lb></lb>plicata ſunt, replicare, videlicet, gravium reſiſtentiam, ne mo<lb></lb>veantur, non eſſe attendendam penès ipſam gravitatem dum<lb></lb>taxat, verùm etiam motús, qui ſitum ipſum atque poſitionem <lb></lb>conſequeretur, velocitate aut tarditate dimetiendam; hanc ve<lb></lb>rò unius tarditatem cum alterius velocitate comparari non poſ<lb></lb>ſe niſi ex longitudine ſpatiorum, quæ utrumque codem tempo<lb></lb>ris intervallo percurreret. </s> <s id="s.002672">Ex quo manifeſtâ conſequutione con<lb></lb>ficitur ſatis eſſe, ſi ſpatiorum inæqualitas aut æqualitas oſtenda<lb></lb>tur; ut præponderatio aut æquilibritas innoteſcat: ac propterea <lb></lb>ſatis eſt hîc ſecantium exceſſus cum Tangente comparare; hæc <lb></lb>enim ponderis intermedij, illi ponderum extremorum motum <lb></lb>definiunt. </s> </p> <p type="main"> <s id="s.002673">Quapropter animum in rem ipſam attentiùs intendentes ob<lb></lb>ſervamus deſcendentis ponderis intermedij funiculum BFA <pb pagenum="350" xlink:href="017/01/366.jpg"></pb>cum horizontali lineá BA angulos conſtituere ad B & A pri<lb></lb>mum quidem acutiſſimos, deinde majores & majores; ac <lb></lb>propterea Tangentis ad Exceſſum ſecantis Rationem ſemper mi<lb></lb>nui ex propoſ. </s> <s id="s.002674">3. ideóque tandem ad eam deveniri Rationem, <lb></lb>quæ non ſit major Ratione ponderum reciprocè ſumptorum. </s> <lb></lb> <s id="s.002675">Quid igitur mirum, ſi tandem fiat quies, ubi non eſt Ratio<lb></lb>num inæqualitas? </s> <s id="s.002676">Viciſſim autem quia ponderum certa eſt Ra<lb></lb>tio; certa eſt etiam Ratio Tangentis ad Exceſſum ſecantis certi <lb></lb>cujuſdam anguli; igitur ex eádem prop.3. minoris anguli Tan<lb></lb>gens ad Exceſſum ſuæ ſecantis majorem habet Rationem, quam <lb></lb>ſit Ratio ponderum reciprocè: ideóque pondus in E conſtitu<lb></lb>tum poſitionem habens, ex quâ aliquis major motus deorſum <lb></lb>conſequi poteſt, quàm aſcendant extrema pondera, deſcendit, <lb></lb>& ſuperat eorum reſiſtentiam. </s> <s id="s.002677">Sed quoniam ſuppoſita extre<lb></lb>mis ponderibus manu ita elevare ea poſſumus, ut pondus inter<lb></lb>medium deſcendens funiculumque intendens conſtituat ad B <lb></lb>& A angulos, quorum communis Tangens EF habeat ad Ex<lb></lb>ceſſum ſecantium HFG Rationem minorem, quàm ſit reci<lb></lb>procè Ratio ponderum extremorum ad pondus intermedium, <lb></lb>ſatis conſtat, cur illa extrema præponderent, cùm & plus gra<lb></lb>vitatis & majora momenta, hoc eſt propenſionem ad majorem <lb></lb>motum, obtineant. </s> <s id="s.002678">Quamvis enim ex prop.4. differentia inter <lb></lb>Tangentes duorum in eodem circulo arcuum inæqualium ma<lb></lb>jor ſemper ſit differentia, quæ inter eorumdem ſecantes inter<lb></lb>cedit; quia tamen ex prop.5. Ratio hæc ſemper fit minor, quò <lb></lb>anguli augentur, idcircò ſi Tangens ſit duobus circulis com<lb></lb>munis, fieri poteſt, ut utriuſque circuli ſecantium differentiæ <lb></lb>ſimul ſumptæ majores ſint ipsá Tangente, vel ſaltem Tangens <lb></lb>ad illas ſimul ſumptas eam habeat Rationem, quæ minor ſit Ra<lb></lb>tione ponderum reciprocè. </s> </p> <p type="main"> <s id="s.002679">Et ut veritas exemplis ante omnium oculos poſita nullum du<lb></lb>bitationi locum relinquat, data ſit Ratio extremorum ponde<lb></lb>rum ad pondus intermedium, & inquiratur Tangens ſimilem <lb></lb>Rationem habens ad utriuſque ſecantis Exceſſum: intelligatur <lb></lb>autem hîc facilitatis gratià punctum E omninò æqualiter <lb></lb>diſtans ab A & B ita, ut æquales etiam ſint ſecantium exceſſus <lb></lb>HF & GF. </s> <s id="s.002680">Et primò quidem ponatur pondus medium æqua<lb></lb>le ſingulis extremis. </s> <s id="s.002681">Eſt igitur quæſita Ratio dupla Tangentis <pb pagenum="351" xlink:href="017/01/367.jpg"></pb>EF ad Exceſſuum ſummam HFG, cujus ſummæ ſemiſſis eſt <lb></lb>GF, atque adeò Ratio Tangentis EF ad GF eſt quadrupla, <lb></lb>hoc eſt ut 4 ad 1. </s> <s id="s.002682">Ergo ex corollar. prop. 2. ut quadratum Ex<lb></lb>ceſsus ad differentiam inter quadrata Exceſsûs & Tangentis <lb></lb>(ſunt autem quadrata 1 & 16) hoc eſt ut 1 ad 15, ita exceſſus <lb></lb>ſecantis ad duplum Radij BE. </s> <s id="s.002683">Quare Exceſſus ſecantis ad Ra<lb></lb>dium BE eſt ut 1 ad 7 1/2. Poſito igitur Radio BE 100000, Ex<lb></lb>ceſſus ſecantis GF eſt 13333 1/3, & ejus quadrupla Tangens EF <lb></lb>53333 1/3 dat angulum EBF gr. 28. 4′. </s> <s id="s.002684">21″, cujus ſecans BF eſt <lb></lb>113333 1/3. Diſtantia AB ſtatuatur pedum quatuor, hoc eſt di<lb></lb>gitorum 64: eſt BE dig. </s> <s id="s.002685">32. Igitur ut BE 100000 ad GF <lb></lb>13333, ita BE dig. </s> <s id="s.002686">32. ad GF dig. </s> <s id="s.002687">4 1/4. & Tangens hujus Ex<lb></lb>ceſsus quadrupla erit deſcenſus EF dig. </s> <s id="s.002688">17, aſcenſus verò DK <lb></lb>aut CI dig. </s> <s id="s.002689">4 1/4 ſinguli, & ambo ſimul 8 1/2. In omnibus igitur <lb></lb>angulis minoribus angulo gr. 28. 4′. </s> <s id="s.002690">21″. </s> <s id="s.002691">Ratio Tangentis ad <lb></lb>Exceſſuum ſecantium ſummam major eſt Ratione duplâ, quæ eſt <lb></lb>ponderum Ratio, in angulis verò majoribus minor eſt Ratione <lb></lb>duplâ: ac propterea ibi pondus intermedium ſuperat extrema, <lb></lb>hic ſuperatur ab illis, & quieſcunt in invento angulo gr. 28. <lb></lb>4′. </s> <s id="s.002692">21″. </s> </p> <p type="main"> <s id="s.002693">Generaliter autem ut invenias, quantum aſcendere poſſint <lb></lb>extrema pondera vi ponderis medij deſcendentis, ſit nota Ra<lb></lb>tio ponderum: tùm minoris termini Rationis datæ ſemiſſem ac<lb></lb>cipe (quia unicus Exceſſus hîc ſumitur, & pondus medium <lb></lb>æquali intervallo diſtat ab A & B) & hujus ſemiſſis quadratum <lb></lb>deme ex quadrato termini majoris: Deinde fiat ut hæc quadra<lb></lb>torum differentia ad quadratum illius ſemiſſis, ita duplum Ra<lb></lb>dij, hoc eſt tota claviculorum diſtantia AB ad aliud, & erit Ex<lb></lb>ceſſus unius ſecantis, quæ eſt menſura aſcensûs æqualis pon<lb></lb>derum DK aut CI. </s> </p> <p type="main"> <s id="s.002694">Ponderum extremorum Ratio ſimul ſumptorum ad interme<lb></lb>dium ſit ex. </s> <s id="s.002695">gr. ut 7 ad 6: termini minoris 6 ſemiſſis eſt 3, cu<lb></lb>jus quadratum 9 ex 49 quadrato termini majoris 7 deme, & eſt <lb></lb>differentia 40. Diſtantia claviculorum A & B ſit digitorum 80; <lb></lb>fiat igitur ut 40 ad 9 ita 80 ad 18, & vi ponderis illius interme<lb></lb>dij poterunt extrema pondera aſcendere dig. </s> <s id="s.002696">18. Ut verò inno<lb></lb>teſcat, quantum deſcendat pondus medium, inter Exceſſum <pb pagenum="352" xlink:href="017/01/368.jpg"></pb>ſecantis 18, & 98 ſummam ſecantis & Radij, quære mediam <lb></lb>proportionalem, & ex prop.2. hæc eſt Tangens dig.42: dupli<lb></lb>catus autem 18 pro utroque exceſſu ſecantis dat 36, atque mo<lb></lb>tuum Ratio 42 ad 36 eadem eſt cum reciprocá Ratione ponde<lb></lb>rum 7 ad 6. Quòd ſi angulum EBF tantummodo quæris, quem <lb></lb>funiculus FB conſtituit cum horizontali AB, fiat ſimiliter ut <lb></lb>40 ad 9 ita Radij duplum 200000 ad 45000 Exceſſum Radio <lb></lb>addendum, ut habeatur ſecans 145000 gr.46. 24′. </s> </p> <p type="main"> <s id="s.002697">Ex his facilè intelligitur cur pro majore claviculorum A & B <lb></lb>intervallo pondus medium magis deſcendat, quia ſcilicet atten<lb></lb>denda eſt anguli magnitudo, ex quâ pendet Tangentis & ſe<lb></lb>cantis Ratio; ubi verò major eſt Radius, majorem quoque eſſe <lb></lb>ſimilis anguli Tangentem atque ſecantem manifeſtum eſt. </s> <lb></lb> <s id="s.002698">Quare ſi exiguum ſit pondus medium, & vix appareat, an ab <lb></lb>illo extrema pondera eleventur, atque dubitetur, an ideò ſo<lb></lb>lùm illud deſcendat, quia funiculum magis intendit; adhibe <lb></lb>longiorem funiculum, cui eadem pondera adnectas, & augea<lb></lb>tur, quantum opus fuerit, claviculorum A & B intervallum; <lb></lb>demum enim apparebit extremorum ponderum aſcendentium <lb></lb>motus: acutiſſimus ſcilicet angulus in majore circulo habet ſe<lb></lb>cantis Exceſſum ſuprà Radium faciliùs notabilem quàm in mi<lb></lb>nore. </s> <s id="s.002699">Sic vides poſito Radio habente unitatem cum ſeptem <lb></lb>cyphris, non inveniri Exceſſum ſecantis niſi gr.0. 1′. </s> <s id="s.002700">10. uni<lb></lb>tatem: at poſito Radio cum quindecim cyphris, habetur ejuſ<lb></lb>dem anguli ſecantis Exceſſus ſupra Radium partium 57585857: <lb></lb>immò habetur etiam unius ſecundi ſecans, cujus Exceſſus ſu<lb></lb>pra Radium eſt 11752. </s> </p> <p type="main"> <s id="s.002701">Hinc etiam deſines mirari, cur longiores funes aut catenæ <lb></lb>nullâ vi ita intendi poſſint, ut in lineâ horizonti parallelâ <lb></lb>rectam poſitionem habentes conſiſtant, ſed aliquantulum ſal<lb></lb>tem inflectantur; quia nimirum inſitum funi aut catenæ pon<lb></lb>dus idem præſtat, quod in hoc experimento pondus in medio <lb></lb>appenſum. </s> <s id="s.002702">Id quod nautæ non ignorantes ſæpius malunt uni <lb></lb>anchoræ funem duplo longiorem adnectere, quàm duabus an<lb></lb>choris ſimplici & ſubduplo fune inſtructis navem firmare: nô<lb></lb>runt ſiquidem longè majore vi opus eſſe ut funis longitudinem <lb></lb>habens ducentorum cubitorum intendatur, quàm ſi centum <lb></lb>tantummodo cubitorum longitudo eſſet; ac proinde undarum <pb pagenum="353" xlink:href="017/01/369.jpg"></pb>impetum longior funis faciliùs eludit, eóque minùs timendum <lb></lb>eſt, ne dirumpatur, quò difficiliùs intendi poteſt. </s> </p> <p type="main"> <s id="s.002703">Simile quiddam dicendum videtur, cùm longiorum priſma<lb></lb>tum aut cylindrorum extremitates ſubjectis fulcris totam longi<lb></lb>tudinem horizonti parallelam in aëre quaſi ſuſpenſam ſuſtinent; <lb></lb>ſuo enim pondere ſi non franguntur, ſaltem curvantur; id quod <lb></lb>brevioribus cylindris aut priſmatis non contingit. </s> <s id="s.002704">Quia vide<lb></lb>licet ex ipsá poſitione partes, quæ in mediá longitudine locum <lb></lb>obtinent, & quæ his proximæ ſunt, aptæ ſunt velociùs moveri <lb></lb>quàm remotiores: & quemadmodum pondus in medio poſitum <lb></lb>deſcendens vincit reſiſtentiam extremorum ponderum aſcen<lb></lb>dentium, ita vis harum partium mediarum ſuperat vim, quâ <lb></lb>partes invicem nectuntur, ac proinde diſtractæ flectuntur ſal<lb></lb>tem, & demum ſeparantur. </s> </p> <p type="main"> <s id="s.002705">Sed antequam planè ex animo effluat, unum hîc obſervan<lb></lb>dum (de quo fortaſſe maluiſſes initio præmoneri) aliud eſſe <lb></lb>quod ex naturæ inſtituto, aliud quod ex iis, quæ accidunt, con<lb></lb>tingit. </s> <s id="s.002706">Quæ hactenus diximus de Ratione motuum ſpectatis <lb></lb>ponderum gravitatibus, intelligenda ſunt, niſi quid interveniat, <lb></lb>quod legem hanc infringat; cujuſmodi eſt aliqua funiculi re<lb></lb>miſſio, vel minor intenſio, ita ut hic faciliùs à medio pondere <lb></lb>deſcendente adhuc intendatur, quàm extrema pondera eleven<lb></lb>tur; ubi enim eò devenerit pondus medium, ut intentus funi<lb></lb>culus cum lineá horizonti parallelá angulum faciat, cujus Tan<lb></lb>gens ad ſecantium Exceſſus Rationem habet reciprocam pon<lb></lb>derum, ibi ſubſiſtit, etiamſi extrema pondera elevata non ſue<lb></lb>rint niſi juxtâ menſuram differentiæ ſecantium duorum angu<lb></lb>lorum, ejus videlicet quem demum funiculus conſtituit, & ejus <lb></lb>qui funiculi remiſſionem ipſo motûs initio conſequitur: quia <lb></lb>ulterior deſcenſus ad ulteriorem aſcenſum non haberet majo<lb></lb>rem Rationem, ſed minorem Ratione ponderum reciprocè <lb></lb>ſumptorum. </s> <s id="s.002707">Quòd ſi valde inæqualia fuerint pondera, eveni<lb></lb>re poteſt totam vim deſcendendi, quam pondus medium habet, <lb></lb>abſumi in funiculo intendendo, nec quicquam virium ſupereſ<lb></lb>ſe ad extrema pondera attollenda. </s> </p> <p type="main"> <s id="s.002708">Húc etiam ſpectat impedimentum, quod ex funiculi clavi<lb></lb>culos terentis conflictu oritur; cùm enim deſcendentis ponde<lb></lb>ris medij momentum ſemper decreſcat, ut ex prop.5. conſtat, <pb pagenum="354" xlink:href="017/01/370.jpg"></pb>adeò extenuari poteſt, ut jam ſuperare non valeat extremorum <lb></lb>ponderum aſcendentium momenta aucta momento, quod ex <lb></lb>partium conflictu oritur; qui conflictus ſi non adeſſet, pergeret <lb></lb>illud adhuc deſcendendo. </s> <s id="s.002709">Propterea ſi claviculos ipſos con<lb></lb>gruentibus rotulis inſeras, adeò ut funiculus excavatæ abſidi <lb></lb>inſideat, longè majorem motum faciliúſque perfici videbis; <lb></lb>minùs enim rotula cum ſuo axe confligit, quàm funiculus <lb></lb>cum claviculo, ſi illum terat; & quidem quò major fuerit ro<lb></lb>tula, circa eundem axem faciliùs volvitur, minor ſiquidem <lb></lb>partium tritus ſit, ſi cætera omnia ſint paria. </s> <s id="s.002710">Simili modo ſi <lb></lb>pondus medium plus æquo per vim deprimas, faciliùs ſuum <lb></lb>in locum redibit adhibitis rotulis, quàm ſi funiculus clavicu<lb></lb>lis inſiſteret: quia pondera extrema ſuperare non valent & gra<lb></lb>vitatem ponderis medij & impedimentum, quod oritur ex ma<lb></lb>jori tritu funiculi & claviculorum, quàm rotularum & axium. </s> <lb></lb> <s id="s.002711">Obſervabis etiam adhibitis rotulis pondus medium ſibi re<lb></lb>lictum tanto impetu à lineâ horizonti parallelâ deſcendere, ut <lb></lb>ex concepto impetu fines ſuos tranſiliat, ac idcirco deſinente <lb></lb>impetu, quem in motu acquiſivit, iterum ſurſum trahi ab ex<lb></lb>tremis ponderibus, quæ ſicut minorem Rationem habebant ad <lb></lb>gravitatem ponderis medij auctam impetu acquiſito, ita ma<lb></lb>jorem Rationem habent ad eandem ſpoliatam illo impetu. </s> </p> <p type="main"> <s id="s.002712">Porrò hæc quæ hactenus de pondere in mediâ planè diſtan<lb></lb>tiâ inter claviculos aut rotulas conſtituto dicta ſunt, intelli<lb></lb>genda ſunt pariter de pondere claviculorum intervallum inæ<lb></lb>qualiter dividente, quod quidem ſpectat ad æquilibrium aut <lb></lb>præponderationem propter Rationum æqualitatem aut inæqua<lb></lb>litatem. </s> <s id="s.002713">Peculiare tamen aliquid obſervandum eſt, videlicet <lb></lb>aliquando contingere, ut hoc pondere medio deſcendente <lb></lb>pondus proximum aſcendat, remotum verò deſcendat, utró<lb></lb>que autem pondere extremo aſcendente magis aſcendere quod <lb></lb>proximum eſt, minùs quod remotum. </s> <s id="s.002714">Hujus inæqualis aſcen<lb></lb>sûs (ſi pondus medium rectâ ad perpendiculum deſcendat) <lb></lb>cauſa in promptu eſt ex iis, quæ prop. 8. indicata ſunt, nam <lb></lb>ejuſdem Tangentis quadrato æqualia ſunt, atque adeò & in<lb></lb>ter ſe æqualia, rectangula, quæ fiunt ſub Exceſſu ſecantis & <lb></lb>aggregato ſecantis & Radij: ſunt igitur ex 14. lib.6. Exceſſus <lb></lb>ſecantium reciprocè in Ratione aggregatorum ſecantis & Ra-<pb pagenum="355" xlink:href="017/01/371.jpg"></pb>dij: quapropter ubi major eſt Radius & ſecans, ibi minor eſt <lb></lb>ſecantis Exceſſus, hoc eſt remoti ponderis aſcenſus, & <lb></lb>contra ubi minor eſt Radius & ſecans, ibi major eſt ſecan<lb></lb>tis Exceſſus, hoc eſt ponderis proximi aſcenſus. </s> </p> <p type="main"> <s id="s.002715">Cur autem aliquando proximum pondus aſcendat, atque <lb></lb>remotum deſcendat, quando nimirum valde inæquales ſunt <lb></lb>ponderis medij à claviculis diſtantiæ, hinc fit, quod idem pon<lb></lb>dus ex longiore funiculo majorem habet vim deſcendendi, quàm <lb></lb>ex breviore; cui majori momento cum reſiſtere debeat pondus <lb></lb>proximum, faciliùs cedit deſcendenti, atque adeò non rectâ de<lb></lb>orſum tendit pondus medium, ſed obliquè, accedendo ad pon<lb></lb>dus remotum, quod propterea deſcendit. </s> <s id="s.002716">Sic poſitum pondus in <lb></lb>E valde inæqualia habet mo<lb></lb><figure id="id.017.01.371.1.jpg" xlink:href="017/01/371/1.jpg"></figure><lb></lb>menta comparatum cum ex<lb></lb>tremis ponderibus D & C, <lb></lb>quæ in punctis B & A exer<lb></lb>cent ſuas vires adversùs pon<lb></lb>dus medium; quod ubi infrà <lb></lb>horizontalem AB <expan abbr="deſcẽderit">deſcenderit</expan>, <lb></lb>illico inæquales angulos cum <lb></lb>horizontali linea AB conſti<lb></lb>tuit inflexus funiculus; ut ſi intelligatur pondus ex E veniſſe in <lb></lb>F, angulus FBA major eſt angulo FAB ex 18. lib.1. quia latus <lb></lb>AF eſt majus latere FB. </s> <s id="s.002717">Igitur angulus FBD, quem funiculus <lb></lb>inflexus FB facit cum perpendiculari BD minor eſt angulo <lb></lb>FAC; ergo ex dictis lib.1. cap. 15. pondus in F minora habet <lb></lb>momenta ad deſcendendum versùs perpendiculum BD, quàm <lb></lb>ad deſcendendum versùs AC, & quidem duplici titulo, ſcilicet <lb></lb>anguli FBD minoris, & funiculi FB brevioris. </s> <s id="s.002718">Cum itaque pon<lb></lb>dus illicò ac ex E deſcendit magis pronum ſit ad deſcendendum <lb></lb>versùs perpendiculum AC, non per rectam EF <expan abbr="perpendicularẽ">perpendicularem</expan> <lb></lb>deſcendit; ſed obliquè per lineam EG, ita ut funiculus GA bre<lb></lb>vior ſit funiculo EA, ac propterea cedit ponderi C deorſum tra<lb></lb>henti. </s> <s id="s.002719">Et quia funiculus GB longior eſt funiculo FB, & multo <lb></lb>magis funiculo EB, propterea aliquando contingere poteſt pon<lb></lb>dus D magis aſcendere, quàm aſcenderet, ſi E fuiſſet planè in <lb></lb>mediâ diſtantiâ inter A & B. </s> <s id="s.002720">Ex quo etiam fit deſcenſum per<lb></lb>pendicularem ponderis medij minorem eſſe; nam punctum G <pb pagenum="356" xlink:href="017/01/372.jpg"></pb>minùs diſtat ab horizontali AB, quàm punctum F, & tamen <lb></lb>major eſt differentia inter EB & GB; ideò minor eſt Ratio IG <lb></lb>ad Exceſſum GL, quàm EF ad Exceſſum FO. </s> </p> <p type="main"> <s id="s.002721">Hanc momentorum inæqualitatem perſpicies, ſi pondus me<lb></lb>dium ſingulis extremis æquale inter claviculos æqualiter conſti<lb></lb>tutum deſcendere permittas, ſuóque in loco <expan abbr="cõſiſtere">conſiſtere</expan>; cùm enim <lb></lb>æqualis ſit funiculorum illud ſuſtinentium longitudo, & æqua<lb></lb>les faciat angulos tùm cum horizontali, tùm cum perpendicula<lb></lb>ribus, contra utrumque extremum æqualibus momentis pugnat, <lb></lb>ac rectâ ad perpendiculum deſcendit. </s> <s id="s.002722">Tum alteri extremorum <lb></lb>aliquid adde ponderis; hoc utique deſcendens ſecum rapit & <lb></lb>ponderis medij & reliqui extremi gravitates, quas cogit aſcen<lb></lb>dere, donec ea fiat funiculorum inæqualitas, ut momenta, quæ <lb></lb>pondus medium habet ad deſcendendum ratione diſtantiæ á cla<lb></lb>viculo remotiori, jam ſuperari non valeant à pondere illo ex<lb></lb>tremo cum ſuo additamento. </s> </p> <p type="main"> <s id="s.002723">Nec diſpar eſt philoſophandi methodus, cum funiculi extre<lb></lb>mitas alterutri claviculo alligatur, unico pondere in alterâ extre<lb></lb>mitate pendente ex altero claviculo: pondus enim inter clavi<lb></lb>culos funiculo adnexum, quia velociùs movetur deſcendendo, <lb></lb>quàm reliquum pondus aſcendendo, ſuperare poteſt illius gra<lb></lb><figure id="id.017.01.372.1.jpg" xlink:href="017/01/372/1.jpg"></figure><lb></lb>vitatem. </s> <s id="s.002724">Sit enim funiculus alligatus <lb></lb>in A, & pendeat pondus D ex clavicu<lb></lb>lo B: pondus (utrùm æquale ſit, an ma<lb></lb>jus, an minus, parum refert) adnectatur <lb></lb>in C: utique deſcendens deſcribit ar<lb></lb>cum CI circa centrum A; eſt autem <lb></lb>funiculus IB longior quàm CB ex 8. <lb></lb>lib.3. Sed quoniam duo latera BC & <lb></lb>CI ſimul majora ſunt reliquo latere IB ex 20.lib. 1. major eſt <lb></lb>recta CI, & multo magis arcus CI ſpatium quod percurrit pon<lb></lb>dus medium deſcendens) quàm IE Exceſſus lateris IB ſupra <lb></lb>CB, hoc eſt menſura motûs ponderis D aſcendentis. </s> <s id="s.002725">Quia verò <lb></lb>ponderis medij deſcendentis circa centrum A momenta decreſ<lb></lb>cunt ex dictis lib.1. cap.15. circa centrum autem B decreſcunt <lb></lb>quidem, quia minor fit angulus declinationis à perpendiculo <lb></lb>GBD, ſed decrementum hoc temperatur, quia momenta creſ<lb></lb>cunt ratione longitudinis funiculi, quæ ſemper augetur ex 8. <pb pagenum="357" xlink:href="017/01/373.jpg"></pb>lib.3.propterea ad momentorum æqualitatem venit, ubi demùm <lb></lb>quieſcit. </s> <s id="s.002726">Quantum autem deſcendat, pendet ex ipſius ponderis <lb></lb>gravitate abſolutâ ſive majori, ſive minori, ſive æquali compara<lb></lb>tâ cum pondere D, & ex diſtantiâ à centro A: ſi enim valde pro<lb></lb>pinquum ſit centro, parùm deſcendit, etiamſi cæteroqui gravius <lb></lb>ſit; & ſi per vim adhuc deprimatur, ut veniat in G, ceſſante vi <lb></lb>extrinſecùs illatâ pondus D deſcendens illud iterum attollit. </s> </p> <p type="main"> <s id="s.002727">Cave tamen ponderis medij deſcendentis momenta metiaris <lb></lb>ex arcu, quem deſcribit, ſed potiùs illa definienda ſunt ex ipſo <lb></lb>deſcenſu perpendiculari, cum moveatur vi ſuæ gravitatis. </s> <s id="s.002728">Quo<lb></lb>niam verò æqualibus arcubus deſcriptis non reſpondent paria <lb></lb>perpendicularium linearum incrementa ex prop.10.ſed ſemper <lb></lb>minora fiunt; contra verò incrementa ſecantium augentur, hinc <lb></lb>eſt deveniri ad momentorum æqualitatem, ita ut pondus me<lb></lb>dium gravius pondere extremo aptum ſit minùs deſcendere <lb></lb>quàm illud aſcenderet ſecundùm <expan abbr="reciprocã">reciprocam</expan> <expan abbr="Rationẽ">Rationem</expan> gravitatum. </s> </p> <p type="main"> <s id="s.002729">Hinc elici poteſt compendium aliquod in attollendo ponde<lb></lb>re cæteroqui valde gravi; ſit enim pondus P attollendum fune <lb></lb>circumducto rotulæ A: quò longior <lb></lb><figure id="id.017.01.373.1.jpg" xlink:href="017/01/373/1.jpg"></figure><lb></lb>funis poteſt alligari in B, eò faciliùs <lb></lb>ſequetur motus, ſi ad ſervandam in <lb></lb>mediâ diſtantiâ poſitionem poten<lb></lb>tiæ moventis ſimplicem trochleam <lb></lb>aut annulum in C addideris, cui in<lb></lb>ſeratur funis BA: nam applicata po<lb></lb>tentia in D deorſum trahens multo <lb></lb>faciliùs attollet pondus P, quàm ſi <lb></lb>arreptâ funis extremitate B idem <lb></lb>onus elevare conaretur ad eam al<lb></lb>titudinem, ad quam attolleretur à pondere in C adnexo, quod <lb></lb>æqualibus viribus præditum eſſet cum potentiâ in D trahente. </s> <lb></lb> <s id="s.002730">Ubi jam ſit attollendi difficultas, ſuppone aliquid ponderi P, cui <lb></lb>illud incumbat, nec contra funem conetur: tùm iterum funem <lb></lb>intende, & alliga in B, ut ſit AB horizonti parallelus, & ite<lb></lb>rum in D deorſum trahens priorem facilitatem experieris: id <lb></lb>quod toties iterari poterit, quoties opus fuerit. </s> </p> <p type="main"> <s id="s.002731">Ex his omnibus, quæ toto hoc capite diſputata ſunt, mani<lb></lb>feſtum eſt non referendas eſſe machinarum vires ad Rationes <pb pagenum="358" xlink:href="017/01/374.jpg"></pb>circuli aut Vectis, quandoquidem hic videmus minori pondere <lb></lb>majus pondus moveri abſque ullo motu circulari. <lb></lb></s> </p> <p type="main"> <s id="s.002732"><emph type="center"></emph>CAPUT XIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002733"><emph type="center"></emph><emph type="italics"></emph>An aliqua ſit Libræ obliquæ utilitas.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002734">LIbram obliquam vocat Simon Stevinus Static. lib.3.prop.6. <lb></lb>rotulam L funiculi in excavatâ apſide capacem pondus <lb></lb><figure id="id.017.01.374.1.jpg" xlink:href="017/01/374/1.jpg"></figure><lb></lb>cum æquipondio jungentis, & in ſuo lo<lb></lb>culamento facillimè verſatilem, cujus par<lb></lb>ticula extans E poſſit pro re natâ eximi, at<lb></lb>que iterum inſeri foraminibus, quibus <lb></lb>exactè congruat, tigilli P firmè infixi pedi <lb></lb>ſatis gravi, ne valeat à ponderis examinan<lb></lb>di gravitate rapi & inclinari. </s> <s id="s.002735">Hanc ille ad <lb></lb>ponderum obliquorum momenta inveſti<lb></lb>ganda utilem exiſtimavit, eamque ſæpiùs <lb></lb>ingerit Static.lib.1.prop.19.& ſeqq quam<lb></lb>vis ſemper illam cum elevante directo conjunctam adhibeat. </s> <lb></lb> <s id="s.002736">Propterea, an aliquid ex illâ emolumenti, ſi ſolitaria adhibeatur, <lb></lb>capere poſſimus in ponderum momentis inveſtigandis ſivè ſuſ<lb></lb>penſorum, ſivè in plano inclinato jacentium, hîc examinare ope<lb></lb>ræ pretium fuerit; nam & à ſuperioris capitis argumento non <lb></lb>aliena videtur eſſe præſens diſputatio. </s> </p> <p type="main"> <s id="s.002737">Antequam verò rem aggrediar, monendum te cenſeo, Ami<lb></lb>ce Lector, opportunius accidere, ſi tigilli perforati loco cylin<lb></lb>drum in cochleam efformatum ſtatueris, cui congruat in ſimi<lb></lb>lem helicem excavatum foramen S in rotulæ L loculamento: <lb></lb>ſic enim faciliùs elevabitur aut deprimetur rotula, prout exiget <lb></lb>ipſius ponderis poſitio. </s> </p> <p type="main"> <s id="s.002738">Dupliciter itaque contingere poteſt ponderis obliquitas, ſeu <lb></lb>quia ſuſpenſum non in codem perpendiculo, in quo eſt punctum <lb></lb>ſuſpenſionis, habet centrum ſuæ gravitatis, ſeu quia plano incli<lb></lb>nato incumbit; utroque enim in caſu momenta habet ad deſcen<lb></lb>dendum, quæ communi librâ aut ſtaterâ veſtigare utique non <lb></lb>poſſumus: an libræ obliquæ ope id aſſequemur? </s> <s id="s.002739">Et primò qui<lb></lb>dem ſi pondus examinandum è funiculo ſuſpenſum fuerit, ejuſ<lb></lb>que momenta pro variá declinatione à ſuo perpendiculo inqui-<pb pagenum="359" xlink:href="017/01/375.jpg"></pb>rantur, res manet incerta, ſi in praxim deducatur, quia plurimum <lb></lb>intereſt, quâ obliquitate inclinetur, atque à ſuo perpendiculo de<lb></lb>flectat funiculus libræ obliquæ, ſi maxime cum diversa obliqui<lb></lb>tate jungatur diſpar funiculi illius longitudo. </s> <s id="s.002740">Nam ex A ſuſ<lb></lb>pendatur pondus B habens BAC <expan abbr="angulũ">angulum</expan> <lb></lb><figure id="id.017.01.375.1.jpg" xlink:href="017/01/375/1.jpg"></figure><lb></lb>declinationis à ſuo perpendiculo AC; & <lb></lb>primùm ſit libra obliqua D, itaut <expan abbr="æquipẽ-dium">æquipen<lb></lb>dium</expan> E retineat pondus B in eodem ſitu: <lb></lb>deinde transferatur libra obliqua ex D in <lb></lb>F, & æquipondium G retineat pariter in <lb></lb>codem ſitu pondus B cum declinationis <lb></lb>angulo BAC. </s> <s id="s.002741">Si in eâdem rectâ lineâ ſint <lb></lb>BDF, nulla eſt momentorum inæqualitas, <lb></lb>quamvis diſparitas intercedat inter funi<lb></lb>culi longitudines BD, & BF. </s> <s id="s.002742">Sin autem F <lb></lb>paulo ſuperior fuerit aut paulo inferior, jam BD & BF angulum <lb></lb>in B conſtituunt, & momenta mutantur. </s> <s id="s.002743">Quoniam enim IE & <lb></lb>HG <expan abbr="perpẽdiculares">perpendiculares</expan> ſunt parallelæ, in eaſque incidit recta BDF <lb></lb>producta, anguli BIE, & BHG ſunt æquales per 29.lib.1. at verò <lb></lb>ſi libra obliqua F non planè in eâdem rectâ lineâ, ſed ſuperiore <lb></lb>loco collocaretur, angulum conſtitueret cum perpendiculo HG <lb></lb>acutiorem, & inferiùs poſita angulum efficeret minùs acutum. </s> <lb></lb> <s id="s.002744">Quare pondus B, quò acutior eſt angulus, & magis accedit ad <lb></lb><expan abbr="perpendiculũ">perpendiculum</expan> FG, eò etiam magis conatur contra F, & ad æqui<lb></lb>librium exigit majorem gravitatem in G, quàm cum angulus eſt <lb></lb>minù, acutus. </s> <s id="s.002745">Id quod experimento allato ſuperiori capite ma<lb></lb>nifeſtum ſit; ſi enim funiculi extremitates jungant pondera inæ<lb></lb>qualia, pondus intermedium magis accedit ad perpendiculum, in <lb></lb>quo eſt major gravitas. </s> <s id="s.002746">Hinc quia valde incertum eſt in praxi <lb></lb>utrùm B, D, & F in eâdem ſint rectá lineâ, propterea <expan abbr="etiã">etiam</expan> <expan abbr="incertũ">incertum</expan> <lb></lb>erit ex gravitate ponderis G inferre, quanta ſint ponderis B <expan abbr="mo-mẽta">mo<lb></lb>menta</expan> cum declinatione BAC: Niſi fortè <expan abbr="duplicẽ">duplicem</expan> inſtituas libræ <lb></lb>obliquæ poſitionem in D, & in F atque <expan abbr="eodẽ">eodem</expan> ſemper <expan abbr="põdere">pondere</expan> tam <lb></lb>in E quàm in G retineatur pondus B in poſitione <expan abbr="câdẽ">eâdem</expan>. </s> <s id="s.002747">Ita <expan abbr="tamẽ">tamen</expan> <lb></lb>collocanda eſt libra obliqua, ut angulus ABD ſit rectus; ex illo <lb></lb>quippe æſtimatur <expan abbr="planũ">planum</expan> <expan abbr="inclinatũ">inclinatum</expan>, in quo pondus B conatur deſ<lb></lb>cendere, ut <expan abbr="dictũ">dictum</expan> eſt lib.1.cap 15.alioquin ſi acutus fuerit aut ob<lb></lb>tuſus ille angulus, quamvis in eâdem declinatione BAC reti<lb></lb>neatur, valde inæqualia apparebunt momenta. </s> <s id="s.002748">Quis autem de <pb pagenum="360" xlink:href="017/01/376.jpg"></pb>anguli illius rectitudine certus fuerit? </s> <s id="s.002749">cùm maximè rectam DB <lb></lb>oporteat ad perpendiculum inſiſtere lineæ jungenti punctum A <lb></lb>ſuſpenſionis cum centro gravitatis ponderis B. </s> <s id="s.002750">Ex pondere ita<lb></lb>que, quod eſt in E, aut in G, nemo poteſt certò definire mo<lb></lb>menta ponderis B ſuſpenſi. </s> </p> <p type="main"> <s id="s.002751">At ſi dato quopiam plano inclinato jaceat pondus, veliſque li<lb></lb>brâ hujuſmodi obliquâ explorare, quanta habeat pro eâ plani in<lb></lb>clinatione ad deſcendendum momenta, ego ſanè nihil certi affir<lb></lb>mare auderem; quippè qui ſemper incertus <expan abbr="hærerẽ">hærerem</expan>, an æquipon<lb></lb>dium libræ obliquæ indicaret ipſa <expan abbr="momẽta">momenta</expan> ponderis in plano in<lb></lb>clinato pro ratione inclinationis; nam plani ſubjecti non omnino <lb></lb>lubrica ſuperficies, & ponderis illi incumbentis aſperitas impe<lb></lb>dientes motum, non nihil detrahunt momenti ad <expan abbr="deſcendendũ">deſcendendum</expan>. </s> <lb></lb> <s id="s.002752">Cum verò pro diversa inclinatione <expan abbr="planũ">planum</expan> inæqualiter prematur <lb></lb>ab inſiſtente <expan abbr="põdere">pondere</expan>, adhuc eadem <expan abbr="ſuperficierũ">ſuperficierum</expan> ſe <expan abbr="contingentiũ">contingentium</expan> <lb></lb>aſperitas magis obſiſtit motui, quò major eſt plani inclinatio de<lb></lb>clinans à perpendiculo. </s> <s id="s.002753">Quare adhuc magis incerta eſſent mo<lb></lb>menta, quæ ab æquipondio libræ obliquæ indicarentur. </s> </p> <p type="main"> <s id="s.002754">Nihil aliud itaque commodi hinc ſperari poteſt præter <expan abbr="notitiã">notitiam</expan> <lb></lb>momenti, quod planorum aſperitas detrahit <expan abbr="momẽto">momento</expan> deſcenden<lb></lb>di. </s> <s id="s.002755">Si enim nota ſit ponderis dati gravitas abſoluta, & plani incli<lb></lb>natio innotuerit, videlicet angulus, quem planum <expan abbr="inclinatũ">inclinatum</expan> cum <lb></lb>plano horizontali conſtituit, fiat ut Radius ad Sinum noti anguli <lb></lb>inclinationis, ita gravitas abſoluta dati ponderis ad <expan abbr="momẽta">momenta</expan>, quæ <lb></lb>habet in plano inclinato: Tum librâ obliquâ exploretur, quanto <lb></lb>æquipondio opus ſit ad <expan abbr="retinẽdum">retinendum</expan> pondus in plano inclinato, ne <lb></lb>deorſum labatur: nam differentia inter gravitatem æquipondij, & <lb></lb>momenta inventa pro tali inclinatione indicabit, quantum impe<lb></lb>dimenti oriatur ex <expan abbr="planorũ">planorum</expan> ſe contingentium aſperitate, ſi æqui<lb></lb>pondij gravitas minor ſit momentis, quæ ab hujuſmodi inclina<lb></lb>tione exiguntur. </s> <s id="s.002756">Sic ex.gr. ſit ponderis dati abſoluta gravitas un<lb></lb>ciarum 30, inclinationis angulus dati plani cum plano <expan abbr="horizõtali">horizontali</expan> <lb></lb>ſit gr.60.fiat ut 10000 Radius ad 86603 Sinum gr.60.ita 30 ad <lb></lb>25.98. Si applicata libra obliqua æquipondium habeat ſolùm <lb></lb>unc. 24, manifeſtum eſt à planorum aſperitate detrahi momenti <lb></lb>partem ferè decimam tertiam, cùm deſint juſto æquipondio ferè <lb></lb>unciæ 2. Verùm & hîc obſervandum, opus eſſe funiculi, à quo <lb></lb>pondus retinetur, paralleliſmum cum plano inclinato, prout ex <lb></lb>iis, quæ de obliquis tractionibus lib. 1. cap.16. dicta ſunt, ſatis <lb></lb>conſtat. </s> </p> <pb pagenum="361" xlink:href="017/01/377.jpg"></pb> <figure id="id.017.01.377.1.jpg" xlink:href="017/01/377/1.jpg"></figure> <p type="main"> <s id="s.002757"><emph type="center"></emph>MECHANICORUM<emph.end type="center"></emph.end><emph type="center"></emph>LIBER QUARTUS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002758"><emph type="center"></emph><emph type="italics"></emph>De Vecte.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002759">HACTENUS de inſtrumentis ad movenda pondera <lb></lb>idoneis nihil, niſi fortaſsè obiter, dictum eſt: Jam <lb></lb>ad illa explicanda accedimus, quibus veteres facul<lb></lb>tatibus nomen indiderunt. </s> <s id="s.002760">Quamvis autem in <lb></lb>quinque facultatibus enumerandis primum locum <lb></lb>Vecti Pappus lib. 8. Collect. Math. non tribuat, placuit tamen <lb></lb>de Vecte ante cæteras facultates diſſerere, eſt ſiquidem paratu <lb></lb>facillimus, & ad ſubitum uſum promptiſſimus, atque cenſeri <lb></lb>poteſt, ut idem Pappus loquitur, <emph type="italics"></emph>fortaſſe præmeditatio motús cir<lb></lb>ca excedentia pondera: ſtatuentes enim quidam magna pondera mo<lb></lb>vere [quoniam primùm à terrâ attollere oportet, anſas autem non <lb></lb>habebant) quòd omnes partes baſis ipſius ponderis ſolo incumberent, <lb></lb>paulum ſuffodientes, & ligni longi extremitatem ſubjicientes ſub <lb></lb>onus, adducebant ex alterâ extremitate, ſupponentes ligno propè <lb></lb>ipſum onus lapidem, qui Hypomochlium appellatur. </s> <s id="s.002761">Cúmque illis vi<lb></lb>ſus eſſet hic motus valde facilis, exiſimaverunt fieri peſſe, ut hoc <lb></lb>pacio magna pondera moveretur. </s> <s id="s.002762">Vocatur autem tale lignum Vectis, <lb></lb>ſive quadratum ſit, ſive rotundum, & quanto propinquius oneri poni<lb></lb>tur hypomochlium, tanto faciliùs pondus movetur.<emph.end type="italics"></emph.end> Hæc illo vectis <lb></lb>ortum & procreationem quodammodo indigitans. </s> </p> <p type="main"> <s id="s.002763">Contingere quidem poteſt, ut Vecte aliquando utamur ad <lb></lb>ſuſtinendum ingens pondus, non autem ad movendum, adeò <lb></lb>ut potentia exigua ſuſtinens, in alterâ vectis extremitate poſita. <pb pagenum="362" xlink:href="017/01/378.jpg"></pb>habeat rationem æquipondij retinentis pondus in oppoſitá ex<lb></lb>tremitate collocatum: & tunc locum habet Ariſtotelis ſenten<lb></lb>tia Mechan. quæ ſt.3.dicentis, <emph type="italics"></emph>Ipſe vectis eſt in causá libra exiſtens, <lb></lb>ſpartum inferne habens, in inæqualia diviſa; hypomoclion enim eſt <lb></lb>ſpartum, ambo namque ſunt ut centrum.<emph.end type="italics"></emph.end></s> <s id="s.002764"> Verùm cùm propriè, & <lb></lb>preſsè tunc facultas eſſe non videatur, neque exerceat munus <lb></lb>vectis, quia non movet, ſed ſit quaſi jugum ſtateræ; fruſtra <lb></lb>Vectis quâ vectis eſt, ad libram revocatur: præſertim cùm ali<lb></lb>quod vectis genus ſit, in quo nullum libræ veſtigium depre<lb></lb>hendi poteſt, etiamſi pondus cæteroqui ruiturum ſuſtineat; ſi <lb></lb>nimirum pondus ipſum inter vectis extremitates conſtitutum <lb></lb>ſuſtineatur, aut potentia ipſa ſuſtentans medium locum occu<lb></lb>pet inter pondus & hypomochlium, ut infra dicetur. </s> <s id="s.002765">Quid <lb></lb>enim pariter non revocetur libra aut ſtatera ad Vectem, ſi ex <lb></lb>altera jugi extremitate pondus addatur, quod ad oppoſitum <lb></lb>pondus majorem habeat Rationem, quàm libræ, aut ſtateræ <lb></lb>brachia reciprocè ſumpta? </s> <s id="s.002766">tunc enim (quaſi ſtateræ aut libræ <lb></lb>centrum motûs eſſet hypomochlium) ſequitur motus prout ex <lb></lb>vecte. </s> <s id="s.002767">Quemadmodum igitur libra aut ſtatera ad ponderum <lb></lb>æquilibrium inſtitute, non verò ad eorum motum, libræ aut <lb></lb>ſtateræ munus non exercent in motu, quâ motus eſt; ita pari<lb></lb>ter vectis hypomochlium inter extremitates habens non exer<lb></lb>cet munus vectis in quiete: alioquin & vectis ad libram, & vi<lb></lb>ciſſim libra ad vectem abſurdo circulo revocaretur. </s> <s id="s.002768">Adde verò <lb></lb>genus hoc vectis hypomochlium inter extremitates habentis, ſi <lb></lb>adhibeatur ad onus in plano horizontali movendum, non verò <lb></lb>ad illud ſuſtentandum, nihil habere commercij cum librâ, onus <lb></lb>ſi quidem nullam exercet vim ſuæ gravitatis adversùs ipſum <lb></lb>vectem, nam ceſſante potentia onus illicò quieſcit; at in libra <lb></lb>ſublato æquipondio pondus deſcendit. </s> <s id="s.002769">Quid ſi vecte utamur <lb></lb>ad corpus leve infra aquam deprimendum? </s> <s id="s.002770">an erit illa libra in<lb></lb>verſa? </s> <s id="s.002771">Non igitur me fruſtra conficiam labore enitens rationes <lb></lb>libræ in vecte recognoſcere, ſed ipſum per ſe conſiderans, quæ <lb></lb>opportuniora cenſuero, diſputabo. <pb pagenum="363" xlink:href="017/01/379.jpg"></pb> </s> </p> <p type="main"> <s id="s.002772"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002773"><emph type="center"></emph><emph type="italics"></emph>Vectis forma, & vires explicantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002774">VEctis ob id ipſum quia Vectis eſt & Facultas mechanica, <lb></lb>longitudo quædam eſt, in qua tria puncta aſſignantur, pri<lb></lb>mum Potentiæ moventi, alterum Ponderi movendo, tertium <lb></lb>Fulcro, ſeu Hypomochlio, cui innixus vectis tanquam ex <lb></lb>centro duos arcus deſcribens duplicem motum definit, Poten<lb></lb>tiæ videlicet & Ponderis, pro variâ illorum ab codem fulcro <lb></lb>diſtantiâ. </s> <s id="s.002775">Hinc quia tripliciter in hac longitudine tria hæc <lb></lb>puncta diſponi poſſunt, tria oriuntur vectis genera. </s> <s id="s.002776">Primum <lb></lb>eſt vectis genus, cùm extremi<lb></lb>tates occupantur à Potentia A <lb></lb><figure id="id.017.01.379.1.jpg" xlink:href="017/01/379/1.jpg"></figure><lb></lb>& Pondere B, medius locus <lb></lb>Hypomochlio C cedit. </s> <s id="s.002777">Secun<lb></lb>dum genus eſt, cum extremi<lb></lb>tati alteri F innititur vectis, al<lb></lb>teri Potentia D adjungitur, & <lb></lb>inter utramque extremitatem <lb></lb>collocatur Pondus E. </s> <s id="s.002778">Tertium <lb></lb>genus eſt, cum Potentia & Pondus loca ſecundi generis invi<lb></lb>cem permutant, Potentia G videlicet in medio, Pondus H in <lb></lb>extremitate conſtituitur, manente alterâ extremitate I tan<lb></lb>quam motuum centro. </s> <s id="s.002779">Cum itaque nulla alia fieri poſſit trium <lb></lb>hujuſmodi punctorum diverſa diſpoſitio, patet tria ſolùm <lb></lb>Vectis genera excogitari potuiſſe. </s> <s id="s.002780">quod enim quartum Vectis <lb></lb>genus, ſcilicet inflexum RSV comminiſci quibuſdam placuit, <lb></lb>omnino ineptum eſt, quippe quod à primo genere nihil differt, <lb></lb>niſi quia, loco ſubjecti fulcri, adnexum habet hypomochlium <lb></lb>inter extremitates conſtitutum in S, ubi ſinuatur in angulum, <lb></lb>cui in motu innititur. </s> </p> <p type="main"> <s id="s.002781">Quemadmodum autem inter hæc tria Vectis genera diſſimi<lb></lb>litudo, ita non modica inter eorum vires diſcrepantia interce-<pb pagenum="364" xlink:href="017/01/380.jpg"></pb>dit. </s> <s id="s.002782">Primum enim genus, ſi ab hypomochlio inæqualiter di<lb></lb>vidatur longitudo vectis, ut ab eo plus diſtet Potentia, quàm <lb></lb>Pondus, juvat Potentiam; ſecus verò, ſi Potentia & Pondus <lb></lb>æqualibus intervallis ab hypomochlio abſint, aut propior ſit <lb></lb>Potentia quàm Pondus; Potentiæ etenim tunc vectis vel nihil <lb></lb>affert adjumenti, vel plurimum detrimenti. </s> <s id="s.002783">Secundum genus <lb></lb>Potentiæ laborem ſemper minuit, Tertium ſemper auget. </s> <s id="s.002784">Quo<lb></lb>nam id pacto contingat, manifeſtum fiet, ſi vectis vires unde <lb></lb>ortum habeant, aperiamus. </s> </p> <p type="main"> <s id="s.002785">Certum eſt fieri non poſſe, ut pondus aliquod per vim mo<lb></lb>veatur, niſi potentiæ moventis virtus ſuperet ponderis reſiſten<lb></lb>tiam; ſi enim pari conatu confligerent, anceps eſſet victoria, <lb></lb>& nullus eſſet motus; multo minùs à potentiâ infirmiore, quàm <lb></lb>par ſit, vinci poterit innata ponderis propenſio. </s> <s id="s.002786">Hoc igitur <lb></lb>ipſo quod motus efficitur, argumento eſt potentiæ virtutem re<lb></lb>ſiſtentiâ ponderis eſſe majorem: Quod verò pondus eodem <lb></lb>temporis intervallo plus ſpatij aut minus decurrat, pro ratione <lb></lb>exceſsûs virium potentiæ ſupra ponderis reſiſtentiam definitur; <lb></lb>nam ſi perexiguus fuerit exceſſus, movebitur quidem pondus, <lb></lb>ſed tardè; ſin autem potentiæ virtus longè excedat ponderis <lb></lb>vires, eam celerior motus conſequetur. </s> <s id="s.002787">Et hæc quidem intel<lb></lb>ligi hactenus velim, quando potentia & pondus juxta æqualem <lb></lb>ſpatij longitudinem pari velocitate promoventur, ut ipſa expe<lb></lb>rientia omnibus manifeſtum facit; nemo ſiquidem dubitat, an <lb></lb>currus à validioribus equis celerius quàm à debilibus canthe<lb></lb>riis trahatur; & à robuſtiore bajulo citiùs quàm ab imbecillio<lb></lb>re onus in deſtinatum locum transferri quotidie videmus. </s> </p> <p type="main"> <s id="s.002788">Ut igitur vecte pondus moveri valeat, lex hæc eadem ſtabi<lb></lb>lis & firma permaneat, neceſſe eſt, ut ponderis reſiſtentia mi<lb></lb>nor ſit virtute potentiæ moventis. </s> <s id="s.002789">Quia verò reſiſtentia com<lb></lb>ponitur ex innatâ ponderis gravitate, & ex motûs violenti tar<lb></lb>ditate aut velocitate, hoc eſt ex motûs hujuſmodi quantitate <lb></lb>intra datam temporis menſuram; propterea ita duo hæc tempe<lb></lb>rari oportet, ut quod alteri additur, alteri dematur; ne adeò <lb></lb>reſiſtentia augeatur, ut jam minor non ſit virtute potentiæ. </s> <lb></lb> <s id="s.002790">Quare in vecte, cujus extremitati A potentia applicatur certæ <lb></lb>virtutis, ita ſtatuendus eſt hypomochlio C locus, ut compara<lb></lb>to motu potentiæ in A cum motu ponderis in B, ea ſit motûs B <pb pagenum="365" xlink:href="017/01/381.jpg"></pb>tarditas; quæ addita gravitati ponderis B reſiſtentiam compo<lb></lb>nat minorem virtute movendi potentiæ A. </s> <s id="s.002791">Quoniam enim, <lb></lb>manente puncto C tanquam centro motús potentiæ deſcenden<lb></lb>tis & ponderis aſcendentis, manifeſtum eſt eam eſſe motuum <lb></lb>Rationem, quæ eſt Radiorum CA & CB idcirco quò major <lb></lb>erit hujuſmodi Radiorum inæqualitas, eò etiam major erit Ra<lb></lb>tio motûs potentiæ ad motum ponderis, cujus tarditas gravi<lb></lb>tatem compenſans minuet reſiſtentiam, ut virtuti potentiæ, pro<lb></lb>portione reſpondeat. </s> </p> <p type="main"> <s id="s.002792">Hic verò, ſi rem paulò attentiùs introſpicias, deprehendes <lb></lb>tamdiu ſolùm admirationi eſſe machinarum vires, quamdiu <lb></lb>cauſa occulta manet; quæ ſi in medium proferatur, admiratio<lb></lb>ni nobis eſt ipſa noſtra admiratio. </s> <s id="s.002793">Aio igitur potentiam tan<lb></lb>tumdem plane motûs in pondere efficere cum vecte conjunctam <lb></lb>(idem de cæteri pariter Facultatibus intelligatur, ne idem ſæ<lb></lb>piùs ad nauſeam inculcare oporteat) ac ſi ſolitaria eodem cona<lb></lb>tu pondus aliquod ſecum pari velocitate adduceret, aut eleva<lb></lb>ret. </s> <s id="s.002794">Sit potentia A æqualiter, ac pondus B, diſtans à fulcro C; <lb></lb>& quo conatu movetur potentia deſcendens ſpatio digitorum <lb></lb>decem, dum arteria bis pulſat; cogat oppoſitum pondus libræ <lb></lb>unius aſcendere pariter eodem tempore per digitos decem; eſſe <lb></lb>enim æquales oppoſitos hujuſmodi motus, qui ex æqualibus <lb></lb>Radiis arcus æquales deſcribunt, certum eſt. </s> <s id="s.002795">Jam manente Ra<lb></lb>dio CA, finge Radium CB mutilum atque decurtatum adeò, <lb></lb>ut ſola ejus pars decima reliqua ſit, & CB ponderis diſtantia <lb></lb>ab hypomochlio ſit ſubdecupla diſtantiæ CA potentiæ ab eo<lb></lb>dem hypomochlio: erit igitur motus in B ſubdecuplus motûs <lb></lb>in A. </s> <s id="s.002796">Quare pondus unius libræ in hac ſubdecuplâ diſtantiâ <lb></lb>cùm ſubdecuplo tardius moveatur (percurrit enim tempore eo<lb></lb>dem ſpatium ſubdecuplum) indiget ſolùm ſubdecuplo impetu <lb></lb>ejus, quem prius exigebat, ut æqualiter cum potentiâ move<lb></lb>retur. </s> <s id="s.002797">Totus igitur impetus ille, quem potentia ponderi unius <lb></lb>libræ imprimebat, ut æquali velocitate pariter moverentur, illa <lb></lb>deſcendendo, hoc aſcendendo, ſi decem ponderibus ſimilibus <lb></lb>diſtribuatur, ſatis eſt, ut omnia illa moveantur ſubdecuplâ ve<lb></lb>locitate. </s> <s id="s.002798">Quia autem duorum arteriæ pulſuum ſpatio ſingula <lb></lb>aſcendunt digitum unum, & ſunt decem aſcenſus digitales, <lb></lb>dum potentia deſcendit digitos decem, & dum potentia primo <pb pagenum="366" xlink:href="017/01/382.jpg"></pb>arteriæ pulſu decurrit digitos quinque, decem illa pondera <lb></lb>motum quinque digitorum perficiunt, ſingula videlicet per ſe<lb></lb>midigitum (id quod pariter obſervari facilè poterit in ſingulis <lb></lb>minutioribus temporis particulis) tantumdem motus perficit <lb></lb>potentia ac pondus, ſive toto impetu uni libræ impreſſo libra <lb></lb>una habeat motum decem digitorum, ſive decimâ impetûs par<lb></lb>te ſingulis libris impreſsâ, ſingulæ habeant motum digitalem: <lb></lb>utrobique ſcilicet ſunt decem motus digitales, ſive unius pon<lb></lb>deris, ſive decem ponderum eodem tempore. </s> <s id="s.002799">Quis verò mi<lb></lb>retur, ſi ille idem, qui decem aureis nobili hoſpiti ſplendidio<lb></lb>res epulas parare poſſet, decem hominibus frugalem menſam <lb></lb>inſtrueret ſingulis aureis in ſingulos homines tributis? </s> <s id="s.002800">Deſinat <lb></lb>igitur pariter mirari, ſi potentia eadem, quæ decem impetûs <lb></lb>particulis libram unam ſecum pari velocitate movet, ſingulis <lb></lb>particulis in ſingulas libras tributis moveat decem libras, ſin<lb></lb>gulas ſubdecupla velocitate; neque enim hic plus conatûs, <lb></lb>quàm ibi, requiritur. </s> </p> <p type="main"> <s id="s.002801">In hoc itaque Vectis vires ſitæ ſunt, quod ex Potentiæ & <lb></lb>Ponderis poſitione ita temperantur motus, ut impetûs quem <lb></lb>potentia ponderi imprimere valet, aut re ipſa imprimit, inten<lb></lb>ſio reſpondeat tarditati aut velocitati motûs ipſius ponderis. </s> <lb></lb> <s id="s.002802">Hinc ſi Potentia, & Pondus æqualibus intervallis ab hypomo<lb></lb>chlio diſtent; motus æquales ſunt; & perinde ac ſi potentia ſo<lb></lb>litaria ſine vecte (ſi illa quidem vivens ſit) attolleret pondus, <lb></lb>vectis nihil juvat potentiam, quia pondus hoc recipit totam <lb></lb>impetûs intenſionem, quam illa efficere poteſt. </s> <s id="s.002803">Sin autem <lb></lb>Potentia quidem magis, Pondus verò minùs à fulcro abſit, tar<lb></lb>dior ponderis motus minorem exigit impetûs intenſionem; ac <lb></lb>proinde entitas eadem impetûs, quæ eſt intenſivè minor, po<lb></lb>teſt fieri extenſivè major, & communicari ponderi majori, ac <lb></lb>priùs. </s> <s id="s.002804">Quare pro Ratione tarditatis motûs extenuatur impetûs <lb></lb>intenſio, atque ideò pro eadem Ratione augeri poteſt ponderis <lb></lb>extenſio, hoc eſt gravitas; ut quæ Ratio eſt velocitatis motûs <lb></lb>in pondere æqualis velocitati motûs in potentiâ, ad tarditatem <lb></lb>motûs in pondere minoris motu in potentiâ, eadem ſit directè <lb></lb>Ratio intenſionis impetûs in pondere æquè veloci ad intenſio<lb></lb>nem impetûs in pondere tardiori, & reciprocè eadem ſit Ratio <lb></lb>ponderis tardioris majoris ad pondus illud minus, quod æquè <pb pagenum="367" xlink:href="017/01/383.jpg"></pb>velociter cum potentia moveri poteſt. </s> <s id="s.002805">Quòd ſi potentia pro<lb></lb>pior fuerit hypomochlio, quàm pondus, potentia tardiùs, pon<lb></lb>dus movetur velocius: plus igitur intenſionis impetûs requiri<lb></lb>tur in pondere quàm in potentiâ, adeò ut impetus, qui in po<lb></lb>tentiâ non vivente eſt extenſivè major, intenſivè minor, con<lb></lb>tra in pondere ſit extenſivè minor, intenſivè major: ac <lb></lb>propterea pondus tantò levius eſſe oportet pondere, quod æquè <lb></lb>velociter cum potentiâ moveretur, quantò velociùs movetur <lb></lb>præ illo æquè veloci. </s> <s id="s.002806">Non igitur vectis juvat potentiam, ut <lb></lb>faciliùs moveat, ſed movendi difficultatem auget. </s> <s id="s.002807">Id quod in <lb></lb>Tertio vectis genere ſemper contingit, in quo potentia G mi<lb></lb>nus ab hypomochlio I diſtat, quam pondus H, & tardiùs mo<lb></lb>vetur. </s> <s id="s.002808">Accidit autem hoc idem etiam in Primo genere, cum <lb></lb>vectis inæqualiter ab hypomochlio diſtinguitur in partes, ſi lo<lb></lb>ca permutentur, ut potentia propior ſit, quàm pondus. </s> <s id="s.002809">His <lb></lb>tamen uti poſſumus, quoties quidem viribus abundamus, ſed <lb></lb>ſpatium, in quo potentia moveatur, anguſtum eſt, oportet au<lb></lb>tem ponderi velocem motum conciliare. </s> <s id="s.002810">Contra verò in vecte <lb></lb>Secundi generis potentia à fulcro ſemper remotior eſt, quàm <lb></lb>pondus; idcirco ſemper juvat potentiam; quia quo tardior eſt <lb></lb>ponderis motus, eò minorem ponderis pars, quæ æqualis ſit <lb></lb>ponderi æquè veloci, exigit impetûs intenſionem; ac propterea <lb></lb>quod reliquum eſt impetûs à potentia producendi, pluribus <lb></lb>aliis ſimilibus ponderis partibus impertiri poteſt; atque adeò <lb></lb>abſolutè majus eſt pondus, quàm quod æquè velociter mo<lb></lb>veretur. </s> </p> <p type="main"> <s id="s.002811">Hæc eadem, quæ de ponderibus vecte movendis dicta ſunt, <lb></lb>intelligi pariter oportet de ponderibus vecte ſuſtentandis citra <lb></lb>motum; eo tantùm obſervato diſcrimine, quod ad motum ma<lb></lb>jor requiritur potentiæ virtus, quàm ſit ponderis reſiſtentia, in <lb></lb>ſuſtentatione verò par reſiſtentiæ ponderis eſt virtus potentiæ. </s> <lb></lb> <s id="s.002812">Reſiſtentia autem in ſuſtentatione non ex motûs tarditate aut <lb></lb>velocitate, quæ re ipſa ſit, ſed ex eâ, quæ eſſet, ſi motus fieret, <lb></lb>quatenus pondus eſt vecti connexum, definienda eſt; & pro <lb></lb>hujuſmodi momentorum Ratione, quibus pondus deorſum co<lb></lb>natur, etiam impetûs contranitentis intenſionem dimetiri ne<lb></lb>ceſſe eſt. </s> <s id="s.002813">Quia igitur pondus cum vecte connexum quo propiùs <lb></lb>ad hypomochlium accedit, eo tardiùs ſibi relictum deſcenderet; <pb pagenum="368" xlink:href="017/01/384.jpg"></pb>propterea etiam minorem contranitentis impetûs intenſionem <lb></lb>requirit: Ex quo fit eodem potentiæ conatu, quo illa pondus ſi<lb></lb>ne vecte ſuſtineret, poſſe majorem ponderis gravitatem ſuſtineri <lb></lb>adhibito vecte, eóque majorem, quo major eſt Ratio diſtantiæ <lb></lb>potentiæ ad <expan abbr="diſtantiã">diſtantiam</expan> ponderis à fulcro; & viciſſim potentia mi<lb></lb>nore conatu idem pondus ſuſtinebit, ſi hoc propius admoveatur <lb></lb>ad hypomochlium, quàm priùs, cum opus erat majore conatu. </s> </p> <p type="main"> <s id="s.002814">Porrò conatum potentiæ de induſtria dixi, ut vocabulo ute<lb></lb>rer, quo tum potentia vivens, tùm inanimata æquè comprehen<lb></lb>deretur; quia aliquando quidem potentia conatum adhibet in<lb></lb>natâ ſuâ gravitate, aliquando autem præter, aut contra gravitatis <lb></lb>propenſionem. </s> <s id="s.002815">Gravitate utitur, quæ inanima eſt, & vires ſuas <lb></lb>exerit totas, quodcunque demum pondus vecte movendum aut <lb></lb>ſuſtentandum proponatur. </s> <s id="s.002816">Potentia verò vivens ſuo conſulens <lb></lb>commodo, ne ſe inani conficiat labore, non plus operæ confert, <lb></lb>quàm opus fuerit, ſed vires ex opportunitate adminiſtrat, modò <lb></lb>majores, modò minores impendens, quippe quæ muſculorum <lb></lb>contentione voluntarios motus perficit, & non ſolùm deorſum <lb></lb>premendo, ſed etiam ſurſum connitendo, aut in tranſverſum ur<lb></lb>gendo, vecte uti poteſt: At inanimata potentia non niſi deſcen<lb></lb>dendo vi ſuæ gravitatis cogere poteſt adverſum pondus ad aſcen<lb></lb>dendum; atque ſi primum vectis genus demas, cui poteſt illa <lb></lb>proximè admoveri, in cæteris generibus, ſi attollendum ſit pon<lb></lb>dus, artificium aliquod excogitandum eſt, quo interjecto, aut <lb></lb>potentiæ virtus, aut ipſum pondus ad vectem applicetur, ut <lb></lb>propoſitum finem aſſequamur; conatus enim potentiæ & pon<lb></lb>deris, licet inæquales, non tamen oppoſiti ſunt, ſed ad eandem <lb></lb>partem ſua gravitate contendunt. </s> </p> <p type="main"> <s id="s.002817">Sic vecte RS, cujus fulcrum ſit in extremitate R, non poteſt <lb></lb><figure id="id.017.01.384.1.jpg" xlink:href="017/01/384/1.jpg"></figure><lb></lb>pondus V attolli à po<lb></lb>tentia inanimata P, ſi <lb></lb>proximè illi adjungatur <lb></lb>in S; ac propterea rotu<lb></lb>la in T figenda eſt ver<lb></lb>ſatilis, & funiculo STP <lb></lb>jungenda potentia P, <lb></lb>quæ deorſum connitens <lb></lb>elevat vectem in S, at-<pb pagenum="369" xlink:href="017/01/385.jpg"></pb>que adeò etiam pondus V. </s> <s id="s.002818">Simili ratione ſit vectis Secundi <lb></lb>generis MN, & hypomochlium in M, locus autem ponderis <lb></lb>in H: ſi potentia N inanimata vecti proximè adnectatur, uti<lb></lb>que elevare non poterit pondus in H collocatum: quare ſta<lb></lb>tuatur in loco ſuperiore rotula K, & funiculo HKL jungatur <lb></lb>pondus L cum puncto H; nam potentia N ſua gravitate deſcen<lb></lb>dens deprimendo punctum H vectis elevabit pondus L. </s> <s id="s.002819">Idem <lb></lb>continget, ſi vectis MN ſit tertij generis, & N ſit pondus at<lb></lb>tollendum, potentia verò inanimata collocanda ſit in H. </s> <s id="s.002820">Nihil <lb></lb>utique præſtabit deſcendendo in H; ut igitur punctum H <lb></lb>aſcendat, rotula K adhibeatur, & à potentia L deſcendente <lb></lb>elevabitur idem punctum H, ac proinde etiam pondus N. </s> <s id="s.002821">Vel <lb></lb>ſi in vecte RS tertij generis ſtatuatur potentia V, illa deſcen<lb></lb>dens deprimet velociter extremitatem S, & pari velocitate <lb></lb>aſcendet pondus P. </s> <s id="s.002822">Quid hoc ſimplex artificium aliquando in <lb></lb>ſcenicis motionibus præſtare poſſit emolumenti, facilè prudens <lb></lb>machinator intelligit. </s> </p> <p type="main"> <s id="s.002823">Ex his, quæ de Vectis viribus explicata ſunt, apertè liquet <lb></lb>omnino veritati conſentanea eſſe ea, quæ lib. 2. cap. 8. diximus, <lb></lb>in rotis curruum inveniri non poſſe rationem vectis, quia duo <lb></lb>tantummodo ſunt puncta, ſcilicet extremitas Radij ſubjectam <lb></lb>tellurem tangentis, & rotæ centrum, cui & innititur pondus, <lb></lb>& medio temone applicatur potentia. </s> <s id="s.002824">Cum igitur potentia & <lb></lb>pondus eandem habeant poſitionem, & æquali velocitate mo<lb></lb>veantur, nullum habetur ex Vectis rationibus compendium. </s> <lb></lb> <s id="s.002825">Eatenus enim Vectis in Mechanicarum Facultatum cenſu nu<lb></lb>meratur, quoad potentia & pondus diſpari celeritate moventur, <lb></lb>vel quia potentia ſe velociter movens exiguo conatu tardè mo<lb></lb>vet pondus, ut in primo & ſecundo genere vectis, vel quia po<lb></lb>tentia ſe tardè movens multo conatu celeriter movet pondus, <lb></lb>ut in tertio genere. </s> <s id="s.002826">Quare ſemper in motu ponderis per vectem <lb></lb>aliquid lucri habetur, nimirum aut major ponderis gravitas, <lb></lb>quæ movetur, aut ſaltem major velocitas, qua movetur. <pb pagenum="370" xlink:href="017/01/386.jpg"></pb> </s> </p> <p type="main"> <s id="s.002827"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002828"><emph type="center"></emph><emph type="italics"></emph>Quid in hypomochlij collocatione ſit ohſervandum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002829">TRia in Vecte, ut dictum eſt, puncta conſtituuntur & de<lb></lb>ſignantur duo quæ moventur, tertium illorum motuum <lb></lb>centrum, quod alicui corpori innititur, ut vectis conſiſtat, nec <lb></lb>à ponderis gravitate, aut à potentiæ vi abripiatur: huic cor<lb></lb>pori <emph type="italics"></emph>Hypomochlio<emph.end type="italics"></emph.end> nomen inditum eſt à Græcis, quaſi (ſi verbum <lb></lb>è verbo volumus) <emph type="italics"></emph>ſubvectis,<emph.end type="italics"></emph.end> nam ut plurimum vecti ſubjicitur, <lb></lb>nos <emph type="italics"></emph>Fulcrum<emph.end type="italics"></emph.end> dicimus, quia vectem ſibi incumbentem fulcit. </s> <lb></lb> <s id="s.002830">Cæterùm non eſt hæc conſtans, & perpetua hujus corporis <lb></lb>poſitio, ut ſub vecte ſit, quamvis ſemper Hypomochlij aut Ful<lb></lb>cri nomine donetur; quandoquidem in vecte tertij generis, ubi <lb></lb>pondus in extremitate eſt, potentia medium locum obtinet, ſi <lb></lb>infra alteram vectis extremitatem eſſet corpus hujuſmodi, uti<lb></lb>que à potentia nequiret attolli pondus, ut patet: in ſuperiore <lb></lb>igitur parte ſit oportet, ut potentiâ ſurſum conante, pondere <lb></lb>deorſum contranitente, impediatur altera vectis extremitas, ne <lb></lb>fiat totius vectis converſio obſecundans aut potentiæ conatui, <lb></lb>aut gravitati ponderis, quod eſſet attollendum. </s> <s id="s.002831">Quod ſi hoc <lb></lb>vecte tertij generis deprimendum eſſet infra aquam per vim <lb></lb>corpus aliquod leve, tunc ſub vecte conſtitueretur hypomo<lb></lb>chlium: contrà vectis primi & ſecundi generis ſi ad premen<lb></lb>dum aut deprimendum adhibeatur, exigit hypomochlium in <lb></lb>ſuperiori parte. </s> <s id="s.002832">Similiter non eſt ſub vecte, ſed ad latus adja<lb></lb>cet, quoties pondus eſt movendum in plano horizontali, ſive in <lb></lb>eodem plano ſit vectis, ſive in plano verticali, ut cùm duo mar<lb></lb>mora non elevanda ſunt, ſed immiſſo inter illa vecte invicem <lb></lb>disjungenda. </s> <s id="s.002833">Quemadmodum igitur lapis à lædendo pedem <lb></lb>vocabulum habet, etiamſi non lapides omnes pedem lædant; <lb></lb>ita corpus illud, cui punctum vectis quieſcens innititur, hypo<lb></lb>mochlij & fulcri nomen retinet, quamvis non ſemper ſub vecte <lb></lb>ſit, illúmque ſuffulciat. </s> <s id="s.002834">Quid autem profuerit immutare vo-<pb pagenum="371" xlink:href="017/01/387.jpg"></pb>cabula, ubi rem ipſam tenemus? </s> <s id="s.002835">Immò punctum ipſum vectis <lb></lb>quieſcens, quod hypomochlio reſpondet, non raro ab iis hy<lb></lb>pomochlium dicitur, aut fulcrum, qui verborum compendio <lb></lb>claritati conſultum volunt; mihíque hanc loquendi facultatem, <lb></lb>ubi res tulerit, reſervo. </s> </p> <p type="main"> <s id="s.002836">Quieſcens autem voco punctum vectis, quod eſt centrum <lb></lb>motuum potentiæ, & ponderis; non quia ſemper omnino <lb></lb>quieſcat, ſed quia ſi aliquo motu moveatur, tardiſſimum certè <lb></lb>eſt omnium punctorum; cætera quippe vectis puncta circa hoc <lb></lb>tanquam circa centrum deſcribunt lineam inflexam ac recur<lb></lb>vam: alioquin ſi punctum hoc plus moveretur quàm pondus, <lb></lb>mutatæ fuiſſent vices, & quod pondus dicitur, eſſet reipſa hy<lb></lb>pomochlium, corpus verò, quod hypomochlium dicitur, eſſet <lb></lb>pondus, quod à potentiâ potiſſimum moveretur. </s> <s id="s.002837">Obſervandum <lb></lb>enim eſt non pondus ſolum, verùm etiam hypomochlium acci<lb></lb>pere vim externam potentiæ vectem agitantis, reſiſtente vide<lb></lb>licet pondere, ex quo fit illud premi; quod ſi inæqualiter re<lb></lb>ſiſtant, licet utrumque moveatur, in illud potiùs exercet vir<lb></lb>tutem ſuam potentia, quod languidiùs reſiſtit, altero validiore <lb></lb>hypomochlij rationem habente. </s> <s id="s.002838">Sic vecti ad attollendum mar<lb></lb>mor applicato ſi glebam, hypomochlij loco, ſuppoſueris, non <lb></lb>marmor attolles, ſed glebam vecte conteres: marmor igitur eſt <lb></lb>hypomochlium vecti ſuperpoſitum, & glebæ eſt pondus contri<lb></lb>tum vecte ſecundi generis: At ſi pro gleba lignum ſubjicias, <lb></lb>quod non frangatur, ſed aliquantulum cedens comprimatur, & <lb></lb>vectis veſtigium recipiat, ita tamen, ut marmor moveatur, du<lb></lb>plex vectis genus hic intercedit, prout duplex effectus poten<lb></lb>tiæ conatum conſequitur; ad comprimendum ſcilicet lignum <lb></lb>vectis eſt ſecundi generis hypomochlium habens impoſitum <lb></lb>marmor, ad elevandum autem marmor vectis eſt primi generis, <lb></lb>cujus hypomochlium eſt ſubjectum lignum. </s> <s id="s.002839">Cujuſmodi ſit hy<lb></lb>pomochlium, ſive ſit funis vectem retinens, ſive axis infixus, <lb></lb>circa quem volvatur vectis, ſive quodcumque aliud corpus, cui <lb></lb>ille incumbat, aut innitatur, modò abſit incommodi periculum <lb></lb>ex ejus fragilitate, parum refert: ſatis eſt, ſi par fuerit ferendo <lb></lb>oneri, quod vecte elevatur. </s> <s id="s.002840">Ex ponderis autem gravitate hy<lb></lb>pomochlij ſoliditas atque materies definienda eſt; ex motùs <pb pagenum="372" xlink:href="017/01/388.jpg"></pb>qualitate (ſpectatâ loci, in quo perficiendus eſt, poſitione) <lb></lb>forma hypomochlij ſtatuatur. </s> </p> <p type="main"> <s id="s.002841">Illud examinandum videtur, quandó nam præſter uti vecte <lb></lb>primi generis, quando vecte Secundi generis, hoc eſt an plus <lb></lb>commodi aſſerat fulcrum in vectis extremitate collocatum, ut <lb></lb>in ſecundo genere, an verò inter pondus atque potentiam in<lb></lb>terjectum, ut in primo genere. </s> <s id="s.002842">Propoſita ſit vectis longitudo <lb></lb>decem palmorum, quo oporteat pondus ita attollere, ut ejus <lb></lb>motus ſit reſpondens arcui deſcripto ex Radio duorum palmo<lb></lb>rum. </s> <s id="s.002843">Si vectis ſit primi generis, pondus & potentia ſunt in <lb></lb>vectis extremitatibus, hypomochlium dividit totam longitudi<lb></lb>nem in partes duas, quarum major ad potentiam ſpectans eſt <lb></lb>quadrupla minoris ſpectantis ad pondus; eſt ſcilicet illa octo, <lb></lb>hæc duorum palmorum. </s> <s id="s.002844">At ſi vectis fuerit ſecundi generis, <lb></lb>hypomochlium & potentia illius extremitates occupant, pon<lb></lb>dus ab hypomochlio diſtat palmos duos: quare potentiæ diſtan<lb></lb>tia ab hypomochlio cum ſit tota vectis longitudo, eſt quintu<lb></lb>pla diſtantiæ ponderis. </s> <s id="s.002845">Cum igitur ponderis motus cum po<lb></lb>tentiæ motu comparatus hic quintuplo tardior ſit, ibi verò ſo<lb></lb>lum quadruplo tardior, minore impetu indiget, ut moveatur <lb></lb>vecte ſecundi generis. </s> <s id="s.002846">Cæterùm conſiderato hoc duplici vectis <lb></lb>genere, obſervandum eſt in ſecundo genere à potentia elevan<lb></lb>dum non ſolum pondus ſed etiam vectem ipſum, qui ſi valde <lb></lb>gravis ſit (ut aliquando contingere poteſt trabem fungi vectis <lb></lb>munere) auget potentiæ movendi difficultatem: Contra verò <lb></lb>in vecte primi generis ipſa vectis gravitas juvat potentiam; & <lb></lb>quidem ſi homo ſit, qui vectem premat, ipſa corporis gravitas <lb></lb>acceſſionem facit, ad impetum, qui à vitali conatu oritur: præ<lb></lb>terquam quod hic liberè & facillimè potentiam inanimatam <lb></lb>adhibere poſſumus, & aliam atque aliam adjicere prout opus <lb></lb>fuerit; at non item in vecte ſecundi generis, niſi adhibito arti<lb></lb>ficio, de quo ſuperiori capite dictum eſt. </s> </p> <p type="main"> <s id="s.002847">Datâ igitur ponderis movendi gravitate, & datâ potentiæ <lb></lb>virtute (quæ videlicet tanto conatu adhibito poteſt certam gra<lb></lb>vitatem ſola ſine vecte movere in ſimili plano ſive horizontali, <lb></lb>ſive inclinato, ſive verticali) diſtinguatur vectis in duas partes <lb></lb>ita, ut vel pars ad partem, ſi ſit primi generis, vel totus ad par-<pb pagenum="373" xlink:href="017/01/389.jpg"></pb>tem, ſi ſit ſecundi generis, eandem Rationem habeat, quæ <lb></lb>eſt dati ponderis ad pondus, quod à potentia ſolâ ſine vecte po<lb></lb>teſt moveri. </s> <s id="s.002848">Sic data Potentia virtutem habeat movendi pon<lb></lb>dus lib. 6. certo conatu, oporteat autem hoc codem conatu <lb></lb>movere lib. 30: quia virtus potentiæ eſt ſubquintupla pon<lb></lb>deris dati, propoſitus vectis intelligatur primùm diſtinctus <lb></lb>in partes ſex, quarum una tribuatur diſtantiæ ponderi, ab <lb></lb>hypomochlio, reliquæ quinque tribuantur diſtantiæ poten<lb></lb>tiæ, ita ut reciprocè ſit diſtantia potentiæ ad diſtantiam <lb></lb>ponderis, ut pondus datum ad virtutem potentiæ: & hic <lb></lb>eſt vectis primi generis. </s> <s id="s.002849">Deinde ut habeatur vectis ſecun<lb></lb>di generis, diſtinguatur totus vectis in partes quinque, & <lb></lb>una ex illis ſit diſtantia ponderis ab hypomochlio in vectis <lb></lb>extremitate conſtituto. </s> <s id="s.002850">In utroque enim caſu motus poten<lb></lb>tiæ eſt quintuplus motûs ponderis, atque adeò potentia <lb></lb>poterit vecte movere pondus quintuplum ponderis, quod ſo<lb></lb>la poteſt movere. </s> </p> <p type="main"> <s id="s.002851">Potentiæ virtutem dixi, non potentiæ gravitatem, tùm <lb></lb>quia non omnis potentia vim movendi habet ex gravitate, <lb></lb>tum quia potentiæ gravitas movere non poteſt gravitatem <lb></lb>æqualem, ſed minorem, nam cum æquali facit æquili<lb></lb>brium, & ſolùm poteſt illam ſuſpendere. </s> <s id="s.002852">Quare ſi poten<lb></lb>tia vi ſuæ gravitatis moveat, non ſatis erit, ſi fiat ut po<lb></lb>tentiæ gravitas ad ponderis gravitatem, ita reciprocè pon<lb></lb>deris diſtantia à centro motús ad diſtantiam potentiæ ab eo<lb></lb>dem centro; ſed diſtantia ponderis ad diſtantiam potentiæ <lb></lb>exigit habere minorem Rationem. </s> <s id="s.002853">Hinc ſi potentia ſit pon<lb></lb>deris ſubquintupla ratione ſuarum gravitatum, pondus ab <lb></lb>hypomochlio diſtare debet minus quàm parte quinta diſtan<lb></lb>tiæ potentiæ ab eodem hypomochlio. </s> <s id="s.002854">Quod ſi vectis is eſſet, <lb></lb>cujus gravitas notabile momentum adderet potentiæ, tunc <lb></lb>diſtantia ponderis, quæ eſſet ſubquintupla diſtantiæ potentiæ, <lb></lb>ſufficeret, minor enim eſſet Ratione potentiæ adæquatè ac<lb></lb>ceptæ ad Pondus. </s> </p> <p type="main"> <s id="s.002855">Ubi verò ponderis gravitatem conſiderare oportet, non <lb></lb>ſatis eſt illam notam habere, ac ſi ſtaterâ expenderetur, <lb></lb>ſed conſiderandum eſt planum, in quo illud movendum eſt; <lb></lb>neque enim eadem habet momenta, ſi ſurſum elevandum ſit <pb pagenum="374" xlink:href="017/01/390.jpg"></pb>in plano Verticali, ac ſi urgendum ſit in plano inclinato, <lb></lb>aut propellendum in horizontali: propterea in Ratione aſ<lb></lb>ſignandà partibus vectis non eſt attendenda gravitas abſoluta <lb></lb>ponderis, ſed quatenus in propoſito plano. </s> <s id="s.002856">Idem eſt de gravi<lb></lb>tate potentiæ dicendum. </s> </p> <p type="main"> <s id="s.002857">Ex dictis patet non quamcumque vectis longitudinem ſem<lb></lb>per opportunam eſſe, quamvis verum ſit quemlibet vectem <lb></lb>poſſe ſecundùm quamcumque Rationem in partes diſtingui, <lb></lb>atque proinde quodcumque pondus à quacumque datâ po<lb></lb>tentiâ poſſe moveri, ſi ritè applicari poſſet. </s> <s id="s.002858">Unum enim <lb></lb>eſt incommodum, quod, quo propiùs ad centrum motuum <lb></lb>admovetur pondus, eo minor eſt illius motus: & continge<lb></lb>re poteſt adeò exiguam eſſe ponderis ab hypomochlio diſtan<lb></lb>tiam, ut motus adeò tenuis nulli futurus ſit uſui. </s> <s id="s.002859">Qua<lb></lb>propter longiori vecte utendum erit, ut, ſervatâ eâdem <lb></lb>diſtantiarum Ratione, intervallum inter pondus & centrum <lb></lb>motuum ſit notabile & conſpicuum, ex quo motus ſuffi<lb></lb>ciens obtineri poſſit. </s> <s id="s.002860">Quid enim juvaret, ſi vecte palmo<lb></lb>rum 25 tentares attollere pondus centuplum virtutis poten<lb></lb>tiæ? </s> <s id="s.002861">an ut pondus ab hypomochlio diſtans per digitum (ſu<lb></lb>mo digitos quatuor pro ſingulis palmis) elevaretur ad altitu<lb></lb>dinem unius aut alterius grani hordei? </s> <s id="s.002862">Præterquam quod <lb></lb>tam ingens pondus ægrè poſſet in tantillo ſpatio ad vectem <lb></lb>opportunè applicari. </s> </p> <p type="main"> <s id="s.002863">Quod autem ad hypomochlium attinet, curandum maxi<lb></lb>mè eſt, ut qua parte vectem contingit, minimum ſit, &, ſi <lb></lb>fieri poteſt, proximè in aciem deſinat; ut ſcilicet eandem <lb></lb>ſemper in motu vectis partem contingat; ſi enim alia atque <lb></lb>alia vectis pars hypomochlio inſiſtat, mutantur ponderis at<lb></lb>que potentiæ momenta, ideoque augeri poteſt movendi diffi<lb></lb><figure id="id.017.01.390.1.jpg" xlink:href="017/01/390/1.jpg"></figure><lb></lb>cultas. </s> <s id="s.002864">Sit vectis ſecundi generis AB <lb></lb>innixus ſaxo, quod contingit in C, <lb></lb>& centri gravitatis ponderis locus ſit <lb></lb>D: utique quia DC minoreſt quàm <lb></lb>DB, major eſt Ratio AB ad DC mi<lb></lb>norem, quàm ejuſdem AB ad DB <lb></lb>majorem, per 8. lib. 5. At elevato <lb></lb>vecte, ut habeat poſitionem FE, ſi-<pb pagenum="375" xlink:href="017/01/391.jpg"></pb>cut A venit in F, ita B venit in E, ubi ſaxo innititur, & pon<lb></lb>dus D venit in G. </s> <s id="s.002865">Eſt igitur FE ad GE, ut AB ad DB; ergo <lb></lb>etiam FE ad GE habet minorem Rationem quàm AB ad DC. </s> <lb></lb> <s id="s.002866">Quo autem minor eſt motuum Ratio, eò etiam minus eſt po<lb></lb>tentiæ momentum ad momentum ponderis; igitur ſi Ratio AB <lb></lb>ad DB minor ſit, quàm AC ad DC, etiam Ratio FE ad GE <lb></lb>minor erit quàm Ratio AC ad DC. </s> <s id="s.002867">Quare tunc ſolùm ea<lb></lb>dem movendi facilitas manebit (quod quidem ſpectat ad ra<lb></lb>tionem hypomochlij quicquid ſit an ex alio capite mutetur, ut <lb></lb>infra) quando CB pars extrema vectis, quæ innititur hypo<lb></lb>mochlio, ea eſt, ut eadem ſit Ratio AB ad DB, quæ eſt AC <lb></lb>ad DC: Hoc autem fieri omnino non poteſt, quia AB & DB <lb></lb>ſunt idem ac AC, atque DC, ſi his utriſque addatur eadem <lb></lb>pars CB. </s> <s id="s.002868">Si ergo ut AC plus CB ad DC plus CB eſſet ut <lb></lb>AC ad DC, etiam permutando, & dividendo, & iterum per<lb></lb>mutando, per 16. & 17. lib. 5. eſſet ut AC ad DC ita CB ad <lb></lb>CB, ac propterea AC totum æquale eſſet parti DC. </s> <s id="s.002869">Non <lb></lb>igitur fieri poteſt, ut maneat in motu eadem facilitas ratione <lb></lb>hypomochlij, ſi accidat, ut vectis poſitiones in motu ſe de<lb></lb>cuſſent; id quod evenit, ſi alia atque alia pars vectis hypomo<lb></lb>chlium tangat. </s> <s id="s.002870">Et quia major eſt Ratio totius AB ad totam <lb></lb>DB, quàm ſit ablatæ CB ad ablatam CB, erit etiam, per 33. <lb></lb>lib. 5. reliquæ AC ad reliquam DC major Ratio quàm totius <lb></lb>AB ad totam DB, hoc eſt major Ratio quàm FE ad GE. </s> </p> <p type="main"> <s id="s.002871">Similiter in vecte primi generis, ſi fulcrum ſit cylindricum, <lb></lb>tangit quidem in puncto, ſed dum vectis deorſum urgetur, <lb></lb>aliud atque aliud ejus punctum aliis cylindri punctis congruit: <lb></lb>nam ſi fuerit potentia in C, & pondus <lb></lb><figure id="id.017.01.391.1.jpg" xlink:href="017/01/391/1.jpg"></figure><lb></lb>in E, vectis autem tangat in F, in con<lb></lb>verſione cum E venerit in I, & C in L, <lb></lb>jam contactus fit in H ita, ut HL minor <lb></lb>ſit quàm FC, contrà verò HI major ſit <lb></lb>quàm FE. </s> <s id="s.002872">Decreſcunt ergo potentiæ <lb></lb>momenta, cujus diſtantia à motûs centro <lb></lb>minuitur, augentur autem ponderis momenta, cujus diſtan<lb></lb>tiæ à motûs centro aliquid ſemper accedit. </s> <s id="s.002873">Et quidem quò <lb></lb>craſſior fuerit cylindrus, factâ pari vectis inclinatione, major <lb></lb>etiam oritur diſtantiarum differentia; ut facilè demonſtratur, <pb pagenum="376" xlink:href="017/01/392.jpg"></pb>ſi duo circuli ſe intus contingant in O, ubi vectem ſuſtinent, <lb></lb>& deinde vectis inclinetur, ut faciat angulum OIG tangens <lb></lb><figure id="id.017.01.392.1.jpg" xlink:href="017/01/392/1.jpg"></figure><lb></lb>cylindrum minorem <lb></lb>in G, aut faciat angu<lb></lb>lum OHS illi æqua<lb></lb>lem tangens cylin<lb></lb>drum majorem in S: <lb></lb>duo ſi quidem trian<lb></lb>gula IRG & HMS <lb></lb>ſunt æquiangula, quia <lb></lb>vectes CK & BD ſunt <lb></lb>paralleli ex hypothe<lb></lb>ſi, lineæ verò à centris R & M ad puncta contactuum G & S <lb></lb>ductæ cadunt ad angulos rectos, ex 18. lib. 3. quapropter & an<lb></lb>guli ad centra R & M ſunt æquales: igitur etiam arcus OG <lb></lb>& OS ſunt ſimiles in Ratione ſuarum ſemidiametrorum OR <lb></lb>& OM: major ergo eſt arcus OS quàm arcus OG, ac <lb></lb>propterea illi major quàm huic vectis pars in converſione apta<lb></lb>tur, adeóque diſtantia ponderis ab hypomochlio minùs auge<lb></lb>tur ab O in G, quàm ab O in S, factâ æquali vectis inclina<lb></lb>tione. </s> <s id="s.002874">Illud tamen habetur compendij, ſi craſſior cylindrus <lb></lb>vecti ſupponatur, quod non adeò inclinandus ſit vectis, ut ad <lb></lb>certam altitudinem attollatur pondus, ac illum inclinare opor<lb></lb>teret, ſi exilior cylindrus fulcri munere fungeretur. </s> </p> <p type="main"> <s id="s.002875">Quæ de cylindro dicta ſunt, manifeſta quoque apparent, ſi <lb></lb>hypomochlium planum ſit, ut OS: eſt nimirum longè alia <lb></lb><figure id="id.017.01.392.2.jpg" xlink:href="017/01/392/2.jpg"></figure><lb></lb>Ratio VO ad OR atque XS ad ST; <lb></lb>nam additur ipſi OR longitudo OS, ut <lb></lb>habeatur ST. </s> <s id="s.002876">Cum ergo minor ſit poten<lb></lb>tiæ diſtantia XS, quàm VO, minora ſunt <lb></lb>potentiæ momenta: contra verò cum ma<lb></lb>jor ſit ponderis diſtantia TS, quàm RO, <lb></lb>majora pariter ſunt ponderis momenta. </s> <s id="s.002877">Ut itaque in vectis mo<lb></lb>tu momentorum Ratio ſtabilis ac firma perſeveret, ſatius eſt <lb></lb>hypomochlium vecti objicere aciem anguli, in quem duæ ſub<lb></lb>jecti corporis facies concurrunt, aut vecti axem infigi, circa <lb></lb>quem ille convolvatur. <pb pagenum="377" xlink:href="017/01/393.jpg"></pb> </s> </p> <p type="main"> <s id="s.002878"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002879"><emph type="center"></emph><emph type="italics"></emph>Qua Ratione ſtatuendus ſit ponderi locus in Vecte <lb></lb>primi generis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002880">QUoniam pondus vecte movendum non eſt corpus aliquod <lb></lb>planè individuum, ſed partes habet, quarum aliæ ſunt <lb></lb>puncto fulcri, hoc eſt, centro motûs, propiores, aliæ remotio<lb></lb>res; animum diligenter advertere opus eſt, cuinam vectis <lb></lb>puncto intelligendum ſit adjunctum onus, ut ex eo ad fulcrum <lb></lb>diſtantia determinetur. </s> <s id="s.002881">Et quidem vix cuiquam dubium eſſe <lb></lb>poteſt, an inter omnia ponderis puncta illud unum eligendum <lb></lb>ſit, in quo gravitas vires ſuas omnes exercere intelligitur, vi<lb></lb>delicet circa quod paribus momentis deorſum nititur, ſi ipſa ſibi <lb></lb>relinquatur: hoc autem eſt Gravitatis centrum ipſi ponderi in<lb></lb>ſitum, in quod ſingularum partium conatus confluere, & ſe<lb></lb>cundùm quod per directionis lineam deorſum vectem urgeri <lb></lb>concipimus. </s> </p> <p type="main"> <s id="s.002882">Sit enim pondus P, quod vecti AB infixum, & longitudini <lb></lb>AC congruens, ſuo gravitatis centro I deorſum nititur per li<lb></lb>neam directionis IH. </s> <s id="s.002883">Dico vectem <lb></lb>perinde à toto pondere urgeri, atque <lb></lb><figure id="id.017.01.393.1.jpg" xlink:href="017/01/393/1.jpg"></figure><lb></lb>ſi tota ejus gravitas eſſet in puncto I, <lb></lb>atque ideò diſtantiam ponderis ab <lb></lb>hypomochlio D eſſe, neque AD ma<lb></lb>ximam, neque CD minimam, ſed <lb></lb>ID mediam: quia, etſi partibus ſin<lb></lb>gulis ſua inſit gravitas, & ſingula pro ſuâ à puncto D diſtantia <lb></lb>ſua habeant momenta, ita majora momenta remotiorum parti<lb></lb>cularum à minoribus vicinarum compenſantur, ut intelligenda <lb></lb>ſit vel tota gravitas in media diſtantia ID vel ſemiſſis gravitatis <lb></lb>in extrema diſtantia AD, prout lib. 3. cap. 2. de momentis bra<lb></lb>chiorum inæqualium libræ oſtenſum eſt. </s> <s id="s.002884">Hoc autem, quod <lb></lb>de pondere ſecundùm molem & gravitatem æquabili dicitur, <lb></lb>etiam de ponderibus, quorum anomala eſt figura, vel ex diver-<pb pagenum="378" xlink:href="017/01/394.jpg"></pb>ſis ſecundùm ſpeciem gravitatibus compoſita, intelligendum <lb></lb>eſt, ſi eorum centro gravitatis congruat vectis longitudo; nam <lb></lb>ponderis diſtantia non eſt Arithmeticè media inter maximam & <lb></lb>minimam, ſed eſt intervallum, quod inter fulcrum & centrum <lb></lb>gravitatis interjicitur. </s> </p> <p type="main"> <s id="s.002885">Sed quia non rarò pondus aut vecti totum incumbit, aut plu<lb></lb>ribus funiculis firmiter alligatum ex illo ſuſpenditur, propterea <lb></lb>obſervandum eſt, in quod vectis punctum incidat Directionis <lb></lb>linea ex centro gravitatis ponderis ducta; hæc enim definiet <lb></lb>diſtantiam ponderis ab hypomochlio, & innoteſcent momenta, <lb></lb>quibus illud reſiſtit potentiæ elevanti. </s> <s id="s.002886">Id quod per libram <lb></lb>æqualium brachiorum (ne illorum inæqualitas aliquam pariat <lb></lb>difficultatem) inſtituto æquilibrio facillimè experiri poteris, ſi <lb></lb>laminas ligneas, aut metallicas, in varias figuras conformave<lb></lb>ris, in quibus centrum gravitatis inventum fuerit, & ita ſingu<lb></lb>las ſecundùm unum latus immobiliter uni brachio aptaveris, ut <lb></lb>illi congruant, atque in oppoſitâ jugi extremitate æquipon<lb></lb>dium addideris; facto enim æquilibrio, & demiſſo perpendicu<lb></lb>lo per centrum gravitatis notatum tranſeunte, apparebit, cui<lb></lb>nam libræ puncto reſpondeat; atque inter hoc punctum, & cen<lb></lb>trum motús libræ, diſtantia erit ad reliqui brachij totam lon<lb></lb>gitudinem, ut æquipondij gravitas ad ponderis examinati gra<lb></lb>vitatem. </s> </p> <p type="main"> <s id="s.002887">Quod ſi pondus ex unico fune pendulum adnectatur vecti, <lb></lb>ſatis conſtat, ex quo vectis puncto deſumatur ejus diſtantia, ni<lb></lb>mirum ex puncto ſuſpenſionis; intentus enim funis à pendente <lb></lb>gravitate lineam Directionis oſtendit. </s> <s id="s.002888">Quamvis autem ſi hujus <lb></lb>puncti tantummodo ratio habeatur, eadem videantur futura <lb></lb>ponderis momenta, quæcumque tandem fuerit vectis poſitio <lb></lb>ſive horizonti parallela, ſive obliqua, examinandum tamen <lb></lb>erit inferius cap.8. utrum ratione anguli, ſecundùm quem pon<lb></lb>dus deorſum trahere conatur vectem, ejus momenta mutentur. </s> </p> <p type="main"> <s id="s.002889">Nunc autem pondus firmiter vecti adnexum, non verò ex <lb></lb>unico fune pendulum, conſideremus, ſive vecti incumbat, ſive <lb></lb>infra vectem collocetur; hoc nimirum eſt illud, in quo, propo<lb></lb>ſitis majoribus ponderibus, non videtur connivendum; neque <lb></lb>enim nihil refert, utrùm infra, an ſupra vectem ſit movendæ <lb></lb>gravitatis centrum, quantóque intervallo hoc ab illo abſit, ibi <pb pagenum="379" xlink:href="017/01/395.jpg"></pb>ſi quidem gravitas collocata intelligitur, ubi ſuas omnes vires <lb></lb>omnium partium conſpiratione exercet. </s> <s id="s.002890">Quapropter, ut pon<lb></lb>deris momenta innoteſcant, centri gravitatis motum perpen<lb></lb>dere, ac dimetiri oportet. </s> <s id="s.002891">Hinc eſt pondus firmiter adnexum <lb></lb>vecti perinde ſe habere, atque ſi vectis quidam curvus in an<lb></lb>gulum inflexus ad punctum hypomochlij, ſi ſit vectis primi ge<lb></lb>neris, extremitatem alteram in centro gravitatis ponderis, al<lb></lb>teram in potentiâ haberet. </s> </p> <p type="main"> <s id="s.002892">Sit Vectis rectus AB horizonti parallelus, hypomochlium <lb></lb>habens in C, & in parte inferiore ſtabili nexu adjungatur pon<lb></lb>dus, cujus gravitatis centrum I. <lb></lb><figure id="id.017.01.395.1.jpg" xlink:href="017/01/395/1.jpg"></figure><lb></lb>Ex I in vectem horizontalem cadat <lb></lb>perpendicularis linea directionis <lb></lb>IE; hoc enim perpendiculum de<lb></lb>finit diſtantiam gravitatis à vecte. </s> <lb></lb> <s id="s.002893">Eſt igitur potentia in A, & pondus <lb></lb>in I perinde, atque ſi eſſet vectis <lb></lb>ACI; & ut pondus atque potentia <lb></lb>in eâdem linea horizontali conſiſtant, non eſt attendenda <lb></lb>vectis poſitio AB, ſed rectæ lineæ AI jungentis centrum po<lb></lb>tentiæ A cum centro gravitatis ponderis I; quæ linea AI ſimul <lb></lb>ut æquè ab horizonte diſtabit, & linea CH ad angulos rectos <lb></lb>cadens in eandem lineam AI congruens erit rectæ lineæ jun<lb></lb>genti punctum hypomochlij C cum centro terræ, æquilibrium <lb></lb>indicabit; eademque definiet Rationem ponderis ad potentiam <lb></lb>ſuſtinentem horizontaliter, juxta reciprocam eorumdem <lb></lb>diſtantiam à puncto H; pro ut lib.3. cap.5. de librâ curvâ ex<lb></lb>plicatum eſt. </s> <s id="s.002894">In poſitione autem obliqua AI, quando recta ex <lb></lb>C ad centrum terræ ducta eſt CG cadens ſuper AI ad angu<lb></lb>los inæquales, potentia ſuſtinens eſt ad pondus, ut IG ad GA. </s> <lb></lb> <s id="s.002895">Cum igitur ſit IG minor quàm IH, contrà verò GA ſit ma<lb></lb>jor quàm HA, erit minor Ratio IG ad GA, quàm IH <lb></lb>ad HA. </s> </p> <p type="main"> <s id="s.002896">Quoniam verò linea directionis ponderis IE perpendicula<lb></lb>ris eſt ad vectem AB horizontalem ex hypotheſi, & parallela <lb></lb>lineæ CG, eſt ut AG ad GI, ita AC ad CE, per 2. lib.6. ac <lb></lb>propterea, in ſitu vectis parallelo horizonti, locus ponderis eſt <lb></lb>in vecte determinatus à lineâ directionis ponderis occurrente <pb pagenum="380" xlink:href="017/01/396.jpg"></pb>ipſi vecti. </s> <s id="s.002897">Et quia major eſt Ratio AG ad GI, quàm ſit AH <lb></lb>ad HI, etiam major eſt Ratio AC ad CE, quàm ſit AH ad <lb></lb>HI: Ergo convertendo EC ad CA minorem habet Ratio<lb></lb>nem, quàm IH ad HA, per 26. lib. 5. Atqui potentia ſuſti<lb></lb>nens pondus datum, quando recta AI æquè diſtat ab horizon<lb></lb>te, eſt ad pondus ut IH ad HA; quando autem pondus eſt in<lb></lb>fra lineam BA illud cum potentiâ jungentem horizonti paral<lb></lb>lelam eſt ut EC ad CA. </s> <s id="s.002898">Igitur potentia ſuſtinens in horizon<lb></lb>tali pondus habet majorem Rationem ad illud, quàm ad idem <lb></lb>pondus habeat potentia ſuſtinens illud infra horizontalem. </s> <lb></lb> <s id="s.002899">Ergo, ex 8. lib. 5. potentiâ ſuſtinens pondus infra horizonta<lb></lb>lem minor eſt potentiâ illud ſuſtinente in horizontali. </s> <s id="s.002900">Finge <lb></lb>enim eſſe libram curvam ACI habentem ſpartum in C: uti<lb></lb>que ſi in A eſſet æquipondium, quod ad pondus I eſſet ut IH <lb></lb>ad HA, non maneret in eadem poſitione obliqua, ſed A deſcen<lb></lb>deret ad poſitionem horizontalem, ut dictum eſt lib. 3. cap.4. <lb></lb>ut igitur obliqua maneat, æquipondium A debet eſſe minus. </s> <lb></lb> <s id="s.002901">Ad ſuſtinendum autem pondus, hîc in vecte idem à Potentiâ <lb></lb>præſtatur, ac ab æquipondio in librâ brachiorum inæqualium. </s> </p> <p type="main"> <s id="s.002902">Simili omnino methodo oſtendetur pondus idem vecti AB <lb></lb>horizontali impoſitum, cujus centrum gravitatis ſit D, linea <lb></lb>directionis DI occurrens vecti in E, eſſe ad potentiam A, ut <lb></lb>eſt AC ad CE; at ſi recta AD jungens potentiam cum cen<lb></lb>tro gravitatis D eſſet horizonti parallela, pondus ad potentiam <lb></lb>eſſet ut AL ad LD, quam Rationem determinat CL cadens <lb></lb>ad angulos rectos in rectam AD. </s> <s id="s.002903">Quia enim DE & IE ſunt <lb></lb>æquales ex hypotheſi, cum ſit idem pondus, & latus EA eſt <lb></lb>commune, anguli verò ad E ſunt recti, etiam, per 4. lib. 1. <lb></lb>lineæ AD & AI, item anguli EAD & EAI ſunt æquales. </s> <lb></lb> <s id="s.002904">Præterea in triangulis CHA, CLA rectangulis ad H & L, <lb></lb>latus CA eſt commune, & anguli ad A ſunt æquales; igitur, <lb></lb>per 26. lib. 1. lineæ AL & AH ſunt æquales, igitur & reſi<lb></lb>duæ LD & HI ſunt æquales. </s> <s id="s.002905">Quapropter ut AH ad HI, ita <lb></lb>AL ad LD: quia igitur Ratio AH ad HI oſtenſa eſt ſuperiùs <lb></lb>minor Ratione AC ad CE, etiam minor eſt Ratio AL ad LD, <lb></lb>quàm AC ad CE. </s> <s id="s.002906">Sed ut AC ad CE, ita AO ad OD, per <lb></lb>2. lib. 6. propter paralleliſmum linearum CO & ED; ergo mi<lb></lb>nor eſt Ratio AL ad LD, quàm AO ad OD. </s> <s id="s.002907">Atqui cùm AD <pb pagenum="381" xlink:href="017/01/397.jpg"></pb>parallela eſt horizonti, pondus D impoſitum vecti ad poten<lb></lb>tiam A ſuſtinentem eſt ut AL ad LD, in poſitione verò obli<lb></lb>quâ AD eſt idem pondus ad potentiam ſuſtinentem ut AO <lb></lb>ad OD; ergo in priori poſitione horizontali pondus ad poten<lb></lb>tiam habet minorem Rationem, quàm in poſteriori poſitione <lb></lb>obliqua: ergo per 8. lib. 5. in priori eſt major potentia, quàm <lb></lb>in poſteriori. </s> </p> <p type="main"> <s id="s.002908">Quamvis autem, cùm vectis eſt horizonti parallelus, pon<lb></lb>dus ſive illi impoſitum, ſive ſuppoſitum fuerit, iiſdem momen<lb></lb>tis reluctetur potentiæ ſuſtinenti, non ita tamen ſe res habet, <lb></lb>ſi idem vectis <lb></lb><figure id="id.017.01.397.1.jpg" xlink:href="017/01/397/1.jpg"></figure><lb></lb>AB, fulcrum <lb></lb>habens in C, <lb></lb>elevetur ſupra <lb></lb>lineam hori<lb></lb>zontalem RT: <lb></lb><expan abbr="plurimũ">plurimum</expan> enim <lb></lb>intereſt, <expan abbr="utrũ">utrum</expan> <lb></lb>ponderi ſub<lb></lb>jectus ſit ve<lb></lb>ctis, an vecti pondus. </s> <s id="s.002909">Sint, ut prius, gravitatis ponderis cen<lb></lb>tra D ſuperius, & I inferius, ex quibus in vectem perpendi<lb></lb>culares cadunt DE & IE, quæ, ex 14. lib. 1. ſunt una recta li<lb></lb>nea DI. </s> <s id="s.002910">Jungantur centra potentiæ & ponderis rectâ AD, <lb></lb>quæ ſecat rectam tranſeuntem per fulcrum C & terræ centrum <lb></lb>in puncto M. </s> <s id="s.002911">Quare ex dictis de librâ curva, ſi ſint æqualia <lb></lb>momenta ponderis atque potentiæ, erit ut AM ad MD, ita <lb></lb>pondus D ad potentiam A. </s> <s id="s.002912">Ducatur ex D linea directionis <lb></lb>DN parallela perpendiculari MC; & per 2. lib. 6; eſt ut AM <lb></lb>ad MD, ita AC ad CN: eſt autem CN minor quàm CE, <lb></lb>ergo, ex 8. lib. 5. major eſt Ratio AC ad CN, quàm AC ad <lb></lb>CE. </s> <s id="s.002913">Atqui in vecte horizontali potentia ad pondus eſt ut EC <lb></lb>ad CA; hic autem ut NC ad CA; igitur minor eſt potentia <lb></lb>ſuſtinens pondus impoſitum vecti obliquo ſupra horizontem, <lb></lb>quàm potentia ſuſtinens pondus idem vecte parallelo hori<lb></lb>zonti. </s> </p> <p type="main"> <s id="s.002914">At ſi pondus vecti ſubjiciatur, & ſit ejus gravitatis centrum <lb></lb>I, ducatur recta AI ſecans perpendiculum ex C ductum ad <pb pagenum="382" xlink:href="017/01/398.jpg"></pb>centrum terræ in V. </s> <s id="s.002915">Igitur ſi æqualia ſunt momenta ponde<lb></lb>ris I & potentiæ A, eſt pondus ad potentiam ut AV ad VI. </s> <lb></lb> <s id="s.002916">Ex I centro gravitatis linea directionis IB parallela lineæ CV <lb></lb>occurrat vecti in B; igitur, ex 2. lib.6. ut AV ad VI, ita AC <lb></lb>ad CB: eſt autem CB major quàm CE; ergo AC ad CB ha<lb></lb>bet, ex 8. lib. 5. minorem Rationem, quàm AC ad CE. </s> <s id="s.002917">Cum <lb></lb>itaque in vecte horizontali potentia ad pondus eſſet ut EC ad <lb></lb>CA, hic autem in vecte obliquo ſit ut BC ad eandem CA, <lb></lb>major potentia ſuſtinens hîc requiritur. </s> <s id="s.002918">Quare tantumdem <lb></lb>creſcit ſuſtinendi difficultas in pondere infra vectem adjuncto, <lb></lb>quantum decreſcit in ſuſtinendo pondere ſupra vectem poſito. </s> <lb></lb> <s id="s.002919">Cum enim triangula BEI, DEN ſint æquiangula (quia BI <lb></lb>& DN, per 30. lib. 1. ſunt parallelæ, adeóque per 29. lib. 1. <lb></lb>alterni anguli ad B & N, & alterni ad I & D ſunt æquales, & <lb></lb>reliquus reliquo, per 32. lib.1.) eſt, per 4. lib. 6. ut IE ad ED, <lb></lb>ita BE ad EN: ſunt autem ex hypotheſi DE & IE æquales, <lb></lb>igitur & BE æqualis eſt ipſi EN, illa refert incrementum po<lb></lb>tentiæ, hæc decrementum; ergo æqualiter ibi creſcit, hic de<lb></lb>creſcit difficultas ſuſtinendi pondus. </s> </p> <p type="main"> <s id="s.002920">Contraria ſunt momenta, quæ ponderibus accidunt, vecte <lb></lb>cum pondere infra horizontalem lineam inclinato: concipe <lb></lb>enim hoc idem ſchema ita converſum, ut potentia A ſit in ſu<lb></lb>periore loco, pondera autem I & D ſint infra horizontalem <lb></lb>RT. </s> <s id="s.002921">Jam pondus I incumbit vecti, pondus verò D illi ſub<lb></lb>jectum adnectitur. </s> <s id="s.002922">Igitur pondus I vecti impoſitum majora mo<lb></lb>menta habet vecte cum pondere infra horizontem inclinato, <lb></lb>quàm vecte horizonti parallelo: in hoc autem eodem ſitu in<lb></lb>clinato pondus ſubjectum D minora habet momenta, nam pon<lb></lb>dus I ad potentiam A ſuſtinentem eſt ut AC ad CB majorem, <lb></lb>quæ eſt minor Ratio quàm AC ad CE minorem, ex 8. lib. 5: <lb></lb>è contrario D pondus ad potentiam A ſuſtinentem eſt ut AC <lb></lb>ad CN minorem, quæ eſt major Ratio, quàm AC ad CE ma<lb></lb>jorem. </s> <s id="s.002923">Hinc eſt momenta ponderis vecti ex primo genere im<lb></lb>poſiti infra horizontem majora eſſe, ſupra horizontem minora; <lb></lb>contrà autem ponderis vecti ſubjecti infra horizontem minora <lb></lb>eſſe, ſupra horizontem majora. </s> </p> <p type="main"> <s id="s.002924">Et hæc <expan abbr="quidẽ">quidem</expan> eatenus dicta intelligantur, quatenus concipitur <lb></lb>Potentia vi ſuæ gravitatis rectâ deorſum connitens, adeò ut Di-<pb pagenum="383" xlink:href="017/01/399.jpg"></pb>rectione, <expan abbr="Potẽtiæ">Potentiæ</expan> atque Ponderis ſint parallelæ, propterea enim <lb></lb>conſiderata eſt linea per <expan abbr="centrũ">centrum</expan> motûs, hoc eſt punctum fulcri, <lb></lb>ducta ad <expan abbr="centrũ">centrum</expan> terræ utrique Directioni parallela. </s> <s id="s.002925">At ſi linea Di<lb></lb>rectionis Potentiæ non eſſet parallela Directioni gravitatis Pon<lb></lb>deris ſi res ſcrupuloſius agatur, paulo aliter conſideranda vide<lb></lb>tur linea per punctum fulcri tranſiens, quæ determinet partes li<lb></lb>neæ jungentis Potentiam & Centrum gravitatis ponderis, linea <lb></lb>videlicet per fulcrum ducta ex puncto, in quo concurrunt di<lb></lb>rectiones Potentiæ atque <lb></lb><figure id="id.017.01.399.1.jpg" xlink:href="017/01/399/1.jpg"></figure><lb></lb>Ponderis. </s> <s id="s.002926">Sit Vectis AB <lb></lb>inſiſtens fulcro C depreſ<lb></lb>ſus in A infra horizontem, <lb></lb>ut ſuſtineat pondus D in<lb></lb>cumbens vecti, à quo diſtat <lb></lb>per lineam DE. </s> <s id="s.002927">Directio <lb></lb>gravitatis ponderis eſt per<lb></lb>pendicularis DR, at di<lb></lb>rectio Potentiæ non ſit per<lb></lb>pendicularis AT, verùm <lb></lb>obliqua AR faciens cum <lb></lb>vecte angulum BAR. </s> <lb></lb> <s id="s.002928">Concurrunt itaque di<lb></lb>rectiones Ponderis, & Potentiæ in R. </s> <s id="s.002929">Quare ſicuti quando <lb></lb>ſunt directiones DR & AT parallelæ, premunt fulcrum C <lb></lb>juxta perpendicularem CV, quæ rectam AD ſecat in M, ita <lb></lb>directiones DR & AR videntur premere fulcrum C juxta <lb></lb>rectam CR, quæ producta ſecat rectam AD in S: ac propterea <lb></lb>Ratio Potentiæ ſuſtinentis ad Pondus non eſt ut DM ad MA, <lb></lb>ſed ut DS ad SA. </s> </p> <p type="main"> <s id="s.002930">Hinc eſt lineam directionis Potentiæ, quò majorem angu<lb></lb>lum conſtituit cum vecte in A, eò minorem angulum efficere <lb></lb>cum perpendiculari lineâ directionis ponderis DR productâ, <lb></lb>atque proinde cum illa concurrere multo remotiùs quàm in R, <lb></lb>& lineam ex puncto concursûs directionum ductam ad C, & <lb></lb>ulterius productam ſecare lineam AD inter M & S, adeò ut <lb></lb>aliquando facilè citra notabilem errorem aſſumi poſſit punctum <lb></lb>M: Cum enim DR & MV ſint parallelæ, angulus DRC in<lb></lb>ternus æqualis eſt externo MCS, ex 29. lib. 1. idémque di-<pb pagenum="384" xlink:href="017/01/400.jpg"></pb>cendum de quolibet angulo conſtituto cum perpendiculari <lb></lb>DR à lineâ ex puncto concurſus directionum ducta per C <lb></lb>punctum fulcri: ideò quo minor fit angulus ad B, minor quo<lb></lb>que eſt ad C, & punctum in lineâ AD notatum magis acce<lb></lb>dit ad M. </s> </p> <p type="main"> <s id="s.002931">Hinc pro determinanda Ratione momentorum potentiæ ad <lb></lb>momenta ponderis pro diversâ vectis inclinatione duplici me<lb></lb>thodo uti poteris. </s> <s id="s.002932">Prima eſt, fi ex centro gravitatis ponderis <lb></lb>lineam directionis ducas, punctum enim, in quo hæc occurrit <lb></lb>vecti, illud eſt, quod definit locum ponderis, in quo ſua exer<lb></lb>cet momenta. </s> <s id="s.002933">Secunda eſt, ſi tam ex Potentiæ quàm ex Pon<lb></lb>deris centro gravitatis lineam ducas ad perpendiculum in li<lb></lb>neam horizontalem, quæ tranſit per C punctum fulcri; nam <lb></lb>partes hujus lineæ horizontalis interceptæ inter puncta, in quæ <lb></lb>cadunt perpendiculares, & punctum C, illæ ſunt, quæ reci<lb></lb>procè ſumptæ oſtendunt Rationem ponderis ad potentiam. </s> <s id="s.002934">In <lb></lb>ſitu namque horizontali vectis punctum E congruit puncto S, <lb></lb>& potentia A congruit puncto X: eſt igitur ut AC ad CE ita <lb></lb>XC ad CS: in poſitione autem obliquá ex A in horizontalem <lb></lb>perpendicularis cadit in Z, ex D cadit in K, ex I verò in O. </s> <lb></lb> <s id="s.002935">Quia igitur triangula AZC & NKC ſunt æquiangula, vide<lb></lb>licet rectangula ad Z & K, angulos ad verticem C, ex 15.lib.1; <lb></lb>æquales habent, &, ex 32 lib. 1. reliquum reliquo, eſt per 4. <lb></lb>lib. 6. ut AC ad CN ita ZC ad CK. </s> <s id="s.002936">Similiter triangula <lb></lb>BOC & AZC rectangula ad O & Z angulos ad verticem C <lb></lb>æquales habent, & reliquum reliquo, adeóque ſunt ſimilia, & <lb></lb>ut AC ad CB, ita ZC ad CO, Quare in hac obliquâ vectis <lb></lb>poſitione momentum ponderis D ad momentum potentiæ ſuſti<lb></lb>nentis eſt ut ZC ad CK, & momentum ponderis I ad momen<lb></lb>tum potentiæ ſuſtinentis eſt ut ZC ad CO. </s> </p> <p type="main"> <s id="s.002937">Ex his, quæ de potentia ſuſtentante dicta ſunt, ſatis apparet <lb></lb>potentiam paulo validiorem ſatis eſſe ad pondus movendum. </s> <lb></lb> <s id="s.002938">Verùm licèt in vecte primi generis ad pondus ſuſtentandum <lb></lb>opportunè animum adverterimus ad libram curvam, hæc ta<lb></lb>men in vecte ſecundi generis locum habere non poſſunt; <lb></lb>propterea ad aliam explicandi rationem confugiendum eſt, <lb></lb>quæ utrique generi communis ſit; nec difficile erit ea, quæ ſta<lb></lb>tim capite ſequenti ſubjiciam pro ſecundo vectis genere ad pri-<pb pagenum="385" xlink:href="017/01/401.jpg"></pb>mum traducere. </s> <s id="s.002939">Conſideratur nimirum motus ponderis com<lb></lb>paratus cum eodem motu potentiæ: ſi enim potentia ſit ſuâ <lb></lb>gravitate deſcendens, ejus deſcenſum metitur ZA: pondus <lb></lb>vecti impoſitum aſcendit, ut ſit ſupra horizontalem altitudine <lb></lb>KD; ſed ex hac demenda eſt centri gravitatis diſtantia DE, <lb></lb>qua eminebat ſupra horizontalem, ut habeatur ejus motus <lb></lb>DK minùs DE, hoc eſt GK. </s> <s id="s.002940">Contra verò pondus vecti ſub<lb></lb>jectum erat infra horizontalem diſtantiâ IE, quæ ſi addatur al<lb></lb>titudini OI, dabit OH motum ipſius ponderis. </s> <s id="s.002941">Major eſt au<lb></lb>tem motus OI plus IE, hoc eſt plus DE, quàm ſit motus KD <lb></lb>minùs DE; nam poſita obliquitate lineæ DI, facto centro D, <lb></lb>intervallo DE circulus deſcriptus tranſit per G punctum de<lb></lb>preſſius quàm E, & ex I intervallo IE deſcriptus tranſit per H <lb></lb>punctum altius quàm E: ergo motus ZA ad minorem motum <lb></lb>habet majorem Rationem, quàm ad majorem motum, atque <lb></lb>adeò major eſt movendi facilitas. <lb></lb></s> </p> <p type="main"> <s id="s.002942"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002943"><emph type="center"></emph><emph type="italics"></emph>Momenta ponderis in Vecte ſecundi generis <lb></lb>conſiderantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.002944">IN Vecte ſecundi generis circa extremitatem, ubi eſt ful<lb></lb>crum, deſcribuntur à pondere proximo & à potentiâ remotâ <lb></lb>duo circulorum arcus tanquam circa commune centrum. </s> <s id="s.002945">Et <lb></lb>quidem ſi in eadem rectâ lineâ ſint punctum fulcri, centrum <lb></lb>gravitatis ponderis, & ipſa virtus potentiæ ſurſum aſcendentis, <lb></lb>motus potentiæ & ponderis ſunt in eadem Ratione, in qua ſunt <lb></lb>diſtantiæ ab hypomochlio, ſive pondus ſupra horizontalem <lb></lb>tranſeuntem per fulcrum, ſive à loco inferiore ad horizontalem <lb></lb>elevetur; quia videlicet tam pondus quàm potentia per ſimi<lb></lb>les arcus ab horizontali æqualiter remotos moventur; ac pro<lb></lb>inde eorum arcuum Sinus, qui metiuntur elevationem, ha<lb></lb>bent inter ſe Rationem eandem, quæ eſt radiorum, ſive di<lb></lb>ſtantiarum. </s> </p> <pb pagenum="386" xlink:href="017/01/402.jpg"></pb> <p type="main"> <s id="s.002946">At verò ſi centrum gravitatis ponderis ſit extra lineam <lb></lb>rectam jungentem punctum fulcri cum puncto virtutis poten<lb></lb>tiæ exiſtentis in alterâ vectis extremitate, ſive ſupra vectem, <lb></lb>ſive infra illum ſit, non manet eadem Ratio motuum, quæ eſt <lb></lb>diſtantiarum potentiæ & ponderis (quatenus ponderis diſtan<lb></lb>tia ſumitur à puncto, in quod à centro gravitatis cadit in <lb></lb>vectem perpendicularis) quia aſcenſus & elevationes non ſer<lb></lb>vant eandem Rationem; ex eo quod, licèt in vectis converſio<lb></lb>ne tam centrum gravitatis ponderis quàm centrum potentiæ <lb></lb>deſcribant in motu arcus ſimiles, hi tamen arcus non ſunt ſi<lb></lb>militer poſiti, hoc eſt ſimili modo ab horizontali diſtantes: ac <lb></lb>propterea (ut patet ex doctrina Sinuum) differentiæ Sinuum, <lb></lb>qui conveniunt arcubus ſupra vel infra horizontem, ubi incipit <lb></lb>quadrans circuli, æqualiter creſcentibus, non ſunt æquales: hæ <lb></lb>autem differentiæ metiuntur motum elevationis, qui maximè <lb></lb>attenditur, quatenus opponitur innatæ propenſioni gravitatis. </s> </p> <p type="main"> <s id="s.002947">Sit in C fulcrum vectis CA, & in A ſit potentia movens. </s> <lb></lb> <s id="s.002948">Si centrum gravitatis ponderis ſit in eadem rectâ CBA, ſem<lb></lb><figure id="id.017.01.402.1.jpg" xlink:href="017/01/402/1.jpg"></figure><lb></lb>per motus ponderis & <lb></lb>potentiæ ſunt omnino <lb></lb>ſimiles, & ut CB ad <lb></lb>ad CA; illud enim deſ<lb></lb>cribit arcum BG, hæc <lb></lb>verò arcum AS, & ele<lb></lb>vatio ponderis ex B in <lb></lb>G eſt BR, aſcenſus po<lb></lb>tentiæ eſt AP; & prop<lb></lb>ter triangulorum rectan<lb></lb>gulorum CRB & CPA ſimilitudinem eſt ut CB ad CA, <lb></lb>ita BR ad AP. </s> <s id="s.002949">Et quamvis, diviſo arcu BG in partes ali<lb></lb>quot æquales, & in totidem æquales partes diviſo arcu <lb></lb>ſimili AS, non ſint in ſingulis ejuſdem arcûs partibus æquales <lb></lb>aſcenſus) nam BH minor eſt quàm HI, hic minor quàm IK, <lb></lb>& hic minor quàm KR, ſimiliterque AL minor quàm LM, <lb></lb>hic minor quàm MN, & hic minor quàm NP) comparatis ta<lb></lb>men ſingulis aſcenſibus in minore arcu BG, cum ſingulis <lb></lb>aſcenſibus in arcu majore AS ſibi invicem reſpondentibus, ma<lb></lb>net eadem Ratio, & ut BH ad AL, ita HI ad LM, & ſic de <pb pagenum="387" xlink:href="017/01/403.jpg"></pb>reliquis (ut ex Sinuum doctrinâ manifeſtum eſt, nec opus eſt <lb></lb>hic oſtendere) ſunt enim omnes in Ratione Radij CB ad Ra<lb></lb>dium CA. </s> </p> <p type="main"> <s id="s.002950">Longè aliter ſe res habet, quando extra rectam lineam jun<lb></lb>gentem punctum fulcri cum potentiâ eſt centrum gravitatis <lb></lb>ponderis. </s> <s id="s.002951">Nam ſi Vecti CBA impoſitum ſit pondus, cujus <lb></lb>centrum gravitatis ſit D, potentiâ A deſcribente arcum AQ <lb></lb>centrum gravitatis ponderis deſcribit arcum DE, qui licèt <lb></lb>æqualis ſit arcui BD; habet tamen aſcenſum HI majorem <lb></lb>quàm BH: igitur aſcenſus AL ad HI majorem, habet mino<lb></lb>rem Rationem quàm ad BH minorem, ex 8.lib.5. igitur in hoc <lb></lb>motu Potentia ad Ponderis motum habet minorem Rationem, <lb></lb>quàm ſi centrum gravitatis ponderis eſſet in B; ergo majorem <lb></lb>experitur in movendo difficultatem. </s> </p> <p type="main"> <s id="s.002952">Contrà verò ſi pondus ſit vecti CBA ſubjectum, ejúſque <lb></lb>centrum gravitatis ſit O; dum potentia A deſcribit arcum AQ, <lb></lb>centrum gravitatis O deſcribit arcum OB, ejuſque aſcenſus <lb></lb>eſt OV; atqui OV minor eſt quàm BH; ergo AL aſcenſus <lb></lb>potentiæ ad OV minorem eſt in majori Ratione quàm ad BH <lb></lb>majorem; eſt autem HI major quàm BH; ergo AL ad OV <lb></lb>multo majorem Rationem habet quàm ad HI. </s> <s id="s.002953">Ergo datâ eâ<lb></lb>dem vectis poſitione, eodemque motu, major facilitas erit in <lb></lb>elevando pondere habente centrum gravitatis infra vectem in <lb></lb>O, quàm ſi illud habeat ſupra vectem in D. </s> </p> <p type="main"> <s id="s.002954">Eadem erit demonſtrandi methodus in cæteris aſcenſibus: <lb></lb>nam potentia percurrens arcum AT habet aſcenſum AM, <lb></lb>centrum D percurrit arcum DF, cujus aſcensûs menſura eſt <lb></lb>HK; centrum autem O percurrens arcum OD habet aſcen<lb></lb>ſum OX: cùm igitur OX minor ſit quàm BI, & hic minor <lb></lb>quàm HK, etiam AM ad OX minorem eſt in majore Ratione <lb></lb>quàm ad HK majorem. </s> </p> <p type="main"> <s id="s.002955">Et hæc quidem hactenus dicta intelliguntur de vecte infra <lb></lb>lineam horizonti parallelam depreſſo; nam vecte ſupra hori<lb></lb>zontalem lineam elevato, contraria prorſus accidere ex dictis <lb></lb>demonſtratur. </s> <s id="s.002956">Concipe vectem AC elevatum ſupra horizon<lb></lb>tem, pondus OB eſt illi impoſitum, pondus DB eſt ſubjectum: <lb></lb>quando potentia aſcendens per arcum QA habet aſcenſum <lb></lb>LA, centrum gravitatis O deſcribit arcum BO, & aſcensûs <pb pagenum="388" xlink:href="017/01/404.jpg"></pb>menſura eſt VO; at centrum gravitatis D deſcribens arcum <lb></lb>ED habet aſcenſum IH. </s> <s id="s.002957">Cum igitur oſtenſum ſit majorem <lb></lb>Rationem eſſe LA ad VO, quàm ad IH, ctiam ſupra horizon<lb></lb>tem elevato vecte major erit facilitas in movendo pondere vecti <lb></lb>impoſito, quàm in elevando pondus habens centrum gravita<lb></lb>tis infra vectem. </s> </p> <p type="main"> <s id="s.002958">Ut autem innoteſcat, qua Ratione in progreſſu motûs creſ<lb></lb>cat difficultas, aut minuatur, obſerva ex Canone in arcubus <lb></lb>æqualiter creſcentibus Sinuum differentias ab initio quadran<lb></lb>tis progrediendo uſque ad finem Quadrantis ſemper decreſce<lb></lb>re, harum verò differentiarum differentias, hoc eſt differen<lb></lb>tias ſecundas, ſemper augeri. </s> <s id="s.002959">Hinc eſt ita RK Sinum arcûs <lb></lb>GF majorem eſſe quàm differentiam KI, & KI majorem quàm <lb></lb>IH, & IH majorem quàm HB, ut differentia inter Sinum <lb></lb>RK & differentiam KI minor ſit quàm differentia inter KI <lb></lb>& IH, hæc verò differentia minor ſit quàm differentia inter <lb></lb>IH & HB. </s> <s id="s.002960">Idem dicendum de ſimilibus differentiis inter Si<lb></lb>num PN, & differentias NM, & ML, & LA. </s> <s id="s.002961">In iiſdem li<lb></lb>neis PA & RB particulas aſſumptas donavi vocabulo Sinuum <lb></lb>aut differentiarum, non quaſi ignorans illas particulas non eſſe <lb></lb>Sinus aut differentias Sinuum arcubus æqualiter creſcentibus <lb></lb>reſpondentium, ſed claritatis gratia abutens vocabulo; quan<lb></lb>doquidem illis æquales ſunt, cum aſſumantur per lineas Radio <lb></lb>CS parallelas. </s> </p> <p type="main"> <s id="s.002962">His poſitis intelligatur vectis totus CA cum pondere B intrà <lb></lb>aquam, potentia verò ſit cortex ſuberis, aut uter inflatus, ſeu <lb></lb>veſica, aut quid hujuſmodi levitans. </s> <s id="s.002963">Potentiæ motum metiri <lb></lb>oportet ex naturalibus aſcenſibus AL, LM, & reliquis. </s> <s id="s.002964">Quia <lb></lb>autem eſt ut AL ad LM, ita BH ad HI; etiam viciſſim, per <lb></lb>16. lib. 5. ut AL ad BH, ita LM ad HI, & ſic de cæteris, ſive <lb></lb>infra, ſive ſupra horizontalem: propterea eadem ſemper manet <lb></lb>facilitas aut difficultas elevandi pondus in aquâ gravitans, cu<lb></lb>jus gravitatis centrum congruat vecti CA. </s> <s id="s.002965">Idem dic ſi Poten<lb></lb>tia S in aqua gravitans deprimeret per vim pondus G, quod in <lb></lb>aquâ levitaret: nam PN deſcenſus naturalis potentiæ ad RK <lb></lb>depreſſionem ponderis, eandem Rationem haberet, quam <lb></lb>deſcenſus NM ad depreſſionem KI. </s> </p> <p type="main"> <s id="s.002966">Si vectis ſit CA, cui pondus incumbat habens centrum gra-<pb pagenum="389" xlink:href="017/01/405.jpg"></pb>vitatis D, atque tam pondus quàm potentia ſint in medio, in <lb></lb>quo alterum levitet, alterum gravitet, utriuſque motum qua<lb></lb>tenus naturalis eſt auz violentus, metitur linea perpendicularis <lb></lb>in horizontalem cadens: & ut particulæ ipſæ invicem compa<lb></lb>rentur, Sinuum differentiæ AL, LM &c. </s> <s id="s.002967">BH, HI &c. </s> <s id="s.002968">con<lb></lb>ſiderandæ ſunt. </s> <s id="s.002969">Cum itaque differentia inter BH, & HI ma<lb></lb>jor ſit quàm differentia inter HI, & IK, utique BH magis de<lb></lb>ficit ab æqualitate cum HI, quàm HI cum IK; ideóque mi<lb></lb>nor eſt Ratio BH ad HI, quàm HI ad IK: Atqui eadem eſt <lb></lb>Ratio BH ad HI, quæ eſt AL ad LM; igitur minor eſt etiam <lb></lb>Ratio AL ad LM, quàm HI ad IK, & viciſſim, per 27. lib. 5. <lb></lb>minor eſt Ratio AL ad HI, quàm ſit LM ad IK. </s> <s id="s.002970">Igitur ſi po<lb></lb>tentia A levitet, & pondus, cujus centrum gravitatis D, gravi<lb></lb>tet, aſcendendo ad horizontalem, quæ per fulcrum C tranſit, <lb></lb>acquirit movendi facilitatem. </s> </p> <p type="main"> <s id="s.002971">Jam figuram inverte, ut vectis moveatur ſupra horizontalem: <lb></lb>vecte congruente lineæ horizontali CS, ponderis impoſiti cen<lb></lb>trum gravitatis erit in F, & aſcendet juxta menſuram KI & IH, <lb></lb>cum potentiæ aſcenſus erit PN & NM. </s> <s id="s.002972">Quia igitur differentia <lb></lb>inter Sinum RK & differentiam KI minor eſt, quàm differentia <lb></lb>inter KI & IH, utique RK minùs excedit æqualitatem cum <lb></lb>KI, quàm KI cum IH: ideóque minor eſt Ratio RK ad KI, <lb></lb>quàm KI ad IH. </s> <s id="s.002973">Eſt autem eadem Ratio RK ad KI, quæ eſt <lb></lb>PN ad NM; igitur minor eſt Ratio PN ad NM, quàm KI <lb></lb>ad IH, & viciſſim minor eſt Ratio PN ad KI, quàm NM ad <lb></lb>IH: Igitur aſcendendo magis & recedendo ab horizontali <lb></lb>creſcit movendi facilitas. </s> </p> <p type="main"> <s id="s.002974">Demum ſi vecti CA ſubjectum ſit pondus, cujus centrum <lb></lb>gravitatis O, & potentiæ motum metiatur perpendicularis AP <lb></lb>aſcendendo versùs horizontalem; quia differentia inter OV, & <lb></lb>VX major eſt quàm differentia inter VX & HI, adeóque OV <lb></lb>magis deficit ab æqualitate cum VX, quàm VX cum HI, prop<lb></lb>terea OV ad VX habet minorem Rationem quàm VX ad HI: <lb></lb>ſed ut VX, hoc eſt BH, ad HI, ita AL ad LM; ergo minor eſt <lb></lb>Ratio OV ad VX quàm AL ad LM; & viciſſim minor eſt Ra<lb></lb>tio OV ad AL quàm VX ad LM; ideóque faciliùs elevatur ex <lb></lb>O in B, quàm ex B in D. </s> <s id="s.002975">Factâ autem figuræ converſione, ut <lb></lb>aſcenſus Potentiæ ſit PA, & aſcenſus Ponderis ſit RB, ſi poten-<pb pagenum="390" xlink:href="017/01/406.jpg"></pb>tia ſit in Z, centrum gravitatis ponderis ſubjecti eſt in G, & <lb></lb>dum potentia aſcendit per NM & ML deſcribens arcum ZQ, <lb></lb>pondus aſcendit per RK & KI. </s> <s id="s.002976">Atqui RK ad KI habet mi<lb></lb>norem Rationem quàm KI ad IH, ut ſuperiùs oſtenſum eſt, & <lb></lb>ut KI ad IH, ita NM ad ML; ergo minor eſt Ratio RK ad <lb></lb>KI, quàm NM ad ML, & viciſſim minor eſt Ratio RK ad <lb></lb>NM quàm KI ad ML; ergo faciliùs movetur per RK aſcen<lb></lb>dendo, quàm per KI, adeóque creſcit difficultas elevandi <lb></lb>pondus ſubjectum vecti ſuprà horizontalem, ſi comparentur <lb></lb>inter ſe partes elevationis. </s> </p> <p type="main"> <s id="s.002977">Quare, ut in ſummam ea, quæ dicta ſunt, referantur, ſi pon<lb></lb>dus ſit infra vectem ſecundi generis, faciliùs elevatur eodem <lb></lb>vectis motu versùs horizontalem, quàm ſi fuerit ſupra vectem: <lb></lb>Contrà verò ſupra horizontalem faciliùs eodem vectis motu <lb></lb>elevatur pondus vecti impoſitum, quàm vecti ſubjectum. </s> <s id="s.002978">Con<lb></lb>ſideratis autem particulatim ſingulis elevationibus, diviſo ſcili<lb></lb>cet in æquales particulas univerſo motu ejuſdem ponderis, ſi <lb></lb>pondus ſit in eâdem rectâ lineâ cum fulcro & potentia, eadem <lb></lb>ſemper eſt movendi facilitas aut difficultas: Si pondus ſit ſupra <lb></lb>vectem, & motus infra horizontalem incipiat, ſemper creſcit <lb></lb>movendi facilitas non ſolùm uſque ad horizontalem, verùm <lb></lb>etiam ſupra illam: At ſi pondus ſit infra vectem, motúſque in<lb></lb>fra horizontalem incipiat, augetur ſemper difficultas movendi <lb></lb>tùm uſque ad horizontalem, tùm ſupra illam. </s> </p> <p type="main"> <s id="s.002979">Hæc omnia confirmari poſſunt, ſi lineam directionis per cen<lb></lb>trum gravitatis ponderis ductam produci intelligamus uſque ad <lb></lb>horizontalem lineam, quæ per fulcrum tranſit; Secabit enim <lb></lb>vectem, & in ſectionis puncto quodammodo conſtitutum pon<lb></lb><figure id="id.017.01.406.1.jpg" xlink:href="017/01/406/1.jpg"></figure><lb></lb>dus concipere poſſumus. </s> <s id="s.002980">Sit enim <lb></lb>infra horizontalem CR, vectis <lb></lb>CA, & ad punctum B illi inſiſtat <lb></lb>perpendiculariter linea à centro <lb></lb>gravitatis ducta, ſcilicet DB ſu<lb></lb>pra, & OB infra. </s> <s id="s.002981">Quando vectis <lb></lb>CA congruet lineæ CR, & erit <lb></lb>horizonti parallelus, pondus con<lb></lb>cipietur niti in B contra vectem: <lb></lb>at infra horizontalem centrum D nititur in S, & centrum O <pb pagenum="391" xlink:href="017/01/407.jpg"></pb>in T, juxta lineas directionis DS & OT. </s> <s id="s.002982">Quia igitur punctum <lb></lb>S magis diſtat à fulcro C quàm punctum T, pondus infra <lb></lb>vectem faciliùs ſuſtinetur ſub horizontali, quàm pondus ſupra <lb></lb>vectem. </s> <s id="s.002983">Contra autem ſupra horizontalem centrum O nititur <lb></lb>in I remotiùs à fulcro C, & centrum D in H propiùs; ergo <lb></lb>ſupra horizontalem faciliùs ſuſtinetur pondus vecti impoſitum, <lb></lb>quàm illi ſubjectum. </s> </p> <p type="main"> <s id="s.002984">Quoniam verò triangula rectangula CNT, & OBT, an<lb></lb>gulos ad verticem T æquales habent, & reliquum reliquo <lb></lb>æqualem, erit, ex 4. lib. 6. ut CT ad TN, ita OT ad TB. </s> <lb></lb> <s id="s.002985">Igitur prout ex elevatione vectis minuitur angulus ACN, <lb></lb>etiam minuitur angulus TOB, ac propterea T recedit à ful<lb></lb>cro C verſus B, & augetur ſuſtinendi atque movendi difficul<lb></lb>tas. </s> <s id="s.002986">Iſti autem acceſſus versùs B ſunt inæquales, etiam ſi æqua<lb></lb>lia ſint anguli TOB decrementa, prout decreſcunt angulo<lb></lb>rum ad O factorum Tangentes, poſito Radio OB. </s> <s id="s.002987">Porrò ex <lb></lb>Canone Tangentium conſtat illarum differentias ſemper ma<lb></lb>jores fieri, ſi augeatur angulus, minores fieri, ſi minuatur an<lb></lb>gulus. </s> <s id="s.002988">Igitur recedente lineâ directionis Centri gravitatis O à <lb></lb>fulcro C, augetur difficultas ſuſtinendi & elevandi pondus <lb></lb>vecti ſubjectum: & quia ſupra horizontalem ſemper magis re<lb></lb>cedit ab eodem fulcro C ultrà punctum B versùs A potentiam, <lb></lb>puta, ut ſit OI, multo adhuc major eſt ſuſtinendi atque mo<lb></lb>vendi difficultas. </s> <s id="s.002989">Conſideratis autem particulatim motibus, <lb></lb>quia infra horizontalem differentiæ receſſuum à puncto C fiunt <lb></lb>ſemper minores; propterea creſcit quidem difficultas, ſed inæ<lb></lb>qualibus & minoribus incrementis; quia verò ſupra horizon<lb></lb>talem differentiæ receſſuum à fulcro C fiunt ſemper majores, <lb></lb>creſcit adhuc difficultas, & quidem ſemper majoribus incre<lb></lb>mentis. </s> <s id="s.002990">At ſi pondus ſit D vecti impoſitum, linea directionis <lb></lb>DS accedit versùs B uſque ad horizontalem, ſupra quam re<lb></lb>cedit à B versùs C, ut ſit ex. </s> <s id="s.002991">gr. DH: ſemper igitur faciliùs <lb></lb>movetur, quamquam non æqualibus facilitatis incrementis; <lb></lb>fiunt enim incrementa infra horizontalem ſenſim minora, ſu<lb></lb>pra autem fiunt ſemper majora. </s> <s id="s.002992">Sed hic unum explicandum <lb></lb>eſt, quod fortaſſe alicui animum minùs attentè advertenti dif<lb></lb>ficultatem pariat adversùs ea, quæ ſuperiùs dicta ſunt: videlicet <lb></lb>oſtenſum eſt pondus vecti impoſitum, ſi motus incipiat infra <pb pagenum="392" xlink:href="017/01/408.jpg"></pb>horizontalem, majori difficultate moveri, quàm pondus vecti <lb></lb>ſubjectum. </s> <s id="s.002993">Si enim, inquis, linea Directionis DS magis ac <lb></lb>magis accedit ad B, utique creſcit movendi facilitas; contra <lb></lb>verò lineâ directionis OT accedente ad B creſcit movendi <lb></lb>difficultas. </s> </p> <p type="main"> <s id="s.002994">Ut nodum hunc ſolvas, obſerva triangula SBD, & TBO <lb></lb>rectangula ad B, quia DS & TO ſunt parallelæ, eſſe æquian<lb></lb>gula & ſimilia, immò æqualia, quia ut DB ad OB ſibi ex hy<lb></lb>potheſi æqualem, ita SB ad TB. </s> <s id="s.002995">Igitur qua Ratione minuitur <lb></lb>angulus ACR, etiam minuitur anguius SDB, & angulus <lb></lb>TOB: igitur Tangentium differentiæ fiunt ſemper minores. </s> <lb></lb> <s id="s.002996">Quare in primo motu tam linea directionis DS, quàm linea <lb></lb>directionis OT, magis accedit ad B quàm in ſecundo motu, <lb></lb>& magis in ſecundo, quàm in tertio; acceſſus tamen utriuſ<lb></lb>que lineæ directionis ex eodem vectis motu ſunt æquales; & <lb></lb>qua menſurâ augetur receſſus ponderis D vecti impoſiti, à Po<lb></lb>tentia A, eâdem pariter menſura augetur receſſus ponderis O <lb></lb>vecti ſubjecti, à fulcro C. </s> <s id="s.002997">Itaque creſcit quidem illius facili<lb></lb>tas, hujus difficultas, ſi ponderum ſingulorum motus particu<lb></lb>latim accipiantur, ejuſdémque ponderi, motûs pars cum par<lb></lb>te conferatur: at verò ſi utriuſque ponderis motus invicem <lb></lb>comparentur, utique pondus D difficiliùs movetur, cùm ejus <lb></lb>linea directionis eſt citra punctum B versùs potentiam, quàm <lb></lb>moveatur pondus O, quamdiu ejus linea directionis eſt ultra <lb></lb>idem punctum B. </s> </p> <p type="main"> <s id="s.002998">Ex his, quæ de vecte ſecundi generis dicta ſunt, quid de <lb></lb>vecte tertij generis dicendum ſit, faciliùs innoteſcit, quàm ut <lb></lb>illud pluribus explicari oporteat; potentia ſi quidem & pondus <lb></lb>invicem loca commutant, ſed motuum Ratio eadem eſt, & quæ <lb></lb>in vecte ſecundi generis eſt Ratio motûs Potentiæ ad motum <lb></lb>Ponderis, vice versâ in vecte tertij generis eſt Ratio motûs <lb></lb>Ponderis ad motum Potentiæ. </s> </p> <p type="main"> <s id="s.002999">Hoc te monitum velim, Amice Lector, conſideratum hacte<lb></lb>nus vectem ad movenda ſurſum pondera gravia, aut deprimen<lb></lb>da deorſum levia, & quidem à Potentia, quæ vi ſuæ gravitatis <lb></lb>aut levitatis moveatur, cujus propterea aſcenſum aut deſcen<lb></lb>ſum conſideravimus. </s> <s id="s.003000">Nam ſi in plano horizontali à Potentia <lb></lb>vivente movendum ſit pondus, utique Potentiæ motus circu-<pb pagenum="393" xlink:href="017/01/409.jpg"></pb>laris obſervatur, & attendendum eſt vectis punctum, in quod <lb></lb>cadit linea, quæ à centro gravitatis ponderis in vectem per<lb></lb>pendicularis ducitur, ut ponderis locus ſtatuatur, & momen<lb></lb>ta definiantur. </s> <s id="s.003001">Naturâ quippe comparatum eſt, ut ſi vectis non <lb></lb>occurrat huic perpendiculari, non moveatur totum pondus, <lb></lb>ſed fiat ponderis converſio vel circa gravitatis centrum, vel <lb></lb>circa aliud punctum quod maneat immotum, aut ſaltem mino<lb></lb>re motu moveatur. <lb></lb> </s> </p> <p type="main"> <s id="s.003002"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003003"><emph type="center"></emph><emph type="italics"></emph>Quæ ſit Ratio Vectis Hypomochlium mobile <lb></lb>habentis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003004">NOn hîc hypomochlium mobile illud intelligo, quod ſimul <lb></lb>cum pondere à potentiâ ſuſtentato ad eaſdem partes pro<lb></lb>movetur; cujuſmodi ſunt manualia bajulorum vehicula, quæ <lb></lb>unicâ rotâ inſtruuntur, & habentia rationem vectis ſecundi ge<lb></lb>neris; nam fulcrum habent in axe rotæ, & potentiam in extre<lb></lb>mitate manubriorum, quibus illa ſuſtinet pondus transferen<lb></lb>dum: cui propterea addita eſt rota illa verſatilis, ut etiam hy<lb></lb>pomochlium citra difficultatem, quin atterat ſubjectam plani<lb></lb>tiem, ſimul cum pondere jam elevato, atque ſuſtentato pro<lb></lb>moveatur. </s> </p> <p type="main"> <s id="s.003005">Hujuſmodi pariter eſt novitium vehiculi genus, cui Sellæ <lb></lb>Rotatæ nomen fecerunt, hoc uno à lecticâ viatoriâ diſcrepans, <lb></lb>quòd loco poſterioris jumenti ſuſtinentis additus eſt axis dua<lb></lb>bus rotis infixus, cui innituntur vectes ab anteriore equo <lb></lb>ſuſtentanti unâ cum pondere intermedio. </s> <s id="s.003006">Hic eſt vectis ſecun<lb></lb>di generis, cujus hypomochlium ſequitur potentiam trahen<lb></lb>tem pariter ac ſuſtentantem impoſitum pondus, non mutatâ <lb></lb>Ratione momenti potentiæ ſuſtinentis, ſive hypomochlium <lb></lb>moveatur, ſive ſtabile ſit ac fixum. </s> <s id="s.003007">Cæterùm quò pondus ma<lb></lb>gis à rotis diſtat, magis equum gravat, minùs autem ſubſilit, <lb></lb>cùm rotæ in offendiculum incurrunt. </s> </p> <p type="main"> <s id="s.003008">Nomine igitur hypomochlij mobilis illud intelligo, quod <pb pagenum="394" xlink:href="017/01/410.jpg"></pb>movente potentiâ atque conante adversùs pondus, reſiſtit qui<lb></lb>dem vecti, ſed & ſimul loco cedit ita, ut pondus & hypomo<lb></lb>chlium in oppoſitas partes immiſſo inter illa vecte moveantur. </s> <lb></lb> <s id="s.003009">Sic contingere poteſt fulcrum deprimi, dum pondus elevatur, <lb></lb>aut fulcrum elevari, dum pondus deprimitur, aut ſi utrumque <lb></lb>in plano horizontali moveatur, in oppoſitas plagas recedere. </s> <lb></lb> <s id="s.003010">Loquor autem de vecte primi & ſecundi generis, quibus com<lb></lb>muniter utimur; nam in vecte tertij generis, ſi hypomochlium <lb></lb>cedat, movetur ad eaſdem partes cum pondere & potentia, ſed <lb></lb>tardiùs. </s> <s id="s.003011">Hinc ſi vecte inter duo pondera non immodicè inæ<lb></lb>qualia interjecto alterutrum movere coneris, reliquum etiam <lb></lb>movetur; ita tamen ut neutrum tantum motûs perficiat, quan<lb></lb>tum haberent ſingula, ſi ſolitariè moverentur, reliquo manen<lb></lb>te immoto. </s> </p> <p type="main"> <s id="s.003012">Sit vectis AB inter duos lapides C & D interjectus, qui la<lb></lb>pidem C non dimovebit, niſi eum tangat in puncto cui occur<lb></lb><figure id="id.017.01.410.1.jpg" xlink:href="017/01/410/1.jpg"></figure><lb></lb>rit linea ex C gravitatis centro <lb></lb>ducta (aut potiùs planum per <lb></lb>idem gravitatis centrum C <lb></lb>tranſiens) ad perpendiculum <lb></lb>in vectem, & ſit linea CE; niſi <lb></lb>enim in E lapis à vecte tanga<lb></lb>tur, movebitur quidem lapis circa centrum C, donec congruat <lb></lb>vecti, ſed non propelletur totus lapis. </s> <s id="s.003013">Idem dic de lapide D, <lb></lb>niſi tangatur in F occurrente lineæ perpendiculari DF. </s> <s id="s.003014">Qua<lb></lb>re pondera intelligantur in E & F: & quoniam F reſiſtit vecti, <lb></lb>ut E propellatur versùs C, & viciſſim E reſiſtit vecti, ut F <lb></lb>propellatur versùs D, propterea ad movendum pondus C, <lb></lb>vectis AE eſt primi generis, & ad movendum pondus D, <lb></lb>vectis AE eſt ſecundi generis; atque pondera illa viciſſim ha<lb></lb>bent rationem hypomochlij, quia vectis alteri innititur, ut al<lb></lb>terum moveat. </s> </p> <p type="main"> <s id="s.003015">Cæterùm ſingulorum lapidum abſoluta & ſimpliciter ſumpta <lb></lb>reſiſtentia tum ex eorum ingenitâ gravitate, tum ex ſuperfi<lb></lb>cierum ſe tangentium aſperitate atque conflictu definitur: <lb></lb>Comparatè verò ad vectem non ſic accipienda eſt ſingulorum <lb></lb>reſiſtentia, quaſi motûs centra eſſent E aut F: experimento <lb></lb>enim manifeſto deprehenditur motum potentiæ A ad motum <pb pagenum="395" xlink:href="017/01/411.jpg"></pb>ponderis C non eſſe ut AF ad FE, neque ejuſdem potentiæ A <lb></lb>æqualem motum eſſe ad motum ponderis D ut AE ad FE. </s> <lb></lb> <s id="s.003016">Nam ſi punctum E vectis fixum eſſet, & potentiæ motus eſſet <lb></lb>AL, motus ponderis F eſſet FH: Si verò punctum F mane<lb></lb>ret immotum, & potentiæ <lb></lb><figure id="id.017.01.411.1.jpg" xlink:href="017/01/411/1.jpg"></figure><lb></lb>motus eſſet AI æqualis ipſi <lb></lb>AL, motus ponderis eſſet <lb></lb>EG. </s> <s id="s.003017">Tunc autem motus AI <lb></lb>æqualis eſt motui AL, quan<lb></lb>do ut AF ad AE, ita viciſſim <lb></lb>angulus AEL ad angulum <lb></lb>AFI: æqualium ſi quidem <lb></lb>angulorum in circulis inæ<lb></lb>qualibus arcus ſunt ut Radij; <lb></lb>ergo ſi fuerint anguli reciprocè ut Radij, ſcilicet minor in ma<lb></lb>jore circulo, & major angulus in minore, erunt æquales arcus <lb></lb>illis oppoſiti: Sic anguli AFR æqualis angulo AEL arcus AR <lb></lb>eſt ad AL, ut Radius FA ad Radium EA; ſed ut FA ad EA, <lb></lb>ita arcus AR ad arcum AI ex conſtructione; ergo ut AR ad <lb></lb>AL ita AR ad AI: ergo per 9.lib.5. AI & AL ſunt æquales. </s> </p> <p type="main"> <s id="s.003018">Quoniam igitur tam E quàm F ex hypotheſi in oppoſitas <lb></lb>partes moventur circumacto vecte, punctum aliquod eſt inter E <lb></lb>& F, quod eſt veluti centrum motuum tam potentiæ quàm pon<lb></lb>derum, in quo centro quodammodo diviſa intelligitur re<lb></lb>ſiſtentia, quæ componitur tùm ex eorum innatâ gravitate, <lb></lb>tùm ex eorum motu, ſpectatâ poſitione ad vectem. </s> <s id="s.003019">Hinc ma<lb></lb>nifeſtum eſt ſingula pondera minùs moveri, quàm ſi ſingula <lb></lb>moverentur reliquo manente immoto; quia videlicet ſingula <lb></lb>minùs diſtant à centro, circa quod moventur. </s> <s id="s.003020">Sic ponderum <lb></lb>E & F gravitas ponatur æqualis: ſi intelligatur centrum mo<lb></lb>tûs ab utroque æqualiter diſtare, ut ſit KE æqualis ipſi KF, <lb></lb>motus potentiæ factus intervallo AK æqualem habet Ratio<lb></lb>nem ad motum, qui fit à ſingulis ponderibus. </s> </p> <p type="main"> <s id="s.003021">Quare potentiæ momentum perinde ſe habet, atque ſi utrum<lb></lb>que pondus eſſet in E, aut utrumque in F, hypomochlium verò <lb></lb>in K. </s> <s id="s.003022">Ponamus enim EF eſſe partium 6, quarum partium 7 eſt <lb></lb>FA: igitur EK eſt 3, & KA 10; & potentia ſine vecte movens <lb></lb>lib.3, vecte AKE movebit lib.10 in E. </s> <s id="s.003023">Similiter KF eſt 3, & <pb pagenum="396" xlink:href="017/01/412.jpg"></pb>KA eſt 10; igitur potentia ut 3 in A, movebit in F pondus ut <lb></lb>10: igitur etiam in A potentia ut 6, facto motûs centro K, mo<lb></lb>vebit vel utrumque pondus ut 10 in E & F, vel unicum pondus <lb></lb>ut 20 ſive in E, ſive in F. </s> <s id="s.003024">Conſtituatur itaque potentiæ virtus <lb></lb>ut 6, ſi hypomochlium eſſet F immotum, non moveret niſi pon<lb></lb>dus grave ut 7 poſitum in E; & facto hypomochlio ſtabili E <lb></lb>moveret pondus grave 13 poſitum in F; adeóque univerſum <lb></lb>pondus eſſet librarum 20. Quare idem pondus lib. 20 movetur <lb></lb>ab eâdem potentia, ſed non eodem motu: Nam hîc amborum <lb></lb>ſimul ponderum motus circa centrum K eſt ut 6; at ſi potentiæ <lb></lb>motus AI ſit 10 (quemadmodum motus potentiæ circa cen<lb></lb>trum K eſt 10) circa F centrum, motus EG eſt 8 4/7; & ſi po<lb></lb>tentiæ motus AL ſit pariter 10 circa centrum E motus FH <lb></lb>eſt (4 8/13). </s> </p> <p type="main"> <s id="s.003025">Hinc patet ſingulorum ponderum motum, quando utrum<lb></lb>que ſimul movetur, minorem eſſe, quàm ſi ſingula ſolitariè <lb></lb>moverentur, adeóque totum motum, qui ex duobus motibus <lb></lb>coaleſcit, minorem eſſe ſummâ, quæ conflatur ex motu EG <lb></lb>& motu FH. </s> <s id="s.003026">Præterea manifeſtum eſt cæteris paribus move<lb></lb>ri faciliùs pondus F, quod eſt Potentiæ A proximum, quàm <lb></lb>pondus E ab eádem remotum; minor enim differentia eſt in<lb></lb>ter (4 8/13) & 3, quàm inter 8 4/7 & 3. </s> </p> <p type="main"> <s id="s.003027">Quod ſi duorum ponderum E & F abſoluta reſiſtentia, quæ <lb></lb>ex gravitate oritur, inæqualis fuerit, inæqualem pariter eſſe <lb></lb>oportet reſiſtentiam ex motûs velocitate, quæ unicuique pon<lb></lb>deri conveniat, ſed reciprocè, ut fiat totius reſiſtentiæ æquali<lb></lb>tas. </s> <s id="s.003028">Cum enim utrumque pondus movendum ſit, par eſt ita re<lb></lb>ſiſtentiam dividi, ut æqualibus momentis adverſentur poten<lb></lb>tiæ contranitenti; quod ſcilicet gravius eſt, difficiliùs movetur, <lb></lb>quod minus grave, faciliùs: igitur illius motus minor eſt, hu<lb></lb>jus major. </s> <s id="s.003029">Propterea centrum motuum iis intervallis ab utro<lb></lb>que pondere aberit, ut quæ Ratio eſt gravioris ponderis ad mi<lb></lb>nus grave, ea ſit Ratio diſtantiæ centri motûs à minùs gravi ad <lb></lb>diſtantiam ejuſdem centri à graviore. </s> <s id="s.003030">Sit ex. </s> <s id="s.003031">gr. pondus E lib.8. <lb></lb>& pondus F lib.12; diſtantia EF eadem quæ priùs, hoc eſt, 6; & <lb></lb>FA 7. Cum igitur pondera ſint ut 2 ad 3, dividatur EF in quin<lb></lb>que partes, & propè gravius F aſſumantur duæ FM, reliquæ <pb pagenum="397" xlink:href="017/01/413.jpg"></pb>tres ME ſpectent ad minus grave E. </s> <s id="s.003032">Si itaque circa centrum <lb></lb>M moveantur pondera E & F, habent æqualia reſiſtentiæ mo<lb></lb>menta; nam lib. 12 moventur ut 2, & lib. 8 moventur ut 3. <lb></lb>Quare AM eſt ad ME ut 9 2/5 ad 3 3/5, & AM ad MF eſt ut 9 2/5 <lb></lb>ad 2 2/5. Fiat igitur ut AM ad ME, ita reciprocè pondus E <lb></lb>lib. 8 ad virtutem potentiæ A movendi ſine vecte libras (3 3/47): & <lb></lb>ut AM ad MF, ita reciprocè pondus F lib.12 ad ejuſdem po<lb></lb>tentiæ A virtutem movendi ſinè vecte libras (3 3/47). In hac ita<lb></lb>que ponderum inæqualium diſpoſitione paulo plus virium re<lb></lb>quiritur in potentia (hoc eſt vis movendi lib. (6 6/47)) quàm ſi eſ<lb></lb>ſent æqualia, eandemque gravitatis ſummam lib. 20 conſti<lb></lb>tuerent. </s> </p> <p type="main"> <s id="s.003033">At ſi vice versâ pondus E eſſet lib. 12, & F lib. 8, centrum <lb></lb>motuum eſſet N, atque AN eſſet 10 3/5: ac propterea ut AN <lb></lb>10 3/5 ad NE 2 2/5, ita pondus E lib. 12 ad virtutem potentiæ ſi<lb></lb>ne vecte moventis libras (2 38/53); & ut AN 10 3/5 ad NF 3 3/5, ita <lb></lb>pondus F lib. 8 ad virtutem potentiæ A moventis ſine vecte li<lb></lb>bras (2 38/53). Tota igitur virtus potentiæ in hac eorumdem pon<lb></lb>derum inæqualium collocatione ſufficiet, ſi fuerit vis movendi <lb></lb>lib. (5 23/53), quæ minor eſt eâ, quæ requiritur; quando pondera <lb></lb>ſunt æqualia, & differt à virtute, quæ requiritur, quando F <lb></lb>gravius eſt quàm E, vi movendi ferè uncias 8 1/3. </s> </p> <p type="main"> <s id="s.003034">Simili argumentatione ratiocinando deprehendes, quo mi<lb></lb>nus fuerit intervallum inter E & F, etiam faciliùs duo illa pon<lb></lb>dera eodem vecte moveri. </s> <s id="s.003035">Nam ſi idem vectis AE 13 adhi<lb></lb>beatur, atque pondera E & F æqualia fuerint, intervallum ve<lb></lb>rò EF ſit 4, centrum motuum diſtabit ab A intervallo 11, & <lb></lb>à ſingulis ponderibus intervallo 2: Quare potentia ut 4 move<lb></lb>bit pondera ſingula ut 11: vel ſi ponantur ut prius ſingula <lb></lb>lib. 10, fiat ut 11 ad 2, ita lib. 10 ad potentiam ſine vecte mo<lb></lb>ventem lib. (1 9/11); atque ideò tota potentia ſufficiens ad movenda <lb></lb>duo pondera æqualia ſimul ſumpta lib. 20, erit vis movendi ſine <lb></lb>vecte lib. (3 7/11). Quod ſi E fuerit lib. 8, & F lib.12, E diſtabit à <lb></lb>à centro motuum partibus 2 2/5, F verò part. </s> <s id="s.003036">1 3/5, & potentia A <lb></lb>diſtabit part. </s> <s id="s.003037">10 3/5: Ex quo fit ſingula moveri poſſe à potentia <pb pagenum="398" xlink:href="017/01/414.jpg"></pb>habente virtutem movendi ſine vecte lib. (1 43/53), & ambo ſimul à <lb></lb>potentia habente vim movendi lib. (3 33/53). At verò ſi viciſſim E <lb></lb>fuerit lib. 12, & F lib. 8, diſtantia potentiæ à centro motuum <lb></lb>erit part. </s> <s id="s.003038">11 2/5, ac propterea ſingula pondera exigent virtutem <lb></lb>movendi lib. (1 39/57), & tota potentia ad utrumque ſimul moven<lb></lb>dum ſufficiens erit vis movendi ſine vecte lib. (3 21/57), quæ deficit <lb></lb>à vi movendi lib. (3 33/53), ea virtute, quæ requireretur ad moven<lb></lb>dum uncias (3 1/20), atque à vi movendi lib. (3 7/11) deficit per uncias <lb></lb>3 1/5 ferè. </s> </p> <p type="main"> <s id="s.003039">Que de corpore gravi dimovendo dicta ſunt, intelligantur <lb></lb>pariter, ſi vectis inter duo corpora flectenda, aut divellenda, <lb></lb>interjiceretur; quemadmodum objectos caveæ ſi quæras fran<lb></lb>gere clathros: quod enim hîc gravitas, ibi ferreæ virgæ aut <lb></lb>lignei tigilli ſoliditas impedimentum affert motui. </s> </p> <p type="main"> <s id="s.003040">Porrò in vecte tertij generis, quando potentia inter utrum<lb></lb>que pondus mobile conſtituitur, aliter res ſe habet: adhoc ſci<lb></lb>licet, ut aliquam vectis Rationem habeat, requiritur aut inæ<lb></lb>qualitas ponderum, aut ſaltem inæqualitas diſtantiarum po<lb></lb>tentiæ à ponderibus in utrâque extremitate conſtitutis, ita ta<lb></lb>men ut hæ diſtantiæ non ſint in reciprocâ Ratione ponderum: <lb></lb>nam ſi planè æqualiter diſtaret potentia ab æqualibus ponderi<lb></lb>bus, aut inæquales diſtantiæ eſſent in reciprocâ Ratione inæ<lb></lb>qualium ponderum, ita utrumque traheretur, aut impellere<lb></lb>tur, ut pondera ſingula æquè moverentur ac potentia: ad Ra<lb></lb>tiones autem vectis ſpectat inæqualiter moveri potentiam ac <lb></lb>pondus, ſi vectis quidem obtineat vim Facultatis Mechanicæ. </s> </p> <p type="main"> <s id="s.003041">Quoniam igitur in hujuſmodi vecte tertii generis oportet <lb></lb>utrumque pondus opponi motui potentiæ; vel quia utrumque <lb></lb>impellitur, vel quia alterum trahitur, alterum impellitur, ſit <lb></lb><figure id="id.017.01.414.1.jpg" xlink:href="017/01/414/1.jpg"></figure><lb></lb>vectis AB, in cujus extremitati<lb></lb>bus pondera reſpondeant punctis <lb></lb>A & B: Si potentia fuerit in C <lb></lb>æquè diſtans ab A & B, pondera <lb></lb>autem fuerint æqualia; potentia <lb></lb>ex C versùs D mota nullum ha<lb></lb>beret ſui motûs centrum, ſed pariter traheret aut impelleret <lb></lb>ad partes D utrumque pondus; nam æquè utrumque reſiſteret <pb pagenum="399" xlink:href="017/01/415.jpg"></pb>tùm ratione gravitatis, tùm ratione poſitionis & diſtantiæ, quæ <lb></lb>legem daret motui, ac proinde utrumque æqualiter cederet <lb></lb>virtuti potentiæ. </s> <s id="s.003042">At ſi pondus A minus fuerit, quàm pondus B, <lb></lb>ſed reciprocam Rationem habeant diſtantiæ potentiæ exiſtentis <lb></lb>in E, ut ſit EB ad EA, in Ratione ponderis A ad pondus B; <lb></lb>adhuc æquales ſunt reſiſtentiæ; ſicut enim in plano Verticali <lb></lb>potentiæ in E ſuſtineret utrumque pondus in æquilibrio, ita in <lb></lb>plano horizontali trahens aut impellens utrumque æqualiter <lb></lb>moveret. </s> </p> <p type="main"> <s id="s.003043">Sint igitur pondera A & B ſive æqualis gravitatis, ſive inæ<lb></lb>qualis, & ita potentia ſit in E, ut EB ad EA non ſit in Ratio<lb></lb>ne ponderis A ad pondus B: utique ſi B moveri non poſſet, <lb></lb>potentia E circa B, tanquam circa centrum, deſcriberet ar<lb></lb>cum EI, & pondus A arcum AF: ſimiliter ſi pondus A immo<lb></lb>tum maneret, potentia circa A, tanquam circa centrum, deſ<lb></lb>criberet arcum EH, & pondus B arcum BG, ex hypotheſi <lb></lb>æqualem arcui AF. </s> <s id="s.003044">Potentia igitur in E faciliùs cæteris pari<lb></lb>bus moveret pondus B ſibi proximum, quàm pondus A remo<lb></lb>tum, ſi ſingula ſingillatim movenda eſſent; quia, cum arcus <lb></lb>EH major ſit arcu EI, arcus autem BG, & AF ſint æquales, <lb></lb>major eſt Ratio EH ad EI quàm BG ad AF; & per 27. lib.5. <lb></lb>viciſſim EH ad BG habet majorem Rationem quàm EI ad <lb></lb>AF. </s> <s id="s.003045">Cum itaque neutra extremitas immota maneat, ſed ambo <lb></lb>pondera moveantur, minùs movetur A, quod difficiliùs, ma<lb></lb>gis B, quod faciliùs: ac propterea A ſimpliciter fungitur mu<lb></lb>nere hypomochlij ad motum ponderis B: hoc verò viciſſim ad <lb></lb>ponderis A motum, quamvis minorem, ſubit vicem fulcri: <lb></lb>Neque enim hic unum tribus motibus, potentiæ videlicet & <lb></lb>duorum ponderum, commune centrum reperire eſt, quia ad <lb></lb>eandem partem omnium motus dirigitur. </s> <s id="s.003046">Hinc ſi fune alligato <lb></lb>in E trahas vectem cum ponderibus, punctum E neque omni<lb></lb>no versùs I, neque omnino versùs H dirigetur, quamquam ad <lb></lb>H potiùs, quàm ad I inclinabitur; quia faciliùs A vectis <lb></lb>punctum reſpondens ponderi convertitur circa centrum gravi<lb></lb>tatis ponderis, quàm propellat aut trahat totum pondus, pro <lb></lb>ut ferunt, & ipſius gravitas, & ejuſdem diſtantia ab E, quæ il<lb></lb>lius reſiſtentiam componunt. <pb pagenum="400" xlink:href="017/01/416.jpg"></pb></s> </p> <p type="main"> <s id="s.003047"><emph type="center"></emph>CAPUT VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003048"><emph type="center"></emph><emph type="italics"></emph>Quænam ſint momenta Vectis pondus fune <lb></lb>connexum trahentis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003049">COntingere poteſt oblato ponderi ſuper planum horizonta<lb></lb>le, aut inclinatum, trahendo non eſſe parem Potentiam: <lb></lb>hujus imbecillitati opem ferre licebit Vecte potiſſimùm ſecun<lb></lb>di generis, cujus extremitas altera fixa & ſtabilis maneat in pla<lb></lb>no, in quo pondus jacet, alteram extremitatem Potentia mo<lb></lb>veat, & loco intermedio alligetur funis cum pondere connexus, <lb></lb>qui dum vecte movetur, ſecum rapiat & pondus. </s> <s id="s.003050">Verùm non <lb></lb>leviter hallucinaretur, quiſquis momenta vectis ex alligati fu<lb></lb>nis loco ſimpliciter & abſolutè definiret; cum potiùs ponderis <lb></lb>reſiſtentiam ex ipſius motu computare oporteat. </s> <s id="s.003051">Quoniam verò <lb></lb>duplex eſſe poteſt vectis motus, nimirum aut in plano Verticali, <lb></lb>aut in Horizontali, propterea uterque ſeorſim conſiderandus <lb></lb>eſt; diverſas enim lineas in plano, in quo jacet, pondus percur<lb></lb>rit; rectam ſcilicet, ſi vectis in plano Verticali agitatur; curvam <lb></lb>verò, ſi in plano horizontali aut inclinato eodem, cui pondus <lb></lb>incumbit etiam vectis moveatur. </s> </p> <p type="main"> <s id="s.003052">Sit in plano, in quo pondus jacet, linea AB, cui vectis con<lb></lb>gruere intelligatur, & concipiatur pondus in puncto C; vecti <lb></lb><figure id="id.017.01.416.1.jpg" xlink:href="017/01/416/1.jpg"></figure><lb></lb>autem in D alligetur funis ita <lb></lb>connectens pondus cum vecte, <lb></lb>ut parti vectis DA æqualis ſit <lb></lb>funis DC. </s> <s id="s.003053">Attollatur in plano <lb></lb>Verticali vectis, ut ſit AE <lb></lb>deſcribens arcum BE; etiam <lb></lb>punctum D aſcendit in F, ac <lb></lb>propterea funis eſt FG, & <lb></lb>ponderis motus eſt CG. </s> <s id="s.003054">Ite<lb></lb>rum attollatur æqualiter vectis <lb></lb>ex E in H; funis caput venit in I, & pondus in K. </s> <s id="s.003055">Similiter <pb pagenum="401" xlink:href="017/01/417.jpg"></pb>vecte in L ſublato, funis venit in M, & pondus in N. </s> <s id="s.003056">Sunt <lb></lb>igitur tre, ponderis motus, CG, GK, KN, inter ſe inæqua<lb></lb>les, qui ſemper majores fiunt; motus autem potentiæ BE, EH, <lb></lb>HL ex hypotheſi ſunt æquales; igitur major eſt Ratio motûs <lb></lb>BE ad motum CG, quàm motûs EH ad motum GK, & hæc <lb></lb>Ratio major eſt Ratione motûs HL ad motum KN. </s> <s id="s.003057">Cum ita<lb></lb>que motibus BE, EH, HL ſimiles ſint motus DF, FI, IM, <lb></lb>manifeſtum eſt motum ponderis non ſervare Rationem ſecun<lb></lb>dùm quam dividitur vectis ab alligati funis capite, eadem quip<lb></lb>pe ſemper eſt Ratio EA ad AF, & HA ad AI, & LA <lb></lb>ad AM. </s> </p> <p type="main"> <s id="s.003058">Motus autem illos ponderis CG, GK & KN ſemper eſſe <lb></lb>majores hinc conſtat, quia pars vectis inter funem alligatum <lb></lb>atque hypomochlium A ex hypotheſi eſt æqualis ipſi funi con<lb></lb>nectenti pondus: ſunt igitur triangula Iſoſcelia æqualium ſem<lb></lb>per laterum, ſed quæ majores & majores angulos ad baſim ha<lb></lb>bent, ideóque minorem & minorem angulum verticalem con<lb></lb>tinent. </s> <s id="s.003059">Atqui angulorum ad centrum in circulis æqualibus, <lb></lb>vel in eodem circulo, ſemper æqualiter decreſcentium ſubten<lb></lb>ſæ minores fiunt eâ lege, ut decrementa illa, hoc eſt, ſubtenſa<lb></lb>rum diminutarum differentiæ augeantur, ut ex Canone Si<lb></lb>nuum conſtat. </s> <s id="s.003060">Cum itaque AG ſit ſubtenſa anguli AFG, & <lb></lb>AK ſit ſubtenſa anguli AIK minoris, & AN ſit ſubtonſa an<lb></lb>guli AMN adhuc minoris; harum ſubtenſarum differentiæ, <lb></lb>videlicet CG (differentia inter diametrum circuli AC & ſub<lb></lb>tenſam AG) GK & KN motus ponderis ſemper augentur. </s> </p> <p type="main"> <s id="s.003061">Id quod ut manifeſtum fiat, triangula ipſa ad calculos revo<lb></lb>cemus ſingulorum baſim inquirentes: ponamus verò ex. </s> <s id="s.003062">gr. ar<lb></lb>cus BE, EH, HL ſingulos grad. 15, & latera ſingula AF & <lb></lb>GF eſſe partium 100. Igitur angulus AFG eſt grad. 150, & <lb></lb>baſis AG deprehenditur partium (193 18/100). Eſt autem AC ex <lb></lb>hypotheſi 200, adeóque CG part. (6 82/100). In triangulo AIK la<lb></lb>tera ſunt eadem, anguli ad baſim ſinguli grad. 30, angulus ver<lb></lb>ticalis grad. 120; ergo baſis AK part. (173 20/100): & inter AK at<lb></lb>que AG differentia GK eſt (19 98/100). Deinde in triangulo AMN <lb></lb>anguli ad baſim ſinguli ſunt grad. 45; igitur angulus vertica<lb></lb>lis grad. 90, & baſis AN part. (141 42/100), & inter AN & AK dif-<pb pagenum="402" xlink:href="017/01/418.jpg"></pb>ferentia KN eſt (31 78/100). Et ſi vectem elevando pergas, idem in <lb></lb>conſequentibus triangulis deprehendes, augeri ſcilicet diffe<lb></lb>rentia, uſque ad A. </s> </p> <p type="main"> <s id="s.003063">Hinc patet eò faciliorem eſſe, cæteris paribus, motum, quò <lb></lb>majorem angulum funis cum vecte conſtituit, nam ab æquali <lb></lb>potentiæ motu minor ponderis motus efficitur, quam ſi major <lb></lb>eſſet angulus elevationis vectis. </s> <s id="s.003064">Quare faciliùs promovebitur <lb></lb>ad deſtinatum locum pondus quod trahitur, ſi poſt aliqualem <lb></lb>vectis elevationem, iterum inclinato, quàm maximè fieri po<lb></lb>terit, vecte, extremitatem A, hoc eſt hypomochlium ſubinde <lb></lb>promoveas, quantum feret funis longitudo: tunc enim fune <lb></lb>maximè inclinato tractio ponderis minùs obliqua juvabit mo<lb></lb>tum, qui etiam minor eſt, quàm ſi pergeres vectem elevando. </s> </p> <p type="main"> <s id="s.003065">Non eſt tamen neceſſe ſervari hanc, quam claritatis gratiâ <lb></lb>propoſui, æqualitatem funis FG, & partis vectis FA; ſed aſſu<lb></lb>mi poteſt vel longior, vel brevior funis, adeò ut ex vectis par<lb></lb>te, ex fune, & ex diſtantiâ ponderis ab hypomochlio fiat trian<lb></lb>gulum ſcalenum: in quo ſi funis fuerit longior parte vectis, eo<lb></lb>dem potentiæ motu minùs accedet pondus ad hypomochlium, <lb></lb>quàm ſi funis fuerit brevior eâdem vectis parte; atque quo lon<lb></lb>gior fuerit funis, etiam minor erit, adeóque facilior, ponderis <lb></lb>motu, cæteris paribus, nam & tractio minùs obliqua erit. </s> <lb></lb> <s id="s.003066">Statue igitur ex. </s> <s id="s.003067">gr. AD eſſe partium 73, & DC part. </s> <s id="s.003068">100: <lb></lb>quare vecte jacente, diſtantia AC eſt 173. </s> </p> <p type="main"> <s id="s.003069">Sit angulus FAG iterum gr. 15; invenitur angulus AFG <lb></lb>gr. 154. m. </s> <s id="s.003070">7, & baſis AG part. (168 66/100); igitur CG (4 34/100). In <lb></lb>triangulo AIK latera AI 73, IK 100 ut priùs, angulus IAK <lb></lb>gr. 30: invenitur angulus verticalis AIK gr. 128. m. </s> <s id="s.003071">36, & <lb></lb>baſis AK part. (156 30/100): igitur GK eſt (12 36/100). Demum in trian<lb></lb>gulo AMN latera ſunt eadem ut priùs, angulus MAN gr.45: <lb></lb>ex quibus datis invenitur angulus AMN gr.103. m. </s> <s id="s.003072">55, & ba<lb></lb>ſis AN part. (137 27/100): igitur KN (19 3/100), & totus motus CN eſt <lb></lb>part. (35 73/100). </s> </p> <p type="main"> <s id="s.003073">Sed viciſſim ſtatue AD partium 100, DC verò funem <lb></lb>part. </s> <s id="s.003074">73; quibus æqualia ſunt trianguli AFG latera AF 100 <lb></lb>& FG 73; angulus autem FAG eſt gr. 15: invenitur angulus <lb></lb>FGA gr. 20. m. </s> <s id="s.003075">46, & angulus AFG gr. 144. m. </s> <s id="s.003076">46: quare <pb pagenum="403" xlink:href="017/01/419.jpg"></pb>AG eſt part. (164 85/100), & motus CG part. (8 15/100), qui tamen ſu<lb></lb>periùs, quando DC major erat, quàm AD, deprehenſus eſt <lb></lb>ſolum (4 34/100). In Triangulo AIK ſimiliter datur AI 100, IK 73, <lb></lb>angulus IAK gr. 30: invenitur angulus IKA gr. 43. m. </s> <s id="s.003077">14, <lb></lb>ac proinde angulus verticalis AIK gr. 106. m. </s> <s id="s.003078">46, & baſis AK <lb></lb>part. (139 79/100): igitur GK part. (25 6/100), quæ tamen priùs erat <lb></lb>(12 36/100). Deinde in triangulo AMN latera ſint eadem, & an<lb></lb>gulus MAN gr. 45: invenitur angulus MNA gr. 75. m. </s> <s id="s.003079">37, <lb></lb>& verticalis AMN gr.59.m.23: quare baſis AN eſt par. (88 85/100), <lb></lb>& motus KN part. (50 94/100), qui priùs erat (19 3/100). Verùm ele<lb></lb>vari vectis poterit ſolùm, ut funis fiat perpendicularis horizon<lb></lb>ti, ſcilicet facto angulo ad A gr. 46. m. </s> <s id="s.003080">53; & baſis erit diſtan<lb></lb>tia ab A part. (68 35/100). </s> </p> <p type="main"> <s id="s.003081">Ut autem innoteſcat, quid contingat fune adhuc longiore <lb></lb>quàm part. </s> <s id="s.003082">100, poſitâ eádem vectis parte AD part. </s> <s id="s.003083">73. non <lb></lb>pigeat iterum examinare triangula. </s> <s id="s.003084">Sit ergo funis DC <lb></lb>part. </s> <s id="s.003085">200, quarum AD eſt 73; anguli elevationis vectis ſint <lb></lb>iidem, qui ſuperiùs. </s> <s id="s.003086">In Triangulo AFG, angulus FAG eſt <lb></lb>gr. 15, latus AF 73, latus FG 200: invenitur angulus FGA <lb></lb>gr. 5. m. </s> <s id="s.003087">25, & angulus AFG gr. 159. m. </s> <s id="s.003088">35: ac proinde baſis <lb></lb>AG part. (269 56/100), & motus CG part. (3 44/100), qui, poſito fune <lb></lb>FG 100, erat (4 34/100). In triangulo AIK latera ſunt eadem, <lb></lb>angulus IAK eſt gr. 30; ergo angulus IKA gr. 10. m. </s> <s id="s.003089">31, & <lb></lb>verticalis AIK gr. 139. m. </s> <s id="s.003090">29; atque baſis AK part. (259 87/100); ac <lb></lb>propterea GK part. (9 69/100), quæ priùs fuit (12 36/100). Denique in <lb></lb>Triangulo AMN eadem latera 73 & 200 cum angulo MAN <lb></lb>gr. 45, dant angulum MNA gr. 14. m. </s> <s id="s.003091">57, & verticalem <lb></lb>AMN gr. 120. m. </s> <s id="s.003092">3; atque baſim AN part. (244 82/100): quare <lb></lb>motus KN eſt (15 5/100), qui in priore hypotheſi erat (19 3/100). Lon<lb></lb>gior itaque funis dat minorem & faciliorem motum pon<lb></lb>deris. </s> </p> <p type="main"> <s id="s.003093">Quemadmodum verò elevando vectem à poſitione hori<lb></lb>zontali uſque ad perpendiculum difficultas trahendi augetur, <lb></lb>quia pondus velociùs movetur, ita ex adverſo, ſi vectis hori<lb></lb>zonti perpendicularis inclinetur ad partem oppoſitam ponderi <pb pagenum="404" xlink:href="017/01/420.jpg"></pb>(adeò ut vectis ſit inter potentiam & pondus) creſcit trahendi <lb></lb>facilitas, quia pondus ſemper tardiùs movetur, quo magis <lb></lb><figure id="id.017.01.420.1.jpg" xlink:href="017/01/420/1.jpg"></figure><lb></lb>vectis ad horizontem de<lb></lb>primitur. </s> <s id="s.003094">Sit enim pon<lb></lb>dus in P, vectis perpendi<lb></lb>cularis CB, funis OP <lb></lb>utique longior parte vectis <lb></lb>OC. </s> </p> <p type="main"> <s id="s.003095">Inclinetur vectis per <lb></lb>Quadrantis trientem, ut O <lb></lb>veniat in S; funis erit ST, <lb></lb>& pondus veniet ex P in T. Æquali inclinatione deprimatur <lb></lb>vectis, ut S veniat in M; funis erit MV, & ponderis motus <lb></lb>TV. </s> <s id="s.003096">Demum vectis horizonti congruat, ut M veniat in N; <lb></lb>funis erit NI, motúſque ponderis VI. </s> <s id="s.003097">Cum igitur ſemper tar<lb></lb>diùs moveatur pondus, quia ſpatia PT, TV, VI ſemper de<lb></lb>creſcunt, æquales autem potentiæ motus ill's reſpondeant, <lb></lb>etiam creſcit trahendi facilitas. </s> </p> <p type="main"> <s id="s.003098">Illa autem baſium CP, CT, CV decrementa in triangulis <lb></lb>COP, CST, CMV ſemper minui conſtabit ex Trigonome<lb></lb>tria; dantur enim in omnibus eadem duo latera, ſcilicet funis <lb></lb>longitudo, & pars vectis, datúrque in ſingulis æqualiter <lb></lb>creſcens angulus funi oppoſitus; quare inveniuntur & baſes, <lb></lb>quarum differentiæ ſemper minores fiunt. </s> <s id="s.003099">Sic in triangulo <lb></lb>OCP rectangulo ſit perpendiculum OC partium 73, & hypo<lb></lb>thenuſa OP part. </s> <s id="s.003100">100; igitur CP baſis eſt part. (68 34/100). Deinde <lb></lb>quia CS eſt part. </s> <s id="s.003101">73, ST part. </s> <s id="s.003102">100, & angulus SCT gr. 120, <lb></lb>invenitur CT part. (40 97/100): ergo PT eſt part. (27 37/100). Similiter <lb></lb>MC eſt 73, MV 100, angulus MCV gr. 150; igitur CV in<lb></lb>venitur part. (29 91/100); ac proinde TV eſt part. (11 6/100). Demum <lb></lb>quia NC eſt 73, & NI eſt 100, remanet CI part. </s> <s id="s.003103">27: & <lb></lb>ablatâ CI ex CV, relinquitur IV part. (2 91/100). Totus itaque <lb></lb>motus PI eſt part. (41 34/100). Fac autem OC 73 eſſe quartam to<lb></lb>tius vectis partem, qui proinde erit part. </s> <s id="s.003104">292. Et quia Radius <lb></lb>ad Quadrantem peripheriæ circuli eſt ut 7 ad 11, ſi fiat ut 7 <lb></lb>ad 11, ita 292 ad 458 6/7, potentia in vectis extremitate poſita <pb pagenum="405" xlink:href="017/01/421.jpg"></pb>motum habet part. </s> <s id="s.003105">458, dum pondus movetur ſolum per ſpa<lb></lb>tium part. </s> <s id="s.003106">41. </s> </p> <p type="main"> <s id="s.003107">Jam verò finge omnia eadem, præter funis longitudinem, <lb></lb>quam ſtatuamus OP ex. </s> <s id="s.003108">gr. partium 200, quarum OC eſt 73. <lb></lb>igitur CP eſt 186 1/5; & quia NC eſt 73, atque NI ex hypo<lb></lb>theſi eſt 200, remanet CI part. </s> <s id="s.003109">127; atque adeò totus motus <lb></lb>PI eſt part. </s> <s id="s.003110">59 1/5: ad quem motum idem potentiæ motus 458 <lb></lb>habet minorem Rationem quàm ad motum part.41, quem dat <lb></lb>minor funis longitudo. </s> </p> <p type="main"> <s id="s.003111">Supereſt adhuc tertia quædam ponderis poſitio, vecte agita<lb></lb>to in plano verticali; quando nimirum initio motûs ſtatuitur <lb></lb>pondus proximum hypomochlio, à quo in motu recedat: hu<lb></lb>juſmodi ſcilicet vecte uti poſſumus, cùm aliquid modicè qui<lb></lb>dem movendum in plano horizontali proponitur, ſed multa eſt <lb></lb>difficultas. </s> <s id="s.003112">Similiter ſi longiuſcula ferrea bractea eſſet ſuis ex<lb></lb>tremitatibus validè connexa cum aliquo corpore, & circa me<lb></lb>dium eam flecti oporteret, ut cuneus vel aliquid ſimile inter <lb></lb>corpus & bracteam inſeri poſſet; funi adnecteretur uncus <lb></lb>bracteam apprehendens, qui elevato vecte, ſive depreſſo il<lb></lb>lam aliquantulum flecteret. </s> </p> <p type="main"> <s id="s.003113">Sit vectis hypomochlium in R, funis in K alligatus, & funis <lb></lb>longitudo KS 73; pars verò vectis RK 100: quare RS diſtan<lb></lb>tia ponderis S ab hypomo<lb></lb><figure id="id.017.01.421.1.jpg" xlink:href="017/01/421/1.jpg"></figure><lb></lb>chlio R eſt 27. Moveatur <lb></lb>vectis ſurſum, & faciat angu<lb></lb>lum LRK gr. 15 funis eſt LI <lb></lb>part. </s> <s id="s.003114">73, & LR part. </s> <s id="s.003115">100; igi<lb></lb>tur ex his datis invenitur an<lb></lb>gulus RIL gr. 159. m. </s> <s id="s.003116">14, & <lb></lb>RLI gr. 5. m. </s> <s id="s.003117">46; adeóque <lb></lb>baſis RI part. (28 34/100); igitur mo<lb></lb>tus ex S in I eſt (1 34/100). Quod <lb></lb>ſi ponatur RL eſſe quarta pars vectis, totus Radius eſt part.400, <lb></lb>& arcus ab extremitate vectis deſcriptus gr. 15, eſt part. <lb></lb>(104 32/100): Ex quo vides motum potentiæ ad motum ponderis eſſe <lb></lb>proximè ut 78 ad 1. At verò ſi funis LI longior ponatur, ut ſit <lb></lb>part. </s> <s id="s.003118">90, & reliqua ſint ut priùs, invenitur angulus RIL gr. 163. <pb pagenum="406" xlink:href="017/01/422.jpg"></pb>m. </s> <s id="s.003119">17, & angulus RLI gr. 1. m. </s> <s id="s.003120">43′: quare baſis RI eſt part. <lb></lb>(10 4s/100): & quia RS ex hypotheſi eſt ſolum part. </s> <s id="s.003121">10, motus SI <lb></lb>eſt (42/100) multo minor quàm cum funis brevior ponitur, ac prop<lb></lb>terea etiam facilior eſt motus, quippe qui minorem Rationem <lb></lb>habet ad motum potentiæ. </s> </p> <p type="main"> <s id="s.003122">Pergendo autem in elevatione vectis adhuc per gr. 15, ita ut <lb></lb>angulus PRO ſit gr. 30, PR eſt 100, PO eſt 73: invenitur <lb></lb>angulus ROP gr. 136. m. </s> <s id="s.003123">46, & angulus RPO gr. 13. m. </s> <s id="s.003124">14, <lb></lb>atque baſis RO part. (33 42/100): quare motus IO eſt part. (5 8/100). Et <lb></lb>iterum elevando vectem per gr. 15, ita ut angulus QRT ſit <lb></lb>gr. 45, invenitur angulus QTR gr. 104. m. </s> <s id="s.003125">23, & angulus <lb></lb>RQT gr. 30. m. </s> <s id="s.003126">37, baſis autem RT part. (52 57/100): ex quo fit <lb></lb>motum OT eſſe partium (19 15/100). Hinc patet æqualibus poten<lb></lb>tiæ motibus inæquales, ſempérque majores ponderis motus <lb></lb>reſpondere, ac proinde creſcere movendi difficultatem; cum <lb></lb>enim pondus ſuâ gravitate inſiſtat ſubjecto plano, in quo trahi<lb></lb>tur, quò magis elevatur vectis, etiam funis magis obliquus eſt, <lb></lb>minuſque valida fit tractio, quæ magis obliqua eſt. </s> </p> <p type="main"> <s id="s.003127">Quas hactenus recenſuimus tractiones, fieri per lineam <lb></lb>rectam vel accedendo ad punctum hypomochlij, vel ab eo re<lb></lb>cedendo, ſatis conſtat; quia, dum vectis in plano Verticali <lb></lb>movetur, pondus non recedit ab illo eodem plano Verticali <lb></lb>ſemper ſuâ gravitate inſiſtens plano horizontali, atque idcirco <lb></lb>motus illius eſt in communi horum planorum ſectione, hoc <lb></lb>eſt, in lineâ rectâ. </s> <s id="s.003128">Sin autem motus vectis fuerit in eodem <lb></lb>plano horizontali, in quo eſt pondus fune trahendum, quia <lb></lb>vectis circulariter movetur, illum ſequitur pondus per lineam <lb></lb>curvam, ſed quo ad ejus fieri poſſit, breviſſimam, ut quàm mi<lb></lb>nimam patiatur violentiam. </s> <s id="s.003129">Certum quippe eſt oportere fu<lb></lb>nem vecti congruentem citra quamlibet anguli inclinationem, <lb></lb>eſſe breviorem parte illâ vectis, quæ inter hypomochlium, & <lb></lb>locum alligati funis, intercipitur; ſi enim æqualis eſſet, cir<lb></lb>cumducto vecte pondus fulcro proximum non moveretur; <lb></lb>multo minùs, ſi longior eſſet funis. </s> <s id="s.003130">Cum itaque brevior ſit, <lb></lb>neceſſe eſt pondus quoque circumduci, ſed non eâ ratione, <lb></lb>qua moveretur, ſi funis eundem ſemper angulum cum vecte <lb></lb>conſtitueret; quemadmodum contingeret, ſi vecti loco funis <pb pagenum="407" xlink:href="017/01/423.jpg"></pb>flexilis rigidum brachium adjaceret, cur pondus adnecteretur. </s> <lb></lb> <s id="s.003131">Verùm quia pondus ſuâ gravitate reſiſtit, dum vectis movetur, <lb></lb>recinetur aliquantulum funis à pondere, & angulum ſubinde <lb></lb>majorem cum vecte efficit; trahitur tamen pondus, ſed ita, ut <lb></lb>violentiam ſubeat quàm minimam pro ratione poſitionis; ac <lb></lb>propterea lineam curvam helici ſimilem deſcribit, quo ad fu<lb></lb>nis certum angulum acutum (pro ut funis, aut pars vectis lon<lb></lb>giores fuerint, ſive breviores) cum vecte conſtituat; quo de<lb></lb>inde angulo manente pondus in gyrum ducitur per circuli am<lb></lb>bitum. </s> <s id="s.003132">Obſervabis enim poſitâ eadem vectis parte, quò bre<lb></lb>vior fuerit funis, eò majorem eſſe angulum illum, ad quem de<lb></lb>venitur, & in quo conſiſtitur nec illum augendo, nec minuendo. </s> <lb></lb> <s id="s.003133">Quod ſi in funem eâdem vectis parte longiorem ita diſponas, ut <lb></lb>non vecti congruat, ſed cum illo angulum efficiat, circumducto <lb></lb>vecte ita pondus per helicem moveri videbis, ut diminuto ſub<lb></lb>inde angulo, demum funis vecti in eadem rectâ lineâ congruat, <lb></lb>& ponderis ultra hypomochlium manentis tractio deſinat: quia <lb></lb>videlicet ponderis gravitas reſiſtens licèt trahatur, retinet ta<lb></lb>men funem, & minuitur angulus, uſque dum omnis angulus <lb></lb>pereat. </s> </p> <p type="main"> <s id="s.003134">Hujuſmodi motûs cauſam deprehendes, ſi attentè inſpicias <lb></lb>pondus, cum vecti, in gyrum moveri incipit, ita trahi, ut etiam <lb></lb>aliquantulum circa ſuum centrum gravitatis, aut circa aliud <lb></lb>punctum (neque enim hic locus eſt punctum illud definiendi) <lb></lb>volvatur; ex qua converſione fit minore motu opus eſſe, ut <lb></lb>pondus conſequatur trahentem: ubi verò tanta facta fuerit <lb></lb>ponderis circa ſuum centrum converſio, ut ſi in hanc, vel in <lb></lb>illam partem adhuc converteretur, majorem ſubiret in tractio<lb></lb>ne violentiam, hoc eſt, cogeretur majorem motum perficere, <lb></lb>quàm ſit motus ejuſdem nulla factâ circa ſuum illud centrum <lb></lb>converſione, tunc manet angulus funis cum vecte, nec jam au<lb></lb>getur aut minuitur. </s> <s id="s.003135">Quia autem, cæteris paribus, quò bre<lb></lb>vior eſt funis, cò major fit ponderis circa ſuum centrum con<lb></lb>verſio; propterea ad majorem angulum demum inclinatur fu<lb></lb>nis in vectem. </s> <s id="s.003136">Sed in hoc non eſt diutiùs immorandum; rarus <lb></lb>quippe eſt hujuſmodi tractionis uſus. <pb pagenum="408" xlink:href="017/01/424.jpg"></pb></s> </p> <p type="main"> <s id="s.003137"><emph type="center"></emph>CAPUT VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003138"><emph type="center"></emph><emph type="italics"></emph>Quid conferat Potentiæ moventis applicatio <lb></lb>ad vectem.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003139">QUoniam duplex eſt Potentiæ genus, alia ſiquidem inani<lb></lb>ma eſt, cujus conatus ſecundùm rectam lineam in cen<lb></lb>trum, vel à centro, dirigitur, prout gravis eſt aut levis, alia <lb></lb>eſt vivens, quæ pro variâ muſculorum intentione, ac mem<lb></lb>brorum inflexione in aliam atque aliam partem dirigi poteſt; <lb></lb>idcirco, cujuſmodi potentiâ uti liceat, conſiderandum eſt, ut <lb></lb>opportunum vectis genus eligatur. </s> <s id="s.003140">Nam ſi vectis primi gene<lb></lb>ris depreſſione attollendum ſit pondus, & potentia ſit vivens, <lb></lb>ut homo, major eſt movendi facilitas, tum quia ipſum vectis <lb></lb>pondus potentiam juvat ſuâ gravitate, tum quia in hujuſmodi <lb></lb>depreſſione vectis non ſolùm brachiorum, ſed etiam quando<lb></lb>que totius humani corporis vecti incumbentis, vel ex vecte <lb></lb>pendentis gravitas momentum non leve addit contentioni, qua <lb></lb>virtus movendi impetum vecti imprimen connititur. </s> <s id="s.003141">Sin au<lb></lb>tem vecte ſecundi, aut tertij generis elevandum ſit pondus <lb></lb>idem, ipſa vectis gravitas officit, quam pariter cum ipſo pon<lb></lb>dere attollere oportet, & majore virium contentione opus eſt, <lb></lb>ut experientiâ docemur. </s> </p> <p type="main"> <s id="s.003142">Verùm illud, quod hîc potiſſimum examinandum proponi<lb></lb>tur, eſt ipſa potentiæ, quæcumque illa ſit, applicatio ad <lb></lb>vectem: neque enim ſatis eſt, ſi illa extremitati vectis adjun<lb></lb>gatur, aut certo quodam loco in vecte tertij generis collocata <lb></lb>intelligatur; ſed maximè attendendum eſt, ſecundùm quam <lb></lb>lineam potentiæ motus dirigatur; diverſa quippe ſunt poten<lb></lb>tiæ momenta pro alia atque aliâ hujuſmodi motûs directione, <lb></lb>quatenus cum vecte comparatur. </s> <s id="s.003143">Quemadmodum enim ſi po<lb></lb>tentia vectem urgeat, aut trahat, juxta ejuſdem vectis in hy<lb></lb>pomochlij puncto firmati longitudinem, nihil prorsùs in pon<lb></lb>dere efficit; ita quoquè ſi in vectis longitudinem obliquè inci-<pb pagenum="409" xlink:href="017/01/425.jpg"></pb>dat impetûs a potentiâ concepti directio, pro ratione obliqui<lb></lb>tatis minuitur potentiæ momentum; quod integrum manet, ſi <lb></lb>ad angulos rectos vecti occurrat linea motus, quam init po<lb></lb>tentia. </s> </p> <p type="main"> <s id="s.003144">Sit vectis primi generis AB habens hypomochlium in C, ſi<lb></lb>ve ſecundi aut tertij generis DE habens hypomochlium in D. </s> <lb></lb> <s id="s.003145">Si potentia conſtituta in B, aut <lb></lb>E aut F, motum ſuum dirigat <lb></lb><figure id="id.017.01.425.1.jpg" xlink:href="017/01/425/1.jpg"></figure><lb></lb>ſecundùm eandem rectam li<lb></lb>neam BA aut ED, ſive urgen<lb></lb>do vectem versùs C aut D, ſive <lb></lb>illum inde retrahendo, mani<lb></lb>feſtum eſt, puncto hypomo<lb></lb>chlij C aut D manente, pondus <lb></lb>in A, aut in F, aut in E conſti<lb></lb>tutum nihil prorſus moveri, nam totus potentiæ conatus irri<lb></lb>tus eſt, nec vectem movet. </s> <s id="s.003146">Oportet igitur lineam, ſecundùm <lb></lb>quam dirigitur motus potentiæ, conſtituere cum vecti, longi<lb></lb>tudine angulum aut rectum, aut recto minorem, aut majorem. </s> <lb></lb> <s id="s.003147">Si angulum acutum HBA efficiat, movetur quidem potentia <lb></lb>& vectis, ſed cùm urgeatur vectis versùs hypomochlium C, <lb></lb>impeditur potentia, nec movet vectem pro ratione impetûs, <lb></lb>quem illa concipit. </s> <s id="s.003148">Similiter ſi directio motûs potentiæ ſit ſe<lb></lb>cundùm lineam BG, & fiat angulus obtuſus GBA, quamvis <lb></lb>vectem moveat, minùs tamen illum flectit aut deprimit, quàm <lb></lb>requirat impetûs concepti intenſio, quia conatur vectem re<lb></lb>trahere ab hypomochlio C, & ab illo retinetur. </s> <s id="s.003149">Cùm autem, <lb></lb>quò acutior aut obtuſior eſt angulus, eò etiam majus ſit impe<lb></lb>dimentum, hinc eſt pariter plus laboris à potentiâ impendi. </s> <lb></lb> <s id="s.003150">Quare, cùm nullum ſit hujuſmodi impedimentum, quando ad <lb></lb>rectos cum vecte angulos potentiæ motus dirigitur, ut IBA, <lb></lb>propterea tunc ſolùm potentia obtinet omnia momenta, quæ <lb></lb>concepto impetui reſpondent: nihil enim impetûs deteritur ab <lb></lb>impedimento, quod vectis inferat, quippe qui nec versùs hy<lb></lb>pomochlium urgetur, nec ab illo retrahitur. </s> </p> <p type="main"> <s id="s.003151">Porrò obſerva longè aliam eſſe lineam motûs potentiæ, à li<lb></lb>neâ ſecundùm quam ejuſdem potentiæ motus dirigitur; nam <lb></lb>potentia in B applicata movetur deſcribendo arcum circa C <pb pagenum="410" xlink:href="017/01/426.jpg"></pb>punctum hypomochlij, ſed pro varià directione modò majo<lb></lb>rem, modò minorem arcum deſcribit eodem tempore ex vi <lb></lb>ejuſdem impetûs concepti. </s> <s id="s.003152">Hinc eſt, niſi potentia ſuum mo<lb></lb>tum in gyrum dirigat, fieri non poſſe, ut in motu eadem ſer<lb></lb>vet virium momenta: Nam licèt eandem directionem ſervaret, <lb></lb>quatenus horizontem reſpicit, aut certum aliquod punctum, <lb></lb>non eſſet tamen eadem directio comparata cum vecte; alium <lb></lb>quippe atque alium cum vecte angulum conſtitueret illa ea<lb></lb>dem directionis linea: id quod manifeſtò conſtat, cùm vectis <lb></lb>à potentiâ gravi deprimitur; linea enim directionis in centrum <lb></lb>gravium directa ſemper obliquior incidit in vectem, qui depri<lb></lb>mitur. </s> </p> <p type="main"> <s id="s.003153">Voco autem <emph type="italics"></emph>Directionem motûs<emph.end type="italics"></emph.end> lineam illam, quam potentia <lb></lb>ex vi concepti impetûs ſponte percurreret, niſi ab illâ deflecte<lb></lb>re cogeretur, quia cum vecte connectitur. </s> <s id="s.003154">Sic potentia B li<lb></lb>neam BH ex. </s> <s id="s.003155">gr. percurreret, niſi vectis in C firmati ſoliditas <lb></lb>obſtaret, cogerétque arcum BR deſcribere: idem de cæteris <lb></lb>lineis dicendum. </s> <s id="s.003156">Hinc ſi longitudo BH concipiatur ſpatium, <lb></lb>quod à potentiâ libera vi ſui impetûs certo tempore perficere<lb></lb>tur, illa utique non recederet à lineâ AB niſi pro ratione Si<lb></lb>nûs Recti angulo HBA convenientis poſito Radio BH, ſcili<lb></lb>cet per HO. </s> <s id="s.003157">Similiter ſi directio motûs ſit BG angulum obtu<lb></lb>ſum GBA conſtituens, potentia non recederet ab eâdem lineâ <lb></lb>AB niſi pro ratione GK ſinus Recti ejuſdem anguli GBA ob<lb></lb>tuſi poſito Radio BG, qui ex hypotheſi æqualis eſt Radio BH, <lb></lb>ponitur enim utrobique æqualis impetus potentiæ. </s> <s id="s.003158">Quare cùm <lb></lb>idem ſit Sinus Rectus anguli acuti, atque obtuſi, quorum ſum<lb></lb>ma æquatur duobus rectis, eadem pariter momenta virium <lb></lb>exercet potentia, ſive ad acutum, ſive ad obtuſum cum vecte <lb></lb>angulum dirigatur. </s> <s id="s.003159">Hoc tamen intercedit diſcrimen, quando <lb></lb>potentia eandem ſervat ad horizontem directionem, quod acu<lb></lb>tus angulus procedente motu fit major accedens ad Rectum, <lb></lb>augetúrquo ejus Sinus; obtuſus verò angulus fit obtuſior magis <lb></lb>recedens à Recto, minuitúrque ejus Sinus; ac proinde ibi au<lb></lb>getur, hîc minuitur movendi facilitas. </s> </p> <p type="main"> <s id="s.003160">Potentia itaque motum ſuum dirigens ad acutum angulum <lb></lb>per lineam BH, vi ſui impetûs deſcribit circa centrum C ar<lb></lb>cum BR; ad angulum rectum per lineam BI deſcribit arcum <pb pagenum="411" xlink:href="017/01/427.jpg"></pb>BN; ad anguium demum obtuſum per lineam BG deſcribit <lb></lb>arcum BL, qui eſt æqualis arcui BR, ſi angulu; obtuſus GBA <lb></lb>ſit ſupplementum ad duo, recto, anguli acuti HBA, major <lb></lb>autem, aut mino, eodem arcu BR, ſi angulus obtuſus ſit mi<lb></lb>nor aut major codem Supplemento ad duos recto. </s> <s id="s.003161">Hæc tamen <lb></lb>ita dicta intelligas velim, ut hujuſmodi arcus toti atque integri <lb></lb>non motum ipſum exprimant, qui revera fiat, ſed virium Ra<lb></lb>tionem pro diversa potentiæ applicatione in minimâ arcû deſ<lb></lb>cripti particula; neque enim ſingulis temporis momentis æqua<lb></lb>lis pars arcus eidem impetui reſpondet, ſingulis nimirum mo<lb></lb>mentis mutatur vectis inclinatio, & manet eadem motús di<lb></lb>rectio, atque varia eſt potentiæ ad vectem applicatio, niſi il<lb></lb>la impetum concipiat, quo ſua ſponte in gyrum ageretur, <lb></lb>etiamſi ad motum circularem non determinaretur à vecte. </s> <lb></lb> <s id="s.003162">Sed quoniam arcus eodem Radio CB à potentiâ deſcriptus <lb></lb>eſt ſimilis arcui eodem Radio CA deſcripto à pondere, <lb></lb>proinde non mutatur Ratio motuum, ſive potentia deſcri<lb></lb>bat codem impetu minorem, ſive majorem arcum ejuſdem <lb></lb>circuli: propterea non mutatur quidem momentum poten<lb></lb>tiæ cum pondere abſolutè comparatæ, mutatur tamen ſubinde <lb></lb>momentum potentiæ, quatenus ſecum ipſa comparatur, faci<lb></lb>liúſque movere pondus tunc dicitur, quando eodem conatu <lb></lb>majorem motum ponderi æquali tempore conciliat; id quod <lb></lb>fit, cùm ad angulum rectum vecti applicatur. </s> </p> <p type="main"> <s id="s.003163">Hæc eadem, quæ in vecte primi generis explicata ſunt, <lb></lb>in reliquis pariter duobus generibus locum habent, nec opus <lb></lb>eſt illa iterum inculcare. </s> <s id="s.003164">Unum in his obſervandum vide<lb></lb>tur, quando potentia movens eſt à vecte ſejuncta, illúm<lb></lb>que trahendo movet certo in loco firmiter conſtituta, vectem <lb></lb>in motu propiùs accedere ad potentiam trahentem, ac proin<lb></lb>de diligenter attendendam eſſe ipſius potentiæ poſitionem, ut <lb></lb>innoteſcat, utrùm angulus, quem ſubinde cum vecte funicu<lb></lb>lus efficit, accedat magis ad rectum, an verò recedat à recto, <lb></lb>quia in motu acutior aut obtuſior evadat. </s> <s id="s.003165">Id quod ſatis fuerit <lb></lb>ſubindicáſſe; præſtat ſiquidem laborem in motu minui, quàm <lb></lb>augeri. </s> </p> <p type="main"> <s id="s.003166">Sic dato vecte CB ſecundi generis habente hypomo<lb></lb>chlium in C, ſtatue quantum moveri debeat, ex. </s> <s id="s.003167">gr. per <pb pagenum="412" xlink:href="017/01/428.jpg"></pb>arcum BD. </s> <s id="s.003168">Erit igitur vectis poſitio CD. </s> <s id="s.003169">Excitetur ex D <lb></lb><figure id="id.017.01.428.1.jpg" xlink:href="017/01/428/1.jpg"></figure><lb></lb>perpendicularis DE, & in ali<lb></lb>quo rectæ lineæ DE puncto, pu<lb></lb>ta in E, ſtatuatur potentia, quæ <lb></lb>funiculo EB trahens vectem ita <lb></lb>vecti intelligatur applicata, ut <lb></lb>majorem ſubinde angulum effi<lb></lb>ciat, donec ad rectum CDE de<lb></lb>veniat. </s> <s id="s.003170">Sic facilior erit motus, & <lb></lb>labor minuetur. </s> <s id="s.003171">Quod ſi poten<lb></lb>tia movendo pergeret adhuc trahens vectem, jam augeretur <lb></lb>labor, quia applicaretur ad angulum obtuſum. </s> <s id="s.003172">Porrò in recta <lb></lb>DE eligendum eſſe punctum, quoad fieri poterit, proximum <lb></lb>puncto D, ut funiculus EB minùs acutum angulum cum vecte <lb></lb>CB conſtituat, apertius eſt, quàm ut oporteat id pluribus <lb></lb>hic oſtendere. </s> <s id="s.003173">Eximendus tamen eſt omnis ſcrupulus, oſten<lb></lb>dendo angulum ſemper majorem fieri, quando diſtantia po<lb></lb>tentiæ E ab hypomochlio C major eſt longitudine dati <lb></lb>vectis CB. </s> </p> <p type="main"> <s id="s.003174">Intelligatur deſcriptus integer circulus BGIL à vecte CB <lb></lb>circumducto, & producatur BC in I, atque funiculus EB ſe<lb></lb><figure id="id.017.01.428.2.jpg" xlink:href="017/01/428/2.jpg"></figure><lb></lb>cet peripheriam in G. </s> <s id="s.003175">Tum vectis <lb></lb>poſitio fiat CO, & funiculus EO <lb></lb>ſecet peripheriam in H. </s> <s id="s.003176">Mani<lb></lb>feſtum ex 33. lib. 6. angulum <lb></lb>COE majorem eſſe angulo CBE: <lb></lb>nam, productâ OC in L, angulus <lb></lb>CBE inſiſtit arcui GI, angulus <lb></lb>autem COE inſiſtit arcui HL, qui <lb></lb>major eſt arcu GI: omnes autem <lb></lb>acuti ſunt, quia inſiſtunt periphe<lb></lb>riæ minori, quàm ſit ſemicirculus <lb></lb>(ſunt enim, ex 20.lib.3, ſubdupli <lb></lb>ſuorum angulorum ad centrum <lb></lb>iiſdem peripheriis inſiſtentium) <lb></lb>donec funiculus ED ſit Tangens circuli, & ex 18.lib.3. angu<lb></lb>lum rectum conſtituat in D. </s> <s id="s.003177">Quòd ſi vectis adhuc trahatur à <lb></lb>potentia E, & veniat in H & in G, conſtat ex 21.lib.1. angu-<pb pagenum="413" xlink:href="017/01/429.jpg"></pb>lum CDE minorem eſſe angulo CHE, hunc verò minorem <lb></lb>angulo CGE, atque ita deinceps. </s> </p> <p type="main"> <s id="s.003178">Idem contingit, ſi diſtantia potentiæ R ab hypomochlio C <lb></lb>omnino æquali ſit longitudini vectis CB; nimirum trahendo <lb></lb>vectem ex B in S, angulus RSC major eſt angulo RBC, & ſic <lb></lb>deinceps trahendo ex S versùs R: quamvis enim ſemper ſit an<lb></lb>gulus acutus, major tamen ſubinde fit & major; quia manente <lb></lb>eadem diſtantia RC æquali longitudini vectis, tam triangulum <lb></lb>CBR quàm CSR, & reliqua omnia ſunt Iſoſcelia; quò ergo <lb></lb>minor fit angulus ad C, eò major fit angulus ad baſiin in R, <lb></lb>cui, per 5. lib. 1. æqualis eſt reliquus angulus ad eandem ba<lb></lb>ſim. </s> <s id="s.003179">At ſi diſtantia potentiæ ab hypomochlio minor fuerit lon<lb></lb>gitudine vectis, utique locus potentiæ eſt intra circulum à <lb></lb>vecte circumducto deſcriptum. </s> <s id="s.003180">Conſideranda eſt igitur varia <lb></lb>funiculi ad vectem inclinatio: pro qua explicanda hæc præ<lb></lb>mitto lemmata. </s> </p> <p type="main"> <s id="s.003181">LEMMA I. </s> <s id="s.003182">Si intra circulum aſſumptum fuerit punctum <lb></lb>E, in quo duæ rectæ lineæ BF & GH æqualiter à cen<lb></lb>tro diſtantes, ideóque ex 14.lib.3.æquales, ſe invicem ſe<lb></lb>cent; & à centro C ducantur Radij CB & CG; anguli <lb></lb>CBF & CGH ſunt æquales. </s> </p> <p type="main"> <s id="s.003183">Producatur BC in M, & <lb></lb><figure id="id.017.01.429.1.jpg" xlink:href="017/01/429/1.jpg"></figure><lb></lb>GC in N, ducantúrque rectæ <lb></lb>FM & HN. </s> <s id="s.003184">Quia MB & NG <lb></lb>ſunt diametri, anguli in ſemicir<lb></lb>culo BFM & GHN, ex 31. <lb></lb>lib.3. ſunt recti: igitur quadra<lb></lb>ta BF & FM ſimul ſumpta <lb></lb>ſunt æqualia quadratis GH & <lb></lb>HN ſimul ſumptis, cum, ex <lb></lb>47. lib. 1. æqualia ſint qua<lb></lb>drato diametri. </s> <s id="s.003185">Eſt autem qua<lb></lb>dratum BF æquale quadrato <lb></lb>GH, nam rectæ BF & GH <lb></lb>ex hypotheſi ſunt æquales; <lb></lb>igitur quadrata FM & HN ſunt æqualia, ideoque rectæ <pb pagenum="414" xlink:href="017/01/430.jpg"></pb>FM & HN ſunt æquales; ergo ex 28. lib. 3. ſubtendunt <lb></lb>æquales peripherias, FHM & HMN; ergo ex 27.lib.3. <lb></lb>anguli FBM & HGN æqualibus peripheriis inſiſtentes <lb></lb>æquales ſunt. </s> </p> <p type="main"> <s id="s.003186">Invenitur autem recta linea tranſiens per E, quæ æqua<lb></lb>lis ſit rectæ BF, ſi facto centro E, intervallo EF, deſcri<lb></lb>batur circulus FRG ſecans datum circulum in G; nam ex <lb></lb>G per E ducitur recta GH quæſita: eſt enim, per 35.lib.3, <lb></lb>rectangulum GEH æquale rectangulo FEB; ſunt autem GE <lb></lb>& FE æquales Radij ejuſdem circuli ex conſtructione; igitur <lb></lb>per 1. lib. 6, etiam EH & EB ſunt æquales; ergo tota GH <lb></lb>toti FB eſt æqualis. </s> </p> <p type="main"> <s id="s.003187">LEMMA II. </s> <s id="s.003188">Si in puncto E intra circulum aſſump<lb></lb>to ſecent ſe invicem duæ rectæ BF & IO inæquales, <lb></lb>ac proinde ut colligitur ex 15.lib. 3. inæqualiter à cir<lb></lb>culi centro diſtantes, ducantúrque ex centro Radij <lb></lb>CB, & CI; angulus factus à Radio cum lineâ remo<lb></lb>tiore major eſt angulo facto à Radio cum lineâ pro<lb></lb>pinquiore. </s> </p> <p type="main"> <s id="s.003189">Perficiantur triangula BFM & IOS rectangula ad F & O <lb></lb>ex 31.lib.3, quia MB & SI ſunt diametri. </s> <s id="s.003190">Quadrata BF & FM <lb></lb>ſimul ſumpta, ex 47. lib. 1, ſunt æqualia quadratis IO & OS <lb></lb>ſimul ſumptis: Quia autem ex hypotheſi recta IO remotior eſt <lb></lb>à centro quàm BF, eſt etiam minor, ut conſtat ex 15. lib. 3: <lb></lb>igitur quadratum IO minus eſt quadrato BF, adeóque qua<lb></lb>dratum reliquum OS majus eſt reliquo quadrato FM, & <lb></lb>linea OS major eſt linea FM. </s> <s id="s.003191">Quapropter etiam OS ſub<lb></lb>tendit majorem arcum OMS, & FM ſubtendit minorem <lb></lb>arcum FOM, & angulus SIO factus à Radio cum lineâ re<lb></lb>motiore major eſt angulo MBF facto à Radio cum lineâ pro<lb></lb>pinquiore. </s> </p> <p type="main"> <s id="s.003192">LEMMA III. </s> <s id="s.003193">Si in circulo ab extremitate diametri B <lb></lb>exeat recta linea BC circulum ſecans, in qua aſſumatur <lb></lb>punctum D eam bifariam æqualiter dividens, & per <pb pagenum="415" xlink:href="017/01/431.jpg"></pb>punctum D alia recta circulum ſecans ducatur, hæc <lb></lb>vicinior eſt centro, & major. </s> </p> <p type="main"> <s id="s.003194">Ducatur ex centro S <lb></lb><figure id="id.017.01.431.1.jpg" xlink:href="017/01/431/1.jpg"></figure><lb></lb>recta SD, quæ per 3.lib.3. <lb></lb>facit angulum SDC rec<lb></lb>tum: Tum per D alia <lb></lb>quædam linea EF tran<lb></lb>ſeat, quæ utique cum rec<lb></lb>ta SD facit angulum <lb></lb>SDF minorem recto, & <lb></lb>SDE majorem recto: <lb></lb>nam ſi angulos faceret <lb></lb>rectos, eſſet SD utrique <lb></lb>lineæ BC, & EF perpendicularis, ſecaret EF bifariam in D <lb></lb>per 3.lib. 3. adeóque duæ rectæ BC & EF ſe mutuo bifariam <lb></lb>ſecarent, contra 4. lib. 3. Igitur in rectam EF perpendicu<lb></lb>laris ducta ex centro S erit SG cadens ad partes anguli acu<lb></lb>ti. </s> <s id="s.003195">Quapropter in triangulo SGD rectangulo ad G ma<lb></lb>jor eſt hypothenuſa SD, quàm perpendiculum SG. </s> <s id="s.003196">Ma<lb></lb>gis ergo diſtat linea BC quàm linea EF à centro, ac proinde <lb></lb>per 15.lib.3. illa eſt minor, hæc major. </s> </p> <p type="main"> <s id="s.003197">LEMMA IV. </s> <s id="s.003198">Si in eâdem rectâ BC aſſumatur punctum <lb></lb>I inter extremitatem B & punctum medium D, atque, <lb></lb>ex centro directâ rectâ SIV, inter V & B alia quæ<lb></lb>piam per I tranſeat recta HL circulum ſecans, quæ <lb></lb>& ſecet perpendicularem SD, ex. </s> <s id="s.003199">gr. in puncto K; <lb></lb>hæc pariter HL centro propinquior eſt quàm BC, ac <lb></lb>proinde major. </s> </p> <p type="main"> <s id="s.003200">Angulus KDI eſt rectus, angulus DKI, & qui eſt illi ad <lb></lb>verticem, SKL eſt acutus, igitur perpendicularis ex centro S <lb></lb>in rectam HL ducta cadit inter K & L, puta in M. </s> <s id="s.003201">In trian<lb></lb>gulo igitur rectangulo SMK major eſt SK quàm SM, ex <lb></lb>18. lib. 1: ergo multò major eſt SD quàm SM, ac prop<lb></lb>terea ex 15. lib. 3. HL vicinior eſt centro, & major <lb></lb>quàm BC. </s> </p> <pb pagenum="416" xlink:href="017/01/432.jpg"></pb> <p type="main"> <s id="s.003202">LEMMA V. </s> <s id="s.003203">Si in rectà BC ducta ab extremitate dia<lb></lb><figure id="id.017.01.432.1.jpg" xlink:href="017/01/432/1.jpg"></figure><lb></lb>metri aſſumatur punctum N <lb></lb>ultra punctum medium D, at<lb></lb>que ex S centro ductâ per N <lb></lb>lineâ rectâ SO, productâque <lb></lb>perpendiculari SD in P, tran<lb></lb>ſeat per N alia quæpiam recta <lb></lb>QR inter P & B circulum ſe<lb></lb>cans in <expan abbr="q;">que</expan> hæc pariter ſecat in <lb></lb>T perpendicularem productam, <lb></lb>& eſt à centro S remotior quàm recta BC atque pro<lb></lb>inde minor. </s> </p> <p type="main"> <s id="s.003204">Quia in Triangulo NDT rectangulo ad D, angulus DTN <lb></lb>eſt acutus, utique in lineâ TS aſſumpto puncto S, ex hoc ca<lb></lb>det in lineam QR perpendicularis inter puncta T, & R; quam <lb></lb>dico majorem eſſe perpendiculari SD. </s> <s id="s.003205">Nam ſi ipſa recta SN <lb></lb>perpendicularis fuerit ad RQ, eſt triangulum SDN rectan<lb></lb>gulum, adeóque hypothenuſa SN major eſt quàm latus SD: <lb></lb>Sin autem perpendicularis ad RQ cadat in V ſecans rectam <lb></lb>BC in Z, utique SZ ſubtendens angulum rectum SDZ ma<lb></lb>jor eſt quàm SD; eſt autem SV major quàm SZ, ergo & <lb></lb>multo maior, quàm SD: ergo linea QR remotior eſt quàm <lb></lb>BC, & minor. </s> </p> <p type="main"> <s id="s.003206">Deinde ſi linea per N tranſiens, & circulum ſecans, extre<lb></lb>mitatem alteram habeat non inter punctum P terminum per<lb></lb>pendicularis SD productæ, atque B terminum rectæ BC; Vel <lb></lb>dividitur in N bifariam, & linea SN per 3.lib.3. eſt perpendi<lb></lb>cularis ad illam, quæ major eſt quàm SD, ut pote oppoſita an<lb></lb>gulo recto SDN: Vel dividitur inæqualiter. </s> <s id="s.003207">Si ſegmentum <lb></lb>majus ſit in parte ſuperiori, hoc inter N & arcum OP, utique <lb></lb>perpendicularis ex S centro ducta in illam lineam cadens ſeca<lb></lb>bit lineam BC inter puncta N & D, ac propterea oſtendetur <lb></lb>major quàm SD, ut ſupra oſtenſum eſt de linea RQ At ſi in <lb></lb>parte ſuperiori, hoc eſt inter N & arcum OP ſit ſegmentam <lb></lb>minus, perpendicularis ex S in lineam ducta cadet inſra <pb pagenum="417" xlink:href="017/01/433.jpg"></pb>punctum N, & à ſegmento majore abſcindet particulam in<lb></lb>ter N & punctum perpendiculi interceptam. </s> <s id="s.003208">Hæc particula ſi <lb></lb>fuerit æqualis particulæ ND, linea BC & linea ducta ſunt <lb></lb>æqualiter à centro remotæ; ſin illa particula minor fuerit quàm <lb></lb>ND, linea ducta remotior erit quàm BC; ſi demùm major fue<lb></lb>rit quàm ND, linea ducta propinquior centro erit quam BC. </s> <lb></lb> <s id="s.003209">Finge ſcilicet ductam eſſe rectam PNX, & ſegmentum majus <lb></lb>eſſe NX; utique perpendicularis ex S bifariam ſecans totam <lb></lb>PX cadit inter N & X, puta in Y. </s> <s id="s.003210">Eſt igitur SYN triangu<lb></lb>lum rectangulum in Y, & per 47. lib.1. quadratum SN æqua<lb></lb>le eſt quadratis NY & YS; atqui etiam triangulum SDN eſt <lb></lb>rectangulum ex hypotheſi, eandemque habet hypothenuſam <lb></lb>SN; igitur quadrata ND & DS æqualia ſunt quadratis NY <lb></lb>& YS. </s> <s id="s.003211">Quare ſi particulæ NY & ND æquales ſunt, æqualia <lb></lb>ſunt & earum quadrata, ac idcircò etiam æqualia ſunt quadra<lb></lb>ta YS & DS, atque eorum latera æqualia ſunt, & lineæ BC <lb></lb>atque PX ſunt æqualiter remotæ. </s> <s id="s.003212">Quod ſi particula NY mi<lb></lb>nor eſt quàm ND, etiam illius quadratum minus eſt quadrato <lb></lb>hujus; ergo reliquum quadratum YS majus eſt reliquo qua<lb></lb>drato DS, atque adeò linea SY major eſt quàm linea SD, & <lb></lb>linea ducta PX remotior eſt atque minor quàm BC. </s> <s id="s.003213">Si demum <lb></lb>NY major eſt quàm ND, etiam illius quadratum majus eſt hu<lb></lb>jus quadrato, & reliquum quadratum YS minus eſt reliquo <lb></lb>quadrato DS: igitur linea YS minor eſt quàm linea DS, ac <lb></lb>propterea linea ducta PX propinquior eſt centro, & major <lb></lb>quàm BC. </s> </p> <p type="main"> <s id="s.003214">His præmiſſis facilis eſt ſolutio propoſitæ difficultatis, ut in<lb></lb>noteſcat, utrùm in tractione minuatur labor, an augeatur, <lb></lb>quando potentiæ trahentis diſtantia ab hypomochlio eſt minor <lb></lb>longitudine vectis. </s> <s id="s.003215">Dato ſi quidem loco potentiæ datur ejuſ<lb></lb>dem diſtantia tùm ab hypomochlio, tum ab extremitate vectis, <lb></lb>cum qua funiculus connectitur; ſed & datur ipſius vectis longi<lb></lb>tudo: quare per Trigonometriam innoteſcit quantitas anguli, <lb></lb>cui opponitur vectis. </s> <s id="s.003216">Nam ſi ille rectus eſt, ut SDB, per lem<lb></lb>ma 3. in tractione funiculus fit pars lineæ centro propinquio<lb></lb>ris, quàm primò aſſumpta DB: igitur in tractione angulus fu<lb></lb>niculi cum vecte fit ſenſim acutior ex lemm. </s> <s id="s.003217">2. augeturque <lb></lb>difficultas trahendi. </s> <s id="s.003218">Si angulus vecti oppoſitus ſit obtuſus, ut <pb pagenum="418" xlink:href="017/01/434.jpg"></pb>SIB, in tractione funiculus evadit pars lineæ propinquioris <lb></lb>centro, quam prima IB ex lemm. </s> <s id="s.003219">4. & ſimiliter ex lemm. </s> <s id="s.003220">2. <lb></lb>fit angulus magis acutus, atque trahentis labor augetur. </s> <s id="s.003221">Si de<lb></lb>mum angulus vecti oppoſitus ſit acutus, ut SNB, ex lemm.3. <lb></lb>minuitur labor trahentis uſque ad certum terminum, quandiu <lb></lb>ſcilicet vectis non ſecat perpendiculariter primam funiculi po<lb></lb>ſitionem NB, hoc eſt vectis circumductus nondum eſt SP; <lb></lb>tandiu enim funiculus eſt pars lineæ à centro remotioris, & fa<lb></lb>cit per lemm. </s> <s id="s.003222">2. cum vecte angulum majorem. </s> <s id="s.003223">Ubi autem <lb></lb>vectis fuerit SP, tunc obſervandum eſt, utrùm angulus SNP <lb></lb>rectus ſit, an obtuſus, an acutus; & eadem methodo proce<lb></lb>dendum eſt, quaſi prima funiculi poſitio eſſet NP, ut innoteſ<lb></lb>cat, utrùm funiculus in ulteriori tractione fiat pars lineæ re<lb></lb>motioris, an verò propinquioris, ac proinde fiat angulus ſub<lb></lb>inde major, an verò minor. </s> </p> <p type="main"> <s id="s.003224">Quæ de Vecte in alterâ extremitate hypomochlium, in alte<lb></lb>râ potentiam habente hactenus exempli gratia explicata ſunt, <lb></lb>facilè referuntur ad vectem, quando hypomochlium, aut po<lb></lb>tentia inter extremitates collocantur; ſemper enim attendenda <lb></lb>eſt hypomochlij diſtantia à potentia trahente, ut potentiæ obli<lb></lb>què trahentis momenta innoteſcant; angulus ſcilicet funiculi <lb></lb>cum vecte pendet ab hypomochlij puncto, circa quod fit vectis <lb></lb>converſio. </s> </p> <p type="main"> <s id="s.003225">Quoniam autem hujus capitis initio momentorum Rationem <lb></lb>juxta diverſam potentiæ applicationem ex arcubus vi ejuſdem <lb></lb>impetùs deſcriptis æſtimandam eſſe dictum eſt, & quis fortaſſe <lb></lb>ſuſpicetur arduum eſſe hujuſmodi arcus inter ſe comparare; <lb></lb>animadvertat ex Tabulis Trigonometricis ejuſdem arcûs Si<lb></lb>num & Tangentem iiſdem planè numeris definiri, quando ar<lb></lb>cus valde exiguus eſt. </s> <s id="s.003226">Quapropter cum quilibet arcus minor <lb></lb>ſit ſuâ Tangente, & major Sinu, arcuum minorum Rationem <lb></lb>citra ullum erroris periculum explicare poſſumus per eorum <lb></lb>Sinus. </s> <s id="s.003227">Cum verò hîc, ubi de potentiæ ad vectem ſecundùm <lb></lb>diverſos angulos applicatæ momentis ſermo eſt, non niſi mini<lb></lb>mi arcus aſſumendi ſint, eorum Ratio eadem aſſumitur, quæ <lb></lb>Sinuum. </s> </p> <p type="main"> <s id="s.003228">Quare ſi vectis ſit AB, hypomochlium C, vis potentiæ & <lb></lb>directio motûs potentiæ BH: loco arcûs BK, qui in motu vi <pb pagenum="419" xlink:href="017/01/435.jpg"></pb>talis impetús cum hac directione deſcribitur; aſſumi poteſt an<lb></lb>guli HBO, Radio BH, Sinus HO, <lb></lb><figure id="id.017.01.435.1.jpg" xlink:href="017/01/435/1.jpg"></figure><lb></lb>qui eſt æqualis Sinui arcus BK Ra<lb></lb>dio CB. </s> <s id="s.003229">Eſt autem minimus ar<lb></lb>cus longe minor quàm arcus BK, <lb></lb>ſed claritatis gratia arcum notabi<lb></lb>lem & conſpicuum aſſumere opor<lb></lb>tuit. </s> <s id="s.003230">Eſt igitur potentiæ ad an<lb></lb>gulum rectum in B applicatæ mo<lb></lb>mentum, ad ejuſdem potentiæ ad <lb></lb>angulum HBO acutum applicatæ momentum, ut Radius BH <lb></lb>ad acuti anguli Sinum HO. </s> </p> <p type="main"> <s id="s.003231">Jam intellige vectem AB converti, & lineam BH produci, <lb></lb>donec in D ad angulos rectos occurrat vecti habenti poſitio<lb></lb>nem EF. </s> <s id="s.003232">Dico potentiæ ad perpendiculum & obliquè appli<lb></lb>catæ momenta invicem comparata ita eſſe, ac ſi eadem poten<lb></lb>tia tam in F quàm in D ad angulum rectum applicaretur, quia <lb></lb>ut BH ad HO, ita eſt FC ad CD. </s> <s id="s.003233">Ducatur enim ex D ad CB <lb></lb>perpendicularis DG, quæ eſt parallela ipſi HO. quare per 4. <lb></lb>lib. 6. ut BH ad HO, ita BD ad DG, & per 8. lib. 6. ut BD <lb></lb>ad DG, ita BC, hoc eſt FC, ad CD: igitur per 11. lib. 5. ut <lb></lb>BH ad HO, ita FC ad CD; hoc eſt ut Radius ad Sinum angu<lb></lb>li, ſecundùm quem potentia dirigitur, ita momentum poten<lb></lb>tiæ perpendiculariter applicatæ ad momentum ejuſdem obli<lb></lb>què ad angulum acutum, vel obtuſum applicatæ. </s> <s id="s.003234">Nam ſi po<lb></lb>tentia in B dirigat ſuum motum ſecundum lineam BI, utique <lb></lb>poſito Radio BI, Sinus anguli ABI obtuſi eſt IL, & Ratio mo<lb></lb>menti potentiæ in B applicatæ ſecundùm angulum rectum, ad <lb></lb>momentum ejuſdem potentiæ in B applicatæ ſecundùm angu<lb></lb>lum obtuſum ABI, eſt ut BI ad IL. </s> <s id="s.003235">Producatur IB, donec in <lb></lb>D perpendicularis cadat ſupra CF rectam æqualem ipſi CB. </s> <lb></lb> <s id="s.003236">Quia <expan abbr="triāgula">triangula</expan> BIL & BCD rectangula ad L & D, & æquales <lb></lb>angulos ad verticem B habentia, ſimilia ſunt, eſt ut BI ad IL, <lb></lb>ita BC ad CD per 4. lib. 6. Perinde igitur in extremitate B ad <lb></lb>angulum obtuſum ABI applicata potentia operatur, atque ſi ad <lb></lb>angulum rectum applicaretur in D puncto vectis EF, qui idem <lb></lb>ponitur eſſe ac vectis AB: & momenta potentiæ ſunt ut FC <lb></lb>ad CD. </s> </p> <pb pagenum="420" xlink:href="017/01/436.jpg"></pb> <p type="main"> <s id="s.003237">Dato itaque angulo, ſecundum quem potentia applicatur <lb></lb>ad vectem, ſi angulus ſit Rectus, momentum eſt ut Radius; ſin <lb></lb>autem angulus acutus ſit vel obtuſus, momentum eſt ut Sinus <lb></lb>ejuſdem anguli; atque adeò comparando inter ſe hujuſmodi <lb></lb>angulos, Ratio illorum erit eadem, quæ eſt Sinuum. </s> <s id="s.003238">Hinc <lb></lb><figure id="id.017.01.436.1.jpg" xlink:href="017/01/436/1.jpg"></figure><lb></lb>datus vectis AB hypomochlium ha<lb></lb>bens in C, ſi fuerit ita inclinatus, ut <lb></lb>poſitionem habeat EF, potentia in E <lb></lb>deorſum premens per rectam EG per<lb></lb>pendicularem ad CB, momentum ha<lb></lb>bet ut CG; & ſi <expan abbr="poſitionẽ">poſitionem</expan> habeat IH, <lb></lb>potentia in I deorſum premens aut tra<lb></lb>hens juxta rectam IK, quæ producta <lb></lb>incidat perpendicularis ad rectam AB <lb></lb>in L, momentum habet ut CL. </s> <s id="s.003239">Quare <lb></lb>ex E in B augentur prementis aut trahentis momenta, quæ ex <lb></lb>B in I minuuntur. </s> </p> <p type="main"> <s id="s.003240">Id quod iis etiam, qui campanas pulſant, manifeſtum eſt: ſi <lb></lb>enim intelligatur vecti CB adhærere campanam, cujus cen<lb></lb>trum gravitatis ſit O, utique dum B deprimitur, O elevatur, <lb></lb>ſed elevandi difficultas creſcit, tum quia centrum gravitatis O <lb></lb>arcum deſcribens circa punctum C, æqualibus temporibus in <lb></lb>quales, atque ſemper majores habet aſcenſus juxta incremen<lb></lb>ta Sinuum Verſorum, tum quia ex depreſſione vectis ex B in I <lb></lb>facto angulo funis & vectis ſemper obtuſiore, momenta poten<lb></lb>tiæ minuuntur: & licet in reditu ex I in B creſcerent, ſi quis <lb></lb>vectem ſurſum traheret, hoc nihil juvat potentiam deorſum <lb></lb>trahentem ad elevandam campanam, quæ ſponte ſua deſcen<lb></lb>dens elevat vectis caput, cui funis adnectitur. </s> <s id="s.003241">Propterea majo<lb></lb>ribus gravioribúſque campanis non ſimplicem vectem CB ſed <lb></lb>rotam, aut rotæ ſegmentum adjungunt, cujus excavatæ peri<lb></lb>metro funis inſeritur; qui dum trahitur, ſemper eſt Tangens <lb></lb>circuli; atque ideo ad Radium circuli, quaſi eſſet novus atque <lb></lb>novus vectis, applicatur potentia trahens ad angulum rectum. <pb pagenum="421" xlink:href="017/01/437.jpg"></pb></s> </p> <p type="main"> <s id="s.003242"><emph type="center"></emph>CAPUT VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003243"><emph type="center"></emph><emph type="italics"></emph>Oneris ex Vecte pendentis momentum inquiritur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003244">COntingit aliquando pondus vecte elevandum fune con<lb></lb>necti, & pendulum ex vecte ſuſpendi. </s> <s id="s.003245">Nemo dubitat, an <lb></lb>gravitas ponderis ibi ſua exerceat momenta, ubi cum vecte <lb></lb>connectitur; funis ſi quidem intentus congruit lineæ directio<lb></lb>nis, qua pondus ipſum nititur in centrum gravium: verùm non <lb></lb>eandem percipi in elevando difficultatem experientia teſtatur <lb></lb>pro varia vectis inclinatione. </s> <s id="s.003246">Si enim ex vecte AB horizontali <lb></lb>hypomochlium in extremitate B ha<lb></lb><figure id="id.017.01.437.1.jpg" xlink:href="017/01/437/1.jpg"></figure><lb></lb>bente, pondus D ſuſpenſum ex C <lb></lb>pendeat ad angulos rectos, omnia ſua <lb></lb>momenta exercet pro ratione diſtan<lb></lb>tiæ CB ab hypomochlio. </s> <s id="s.003247">At ſi ele<lb></lb>vatus vectis poſitionem habeat EB, <lb></lb>& C venerit in F, pondus verò pen<lb></lb>dulum D venerit in G, ita ut linea <lb></lb>Directionis in centrum gravium congruat funiculo ſuſpenden<lb></lb>ti FG; etiamſi FB æqualis ſit ipſi CB, non eadem tamen mo<lb></lb>menta habet pondus adversùs eandem <expan abbr="potẽtiam">potentiam</expan> ex A <expan abbr="tranſlatã">tranſlatam</expan> <lb></lb>in E; quia ſcilicet angulus GFB eſt acutus, DCB autem rectus. </s> </p> <p type="main"> <s id="s.003248">Id explicare ex iis, quæ ſuperiori capite diſputata ſunt, non <lb></lb>erit difficile, ſi animadvertamus in vectibus ſecundi & tertij <lb></lb>generis utrumque genus conjungi: quemadmodum enim po<lb></lb>tentia conatur adversùs gravitatem ponderis, ita pondus cona<lb></lb>tur adversùs vim potentiæ: & in hoc conatu viciſſim exercent <lb></lb>munus potentiæ & ponderis. </s> <s id="s.003249">Finge ſiquidem duos homines ap<lb></lb>plicari vecti AB, alterum quidem in A, alterum verò in C, ſed <lb></lb>in adverſa conantes; uterque eſt potentia, uterque eſt pondus, <lb></lb>dum ſibi reluctantur. </s> <s id="s.003250">Anne ita hæc vocabula intra certos fines <lb></lb>coërceri exiſtimas, ut potentiæ nomine illum ſolum donandum <lb></lb>putes, qui reliquum vincit? </s> <s id="s.003251">ſed quid, ſi horum hominum co-<pb pagenum="422" xlink:href="017/01/438.jpg"></pb>natus ſint reciprocè ut eorum diſtantiæ ab hypomochlio B, & <lb></lb>A quidem conetur ut CB, C autem conetur ut AB; utique <lb></lb>neuter ſuperat; nec tamen negari poteſt ideò contingere mo<lb></lb>mentorum æqualitatem inter inæquales conatus, quia vecti ap<lb></lb>plicantur: ſunt igitur ſibi viciſſim potentia & pondus. </s> <s id="s.003252">Si ita<lb></lb>que A eſt potentia, & C pondus; vectis eſt ſecundi generis: <lb></lb>Si verò C eſt potentia, & A pondus, vectis eſt tertij generis. </s> <lb></lb> <s id="s.003253">Illud igitur quod de hominibus dicitur, de reliquis omnibus <lb></lb>vim movendi habentibus dictum intelligitur: nihil ſi quidem <lb></lb>intereſt, utrùm animata ſint, an inanima, quæ vecti applican<lb></lb>tur, & in oppoſita partes conantur. </s> <s id="s.003254">Et quamvis non interce<lb></lb>dat inter ipſos conatus momentorum æqualitas, quam conſe<lb></lb>quatur quies, ſed efficiatur motus; ita tamen id quod prævalet, <lb></lb>eſt potentia ad motum efficiendum, ut id quod vincitur, & re<lb></lb>ſiſtit, ſit potentia ad motum retardandum. </s> <s id="s.003255">In omni itaque <lb></lb>vecte ſive ſecundi, ſive tertij generi ſit, utrumque genus ami<lb></lb>co, nec ſolubili fœdere copulantur. </s> <s id="s.003256">In vecte autem primi ge<lb></lb>neris idem genus manet, licèt viciſſim habeant rationem po<lb></lb>tentiæ & ponderis ad movendum & retardandum, ſimiliter <lb></lb>enim potentiæ & ponderi, licet inæqualibus intervallis, inter<lb></lb>jacet hypomochlium. </s> </p> <p type="main"> <s id="s.003257">Hîc itaque, ubi oneris ex vecte pendentis momentum inqui<lb></lb>ritur, conſiderandus eſt vectis tertij generis, in quo gravitas in <lb></lb>C, aut in F poſita exercet munus potentiæ conantis deprimere <lb></lb>vim ſurſum connitentem in A, aut in E. </s> <s id="s.003258">Quare in poſitione vectis <lb></lb>horizontali, cum ſit angulus rectus DCB, neque gravitas illa <lb></lb>vectem versùs hypomochlium B urgeat, aut cum ab illo re<lb></lb>trahat, omnia ſua momenta obtinet, quæ in hac à fulcro diſtantiâ <lb></lb>gravitati huic convenire poſſunt. </s> <s id="s.003259">At elevato vecte ita, ut fiat an<lb></lb>gulus acutus GFB, licet eadem maneat gravitas, eadémque ab <lb></lb>hypomochlio diſtantia, non tamen eadem manent momenta, ſed <lb></lb>decreſcunt pro ratione Sinûs anguli, ut ſuperiori capite dictum <lb></lb>eſt. </s> <s id="s.003260">Producta igitur intelligatur linea directionis FG uſque ad <lb></lb>horizontalem in H: poſito Radio BF, hoc eſt BC, eſt BH Si<lb></lb>nus anguli GFB; ac proinde ut BC ad BH, ita momentum <lb></lb>oneris pendentis ex vecte horizontali, ad momentum ejuſdem <lb></lb>oneris pendentis ex codem vecte inclinato. </s> <s id="s.003261">Hinc eſt, inclinato <lb></lb>vecte EB, tantumdem conatûs adhibendum eſſe in E ad ſuſti-<pb pagenum="423" xlink:href="017/01/439.jpg"></pb>nendum onus G, quanto conatu opus eſſet in vecte horizontali <lb></lb>AB ad ſuſtinendum idem onus, ſi penderet ex H. </s> <s id="s.003262">Quoniam igi<lb></lb>tur diſtantia BH minor eſt quàm BC, major eſt Ratio AB ad <lb></lb>BH, quàm ejuſdem AB ad BC, ex 8.lib.5. ideóque faciliùs ſuſti<lb></lb>netur idem onus vecte inclinato, quàm vecte horizontali. </s> </p> <p type="main"> <s id="s.003263">Quod ſi ex G centro gravitatis oneris ductam intelligas ad <lb></lb>vectem EB rectam perpendicularem GI, habes ſimiliter mo<lb></lb>mentorum differentiam, quæ ſcilicet intercedit inter FG, & GI, <lb></lb>ſi FG repræſentet omnia momenta in vecte horizontali: ſunt <lb></lb>enim triangula FIG & FHB rectangula, communem angulum <lb></lb>ad F habentia, adeóque ſimilia, & ut FB ad BH, ita FG ad GI. </s> <lb></lb> <s id="s.003264">Cave autem ne putes (ut non pauci hallucinantur) ita ex I ter<lb></lb>mino rectæ CI perpendicularis deſumendam eſſe menſuram <lb></lb>decrementi momentorum, ut perinde ſe habeat, quaſi pondus <lb></lb>eſſet in I: hoc enim à veritate longiſſimè abeſſe deprehendes, ſi <lb></lb>manente eadem vectis inclinatione, & eadem oneris gravitate, <lb></lb>funiculo longiore onus ſuſpenderis; quandoquidem ctiam <lb></lb>punctum I magis accedet ad hypomochlium B, nec tamen adhi<lb></lb>bito longiore funiculo adeò minuuntur momenta; alioquin <lb></lb>tam longo funiculo ſuſpendere poſſes onus, ut recta ex one<lb></lb>ris centro ducta ad vectem EB perpendicularis caderet in B, <lb></lb>atque ideo nullum eſſet gravitatis momentum, quaſi onus eſſet <lb></lb>in B: id autem omnino falſum eſt. </s> </p> <p type="main"> <s id="s.003265">Quando autem dicitur faciliùs à potentiâ ſuſtineri idem onus <lb></lb>ſuſpenſum vecte inclinato, quàm vecte horizontali, ita intelli<lb></lb>gendum eſt, ut linea directionis motus potentiæ ſuſtinentis <lb></lb>eundem ſemper faciat cum vecte angulum: nam ſi hæc linea <lb></lb>alium atque alium efficiat angulum, etiam potentiæ momenta <lb></lb>variantur, quæ cum oneris momentis comparanda ſunt. </s> <s id="s.003266">Hinc <lb></lb><figure id="id.017.01.439.1.jpg" xlink:href="017/01/439/1.jpg"></figure><lb></lb>eſt in vecte primi generis CD, <lb></lb>cujus hypomochlium O, ſi po<lb></lb>tentia & pondus ſint gravia M <lb></lb>& N, licèt inclinato vecte, ut <lb></lb>habeat poſitionem RS, receden<lb></lb>tibus angulis à rectitudine, ſin<lb></lb>gulorum momenta minora fiant, <lb></lb>non tamen mutari momentorum <lb></lb>potentiæ & ponderis invicem <pb pagenum="424" xlink:href="017/01/440.jpg"></pb>comparatorum Rationem; quia ſcilicet ſingulorum momenta <lb></lb>proportionaliter minuuntur. </s> <s id="s.003267">Cum enim gravia ſemper nitan<lb></lb>tur juxta ſuas lineas directionis in centrum gravium, hujuſmo<lb></lb>di lineæ parallelæ cenſentur, & cum vecte duos angulos effi<lb></lb>ciunt duobus rectis æquales, ac proinde ſi alter acutus fuerit, <lb></lb>alter eſt obtuſus ſupplementum acuti ad duos rectos. </s> <s id="s.003268">Sicut au<lb></lb>tem in eodem circulo idem eſt Sinus anguli acuti, atque obtuſi, <lb></lb>qui compleat duos rectos; ita in diverſis circulis hujuſmodi an<lb></lb>gulorum Sinus proportionales ſunt ſuis Radiis. </s> <s id="s.003269">Quapropter in<lb></lb>clinato vecte, ut ſit RS, gravia nituntur deorſum juxta lineas <lb></lb>directionis ST & RV parallelas, quæ occurrunt perpendicu<lb></lb>lares horizontali in Z & V. </s> <s id="s.003270">Momentum igitur gravis T ad <lb></lb>momentum æqualis, ſeu ejuſdem gravis N eſt ut OZ ad OD, <lb></lb>& momentum gravis V ad momentum æqualis, ſeu ejuſdem <lb></lb>gravis M eſt ut OV ad OC. </s> <s id="s.003271">Quare in vecte RS inclinato <lb></lb>momenta gravium pendentium ſunt ut OZ ad OV. </s> <s id="s.003272">Quia ve<lb></lb>rò triangula RVO, SZO rectangula, & angulos ad verticem <lb></lb>O æquales habentia, ſunt ſimilia, per 4. lib. 6. ut OS ad OR, <lb></lb>hoc eſt ut OD ad OC, ita OZ ad OV. </s> <s id="s.003273">Manet itaque ea<lb></lb>dem momentorum Ratio invicem comparatorum, ſive integra <lb></lb>in vecte horizontali, ſive diminuta in vecte inclinato ſint ſingu<lb></lb>lorum momenta. </s> </p> <p type="main"> <s id="s.003274">At in vecte ſecundi aut tertij generis, ſi potentia non fuerit <lb></lb>vivens, fieri non poteſt ut eadem ſervetur momentorum Ratio <lb></lb>inter potentiam & pondus, niſi fortè in eodem medio ho<lb></lb>rum alterutrum grave eſſet, alterum leve; ut ſi vectis AK <lb></lb><figure id="id.017.01.440.1.jpg" xlink:href="017/01/440/1.jpg"></figure><lb></lb>intra aquam conſtitutus adnexum <lb></lb>haberet in K inflatum ut rem V, <lb></lb>in L verò pendulus eſſet lapis: <lb></lb>tunc enim, ſi uter aſcendens trahat <lb></lb>vectem in B, elevabit lapidem <lb></lb>pendulum, ut ſit angulus ICA <lb></lb>acutus, & angulus ABF obtu<lb></lb>ſus; qui cum æqualis ſit alterno <lb></lb>BCI (ſunt enim FBE & CI paral<lb></lb>lelæ, quia utraque ad horizontem <lb></lb>perpendicularis eſt) ſimilem habet Sinum Sinui acuti ICA ſe<lb></lb>cundum Rationem Radiorum BA & CA, hoc eſt KA & LA; <pb pagenum="425" xlink:href="017/01/441.jpg"></pb>atque ut BA ad CA, ita eſt EA ad IA; ſimilia quippe ſunt <lb></lb>triangula ABE & ACI. </s> <s id="s.003275">Cæterùm ſi rotulæ X inſiſtens funi<lb></lb>culus jungeret vectis extremitatem K & pondus aliquod ad<lb></lb>nexum in fungens munere potentiæ elevantis; hoc deſcen<lb></lb>dens ex S in M elevaret vectem in B, & lapidem, qui penderet <lb></lb>ex C: ſed angulus ABX eſſet multò obtuſior quàm Supple<lb></lb>mentum acuti ICA ad duos rectos, ac proinde Sinus anguli <lb></lb>ABX eſſet multo minor quàm EA Sinus anguli obtuſi ABF: <lb></lb>igitur multo minor eſſet Ratio momenti potentiæ applicatæ in <lb></lb>B ad momentum ponderis in C, quàm ſit Ratio BA ad CA, <lb></lb>hoc eſt KA ad LA. </s> <s id="s.003276">Sola igitur potentia vivens poteſt ita ſui <lb></lb>motûs directionem inflectere, ut eundem faciat cum vecte an<lb></lb>gulum, ideóque elevans vectem acquirat majorem ſuſtinendi <lb></lb>facilitatem. </s> </p> <p type="main"> <s id="s.003277">His conſequens eſt, quantò altiùs ſupra horizontem eleva<lb></lb>tur vectis cum pondere pendulo, tantò validiùs a pondere pre<lb></lb>mi aut urgeri hypomochlium A. </s> <s id="s.003278">Nam quemadmodum in vecte <lb></lb>horizontali AK pondus in L ſuſpenſum magis premit hypomo<lb></lb>chlium A vicinum quàm potentiam K remotam ex hypotheſi, <lb></lb>ita elevato vecte multo magis premitur hypomochlium, quia <lb></lb>quodammodo propiùs illi admovetur pondus in I quàm in L, <lb></lb>ſuáque innatá gravitate in vecte elevato conatur versù hypo<lb></lb>mochlium quaſi ſecedens à potentia; ut nihil dicam de vecte <lb></lb>ipſo, cujus gravitas, maximam partem, innititur fulcro. <lb></lb></s> </p> <p type="main"> <s id="s.003279"><emph type="center"></emph>CAPUT IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003280"><emph type="center"></emph><emph type="italics"></emph>An duo pondus geſtantes æqualiter premantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003281">HActenus diſputatis proximè affinis eſt præſens quæſtio, qua <lb></lb>inquirimus, utrùm æqualis ſit labor duorum in codem pon<lb></lb>dere geſtando conſentientium. </s> <s id="s.003282">Et quidem ſi movendum ſit <lb></lb>pondus atque trahendum, cur duo ſimul faciliùs illud mo<lb></lb>veant, quàm ſinguli, omnes intelligunt; quia plus impetûs à <lb></lb>duobus producitur, quàm à ſingulis; & quem impetum mo-<pb pagenum="426" xlink:href="017/01/442.jpg"></pb>vendo oneri parem ſinguli multo conatu producerent, ſingu<lb></lb>lis in parte impetûs efficiendâ minùs conantibus, totus produ<lb></lb>citur, totique oneri imprimitur. </s> <s id="s.003283">At in pondere ſuſtentando, <lb></lb>cujus gravitas in partes non dividitur, quomodo hæc ſingulis <lb></lb>levior accidat, ſi plures in ſuſtentando conſpirent, quàm ſi ſin<lb></lb>gulis imponeretur, tunc maximè cùm nullum impetum ſurſum <lb></lb>gravitatis conatui adverſantem producant, non ita explicatu <lb></lb>facile exiſtimant aliqui. </s> <s id="s.003284">Verum ex rationibus vectis ſatis ma<lb></lb>nifeſta ſolutio eruitur. </s> <s id="s.003285">Claritatis autem gratiâ, obſervandum <lb></lb>eſt, an onus palangâ (ut cum bajuli dolium ex funibus ſuſpen<lb></lb>ſum transferunt) an verò ſubjectis humeris ſuſtineatur. </s> </p> <p type="main"> <s id="s.003286">Et primo ſit palanga AB, cujus extremitates à geſtatoribus <lb></lb>ſuſtineantur; ſit autem onus in C. </s> <s id="s.003287">Duplex effectus hîc conſi<lb></lb><figure id="id.017.01.442.1.jpg" xlink:href="017/01/442/1.jpg"></figure><lb></lb>derandus eſt, videlicet oneris ſuſten<lb></lb>tatio, & geſtatorum preſſio; Si pri<lb></lb>mum reſpicias, geſtatores A & B ratio<lb></lb>nem habent potentiæ efficientis ſu<lb></lb>ſtentationem, atque impedientis motum oneris ſuá gravitate <lb></lb>deorſum conantis: Si ſecundum, idem onus C munus potentiæ <lb></lb>preſſionem efficientis exercet, dum ſecum palangam deorſum <lb></lb>trahens, oppoſitos geſtatorum humeros comprimit, aut ſi ma<lb></lb>nibus palanga geſtatur contentos contractoſque brachiorum <lb></lb>muſculos, quantum poteſt, diſtrahit, atque relaxat. </s> <s id="s.003288">Sunt <lb></lb>enim duo conatus, geſtatorum ſcilicet & oneris, motum in op<lb></lb>poſitas partes efficere valentes, niſi ſibi mutuo impedimento <lb></lb>eſſent: hinc ſi geſtatores conari ceſſent, onus deſcendit; Si ex <lb></lb>improviſo abruptis funibus onus à palangâ ſejungatur, geſtato<lb></lb>res palangam ſurſum attollunt, ſive æqualiter, ſive inæqualiter, <lb></lb>pro ut æquales aut inæquales ſunt eorum conatus. </s> <s id="s.003289">Quare æſti<lb></lb>manda res eſt ex motu, quem ſinguli conantes efficerent tum <lb></lb>in ſe, tum in oppoſito conante, niſi prohiberentur momento<lb></lb>rum æqualitate. </s> <s id="s.003290">Sic potentia in A ſuo conatu elevaret pondus <lb></lb>in C poſitum, & circa centrum B arcum deſcriberent; ſimili<lb></lb>ter potentia in B ſuo conatu elevaret pondus idem in C poſi<lb></lb>tum, & circa centrum A ſuos motus perficerent. </s> <s id="s.003291">Quod ita<lb></lb>que ad ſuſtentationem ſpectat, geſtatores A & B viciſſim ha<lb></lb>bent rationem potentiæ & fulcri; nam ſi A eſt potentia, ful<lb></lb>crum eſt B; atque viciſſim ſi B ſit potentia, fulcrum eſt A; & <pb pagenum="427" xlink:href="017/01/443.jpg"></pb>eſt duplex vectis ſecundi generis, ſcilicet AB & BA. Q: od <lb></lb>verò ad preſſionem attinet, in qua onus C eſt potentia premens, <lb></lb>geſtatores viciſſim habent rationem fulcri atque ponderi preſ<lb></lb>ſi; & eſt duplex vectis tertij generis, quo utitur unica poten<lb></lb>tia, ſicut in duplici vecte ſecundi generis unicum eſt pondus, <lb></lb>quod ſuſtinent duæ potentiæ. </s> </p> <p type="main"> <s id="s.003292">In vecte igitur ſecundi generis poſito fulcro B, momentum <lb></lb>potentiæ A ſurſum nitentis, ad momentum ponderis C deor<lb></lb>ſum conantis, eſt ut AB ad CB; ac propterea potentia ſuſtinere <lb></lb>valens ſine vecte pondus C, ad potentiam vecte AB ſuſtinentem <lb></lb>idem pondus C, eſt ut AB ad CB; quanto igitur CB minor eſt <lb></lb>quàm AB, tanto minor potentia requiritur in A, quàm require<lb></lb>retur in C, ſi in C pondus ſine vecte ſuſtineretur. </s> <s id="s.003293">Idem quod <lb></lb>de potentiâ A, poſito fulcro B, dictum eſt, dic viciſſim de poten<lb></lb>tiâ B; poſito fulcro A; Requiritur enim in B potentia ut CA, ad <lb></lb>potentiam, quæ eſſet ut BA, ſi ſine vecte pondus in C ſuſtinere<lb></lb>tur. </s> <s id="s.003294">Hinc eſt vires ſuſtinendi requiri reciprocè tantas, quanta <lb></lb>eſt Ratio diſtantiarum à pondere ipſorum ſuſtinentium: vires ſi <lb></lb>quidem in A requiruntur ut CB, & vires in B ut CA. </s> <s id="s.003295">Si itaque <lb></lb>æquali intervallo pondus medium diſtet à geſtatoribus, æqua<lb></lb>liter eos conari oportet, ut illud ſuſtineant in C: at ſi inæquali<lb></lb>ter ab iis remotum ſit, ut in D, requiruntur in A vires tam eò ma<lb></lb>jores quàm in B, quantò major eſt diſtantia DB quàm DA. </s> <lb></lb> <s id="s.003296">Quapropter datá virium inæqualitate; ſtatim innoteſcet, in quo <lb></lb>palangæ puncto adnectendum ſit onus; ſi nimirum palangæ lon<lb></lb>gitudo dividatur ſecundùm Rationem virium, & geſtatores re<lb></lb>ciprocè collocentur. </s> <s id="s.003297">Sint enim ex. </s> <s id="s.003298">gr. duo, quorum alter vires <lb></lb>habeat ut 3, alter ut 2: concipe totam longitudinem AB in <lb></lb>quinque partes diſtinctam, & hinc accipe duas AD, hinc verò <lb></lb>tres BD: locus ponderi debitus eſt punctum D, in quod cadit <lb></lb>diviſio in duas partes juxta datam Rationem: locus debilioris <lb></lb>geſtatoris eſt in palangæ extremitate B, ad quam ſpectat major <lb></lb>diſtantia ab onere in C poſito. </s> <s id="s.003299">Similiter virium inæqualitatem <lb></lb>deprehendes, ſi pondere in medio puncto C poſito, alter ſe præ<lb></lb>gravari ſentiat: palanga enim ita promota, ut pondere in D <lb></lb>conſtituto neuter ſe ultra vires prægravatum experiatur, indi<lb></lb>cabit vires geſtatoris A eſſe ut DB, ad vires geſtatoris B, quæ <lb></lb>ſunt ut DA. </s> </p> <pb pagenum="428" xlink:href="017/01/444.jpg"></pb> <p type="main"> <s id="s.003300">At ſi vectem tertij generis, quatenus geſtatorum preſſio ab <lb></lb>onere efficitur, conſideremus; poſito fulcro in A, potentia in C <lb></lb>aut in D exiſtens non premit geſtatorem B perinde, atque ſi <lb></lb>nullo intercedente vecte geſtator eſſet pariter in C aut in D, ſed <lb></lb>tanto minus, quanto minor eſt CA aut DA, quam BA: idem<lb></lb>que de geſtatore A, poſito fulcro in B, dicendum eſt. </s> <s id="s.003301">Quare re<lb></lb>ciprocæ ſunt preſſiones diſtantiis geſtatorum ab onere, & A <lb></lb>premitur ut CB aut DB, B autem premitur ut CA aut DA. </s> <lb></lb> <s id="s.003302">Cum enim vis ipſa gravitatis oneris deorſum conantis apta ſit <lb></lb>circa centrum A moveri pro ratione diſtantiæ CA aut DA, uti<lb></lb>que in B motum efficere debet pro ratione diſtantiæ BA: eſt <lb></lb>autem CA major quam DA ex hypotheſi, igitur, ex 8.lib.5.ma<lb></lb>jor eſt Ratio momenti CA quàm momenti DA ad idem mo<lb></lb>mentum BA, ac proinde major preſſio geſtatoris B efficitur ab <lb></lb>onere in C, quam in D, collocato. </s> <s id="s.003303">Contra verò geſtatorem A <lb></lb>magis premit onus in D quam in C poſitum, quia motus reſpi<lb></lb>cit centrum B, atque adeo ad eandem diſtantiam AB major <lb></lb>eſt Ratio diſtantiæ DB majoris, quam CB minoris diſtantiæ: <lb></lb>eadem autem eſt diſtantiarum, & motuum Ratio, ac proinde <lb></lb>momentorum. </s> </p> <p type="main"> <s id="s.003304">Obſerva autem mihi ideò de geſtatoribus oneris ſermonem <lb></lb>fuiſſe, ut duplicem effectum ſuſtentationis atque preſſionis ex<lb></lb>preſſius recognoſcerem; qui enim onus geſtando ſuſtentant, <lb></lb>muſculorum contentione conantes elidunt impetum oneris de<lb></lb>orſum nitentis, & aliquid efficientes, dum Activè reſiſtunt, no<lb></lb>men Potentiæ merentur. </s> <s id="s.003305">At ſi onus palangæ connexum ſuſti<lb></lb>neretur à duobus fulcris in extremitate poſitis, hæc utique cùm <lb></lb>oneris gravitati nullo conatu adverſarentur, ſolam reſiſtentiam <lb></lb>Formalem ſuâ ſoliditate exercerent, impediendo ne onus cum <lb></lb>palangâ deſcenderet, ſed nullam haberent <expan abbr="Reſiſtentiã">Reſiſtentiam</expan> Activam, <lb></lb>quæ illis Potentiæ vocabulum tribueret. </s> <s id="s.003306">In his unicus preſſionis <lb></lb>effectus attendendus eſt, & validiori fulcro propiùs admoven<lb></lb>dum eſt onus, ne fortè fulcrum infirmius nimiâ preſſione coga<lb></lb>tur ſuccumbere. </s> </p> <p type="main"> <s id="s.003307">Unum adhuc in palangæ geſtatoribus attendendum eſt, ſi vi<lb></lb>rium inæqualium fuerint, & onus non ita ſit palangæ applica<lb></lb>tum, ut ejus diſtantiæ à geſtatoribus ſint permutatim ut eorum<lb></lb>dem vires; nimirum contingere poſſe, ut validior geſtator dum <pb pagenum="429" xlink:href="017/01/445.jpg"></pb>juxta ſuas vires conatur adversùs onus, magis premat infirmio<lb></lb>rem geſtatorem, quam premeretur fulcrum infirmius, ſi unâ <lb></lb>cum validiore fulcro eandem gravitatem ſuſtineret. </s> <s id="s.003308">Quia vide<lb></lb>licet in eâdem palanga AB vectem primi generis conſiderare <lb></lb>poſſumus, in quo onus C deorſum nitens contra vim geſtatoris <lb></lb>habeat rationem hypomochlij, & validior geſtator A ſit poten<lb></lb>tia repellens geſtatorem infirmiorem B conantem adverſus <lb></lb>onus, ac proinde illum premat: quemadmodum ſi funi deorſum <lb></lb>firmiter alligato inſereretur palanga, cui humeros ſubjicerent <lb></lb>duo inæqualibus viribus ſurſum conantes; conſtat enim infir<lb></lb>miorem à validiore premi, & eſſe vectem primi generis. </s> <s id="s.003309">Ex quo <lb></lb>vides, cur bajuli dato invicem ſigno curent, ne alter alterum <lb></lb>præveniat in elevanda palanga; ne ſcilicet qui ſegnior fuerit, <lb></lb>preſſionem, non ab onere adhuc jacente & nondum elevato, ſed <lb></lb>à ſocio diligentiùs ſuam palangæ extremitatem elevante, reci<lb></lb>piat. </s> </p> <p type="main"> <s id="s.003310">Hæc eadem, quæ de onere ſublevando ſunt dicta, de eodem <lb></lb>trahendo pariter intelligantur, ſi vecti illigatum ſit onus, & <lb></lb>vectis extremitatibus jungantur trahentes: horum enim cona<lb></lb>tus eſſe oportet permutatim in Ratione diſtantiarum ab onere, <lb></lb>hoc eſt à vectis puncto, cui onus adnectitur. </s> <s id="s.003311">Id non ſine jucun<lb></lb>dâ quadam animi titillatione vidi aliquando obſervatum à ruſti<lb></lb>co, qui alterius equorum currum trahentium defatigati laborem <lb></lb>miſeratus, tranſverſarium, cui ambo adjungebantur, ita tranſtu<lb></lb>lit, ut in partes inæquales à temone diſtingueretur, & longior <lb></lb>tranſverſarij pars ad debiliorem equum ſpectaret. </s> <s id="s.003312">Inerant ſiqui<lb></lb>dem tranſverſario tria foramina, per quæ temoni nectebatur fer<lb></lb>reo clavo; unum quidem plane æqualiter ab extremitatibus abe<lb></lb>rat, reliqua duo hinc & hinc à medio diſtabant modico quidem <lb></lb>ſed congruo intervallo, ut ſi equus dexter defatigaretur, clavus <lb></lb>immitteretur ſiniſtro foramini, aut contra dextro, ſi ſiniſter <lb></lb>equus languidiùs traheret. </s> <s id="s.003313">Verùm cautè modica intervalla de<lb></lb>finierat, ne nimia fieret momentorum inæqualitas; quod enim <lb></lb>alteri equorum laboris demebatur, addebatur reliquo. </s> </p> <p type="main"> <s id="s.003314">Quando autem non palangâ defertur onus, ſed ipſum imme<lb></lb>diate à duobus ſuſtinetur, eadem prorſus eſt philoſophandi ra<lb></lb>tio, quandoquidem eſt quodammodo onus vecti conjunctum, <lb></lb>atque juxta vectis longitudinem diſtributum. </s> <s id="s.003315">Quamvis verò <pb pagenum="430" xlink:href="017/01/446.jpg"></pb>ſingulis partibus ſua gravitas inſit, quia tamen in unam coaleſ<lb></lb>cunt gravitatem, ideò totius molis gravitas ibi intelligenda eſt, <lb></lb>ubi eſt centrum gravitatis; vectis autem longitudo æſtimanda <lb></lb>eſt in lineá jungente puncta, quibus geſtatores aut ſuſtinentes <lb></lb>applicantur: Ex quibus punctis ſi ponamus exire lineas paral<lb></lb>lelas lineæ directionis exeunti ex centro gravitatis, cadent om<lb></lb>nes ad perpendiculum in lineam horizontalem tranſeuntem per <lb></lb>centrum gravitatis, aut illi parallelam. </s> <s id="s.003316">Harum igitur parallela<lb></lb>rum, quæ directionem conatûs oppoſiti gravitationi ponderis <lb></lb>referunt, diſtantia à lineâ directionis centri gravitatis, ipſorum <lb></lb>deferentium conatum in ſuſtinendo, permutatim ſumpta de<lb></lb>finiet; quæcumque demùm ſit oneris figura. </s> </p> <p type="main"> <s id="s.003317">Sit onus deferendum, cujus centrum gravitatis E; linea per <lb></lb><figure id="id.017.01.446.1.jpg" xlink:href="017/01/446/1.jpg"></figure><lb></lb>geſtatores tranſiens, habenſque rationem vectis, ſit BC, quæ <lb></lb>intelligatur horizonti parallela. </s> <s id="s.003318">In hanc igitur ad angulos <lb></lb>rectos cadit linea directionis EF; & geſtatores in quocumque <lb></lb>puncto lineæ BC fuerint, permutatim habent momenta ſuſti<lb></lb>nendi, aut recipiunt momenta preſſonis pro Ratione diſtan<lb></lb>tiarum à puncto F, in quod cadit linea directionis: cùm enim <lb></lb>linea BC ex hypotheſi ſit horizonti parallela, omnium ipſi FE <pb pagenum="431" xlink:href="017/01/447.jpg"></pb>parallelarum diſtantia ab eádem FE, deſumenda eſt ex inter<lb></lb>valli, geſtatorum & puncti F. </s> </p> <p type="main"> <s id="s.003319">At verò ſi recta BC non fuerit horizonti parallela, vel quia <lb></lb>deferentes onus non ſunt æquè alti, vel quia in clivo conſiſtunt, <lb></lb>utique linea directionis centri gravitatis E non cadit amplius in <lb></lb>rectam BC ad angulos rectos in F, ſed obliquè incidit in H. </s> <lb></lb> <s id="s.003320">Concipiatur itaque per H tranſiens linea GI horizonti paral<lb></lb>lela, & ipſi EH ſint parallelæ CD & BK: ſunt igitur diſtan<lb></lb>tiæ HD & HK, quæ eandem inter ſe habent Rationem, quæ <lb></lb>reperitur inter HC & HB; ſunt enim triangula HDC & <lb></lb>HKB rectangula ad D & K, æquales angulos ad verticem H <lb></lb>habentia, ac proinde ſimilia, & per 4. lib.6 ut HD ad HK, <lb></lb>ita HC ad HB. </s> <s id="s.003321">Quo igitur magis ab horizonte removetur <lb></lb>punctum B præ puncto C, etiam linea directionis ex E propior <lb></lb>cadit puncto C; atque adeò qui inferior eſt, magis gravatur <lb></lb>ab onere. </s> </p> <p type="main"> <s id="s.003322">Id quod ex iis, quæ hujus libri cap.4. dicta ſunt, confirma<lb></lb>tur: Nam vectis CB habens hypomochlium B, & pondus E <lb></lb>vecti impoſitum, eſt infra horizontem inclinatus; igitur plus la<lb></lb>boris potentia impendit, quàm in horizontali poſitione vectis. </s> <lb></lb> <s id="s.003323">Similiter vectis BC habens hypomochlium C & pondus E <lb></lb>vecti impoſitum, eſt elevatus ſupra horizontalem; igitur mi<lb></lb>nus laborat potentia quàm in poſitione horizontali. </s> <s id="s.003324">Itaque ſi <lb></lb>poſita lineá BC horizonti parallelâ æqualiter premebantur <lb></lb>geſtatores in B & in C, facta inclinatione ad horizontem, mi<lb></lb>nùs premitur B ſuperior quàm C loco inferior. </s> </p> <p type="main"> <s id="s.003325">Quòd ſi geſtatores non ſuſtineant onus ſubjectis humeris, ſed <lb></lb>illud manibus arreptum quaſi ſuſpenſum retineant in M & N; <lb></lb>ſimili ratione attendenda eſt diſtantia illorum à puncto, in quod <lb></lb>cadit linea directionis centri gravitatis E; quæ utique ad angu<lb></lb>los recto incidit in rectam MN, ſi hæc ſuerit horizonti pa<lb></lb>rallela, & labor geſtatorum eſt permutatim ut eorum diſtantia <lb></lb>à pancto S. </s> <s id="s.003326">At verò ſi linea MN fuerit ad horizontem incli<lb></lb>nata, & linea directionis ſit EO; utique minor eſt diſtantia à <lb></lb>ſuperiore M, quam ab inferiore N, ideoque plus laborabit ſu<lb></lb>perior retinendo, quam inferior. </s> <s id="s.003327">Id quod pariter ex dictis <lb></lb>cap.4. confirmatur; nam pondus eſt vecti ſubjectum, & vectis <lb></lb>MN habens hypomochlium N eſt ſupra horizontalem lineam, <pb pagenum="432" xlink:href="017/01/448.jpg"></pb>ac propterea potentia plus laborat quam in horizontali: contra <lb></lb>autem vectis NM habens hypomochlium M eſt inſta horizon<lb></lb>talem lineam depreſſus, ideoque minus potentia laborat quam <lb></lb>in horizontali. </s> <s id="s.003328">Quæ omnia tam aperte reſpondent quotidia<lb></lb>no experimento, ut mirum videatur potuiſſe aliquos authores <lb></lb>idem planè opinari, ſive geſtatores ſuſtineant impoſitum onus, <lb></lb>ſive illud ſuſpenſum retineant in poſitione vectis declivi; Si <lb></lb>enim ducti per O lineâ horizonti parallelâ, ducantur ex M & <lb></lb>N rectæ MT, & NV parallelæ lineæ directionis centri gravi<lb></lb>tatis EO, utique diſtantiæ ſunt TO & VO: atqui TO ad <lb></lb>VO eſt ut MO ad NO propter triangulorum OTM & <lb></lb>OVN ſimilitudinem; & MO ad ON habet minorem Ratio<lb></lb>nem quàm MS ad SN ex 8.lib.5. igitur etiam TO ad OV <lb></lb>habet minorem Rationem quàm MS ad SN: igitur in poſitio<lb></lb>ne vectis declivi, M ſuperior laborabit ut ON, atque N infe<lb></lb>rior laborabit ut OM. </s> </p> <p type="main"> <s id="s.003329">Ex his unuſquiſque intelligit non ad duos tantùm geſtato<lb></lb>res, ſed etiam ad plures referenda eſſe, quæ hactenus diximus, <lb></lb>habitâ ſcilicet diſtantiarum ratione, quibus ſinguli abſunt a <lb></lb>pondere, adeò ut qui æqualibus intervallis à pondere diſtant, <lb></lb>æqualem conatum impendant in eo ſuſtinendo. </s> <s id="s.003330">Sic ſi à pon<lb></lb><figure id="id.017.01.448.1.jpg" xlink:href="017/01/448/1.jpg"></figure><lb></lb>dere P æqualiter diſtent A & B, æqua<lb></lb>liter premuntur: item C & D æquali<lb></lb>ter diſtantes ab eodem pondere P æqua<lb></lb>lem preſſionem recipiunt: Et ſi compa<lb></lb>rentur invicem D & B, aut C & A, <lb></lb>manifeſtum eſt propinquiores premi præ remotioribus; ac <lb></lb>propterea, ſi ſolùm poſitionis ratio haberetur, qui robuſtiores <lb></lb>ſunt, collocandi eſſent in C & D, infirmiores verò in A & B: <lb></lb>ſed quoniàm contingit inter plures ſodales aliquem aliquando <lb></lb>connivere, ideò ut plurimum extremi A & B validiores ſunt, <lb></lb>ut ſi fortè mediorum aliquis languidiùs conetur ſuſtinendo, <lb></lb>illi faciliùs muneri ſuo ſatisfaciant. <pb pagenum="433" xlink:href="017/01/449.jpg"></pb></s> </p> <p type="main"> <s id="s.003331"><emph type="center"></emph>CAPUT X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003332"><emph type="center"></emph><emph type="italics"></emph>An vis Elaſtica ad aliquod Vectis genus <lb></lb>pertineat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003333">QUoniam Græcis <foreign lang="grc">ἐλασμὰ</foreign> tùm laminam, tùm plicam ſeu <lb></lb>flexum ſignificat, atque <foreign lang="grc">ἔλαστρῶν</foreign> eſt id, quod impellit; ſæ<lb></lb>pius autem chalybeas laminas in machinulis ita diſponimus, ut <lb></lb>primùm flexæ, deinde ſibi dimiſſæ, dum ſeſe reſtituunt, aliud <lb></lb>corpus impellant, cui motum concilient; propterea Elaſmata, <lb></lb>ſeu Elaſmos, hujuſmodi laminas dicimus, quas Itali <emph type="italics"></emph>Suſte <emph.end type="italics"></emph.end> aut <lb></lb><emph type="italics"></emph>molle<emph.end type="italics"></emph.end> vocamus; & facultatem illam, qua ſibi congruentem fi<lb></lb>guram atque poſitionem hæ laminæ reparant, Vim Elaſticam <lb></lb>appellamus. </s> <s id="s.003334">Quamquam non ſolis laminis, ſed cæteris quoque <lb></lb>corporibus per vim inflexis, & ad ſibi debitam rectitudinem <lb></lb>redeuntibus, facultas hæc Elaſtica tribuenda eſt, quemadmo<lb></lb>dum flexilibus virgultorum ramis, à quibus ſecundus in ſylvâ <lb></lb>ſibi cavere debet, & perticæ, quâ toreutæ utuntur in toreu<lb></lb>mate elaborando, dum tornum circumagunt circumducto fu<lb></lb>niculo, qui depreſſo ſuppedanco perticam flectit, hæc enim, <lb></lb>ceſſante pedis preſſione, funiculum retrahens ſuam ſibi repa<lb></lb>rat rectitudinem. </s> <s id="s.003335">An verò ignis atque aër ſive externá com<lb></lb>preſſione, ſive alieno frigore concretus, & in exigua ſpatia con<lb></lb>tractus, ubi ceſſante vi, aut abeunte frigore, extenuatus am<lb></lb>pliorem locum occupat, proximumque corpus pellens à ſuá ſe<lb></lb>de removet, facultate Elaſticâ præditus dicendus ſit, quæſtio <lb></lb>Grammaticis dirimenda relinquatur: hæc enim fluida corpora, <lb></lb>nullam partium texturam habentia, nec certis figuræ, quam <lb></lb>expetant, terminis ſuapte naturà circumſcripta, vix quicquam <lb></lb>cum Elaſmate commune habere videntur. </s> </p> <p type="main"> <s id="s.003336">Cum itaque inæquales deprehendantur elaſmatis ejuſdem <lb></lb>vires pro diversâ ſuarum partium poſitione juxta longitudinem, <lb></lb>animum ſubiit cupido examinandi, an forte in eo aliqua vectis <lb></lb>ſpecies reperiatur, ut propterea & vectis rationibus illa virium <pb pagenum="434" xlink:href="017/01/450.jpg"></pb>inæqualitas definienda ſit. </s> <s id="s.003337">Et quidem manifeſtum eſt aliquam <lb></lb>elaſmatis partem fixam eſſe atque manentem, ſive illa extrema <lb></lb>ſit, ut in perticâ toreutæ, ſive media, ut in arcu baliſtæ, ſive <lb></lb>utraque extremitate manente pars media flectatur in ſinum, ut <lb></lb>citharæ nervis contingit. </s> <s id="s.003338">Quî enim fieri poſſet, ut per vim la<lb></lb>mina flecteretur, ſi partes omnes æqualiter moverentur? </s> <s id="s.003339">Ut igi<lb></lb>tur externam vim recipiat, & flectatur, aliquam ejus partem <lb></lb>oportet aut omnino immotam manere, aut ſaltem languidiùs <lb></lb>moveri. </s> </p> <p type="main"> <s id="s.003340">Hinc eſt elaſmatis motum, dum inflectitur, circa partem <lb></lb>manentem perfici, ac proinde particulas, quæ ad cavam qui<lb></lb>dem ſuperficiem ſpectant, per vim comprimi, quæ verò ad con<lb></lb>vexam, intendi. </s> <s id="s.003341">Quòd ſi particulæ illæ non ita tenaci nexu in<lb></lb>ter ſe invicem cohærerent, ut facilè diſtraherentur contentæ, <lb></lb>& exprimerentur compreſſæ, quemadmodum plumbeæ laminæ, <lb></lb>quæ in figuram quamlibet conformatur, accidit, amiſſam recti<lb></lb>tudinem non recuperarent. </s> <s id="s.003342">Sed quoniam arctiſſimo vinculo <lb></lb>conjunguntur, quod niſi validioribus viribus revelli non poteſt, <lb></lb>ut in chalybeâ laminâ obſervamus; ceſſante externâ vi, quæ <lb></lb>contentæ fuerant, ſe contrahunt, quæ compreſſæ, ſelatiùs ex<lb></lb>plicant; atque adeò his debitam poſitionem ſibi reparantibus, <lb></lb>lamina ad priſtinam formam eò vehementiùs redit, quò majo<lb></lb>rem violentiam patiebatur. </s> <s id="s.003343">Quare Potentia movens ſunt ipſæ <lb></lb>particulæ illatam vim excutientes, & ad ſibi debitam poſitio<lb></lb>nem redeuntes. </s> </p> <p type="main"> <s id="s.003344">Licet igitur in arcu baliſtæ intento duplex elaſma hinc atque <lb></lb>hinc conſiderare; media ſi quidem pars arcûs baliſtæ manubrio <lb></lb>infixa manet, & ſingula cornua ſinuantur, ſed eò difficiliùs, <lb></lb>quò breviora ſunt, cæteris paribus, attentâ eorum craſſitie, & <lb></lb>ferri temperatione; pari enim flexione paucioribus minoris ar<lb></lb>cûs particulis major violentia inferenda eſt, quippe quas ma<lb></lb>gis comprimi, magíſque intendi oportet, quàm in longiore ar<lb></lb>cu, ubi minore plurium partium compreſſione & intentione <lb></lb>flexio eadem habetur. </s> <s id="s.003345">Præterquam quod in ipſa flexione adhi<lb></lb>betur quodammodo vectis ſecundi generis, quem ipſa longitu<lb></lb>do repræſentat, pars manens vicem hypomochlij ſubit, & par<lb></lb>tes intermediæ, quas per vim coarctari aut dilatari oportet, lo<lb></lb>cum obtinent ponderis: nihil igitur mirum, ſi Potentia extre-<pb pagenum="435" xlink:href="017/01/451.jpg"></pb>mitatem arcûs ad ſe nervo adnexo trahens faciliùs moveat par<lb></lb>ticulas eaſdem, quò, longiùs abſens ab hypomochlio, faciliùs <lb></lb>movetur. </s> </p> <p type="main"> <s id="s.003346">Hîc autem ubi arcûs mentio incidit, in ipſo nervo illud elaſ<lb></lb>matis genus occurrit, quod utramque extremitatem habet ma<lb></lb>nentem; curvato enim arcu nervus inflexus intenditur; poſtea <lb></lb>cùm dimittitur, pars media, cui ſagitta aut globus excutiendus <lb></lb>aptatur, plus movetur quàm ejus extremitates arcûs cornibus <lb></lb>cohærentes. </s> <s id="s.003347">Univerſa autem violentia, quam nervus conten<lb></lb>tus ſubit, conſiſtit in ſuarum particularum intentione, quæ, <lb></lb>dum ſe contrahentes aliquid juvant ad nervum ipſum juxta <lb></lb>rectam lineam extendendum, aliquid etiam impetûs ſagittæ ex<lb></lb>cuſſæ imprimunt. </s> </p> <p type="main"> <s id="s.003348">Quod verò ad ipſa arcûs cornua attinet, ſatis liquet illa ſi<lb></lb>milis craſſitiei, paris longitudinis, æqualiſque temperationis <lb></lb>eſſe debere, ut æqualis fiat hinc & hinc compreſſio atque in<lb></lb>tentio partium, ex qua æquales oriantur vires ſeſe in priſtinam <lb></lb>formam reſtituendi. </s> <s id="s.003349">Si enim alterutra pars arcûs majorem <lb></lb>violentiam paſſa velociùs atque validiùs præ reliquâ ſe move<lb></lb>ret, à deſtinato ſcopo ſagitta aberraret in dexteram aut in ſi<lb></lb>niſtram declinans. </s> </p> <p type="main"> <s id="s.003350">Ut igitur hiſce prænotatis ad propoſitam quæſtionem acce<lb></lb>damus, non eſt hîc ſermo de laminâ in ſpiram multiplicem in<lb></lb>flexâ, atque ſpiſsè per vim contorta, quæ amoto repagulo ſeſe <lb></lb>in ampliores gyros explicans ſecum rapit aliud corpus extremi<lb></lb>tati mobili adnexum; cujuſmodi eſt Elaſma in Automatis ho<lb></lb>ras indicantibus, cujus extremitati adnectitur tympanum ſpi<lb></lb>ram illam includens; dum enim ex dilatatione Elaſmatis in am<lb></lb>pliorem ſpiram, circumagitur tympanum, adnexam catenu<lb></lb>lam conum circumplexam trahit, totique machinulæ motum <lb></lb>conciliat. </s> <s id="s.003351">Hîc ſiquidem, uti nulla longitudo in conſideratio<lb></lb>nem cadere poteſt, nullam vectis ſpeciem habere poſſu<lb></lb>mus; nam facultas movendi non ratione poſitionis exte<lb></lb>nuatur, ut in vecte, ſed vires initio validæ ſenſim lan<lb></lb>gueſcunt, quia elaſmatis partes compreſſæ atque conten<lb></lb>tæ, pro ratione violentiæ, quam ſubeunt, excutiendæ, ve<lb></lb>hementiùs primùm, deinde remiſſiùs conantur. </s> <s id="s.003352">Quare con<lb></lb>troverſia in illo eſt, utrùm in elaſmate, cujus aliqua <pb pagenum="436" xlink:href="017/01/452.jpg"></pb>longitudo deſignari poteſt, aliqua vectis ſpecies repe<lb></lb>riatur. </s> </p> <p type="main"> <s id="s.003353">Et ut majori in luce quæſtio verſetur, perticam toreutæ, <lb></lb><figure id="id.017.01.452.1.jpg" xlink:href="017/01/452/1.jpg"></figure><lb></lb>oculis ſubjiciamus, quæ ſit AB, & <lb></lb>in A fixa atque immota perſeveret, <lb></lb>quamvis extremitas B deprimatur, <lb></lb>ut veniat in C. </s> <s id="s.003354">In hac perticæ <lb></lb>flexione partes, quæ circa D ex. </s> <lb></lb> <s id="s.003355">gr. intelliguntur, maximam violentiam patiuntur, nam inter <lb></lb>eas, quæ ad cavitatem ſpectantes compreſſione coarctantur, <lb></lb>illæ præ cæteris hinc atque hinc cohærentibus urgentur ma<lb></lb>gis; inter eas verò, quæ convexitatem reſpicientes diſtentu <lb></lb>explicantur, quæ ibi ſunt, præ reliquis à ſummo flexu paulò <lb></lb>remotioribus vehementiùs tenduntur. </s> <s id="s.003356">Hinc licèt particulæ <lb></lb>omnes in hac flexione vim paſſæ, dum nituntur ſingulæ priſti<lb></lb>num ſtatum ſibi reparare, conatus ſuos exerant, majores aut <lb></lb>minores pro ratione majoris aut minoris violentiæ; potiſſima ta<lb></lb>men vis elaſtica ibi conſideranda eſt, ubi ſumma inflexio ſum<lb></lb>mam vim particulis infert; ibi enim majore conatu quàm alibi <lb></lb>violentiam excutit natura. </s> <s id="s.003357">Quamvis igitur vis elaſtica per uni<lb></lb>verſam elaſmatis longitudinem, quàm particulæ compreſſæ at<lb></lb>que contentæ obtinent, extendatur, ibi tamen potiſſimùm col<lb></lb>locata intelligitur, ubi in ſummo flexu puta in D, validiùs co<lb></lb>natur. </s> </p> <p type="main"> <s id="s.003358">Jam verò quis ignorat in tornando plurimum intereſſe, utrùm <lb></lb>funiculus in ipsâ extremitate B, an verò in E adnectatur? </s> <s id="s.003359">Si<lb></lb>quidem, ſicut ex E difficiliùs flectitur pertica, quàm ex B, æqua<lb></lb>li flexione, ita cæteris paribus in E validiùs retrahitur funicu<lb></lb>lus, & minor motus perficitur quàm in B. </s> <s id="s.003360">Eſt igitur hîc ratio <lb></lb>Vectis tertij generis, in quo hypomochlium eſt A pars fixa & <lb></lb>immota; Potentia movens (ſcilicet particulæ vim illatam ex<lb></lb>cutientes) eſt potiſſimùm in D; pondus, quod movetur, eſt <lb></lb>ultra D, ſive in extremitate B, ſive in aliqua ex partibus inter<lb></lb>mediis, ut in E. </s> <s id="s.003361">Quare in collocatione corporis, quod ope <lb></lb>elaſmatis movendum eſt, attendere oportet, quanto motu opus <lb></lb>ſit, ut in majore ſeu minore diſtantiâ à puncto elaſmatis ma<lb></lb>nente, & immoto applicetur: quo enim minor eſt diſtantia, <lb></lb>minus ſpatium percurrit. </s> </p> <pb pagenum="437" xlink:href="017/01/453.jpg"></pb> <p type="main"> <s id="s.003362">Quamvis autem elaſmatis vires ad impellendum vel trahen<lb></lb>dum corpus ex hujuſmodi diſtantiâ pendeant, & comparatis <lb></lb>inter ſe duabus poſitionibus E atque B, validiùs operetur in E <lb></lb>quàm B, non tamen eâdem vi motus (quicumque demum ille <lb></lb>ſit ſive major, ſive minor) inchoatur, atque procedit, ut ſupra <lb></lb>innuimus; natura quippe remiſſiore niſu reluctatur, ubi mino<lb></lb>rem patitur violentiam, ac proinde ſenſim attenuatur conatus, <lb></lb>quatenus particularum violenta compreſſio atque contentio di<lb></lb>minuitur. </s> </p> <p type="main"> <s id="s.003363">Hæc quæ de elaſmate prorſus recto explicata ſunt, etiam de <lb></lb>incurvo intelliguntur; cujuſmodi eſſet lamina chalybea RT <lb></lb>inflexa in S, cujus manens & immota ex<lb></lb><figure id="id.017.01.453.1.jpg" xlink:href="017/01/453/1.jpg"></figure><lb></lb>tremitas eſſet R: dum enim pars ST pro<lb></lb>pellitur versùs R, particulæ, potiſſimùm <lb></lb>quæ in S, comprimuntur atque intendun<lb></lb>tur. </s> <s id="s.003364">Quod ſi pars RS paulo longior fuerit, <lb></lb>contingere poteſt, ut facilius ſit illam inflecti ſaltem leviter, <lb></lb>quàm particulas in S ulteriùs comprimi, aut intendi. </s> <s id="s.003365">Quare <lb></lb>particulæ ipſius RS ſeſe reſtituentes impellunt S, particulæ au<lb></lb>tem ipſius ST impellunt T. </s> <s id="s.003366">Semper autem Potentiam minùs <lb></lb>moveri, quàm corpus, quod impellitur, conſtat, quemadmo<lb></lb>dum ratio vectis tertij generis exigit. </s> </p> <p type="main"> <s id="s.003367">Neque his, quæ dicta ſunt, adverſantur percuſſiones, quæ <lb></lb>in extremitate longioris elaſmatis per vim inflexi, ſtatimque di<lb></lb>miſſi, validiores fiunt, quam in partibus mediis; ſicut ipſe te <lb></lb>docere potes, ſi longiuſculi virgulti inflexi atque dimiſſi pri<lb></lb>mùm parti mediæ deinde extremitati manum in eodem plano <lb></lb>verticali conſtitutam opponas, quam percutiat, magis enim ex <lb></lb>ſecundâ quàm ex primâ percuſſione dolebis. </s> <s id="s.003368">Quia ſcilicet non <lb></lb>impetus ſolùm primo productus, ſed & velocitas percutientis <lb></lb>cum impetu acquiſito ex motu ante percuſſionem (ut ſuo loco <lb></lb>dicetur) attenditur, ut validior ſit ictus: majorem autem eſſe <lb></lb>partis extremæ quàm mediarum velocitatem conſtat, quamvis <lb></lb>initio illæ impetu eodem, aut æquali moveantur. </s> <s id="s.003369">Quando ve<lb></lb>rò elaſmatis vires prope partem manentem majores eſſe, quam <lb></lb>procul ab illâ, dictum eſt, non eſt habita ratio percuſſionis, <lb></lb>quæ prævium percutientis motum requirit, ſed tractionis aut <lb></lb>impulſionis, quæ nullum trahentis aut impellentis prævium <pb pagenum="438" xlink:href="017/01/454.jpg"></pb>motum exigunt, eóque faciliores accidunt, quò tardiores ſunt; <lb></lb>minùs enim reſiſtit corpus, quod tardè movetur, ac proinde <lb></lb>validiùs trahitur aut impellitur, quo minorem potentia invenit <lb></lb>reſiſtentiam: Contrà quàm accidat in percuſſione, quæ vali<lb></lb>diorem facit ictum, quo majorem invenit reſiſtentiam; hæc <lb></lb>autem major eſt, quò velociùs moveri deberet corpus percuſ<lb></lb>ſum, ut percutientis motui obſecundaret, cui magis reſiſtens <lb></lb>majorem ictum recipit; cum tamen hic languidior eſſet atque <lb></lb>infirmior, ſi manum ſenſim ſubduceres virgulto percutienti. </s> <lb></lb> <s id="s.003370">Quare pars elaſmatis extrema validiùs percutit, quia majorem <lb></lb>invenit reſiſtentiam, pars media validiùs trahit aut impellit, <lb></lb>quia minùs illi reſiſtitur. <lb></lb></s> </p> <p type="main"> <s id="s.003371"><emph type="center"></emph>CAPUT XI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003372"><emph type="center"></emph><emph type="italics"></emph>Cur longiora corpora faciliùs flectantur, difficiliùs <lb></lb>ſuſtineantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003373">PRæſens diſputatio non diſtat ab iis, quæ ab Ariſtotele in<lb></lb>quiruntur in Mechanicis quæſt.14. <emph type="italics"></emph>Cur ejuſdem magnitu<lb></lb>dinis lignum faciliùs genu frangitur, ſi quiſpiam æquè diductis ma<lb></lb>nibus extrema comprehendens fregerit, quàm ſi juxta genu: & ſi <lb></lb>terræ illud applicans pede ſuperimpoſito manu longè diductâ confre<lb></lb>gerit, quàm propè.<emph.end type="italics"></emph.end> & quæſt. </s> <s id="s.003374">16. <emph type="italics"></emph>Cur quanto longiora ſunt ligna, <lb></lb>tanto imbecilliora fiunt: & ſi tollantur, inflectuntur magis; tametſi <lb></lb>quod breve quidem eſt, ceu bicubitum, fuerit tenue; quod verò cu<lb></lb>bitorum centum, craſſum?<emph.end type="italics"></emph.end> & quæſt. </s> <s id="s.003375">26. <emph type="italics"></emph>Cur difficilius eſt longa <lb></lb>ligna ab extremo ſuper humeros ferre, quàm ſecundùm medium, <lb></lb>æquali exiſtente pondere?<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.003376">Et quidem quod ad primum, ſcilicet ad flexionem ſpectat, <lb></lb>quam demum conſequitur fractio, quemadmodum flectendi <lb></lb>corpus aliquod longum aut frangendi difficultas oritur ex com<lb></lb>plexione atque copulatione particularum, quibus conſtat, ægrè <lb></lb>diſſolubili, ita illud inflectitur, atque frangitur, cum earum<lb></lb>dem particularum coagmentatio vehementi impulſione labe-<pb pagenum="439" xlink:href="017/01/455.jpg"></pb>factatur, his quidem per vim compreſſis, his verò validè in di<lb></lb>verſa diſtractis. </s> <s id="s.003377">Quo igitur faciliùs compreſſio hæc atque di<lb></lb>ſtractio perficitur, eò etiam faciliùs flectitur corpus, aut fran<lb></lb>gitur. </s> <s id="s.003378">Hanc autem particularum compreſſionem atque di<lb></lb>ſtractionem faciliùs contingere longiori corpori quàm breviori <lb></lb>manifeſtum eſt; quia videlicet motus ille particularum ad fle<lb></lb>xionem aut fractionem neceſſarius minorem Rationem habet <lb></lb>ad motum Potentiæ longiùs applicatæ, quàm ad motum Po<lb></lb>tentiæ propioris. </s> </p> <p type="main"> <s id="s.003379">Potentia movens bifariam conſiderari poteſt, ſivè in ipſo cor<lb></lb>pore incluſa, cujuſmodi eſt illi inſita atque ingenita gravitas, <lb></lb>vi cujus ſponte ſuâ flectitur; ſivè extrinſecùs adhibita, ut ſi <lb></lb>onus aliquod grave deorſum premens adjiciatur, aut potentia <lb></lb>vivens ad motum in quamcumque poſitionis differentiam apta: <lb></lb>utrobique tamen eſt eadem ratio; ubi ſcilicet aſſumpta atque <lb></lb>adventitia potentia applicatur, ibi operatur; atque ibi innata <lb></lb>gravitas intelligitur ſua exercere momenta, ubi partis ultra ſub<lb></lb>jectum fulcrum extantis centrum gravitatis reperitur; illáque <lb></lb>eſt à fulcro diſtantia potentiæ flectentis, aut etiam frangentis. </s> <lb></lb> <s id="s.003380">Sit enim priſma AB, cujus pars AC <lb></lb>infixa ſit parieti, extra quem emineat <lb></lb><figure id="id.017.01.455.1.jpg" xlink:href="017/01/455/1.jpg"></figure><lb></lb>horizonti parallela pars CB ſuâ gravi<lb></lb>tate deorſum connitens; quæ ſane non <lb></lb>eſt intelligenda in B, ſed quaſi tota <lb></lb>conſtituta eſſet in D, ubi eſt centrum gravitatis non totius cor<lb></lb>poris AB, ſed partis extantis CB. </s> <s id="s.003381">Quod ſi brevius eſſet priſ<lb></lb>ma AE, partis CE minor eſſet gravitas, quàm partis CB, & <lb></lb>prætereà minus abeſſe à fulcro C intelligeretur, quippe cujus <lb></lb>centrum gravitatis eſſet F multo propius quàm D. </s> <s id="s.003382">Plura igi<lb></lb>tur momenta habet CB quàm CE ad flectendum priſma parie<lb></lb>ti infixum, juxta ea, quæ uberiùs dicta ſunt lib.2. cap.6. ubi ſo<lb></lb>lidorum Reſiſtentiam reſpectivam conſideravimus, nec vacat <lb></lb>hic iterum inculcare. </s> </p> <p type="main"> <s id="s.003383">Unum hîc conſiderandum eſt, quod ad rationes vectis atti<lb></lb>net, videlicet, ſi ſuperiori priſmatis parti, quæ reſpondet ipſi <lb></lb>AC, incumberet onus, quod faciliùs loco moveri poſſit, quàm <lb></lb>particularum complexio labefactari, aut omnino diſſolvi, priſ<lb></lb>ma neque frangi, aut fortaſſe ne flecti quidem contingeret; <pb pagenum="440" xlink:href="017/01/456.jpg"></pb>ſed rationem vectis primi generis haberet, cujus fulcrum eſſet <lb></lb>in C, pondus in A, potentia in D. </s> <s id="s.003384">At ſi onus impoſitum nul<lb></lb>latenus dimoveri queat, quemadmodum cùm priſma parieti in<lb></lb>figitur, ſi CB ejus ſit longitudinis, ut vis gravitatis ad deſcen<lb></lb>dendum tali intervallo CD ſejuncta à fulcro C plus habeat <lb></lb>momenti, quàm particularum coagmentatio, ne compriman<lb></lb>tur, aut diſtrahantur; tunc vectis eſt ſecundi generis, fulcrum <lb></lb>quidem habens in C, quatenus totum ſegmentum AC retine<lb></lb>tur prorſus immotum, & potentia in D, pondus verò, cujus <lb></lb>vires vincuntur, eo loco, ubi maxima fit particularum com<lb></lb>preſſio atque diſtractio. </s> </p> <p type="main"> <s id="s.003385">Hinc factum videtur ſatis Ariſtoteli quærenti, cur faciliùs <lb></lb>flectatur lignum craſſum cubitorum centum, quàm tenue bi<lb></lb>cubitum; quia nimirum in craſſo ligno cubitorum centum è <lb></lb>pariete extantium, ſi ponatur ſimilem atque æquabilem craſſi<lb></lb>tiem juxta totam longitudinem habere, gravitatis centrum <lb></lb>diſtat à fulcro cubitis quinquaginta, tenue verò atque exile <lb></lb>lignum ſimilis figuræ atque materiæ centrum habet uno tantùm <lb></lb>cubito diſtans à fulcro, & gravitas illius ad hujus gravitatem in <lb></lb>eâ eſt Ratione, quam habent inter ſe ipſorum lignorum moles, <lb></lb>quæ ſcilicet ex Rationibus baſium atque longitudinum compo<lb></lb>nitur. </s> <s id="s.003386">Cum itaque longitudo ad longitudinem ſit ut 50 ad 1, <lb></lb>ſi baſium ſimilium latera homologia ſint ut 10 ad 1, baſium <lb></lb>Ratio eſt ut 100 ad 1: quare cùm longioris ligni gravita<lb></lb>tis Ratio ad gravitatem brevioris componatur ex Ratione ba<lb></lb>ſium ut 100 ad 1, & ex ratione longitudinum ut 50 ad 1, gravi<lb></lb>tas longioris ad gravitatem brevioris eſt ut 5000 ad 1. Atqui <lb></lb>momenta ad deſcendendum componuntur ex gravitate & di<lb></lb>ſtantia à fulcro; igitur momenta longioris ad momenta brevio<lb></lb>ris ſunt ut 250000 ad 1. At verò reſiſtentia abſoluta, ne flectan<lb></lb>tur, aut frangantur hujuſmodi ligna, eſt in Ratione compoſitâ <lb></lb>ex Rationibus baſium atque craſſitierum; ac proinde ſi baſes <lb></lb>ſint ſimiles, & ſimiliter poſitæ, Ratio eſt triplicata Rationis la<lb></lb>terum homologorum, hoc eſt Rationis 10 ad 1; atque adeò re<lb></lb>ſiſtentia longioris ad reſiſtentiam brevioris eſt ut 1000 ad 1. <lb></lb>Patet igitur momenta ut 250.000 ad reſiſtentiam ut 1000 ma<lb></lb>jorem habere Rationem, quàm momenta ut 1 ad reſiſtentiam <lb></lb>ut 1: faciliùs ergo illa quàm hæc momenta reſiſtentiam ſibi con-<pb pagenum="441" xlink:href="017/01/457.jpg"></pb>gruentem ſuperant, atque faciliùs lignum craſſum longius <lb></lb>flectitur, aut frangitur, quàm brevius. </s> </p> <p type="main"> <s id="s.003387">Quæ autem de ligno parieti ſecundum alteram extremitatem <lb></lb>infixo dicta ſunt, ſervatâ analogiâ de eodem dicantur, ſi circa <lb></lb>medium fulcro alicui inſiſtat ita, ut hinc atque hinc habeat <lb></lb>gravitatis momenta compoſita ex ipſarum partium gravitate & <lb></lb>ex diſtantia centrorum gravitatis à fulcro, cui innititur: eâdem <lb></lb>enim ratiocinatione colligitur in longiore ligno majorem eſſe <lb></lb>Rationem momentorum gravitatis ad reſiſtentiam ortam ex <lb></lb>partium complexione, ne flectatur, quàm in breviore. </s> <s id="s.003388">Quod <lb></lb>verò ſpectat ad longioris ligni faciliorem flexionem, quando <lb></lb>utraque extremitas innixa eſt ſubjecto fulcro, non videtur pro<lb></lb>priè hujus loci, ſed de eâ dictum eſt ſuperiùs lib.3. cap.12. </s> </p> <p type="main"> <s id="s.003389">Ex his, quæ de priſmate extra parietem extante, quod faci<lb></lb>liùs flectitur, hactenus diximus, ulteriùs patet, cur ex contra<lb></lb>rio longius lignum ut AB, etiamſi parem cum breviore AE <lb></lb>craſſitiem habeat, alterâ extremitate æqualiter in AC ap<lb></lb>prehenſum difficiliùs ſuſtineatur. </s> <s id="s.003390">Nam quod longius eſt ad il<lb></lb>lud, quod brevius eſt, ſecundùm gravitatem, quæ deorſum ni<lb></lb>titur, eam habèt Rationem, quæ eſt longitudinis majoris CB <lb></lb>ad longitudinem minorem CE: & præterea momenta, quæ ex <lb></lb>diſtantia oriuntur, ſunt ut CD ad CF, hoc eſt ut CB ad CE, <lb></lb>ſi quidem ex hypotheſi centrum gravitatis intelligatur in me<lb></lb>diâ longitudine; ſecus autem, univerſaliter juxta diſtantias cen<lb></lb>tri gravitatis à fulcro. </s> <s id="s.003391">Quare tota momentorum Ratio ea eſt <lb></lb>quæ componitur ex Rationibus gravitatum reſpondentium <lb></lb>moli ultrà fulcrum protenſæ, & diſtantiarum centri gravitatis. </s> <lb></lb> <s id="s.003392">Cum itaque in longiore ligno plus inveniatur gravitatis, & ma<lb></lb>gis à fulcro diſtet centrum gravitatis, quàm in breviore ligno, <lb></lb>nil mirum, ſi vis in A poſita, ut contranitatur momentis lon<lb></lb>gioris ligni innixi fulcro C, major eſſe debeat, quàm ut reſiſte<lb></lb>ret momentis ligni brevioris. </s> </p> <p type="main"> <s id="s.003393">Deſinant igitur mirari, qui ſariſſam decem cubitorum per<lb></lb>pendicularem extremo digiti apice ſuſtineri, eandem verò ho<lb></lb>rizontaliter jacentem non niſi valido conatu elevari vident. </s> <lb></lb> <s id="s.003394">Res enim ex dictis perſpicua eſt; quia dum haſta perpendicu<lb></lb>laris digito incumbit, centrum gravitatis rectá deorſum urgens <lb></lb>digito motum ſibi æqualem præſcribit, ac proinde viciſſim <pb pagenum="442" xlink:href="017/01/458.jpg"></pb>æqualis eſt digiti & haſtæ motus ſurſum, ſi digitus ſurſum co<lb></lb>netur: hinc eſt ſolam Rationem gravitatis comparatæ ad vires <lb></lb>ſuſtinendi attendendam eſſe, ideóque ſi ſariſſæ pondus ſit ex. </s> <lb></lb> <s id="s.003395">gr. lib.10, ſolo niſu opus eſt, quo libræ 10 ſuſtineantur. </s> <s id="s.003396">Cum <lb></lb>verò haſta obliqua eſt, & horizonti parallela, ſive ad illum in<lb></lb>clinata, jam non idem ſeu æqualis convenit motus manni <lb></lb>haſtam elevanti, atque centro gravitatis, ſed hoc ad motum <lb></lb>multo majorem incitatur; ac propterea momentorum Ratio <lb></lb>non ex ſolâ gravitate pendet, verùm etiam ex motuum Ratio<lb></lb>ne componitur. </s> </p> <p type="main"> <s id="s.003397">Sit haſta horizontaliter jacens AI cubitorum 10; pars manu <lb></lb><figure id="id.017.01.458.1.jpg" xlink:href="017/01/458/1.jpg"></figure><lb></lb>apprehenſa ſit IC quinta <lb></lb>fermè pars cubiti adeò ut <lb></lb>IC ad CA ſit ut 1 ad 49: <lb></lb>punctum I reſpondet ex<lb></lb>tremæ parti metacarpij, <lb></lb>quâ carpo adhæret arti<lb></lb>culatio minimi digiti: <lb></lb>punctum autem C reſpondet ſecundo indicis articulo; motúſ<lb></lb>que elevationis haſtæ perficitur deprimendo I & elevando C, <lb></lb>ac motûs centrum eſt in juncturâ manûs cum oſſe cubiti; quod <lb></lb>centrum propterea intelligitur reſpondere ſariſſæ ex. </s> <s id="s.003398">gr. in O <lb></lb>inter C & I. </s> <s id="s.003399">Quapropter ſi facultas in I deprimens conſidere<lb></lb>tur, vectis eſt primi generis, ſin autem vis in C elevans atten<lb></lb>datur, vectis eſt tertij generis; pondus verò movendum eſt ſi<lb></lb>ve tota gravitas longitudinis OA in centro gravitatis E, ſive <lb></lb>ſemiſſis gravitatis in extremitate A, ut conſtat ex iis, quæ diſ<lb></lb>putata ſunt lib.3. cap.2. de brachiis libræ. </s> </p> <p type="main"> <s id="s.003400">Intelligatur itaque, facilioris explicationis gratiâ, centrum <lb></lb>motû, in O planè medium inter C & I; eritque tam AO ad <lb></lb>OC, quàm AO ad OI, ut 99 ad 1. Gravitas igitur partis OA <lb></lb>eſt lib. (9 9/10) ex hypotheſi; illius ſemiſſis eſt lib.(4 19/20); cujus mo<lb></lb>mentum in A ad momentum, quod haberet illa eadem in Caut <lb></lb>in I, eſt ut 99 ad 1. Cùm autem potentia in I deprimens æqui<lb></lb>valeat potentiæ elevanti in C, quippe illarum diſtantia ab O <lb></lb>centro motûs ex hypotheſi eſt æqualis, perinde eſt atque ſi in C <lb></lb>unica potentia totum pondus elevans poſita eſſet æquivalens <lb></lb>duplici illi potentiæ in I & in C. </s> <s id="s.003401">Quare potentia in C elevans <pb pagenum="443" xlink:href="017/01/459.jpg"></pb>pondus perpendiculare lib.(9 9/10) ad potentiam in C pariter conſti<lb></lb>tutam elevantem lib.(4 10/20) in diſtantia, quæ exigat motum unde<lb></lb>centuplum erit ut (9 9/10) ad 490, hoc eſt, tàm valida eſſe debet, ut <lb></lb>poſſet perpendiculariter elevare libras 490. </s> </p> <p type="main"> <s id="s.003402">Porrò elevatâ haſtâ ita ut A veniat in F, jam non intelligi<lb></lb>tur ſemiſſis gravitatis in A, ſed in G puncto, quod definitur à <lb></lb>perpendiculari cadente ex F in horizontalem: & idcirco gra<lb></lb>vitas (4 19/20) ducenda eſt in diſtantiam GO minorem quàm AO; <lb></lb>atque ita deinceps minuitur, uſque dum haſta fiat in O hori<lb></lb>zonti perpendicularis, & facillimè ſuſtineatur, aut attollatur. </s> <s id="s.003403">Si <lb></lb>autem in hac ratiocinatione tibi, Lector, placuerit non negli<lb></lb>gere momentum illud exiguum, quod potentiæ elevanti addi<lb></lb>tur à gravitate particulæ OI, non abnuo, ſi operæ pretium te <lb></lb>facturum exiſtimes. </s> </p> <p type="main"> <s id="s.003404">Quòd ſi punctum I concipiatur omnino immotum, illud eſt <lb></lb>centrum motûs, & vis elevans in C aliam habet Rationem; <lb></lb>nam potentiæ motus ad motum ſemiſſis ponderis haſtæ in A eſt <lb></lb>ut 1 ad 50; ſunt igitur lib.5 ex hypotheſi, quæ moventur motu <lb></lb>quinquagecuplo; ac propterea vis elevandi datam haſtam poſi<lb></lb>ta in C, quando haſta eſt horizonti parallela, ea eſſe debet, quæ <lb></lb>poſſit elevare libras 250 perpendiculares. </s> <s id="s.003405">Hinc eſt quod, ſi <lb></lb>haſtam eandem lib.10. humero ita imponas in C, ut apprehen<lb></lb>ſum calcem in I manus retineat, & CI ſit pars decima totius <lb></lb>longitudinis haſtæ parallelæ horizonti, ſemiſſis (ſcilicet lib.4 1/2) <lb></lb>reliquæ haſtæ ultra humerum intelligitur in A, & ut IC ad <lb></lb>CA, hoc eſt ut 1 ad 9, ita lib.4 1/2 ad lib.40 1/2, quibus æquiva<lb></lb>lere debet partis CI momentum & vis manûs deorſum urgen<lb></lb>tis, atque in I retinentis haſtam horizonti parallelam. </s> <s id="s.003406">Perinde <lb></lb>itaque humerus in C premitur ab haſtâ ſic poſitâ, & à manu <lb></lb>deorſum urgente, atque ſi ponderis librarum 81 centrum gra<lb></lb>vitatis immineret humero; nam ſi loco manûs deorſum trahen<lb></lb>tis adderes in I pondus faciens æquilibrium, eſſe oporteret <lb></lb>lib.40; ſiquidem partis CI momentum eſt lib. 1/2 in I. </s> <s id="s.003407">At ſi ex<lb></lb>tremitas I retineatur quidem, ſed nemine deorſum urgente <lb></lb>(quemadmodum ſi in parietis foramen inferatur, & à ſuperio<lb></lb>re foraminis ſaxo impediatur, ne poſſit elevari) in C verò ſuſti<lb></lb>neatur ab humero; tunc humeri preſſio ſoli gravitati haſtæ tri-<pb pagenum="444" xlink:href="017/01/460.jpg"></pb>buenda eſt; haſta quippe eſt vectis ſecundi generis hypomo<lb></lb>chlium in I habens, pondus movendum, hoc eſt, humerum <lb></lb>premendum in C, potentiam verò, hoc eſt lib.5. ſemiſſem gra<lb></lb>vitatis haſtæ, in A, ita ut AI diſtantia ſit decupla diſtantiæ CI: <lb></lb>premitur ergo humerus, quaſi ſuſtineat libras 50. </s> </p> <p type="main"> <s id="s.003408">Demum, ne intacta relinquatur Ariſtotelis quæſtio 14. de <lb></lb>ligno, quod terræ applicatum pede impoſito faciliùs frangitur <lb></lb>manu longè diducta quàm prope, dic ligni partem, quæ inter <lb></lb>pedem impoſitum, & terram ſubjectam interjicitur, eſſe pror<lb></lb>ſus ſimilem parti priſmatis infixi parieti, ne moveatur, manum <lb></lb>verò eſſe potentiam, quæ longiùs applicata majora habet mo<lb></lb>menta ad vincendum nexum particularum ligni; eſt enim lon<lb></lb>gior vectis. </s> <s id="s.003409">Similiter applicato ad genu ligno, & æquè di<lb></lb>ductis manibus; duo ſunt vectes hinc atque hinc, fulcrum ad <lb></lb>genu, ſcilicet ad duo puncta contactuum, habentes, eóque <lb></lb>longiores, quò magis diductæ fuerint manus, ac proinde faci<lb></lb>liùs diſtrahentes particulas extimas ligni, quod circa genu <lb></lb>curvatur, faciliúſque comprimentes particulas ejuſdem ligni <lb></lb>ad cavam faciem pertinentes; quæ dum ſibi viciſſim obſiſtunt, <lb></lb>uberiorem reliquarum diſtractionem juvant: longiorem autem <lb></lb>vectem præ breviori eligendum eſſe quis neſciat? </s> <s id="s.003410">ac propterea <lb></lb>ſi ad genu propiùs admoverentur manus ligno, cùm minor eſ<lb></lb>ſet illarum motûs Ratio ad motum particularum ligni diſtrahen<lb></lb>darum, quàm ſit Ratio motûs illarum longiùs diductarum, uti<lb></lb>que difficiliùs frangeretur lignum; ideóque longiùs diducun<lb></lb>tur manus, ut longiores ſint vectes. <lb></lb></s> </p> <p type="main"> <s id="s.003411"><emph type="center"></emph>CAPUT XII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003412"><emph type="center"></emph><emph type="italics"></emph>Vnde oriantur forcipum & forficum vires.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003413">FOrcipum duplex eſt uſus; primus quidem ad corpus ali<lb></lb>quod firmiter apprehendendum, ſecundus verò ad evel<lb></lb>lendum illud faciliùs, vel ſuâ è ſede dimovendum; id quod <lb></lb>adhibito hujuſmodi inſtrumento faciliùs perficitur, quàm nudâ <pb pagenum="445" xlink:href="017/01/461.jpg"></pb>manu. </s> <s id="s.003414">Hinc Ariſtoteles mechan. </s> <s id="s.003415">quæſt. </s> <s id="s.003416">21. quærit, <emph type="italics"></emph>Cur medi<lb></lb>ci faciliùs dentes extrahunt denti forcipis onere adjecto, quàm ſi ſo<lb></lb>lâ utantur manu?<emph.end type="italics"></emph.end> Quia nimirum infixum mandibulæ dentem <lb></lb>extrahendum vix ſummis duobus digitis, quibus non multa vis <lb></lb>ineſt, arripere valent, & ob carnis mollitudinem facilè è tra<lb></lb>hentibus digitis elabitur lubricus dens: at forcipulam in os im<lb></lb>mittere potiùs, quàm digitos, ſæpe facilius eſt, validiûſque <lb></lb>trahit manus in pugnum conſtricta forcipi dentem per vim <lb></lb>educenti applicata, quàm digitorum extremitates dentem <lb></lb>adhuc in gingivâ hærentem evellere valeant. </s> <s id="s.003417">Præterquam <lb></lb>quod in dentiforcipe, cujuſcumque tandem figuræ ſit, ratio <lb></lb>vectis intercedit ad dentem firmiùs apprehendendum, dum <lb></lb>preſſo manubrio arctiùs conſtringitur: nec facilè Chirurgus <lb></lb>operam ludit, ubi dens forcipem ſubterfugere nullatenus po<lb></lb>teſt. </s> <s id="s.003418">Totam igitur vectis vim in dentiforcipe agnoſco ad ſtrin<lb></lb>gendum dentem, ut medica manus illum faciliùs evellat: ne<lb></lb>que enim eâdem ratione à medicis (niſi fortè veterinariis) ex<lb></lb>trahuntur dentes, quâ fabri lignarij revellunt infixos tabulæ <lb></lb>clavos, de quibus mox erit ſermo. </s> </p> <p type="main"> <s id="s.003419">Similiter quia ad ſtringendam exilem aliquam materiam <lb></lb>inepta eſſet digitorum craſſitudo, minutorum opuſculorum fa<lb></lb>bricatores forcipulis utuntur, quibus illam apprehendentes fir<lb></lb>miter, aut limæ ſubjiciunt, aut opportunè collocant. </s> <s id="s.003420">Et quia <lb></lb>candens ferrum manu tractari nequit, ut in quamcumque par<lb></lb>tem verſetur, incudique impoſitum niſi retineretur, ſæpè ſe<lb></lb>cundis aut tertiis malleorum ictibus ſe ſubduceret, propterea <lb></lb>fabri ferrarij forcipes adhibent, quarum author & inventor <lb></lb>Cinyra Cyprius Agriopæ filius ſcribitur à Plinio lib. 7 cap. 56; <lb></lb>ideóque forcipes, quaſi forvicapes, dictæ ſunt, quòd iis for<lb></lb>va, ideſt calida, capiantur. </s> </p> <p type="main"> <s id="s.003421">Vis autem forcipum in eo ſita eſt, quòd duo vectes primi <lb></lb>generis AB, & CD in E connexi <lb></lb><figure id="id.017.01.461.1.jpg" xlink:href="017/01/461/1.jpg"></figure><lb></lb>commune hypomochlium E habent; <lb></lb>potentia verò in B & D longiorum <lb></lb>brachiorum extremitates adducens eò <lb></lb>validius ſtringit ferrum brevioribus brachiis EA & EC ap<lb></lb>prehenſum, quo major fuerit Ratio BE ad EA: támque firma <lb></lb>retentio eſſe poteſt, ut modico pueri conatu extremitates B & D <pb pagenum="446" xlink:href="017/01/462.jpg"></pb>viciſſim comprimentis, robuſtiſſimi cujuſque vires elidantur, <lb></lb>ne arreptum ferrum ex AC poſſit eximere. </s> <s id="s.003422">Quòd ſi forcipibus <lb></lb>BA & DC utatur aliquis veterinarius vice Poſtomidis (ſeu, ut <lb></lb>aliquibus Grammaticis placet Paſtomidis) equi nares, ad frænan<lb></lb>dam ejus tenaciam, ut loquitur Feſtus, inter longiora brachia <lb></lb>BE & DE contingens; jam BE & DE vectes ſunt ſecundi ge<lb></lb>neris, cum illud, quod ponderis vicem ſubit, inter hypomo<lb></lb>chlium & potentiam interjiciatur. </s> </p> <p type="main"> <s id="s.003423">Hujuſmodi forcipibus vectis in EC & EA non abſimile fuiſſe <lb></lb>exiſtimo inſtrumentum antiquioribus Græcis ad frangendas <lb></lb>abſque ictu percutientis mallei nuces familiare, ut ex Ariſtote<lb></lb>le Mechan. quæſt. </s> <s id="s.003424">22. colligitur: quod fortaſſe vel in alterutrâ, <lb></lb>vel in utraque interiori facie breviorum brachiorum modicè <lb></lb>excavatá frangendæ nuci locum deſignabat; adducto enim in <lb></lb>oppoſitas partes utroque vecte BA & DC, quo propior erat <lb></lb>nux communi hypomochlio, puncto ſcilicet connexionis E, eò <lb></lb>faciliùs frangebatur, quia eò major erat Ratio motûs poten<lb></lb>tiæ ad motum particularum nucis ex compreſſione dividenda<lb></lb>rum, quàm eſſet Ratio reſiſtentiæ ex earumdem particularum <lb></lb>complexione ortæ, ad vim motivam potentiæ. </s> <s id="s.003425">Et quoniam in <lb></lb>nucum mentionem incidi, ne levitati mihi tribuas, quòd hîc <lb></lb>puerile inventum à me puero, & tunc quidem admiratione <lb></lb>obſtupefacto, obſervatum commemorare non erubeſcam. </s> <s id="s.003426">Vide<lb></lb>bam pueros clandeſtinis jentaculis indulgentes, ut citra multi<lb></lb>plicis percuſſionis ſtrepitum nuces confringerent, eas inter <lb></lb>poſtium angulos & fores collocare; tum adductis foribus leviſ<lb></lb>ſimo negotio unâ operâ confringere. </s> <s id="s.003427">Erat ſcilicet vectis primi <lb></lb>generis, cuius majorem longitudinem definiebat foris latitudo, <lb></lb>minorem ipſius foris craſſitudo, ita ut vectis eſſet in angulum <lb></lb>inflexus, cujus hypomochlium cardinibus reſpondebant. </s> <lb></lb> <s id="s.003428">Uſque adeò natura ipſa Mechanicen, uſumque vectis, vel pue<lb></lb>ros docet. </s> </p> <p type="main"> <s id="s.003429">His adde acutas forcipulas, quibus catenularum fabricatores <lb></lb>extremitatem fili ferrei inflectunt: ratio enim vectis potiſſi<lb></lb>mùm conſiſtit in validâ & firmâ ipſius fili ferrei apprehenſio<lb></lb>ne; nam quo ad ejuſdem inflexionem ſpectat, non eſt, cur nos <lb></lb>torqueamus, ut aliquam demum vectis umbram venemur: ſa<lb></lb>tis eſt, ſi manubrij amplitudinem conſiderantes, eámque cum <pb pagenum="447" xlink:href="017/01/463.jpg"></pb>tenui apice forcipulæ, circa quem filum ferreum contorquetur, <lb></lb>comparantes motum potentiæ manubrio applicatæ longè ma<lb></lb>jorem motu particularum fili ferrei, quod flectitur, deprehen<lb></lb>damus; hinc quippe aucta potentiæ momenta cognoſcimus. </s> </p> <p type="main"> <s id="s.003430">Aliud forcipum genus frequentiùs uſurpatur, quarum potiſ<lb></lb>ſimus uſus eſt in eximendis clavis, <lb></lb><figure id="id.017.01.463.1.jpg" xlink:href="017/01/463/1.jpg"></figure><lb></lb>& minora brachia AE & CE non <lb></lb>recta ſunt, ſed curva; non ſolùm ut <lb></lb>clavus tenaciùs apprehendatur ex<lb></lb>cepto ejus capite intra forcipum ſi<lb></lb>num, verùm etiam ut forcipes aliam <lb></lb>exerceant vectis curvi rationem: cùm enim arrepto inter A & <lb></lb>C clauo inclinantur forcipes, ut punctum H tangat ſubjectum <lb></lb>planum, ſive paries ſit, ſive tabula, jam hypomochlium eſt <lb></lb>in H, & momenta potentiæ in B ad reſiſtentiam clavi evellen<lb></lb>di, ſunt ut BH ad HA, cùm circa punctum H perficiatur <lb></lb>motus. </s> <s id="s.003431">Quare ad conſtringendum clavum momentorum Ra<lb></lb>tio eſt ut BE ad EA (perinde atque ſi ab E ad A ducta eſſet <lb></lb>recta linea) ad revellendum verò momentorum Ratio eſt ut <lb></lb>recta ex B ad H ducta ad rectam, quæ ex H ad A ducitur; ne<lb></lb>que enim curva linea ex H ad B, aut ex H ad A, ſed recta le<lb></lb>gem conſtituit motibus potentiæ in B, & ponderis in A. </s> </p> <p type="main"> <s id="s.003432">Id quod pariter contingit cùm averſam mallei partem ſubti<lb></lb>liorem clavo ſubmittimus, & in oppoſitam partem manu<lb></lb>brium retrahimus, ut clavus extrahatur: eſt ſiquidem curvus <lb></lb>quidam vectis fulcrum habens in E, circa <lb></lb><figure id="id.017.01.463.2.jpg" xlink:href="017/01/463/2.jpg"></figure><lb></lb>quod punctum manens uterque motus perfi<lb></lb>citur; & motus potentiæ in H ad motum <lb></lb>clavi in I habet Rationem rectæ HE ad <lb></lb>rectam EI. </s> <s id="s.003433">Ex quo patet pro majori manu<lb></lb>brij longitudine augeri etiam potentiæ mo<lb></lb>menta. </s> </p> <p type="main"> <s id="s.003434">Quoniam verò aliquando forcipes hujuſ<lb></lb>modi curvæ aciem habent in A & C, ut id, <lb></lb>quod conſtringitur vehementiùs, etiam ſcin<lb></lb>datur, non eſt alia philoſophandi ratio, quod quidem ſpectat <lb></lb>ad momenta potentiæ duplici illi vecti applicatæ, hoc uno dif<lb></lb>ferunt, quod vis ſcindendi orta ex acie ferri pertinet ad ratio-<pb pagenum="448" xlink:href="017/01/464.jpg"></pb>nes Cunei, de quo inferiùs ſuo loco. </s> <s id="s.003435">Idem dicendum de for<lb></lb>ficibus, quarum acies pariter ex rationibus Cunei vim ſcin<lb></lb>dendi habent; majora autem momenta potentiæ, quæ faci<lb></lb>liùs ſcindat, petenda ſunt ex rationibus vectis; ſunt enim <lb></lb>hîc pariter duo vectes in oppoſitas partes commoti, commu<lb></lb>ne hypomochlium in puncto connexionis habentes; & quo <lb></lb>majorem Rationem manubriorum longitudo habet ad diſtan<lb></lb>tiam rei ſcindendæ à puncto connexionis, eò etiam facilior <lb></lb>contingit ſciſſio. </s> <s id="s.003436">Idcirco quæ duriora ſunt, prope connexio<lb></lb>nis punctum applicantur, quia eadem manubriorum longi<lb></lb>tudo ad minorem diſtantiam habet majorem Rationem quàm <lb></lb>ad diſtantiam majorem; & quæ ad hæc duriora ſcindenda <lb></lb>inſtitutæ ſunt forfices, breviora habent brachia, quæ ad <lb></lb>ſcindendum exacuuntur, longiora verò ea, quibus poten<lb></lb>tia movens applicatur; cujuſmodi ſunt forfices, quibus fa<lb></lb>bri ferrarij ad æreas aut ferreas laminas ſcindendas utuntur. </s> <lb></lb> <s id="s.003437">In harum uſu illud etiam obſervare poteris, ſatis eſſe, ſi <lb></lb>duorum vectium communi fulcro connexorum, ita ut de<lb></lb>cuſſati exiſtant, alterum moveatur manente altero: hoc <lb></lb>enim potiſſimum attenditur, quo pacto potentia validiùs <lb></lb>applicetur, ubi multâ opus eſt virtute; cum autem uni<lb></lb>cum hujuſmodi forficum brachium movetur, tota illi ma<lb></lb>nus applicatur, & reliquo deorſum connitente corpore va<lb></lb>lidè premit. <lb></lb></s> </p> <p type="main"> <s id="s.003438"><emph type="center"></emph>CAPUT XIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003439"><emph type="center"></emph><emph type="italics"></emph>Cur Tollenones juxta puteos conſtituantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003440">QUi Tollenones Latinis (Ciconias aliqui vocant) Græcis <lb></lb><foreign lang="grc">Κελόνια</foreign> dicuntur, familiaria ruſticis & olitoribus inſtru<lb></lb>menta ad hauriendas ex puteis non admodum altis aquas, ali<lb></lb>qua habent explicatu digna, quæ ex Vectis doctrinâ petenda <lb></lb>ſunt, nec viſum eſt Ariſtoteli quæſtione 28. hanc eandem diſ<lb></lb>putationem inſtituere indecorum, aut homini Philoſopho mi-<pb pagenum="449" xlink:href="017/01/465.jpg"></pb>nùs conveniens. </s> <s id="s.003441">Et primùm quidem ipſa Tollenonis con<lb></lb>ſtructio pendet ex rationibus vectis primi generis, habet ſi qui<lb></lb>dem fulcrum medium inter potentiam moventem & pondus <lb></lb><figure id="id.017.01.465.1.jpg" xlink:href="017/01/465/1.jpg"></figure><lb></lb>elevatum. </s> <s id="s.003442">Erecto enim tigno DA imponitur tranſverſa haſta <lb></lb>CE, infigitúrque axi in A, circa quem liberè converti poſſit. </s> <lb></lb> <s id="s.003443">Tum extremitati E puteo appoſitæ alligatur plumbum, aut ſa<lb></lb>xum, ſive grave aliud quodpiam; ab extremitate autem C, <lb></lb>quæ puteo reſpondet, funis CF pendet (ſeu haſta fune con<lb></lb>vexa in C, ſed tamen facilè mobilis) cui in F ſitula adnectitur. </s> </p> <p type="main"> <s id="s.003444">Jam, verò duplex motus in hauriendâ aquá conſiderandus <lb></lb>eſt, alter, quo hydria vacua in puteum demittitur, alter, quo <lb></lb>eadem hydria aquæ plena è puteo extrahitur. </s> <s id="s.003445">Priori motui <lb></lb>utique non favet tolleno, faciliùs quippe hydria deſcende<lb></lb>ret, ſi nullum eſſet onus in E, quod depreſsâ hydria eſſet <lb></lb>elevandum; hujus enim gravitas major eſt hydriæ gravita<lb></lb>te; ac propterea præter ejuſdem hydriæ gravitatem alia po<lb></lb>tentia deprimens requiritur in F, ut major ſit Ratio gra<lb></lb>vitatis & potentiæ in F ad gravitatem ponderis in E, quàm <lb></lb>ſit reciprocè Ratio diſtantiæ AE ad diſtantiam AC. </s> <s id="s.003446">Quare <lb></lb>ſi AC longitudo multo major ſit longitudine AE, facilior <pb pagenum="450" xlink:href="017/01/466.jpg"></pb>erit hydriæ vacuæ depreſſio; contra verò deprimendi difficul<lb></lb>tas augebitur, quo magis pondus E diſtabit à fulcro A. </s> <lb></lb> <s id="s.003447">Sed hæc eadem, quæ deprimendi difficultatem augent, ju<lb></lb>vant ad extrahendam faciliùs hydriam: pondus enim E quò <lb></lb>longiùs aberit à fulcro A, eò plura habebit momenta adversùs <lb></lb>gravitatem aquæ & hydriam pendentes ex C. </s> <s id="s.003448">Hinc eſt poten<lb></lb>tiæ atque ponderis vices permutari; in depreſſione nimirum <lb></lb>pondus in E exiſtens attollitur, & potentia in C deſcendit; at <lb></lb>in elevatione viciſſim pondus elevatur in C, & potentia in E <lb></lb>deſcendit. </s> <s id="s.003449">Prudenter itaque providere oporteτ, ut & haſtæ <lb></lb>CE longitudo opportunè diſtinguatur in partes CA, AE, & <lb></lb>pondus in E neque ita leve ſit, ut parum adjumenti afferat in <lb></lb>extrahendâ aquâ, neque ita grave, ut detrimento ſit in depri<lb></lb>mendâ hydriâ. </s> <s id="s.003450">Præſtat tamen plus aliquid laboris ſuſcipere in <lb></lb>deprimenda hydriâ, ut ea deinde elevetur majore compen<lb></lb>dio: nemo quippe dubitat, quin longè faciliùs ſit homini <lb></lb>funem FC deorſum trahenti attollere pondus E, quàm pa<lb></lb>rium momentorum aquam è puteo extrahere. </s> </p> <p type="main"> <s id="s.003451">Porrò non abs re fuerit monere hîc aliquem, ne ſe ruſticis <lb></lb>ridendum præbeat, ubi pro altitudine putei aſſumpto fune CF, <lb></lb>haſtam CE æquo longiorem conſtituerit præter rationem inter<lb></lb>valli inter tigillum DA & puteum; contingeret enim, ut <lb></lb>haſta in putei labra incurrens neceſſariam funis longitudinem <lb></lb>minueret. </s> <s id="s.003452">Quapropter tria hæc neceſſe eſt ſibi invicem pro<lb></lb>portione reſpondere, videlicet haſtæ CE longitudinem, tigilli <lb></lb>AD altitudinem, ejúſque à puteo diſtantiam; ut erecta ferè ad <lb></lb>perpendiculum haſta eam admittat funis longitudinem, quæ <lb></lb>& facile hydriæ jungi poſſit, & putei altitudinem exæquet. </s> </p> <p type="main"> <s id="s.003453">Cur autem tùm in deprimendo Tollenone, ut hydria im<lb></lb>mergatur, tùm in attollendo, ut aqua è puteo eximatur, non <lb></lb>parem ſemper & æquabilem experiamur facilitatem, ratio in <lb></lb>promptu eſt; quia ſcilicet varia eſt potentiæ medio fune FC <lb></lb>tollenonem agitantis applicatio; quo enim acutior fuerit angu<lb></lb>lus FCA, eò minora ſunt potentiæ trahentis momenta, quæ <lb></lb>creſcente angulo pariter augentur, ut tunc maxima ſint, cùm <lb></lb>funis FC, & haſta CA angulum rectum conſtituerint. </s> <s id="s.003454">Et qui<lb></lb>dem licèt, ubi funis ab angulo recto ad obtuſum deſci erit, ite<lb></lb>rum momenta potentiæ decreſcant, ſi applicationis potentiæ <pb pagenum="451" xlink:href="017/01/467.jpg"></pb>ejuſdem tantummodo habeatur ratio; fieri tamen poteſt, ut <lb></lb>ponderis in E momenta minuantur, quo altiùs attollitur, ſi il<lb></lb>lud fuerit haſtæ impoſitum, cum ejuſdem linea directionis ca<lb></lb>dat in haſtæ punctum, quod magis ad fulcrum A accedat, jux<lb></lb>ta ea, quæ hujus libri cap. 3. dicta ſunt; atque adeò deprimendi <lb></lb>facilitas, quæ hinc ſumit incrementum, diminutâ ponderis E re<lb></lb>ſiſtentiâ ſuppleat decrementum, quod obliquam potentiæ ap<lb></lb>plicationem conſequitur. </s> </p> <p type="main"> <s id="s.003455">Nec abſimilis momentorum varietas contingit ex diſparili <lb></lb>angulorum amplitudine, quos lineæ directionis gravitatum <lb></lb>tum aquæ attollendæ, tum ponderis E, cum haſtâ CE conſti<lb></lb>tuunt. </s> <s id="s.003456">Nam depreſsâ haſtâ, & pondere maximè elevato, hu<lb></lb>jus momenta initio minora ſunt, & ſubinde augentur receden<lb></lb>te à fulcro A lineâ directionis centri gravitatis, ſi illud quidem <lb></lb>haſtæ incumbat: Pondere igitur E minùs conante adversùs <lb></lb>aquam cum hydriâ attollendam, plus laborandum eſt homini <lb></lb>funem ſurſum trahenti; cujus deinde labor minuitur auctis gra<lb></lb>vitatis E momentis; & tunc potiſſimùm præſtare videntur, cum <lb></lb>angulus FCA ex recto in acutum tranſit; tunc enim aquæ de<lb></lb>orſum connitentis ac oppoſito ponderi reſiſtentis momenta de<lb></lb>creſcere incipiunt, ac infirmiora fieri. </s> </p> <p type="main"> <s id="s.003457">Ex his non parum lucis affulget ſcenicis machinationibus, <lb></lb>in quibus non planè ad perpendiculum, ſed obliquè aſcenden<lb></lb>dum eſt aut deſcendendum, ſi enim ſtatuatur vectis ZX ha<lb></lb>bens in P fulcrum, & fune <lb></lb><figure id="id.017.01.467.1.jpg" xlink:href="017/01/467/1.jpg"></figure><lb></lb>ZN pendeat corpus demit<lb></lb>tendum, utique obliquus erit <lb></lb>deſcenſus ex N in M, & viciſ<lb></lb>ſim obliquus aſcenſus ex M in <lb></lb>N: momenta autem ponderis <lb></lb>X, aut S, pro variâ poſitione, <lb></lb>ut dictum eſt, diſſimilia atque <lb></lb>diſparia ſunt: Quapropter <lb></lb>temperanda ſunt pro motûs <lb></lb>inſtituendi opportunitate; atque ſi pondus X levius ſit corpore <lb></lb>demittendo ex N, hoc ſponte deſcendet; ſi verò in S augea<lb></lb>tur pondus, ut corporis in M gravitatem ſuperet, hoc ex M <lb></lb>in N elevabitur. </s> <s id="s.003458">Quod autem de ſcenicâ machinatione hîc <pb pagenum="452" xlink:href="017/01/468.jpg"></pb>innui, ad alias motiones corporum elevandorum (ut ſi ex navi <lb></lb>in altiorem fluminis ripam onus transferendum eſſet) facilè <lb></lb>traduci poſſe ita manifeſtum eſt, ut pluribus non ſit opus, ſi <lb></lb>accuratè examinetur altitudo, ad quam deducendum eſt, & <lb></lb>amplitudo ſeu diſtantia parallelarum, intra quas obliquus mo<lb></lb>tus perficiendus eſt, ut vecti congrua longitudo ſtatuatur, & <lb></lb>opportuno loco collocetur, ubi eam anguli RPZ inclinatio<lb></lb>nem habeat, cui Sinus Verſus RN reſpondeat. <lb></lb></s> </p> <p type="main"> <s id="s.003459"><emph type="center"></emph>CAPUT XIV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003460"><emph type="center"></emph><emph type="italics"></emph>Remorum vires in agendâ navi expenduntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003461">REmum, quo naves aguntur, Copenſibus, & Platæenſibus <lb></lb>debemus, ut Plinius lib. 7 cap. 56. ſcribens ait, <emph type="italics"></emph>Remum <lb></lb>Copæ, latitudinem ejus Platææ<emph.end type="italics"></emph.end> (utraque eſt Bœotiæ urbs) inve<lb></lb>nerunt; & nomen ipſum ab inventoribus inditum videtur, nam <lb></lb>Græcis <foreign lang="grc">χώπη</foreign> Remus, <foreign lang="grc">πλάτη</foreign> Palmula, latior ſcilicet remi pars <lb></lb>dicitur. </s> <s id="s.003462">Ratem ſiquidem conto propellere rudis adhuc ars nau<lb></lb>tica noverat, ubi fluminis non admodum alti fundum perticâ <lb></lb>pertentare licebat; at ubi uberior unda prohibet, ne fundum <lb></lb><figure id="id.017.01.468.1.jpg" xlink:href="017/01/468/1.jpg"></figure><lb></lb>attingatur, operam luderet, qui navim <lb></lb>conto AB impellere ſe poſſe ſibi per<lb></lb>ſuaderet, niſi fortè extremitati B <expan abbr="ligneã">ligneam</expan> <lb></lb>tabellam adjungeret, ita levem, ut <lb></lb>ſponte ſuâ innataret; illam enim per <lb></lb>vim velociter immergenti obliquam, <lb></lb>aqua reſiſteret, & navis aliquantulum <lb></lb>promoveretur: cæterum ingens eſſet <lb></lb>labor in conto retrahendo, & tabellâ ex <lb></lb>aquis extrahendâ, etiamſi ſcalmo tigil<lb></lb>lus longiuſculus EF ad perpendiculum <lb></lb>infigeretur, ex cujus ſummo vertice <lb></lb>funis EI penderet, fune autem contus <lb></lb>medius in I ſuſpenderetur. </s> <s id="s.003463">Quare opus fuit inſtrumentum <pb pagenum="453" xlink:href="017/01/469.jpg"></pb>moliri, quo & facile uteremur, & aliquod laboris compen<lb></lb>dium inveniremus. </s> </p> <p type="main"> <s id="s.003464">Ratione ſuæ longitudinis ad unum aliquod vectis genus re<lb></lb>ferendus eſt remus, ad cujus caput applicatur potentia, videli<lb></lb>cet remex; extrema palmula immergitur aquæ, & circa me<lb></lb>dium innititur ſcalmo: ſed aquæ ne? </s> <s id="s.003465">an ſcalmo? </s> <s id="s.003466">ratio fulcri <lb></lb>conveniat, diſputatur. </s> <s id="s.003467">Si Ariſtoteles audiendus eſſet mechan. </s> <lb></lb> <s id="s.003468">quæſt. </s> <s id="s.003469">4. <emph type="italics"></emph>hypomochlion fit ſcalmus, ſtat enim ille, pondus verò mare <lb></lb>eſt, quod propellit remus, vectem autem movens ipſe eſt remex.<emph.end type="italics"></emph.end></s> <s id="s.003470"> Id <lb></lb>quidem verum eſſet, ſi quis anchoris nondum ſolutis, & ſtante <lb></lb>navi, adumbratâ ad ſpeciem remigatione ſe exerceret; nil enim <lb></lb>præſtaret præter aquarum impulſionem. </s> <s id="s.003471">Cæterùm nautæ re<lb></lb>morum pulſu non aquam verberare, ſed navim impellere con<lb></lb>tendunt. </s> <s id="s.003472">Igitur aqua, cui remi palmula immergitur, diviſio<lb></lb>ni reſiſtens, atque impediens motum palmulæ, hypomochlij, <lb></lb>cui vectis, hoc eſt remus, innititur, rationem habet; navis ve<lb></lb>rò ipſa, quæ promovetur, quatenus eſt ſcalmo conjuncta, uti<lb></lb>que eſt pondus, ex cujus movendi, non ex aquæ repellendæ <lb></lb>difficultate æſtimandus eſt nautarum labor: alioquin eodem <lb></lb>remo, qui ſcalmo ſimiliter inſiſteret, æqualis labor eſſet, ſive <lb></lb>actuarium, ſive corbitam impellere oporteat; pari ſiquidem <lb></lb>aquæ occurrit utrobique palmula. </s> <s id="s.003473">Manifeſtum eſt igitur pon<lb></lb>dus vecte promovendum navim eſſe, non aquam, ac propterea <lb></lb>hypomochlij vices aquam ſubire, adeóque remum cenſendum <lb></lb>eſſe vectem ſecundi generis, cujus extremitates potentia & <lb></lb>fulcrum occupant. </s> </p> <p type="main"> <s id="s.003474">Hinc eſt aliquod ſemper haberi laboris compendium, pon<lb></lb>deris enim motus, qui vecte perficitur, minor eſt motu poten<lb></lb>tiæ remi capiti applicatæ, illud enim minùs, hæc magis ab hy<lb></lb>pomochlio diſtat. </s> <s id="s.003475">Motus, inquam, qui vecte perficitur, mi<lb></lb>nor eſt motu potentiæ; fieri enim contingit, ut vi impreſſi im<lb></lb>petûs, etiam ceſſante remigis impulſione, navis promoveatur, <lb></lb>adeò ut pro ratione impetûs multo major ſit navis motus, quàm <lb></lb>potentiæ impellentis. </s> <s id="s.003476">Verum hoc non ex vecte ob idipſum, <lb></lb>quia vectis eſt, oritur, ſed quia navis innatans aquæ non eam <lb></lb>invenit à corpore fluido reſiſtentiam, quam cæteroqui ex mu<lb></lb>tuo tritu inveniunt pondera corpori ſolido inſiſtentia, etiamſi <lb></lb>vecte horizontaliter moveantur; ac proinde impreſſus impetus, <pb pagenum="454" xlink:href="017/01/470.jpg"></pb>ceſſante vi externâ, non ſtatim perit. </s> <s id="s.003477">Id autem intelligendum <lb></lb>eſt, cum in lacu vel tranquillo mari navigatur, cum ſcilicet <lb></lb>aqua ſuo curſu non adverſatur motui navis: nam ſi adverſo flu<lb></lb>mine promovendum ſit navigium, contrarius aquæ impulſus <lb></lb>impetum à remige impreſſum elidit, fieríque poteſt, ut utilius <lb></lb>accidat navim trahere, quàm remigando impellere, ne ſubla<lb></lb>tis ex aquâ remis navigium vi aquæ fluentis retro-actum eò re<lb></lb>deat, unde diſceſſit, & alternâ remorum immerſione atque ex<lb></lb>tractione opera ludatur: præterquam quod quò magis immerſa <lb></lb>remi palmula ab adverſo flumine repellitur, eò amplius detrahi<lb></lb>tur motui navis. </s> <s id="s.003478">Ideò quamvis navim trahens plus laboris ſim<lb></lb>pliciter impendat, quàm remigans, facit tamen operæ pretium, <lb></lb>qui enim navim adversùs profluentem trahit, etiam retinet, ne <lb></lb>retrorſum agatur; at qui remo impellit, ſublatâ ex undis pal<lb></lb>mulâ, receſſum impedire non valet. </s> </p> <p type="main"> <s id="s.003479">Cum itaque remus vectis ſit ſecundi generis, remigis vires <lb></lb>æſtimandæ ſunt ex Ratione, quam longitudo remi habet ad il<lb></lb>lam ejuſdem remi partem, quæ aquæ & ſcalmo interjecta eſt; <lb></lb>hæc ſiquidem Ratio eſt motuum, ac proinde & momentorum, <lb></lb>ut ſæpiùs dictum eſt. </s> <s id="s.003480">Remi autem longitudinem non abſolutam <lb></lb>intelligas; ſed primùm ea demenda eſt palmulæ particula, quæ <lb></lb>aquæ immergitur; quippe quæ aquam repellens quaſi hypomo<lb></lb>chlio incumbit. </s> <s id="s.003481">Deinde attendendum eſt, quam remi partem <lb></lb>remex apprehendat; ſi enim plures eundem remum agitent, ut <lb></lb>in triremibus, non ſunt æqualia momenta ſingulorum, ſed ejus, <lb></lb>qui ſcalmo propior eſt, minora ſunt (perinde atque ſi remo <lb></lb>adeò brevi uteretur) ejus, qui remi caput apprehendit, maxi<lb></lb>ma ſunt momenta; medij autem medio modo ſe habent. </s> </p> <p type="main"> <s id="s.003482">Quare longitudo vectis in remo definitur intervallo, quod in<lb></lb>ter aquam, & remigis manum interjectum eſt; ponderis diſtan<lb></lb>tiam ab hypomochlio metitur intervallum, quo ſcalmus ab aquâ <lb></lb>palmulam excipiente disjungitur. </s> <s id="s.003483">Si igitur intervallum illud <lb></lb>eſt hujus intervalli duplum aut ſeſquialterum, momenta Po<lb></lb>tentiæ ad momenta ponderis Rationem habent duplam aut ſeſ<lb></lb>quialteram, & quatuor remiges ad promovendam navim tan<lb></lb>tumdem ferè valent ac ſex aut octo, qui pari conatu navim <lb></lb>eandem ſine remis propellerent, aut traherent. </s> <s id="s.003484">Dixi, <emph type="italics"></emph>ferè,<emph.end type="italics"></emph.end> quia <lb></lb>cum motus cujuſlibet vectis ſit circularis circa punctum hypo-<pb pagenum="455" xlink:href="017/01/471.jpg"></pb>mochlij, remex, qui in dextero navis latere remigat, ſecun<lb></lb>dùm vectis naturam arcum deſcribit ad dexteram inclinatum, <lb></lb>id quod pariter contingit ſiniſtro remigi arcum ſiniſtrorſum <lb></lb>deſcribenti: Cum autem navis non niſi unico motu moveri poſ<lb></lb>ſit, ex his duobus circularibus ſibi adverſantibus reſultat tertius <lb></lb>medius, ſcilicet rectus, qui proinde tantus eſſe non poteſt, <lb></lb>quantus eſſet, ſi ſex aut octo homine, æquali niſu navim ſine <lb></lb>remis impellerent aut traherent; quia contrariæ illæ directio<lb></lb>nes ad dexteram & ad ſiniſtram nequeant in tertiam mixtam <lb></lb>directionem coaleſcere, ſinè aliquo impetûs detrimento. </s> </p> <p type="main"> <s id="s.003485">Quod ſi remiges omnes non conſentirent in deprimendo, im<lb></lb>pellendo, atque extrahendo remo, ſed alij alios præverterent, <lb></lb>non ſolùm id incommodi accideret, quod ab inſtituto itinere de<lb></lb>flecteret navis in alterutram partem, niſi æqualis utrinque eſſet <lb></lb>impulſus, verùm etiam retardaretur motus, tùm quia minor im<lb></lb>pulſus à paucioribus navi imprimitur, tum quia remi tardiores, <lb></lb>reliquis elevatis, adhuc immerſi dum communi navis motu <lb></lb>moventur, minùs impellunt aquam poſt ſe fugientem, & pal<lb></lb>mulæ latitudo occurrenti aquæ obverſa moram infert, ut eam <lb></lb>dividat; ex quo fit, ut aliquid impetûs ab aliis remigibus im<lb></lb>preſſi deteratur, qui citra hoc impedimentum adhuc perſevera<lb></lb>ret. </s> <s id="s.003486">Sunt ſcilicet plures remi plures vectes, quibus idem pon<lb></lb>dus movetur; & niſi remiges omnes conſpiraverint, aut navis <lb></lb>tardiùs movetur, aut aliquorum labor augetur: haud ſecus ac <lb></lb>ſi plures homines uni vecti ad pondus aliquod elevandum appli<lb></lb>carentur, uno aut altero ceſſante reliquorum niſus augendus <lb></lb>eſſet, ſupplementum deſidioſorum. </s> </p> <p type="main"> <s id="s.003487">Ex rationibus igitur vectis ſatisfit quæſtioni ab Ariſtotele <lb></lb>propoſitæ, <emph type="italics"></emph>Cur <02>, qui in navis medio ſunt remiges, maxime navim <lb></lb>movent?<emph.end type="italics"></emph.end> Allatam à Philoſopho reſponſionem intactam relinquo; <lb></lb>an ſatis commoda ſit, alij examinent. </s> <s id="s.003488">Remigum alij in puppi <lb></lb>conſtituuntur, qui Thranitæ dicebantur, ut eſt apud Suidam, <lb></lb>alij in prorâ, qui Thalamij ſeu Thalamitæ, alij in navis medio, <lb></lb>qui Zygitæ: & quamvis omnes ad promovendam navim ſuum <lb></lb>conatum conferant, non tamen omnium æqualis eſt labor, aut <lb></lb>par in movendo efficacitas; quia non ſecundùm eandem Ratio<lb></lb>nem ſingulorum remorum longitudo in partes à ſcalmo diſtin<lb></lb>guitur; ſed quia puppis altior eſt, & ſpatium anguſtum, major <pb pagenum="456" xlink:href="017/01/472.jpg"></pb>remi pars extra navim eſt, parúmque à ſcalmo diſtat remex; <lb></lb>ideò motus potentiæ ad motum ponderis minorem habet ratio<lb></lb>nem, quàm ſi brevior eſſet inter palmulam, & ſcalmum, lon<lb></lb>giórque inter ſcalmum & remigem diſtantia, ut contingit in <lb></lb>medio, ubi navis depreſſior eſt, & maximam habet latitudi<lb></lb>nem; pondere enim minùs diſtante ab hypomochlio, majora <lb></lb>ſunt potentiæ momenta, cum eadem ponatur utrobique vectis <lb></lb>longitudo. </s> <s id="s.003489">Quæ autem de puppi dicta ſunt, ſaltem quo ad ſpa<lb></lb>tij anguſtias, etiam de prorâ intelligenda ſunt, quæ quia de<lb></lb>preſſior eſt puppi, & aliquanto altior quàm circa medium, <lb></lb>propterea Thalamiorum labor medius eſt inter Thranitarum & <lb></lb>Zygitarum laborem. </s> <s id="s.003490">Dicuntur autem remiges, qui in navis <lb></lb>medio ſunt, maximè movere vim, non quia navis motus, qui <lb></lb>circa hypomochlium tanquam circa centrum fit, ibi ſit major <lb></lb>motu, qui fit in puppi, ſi remiges parem arcum deſcribant, <lb></lb>nam potiùs oppoſitum contingit; ſed quia remex in medio mi<lb></lb>norem inveniens ponderis movendi reſiſtentiam plus navim <lb></lb>impellit, quàm ſi in puppi pariter conaretur, ubi eodem niſu <lb></lb>non poteſt eodem temporis ſpatio tam amplum arcum deſcribe<lb></lb>re. </s> <s id="s.003491">Propterea fortiſſimi remiges ad puppim ſtatuuntur, ut ma<lb></lb>jore impetu producto vincant majorem reſiſtentiam; ideóque <lb></lb>Thranitis præter publicum ſtipendium etiam extraordinarium <lb></lb>datum commemorat Thucydides lib. 6. Hæc verò, quæ de An<lb></lb>tiquorum navibus magis propriè dicuntur, quarum forma à <lb></lb>noſtris diſſidebat, noſtris tamen celocibus aut triremibus ſerva<lb></lb>tâ analogiâ accommodari poſſunt; nam etiam apud nos ſcal<lb></lb>mus ad proram & ad puppim aſcendit, & in medio major eſt <lb></lb>navigij amplitudo, ita ut, licèt remorum capita in eâdem rectâ <lb></lb>lineâ juxta navigij longitudinem conſtituantur, diſpari tamen <lb></lb>Ratione à ſcalmo diſtinguantur in partes. </s> </p> <p type="main"> <s id="s.003492">Sed præſtat ipſum navis motum paulo attentiùs conſiderares <lb></lb>quandoquidem ſi hypomochlium eſſet prorſus immobile, & <lb></lb>aqua locum non daret palmulæ urgenti, utique motus navis <lb></lb>ad motum capitis remi in eâ eſſet Ratione, quæ intercedit <lb></lb>inter diſtantias ſcalmi, & capitis remi ab aquâ. </s> <s id="s.003493">Nam ſi pal<lb></lb>mula B immota maneret, & ſcalmus eſſet in C, motus remi<lb></lb>gis AD ad motum navis CE eſſet ut AB ad CB. </s> <s id="s.003494">Contra <lb></lb>verò ſi aqua nihil prorſus obſiſteret remo (ſicuti continge-<pb pagenum="457" xlink:href="017/01/473.jpg"></pb>ret, ſi ille admodum lentè moveretur, aut palmula nimis <lb></lb>obliqua aquam finderet) tantúmque palmula retrogrederetur <lb></lb>per BH, quantum remex per AD <lb></lb><figure id="id.017.01.473.1.jpg" xlink:href="017/01/473/1.jpg"></figure><lb></lb>progreditur, immota maneret na<lb></lb>vis in C: id quod etiam continge<lb></lb>ret, ſi regreſſus BH ad progreſ<lb></lb>ſum AD eſſet in Ratione CB ad <lb></lb>CA. </s> <s id="s.003495">Quod ſi palmula à profluen<lb></lb>te rapta ex B in L majus ſpatium <lb></lb>conficeret, quàm remex ex A in D <lb></lb>(aut ſaltem BL ad AD eſſet in <lb></lb>majore Ratione quàm CB ad CA) <lb></lb>utique navis ipſa retrocederet, & <lb></lb>ſcalmus ex C veniret in F. </s> <s id="s.003496">Cum <lb></lb>igitur promoveatur navis, & aqua <lb></lb>palmulæ obſiſtens ſit hypomochlium mobile, neceſſe eſt pro<lb></lb>greſſu remigis AD minorem eſſe palmulæ regreſſum BI, ut <lb></lb>ſcalmus ex C propellatur in E. </s> <s id="s.003497">Quare quo magis aqua reſiſtit, <lb></lb>minúſque palmula movetur in oppoſitam remigis motui par<lb></lb>tem, magis promovetur navis, quia majorem impulſum recipit. </s> <lb></lb> <s id="s.003498">Majorem autem aquæ reſiſtentiam efficere poteſt aut velocior <lb></lb>remi motio, aut major palmulæ immerſio: conſtat ſi quidem, ſi <lb></lb>baculo aquam lentè dividas, vix percipi in illa ſcindendâ labo<lb></lb>rem; at ſi velociter baculum immerſum agitare libeat, multò <lb></lb>validiùs illam reſiſtere: ſimiliter quò major palmulæ immerſæ <lb></lb>pars plus aquæ propulſat, eò majorem invenit reſiſtentiam, dif<lb></lb>ficiliùs enim multa, quàm modica aqua dividitur. </s> <s id="s.003499">Verùm cum <lb></lb>feſtinato opus eſt, ſatius eſt velociter remum movere, & parum <lb></lb>immergere palmulam, ut frequentiori remorum percuſſione <lb></lb>plus impetûs navi imprimatur. </s> </p> <p type="main"> <s id="s.003500">Quod demum ſpectat ad remi motum, unum ſupereſt obſer<lb></lb>vandum, videlicet, non eum tantummodo motum capiti remi <lb></lb>tribuendum, qui reſpondet partibus navis, quatenus ex remigis <lb></lb>muſculorum contentione atque membrorum inclinatione pen<lb></lb>det, cujus menſuram definiret perpendiculum à capite remi in <lb></lb>ſubjectum navis planum deſcendens, & in eo remi iter deſcri<lb></lb>bens; ſed præterea addendus eſt motus navis, qui omnibus in <lb></lb>navi exiſtentibus communis eſt, adeò ut navis vi remorum acta <pb pagenum="458" xlink:href="017/01/474.jpg"></pb>moveatur à motore tranſlato. </s> <s id="s.003501">Quapropter ſi AD eſt univerſus <lb></lb>capitis remi, ſeu manús remigis motus, demendus ex illo eſt <lb></lb>navis progreſſus CE, & reſiduus motus à remigis conatu, qua<lb></lb>tenus remum impellit, pendet. </s> </p> <p type="main"> <s id="s.003502">Sed antequàm ab hac remorum contemplatione animum <lb></lb>avertamus, placet innuere, quæ de Sinenſium remis attigit <lb></lb>Atlas Sinicus in Præfatione pag. </s> <s id="s.003503">10, ubi de Præfectorum navi<lb></lb>bus, quæ noſtris triremibus æquales ſunt, hæc habet. <emph type="italics"></emph>Dum <lb></lb>ceſſant ventorum flatus, adſunt deſtinati, qui remulco trahant, aut <lb></lb>remis moles tota impellitur motis ad modum caudæ piſcium, methodo <lb></lb>facili, & compendiosâ; quippe ſine ulla aquæ percuſſione, extractio<lb></lb>neve remi, vel remo unico propellitur & dirigitur navis; adeòque <lb></lb>unus hic ſex aut octo noſtratibus nautis æquivalet.<emph.end type="italics"></emph.end></s> <s id="s.003504"> Poſtremum hoc <lb></lb>de uno remige ſex aut octo noſtratibus nautis æquivalente, <lb></lb>adeò magnificè dictum videtur, ſed & adeò jejunè expoſitum, <lb></lb>ut verba mihi dari non facilè patiar, nec me libenter præbeam <lb></lb>credulum: fundamentum conſtituendæ fidei fuiſſet remigan<lb></lb>di ordo deſcriptus, remorum forma atque poſitio verbis aut <lb></lb>iconiſmo propoſita, ut, quanta ſint remigis Sinici momenta, <lb></lb>innoteſcerent; aliam enim utique à noſtratibus remorum for<lb></lb>mam eſſe neceſſe eſt, quippe quos flexiles eſſe oporteat, ut ad<lb></lb>modum caudæ piſcium moveantur; hi ſcilicet poſtremam cor<lb></lb>poris ſui partem flectunt priùs atque contorquent, ut caudam <lb></lb>poſtmodum velociter porrigentes aquam verberent, qua re<lb></lb>ſiſtente conceptus impetus totum corpus promoveat, quandiu <lb></lb>ille perſeverat. </s> <s id="s.003505">Ubi animadvertendum eſt, quàm ſapienti na<lb></lb>turæ inſtituto factum ſit, ut piſces caudam lentiùs inflectant, <lb></lb>ſed velociùs explicent, inflectant obliquam, explicent erectam; <lb></lb>ſi enim erectam caudam velociter flecterent, ita aqua reſiſte<lb></lb>ret, ut potiùs retrocederent, quemadmodum Aſtaci fluviati<lb></lb>les (cammaros, alij cancros fluviatiles vocant, rectè ne? </s> <s id="s.003506">an <lb></lb>perperam? </s> <s id="s.003507">non eſt hujus loci examinare) quando timent, cau<lb></lb>dâ aquam validè percutientes, ac quaſi ad ſe velociter trahen<lb></lb>tes non procedunt, ſed retrorſum curvatâ caudâ ſecedunt. </s> <s id="s.003508">Sic <lb></lb>etiam contingeret cymbæ, ſi quis in puppi ſtans ligneam tabel<lb></lb>lam extremæ perticæ infixam aquæ à tergo poſitæ immitteret, <lb></lb>perticámque ad ſe velociter traheret, nam cymba retrorſum <lb></lb>agi videretur. </s> <s id="s.003509">Cùm autem piſces caudam & obliquè & lentiùs <pb pagenum="459" xlink:href="017/01/475.jpg"></pb>inflectant, minorem aquæ reſiſtentiam percipiunt. </s> <s id="s.003510">Quare, ut <lb></lb>remus ſuo motu imitetur motum caudæ piſcium, opus eſt <lb></lb>erectam palmulam (hoc eſt, in plano verticali longitudinem <lb></lb>navis obliquè, aut ad rectos angulos, ſecante exiſtentem) aquæ <lb></lb>occurrere, ut aquâ reſiſtente propellatur navis, eandem verò <lb></lb>palmulam poſteà obliquam fieri, ne dum, intra aquam retrahi<lb></lb>tur ad iterandum impulſum, tantam inveniat reſiſtentiam, ſed <lb></lb>aquam faciliùs findat. </s> <s id="s.003511">Hinc conjecturâ aliquâ ducebar ali<lb></lb>quando ad ſuſpicandum, an ita remi palmula reliquæ remi lon<lb></lb>gitudini adnexa eſſet fibulâ plicatili, ut, cùm remi caput pup<lb></lb>pim versùs, palmula verò in oppoſitam partem impelleretur, <lb></lb>hæc occurrenti aquæ cederet, eámque obliquè finderet modi<lb></lb>câ manûs remigis deflexione remum interim contorquentis. </s> <lb></lb> <s id="s.003512">Sed, ut quod res eſt eloquar, vereor, ne argutum nimis, víx<lb></lb>que aliquid habens compendij, artificium hoc videatur: nam <lb></lb>& noſtrates cymbularij communi remo cymbam ex puppi <lb></lb>agentes eam propellunt, & dirigunt aquam non percutientes, <lb></lb>nec remum extrahentes, cujus varia inclinatione, loco guber<lb></lb>naculi, cymbæ motum temperant remigando. </s> <s id="s.003513">Ut quid ergo <lb></lb>remum in duas partes, quæ fibulâ jungantur, diviſum adhibe<lb></lb>re: quippe qui noceat potiùs; nam remigis motus in proram <lb></lb>directus nullum impulſum imprimit navi, niſi quando iterum <lb></lb>rectus factus fuerit remus. </s> <s id="s.003514">Sed flectatur & dirigatur remus in <lb></lb>morem caudæ piſcium; quid hoc, ut unus remex ſex aut octo <lb></lb>noſtratibus nautis æquivaleat? </s> <s id="s.003515">Hac autem oblatâ occaſione <lb></lb>cùm varias excogitaverim rationes utendi remis aquæ ſemper <lb></lb>immerſis, liceat mihi per lectoris patientiam unum proponere, <lb></lb>quod fortaſſe nec incommodum, nec inutile accideret, ſi in <lb></lb>uſum deduceretur, tunc maximè, cum plures hinc & hinc re<lb></lb>miges adhibentur, qui navis æquilibrio non officerent: neque <lb></lb>enim facilè author eſſem, ut levioribus cymbis methodus hæc <lb></lb>communis eſſet: quippe quæ deficiente ponderis hinc & hinc <lb></lb>æqualitate in alterutram partem nimis inclinarentur, nec citra <lb></lb>casûs nautæ, aut everſionis naviculæ periculum. </s> <s id="s.003516">Remiges <lb></lb>ſtatuo hinc & hinc ſcalmo inſiſtentes; id quod incommo<lb></lb>dum non erit; quandoquidem additis extrinſecùs opportu<lb></lb>nis fulcris craſſiorem ſatíſque firmam tabulam impono me<lb></lb>diocris latitudinis à ſcalmo diſtantem tanto intervallo, quan-<pb pagenum="460" xlink:href="017/01/476.jpg"></pb>to opus eſt, ut interjici valeat remus, commodéque agitari: <lb></lb>externam autem additæ tabulæ oram ambiat limbus, ne facilè <lb></lb>pes labatur; alterum enim pedem tabulæ, alterum ſcalmo com<lb></lb>mode imponere poterit remex. </s> <s id="s.003517">Remi longitudinem definit al<lb></lb><figure id="id.017.01.476.1.jpg" xlink:href="017/01/476/1.jpg"></figure><lb></lb>titudo ſcalmi ferè ſupra navis fundum, <lb></lb>addita mediocri hominis altitudine; <lb></lb>ipsíque remi capiti cylindrulus tranſ<lb></lb>verſarius injicitur, ita tamen, ut in eo<lb></lb>dem plano inveniatur cum remi pal<lb></lb>mulâ: apprehenſo ſi quidem utrâque <lb></lb>remigis manu hinc & hinc cylindrulo, <lb></lb>palmulæ planities aquæ obvertitur, <lb></lb>eámque impellit; apprehensâ autem <lb></lb>alterâ tantum extremitate, ſive A, ſi<lb></lb>ve B, quando retrahitur remus, pal<lb></lb>mula DE aquam findit, & eſt quo<lb></lb>dammodo parallela carinæ. </s> <s id="s.003518">Duplicem <lb></lb>igitur motum remo conciliare oportet, <lb></lb>alterum quidem à puppi ad proram, & viciſſim, cùm ſcilicet im<lb></lb>pellitur, & retrahitur, alterum verò circa ſuum axem longitu<lb></lb>dinis, ut convertatur nunc ad impellendam, nunc ad findendam <lb></lb>aquam. </s> <s id="s.003519">Primus motus perficitur, ſi ferreo circulo HI remus in<lb></lb>ſeratur duos polos habenti; quorum alter ſcalmum, alter additam <lb></lb>tabulam ingrediatur (ſivè potiùs excavatæ congruæ crenæ in<lb></lb>cumbant, ut extrahi pro libito poſſint) síntque facilè verſatiles: <lb></lb>remus enim in eodem plano verticali ſemper exiſtens deprimi <lb></lb>poteſt, ut horizontem versùs inclinetur, atque iterum elevari ac <lb></lb>etiam in oppoſitam partem inclinari. </s> <s id="s.003520">Secundus autem motus fa<lb></lb>cillimè habetur, ſi remo ferreus rotundus claviculus F adjicia<lb></lb>tur circuli HI ſuperiorem partem contingens; impedit enim, <lb></lb>ne remus deorſum prolabatur, adeóque nullo labore converti<lb></lb>tur circa axem ſuæ longitudinis remus, dimiſsâ alterutrâ cylin<lb></lb>druli AB extremitate, quando retrahitur. </s> <s id="s.003521">Inventum hoc rudi<lb></lb>ter propoſitum expolire, atque accuratiùs excolere poteris, in<lb></lb>genioſe Lector, qui fortaſſe tuâ induſtriä conſequeris artem <lb></lb>mihi ignotam remos ita diſponendi, ut remex unus pluribus <lb></lb>nautis æquivaleat, quemadmodum de Sinenſibus narratur. <pb pagenum="461" xlink:href="017/01/477.jpg"></pb></s> </p> <p type="main"> <s id="s.003522"><emph type="center"></emph>CAPUT XV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003523"><emph type="center"></emph><emph type="italics"></emph>Quomodo Naves à gubernaculo moveantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003524">REs eſt, cui aſſiduus uſus admirationem detraxiſſe videtur, <lb></lb>non tamen propterea minus habet admirabilitatis, motus <lb></lb>ſcilicet navium, quæ à gubernaculo reguntur, cùm magnum <lb></lb>pondus temporis momento moveatur. </s> <s id="s.003525">Nam & Apoſtolus S. Ja<lb></lb>cobus in Canonicâ Epiſtola cap.3.ait, <emph type="italics"></emph>Ecce & naves cum magnæ <lb></lb>ſint, & à ventis validis minentur, circumferuntur à modico guber<lb></lb>naculo, ubi impetus dirigentis voluerit.<emph.end type="italics"></emph.end></s> <s id="s.003526"> Et Ariſtoteles Mechan. <lb></lb>quæſt. </s> <s id="s.003527">5. inquirit. <emph type="italics"></emph>Cur parvum exiſtens gubernaculum, & in ex<lb></lb>tremo navigio, tantas habet vtres, ut ab exiguo temone, & ab homi<lb></lb>nis unius viribus alioquin modicè utentis, magnæ navigiorum movean<lb></lb>tur moles?<emph.end type="italics"></emph.end> Partes duas in gubernaculo invenimus; alteram ex<lb></lb>trinſecùs navi adjectam, ligneam videlicet alam, ſive cardini<lb></lb>bus, circa quos converti poteſt, poſtremæ puppis parti affixam, <lb></lb>ſive ad latus puppi adjacentem, tignóque, quod ex ſcalmo aſ<lb></lb>ſurgit, adalligatam, prout maritimo vel fluviatili itineri deſti<lb></lb>nata ſunt navigia; alteram intra navim, temonis in morem; ex <lb></lb>cujus converſione aut pars illa externa in oppoſitam plagam <lb></lb>convertitur, ita ut deductâ temonis extremitate ad dexteram, <lb></lb>gubernaculi ala extremæ puppi adjacens in ſiniſtram circa ſuos <lb></lb>cardines deflectat, & viciſſim hæc ad dexteram, temone in ſi<lb></lb>niſtram converſo: aut in navigiis, quorum in fluminibus uſus <lb></lb>eſt, gubernaculum ad puppis latus habentibus, depreſſo temo<lb></lb>ne ſuperior extremitas alæ in triangulum ſubjecto cylindro <lb></lb>infixum conformatæ propiùs accedit ad navim, temone autem <lb></lb>elevato ab illa recedit: contra verò ſi ala infra cylindrum conſti<lb></lb>tuatur, ejus extremitas inferior ad navim accedit temone ele<lb></lb>vato, à navi recedit temone depreſſo. </s> </p> <p type="main"> <s id="s.003528">Quemadmodum autem ad propellendam navim inſtituti ſunt <lb></lb>remi, ita ad ejuſdem curſum dirigendum, atque pro opportu<lb></lb>nitate in dexteram aut in ſiniſtram inclinandum, clavus ad-<pb pagenum="462" xlink:href="017/01/478.jpg"></pb>jectus eſt. </s> <s id="s.003529">Quamquam enim remorum pulſu navis acta proram <lb></lb>in dexteram obvertere poſſit, ſi remiges ſiniſtri ceſſent, atque <lb></lb>è contrario in ſiniſtram ceſſantibus dexteris; aut etiam vento <lb></lb>navim impellente fieri poſſit hæc in alterutram partem decli<lb></lb>natio modò paſſo, modò contracto velo, ut me obſervaſſe me<lb></lb>mini, cum ex inſula Seelandia per fretum Oreſunticum in Sca<lb></lb>niam (Elſingorâ ſcilicet Elſemburgum) navigiolo transfreta<lb></lb>rem: id tamen longè faciliùs, atque ad unius gubernatoris ar<lb></lb>bitrium perficitur converſo opportunè clavo, ut quotidiano <lb></lb>experimento docemur. </s> </p> <p type="main"> <s id="s.003530">Porrò dupliciter gubernaculi motum conſiderare poſſumus; <lb></lb>neque enim eadem eſt ratio, cùm navis quieſcit, nulluſque eſt <lb></lb>aquæ motus, atque cùm navis vento ſeu remis agitur, aut aqua <lb></lb>ipſa movetur. </s> <s id="s.003531">Et quidem ſi navigium in aquà immotâ quieſ<lb></lb>cat, qui gubernaculi temonem movet, eſt potentia applicata <lb></lb>vecti, cujus hypomochlium eſt aqua, ſi navis non ſit tantæ gra<lb></lb>vitatis, ut faciliùs ipſa moveatur, quam tota aqua propellatur <lb></lb>ab alâ gubernaculi; & tunc eſt vectis ſecundi generis, nam <lb></lb>puppis, aquâ reſiſtente, ſecedit ad dexteram aut ad ſiniſtram <lb></lb>ſequens temonis converſionem. </s> <s id="s.003532">At ſi tanta ſit navis gravitas, <lb></lb>ut multo faciliùs tota aqua propellatur, quàm navis loco mo<lb></lb>veatur, vectis eſt primi generis habens hypomochlium in car<lb></lb>dinibus, circa quos gubernaculi ala convertitur, pondus au<lb></lb>tem, quod movetur, eſt aqua, quæ eò faciliùs, minori ſcilicet <lb></lb>labore, propellitur, quò longior eſt temo; tunc enim potentia <lb></lb>plus habet momenti. </s> <s id="s.003533">Hinc duplex vectis ratio invenitur, cùm <lb></lb>aliquâ ex parte aqua, aliquâ ex parte puppis movetur; quo in <lb></lb>motu ſatis conſtat neque motum puppis fieri circa aquam extre<lb></lb>mæ gubernaculi alæ reſpondentem, neque motum aquæ reſpon<lb></lb>dentis extremo gubernaculo fieri circa cardines puppi inhæren<lb></lb>tes; ſed converſionem fieri circa punctum aliquod intermedium <lb></lb>reciprocè acceptum pro Ratione reſiſtentiarum aquæ & navis, <lb></lb>juxta dicta cap.5. hujus libri. </s> <s id="s.003534">Cùm autem reſiſtentia aquæ æſti<lb></lb>manda ſit ex magnitudine & figura alæ gubernaculi aquam <lb></lb>ipſam impellentis, & reſiſtentia navis pariter definienda ſit tùm <lb></lb>ex ejus gravitate, tum ex aquæ propellendæ quantitate, dum <lb></lb>navis in dexteram aut in ſiniſtram convertitur, patet nul<lb></lb>lum certum punctum navibus omnibus commune ſtatui poſſe; <pb pagenum="463" xlink:href="017/01/479.jpg"></pb>ſunt ſiquidem hujuſmodi reſiſtentiæ multiplici varietati ob<lb></lb>noxiæ. </s> </p> <p type="main"> <s id="s.003535">At verò cùm navis in motu eſt, & vento impellente ſeu re<lb></lb>mis agitur, potentia quidem pro temonis longitudine ſua ha<lb></lb>bet momenta, & ad navis converſionem juvat, magis tamen <lb></lb>accipiendo vim externam & ferendo, quàm agendo & facien<lb></lb>do, hoc eſt retinendo gubernaculum in illa obliquâ poſitione <lb></lb>adversùs vim aquæ in contrarium nitentis, aut reſiſtentis. </s> <s id="s.003536">Quò <lb></lb>enim velociùs fertur navis, obviam aquam prorâ ſcindens illam <lb></lb>ità dividit, ut ad navis latera hinc atque hinc velociùs refluat <lb></lb>in puppim; ubi ſi gubernaculi alam inveniat rectam, pergit na<lb></lb>vis recto itinere; Sed ſi aqua refluens obli<lb></lb>quum gubernaculum offendat, ut ſi exiſten<lb></lb><figure id="id.017.01.479.1.jpg" xlink:href="017/01/479/1.jpg"></figure><lb></lb>te carina AB fuerit gubernaculum obli<lb></lb>quum CD, aqua in alam AD incurrens <lb></lb>dum illam urget, puppim cogit declinare <lb></lb>ex A in E, & prora obvertitur versùs F. </s> </p> <p type="main"> <s id="s.003537">Hæc tamen, quæ de aquá ad navis late<lb></lb>ra refluente dicta ſunt, non ita accipi velim, <lb></lb>ut non niſi ab ejus impetu flecti navis cur<lb></lb>ſum exiſtimes; ſed hæc deflexio præcipuè <lb></lb>tribuenda eſt reſiſtentiæ ipſius aquæ, in <lb></lb>quam incurrit gubernaculum obliquum, <lb></lb>dum navis tota impellitur; eò autem major <lb></lb>eſt aquæ reſiſtentia, quò velociùs illam ſcindi oportet, ut ſæ<lb></lb>piùs dictum eſt. </s> <s id="s.003538">Ideò quo validiore venti aut remorum impul<lb></lb>ſu agitur navis, faciliùs flectitur ope gubernaculi, majorem <lb></lb>quippe invenit reſiſtentiam. </s> <s id="s.003539">Cum verò reſiſtentia hæc ex al<lb></lb>terutra tantùm parte inveniatur, neceſſe eſt proram in eandem <lb></lb>obverti plagam. </s> <s id="s.003540">Cujus rei obvium experimentum ſumere quiſ<lb></lb>que poteſt, ſi corpus aliquod angulatum (cujuſmodi eſſet nor<lb></lb>ma, qua ad angulum rectum deſcribendum utimur) in plano <lb></lb>inclinato æquabili ac polito deſcendere permittat: nam ſi, quod <lb></lb>ponè ſequitur, brachium in offendiculum aliquod incurrat, il<lb></lb>lico reliquum brachium ad eam partem inclinari videbit, im<lb></lb>petu ſcilicet promovente corpus, atque objectum impedimen<lb></lb>tum declinante. </s> </p> <p type="main"> <s id="s.003541">Quare id, quod navim maximè movet in dexteram aut ſi-<pb pagenum="464" xlink:href="017/01/480.jpg"></pb>niſtram, eſt impetus ab ipſo vento aut à remigibus navi impreſ<lb></lb>ſus; gubernaculum autem infert moram & impedimentum, ne <lb></lb>motus omnino fiat juxta directionem impetûs ab impellente im<lb></lb>preſſi: quamdiu verò impedimentum perſeverat, navis magis <lb></lb>aut minùs obliquè fertur, pro ut modificata impetûs directio <lb></lb>exigit. </s> <s id="s.003542">Juvat autem, ut dictum eſt, aqua, quæ à prorâ dividi<lb></lb>tur, & ad latera refluit, maximè ſi adverſo flumine, aut contra <lb></lb>marini æſtus curſum navigatio inſtituatur; aucto enim impedi<lb></lb>mento faciliùs flectitur inſtituta progreſſio; ſed idcircò etiam <lb></lb>navis motus retardatur magis. </s> </p> <p type="main"> <s id="s.003543">Quod quidem ſpectat ad gubernaculum extremæ puppis pla<lb></lb>næ faciei adhærens, ut in majoribus navigiis maritimo itineri <lb></lb>deſtinatis, ſatis jam explicatum eſt: unum addendum videtur, <lb></lb>quod in navigiis ad devehendas merces fabricatis in foſsâ qua<lb></lb>dam manufactâ aliquando obſervaſſe me memini; ex puppi vi<lb></lb>delicet extremâ in acutum aſſurgente, quaſi caudæ in morem, <lb></lb>clavus longiùs protendebatur apici puppis inſiſtens remo abſi<lb></lb>milis tantùm, quatenus palmula paulò latior, nec juxta ſcapi <lb></lb>longitudinem directa, ſed inflexa intra aquam immergebatur; <lb></lb>caput autem temonis fune jungebatur navigij plano ita, ut gu<lb></lb>bernaculi pars externa ſuo pondere recidere nequiret, ac fun<lb></lb>dum alvei non peteret, ſed palmula paulò infra aquæ ſuperfi<lb></lb>ciem conſiſteret. </s> <s id="s.003544">Hinc enim fiebat, ut temonis capite in alteru<lb></lb>tram partem adducto in eandem puppis recederet aquâ reſiſten<lb></lb>te palmulæ, ac proinde prora in oppoſitam partem obvertere<lb></lb>tur: perinde atque in cymbis gubernaculo deſtitutis, cùm remi <lb></lb>ad latus extremæ puppis directè immerſi caput ad ſe retrahit <lb></lb>nauta, puppim ipſam impellit, ac proram in oppoſitam partem <lb></lb>convertit. </s> <s id="s.003545">Huc ſpectare poſſunt, quæ habet Atlas Sinicus <lb></lb>pag.123 in XI Provincia Fokien loquens de flumine Min, quod <lb></lb>ex Puching ad oppidum uſque Xuiken per valles & ſaxa ingen<lb></lb>ti impetu ac violentiâ volvitur, inde placidiſſimum flumen eſt; <lb></lb>& quantumcumque violentum enavigatur tamen à Sinis con<lb></lb>ſuetâ illorum induſtriâ, ac parvarum navicularum artificio: hæ <lb></lb>enim naves clavum, ut aliæ non habent, ſed duos longiſſimè <lb></lb>porrectos, ad puppim unum, ad proram alterum: his per ſaxa <lb></lb>ac ſcopulos prominentes facillime ac velociſſimè naves, ac ſi <lb></lb>fræno equos continerent, dirigunt. </s> <s id="s.003546">Hæc ibi. </s> </p> <pb pagenum="465" xlink:href="017/01/481.jpg"></pb> <p type="main"> <s id="s.003547">Sed ut aliquid etiam de guberna culo ad puppis latus conſti<lb></lb>tuto in navigiis, quorum potiſſimus uſus eſt in fluviis, dicatur, <lb></lb>animadvertendum eſt hujuſmodi navigia non ſolùm proram, <lb></lb>ſed & puppim habere, quæ obliquè aſſurgentes in acutum de<lb></lb>ſinunt, gubernaculum autem conſtare ex cylindro obliquè <lb></lb>deſcendénte juxta puppis longitudinem, atque ex alâ triangu<lb></lb>lari ut plurimùm ſurſum reſpiciente, cujus latus unum cylin<lb></lb>dro congruit, cui ïnfixum eſt. </s> <s id="s.003548">Quandiu ala ſurſum reſpicit, <lb></lb>nihil impedit navis motum, æqualiter enim aqua hinc & hinc <lb></lb>fluit, ac proinde navis fertur juxta impetûs à vento aut à remi<lb></lb>gibus impreſſi directionem (idem dic, cùm navis trahitur) <lb></lb>quam ſequitur, niſi aliquid fortuito interveniat, à quo turbe<lb></lb>tur motus, & præter nautarum voluntatem aliò flectatur. </s> <s id="s.003549">Quod <lb></lb>ſi convoluto circà ſuum axem cylindro, ala in hanc aut illam <lb></lb>partem vertatur, jam occurrit aquæ, ex cujus eſiſtentiâ impe<lb></lb>dimentum objicitur navi, ne recta feratur, ſed in alteram par<lb></lb>tem detorquetur: nam ſi depreſſo temone, qui priùs erat hori<lb></lb>zonti parallelus, ala versùs navim inclinetur, aqua inter guber<lb></lb>naculum & navim intercepta reſiſtit, atque interfluens conatur <lb></lb>alam gubernaculi in directum reſtituere: quapropter puppim in <lb></lb>dexteram trahens, illíque ad dexteram reſiſtens (clavus ſcilicet <lb></lb>dextero puppis lateri adjacet) proram obvertit ad ſiniſtram. </s> <s id="s.003550">At <lb></lb>ſi gubernaculi ala in oppoſitam navi partem extrorſum verta<lb></lb>tur, obviam habet aquam externam, qua reſiſtente repellitur <lb></lb>puppis in ſiniſtram, & prora in dexteram convertitur. </s> <s id="s.003551">Quod ſi <lb></lb>alam triangularem placeat potiùs cylindro ſubjicere, elevato te<lb></lb>mone ala accedit ad navim, & depreſſo temone ala recedit à na<lb></lb>vi: quapropter ibi puppis repellitur in ſiniſtram, hîc ab aquâ in<lb></lb>tercurrente trahitur in dexteram, motùſque oppoſiti proræ <lb></lb>conveniunt. </s> </p> <p type="main"> <s id="s.003552">Ex his facilè innoteſcit, quid præſtet gubernaculum inter <lb></lb>puppes duorum pontonum, quos impoſitus pons jungit, vali<lb></lb>dúſque rudens congruæ longitudinis retinet, ne ſecundo flu<lb></lb>mine rapiantur; prout enim in hanc vel illam partem guberna<lb></lb>culi ala vertitur, obvium habet interjectarum aquarum impe<lb></lb>tum, quo propellitur in adverſam partem, eáque ratione traji<lb></lb>citur flumen, ut in Pado & aliis Galliæ Ciſalpinæ fluviis paſſim <lb></lb>videre eſt. </s> </p> <pb pagenum="466" xlink:href="017/01/482.jpg"></pb> <p type="main"> <s id="s.003553">Illud poſtremò conſideratione dignum eſt, quod ad ipſius <lb></lb>navis converſionem attinet nimirùm quodnam ſit punctum <lb></lb>circa quod convertitur: Manifeſtum eſt enim neque circa pup<lb></lb>pim tanquam circa centrum deſcribi arcum à prorâ, neque vi<lb></lb>ciſſim circa proram quaſi centrum arcum à puppi deſcribi, quia <lb></lb>aquæ quantitas reſpondens longitudini carinæ plurimum re<lb></lb>ſiſtit, ne circulariter moveatur tota ad eandem partem, coge<lb></lb>retur ſcilicet nimis amplum arcum deſcribere, nimíſque veloci<lb></lb>ter moveri in latus, ut per deſtinatum navigationis Rumbum <lb></lb>nova loxodromia inſtitueretur: faciliùs igitur convertitur na<lb></lb>vis, ſi dum pars anterior proræ aquam in dexteram propellit, <lb></lb>reliqua pars poſterior puppi proxima aquam repellat in ſi<lb></lb>niſtram, utraque enim extremitas minore arcu deſcripto ad ma<lb></lb>jorem angulum carinam inclinat atque deflectit à lineâ prioris <lb></lb>cursûs, & minore velocitate aquam urgens minorem invenit <lb></lb>reſiſtentiam. </s> <s id="s.003554">Fit igitur converſio circa punctum aliquod me<lb></lb>dium inter proram & puppim; illud autem eſt, circa quod na<lb></lb>tura faciliùs aſſequitur propoſitum, & minore motu removetur <lb></lb>impedimentum, quod ab aquâ occurrente infertur, quæ cùm <lb></lb>dividatur à prorâ, refluátque juxtà navis latera, æqualiter qui<lb></lb>dem à prorâ diſpertitur, ſed ubi navis ventrem, hoc eſt ampliſ<lb></lb>ſimam navigij partem prætergreſſa eſt, offendens ex alterâ <lb></lb>parte gubernaculi alam fluere non poteſt, qua velocitate flue<lb></lb>ret nullo objecto offendiculo; propterea aquæ refluenti ex ad<lb></lb>verſo navis latere objiciendum eſt obliquè puppis latus, ut illa <lb></lb>pariter lentiùs fluat, divisóque impedimento æquales aquæ por<lb></lb>tiones ex utroque puppis latere fluant. </s> <s id="s.003555">Quare probabili con<lb></lb>jecturâ exiſtimo converſionem fieri circa illud carinæ punctum, <lb></lb>quod reſpondet maximæ navigij amplitudini; pars quippe na<lb></lb>vigij anterior juxta ſuam latitudinem occurrens aquæ invenit <lb></lb>reſiſtentiam; aqua igitur incurrens in gubernaculum movet <lb></lb>partem poſteriorem in latus, ubi non eſt tam valida aquæ re<lb></lb>ſiſtentia. </s> <s id="s.003556">Cum autem in majoribus navigiis præcipuus malus <lb></lb>ſtatuatur in maxima navigij amplitudine, hoc eſt, ubi carinæ <lb></lb>longitudo beſſem relinquit puppim verſus, & trientem versùs <lb></lb>proram, carina ad proram ſpectans minùs movetur quàm quæ <lb></lb>ad puppim; ſed propter notabilem proræ projecturam ſi pars na<lb></lb>vis ſuprema inſpiciatur, malus ille eſt circa mediam totius na-<pb pagenum="467" xlink:href="017/01/483.jpg"></pb>vis longitudinem, ibíque fit converſio. </s> <s id="s.003557">Cæterùm quicumque <lb></lb>navis formam, tormentorumque bellicorum diſpoſitionem ac <lb></lb>numerum obſervet, utique centrum gravitatis inter puppim & <lb></lb>malum præcipuum interjectum eſſe affirmabit; præſertim cum <lb></lb>id neceſſe ſit, ne ventorum vi prora nimis deprimatur; id quod <lb></lb>multo manifeſtiùs innoteſcit in minoribus navigiis, ſi fortè ve<lb></lb>lo uti contingat, malus enim maximè ad proram accedit. </s> </p> <p type="main"> <s id="s.003558">Sit igitur carina AB, maxima navis latitudo HI, malus pri<lb></lb>marius in G, centrum gravitatis navigij <lb></lb><figure id="id.017.01.483.1.jpg" xlink:href="017/01/483/1.jpg"></figure><lb></lb>ex. </s> <s id="s.003559">gr. in K. </s> <s id="s.003560">Duo ſunt principia moven<lb></lb>tia; unum eſt ventus in G, alterum eſt <lb></lb>aqua refluens in AD: duo pariter ſunt hy<lb></lb>pomochlia, ſeu impedimenta; vento reſiſtit <lb></lb>gubernaculum AD, propterea navim mo<lb></lb>tione tranſversâ promovens transfert cen<lb></lb>trum gravitatis K verſus H: aquæ refluenti <lb></lb>reſiſtit vis venti in G, ita ut non valeat na<lb></lb>vim retrorſum agere, propterea puppim ex <lb></lb>A transfert in E, & centro gravitatis K im<lb></lb>petum imprimit dirigentem versùs I, cui <lb></lb>tamen prævalente impetu venti dirigente <lb></lb>versùs H, obliquus navis motus efficitur. </s> <s id="s.003561">Quare duplex eſt <lb></lb>vectis ſecundi generis; aqua in AD ad reſiſtentiam centri gra<lb></lb>vitatis K habet momentum ut AG, ſeu DG, ad KG: Ventus <lb></lb>in G ad ejuſdem centri gravitatis K reſiſtentiam habet momen<lb></lb>tum ut GA ad KA. </s> <s id="s.003562">Ex quo conſtat majorem quidem eſſe Ra<lb></lb>tionem AG ad KG minorem, quàm ad KA majorem; ſed <lb></lb>multo validiorem potentiam eſſe ventum, quàm aquam re<lb></lb>fluentem, niſi fortè addatur naturalis fluxus aquæ, qui aliquan<lb></lb>do prævalere dignoſcitur ex occultis Maris Currentibus, quæ <lb></lb>navim aliquando retrorſum agunt contrà vim venti. </s> <s id="s.003563">Sed quo<lb></lb>niam tam varia & multiplex eſt navigiorum forma, nec in iis <lb></lb>conſtruendis omnes artifices eandem ſervant partium membro<lb></lb>rúmque Rationem, nulla aſſignari poteſt certa Ratio, quæ in<lb></lb>tercedat inter diſtantiam centri gravitatis ab extremitate pup<lb></lb>pis, atque diſtantiam puncti, circa quod fit converſio, ab ea<lb></lb>dem extremitate. </s> </p> <p type="main"> <s id="s.003564">Hìc autem (ne quis facilè ſimiliter labatur) fateor me ali-<pb pagenum="468" xlink:href="017/01/484.jpg"></pb>quando veri quadam ſpecie deceptum exiſtimaſſe intervallum <lb></lb>inter extremam proram & punctum converſionis ad quartam <lb></lb>totius longitudinis partem proximè ſtatuendum eſſe; ducebar <lb></lb>ſcilicet quadam analogiâ deſumpta ex cylindro ligneo innatan<lb></lb>te, cujus quieſcentis extremitatem ſi tanto impetu percuſſeris, <lb></lb>quo certum ſpatium percurrat, videbar mihi ritè inferre <lb></lb>punctum, circa quod convertitur, diſtare ab extremitate per<lb></lb>cuſsâ ad totius longitudinis dodrantem: ſatis enim ipſo uſu in<lb></lb>noteſcebat, nec punctum medium, ſcilicet centrum gravitatis, <lb></lb>nec oppoſitam extremitatem eſſe centrum converſionis. </s> <s id="s.003565">Vide<lb></lb>batur autem naturæ ſua jura tueri conanti valde conſenta<lb></lb>neum, ſi corpus amans quietis externo impulſui ita obſecun<lb></lb>det, ut quam minimo totius corporis motu impreſſus impetus <lb></lb>partem percuſſam pro ſuæ intenſionis modo transferat. </s> </p> <p type="main"> <s id="s.003566">Sit enim Cylindrus AB, cujus medium atque centrum gra<lb></lb><figure id="id.017.01.484.1.jpg" xlink:href="017/01/484/1.jpg"></figure><lb></lb>vitatis C: AE verò ſit <lb></lb>totius longitudinis do<lb></lb>drans: percutiatur extre<lb></lb>mitas A tanto impetu, <lb></lb>quanto illa ferri poſſet <lb></lb>per ſpatium AD, ſi mo<lb></lb>veretur circa centrum C. </s> <lb></lb> <s id="s.003567">Ducatur igitur per C <lb></lb>recta DO æqualis toti <lb></lb>cylindro; qui ſi movere<lb></lb>tur circa punctum C, utique ſuo motu deſcriberet duos Secto<lb></lb>res, ACD, & BCO. </s> <s id="s.003568">Item per E ducatur ipſi DO parallela <lb></lb>FI ita, ut ipſi EA æqualis ſit EF, & ipſi EB æqualis ſit EI. </s> <s id="s.003569">Cy<lb></lb>lindrus igitur AB converſione factâ circa punctum E deſcribe<lb></lb>ret Sectores AEF & BEI duobus prioribus ſimiles. </s> <s id="s.003570">Sunt au<lb></lb>tem Sectores ſimiles, ut quadrata Radiorum; quemadmodum <lb></lb>facilè colligitur ex 2 lib. 12: atque ideò, cum Radius AC ad <lb></lb>Radium AE ſit ut 2 ad 3, Sector ACD ad Sectorem AEF eſt <lb></lb>ut 4 ad 9: & quia Radius BC ad Radium BE eſt ut 2 ad 1, <lb></lb>Sector BCO ad Sectorem BEI eſt ut 4 ad 1. Motus igitur cy<lb></lb>lindri circa centrum C ad motum circa centrum E eſt ut 8 ad <lb></lb>10, ſi Sectores ſimiles deſcribantur. </s> <s id="s.003571">Atqui impetus impreſſus <lb></lb>ſolùm poteſt extremitatem A transferre per ſpatium æquale <pb pagenum="469" xlink:href="017/01/485.jpg"></pb>ipſi AD (eſt autem arcus AD ad arcum AF ſimilem, ut Ra<lb></lb>dius AC ad Radium AE, hoc eſt ut 2 ad 3) igitur eandem <lb></lb>transfert ſolùm per AG beſſem arcûs AF, ac proinde motus eſt <lb></lb>per Sectores AEG & BEH, qui ex ult. </s> <s id="s.003572">lib. 6. ſunt bes duo<lb></lb>rum Sectorum AEF & BEI, quorum ſumma eſt 10; ipſius au<lb></lb>tem 10 bes eſt 6 2/3. Motus igitur circa centrum E minor eſt <lb></lb>motu circa centrum C, & impetus impreſſus æqualiter transfert <lb></lb>extremitatem A. </s> </p> <p type="main"> <s id="s.003573">Fateor potuiſſe ſtatui AE mediam proportionalem inter to<lb></lb>tam longitudinem AB & ejus ſemiſſem AC, & motus fuiſſet <lb></lb>paulo minor. </s> <s id="s.003574">Ponatur enim tota AB 200, AC 100, eſt AE <lb></lb>141 2/5 proximè: igitur ut quadratum AC ad quadratum AE <lb></lb>mediæ proportionalis, hoc eſt ut 10000 ad 19994, ita Sector <lb></lb>ACD ad ſectorem AEF ſimilem; & ut ipſius CB 100 qua<lb></lb>dratum 10000, ad ipſius EB 59 proximè quadratum 3481, ita <lb></lb>Sector BCO ad Sectorem ſimilem BEI. </s> <s id="s.003575">Quare ſumma Secto<lb></lb>rum ACD, BCO eſt 20000, Sectorum verò AEF, BEI eſt <lb></lb>23475. Sed quia ut AC ad AE, ita arcus AD ad arcum AF, <lb></lb>quarum partium AD eſt 100, AF eſt 141 proximè: & aſſump<lb></lb>to arcu AG 100, ſumma Sectorum AEG & BEH, ad ſum<lb></lb>mam Sectorum AEF & BEI erit ut 100 ad 141, hoc eſt, ut <lb></lb>16649 ad 23475: minor igitur eſt quàm ſumma Sectorum <lb></lb>ACD & BCO, quæ eſt 20000. At ſi AE ſit 150, & EB 50, <lb></lb>ſumma Sectorum AEF, BEI ut 25000, bes autem 16666 2/3, <lb></lb>qui excedit numerum ſuperiùs inventum 16649 adeò modico <lb></lb>intervallo, ut contemnendum ſit; cùm maximè impetus per ar<lb></lb>cum AF aliquantulo majorem motum efficiat quàm per cir<lb></lb>cumferentiam circuli minoris, ac propterea cenſendus ſit arcus <lb></lb>AG aliquantulum major quàm arcus AD; idcircò vero pro<lb></lb>pior eſt AE dodrans totius longitudinis AB, quàm AE media <lb></lb>proportionalis inter AC & AB. </s> </p> <p type="main"> <s id="s.003576">Verùm quæ de cylindro in aquâ quieſcente dicuntur ſatis <lb></lb>probabiliter, non omninò congruere poſſunt motui navis, quæ <lb></lb>præter motum aquæ percutientis gubernaculum promovetur à <lb></lb>vento aut à remigibus, & præterea non habet æquabili ductu <lb></lb>conſtitutam figuram, quemadmodum cylindrus: propterea <lb></lb>huic analogiæ non acquieſcendum duxi. </s> <s id="s.003577">Sed & illud adden-<pb pagenum="470" xlink:href="017/01/486.jpg"></pb>dum, quod neque de cylindro ſatis certus eſſe poſſum; nam ſi <lb></lb>alia fiat hypotheſis, & ad totius longitudinis beſſem ſtatuatur <lb></lb>punctum converſionis ita ut ſemiſſis AC ſit 3, AE verò ſit 4, <lb></lb>& EB ſit 2; Sector ACD ad Sectorem AEF eſt ut 9 ad 16, <lb></lb>& Sector BCO ad Sectorem BEI eſt ut 9 ad 4; igitur ſumma <lb></lb>priorum ad ſummam poſteriorum eſt ut 18 ad 20. Atqui Sector <lb></lb>AEG ad Sectorem AEF eſt ut 3 ad 4 (id quod de ſimili <lb></lb>Sectore BEH ad Sectorem BEI intelligendum eſt) igitur, cum <lb></lb>AEF ſit 16, AEG eſt 12, & cum BEI ſit 4, BEH eſt 3, ac <lb></lb>propterea ſumma Sectorum AEG & BEH eſt ut 15 ad ſum<lb></lb>mam Sectorum ACD & BCO ut 18. In prima autem hypo<lb></lb>theſi quando erat AC ut 2 & AE ut 3, erat motuum Ratio ut <lb></lb>8 ad 6 2/3, quæ eſt planè eadem cum Ratione 18 ad 15. Cum <lb></lb>itaque eadem motuum Ratio ſequatur, ſive AE ſit bes, ſive <lb></lb>dodrans totius longitudinis AB, cur dodrantem potiùs quàm <lb></lb>beſſem pronunciemus, niſi aliunde doceamur? <lb></lb></s> </p> <p type="main"> <s id="s.003578"><emph type="center"></emph>CAPUT XVI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003579"><emph type="center"></emph><emph type="italics"></emph>An malus in motu navis habeat Rationem vectis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003580">NAvim impelli ventorum vi certum eſt, qui velum implent <lb></lb>ex antennâ ſuſpenſum atque expaſſum, funibùſque, quos <lb></lb>Propedes vocant, poſteriori navis parti alligatum. </s> <s id="s.003581">Quoniam <lb></lb>verò, ut quotidiano uſu didicimus, quò altiùs evecta fuerit an<lb></lb>tenna, eò validiùs, cæteris paribus, navis à vento impellitur, <lb></lb>quæritur ab Ariſtotele quæſt.6. <emph type="italics"></emph>Cur quando antenna ſublimior <lb></lb>fuerit, iiſdem velis, & codem vento, celeriùs feruntur navigia?<emph.end type="italics"></emph.end><lb></lb>Cauſam ille ex vectis rationibus petendam opinatur, quaſi ma<lb></lb>lus ſit vectis habens hypomochlium in ea carinæ parte, cui in<lb></lb>figitur; potentia movens ſit ventus velum implens ſupremæ <lb></lb>mali parti applicatus, ubi antenna cum malo connectitur; pon<lb></lb>dus verò ſit navigium: quò igitur potentia magis ab hypomo<lb></lb>chlio abeſt, plus habere momenti manifeſtum eſt. </s> <s id="s.003582">Verùm non <lb></lb>ego hîc inani labore ſuſcepto, ut Philoſophi dicto aliquam veri <pb pagenum="471" xlink:href="017/01/487.jpg"></pb>ſimilitudinem adjiciam, tempus conteram: Cui otium eſt, <lb></lb>Authores legat. </s> </p> <p type="main"> <s id="s.003583">Si quæſtio eſſet, Cur longiores mali ſint magis obnoxij peri<lb></lb>culo fractionis, facilè invenirem rationem vectis, quia ponde<lb></lb>ris vicem ſubeunt particulæ ipſæ, quarum nexus per vim ſol<lb></lb>vendus eſt in fractione; quò autem longior eſt malus, ad mo<lb></lb>tum partium, quæ dividuntur, majorem Rationem habet mo<lb></lb>tus venti applicati longiori malo, quàm motus ejuſdem brevio<lb></lb>ri malo applicati. </s> <s id="s.003584">Sed hîc, ubi de navis motu quæſtio eſt, ſive <lb></lb>altè aſſurgat malus, ſive brevis ſit, ſemper eadem eſt Ratio mo<lb></lb>tûs venti velo excepti, atque navis. </s> <s id="s.003585">Quomodo enim pterna, <lb></lb>ideſt ima mali calx, eſſe poteſt extremus vectis in carinâ, cui <lb></lb>inſeritur, velut in hypomochlio quieſcens, navis verò tota ſi<lb></lb>mul mota æquali motu, rationem habet ponderis à vecte impul<lb></lb>ſi? </s> <s id="s.003586">nonne hypomochlium, pondus, & potentia æquali planè <lb></lb>motu moventur? </s> <s id="s.003587">neque enim velociùs movetur ventus velo ex<lb></lb>ceptus, quàm ipſum velum, nec velum velociùs quàm navis; <lb></lb>& cum ipſa navi planè æqualiter movetur carinæ punctum, cui <lb></lb>malus infigitur. </s> <s id="s.003588">Quis autem motus per Vectem, qua Vectis eſt <lb></lb>Facultas mechanica, hujuſmodi æqualitatem admittit? </s> <s id="s.003589">Non <lb></lb>igitur malus in motu, quo navis progreditur, Rationem vectis <lb></lb>habere dicendus eſt. </s> </p> <p type="main"> <s id="s.003590">Sit malus CD cujus pterna C inſeratur carinæ AB, car<lb></lb>cheſio autem D applicetur an<lb></lb><figure id="id.017.01.487.1.jpg" xlink:href="017/01/487/1.jpg"></figure><lb></lb>tenna cum velo pendente, cujus <lb></lb>imæ extremitates navis lateribus <lb></lb>opportunè ad ventum excipien<lb></lb>dum jungantur. </s> <s id="s.003591">Certum eſt ma<lb></lb>lum CD moveri ſemper ſibi pa<lb></lb>rallelum (niſi fortè aliquanto <lb></lb>plus extremitas D moveatur, ſi<lb></lb>cut in homine plus caput move<lb></lb>tur quàm pedes ſupra ſphæricam <lb></lb>terræ vel aquæ ſuperficiem, ſed <lb></lb>hoc nihil refert) neque poſſe obtinere rationem vectis niſi <lb></lb>comparatè ad eum motum, quo circa C tanquam circa centrum <lb></lb>fieret converſio; quemadmodum ſi deprimenda eſſet prora & <lb></lb>elevanda puppis, ut carina AB non eſſet horizonti parallela, <pb pagenum="472" xlink:href="017/01/488.jpg"></pb>ſed B deprimeretur infra planum horizontale, & A ſupra illud <lb></lb>elevaretur. </s> <s id="s.003592">Verùm (præterquam quod non hic eſt navis mo<lb></lb>tus, de quo diſputatur) obſervandum eſt in navigiis minoribus, <lb></lb>quibus movendis unicus malus adhibetur, hunc ſtatui non in <lb></lb>medio navigio, ſed magis accedere ad proram, in majoribus <lb></lb>autem navibus, quæ plures malos habent, maximum quidem <lb></lb>malum, cujus validiſſimæ ſunt vires, aliquanto quidem magis <lb></lb>ad puppim quàm ad proram accedere (ſi longitudo in ſuperiori <lb></lb>parte attendatur) ut in Oceano videre eſt Anglicas, Gallicas, <lb></lb>Hollandicas naves, comparatè tamen ad carinam, majorem <lb></lb>carinæ partem puppim, minorem proram reſpicere. </s> <s id="s.003593">Id autem <lb></lb>eo conſilio factum eſt, ne malus centro gravitatis navis reſpon<lb></lb>deat, neque exercere poſſit munus vectis deprimendo proram, <lb></lb>puppímque elevando. </s> <s id="s.003594">Quando enim magis ad proram accedit <lb></lb>malus, major pars navigij inter malum & puppim interjecta re<lb></lb>nititur ſua gravitate, ne elevetur, quando verò à medio pup<lb></lb>pim versùs recedit, major pars navis, quæ deprimenda eſſet, <lb></lb>majorem aquæ reſiſtentiam invenit; ac proinde ſervatà carinæ <lb></lb>poſitione horizontali faciliùs navis movetur. </s> <s id="s.003595">Hinc tardiorem <lb></lb>fieri navigij curſum contingit, vel quia perperam collocatus eſt <lb></lb>malus, vel quia pondera in navi non ſunt ritè diſtributa, adeò <lb></lb>ut à malo vix abſit centrum gravitatis navigij onuſti; tunc enim <lb></lb>depreſsâ prorâ & carinâ ad horizontem inclinatâ major vis ob<lb></lb>viæ aquæ reſiſtit. </s> <s id="s.003596">Quare tantum abeſt malus à ratione vectis, <lb></lb>vi cujus progrediatur navigium, ut potius caveatur, ne vectis <lb></lb>munus ille exerceat, motum aliquem efficiendo, qui celeritati <lb></lb>non parum officeret. </s> </p> <p type="main"> <s id="s.003597">In motu autem majoris navigij pluribus malis inſtructi non <lb></lb>ſolus malus, qui præcipuus eſt & maximus, attenditur, ſed etiam <lb></lb>reliqui: potior tamen ad provehendam navim eſt malus, qui à <lb></lb>medio ad proram accedit, quippe qui navim trahit; nam qui â <lb></lb>centro gravitatis puppim versùs recedit, navim impellit potiùs, <lb></lb>quàm trahat: quamquam ille, qui ad puppim proximè ſpectat, <lb></lb>& velum habet triangulare, maximè juvat, ut gubernatoris pro<lb></lb>poſito, qui clavum regit, obſecundet ad navis curſum in alteru<lb></lb>tram partem dirigendum. </s> <s id="s.003598">Verum quicumque malus conſide<lb></lb>retur, in nullo rationem vectis reperies, ſive ad impellendam, <lb></lb>ſive ad trahendam navim. </s> </p> <pb pagenum="473" xlink:href="017/01/489.jpg"></pb> <p type="main"> <s id="s.003599">At, inquis, ſi adverſo flumine deducendum ſit navigium ſi<lb></lb>ve à nautica turbâ ſive ab equis trahentibus, cur funis ma<lb></lb>lo, non autem proræ, alligatur, ſi nihil confert facilitatis appli<lb></lb>catio potentiæ trahentis medio fune ad majorem altitudinem à <lb></lb>carinâ? </s> <s id="s.003600">Ego verò ex te, quiſquis hæc objicis, quæro, cur jidem <lb></lb>nautæ ſi remulco navim trahere aggrediantur, funem navi non <lb></lb>tam altè alligant; ſi ex vectis rationibus illa altitudo aliquod af<lb></lb>fert compendium laboris in trahendo. </s> <s id="s.003601">Sed ſatis utrique quæſtio<lb></lb>ni factum videbis, ſi obſerves non planè æqualem eſſe in uni<lb></lb>verſo alveo aquæ altitudinem, ac proinde neque poſſe navim <lb></lb>æquè ſemper abeſſe à fluminis ripâ, in qua trahentes progre<lb></lb>diuntur; idcircò longiore fune opus eſt, qui ſuo pondere ſpon<lb></lb>te curvatus aquam ſecaret, & trahentium laborem augeret, aut <lb></lb>in occultum aliquem ſub aquis latentem obicem incurreret non <lb></lb>ſine gravi incommodo, ſi funis extremitas depreſſiori loco navi<lb></lb>gij alligaretur; propterea malo altiùs adnectitur, eo quoque <lb></lb>conſilio, ut ſi quæ virgulta aut arbuſculæ ſecundùm fluminis <lb></lb>ripam occurrant, minori impedimento ſint funi obliquè incli<lb></lb>nato, quàm ſi horizonti eſſet ferè parallelus. </s> <s id="s.003602">Qui verò navim <lb></lb>remulco trahunt, non adeò longè ab illa abeſſe coguntur, nec <lb></lb>hujuſmodi impedimentis obnoxij ſunt; ideò breviore fune <lb></lb>utuntur, quem proræ alligant. </s> <s id="s.003603">Cæterùm nullæ vectis vires <lb></lb>exercentur; non enim prora infra aquam deprimi, & puppis <lb></lb>elevari poteſt: id quod ſi contingeret, prora magis demerſa <lb></lb>plus inveniret reſiſtentiæ ab aquâ dividendâ. </s> </p> <p type="main"> <s id="s.003604">Quid igitur, ais, cauſæ eſt, quòd antennâ uſque ad carche<lb></lb>ſium D elevatâ, magis promovetur navis, quàm ſi tantummodo <lb></lb>uſque ad E attolleretur? </s> <s id="s.003605">quandoquidem nulla vectis ratio hîc <lb></lb>habetur. </s> <s id="s.003606">Eos, qui cum Ariſtotele ſentiunt, æquivocatione la<lb></lb>borare facilè oſtenditur: quid enim refert, utrùm antenna ma<lb></lb>gis an minùs elevetur, ſi potentia, videlicet ventus velum im<lb></lb>plens, illi mali parti applicata intelligeretur, cui antenna ad<lb></lb>nectitur? </s> <s id="s.003607">hæc autem funibus, quos <foreign lang="grc">μεσου<gap></gap>ίας</foreign> vocant, ſurſum <lb></lb>trahitur, ſemperque, ſive altior, ſive depreſſior ſit, adnectitur <lb></lb>carcheſio in D: quemadmodum nauta fune in D alligato na<lb></lb>vim trahens, ſemper in D applicatus intelligitur, quamvis hu<lb></lb>miliore in loco, quàm D, conſtituatur. </s> <s id="s.003608">Verùm non ibi vis <lb></lb>venti præcisè intelligenda eſt, ubi antenna eſt, ſed toti malo <pb pagenum="474" xlink:href="017/01/490.jpg"></pb>aut ejus parti applicatur, quæ reſpondet velo non ſolùm anten<lb></lb>næ cornibus, ſed etiam navis lateribus alligato: velum autem <lb></lb>in humiliore loco minus recipit venti, quia alta majorum na<lb></lb>vium puppis (niſi ventus ex latere ſpiret) vento oppoſita illum <lb></lb>ſubtrahit velo, & præterea ventus, qui in navis puppim & la<lb></lb>tera illiditur, reflectitur, & proximas venti partes turbat, atque <lb></lb>aliorſum dirigit, vel ſaltem illarum impetum imminuit; ex quo <lb></lb>oritur minori vi impelli velum. </s> <s id="s.003609">At partes venti ſublimiores ab <lb></lb>his inferioribus reflexis, vel nihil, vel mitiùs turbantur, atque <lb></lb>adeò plures ad implendum velum majore vi accurrunt. </s> </p> <p type="main"> <s id="s.003610">Adde (his etiam mente ſecluſis) ventum in ſublimiore loco <lb></lb>multo validiorem eſſe, quàm in inferiore, ac propterea quò al<lb></lb>tius attollitur velum, non ſolùm majorem, ſed etiam validio<lb></lb>rem ventum excipit, quo fit, ut incitatior ſit navigij motus. </s> <lb></lb> <s id="s.003611">Neque de hoc venti diſcrimine dubitare poterit cui contin<lb></lb>gat iter habere in ampla planitie arboribus & ædificiis va<lb></lb>cuâ vento flante; ſi enim ex equo deſiliat, & humi ſedeat, <lb></lb>manifeſtè percipiet, quanto minore vi impetatur à vento. </s> <s id="s.003612">Id <lb></lb>quod pariter ex ipsâ veli figurâ arguitur; ſive enim velum trian<lb></lb>gulare fuerit, & obliquâ antennâ erigatur ita ut quaſi aurem <lb></lb>leporis imitetur, altiori vento, utpote vehementiori, pars veli <lb></lb>ſtrictior objicitur; ſive pluribus quadrangularibus velis inſtrua<lb></lb>tur navigium ita, ut alia ſuperiora, ſcilicet dolones, alia infe<lb></lb>riora ſint, videlicet Acatia; quæ ſupra Corbem ſtatuuntur, <lb></lb>non ſolum minora ſunt inferiore velo, ſed etiam eorum ſupre<lb></lb>ma pars longè ſtrictior eſt baſi, ut nimirum minus recipiat <lb></lb>venti validioris: propterea ingruente tempeſtate primùm ſu<lb></lb>periora vela deprimuntur, at majori ventorum vi ſubducan<lb></lb>tur; eriguntur autem celeritatis cauſa, ut ſi quando effusè fu<lb></lb>gere opus ſit. </s> </p> <p type="main"> <s id="s.003613">Ecce igitur citra omnem vectis rationem, <emph type="italics"></emph>Cur quando anten<lb></lb>na ſublimior fuerit, iiſdem velis, & vento codem, celeriùs feruntur <lb></lb>navigia:<emph.end type="italics"></emph.end> quia ſcilicet velum altiùs ſublatum & plus venti, & <lb></lb>validiorem ventum recipit. </s> <s id="s.003614">Quod ſi ad vectis rationes confu<lb></lb>giendum eſſet, non quæreretur, cur celeriùs ferantur navigia, <lb></lb>ſed, cur faciliùs? </s> <s id="s.003615">Nam vectis longitudo (niſi fortè in vecte ter<lb></lb>tij generis, cujus nullum veſtigium deprehenditur in malo na<lb></lb>vis) non celeritatem motûs ponderi conciliat, ſed facilitatem, <pb pagenum="475" xlink:href="017/01/491.jpg"></pb>ita ut poſito longiore vecte potentia ſervans eandem ſui motûs <lb></lb>velocitatem faciliûs quidem moveat propoſitum pondus ſed <lb></lb>tardiùs quàm breviore vecte, poſità eádem ponderis ab hypo<lb></lb>mochlio diſtantiâ: Ac propterea, ſi in hoc navis motu, de quo <lb></lb>quæſtio eſt, intercederet ratio vectis, idem ventus eadem vela <lb></lb>altiùs ſublata implens eâdem quidem velocitate moveretur, ſed <lb></lb>tardiùs navim moveret, quamquam faciliùs, hoc eſt magis <lb></lb>onuſtam. </s> <s id="s.003616">Id autem à vero longiſſimè abeſſe teſtatur experien<lb></lb>tia; quæ idcirco confirmat navigij malo nihil eſſe cum vecte <lb></lb>commercij ad navim promovendam. <lb></lb></s> </p> <p type="main"> <s id="s.003617"><emph type="center"></emph>CAPUT XVII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003618"><emph type="center"></emph><emph type="italics"></emph>An ex vectis rationibus pendeat uſus anchoræ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003619">QUandoquidem nauticas aliquot quæſtiones cum Ariſtote<lb></lb>le ſuperioribus capitibus examinare placuit, liceat & hanc <lb></lb>addere, quæ ad uſum anchoræ ſpectat in firmanda navi, ne à <lb></lb>fluctibus, aut à vento abripiatur: tranquillo enim mari, aut in <lb></lb>lacu quieſcente, ſua ſponte ſubſiſtit navis, nec anchoræ ope in<lb></lb>diget, ut ſua in ſtatione permaneat. </s> <s id="s.003620">Et quidem ipſa navis gra<lb></lb>vitas cum ſuis inſtrumentis, & onus quod illa ferre poteſt (cu<lb></lb>jus gravitas æquat navigij gravitatem) ſatis per ſe reſiſtunt, <lb></lb>nec facilè cuilibet auræ aut fluctui cedunt. </s> <s id="s.003621">Quare major eſſe <lb></lb>debet vis venti, aut fluctuum, aut profluentis, quàm ut illi ob<lb></lb>ſiſtere valeat univerſum navigij pondus, ad hoc ut ſit opus an<lb></lb>chorâ, qua navigium firmetur. </s> </p> <p type="main"> <s id="s.003622">In anchorâ autem ſpectanda eſt & gravitas, & figura; utra<lb></lb>que enim juvat: aliquando ſi quidem ſolum anchoræ pondus <lb></lb>ſufficit, ne placidiores fluctus, aut fluminis impetus, aut lenis <lb></lb>flatus navim ſecum rapiant. </s> <s id="s.003623">Sic legiſſe me memini navim à <lb></lb>naufragio anchoris omnibus deſtitutam in ſtatione totam <lb></lb>noctem quieviſſe ſecuram firmatam ſacculo, quo mille trecen<lb></lb>ti Hiſpani Crucigeri (octo Reales ſingulis Crucigeris tribuun<lb></lb>tur) continebantur, rudentis autem munus ſupplebat evolutus <pb pagenum="476" xlink:href="017/01/492.jpg"></pb>telæ ſcapus: qui enim fluctus navim aliò propellere potuiſſent, <lb></lb>non ſatis habebant virium, ut etiam illud argenti pondus ma<lb></lb>ris fundo incumbens & navi connexum pariter trahere poſſent. </s> <lb></lb> <s id="s.003624">Simili igitur ratione anchora, licèt duriori ſolo dentem infige<lb></lb>re non valeat, aliquando ſuo pondere navim firmabit. </s> <s id="s.003625">Reſpon<lb></lb>det autem anchoræ gravitas oneri, quod ferre poteſt navis, ea <lb></lb>Ratione, ut pro oneris libris 40000 (hoc eſt 20 Amphoris aut <lb></lb>doliis, ſingulorum quippe doliorum gravitas ſtatuitur librarum <lb></lb>bis mille, & ſingulis libris unciæ ſexdecim tribuuntur) ferri li<lb></lb>bras centum & decem habeat primaria & maxima anchora, <lb></lb>ſecunda habeat primæ dodrantem, tertia beſſem, quarta ſemiſ<lb></lb>ſem. </s> <s id="s.003626">Rudentis vero, cui anchora adnectitur, pondus ferè <lb></lb>ponitur duplum ſeſquiquartum gravitatis ſuæ anchoræ. </s> <s id="s.003627">Quam<lb></lb>quam non omnino ſervetur hæc Ratio ponderis anchoræ in in<lb></lb>gentibus navigiis, quæ nimirum ſuâ gravitate maximè re<lb></lb>ſiſtunt fluctuum impulſioni, ac proinde minore anchorâ opus <lb></lb>habent. </s> </p> <p type="main"> <s id="s.003628">Primariæ anchoræ potiſſimus uſus communiter eſt, cùm va<lb></lb>lidior tempeſtas navim aggreditur; ſecundæ, ut navis in ſta<lb></lb>tione quieſcat; tertiam adhibent nautæ, ut duabus anchoris ad <lb></lb>diverſa plagas conſtitutis (puta, alterâ ad Subſolanum, alterâ <lb></lb>ad Boream aut ad Borrhapeliotem,) vento & fluctibus navis re<lb></lb>ſiſtat validiùs, nec abrepta à fluctibus anchoram pariter ſecum <lb></lb>rapiat, ſed tantum alternis motibus quaſi circa centrum agite<lb></lb>tur: Quartam demum lintre transferunt procul à navi juxta lon<lb></lb>gitudinem funis adnexi ducentorum circiter cubitorum, quem <lb></lb>machinâ ad id deſtinatâ colligentes accedunt ad anchoram, & <lb></lb>ſtationem commutant, aut portum intrant, ſeu ab illo exeunt, <lb></lb>ubi ceſſat ventus, aut adverſus ſpirat. </s> </p> <p type="main"> <s id="s.003629">Ad firmandam verò navim plurimum habet momenti longi<lb></lb>tudo ipſa rudentis; ſatis enim manifeſtum eſt, quantâ vi opus <lb></lb>ſit, ut longior funis intendatur, qui ceſſante externâ vi illico <lb></lb>ſinuatur; ac propterea vehementi conatu ventorum ac fluctuum <lb></lb>navis impellenda eſt, ut rudens intentus anchoram trahat. </s> <lb></lb> <s id="s.003630">Varia eſt autem Rudentis longitudo pro anchorarum Ratione; <lb></lb>longitudo ſi quidem rudentis anchoræ primariæ cubitos habet <lb></lb>centum viginti, ſecundæ cubitos centum, tertiæ cubitos octo<lb></lb>ginta: quò enim adversùs validiorem impetum repugnandum <pb pagenum="477" xlink:href="017/01/493.jpg"></pb>eſt, eò longior adhibetur rudens, ut difficiliùs intendatur, ac <lb></lb>idcirco fractionis periculo minùs obnoxius ſit, & venti <lb></lb>fluctuumve impetus in rudente intendendo eliſus minus vi<lb></lb>rium habeat ad rapiendam ſimul cum navi anchoram. </s> <s id="s.003631">Hine <lb></lb>ingentes bellicæ naves in Oceano ferè ſemper primariam an<lb></lb>choram demittunt, & tres aut quatuor rudentes capitibus in<lb></lb>vicem firmiter colligatis in unam longitudinem productos adji<lb></lb>ciunt; vix enim tanta eſſe poteſt fluctuum aut venti vis, quæ <lb></lb>valeat tam longum rudentem intendere atque dirumpere, niſi <lb></lb>fortè ad navis latus aut ad ſcopulum colliſus atteratur. </s> <s id="s.003632">Id quod <lb></lb>aliis quoque nautis placet tùm propter eandem cauſam, tùm ut <lb></lb>longiùs à littore conſiſtere poſſit navis, & anchora arenæ infi<lb></lb>gi, etiamſi altior ſit aqua. </s> <s id="s.003633">Mihi ſanè contingit nautarum incu<lb></lb>riam experiri in Albi fluvio; cùm enim anchoram breviore ru<lb></lb>dente demiſiſſent, nocturno maris æſtu intumeſcentibus undis <lb></lb>ita elevatum eſt navigium, ut ex prorâ penderet ſuſpenſa an<lb></lb>chora, nóſque dormientes æſtus abriperet; quos demum exci<lb></lb>tavit fragor ex colliſione cum altero navigio, in quod tanto im<lb></lb>petu impacti fuimus, ut abrupto fune ſcapham amiſerimus. </s> </p> <p type="main"> <s id="s.003634">Sed quod ad anchoræ formam attinet, non eadem omnibus <lb></lb>eſt figura; navigia enim, quorum in majoribus fluminibus uſus <lb></lb>eſt, ut noctu in medio alveo ſubſiſtant, anchoram habent qua<lb></lb>tuor aduncis brachiis inſtructam; cujuſmodi pariter ſunt trire<lb></lb>mium anchoræ. </s> <s id="s.003635">At in Oceano navium anchoræ non niſi duo <lb></lb>habent brachia ad angulum acutum inflexa cum ſcapo; ne ve<lb></lb>tò demiſſa anchora prorſus jaceat in maris fundo, ſcapo prope <lb></lb>annulum adnectitur ligneum tranſverſarium (cujus gravitas <lb></lb>eſt ferè ſubquintupla gravitatis anchoræ, ſi tamen etiam fer<lb></lb>reos clavos, quibus firmatur, in computationem admittas) ejuſ<lb></lb>dem cum Scapo longitudinis, adeò ut jacente utroque brachio <lb></lb>Scapus tranſverſario ſecundum extremitatem innixus obliquè <lb></lb>inclinetur. </s> <s id="s.003636">Ex quo etiam fit, ut extremæ brachiorum palmulæ <lb></lb>obliquè occurrentes maris fundo faciliùs in ſubjectum ſolum <lb></lb>penetrent. </s> <s id="s.003637">Quando igitur vehementior eſt fluctuum impetus, <lb></lb>aut venti impulſus validior, quàm ut illi reſiſtere poſſit ipſa an<lb></lb>choræ gravitas, intento rudente tantiſper abripit cum navi an<lb></lb>choram, quæ maris fundum ſulcans, ubi brachiorum palmu<lb></lb>læ arenis aliquantulum immerſæ inæqualem invenerint ſubjecti <pb pagenum="478" xlink:href="017/01/494.jpg"></pb>ſoli reſiſtentiam (quocumque tandem ex capite oriatur hæc re<lb></lb>ſiſtentiæ inæqualitas) poſitionem mutat, nec ampliùs jacet <lb></lb>utrumque brachium, ſed illud, cui minùs obſiſtitur, elevatur, <lb></lb>adnitente etiam ligneo tranſverſario, cui naturalis eſt in aquâ <lb></lb>poſitio horizonti parallela, quam acquirens ita anchoram con<lb></lb>vertit, ut Dens maris fundo inhærens magis in illud infigatur <lb></lb>tùm urgente deorſum ipſius anchoræ gravitate, tùm trahente <lb></lb>ipſa navi, quam fluctus aut ventus impellit; cùm etenim bra<lb></lb>chium cum ſcapo acutum angulum conſtituat, non ad perpen<lb></lb>diculum, ſed obliquè fundum ingreditur, & idcirco in illud <lb></lb>profundiùs penetrat. </s> </p> <p type="main"> <s id="s.003638">Cum itaque anchoræ ſcapo alij duplicem, alij triplicem tri<lb></lb>buant alterius brachij longitudinem, hæc utique major eſt, <lb></lb>quàm diſtantia inter extremos anchoræ dentes, non enim bra<lb></lb>chia cum ſcapo rectum ſed acutum angulum, ut dictum eſt, <lb></lb>conſtituunt. </s> <s id="s.003639">Ad hanc igitur extremorum dentium diſtantiam <lb></lb>major tranſverſarij longitudo majorem habet Rationem, quàm <lb></lb>minor; eſt autem longitudini ſcapi par tranſverſarij longitudo; <lb></lb>quare longioris anchoræ tranſverſarium longius eſt, ejúſque <lb></lb>converſio, ut ſe horizonti parallelum ſtatuat, magis juvat an<lb></lb>choræ converſionem, ut dens inferior profundiùs in arenam <lb></lb>infigatur. </s> </p> <p type="main"> <s id="s.003640">Sit primùm anchoræ ſcapus AB duplex longitudinis bra<lb></lb><figure id="id.017.01.494.1.jpg" xlink:href="017/01/494/1.jpg"></figure><lb></lb>chij AC, & prope an<lb></lb>nulum in B æquale tranſ<lb></lb>verſarium EF adjiciatur <lb></lb>ad angulos rectos, ea ta<lb></lb>men conditione, ut ja<lb></lb>centibus brachiis AC <lb></lb>& AD in plano hori<lb></lb>zontali, tranſverſarium <lb></lb>ſit in plano verticali, ejúſ<lb></lb>que altera extremitas, ex. </s> <lb></lb> <s id="s.003641">gr.F.maris fundum con<lb></lb>tingat, altera E ſublimis <lb></lb>ſit, ac proinde ſcapus AB <lb></lb>inclinetur ad horizon<lb></lb>tem grad. 30. Vento, aut fluctu, navim impellente intenditur <pb pagenum="479" xlink:href="017/01/495.jpg"></pb>rudens, & ſcapi extremitas B annulo proxima elevatur, nec <lb></lb>ampliùs tranſverſarium in F incumbit arenæ; propterea bra<lb></lb>chiorum palmulæ C & D in triangulum conformatæ, dum ſi<lb></lb>mul cum navi trahuntur, ſe ſe profundiùs in arenam inſinuant: <lb></lb>ſed ſi inæqualem offendant reſiſtentiam, aut altera, ex.gr. C, <lb></lb>profundiùs infigatur præ reliquâ (ſive ex ſubjecti ſoli diverſi<lb></lb>tate, ſive quia navis in tranſverſum acta trahit anchoræ caput B <lb></lb>in latus, & brachij alterius extremitas deſcribens circa A in <lb></lb>ſolo arcum versùs navim profundiùs infigitur, atque adeò re<lb></lb>liqua extremitas oppoſiti brachij in contrarium mota circa A, <lb></lb>vix terram mordet) vis in B trahens, neque valens pariter <lb></lb>utrumque brachium trahere, cogitur circa C, tanquam circa <lb></lb>centrum, ſeu potiùs tanquam circa polum, moveri. </s> <s id="s.003642">Et quia <lb></lb>punctum B ſublimius eſt puncto C, neceſſe eſt ita hujuſmodi <lb></lb>converſionem fieri, ut oppoſita extremitas D elevetur, atque <lb></lb>ex fundo extrahatur. </s> <s id="s.003643">Cúmque jam tranſverſarium non æqua<lb></lb>liter hinc & hinc retineatur per vim in plano verticali, ſed ejus <lb></lb>ſuperior pars BE versùs C inclinetur, conatur poſitionem ho<lb></lb>rizontalem acquirere, ejúſque inferior pars BF ad latus decli<lb></lb>nans aſcendit, juvátque ipſius brachij AD aſcenſum; ex quo <lb></lb>fit demum centrum gravitatis totius anchoræ imminere palmu<lb></lb>læ C, quæ propterea etiam urgente gravitate profundiùs in<lb></lb>figitur. </s> </p> <p type="main"> <s id="s.003644">In hac lignei tranſverſarij converſione obſervandum eſt par<lb></lb>tem alteram ſublimiorem BE per vim in aquâ deprimi, partem <lb></lb>autem inferiorem BF in aquâ ſponte aſcendere, ac proinde, <lb></lb>propter intermediam gravitatem in B, illam reſiſtere huic ſur<lb></lb>ſum conánti, atque ideò illam habere rationem hypomochlij, <lb></lb>hanc potentiæ, pondus verò eſſe in B, quod & convertitur: <lb></lb>non quidem quia totum pondus ſit in B, ſed quia totius ancho<lb></lb>ræ centrum gravitatis eſt in ſcapo AB, adeóque intelligitur <lb></lb>applicatum puncto B, quamvis ipſius centri gravitatis conver<lb></lb>ſio fiat circa extremitatem C manentem. </s> </p> <p type="main"> <s id="s.003645">At verò ſi ſcapus AK fuerit triplex brachij AC, etiam tranſ<lb></lb>verſarium HI ſcapo æquale eſt ejuſdem brachij triplex: hinc <lb></lb>fit ipſius longioris tranſverſarij HI vim, qua ſe horizontale <lb></lb>ſtatuat in aquâ, majorem eſſe, quàm brevioris EF; lignum <lb></lb>enim longiùs difficiliùs in aquâ erectum retinetur. </s> <s id="s.003646">Quamvis <pb pagenum="480" xlink:href="017/01/496.jpg"></pb>autem eadem ſit Ratio FE ad BE, quæ eſt IH ad KH, ta<lb></lb>men major eſt Ratio motûs ipſius K ad motum centri gravitatis <lb></lb>circa extremitatem C manentem, quàm ſit Ratio motûs ipſius B <lb></lb>ad motum centri gravitatis circa idem punctum C: in illa enim <lb></lb>converſione centrum gravitatis exiſtens in aliquo puncto lon<lb></lb>gitudinis AK elevari vix poteſt ad majorem altitudinem, quàm <lb></lb>ſit CL; quia in longiore anchorâ AK centrum gravitatis ma<lb></lb>gis recedens ab extremitate A, quàm in anchorâ breviore AB, <lb></lb>magis imminet palmulæ C, eámque profundiùs in arenam in<lb></lb>figit; ideóque ſi fortè ſit inter L & K, atque ex inclinatione <lb></lb>ſcapi ad horizontem paulò altius exiſteret quàm CL, ſi C ma<lb></lb>neret in ſuperficie fundi maris, ipſa depreſſio puncti C infra il<lb></lb>lam ſuperficiem demit aliquid ex altitudine. </s> </p> <p type="main"> <s id="s.003647">Nam quod ſpectat ad centrum gravitatis anchoræ longioris, <lb></lb>certum eſt illud non removeri ab extremitate Scapi A ſecun<lb></lb>dùm eandem Rationem, ſecundùm quam ejus longitudo pro<lb></lb>ducitur: ſi enim ſcapus eſſet longitudo pari & æquabili craſſitie <lb></lb>ducta, utique ſicut AK eſt ipſius AB ſeſquialtera, etiam cen<lb></lb>tri gravitatis diſtantia ab A in ſcapo longiore eſſet ſeſquialtera <lb></lb>diſtantiæ centri gravitatis ab A in Scapo breviore. </s> <s id="s.003648">Quoniam <lb></lb>verò & pars BK aliquanto decremento deficit à craſſitie reli<lb></lb>quæ partis BA, & pro centro gravitatis totius anchoræ atten<lb></lb>denda eſt non ſolius ſcapi gravitas, ſed & brachiorum, mani<lb></lb>feſtum eſt centrum gravitatis anchoræ longioris removeri ab A <lb></lb>minùs, quàm in Ratione ſeſquialtera. </s> <s id="s.003649">Atqui circa punctum A <lb></lb>(quando jacent brachia, & elevari incipit extremitas altera ſca<lb></lb>pi) moventur B & K pro Ratione diſtantiarum, hoc eſt in Ra<lb></lb>tione ſeſquialtera; igitur motus ipſius K ad motum ſui centri <lb></lb>gravitatis eſt in majore Ratione, quàm motus puncti B ad mo<lb></lb>tum ſui centri gravitatis. </s> </p> <p type="main"> <s id="s.003650">Hinc eſt intento rudente faciliùs pro rata portione elevari <lb></lb>extremitatem K longioris ſcapi, quam B brevioris, & centrum <lb></lb>gravitatis inter A & K, hoc eſt inter hypomochlium & poten<lb></lb>tiam, habere rationem ponderis, quod elevatur vecte ſecundi <lb></lb>generis AK. </s> <s id="s.003651">Quia autem facta elevatione puncti K jacentibus <lb></lb>adhuc brachiis, poſtea fieri debet converſio circa palmulam C <lb></lb>manentem, tunc punctum C habet rationem hypomochlij, & <lb></lb>pondus intelligitur eſſe centrum gravitatis interjectum inter K <pb pagenum="481" xlink:href="017/01/497.jpg"></pb>& C, ſi minus ſit intervallum inter K & centrum gravitatis, <lb></lb>quàm inter K & hypomochlium C, cujuſmodi eſſet, ſi cen<lb></lb>trum gravitatis eſſet citra L versùs K, & eſſet vectis curvus ſe<lb></lb>cundi generis. </s> <s id="s.003652">Quòd ſi magis diſtat centrum gravitatis à <lb></lb>puncto K, quàm ab eodem puncto K diſtet punctum C, vectis <lb></lb>eſt curvus primi generis. </s> <s id="s.003653">Quid autem, inquis, ſi pari interval<lb></lb>lo diſter punctum K à puncto C, atque à centro gravitatis? </s> <s id="s.003654">cu<lb></lb>juſmodi generis vectis erit? </s> <s id="s.003655">primi-ne? </s> <s id="s.003656">an ſecundi? </s> </p> <p type="main"> <s id="s.003657">Reſpondeo in vecte hoc curvo, cujus altera extremitas ma<lb></lb>net, & pondus non ad perpendiculum, neque motu recto in <lb></lb>plano verticali, ſed converſione elevatur, attendenda eſſe pla<lb></lb>na, in quibus tùm potentia, tùm pondus propriam converſio<lb></lb>nem perficiunt; his autem planis parallelum concipe aliud pla<lb></lb>num, quod per extremitatem C manentem tranſeat, quod pla<lb></lb>num ſi interjectum fuerit inter illa plana converſionum, vectis <lb></lb>erit primi generis, quia hypomochlium eſt inter potentiam & <lb></lb>pondus; ſin autem hoc extremum fuerit, & medium locum ob<lb></lb>tineat planum, in quo convertitur centrum gravitatis, vectis <lb></lb>erit ſecundi generis. </s> </p> <p type="main"> <s id="s.003658">Facta demùm converſione ita, ut tranſverſarium ligneum <lb></lb>poſitionem habeat horizontalem, & utrumque brachium in <lb></lb>eodem ſit plano verticali; quia faciliùs elevatur K quàm B, & <lb></lb>tranſverſarium HI longius majorem habet vim ſuſtinendi, <lb></lb>quàm tranſverſarium EF brevius, hinc eſt brachium AC ma<lb></lb>gis inclinari ad ſubjectum maris planum horizontale, ac prop<lb></lb>terea etiam validiùs in arenam infigi, quando à navi trahitur <lb></lb>anchora. <lb></lb></s> </p> <p type="main"> <s id="s.003659"><emph type="center"></emph>CAPUT XVIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003660"><emph type="center"></emph><emph type="italics"></emph>Plures Vectis uſus exponuntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003661">QUod ſuperiore libro præſtitimus libræ atque ſtateræ uſum <lb></lb>extendentes, & hîc præſtare operæ pretium fuerit, tum <lb></lb>ut vectis natura ex uberiori utilitate innoteſcat, tum ut fax ali-<pb pagenum="482" xlink:href="017/01/498.jpg"></pb>qua tyronibus præferatur viam commonſtrando, qua ſimiles <lb></lb>uſus poſſint pro opportunitate excogitare. </s> </p> <p type="main"> <s id="s.003662"><emph type="center"></emph>PROPOSITIO I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003663"><emph type="center"></emph><emph type="italics"></emph>Duplex Vectis genus in uno vecte conjungere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003664">SÆpiſſime contingit unico quidem vecte nos uti, re tamen <lb></lb>vera duplicem eſſe vectem; quemadmodum cùm ingentis <lb></lb>alicujus ſaxi extremitati vectem ſubjicimus, quo extremitatem <lb></lb>illam attollimus. </s> <s id="s.003665">Sit enim ſaxum, cujus centrum gravitatis S, <lb></lb><figure id="id.017.01.498.1.jpg" xlink:href="017/01/498/1.jpg"></figure><lb></lb>& ſubjecto vecte AB ha<lb></lb>bente hypomochlium in <lb></lb>C attollatur extremitas F, <lb></lb>manente extremitate E: <lb></lb>utique vectis primi gene<lb></lb>ris eſt AB; ſed ſi rem at<lb></lb>tentiùs perpendamus, <expan abbr="etiã">etiam</expan> <lb></lb>longitudo FE, aut potiùs <lb></lb>BE vectis eſt ſecundi ge<lb></lb>neris habens impoſitum pondus S, & fulcrum in E; at<lb></lb>que quò magis ſupra horizontem elevatur, linea Directio<lb></lb>nis SD magis accedit versùs E, ex quo oritur movendi fa<lb></lb>cilitas; quam juvat Potentiæ A depreſſio, ex qua fit ut B ma<lb></lb>gis accedens ad F, magis etiam recedat ab hypomochlio E. </s> <lb></lb> <s id="s.003666">Manifeſtum eſt autem pondere accedente ad hypomochlium, <lb></lb>& potentiâ ab eodem recedente, majorem fieri Rationem mo<lb></lb>tûs Potentiæ ad motum Ponderis, atque adeò augeri movendi <lb></lb>facilitatem. </s> <s id="s.003667">Quare momenta potentiæ in A ſuſtinentis ſaxum <lb></lb>ea ſunt, quæ componuntur ex Ratione AC ad CB, & Ratio<lb></lb>ne BE ad EI. </s> <s id="s.003668">Sed de hoc nullus mihi hîc ſermo; quia vel duo <lb></lb>vectes ſunt, ut explicatum eſt, alter quidem ab ipſo pon<lb></lb>dere non ſejunctus FE, alter verò ab eo diſtinctus AB; vel <lb></lb>ſi unicus intelligatur vectis, qui ponderi applicatur, hic <lb></lb>ſanè ad unum pertinet genus non ad duo, ut hæc propo<lb></lb>ſitio exigit. </s> </p> <p type="main"> <s id="s.003669">Sit igitur dati vectis longitudo CD, in cujus medio hypo<lb></lb>mochlium O bifariam æqualiter dividat totam longitudinem, <pb pagenum="483" xlink:href="017/01/499.jpg"></pb>& ſit pondus in P. Erit, peri 7. lib. 5. eadem Ratio CO ad <lb></lb>OP, atque DO ad OP; & <lb></lb><figure id="id.017.01.499.1.jpg" xlink:href="017/01/499/1.jpg"></figure><lb></lb>ſi in C ſit potentia depri<lb></lb>mens, in D autem potentia <lb></lb>elevans, æqualia habent mo<lb></lb>menta ad elevandum pondus <lb></lb>in P. </s> <s id="s.003670">Eſt ergo CP vectis primi generis, & DO vectis ſe<lb></lb>cundi generis, cui cum primo commune eſt hypomochlium <lb></lb>O, & communis pars OP. </s> <s id="s.003671">Quòd ſi Potentiæ inæquales fue<lb></lb>rint, utraque autem valeat ſive deprimere, ſive elevare, di<lb></lb>vidatur longitudo CD in duas partes, quarum Ratio eadem <lb></lb>ſit ac Potentiarum, & in puncto diviſionis ſtatuatur fulcrum: <lb></lb>tum in extremitatibus reciprocè collocentur Potentiæ, vali<lb></lb>dior ſcilicet propior ſit fulcro, debilior verò remotior, ut æqua<lb></lb>lium ſint momentorum. </s> </p> <p type="main"> <s id="s.003672">Datæ Potentiæ ſint ut 5 ad 3. Dividatur CD partium <lb></lb>octo ita in P (ubi ſtatuendum eſt fulcrum) ut CP ſit 5, <lb></lb>PD ſit 3; & Potentia robuſtior, quæ eſt ut 5 ſit in D; <lb></lb>infirmior verò, quæ eſt ut 3, ſit in C; & pondus ſit in R, <lb></lb>quoniam CP ad PR eſt ut 5 ad 1, & DP ad PR eſt ut <lb></lb>3 ad 1. Igitur ſi pondus R ſit lib. 30, attolletur à Poten<lb></lb>tia C potente ſine vecte attollere lib. 3, & à Potentia D <lb></lb>potente ſine vecte elevare lib. 5: utriuſque enim momenta <lb></lb>ſingillatim accepta ſunt 15 compoſita ex virtute movendi & <lb></lb>motûs velocitate. </s> <s id="s.003673">At ſi pondus P ſit lib. 30, & fulcrum in <lb></lb>O, ſit autem CO ad OP, atque DO ad OP ut 4 ad 1, <lb></lb>ſatis eſt ſi ſingulæ Potentiæ æquales C & D poſſint ſine vecte <lb></lb>attollere lib. 3. unc. </s> <s id="s.003674">9. </s> </p> <p type="main"> <s id="s.003675">Porrò ſi inæqualium potentiarum altera poſſit ſolùm depri<lb></lb>mendo vectem elevare pondus, manifeſtum eſt ad illam per<lb></lb>tinere vectem primi. </s> <s id="s.003676">generis: ac propterea ſi illa ſit potentia <lb></lb>validior, eidem tribuetur minor diſtantia ab hypomochlio; ſin <lb></lb>autem illa ſit imbecillior, ipſi tribuetur diſtantia major, atque <lb></lb>illam inter ac pondus ſtatuetur fulcrum. </s> <s id="s.003677">Hinc facilè pote<lb></lb>rit potentia vivens uti ope potentiæ inanimatæ, quæ vi ſuæ <lb></lb>gravitatis deorſum premat oppoſitam extremitatem propoſiti <lb></lb>vectis. </s> </p> <p type="main"> <s id="s.003678">Huc ſpectare videtur facillimum genus antliæ ſimplicis, <pb pagenum="484" xlink:href="017/01/500.jpg"></pb>qua ex depreſſiore loco in altiorem aquas attollimus. </s> <s id="s.003679">Sit enim <lb></lb>modiolus B, cui aptè inſeratur congruens embolus medio <lb></lb><figure id="id.017.01.500.1.jpg" xlink:href="017/01/500/1.jpg"></figure><lb></lb>haſtili CD connexus cum tranſverſario EF verſatili circa <lb></lb>axem infixum in I: cujus tranſverſarij extremitatem E occu<lb></lb>pet maſſa plumbea opportunæ gravitatis ad deprimendum em<lb></lb>bolum intra modiolum, poſtquam elevatus fuerit à potentia fu<lb></lb>nem FG trahente adnexum in altera extremitate F. </s> <s id="s.003680">Vectis FE <lb></lb>eſt primi generis duplex habens pondus, alterum in E, alterum <lb></lb>in D, utrumque enim per vim elevatur. </s> <s id="s.003681">At vectis EI eſt ſe<lb></lb>cundi generis, in quo E eſt potentia deprimens embolum, & <lb></lb>quo magis diſtabit ab hypomochlio I, minor maſſa plumbea <lb></lb>eadem obtinebit momenta. </s> <s id="s.003682">Quòd ſi IF conſtet materiâ ſatis <lb></lb>gravi, jam habet rationem ponderis, ac propterea diſtantia <lb></lb>centri gravitatis illius H à puncto I determinabit ejus momen<lb></lb>ta. </s> <s id="s.003683">Quare potentia E vecte EI ſecundi generis deprimet em<lb></lb>bolum, & vecte EH primi generis attollet pondus brachij IF. </s> <lb></lb> <s id="s.003684">Hinc eſt commodius accidere, ſi longitudo EK ferrea ſit, in K <pb pagenum="485" xlink:href="017/01/501.jpg"></pb>verò inſeratur, ut firmiter cohæreat, ſatis validus baculus <lb></lb>ligneus KF; poterit enim longior eſſe, & faciliorem efficere <lb></lb>antliæ agitationem, quin gravitas nimia indigeat multo plum<lb></lb>bo in E, ut præponderetur. </s> </p> <p type="main"> <s id="s.003685">Quòd ſi non placeret addere plumbum in E, & ſolo vecte <lb></lb>primi generis FD uti velles, recurrendum eſſet ad vim elaſti<lb></lb>cam, qua vel arcûs X in ſuperiore loco firmati nervo, vel <lb></lb>extremæ perticæ AM longiuſculæ (ut Toreuticen exercen<lb></lb>tibus ſolemne eſt) adnecteretur funis pertingens ad F, ut <lb></lb>ex tractione Potentiæ GF curvatus arcus, vel inflexa per<lb></lb>tica, ceſſante potentiâ, iterum ſe ſuum in ſtatum reſtitueret, <lb></lb>ſursúmque traheret extremitatem F, ac proinde embolum <lb></lb>intra modiolum B deprimeret. </s> <s id="s.003686">Tunc enim eſſet FD vectis <lb></lb>primi generis, cujus extremitati F applicarentur duæ Poten<lb></lb>tiæ, altera deorſum, altera viciſſim alterno conatu ſurſum <lb></lb>trahens. </s> </p> <p type="main"> <s id="s.003687">At ſi fortè duplicem antliam velis ſimul componere, dupli<lb></lb>cémque potentiam viventem alternis operis conantem adhi<lb></lb>bere, jugo RS verſatili cir<lb></lb><figure id="id.017.01.501.1.jpg" xlink:href="017/01/501/1.jpg"></figure><lb></lb>ca axem X adde duo, le<lb></lb>viora quidem, ſed ſatis fir<lb></lb>ma, manubria RO & SM, <lb></lb>quorum extremitates aut <lb></lb>premi, aut adjecto fune <lb></lb>trahi deorſum valeant: <lb></lb>nam depreſsâ extremitate <lb></lb>O deprimitur pariter haſti<lb></lb>le infixum in R, & eſt OX <lb></lb>vectis ſecundi generis, at<lb></lb>que attollitur haſtile ad<lb></lb>nexum in S, & eſt OS <lb></lb>vectis primi generis. </s> <s id="s.003688">Simi<lb></lb>liter MX vectis eſt ſecundi generis, movens pondus poſitum <lb></lb>in S, atque MR eſt vectis primi generis attollens pondus po<lb></lb>ſitum in R. </s> <s id="s.003689">Propterea autem leviora dixi adjecta manubria <lb></lb>RO & SM, ne ſuâ gravitate movendi difficultatem augeant. </s> <lb></lb> <s id="s.003690">Verùm ſi ſolus volueris antliam utramque agitare, unus ſit <lb></lb>continuus funis ex O per rotulas P & Q tranſiens, atque in M <pb pagenum="486" xlink:href="017/01/502.jpg"></pb>connexus: quacumque enim in parte conſtitutus fueris, tan<lb></lb>tumdem funis ſequitur aſcendentem extremitatem, quantum <lb></lb>trahitur deprimendo alteram extremitatem: ſic trahendo fu<lb></lb>nem PO deprimitur extremitas O, deinde trahendo funem <lb></lb>PQ deprimitur extremitas M. </s> <s id="s.003691">Aut etiam ſit unicum ma<lb></lb>nubrium SM, & erit RM: atque premens in M attollet <lb></lb>haſtulam R, elevans aut in M attollet haſtulam S. </s> <s id="s.003692">Ut au<lb></lb>tem faciliùs attollatur M, ſit in ſuperiore loco orbiculus, <lb></lb>per quem tranſeat funis connexus in M, alteram ením ex<lb></lb>tremitatem deorſum trahens attollet manubrium M. </s> <s id="s.003693">Quod <lb></lb>ſi ab oculis remotam volveris antliam, fac per parietis fora<lb></lb>men in proximum conclave exire funem MI orbiculi H <lb></lb>excavatæ abſidi inſertum, & per orbiculos P, Q, tranſire <lb></lb>funem OP QL; connexis enim funium extremitatibus I <lb></lb>& L modò hunc modò illum funem trahendo utramque <lb></lb>antliam agitabis. </s> </p> <p type="main"> <s id="s.003694"><emph type="center"></emph>PROPOSITIO II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003695"><emph type="center"></emph><emph type="italics"></emph>Antliam opportuno vecte inſtruere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003696">UT aliquam ſpeciem vectis curvi oneri movendo deſtina<lb></lb>ti exhibeam, placet in antlia, qua ad hauriendas aquas <lb></lb>utimur, exemplum ponere, quod facilè in reliquis pro re <lb></lb>nata imitari poſſimus. </s> <s id="s.003697">Eſt in antliâ loco ponderis aqua, quæ <lb></lb>adducto embolo attrahitur in modiolum, eóque reducto ex<lb></lb>primitur, & prætereà conflictus ipſe emboli cum modiolo; <lb></lb>ſuperanda quippe eſt difficultas, quæ ex mutuo horum con<lb></lb>tactu oritur, & aqua per vim elevanda eſt, ſive ſolùm at<lb></lb>trahatur, ut ex modiolo per emboli reducti foramen ſubin<lb></lb>de erumpens effluat, ſive in modiolo compreſſa ab embolo, <lb></lb>cùm reducitur, exprimatur in tubum, ut adhuc altiùs aſcen<lb></lb>dat, juxta ea, quæ in Hydrotechnicis fuſiùs dicuntur. </s> <s id="s.003698">Id <lb></lb>quidem fieret ſi haſtili, quod embolo infigitur, ipſa poten<lb></lb>tia proximè applicaretur; ſed ut minus laboris illa ſubeat, <lb></lb>additur vectis, ut multo major ſit potentiæ motus, quàm <lb></lb>emboli. </s> </p> <pb pagenum="487" xlink:href="017/01/503.jpg"></pb> <p type="main"> <s id="s.003699">Sit enim embolus A congruens modiolo B, illíque in<lb></lb>fixum haſtile CA; quo elevato aqua <lb></lb><figure id="id.017.01.503.1.jpg" xlink:href="017/01/503/1.jpg"></figure><lb></lb>per ſubjectum modiolo tubum F at<lb></lb>trahitur in modiolum ipſum B, quo <lb></lb>depreſſo aqua cogitur ex eodem mo<lb></lb>diolo exire. </s> <s id="s.003700">Sed ut minore operâ id <lb></lb>totum perficiatur, additur in C vectis <lb></lb>curvus CDE verſatilis circa axem D <lb></lb>infixum parieti interjecto inter an<lb></lb>tliam & potentiam moventem. </s> <s id="s.003701">Nam <lb></lb>extremitatem E arripiens potentia ſi <lb></lb>vectem urgeat versùs parietem inter<lb></lb>medium, elevatur embolus, & aqua <lb></lb>modiolum implet, ſi verò vectis ex<lb></lb>tremitatem à pariete removeat, deprimitur embolus, & <lb></lb>compreſſa aqua exprimitur. </s> <s id="s.003702">Hìc vectem primi generis agno<lb></lb>ſcis habentem hypomochlium in D, ſcilicet in axe, circa <lb></lb>quem verſatur vectis; & pro Ratione longitudinis DE ad <lb></lb>longitudinem DC eſt Ratio momentorum potentiæ ad re<lb></lb>ſiſtentiam ponderis, hoc eſt tantò magis augentur potentiæ <lb></lb>vires, quò major eſt Ratio DE ad DC: ſumitur autem DE <lb></lb>recta linea non computato flexu DGE, qui eatenus adſtrui<lb></lb>tur, quatenus parietis craſſities obſtaret, ne commodè utere<lb></lb>mur vecte EDC inflexo in D. </s> </p> <p type="main"> <s id="s.003703">Quia verò faciliùs ab homine urgetur vectis in E, quàm <lb></lb>ipſa extremitas E retrahatur, ideò in antliâ ſolùm attrahen<lb></lb>te utendo hoc vecte primi generis curvo minus eſt laboris, <lb></lb>nam in deprimendo embolo minus eſt difficultatis quàm in <lb></lb>elevando. </s> <s id="s.003704">At ſi aqua altiùs elevanda eſſet ſupra antliam non <lb></lb>attrahentem ſolum, ſed etiam expellentem, faciliùs attol<lb></lb>leretur embolus, quàm deprimeretur, propter majorem aquæ <lb></lb>reſiſtentiam, cùm exprimitur, juxtà altitudinem perpendi<lb></lb>cularem, ad quam expellitur: propterea tunc mutanda eſ<lb></lb>ſet poſitio, ut eſſet vectis ſecundi generis; hypomochlium <lb></lb>enim ſtatuendum eſſet in C, & haſtile emboli adnecten<lb></lb>dum in D. </s> </p> <p type="main"> <s id="s.003705">Quod ſi potentia viribus abundet, poterit duplicem an<lb></lb>tliam agitare, cujuſmodi eſſet ſi jugum RS bifariam diviſum <pb pagenum="488" xlink:href="017/01/504.jpg"></pb>in X jungeretur in R & S duplici haſtili, centrum autem <lb></lb><figure id="id.017.01.504.1.jpg" xlink:href="017/01/504/1.jpg"></figure><lb></lb>motûs reſponderet puncto X, cui <lb></lb>firmiter adnecteretur manu<lb></lb>brium XZ, quod agitaretur pa-<lb></lb>rallelum plano, per quod tranſit <lb></lb>axis jungens jugum RS cum <lb></lb>manubrio ipſo: dum enim Z <lb></lb>versùs P movetur, deprimitur <lb></lb>R & attollitur S, atque viciſſim <lb></lb>remeans in Q deprimit embo<lb></lb>lum reſpondentem jugi extremi<lb></lb>tati S, & oppoſitum attollit. </s> <s id="s.003707">Sunt <lb></lb>autem duo vectes curvi ZXR <lb></lb>& ZXS primi generis partem <lb></lb>unam, videlicet manubrium XZ, <lb></lb>habentes communem. </s> </p> <p type="main"> <s id="s.003708">Sed quoniam poſito longiore manubrio ZX, vel DE, faci<lb></lb>liùs quidem attollitur aqua, quàm ſi illud brevius eſſet, major <lb></lb>tamen corporis agitatio requiritur, & multâ membrorum incli<lb></lb>natione laborioſa exercitatio ſuſcipienda eſt, propterea ſatius <lb></lb>eſt uti vecte recto, ut prop. 1. dictum eſt, quem etiam ſedens <lb></lb>modico labore commovere poteris adnexum extremitati fu<lb></lb>nem deorſum trahendo. </s> </p> <p type="main"> <s id="s.003709"><emph type="center"></emph>PROPOSITIO III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003710"><emph type="center"></emph><emph type="italics"></emph>Rotam in profluente poſitam, quæ aquam faciliùs elevet ex <lb></lb>vectis Rationibus, conſtituere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003711">AQuam ex depreſſiore loco in altiorem provehi vaſculis ab<lb></lb>ſidi rotæ circum circa alligatis, quæ in infimâ rotæ parte <lb></lb>ſubjectam aquam immerſa hauriunt, & circumactâ rotâ, ubi <lb></lb>circuli ſemiſſem aſcendendo perfecerint, deſcendendo effun<lb></lb>dunt, quibus per Helvetios iter facere contigit, perſpectum <lb></lb>eſt; ſi in Tigurinâ Urbe, quam lacus Limagum fluvium exci<lb></lb>piens interluit, obſervârunt ab utrâque ripâ ductum ex palis <lb></lb>confertim denſatis obicem obliquum uſque ad medium alveum, <pb pagenum="489" xlink:href="017/01/505.jpg"></pb>ut ex anguſtiis erumpens aqua cæteroqui leniter defluens, ve<lb></lb>lociùs fluere cogatur, & validiùs in prominentes rotæ palmulas <lb></lb>incurrens ingentem illam rotam cum adjunctis vaſculis aquâ <lb></lb>plenis faciliùs circumagat, atque adeò in ſubjectum vas ponti <lb></lb>impoſitum effuſa aqua per urbem univerſam dividatur. </s> <s id="s.003712">Verùm <lb></lb>quia pondus, ſcilicet aqua vaſculis contenta, ſemper à centro <lb></lb>rotæ intervallo eodem abeſt, aliud rotæ genus excogitari po<lb></lb>teſt, quod aquam facilius elevet, nec adnexa, ſed congenita <lb></lb>habeat vaſcula. </s> </p> <p type="main"> <s id="s.003713">In plano ex aſſeribus rite conjunctis compacto, centro A, in<lb></lb>tervallo AB, intelli<lb></lb><figure id="id.017.01.505.1.jpg" xlink:href="017/01/505/1.jpg"></figure><lb></lb>gatur deſcriptus cir<lb></lb>culus, cujus ſemidia<lb></lb>meter aliquanto ma<lb></lb>jor ſit altitudine, ad <lb></lb>quam aqua evehen<lb></lb>da eſt, dividatúrque <lb></lb>deſcripti circuli peri<lb></lb>pheria in quotlibet <lb></lb>æquales partes, ex. </s> <lb></lb> <s id="s.003714">gr. duodecim, aut <lb></lb>plures. </s> <s id="s.003715">Tum aſſump<lb></lb>tâ palmulæ congruâ <lb></lb>altitudine BD, alius <lb></lb>interior circulus eo<lb></lb>dem centro A, in<lb></lb>tervallo AD deſcribatur, qui à ductis per centrum A dia<lb></lb>metris ſimiliter in totidem æquales partes dividitur. </s> <s id="s.003716">Aſſump<lb></lb>tâ itaque CF æquali ipſi DB, ſtatuatur CE intervallum op<lb></lb>portunæ amplitudinis, ut aqua facilè ingredi poſſit. </s> <s id="s.003717">Et ductâ <lb></lb>rectâ lineâ BE, reſecetur particula exterior, ut ſit BE CF: <lb></lb>idémque de cæteris partibus intelligatur, prout adjectum <lb></lb>ſchema refert. </s> </p> <p type="main"> <s id="s.003718">Duo hujuſmodi plana parentur omnino æqualia, ſimilitérque <lb></lb>denticulata, quæ cylindro (ſive priſmati ſimilem baſim haben<lb></lb>ti cum polygono ab initio deſcripto) hoc eſt axi inſerantur in <lb></lb>A, & parallela ſint. </s> <s id="s.003719">Planorum autem intervallum definiant aſ<lb></lb>ſeres æquè lati, qui perpendiculares inſiſtant lineis GBE, & <pb pagenum="490" xlink:href="017/01/506.jpg"></pb>ſimilibus; quorum aſſerum latitudo palmulis quoque DB, CF, <lb></lb>& reliquis latitudinem ſtatuet. </s> <s id="s.003720">Omnibus ritè firmatis, ac ob<lb></lb>ſtructis accuratè rimulis, rota ſuper polos axi infixos collocetur <lb></lb>in profluente, ità ut palmula tota in aquam immergatur, quæ per <lb></lb>apertum oſculum CE ingrediens impleat ſpatium EBD. </s> </p> <p type="main"> <s id="s.003721">Impetu igitur profluentis dum rota convertitur, aqua incluſa <lb></lb>paulatim versùs rotæ centrum ſecedit, donec quadrantem cir<lb></lb>culi aſcendendo tranſgreſſa proxima fiat axi: cùm enim B vene<lb></lb>rit in H, aqua erit in I, cùm verò ex H in S venerit, jam aqua <lb></lb>in ſubjectum vas effluet. </s> <s id="s.003722">Quare, licèt æqualium converſionum <lb></lb>non ſint æquales aſcenſus in eâdem circuli peripheriâ, ſed ab <lb></lb>imo puncto uſque ad finem Quadrantis creſcant, quia tamen <lb></lb>centrum gravitatis aquæ ſe in æquilibrio ſtatuentis ſenſim cen<lb></lb>trum versùs recedit, ejus aſcenſus minor eſt, quàm ſi eodem <lb></lb>ſemper intervallo abeſſet à centro rotæ. </s> </p> <p type="main"> <s id="s.003723">Eſt itaque vectis curvus primi generis, cujus hypomochlium <lb></lb>reſpondet centro A, Potentia movens duplex eſt, ſcilicet vis <lb></lb>profluentis applicata in B, atque vis aquæ deſcendentis exiſtens <lb></lb>in S: pro variâ autem centri gravitatis aquæ elevatæ diſtantiâ <lb></lb>ab hypomochlio A, diverſa etiam eſt motuum Ratio & momen<lb></lb>torum. </s> <s id="s.003724">Aqua enim in ſuperiore ſemicirculo ſupra RS in ſingu<lb></lb>lis loculamentis ſibi invicem hinc atque hinc reſpondentibus <lb></lb>æqualiter diſpoſita obtinet æqualia gravitatis momenta. </s> <s id="s.003725">Qua<lb></lb>propter totus profluentis conatus impenditur in elevandâ aquâ, <lb></lb>quæ loculamentis inter B & R interceptis continetur. </s> <s id="s.003726">Quare <lb></lb>ſi multæ ſint profluentis vires, craſſior rota ſtatui poteſt, ut, <lb></lb>planis magis diſtantibus, major aquæ copia ſingulis loculamen<lb></lb>tis hauriatur: quo fiet, ut palmula latior majorem incurrentis <lb></lb>aquæ impetum recipiat. </s> <s id="s.003727">Quòd ſi placuerit palmulas addere la<lb></lb>tiores, quàm ſit rotæ craſſitudo, non abnuo: hæc enim, & cæ<lb></lb>tera, quæ conſtructionis facilitatem juvent, prudentis machi<lb></lb>natoris arbitrio relinquuntur: mihi ſatis eſt innuiſſe, quid com<lb></lb>pendij ex vectis rationibus peti poſſit. </s> </p> <pb pagenum="491" xlink:href="017/01/507.jpg"></pb> <p type="main"> <s id="s.003728"><emph type="center"></emph>PROPOSITIO IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003729"><emph type="center"></emph><emph type="italics"></emph>A pluribus hominibus ingens pondus transferri poſſe ita, ut <lb></lb>omnes æqualiter ferant.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003730">ONus ingens palangâ transferri pluribus hinc atque hinc <lb></lb>longo ordine ſuccollantibus, notum eſt: ſed quoniam non <lb></lb>omnium æqualis eſt diſtantia à pondere (niſi fortè bini & bini <lb></lb>æquè diſtarent à centro gravitatis) non ſunt æqualia momenta; <lb></lb>ſed qui propiores ſunt magis premuntur, cæteris paribus, quàm <lb></lb>remotiore, maximè ſi quis ſenſim ſe ſubducat oneri adeò, ut <lb></lb>inæqualis fiat oneris diſtantia ab iis, qui illud ſuſtentant. </s> <s id="s.003731">Prop<lb></lb>terea methodus aliqua excogitanda eſt, qua fiat ut ſinguli pa<lb></lb>rem experiantur in deferendo onere difficultatem. </s> </p> <p type="main"> <s id="s.003732">Sit ponderi dato alligatus vectis AB, & gravitatis centro <lb></lb>reſpondeat punctum C, <lb></lb><figure id="id.017.01.507.1.jpg" xlink:href="017/01/507/1.jpg"></figure><lb></lb>atque æqualia ſint in<lb></lb>tervalla AC & BC. </s> <s id="s.003733">Si <lb></lb>centrum gravitatis pon<lb></lb>deris reſpondens puncto <lb></lb>C vectis non fuerit pla<lb></lb>nè in mediâ ejuſdem <lb></lb>ponderis longitudine, <lb></lb>neque fuerit vectis val<lb></lb>dè longior ipſo ponde<lb></lb>re, non poterunt plu<lb></lb>res ita æqualiter diſpo<lb></lb>ni, ut ad ferendum <lb></lb>æqualiter pondus ſin<lb></lb>gulis anterioribus ſin<lb></lb>guli poſteriores reſpon<lb></lb>deant æquè à puncto C <lb></lb>diſtantes, impediente <lb></lb>videlicet ipsâ ponderis <lb></lb>longitudine. </s> </p> <pb pagenum="492" xlink:href="017/01/508.jpg"></pb> <p type="main"> <s id="s.003734">Quare tam in A quàm in B duo alij vectes DE, & FG <lb></lb>bifariam æqualiter diviſi ſuſtineant vectem AB: atque adeò <lb></lb>quemadmodum in A (idem dic de B) ſuſtinetur ſemiſſis <lb></lb>totius gravitatis, in D ſuſtinetur tantùm quadrans, ſicut <lb></lb>& in E. </s> <s id="s.003735">Suſtineatur ſimiliter extremitas D alio vecte HI <lb></lb>(id quod præſtitum intellige pariter in extremitatibus E, F, G) <lb></lb>& in H ſuſtinetur octava pars: item extremitas H ſuſti<lb></lb>neatur vecte KL, & extremitas K vecte MN; percipitur <lb></lb>in K gravitatis pars decima ſexta, & in M pars trigeſima <lb></lb>ſecunda. </s> <s id="s.003736">Poterunt igitur hac ratione diſponi homines 32, <lb></lb>qui ſi poſſint ſinguli deferre lib. 100, transferent pondus <lb></lb>lib. 3200, & erit æqualiter inter illos diſtributa gravitas. </s> <lb></lb> <s id="s.003737">Quòd ſi ſpatium non prohibeat adhuc vectem ſingulis ex<lb></lb>tremitatibus adjungere, numerus hominum deferentium du<lb></lb>plicabitur, & vel ſingulorum labor dimidiatus erit, vel du<lb></lb>plicatum pondus transferre poterunt. </s> <s id="s.003738">Porrò vectem vecti <lb></lb>eſſe firmo vinculo connectendum, ne fortè in motu, vecte <lb></lb>aliquo ſe ſubducente, luxetur machina, non opus eſt mo<lb></lb>nere, cum per ſe res ipſa loquatur. </s> <s id="s.003739">Illud obſerva, quod <lb></lb>vectium inter ſe æqualitatem, ſive longitudo, ſive craſſi<lb></lb>ties ſpectetur, non opus eſt ſtudiosè accurare, dum<lb></lb>modo ſinguli vectes æqualiter bifariam dividantur: im<lb></lb>mò poſtremi, & breviores eſſe poſſunt, ut minus ſpa<lb></lb>tij requiratur, & graciliores, minùs quippe urgentur à <lb></lb>pondere. </s> </p> <p type="main"> <s id="s.003740">In Atlante Sinico hæc lego pag. </s> <s id="s.003741">125. <emph type="italics"></emph>In ferendis oneri<lb></lb>bus ſcitißimi ſunt Sinæ, ac rustici illic non parvum ſanè ſta<lb></lb>ticis noſtris ſpeculatoribus faceſſerent negotium ad cauſas inve<lb></lb>niendas ac rationes, ſi viderent illos tormenta etiam majora, <lb></lb>ac ſimilia pondera, ita vectibus utrinque ſuſpendentes, ut per <lb></lb>arctiſſimas etiam montium fauces facillimè transferant; ac li<lb></lb>cèt præcedant alij, alij ſubſequantur, multíſque paſſibus à pon<lb></lb>dere ſuſpenſo diſtent, ita tamen illud vectibus ac funibus ex æquo <lb></lb>nôrunt dividere, ut quilibet æquale ferè ſentiat onus, ſeu paulò re<lb></lb>motior ſit, ſive vicinior. </s> <s id="s.003742">Hoc pacto ingentia marmora, atque inte<lb></lb>gras etiam arbores facilè videas humeris geſtare Sinas.<emph.end type="italics"></emph.end> Hæc ibi. </s> <lb></lb> <s id="s.003743">Sed quonam id artificio in praxim deducatur, nullum planè <lb></lb>apparet veſtigium. </s> </p> <pb pagenum="493" xlink:href="017/01/509.jpg"></pb> <p type="main"> <s id="s.003744">Si igitur funibus ſuſpenditur pondus, & deferentes alij <lb></lb>propiores ſunt, alij remotiores, duo obſervanda ſunt. </s> <s id="s.003745">Pri<lb></lb>mum eſt, quòd ſuſpenſio non eſt perpendicularis ſed obli<lb></lb>qua, ac proinde plus virium requiritur, ut conſtat ex iis, <lb></lb>quæ dicta ſunt tum lib. 1. cap. 16. de elevationibus obli<lb></lb>quis, tum lib. 3. cap. 12, de præponderatione, & æquili<lb></lb>britate gravium fune ſuſpenſorum. </s> <s id="s.003746">Verùm hoc momento<lb></lb>rum augmentum in elevatione & ſuſpenſione obliquâ, ubi <lb></lb>operis abundamus, non conſideratur; videtur quippe ſatis <lb></lb>leve incommodum, quod facilitate transferendi onus com<lb></lb>penſatur. </s> <s id="s.003747">Secundum eſt, quòd ſi omnes ferè æqualiter la<lb></lb>borant, non diſſimiles eſſe oportet, ſed proximè eaſdem <lb></lb>obliquitates funium, ex quibus onus ſuſpenſum defer<lb></lb>tur: manifeſtum enim eſt in minori obliquitate ſuſpen<lb></lb>ſionis minus virium requiri, quàm in majori obliqui<lb></lb>tate. </s> </p> <p type="main"> <s id="s.003748">Quare ſi hanc Sinarum induſtriam æmulari conarer, pri<lb></lb>mùm oneris transferendi extremitatibus (vel ſaltem in pa<lb></lb>ri diſtantiâ à centro gravitatis, quantùm conjecturâ aſſe<lb></lb>qui poſſem) vectes tranſverſos firmin mè alligarem, ut <lb></lb>vectium horum capitibus jungerem funes, quibus ſuſpen<lb></lb>ſum onus deferatur. </s> <s id="s.003749">Horum autem tranſverſorum vectium <lb></lb>longitudinem ita definirem, ut in lineâ vectibus parallelâ, <lb></lb>& æquali quatuor ſaltem homines commodè collocari <lb></lb>queant, quin ſibi ullum impedimentum progredientes in<lb></lb>ferant. </s> <s id="s.003750">Deinde ſatis validos funes utrique vectium extre<lb></lb>mitati adnexos tantæ longitudinis ſtatuerem, quantâ opus <lb></lb>ſit, ut (tribus hominibus ante onus ſibi ordine recto ſuc<lb></lb>cedentibus ac mediocriter diſtantibus, quin poſterior prio<lb></lb>ris calcem progrediendo feriat) ad tertij humerum pertin<lb></lb>gere poſſit; hæc enim videtur minima obliquitas ſuſpenſio<lb></lb>nis, & quæ proximè accedat ad ſuſpenſionem perpendicu<lb></lb>larem: Si verò major fuerit funium longitudo, majori labo<lb></lb>re deferetur onus, ſi maximè ita elevetur, ut multum diſtet <lb></lb>à ſubjecto ſolo, major enim erit obliquitas ſuſpenſionis. </s> <lb></lb> <s id="s.003751">Tum extremitati funis alius vectis alligetur, qui vecti<lb></lb>bus aliis ſuſtentetur eâ methodo, quam paulo ſuperiùs in<lb></lb>dicavi. </s> </p> <pb pagenum="494" xlink:href="017/01/510.jpg"></pb> <p type="main"> <s id="s.003752">Sit onus transferendum P; extremitati anteriori (omnia <lb></lb><figure id="id.017.01.510.1.jpg" xlink:href="017/01/510/1.jpg"></figure><lb></lb>eadem in alterâ extre<lb></lb>mitate poſita intelli<lb></lb>gantur) adnectatur vec. </s> <lb></lb> <s id="s.003753">tis AB, cui in A & B <lb></lb>jungantur funes AD <lb></lb>& BC ſufficientis lon<lb></lb>gitudinis, quibus in D <lb></lb>& C alligetur vectis <lb></lb>ab aliis vectibus, ut <lb></lb>paulo ſuperiùs indica<lb></lb>tum eſt, ſuſtentatus, <lb></lb>adeò ut quarto vecti <lb></lb>duo homines facilè <lb></lb>humeros ſupponere va<lb></lb>leant, & ſingulorum <lb></lb>funium extremitates D <lb></lb>& C à ſexdecim ho<lb></lb>minibus ſuſtineantur. </s> <lb></lb> <s id="s.003754">Quare ſi totidem fu<lb></lb>nes atque homines po<lb></lb>ſteriori ponderis parti <lb></lb>ſimili ratione applicen<lb></lb>tur, totum pondus ab hominibus 64 æqualiter laborantibus <lb></lb>ſuſtinetur. </s> </p> <p type="main"> <s id="s.003755">Ex quo fit non adeò difficile eſſe in exercitu, ubi non eſt <lb></lb>hominum ſuccollantium inopia, bombardas ex loco in locum <lb></lb>transferre, ſi nimis arduum ſit iter, nec equis trahi poſſint: <lb></lb>Nam majoribus bombardis pro ſingulis globi ferrei libris me<lb></lb>talli libræ 150 aut 160 dimidiatis Cartois, ut vocant, in ſin<lb></lb>gulas globi libras, metalli libræ 180 aut 190, campeſtribus & <lb></lb>minoribus bombardis metalli libræ 238 uſque ad 266 in ſingu<lb></lb>las globi libras communiter tribuuntur. </s> </p> <p type="main"> <s id="s.003756">Quòd ſi eæ ſint viarum anguſtiæ, quæ octo homines pariter <lb></lb>incedentes non capiant, adhibeatur longior funis, duos, aut <lb></lb>etiam tres, aut plures vectes connectens ita invicem diſtan<lb></lb>tes, ut intentus funis rectus ſit, & propiores quidem ſuum <lb></lb>vectem aut manu apprehenſum ſuſtentent, aut fune ſuſpenſum <pb pagenum="495" xlink:href="017/01/511.jpg"></pb>alio vecte parallelo humeris geſtent, remotiſſimi verò humeros <lb></lb>ſuo vecti ſubjiciant. </s> <s id="s.003757">Sic diſponatur funis AH, ut intentus <lb></lb>pertingat ad humeros <lb></lb><figure id="id.017.01.511.1.jpg" xlink:href="017/01/511/1.jpg"></figure><lb></lb>eorum, qui in E & F <lb></lb>ſuſtentant vectes DE <lb></lb>& FI. </s> <s id="s.003758">Quoniam ve<lb></lb>rò vectis MN longè <lb></lb>depreſſior eſt, quàm <lb></lb>humeri eorum, qui tam <lb></lb>propè abſunt à ponde<lb></lb>re; propterea vel ſolis <lb></lb>manibus apprehenſum <lb></lb>vectem ſuſtentent, vel, <lb></lb>quod ſatius eſt, alium <lb></lb>præterea vectem hu<lb></lb>meris geſtent paralle<lb></lb>lum vecti MN, ita ut <lb></lb>ex illo funibus ad per<lb></lb>pendiculum intentis <lb></lb>ſuſpendantur extremi<lb></lb>tates M & N. </s> <s id="s.003759">Id quod <lb></lb>etiam de reliquis, at<lb></lb>que de conſequentibus <lb></lb>vectibus dictum intel<lb></lb>ligatur. </s> <s id="s.003760">Omnes autem <lb></lb>æqualiter conari palàm <lb></lb>eſt, quia intento fune <lb></lb>AH eadem eſt obliqua <lb></lb>ſuſpenſio ponderis, & <lb></lb>paria ſunt momenta ad<lb></lb>versùs ſingulos vectes, <lb></lb>quos funis connectit. </s> <lb></lb> <s id="s.003761">Illud tamen negari non <lb></lb>poteſt, quod pro majore <lb></lb>funis AH longitudine <lb></lb>major eſt ſuſpenſionis <lb></lb>obliquitas, ac proinde, <lb></lb>& major ſuſtentandi labor. </s> </p> <pb pagenum="496" xlink:href="017/01/512.jpg"></pb> <p type="main"> <s id="s.003762">Unum adhuc hìc addere (ne quid intactum relinquatur) fuerit <lb></lb>operæ pretium, videlicet, ſi ponderis transferendi craſſities <lb></lb>ſeu altitudo mediocris ſaltem fuerit, ita ut non ſolùm infi<lb></lb>mo plano ſubjici vectes poſſint, ſed etiam ſupremæ aut me<lb></lb>diæ parti adnecti, poſſe eidem lateri duos aut etiam tres fu<lb></lb>nes, non quidem omninò, ſed proximè parallelos alligari, <lb></lb>quibus duæ, aut tres, ferè ſimiles obliquæ ſuſpenſiones fiant, <lb></lb>& deferentes pondus alij aliis remotiores ſint, ferè tamen <lb></lb>æqualiter conantes. </s> <s id="s.003763">Sic ingentis ſaxi altitudo ſit FG, & al<lb></lb><figure id="id.017.01.512.1.jpg" xlink:href="017/01/512/1.jpg"></figure><lb></lb>ligatus in F funis connectantur cum vecte in S aliis vectibus <lb></lb>ſuſtentato, ut ſupra. </s> <s id="s.003764">Item in E & in G alij funes paralleli ſimili<lb></lb>ter jungantur cum vectibus in T & V, ut homines ibi ſuccollan<lb></lb>tes vectibúſque ſubjecti ſibi invicem impedimento non ſint. </s> <s id="s.003765">Si <lb></lb>igitur ſingulis lateribus ad B, C, D tres funes hac ratione addan<lb></lb>tur, erunt 12 funes, & ſi homines 16 ſingulis funibus applicen<lb></lb>tur methodo ſuperiùs indicatâ, pondus geſtabitur à viris 192: <lb></lb>conſtat igitur quàm ingens onus facilè transferri vectibus <lb></lb>queat. </s> </p> <p type="main"> <s id="s.003766"><emph type="center"></emph>PROPOSITIO V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003767"><emph type="center"></emph><emph type="italics"></emph>Multiplici vecte moventis vires augere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003768">PRo vectis longitudine majori, in eâdem ab hypomochlio <lb></lb>diſtantiâ ponderis, potentiæ momenta augeri, quia Ratio <pb pagenum="497" xlink:href="017/01/513.jpg"></pb>motûs potentiæ ad motum ponderis augetur, ſatis manifeſtum <lb></lb>eſt ex dictis. </s> <s id="s.003769">Verùm quia non rarò tam longus vectis, quanto <lb></lb>opus eſſet, in promptu non eſt, aut ipſa longitudo illum redde<lb></lb>ret fractioni, aut ſaltem flexioni, magis obnoxium, aut, ſi peri<lb></lb>culo huic occurratur, tam immanis eſt vectis moles, ut non levi <lb></lb>incommodo ſit eo utentibus: propterea ars aliqua excogitanda <lb></lb>eſt, qua oblati vectis brevitatem compenſatione aliquâ ſup<lb></lb>pleamus. </s> </p> <p type="main"> <s id="s.003770">Et primò quidem ſi oblatus ſit vectis AB, habens hypomo<lb></lb>chlium in C, & pondus in B tam <lb></lb><figure id="id.017.01.513.1.jpg" xlink:href="017/01/513/1.jpg"></figure><lb></lb>grave, ut unica potentia in A non <lb></lb>ſatis ſit ad vincendam oneris reſi<lb></lb>ſtentiam, utique ſi altero, aut ter<lb></lb>tio movente opus ſit, non omnes in <lb></lb>extremitate A vectem apprehen<lb></lb>dere valent, ſed alter in D, tertius <lb></lb>in E; qui propterea, licèt ſinguli <lb></lb>æquali robore polleant, non tamen <lb></lb>æqualia habent momenta, ſed pri<lb></lb>mus ut AC, ſecundus ut DC, ter<lb></lb>tius ut EC. </s> <s id="s.003771">Quapropter alter vectis GH adnectatur extremi<lb></lb>tati A ad angulos rectos, ut huic applicati motores plus ha<lb></lb>beant momenti. </s> <s id="s.003772">Si enim AC ad CB fuerit ut 10 ad 1, perinde <lb></lb>eſt, atque ſi decima ponderis pars à duabus in G & H æquali<lb></lb>ter ab A diſtantibus movenda eſſet, ac propterea ſinguli ſemiſ<lb></lb>ſem decimæ partis reſiſtentiæ percipiunt, hoc eſt, habent ſimul <lb></lb>ſumpti momentum ut 20 ad 1: qui autem in A eſſet ſolus, habe<lb></lb>ret momentum ut 10, & qui in D haberet momentum ex.gr. ut <lb></lb>9; qui idcircò ſimul ſumpti minùs poſſunt quàm G & H. </s> </p> <p type="main"> <s id="s.003773">At ſi volueris tres homines in extremitatibus vectis GH di<lb></lb>ſtribuere in potentias æqualiter conantes, diſtingue GH in tres <lb></lb>partes, & ſit AH triens totius longitudinis GH: tum duo ap<lb></lb>plicentur extremitati H, tertius verò extremitati G: ut enim <lb></lb>potentia duplex in H ad potentiam in G, ita reciprocè duplex <lb></lb>diſtantia GA ad diſtantiam AH. </s> <s id="s.003774">Nam quemadmodum de <lb></lb>ſuſtinentibus pondus vecte ſive æqualiter, ſive inæqualiter di<lb></lb>viſo dictum eſt, ita hìc pariter de Prementibus dicendum, qui <lb></lb>in H & in G ſunt viciſſim Potentia & Hypomochlium: ex eo <pb pagenum="498" xlink:href="017/01/514.jpg"></pb>ſcilicet quòd potentia in H premit, habet rationem hypomo<lb></lb>chlij, dum reſiſtit, ne potentia in G premens elevet ipſam ex<lb></lb>tremitatem H, ac propterea deprimat pondus in A exiſtens; & <lb></lb>viciſſim potentia in G premens habet rationem hypomochlij <lb></lb>reſiſtendo, ne elevetur à potentiâ premente in H; quæ prop<lb></lb>terea deprimit pondus in A. </s> <s id="s.003775">Eſt itaque veluti duplex vectis ſe<lb></lb>cundi generis; & AG ad AH ſi fuerit ut 2 ad 1, momentum <lb></lb>potentiæ in G ad reſiſtentiam ponderis in A eſt ut 3 ad 1, hoc <lb></lb>eſt ut GH ad AH: & momentum unius potentiæ in H ad re<lb></lb>ſiſtentiam ponderis in A eſt ut 3 ad 2, hoc eſt, ut HG ad AG. </s> <lb></lb> <s id="s.003776">Sed quia in H ex hypotheſi ſunt duæ potentiæ, duplicatæ po<lb></lb>tentiæ in H momentum erit ut 6 ad 2, hoc eſt, ſingularum <lb></lb>momentum ut 3 ad 1. Igitur qui eſt in G habet momentum at<lb></lb>que conatum, quaſi ſine vecte moveret trigeſimam partem pon<lb></lb>deris in B exiſtentis; & duo, qui in H, ſinguli habent momen<lb></lb>tum æquale atque conatum ſimilem. </s> <s id="s.003777">Ponatur pondus B lib.60; <lb></lb>in A percipitur ponderis (1/10), hoc eſt, lib. 6. Igitur potentia in <lb></lb>G percipit reſiſtentiam lib.2, & duæ potentiæ in H ſimul lib.4, <lb></lb>hoc eſt ſingulæ lib.2. </s> </p> <p type="main"> <s id="s.003778">Quod ſi non placuerit longitudinem GH habere tanquam <lb></lb>vectem, qui non alternâ quadam motione & quiete extremita<lb></lb>tum perficitur motus, ſed G, & A, & H omnino ſimul & <lb></lb>æquali motu moventur, non admodum contendo: perinde erit, <lb></lb>atque ſi tres potentiæ in A eſſent conſtitutæ, quarum ſingulæ <lb></lb>tertiam partem ponderis moveant conatu ſubdecuplo illius co<lb></lb>natus, quo tertia illa pars ſine vecte movenda eſſet. </s> </p> <p type="main"> <s id="s.003779">Hinc ſaltem conſtat, quo virium atque conatûs compendio <lb></lb>valeat unicus homo oblati vectis momenta augere: Nam ſi idem <lb></lb>ſit primi generis vectis AB, & AC ad CB ſit ut 10 ad 1, adhi<lb></lb>be vectem ſecundi generis GH, & alterâ extremitate fixâ, ut <lb></lb>ibi ſit hypomochlium, idem augebit momenta juxta Rationem <lb></lb>totius longitudinis GH ad diſtantiam ipſius A ab hypomo<lb></lb>chlio: Quare ſi Ratio ſit dupla, aut tripla, æquivalebit duobus <lb></lb>aut tribus, qui in A moventes haberent momentum decuplum; <lb></lb>nam A movetur decuplo velociùs quàm B, & poſito hypomo<lb></lb>chlio H, movetur potentia G duplo aut triplo velociùs quàm A, <lb></lb>hoc eſt vigecuplo aut trigecuplo velociùs quàm B. </s> <s id="s.003780">Id quod <lb></lb>uſum habet non ſolùm, quando vectis AB movendus eſt in pla-<pb pagenum="499" xlink:href="017/01/515.jpg"></pb>no Verticali, ſed etiam in plano horizontali, ut ſi duo marmora <lb></lb>disjungenda eſſent, aut clathri diſſipandi. </s> <s id="s.003781">Juxta autem loci op<lb></lb>portunitatem adjungendus eſt ſecundus vectis GH aut proxi<lb></lb>mè ipſi primo vecti, aut remotè medio fune extremitatem A <lb></lb>connectente cum ſecundo <lb></lb>vecte. </s> <s id="s.003782">Sic inter duo mar<lb></lb><figure id="id.017.01.515.1.jpg" xlink:href="017/01/515/1.jpg"></figure><lb></lb>mora immiſſus ferreus cla<lb></lb>vus SR jungitur vecti TV <lb></lb>fune SO, & potentia in <lb></lb>T habet momentum com<lb></lb>poſitum ex Rationibus <lb></lb>TV ad VO, & SX ad XR. </s> </p> <p type="main"> <s id="s.003783">Neque duos tantummodo, verùm etiam plures vectes adhi<lb></lb>bere poſſumus, tunc maximè, cùm ingenti oneri exiguus <lb></lb>motus tribuendus eſt. </s> <s id="s.003784">Sit enim marmor P attollendum ſub<lb></lb>jecto vecte AB ſecundi <lb></lb><figure id="id.017.01.515.2.jpg" xlink:href="017/01/515/2.jpg"></figure><lb></lb>generis habente hypo<lb></lb>mochlium in B, ac pon<lb></lb>dere incumbente illi in <lb></lb>C: & AB ad CB ſit <lb></lb>ut 7 ad 1. Quia vectis <lb></lb>attollendus eſt, ſubji<lb></lb>ce illi in A vectem al<lb></lb>terum DE, ut. </s> <s id="s.003785">ED ad <lb></lb>AD ſit in Ratione 3 ad 1. Item extremitati E ſubjice tertium <lb></lb>vectem FG, & ſit GF ad EF ut 8 ad 1. Igitur A movetur ſep<lb></lb>tuplò velociùs quàm C, & E triplo velociùs quàm A, atque <lb></lb>G octuplo velociùs quàm E. </s> <s id="s.003786">Quare motus potentiæ in G ad <lb></lb>motum ponderis in C eſt ut 168 ad 1. Quàm difficile autem <lb></lb>accideret, ſi tam longum vectem parare oporteret, cujus lon<lb></lb>gitudo eſſet ad CB ut 168 ad 1! </s> </p> <figure id="id.017.01.515.3.jpg" xlink:href="017/01/515/3.jpg"></figure> <p type="main"> <s id="s.003787">Adde non ſolùm vectibus rectis hoc <lb></lb>momentorum incrementum acquiri <lb></lb>poſſe, ſed etiam pro loci opportunita<lb></lb>te vectibus curvis aut angulatis. </s> <s id="s.003788">Si <lb></lb>enim in ſuperiore loco fuerit vectis <lb></lb>ſecundi generis MN oneri ſubjectus, <lb></lb>aut oneri inferiùs poſito junctus fune <pb pagenum="500" xlink:href="017/01/516.jpg"></pb>in O, non ſolùm poſſumus extremitatem M fune connectere <lb></lb>cum vecte recto ſuperiùs poſito, ſed etiam ſubjicere illi poſſu<lb></lb>mus vectem curvum KL fixum in I, & extremitas L fune LR <lb></lb>trahi poteſt deorſum, ut IK elevetur, atque illo motu attol<lb></lb>lat extremitatem M, quantum ferre poteſt flexus IK. </s> <s id="s.003789">Non <lb></lb>eſt autem opus monere inæqualia ſenſim fieri momenta, prout <lb></lb>ſubjectus vectis curvus KIL in alio atque alio puncto contingit <lb></lb>vectem MN, pro variâ ſcilicet diſtantiâ ab hypomochlio. </s> </p> <p type="main"> <s id="s.003790">In vecte tertij generis majorem eſſe ponderis motum mo<lb></lb>tu potentiæ, ac proinde majores requiri potentiæ vires ad at<lb></lb>tollendum onus, ſi illa conjuncta ac ſociata ſit cum hujuſmo<lb></lb>di vecte, quàm ſi ipſa ſolitaria manum admoveret ponderi <lb></lb>ſublevando, manifeſtum eſt; propterea infirmiori potentiæ ſub<lb></lb>ſidium aliquod induſtriâ comparare poſſumus, & propoſitum <lb></lb>vectem in aliam vectis ſpeciem quaſi convertere, <expan abbr="etiã">etiam</expan> ſi ſpatij an<lb></lb>guſtiis coarctemur, modò liceat proximum parietem perfodere. </s> </p> <p type="main"> <s id="s.003791">Sit parieti AB innixus vectis CD, cujus extremitati D <lb></lb><figure id="id.017.01.516.1.jpg" xlink:href="017/01/516/1.jpg"></figure><lb></lb>adnectendum ſit <lb></lb>pondus ex. </s> <s id="s.003792">gr. <lb></lb>lib. 200: poten<lb></lb>tia autem appli<lb></lb>cari nequeat ni<lb></lb>ſi in E, ita ut <lb></lb>EC ſit quarta <lb></lb>pars totius vectis <lb></lb>CD. Igitur, cum <lb></lb>motus in D ſit quadruplus motûs in E, ut potentia ſublevet onus <lb></lb>D, tanta ſit, oportet, ut ipſa ſe ſola valeat quadruplum onus, ſci<lb></lb>licet lib.800 attollere: id quod valdè incommodum accideret, <lb></lb>ſi adeò validam potentiam invenire opus eſſet. </s> <s id="s.003793">Perfode igitur in <lb></lb>ſuperiore parte B parietem, illíque immitte vectem FG facile in <lb></lb>B hypomochlio verſatilem, ita ut BF pars imminens ſubjecto <lb></lb>vecti ſit æqualis parti EC, hoc eſt, diſtantiæ potentiæ E ab hy<lb></lb>pomochlio C, & fune FE connectantur: pars verò ultra parie<lb></lb>tem in proximum conclave extans BG ad partem BF ſit in qua<lb></lb>cumque Ratione. </s> <s id="s.003794">Tum in inferiore loco, prout opportunius ac<lb></lb>ciderit, vectem alium ſtatue HI, cui junge ſuperioris Vectis ex<lb></lb>tremitatem G fune GM: nam Ratio compoſita ex Rationibus <pb pagenum="501" xlink:href="017/01/517.jpg"></pb>IH ad MH, & GB ad BF dabit momentum potentiæ in I po<lb></lb>ſitæ ad attollendum pondus in D conſtitutum per vectem da<lb></lb>tum CD habentem potentiam in E. </s> </p> <p type="main"> <s id="s.003795">Hîc habes tria vectis genera; nam IH eſt ſecundi generis, quia <lb></lb>pondus intelligitur in M inter potentiam I & hypomochlium H; <lb></lb>GF eſt primi generis, quia hypomochlium B eſt inter potentiam <lb></lb>G & pondus in F; CD eſt tertij generis, quemadmodum ab ini<lb></lb>tio conſtitutum eſt. </s> <s id="s.003796">Si itaque in E requireretur vis attollendi <lb></lb>lib.800, & ſit GB dupla ipſius BF, requiritur in G vis attollendi <lb></lb>lib.400. Si verò I H ad MH ſit quadrupla, requiritur in I vis <lb></lb>elevandi lib.100. Quare & uteris vecte tertij generis CD, quo <lb></lb>ſatis notabiliter movetur pondus D; & potentiæ momenta <lb></lb>auxiſti adeò, ut non ſolùm non requiratur potentia major <lb></lb>pondere attollendo, ſed ſufficiat potentia minor, habet quippe <lb></lb>motum duplo majorem, quàm ſit motus ponderis D; nam mo<lb></lb>tus extremitatis F & puncti E ſunt æquales; motus potentiæ I <lb></lb>eſt quadruplus motus ipſius M; hoc eſt extremitatis G; hæc ve<lb></lb>rò motum habet duplum motus ipſius F: igitur motus poten<lb></lb>tiæ I eſt octuplus motûs puncti E, quod movetur motu ſubqua<lb></lb>druplo extremitatis D: Motus igitur potentiæ I ad motum <lb></lb>ponderis D eſt ut 8 ad 4, hoc eſt ut 2 ad 1. </s> </p> <p type="main"> <s id="s.003797"><emph type="center"></emph>PROPOSITIO VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003798"><emph type="center"></emph><emph type="italics"></emph>Stateræ vires addito Vecte augere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003799">PAretur haſta AB, atque extremitati A addatur annulus, cui <lb></lb>inſeri valeat ſtateræ CD uncus, & extremitas B ita confor<lb></lb><figure id="id.017.01.517.1.jpg" xlink:href="017/01/517/1.jpg"></figure><lb></lb>metur, ut notabile ſit & conſpicuum punctum, quod hypomo<lb></lb>chlio reſpondeat; ſitque certa nota, qua dignoſcatur vec is pa<lb></lb>rallelúſne ſit horizonti, an inclinatus. </s> <s id="s.003800">Tum diſtantia BA di-<pb pagenum="502" xlink:href="017/01/518.jpg"></pb>vidatur primùm in duas partes, deinde in tres, & ſic deinceps, <lb></lb>quatenus commodè fieri id poterit citra confuſionem, quando <lb></lb>opus fuerit huic aut illi puncto adnectere onus expendendum, <lb></lb>adeò ut certi ſcimus, quotuplex ſit totius vectis AB longitudo <lb></lb>comparata cum diſtantiâ ponderis ab hypomochlio B. </s> </p> <p type="main"> <s id="s.003801">Hoc vecte ad uſum parato, examinetur ſtaterâ communi, <lb></lb>quantum ille gravitet parallelus horizonti: & ſit æquipondium <lb></lb>ſtateræ ex. </s> <s id="s.003802">gr. in H indicans lib. 2: id quod memoriâ retinen<lb></lb>dum eſt, ut, cùm ponderis gravitas explorabitur, ex numero, <lb></lb>qui in ſtateræ jugo indicabitur ab æquipondio, dematur ipſa <lb></lb>vectis gravitas deprehenſa, ſcilicet lib. 2. </s> </p> <p type="main"> <s id="s.003803">Propoſita igitur gravitate majori, quàm ut expendi valeat <lb></lb>communi ſtaterâ CD, adnecte onus vecti in aliquo ex adno<lb></lb>tatis punctis, ex. </s> <s id="s.003804">gr. in puncto 6, prout commodius acciderit: <lb></lb>tum reduc tantiſper æquipondium ſtateræ, dum ejuſdem ſta<lb></lb>teræ jugum & vectis æquè ab horizonte diſtent, & conſiſtat <lb></lb>æquipondium, puta, in puncto I indicante lib.12. unc. </s> <s id="s.003805">8. de<lb></lb>me lib. 2. gravitatem vectis, remanent lib. 10. unc. </s> <s id="s.003806">8. Quia <lb></lb>autem onus ex hypotheſi adnexum eſt in puncto 6, multiplica <lb></lb>per 6 lib. 10. unc. </s> <s id="s.003807">8, & habebis lib. 64 gravitatem oneris quæ<lb></lb>ſitam. </s> <s id="s.003808">Quod ſi plane in medio puncto 2 conſtitutum fuiſſet <lb></lb>pondus, duplicanda eſſet gravitas indicata à ſtaterâ. </s> <s id="s.003809">Mani<lb></lb>feſta eſt hujus operationis ratio; ſiquidem æquipondium ſta<lb></lb>teræ in puncto I ſuſtinet lib. 12. unc. </s> <s id="s.003810">8 adnexas extremitati D. </s> <lb></lb> <s id="s.003811">At vis ſuſtinendi in vecte ſecundi generis poſita in A ſuſtinet <lb></lb>vectem, cujus momentum eſt lib. 2, & præterea ſuſtinet pon<lb></lb>dus in puncto 6 poſitum, quod ad reliquam potentiæ virtutem <lb></lb>in A, hoc eſt lib.10. unc. </s> <s id="s.003812">8, habet Rationem, quæ ſit AB ad <lb></lb>B 6. igitur convertendo ut 1 ad 6, ita lib.10. unc.8. ad lib.64. </s> </p> <p type="main"> <s id="s.003813">Quoniam verò accidere poteſt, ut oblatum pondus exce<lb></lb>dat quidem datæ ſtateræ vires, ſed ejus gravitas minor ſit quàm <lb></lb>dupla ejus, cui æquipondium in extremo ſtateræ jugo reſpon<lb></lb>det; propterea diviſiones eædem, quæ ex 2 ad B adnotatæ ſunt, <lb></lb>transferantur ex 2 versùs A, ut habeamus diverſa puncta in <lb></lb>vecte, quibus applicari poſſit onus ponderandum. </s> <s id="s.003814">Ex numero <lb></lb>igitur adnotato, cui adnectitur pondus, fiat numerator fractionis, <lb></lb>cujus Denominator ſit unitate minor ipſo numeratore; & per <lb></lb>hanc fractionem multiplicetur numerus à ſtaterâ indicatus <pb pagenum="503" xlink:href="017/01/519.jpg"></pb>(demptâ priùs vectis gravitate, ut ſuperiùs dictum eſt) & habe<lb></lb>bitur oneris gravitas. </s> <s id="s.003815">Sit igitur inter A & 2 adnexum pondus in <lb></lb>puncto 7, & ſtatera indicet lib.13. unc.7: demo vectis gravita<lb></lb>tem, quæ eſt lib. 2 ex hypotheſi, remanent lib.11. unc.7. mul<lb></lb>tiplicandæ per 7/6, & dabitur oneris gravitas lib. 13. unc. </s> <s id="s.003816">6 1/6. <lb></lb>Quare in hoc caſu fortaſſe nullum habetur ex vecte adjecto <lb></lb>compendium, potuiſſet enim ex ipsâ ſtatera immediatè cogno<lb></lb>ſci eadem gravitas. </s> <s id="s.003817">Quod ſi eundem numerum indicaſſet ſta<lb></lb>tera, ſed onus adjunctum fuiſſet in puncto 3, per 3/2 multiplica<lb></lb>tis lib. 11. unc. </s> <s id="s.003818">7, proveniſſet gravitas oneris quæſita lib. 17 <lb></lb>unc. </s> <s id="s.003819">4 1/2, quæ ex hypotheſi major eſt, quàm ut ſolâ ſtaterâ <lb></lb>oblatâ expendi poſſit. </s> <s id="s.003820">Vel ſi rem breviùs expedire placuerit, <lb></lb>numeri ſtaterâ inventi accipe partem denominatam à numero <lb></lb>vectis unitate minore, eámque illi numero invento adde, & <lb></lb>idem obtinebis. </s> <s id="s.003821">Sic quia in puncto 3 appenſum fuit onus, ac<lb></lb>cipe librarum 11. unc. </s> <s id="s.003822">7. partem denominatam à 2, ſcilicet lib.5. <lb></lb>unc. </s> <s id="s.003823">9 1/2, eámque adde libris 11. unc. </s> <s id="s.003824">7 inventis, & habebis, <lb></lb>ut priùs, lib. 17 unc. </s> <s id="s.003825">4 1/2. Cur hac methodo operandum ſit, <lb></lb>manifeſtò conſtat ex ipſa vectis diviſione; nam AB ad A 3 eſt <lb></lb>ut 3 ad 1 ex conſtructione, atque ideò AB ad 3 B eſt ut 3 ad 2: <lb></lb>igitur ut 2 ad 3, ita numerus à ſtatera indicatus (demptâ <lb></lb>vectis gravitate) ad numerum quæſitum, quo ponderis gravitas <lb></lb>innoteſcit. </s> </p> <p type="main"> <s id="s.003826">Generatim itaque atque universè oblato quocumque vecte <lb></lb>ad ſubitum uſum properato utere, etiamſi nullæ in eo diviſio<lb></lb>nes adnotatæ fuerint, examinato tamen priùs ipſius vectis ho<lb></lb>rizonti paralleli gravitatis momento, quatenus ad ſtateram <lb></lb>comparatur: Tum datum pondus ibi alliga, ubi commodè à <lb></lb>ſtaterâ extremo vecti applicatâ elevari poſſit. </s> <s id="s.003827">Facto demum <lb></lb>æquilibrio, ſtateræ numerum (dempto priùs vectis momen<lb></lb>to) multiplica per Rationem, quam habet vectis longitudo ad <lb></lb>diſtantiam ponderis ab hypomochlio; & propoſitum obtine<lb></lb>bis. </s> <s id="s.003828">Hìc habes maximum compendium ad ingentium ponde<lb></lb>rum gravitatem explorandam: etiamſi enim vectis non ſit <lb></lb>adeò craſſus, quia tamen non procul ab extremitate illius, ubi <lb></lb>eſt hypomochlium, alligatur onus, validè reſiſtit fractioni; <lb></lb>& quo major eſt Ratio longitudinis vectis ad diſtantiam pon-<pb pagenum="504" xlink:href="017/01/520.jpg"></pb>eris ab hypomochlio, tanto majore incremento augentur ſta<lb></lb>teræ vires. </s> </p> <p type="main"> <s id="s.003829">Quod ſi fortè unicus vectis ſatis non fuerit, nihil prohibet <lb></lb>plures adhiberi vectes multo majore compendio, quàm ſi uni<lb></lb>cum longiorem adhiberes. </s> <s id="s.003830">Nam ſi vectis AB non ita ſtateræ <lb></lb><figure id="id.017.01.520.1.jpg" xlink:href="017/01/520/1.jpg"></figure><lb></lb>vires multiplicet, <lb></lb>ut tormentum æ<lb></lb>neum in C <expan abbr="alli-gatũ">alli<lb></lb>gatum</expan> elevari poſ<lb></lb>ſit ab æquipondio <lb></lb>ſtateræ, alium <lb></lb>vectem EF ſtatue ipſi AB parallelum, habeátque in E hypo<lb></lb>mochlium, & ſtatera in F adnectatur, qua primùm ipſorum <lb></lb>vectium fune HI conjunctorum & poſitionem horizonti paral<lb></lb>lelam habentium gravitatis momentum expendatur. </s> <s id="s.003831">Deinde <lb></lb>facto æquilibrio dematur vectium momentum, & reliquus li<lb></lb>brarum numerus à ſtaterâ indicatus multiplicetur primò per <lb></lb>Rationem FE ad HE, & quod ex hac multiplicatione conſur<lb></lb>git, ſecundò multiplicetur per Rationem IB ad CB; habebitur <lb></lb>enim demum tormenti ænei gravitas quæſita. </s> <s id="s.003832">Sit ex. </s> <s id="s.003833">gr. IB ad <lb></lb>CB ut 10 ad 1, & FE ad HE ut 12 ad 1, atque ſtatera, dempto <lb></lb>vectium momento, indicet libras 100: igitur 100 per 12 dat 1200, <lb></lb>& 1200 per 10 dat lib.12000 gravitatem ænei tormenti. </s> </p> <p type="main"> <s id="s.003834">His autem indicatis ſtatim occurit animo non duos tantum<lb></lb>modo ſed plures vectes poſſe ita diſponi, ut ſemper fiat major <lb></lb>Ratio, quæ ex illorum Rationibus componitur: ſi nimirum in<lb></lb>ter duas trabes in ſolo ad perpendiculum firmatas, & æquali in<lb></lb>tervallo à ſe invicem diſſitas interjiciantur vectes alterna hypo<lb></lb>mochlia habentes in axibus, circa quos facilè converti poſſint, <lb></lb>& ſimili ratione jungantur, ac de duobus vectibus AB & EF <lb></lb>dictum eſt: Ex ſingulorum enim vectium Rationibus una Ra<lb></lb>tio componitur, per quam multiplicandus eſt numerus à ſtate<lb></lb>râ indicatus, dempto priùs vectium momento. </s> <s id="s.003835">Id quod paulo <lb></lb>latiùs explicatum eſt in <emph type="italics"></emph>Terra machinis mota.<emph.end type="italics"></emph.end> diſſert.-1-n. </s> <s id="s.003836">16. nec <lb></lb>opus eſt hìc tranſcribere. </s> </p> <pb pagenum="505" xlink:href="017/01/521.jpg"></pb> <figure id="id.017.01.521.1.jpg" xlink:href="017/01/521/1.jpg"></figure> <p type="main"> <s id="s.003837"><emph type="center"></emph>MECHANICORUM <emph.end type="center"></emph.end><emph type="center"></emph>LIBER QUINTUS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003838"><emph type="center"></emph><emph type="italics"></emph>De Axe in Peritrochio.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003839">QUÆ de Vecte ejúſque viribus ſuperiore libro <lb></lb>diſputata ſunt, illa quidem vera ſunt, & ad<lb></lb>mirabilia, ſed, niſi vectis admodum longus <lb></lb>ſit, exiguus motus conciliatur ponderi, adeò <lb></lb>ut, ſi ad notabilem aliquam altitudinem attol<lb></lb>lendum illud ſit, oporteat ſubinde & ipſi ponderi fulcrum <lb></lb>ſupponere, ne recidat, & ipſi vecti hypomochlium altiùs <lb></lb>ſubjicere, ut congruo loco ſtatuatur. </s> <s id="s.003840">Præterquam quod pro <lb></lb>variâ ipſius vectis inclinatione, oneríſque illi impoſiti, aut <lb></lb>ſubjecti poſitione, varia quoquè ſunt momenta potentiæ <lb></lb>vectem urgentis. </s> <s id="s.003841">Hinc alia Facultas excogitata eſt, quæ, <lb></lb>ut pluribus placet, vectis quidam ſit perpetuus, citrà in<lb></lb>commoda, quæ in ſimplici Vecte, ut innuebam, occur<lb></lb>runt. <emph type="italics"></emph>Vectem<emph.end type="italics"></emph.end> autem appellant, quia ad vectis Rationes il<lb></lb>lius vim revocant; <emph type="italics"></emph>perpetuum<emph.end type="italics"></emph.end> verò, quia nullâ opus eſt <lb></lb>hypomochlij mutatione: proprio tamen, tritóque jam ve<lb></lb>tuſtate vocabulo, communiter dicitur <emph type="italics"></emph>Axis in Peritrochio,<emph.end type="italics"></emph.end><lb></lb>quaſi <emph type="italics"></emph>Axis in Rota,<emph.end type="italics"></emph.end> ut quidam interpretantur; ſed fortaſsè <lb></lb>clariùs, pleniúſque vocabuli vim aſſequeremur, ſi <emph type="italics"></emph>Axem <lb></lb>Convolutum<emph.end type="italics"></emph.end> vocaremus; neque enim ſemper adeſt Rota, <lb></lb>cum tamen ſemper interſit Convolutio, ſimul quippe vol<lb></lb>vitur, & Axis ipſe, & id, cum quo Axis conjungitur. <pb pagenum="506" xlink:href="017/01/522.jpg"></pb>Neque hic ſumitur Axis quemadmodum in Cono, Cylin<lb></lb>dro, atque Sphærâ, pro linea rectâ, circa quam immo<lb></lb>tam corpora illa in gyrum aguntur; ſed eſt corpus ſuâ <lb></lb>præditum craſſitie, cui Axis nomen inditum eſt, quia ro<lb></lb>tarum axem imitatur, non tamen circà illum fit convolu<lb></lb>tio, ſed ipſe circa idem centrum volvitur minore motu, <lb></lb>circa quod potentia motu majore rotatur, quatenus illi ap<lb></lb>plicatur, ut ex his, quæ dicentur, manifeſtum fiet. <lb></lb></s> </p> <p type="main"> <s id="s.003842"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003843"><emph type="center"></emph><emph type="italics"></emph>Axis in Peritrochio forma, & vires <lb></lb>deſcribuntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003844">AXis in Peritrochio forma à Pappo Alexandrino circa <lb></lb>finem lib. 8. Collect. Mathem. deſcribitur, quadratum <lb></lb><figure id="id.017.01.522.1.jpg" xlink:href="017/01/522/1.jpg"></figure><lb></lb>ſcilicet lignum tympano quadra<lb></lb>to foramen AB eidem ligno con<lb></lb>gruens circa ſuum centrum ha<lb></lb>benti inſeritur, ut ſimul verti <lb></lb>poſſint: ligni autem partes è <lb></lb>tympano prominentes in cylin<lb></lb>dricam rotunditatem conforman<lb></lb>tur; & lignum horizonti parallelum ſuper polos æreos, aut <lb></lb>ferreos (choinicidas Pappus vocat) congruis fulcris inſiſten<lb></lb>tes ſtatuitur. </s> <s id="s.003845">Extremæ verò tympani orbitæ infiguntur Ra<lb></lb>dij CD, EF, &c, quos Pappus <emph type="italics"></emph>Scytalas,<emph.end type="italics"></emph.end> Ariſtoteles <emph type="italics"></emph>Col<lb></lb>lopes<emph.end type="italics"></emph.end> nominat, longiores ſcilicet paxilli, quibus arreptis <lb></lb>verſatur tympanum, & cum eo Axis, quem ductarius fu<lb></lb>nis HI in convolutione circumplectens attollit adnexam in <lb></lb>I ſarcinam; atque hæc tantumdem attollitur, quantus funis <lb></lb>Axem circumplicat ex convolutione. </s> </p> <p type="main"> <s id="s.003846">Ut Axis hujuſmodi vires explicentur, communiter in eo <lb></lb>agnoſcunt Vectis Rationes: cum enim CB ſit ſemidiame<lb></lb>ter cylindri, quem funis complectitur, & CE ſemidiame-<pb pagenum="507" xlink:href="017/01/523.jpg"></pb>ter tympani circumpoſiti, EA verò longitudo Radij, conci<lb></lb>piunt AB quaſi Vectem <lb></lb>primi generis habentem hy<lb></lb><figure id="id.017.01.523.1.jpg" xlink:href="017/01/523/1.jpg"></figure><lb></lb>pomochlium in C, adeò ut <lb></lb>ex Ratione AC ad CB mo<lb></lb>mentum potentiæ in A ap<lb></lb>plicatæ computetur. </s> <s id="s.003847">Quæ <lb></lb>quidem vera eſſe non nega<lb></lb>verim, ſi hoc unum intelli<lb></lb>gatur, quòd Ratio AC ad <lb></lb>CB ſimilis ſit Rationi, quam <lb></lb>haberet æqualis Vectis ſimi<lb></lb>lem habens poſitionem Po<lb></lb>tentiæ, Hypomochlij, & <lb></lb>Ponderis. </s> <s id="s.003848">Verùm cur primi <lb></lb>potiùs quàm ſecundi gene<lb></lb>ris vectis dicatur Axis in Pe<lb></lb>ritrochio, cùm æquè attolla<lb></lb>tur pondus P, ſi Radij extre<lb></lb>mitas D elevetur ſurſum, ac ſi extremitas Radij A deprimatur <lb></lb>deorſum? </s> <s id="s.003849">Eſto facilior ſit depreſſio, quàm elevatio. </s> <s id="s.003850">Quid, <lb></lb>ſi Axis ſtatueretur horizonti perpendicularis, tympanum au<lb></lb>tem horizonti parallelum, non ad attollendum, ſed ad trahen<lb></lb>dum pondus? </s> <s id="s.003851">Utique par eſſet trahendi facilitas, ſive impel<lb></lb>latur D versùs H, ſive A versùs I: adeóque nulla eſſet ratio, <lb></lb>cur primi potiùs quàm ſecundi generis Vectis diceretur: an <lb></lb>utrique generi aſcribendus eſt? </s> </p> <p type="main"> <s id="s.003852">Sed quid Axem ad Vectem revocare opus eſt? </s> <s id="s.003853">cùm eodem <lb></lb>ex fonte ita utriuſque vires emanent, ut etiamſi Vectem extra <lb></lb>omnem Naturæ facultatem poſitum, atque inter <foreign lang="grc">ἄδυνατα</foreign> re<lb></lb>cenſendum eſſe fingeremus, adhuc Axi ſua permanerent mo<lb></lb>menta: Eſt nimirum, ſi ſecundum velocitatem comparentur, <lb></lb>motûs potentiæ ad motum Ponderis Ratio major, quàm gravi<lb></lb>tatis ponderis ad virtutem potentiæ: dum enim funis ductarius <lb></lb>ſemel cylindrum circumplectitur, potentia ſemel percurrit ſpa<lb></lb>tium æquale peripheriæ circuli ab extremo Radio deſcripti; <lb></lb>cùm autem ſint peripheriæ circulorum in Ratione ſemidiame<lb></lb>trorum, motus potentiæ A ad motum ponderis P eſt ut AC ad <pb pagenum="508" xlink:href="017/01/524.jpg"></pb>CB. </s> <s id="s.003854">Quare potentiæ Peritrochium verſantis conatus, ad cona<lb></lb>tum potentiæ ſine machinâ attollentis pondus P, erit in Ratio<lb></lb>ne CB ad AC; quò enim minor ſecundùm velocitatem eſt <lb></lb>motus ponderis comparatus cum motu potentiæ, eo minor eſt <lb></lb>ejuſdem reſiſtentia; minorem autem reſiſtentiam minor cona<lb></lb>tus ſuperat. </s> <s id="s.003855">Quæ ita ex dictis tùm lib.2.cap.5. tum lib.4.cap.1. <lb></lb>clara ſunt, ut uberior explicatio ſupervacanea cenſenda ſit. </s> </p> <p type="main"> <s id="s.003856">Hinc apparet, quid juvet ipſius rotæ adjunctæ magnitudo, <lb></lb>aut infixarum ſcytalarum longitudo; quò enim fuerit major <lb></lb>Potentiæ diſtantia à centro motûs, eò pariter major erit mo<lb></lb>vendi facilitas. </s> <s id="s.003857">Quo circa ſi eidem Peritrochio placuerit dupli<lb></lb>cem applicare potentiam, atque ideò ſcytalas non exteriori ab<lb></lb>ſidi tympani infigas, ſed potiùs extremam tympani oram ſcy<lb></lb>talis ad ejus planum perpendicularibus transfigas; tunc ad au<lb></lb>genda Potentiæ momenta nequicquam prodeſt ſcytalæ longitu<lb></lb>do, ſed à foramine, cui illa infigitur, uſque ad centrum deſu<lb></lb>menda eſt potentiæ diſtantia, quæ ut major fiat, tympani dia<lb></lb>meter augenda eſt. </s> <s id="s.003858">Id quod pariter dicendum eſt, quando ma<lb></lb>nubrium (unicus ſcilicet paxillus tympani plano infixus) appo<lb></lb>nitur, quod moventis manu perpetuò in converſione retine<lb></lb>tur; ejus enim diſtantia à centro perinde conſideratur, atque ſi <lb></lb>potentia illi tympani parti fuiſſet proximè applicata, cui manu<lb></lb>brium infigitur. </s> </p> <p type="main"> <s id="s.003859">Cavendum tamen hìc videtur, ne quis majorem aliquam ro<lb></lb>tam ultrà manubrium excurrentem cylindro circumpoſitam <lb></lb>conſiderans, quæ aliquando plus habere videtur momenti, <lb></lb>quàm ſi rota non major eſſet, quàm ferat manubrij à centro <lb></lb>diſtantia, exiſtimet non ex hac diſtantiâ computandum eſſe po<lb></lb>tentiæ manubrio applicatæ momentum. </s> <s id="s.003860">Obſervet, oportet, hoc <lb></lb>non contingere in immanibus & coloſſicoteris ponderibus, im<lb></lb>mò neque in mediocribus movendis, ſed in iis tantummodo, <lb></lb>quæ leviore negotio & velociter moveri poſſunt: Rota enim, <lb></lb>cujus ſemidiameter major eſt, quàm manubrij à centro diſtan<lb></lb>tia, impreſſum à movente potentiâ impetum concipit, qui le<lb></lb>vem nactus reſiſtentiam non ſtatim perit, ſed aliquantiſper per<lb></lb>ſeverans motum rotæ unà cum novo potentiæ conatu efficit <lb></lb>majorem, quàm pro ſolitariis potentiæ viribus: immò tanta fie<lb></lb>ri poteſt impetûs impreſſi acceſſio, ut poſt aliquod tempus, <pb pagenum="509" xlink:href="017/01/525.jpg"></pb>etiam dimiſſo à potentiâ manubrio, vi ejuſdem impreſſi impe<lb></lb>tûs adhuc ſe rota in gyrum contorqueat. </s> <s id="s.003861">Hinc eſt aliquando <lb></lb>ejuſdem rotæ diametrorum extremitatibus addi plumbeas maſ<lb></lb>ſas, quæ plus impetûs concipientes, atque diutiùs retinentes, <lb></lb>rotæ converſionem validiùs promoveant, etiam ceſſante poten<lb></lb>tiâ. </s> <s id="s.003862">Sed hìc non unica eſt potentia, quæ manubrio applicatur, <lb></lb>cujus momenta ex diſtantiâ manubrij à centro definimus; ſed <lb></lb>præterea impetus ille perſeverans rationem habet alterius po<lb></lb>tentiæ applicatæ illis rotæ partibus, quibus ineſt; & pro variâ <lb></lb>à centro diſtantiâ, alia pariter atque alia ſunt particularum ip<lb></lb>ſius impetûs impreſſi momenta ad rotam convertendam. </s> <s id="s.003863">Quo<lb></lb>niam verò rotæ ſemidiameter ex hypotheſi major eſt, quàm <lb></lb>manubrij à centro diſtantia, nil mirum, ſi particulæ impetûs ex<lb></lb>tremæ rotæ impreſſi multum habeant momenti, quippe quæ <lb></lb>magis diſtant, & velociorem motum efficiunt. </s> </p> <p type="main"> <s id="s.003864">Quod verò ad cylindrum ſpectat, quem funis ductarius cir<lb></lb>cumplicat, non eſt neceſſe illum eſſe exactè & Geometricè ro<lb></lb>tundum, ſed ſatis eſt ſi cylindricam figuram æmuletur: catenus <lb></lb>ſiquidem rotundum axem conſtruimus, quatenus eadem volu<lb></lb>mus in convolutione ſervari momenta: ſi verò angulatus eſſet <lb></lb>axis, perpendiculum, in quo eſſet pondus, modò vicinum cen<lb></lb>tro eſſet, modò ab eo remotum, ac propterea ejuſdem remoti <lb></lb>majora eſſent momenta, quàm vicini. </s> <s id="s.003865">Sit enim ex. </s> <s id="s.003866">gr. qua<lb></lb>dratus Axis BDHG: utique per<lb></lb>pendiculum, in quo eſt funis reti<lb></lb><figure id="id.017.01.525.1.jpg" xlink:href="017/01/525/1.jpg"></figure><lb></lb>nens pondus quod attollitur, va<lb></lb>riam habet à centro C diſtantiam; <lb></lb>nam quando latus BD congruit fu<lb></lb>ni perpendiculari, diſtantia à cen<lb></lb>tro C æqualis eſt ſemiſſi lateris GB, <lb></lb>& eſt CI; cum verò latus BD in <lb></lb>converſione fit obliquum, diſtan<lb></lb>tia perpendiculi fit major, & eſt <lb></lb>CE, ita ut demùm diſtantia maxi<lb></lb>ma ſit æqualis ipſi CB; quæ iterum decreſcit, donec funis <lb></lb>congruat lateri BG. </s> <s id="s.003867">Potentiæ autem à centro diſtantia eadem <lb></lb>ſemper manet AC, ideòque momentorum potentiæ ad mo<lb></lb>menta ponderis Ratio ſubinde mutatur. </s> <s id="s.003868">Quòd ſi non quadra-<pb pagenum="510" xlink:href="017/01/526.jpg"></pb>tus ſit Axis, ſed plurium angulorum, ita ut latera breviſſima <lb></lb>ſint, ſicuti vix diſtat à rotunditate cylindri, ita vix momento<lb></lb>rum diſparitatem infert. </s> </p> <p type="main"> <s id="s.003869">Illud quidem animadverſione dignum eſt, quòd non te<lb></lb>merè ſtatuenda ſit Axi craſſitudo, ſed adeò validus eſſe de<lb></lb>bet ac firmus, ut ponderis gravitati obſiſtere poſſit, quin <lb></lb>flectatur, aut diſſiliat; ſi enim incurveſceret, augeretur mo<lb></lb>vendi difficultas, quia nimirum in converſione majorem <lb></lb>ambitum deſcriberet, quàm pro ejus ſoliditate. </s> <s id="s.003870">Sed neque <lb></lb>idcircò præter modum craſſus Axis eligi debet, quia quò <lb></lb>major ille eſt atque craſſior, eò major etiam eſt potentiæ <lb></lb>moventis labor, niſi pariter majus illi addatur Peritro<lb></lb>chium. </s> <s id="s.003871">Hinc fit contingere poſſe, ut in attollendo ponde<lb></lb>re augeatur labor potentiæ circa finem motûs; quia vide<lb></lb>licet, ſi ductarij funis ſpiræ jam univerſam cylindri faciem <lb></lb>circumplectantur, & ſequentes ſpiræ non cylindro cohæ<lb></lb>reant, ſed ſubjecto funi, jam intelligitur ſemidiameter axis <lb></lb>aucta craſſitudine funis ſubjecti, ac proinde ſecundus hic <lb></lb>ſpirarum ordo majorem funis longitudinem exigit, adeóque <lb></lb>etiam infert majorem ponderis motum, quo tempore poten<lb></lb>tia motum non majorem perficit: quare diminutâ Ratione mo<lb></lb>tûs potentiæ ad motum ponderis, minora fiunt illius momenta <lb></lb>ad attollendum pondus. </s> </p> <p type="main"> <s id="s.003872">Porrò non eſt omnino neceſſe, ut ad pondus attollendum <lb></lb>Axis ſtatuatur in ſuperiore loco, ſed fieri poteſt, ut longè al<lb></lb>tiùs elevetur pondus ſupra locum Axis; ſi nimirum funis <lb></lb>ductarius tranſeat per orbiculum ſuperiùs firmatum: Ve<lb></lb>rùm ita firmiter ſtabilienda eſt machina, ut hæc à nimiâ <lb></lb>ponderis gravitate non rapiatur ſurſum. </s> <s id="s.003873">Cæterùm cùm fu<lb></lb>nis immediatè nectitur ponderi inferiùs poſito, ipſa ponde<lb></lb>ris gravitas ſtabilit machinam ſuis fulcris inſiſtentem ſolo. </s> <lb></lb> <s id="s.003874">Hactenus quidem Axem rectum, prout magis communiter <lb></lb>uſurpatur, ſtatuimus; pro opportunitate tamen adhiberi etiam <lb></lb>poteſt curvatus. </s> <s id="s.003875">Quemadmodum ſi ex profluente aquam ſur<lb></lb>ſum antliâ propellere velimus, rotæ BC congruis prunis <lb></lb>inſtructæ, in quas aqua incurrens vim ſuam exerceat, addi<lb></lb>tur craſſior ferreus ſtylus centro A infixus, curvatúſque <lb></lb>ADEFGHIK (ſi IK ſit alter polus, cui machina incum-<pb pagenum="511" xlink:href="017/01/527.jpg"></pb>bit, nam ſi fulcrum ſit propè A inter A & D, ſufficit ſi in H <lb></lb>terminetur) ita ut ipſi DE <lb></lb>æqualis ſit particula HI, utri<lb></lb><figure id="id.017.01.527.1.jpg" xlink:href="017/01/527/1.jpg"></figure><lb></lb>uſque autem dupla FG, atque <lb></lb>inter EF & GH annulo inſe<lb></lb>ritur haſta adnexa embolo, ita <lb></lb>ut dum alter embolus attolli<lb></lb>tur, alter deprimatur. </s> <s id="s.003876">Hìc <lb></lb>attendenda eſt Ratio ſemidia<lb></lb>metri rotæ, ſeu diſtantiæ poten<lb></lb>tiæ à centro, ad DE, quæ eſt <lb></lb>ſemidiameter cylindri, qui ex <lb></lb>ejus convolutione gignitur; <lb></lb>perinde atque ſi eſſet cylin<lb></lb>drus, cujus tota diameter eſſet <lb></lb>FG: atque ideo non ex ipſius <lb></lb>ferrei ſtyli craſſitudine, ſed ex flexu æſtimanda eſt Axis ſe<lb></lb>midiameter; eatenus quippe craſſior, aut exilis ferreus ſtylus <lb></lb>eligitur, quatenus majore aut minore vi opus eſt in attollendo <lb></lb>atque deprimendo embolo. <lb></lb></s> </p> <p type="main"> <s id="s.003877"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003878"><emph type="center"></emph><emph type="italics"></emph>Succulæ & Ergatæ uſus conſideratur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003879">PEritrochij uſus quidem frequens eſt, ſed & ſæpiùs ſine rotâ <lb></lb>idem præſtatur, vel addito Axi manubrio, vel Radiis Axi in<lb></lb>fixis; & machina Latinis <emph type="italics"></emph>Succula<emph.end type="italics"></emph.end> dicitur; parvulus autem paxil<lb></lb>lus, cui funis ductarij caput adnectitur, <emph type="italics"></emph>Porculus<emph.end type="italics"></emph.end> nominatur: Si <lb></lb>tamen paxilli loco annulum cylindro adnectas, cui funis caput <lb></lb>inſertum firmetur, perinde eſt. </s> <s id="s.003880">Hu<lb></lb><figure id="id.017.01.527.2.jpg" xlink:href="017/01/527/2.jpg"></figure><lb></lb>juſmodi eſt cylindrus AB ſuis polis <lb></lb>inſiſtens congruo pegmati, infixoſ<lb></lb>que habens Radios CD, EF; quibus <lb></lb>manu arreptis cylindrus volvitur, & <lb></lb>funis adnexus paxillo I circumduci<lb></lb>tur cylindro, atque connexum in H <lb></lb>onus attollitur. </s> <s id="s.003881">Dupliciter autem <pb pagenum="512" xlink:href="017/01/528.jpg"></pb>ſucculâ uti licet, aut Radiis perpetuò infixis, aut qui cylindri <lb></lb>foraminibus dum verſatur, ſubinde inſerantur: Si perpetuò in<lb></lb>fixi maneant, unus poteſt Axem convertere alio poſt alium Ra<lb></lb>dio arrepto: at verò ſi idem Radius in aliud atque aliud fora<lb></lb>men immittendus ſit, duo ſint, oportet, qui alternâ operâ ſuum <lb></lb>Radium deprimentes Axem convolvant; alioquin, niſi artificio <lb></lb>aliquo retineatur, dum ex uno foramine extrahitur Radius, ut <lb></lb>in aliud immittatur, pondus ſuâ gravitate deorſum relaberetur. </s> <lb></lb> <s id="s.003882">Licet tamen hanc duplicis potentiæ neceſſitatem utilitate aliâ <lb></lb>compenſare; ubi enim duo ſint, quorum viciſſitudine circum<lb></lb>agatur Axis immiſſitio hujuſmodi Radio, hic poteſt eſſe multo <lb></lb>longior, quàm ſi eidem Axi infixus maneret; oporteret ſiqui<lb></lb>dem plures Radios perpetuò manentes infigere; id quod, ſi lon<lb></lb>giores eſſent, non careret incommodo. </s> <s id="s.003883">Quo autem longior <lb></lb>Radius fuerit, eò pariter faciliùs potentia movebit, quippe quæ <lb></lb>motum multò velociorem motu ponderis habebit, pro Ratione <lb></lb>longitudinis Radij plus ſemidiametro cylindri, ad eandem cy<lb></lb>lindri ſemidiametrum. </s> </p> <p type="main"> <s id="s.003884">Ad hoc fortaſſe genus revocari poſſunt Scytalæ oneribus <lb></lb>promovendis ſubjectæ, de quibus dictum eſt lib. 2. cap.9; cùm <lb></lb>harum capitibus aptè perforatis immittuntur ferrei aut lignei <lb></lb>vectes, quorum ope ſcytalæ ipſæ convertuntur, atque incum<lb></lb>bens onus dum ad aliam atque aliam orbitæ partem accommo<lb></lb>datur, promovetur. </s> <s id="s.003885">Quo enim longioribus vectibus utimur, <lb></lb>potentia circa cylindri centrum multò velociùs movetur <lb></lb>quàm impoſitum ſaxum, cujus motus æqualis eſt converſioni <lb></lb>peripheriæ. </s> <s id="s.003886">Nam quod motus abſolutè ſumptus ſit aliquantulo <lb></lb>major, quia centrum ipſum promovetur, nihil refert, quia <lb></lb>motus hic & cylindro ſubjecto, & oneri, & Potentiæ commu<lb></lb>nis eſt. </s> </p> <p type="main"> <s id="s.003887">Præter Succulam Radiis infixis inſtructam, cujuſmodi ea <lb></lb>eſt, quæ ad hauriendas è puteis aquas vulgò uſurpatur (quam<lb></lb>quam ob radiorum brevitatem & ipſius Axis craſſitudinem non <lb></lb>admodum potentiæ momenta augeantur) forma alia cæmenta<lb></lb>riis maximè familiaris eſt ad attollenda ſaxa, lateres, & calcem, <lb></lb>duplici manubrio in oppoſitas partes diſpoſito, ut quædam co<lb></lb>natuum conſtans ſimilitudo ſervetur, dum altero ſuum manu<lb></lb>brium deprimente, ſuum alter elevat: cùm enim vi brachio <pb pagenum="513" xlink:href="017/01/529.jpg"></pb>deorſum connitentium facilior contingat depreſſio, quàm ele<lb></lb>vatio, ſi manubriorum inflexio ad eandem partem collocaretur, <lb></lb>uterque ſimul deprimendo faciliùs axem converteret, at <lb></lb>uterque ſimul elevans aliquid amplius laboris ſubiret; alternis <lb></lb>autem elevationibus atque depreſſionibus labor temperatur. </s> <lb></lb> <s id="s.003888">Cæterum quod ad potentiæ momenta attinet, parum intereſt, <lb></lb>quam poſitionem manubria habeant viciſſim comparata; <lb></lb>ſpectatur videlicet ſingulorum longitudo & cujuſmodi motum <lb></lb>potentia manubrio applicata deſcribat: Sic manubrij longitu<lb></lb>do GH, hoc eſt potentiæ apprehen<lb></lb>dentis HO diſtantia perpendicula<lb></lb>ris ab axe cylindri, qui convolvitur, <lb></lb><figure id="id.017.01.529.1.jpg" xlink:href="017/01/529/1.jpg"></figure><lb></lb>attendenda eſt, & cùm ipſius cylin<lb></lb>dri, ſemidiametro comparanda, ut <lb></lb>Ratio motûs Potentiæ ad ponderis <lb></lb>motum innoteſcat, ac proinde Po<lb></lb>tentiæ momentum deſiniatur. </s> </p> <p type="main"> <s id="s.003889">Hinc apparet pro ipſius GH longitudine ad cylindri axem <lb></lb>productum perpendiculari augeri momenta potentiæ; perinde <lb></lb>namque ſe habet, ac ſi infixus eſſet cylindro Radius KI ipſi <lb></lb>GH æqualis; quia ut KI ad KE ſemidiametrum, ita GH ad <lb></lb>KE, & ambitus à potentia in H deſcriptus ad ejuſdem cylin<lb></lb>dri ambitum. </s> <s id="s.003890">Quare non leviter allucinantur, qui manubrij <lb></lb>longitudinem GH non rectam, ſed in hemicyclum curvatam <lb></lb>volunt quaſi hinc plus aliquid momenti potentiæ conferretur; <lb></lb>quamvis enim circuli ſemiperipheria ſit <expan abbr="ſaltẽ">ſaltem</expan> diametri ſeſquial<lb></lb>tera, potentiæ applicatæ motui non ſemiperipheria, ſed diame<lb></lb>ter legem ſtatuit: alioquin ſi ex ipſa manubrij inflexione mo<lb></lb>menta augerentur, ſatius eſſet non tantùm ſemiperipheriæ, ſed <lb></lb>majori circuli ſegmento ſimile eſſe <expan abbr="manubriũ">manubrium</expan>; id quod ſi expe<lb></lb>riri voluerint, tantum abeſt, ut movendi facilitatem acquirant, ut <lb></lb>potiùs momenta minui ſentiant; nam in circulo maximam <expan abbr="lineã">lineam</expan> <lb></lb>eſſe diametrum, & quò <expan abbr="majorũ">majorum</expan> <expan abbr="egmentorũ">ſegmentorum</expan> arcus majores fiunt, <lb></lb>minui ſubtenſas chordas, ex 15. lib.3. nôrunt ipſi Elementarij. </s> </p> <p type="main"> <s id="s.003891">Hac igitur manubrij longitudine perpensâ, non ſolùm non <lb></lb>eſt eorum æqualitas religiosè ſervanda, verùm author eſſem <lb></lb>cætementariis, ut manubriorum alterum paulò longius conſti<lb></lb>tuerent; cùm enim ut plurimum inæquales ſint operarum vi-<pb pagenum="514" xlink:href="017/01/530.jpg"></pb>res, ſi æqualia ſint manubria, qui infirmior eſt, plus ſubit <lb></lb>laboris, quàm ferre poſſit: at ſi alterum paulo longius ſit, de<lb></lb>biliorem illi applicari oportebit, ut minore incommodo præ<lb></lb>ſcriptum opus perficiat. </s> <s id="s.003892">Quòd ſi contingat ab unico homine <lb></lb>convertendam eſſe ſucculam, non erit contemnendum laboris <lb></lb>compendium, ſi poſſit longiore manubrio uti. </s> </p> <p type="main"> <s id="s.003893">Quamvis autem nullus ſtatuatur finis converſioni, quia funis <lb></lb>ductarius ſucculam non circumplectatur, eadem manet Ratio. </s> <lb></lb> <s id="s.003894">Si enim axi polygono inſiſtens catena ſingulis palmaribus, aut <lb></lb>majoribus intervallis adnexos globulos aut diſcos habeat tubo, <lb></lb>per quem tranſeunt, congruentes, qui intra tubum aquam <expan abbr="in-tercipiẽtes">in<lb></lb>tercipientes</expan> dum ex ſucculæ converſione attolluntur, aquam pa<lb></lb>riter elevant, ſecúmque rapiunt, perpetua fieri poteſt conver<lb></lb>ſio; pondus autem, quod movetur, eſt aqua tubum implens. </s> <lb></lb> <s id="s.003895">Ubi aliquorum imperitiam caſtigare oporteret, qui manubrij <lb></lb>longitudinem (quæ ipſe non ſine inſcitiæ admiratione vidi, <lb></lb>narro) minorem ſemidiametro axis, cui catena inſiſtit, conſti<lb></lb>tuunt, & operarum laborem fruſtra augent, dum minor eſt <lb></lb>potentiæ motus, quàm ponderis. </s> <s id="s.003896">Quid enim paulo majorem <lb></lb>longitudinem manubrio non tribuunt? </s> <s id="s.003897">minùs ſcilicet laboran<lb></lb>tes operæ concitatiùs axem volverent, & globuli celeriùs ele<lb></lb>vati minus aquæ elabi ſinerent. </s> </p> <p type="main"> <s id="s.003898">Jam verò ad Ergatam, quæ modicum à ſucculâ differt, tran<lb></lb>ſeamus, cujus uſus potiſſimùm eſt in trahendis oneribus, quan<lb></lb>quam illâ etiam, adhibitâ videlicet trochleâ, ad onera attollen<lb></lb>da uti poſſimus, & frequenter utamur. </s> <s id="s.003899">Quemadmodum autem <lb></lb>in ſucculæ poſitione eſt cylindrus ut plurimùm horizonti pa<lb></lb>rallelus, ita in Ergatâ ſtatuitur horizonti perpendicularis. </s> <s id="s.003900">Cy<lb></lb><figure id="id.017.01.530.1.jpg" xlink:href="017/01/530/1.jpg"></figure><lb></lb>lindro enim DC ita fir<lb></lb>mato, ut vel circa extre<lb></lb>mos polos, vel in locula<lb></lb>mento congruo converti <lb></lb>poſſit, additur vectis GEF <lb></lb>(aut etiam plures vectes <lb></lb>eidem cylindro infigun<lb></lb>tur) cui applicata Poten<lb></lb>tia dum cylindrum circa <lb></lb>ſuum axem verſat, fu-<pb pagenum="515" xlink:href="017/01/531.jpg"></pb>nemque convolvit, adnexam ſarcinam adducit. </s> <s id="s.003901">Æſtimatur au<lb></lb>tem potentiæ momentum ex ejuſdem Potentiæ diſtantiâ ab axe <lb></lb>cylindri, comparata cum ipſius cylindri ſemidiametro: quan<lb></lb>tus nimirum funis cylindrum circumplectitur, tantus eſt one<lb></lb>ris adducti motus, qui ad potentiæ motum eam habet Ratio<lb></lb>nem, quæ inter cylindri ambitum circularem, & peripheriam <lb></lb>Radio EF, aut EG, deſcriptam intercedit. </s> <s id="s.003902">Cum verò hujuſ<lb></lb>modi vecti EF tanta tribui poſſit longitudo, quantam ferre poſ<lb></lb>ſit ſpatium; in quo Potentia movetur, patet longiore vecte mo<lb></lb>mentum potentiæ pro arbitratu augeri poſſe. </s> <s id="s.003903">Verum quidem <lb></lb>eſt plures potentias eidem vecti EE applicatas inæqualia ha<lb></lb>bere momenta pro Ratione inæqualium diſtantiarum ab axe <lb></lb>cylindri. </s> </p> <p type="main"> <s id="s.003904">Sed illud maximè commodum accidit in Ergatâ, quod hìc <lb></lb>jumentorum ope hominum laborem minuere licet, dum illa <lb></lb>extremo vecti alligata, & in gyrum acta cylindrum convolvunt; <lb></lb>à quibus tamen ſubſidium petere in ſucculæ convolutione non <lb></lb>poſſumus; niſi fortè cylindrum horizontalem Verticali peritro<lb></lb>chio inſeramus, & extremam craſſioris peritrochij orbitam fu<lb></lb>nis circumplectatur; qui dum jumento trahente evolvitur, co<lb></lb>gat cylindrum converti, funémque, cui ſarcina adnectitur, cir<lb></lb>ca cylindri orbitam convolutum attollere pondus. </s> <s id="s.003905">Id quod <lb></lb>etiam præſtare valemus, ſi <lb></lb><figure id="id.017.01.531.1.jpg" xlink:href="017/01/531/1.jpg"></figure><lb></lb>trahendum ſit onus, ne<lb></lb>que in locum inducere li<lb></lb>ceat jumentum: nam per<lb></lb>pendiculari cylindro HI <lb></lb>peritrochium, ſeu tympa<lb></lb>num LM horizonti paral<lb></lb>lelum circumponitur, & <lb></lb>pluribus ſpiris tympano cir<lb></lb>cumducitur funis, quem <lb></lb>in O jumentum trahens <lb></lb>quamvis procul poſitum <lb></lb>explicat, atque cylindrum <lb></lb>convertit, ac propterea <lb></lb>onus in N adnexum ad<lb></lb>ducit. </s> </p> <pb pagenum="516" xlink:href="017/01/532.jpg"></pb> <p type="main"> <s id="s.003906">Porrò in funis ductarij circumvolutione circa ſucculæ aut <lb></lb>Ergatæ cylindrum obſervandum eſt, non eſſe neceſſe totum <lb></lb>funem circumvolvi, illique adnecti; nimis enim multus ali<lb></lb>quando eſſet, & non leve afferret incommodum; ut ſatis <lb></lb>conſtat, cùm ſolvendæ ſunt anchoræ, ſi craſſum illum ruden<lb></lb>tem totum cylindro circumduci opus eſſet, ut anchora è maris <lb></lb>fundo extrahatur. </s> <s id="s.003907">Satis igitur eſt, ſi funis duplici aut triplici <lb></lb>ſpirâ cylindrum circumplectatur, quando ingentia pondera <lb></lb>movenda ſunt; hæc ſiquidem valdè reſiſtunt, & ita funis circa <lb></lb>ipſum cylindrum conſtringitur, ut illum validè premat, nec fa<lb></lb>cilè poſſit excurrere, maximè ſi cylindrus non fuerit exquiſitè <lb></lb>tornatus; nimius ſcilicet partium ſe ſe mutuo contingentium <lb></lb>affrictus, qui cum cylindri ſuperficie fieri deberet, perinde re<lb></lb>ſiſtit, atque ſi funis paxillo aut annulo eſſet idem cylindro ad<lb></lb>nexus. </s> <s id="s.003908">Quare ſatis fuerit, ſi puer funem in converſione expli<lb></lb>catum colligat. </s> </p> <p type="main"> <s id="s.003909">Ex dictis tùm hoc, tùm ſuperiori capite, ſatis conſtat, quæ<lb></lb>nam longitudo ſtatuenda ſit Radio, cui potentia data applican<lb></lb>da eſt, ſi pariter cylindri ſemidiameter, & oneris gravitas detur. </s> <lb></lb> <s id="s.003910">Nam ſi fiat ut data Potentia ad datam ponderis gravitatem, ita <lb></lb>data cylindri ſemidiameter ad quæſitam Radij longitudinem, <lb></lb>habetur longitudo ſufficiens ad ſuſtinendum pondus in aëre <lb></lb>ſuſpenſum. </s> <s id="s.003911">Quare pro arbitratu augeatur longitudo Radij, &, <lb></lb>cùm facta jam ſit major Ratio motûs potentiæ ad motum pon<lb></lb>deris, quàm ſit Ratio gravitatis ponderis ad virtutem potentiæ <lb></lb>ſuſtinentis, illa poterit propoſitum pondus movere. </s> <s id="s.003912">Sic quo<lb></lb>niam in navibus ad proram jacet horizonti parallelus verſatilis <lb></lb>cylindrus (aut potiùs hexagonum ſeu octogonum priſma) cu<lb></lb>jus extremitas decreſcentibus crenis denticulata incumbentem <lb></lb>ligneam regulam ſingulis ſubinde crenis excipit, ne ponderis <lb></lb>vi in contrariam partem retroagi valeat, & cylindro circumdu<lb></lb>citur rudens (<emph type="italics"></emph>Piſma<emph.end type="italics"></emph.end> ab aliquibus dicitur) ex quo anchora pen<lb></lb>det; nec habere poteſt plures Radios perpetuò adnexos, quos <lb></lb>videlicet ſpatij anguſtiæ ferre non poſſent, ideò foramina quæ<lb></lb>dam habet, quibus, ubi opus fuerit, inſeruntur vectes. </s> <s id="s.003913">Ut <lb></lb>vectium longitudo ſtatuatur, anchoræ gravitas cum adjecto <lb></lb>ligneo tranſverſario conſideranda eſt, quæ eſt ferè ſub trecen<lb></lb>tupla gravitatis navis vacuæ, ut conſtat ex iis, quæ lib.4.cap.17. <pb pagenum="517" xlink:href="017/01/533.jpg"></pb>innuimus. </s> <s id="s.003914">Navis autem capacitas (hoc eſt pondus, quod <lb></lb>navis geſtare valet, & æquale eſt gravitati navis in aëre) <lb></lb>vel per dolia, ſeu amphoras aquæ, quam ſine incommodo <lb></lb>ferre poteſt, numeratur, ut ſolent Galli & Angli ſingulis do<lb></lb>liis navalibus libras bis mille tribuentes, vel per pondera, <lb></lb>quæ Hollandis atque Germanis <emph type="italics"></emph>Laſt<emph.end type="italics"></emph.end> dicuntur, ſingula li<lb></lb>brarum ſaltem quatuor millibus definita (nam <emph type="italics"></emph>Laſt<emph.end type="italics"></emph.end> Ham<lb></lb>burgi continet libras 4554, Amſtelodami, ſi ſit triticum <lb></lb>habet lib. 4800, ſin autem ſiligo lib. 4200, Stevinus verò <lb></lb>lib. 3. ſtaticæ pop. </s> <s id="s.003915">10 ſingulos modios definit lib. 360) & <lb></lb>ſingulis libris unciæ ſexdecim, ſeu <emph type="italics"></emph>Lotones<emph.end type="italics"></emph.end> 32, hoc eſt ſe<lb></lb>munciæ tribuendæ ſunt. </s> <s id="s.003916">Quare data navis capacitas ex. </s> <s id="s.003917">gr. <lb></lb>doliorum 400, multiplicetur per lib. 2000; & ſunt. </s> <s id="s.003918">lib. <lb></lb>800000, quarum pars trecenteſima lib. 2666 eſt ferè pon<lb></lb>dus anchoræ cum ligneo tranſverſario. </s> <s id="s.003919">Poſſunt autem non <lb></lb>plures applicari vectes quàm quatuor, ideóque ſinguli quar<lb></lb>tam ponderis partem elevare debent, hoc eſt lib. 666. Si <lb></lb>fuerit igitur cylindri ſemidiameter 3/4 pedis, & vis potentiæ <lb></lb>(quia ipſa corporis gravitas vectem premit) ſit elevandi <lb></lb>lib. 100, fiat ut 100 ad 666, ita 3/4 pedis ad pedes ferè <lb></lb>quinque; & hæc erit quæſita longitudo Radij, cui poten<lb></lb>tia applicanda eſt. <lb></lb></s> </p> <p type="main"> <s id="s.003920"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003921"><emph type="center"></emph><emph type="italics"></emph>Tympani à calcante circumacti vires <lb></lb>expenduntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003922">TYmpana, quæ Græcis <foreign lang="grc">γερανοι</foreign>, Latinis retentâ vocabuli in<lb></lb>terpretatione <emph type="italics"></emph>Grues<emph.end type="italics"></emph.end> dicuntur, hoc differunt à Succulâ, <lb></lb>quòd ab hominibus non brachiorum contentione, ſed cor<lb></lb>poris calcantis gravitate moventur. </s> <s id="s.003923">Horum autem frequen<lb></lb>tiſſimus eſt uſus tum in Hollandiâ tùm in Germaniâ juxta <lb></lb>fluvios navigabiles, ut ingentia pondera è navibus extra<lb></lb>hant, & in ripa deponant: quamquam & ad alios uſus fa-<pb pagenum="518" xlink:href="017/01/534.jpg"></pb>cilè traduci poſſint, ſi apto loco collocentur. </s> <s id="s.003924">Cylindro AB <lb></lb><figure id="id.017.01.534.1.jpg" xlink:href="017/01/534/1.jpg"></figure><lb></lb>verſatili, & horizon<lb></lb>ti parallelo; ac ritè <lb></lb>firmato, ampliorem <lb></lb>rotæ peripheriam <lb></lb>CDE circumponi<lb></lb>mus ex latioribus aſ<lb></lb>ſeribus compactam, <lb></lb>ut unus ſaltem ho<lb></lb>mo ingredi valeat; <lb></lb>qui dum ex C in D <lb></lb>aſcendere conatur, ſua gravitate deprimens tympanum, cy<lb></lb>lindrum pariter convertit, ductariúmque funem convolvit, <lb></lb>qui per orbiculos F & G ſuperiori trabi exporrectæ connexos <lb></lb>tranſiens, cum onere in P connectitur, atque adeò ex Cylin<lb></lb>dri converſione attollitur pondus, etiamſi à machinâ ipſa ab<lb></lb>ſit, quantum trabs exporrigitur. </s> <s id="s.003925">Si placuerit eidem cylindro <lb></lb>duplex tympanum, aut unicum valdè amplum apponere, lice<lb></lb>bit, ut plurium hominum operâ in elevando onere uti poſ<lb></lb>ſimus. </s> </p> <p type="main"> <s id="s.003926">Machinæ hujus vires eò majores eſſe, quò major eſt ſemi<lb></lb>diametri rotæ ad cylindri ſemidiametrum Ratio, ſatis mani<lb></lb>feſtum eſt ex iis, quæ ſæpiſſimè dicta ſunt; major eſt enim <lb></lb>potentiæ motus, quò amplior eſt rota. </s> <s id="s.003927">Cavendum tamen ne, <lb></lb>quemadmodum in ſucculâ atque Ergatâ, ita etiam hìc omnino <lb></lb>ex ipſa ſemidiametrorum rotæ & Cylindri Ratione definiantur <lb></lb>potentiæ momenta: hìc ſcilicet potentia tympanum movens eſt <lb></lb>inſita homini gravitas deorſum connitens; in ſucculâ autem <lb></lb>atque in Ergatâ potentia movens eſt impulſus ab animali facul<lb></lb>tate impreſſus, ac in gyrum directis. </s> <s id="s.003928">Quapropter in ſucculâ, at<lb></lb>que in Ergatâ cùm eadem ſit potentiæ directio ſimiliter appli<lb></lb>catæ in quocumque ſitu, eadem manent in converſione poten<lb></lb>tiæ momenta: at hominis tympanum calcantis non eædem ſem<lb></lb>per ſunt vires, ſed quo magis aſcendit versùs D, augentur ejus <lb></lb>momenta; quia videlicet perinde eſt, atque ſi à centro ad <lb></lb>punctum orbitæ, in quo eſt gravitas calcans, ducta eſſet linea; <lb></lb>ibi enim momentum deſcendendi eſt ut Sinus declinationis à <lb></lb>perpendiculo, juxta dicta lib.1.cap.15. </s> </p> <pb pagenum="519" xlink:href="017/01/535.jpg"></pb> <p type="main"> <s id="s.003929">Sit enim rotæ CHG ſemidiameter EB, cylindri verò ſe<lb></lb>midiameter EO. </s> <s id="s.003930">Si homo <lb></lb><figure id="id.017.01.535.1.jpg" xlink:href="017/01/535/1.jpg"></figure><lb></lb>tympanum ingreſſus con<lb></lb>ſiſtat in infimo loco H, in <lb></lb>quem ſcilicet cadit per<lb></lb>pendiculum EH, utique <lb></lb>machinam non movet, à <lb></lb>qua ipſe ſuſtinetur: Simi<lb></lb>liter ſi funi, OC per ſupe<lb></lb>rioris trabis orbiculos tran<lb></lb>ſiens fuerit intentus, etiam<lb></lb>ſi ex H aſcendat in D, in <lb></lb>quod punctum cadit recta <lb></lb>CO cylindrum tangens, <lb></lb>non movetur pondus, quod ex hypotheſi excedit gravitatem <lb></lb>hominis tympanum calcantis: quia nimirum in D homo ra<lb></lb>tione poſitionis non habet deſcendendi momenta majora, <lb></lb>quàm ſit ſemidiameter OE, quæ pariter ſunt momenta one<lb></lb>ris funi OC adnexi: Cùm autem ratione poſitionis momen<lb></lb>ta ponderis atque potentiæ æqualia ſint, ſed ratione gravita<lb></lb>tis potentia infirmior ſit ex hypotheſi quàm pondus, illa uti<lb></lb>que elevare pondus non valebit. </s> <s id="s.003931">Procedet igitur aſcenden<lb></lb>do ex. </s> <s id="s.003932">gr. uſque ad I, ubi obtinebit momenta ut VE (hoc <lb></lb>eſt IK ſinus anguli declinationis IEH) ad momenta one<lb></lb>ris ut OE, adeò ut quæ Ratio eſt VE ad OE, eadem ſit <lb></lb>Ratio gravitatis oneris ad gravitatem hominis; ac proinde <lb></lb>aſcendentis ex I in G momenta augebuntur, & in G erunt <lb></lb>ut SE. </s> <s id="s.003933">Cùm ergo SE ad OE Ratio major ſit quàm VE <lb></lb>ad OE, hoc eſt gravitatis oneris ad gravitatem hominis, <lb></lb>jam prævalet potentia, & tympanum convertitur, deſcen<lb></lb>dítque illius punctum G in locum, ubi erat punctum I, in <lb></lb>quo fit conſiſtentia & quoddam æquilibrium ſuſtentando <lb></lb>pondus, quod ut porrò moveatur, pergendum eſt in per<lb></lb>currendâ tympani orbitâ. </s> <s id="s.003934">Numquam igitur ratione poſitio<lb></lb>nis potentiæ momentum eſt ut ſemidiameter rotæ, niſi ho<lb></lb>mo ita aſcenderet, ut ejus centrum gravitatis reſponderet <lb></lb>puncto B; ex momento enim, quod potentia obtineret in <lb></lb>B, demendum eſt, quantum ab ipſo centro retrahitur: in G <pb pagenum="520" xlink:href="017/01/536.jpg"></pb>autem retrahitur juxta menſuram BS, & in I juxta menſuram <lb></lb>BV, ac propterea ibi momentum remanet ut SE, hìc ut VE. </s> <lb></lb> <s id="s.003935">Perinde autem ſe habere momentum in G ad pondus, atque ſi <lb></lb>eſſet libra curva GEF, & ab alterâ extremitate F diametri cy<lb></lb>lindri, duceretur recta FG ſecans in R perpendicularem EH, <lb></lb>manifeſtum eſt, quia ex 2. lib. 6. ut SE ad EF, hoc eſt OE, <lb></lb>ita GR ad RF. </s> <s id="s.003936">Quòd ſi tympani orbitam limbus hinc & hinc <lb></lb>ambiat, cui teretes paxilli inſerti veluti gradus ſcalas conſti<lb></lb>tuant, quibus homo non ſolùm inſiſtat pedibus, ſed quos etiam <lb></lb>manibus apprehendat; obſervare oportet, pedibuſne tantum <lb></lb>premat ſubjectum tympanum, an ex manibus quaſi ſuſpenſus <lb></lb>pendeat. </s> <s id="s.003937">Nam ſi in eodem perpendiculo non ſint paxillus, cui <lb></lb>inſiſtit, & is ex quo dependet, valde diſparia ſunt momenta. </s> <s id="s.003938">Si <lb></lb>verò non planè rectum ſit corpus, ſed quaſi procumbens incli<lb></lb>netur, tunc potentiæ locus definitur à perpendiculo tranſeunte <lb></lb>per centrum gravitatis ipſius hominis. </s> <s id="s.003939">Id quod dicendum pari<lb></lb>ter, quando tympano includitur canis (nam & à cane ingens <lb></lb>tympanum verſari vidi, quo in Solarium attollebatur non me<lb></lb>diocris ciſta linteis recens ablutis plena) cujus gravitatis cen<lb></lb>trum ſpectandum eſt, ejúſque diſtantia à perpendiculo tran<lb></lb>ſeunte per tympani centrum. </s> </p> <p type="main"> <s id="s.003940">Hinc ſi ex navi aliæ atque aliæ ſarcinæ hac machinâ extrahan<lb></lb>tur, is qui tympanum verſat, etiamſi omnino non videat onus ex<lb></lb>tra machinæ domunculam poſitum, facilè pronuntiabit major<lb></lb>ne? </s> <s id="s.003941">an minor ſit ſecundæ ſarcinæ gravitas comparata cum prio<lb></lb>re: quò enim magis aſcendere cogitur in tympano, eò major eſt <lb></lb>oneris gravitas; quærenda nimirum ſunt momenta majora ex <lb></lb>poſitione, ut majore intervallo abſit à perpendiculo EH tran<lb></lb>ſeunte per E centrum. </s> <s id="s.003942">Simili ratione, ſi inter duos homines <lb></lb>quæſtio oriatur uter illorum gravior ſit, facilè litem dirimes, ſi <lb></lb>alter poſt alterum ingrediatur tympanum, ut idem onus attollat; <lb></lb>qui enim magis aſcendere cogitur, minus habet gravitatis, ideò<lb></lb>que majora momenta quærit ex poſitione. </s> <s id="s.003943">Quanta autem ſit one<lb></lb>ris gravitas, dignoſcetur ex artificio ſtatim indicando. </s> <s id="s.003944">Unum hìc <lb></lb>quaſi per anticipationem addendum, quod ad funem ductarium <lb></lb>ſpectat; præſtat ſcilicet ejus extremitatem unco extremæ trabi <lb></lb>infixo adnecti, & per orbiculum cum onere conjunctum tranſi<lb></lb>re, atque hinc per orbiculos G & F ad cylindrum deduci: hac <pb pagenum="521" xlink:href="017/01/537.jpg"></pb>enim ratione attollendi facilitas geminatur, ut clariùs patebit ex <lb></lb>iis, quæ ſequenti libro de Trochleâ dicentur. </s> </p> <p type="main"> <s id="s.003945">Ut igitur innoteſcat, quanta ſit proximè oneris gravitas ob<lb></lb>ſervandus eſt in tympano locus, ubi homo illud calcans facit <lb></lb>cum pondere æquilibrium: quando ſcilicet eò venerit, ut paulo <lb></lb>altiùs aſcendens incipiat attollere pondus. </s> <s id="s.003946">Quoniam verò hu<lb></lb>juſmodi pondera ea ſunt, ut in iis exiguæ differentiæ contem <lb></lb>nantur, exquiſita quædam accuratio omnino ſupervacanea eſ<lb></lb>ſet, ſi ſingulas, aut pauculas libras ad calculos revocandas eſſe <lb></lb>cenſeremus, cum ſæpè non niſi per centenas libras eorum gra<lb></lb>vitas definiatur. </s> <s id="s.003947">Primùm ex centro E in ipsâ limbi craſſitudine <lb></lb>deſcribatur circuli peripheria BCH: id quod facilè fiet funi<lb></lb>culo extento, & axem FRO complectente; quo funiculo cir<lb></lb>cumducto ſtylus in extremitate colligatus deſcribet <expan abbr="peripheriã">peripheriam</expan>. </s> </p> <p type="main"> <s id="s.003948">Deinde ſi nota non ſit accurata ſemidiametri menſura, quæ <lb></lb>peripheriæ ſextantem accipiat, punctum unum, quod placuerit, <lb></lb>ſtatue, ex quo peripheriam in partes aliquotas (quaſcumque <lb></lb>tandem opportunitas dederit) dividere incipias: nam per nu<lb></lb>merum partium diviſis gradibus 360, ſtatim patebit, quot gra<lb></lb>dus ſingulæ partes contineant, quas aliquotas aſſumpſiſti. </s> <s id="s.003949">Par<lb></lb>tem igitur unam in gradus ſibi congruentes tribue; eorum enim <lb></lb>menſura in conſequentem arcum tranſlata, quoties oportuerit, <lb></lb>demùm integrum Quadrantem in gradus 90 diviſum dabit. </s> <s id="s.003950">Po<lb></lb>namus commodam accidiſſe diviſionem peripheriæ in partes 15: <lb></lb>diviſis gr. 360 per 15, quotiens 24 indicat numerum graduum <lb></lb>parti decimæ quintæ tribuendorum. </s> <s id="s.003951">Quare partem unam bifa<lb></lb>riam divide, & intervallum gr. 12 inter punctum diviſionis & <lb></lb>aſſumptum punctum, ex quo diviſio incipit, iterum divide bifa<lb></lb>riam, ut parti uni cedant gradus 6: his iterum bifariam diviſis, <lb></lb>habetur partis aliquotæ primò aſſumptæ pars octava gr.3. hanc <lb></lb>in tres æquales partes diſtribue, & ſingulorum graduum men<lb></lb>ſura manifeſta eſt. </s> <s id="s.003952">Acceptis itaque tribus partibus decimis <lb></lb>quintis addantur gradus 18, & habebitur integer peripheriæ <lb></lb>Quadrans in gradus 90 diſtributus, qui adeò notabiles erunt, <lb></lb>ut etiam gradûs partes, cujuſmodi eſt ſemiſſis, triens, & qua<lb></lb>drans ſatis clarè dignoſci queant. </s> </p> <p type="main"> <s id="s.003953">Tertiò. </s> <s id="s.003954">Quia non arcus HG, ſed ſemidiametri pars ES con<lb></lb>ſideratur, ut dictum eſt, concipe ſemidiametrum EB in partes <pb pagenum="522" xlink:href="017/01/538.jpg"></pb>aliquotas diſtributam, primùm in duas, deinde in tres, in qua<lb></lb>tuor, & deinceps, prout opportunum accidet, ita tamen, ut non <lb></lb>venias ad partem aliquotam minorem ſemidiametro Axis: Id <lb></lb>quod deprehendes, ſi aſſumptam chordam ſubtenſam gradibus <lb></lb>60, in illud genus partium aliquotarum, de quo dubitas, diviſe<lb></lb>ris, & in ſemidiametro BE incipiendo ab extremitate B illas ac<lb></lb>ceperis; ſi enim poſtrema pars aliquota reſidua major ſit ſemi<lb></lb>diametro Axis, aut illi æqualis, non eſt juſto minor. </s> <s id="s.003955">Igitur ex <lb></lb>Canone Sinuum exquire arcum ſingulis partibus, incipiendo à <lb></lb>centro tympani, congruentem, & in peripheriâ deſcriptâ atque <lb></lb>in gradus diſtributa arcum inventum ex Canone deſigna notâ <lb></lb>partis aliquotæ 1/2, 1/3, 1/4 &c. </s> <s id="s.003956">ut ſtatim appareat, quo loco intelli<lb></lb>gatur poſita potentia ſive in ſemiſſe, ſive in triente, ſive in qua<lb></lb>drante ſemidiametri, ſive in ejuſdem beſſe aut dodrante &c. </s> <lb></lb> <s id="s.003957">Factâ ſiquidem comparatione inter diſtantiam potentiæ ˊ cen<lb></lb>tro, & Axis ſemidiametrum, innoteſcet Ratio ponderis ad po<lb></lb>tentiam in tympano calcantem. </s> <s id="s.003958">Ponamus itaque tympani ſemi<lb></lb>diametrum diſtinctam in partes 10, ita ut Axis ſemidiameter <lb></lb>EO ad EB ſit ut 1 ad 10: poſſunt commodè omnes partes intra <lb></lb>decimas reperiri, pro ut in adjectâ tabellâ oculis ſubjicio, in <lb></lb>qualibèt minuta ſecunda exprimantur, ut innoteſcat etiam alios <lb></lb>in uſus, quibus Sinubus quinam arcus reſpondeant: in præſenti </s> </p> <p type="table"> <s id="s.003959">TABELLE WAR HIER <pb pagenum="523" xlink:href="017/01/539.jpg"></pb>tamen opere prorſus inutilis accideret tam exquiſita accuratio: <lb></lb>ſatis quippe eſt circiter illum gradum ejúſque minuta prima <lb></lb>rotam appingere, indicem partis, vel partium ſemidiametri <lb></lb>tympani. </s> </p> <p type="main"> <s id="s.003960">Quartò. </s> <s id="s.003961">Funiculum Axi inſiſtentem, & facilè excurrentem <lb></lb>ita diſpone, ut plumbeus globus in ejus extremitate pendulus <lb></lb>intendat funiculum ipſum, qui in tympani limbo deſignet <lb></lb>punctum, per quod tranſit linea perpendicularis ab Axis cen<lb></lb>tro in horizontem deſcendens. </s> <s id="s.003962">Tum ab hoc puncto uſque ad <lb></lb>punctum, ubi fit æquilibrium, ſumatur intervallum, atque <lb></lb>transferatur in Quadrantem gradibus diſtinctum: Nam <lb></lb>punctum, in quod ab initio Quadrantis cadit altera obſervati <lb></lb>intervalli extremitas, indicabit notâ in limbo prænotatâ, quotâ <lb></lb>ſemidiametri parte diſtet à tympani centro gravitas calcans <lb></lb>ipſum tympanum, ex. </s> <s id="s.003963">gr. 3/5 aut 4/7. Cum igitur jam innotuerit <lb></lb>Ratio ſemidiametri Axis ad ſemidiametrum tympani, ſcilicet <lb></lb>ex hypotheſi (1/10), fiat ut fractio index Rationis ſemidiametro<lb></lb>rum, ad fractionem in limbo notatam, ita gravitas calcantis <lb></lb>tympanum ad gravitatem ponderis, cum quo facit æquilibrium, <lb></lb>videlicet ut (1/10) ad 3/5, ita gravitas hominis, puta lib. 250, ad gra<lb></lb>vitatem oneris lib. 1500. Hinc patet in puncto D, per quod <lb></lb>tranſit linea OD tangens Axem & parallela perpendiculari <lb></lb>EH ex centro demiſſæ, æquilibrium eſſe inter gravitates om<lb></lb>nino æquales, ac proinde minimum pondus eſſe æquale gravi<lb></lb>tati hominis calcantis: Nam ſi inter H & D fieret æquilibrium, <lb></lb>pondus levius eſſet quàm homo, & communi ſtaterá facile po<lb></lb>tes aſſequi illius gravitatem. </s> <s id="s.003964">Maximum autem pondus eſt il<lb></lb>lud, quod indicat ſemidiameter tympani ad ſemidiametrum <lb></lb>Axis, homine nimirum ſuæ gravitatis vires exercente in B, ac <lb></lb>propterea gravitas ponderis ad gravitatem hominis in B eſſet <lb></lb>ex. </s> <s id="s.003965">gr. in Ratione decuplâ. </s> </p> <p type="main"> <s id="s.003966">Illud tamen hìc perpende, quòd, ſi homo calcans in B, aut <lb></lb>indè pendens, non volvit Axem, atque adeò non attollit pon<lb></lb>dus adnexum, non conſtat, an ſit æquilibrium, idem enim ac<lb></lb>cideret etiam, ſi pondus eſſet multò majus; ac proinde neque <lb></lb>conſtat de ejuſdem ponderis gravitate niſi hoc, quod ſit ut mi<lb></lb>nimum decupla gravitatis hominis; quia nimirum nunquam il-<pb pagenum="524" xlink:href="017/01/540.jpg"></pb>lud movebit; nam aſcendens homo ex B versùs C minora ſem<lb></lb>per obtinet momenta, quàm in B. </s> <s id="s.003967">Hoc autem ubi contigerit, <lb></lb>& velis exploratam habere oneris gravitatem, aſſume pondus <lb></lb>aliquod notæ gravitatis, quod adnectere valeas oræ tympani <lb></lb>aut in B, aut eo loco, ut deinde homo calcans infra B, attollat <lb></lb>pondus: ubi enim demum fiat æquilibrium, duplex inſtitue ra<lb></lb>tiocinium, alterum quidem ratione hominis, alterum verò ra<lb></lb>tione gravitatis additæ: inventi ſiquidem termini ſimul additi <lb></lb>indicabunt quæſitam oneris gravitatem. </s> <s id="s.003968">Sic ſi aſſumptum pon<lb></lb>dus ſit lib.36, & homo calcans ſit lib. 250, fiat autem æquili<lb></lb>brium homine exiſtente in X, pondere verò in G; ſumptis in<lb></lb>tervallis XH & GH, atque tranſlatis in Quadrantem invenia<lb></lb>tur pro homine (9/10), & pro pondere 4/5. Fiat primò ut (1/10) ad (9/10), ita <lb></lb>lib. 250 ad lib. 2250: deinde ut (1/10) ad 4/5, ita lib.36 ad lib. 288: <lb></lb>igitur ſumma lib. 2538 indicat ponderis gravitatem. </s> <s id="s.003969">Quòd ſi <lb></lb>funis ductarij extremitas ſit adnexa extremæ trabi, ut indica<lb></lb>tum eſt, atque tranſeat per orbiculum ponderi conjunctum, <lb></lb>inventus numerus lib.2538 duplicandus eſt & ſunt lib.5076. </s> </p> <p type="main"> <s id="s.003970">Ex his fortaſſe alicui placeat ſtateræ L M vires augere addi<lb></lb>to tympano hujuſmodi EB ut ſupra prænotato. </s> <s id="s.003971">Nam firmatâ <lb></lb><figure id="id.017.01.540.1.jpg" xlink:href="017/01/540/1.jpg"></figure><lb></lb>in ſuperiore loco ſtaterâ <lb></lb>LM, cujus anſa ſit in N, <lb></lb>primùm obſervetur, quan<lb></lb>tum ponderis requiratur <lb></lb>in M, ut fiat æquilibrium <lb></lb>cum ſolo brachio NL; <lb></lb>hæc enim gravitas ſemper <lb></lb>addenda erit gravitati, quæ <lb></lb>invenietur ex ratiocina<lb></lb>tione, qua componuntur <lb></lb>Rationes ſtateræ & tym<lb></lb>pani. </s> <s id="s.003972">Deinde in tympani <lb></lb>limbo notetur punctum C, <lb></lb>cui congruit funis OL, quando ſtatera eſt horizonti parallela; <lb></lb>ut hinc dignoſcatur, quo in loco tympani, dum convertitur, <lb></lb>contingat æquilibrium. </s> <s id="s.003973">Demum cùm nota ſit Ratio MN ad <lb></lb>NL, componatur cum Ratione ſemidiametri Axis ad partem <pb pagenum="525" xlink:href="017/01/541.jpg"></pb>ſemidiametri tympani, ex. </s> <s id="s.003974">gr. EO ad ES; & habetur Ratio <lb></lb>gravitatis hominis tympanum calcantis ad gravitatem oneris, <lb></lb>quod in M expenditur. </s> <s id="s.003975">Sit EO ad ES ut (1/10) ad 4/5, & MN ad <lb></lb>NL ut 2 ad 25; Ratio compoſita eſt ut 1/5 ad 20, hoc eſt ut 1 ad <lb></lb>100. Igitur pondus in M expenſum eſt, ut minimum, centu<lb></lb>plum gravitatis hominis; nam addenda præterea eſt gravitas <lb></lb>reſpondens gravitati brachij LN longioris ipſius ſtateræ. </s> </p> <p type="main"> <s id="s.003976">Si verò neque tympano, quod ab homine intùs calcante <lb></lb>premitur, neque adeò incerto ſacomate, cujuſmodi eſt varia <lb></lb>hominum gravitas, uti volueris, aut potius non ingentes ſarci<lb></lb>nas, ſed onera mediocria expendere placuerit, paretur CBH <lb></lb>diſcus ligneus parvulum axem habens ad centrum E, & in eo <lb></lb>deſcripta ſit peripheria circuli CBH, atque adnotatum <lb></lb>punctum C, per quod funiculus OL tranſit, quando ſtateræ <lb></lb>jugum LM eſt horizonti parallelum. </s> <s id="s.003977">Tum converſo diſco ita, <lb></lb>ut LO tranſeat per C, dimiſſo perpendiculo inſiſtente Axi, <lb></lb>notetur in peripheria punctum H, per quod tranſit perpendi<lb></lb>cularis à centro E. </s> <s id="s.003978">Deinde ex H versùs B aſcendendo acci<lb></lb>piantur gradus juxta ſuperiorem tabellam, affixis notis indici<lb></lb>bus partium, ſimiliter ac de tympano dictum eſt. </s> <s id="s.003979">Demum ſa<lb></lb>coma certæ gravitatis, puta unius aut alterius libræ, ita diſpo<lb></lb>natur, ut per diſci ambitum ex H versùs B excurrere poſſit, & <lb></lb>cochleâ firmari, ubi æquilibrium contigerit: Aut potiùs ſingu<lb></lb>lis Quadrantis gradibus, aut ſaltem punctis partium notatis, <lb></lb>claviculos infige, qui inſeri poſſint annulo ſacomatis. </s> <s id="s.003980">Nota <lb></lb>enim Ratio ſemidiametri axis EO ad diſci ſemidiametrum EB <lb></lb>indicabit, quid faciendum ſit juxta dicta, ut gravitas ponderis <lb></lb>in M ſuſpenſi innoteſcat. </s> <s id="s.003981">At ſi volueris Quadrantem HB in <lb></lb>ſuos 90 gradus diſtribuere, & uti Canone Sinuum, priùs inno<lb></lb>teſcat Ratio ſemidiametri axis ad Radium in partibus Radij: <lb></lb>deinde fiat ut partes Radij axi congruentes, ad Sinum Rectum <lb></lb>graduum, ubi fit æquilibrium, ita Sacoma appenſum ad gra<lb></lb>vitatem ponderis, quod expenditur. <pb pagenum="526" xlink:href="017/01/542.jpg"></pb></s> </p> <p type="main"> <s id="s.003982"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003983"><emph type="center"></emph><emph type="italics"></emph>An Axis in Peritrochio inveniatur etiam ſine <lb></lb>tractione.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.003984">HActenus ductarij funis converſionem circa Axem convolu<lb></lb>tum conſideravimus, ex quo oritur ponderis fune connexi <lb></lb>tractio: ſed numquid non etiam ad hoc genus machinæ aliquæ <lb></lb>revocari poſſunt, quibus non quidem trahitur pondus, ſed ali<lb></lb>qua reſiſtentia ſuperatur? </s> </p> <p type="main"> <s id="s.003985">Occurrit autem primo loco antiquus ſervorum metus Piſtri<lb></lb>num, in quod detruſi frumentum tundere cogebantur molâ <lb></lb>verſatili, ſive in noſtrarum Moletrinarum ſpeciem ac ſimili<lb></lb>tudinem metam congruo Catillo impoſitam manu truderent, ac <lb></lb>circumagerent, ſive ingentem lapideum diſcum perpendicula<lb></lb>ri cylindro coagmentatum verſarent aſellorum vicarij laborio<lb></lb>ſo muneri ſuccedentes, cum Vectis cylindro ad angulos rectos <lb></lb>infixi extremitatem aut traherent, aut propellerent: Cujuſmo<lb></lb>di machinæ genere nos quoque utimur in frendendis legumi<lb></lb>nibus, & in contundendis ſeminibus, ex quibus demùm oleum <lb></lb>prælo exprimitur. </s> <s id="s.003986">Et hìc quidem non ipſius molaris lapidis <lb></lb>gravitatem movendam attendimus, quippe qui ipſius machinæ <lb></lb>pars eſt; ſed potiſſimum corporis à molâ compreſſi reſiſtentia <lb></lb>conſideranda eſt, quæ nimirum vincenda proponitur. </s> <s id="s.003987">Oritur <lb></lb>autem hæc reſiſtentia ex corporis obterendi aut contundendi <lb></lb>duritiæ, in quod incurrit ſcabra molæ circumactæ ſuperficies; <lb></lb>cum verò illud incumbenti lapidi ſe ſubducere non poſſit, à la<lb></lb>pidis gravitate & potentiæ impetu cogitur diſſilire in partes. </s> <lb></lb> <s id="s.003988">Quia igitur potentiæ cum machinâ connexæ motum impedit <lb></lb>illa granorum aut nucleorum frangendorum durities, compa<lb></lb>randa eſt diſtantia potentiæ moventis à centro motûs, cum <lb></lb>diſtantiâ corporis comminuendi; & quò major eſt hujuſmodi <lb></lb>intervallorum Ratio, majora pariter ſunt potentiæ momenta. </s> </p> <p type="main"> <s id="s.003989">Hinc vides, cur in truſatili mola (quam mediocrem eſſe <pb pagenum="527" xlink:href="017/01/543.jpg"></pb>oportet, ne nimio labore frangatur molitor in immani ſaxo ver<lb></lb>ſando, catillus quidem planus eſt, meta verò, quâ catillum <lb></lb>reſpicit, non omnino ſubjecto plano congruit, ſed cavam ob<lb></lb>tuſiſſimi coni ſuperficiem æmulatur: ut ſcilicet integra grana <lb></lb>per medium foramen immiſſa inter utrumque lapidem interci<lb></lb>piantur non procul à centro, à quo potentia abeſt, comminu<lb></lb>ta autem peripheriam versùs accedant ad anguſtiora ſpatia, <lb></lb>quò magis obterantur: cùm enim integra grana magis fractio<lb></lb>ni obſiſtant, quàm comminuta, integris frangendis majora de<lb></lb>bentur potentiæ momenta, comminutis in minuſculas particu<lb></lb>las redigendi, minores vires ſufficiunt. </s> <s id="s.003990">In molâ autem Aſina<lb></lb>riâ ubi lapideus diſcus in plano Verticali conſtitutus ſub<lb></lb>jectum catillum modicè excavatum vix extremo ambitu con<lb></lb>tingit, eadem ferè eſt ſemper diſtantia à centro motûs, niſi <lb></lb>quatenus ipſius molæ craſſitudo partem aliam centro motûs pro<lb></lb>piorem, aliam remotiorem habet: porrò grana illa, quæ lapi<lb></lb>dum contactui, vel quaſi contactui, propiora ſunt, validiùs te<lb></lb>runtur, quàm quæ magis ab eodem contactu recedunt: ſed <lb></lb>hoc nihil ad præſentem diſputationem attinet, niſi quatenus la<lb></lb>pidis partes remotiores ſubjecta grana agitantes, atque tunden<lb></lb>tes craſſiùs, majorem Rationem ad potentiæ motum habent in <lb></lb>ſuâ convolutione, quàm partes ejuſdem minùs à centro <lb></lb>remotæ. </s> </p> <p type="main"> <s id="s.003991">Haud diſſimili ratione, ſi ex chalybe ellipticum ſphæroides <lb></lb>obliquis ſtriis modicè aſperum, quaſi in limæ ſpeciem, congruo <lb></lb>loculamento interiùs pariter aſperato includatur, ita tamen, ut <lb></lb>ſpatium, quo ſphæroides à loculamento diſtat, paulatim à lati<lb></lb>tudine in anguſtias ſe ſe contrahat; axi verò ſphæroidis ſupe<lb></lb>riùs producto addatur manubrium, quo arrepto converti poſ<lb></lb>ſit in gyrum; grana piperis, aut ſimilia ſuperiùs immiſſa leviſ<lb></lb>ſimo negotio comminuentur: quorum ſcilicet durities ſi cum <lb></lb>potentiæ viribus conferatur, reſiſtentiam habet maximam pro <lb></lb>Ratione ſemidiametri tranſverſæ Ellipſis ad manubrij longitu<lb></lb>dinem: initio autem, quia grana minùs ab axe diſtant, minùs <lb></lb>reſiſtunt, ſi cætera fuerint paria, hoc eſt, ſi modicè comminu<lb></lb>torum durities, integrorum duritiei omnino reſpondeat; nam <lb></lb>minor diſtantia à centro motûs minorem habet Rationem, <lb></lb>quàm diſtantia major ad eandem manubrij longitudinem. </s> </p> <pb pagenum="528" xlink:href="017/01/544.jpg"></pb> <p type="main"> <s id="s.003992">Par erit philoſophandi ratio, ſi tympanis non idem centrum <lb></lb>habentibus incluſa aqua ex interioris tympani converſione ad <lb></lb>anguſtias redigatur, atque compreſſa exprimatur ex tubo; cu<lb></lb>juſmodi fortaſſe fuit veterum Hydracontiſterium; de cujus for<lb></lb>mâ non eſt hìc diſputandi locus; nam manubrij à potentiâ <lb></lb>commoti longitudo comparanda eſt cum diſtantia peripheriæ <lb></lb>tympani aquam comprimentis à centro, circa quod perficitur <lb></lb>motus, ut momentorum Ratio perſpecta ſit; aqua ſcilicet dum <lb></lb>impellitur, atque exprimitur, reſiſtit. </s> </p> <p type="main"> <s id="s.003993">Ad hæc porrò inverſus quidam Axis in peritrochio uſus con<lb></lb>ſiderandus eſt, quando videlicet potentia non peritrochio, ſed <lb></lb>ipſi Axi, applicatur; id quod tunc potiſſimum contingit, cùm <lb></lb>potentia vitibus abundat, motui autem, qui efficiendus eſt, <lb></lb>non admodum reſiſtit corpus, quod vel modicè impellendum <lb></lb>eſt, vel in orbem circumducendum. </s> <s id="s.003994">Certum quippe eſt poten<lb></lb>tiam Axi applicatam tardiùs multò moveri, quàm peritrochij <lb></lb>peripheriam, pro Ratione ſemidiametrorum Axis & Peritro<lb></lb>chij, ac proinde licèt impetus amplioris peripheriæ partibus <lb></lb>impreſſus imbecillior quodammodo ſit, ut pote diſtractus, ſatis <lb></lb>tamen eſſe ad vincendam levem reſiſtentiam. </s> <s id="s.003995">Hinc quoniam <lb></lb>ferrum cotis tritu extenuatur, eóque faciliùs, quò celeriùs eos <lb></lb>movetur, qui reſtituunt obtuſas cultorum aut novacularum <lb></lb>acies, lapidem ex cotariâ eductum in diſcum rotundant, ut cir<lb></lb>ca axem centro infixum verſatilis circumagi poſſit. </s> <s id="s.003996">Quamvis <lb></lb>autem non rarò eidem axi cohæreat manubrium, quo circum<lb></lb>ducto rotatur lapis, ut tamen minori labore adhuc etiam velo<lb></lb>ciùs rotetur, ſapienter inſtituerunt amplioris rotæ abſidi exca<lb></lb>vatæ funem inſiſtere, qui rotulam eumdem cum lapide axem <lb></lb>habentem circumplectatur, ut minor hæc rotula amplioris ro<lb></lb>tæ ductum ſequens ſecum pariter rapiat cotem; cujus periphe<lb></lb>ria, cùm adnexam rotulam valdè excedat, velociùs quoquè <lb></lb>movetur. </s> <s id="s.003997">Quantùm verò motus hic, celeritate ſuâ, potentiæ <lb></lb>motum ſuperet, facilè conſtabit, ſi motuum ſingulis membris <lb></lb>convenientium ratio ineatur. </s> <s id="s.003998">Sit ex. </s> <s id="s.003999">gr. manubrij longitudo <lb></lb>ad amplioris rotæ ſemidiametrum ſubquadrupla, hæc autem ſe<lb></lb>midiameter ad rotulæ ſemidiametrum ſit octupla: demum ro<lb></lb>tulæ eundem cum cote axem habentis ſemidiameter ſit ſubtri<lb></lb>pla ſemidiametri ipſius cotis. </s> <s id="s.004000">Igitur puncti in cotis peripheriâ <pb pagenum="529" xlink:href="017/01/545.jpg"></pb>notati motus triplo velocior eſt motu ſimilis puncti in rotæ <lb></lb>peripheriâ: rotulæ motum definit funis, qui in convolutione ex<lb></lb>plicatur, hic enim pariter majoris rotæ motum metitur: octies <lb></lb>ergo rotula, & cum eâ lapis rotatur, dum amplior rota ſemel in <lb></lb>gyrum agitur. </s> <s id="s.004001">Quoniam verò rotæ ſemidiameter eſt ad cotis <lb></lb>ſemidiametrum ut 8 ad 3 ex hypotheſi, dum punctum in rotæ, <lb></lb>peripheriá notatum movetur velocitate ut 8, ſimile punctum <lb></lb>cotis movetur velocitate ut 24. Atqui motus rotæ cum motu <lb></lb>potentiæ manubrio applicatæ comparatus eſt ut 4 ad 1, ex hy<lb></lb>potheſi; igitur ſi duæ Rationes 1 ad 4, & 8 ad 24 componan<lb></lb>tur, erit Ratio motus potentiæ ad motum cotis ut 1 ad 12. Sunt <lb></lb>hìc itaque duo Axes in Peritrochiis ſuis compoſiti, & Potentia <lb></lb>Axibus applicata intelligitur; cùm enim in Verticali plano lapis <lb></lb>ipſe verſetur ſuper polos læves atque politos, non admodum re<lb></lb>pugnat motui; impreſſus autem impetus aliquandiu manens <lb></lb>potentiam ipſam juvat. </s> </p> <p type="main"> <s id="s.004002">Simile quiddam obſervandum occurrit in <expan abbr="horologiorũ">horologiorum</expan> motu, <lb></lb>quæ in turribus ſtatuuntur: nam cylindrum horizonti paralle<lb></lb>lum circumplicat funis, quo vi ponderis deſcendentis explica<lb></lb>to, circumagitur rota eidem axi infixa: ex hac in conſequentes <lb></lb>rotas derivatur motus ſemper velocior, qui demum temperatur <lb></lb>ex quadam motûs retardati & breviſſimæ morulæ viciſſitudine, <lb></lb>cum poſtremæ rotæ dentes in ſerræ modum conformati fuſum, <lb></lb>cui Tempus adnectitur alternis motibus agunt. </s> <s id="s.004003">Primùm enim <lb></lb>dens rotæ ſuperior in pin<lb></lb><figure id="id.017.01.545.1.jpg" xlink:href="017/01/545/1.jpg"></figure><lb></lb>nulam A incurrens eam <lb></lb>impellit, axémque HL <lb></lb>convertit unà cum tranſ<lb></lb>verſario CD & adjunctis <lb></lb>globulis plumbeis E & F, <lb></lb>qui ſimilem arcum deſcri<lb></lb>bunt, ac pinnula A, ſed <lb></lb>longè majorem; propterea <lb></lb>pro ratione gravitatis glo<lb></lb>bulorum, eorúmque di<lb></lb>ſtantiæ ab axe, HL, etiam <lb></lb>major vis requiritur, adeóque impeditur, ac retardatur motus <lb></lb>rotæ dentatæ, & cum eâ reliquarum rotarum, atque ipſius pon-<pb pagenum="530" xlink:href="017/01/546.jpg"></pb>deris, à quo totius machinæ motus initium ſumit: qui ſi fuerit <lb></lb>juſto velocior, globuli E & F removentur ab L, ſin autem <lb></lb>juſto tardior, admoventur, ut modò major, modò minor ſit re<lb></lb>ſiſtentia. </s> <s id="s.004004">Deinde quia globuli E & F ex impulſu pinnulæ A <lb></lb>impetum conceperunt ad certam plagam directum, pergerent <lb></lb>illorſum moveri, quamdiu impreſſus impetus perſeveraret, niſi <lb></lb>in eâ converſione inferior pinnula B occurreret inferiori denti <lb></lb>rotæ ſerratæ: hinc fit vi hujus impetûs breviſſimam morulam in<lb></lb>ferri converſioni rotæ, quæ eandem pinnulam urgens ipſos quo<lb></lb>què globulos in contrariam plagam reflectit. </s> <s id="s.004005">Poſſe autem adeò <lb></lb>exilibus viribus morulam inferri tanto ponderi deſcendenti, <lb></lb>paulò inferiùs manifeſtum fiet, ubi de Rotis dentatis in unam <lb></lb>machinam compactis diſſeretur; illud ſaltem palam eſt, ſi mo<lb></lb>rulam nullam admittas, reſiſtentiam eſſe non ſolùm pro gravi<lb></lb>tate globulorum, eorúmque diſtantiâ, verùm etiam pro ratione <lb></lb>impetûs impreſſi in antecedenti impulſione. </s> <s id="s.004006">Quare globuli <lb></lb>iidem quando moventur impulsâ pinnulâ, rationem habent <lb></lb>ponderis peritrochio adnexi, & potentia impellens pinnulæ, <lb></lb>hoc eſt axi, applicatur: Contrà verò ad retardandum, aut tan<lb></lb>tiſper coërcendum motum ponderis, quod pinnulæ applicatum <lb></lb>intelligitur, iidem globuli vi impetûs ſibi impreſſi rationem ha<lb></lb>bent potentiæ peritrochio applicatæ. </s> </p> <p type="main"> <s id="s.004007">Quoniam verò hìc horologiorum mentio incidit, cur in illis, <lb></lb>quæ ſecum quiſque ad perpetuum uſum ferre poteſt, catenula <lb></lb>ſeu nervus cono, non cylindro, circumducatur, manifeſtum <lb></lb>eſt: quia ſcilicet chalybea lamella, à qua motûs origo eſt, initio <lb></lb>in ſpiſſiorem ſpiram contracta ſuam vim elaſticam exerens va<lb></lb>lidiùs conatur ſe reſtituere, trahénſque catenulam, ſeu ner<lb></lb>vum FE, totum conum DEC eíque con<lb></lb><figure id="id.017.01.546.1.jpg" xlink:href="017/01/546/1.jpg"></figure><lb></lb>nexam rotulam dentatam AB in gyrum agit; <lb></lb>cum verò illa fuerit in paulò laxiores ſpiras <lb></lb>explicata, languidiùs conatur, atque catenu<lb></lb>lam, ſeu nervum, trahens jam non propè verti<lb></lb>cem coni, ſed magis ad baſim, eandem rotam <lb></lb>AB circumagit. </s> <s id="s.004008">Cùm igitur motus potentiæ <lb></lb>propè verticem coni, ad motum rotæ AB mi<lb></lb>norem habeat Rationem, quàm ad eundem rotæ motum mo<lb></lb>tus potentiæ in latiore coni parte (ibi enim breviorem, hìc ma-<pb pagenum="531" xlink:href="017/01/547.jpg"></pb>jorem gyrum perficit) ut quædam motûs æquabilitas in horo<lb></lb>logio ſervetur, opportunum fuit potentiæ validiùs conanti ma<lb></lb>jorem opponi reſiſtentiam, minorem verò languidiùs conanti: <lb></lb>nam ſi catenula non conum, ſed cylindrum circumplecte<lb></lb>retur, eadem ſemper eſſet motuum menſura atque Ratio, <lb></lb>ſed inæquales vires elaſticæ motum inæqualiter velocem ef<lb></lb>ficerent. </s> </p> <p type="main"> <s id="s.004009">Huc pariter revocandas eſſe Terebrarum vires vix cui<lb></lb>quam dubium eſſe poteſt, quarum quò ampliora ſunt ma<lb></lb>nubria, majores pariter eſſe vires conſtat; quandoquidem <lb></lb>potentia ampliorem circulum deſcribit, dum terebræ acies <lb></lb>minimo motu ligni aut metalli particulas, in quas incurrit, <lb></lb>abſcindit. </s> <s id="s.004010">Quæcumque demum ſit terebræ forma, five ejus <lb></lb>apex in cochleam ſtriatam exacutam <lb></lb>deſinat, ut AB manubrium habens <lb></lb><figure id="id.017.01.547.1.jpg" xlink:href="017/01/547/1.jpg"></figure><lb></lb>CD rectum, ſive in aciem obli<lb></lb>quam aut planam, aut modicè ex<lb></lb>cavatam exeat ut EF, manubrium <lb></lb>autem LHI inflexum habeat circa <lb></lb>GI verſatile (quam Zerebram Gal<lb></lb>licam aliqui Itali appellant) ſive fer<lb></lb>rea lamina in orbem convoluta, & <lb></lb>inferiùs denticulata, ut MN, ma<lb></lb>nubrio tranſverſo OP coaptetur: Si<lb></lb>militer ſemper eſt momentorum Ra<lb></lb>tio deſumenda aut ex tranſverſarij <lb></lb>CD longitudine ad craſſitiem co<lb></lb>chleæ ſtriatæ B, aut ex diſtantiâ <lb></lb>puncti H à lineâ tranſeunte per GIEF ad integram ſeu di<lb></lb>midiatam aciei F latitudinem (prout extremum punctum F in <lb></lb>mediâ aut in extremâ latitudine poſitionem habet) aut ex ma<lb></lb>nubrij OP longitudine ad diametrum circularis ſerræ NM: <lb></lb>partes autem ſubjecti corporis abſcindendæ, ut illud perfore<lb></lb>tur, habent rationem ponderis movendi eò difficiliùs, quò va<lb></lb>lidiore nexu illæ invicem conjunguntur. </s> </p> <p type="main"> <s id="s.004011">Eadem erit philoſophandi methodus in eo terebræ gene<lb></lb>re, cui nos Itali proximè ad Græcum vocabulum <foreign lang="grc">τρύπανον</foreign><lb></lb>accedentes nomen fecimus. </s> <s id="s.004012">Teretis baculi BC extremitati C <pb pagenum="532" xlink:href="017/01/548.jpg"></pb>additur chalybea cuſpis CD ita in punctum deſinens, ut ad <lb></lb><figure id="id.017.01.548.1.jpg" xlink:href="017/01/548/1.jpg"></figure><lb></lb>aliquam latitudinem obliquè aſcendat pro <lb></lb>ratione ſemidiametri foraminis, quod ma<lb></lb>ximum artificis animus deſtinavit. </s> <s id="s.004013">Inferior <lb></lb>baculi pars infigitur ſphæroidi H, & tranſ<lb></lb>verſarium GI in medio E ita perforatum <lb></lb>eſt, ut facillimè per inſertum baculum ex<lb></lb>currens ſurſum deorſum agitari poſſit: <lb></lb>cum enim extremitates G & I adnexum <lb></lb>funiculum habeant pertingentem uſque <lb></lb>in B, hoc circa baculum contorto tranſverſarium non procul <lb></lb>abeſt à B, quod ſi deprimatur, explicatur funiculus, & bacu<lb></lb>lus in gyrum agitur pariter cum infixa cuſpide; qua ſenſim ac <lb></lb>leniter minimas ſubjectæ laminæ metallicæ particulas abraden<lb></lb>te, demùm ſæpiùs repetitâ tranſverſarij ſurſum deorſum agita<lb></lb>tione, atque adeò celeri terebellæ converſione, foramen patet. </s> <lb></lb> <s id="s.004014">Quamvis autem artificis manus applicetur medio tranſverſario <lb></lb>in E, quod deprimit; potentia tamen intelligitur applicata ſu<lb></lb>perficiei baculi medio funiculo illum circumplexo, perinde at<lb></lb>que ſi inter utramque palmam alternis motibus adductam atque <lb></lb>reductam idem baculus convolveretur: tantóque major eſt po<lb></lb>tentiæ ſic applicatæ motus, quanto exceſſu baculi ambitus ſu<lb></lb>perat terebellæ ſubjectam laminam abradentis gyrum. </s> <s id="s.004015">Quo<lb></lb>niam verò potentia, hoc eſt manus, movetur deſcendendo, <lb></lb>ejus motus comparandus eſt cum multiplici convolutione ba<lb></lb>culi, quæ fit, dum explicatur funis. </s> </p> <p type="main"> <s id="s.004016">Sed & alia potentia hìc conſideranda occurrit: adjunctum <lb></lb>enim ſphæroides H, quod mediocriter grave ſtatuitur, non ſo<lb></lb>lùm ſuo pondere juvat, ut cuſpis paulò preſſiùs adhæreat ſub<lb></lb>jectæ laminæ, verùm etiam concepto in convolutione impetu <lb></lb>dum explicatur funis, pergit ad eaſdem partes moveri, expli<lb></lb>catúmque funem iterum circa baculum contorquens cogit <lb></lb>tranſverſarium GI aſcendere versùs B, quo viciſſim ab artificis <lb></lb>manu depreſſo in contrarias partes volvitur. </s> <s id="s.004017">Impetus igitur <lb></lb>ſphæroidi H impreſſus, dum illud movet, rationem habet po<lb></lb>tentiæ cuſpidem in gyrum contorquentis, cujus momenta ex <lb></lb>diſtantiâ ab axe, circa quem efficitur motus, definienda ſunt. <pb pagenum="533" xlink:href="017/01/549.jpg"></pb></s> </p> <p type="main"> <s id="s.004018"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004019"><emph type="center"></emph><emph type="italics"></emph>Axium in ſuis peritrochiis compoſitione vires <lb></lb>augentur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004020">COntingere poteſt, & quidem non rarò, ad ſervandam pe<lb></lb>ritrochij & axis cum pondere & potentiá analogiam, tam <lb></lb>ingens tympanum aut manubrium Axi coaptandum eſſe, ut <lb></lb>aut loci anguſtiæ commodè illud non patiantur, aut non niſi <lb></lb>majore diſpendio, quàm ſit operæ pretium, tam ampla ma<lb></lb>china parari, aut congruè diſponi queat. </s> <s id="s.004021">Quid enim, ſi ſpecta<lb></lb>tâ potentiæ validioris virtute centuplum onus ſuſtollendum <lb></lb>proponatur? </s> <s id="s.004022">an craſſiori Axi, qui ſatis firmus ſit, rotam aut <lb></lb>tympanum cujus diameter centupla ſit diametri Axis, adjun<lb></lb>gemus? </s> <s id="s.004023">quanto id incommodo futurum eſſet, quantáque ſub<lb></lb>ſidia comparare oporteret, ut tam immanis machina citrà luxa<lb></lb>tionem ſubſiſteret, & congruo pegmati inniteretur, nemo non <lb></lb>videt. </s> <s id="s.004024">Satius itaque fuerit, inſiſtendo iis, quæ lib.2 cap.7. dicta <lb></lb>ſunt, machinam, quam ad centuplam altitudinem augere in<lb></lb>commodum accideret, componere, pluribus Axibus cum ſuis <lb></lb>peritrochiis invicem ritè coaptatis. </s> </p> <p type="main"> <s id="s.004025">Statuendus primùm eſt Axis, cujus ſoliditas oneris gravi<lb></lb>tati ſuſtinendæ reſpondeat, longitudo circumflexas ducta<lb></lb>rij funis ſpiras commodè capiat. </s> <s id="s.004026">Deinde tympanum eligatur, <lb></lb>cujus diameter ad conſtituti cylindri diametrum eam habeat, <lb></lb>quæ placuerit, Rationem, modò illa ſit majoris inæqualitatis, <lb></lb>ut manifeſtum eſt: ſit ex. </s> <s id="s.004027">gr. quintupla, & cylindri craſſities <lb></lb>palmaris ponatur. </s> <s id="s.004028">Sed quoniam propoſito ponderi attollendo <lb></lb>impar eſt potentia applicata machinæ reſiſtentiam gravitatis in <lb></lb>Ratione ſolùm quintuplâ extenuanti, alium adhibe Axem ſuo <lb></lb>Peritrochio infixum (vel quem fors tulerit antè in alios uſus <lb></lb>paratum, vel ſecundùm deſtinatam Rationem elaboratum) cu<lb></lb>jus ductarius funis ipſi quidem Axi congruo loco conjungatur, <lb></lb>ſed tympanum prioris Axis circumplectatur, ut in convolutio-<pb pagenum="534" xlink:href="017/01/550.jpg"></pb>ne ſecundi Axis evolutus tympano illi motum conciliet, adeò<lb></lb>que etiam ponderi. </s> <s id="s.004029">Duas igitur Rationes, quas ſingula Peritro<lb></lb>chia ad ſuos Axes habent, compone, ut potentiæ ſecundo Peri<lb></lb>trochio applicatæ momenta innoteſcant. </s> <s id="s.004030">Sit Ratio hæc poſte<lb></lb>rior ex. </s> <s id="s.004031">gr. quadrupla; & Ratio, quæ ex quadruplâ & quintuplâ <lb></lb>componitur, eſt vigecupla, quæ adhuc minor eſt, quàm opor<lb></lb>teat. </s> <s id="s.004032">Quare, cùm ex Ratione centuplâ Ratio vigecupla ſubducta <lb></lb>relinquat Rationem quintuplam, tertium Axem cum manubrio <lb></lb>quintuplæ longitudinis ad ejuſdem Axis ſemidiametrum ſimili<lb></lb>ter appone, & erit ex his tribus Rationibus compoſita Ratio <lb></lb>centupla quæſita. </s> <s id="s.004033">Quia enim potentia manubrio huic applicata <lb></lb>movetur quintuplo velociùs, quàm punctum, cui illa in ſecundi <lb></lb>Axis tympano applicaretur, hoc verò quadruplo velociùs, quàm <lb></lb>ſimile punctum in tympano prioris Axis, potentia movetur vi<lb></lb>gecuplo velociùs, quàm ſi applicaretur tympano prioris Axis: <lb></lb>huic autem tympano applicata moveretur quintuplo velociùs <lb></lb>quàm pondus: igitur manubrium illud verſans potentia move<lb></lb>tur centuplo velociùs quàm pondus: id quod fieri oportebat, ut <lb></lb>propoſita gravitas in altum attolleretur. </s> </p> <p type="main"> <s id="s.004034">Placeat jam triplicem hunc Axem cum unico illo comparate, <lb></lb>qui ſolus adhiberetur, ſi machina ſimplex eſſet, & non compo<lb></lb>ſita: ille ſiquidem ſi palmaris diametri eſſet, adjunctum tympa<lb></lb>num haberet altitudinis palmorum centum; in quo conſtruen<lb></lb>do quàm multâ materiâ opus eſſet, quantoque artificio compin<lb></lb>genda, ne ſuâ mole labefactata diſſolveretur? </s> <s id="s.004035">Triplex autem hic <lb></lb>Axis cum ſuis duobus tympanis, & manubrio (præterquam quod <lb></lb>multipliciter diſponi poteſt pro loci opportunitate, & potentiæ <lb></lb>moventis commodo) non ſolùm ad altitudinem <expan abbr="palmorũ">palmorum</expan> viginti <lb></lb><expan abbr="nõ">non</expan> aſſurgeret, ſed longè infra <expan abbr="illã">illam</expan> ſubſiſteret, à <expan abbr="quocũque">quocunque</expan> artifice <lb></lb>nullo negotio conſtrueretur, ab alio in <expan abbr="aliũ">alium</expan> <expan abbr="locũ">locum</expan> facillimè <expan abbr="trãsfer-retur">transfer<lb></lb>retur</expan>, levíque <expan abbr="diſpẽdio">diſpendio</expan> pararetur, ut cuique <expan abbr="cõſiderãti">conſideranti</expan> <expan abbr="obviũ">obvium</expan> eſt. </s> </p> <p type="main"> <s id="s.004036">Hocautem, quod in tribus Axibus explicatum eſt, de pluribus <lb></lb>etiam <expan abbr="dictũ">dictum</expan> intelligatur: nam ſi Rationes ſingulæ peritrochij ad <lb></lb>ſuum axem conſiderentur, & ſimul componantur multiplicando <lb></lb>invicem omnes Rationum terminos Antecedentes, item omnes <lb></lb>Conſequentes, ut habeatur novus Antecedens & novus Conſe<lb></lb>quens, apparebit Ratio motûs potentiæ ad <expan abbr="motũ">motum</expan> ponderis, adeó<lb></lb>que gravitatis ponderis ad virtutem potentiæ. </s> <s id="s.004037">Ex quo patet quoſ-<pb pagenum="535" xlink:href="017/01/551.jpg"></pb>cumque Axes oblatos utiles eſſe poſſe, modò Peritrochiorum ad <lb></lb>ſuos Axes Ratio innoteſcat, ſive ſimiles ſint, ſive diſſimiles Ratio<lb></lb>nes, ſive multiplices, ſive ſuperparticulares, ſive ſuperpartientes: <lb></lb>demum enim, ſi quid deſit ad quæſitam Rationem, addi poteſt <lb></lb>certus Axis cum manubrio ita, ut Ratio quæſita expleatur. </s> <s id="s.004038">Sint <lb></lb>quinque Axes in ſuis Peritrochiis omnino ſimiles, & ſinguli con<lb></lb>tineant <expan abbr="Rationẽ">Rationem</expan> <expan abbr="decuplã">decuplam</expan>: quinque Rationes 10 ad 1 invicem du<lb></lb>cantur, & erit motus <expan abbr="Potẽtiæ">Potentiæ</expan> ad <expan abbr="motũ">motum</expan> ponderis, ut 100000 ad 1; <lb></lb>ac propterea quo conatu Potentia ſolitaria moveret talentum, <lb></lb>ac machinâ compoſitâ movebit centum millia talentorum. </s> <s id="s.004039">Sint <lb></lb>item quinque Rationes, 10 ad 1, 20 ad 7, 8 ad 3, 9 ad 2, 4 ad 1 <lb></lb>(quocumque ordine inter ſe diſponantur) omnes Antecedentes <lb></lb>invicem ducti faciunt novum Antecedentem 57600, omnes au<lb></lb>tem Conſequentes invicem ducti dant novum <expan abbr="Conſequentẽ">Conſequentem</expan> 42; <lb></lb>quare Ratio Compoſita eſt 57600 ad 42, hoc eſt 9600 ad 7: & <lb></lb>potentia valens attollere libras 7, hac machinâ compoſitâ attol<lb></lb>let libras 9600. Quod ſi oporteret moveri libras decies mille ab <lb></lb>hac <expan abbr="eadẽ">eadem</expan> Potentiâ, auferatur Ratio 9600 ad 7 ex Ratione 10000 <lb></lb>ad 7, & relinquitur Ratio 700 ad 672, hoc eſt 25 ad 24: quare <lb></lb>addendus eſſet ſextus Axis cum manubrio, cujus longitudo ad <lb></lb>Axis ſemidiametrum eſſet ut 25 ad 24, & potentia eadem hu<lb></lb>juſmodi manubrio applicata attollere poſſet libras 10000. </s> </p> <p type="main"> <s id="s.004040">Illud habere videtur incom<lb></lb><figure id="id.017.01.551.1.jpg" xlink:href="017/01/551/1.jpg"></figure><lb></lb>modi hæc Axium compoſitio, <lb></lb>quod magnam vim funium tym<lb></lb>pana circumplectentium exi<lb></lb>git, qui ſcilicet ſingulorum tym<lb></lb>panorum motui reſpondeant. </s> <lb></lb> <s id="s.004041">Cum enim tympani A diameter <lb></lb>ſit quintupla Axis BC ex hypo<lb></lb>theſi, ejus motus eſt quintuplo <lb></lb>major motu ponderis P, ac pro<lb></lb>inde funis, qui tympani limbum <lb></lb>complectitur, quintuplo longior <lb></lb>eſſe debet fune PD, hoc eſt mo<lb></lb>tu ponderis; cujus funis tympa<lb></lb>no A circumducti caput cum <lb></lb>Axe EF connectitur, circa <pb pagenum="536" xlink:href="017/01/552.jpg"></pb>quem in motu convolvitur. </s> <s id="s.004042">Quoniam verò tympanum O ex <lb></lb>hypotheſi diametrum habet quadruplam diametri ſui Axis EF <lb></lb>ejuſque motus eſt ad motum ſui Axis quadruplus, funis circum<lb></lb>plicatus tympano O quadruplus eſt funis, qui circa Axem EF <lb></lb>convolvitur, ac propterea etiam vigecuplus funis DP; adeo ut, <lb></lb>ubi totus funis evolutus fuerit, atque circa axem HG convolu<lb></lb>tus, pondus ſublatum uſque in D intelligatur. </s> <s id="s.004043">Ex quo fit poten<lb></lb>tiam manubrio IL applicatam, quia IG longitudo eſt quintu<lb></lb>pla ſemidiametri Axis GH, adhuc quintuplo velociùs moveri <lb></lb>quàm tympanum O, cujus motum metitur evolutio funis illud <lb></lb>circumplectentis; atque idcirco Potentia in I movetur centu<lb></lb>plo velociùs quàm pondus P. </s> <s id="s.004044">Quare ſi adhuc quartum Axem ad<lb></lb>dere oporteret, & loco manubrij GI tympanum ſuo fune in<lb></lb>ſtructum apponeretur, funis ille eſſet ipſius PD centuplus; at<lb></lb>que ita deinceps pro tympanorum & Axium multiplicatione <lb></lb>juxta ſingulorum Rationem augeretur funium longitudo. </s> </p> <p type="main"> <s id="s.004045">Verùm pro tantâ funium longitudine non eſt tympanorum <lb></lb>limbo enormis amplitudo tribuenda, ut eos capiat; quia ſcilicet <lb></lb>quò longiores exiguntur hujuſmodi funes, eò etiam tenuiores <lb></lb>atque exiliores eſſe poſſunt: Si enim funis DP oneri attollendo <lb></lb>par conſtat funiculis contortis invicem ex. </s> <s id="s.004046">gr. centum, funis <lb></lb>qui tympanum A complectitur, non niſi quintam ponderis par<lb></lb>tem reſiſtentem habet, hoc eſt ipſum pondus P ſubquintuplo <lb></lb>minore reſiſtentiâ repugnans potentiæ per tympanum A reti<lb></lb>nenti: ac proinde ſi funium firmitatem funiculorum numerus <lb></lb>metitur, ſatis validus erit funis conſtans ex funiculis viginti. </s> <s id="s.004047">Si<lb></lb>militer funis complectens tympanum O, quia pondus reſiſten<lb></lb>tiam ſubvigecuplo minorem habet, ſatis firmus cenſebitur, ſiex <lb></lb>quinque funiculis invicem contortis confletur. </s> <s id="s.004048">Quod ſi juſto <lb></lb>tenuiores timeas hujuſmodi funes, licebit adhuc paulò craſſio<lb></lb>res adhibere. </s> <s id="s.004049">Illud certè manifeſtum eſt multo minores ſuffi<lb></lb>cere, quàm ſit funis DP. </s> </p> <p type="main"> <s id="s.004050">Ne autem tympanis limbi amplitudinem funis circumducti <lb></lb>capacem temerè conſtituas, ſingulorum funium craſſitudo con<lb></lb>ſideranda eſt, ut eorum diameter innoteſcat, & longitudinis ra<lb></lb>tione habitâ ſpirarum numerus inveniatur, per quem ducta fu<lb></lb>nis diameter dabit neceſſariam limbi amplitudinem; quæ ſi <lb></lb>juſto minor eſſet primum ſpirarum ordinem ſecundus ordo cir-<pb pagenum="537" xlink:href="017/01/553.jpg"></pb>cumplecteretur: & quamvis initio hinc major aliqua movendi <lb></lb>facilitas oriretur (auctâ ſcilicet peritrochij diametro) tamen <lb></lb>evoluto ſecundo hoc ſpirarum ordine diameter peritrochij di<lb></lb>minuta majorem crearet movendi difficultatem; maxime ſi cir<lb></lb>ca Axem convolutus funis ſpirarum ordinem pariter geminaret, <lb></lb>atque adeò Axis diametrum augeret. </s> <s id="s.004051">Funes itaque quaſi cylin<lb></lb>dri conſiderandi ſunt, quorum craſſitudinis diametri ſunt in <lb></lb>ſubduplicatâ baſium Ratione; ac propterea inter ipſas craſſitu<lb></lb>dines numeris definitas inventendus eſt numerus medio loco <lb></lb>proportionalis, & hic indicabit tenuioris funis diametrum, <lb></lb>quemadmodum primus numerus major ponitur pro diametro <lb></lb>craſſioris funis. </s> <s id="s.004052">Sic quoniam funis DP eſt ex hypotheſi ut 100, <lb></lb>& funis circa tympanum A ſubquintuplæ craſſitudinis eſt ut <lb></lb>20, inveniatur inter 100 & 20 medius (44 72/100) proximè, & dia<lb></lb>metri funium erunt proximè in Ratione 100 ad 45. Quare <lb></lb>quam amplitudinem requirunt 45 ſpiræ maximi funis DP, <lb></lb>eandem exigunt 100 ſpiræ minoris funis attributi tympano A, <lb></lb>ſi uteroue funis circa eumdem cylindrum convolvatur. </s> <s id="s.004053">Sed <lb></lb>quia perimeter tympani A quinquies continet perimetrum <lb></lb>Axis BC, unica tympani ſpira quinque Axis ſpiris æquatur ſe<lb></lb>cundùm longitudinem, & centum ſpiræ tympani quingentis <lb></lb>Axis ſpiris reſpondent, ſi lineæ longitudo ſpectetur: ſatis au<lb></lb>tem eſt, ſi longitudinem ſpirarum 225 circa Axem, ille funis <lb></lb>tympani obtineat, quia longitudo illa eſt ad funis DP longitu<lb></lb>dinem 45 quintupla. </s> <s id="s.004054">Propterea tympani limbus minorem exi<lb></lb>git amplitudinem, quàm ſit ſpatium, quod in Axe BC occupa<lb></lb>tur à convoluto fune DP: nimirum à limbo contineri oportet <lb></lb>ſui funis circumplicati ſpiras 45; ſatis igitur fuerit dimidiata <lb></lb>amplitudo. </s> </p> <p type="main"> <s id="s.004055">Simili methodo tympano O limbi amplitudinem definies: <lb></lb>quoniam enim funis craſſitudo ad craſſitudinem funis DP eſt <lb></lb>ſubvigecupla, inter 100 & 5 medio loco proportionalis (22 36/100) <lb></lb>proximè inveniatur; & eſt diameter funis tympani O ad dia<lb></lb>metrum funis DP ut (22 36/100) ad 100. Quare ſi circa eundem cy<lb></lb>lindrum uterque funis convolveretur, quod ſpatium ſpiras fu<lb></lb>nis DP 45 contineret, tenuioris hujus funis ſpiras ſaltem 201 <lb></lb>comprehenderet. </s> <s id="s.004056">Ponamus tympani O perimetrum eſſe ad pe-<pb pagenum="538" xlink:href="017/01/554.jpg"></pb>rimetrum Axis BC ut 4 ad 1: igitur limbus tympani O ſi ean<lb></lb>dem habeat amplitudinem, quam funis DP occupat in ſuo Axe <lb></lb>BC, capiet ſui funis ſpiras 201, quæ in unam longitudinem <lb></lb>extenſæ conſtituunt longitudinem, quæ ad longitudinem ſpi<lb></lb>rarum 45 funis DP eſt ut 804 ad 45. Sed quia longitudo illius <lb></lb>funis eſt vigecupla longitudinis funis DP, debet eſſe ut 900 <lb></lb>ad 45; ideò adhuc majorem exigit amplitudinem, ut adhuc ſpi<lb></lb>ras 24 aut 25 ſupra ducentas obtineat. </s> <s id="s.004057">Quod ſi Axis EF ſubti<lb></lb>lior ſit quàm Axis BC, & tympani O diameter ad ſui axis EF <lb></lb>diametrum quadrupla ſit, jam tympani perimeter ad perime<lb></lb>trum Axis BC habebit minorem Rationem quàm 4 ad 1, ac <lb></lb>proinde ejus limbum adhuc ampliorem conſtitui neceſſe eſt. </s> </p> <p type="main"> <s id="s.004058">Quare ſi Axis BC diameter ſit palmaris, ſpiræ 45 funis DP <lb></lb>convoluti elevabunt pondus P ad altitudinem palmorum circi<lb></lb>ter 141, quanta nimirum eſſet ipſius funis convoluti longitudo: <lb></lb>funis circa tympanum A longitudo eſſet palmorum ſaltem 705, <lb></lb>& funis circa tympanum O longitudo palmorum 2820. Hinc <lb></lb>quamvis præter primum Axem BC oneri ſuſtinendo parem, <lb></lb>reliqui conſequentes Axes EF, & GH, & ſi qui alij adhuc ſint, <lb></lb>poſſint in minorem ſoliditatem extenuari; ſi ponderis reſiſten<lb></lb>tia attendatur, quia tamen, quò ſubtiliores ſunt, frequentiori<lb></lb>bus etiam ſpiris circumplicantur, ex quo fit ut plures ſpirarum <lb></lb>ordines fiant, adeoque Axis diameter aucta minuat momento<lb></lb>rum Rationem; præſtat exiles Axes non ſtudiosè quærere, niſi <lb></lb>fortè neceſſitas aliqua iis uti ſuadeat. </s> <s id="s.004059">Quamquam & huic in<lb></lb>commodo occurri poteſt, ſi, quemadmodum in Ergatæ uſu <lb></lb>funem paucis aliquot ſpiris circumductum, dum in converſio<lb></lb>ne evolvitur, puer agglomerat, ita etiam hîc funem tympano<lb></lb>rum A & O quatuor aut quinque ſpiris circa Axes E & H con<lb></lb>volutum puer colligeret: hoc enim pacto non contingeret, ut <lb></lb>primum ſpirarum ordinem alter ſpirarum ordo ſuperinductus <lb></lb>circumplecteretur. </s> </p> <p type="main"> <s id="s.004060">Porrò ne tantam funium vim comparare cogamur, & am<lb></lb>pliore limbo tympana circumſcribere, haud ſanè ineptum cen<lb></lb>ſerem, ſi pro eorum more, qui novaculas obtuſas acuunt, ut aliàs <lb></lb>innui, funis in ſeſe rediens unâ aut altera (aut etiam triplici, ſi <lb></lb>opus fuerit) ſpirâ tùm Axem, tum ſubjectum tympanum arctè <lb></lb>complecteretur: ſic enim fieret, ut Axe convoluto etiam ſub-<pb pagenum="539" xlink:href="017/01/555.jpg"></pb>jectum tympanum volveretur; idémque funis perpetuo ordine <lb></lb>à tympano in proximum Axem & ab Axe in tympanum ſuc<lb></lb>cedens quantolibet motui perficiendo ſufficeret. </s> <s id="s.004061">Et ut omne <lb></lb>periculum ſubmoveatur, ne funis excurrat, ſatius eſt tùm Axis, <lb></lb>tùm tympani ambitum non in cylindricam ſuperficiem expoli<lb></lb>re, ſed angulis aſperum eſſe. </s> <s id="s.004062">Quod ſi aliquando languidior fu<lb></lb>nis non adeò preſsè complecteretur Axem & tympanum, <lb></lb>ſpongiam aquâ imbutam ipſi funi admove, & intentus fiet. </s> </p> <p type="main"> <s id="s.004063">Demùm in hujuſmodi Axium compoſitione non ſine <lb></lb>animadverſione prætereundæ videntur mutua Axium poſitio, <lb></lb>atque diſtantia, qua ſecundus Axis à primo tympano abeſt. </s> <s id="s.004064">Sit <lb></lb>Axis A in Peritrochio CDE, at<lb></lb><figure id="id.017.01.555.1.jpg" xlink:href="017/01/555/1.jpg"></figure><lb></lb>que ex fune perpendiculari BG <lb></lb>dependeat onus, & GB Tangens <lb></lb>cum Radio AB conſtituat angu<lb></lb>lum rectum ABG. </s> <s id="s.004065">Producatur <lb></lb>recta AB uſque ad tympani peri<lb></lb>pheriam in C, & ſit ad angulum <lb></lb>rectum Tangens CH, cui ad<lb></lb>nexa intelligatur potentia per <lb></lb>axem S trahens, atque tympa<lb></lb>num CDE convertens; ex cu<lb></lb>jus converſione convolvitur Axis <lb></lb>AB versùs I, & pondus aſcendit. </s> <lb></lb> <s id="s.004066">Verùm ſecundus Axis S cum pri<lb></lb>mo tympano comparatus non <lb></lb>hanc ſolùm poſitionem obtinere <lb></lb>poteſt, ut ſuperior ſit, ſed etiam <lb></lb>conſtitui poteſt ad latus ita, ut <lb></lb>funis ductarius cadens in hori<lb></lb>zontem ad perpendiculum ſit KL, Tangens verò HC ſit ho<lb></lb>rizonti parallela; aut ita diſponi poſſunt, ut Axis A ſuperiore <lb></lb>loco, Axis S inferiore loco ſtatuatur, & funis pondus retinens <lb></lb>ſit MN, cui parallelus ſit funis CH. </s> <s id="s.004067">Quamcumque ex his <lb></lb>tribus poſitionem habeat Axis ſecundus S, ſivè ſuperior, ſivè <lb></lb>ad latus, ſive inferior ſit (modò linea CH vel parailela ſit li<lb></lb>neis ductarij funis BG aut MN, vel parallela lineæ AK jun<lb></lb>genti centrum Axis cum puncto contactûs perpendiculi KL) <pb pagenum="540" xlink:href="017/01/556.jpg"></pb>eadem habere videtur momenta; quia punctum C, cui appli<lb></lb>cata intelligitur potentia, juxta potentiæ directionem ſimili <lb></lb>Ratione accedit versùs potentiam comparatè ad aſcenſum <lb></lb>puncti Axis, cui applicatur pondus, ac eſt Ratio ſemidiametri <lb></lb>tympani ad ſemidiametrum Axis. </s> <s id="s.004068">Concipiamus enim in con<lb></lb>volutione punctum C venire in F, punctum verò B in I: <lb></lb>punctum igitur C ſequens potentiæ directionem accedit versùs <lb></lb>potentiam juxta menſuram Sinûs arcûs CF, hoc eſt OF, <lb></lb>quemadmodum punctum B contrà directionem gravitatis pon<lb></lb>deris aſcendit juxta menſuram Sinûs arcûs BI: ſunt autem hi <lb></lb>ſinus arcuum ſimilium ſimiliter poſitorum in Ratione Radio<lb></lb>rum AC ad AB. </s> <s id="s.004069">Atqui ſive in K, ſive in M intelligatur pon<lb></lb>dus, aſcenſus illius eſt æqualis aſcenſui BI; ergo ad illos, ut po<lb></lb>te huic æquales, acceſſus puncti C ad potentiam, qui eſt OF, <lb></lb>eandem habet Rationem, quæ eſt Radij AC ad Radium AB. </s> </p> <p type="main"> <s id="s.004070">At verò ſi Axis ſecundus ſit T, potentia non intelligitur ap<lb></lb>plicata tympano in C, ſed in F, ubi circulum tangit recta TF; <lb></lb>nec ejus directio FT eſt parallela directioni ponderis BG, ſed <lb></lb>obliqua, adeò ut quamvis F veniat in Q per arcum æqualem <lb></lb>arcui CF, quia tamen non eſt ſimiliter poſitus, punctum F ſe<lb></lb>quens directionem potentiæ accedit versùs potentiam acceſſu, <lb></lb>quem metitur RP; eſt autem RP minor quàm AR, hoc eſt <lb></lb>OF, ut ex doctrinâ Sinuum conſtat; igitur acceſſus RP ad <lb></lb>aſcenſum BI habet minorem Rationem, quàm acceſſus OF ad <lb></lb>eundem aſcenſum BI. </s> <s id="s.004071">Potentia igitur volvens Axem T in li<lb></lb>neâ TF obliquâ minora habet momenta, quàm in parallelâ <lb></lb>HC. </s> <s id="s.004072">Similiter ſi Axis fuerit V propior quàm Axis T; linea <lb></lb>VD tangit circulum in D puncto remotiore quàm F, à puncto <lb></lb>C; ac propterea datâ arcûs æqualitate adhuc minor eſt acceſſus <lb></lb>in D quàm in F, multóque minor quàm in C, & idcircò mi<lb></lb>norem habet trahendi facilitatem. </s> <s id="s.004073">Quare quò propior eſt Axis <lb></lb>ſecundus, ſi tractio ſit obliqua, ut TF & VD, plus laboris re<lb></lb>quiritur in movendo. </s> </p> <p type="main"> <s id="s.004074">Neque hoc mihi inconſiderantiæ tribuas, quod aſſumpſerim <lb></lb>arcus CF & BI perinde atque ſi idem eſſet motus, ac quando <lb></lb>funis HC eſſet firmiter alligatus in C, & ejus caput veniret ex <lb></lb>C in F; cum tamen alia ſemper atque alia pars funis aliis ſub<lb></lb>inde peripheriæ tympani partibus reſpondeat, in quibus fit ad <pb pagenum="541" xlink:href="017/01/557.jpg"></pb>angulum rectum cum diametro contactus, dum ille evolvitur. </s> <lb></lb> <s id="s.004075">Eatenus enim notabilem arcum aſſumpſi, quatenus ob oculos <lb></lb>ponenda erat momentorum Ratio: Cæterùm ſatis ſcio non adeò <lb></lb>notabiles arcus, ut CF & BI, conſiderandos eſſe, ſed eorum <lb></lb>particulam minimam, ſive centeſimam dicas, ſive milleſimam <lb></lb>aut decies milleſimam: eadem ſcilicet erit Ratio Sinuum, qui <lb></lb>reſpondent minimis arcubus ſimilibus ac ſimiliter poſitis, qui <lb></lb>nimirum incipiunt à C & B, atque Sinuum reſpondentium <lb></lb>majoribus arcubus ſimilibus ab iiſdem punctis C & B incipien<lb></lb>tibus. </s> <s id="s.004076">Id quod pariter de punctis F & D comparatis cum <lb></lb>puncto B, aut K, aut M dicendum: Nam quæ inito ſemel mo<lb></lb>tu intercedit momentorum Ratio inter potentiam & pondus <lb></lb>ratione poſitionis, eadem in toto motu perſeverat. </s> <s id="s.004077">At ſi funis <lb></lb>non evolveretur, ſed puncto C eſſet firmiter colligatus, in <lb></lb>tractione ex C in F ſubinde mutarentur Potentiæ momenta, <lb></lb>fieréntque ſemper minora, adeò ut demum perirent, & nulla <lb></lb>eſſent, ubi in rectam lineam coaleſcerent Radius AC & fu<lb></lb>nis HC. </s> </p> <p type="main"> <s id="s.004078">Hæc quæ de ſecundo Axe funem primo tympano circum<lb></lb>ductum evolvente dicta ſunt, facilè traduci poſſunt etiam ad <lb></lb>funem in ſe ſe redeuntem, cujuſmodi eſſet funis FTEF, aut <lb></lb>DVXD: niſi enim funis contingat tympanum in puncto ſe<lb></lb>midiametri tranſeuntis per contactum Axis & funis ductarij <lb></lb>(hoc eſt in C extremitate ſemidiametri AC tranſeuntis per B, <lb></lb>aut M) ita ut ſit funi ductario parallelus, aut in puncto ſemi<lb></lb>diametri parallelæ funi ductario KL, conſultius erit, cæteris <lb></lb>paribus, axem ſecundum eſſe remotum ut T, quàm proximum <lb></lb>ut V: in proximo enim lineæ DV & XV productæ coirent <lb></lb>in angulum majorem, quàm lineæ FT & ET, ac propterea <lb></lb>comprehenſus arcus DX minor eſt arcu FE. Cæteris, in<lb></lb>quam, paribus; ſi videlicet in eadem rectâ lineâ intelligan<lb></lb>tur trium Axium centra A, V, T; nam ſi in lineâ eadem jun<lb></lb>gente centra AT non eſſet V, ſed recederet ita, ut funis tym<lb></lb>panum contingens minùs obliquus eſſet, ſed magis accederet <lb></lb>ad paralleliſmum cum lineâ BG, aut cum Radio Axis AK, <lb></lb>quamvis Axis V propior eſſet, quàm Axis T, plus tamen <lb></lb>haberet momenti ratione directionis potentiæ minùs obliquè <lb></lb>trahentis. </s> </p> <pb pagenum="542" xlink:href="017/01/558.jpg"></pb> <p type="main"> <s id="s.004079">Cave autem, ne hîc in latentem quendam æquivocationis <lb></lb>ſcopulum incurras, ſi fortè permixtim accipias ponderis eleva<lb></lb>tionem atque ejuſdem ſuſpenſi retentionem, ne recidat; hæc <lb></lb>enim duo oppoſito modo contingunt, & quæ minor funis obli<lb></lb>quitas cauſa eſt facilioris elevationis, eadem difficiliorem ef<lb></lb>ficit retentionem: nam pondus B retinetur à potentiâ C, <lb></lb>abſque eo quod potentia ullo conatu urgeat polos, quibus <lb></lb>axis & peritrochium incumbit, ideò pondus totas ſuas vi<lb></lb>res exerit adversùs potentiam ſurſum directè trahentem: at <lb></lb>verò in F aut in D potentia ſurſum obliquè trahens tym<lb></lb>panum verſus centrum quodammodo urget, & quidem eò <lb></lb>magis, quò magis obliqua eſt tractio; ac proinde pondus <lb></lb>non ſolùm vincere debet potentiæ vires, ſed etiam reſiſten<lb></lb>tiam ex illâ preſſione ortam; quæ quò major eſt pro majo<lb></lb>re declinatione à paralleliſmo cum lineâ BG, aut Radio AK, <lb></lb>majorem quoque potentiæ tribuit retinendi facilitatem. </s> <s id="s.004080">Hinc <lb></lb>quando ſecundus Axis eſt in inferiore loco, & potentiæ tra<lb></lb>hentis directio deorſum tendit, magis premuntur poli, quin <lb></lb>& à potentia deorſum conante, & à ponderis gravitate ur<lb></lb>gentur, & quidem eò magis, quò magis potentiæ deorſum <lb></lb>trahentis directio accedit ad lineam directioni ponderis MN <lb></lb>parallelam, aut ultra illam excurrit ſe quodammodo invicem <lb></lb>decuſſando. </s> <s id="s.004081">An non exeſa publicorum puteorum marmorea <lb></lb>labra aliquando obſervaſti, quæ diuturno atque frequentiſ<lb></lb>ſimo uſu à funibus, quibus aqua hauritur, detrita ſunt: <lb></lb>Utique aquam in ſitulâ ſurſum trahentis labor minor eſſet, <lb></lb>cæteris paribus, ſi ſolùm ſitulæ & aquæ gravitatem vince<lb></lb>re oporteret, quàm ſi præter hanc etiam ſuperanda ſit re<lb></lb>ſiſtentia, quæ ex funis conflictu cum marmore oritur. </s> <s id="s.004082">Sed <lb></lb>quia deinde hoc eodem conflictu efficitur, ut quando tractio <lb></lb>alternis morulis interciditur, retentio minorem potentiæ co<lb></lb>natum exigat; propterea etiam trahentes facilè patiuntur <lb></lb>reſiſtentiam augeri, ut aliquantulo laboris compendio gau<lb></lb>deant, quoties placuerit quietem aliquam captare. </s> </p> <p type="main"> <s id="s.004083">Quamquam non negaverim rudes fœminas atque pueros hoc <lb></lb>in opere, naturâ duce, quærere etiam in trahendâ ſurſum ſi<lb></lb>tulâ non leve laboris compendium: ſi enim rectâ, intacto pu<lb></lb>tei labro, funis ſurſum trahendus eſſet, id utique ſolâ brachio-<pb pagenum="543" xlink:href="017/01/559.jpg"></pb>rum contentione perfici poſſet; ſed ubi funis labro innititur, <lb></lb>non ſolùm contentis brachiorum muſculis trahunt, ſed etiam <lb></lb>inclinato retrorſum corpore hoc efficiunt, ut ipſa corporis <lb></lb>gravitas nonnihil conferat, quo potentiæ animalis viribus <lb></lb>fiat additamentum. </s> <s id="s.004084">Ex quo manifeſtum eſt reſiſtentiam il<lb></lb>lam ex preſſione ortam & difficiliorem efficere tractionem, & <lb></lb>faciliorem retentionem: ac proinde lapſum putarem, qui tra<lb></lb>hentis potentiæ momenta æſtimaret ex majore retinendi fa<lb></lb>cultate. </s> </p> <p type="main"> <s id="s.004085">In his, quæ in poſteriore hujus capitis parte diſputata ſunt <lb></lb>de hac inæqualitate momentorum pro diverſa poſitione axis <lb></lb>ſecundi, mihi videor ſatis probabiliter philoſophatus: verùm <lb></lb>ſi ad Rationes Vectis (ut pluribus placet) revocanda eſſet vis <lb></lb>Axis in Peritrochio, quamvis aliqua ſatis commodè explica<lb></lb>ri poſſent, ubi Vectis eſt rectus, non omnia tamen, ubi Vectis <lb></lb>curvus intelligendus eſt, congruam patiuntur explicationem, <lb></lb>ut cuilibet rem attentè conſideranti manifeſtum fiet; mihi <lb></lb>enim hìc non videtur operæ pretium in re parùm utili tempus <lb></lb>conterere; placuit tamen id obiter innuere, ut ipſe tibi perſua<lb></lb>deas inanem eſſe laborem, quo quis ſingularum Facultatum <lb></lb>vires ad Vectem revocare conatur. <lb></lb></s> </p> <p type="main"> <s id="s.004086"><emph type="center"></emph>CAPUT VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004087"><emph type="center"></emph><emph type="italics"></emph>Tympanorum dentatorum uſus & vires <lb></lb>exponuntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004088">QUæ hactenus tympana conſideravimus, fune circum<lb></lb>ducto atque evoluto verſantur; nunc genus aliud, cujus <lb></lb>ampliſſimus uſus eſt, contemplari oportet, tympana videlicet <lb></lb>dentata, ſeu Rotas dentatas, in quibus ſivè fuerint ſimplices, <lb></lb>ſivè compoſitæ, aut nullo prorſus fune indigemus aut illo tan<lb></lb>tummodo, quo pondus proximè trahitur, aut attollitur: den<lb></lb>tes enim majoris atque minoris tympani, ubi plura componun<lb></lb>tur, ſe mutuâ collabellatione mordentes ſe viciſſim urgent, <pb pagenum="544" xlink:href="017/01/560.jpg"></pb>pro ut hoc aut illud tympanum habet originem motûs. </s> <s id="s.004089">Sit <lb></lb><figure id="id.017.01.560.1.jpg" xlink:href="017/01/560/1.jpg"></figure><lb></lb>chalybea lamina AB ſatis ſolida, in alte<lb></lb>râ extremitate, quæ pondus reſpicit, mo<lb></lb>dicè ſinuata, ut in A, & in validum un<lb></lb>cum recurva, ut in C; latus autem DE <lb></lb>quaſi ſerræ in morem ſit dentibus aſpe<lb></lb>rum. </s> <s id="s.004090">Tum rotula I paris ſaltem cum la<lb></lb>minâ craſſitudinis paretur dentes habens <lb></lb>ita in orbem diſpoſitos, ut hi in rotulæ <lb></lb>circa ſuum centrum converſione denti<lb></lb>bus laminæ ſubinde congruant: colloca<lb></lb>tis enim in apto loculamento rotulâ, at<lb></lb>que laminâ (cujus tamen pars DC extet) <lb></lb>adeò, ut hæc ex illius converſione liberè <lb></lb>promoveri, illa circa ſuum axem, cui fir<lb></lb>miter infixa ſit, facilè verſari valeat, circumducto axis manu<lb></lb>brio ad latus extra loculamentum extante, urgeri poterit pon<lb></lb>dus, aut trahi: Nimirum ſi rotulæ converſio fiat ex H in I, <lb></lb>propellitur extra loculamentum lamina, ejúſque extremitas A <lb></lb>recedens à rotulâ urget pondus obvium: contrà verò ſi rotula <lb></lb>convertatur ex I in H, laminam ad ſe intra loculamentum re<lb></lb>trahit, & pondus unco C connexum ad ſe rapit. </s> <s id="s.004091">Hinc clariùs <lb></lb>vides, quàm ut monendus ſis, oportere in attollendo, aut pro<lb></lb>pellendo pondere loculamentum aut ponderi ſuppoſitum firmo <lb></lb>ſolo inſiſtere, aut ponderi objectum ſolido repagulo inniti; in <lb></lb>trahendo autem pondere, quod uncus C apprehendit, oppoſi<lb></lb>tam loculamenti extremitatem valido fune retineri. </s> </p> <p type="main"> <s id="s.004092">Illud potiùs attentè perpendendum, quod in ſtatuendis tùm <lb></lb>laminæ, tùm rotulæ dentibus plurimum refert, utrùm rari, an <lb></lb>ſpiſſiores ſint laminæ dentes, ac proinde utrùm pauci, an plures <lb></lb>inſint ipſi rotulæ, cujus peripheria in converſione aptatur lami<lb></lb>næ; hæc enim juxta numerum dentium rotulæ, quibus ſubinde <lb></lb>coaptatur, promovetur, & cum ipsâ pondus pari velocitate aut <lb></lb>tarditate movetur. </s> <s id="s.004093">Præſtare autem pondus tardè; potentiam <lb></lb>velociter moveri, quid opus eſt iterùm inculcare? </s> <s id="s.004094">Igitur quò <lb></lb>minor erit rotula & paucioribus dentibus inſtructa, eodem ma<lb></lb>nente manubrio, faciliùs movebitur pondus; quia ut ſemidia<lb></lb>meter rotulæ ad manubrij longitudinem, ita motus ponderis, <pb pagenum="545" xlink:href="017/01/561.jpg"></pb>ad motum potentiæ, & reciprocè ita potentiæ vis movendi, ad <lb></lb>pondus. </s> <s id="s.004095">Quare ulteriùs manifeſtum eſt, ſi majore potentiæ vir<lb></lb>tute opus fuerit, ſpectatâ ponderis movendi difficultate, poſſe <lb></lb>augeri manubrium, ut majora ſint potentiæ momenta. </s> <s id="s.004096">Quo<lb></lb>niam verò non ſemper in promptu eſt opportunum manubrium, <lb></lb>ſuaderem extremum axis caput, quod manubrio inſeritur, qua<lb></lb>dratum fieri, & longiuſculum eſſe: loco autem vulgaris manu<lb></lb>brij habeatur craſſioris cylindri fruſtulum MN, in cujus imâ <lb></lb>baſi circa centrum excavatum ſit quadratum foramen S ad exci<lb></lb>piendum caput axis, & ipſius cylindri ſcapum penetrent fora<lb></lb>mina rotunda R, T, quibus pro opportunitate inſeri poſſint ba<lb></lb>culi ſive longiores, ſive breviores. </s> <s id="s.004097">Porrò cylindri craſſitiem <lb></lb>nihil obeſſe apertè conſtat, ſi quidem ſola rotulæ ſemidiameter <lb></lb>attenditur ad definiendum ponderis motum comparata cum ba<lb></lb>culi longitudine, quatenus potentia ab axe rotulæ diſtat. </s> </p> <p type="main"> <s id="s.004098">Quod ſi uno eodémque tempore duo pondera in oppoſitas <lb></lb>partes diſpellere, aut ſibi invicem propiora fieri oporteat, ſimi<lb></lb>lem alteram laminam priori parallelam in eodem plano ſed con<lb></lb>trario modo poſitam (ut ſcilicet extremitas ſimilis ipſi ACD <lb></lb>reſpiciat prioris laminæ extremitatem B) in oppoſitâ rotulæ <lb></lb>parte colloca ad I, ut pariter laminæ dentes rotulæ dentibus im<lb></lb>plicentur: Quia enim circuli circa ſuum centrum circumacti <lb></lb>partes adverſæ oppoſitis motibus cientur, etiam laminarum ex<lb></lb>tremitates, quæ pondus propellunt aut trahunt, in contrarias <lb></lb>partes à rotulâ circumactâ moventur, ita ut vel à ſe invicem re<lb></lb>cedant, vel ad ſe mutuò accedant. </s> </p> <p type="main"> <s id="s.004099">Hoc idem quod laminæ rectæ dentatæ tympano ſimiliter den<lb></lb>tato implicitæ contingit, accideret pariter, ſi illius in circulum <lb></lb>inflexæ extremam oram dentes ambirent: quemadmodum enim <lb></lb>recta lamina AB, tympani HI converſi ductum ſequitur, ita il<lb></lb>la in circulum conformata circa ſuum centrum moveretur à <lb></lb>tympani dentibus impulſa; eâ tamen ratione, ut duarum hujuſ<lb></lb>modi rotarum ſe invicem mordentium converſiones in oppoſi<lb></lb>tas plagas tenderent; ſi enim prioris rotæ pars ſuperior Occa<lb></lb>ſum versùs converteretur, poſterioris rotæ pars item ſuperior <lb></lb>Ortum versùs convolveretur; & ſi adhuc tertia rota dentata ad<lb></lb>deretur, hæc iterum proximæ adverſata ad occaſum pergeret; <lb></lb>atque ita deinceps alternis converſionibus ſibi viciſſim reſpon<lb></lb>dentibus. </s> </p> <pb pagenum="546" xlink:href="017/01/562.jpg"></pb> <p type="main"> <s id="s.004100">Obſervandum eſt autem hujuſmodi tympanorum, quæ den<lb></lb>tata vocamus, multiplicem eſſe poſſe formam, eámque eligen<lb></lb>dam, quæ præſtituto motui magis congruere videbitur: non ſo<lb></lb>lùm enim pro majoribus tympanis aſſumi poteſt diſcus extre<lb></lb>mum limbum habens ſerræ in morem denticulatim inciſum, ve<lb></lb>rùm etiam in orbem infigi poſſunt paxilli dentium loco promi<lb></lb>nentes, ſive peripheriam ipſam quaſi radij exeuntes ambiant, ſi<lb></lb>ve ſuprà diſci planum erigantur ad perpendiculum certis inter<lb></lb>vallis diſtributi. </s> <s id="s.004101">Hoc autem dentium inſitorum genus non pa<lb></lb>rum habet utilitatis præ dentibus illis quaſi connatis: nam ſi lon<lb></lb>go uſu dens aliquis atteratur, aut excutiatur, facilè reſtitui po<lb></lb>teſt novo paxillo in prioris locum immiſſo; at non ita facilè re<lb></lb>paratur pars illa limbi denticulata, quæ facta eſt inutilis. </s> <s id="s.004102">Simi<lb></lb>liter pro minoribus tympanis non ſolùm dentatas rotulas adhi<lb></lb>bere poſſumus ſuis axibus infixas, ſed etiam uti licet aut paulò <lb></lb>craſſioribus axibus ſtriatis, quorum excavatæ ſtriæ majoris tym<lb></lb>pani dentibus congruant, aut vertebris pariter ſtriatis ſubtiliori <lb></lb>axi infixis. </s> <s id="s.004103">Quando verò majus tympanum paxillos habet pro <lb></lb>dentibus, tympanum minus illi reſpondens eſt Curriculus (<expan abbr="quẽ">quem</expan> <lb></lb>alij ex Italico idiomate <emph type="italics"></emph>Rocchetum<emph.end type="italics"></emph.end> dicunt) aliquot virgulis, ut <lb></lb>plurimum ferreis ad firmitatem, capita duobus parallelis planis <lb></lb>infixa habentibus conſtans, ita ut in majoris tympani converſio<lb></lb>ne ſingulos paxillos excipiant ſingula virgularum intervalla, <lb></lb>quibus propulſis curriculus convolvitur, & cum eo aut pondus <lb></lb>ipſum, aut aliud tympanum movetur. </s> </p> <p type="main"> <s id="s.004104">Sic contingere poteſt ut Axe AB attollendum ſit pondus, & <lb></lb><figure id="id.017.01.562.1.jpg" xlink:href="017/01/562/1.jpg"></figure><lb></lb>expediat uti jumento, quod tamen <lb></lb>non niſi in plano horizontali moveri <lb></lb>poteſt; tympanum autem CD, in <lb></lb>quo eſt Axis AB horizontalis, eſt in <lb></lb>plano Verticali. </s> <s id="s.004105">Ad tympani CD <lb></lb>planam faciem averſam perpendicu<lb></lb>lares paxillos in ambitu ſtatue, & <lb></lb>Curriculum EF circa ſuum axem <lb></lb>ſuperiùs atque inferiùs firmatum <lb></lb>verſatilem adjice, cujus virgulæ con<lb></lb>gruis intervallis diſtinctæ tympani <lb></lb>dentibus reſpondeant. </s> <s id="s.004106">Aliud item tympanum HG horizonti <pb pagenum="547" xlink:href="017/01/563.jpg"></pb>parallelum dentes habens ex peripheriâ extantes, & Curriculi <lb></lb>EF virgulis aptè congruentes, infigatur axi perpendiculari IK, <lb></lb>cui opportuno loco addatur vectis LM, ita ut in M commodè <lb></lb>jungi poſſit jumentum: hoc enim progrediente, & tympanum <lb></lb>ex G versùs O convolvente, dens H incurrens in virgulam cur<lb></lb>riculi EF illum convertit versùs tympanum CD, cujus pariter <lb></lb>denti occurrens alia virgula, atque impellens cogit infimam <lb></lb>tympani partem D aſcendere, ſimúlque Axem AB converti, & <lb></lb>convoluto fune ductario RS attolli pondus. </s> </p> <p type="main"> <s id="s.004107">Hæc tympanorum duorum & curriculi intermedij complexio <lb></lb>ſi attentè perpendatur, non auget potentiæ momenta præter ea, <lb></lb>quæ obtineret proximè applicata tympano CD ad convolven<lb></lb>dum Axem AB: Nam ſi ponatur vectis LM non longior ſemi<lb></lb>diametro tympani dentati HG, perinde eſt, ac ſi potentia in M <lb></lb>poſita exiſteret in G, æquali ſcilicet motu cum tympani HG <lb></lb>peripheriâ movetur. </s> <s id="s.004108">Curriculi autem EF motus æquè velox eſt <lb></lb>atque motus tympani HG; licèt enim hoc ſit majus, ille mi<lb></lb>nor, tamen dum illud ſemel, hic ſæpiùs convolvitur pro ratione <lb></lb>diametrorum; adeò ut ſi tympanum HG habeat dentes viginti, <lb></lb>curriculus ſtrias quinque, hic quater volvatur ex unicâ tympani <lb></lb>converſione: quapropter quatuor curriculi ſubquadrupli con<lb></lb>volutiones uni converſioni tympani HG æquantur. </s> <s id="s.004109">Similiter <lb></lb>& de tympano CD dicendum, cujus tantummodo dentes quin<lb></lb>que reſpondentes quinque ſtriis aut virgulis curriculi EF ur<lb></lb>gentur unicâ converſione curriculi ejuſdem, & idcircò æqualis <lb></lb>eſt utriuſque motus, ac proinde etiam duo tympana CD & HG <lb></lb>æqualiter moventur, & potentiæ in M applicatæ momenta ea<lb></lb>dem ſunt, quæ forent, ſi tympano CD proximè applicaretur. </s> <lb></lb> <s id="s.004110">Quamobrem, ut aliqua fiat momentorum acceſſio in potentiâ, <lb></lb>oportet vectem LM ſtatuere longiorem ſemidiametro tympani <lb></lb>HG: tunc enim ex Ratione longitudinis LM ad ſemidiame<lb></lb>trum tympani, & Ratione diametri tympani CD ad diametrum <lb></lb>Axis AB, componitur Ratio, quæ definit momenta potentiæ; <lb></lb>eſt ſcilicet Ratio motûs potentiæ ad motum ponderis. </s> </p> <p type="main"> <s id="s.004111">Ex quo ſatis vides eatenus addi tympanum HG, quatenus <lb></lb>quærendus eſt jumento locus, ut in gyrum circumagi valeat: <lb></lb>cæterum ſi tympanum CD cum ſuo Axe AB ita in ſuperiore <lb></lb>aut inferiore loco collocari atque firmari poſſit, ut nulli impedi-<pb pagenum="548" xlink:href="017/01/564.jpg"></pb>mento ſit jumento in inferiore aut ſuperiore plano exiſtenti & <lb></lb>circumacto, ſatius eſt labori & ſumptibus parcere omiſſo tym<lb></lb>pano HG, & circa craſſiorem axem conſtruere curriculum EF, <lb></lb>cui axi opportunè adjungatur vectis LM, ut potentia ipſum <lb></lb>curriculum immediatè convertat; erit ſiquidem major Ratio <lb></lb>ipſius vectis LM ad curriculi ſemidiametrum, quàm ad ſemi<lb></lb>diametrum tympani HG; ac proinde minore conatu indige<lb></lb>bit potentia, ut Curriculum cum adjacente tympano CD con<lb></lb>vertat. </s> </p> <p type="main"> <s id="s.004112">Non alio quàm hujuſmodi artificio videtur uſus Anonymus, <lb></lb>qui de Rebus Bellicis ſcripſit ad Theodoſium Auguſtum ejúſ<lb></lb>que filios Honorium, & Arcadium Cæſares, ubi liburnam pro<lb></lb>ponit <emph type="italics"></emph>navalibus idoneam bellis, quam pro magnitudine ſui virorum <lb></lb>exerceri manibus quodammodo imbecillitas humana prohibeat, & <lb></lb>quocumque utilitas vocet, ad facilitatem cursús ingenij ope ſubnixa <lb></lb>animalium virtus impellit. </s> <s id="s.004113">In cujus alveo, vel capacuate bini boves <lb></lb>machinis adjuncti, adhærentes rotas navis lateribus volvunt; qua<lb></lb>rum ſupra ambitum vel rotunditatem extantes radij currentibus iiſ<lb></lb>dem rotis in modum remorum aquam conatibus elidentes miro quodam <lb></lb>artis effectu operantur, impetu parturiente diſcurſum. </s> <s id="s.004114">Hæc cadens <lb></lb>tamen liburna pro mole ſui, próque machinis in ſemet operantibus <lb></lb>tanto virium fremitu pugnam capeßit, ut omnes adverſarias libur<lb></lb>nas cominùs venientes facili attritu comminuat.<emph.end type="italics"></emph.end></s> <s id="s.004115"> Quamvis, quæ de<lb></lb>mum machinæ eſſent, quibus boves adjungebantur, Author <lb></lb>non exponat, facile tamen eſt opinari boves in ſuperiore tabu<lb></lb>lato circumacto versâſſe axem carinæ perpendiculariter in<lb></lb>ſiſtentem, cui infra tabulatum rota dentata horizonti parallela <lb></lb>infixa eſſet dentes habens in alterutra tympani facie, quibus <lb></lb>ſubinde apprehenderet virgulas curriculi in plano Verticali <lb></lb>convoluti, & infixi axi horizontali, qui utrumque navis latus <lb></lb>permearet, & in extantibus extremitatibus rotas haberet cum <lb></lb>palmulis prominentibus, quæ aquam in converſione verbera<lb></lb>rent. </s> <s id="s.004116">Potuerunt autem hujuſmodi machinæ juxta liburnæ lon<lb></lb>gitudinem multiplicari, prout ipſum ſchema ab Authore pro<lb></lb>poſitum exhibet. </s> <s id="s.004117">An verò hujus liburnæ, quam in Præfatione <lb></lb>dicit <emph type="italics"></emph>velociſſimum liburnæ genus, decem navibus ingenij magiſterio <lb></lb>prævalere,<emph.end type="italics"></emph.end> tantus impetus, tantáque velocitas eſſet, ut adverſa<lb></lb>riæ liburnæ venientes facilè comminuerentur, diſpiciat lector, <pb pagenum="549" xlink:href="017/01/565.jpg"></pb>cui otium fuerit, quemadmodum an noſtris uſibus navalibus <lb></lb>artificium hoc aliquid utilitatis afferre poſſit. </s> </p> <p type="main"> <s id="s.004118">Hinc manifeſta fit illarum machinarum vis, quæ Pancratia <lb></lb>Gloſſocoma, Chariſtia, & ſi quod eſt aliud vocabuli genus, di<lb></lb>cuntur, ex plurium tympanorum complexione minorum & ma<lb></lb>jorum, ita ut à minore tympano, cui manubrium additur, inci<lb></lb>piat motus, & deinceps minora majoribus conſequentibus mo<lb></lb>tum communicent. </s> <s id="s.004119">Quandoquidem rotula minor dentata ſi <lb></lb>comparetur cum rotâ majore, cum qua communem habet <lb></lb>axem, utique tardius movetur, quàm peripheria rotæ majoris, <lb></lb>cum qua connectitur: At verò ſi cum rotâ majore conſequente <lb></lb>comparetur, cujus dentes apprehendit, utique æqualis eſt ipſa<lb></lb>rum motûs velocitas, nam plures minoris converſiones æquales <lb></lb>ſunt uni converſioni majoris, quam efficiunt. </s> <s id="s.004120">Sit rota dentata <lb></lb><figure id="id.017.01.565.1.jpg" xlink:href="017/01/565/1.jpg"></figure><lb></lb>AB, cujus axi firmiter infixo additum ſit manubrium ejuſdem <lb></lb>rotæ ſemidiametri ex. </s> <s id="s.004121">gr. quintuplum, ac propterea potentia <lb></lb>manubrio applicata quintuplo velociùs movetur, quàm <lb></lb>punctum in rotæ AB peripheriâ notatum. </s> <s id="s.004122">Addatur rota ma<lb></lb>jor BC, cujus dentes implicentur dentibus rotulæ BA: ex hu<lb></lb>jus converſione illa pariter convolvitur; ſed ſi diameter BA ſit <lb></lb>diametri BC ſubtripla, ter rotula BA volvitur, ut rota BC <lb></lb>compleat integram converſionem, ac proinde potentia quin<lb></lb>tuplo velociùs movetur, quàm rota BC. </s> <s id="s.004123">Rota hæc major ſibi <lb></lb>connexam habeat in eodem axe minorem SD, quæ apprehen<lb></lb>dat dentes ſecundæ majoris rotæ DE; quæ ſimiliter in eodem <lb></lb>axe conjunctam habeat minorem FR: hujus dentes mor-<pb pagenum="550" xlink:href="017/01/566.jpg"></pb>deant peripheriam tertiæ majoris rotæ FG, in qua eſt Axis <lb></lb>IH, ex cujus convolutione ductarius funis LO trahit <lb></lb>pondus. </s> </p> <p type="main"> <s id="s.004124">Ut potentiæ momenta habeantur, ejus motum cum ponderis <lb></lb>motu collatum ad calculos revoca componendo Rationes, quas <lb></lb>ſingulæ majores rotæ ad ſuas minores habent quo ad diame<lb></lb>trum. </s> <s id="s.004125">Quare ſi manubrium ad ſemidiametrum rotulæ AB ha<lb></lb>beat Rationem quintuplam, diameter BC ad diametrum SD <lb></lb>triplam, DE ad RF item triplam, & FG ad IH ſimiliter tri<lb></lb>plam, compoſitis tribus Rationibus triplis cum Ratione quin<lb></lb>tuplâ, oritur Ratio 135 ad 1: atque adeò ut poſtrema rota FG <lb></lb>& cum eâ Axis IH ſemel volvatur, prima rotula AB facit 27 <lb></lb>converſiones; potentia autem manubrio applicata movetur <lb></lb>quintuplo velociùs quàm ſuæ rotæ AB peripheria; igitur pon<lb></lb>deris motus ad motum potentiæ eſt ut 1 ad 135. Porrò fieri 27 <lb></lb>converſiones manubrij apertè conſtat, quia hoc ter volvi po<lb></lb>nitur, ut rota BC, atque adeò etiam SD illi connexa, ſemel <lb></lb>convolvatur: & quia ex hypotheſi rotula SD tres converſio<lb></lb>nes habet, ut toti peripheriæ rotæ DE congruat, manubrium <lb></lb>novies in gyrum agitur, ut rota major DE & minor RF ſemel <lb></lb>convertatur: demum quia pariter ex hypotheſi rota minor RF <lb></lb>triplici convolutione indiget, ut toti peripheriæ rotæ majoris <lb></lb>FG reſpondeat, ut hæc unicam circuitionem perficiat, viginti <lb></lb>ſeptem manubrij converſionibus opus eſt. </s> <s id="s.004126">Quare momentum <lb></lb>Potentiæ manubrio applicatæ comparatæ cum rotulâ AB eſt <lb></lb>ut 5, cum ſequenti rotulâ SD ut 15, cum rotulâ RF ut 45, <lb></lb>cum Axe IH ut 135. </s> </p> <p type="main"> <s id="s.004127">Ne verò artifex hujuſmodi rotas dentatas majores atque mi<lb></lb>nores parare juſſus inutili demùm labore ſe torqueat, monendus <lb></lb>eſt, ut animum diligenter advertat, utrùm omnes majores ro<lb></lb>tas, item omnes minores, inter ſe æquales ſtatuere velit, an in<lb></lb>æquales; ex hoc enim ipſarum rotarum collocationem definiet, <lb></lb>ne ſibi viciſſim impedimento ſint. </s> <s id="s.004128">Finge enim rotulas DS & <lb></lb>FR æquales eſſe, item majores BC & DE, atque alterno or<lb></lb>dine poſitas, ita ut ſi rotula DS fuerit in parte anteriore ſuæ <lb></lb>rotæ BC, viciſſim rotula FR ſit in parte aversâ rotæ DE: <lb></lb>quemadmodum dentes minoris DS implicantur dentibus <lb></lb>majoris DE, ita pariter dentes majoris BC implicantur <pb pagenum="551" xlink:href="017/01/567.jpg"></pb>dentibus minores FR: igitur unica converſio majoris rotæ <lb></lb>BC ter convolveret minorem rotam FR: atqui cum minore <lb></lb>rotâ FR ſimul converteretur major DE in eodem axe; igitur <lb></lb>dum ſemel converteretur major rota BC, ter convolveretur <lb></lb>rota major DE: hoc autem omnino fieri nequit, quia unica ro<lb></lb>tæ BC converſio eſt etiam unica converſio rotulæ minoris DS, <lb></lb>hujus autem unica converſio reſpondet ſolùm tertiæ parti con<lb></lb>verſionis majoris rotæ DE: plurimum igitur abeſt à trinâ con<lb></lb>verſione. </s> <s id="s.004129">Quare ſi hujuſmodi æqualitas intercederet tùm inter <lb></lb>majores, tùm inter minores rotas dentatas, oporteret minores <lb></lb>rotas ad eandem partem reſpicere, ne rota major conſequenti <lb></lb>minori rotæ motum ullum communicare poſſit. </s> <s id="s.004130">Verùm hoc <lb></lb>fortaſſe alicui videatur incommodum, quod non ita aptè in ſuo <lb></lb>loculamento hujuſmodi rotæ collocari valeant, ſi rotarum ma<lb></lb>jorum conſequentium plana ſuperimponantur planis antece<lb></lb>dentium; id quod exigit ipſa minorum rotularum poſitio, ſi <lb></lb>omnes partem eandem reſpiciant. </s> <s id="s.004131">Propterea inæquales fiant <lb></lb>rotæ ita, ut alternatim poſitæ minores rotulæ occurrant qui<lb></lb>dem ſingulæ peripheriæ conſequentis majoris, non attingantur <lb></lb>autem à dentibus majoris rotæ antecedentis: hoc enim pacto <lb></lb>in ſuo loculamento preſſiùs firmantur, & ſunt quaſi duo plana <lb></lb>parallela, in quibus hinc rota minor inter duas majores, hinc <lb></lb>verò rota major inter duas minores conſpicitur. </s> </p> <p type="main"> <s id="s.004132">Porrò minore, rotæ in eodem axe cum majoribus dupliciter <lb></lb>diſponi poſſunt: primùm ut major rota minori proxima ſit, & <lb></lb>earum plana ſe contingant; deinde ut aliquo inter ſe abſint in<lb></lb>tervallo. </s> <s id="s.004133">Si minor majori cohæreat, ſuaderem minores rotas ex <lb></lb>lamina paulò craſſiore fieri quàm majores; ſic enim in locula<lb></lb>mento ita diſponuntur, ut plana majorum ſe omninò non con<lb></lb>tingant, ac proinde nullus ſit partium ſe viciſſim terentium con<lb></lb>flictus, qui moram inferat motui. </s> <s id="s.004134">Sin autem quæ in eodem axe <lb></lb>ſunt rotæ, major & minor invicem diſtent, nullum quidem ſub<lb></lb>eſt periculum ex mutuo affrictu majorum, verùm cavendum <lb></lb>eſt, ne axis longior, quàm par fuerit, etiam ſit infirmior; ſi ni<lb></lb>mirum axis longioris extremitates loculamento infixæ volvan<lb></lb>tur. </s> <s id="s.004135">Propterea aliter etiam diſponi poſſunt, ita ut loculamentum <lb></lb>validiſſimum ſit, nec fractioni obnoxium, & facilè ex alio in <lb></lb>alium locum transferri valeat. </s> <s id="s.004136">Parentur axes rotundi, ſed utra-<pb pagenum="552" xlink:href="017/01/568.jpg"></pb>que extremitas quadrata ſit, ut inſeratur quadrato foramini, <lb></lb>quod rotarum centro ineſt: tum tigni pars accipiatur craſſitu<lb></lb>dinis tantæ, ut congruis foraminibus rotundi, excipiat axium <lb></lb>rotunditatem, extantibus hinc atque hinc extremitatibus qua<lb></lb>dratis. </s> <s id="s.004137">Deinde quadratas axis extremitates excipiant rotæ den<lb></lb>tatæ alterno ordine, ut qua parte prior axis habet rotam majo<lb></lb>rem, ſecundus axis habet rotam minorem, & viciſſim ille in op<lb></lb>poſitâ tigni parte habeat rotam minorem, hic majorem; illud <lb></lb>ſemper præcavendo, ne rota minor ſecundi axis contingat pe<lb></lb>ripheriam rotæ majoris primi axis; id quod fiet, ſi poſteriores <lb></lb>rotæ majores etiam paulò majorem ſemidiametrum habeant. </s> <lb></lb> <s id="s.004138">Relinquitur autem artificis induſtriæ ita foraminum extremi<lb></lb>tates munire, ut nec axes ultrò citròque commeare valeant, <lb></lb>rotis ipſis illos coërcentibus, nec nimio affrictu tigni faciem <lb></lb>rotæ circumactæ terant, interjecto inter tignum & rotam exiguo <lb></lb>circulo, cum quo tritus omnis atque conflictus exerceatur. </s> </p> <p type="main"> <s id="s.004139">Quamvis autem tria tantummodo tympana dentata præter <lb></lb>primam rotulam manubrio affixam, brevitatis gratiâ, exami<lb></lb>nanda propoſuerim, plura, & plura ſimilia addi poſſe eſt mani<lb></lb>feſtum, adeò ut omni arrogantiæ notâ vacent magnificæ illæ <lb></lb>Mechanicorum propoſitiones, quibus ſe quodcumque etiam <lb></lb>immane pondus moturos ſpondent, immò tellurem ipſam, ſi lo<lb></lb>cus daretur ſtatuendæ machinæ idoneus. </s> <s id="s.004140">Illud tamen incom<lb></lb>modum vitari nullatenus poteſt, quod ex ponderis tarditate ori<lb></lb>tur: quî enim fieri poſſit, ut gravitatis reſiſtentia ex motûs tar<lb></lb>ditate minuatur, quin multo tempore opus ſit ad pondus mo<lb></lb>vendum? </s> <s id="s.004141">Idcirco unâ eadémque operâ, qua potentiæ momen<lb></lb>ta inquiris componendo Rationes, quas majora tympana ha<lb></lb>bent ad minora ſibi adjuncta, etiam motûs tarditatem notam fa<lb></lb>cis; ac proinde conſtituto intra certam temporis menſuram po<lb></lb>tentiæ motu, innoteſcit ponderis motus, quem eodem tempo<lb></lb>re perficit. </s> <s id="s.004142">Fac eſſe decem Rationes quintuplas, quæ compo<lb></lb>nendæ ſunt, ſi manubrium ad ſuæ rotulæ ſemidiametrum ha<lb></lb>beat Rationem quintuplam, & ſimilis ſit Ratio majorum ad ſua <lb></lb>minora tympana. </s> <s id="s.004143">Motus potentiæ manubrio applicatæ eſt ad <lb></lb>motum ponderis ut 9.765625 ad 1: tot igitur ſpatij pedes itera<lb></lb>tis revolutionibus confici à potentiâ neceſſe eſt, ut pondus pe<lb></lb>dem unum percurrat. </s> <s id="s.004144">Quod ſi potentiam tantâ velocitate mo-<pb pagenum="553" xlink:href="017/01/569.jpg"></pb>veri ponamus, ut horis ſingulis pedum quindecim millia per<lb></lb>currat, indigebit horis (651 1/24), hoc eſt diebus 27, horis 3. min. <lb></lb>2 1/2, ut pondus à loco in locum pedis unius intervallo diſtan<lb></lb>tem transferat: atque ideò quis tantâ oculorum acie polleat, ut <lb></lb>ponderis motum dignoſcat, niſi poſt aliquot horas? </s> <s id="s.004145">quando<lb></lb>quidem unius horæ ſpatio vix unius unciæ partem quinquage<lb></lb>ſimam quartam perficit, ſcilicet (1/651) pedis. </s> <s id="s.004146">Verùm tam immane <lb></lb>pondus, quod ad gravitatem reſpondentem potentiæ machinâ <lb></lb>deſtitutæ ſit ut 9. 765625 ad 1, movere, licet tardiſſimè, ſatius <lb></lb>eſt, quàm nullo pacto movere. </s> </p> <p type="main"> <s id="s.004147">Ex his liquet, quid contingat, ſi potentiæ & ponderis loca <lb></lb>ita commutentur, ut potentia extremo tympano applicetur, <lb></lb>pondus verò movendum primæ rotulæ axi aut manubrio reſ<lb></lb>pondeat; exiguus enim validioris potentiæ motus velociſſimè <lb></lb>movet pondus, motúmque diu continuat; ut palam eſt in au<lb></lb>tomatis horas indicantibus, ſive potentia movens ſit vis elaſti<lb></lb>ca laminæ chalybeæ inflexæ, ſivè gravitas ponderis axem ma<lb></lb>ximæ rotæ volvens; niſi enim Tempus alternis motibus objice<lb></lb>ret rotæ ſerratæ dentibus ſui fuſi pinnulas, quæ moram infer<lb></lb>rent, rota ipſa ſerrata velociſſimè volveretur. </s> <s id="s.004148">Sed quoniam ra<lb></lb>rò contingit validiſſimam potentiam adhibere, ut leve pondus <lb></lb>moveatur, propterea non eſt frequens hujuſmodi locorum <lb></lb>commutatio inter pondus & potentiam: uſum tamen aliquan<lb></lb>do habere poſſet in rebus ſcenicis, maximè ſi æquabilis eſſe de<lb></lb>beat motus; gravitas enim, quæ per unius aut alterius palmi <lb></lb>ſpatium deſcendat, non acquirit in motu notabile aliquod velo<lb></lb>citatis incrementum, atque idcirco æquabilis apparet motus <lb></lb>tam ipſius gravitatis deſcendentis, quàm ponderis illius virtu<lb></lb>te aſcendentis: hoc ſi non rectâ ſurſum trahatur, ſed circum<lb></lb>agatur, fortaſſe impreſſus impetus velociorem circumvolutio<lb></lb>nem efficere poſſit. </s> </p> <p type="main"> <s id="s.004149">Quod demum ad ipſos rotarum dentes attinet, ſingulæ qui<lb></lb>dem rotæ à ſuis dentibus in partes æquales tribuuntur; ſed hal<lb></lb>lucinati videntur non pauci fruſtra requirentes in omnibus ro<lb></lb>tis invicem comparatis dentium æqualitatem, & ex dentium <lb></lb>numero potentiæ momenta metientes; quaſi ſervari nequiret <lb></lb>eadem momentorum Ratio, etiamſi minoris tympani dentes <pb pagenum="554" xlink:href="017/01/570.jpg"></pb>non omninò ſimiles eſſent, aut ut veriùs loquar, ſinguli non <lb></lb>eſſent æquales ſingulis dentibus majoris tympani in eodem axe <lb></lb>exiſtentis. </s> <s id="s.004150">Dentium æqualitas in iis tantummodo rotis requiri<lb></lb>tur, quæ ſibi mutuâ collabellatione cohærentes in convolutione <lb></lb>dentem dentibus implicant; niſi enim ab unius rotæ dentium <lb></lb>intervallis alterius dentes ſubinde reciperentur, fieri non poſ<lb></lb>ſet utriuſque rotæ converſio. </s> <s id="s.004151">Cæterùm nil prohibet, quomi<lb></lb>nùs in plurium majorum tympanorum complexione alia rario<lb></lb>res, alia ſpiſſiores dentes habeant, dummodo ſingulis majori<lb></lb>bus ſingula minora, à quibus illa motum recipiunt, reſpondeant <lb></lb>ſimilibus dentibus inſtructa, etiamſi hi diſſimiles ſint dentibus <lb></lb>tympani in eodem axe connexi. </s> <s id="s.004152">Si enim prioris majoris tym<lb></lb>pani peripheria ſit in dentes 24 diſtributa, minoris autem tym<lb></lb>pani eidem axi infixi peripheria ſex dentes habeat, ſed eorum <lb></lb>diametri ſint ut 3 ad 2, utique eorum motus non aliam habent <lb></lb>Rationem quàm ſeſquialteram, licèt dentium numeri ſint in <lb></lb>Ratione quadruplâ. </s> <s id="s.004153">Idem planè dicendum de ſecundo tympa<lb></lb>no majore, quod ad prioris motum convolvitur; hujus enim <lb></lb>motus pariter comparandus eſt cum motu tympani minoris ſibi <lb></lb>coniuncti ſpectatâ diametrorum Ratione, non dentium multi<lb></lb>tudine, ut momenta innoteſcant. </s> <s id="s.004154">Quare illæ dentium multi<lb></lb>tudines invicem comparatæ ſatis quidem faciunt quærenti, <lb></lb>quoties potentia manubrium circumagat, ut ſemel convertatur <lb></lb>Axis, quem ductarius funis complectitur; ſed quibus momen<lb></lb>tis id perficiat ipſa potentia, ſola diametrorum Ratio ſpectata <lb></lb>indicabit. </s> </p> <p type="main"> <s id="s.004155">Sed hìc ubi diametrorum incidit mentio (quamquam res <lb></lb>Mechanicæ in praxim deductæ tantâ ſubtilitate non indigeant) <lb></lb>non eſt diſſimulandum aliquam neceſſariò intercedere momen<lb></lb>torum inæqualitatem in ipſo motu, quando tympanorum am<lb></lb>bitus eſt dentium inciſuris aſperatus: cum enim extremi den<lb></lb>tium apices à centro magis abſint, quàm anguli, in quibus ſibi <lb></lb>dentes occurrunt, non eſt utrobique eadem movendi fa<lb></lb>cultas, quippe quæ in majore à centro diſtantiâ validior <lb></lb>eſt, cæteris paribus. </s> <s id="s.004156">Eatenus ſcilicet rota rotam urget, qua<lb></lb>tenus rota movens ſui dentis apice contingit faciem den<lb></lb>tis rotæ, quæ movetur: hic autem contactus primùm fit <lb></lb>propè angulum, hoc eſt minùs procul à centro, & ſen-<pb pagenum="555" xlink:href="017/01/571.jpg"></pb>ſim dens rotæ moventis ſuo apice excurrens versùs extre<lb></lb>mitatem dentis rotæ, quæ movetur, magis recedit à cen<lb></lb>tro. </s> <s id="s.004157">Cum igitur rota movens ſuam vim exerceat apice den<lb></lb>tis, integra ſemper illius diameter aut ſemidiameter conſi<lb></lb>deranda eſt; at rota, quæ urgetur, cum non in eodem <lb></lb>puncto recipiat moventis impulſionem, non eſt abſolu<lb></lb>tè attendenda integra illius diameter aut ſemidiameter, <lb></lb>ſed potiùs mediocris quædam inter maximam & mini<lb></lb>mam à centro diſtantiam eligenda eſt, ut alter Rationis <lb></lb>terminus habeatur. </s> <s id="s.004158">Ex quo vides (ſi res ſubtiliter elime<lb></lb>tur) non parum intereſſe, utrùm minor rota majorem ur<lb></lb>geat, an è contrario major minorem propellat. </s> <s id="s.004159">Concipe <lb></lb>enim majoris rotæ integram ſemidiametrum eſſe particula<lb></lb>rum 100, & talem eſſe dentium inciſuram, ut angulus, <lb></lb>in quo ipſi dentes coëunt, diſtet à centro particulis 94: <lb></lb>rotæ autem minoris, quæ ſuos dentes illius dentibus impli<lb></lb>cat, ſemidiameter integra ſit ſimilium particularum 20, & <lb></lb>angulus concursûs dentium diſtet à centro particulis 14. Uti<lb></lb>que ſi minor majorem urgeat illius Radius eſt ut 20, hu<lb></lb>jus verò eſt ut 97: contra autem ſi major pellat minorem <lb></lb>illius Radius eſt ut 100, hujus ut 17. Quare ſingulæ com<lb></lb>paratæ cum iis, quæ ſecum communem habent axem, di<lb></lb>verſam conſtituunt Rationem: ſi enim major rota urgea<lb></lb>tur à minore ſibi proximâ, adeò ut ſecunda minor movea<lb></lb>tur ad motum majoris in eodem axe, & Ratio ſit ut 20 ad <lb></lb>97, ſi majore proximâ urgente minorem moveretur major <lb></lb>ad motum minoris in eodem axe, hujus minoris motus ad <lb></lb>motum ſuæ majoris non eſſet pariter ut 20 ad 97, ſed ut 17 <lb></lb>ad 100, quæ eſt minor Ratio. <pb pagenum="556" xlink:href="017/01/572.jpg"></pb></s> </p> <p type="main"> <s id="s.004160"><emph type="center"></emph>CAPUT VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004161"><emph type="center"></emph><emph type="italics"></emph>Moleſtrinarum artificium ex Axe in Peritrochio <lb></lb>pendet.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004162">ARtificia omnia, quæ ex Axe in Peritrochio pendent, re<lb></lb>cenſere res eſſet non quidem injucunda, ſed penè infi<lb></lb>niti laboris, hiſtoriam potiùs redolens, quàm theoriam, cui <lb></lb>potiſſimùm inſervio Machinarum fontes indicans, ex quibus <lb></lb>ingenioſus quiſque Machinas ſuo inſtituto opportunas moliri <lb></lb>queat. </s> <s id="s.004163">Placuit tamen in Molendinorum artificio pauliſper im<lb></lb>morari, ut quam uberem ab Axe in Peritrochio utilitatem ad <lb></lb>vitæ commoda percipiamus, innoteſcat. </s> <s id="s.004164">Quamvis autem po<lb></lb>tiſſimùm inſtituta ſint molendina ad comminuendum triticum <lb></lb>& alia ſemina, ut ex farinâ panis conficiatur, ad alios tamen <lb></lb>uſus pars eorum aliqua deſtinatur: omnibus quippe communis <lb></lb>eſt rota exterior, quam aqua incurrens verſat, & Axis, qui <lb></lb>convolvitur. </s> <s id="s.004165">Si enim tundenda ſit lana, aut Cannabis; ſi in <lb></lb>pollinem redigenda elementa pulveris pyrij carbo, ſulphur, <lb></lb>nitrum; ſi antiqua linteorum reſegmina conterenda, & in mi<lb></lb>nimas particulas diſſipanda ad conficiendam chartam, Axi in<lb></lb>fixæ ſunt pinnulæ, quæ in converſione occurrentes alis piſtil<lb></lb>lorum illos elevant, atque dimittunt, & eorum gravitate reci<lb></lb>dente ſubjecta materia aut contunditur, aut conteritur. </s> </p> <p type="main"> <s id="s.004166">Nec diſſimili methodo diſponi poſſent piſtilli ſuis embolis <lb></lb>congruentes, qui à pinnulis Axis elevati aquam in embolum <lb></lb>attraherent, aut ſponte irruentem admitterent per aſſarium, <lb></lb>tùm dimiſſi vi ſuæ gravitatis aquam exprimerent per tubum, <lb></lb>& in altiorem locum aſcendere cogerent. </s> <s id="s.004167">Vel ſi non adeò gra<lb></lb>ves piſtillos parare placuerit, velíſque certiùs aquam in altio<lb></lb>rem locum pellere, diſpone binos piſtillos fune, aut catenâ, <lb></lb>per excavatum rotæ ſuperiùs poſitæ ambitum tranſeunte con<lb></lb>nexos, aut potiùs tranſverſario, quaſi libræ jugo conjunctos, <lb></lb>ita ut altero depreſſo alter elevetur, pinnula autem Axis de-<pb pagenum="557" xlink:href="017/01/573.jpg"></pb>primat piſtillum, vi cujus aqua in tubum aſcendentem expri<lb></lb>matur, & alter piſtillus attollatur aquam inferiùs poſitam at<lb></lb>trahens, qui pariter ab Axis pinnulâ ejus alæ reſpondente ſub<lb></lb>inde deprimatur. </s> <s id="s.004168">Hinc fit poſſe longiorem Axem addi rotæ, <lb></lb>& plura hujuſmodi piſtillorum paria diſponi pinnulis in ambi<lb></lb>tu Axis ita diſtributis, ut non plures ſimul piſtillos, ſed ſingu<lb></lb>los unum poſt alium premant, ſi non adeò valida fuerit poten<lb></lb>tia rotam verſans; Sin autem validior illa fuerit, plures ſimul <lb></lb>deprimant, iíſque conjugatos attollant. </s> <s id="s.004169">Niſi fortè magis arri<lb></lb>ſerit duobus tantummodo piſtillis conjugatis uti, tot pinnulis <lb></lb>in Axe diſpoſitis, ut in unâ ejuſdem Axis converſione bis aut <lb></lb>ter piſtillus idem deprimatur. </s> </p> <p type="main"> <s id="s.004170">Huc pariter ſpectant, quæ paſſim videre eſt in officinis mal<lb></lb>leatorum cupri aut ferri, ubi & rota exterior vi aquæ labentis <lb></lb>circumacta interiùs in conclavi quaſi manubrium convolvit, <lb></lb>quod ſuperiori Axi horizonti parallelo infixum Radium, ſibí<lb></lb>que regulâ in juncturis plicatili connexum, dum attollit, at<lb></lb>que deprimit, in alterâ ejuſdem Axis extremitate tranſverſa<lb></lb>rium hinc pariter attollens atque hinc deprimens follibus alter<lb></lb>num motum conciliat: Et rota alia validiorem aquæ deciden<lb></lb>tis impetum recipiens, ſuúmque Axem convolvens, pinnulis <lb></lb>axi infixis extremitatem alteram deprimit tigilli, cujus oppoſi<lb></lb>tæ extremitati elevatæ cohæret ingens ferreus malleus, qui præ<lb></lb>terlapsä Axis pinnulâ ſponte recidens tundit ſubjectum cuprum <lb></lb>aut ferrum ignitum. </s> </p> <p type="main"> <s id="s.004171">In his omnibus rotæ quidem ſemidiameter attendenda eſt, <lb></lb>in cujus extantes palmulas aqua incurrens vim potentiæ mo<lb></lb>ventis obtinet; ſed Axis ſemidiameter non ſolitariè accipienda <lb></lb>eſt, verùm & addenda prominentis pinnulæ longitudo, ita ut <lb></lb>ex utrâque conficiatur unica ſemidiameter motûs, qui commu<lb></lb>nicatur piſtillo, aut depreſſæ extremitati mallei. </s> <s id="s.004172">Depreſſæ, in<lb></lb>quam, extremitati mallei, nam mallei elevatio aliquanto major <lb></lb>eſt, quam illa depreſſio, ut validior ictus ſequatur; neque enim <lb></lb>tigillus à ſuo axe, cui innititur, omnino æqualiter dividitur, <lb></lb>ſed ab eo aliquantulo remotior eſt malleus, quàm oppoſita ex<lb></lb>tremitas, quæ deprimitur: ac proinde vis illam deprimens ma<lb></lb>jor eſt, quàm ſi tigillus in partes æquales diſtingueretur. </s> <s id="s.004173">Simi<lb></lb>liter in follium motu primùm comparanda eſt rotæ ſemidiame-<pb pagenum="558" xlink:href="017/01/574.jpg"></pb>ter cum adhærente manubrio, deinde Radius Axi ſuperiori in<lb></lb>fixus comparandus eſt cum ſemiſſe tranſverſarij, cui folles jun<lb></lb>guntur; & ex his duabus Rationibus componitur Ratio mo<lb></lb>mentorum potentiæ ad momenta ponderis movendi. </s> </p> <p type="main"> <s id="s.004174">At verò in molendinis, quibus mola frumentaria plano ho<lb></lb>rizontali parallela circumagenda eſt, & quidem velociter, ut <lb></lb>granum in farinam diſſolvatur, non ſatis eſt exterior rota aquæ <lb></lb>impetum recipiens & Axem ſibi infixum volvens, ſed etiam in<lb></lb>terior rota denticulata in eodem Axe requiritur; & ne Machi<lb></lb>næ membra fruſtrà multiplicentur, ita molares lapides com<lb></lb>muniter diſponuntur, ut ferreus axis metam ſuſtinens, & cur<lb></lb>riculo inſtructus, inferiorem locum obtineat, ac proinde cur<lb></lb>riculus ipſe proximè attingat ſuperiorem partem interioris rotæ <lb></lb>in ſuo plano denticulatæ eundem cum exteriore rotâ axem ha<lb></lb>bentis. </s> <s id="s.004175">Quod ſi molares lapides collocari non poſſint in plano, <lb></lb>infra vel ſupra quod volvatur rota interior denticulata, ſed ſo<lb></lb>lùm paulò infra, aut ſupra axem ejuſdem rotæ; quia Vertebra <lb></lb>ſtriata proximè molari lapidi cohærens; adeóque lapidem ipſum <lb></lb>volvens, diſtat, à rotâ denticulatâ, hæc autem commodè non <lb></lb>admittit tam longos dentes, qui ejuſdem Vertebræ aut curri<lb></lb>culi virgulis aptè commiſceri valeant, propterea exigitur alius <lb></lb>Axis horizonti perpendicularis curriculo & rotæ infixus, quem <lb></lb>convertat rota interior curriculi hujus virgulas ſuis dentibus <lb></lb>impellens; ſimul enim rota dentata horizonti parallela, eidem <lb></lb>Axi perpendicularis infixa volvitur, & curriculum molæ con<lb></lb>junctum circumagit. </s> </p> <p type="main"> <s id="s.004176">Hìc quoque plures Rationes componendæ ſunt; prima eſt <lb></lb>Ratio diametri rotæ exterioris ad diametrum rotæ interioris in <lb></lb>eodem axe; deinde Ratio diametri curriculi molæ adhærentis <lb></lb>ad ipſius molæ circumactæ diametrum (ſive integra diameter <lb></lb>accipienda ſit, ſive illa tantum pars, quæ eſt diameter circuli <lb></lb>in rotatione molæ deſcripti à puncto inter centrum & periphe<lb></lb>riam intermedio) & ſi, ut in ſecundo caſu, interjectus fuerit <lb></lb>Axis perpendicularis, prætereà in compoſitionem venit Ratio <lb></lb>diametri curriculi ad diametrum rotæ denticulatæ in eodem <lb></lb>Axe perpendiculari. </s> <s id="s.004177">Ex quibus apparet præſtare rotæ interio<lb></lb>ris diametrum minorem eſſe diametro rotæ exterioris, ut aquæ <lb></lb>hanc impellentis momenta validiora ſint: ſed & cavendum, ne <pb pagenum="559" xlink:href="017/01/575.jpg"></pb>illa ita minor ſtatuatur, ut ejus dentium numerus vix excedat <lb></lb>numerum virgularum curriculi molæ adhærentis, hæc enim <lb></lb>nimis tardè moveretur; & ſi intermedius fuerit Axis perpen<lb></lb>dicularis, poſitâ hac dentium æqualitate & virgularum cur<lb></lb>riculi, unica rotæ exterioris converſio ſemel tantùm con<lb></lb>volveret rotam denticulatam horizonti parallelam, atque <lb></lb>idcirco eodem tempore mola toties ſolùm converteretur, <lb></lb>quoties numerus virgularum ejus curriculi contineretur in <lb></lb>numero dentium rotæ denticulatæ infixæ Axi perpendi<lb></lb>culari. </s> </p> <p type="main"> <s id="s.004178">Ut autem convolutionem molæ numerum augeas, cave ne <lb></lb>movendi difficultas pariter plus juſto augeatur, ſi nimirum <lb></lb>in axe perpendiculari diameter curriculi ſit immodicè mi<lb></lb>nor diametro rotæ denticulatæ in eodem axe: potentia ſi <lb></lb>quidem curriculo applicata multo tardiùs moveretur, quàm <lb></lb>pondus extremis rotæ dentibus applicatum, ac proinde mo<lb></lb>vendi difficultas augeretur. </s> <s id="s.004179">Quare omnia prudenter admi<lb></lb>niſtranda, ut neque potentiæ moventis vires fruſtra con<lb></lb>terantur, neque mola tardiùs aut velociùs, quàm par ſit, <lb></lb>moveatur. </s> </p> <p type="main"> <s id="s.004180">Quod ſi non placuerit, aut loci diſpoſitio non tulerit, <lb></lb>axem illum intermedium ſtatui perpendicularem, ſed hori<lb></lb>zonti parallelus commodior accidat, tunc rotæ interioris <lb></lb>eundem cum exteriore rotâ axem habentis dentes non pla<lb></lb>no infixi, ſed in extremo ambitu defixi requiruntur, ut ſu<lb></lb>perioris axis curriculum (ſive majorem, ſive minorem, prout <lb></lb>opus fuerit) convertant, & cum eo rotam non in ambitu, ſed <lb></lb>in plano, denticulatam, à qua molæ curriculus convolvatur. </s> <lb></lb> <s id="s.004181">Neque aliter, ac priùs, momentorum Ratio componitur, ex <lb></lb>Rationibus videlicet tympanorum, quæ communem Axem <lb></lb>habent, ut ſatis conſtat ex dictis. </s> </p> <p type="main"> <s id="s.004182">Hinc quoniam potentia movens eſt aqua, obſervamus non <lb></lb>omnino eandem eſſe forman rotæ aquam excipientis; quæ <lb></lb>enim in profluente collocantur rotæ, nimis incommodæ eſ<lb></lb>ſent, ſi valdè amplam diametrum haberent; aut modico aquæ <lb></lb>labentis impetu pellerentur, ſi palmulis exiguis inſtruerentur: <lb></lb>propterea rotæ hujuſmodi mediocrem quidem habent diame-<pb pagenum="560" xlink:href="017/01/576.jpg"></pb>trum, ſed valdè notabilem axis partem occupant palmulis <lb></lb>adeò juxtà axis longitudinem expanſis, ut à multâ aquâ in <lb></lb>illas incurrente validiore impulſu circumagantur. </s> <s id="s.004183">Sic in Pa<lb></lb>do communiter Rotæ hujus longitudo eſt cubitorum 10, <lb></lb>diameter tota cubitorum 6; interior rota diametrum habet <lb></lb>cubit. </s> <s id="s.004184">5 1/2, dentes 108 plano infixos, & molæ curriculus in <lb></lb>fuſos 9 diſtinguitur; lapis autem molaris in craſſitudine nu<lb></lb>merat uncias 6 aut 7, in diametro cubitos 2 1/2. Quia ve<lb></lb>rò aquæ ex alto cadentis motus major eſt quàm profluen<lb></lb>tis, propterea rotarum diameter amplior ſtatui poteſt, ſi <lb></lb>opus fuerit, & palmularum latitudo valde mediocris ſuffi<lb></lb>cit, quippe incluſa canali, per quem aqua decidens labi<lb></lb>tur: modica ſcilicet aqua per planum magis elevatum pro<lb></lb>lapſa majora habet momenta, quàm per planum ferè ho<lb></lb>rizontale: & præterea rota amplioris diametri faciliùs vol<lb></lb>vitur etiam à minore aquâ, nam ad interiorem rotam, <lb></lb>cæteris paribus, habet majorem Rationem. </s> <s id="s.004185">Porrò palmu<lb></lb>læ communiter quidem planæ ſunt, aut non niſi mo<lb></lb>dicè ſinuatæ, ita ut aqua hinc atque hinc diffluat; ali<lb></lb>quando tamen limbo ex utraque parte concluduntur, & <lb></lb>quaſi vaſcula aquam aliquandiu continent, ut ipſius aquæ <lb></lb>incluſæ gravitas converſionem juvet deorſum urgendo. </s> <s id="s.004186">Ad<lb></lb>de in ipſo canali inclinato majores eſſe vires aquæ in parte <lb></lb>inferiore, quàm in ſuperiore propè initium casûs; quia vide<lb></lb>licet aqua naturaliter deſcendens motum habet acceleratum, <lb></lb>& ex antecedente deſcenſu acquiſivit impetum. </s> </p> <p type="main"> <s id="s.004187">Hactenus Molendina, quæ aquarum vi aguntur conſide<lb></lb>ravimus, nihil addentes de iis, quæ ab hominibus, aut ab <lb></lb>animalibus volvuntur, nihil enim hæc habent peculiare præ<lb></lb>terquàm quod axis primæ rotæ, quæ cæteris conſequentibus <lb></lb>membris motum conciliat, eſt horizonti perpendicularis, quia <lb></lb>potentia faciliùs in plano horizontali movetur, quàm in tym<lb></lb>pano Verticali, quod calcaretur, & loco exterioris rotæ ab <lb></lb>aquâ propulſæ vectis axi infigitur, quem aut jumenta trahunt, <lb></lb>aut homines urgent. </s> </p> <p type="main"> <s id="s.004188">Aliquid tamen innuendum de Molendinis, quæ vento <lb></lb>aguntur, ſive ad comminuendas fruges, ſive etiam ad agi-<pb pagenum="561" xlink:href="017/01/577.jpg"></pb>tandas antlias, quibus aquæ depreſſioribus campis, inſiden<lb></lb>tes exhauriuntur. </s> <s id="s.004189">Quod enim attinet ad interius artificium <lb></lb>rotarum & curriculorum, ſimillimum eſt iis, quæ in noſtra<lb></lb>tibus molendinis aquâ urgente commotis reperiuntur, niſi <lb></lb>quod in illis, ut pote à ſubjectâ planitie remotis (locus ſi<lb></lb>quidem amplo ventilabro opportunus tribuendus eſt, & cap<lb></lb>tandus ventus) per ſcalas aſcenditur, & in ſuperiorem lo<lb></lb>cum comportandæ ſunt fruges, quas commolere oportet, <lb></lb>atque farina inde transferenda: quo labore levari poteſt <lb></lb>molitor, ſi operâ eâdem, qua ventus axem primarium cum <lb></lb>rotis verſat, ſaccos tritico aut farinâ plenos attollat, aut de<lb></lb>ponat, fune ductario circa ipſum Axem convoluto, aut evo<lb></lb>luto. </s> <s id="s.004190">Illud potiſſimum in hoc molendinorum genere atten<lb></lb>dendum eſt, quod ad ipſa flabella, quibus ventus excipi<lb></lb>tur, ſpectat; neque enim quemadmodum juxta aquæ cur<lb></lb>ſum rotæ planum dirigitur, etiam ventilabrum flabella habet <lb></lb>ita diſpoſita, ut venti ductum ſequantur: ſed ſuperior do<lb></lb>munculæ pars, qua Axis cum rotâ denticulatâ continetur, <lb></lb>uſque adeò convertitur, ut ventilabrum flanti vento adver<lb></lb>ſum ſtatuatur. </s> </p> <p type="main"> <s id="s.004191">Sunt autem flabella quaſi quatuor ſcalæ in primarij Axis <lb></lb>extremitate conjunctæ, quibus obducitur ſingulis linteum, <lb></lb>ut vento reſiſtat; qui ſi juſto validior fuerit, lintei pars <lb></lb>complicata aliquem vento exitum præbet. </s> <s id="s.004192">Non tamen fla<lb></lb>bella hæc ita ex æquo collocantur, ut in uno eodémque pla<lb></lb>no Verticali conſtituantur, ſed ſingulorum flabellorum pla<lb></lb>num modicè obliquum ſtatuitur latere altero ſe paulatim ſub<lb></lb>ducente à vento. </s> <s id="s.004193">Ex quo fit ventum inter quatuor flabel<lb></lb>lorum intervalla intercurrentem repellere in latus, & quaſi <lb></lb>cubito percutere ipſa flabella, atque adeò Axem converti <lb></lb>juxta flabellorum inclinationem. </s> <s id="s.004194">Nam ſi nulla eſſet flabel<lb></lb>lorum obliquitas, & omnia quaſi unicum planum efficerent, <lb></lb>in quod Axis eſſet perpendicularis, incertum eſſet, quam <lb></lb>in partem fieret converſio. </s> <s id="s.004195">Quod ad latitudinem aut longi<lb></lb>tudinem hujuſmodi flabellorum obliquè poſitorum attinet, <lb></lb>non dubitatur, quin eorum latitudo maximè juvet motum; <lb></lb>quia eâdem obliquitate poſitâ, major aëris pars incurrit in <pb pagenum="562" xlink:href="017/01/578.jpg"></pb>amplius quàm in ſtrictius linteum; & in vehementiori ven<lb></lb>to, ne nimia ſit machinæ velocitas, experimur aliquando <lb></lb>non niſi dimidium velum expandi. </s> <s id="s.004196">An verò fuerit operæ <lb></lb>pretium horum longitudinem augere, incertum eſt: quam<lb></lb>vis enim potentia magis à centro motûs diſtans plus habeat <lb></lb>momenti, tamen quia longiorum flabellorum extremitates <lb></lb>valde inter ſe diſtarent, ventus ampliora ſpatia nactus mi<lb></lb>nus haberet virium; ſicut & aqua fluens, velociùs atque <lb></lb>majore conatu per anguſtias, quàm per patentem alveum <lb></lb>currit. </s> <s id="s.004197">Propterea in hujuſmodi flabellis non auderem omni<lb></lb>no definire, quo loco potentiæ moventis vires ſtatuendæ <lb></lb>ſint quaſi in centro virtutis; nam prope Axem, cui infixa <lb></lb>ſunt, modica eſt diſtantia, & ventus quaſi eorum objectu <lb></lb>compreſſus velociùs ſpirat, procul autem ab Axe in majo<lb></lb>re intervallo faciliùs elabens minùs incitat curſum. </s> <s id="s.004198">Cum <lb></lb>verò non ſit temerè ſtatuendum venti compreſſionem om<lb></lb>nino reſpondere mutuis flabellorum diſtantiis, quæ in eâ<lb></lb>dem Ratione ſunt ac diſtantiæ ab Axe; neque facilè aſſeri <lb></lb>poteſt eâdem Ratione decreſcere vim venti ex compreſſio<lb></lb>ne, qua ejuſdem momenta creſcunt ex diſtantiâ ab Axe: <lb></lb>Ex quo fieret momenta compoſita ex diſtantia ab Axe, <lb></lb>& ex vi compreſſionis, eſſe per totam flabelli longitudi<lb></lb>nem æqualiter diffuſa, ac proinde in mediâ longitudine eſ<lb></lb>ſe Centrum virtutis moventis. </s> <s id="s.004199">Omnibus tamen ritè per<lb></lb>penſis, exiſtimarem centrum hoc virtutis, cui applicata <lb></lb>potentia intelligitur, haud procul abeſſe à mediâ fibelli lon<lb></lb>gitudine: Niſi fortè flabella ipſa talia eſſent, ut eorum la<lb></lb>titudo ab Axe recedens augeretur; ſic enim diminutâ in <lb></lb>extremitatibus flabellorum diſtantiâ, etiam venti compreſſio <lb></lb>augeretur. </s> </p> <p type="main"> <s id="s.004200">Quod ſi occurrendum putares incommodo, quod ſubire <lb></lb>neceſſe eſt ædiculam polo innixam ita convertendo, ut fla<lb></lb>bella adverſum ventum excipiant, haud abs re eſſe duce<lb></lb>rem, ſi quis in ſupremo domûs faſtigio, loco patente & <lb></lb>ventis omnibus expoſito, craſſum ſatíſque validum axem ho<lb></lb>rizonti perpendicularem ſtatueret, quem rota denticulata <lb></lb>horizonti parallela complecteretur, ex cujus converſione de-<pb pagenum="563" xlink:href="017/01/579.jpg"></pb>mum mola circumageretur. </s> <s id="s.004201">At flabellorum latitudo juxta <lb></lb>Axis longitudinem in ejuſdem ſupremo capite extra tectum <lb></lb>collocanda eſſet, ut incurrentis venti impulſum exciperent, <lb></lb>perindè atque fluentis aquæ impetum recipiunt palmulæ ro<lb></lb>tarum. </s> <s id="s.004202">Sed quoniam plana flabella parùm apta videntur ad <lb></lb>converſionem continuandam, quia, quæ ſunt à diametro <lb></lb>oppoſita, demùm venti viribus exponerentur æqualiter, nec <lb></lb>dexterum potiùs quàm ſiniſtrum impellendum eſſet, adeó<lb></lb>que ceſſaret converſio; propterea flabella conſtruenda eſ<lb></lb>ſent modicè incurva; hac enim ratione fieret, ut oppoſita <lb></lb>inæqualiter urgerentur, & dextri quidem convexam, ſi<lb></lb>niſtri verò cavam faciem ventus impeteret inæqualibus vi<lb></lb>ribus, illud ſcilicet quaſi ſe ſubducit vento, nec admo<lb></lb>dum ejus impulſui opponitur extremitas juxtà venti directio<lb></lb>nem inflexa; hoc autem cavo ſinu ventum excipiens to<lb></lb>tum ejus impulſum recipit. </s> <s id="s.004203">Adde quod venti particula in <lb></lb>duo proxima flabella incurrens à convexâ unius facie in ca<lb></lb>vam proximi faciem reflectitur, & auget impulſionem. </s> <lb></lb> <s id="s.004204">Quod ſi placuerit non quatuor, ſed quinque flabella ſta<lb></lb>tuere, ne unquam duo ex diametro opponantur, non ab<lb></lb>nuo. </s> <s id="s.004205">Illud certum eſt hujuſmodi flabellorum tùm longi<lb></lb>tudinem, tùm latitudinem plurimùm juvare, quo enim <lb></lb>ampliora ſunt, plus venti excipiunt, & quò longiora, ut <lb></lb>pote à motûs centro magis ſejuncta, plus habent momen<lb></lb>ti. </s> <s id="s.004206">Quomodo autem ſiſtenda ſit machina, explicanda aut <lb></lb>complicanda vela, ne præter molitoris voluntatem agitentur <lb></lb>flabella, nil refert hìc pluribus diſputare, ubi tantummodo <lb></lb>vis movendi conſideratur. </s> <s id="s.004207">Neque ſolùm hujus molendini <lb></lb>uſus eſſet in comminuendis tritici aut leguminum granis, <lb></lb>ſed etiam in attollendis atque alio derivandis aquis, ut palus <lb></lb>exſiccetur, & cæteris hujuſmodi, quæ præſente ſemper cor<lb></lb>pore movendo, non certo tempori alligantur, quemadmodum, <lb></lb>opus molendi, quod non perpetuò exercetur. <pb pagenum="564" xlink:href="017/01/580.jpg"></pb></s> </p> <p type="main"> <s id="s.004208"><emph type="center"></emph>CAPUT VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004209"><emph type="center"></emph><emph type="italics"></emph>Axis cum Vecte compoſitus auget Potentiæ <lb></lb>momenta.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004210">TAnta eſt aliquando ponderis gravitas, ut datæ poten<lb></lb>tiæ vires illi movendo impares ſint, aut de oblatæ ma<lb></lb>chinæ ſoliditate ac firmitate dubitetur: propterea oppor<lb></lb>tunum accidet Vectem cum Axe in Peritrochio componere. <lb></lb><figure id="id.017.01.580.1.jpg" xlink:href="017/01/580/1.jpg"></figure><lb></lb>Primùm dato Vecte <lb></lb>AB ſecundi gene<lb></lb>ris, cujus hypomo<lb></lb>chlium ſit B, & <lb></lb>pondus conſtitutum <lb></lb>in C, Potentia, quæ <lb></lb>in extremitate A ap<lb></lb>plicanda eſt, minor <lb></lb>ſit, quàm pro gra<lb></lb>vitate ponderis, da<lb></lb>tâ vectis Ratione <lb></lb>CB ad AB. </s> <s id="s.004211">Adhi<lb></lb>beatur ſuccula EF <lb></lb>opportunè collata, ut funis ductarius in A alligatus Vectem <lb></lb>attollat: momenta enim potentiæ componuntur ex Ratio<lb></lb>nibus radiorum ſucculæ ad ſemidiametrum Axis, & diſtan<lb></lb>tiæ AB ad diſtantiam CB in vecte. </s> <s id="s.004212">Hinc ſi Ratio AB <lb></lb>ad CB ſit ut 3 ad 1, Ratio autem radiorum ad Axis ſe<lb></lb>midiametrum ſit ut 4 ad 1, unicus homo Succulam ver<lb></lb>tens momentum habet æquale momentis quatuor hominum <lb></lb>in A vecti applicatorum, quorum ſinguli æquiparantur tri<lb></lb>bus, qui pondus idem ſinè vecte attollere conarentur: atque <lb></lb>adeò unicus homo ſucculam convertens æquat vires duode<lb></lb>cim hominum ponderi ipſi proximè applicatorum citra quod<lb></lb>libet machinæ ſubſidium. </s> </p> <pb pagenum="565" xlink:href="017/01/581.jpg"></pb> <p type="main"> <s id="s.004213">Deinde in vecte primi generis, quando movendo pon<lb></lb>deri velocitas aliqua concilianda eſt, validiore potentiâ opus <lb></lb>eſt, & tamen adjecto Axe infirmæ potentiæ adjumentum <lb></lb>comparare in promptu eſt. </s> <s id="s.004214">Sit enim vectis IG, & hypo<lb></lb>mochlium in H. Uti<lb></lb><figure id="id.017.01.581.1.jpg" xlink:href="017/01/581/1.jpg"></figure><lb></lb>que potentia in I tan<lb></lb>to major requiritur, <lb></lb>quanto major eſſe de<lb></lb>bet ponderis motus ſu<lb></lb>pra motum potentiæ, <lb></lb>hoc eſt in Ratione <lb></lb>HG ad HI. </s> <s id="s.004215">Statua<lb></lb>tur Axis RS, & fu<lb></lb>nis ductarius Vectem <lb></lb>apprehendat in I. </s> <s id="s.004216">Tum axi infigatur Radius VT; nam <lb></lb>pro Ratione longitudinis VT ad Axis ſemidiametrum ita <lb></lb>augeri poſſunt potentiæ momenta, ut non ſolùm ponderis <lb></lb>gravitati paria ſint, ſed & illam excedant. </s> <s id="s.004217">Fac enim IH <lb></lb>ad HG eſſe ut 1 ad 4, pondus verò in G eſſe lib. 200, <lb></lb>certè requireretur in I potentia major libris 800, ut ſuâ <lb></lb>virtute gravitati ponderis præſtaret: At ſi Radius VT ad <lb></lb>Axis RS ſemidiametrum ſit ut 10 ad 1, jam potentia in T <lb></lb>motum habet ad motum ponderis in G ut 10 ad 4: igitur <lb></lb>reciprocè potentia in T ad pondus in G eſſet ut 4 ad 10, <lb></lb>ac proinde potentia habens vires attollendi abſque machinâ <lb></lb>libras 80, applicata in T attollet libras 200. Hæc quæ de <lb></lb>attollendo pondere dicta ſunt, intellige pariter ſi in plano <lb></lb>horizontali aut inclinato movendum eſſet; collocato ſcilicet <lb></lb>Axe non parallelo horizonti, ſed vel perpendiculari, vel in<lb></lb>clinato, pro ut loci opportunitas feret: hìc ſiquidem ſola mo<lb></lb>mentorum incrementa conſiderantur ex harum duarum Fa<lb></lb>cultatum compoſitione. </s> </p> <p type="main"> <s id="s.004218">Quid autem opus eſt monere idem virium compendium <lb></lb>haberi poſſe in Vecte pariter primi generis, quando pon<lb></lb>dus tardè movendum eſt? </s> <s id="s.004219">res enim per ſe clara eſt, hypo<lb></lb>mochlio ſcilicet magis ad extremitatem G accedente, quàm <lb></lb>ad extremitatem I, quæ potentiæ locus eſt, ut ſi eſſet in L: <pb pagenum="566" xlink:href="017/01/582.jpg"></pb>id quod tunc potiſſimùm uſurpari poteſt, cùm elevatio pon<lb></lb>deris ad aliquam non minimam altitudinem requiritur; opor<lb></lb>tet enim hypomochlium à pondere intervallo notabili abeſſe, <lb></lb>unde & major movendi difficultas oritur, atque idcircò addi<lb></lb>tâ ſucculâ potentiam juvari neceſſe eſt. </s> <s id="s.004220">Succulam verò po<lb></lb>tiùs adhibendam proponere cenſui, quippe quæ & parabilior <lb></lb>eſt, & commodior, nec multis impenſis conſtruitur: Cæte<lb></lb>rum nec Ergatam, nec tympana ſeu Grues, nec rotas denta<lb></lb>tas, ſi placuerint, excludo. </s> </p> <p type="main"> <s id="s.004221">Ex his ſatis liquet, quid de Vecte tertij generis dicendum <lb></lb>ſit, in quo Potentia media inter pondus & hypomochlium <lb></lb>collocatur: Succula ſcilicet in ſuperiore loco ſtatuenda eſt, <lb></lb>ita ut funis ductarius vectem apprehendat, ubi potentiæ lo<lb></lb>cus aſſignatur: ſed quoniam minor eſt potentiæ, quàm pon<lb></lb>deris motus, & augenda ſunt potentiæ momenta, ut ponde<lb></lb>ris gravitati elevandæ par ſit, Axi addendus eſt Radius tantæ <lb></lb>longitudinis, ut potentia non jam Vecti, ſed Radio applicata <lb></lb>velociùs moveatur, quàm pondus. </s> </p> <p type="main"> <s id="s.004222">Hactenus Axem in Peritrochio additum Vecti conſide<lb></lb>ravimus, quatenus Vectem ſolitarium infirmior potentia <lb></lb>movere nequit: Nunc Vectem addere opportet Axi in <lb></lb>Peritrochio, ut hujus uſus illo addito facilior accidat. </s> <s id="s.004223">Ma<lb></lb>chinulam ſecum deferunt communiter aurigæ in Germania, <lb></lb>qua rotam currûs, ſi fortè limo profundiùs infixa inhæſe<lb></lb>rit, ſublevant, ac proinde recte <emph type="italics"></emph>pancratium aurigarum<emph.end type="italics"></emph.end> dici <lb></lb>poteſt. </s> <s id="s.004224">Lamina eſt chalybea denticulata, cui rotula pariter <lb></lb>dentata congruit, cujuſmodi initio capitis 6. deſcripſimus: <lb></lb>parvula tamen eſt rotula illa, ſed centrum habens commu<lb></lb>ne cum rotâ majore ſimiliter dentatâ, ex cujus converſione <lb></lb>minor convolvitur, & laminam ſurſum propellit. </s> <s id="s.004225">Majoris <lb></lb>rotæ dentes apprehendit Axis ſtriatus, cujus motûs princi<lb></lb>pium ducitur à manubrio extra loculamentum ad latus ex<lb></lb>tante. </s> <s id="s.004226">Quare duplex eſt Ratio, videlicet manubrij ad ſemi<lb></lb>diametrum axis ſtriati, atque diametri rotæ majoris ad dia<lb></lb>metrum rotulæ minoris concentricæ; ex quibus componi<lb></lb>tur Ratio motûs Potentiæ manubrium verſantis, ad motum <lb></lb>ponderis ſublevati. </s> <s id="s.004227">Quia autem fieri poteſt, ut aut de lami-<pb pagenum="567" xlink:href="017/01/583.jpg"></pb>næ ſoliditate dubitetur, aut ſubjectum rotæ ſolum non ad<lb></lb>mittat congruam machinulæ poſitionem; tunc rotæ elevan<lb></lb>dæ capiti ſubjiciatur validus fuſtis alterâ extremitate incum<lb></lb>bens telluri, alterâ innixus dentatæ laminæ; quæ eò minùs <lb></lb>à plauſtri onere gravabitur, quò major erit Ratio totius lon<lb></lb>gitudinis fuſtis ad ejus partem inter rotæ caput, & ſolum, <lb></lb>cui innititur, interjectam. </s> </p> <p type="main"> <s id="s.004228">Hinc ſi Ratio vectis ſit ut 2 ad 1, machinæ lamina non <lb></lb>niſi à ponderis ſemiſſe gravatur; & Potentiæ manubrio Pan<lb></lb>cratij applicatæ momenta geminantur. </s> <s id="s.004229">Nam ſi manubrij <lb></lb>longitudo ad Axis ſtriati ſemidiametrum ſit ut 8 ad 1, ro<lb></lb>tæ autem majoris diameter ad rotulæ concentricæ diame<lb></lb>trum ſit ut 4 ad 1, potentiæ motus ad motum laminæ den<lb></lb>tatæ eſt ut 32 ad 1: ſed appoſito vecte, cujus Ratio datur ut <lb></lb>2 ad 1, jam motus potentiæ ad motum ponderis elevati eſt ut <lb></lb>64 ad 1, & potentiæ conatus, qui ſatis eſſet ad attollendas <lb></lb>ſine machinâ libras 20, hoc Pancratio unâ cum Vecte attol<lb></lb>leret libras 1280. </s> </p> <p type="main"> <s id="s.004230">Similiter ſi Ergatâ AB raptandum eſſet onus, & poten<lb></lb>tia infirmior eſ<lb></lb><figure id="id.017.01.583.1.jpg" xlink:href="017/01/583/1.jpg"></figure><lb></lb>ſet, quam ut in <lb></lb>extremitate Ra<lb></lb>dij CD valeret <lb></lb>ſuperare oneris <lb></lb>reſiſtentiam, ad<lb></lb>hibe Vectem <lb></lb>EF, & extre<lb></lb>mitate E inni<lb></lb>tente ſubjecto <lb></lb>ſolo, potentia <lb></lb>applicetur ex<lb></lb>tremitati F; nam <lb></lb>ejus momenta <lb></lb><expan abbr="componũtur">componuntur</expan> ex <lb></lb>Rationibus CD <lb></lb>Radij ad ſemidiametrum Axis AB, & vectis FE ad DE. </s> <lb></lb> <s id="s.004231">Poteſt autem poſt aliquantulum motum ſubinde promoveri <pb pagenum="568" xlink:href="017/01/584.jpg"></pb>extremitas E vectis, ut manifeſtum eſt. </s> <s id="s.004232">Quod ſi Ergatâ ipsâ <lb></lb>uteremur ad ſenſim demittendum in plano inclinato onus <lb></lb>quoddam ingens, & timeretur, ne vis gravitatis vinceret co<lb></lb>natum hominum in D reluctantium, ne præceps delabatur <lb></lb>onus; adhibeatur vectis EF, quo ſenſim dimiſſo certiùs reti<lb></lb>netur onus, & lentiùs deſcendit. <lb></lb></s> </p> <p type="main"> <s id="s.004233"><emph type="center"></emph>CAPUT IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004234"><emph type="center"></emph><emph type="italics"></emph>Multiplex rotarum dentatarum uſus innuitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004235">QUanquam ea, quæ ad Mechanicam ſcientiam ſpectant <lb></lb>circa tympana dentata, ſatis in ſuperioribus explicata <lb></lb>ſint, quatenus ex iis ſubſidium petitur ad virium ſupplemen<lb></lb>tum, & fontes indicati ſint, ex quibus unuſquiſque variam <lb></lb>hujuſmodi tympanorum complexionem pro opportunitate ex<lb></lb>cogitare poſſit; placuit tamen auctarium adjicere multiplicis <lb></lb>usûs, etiam aliquando citra momentorum potentiæ moven<lb></lb>tis incrementum. </s> <s id="s.004236">Illud autem generatim obſervandum eſt, <lb></lb>ne pluribus membris diſtinguatur machina, ſi pauciora ſuf<lb></lb>ficiant: fieri ſiquidem non poteſt, quin motui mora aliqua <lb></lb>inferatur, ubi plurium membrorum multiplex conflictus at<lb></lb>que tritus contingit, etiamſi omnia ritè diſponantur, & ſibi <lb></lb>invicem proportione reſpondeant. </s> </p> <p type="main"> <s id="s.004237"><emph type="center"></emph>PROPOSITIO I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004238"><emph type="center"></emph><emph type="italics"></emph>Anemoſcopium, Ventorum flantium indicem diſcribere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004239">SI quis in conclavi manens cognoſcere cupiat, quo vento <lb></lb>impellatur aër externus, & Anemoſcopium conſtruen<lb></lb>dum curet, ſi hoc quidem in fornice, aut in laqueari deſ<lb></lb>cribendum ſit, nullo opus eſt artificio; ſed ſatis eſt intra <lb></lb>laminæ VK foramen erecto axi perpendiculari AB, qui <pb pagenum="569" xlink:href="017/01/585.jpg"></pb>nodo C laminæ inſiſtat facilè verſatilis, adjicere flabellum AD <lb></lb>ſupra tecti faſtigium loco apto ita eminens, ut directè, citra <lb></lb>reflexionum ſuſpicionem, cujuſli<lb></lb><figure id="id.017.01.585.1.jpg" xlink:href="017/01/585/1.jpg"></figure><lb></lb>bet auræ flantis impulſum exci<lb></lb>piens, & venti ductum ſequens <lb></lb>convertatur, atque ejus extremi<lb></lb>tas D cœli plagam vento oppoſi<lb></lb>tam reſpiciat. </s> <s id="s.004240">In alterâ verò axis <lb></lb>extremitate B infra laquearis aut <lb></lb>fornicis faciem, in qua ritè juxta <lb></lb>horizontis poſitionem deſcripti <lb></lb>ſint ventorum cardines, adnecta<lb></lb>tur index BF, ea lege, ut ex dia<lb></lb>metro contrariam flabello AD <lb></lb>poſitionem BF obtineat: hinc <lb></lb>enim fiet, ut quoniam ventus ex <lb></lb>A in D directus ſpirat, index F <lb></lb>eam horizontis partem, unde flat, <lb></lb>reſpiciat. </s> </p> <p type="main"> <s id="s.004241">Sin autem in plano Verticali <lb></lb>(aut etiam inclinato) deſcriben<lb></lb>dum ſit Anemoſcopium, ſit axis <lb></lb>AH cum flabello AD tranſiens <lb></lb>per C foramen, & acutâ cuſpide <lb></lb>inſiſtens plano H, ut facillimè <lb></lb>converti queat, vertebram ſtria<lb></lb>tam EG habens in octo æquales <lb></lb>ſtrias diſtinctam, quibus ſubinde <lb></lb>exactè congruere poſſint rotæ MN dentes octo, in quos ferrea <lb></lb>lamina diſtributa eſt æqualiter, antequàm in circulum inflecte<lb></lb>retur. </s> <s id="s.004242">Ex hujus rotæ centro infixus exeat axis R parietem per<lb></lb>vadens, & in extremitate adnexum <expan abbr="indicẽ">indicem</expan> convolvens ad indi<lb></lb>candos ventos in interiori, aut exteriori parietis facie deſcriptos. </s> </p> <p type="main"> <s id="s.004243">Verùm in ventorum deſcriptione cavendum, ne, quemad<lb></lb>modum in Mappis Geographicis ſupremus locus Septentrioni, <lb></lb>infimus Auſtro, dexter (qui ſcilicet eſt ad dexteram aſpicien<lb></lb>tis) Subſolano, ſiniſter Favonio tribuitur, ita hìc ordinem eun<lb></lb>dem ſerves: quia enim vento flabellum impellente ſi vertebra <pb pagenum="570" xlink:href="017/01/586.jpg"></pb>ſtriata convertatur ex G in I, rota dentata aſcendit ex N in M, <lb></lb>& ſimiliter index convolvitur ex T in L; propterea ſi ventus ab <lb></lb>Arcto ſpirans in ſupremâ parte deſcriptus ſit in T, & in infimâ <lb></lb>qui à meridie in O, is, qui ab ortu flat, deſcribendus eſt ad ſi<lb></lb>niſtram in L, & qui ab Occaſu, ad dexteram in P. </s> <s id="s.004244">Quare mani<lb></lb>feſtum eſt, quo ordine reliquos intermedios deſcribere oporteat. </s> </p> <p type="main"> <s id="s.004245"><emph type="center"></emph>PROPOSITIO II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004246"><emph type="center"></emph><emph type="italics"></emph>Currûs motum metiri.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004247">QUæ Vitruvius lib. 10. cap. 14. ſcripſit methodum innuens, <lb></lb>qua Veteres navi aut rhedâ vecti peractum iter dimetie<lb></lb>bantur, plurium ingenia excitarunt (quandoquidem non paucis <lb></lb>Vitruvij verba obſcuritate admodum laborare videbantur, quam <lb></lb>tamen notam illi inurere <expan abbr="nõ">non</expan> auſim) ad varias rationes excogitan<lb></lb>das, quibus hoc idem aſſequi ſe poſſe confidant. </s> <s id="s.004248">Maneat ſua cui<lb></lb>que Machinatori laus; neminis inventa improbo, aut aſpernor: <lb></lb>Mihi <expan abbr="planiſſimã">planiſſimam</expan> inire viam ſemper placuit, qua putaverim ad id, <lb></lb>quod volumus, perveniri poſſe: quapropter nec certam rhedæ <lb></lb>formam, nec verſatilem cum affixis rotis axem præſcribo, ſed ali<lb></lb>quid vulgaribus rhedis aut curribus commune comminiſci pla<lb></lb>cuit, modò liceat alterius poſteriorum rotarum (quippe anterio<lb></lb>bus altiores ſunt) modiolo ad partem interiorem infigere bre<lb></lb>viorem paxillum, quo rota ipſa, dum convertitur, motum ma<lb></lb>chinulæ currui alligatæ conciliet. </s> </p> <p type="main"> <s id="s.004249">Unum moneo, quod ad Vitruvium ſpectat (in quo nullam re<lb></lb>perio obſcuritatem) non arguendum eſſe oſcitantiæ, quòd rotæ <lb></lb>diametrum ſtatuerit pedum quaternum & ſextantis, deinde verò <lb></lb>totam rotæ verſationem definiat pedibus duodecim; cùm tamen <lb></lb>ex Rationibus Cyclicis ſint ut minimum tredecim; atque adeò <lb></lb>quadringentæ verſationes perficiant pedes 5200, hoc eſt paſſus <lb></lb>geometricos 40 ſupra milliare; ex quo integrâ die, qua milliaria <lb></lb>30 computarentur, error eſſet paſſum 1200, qui milliaribus 30 ad<lb></lb>dendi eſſent. </s> <s id="s.004250">Contra verò ſi rotæ ambitus ſolùm peragat pedes <lb></lb>12; quadringentæ verſationes dant pedes 4800, & pedes 200 de<lb></lb>ſunt ad milliaris complementum: quare & hìc in milliarium 30 <lb></lb>computatione deeſſent paſſus 1200, qui milliaribus 30 demen<lb></lb>di eſſent. </s> <s id="s.004251">Ipſe tamen Vitruvius quadringentis verſationibus tri<lb></lb>buit ſpatia pedum 5000, hoc eſt integri milliaris. </s> </p> <pb pagenum="571" xlink:href="017/01/587.jpg"></pb> <p type="main"> <s id="s.004252">Non eſt, inquam, oſcitantiæ arguendus Vitruvius, quem iſta <lb></lb>latere non potuerunt, cùm ſint admodum obvia cuique vel levi<lb></lb>ter Geometricis aſperſo; ſed eo conſilio rotæ diametrum ſupra <lb></lb>quatuor pedes ſextante auxit, ut quod deteritur ex aliquâ rotæ <lb></lb>depreſſiore in ſolo, cui impreſſa veſtigia relinquit, hoc augmento <lb></lb>aliquâ ex parte reſtituatur, adeóque tota verſatio conſiſtat intra <lb></lb>pedes 12 & 13; addere autem certam fractiunculam pedibus 12, <lb></lb>temerarium fuiſſet, illa quippe valdè inconſtans & incerta eſt: <lb></lb>ſatis fuit demum in ſummâ 400 verſationum medium eligere <lb></lb>inter 5200, & 4800: neque enim error, qui notabilis eſſet, obre<lb></lb>pere poterat. </s> </p> <p type="main"> <s id="s.004253">Non tamen placet Vitruvianum tympanum, cujus orbita in <lb></lb>quadringentos æquales denticulos eſſet diſtributa, nimia quippe <lb></lb>& incommoda mihi videtur hujuſmodi tympani magnitudo: ſi <lb></lb>enim ligneum fuerit tympanum, ſingulorum <expan abbr="dẽtium">dentium</expan> pedem, quo <lb></lb>orbitæ cohærent, vix puto minorem eſſe poſſe latitudine digitali, <lb></lb>hoc eſt quatuor granorum hordei, ſi <expan abbr="quidẽ">quidem</expan> ſatis validi, & ad pe<lb></lb>rennitatem conſtructi intelligantur: ſin autem ferreum fuerit <lb></lb>tympanum, latitudo ſingulorum ſaltem æquabit duo grana hor<lb></lb>dei. </s> <s id="s.004254">Quare orbita tympani in 400 hujuſmodi dentes diſtributa, <lb></lb>erit digitorum 400, aut 200, hoc eſt, palmorum 100, aut 50; ac <lb></lb>proinde diameter erit palmorum ferè 32, aut 16. Commodius <lb></lb>igitur acciderit minora tympana componere, quàm adeò in<lb></lb>gens tympanum conſtruere in tot dentes diviſum. </s> </p> <p type="main"> <s id="s.004255">Sit it a que primùm rota denticulata A, cujus denti inſiſtens <lb></lb>haſtula CI axiculo in I jun<lb></lb><figure id="id.017.01.587.1.jpg" xlink:href="017/01/587/1.jpg"></figure><lb></lb>gatur laminæ HI ita fixæ <lb></lb>in H, ut elateris, non tamen <lb></lb>admodùm validi, vice fun<lb></lb>gatur. </s> <s id="s.004256">Tum haſtulæ CI <lb></lb>ſubjiciatur elaſma D ali<lb></lb>quanto validius, quantum <lb></lb>ſatis fuerit ad efficiendum <lb></lb>motum, quem ſtatim indi<lb></lb>cabo. </s> <s id="s.004257">Alia pariter haſtula <lb></lb>FG cum ſuo elaſmate E ita <lb></lb>diſponatur, ut denti G occurrens non permittat rotam retroagi <lb></lb>ex G versùs B, ſed ſolùm converti poſſe ex G in C, atque à ſin-<pb pagenum="572" xlink:href="017/01/588.jpg"></pb>gulis dentibus elevata ſtatim vi elaſmatis E recidat, séque illis <lb></lb>objiciat, ne retrocedant. </s> <s id="s.004258">Additus igitur ſuniculus CS ſi trahatur, <lb></lb>dentem rotæ convertit haſtula CI impellens ſubjectum elaſma <lb></lb>D, eademque operâ dens unus tranſgreditur haſtulam FG, quæ <lb></lb>vi elaſmatis E recidens prohibet, ne in contrarium fieri poſſit ro<lb></lb>tæ converſio. </s> <s id="s.004259">Quia verò haſtula CI dum trahitur, dentem quo<lb></lb>què ſecum rapit, & ab eo inclinato demum liberatur, dimiſſo fu<lb></lb>niculo, vi elaſmatis D ſurſum validè propellitur, & per obliquum <lb></lb>dentis latus excurrens extremitas C, obliquè pariter deſinens, <lb></lb>repellit in I elaterem HI, donec haſtula ipſa dentis apicem <lb></lb>tranſgreſſa ab elatere HI ſeſe reſtituente coaptetur lateri ſupe<lb></lb>riori dentis. </s> <s id="s.004260">Quo pacto ſinguli rotæ dentes ſubinde convertun<lb></lb>tur; atquè tandiu hujuſmodi convolutio perſeverat, quandiu <lb></lb>trahitur, & dimittitur funiculus. </s> </p> <p type="main"> <s id="s.004261">Deinde rota altera pariter denticulata paretur, ſuóque axi in<lb></lb>fixa ita diſponatur priori rotæ parallela (ſed citra planorum con<lb></lb>tactum) ut in ejus dentes incurrat paxillus L in rotæ A plano ad <lb></lb>perpendiculum erectus, quo poſt integram prioris rotæ conver<lb></lb>ſionem dens unus ſecundæ rotæ promoveatur. </s> <s id="s.004262">Ex quo fiet tot <lb></lb>prioris rotæ converſiones requiri ad poſteriorem ſemel convol<lb></lb>vendam, quot in poſteriore rotâ dentes numerantur. </s> <s id="s.004263">Simili ra<lb></lb>tione tertia, aut etiam, ſi opus fuerit, quarta rota denticulata pa<lb></lb>retur, & ita pariter parallelæ diſponantur, ut paxillus ſecundæ <lb></lb>rotæ tertiam, & tertiæ quartam convertat, paxillo videlicet den<lb></lb>tium intervalla ſubeunte poſt integram ſuæ rotæ converſionem. </s> </p> <p type="main"> <s id="s.004264">Hinc ut innoteſcat, quoties trahendus, atque dimittendus ſit <lb></lb>funiculus, ut rotæ convertantur, attendendus eſt in ſingulis rotis <lb></lb>dentium numerus: tùm numerus primæ per numerum ſecundæ <lb></lb>ducendus; & qui producitur indicans numerum tractionum fu<lb></lb>niculi, ut ſecunda rota ſemel convertatur, per numerum dentium <lb></lb>tertiæ rotæ eſt multiplicandus, ut ſciamus, quot funiculi tractio<lb></lb>nibus tertia rota gyrum integrum perficiat. </s> <s id="s.004265">Quod ſi hæc poſtre<lb></lb>ma non fuerit, ſed & quarta rota adjiciatur, productus ex ſecun<lb></lb>dâ illâ multiplicatione numerus per numerum dentium quartæ <lb></lb>hujus rotæ multiplicabitur: ac demum innoteſcet, quoties funi<lb></lb>culum trahere oporteat, ut quarta hæc rota totam circuli peri<lb></lb>pheriam percurrat. </s> </p> <p type="main"> <s id="s.004266">Quod ſi paxillis, de quibus dictum eſt, uti non placuerit, ſed po-<pb pagenum="573" xlink:href="017/01/589.jpg"></pb>tiùs libeat ſingulis rotis craſſiuſculos axes inſerere, ex quibus <expan abbr="dẽs">dens</expan> <lb></lb>unus promineat, qui poſt integram ſuæ rotæ converſionem den<lb></lb>tibus ſequentis rotæ implicetur; omnino licebit, & fortaſſe ſuo <lb></lb>commodo non carebit. </s> <s id="s.004267">Illud in rotarum collocatione intra ſuum <lb></lb>loculamentum eſt diligenter animadvertendum, quod prioris ro<lb></lb>tæ paxillus (aut axis dens) non niſi poſt integram ſuæ rotæ con<lb></lb>verſionem incurrat in dentes poſterioris; alioquin in errorem <lb></lb>non ſanè levem inducere nos poſſet index, qui extremitati axis <lb></lb>adnexus in exteriore loculamenti facie indicat ſingularum ro<lb></lb>tarum convolutiones. </s> </p> <p type="main"> <s id="s.004268">Affigatur itaque poſteriori rhedæ parti opportuno loco regula <lb></lb>circa axem verſatilis, cujus ſuperior extremitas conjunctum ha<lb></lb>beat funiculi CS trahendi caput S, inferior <expan abbr="autẽ">autem</expan> extremitas oc<lb></lb>currat paxillo, quem ab initio rotæ modiolo ad partem <expan abbr="interiorẽ">interiorem</expan> <lb></lb>infixiſti: ſic enim fiet, ut paxillo regulam impellente funiculus <lb></lb>trahatur, atque ad ſingulas rotæ currûs converſiones, ſinguli den<lb></lb>tes rotulæ A funiculum trahentem ſequantur: ac propterea in <lb></lb><expan abbr="loculamẽti">loculamenti</expan> facie index cum axe A convolutus indicabit, quoties <lb></lb>rota currûs <expan abbr="cõverſa">converſa</expan> fuerit; & abſolutâ integrâ rotæ A converſio<lb></lb>ne index ſequentis ſecundæ rotulæ oſtendet integras convolu<lb></lb>tiones primæ; atque ita deinceps index tertiæ numerabit convo<lb></lb>lutiones ſecundæ, & index quartæ convolutiones tertiæ. </s> <s id="s.004269">Hinc ſi <lb></lb>rotulæ ſingulæ ſint in dentes <expan abbr="decẽ">decem</expan> diſtributæ, numero, quem in<lb></lb>dicat ſecunda rotula, adde unicam cyphram 0, numero tertiæ ro<lb></lb>tulæ adde duas cyphras 00, & numero à quarta rotula indicato <lb></lb>adde tres cyphras 000; ſtatímque manifeſtus fiet numerus con<lb></lb>verſionum rotæ currûs. </s> <s id="s.004270">Quare ſi poſteriores currûs rotæ <expan abbr="habeãt">habeant</expan> <lb></lb>diametrum quinque pedum, rotæ ambitus eſt trium <expan abbr="paſſuũ">paſſuum</expan> Geo<lb></lb>metricorum (quod eſt ſuper, negligitur, nam ſæpè rota <expan abbr="ſolũ">ſolum</expan> mol<lb></lb>liuſculum penetrans extenuat diametrum) atque adeò, ut ſemel <lb></lb>prima rotula <expan abbr="cõvertatur">convertatur</expan>, currûs rota decies <expan abbr="cõverſa">converſa</expan> percurrit ſpa<lb></lb>tium paſſuum 30; ut ſecunda unicam <expan abbr="converſionẽ">converſionem</expan> perficiat, rota <lb></lb>currûs centies volvitur, & conficit paſſus 300; ut tertia gyrum ab<lb></lb>ſolvat, rota currûs millies vertitur, & tria Italica milliaria percur<lb></lb>rit. </s> <s id="s.004271">Ideò numerus ab indice quartæ rotulæ ſignificatus, indicans <lb></lb>tertiæ rotulæ integras convolutiones, triplicandus eſt, ut peracti <lb></lb>itineris menſura Italicis milliaribus definiatur. </s> <s id="s.004272">Ex quo fit quar<lb></lb>tam rotulam in dentes decem diſtributam ſufficere ad numeran-<pb pagenum="574" xlink:href="017/01/590.jpg"></pb>da milliaria Italica 30: quod ſi plura velis numerare unicâ hu<lb></lb>jus rotæ converſione, in plures dentes, quàm decem, quartam <lb></lb>rotulam diſtingue: ſed non eſt opus, quia unâ convolutione <lb></lb>abſolutâ, milliaribus indicatis addi poſſunt milliaria 30. </s> </p> <p type="main"> <s id="s.004273">Si ad navis curſum dimetiendum machinulam hanc eandem <lb></lb>traducere placeat, <expan abbr="adjiciẽda">adjicienda</expan> eſt ad navis latus rota, ex cujus con<lb></lb>verſione integrâ obſervatum fuerit, <expan abbr="quãtùm">quantùm</expan> navis promoveatur: <lb></lb>nam ſimiliter impellendo regulam, qua funiculus trahitur, rotæ <lb></lb><expan abbr="converſionũ">converſionum</expan> numerus innoteſcet, atque adeò <expan abbr="etiã">etiam</expan> itineris <expan abbr="ſpatiũ">ſpatium</expan>. </s> <lb></lb> <s id="s.004274">Cave tamen, ne in <expan abbr="errorẽ">errorem</expan> incidas, qui facilè obrepere poſſet; cum <lb></lb>enim navis non ſemper æquè mergatur in aquâ (ſeu quia illa non <lb></lb>eſt ſemper æquè onuſta, ſeu quia hæc <expan abbr="nõ">non</expan> eſt ſemper æquè craſſa, <lb></lb>aut tenuis) etiam rota inæqualiter mergitur, ac proinde una rotæ <lb></lb>hujus converſio non ſemper æquali itineris ſpatio reſpondet. </s> </p> <p type="main"> <s id="s.004275">Nec diſſimili ratione pedeſtria itinera metiri licebit, ſi parvu<lb></lb>lam hujuſmodi machinulam ita corpori alligaveris, ut funiculi <lb></lb>extremitas ſub poplite adnectatur: nam ad ſingulos paſſus denti<lb></lb>culus unus convertetur, & demum paſſuum numerus innoteſcet. </s> </p> <p type="main"> <s id="s.004276"><emph type="center"></emph>PROPOSITIO III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004277"><emph type="center"></emph><emph type="italics"></emph>Objecti procul viſi pſeudographam ſpeciem deformare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004278">COntingit aliquando minùs attentos recti ſpecie decipi: pro<lb></lb>pterea hac propoſitione non inutile fuerit abuſum <expan abbr="quendã">quendam</expan> <lb></lb>rotæ dentatæ, quæ facilè fucum faciat imperitis, indicare: ne for<lb></lb>tè ſibi quaſi de præclaro invento inaniter gratulentur. </s> <s id="s.004279">Quadri<lb></lb><figure id="id.017.01.590.1.jpg" xlink:href="017/01/590/1.jpg"></figure><lb></lb>laterum Priſma AB eliga<lb></lb>tur, cujus extremitas in te<lb></lb>nuiorem cylindrum CD <lb></lb>deſinat inſerendum forami<lb></lb>ni ſubjecti plani craſſioris, <lb></lb>ita ut plano ad perpendicu<lb></lb>lum inſiſtat priſma, & ſer<lb></lb>vatâ poſitione perpendicu<lb></lb>lari, facilè converti poſſit in <lb></lb>dexteram, & in ſiniſtram. </s> <lb></lb> <s id="s.004280">Tum rotæ dentatæ ſemiſſis <lb></lb>MFN priſmati ſecundùm <lb></lb>longitudinem excavato inſeratur, atque circa axem A ductum per <pb pagenum="575" xlink:href="017/01/591.jpg"></pb>priſma & rotæ dentatæ centrum circumagi poſſit. </s> <s id="s.004281">In eandem au<lb></lb>tem priſmatis fiſſuram infra rotæ dentatæ ſegmentum immitia<lb></lb>tur regula HI ſuperiùs exaſperata in crenas dentibus rotæ tan<lb></lb>gentis congruentes, adeò ut ex rotæ converſione regula HI ad<lb></lb>ducatur, & reducatur: quæ in I calamum ſcriptorium, aut lapi<lb></lb>dem plumbarium habens (aut ſaltem acutum ſtylum, quo certa <lb></lb>puncta lineis deinde jungenda notari valeant) in ſubjecti char<lb></lb>tâ lineas deſcribit ſequens ductum radij optici per dioptram <lb></lb>MN excepti. </s> <s id="s.004282">Quare quot lineas in objecto procul viſo percurrit <lb></lb>radius opticus, totidem lineæ à ſtylo I deſcribuntur in chartâ. </s> </p> <p type="main"> <s id="s.004283">Quando igitur magis altum, aut longiùs poſitum objecti <lb></lb>punctum per dioptram aſpicitur, dioptræ extremitas oculo pro<lb></lb>xima deprimitur, atque adeò rotæ dentatæ portio ita conver<lb></lb>titur, ut versùs objectum promoveat regulam: contrà verò de<lb></lb>preſſius, aut propius objecti punctum aſpiciens, proximam ocu<lb></lb>lo extremitatem dioptræ elevat, & regulam ab objecto removet; <lb></lb>cuicumque tandem extremitati M, aut N oculum admoveas: Si <lb></lb>enim ex M aſpicias, deprimendo M propellis ſtylum I versùs priſ<lb></lb>ma, hoc eſt versùs objectum; atque ſimiliter ex N <expan abbr="aſpiciẽs">aſpiciens</expan>, depri<lb></lb>mendo N removes ſtylum I à priſmate, & versùs objectum im<lb></lb>pellis. </s> <s id="s.004284">At verò ubi tranſverſum objecti latus aſpiciendum eſt, <lb></lb>factâ circa cylindrulum CD converſione, plurimum intereſt, <lb></lb>utrùm ex M, an ex N aſpicias: Nam ſi oculus ſit in N, & radio <lb></lb>optico percurrat objecti latus à ſiniſtrâ in dexteram, <expan abbr="etiã">etiam</expan> ſtylus I <lb></lb>à ſiniſtrâ in dextram movetur unâ cum extremitate M objectum <lb></lb>reſpiciente. </s> <s id="s.004285">Sin autem oculus ſit in M, atque ſtylus I inter <expan abbr="oculũ">oculum</expan> <lb></lb>& priſma, aut oculus inter ſtylum & priſma interjectus ſit, con<lb></lb>trariam poſitionem habent puncta à ſtylo deſcripta, & ſiniſtra mi<lb></lb>grant in dexteram, atque dextera in ſiniſtram; ſtylus quippe ocu<lb></lb>lum ſequitur, qui motum habet oppoſitum motui alterius extre<lb></lb>mitatis N objectum reſpicientis. </s> <s id="s.004286">Quamobrem expedit oculum <lb></lb>dioptræ in N admovere, & in <expan abbr="objectũ">objectum</expan> <expan abbr="ſtylũ">ſtylum</expan> I obvertere, ut dextra <lb></lb>dextris, & ſiniſtra ſiniſtris <expan abbr="reſpondeãt">reſpondeant</expan>, prout ſub <expan abbr="aſpectũ">aſpectum</expan> cadunt. </s> </p> <p type="main"> <s id="s.004287">Verùm, licèt objecti viſi ſpeciem aliquam hoc artificio adum<lb></lb>brare liceat, <expan abbr="cavendũ">cavendum</expan> tamen, ne ipſi nobis <expan abbr="aſſentãtes">aſſentantes</expan> quaſi <expan abbr="exactã">exactam</expan> <lb></lb>Ichnographiam, & ſubtilem, ſervatis corporis <expan abbr="partiũ">partium</expan> Rationibus, <lb></lb>deſcriptionem nos comparaſſe exiſtimemus: cuique ſcilicet rem <lb></lb>accuratè perpendenti manifeſtum eſt, quandiu ſemicirculus in <pb pagenum="576" xlink:href="017/01/592.jpg"></pb>eodem plano Verticali <expan abbr="cõſiſtit">conſiſtit</expan>, & dioptra elevatur, ſive deprimi<lb></lb>tur, lineam objecti, quam radius opticus percurrit in plano hori<lb></lb>zontali, reſpondere differentiæ Tangentium angulorum, quos <expan abbr="cũ">cum</expan> <lb></lb>perpendiculo AB conſtituit radius opticus: At linea, quam ſty<lb></lb>lus I deſcribit, reſpondet quidem (ſaltem proximè, & quatenus <lb></lb>ſenſu in tantâ parvitate percipi poteſt) differentiæ Tangentium <lb></lb>angulorum æquè differentium, quos cum perpendiculo eodem <lb></lb>AB conſtituere intelligitur linea à centro A ad ſtylum I ducta. </s> <lb></lb> <s id="s.004288">Non tamen fieri poteſt, ut deinde in omnibus poſitionibus muta<lb></lb>to Verticali eadem Ratio ſervetur; quia linea à centro A ad ſty<lb></lb>lum I ducta, non eſt parallela radio optico, ſed angulum multo <lb></lb>minorem conſtituit cum perpendiculo; ac proinde angulorum <lb></lb>minorum differentia, etiamſi æqualis differentiæ angulorum ma<lb></lb>jorum, non infert proportionalem <expan abbr="differentiã">differentiam</expan> Tangentium. </s> <s id="s.004289">Sta<lb></lb>tuatur ex. </s> <s id="s.004290">gr. differentia angulorum duobus gradibus definita, & <lb></lb>in uno Verticali majores anguli à dioptrâ conſtituti ſint gr.88. & <lb></lb>86, minores autem gr.58. & 56: in altero Verticali majores anguli <lb></lb>à dioptrâ conſtituti ſint gr. 73. & 71, minores verò gr.43, & 41. <lb></lb>Quia idem eſt Radius AB, quarum partium 1000 eſt Radius, in <lb></lb>primo Verticali differentia majorum Tangentium eſt 14336, & <lb></lb>differentia Tangentium minorum eſt 118: in ſecuado Vertica<lb></lb>li differentiæ Tangentium ſunt 367 majorum, & 63 minorum <lb></lb>angulorum: inter hos autem terminos non intercedere propor<lb></lb>tionem manifeſtum eſt. </s> </p> <p type="main"> <s id="s.004291">Quando verò, factâ circa cylindrum CD converſione, fit tran<lb></lb>ſitus ab uno plano Verticali ad aliud planum Verticale, linea, <lb></lb>quam radius opticus percurrit, & linea, quam ſtylus I deſcribit, <lb></lb>ſubtendunt quidem ſimiles arcus, opponuntur enim eidem an<lb></lb>gulo Verticalium, ſed ſunt in Ratione diſtantiarum objecti viſi, <lb></lb>atque ſtyli à cylindrulo tanquam centro motûs. </s> <s id="s.004292">Porrò haſce li<lb></lb>neas differentiis illis Tangentium non eſſe analogas perſpicuum <lb></lb>eſt. </s> <s id="s.004293">Quapropter deſcriptum ſchema non ſervans objecti Ratio<lb></lb>nes, cenſendum eſt pſeudographum. </s> </p> <p type="main"> <s id="s.004294">Oporteret plano immobili, cui infigitur priſma, adnectere con<lb></lb>gruis cardinibus aut fibulis, tabellam, quæ ſemper parallela diop<lb></lb>træ cum hac pariter elevaretur & deprimeretur (non tamen cum <lb></lb>eâ convolveretur) ut in chartâ tabellæ affixâ ſpecies magis cum <lb></lb>objecto conveniens deſcriberetur: Qua autem methodo? </s> <s id="s.004295">inge<lb></lb>nioſus lector diſpiciat. </s> </p> <pb pagenum="577" xlink:href="017/01/593.jpg"></pb> <figure id="id.017.01.593.1.jpg" xlink:href="017/01/593/1.jpg"></figure> <p type="main"> <s id="s.004296"><emph type="center"></emph>MECHANICORUM <emph.end type="center"></emph.end><emph type="center"></emph>LIBER SEXTUS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004297"><emph type="center"></emph><emph type="italics"></emph>De Trochlea.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004298">NON ſemper commodum accidit Ergatâ, aut ſuc<lb></lb>culâ, aut Tympano uti ad pondus aliquod moven<lb></lb>dum: ut enim ex iis, quæ ſuperiore libro diſputa<lb></lb>ta ſunt, manifeſtum eſt, ſi in altiorem locum eve<lb></lb>hendum ſit pondus, ibi conſtruere oporteret peg<lb></lb>ma, cui machina inſiſteret: ſæpè autem id fieri non poſſet ſine <lb></lb>magna impenſa, aut citrà incommodum ſive propter loci an<lb></lb>guſtias, ſive propter temporis brevitatem pegmati conſtruendo <lb></lb>imparem. </s> <s id="s.004299">Hinc alia Facultas excogitata eſt, cui <emph type="italics"></emph>Throchleæ<emph.end type="italics"></emph.end> no<lb></lb>men inditum eſt; quippè quæ communiter ex rotulis circà <lb></lb>axem in ſuo loculamento verſatilibus coagmentatur, iíſque cir<lb></lb>cumducitur funis ductarius, quo trahitur pondus trochleæ ad<lb></lb>nexum. </s> <s id="s.004300">Trochleam autem, ut Vitruvius lib. 10 cap.2. teſtatur <lb></lb>nonnulli <emph type="italics"></emph>Rechamum<emph.end type="italics"></emph.end> dicunt. </s> </p> <p type="main"> <s id="s.004301">Ex orbiculorum numero nomen ducit machina; nam ſi uni<lb></lb>cus ſit orbiculus, Trochlea ſimplex, aut Monoſpatos vocatur; <lb></lb>ſi duo fuerint orbiculi, Diſpaſtos; ſi tres Triſpaſtos; atque ita <lb></lb>deinceps. </s> <s id="s.004302">In hac tamen nomenclaturâ obſervandum eſt, non <lb></lb>codem omnes vocabulo uti: aliqui enim cunctos orbiculos <lb></lb>utriuſque loculamenti in unam ſummam referunt, & ex eorum <lb></lb>numero vocabulum ſtatuunt; ut ſi alterius loculamenti duo ſint <lb></lb>orbiculi, alterius verò unicus, Triſpaſton appellant: Alij ta<lb></lb>men nomen indunt ex orbiculis ſingulorum loculamentorum; <lb></lb>nam ſi binos orbiculos ſingula contineant, non Tetraſpaſton, <lb></lb>ſed Diſpaſton vocant, quia communiter ambo loculamenta <lb></lb>æquali orbiculorum numero inſtruuntur, & ex alterius numero <lb></lb>reliqui, pariter numerus innoteſcit. </s> <s id="s.004303">Neque omnino abs re alte-<pb pagenum="578" xlink:href="017/01/594.jpg"></pb>rius tantummodo loculamenti orbiculos numerant, quia hujus <lb></lb>facultatis vires potiſſimùm habentur ex ſolis orbiculis locula<lb></lb>menti, cui pondus trahendum adnectitur; reliquum ſcilicet lo<lb></lb>culamentum cum ſuis rotulis proptereà adjicitur, ut funis ducta<lb></lb>rius ſingulos illius orbiculos complecti poſſit. </s> <s id="s.004304">Ex quo fit, poſito <lb></lb>inæquali orbiculorum numero, modò Monoſpaſton, modò Diſ<lb></lb>paſton dici, prout pondus adnectitur loculamento unum, aut <lb></lb>duos orbiculos habenti. </s> <s id="s.004305">Cæterùm in vocabulis non eſt hæren<lb></lb>dum: Ego Trochleam voco loculamentum unum cum ſuis or<lb></lb>biculis; & quando opus eſt duplici loculamento uti, duplicem <lb></lb>Throchleam dico, atque orbiculos numero, ne ullus ſubeſſe <lb></lb>poſſit æquivocationi locus. </s> </p> <p type="main"> <s id="s.004306">Quantum autem Facultas hæc ſit Axe, aut Vecte utilior, hinc <lb></lb>ſaltem conſtat, quod etiamſi plures potentiæ diverſis funis <lb></lb>ductarij partibus applicentur, æqualia tamen obtinent momen<lb></lb>ta; id quod non contingit pluribus eundem Succulæ Radium, <lb></lb>aut eumdem Vectem urgentibus; neque enim æqualibus à mo<lb></lb>tûs centro intervallis abſunt. <lb></lb></s> </p> <p type="main"> <s id="s.004307"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004308"><emph type="center"></emph><emph type="italics"></emph>Trochlearum forma, & vires exponuntur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004309">ALiquando ſimplicem orbiculum, cujus excavatæ orbitæ <lb></lb>funis ductarius inſiſtit, adhibemus, ut onera ſurſum attol<lb></lb>lamus: & quidem communiter in ſuperiore loco firmatur locu<lb></lb>lamentum cum orbiculo verſatili, & alteram funis extremita<lb></lb>tem apprehendit Potentia, alteri adnectitur pondus ſublevan<lb></lb>dum, quod aſcendendo ſpatium percurrit æquale ſpatio, per <lb></lb>quod Potentia deſcendendo movetur. </s> <s id="s.004310">Id quod eatenus excogi<lb></lb>tatum eſt, quatenus brachia deprimentibus in ponderis eleva<lb></lb>tione inſita brachiorum gravitas vires addit, & minore lacerto<lb></lb>rum contentione opus eſt, quàm ſi pondus ipſum ſurſum trahe<lb></lb>remus brachia elevantes. </s> <s id="s.004311">Factus eſt autem orbiculus circa ſuum <lb></lb>axem verſatilis, ut vitetur difficultas, quæ cæteroqui conſeque<lb></lb>retur mutuum tritum funis cum ſubjecto corpore, cui inſiſteret, <lb></lb>ſi illud non variaretur. </s> <s id="s.004312">Quantus enim ſit hujuſmodi funis cum <lb></lb>ſubjecto corpore (ſi illud non convolvatur) conflictus, <expan abbr="manifeſtũ">manifeſtum</expan> <pb pagenum="579" xlink:href="017/01/595.jpg"></pb>eſt in puteis, quibus ad hauriendam aquam non eſt girgillus, <lb></lb>hoc eſt, orbiculus verſatilis, adjectus, ſed funis tranſverſo fuſti <lb></lb>cylindrico, verùm immobili, inſiſtit; excavatur ſiquidem cylin<lb></lb>der ille diuturno, & frequenti tritu funium. </s> <s id="s.004313">Cæterum ſi non ad <lb></lb>perpendiculum attollendum ſit pondus, ſed in plano horizonta<lb></lb>li, aut inclinato (non tamen lubrico) raptandum, vix, aut ne <lb></lb>vix quidem, ullum compendium conſequeris, ſi funem per or<lb></lb>biculum tranſeuntem trahas in plagam oppoſitam plagæ, ver<lb></lb>sùs quam pondus dirigitur, ac ſi pondus idem arrepto fune ad te <lb></lb>directè rapias: eadem quippe eſt brachiorum contentio, quo<lb></lb>rum inſita gravitas non juvat potentiam, niſi quando hæc deor<lb></lb>ſum tendit. </s> <s id="s.004314">Adhiberi tamen hujuſmodi orbiculus in planitie <lb></lb>poterit, ſi commodiùs Potentia conſiſtat in loco, ubi jacet pon<lb></lb>dus, quàm ibi, quò illud adducendum eſt. </s> </p> <p type="main"> <s id="s.004315">Quamquam verò orbiculus ſtabili loculamento infixus non <lb></lb>ſit aptus ad augendas Potentiæ vires, prout ad Machinæ ratio<lb></lb>nem pertinet; ſi tamen loculamentum ip<lb></lb><figure id="id.017.01.595.1.jpg" xlink:href="017/01/595/1.jpg"></figure><lb></lb>ſum adnectatur ponderi, quod cum illo mo<lb></lb>veatur, geminantur Potentiæ momenta, non <lb></lb>enim æqualis eſt Potentiæ & Ponderis mo<lb></lb>tus, ſed illa duplo velociùs movetur. </s> <s id="s.004316">Sit <lb></lb>pondus attollendum ſivè raptandum A, cui <lb></lb>adnectatur loculamentum orbiculi B; funis <lb></lb>autem ductarius firmetur in C, & funis <lb></lb>extremitatem reliquam apprehendat Po<lb></lb>tentia in D: utique Potentia ut adducat <lb></lb>orbiculum uſque in C, tantumdem pro<lb></lb>gredi debet ultra C, quantum orbiculus <lb></lb>B diſtat à puncto C; oportet ſiquidem <lb></lb>totum funem DBC explicari. </s> <s id="s.004317">Igitur po<lb></lb>tentia ex D venit primùm in E, deinde <lb></lb>in F: eſt autem diſtantia DE æqualis in<lb></lb>tervallo BC; ſed tunc, cùm illa eſt in <lb></lb>E, orbiculus ſolùm eſt in I, & demum <lb></lb>hic eſt in C, quando potentia eſt in F. </s> <lb></lb> <s id="s.004318">Motus itaque potentiæ DF eſt duplus mo<lb></lb>tus orbiculi BC. </s> <s id="s.004319">Porrò cum orbiculo pa<lb></lb>riter trahitur pondus A adnexum; igitur <pb pagenum="580" xlink:href="017/01/596.jpg"></pb>duplo velocior eſt potentiæ motus præ motu ponderis. </s> <s id="s.004320">Qua<lb></lb>re potentia valens trahere motu ſibi æquali pondus aliquod <lb></lb>ſine orbiculo, hoc addito valebit trahere pondus duplo ma<lb></lb>jore gravitate præditum. </s> </p> <p type="main"> <s id="s.004321">Ex quibus manifeſtum eſt, quantum interſit, utrùm ex<lb></lb>tremitati funis adnectatur pondus, & orbiculi loculamen<lb></lb>tum ſtabile ſit, an verò, funis extremitate manente atque <lb></lb>immotâ, ponderi adnectatur loculamentum, quod cum ipſo <lb></lb>pondere moveatur, immò veriùs, cujus motum conſequatur <lb></lb>motus ponderis: nam in ſecundo hoc caſu potentiæ motus <lb></lb>duplus eſt ad motum ponderis; in primâ autem poſitione mo<lb></lb>tus utriuſque ſunt planè æquales. </s> </p> <p type="main"> <s id="s.004322">Hinc ulteriùs conſtat, quando duæ Trochleæ ſimplici <lb></lb>orbiculo inſtructæ adhibentur, ita ut altera fixa maneat, al<lb></lb>tera cum pondere moveatur, nihil addi momenti Potentiæ <lb></lb>ſi funis extremitas alligetur trochleæ ſtabili, aut loco alicui <lb></lb>extra trochleas. </s> <s id="s.004323">Nam ſi in G poſita ſit Trochlea manens <lb></lb>immota H, & altera funis extremitas illi jungatur in O, <lb></lb>ſeu extra illam clavo, aut paxillo in C, Potentia in L ap<lb></lb>plicata æqualiter movetur cum puncto D: at punctum D <lb></lb>movetur duplo velocius, quàm Trochlea B; igitur Poten<lb></lb>tia L movetur ſolum duplo velociùs quàm pondus, perinde <lb></lb>atque ſi non fuiſſet addita trochlea H. </s> <s id="s.004324">Eatenus igitur additur <lb></lb>Trochlea H, quatenus Potentiam & Pondus in oppoſitas pla<lb></lb>gas moveri oportet, aut potentia deorſum conari debet, ut <lb></lb>pondus aſcendat. </s> </p> <p type="main"> <s id="s.004325">Sin autem extremitas funis alligetur Trochleæ mobili, <lb></lb>cui pariter adnectitur pondus, & primùm funis ab unco <lb></lb>trochleæ mobilis deducatus ad orbiculum trochleæ immotæ, <lb></lb>deinde ad orbiculum ejuſdem Trochleæ mobilis, jam Po<lb></lb>tentia triplo velociùs movetur quàm Pondus; quia videli<lb></lb>cet etiam ipſa funis extremitas movetur trahentem ſequens <lb></lb>unâ cum pondere. </s> <s id="s.004326">Concipe enim pondus A ſejunctum à <lb></lb>Trochleâ B, quæ ita firmetur, ut immota maneat, pondus <lb></lb>verò intelligatur tranſlatum in G, atque Trochlea H jam <lb></lb>ſit mobilis: utique Potentia funem in L arreptum trahens <lb></lb>in motu progreditur ultra B, quanta eſt longitudo funis ex<lb></lb>plicati OBDH, quæ longitudo dupla eſt intervalli OB: <pb pagenum="581" xlink:href="017/01/597.jpg"></pb>igitur potentia L accedens ad B ſemel percurrit interval<lb></lb>lum OB, & præterea adhuc duplum ſpatium ultrà B, <lb></lb>dum punctum O venit ad B ſimul cum pondere ad<lb></lb>nexo in G: triplo igitur velociùs movetur Potentia quàm <lb></lb>Pondus. </s> </p> <p type="main"> <s id="s.004327">Simili omnino ratione ac de Trochleis ſimplicibus phi<lb></lb>loſophamur, etiam ratiocinari oportet in Trochleis plures <lb></lb>orbiculos habentibus; ſi enim ſingulæ duos habeant orbi<lb></lb>culos, attendendum eſt, an funis extremitas adnectatur <lb></lb>Trochleæ immotæ, an verò mobili: ſi immotæ, potentia <lb></lb>movetur quadruplo velociùs quàm pondus; ſin autem mo<lb></lb>bili, movetur quintuplo velociùs. </s> <s id="s.004328">Generatim igitur nume<lb></lb>ra orbiculos trochleæ mobilis, cui ſcilicet jungitur pondus, <lb></lb>& pro ſingulis orbiculis duplica potentiæ momenta. </s> <s id="s.004329">Hinc ſi <lb></lb>tres fuerint orbiculi, momentum Potentiæ eſt ſextuplum; ſi <lb></lb>quatuor, octuplum; & ſic deinceps. </s> <s id="s.004330">At ſi eidem Trochleæ <lb></lb>mobili adnectatur extremitas funis, adhuc adde unitatem, <lb></lb>& momentum erit ſeptuplum, aut noncuplum. </s> <s id="s.004331">Funis ſiqui<lb></lb>dem uni trochleæ alligatus primùm inſiſtit orbiculo primo <lb></lb>reliquæ trochleæ; inde flectitur ad orbiculum primum tro<lb></lb>chleæ, cui adnectitur: poſtmodum ad ſecundum orbicu<lb></lb>lum alterius trochleæ tranſit, & rediens ad priorem tro<lb></lb>chleam inſiſtit orbiculo ejus ſecundo; atque ita deinceps, al<lb></lb>terno ex trochleâ in trochleam excurſu, donec orbiculis om<lb></lb>nibus inſiſtat. </s> <s id="s.004332">Quod ſi duabus Trochleis non inſit æqualis <lb></lb>orbiculorum numerus, ſed altera alteram unitate ſuperet, <lb></lb>neceſſe eſt funem alligari trochleæ pauciorum orbiculorum <lb></lb>Quare attendendus pariter eſt numerus orbiculorum trochleæ <lb></lb>mobilis, quæ ſi pauciores habeat orbiculos, utique illi ad<lb></lb>nectitur extremitas funis; atque adeò duplicato ejus orbi<lb></lb>culorum numero addenda eſt unitas: ut, ſi duos habeat or<lb></lb>biculos, motus Potentiæ eſt quintuplus motus Ponderis. </s> <s id="s.004333">At <lb></lb>ſi trochlea mobilis plures habeat orbiculos quàm trochlea im<lb></lb>mota, duplicandus ſolùm eſt illorum numerus, ut habeatur <lb></lb>denominatio momenti; ut, ſi tres fuerint orbiculi, motus po<lb></lb>tentiæ ad ponderis motum eſt ſextuplus. </s> </p> <p type="main"> <s id="s.004334">In hujuſmodi Trochleis plures rotulas habentibus obſer<lb></lb>vandum eſt interiores rotulas minores ſtatui, exteriores verò <pb pagenum="582" xlink:href="017/01/598.jpg"></pb>majores: nam A & C minores ſunt, B & D majores, ne fu<lb></lb>nium ductus ſe invicem intercipiant, ac mo<lb></lb><figure id="id.017.01.598.1.jpg" xlink:href="017/01/598/1.jpg"></figure><lb></lb>tum mutuo tritu retardent, niſi etiam ſeſe viciſ<lb></lb>ſim atterentes funes diſrumpantur. </s> <s id="s.004335">Quare pro<lb></lb>bare non poſſum Trochleas, quæ plures orbi<lb></lb>culos parallelos uni & eidem axi infixos intra <lb></lb>congruum loculamentum habent; quamvis <lb></lb>enim Trochleis hujuſmodi valde inter ſe diſtan<lb></lb>tibus non adeò appareat incommodum funium <lb></lb>ſeſe perfricantium, ubi tamen illæ propiores <lb></lb>factæ fuerint, hoc manifeſtò apparet: præ<lb></lb>terquam quod funis obliquè inſiſtens extremæ <lb></lb>ipſarum rotularum orbitæ, quam contingit, <lb></lb>non adeò facilè movetur, ac ſi illis exactè con<lb></lb>grueret, ut fit, quando ſingulæ rotulæ ſuos ha<lb></lb>bent axes. </s> </p> <p type="main"> <s id="s.004336">Et quidem quod ad axes rotularum ſpectat, <lb></lb>quamvis nec admodum longi ſint, & rotula <lb></lb>ſuo loculamento proximè adhæreat, atque adeò <lb></lb>non ſint facilè obnoxij fractionis periculo, ca<lb></lb>vendum tamen eſt, ne nimis exiles ſint, aut <lb></lb>ex materia non ſatis ſolidâ; ne fortè ponderis <lb></lb>attollendi gravitas illos labefactet. </s> <s id="s.004337">Verum qui<lb></lb>dem eſt non eſſe neceſſe ſingulos axes ſtatuere <lb></lb>ſuſtinendo oneri pares; cum enim plures ſint, adverſus ſingu<lb></lb>los minor conatus ponderis exercetur. </s> <s id="s.004338">Si verò illi exquiſitè <lb></lb>læves atque politi fuerint, faciliorem fore rotularum iis infixa<lb></lb>rum revolutionem apertiùs conſtat, quàm ut moneri artificem <lb></lb>oporteat. </s> </p> <p type="main"> <s id="s.004339">Prætereà rotularum facies optimè lævigatas velim, & lo<lb></lb>culamentum ipſum non placet ita amplum, ut maximam ro<lb></lb>tularum partem includat: ſatis eſt, ſi ità firmum ac ſolidum <lb></lb>ſit, ut axes contineat, & in extremitatibus validos uncos ha<lb></lb>beat, quibus & funis, & onus alligari queant: Quò ſcilicet <lb></lb>minorem rotularum partem tangit, minus cum illis confligit, <lb></lb>adeóque facilior eſt motus: neque enim leviora hæc compen<lb></lb>dia omninò contemnenda ſunt. </s> </p> <p type="main"> <s id="s.004340">Demum funis ductarij craſſitudo ſtatuenda eſt, quæ reti-<pb pagenum="583" xlink:href="017/01/599.jpg"></pb>nendo ponderi reſpondeat: ſed quia plures ſunt funis à trochleâ <lb></lb>in trochleam ductus, ideò quaſi plures funes reputantur, inter <lb></lb>quos quodammodo diſtribuitur ſuſtentatio ponderis, perinde <lb></lb>ferè, atque ſi ex pluribus illis ductibus funis unicus compone<lb></lb>retur. </s> <s id="s.004341">Hinc ſi pondus fuerit adnexum trochleæ I, ſuſtinetur à <lb></lb>quatuor funibus; ſin autem trochlea I in ſuperiore loco firmata <lb></lb>fuerit, & pondus trochleæ H alligatum dependeat, ſuſtinetur <lb></lb>à quinque funibus, nam etiam Potentia in O ſuſtinet fune RO. </s> <lb></lb> <s id="s.004342">Ex funis autem craſſitudine definitur rotularum altitudo, ut ni<lb></lb>mirum orbitæ excavatæ inſiſtere poſſit funis, quin interiorem <lb></lb>loculamenti faciem contingat, ne perpetuo affrictu atteratur <lb></lb>cum diſruptionis periculo, & non levi celeritatis detrimento, <lb></lb>auctâ trahendi difficultate. </s> <s id="s.004343">Porrò cùm excavatam dico rotu<lb></lb>larum orbitam, nolim intelligas quaſi crenam perimetro pro<lb></lb>fundiùs inciſam; ſed ſatius fuerit orbitam ipſam eſſe modicè <lb></lb>ſinuatam; hoc enim pacto facilius excurrit funis, etiamſi paulò <lb></lb>craſſior aliquando adhibendus ſit, qui cæteroqui inter crenæ <lb></lb>inciſæ labra depreſſus non ſine labore ex illis anguſtiis eximere<lb></lb>tur in rotulæ converſione. </s> </p> <p type="main"> <s id="s.004344">Cum itaque ea ſit Trochlearum diſpoſitio, ut pondus tardiùs <lb></lb>moveatur, potentia velocius (ſi videlicet alteri Trochlearum <lb></lb>non Potentia, ſed Pondus adnectatur, alioquin ſi loca permu<lb></lb>tarent, res contrario prorsùs modo ſe haberet) manifeſtum eſt <lb></lb>reſiſtentiam ponderis minui ex tarditate; poterit igitur augeri <lb></lb>ex gravitate: ſæpiùs quippe dictum eſt adæquatum reſiſtentiæ <lb></lb>momentum componi ex inſita gravitate, & ex diſpoſitione ad <lb></lb>motûs velocitatem, aut tarditatem. </s> <s id="s.004345">Potentia igitur valens ſu<lb></lb>perare reſiſtentiam ponderis alicujus certæ gravitatis, ſi cum <lb></lb>illa æqualiter movendum ſit, poterit eodem impetu, atque co<lb></lb>natu ſuperare reſiſtentiam majoris ponderis, ſi ex collocatione, <lb></lb>quatenus cum Potentiâ connectitur, ita minus velociter movea<lb></lb>tur, ut quæ Ratio eſt æqualis illius velocitatis ad minorem ve<lb></lb>locitatem, eadem ſit Ratio majoris ponderis ad pondus illud <lb></lb>æquè velox cum potentiâ; eſt enim omnino par reſiſtentia; <lb></lb>quia quantum addit major velocitas minori ponderi, tantum<lb></lb>dem addit majus pondus minori velocitati. </s> </p> <p type="main"> <s id="s.004346">Quamvis autem ponderis motus non ſit æquè velox ac motus <lb></lb>potentiæ, tamen ponderis motus entitativè acceptus æqualis eſt <pb pagenum="584" xlink:href="017/01/600.jpg"></pb>motui potentiæ, ac proindè mirum non eſt, ſi potentia eadem <lb></lb>impetu eodem æqualem motum producat, atque efficiat. </s> <s id="s.004347">Pone <lb></lb>enim in O gravitatem paulò majorem libris 100; utique ſi in S <lb></lb>ſtatuerentur libræ 100 gravitas O prævaleret, & gravitatem S <lb></lb>elevaret: igitur illa eadem gravitas O elevabit libras 400 in I <lb></lb>adnexas Trochleæ, nam I movetur quadruplo tardiùs quam <lb></lb>O, ex dictis, S autem movetur æqualiter ac O; ergo ratione <lb></lb>motûs tardioris quadruplo minùs reſiſtit pondus lib.400 in I, <lb></lb>licet ratione gravitatis quadruplo magis reſiſtat. </s> <s id="s.004348">Si itaque li<lb></lb>bræ 100 in S intra certum tempus percurrant unà cum Poten<lb></lb>tia O ſpatij pedes 40, eodem tempore ſingulæ libræ 100 gra<lb></lb>vitatis in I adnexæ percurrunt pedes 10: at ſunt libræ 400; <lb></lb>igitur ſunt quatuor motus pedum 10, & illarum omnium mo<lb></lb>tus eſt pedum 40. Quare potentia O idem planè efficit, ac ſi <lb></lb>moveret in S libras 100; id quod præſtare poteſt abſque ulla <lb></lb>machina. </s> <s id="s.004349">Et quidem ſi res attentè perpendatur, nec vulga<lb></lb>ribus vocabulis notionem minus propriam ſubjiciamus, non <lb></lb>eſt dicendum manente eodem conatu, & eadem velocitate Po<lb></lb>tentiæ augeri per Machinam potentiæ momenta, aut vires, <lb></lb>ſemper enim Potentia vincit æqualem reſiſtentiam ſive adhibi<lb></lb>tâ machinâ, ſive abſque illâ, quamvis non ſemper vincat ean<lb></lb>dem gravitatem. </s> <s id="s.004350">Quemadmodum in libra nil refert, utrum <lb></lb>corpus expendendum habeat majorem gravitatem ſecundum <lb></lb>ſpeciem, ſed molem minorem, an verò minorem gravitatem <lb></lb>ſpecificam ſub mole majori, modò reciprocè ſit ut gravitas <lb></lb>ſpecifica ad ſpecificam gravitatem, ita moles ad molem; eſt ſi<lb></lb>quidem par gravitas abſoluta, quæ componitur ex gravitate ſpe<lb></lb>cificâ & mole. </s> <s id="s.004351">Ita pariter æqualis eſt abſoluta ponderis re<lb></lb>ſiſtentia, quæ ex gravitate, & velocitate componitur, ſi fuerit <lb></lb>inter eas reciproca Ratio. <pb pagenum="585" xlink:href="017/01/601.jpg"></pb></s> </p> <p type="main"> <s id="s.004352"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004353"><emph type="center"></emph><emph type="italics"></emph>An Trochlea ad Vectem revocanda ſit.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004354">UT Machinalis motûs cauſa meliùs innoteſcat, neque opus <lb></lb>eſſe Facultates omnes ad Vectem revocare, ut non pauci <lb></lb>hactenus conati ſunt, & adhuc conantur, hìc potiſſimùm <lb></lb>quæſtionem hujuſmodi examinare placuit in Trochleâ. </s> <s id="s.004355">Aiunt <lb></lb>ſiquidem in ſimplici orbiculo, quando ejus centrum immotum <lb></lb>manet, & alteram funis extremitatem potentia apprehendit, ex <lb></lb>alterâ dependet pondus, Vectem eſſe primi generis, cujus hy<lb></lb>pomochlium eſt in centro orbiculi, potentia & pondus in ex<lb></lb>tremitatibus diametri; quæ cùm à centro æqualibus intervallis <lb></lb>abſint, vectis ille nil juvat potentiam. </s> <s id="s.004356">Quando verò ponderi <lb></lb>adnectitur theca, cui orbiculus includitur, adeóque ejus cen<lb></lb>trum unà cum pondere movetur, jam pondus reſpondet orbicu<lb></lb>li centro, & extremitatem alteram diametri obtinet potentia <lb></lb>trahens funem; quapropter hypomochlium cenſendum eſt in <lb></lb>oppoſita diametri extremitate. </s> <s id="s.004357">Quapropter cùm pondus ſit in<lb></lb>ter potentiam, & hypomochlium, vectis eſt ſecundi generis: & <lb></lb>quia pondus eſt in vectis medio, potentiæ momentum duplum <lb></lb>eſt momenti ponderis, ſi poſitio ipſa ſpectetur. </s> <s id="s.004358">Sit orbiculus, <lb></lb>cujus centrum C, ejuſque loculamento <lb></lb>adnexum pondus reſpondeat lineæ CB: <lb></lb><figure id="id.017.01.601.1.jpg" xlink:href="017/01/601/1.jpg"></figure><lb></lb>funis RSDTV ſit alligatus in R, & Poten<lb></lb>tia ſit in V, quæ funem trahens intelligitur <lb></lb>conſtituta in T, & oppoſitum diametri <lb></lb>punctum S cenſetur hypomochlium; at<lb></lb>que adeò momentum Potentiæ ad momen<lb></lb>tum Ponderis eſt ut TS ad CS. </s> <s id="s.004359">Ex quo fit, <lb></lb>ſi reciprocè vis potentiæ ad gravitatem <lb></lb>ponderis ſit ut CS ad TS, ab hujuſmodi <lb></lb>potentiâ ſuſtineri pondus, & potentia ſi augeatur, etiam mo<lb></lb>veri, orbiculo circà ſuum centrum revoluto, & verſus poten-<pb pagenum="586" xlink:href="017/01/602.jpg"></pb>tiam attracto. </s> <s id="s.004360">In converſione autem orbiculi, prout aliæ atque <lb></lb>aliæ ſunt diametri, quas contingunt funis ductus RS, & VT, <lb></lb>alios ſubinde, atque alios vectes eſſe comminiſcuntur. </s> </p> <p type="main"> <s id="s.004361">Verùm hujuſmodi ratiocinationi nunquam aquieſcere po<lb></lb>tui; mihi enim perſpectum eſt, ſi orbiculus non fuerit verſatilis, <lb></lb>ſed omnino fixus in ſuo loculamento, adhuc potentiam V faci<lb></lb>liùs attollere pondus, quod in B intelligitur ſuſpenſum, quàm <lb></lb>illud directè, & immediatè attolleret; & tamen diameter ea<lb></lb>dem TS ſemper maneret horizonti parallela (nam CB ſemper <lb></lb>eſt in perpendiculo) nullumque haberet motum converſionis <lb></lb>circà punctum, quod vocant, hypomochlij S, quo referret mo<lb></lb>tum Vectis proprium. </s> <s id="s.004362">Adde orbiculum in ſuo loculamento <lb></lb>fixum perinde eſſe, atque ſi annulus ponderi adnectatur, & fu<lb></lb>nis alligatus in R inſeratur annulo, atque potentia in V funem <lb></lb>trahat; potentia enim duplo velociùs movetur, quàm annulus <lb></lb>& pondus: hìc autem in annulo, quem nullatenus convolvi cer<lb></lb>tum eſt, quomodo Vectis veſtigium deprehendes? </s> <s id="s.004363">Illud qui<lb></lb>dem incommodi in annulo, & in orbiculo non verſatili, accide<lb></lb>ret, quod funis ob ſuam aſperitatem cum orbiculi orbitá, & cum <lb></lb>annulo confligeret; ex quo tritu non levis movendi difficultas <lb></lb>oriretur: propterea, ad vitandum hujuſmodi incommodum <lb></lb>adhibentur orbiculi circa ſuum axem verſatiles; axis enim poli<lb></lb>tus, aut etiam addito unguine lubricus, ferè nullam creat orbi<lb></lb>culi rotationi difficultatem, funis verò non atterit ejuſdem or<lb></lb>biculi orbitam, quâ revolutâ ille explicatur. </s> <s id="s.004364">Cæterum quod ad <lb></lb>Rationem motuum potentiæ & ponderis ſpectat, eadem eſt Ra<lb></lb>tio dupla, ſive orbiculus verſatilis ſit, ſive fixus, ſive annulus <lb></lb>ponderi adnectatur, ſive etiam ponderi inſeratur funis, ità ut <lb></lb>pondus ipſum excurrere queat. </s> <s id="s.004365">Hoc ſcilicet unicè pendet ex <lb></lb><figure id="id.017.01.602.1.jpg" xlink:href="017/01/602/1.jpg"></figure><lb></lb>ipsâ funis inflexione: nam ſi funis AB ita flectatur, <lb></lb>ut ad extremitatem extremitas accedat, & B veniat <lb></lb>in C propè A; utique non niſi media pars BE mo<lb></lb>vetur; adeò ut, ſi annulus inſeratur funi in B, & per <lb></lb>longitudinem funis, qui complicatur, excurrat, ve<lb></lb>niat ex B in E interea, dum extremitas B, & poten<lb></lb>tiam illam adducens, venit in C: quo in motu ſin<lb></lb>gulæ funis particulæ inter B & E percurrunt ſpa<lb></lb>tium duplum diſtantiæ ſingularum à medio, ante-<pb pagenum="587" xlink:href="017/01/603.jpg"></pb>quam complicarentur: & ſi potentia ex C ulteriùs progredia<lb></lb>tur, ſingulæ funis particulæ inter medium & caput A inter<lb></lb>ceptæ perficiunt ſpatium duplum diſtantiæ ſingularum à capi<lb></lb>te A, ubi funis religatur. </s> </p> <p type="main"> <s id="s.004366">Jam verò ſtatue ampliorem aliquem, & ſatis gravem cylin<lb></lb>drum DEFG, qui rotatu pro<lb></lb><figure id="id.017.01.603.1.jpg" xlink:href="017/01/603/1.jpg"></figure><lb></lb>movendus ſit, aut in plano hori<lb></lb>zontali, aut in ſuperiorem plani <lb></lb>inclinati locum: applicentur au<lb></lb>tem homines in D & G, & quot<lb></lb>quot neceſſarij fuerint juxta cylin<lb></lb>dri longitudinem, qui illum im<lb></lb>pellant. </s> <s id="s.004367">Quæro, an ibi ulla Vectis <lb></lb>ratio intercedat, ita ut ſit quaſi <lb></lb>vectis DE, hypomochlium in E, <lb></lb>& pondus in puncto I, quod <lb></lb>reſpondet centro gravitatis, at<lb></lb>què adeò in cylindri converſione <lb></lb>ſubinde mutetur vectis, & locus <lb></lb>tùm potentiæ, tùm hypomochlij, <lb></lb>prout aliis atque aliis perimetri punctis applicatur potentia im<lb></lb>pellens, quibus ex diametro opponuntur alia atque alia puncta, <lb></lb>in quibus à ſubjecto plano cylinder tangitur. </s> <s id="s.004368">Vix, puto, au<lb></lb>debis Vectem ibi agnoſcere, ubi demum Potentiam impellen<lb></lb>tem, & Pondus, quod in centro gravitatis, ſcilicet in Axe cy<lb></lb>lindri, conſtitutum intelligitur, æqualem motus lineam per<lb></lb>curriſſe deprehenderis, ut manifeſtum eſt in hujuſmodi rotun<lb></lb>dorum corporum revolutione, in qua æqualem lineam percur<lb></lb>runt centrum, & punctum in peripheriâ notatum. </s> <s id="s.004369">Igitur duo<lb></lb>rum funium capita firmiter alliga in M & H, ipsóſque funes <lb></lb>cylindro ſubjice, & in ſuperiorem partem reductos ita diſpo<lb></lb>ne, ut cylindrum complectantur, atque à duabus potentiis, quæ <lb></lb>priùs in D & G impellebant, trahantur capita L & P. </s> <s id="s.004370">Certiſſi<lb></lb>mo conſtat experimento longè faciliùs cylindrum hujuſmodi <lb></lb>funibus convolvi, quàm impulſione potentiarum illi proximè <lb></lb>applicitarum. </s> <s id="s.004371">Si nulla Vectis Ratio agnoſcenda eſt in diame<lb></lb>tro DE, utique facilitas illa movendi non habetur à vecte, qui <lb></lb>nullus eſt: Sin autem Vectem ibi eſſe conſtanter affirmes, igi-<pb pagenum="588" xlink:href="017/01/604.jpg"></pb>tur perindè eſt ſi Potentia proximè, & immediatè applicetur <lb></lb>puncto D, aut H, ad impellendum, atque ſi medio fune MHL <lb></lb>applicetur puncto H trahens funis caput L: atqui longè majo<lb></lb>ra momenta habet funem LH trahens, quàm impellens in H; <lb></lb>cum igitur utrobique idem Vectis; eadem ſcilicet cylindri dia<lb></lb>meter, habeatur, ſed non idem momentum, non ex rationibus <lb></lb>Vectis, ſed aliundè petenda eſt hæc momenti acceſſio: Quia <lb></lb>videlicet fune ſic diſpoſito, potentia duplo velociùs movetur <lb></lb>quàm pondus, nullâ habitâ vectis ratione. </s> <s id="s.004372">Finge jam funem <lb></lb>laxiorem circumplecti cylindrum, & in nodum colligi in X: <lb></lb>utique ſi in X adderetur pondus aliquod raptandum unà cum <lb></lb>cylindro promoto; facilius raptaretur cylindro hujuſmodi fu<lb></lb>nibus revoluto, quàm ſi cylindrus impulſione potentiæ proxi<lb></lb>mè applicatæ promoveretur; & tamen major hæc facilitas ex <lb></lb>nullo vecte addito oriretur. </s> <s id="s.004373">An non ergo cylindrus trochleæ <lb></lb>orbiculum refert, & funis X orbiculi loculamentum, cui pon<lb></lb>dus adnectitur? </s> <s id="s.004374">manifeſto igitur experimento habetur non ex <lb></lb>Vectis rationibus ducendam eſſe majorem movendi facilitatem, <lb></lb>quæ ex ſimplici trochleâ habetur, quando illi adnectitur <lb></lb>pondus. </s> </p> <p type="main"> <s id="s.004375">Sed præſtat examinare, quæ præterea dicuntur, quando ei<lb></lb>dem ſimplici Trochleæ, cui pondus M adnectitur, etiam funis <lb></lb>caput alligatur; tunc enim potentiæ momentum triplex eſt, <lb></lb>adeò ut ad attollendum pondus M ſufficiat potentia ſubtripla <lb></lb><figure id="id.017.01.604.1.jpg" xlink:href="017/01/604/1.jpg"></figure><lb></lb>illius potentiæ, quæ abſque machinâ attol<lb></lb>leret idem pondus. </s> <s id="s.004376">Sic igitur ratiocinantur <lb></lb>apud P. Schott in Magia mechanica Syn<lb></lb>tagm. 4. cap. 2. prop.5. Si fuerit Vectis DE, <lb></lb>in cujus medio C ſit pondus, fuerit autem <lb></lb>quædam potentia in C ſuſtinens, & alia po<lb></lb>tentia illi æqualis ſuſtinens in E, hypomo<lb></lb>chlium verò in D, unaquæque potentia eſt <lb></lb>ſubtripla ponderis ſuſtentati. </s> <s id="s.004378">Quia enim <lb></lb>potentia C diſtat ab hypomochlio D æqua<lb></lb>liter ac pondus in C conſtitutum, ſuſtinet <lb></lb>pondus æquale ſuis viribus; potentia autem <lb></lb>E, quia eſt in duplo majore diſtantiâ quàm <lb></lb>Pondus C, ſuſtinet pondus duplum ſuarum <pb pagenum="589" xlink:href="017/01/605.jpg"></pb>virium. </s> <s id="s.004379">Quoniam ergo Potentiæ ex hypotheſi ſunt æquale, <lb></lb>& totius ponderis duæ partes ſuſtinentur à Potentia E, & una <lb></lb>à Potentia C, illa autem eſt ſubdupla ponderis à ſe ſuſtentati, <lb></lb>unaquæque eſt ejuſdem totius ponderis ſubtripla quo ad vires <lb></lb>ſuſtentandi. </s> <s id="s.004380">Cum igitur in propoſitis Trochleis ſit potentia F <lb></lb>ſuſtinens in medio, & potentia G in alterâ extremitate ſuſti<lb></lb>nens, unaquæque eſt ſubtripla ponderis M ſuſtinendi, ac <lb></lb>propterea Potentia G ſi ſit paulo major quàm ſubtripla, erit <lb></lb>etiam apta ad movendum pondus. </s> </p> <p type="main"> <s id="s.004381">His pariter aſſentiri nequeo, quæ de ponderis ſuſtentatione <lb></lb>dicuntur; nec ſatis video, an Vecti ſecundi generis congruant; <lb></lb>neque enim ſolùm Potentiæ in medio atque in altera extremita<lb></lb>te applicatæ, verùm etiam hypomochlium ipſum exercet vim <lb></lb>ſuſtinendi: ex hoc ſiquidem quod addatur potentia in medio, <lb></lb>ubi eſt pondus, non tollitur omnino preſſio, quâ hypomo<lb></lb>chlium à pondere urgetur. </s> <s id="s.004382">Quare non tota vis ſuſtentandi di<lb></lb>videnda eſt inter duas illas potentias, ſed etiam admittendum <lb></lb>eſt hypomochlij conſortium. </s> </p> <p type="main"> <s id="s.004383">Dic autem, quænam eſt potentia in F retinens pondus? </s> <lb></lb> <s id="s.004384">nonne ſtatim ac potentia in G remiſſiorem conatum adhibet, <lb></lb>etiam F cum pondere deſcendit? </s> <s id="s.004385">ipſa quippe Potentia G dum <lb></lb>intentum funem FI retinet, ſuſtinet etiam pondus; atque adeò <lb></lb>non duæ ſunt potentiæ ſuſtinentes, ſed unica. </s> <s id="s.004386">Et quidem, ſi <lb></lb>res ſincerè exponatur, pondus ſuſtinetur & à potentiâ G ſur<lb></lb>ſum conante, & à clavo S, cui ſuperior trochlea adnectitur, <lb></lb>mediis funibus HD, IF retinente, ità ut centrum gravitatis <lb></lb>ponderis ſit in lineâ Directionis tranſeunte per ipſum clavum S, <lb></lb>ſi funis GE ſit ad perpendiculum, nec in latus retrahat tro<lb></lb>chleam C: eo autem ipſo, quòd Potentia G ſuo conatu prohi<lb></lb>bet, ne funis excurrat, retinet pondus ex eodem clavo S ſuſ<lb></lb>penſum. </s> <s id="s.004387">Quapropter ejuſdem potentiæ G eſt vis illa, quæ & <lb></lb>in F, hoc eſt in C & in E retinet. </s> <s id="s.004388">Quandò verò ſurſum attolli<lb></lb>tur pondus, eadem eſt potentia G, quæ ſurſum trahit F, cui <lb></lb>non minùs applicatur medio fune IF, quàm applicetur ipſi E <lb></lb>medio fune GE; neque enim in F eſt alia potentia ſponte ſur<lb></lb>ſum aſcendens, & ſecum rapiens pondus. </s> </p> <p type="main"> <s id="s.004389">Sed quid fruſtrà confugiamus ad vim ſuſtentandi pondus ex <lb></lb>trochleis dependens? </s> <s id="s.004390">ſi pondus fuerit in plano horizontali tra-<pb pagenum="590" xlink:href="017/01/606.jpg"></pb>hendum, nihil in trochleis reperitur, à quo ſuſtineatur pon<lb></lb>dus omnino incumbens ſubjecto plano, & tamen potentia G <lb></lb>eſt ſubtripla potentiæ, quæ ſine machinâ in eodem plano trahe<lb></lb>ret idem pondus: ratione vectis ED ſolùm eſſe poteſt ſubdu<lb></lb>pla; in F nulla eſt potentia trahens; unde ergo ratione vectis <lb></lb>potentia ad trahendum pondus habet momenti incrementum? </s> <lb></lb> <s id="s.004391">Quod ſi dixeris eandem potentiam, quæ in G trahit, etiam tra<lb></lb>here in F; igitur conatum non adhibet ſubtriplum, ſed ſubſeſ<lb></lb>quialterum; nam conatur & in extremitate E, & in vectis me<lb></lb>dio C, ut tu quidem ais, ita ut utrobique ſit ſubtripla vis mo<lb></lb>vendi: fatendum eſt ergo potentiam trahentem conari ut (2/33) <lb></lb>cum tamen reipſa adhibeat ſolum conatum ut 1/3. </s> </p> <p type="main"> <s id="s.004392">Conſideremus demum Trochleas pluribus in ſtructas orbicu<lb></lb>lis, & videamus, quid ex Vecte ſperari poſſit. </s> <s id="s.004393">Statuunt Autho<lb></lb>res cum eodem P. Schott ibid. </s> <s id="s.004394">prop.7. ſi fuerint duo vectes <lb></lb>BA, & DC, ex quorum me<lb></lb><figure id="id.017.01.606.1.jpg" xlink:href="017/01/606/1.jpg"></figure><lb></lb>dio E & F dependeat pondus G, <lb></lb>duas potentias æquales in B & <lb></lb>D conſtitutas, ſimúlque æqua<lb></lb>liter in ſuſtinendo pondere la<lb></lb>borantes, ſingulas eſſe ſubqua<lb></lb>druplas ponderis. </s> <s id="s.004395">Nam ſi ſola <lb></lb>potentia D ſuſtineret, eſſet pon<lb></lb>deris ſubdupla, ſcilicet ut FC <lb></lb>ad DC; & ſi ſola potentia B <lb></lb>ſuſtineret, eſſet ipſa pariter ſub<lb></lb>dupla, nimirum ut EA ad BA. </s> <lb></lb> <s id="s.004396">Cum igitur ambæ æquales ſint, <lb></lb>& æqualiter conentur, unicui<lb></lb>que reſpondebit ſubduplum <lb></lb>ſubdupli, hoc eſt quarta pars <lb></lb>ponderis. </s> <s id="s.004397">Atqui in Trochleis <lb></lb>binos orbiculos habentibus ſunt duo vectes HI, & PO in me<lb></lb>dio ſuſtinentes pondus, hypomochlia in I & O, atque Po<lb></lb>tentiæ in H & P. </s> <s id="s.004398">Igitur potentia ſuſtinens eſt ponderis ſub<lb></lb>quadrupla, & movens paulò major ſubquadruplâ. </s> </p> <p type="main"> <s id="s.004399">Quæ de duobus Vectibus DC & BA dicuntur, illa quidem <pb pagenum="591" xlink:href="017/01/607.jpg"></pb>eatenus admitto, quatenus ſingulas potentias D & B ſuſtinen<lb></lb>tes ſubquadruplas eſſe ponderis definiunt; nam perinde ſe ha<lb></lb>bent, atque ſi utraque potentia in unius, ejuſdémque vectis ex<lb></lb>tremitate ſimul ſuſtinerent, unicámque potentiam, conſtitue<lb></lb>rent, quæ ſubdupla eſt ponderis: & quia ſingulæ potentiæ ſunt <lb></lb>ad totam & integram potentiam ſubduplæ, ſingulæ ſunt ponde<lb></lb>ris ſubquadruplæ. </s> <s id="s.004400">Cæterum ex hoc quod ambæ potentiæ æqua<lb></lb>les ſint, & ſingulæ ſolitariæ eſſent ſubduplæ arguere, quod uni<lb></lb>cuique reſpondeat ſubduplum ſubdupli, materialiter quidem <lb></lb>verum eſt, non autem formaliter ex modo argumentandi; alio<lb></lb>quin ſi addatur tertius vectis, ſervatâ eadem argumentandi for<lb></lb>mâ, tres eſſent potentiæ, & unicuique reſpondebit ſubduplum <lb></lb>ſubdupli, hoc eſt octava pars ponderis; id quod eſt falſum. </s> <lb></lb> <s id="s.004401">Neque enim ex hoc quod potentia D ſuſtineat pondus in F, <lb></lb>facit illud eſſe minùs grave, quaſi transferatur in E factum <lb></lb>gravitatis ſubduplæ, & potentia B ſubduplam gravitatem pon<lb></lb>deris ſuſtineret ſubduplo conatu, hoc eſt ſubquadruplo ejus, <lb></lb>qui requiritur ad ſuſtinendum totum pondus; alioquin addito <lb></lb>tertio vecte in illius medium transferretur gravitas ſubquadru<lb></lb>pla ponderis, quæ ſuſtineretur à potentiâ illius ſubduplâ, ac <lb></lb>proinde ſuboctuplâ totius ponderis; cum tamen in tribus vecti<lb></lb>bus ſic diſpoſitis tres potentiæ ſuſtinentes ſingulæ ſint ſolum <lb></lb>ſubſextuplæ. </s> <s id="s.004402">Quod ſi pondus alligetur medio primi vectis in F, <lb></lb>tum extremitas D alligetur medio ſecundi vectis in E, & dein<lb></lb>ceps extremitas B alligetur medio tertij vectis, optimè con<lb></lb>cluditur potentiam in F ſuſtinere ſubduplum ſubdupli, & po<lb></lb>tentiam applicatam tertio vecti ſuſtinere ſubduplum ſubdupli <lb></lb>ſubdupli, ac proinde illam eſſe ſubquadruplam, hanc verò ſub<lb></lb>octuplam. </s> <s id="s.004403">Sed hæc diſpoſitio nil juvaret ad explicandum Tro<lb></lb>chlearum momentum. </s> </p> <p type="main"> <s id="s.004404">Verùm in Trochleâ duas illas potentias in H & P non vi<lb></lb>deo; nam unica potentia in X medio fune XSH applicatur <lb></lb>quidem puncto H, ſuóque conatu prohibet ne pondus ſuâ gra<lb></lb>vitate deorſum trahat ipſam Trochleam: at in P quænam alia <lb></lb>Potentia hoc idem efficit? </s> <s id="s.004405">An non eadem Potentia X medio <lb></lb>fune XSHILMP applicatur vecti PO in P? igitur eadem <lb></lb>potentia exhibet conatum duarum potentiarum ſubquadrupla<lb></lb>rum: igitur Potentia non eſt ſubquadrupla, ſed ſolum ſubdu-<pb pagenum="592" xlink:href="017/01/608.jpg"></pb>pla; quemadmodum ſi duos ſimul vectes in D & B idem ſuſti<lb></lb>neret, utique tantumdem virium impenderet in utroque ſimul <lb></lb>ſuſtinendo, quantum ſi unicus eſſet vectis. </s> </p> <p type="main"> <s id="s.004406">Neque dixeris ſuſtineri pondus à funibus inferiores orbicu<lb></lb>los complectentibus: Hoc enim ad propoſitam quæſtionem <lb></lb>nihil eſt, tum quia nulla eſt ſuſtentatio, ſi pondus raptandum <lb></lb>ſit in plano horizontali, & tamen vis Trochleæ exercetur in <lb></lb>motu; tum quia ad pondus retinendum funes vim eandem <lb></lb>exercerent, ſi tam ampla eſſet unius orbiculi orbita, ut funem <lb></lb>utrumque caperet, vel unicus eſſet funis tam validus, ut utri<lb></lb>que illi funi, quibus duo inferiores orbiculi inſiſtunt, æquiva<lb></lb>leret; tum quia vero propius eſt dicere, pondus ſuſtineri à cla<lb></lb>vo, ex quo ſuperior trochlea pendet, quàm à funibus, quemad<lb></lb>modum ipſa potentia ſuſtinet; non autem vis ſuſtinendi tribui<lb></lb>tur funi illi, quem potentia arripit, & quo medio ſuſtinet: <lb></lb>Clavus autem in hujuſmodi trochleis, quando potentia trahens <lb></lb>proximè applicatur trochleæ clavo adnexæ, perinde ſuſtinet <lb></lb>totam atque integram ponderis gravitatem, ſi plures fuerint or<lb></lb>biculi, ac ſi unicus eſſet orbiculus; quamquam potentia minùs <lb></lb>reluctans in pluribus orbiculis, minore impetu conetur adver<lb></lb>sùs pondus, ac proinde illa clavum minùs premat: quando verò <lb></lb>potentia proximè applicatur trochleæ inferiori, atque ſurſum <lb></lb>trahit, clavus nec urgetur ab impetu potentiæ, quem nullum <lb></lb>recipit, nec ipſe ſuſtinet totum pondus. </s> <s id="s.004407">Quod ſi pondus traha<lb></lb>tur in plano horizontali, ſola potentia eſt, quæ adversùs cla<lb></lb>vum ſuam vim exercet ſuperando reſiſtentiam ponderis, quod <lb></lb>nihil agit adversùs clavum, ſed ſuâ gravitate urget ſubjectum <lb></lb>planum. </s> </p> <p type="main"> <s id="s.004408">Ut autem manifeſtè deprehendas nihil eſſe Trochleis cum <lb></lb>Vecte commercij, duo ligna accipe, cujuſcumque tandem fi<lb></lb>guræ: ſingulis tria inſint foramina, quoad ejus fieri poterit, ex<lb></lb>quiſitè polita, ut minore conflictu funis excurrere poſſit: de<lb></lb>inde funis alterno ab uno in alterum lignum ductu per forami<lb></lb>na trajiciatur: Nam ſi alterum lignorum hujuſmodi certo in lo<lb></lb>co firmetur, alteri adnectatur pondus, tum funis extremitatem <lb></lb>arripiens trahas, idem planè præſtabis, quod adhibitis orbicu<lb></lb>lis in communibus Trochleis: & tamen nullum hîc vectis veſti<lb></lb>gium apparet. </s> <s id="s.004409">Certè in majoribus navigiis malus hinc & hinc <pb pagenum="593" xlink:href="017/01/609.jpg"></pb>navis lateribus alligatur, ut rectam poſitionem ſervet: quia au<lb></lb>tem rudentes aliquando remittuntur, ut in majore æſtu, illóſ<lb></lb>que intendi oportet, propterea duo hujuſmodi ligna in Ellipſim <lb></lb>ferè deformata (vel potiùs in ſphæroides Hyperbolium factâ <lb></lb>converſione non circa Axem, ſed circa ordinatim Applicatam) <lb></lb>alterum navis lateri, alterum rudenti adnectunt nautæ, & fu<lb></lb>nem non adeò craſſum per foramina alterno ductu trajiciunt, <lb></lb>quem etiam axungiâ, aut aliâ pinguedine inficiunt, ut faciliùs <lb></lb>excurrat. </s> <s id="s.004410">Cum autem remiſſior factus fuerit rudens, funis illius <lb></lb>caput ſolvunt, & trahentes cogunt ligna illa fieri propiora, ex <lb></lb>quo rudens intenditur exiguo trahentis conatu, ſi animadver<lb></lb>tas quàm operoſum & incommodum eſſet alio artificio ruden<lb></lb>tem remiſſum intendere. </s> <s id="s.004411">Argumentum hoc, quod olim ante <lb></lb>annos vigintiquinque in Collegio Romano meis Auditoribus <lb></lb>inſinuavi, conatus eſt P. Schott ubi ſupra cap.3. eludere dicens <lb></lb><emph type="italics"></emph>ligna illa nullo modo habere rationem trochlearum, quia malus, qui <lb></lb>eſt reſiſtitivum, & debet trahi versùs latera navis, eſt appenſus uni <lb></lb>extremo illorum mediante fune, & potentia trahens eſt applicata <lb></lb>alteri extremo eorumdem, & nihil dependet intermedium. </s> <s id="s.004412">Mirum <lb></lb>ergo non eſt, ſi non habeat Vectis rationem.<emph.end type="italics"></emph.end></s> <s id="s.004413"> Verùm, tanti viri pa<lb></lb>ce dixerim, ligna illa ita habent rationem Trochlearum, ut ſi <lb></lb>illorum loco communes Trochleas ſubſtituas, idem planè & <lb></lb>eodem modo efficias, trochleâ alterâ adnexa navis alteri, alterâ <lb></lb>rudenti intendendo: Neque enim malus eſt reſiſtitivum, quod <lb></lb>ponderis loco ſuccedit, neque ille ad navis latus trahendus eſt, <lb></lb>aut inclinandus, ſed rudentis caput trahendum eſt, ut malo <lb></lb>immoto ad navim accedat, adeóque intendatur: Quare rudens <lb></lb>ipſe intendendus vicem ſubit ponderis, quatenus intentioni <lb></lb>repugnat, & potentia eſt applicata funi per lignorum foramina <lb></lb>trajecto, ſicut applicaretur funi ductario trochlearum orbiculos <lb></lb>complexo. </s> <s id="s.004414">Quod ſi ligna illa non habent rationem Trochlea<lb></lb>rum, & tamen trahendi facilitatem præſtant, ad quam Facul<lb></lb>tatem Mechanicam ſpectant? </s> <s id="s.004415">Non ad Vectem, ut ille quoquè <lb></lb>admittit; non ad Axem, neque ad Cuneum, neque ad Co<lb></lb>chleam, ut manifeſtum eſt, pertinent: igitur vel novam Facul<lb></lb>tatem conſtituunt, vel omnino Trochleæ ſunt. <pb pagenum="594" xlink:href="017/01/610.jpg"></pb></s> </p> <p type="main"> <s id="s.004416"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004417"><emph type="center"></emph><emph type="italics"></emph>An orbiculi magnitudo quicquam conferat.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004418">QUamquam Trochleæ Vires haberi etiam ſinè orbiculis ſu<lb></lb>periùs dictum ſit, communiter tamen rotulas ſuis thecis <lb></lb>incluſas, & verſatiles adhibemus. </s> <s id="s.004419">Quæritur autem, an rotu<lb></lb>larum hujuſmodi magnitudo quicquam conferat ad faciliorem <lb></lb>motum: an verò indiſcriminatim rotulis ſive majoribus, ſive <lb></lb>minoribus uti poſſimus, citra virium notabile diſpendium. </s> <lb></lb> <s id="s.004420">Quæſtioni huic locum fecit Ariſtoteles Mechan. quæſt. </s> <s id="s.004421">9. ubi <lb></lb>inquirit, <emph type="italics"></emph>Cur ea, quæ per majores circulos tolluntur, & trahuntur, <lb></lb>faciliùs & citiùs moveri contingit, veluti majoribus trochleis, quam <lb></lb>minoribus?<emph.end type="italics"></emph.end> & reſpondet, <emph type="italics"></emph>An quoniam quanto major fuerit illa, quæ <lb></lb>à centro eſt, in æquali tempore majus movetur ſpatium? </s> <s id="s.004422">Quamobrem <lb></lb>æquali inexiſtente onere idem faciet, quemadmodum diximus, & <lb></lb>majores libras minoribus exactiores eſſe; ſpartum enim in illis cen<lb></lb>trum eſt.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.004423">Non deſunt, qui negent faciliùs attolli pondus, ex. </s> <s id="s.004424">gr. <lb></lb>ſitulam aquâ plenam è puteo, ſi funis inſiſtat orbiculo majo<lb></lb>ri, quàm ſi minorem complectatur, ac proptereà ab Ariſtotele <lb></lb>fruſtrà quæri cauſam facilitatis, quæ nulla ſit. </s> <s id="s.004425">Si enim diame<lb></lb>ter orbiculi ſumatur ut Vectis primi generis hypomochlium <lb></lb>habens in centro, potentia & pondus in diametri extremitati<lb></lb>bus æqualiter diſtant ab hypomochlio, ac proinde ſive major <lb></lb>ſit, ſive minor diameter, eadem ſemper manet Ratio æqualita<lb></lb>tis momentorum, quatenus ex poſitione pendent; adeóque <lb></lb>nullum eſt facilitatis in movendo diſcrimen. </s> <s id="s.004426">Sin autem nullus <lb></lb>agnoſcatur Vectis, ſed potentiæ motus cum motu ponderis <lb></lb>comparetur, hos ſemper æquales eſſe manifeſtum eſt, ſive ma<lb></lb>jor, ſive minor rotula adhibeatur: atque hinc nullum infert mo<lb></lb>mentorum diſcrimen magnitudo, aut parvitas rotulæ. </s> </p> <p type="main"> <s id="s.004427">Ego tamen, Ariſtotelem omnino temerè majorem hanc mo<lb></lb>vendi facilitatem per majores orbiculos aſſumpſiſſe, affirmare <pb pagenum="595" xlink:href="017/01/611.jpg"></pb>non auſim; neque enim carere potuit experimento aliquo, quo <lb></lb>id ſuaderetur. </s> <s id="s.004428">Difficultas potiùs ſuboriri poteſt, an veram ille <lb></lb>afferat cauſam majoris hujuſce facilitatis: Nam quod innuit de <lb></lb>libris majoribus, quæ exquiſitiores ſunt minoribus, quo pacto <lb></lb>intelligendum ſit, dictum eſt lib. 3. cap. 6: illud autem hìc lo<lb></lb>cum non habere manifeſtum eſt. </s> <s id="s.004429">Nemo negat in majoribus <lb></lb>circulis, quorum major eſt Radius, ab extremitate Radij ma<lb></lb>jorem arcum deſcribi, quàm à Radio minore, ſi tempore eodem <lb></lb>ſimilem arcum deſcribant; ſunt ſcilicet arcus ſimiles in Ratione <lb></lb>Radiorum; ſed quando rotulæ inæquales commune centrum <lb></lb>non habent, neque Radij omnino ſimul moventur, quaſi minor <lb></lb>ſit pars majoris, quid prohibet eodem tempore arcus quídem <lb></lb>æquales, ſed diſſimiles, deſcribi? </s> <s id="s.004430">Nam ſi potentia trahens de<lb></lb>ſcendat per ſpatium palmare, ſive rotula major ſit, ſive minor, <lb></lb>pondus aſcendit per palmum, & punctum in orbitâ rotulæ tam <lb></lb>majoris quàm minoris notatum deſcribit arcum palmarem: hoc <lb></lb>autem tantummodo differunt, quod in univerſo ponderis ele<lb></lb>vati motu rotula minor ſæpiùs convertitur quàm major, & con<lb></lb>verſionum numeri ſunt reciprocè in Ratione Radiorum: ſic ſi <lb></lb>Radius minor ad majorem ſit ut 4 ad 9, novem converſiones <lb></lb>minoris eodem tempore fiunt, ac quatuor converſiones majoris <lb></lb>rotulæ, ſi à Potentiâ æqualiter moveantur. </s> <s id="s.004431">Quare æquali tem<lb></lb>pore major Radius non movetur per majus ſpatium; movetur <lb></lb>ſiquidem æqualiter cum potentiâ trahente & pondere aſcen<lb></lb>dente, quemadmodum & minor Radius. </s> </p> <p type="main"> <s id="s.004432">Ut igitur Ariſtotelis dicto veritatem aliquam conciliemus, <lb></lb>quæ tamen experimentis reſpondeat, illud obſervandum eſt, <lb></lb>quod ſuperiùs innui, videlicet eo conſilio excogitatos eſſe orbi<lb></lb>culos, ut impedimentum ex funis attritu ſubmoveatur, qui ſanè <lb></lb>tantus eſſet cum corpore, cui funis inſiſtit, quanta eſt funis lon<lb></lb>gitudo æqualis motui ponderis, quod trahitur. </s> <s id="s.004433">At in orbiculo <lb></lb>verſatili ſolus axis teritur à cavâ foraminis ſuperficie axis ſuper<lb></lb>ficiei congruente, quæ eò minor eſt, quò minor eſt axis diame<lb></lb>ter diametro ipſius orbiculi: perimetri enim ſunt in Ratione <lb></lb>diametrorum. </s> <s id="s.004434">Quare ſi Axis diameter ad orbiculi diametrum ſit <lb></lb>ex. </s> <s id="s.004435">gr. ſubquadrupla, conflictus axis cum orbiculo eſt ſubqua<lb></lb>druplus ejus, qui eſſet orbitæ orbiculi ſtabilis cum fune mobili; <lb></lb>immò adhuc minor eſt quàm ſubquadruplus, funis enim multò <pb pagenum="596" xlink:href="017/01/612.jpg"></pb>aſperior eſt quàm ſuperficies axis & foraminis ſibi congruentes. </s> <lb></lb> <s id="s.004436">Quoniam verò axis ſoliditas definitur ex pondere, quod ab eo <lb></lb>ſuſtinendum eſt, idem eſſe poteſt axis cum majore, & cum mi<lb></lb>nore orbiculo. </s> <s id="s.004437">Si ergo eidem axi major orbiculus inſeratur, <lb></lb>manifeſtum eſt minorem fieri attritionem datâ motûs æqualita<lb></lb>te. </s> <s id="s.004438">Nam orbiculorum orbitæ ex hypotheſi ſint in Ratione du<lb></lb>plâ, minoris autem orbiculi peripheria ad axis ambitum ſit in <lb></lb>Ratione quadrupla, jam orbita majoris orbiculi ad ambitum axis <lb></lb>eſt in Ratione octupla: ponamus ambitum axis eſſe digitorum <lb></lb>4, orbita minor eſt digitorum 16, orbita major digit. </s> <s id="s.004439">32: igitur ſi <lb></lb>adhibeatur minor orbiculus, dum potentia & pondus pariter <lb></lb>moventur per digitos 16, tritus cum axe eſt per digitos 4 (pono <lb></lb>ſcilicet axem & foramen ſe invicem terere in puncto, in quo <lb></lb>exercetur ſuſtentatio) adhibito autem majore orbiculo, dum <lb></lb>potentia & pondus per digitos 16 moventur, tritus cum axe eſt <lb></lb>ſolum per digitos 2, ſemiſſem ambitûs foraminis. </s> <s id="s.004440">Ubi autem eſt <lb></lb>minus movendi impedimentum, facilior eſt motus; igitur ma<lb></lb>jore orbiculo faciliùs movetur pondus. </s> </p> <p type="main"> <s id="s.004441">Sed ut rem ipſam penitiùs introſpiciamus, animadvertendum <lb></lb>eſt conflictum orbiculi cum Axe non fieri in centro motûs, <lb></lb><figure id="id.017.01.612.1.jpg" xlink:href="017/01/612/1.jpg"></figure><lb></lb>quod idem eſt cum centro <lb></lb>Axis, ſed in ipſius axis ſuper<lb></lb>ficie: quapropter hinc pon<lb></lb>dus repugnans, hinc poten<lb></lb>tia contranitens ſuas exer<lb></lb>cent vires in axem non per <lb></lb>lineam ad ejus centrum <lb></lb>ductam, ſed per lineam à <lb></lb>punctis potentiæ & ponde<lb></lb>ris contingentem ejuſdem <lb></lb>axis ſuperficiem. </s> <s id="s.004442">Sic poſito <lb></lb>Axe, cujus centrum C, ſe<lb></lb>midiameter ad perpendicu<lb></lb>lum CA, ſi in extremitatibus diametri orbiculi DB ſit in B po<lb></lb>tentia, in D pondus, illa ſuas vires in Axem exercet per lineam <lb></lb>contingentem BE, hoc verò per lineam DF. </s> <s id="s.004443">Similiter ſi po<lb></lb>tentia ſit in G, & pondus in I extremitatibus diametri orbiculi <lb></lb>majoris circa eundem Axem, illa vires exercet per contingen-<pb pagenum="597" xlink:href="017/01/613.jpg"></pb>tem GH, hoc per IL. </s> <s id="s.004444">Potentia igitur B trahens, punctum F <lb></lb>orbiculi minoris cogit aſcendere in A, & potentia G cogit <lb></lb>punctum L orbiculi majoris aſcendere pariter in A. </s> <s id="s.004445">Porrò <lb></lb>punctum L propius eſſe puncto A, quàm punctum F, eſt ma<lb></lb>nifeſtum, quia minor Secans CD & minor Tangens DF com<lb></lb>prehendunt arcum SF minorem, quàm ſit arcus SL compre<lb></lb>henſus à majore Secante CI & majore Tangente IL. </s> <s id="s.004446">Hinc ex <lb></lb>Doctrinâ Sinuum conſtat in Radio CA minorem particulam <lb></lb>reſpondere arcui LA, quàm ſit particula reſpondens æquali ar<lb></lb>cui incipienti ab F versùs A. </s> <s id="s.004447">Igitur datâ motûs æqualitate, po<lb></lb>tentiæ ſcilicet trahentis tantum funem, quantus eſt arcus LA, <lb></lb>minùs reſiſtit aſcenſui punctum L, quàm punctum F, & citiùs L <lb></lb>venit in A per breviorem arcum LA, quàm veniat F per lon<lb></lb>giorem arcum FA. </s> <s id="s.004448">Potentia itaque in G faciliùs, hoc eſt mi<lb></lb>nore labore, cæteris paribus movebit pondus in I poſitum, quàm <lb></lb>potentia eadem minori orbiculo in B applicata moveat idem <lb></lb>pondus in D. </s> </p> <p type="main"> <s id="s.004449">Neque dixeris ex æquali funis tractione pondus in ſuâ perpen<lb></lb>diculari lineâ Directionis æqualiter aſcendere. </s> <s id="s.004450">ſive fuerit in D, <lb></lb>ſive in I, ac propterea nullum inveniri facilitatis diſcrimen in <lb></lb>illo attollendo. </s> <s id="s.004451">Quia adhuc conſiderandum eſt pondus, quate<lb></lb>nus eſt <expan abbr="applicatũ">applicatum</expan> Axi medio orbiculo, in quo axe dum aſcendit, <lb></lb>aſcendit pariter in ſuâ <expan abbr="perpẽdiculari">perpendiculari</expan> lineâ Directionis; & quam<lb></lb>vis in hac æqualiter ſe habeat, non tamen eſt æqualiter <expan abbr="applicatũ">applicatum</expan> <lb></lb>axi: lineæ autem GH, IL productæ concurrerent in angulum <lb></lb>magis obtuſum, quàm lineæ BE, DF; quapropter ſibi invicem <lb></lb>minùs adverſantur, quò propiùs accedunt ad rectitudinem. </s> </p> <p type="main"> <s id="s.004452">Verùm facilitas iſta non eſt cum additamento momenti, quod <lb></lb>à machinâ efficitur; Machina enim tribuens movendi facilita<lb></lb>tem eſt pariter cauſa tarditatis motûs; at hìc <emph type="italics"></emph>faciliùs & citiùs<emph.end type="italics"></emph.end> per <lb></lb>majores circulos moveri pondus docet Ariſtoteles; quatenus vi<lb></lb>delicet ſublatâ impedimenti particulâ, quæ ex tritu oriretur, <lb></lb>potentia faciliùs & citiùs movetur, cum qua pariter æquali pla<lb></lb>nè motu etiam pondus movetur, quod tamen per machinam <lb></lb>tardiùs moveretur, quàm potentia. </s> </p> <p type="main"> <s id="s.004453">Quod ſi cylindricam axis ſuperficiem non admittas omnino <lb></lb>congruere cavæ ſuperficiei foraminis, jam contactus Axis eſt ſo<lb></lb>lùm ad punctum A utriuſque orbiculi tam majoris, quàm mino-<pb pagenum="598" xlink:href="017/01/614.jpg"></pb>ris, & tunc refert quandam libræ ſimilitudinem, cujus jugum ſit <lb></lb>aut GI, aut BD, & ſpartum in loco ſuperiore A. </s> <s id="s.004454">Sed non eadem <lb></lb>hìc militat ratio, quæ in libra: nam in brevioribus libræ brachiis, <lb></lb>quando pondera ſunt inæqualis gravitatis, extremitas brachij de<lb></lb>ſcendentis in ſuo motu deflectens à lineâ rectâ, etiam deflectit à <lb></lb>perpendiculo eò magis, quò minor eſt ſemidiameter circuli, cu<lb></lb>jus arcum deſcribit; at in longioribus brachiis majorem arcum <lb></lb>deſcribentibus ſimilem minori, minùs deflectit à perpendiculo; <lb></lb>ac proinde in deſcenſu pauciora deteruntur gravitatis momenta, <lb></lb>cum magis obſecundet naturali gravitatis propenſioni, quæ niti<lb></lb>tur ad perpendiculum. </s> <s id="s.004455">At hìc in orbiculis, ſi Potentia movens ſit <lb></lb>gravitas aliqua major pondere attollendo, non cogitur deflectere <lb></lb>à perpendiculo, ſive major, ſive minor fuerit orbiculus. </s> <s id="s.004456">Quapro<lb></lb>pter non ex Rationibus libræ philoſophandum eſt, ſed conſide<lb></lb>randa eſt preſſio ſuperanda, quæ fit in A, tùm pondere, tùm po<lb></lb>tentiâ deorſum, ex hypotheſi, conantibus; vel ſi pondus in pla<lb></lb>no, cui inſiſtit, raptandum ſit, preſſio fit vi potentiæ trahentis <lb></lb>pondus reſiſtens. </s> <s id="s.004457">Avellenda eſt igitur ab Axe pars orbiculi illum <lb></lb>tangens in A: ſed poſito æquali motu potentiæ tùm in B, tùm in <lb></lb>G, minor motus & tardior particulæ A efficitur, ſi potentia mo<lb></lb>veat in G, quàm ſi moveat in B: facilius igitur illa movet præ iſtâ. </s> <lb></lb> <s id="s.004458">Minorem autem & tardiorem eſſe motum in A, ubi vincenda eſt <lb></lb>vis preſſionis, <expan abbr="cõſtat">conſtat</expan>, quia, ut ſemel foramen orbiculi minoris per<lb></lb>currat axem in A, totus ille convertendus eſt; at orbiculi majo<lb></lb>ris punctum in orbitâ deſignatum ſi moveatur æquali motu ac <lb></lb>punctum minoris orbitæ, non abſolvit integram revolutionem; <lb></lb>atque adeò orbiculus major æquale habens foramen cum orbi<lb></lb>culo minore, ſed multo majorem orbitam, motu æquali non per<lb></lb>currit Axem in A, niſi juxta partem, quæ reſpondeat revolutio<lb></lb>ni orbitæ, quam conſtat non eſſe integram. </s> </p> <figure id="id.017.01.614.1.jpg" xlink:href="017/01/614/1.jpg"></figure> <p type="main"> <s id="s.004459">Hinc conjicere licet poſſe orbiculo con<lb></lb>ſtrui ſatis exactam libram. </s> <s id="s.004460">Fiat ex ligno aut <lb></lb>ex materiâ metallicâ diſcus RST, cujus <lb></lb>centrum V, ejuſque orbita ad tornum mo<lb></lb>dicè excavetur, ut illi inſiſtere poſſint fu<lb></lb>niculi lancium. </s> <s id="s.004461">Tum in V centro fiat fora<lb></lb>men exquiſitè rotundum atque politum, <lb></lb>cui indatur Axis pariter politus & lævis: <pb pagenum="599" xlink:href="017/01/615.jpg"></pb>axis <expan abbr="autẽ">autem</expan> extremitatibus hinc atque hinc eminentibus alligen<lb></lb>tur fila, inter quæ interceptus diſcus poſſit ſuſpendi. </s> <s id="s.004462">Si gravitas <lb></lb>fuerit per univerſam laminam æquabiliter diffuſa, <expan abbr="cõſiſtet">conſiſtet</expan> diſcus <lb></lb>in quacumque poſitione; ſin autem partes fuerint ſecundùm <lb></lb>gravitatem inæquales, ita ſponte convertetur diſcus, ut pars <lb></lb>gravior inferiorem occupatura locum uſque eò deſcendat, dum <lb></lb>centrum gravitatis ſit in lineâ directionis perpendiculari tran<lb></lb>ſeunte per punctum ſuſpenſionis, & punctum contactus orbi<lb></lb>culi cum axe. </s> <s id="s.004463">Hanc perpendicularem lineam refert, atque de<lb></lb>ſignat filum, ex quo ſuſpenditur. </s> <s id="s.004464">Notato igitur diligentiſſimè <lb></lb>puncto S, in quo fila ſuſpendentia tangunt extremam orbitam, <lb></lb>ibi eſt locus apponendæ lingulæ, atque ibi firmandus eſt uter<lb></lb>que funiculus SR & ST. </s> <s id="s.004465">Amotis igitur funiculis, ſeu filis, ex <lb></lb>quibus prius ſuſpendebatur orbiculus, atque adjectâ opportu<lb></lb>nâ lingulâ, apponatur anſa VM, quæ includet lingulam, ſi hæc <lb></lb>fuerit ritè collocata. </s> <s id="s.004466">Demum pendentibus funiculis adnectan<lb></lb>tur lances ita, ut æquilibrium conſtituant, quod à lingula indi<lb></lb>cabitur. </s> <s id="s.004467">Sic parata erit, ut opinor, exactiſſima libra, de qua du<lb></lb>bitari non poſſit, an centrum motûs verè reſpondeat lineæ, in <lb></lb>qua eſt centrum gravitatis: æqualitas brachiorum VR, VT eſt <lb></lb>manifeſta propter faciliorem circuli conſtructionem, quàm bra<lb></lb>chiorum rectorum æqualitatem æquabili & æquali gravitate <lb></lb>præditam: pondera autem ſi inæqualia lancibus imponantur, <lb></lb>ſemper in eodem perpendiculo conſiſtunt, ſive deſcendant, ſive <lb></lb>aſcendant: lingula verò quia ſatis longa eſt, quippe quæ incipit <lb></lb>ab V, quamvis additamentum factum ſit in S, vel modiciſſimam <lb></lb>inclinationem in alterutram partem indicabit. </s> </p> <p type="main"> <s id="s.004468">Cum itaque negari non poſſit in ſimplici orbiculo aliquam <lb></lb>demum movendi facilitatem aquiri, ſi ille major fuerit, quàm <lb></lb>ſi minor, hoc pariter in Trochleis contingere poſſe non nega<lb></lb>rem adhibitis majoribus orbiculis potiùs quàm minoribus. </s> <s id="s.004469">Ve<lb></lb>rùm attendendum eſt, an ſit operæ pretium tam ingentes tro<lb></lb>chleas movendis ponderibus adhibere; illa enim & majore diſ<lb></lb>pendio conſtruerentur, & eſſent valde graves, & ægre trans<lb></lb>ferri poſſent, ſi notabili aliquâ magnitudine præditæ eſſent. </s> <lb></lb> <s id="s.004470">Quare nemini author eſſem, ut rejectis minoribus orbiculis ma<lb></lb>jores quæreret; communiter enim valde mediocribus trochleis <lb></lb>utuntur artifices, & ſatis commodè perficitur motus, ſi orbicu-<pb pagenum="600" xlink:href="017/01/616.jpg"></pb>li facilè convolvantur: commodum verò, quod accederet ex <lb></lb>aliquatenus diminuto orbiculorum cum ſuis axibus conflictu, <lb></lb>non tantum eſt, ut majore incommodo parandum ſit. <lb></lb></s> </p> <p type="main"> <s id="s.004471"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004472"><emph type="center"></emph><emph type="italics"></emph>Qua Ratione Trochlearum vires augeantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004473">EX dictis cap.1. ſatis notum eſt Trochlearum vires augeri <lb></lb>pro multitudine orbiculorum: ſed quoniam non præſtat in<lb></lb>gentes Trochleas conſtruere, proptereà ſatius eſt Trochleas <lb></lb>cum aliâ quapiam Facultate componere, & potiſſimum cum <lb></lb>Axe in Peritrochio, ſive Sucula ſit, ſive Ergata, ſive Tympa<lb></lb>num, quorum Axi dum in converſione circumducitur funis <lb></lb>ductarius, Trochleæ evadunt propiores, & adducitur pondus. <lb></lb><figure id="id.017.01.616.1.jpg" xlink:href="017/01/616/1.jpg"></figure><lb></lb>Ejus rei meminit Lucret. lib.4: Multaque per Trochleas & <lb></lb>Tympana pondere magno commovet, atque levi ſuſtollit ma<lb></lb>china niſu. </s> <s id="s.004474">Et quidem ſuperiore loco, uti de Tympano ageba<lb></lb>tur, indicata eſt methodus geminandi vires Tympani ABCS, <pb pagenum="601" xlink:href="017/01/617.jpg"></pb>ſi nimirum trabis exporrectæ extremitati D alligetur funis <lb></lb>ductarius, qui primùm tranſeat per orbiculum E ponderi attol<lb></lb>lendo adnexum, deinde per orbiculos F & G trabi adhæ<lb></lb>rentes, per quos demum venit ad Axem H, cui circumducen<lb></lb>dus eſt. </s> <s id="s.004475">Quia enim potentia in F duplo velociùs movetur quàm <lb></lb>E, & Potentia in S premens Tympanum movetur velociùs <lb></lb>quam F, in Ratione partis ſemidiametri tympani ad ſemidia<lb></lb>metrum Axis, hoc eſt in Ratione AV ad AH, manifeſtum eſt <lb></lb>geminari momenta tympani ſolitariè accepti. </s> <s id="s.004476">Quod ſi tam ex<lb></lb>tremitati D, quàm Ponderi adnecterentur Trochleæ, adhuc <lb></lb>major eſſet vis Tympani aucta per Trochleas, & viciſſim ma<lb></lb>jor Trochlearum vis aucta per Tympanum. </s> <s id="s.004477">Hinc ſi eſſent duæ <lb></lb>trochleæ binis orbiculis inſtructæ, & funis caput inferiori tro<lb></lb>chleæ adjungeretur, quintuplex fieret tympani momentum, & <lb></lb>viciſſim trochlearum momentum acciperet incrementum in ra<lb></lb>tione AH ad AV. </s> </p> <p type="main"> <s id="s.004478">Diſtinguenda ſunt autem onera, quorum alia ſunt mediocria <lb></lb>(nam minora facilè ſolis trochleis attolluntur, arreptâ ab ho<lb></lb>minibus funis ductarij extremitate) alia majora, & ingentia, <lb></lb>quæ à Vitruvio, ut aliàs innui, Coloſſicotera dicuntur. </s> <s id="s.004479">Pro <lb></lb>mediocribus ponderibus ad operarum numerum minuendum <lb></lb>Trochleis adjungi poteſt Sucula, cui circumducatur funis <lb></lb>ductarius: compoſitis enim Rationibus Suculæ & Trochlea<lb></lb>rum, habetur Ratio momenti potentiæ ad Pondus. </s> <s id="s.004480">Si non ad <lb></lb>multam altitudinem attollendum ſit Pondus, neque proximo <lb></lb>parieti trabem infigi expediat, cui Trochlea adjungatur, & ex <lb></lb>qua onus dependeat, ex Vetruvij præſcripto lib.10. cap.2. tigna <lb></lb>tria parantur longitudine & ſoliditate reſpondentia oneris <lb></lb>magnitudini & gravitati; hæc à capite fibulâ aut funibus con<lb></lb>juncta, in imo divaricata eriguntur, quaſi in pyramidis trian<lb></lb>gularis ſpeciem. </s> <s id="s.004481">Quod ſi timeatur, ne in hanc aut illam par<lb></lb>tem machina inclinetur, funibus in capitibus collocatis, & <lb></lb>circa diſpoſitis in adverſas plagas, atque firmatis, erecta reti<lb></lb>netur. </s> <s id="s.004482">In ſummo, ubi tigna coëunt, alligatur Trochlea; & <lb></lb>inferiùs, ubi commodè applicari poſſit Potentia, exteriori duo<lb></lb>rum tignorum divaricatorum faciei firmiter affiguntur Chelo<lb></lb>nia, hoc eſt fulcra quædam rotundum foramen habentia, in <lb></lb>quæ conjiciuntur Suculæ capita; ut Axis facile verſetur. </s> <s id="s.004483">Sucu-<pb pagenum="602" xlink:href="017/01/618.jpg"></pb>la autem proximè capita aut habet infixos Radios, aut ſaltem <lb></lb>bina foramina ita temperata & diſpoſita, ut vectes in ea immit<lb></lb>ti poſſint variæ longitudinis pro opportunitate atque neceſſita<lb></lb>te, habitâ ratione loci & ponderis. </s> <s id="s.004484">Altera Trochlea adnecti<lb></lb>tur ponderi, prout commodius acciderit, & funis ductarij ex<lb></lb>tremitas ſuperiori trochleæ adnectitur, ejuſque per Trochlea<lb></lb>rum orbiculos trajecti caput ad Suculam religatur, cujus con<lb></lb>verſione attollitur pondus. </s> <s id="s.004485">Machinam hanc aliqui artifices <lb></lb><emph type="italics"></emph>Capram<emph.end type="italics"></emph.end> vocant. </s> </p> <p type="main"> <s id="s.004486">Ex his, ſi data fuerit ponderis gravitas, & nota Potentiæ <lb></lb>virtus, definies trochlearum orbiculos, aut ſaltem vectium lon<lb></lb>gitudinem, qui faciliùs parari poſſunt, & commutari pro re <lb></lb>natâ, quàm aliæ trochleæ inveniri. </s> <s id="s.004487">Sint itaque trochleæ bi<lb></lb>nis orbiculis inſtructæ; harum forma & poſitio non niſi qua<lb></lb>druplum potentiæ motum determinat, ſi cum motu ponderis <lb></lb>comparetur. </s> <s id="s.004488">At potentia univerſa ſint duo homines, ſinguli <lb></lb>valentes attollere libras 25; atque adeò potentia eſt lib. 50; <lb></lb>quæ ſi proximè applicetur funi ductario trochlearum, poterit <lb></lb>ſolùm attollere gravitatem quadruplam, hoc eſt lib.200. Quon<lb></lb>niam verò oblatum pondus eſt ex hypotheſi lib.1000, hoc eſt <lb></lb>quintuplum librarum 200, addenda eſt trochleis Ratio quin<lb></lb>tupla Succulæ, cujus Radij aut Vectes ſint quintupli ſemidia<lb></lb>metri Axis ejuſdem Suculæ. </s> <s id="s.004489">Nam Potentia Vectibus aut Ra<lb></lb>diis applicata quintuplo velociùs movetur, quàm extremitas <lb></lb>funis ductarij Axem complexi; hæc autem quadruplo velociùs <lb></lb>quàm pondus; atque idcircò potentia vigecuplo velociùs mo<lb></lb>vetur quàm pondus, poteritque movere pondus vigecuplum <lb></lb>librarum 50, hoc eſt lib.1000. </s> </p> <p type="main"> <s id="s.004490">Quod ſi ad inſignem aliquam altitudinem evehendum ſit <lb></lb>pondus, non eſt opus tria hujuſmodi tigna compingere, ſed <lb></lb>ut ſumptibus & labori parcatur, ſatis eſt non procul à pondere <lb></lb>longiorem trabem, etiam ex pluribus apte & firmiter con<lb></lb>junctis compoſitam, erigere, atque funibus in oppoſitas vento<lb></lb>rum plagas diſpoſitis ita ejuſdem caput firmare, ut nullam in <lb></lb>partem vi ſuſpenſi ponderis inclinetur. </s> <s id="s.004491">Verùm quidem eſt tra<lb></lb>bem hujuſmodi (Antennam aliqui dicunt) non omnino ad <lb></lb>perpendiculum erigi, ſed modicè inclinatam ſtatui, ut à ſum<lb></lb>mo vertice pendens ad perpendiculum ſarcina, quæ attollitur, <pb pagenum="603" xlink:href="017/01/619.jpg"></pb>non incurrat in trabem. </s> <s id="s.004492">Modicè, inquam, inclinata ſtatuitur <lb></lb>trabs iſta (niſi fortè illa altiùs defodiatur, & circùm fiſtucatio<lb></lb>ne ſolidetur, tunc enim poterit magis inclinata ſtatui) quia <lb></lb>propter notabilem longitudinem ita poteſt inclinari, ut linea <lb></lb>directionis ab illius centro gravitatis ducta cadat intrà (vel cer<lb></lb>tè non admodum ultrà) baſim ſuſtentationis, atque perpen<lb></lb>diculum à ſummo vertice deſcendens & illi lineæ parallelum <lb></lb>tanto abſit intervallo, quod ſatis ſit ad elevandum pondus citra <lb></lb>periculum colliſionis cum trabe: cui periculo occurri non po<lb></lb>teſt in tigno breviore, quo valde inclinato ad vitandum hujuſ<lb></lb>modi periculum colliſionis, linea directionis ab ejus centro <lb></lb>gravitatis cadens multo notabiliùs recederet à baſi ſuſtentatio<lb></lb>nis: propterea ubi brevioribus trabibus fuerit utendum, tres <lb></lb>modo ſuperiùs dicto compinguntur, ut ſe invicem fulcientes <lb></lb>ſponte conſiſtant, & pondus non contingant. </s> <s id="s.004493">Capiti igitur <lb></lb>erectæ trabis longioris altera trochlea alligatur, altera oneri; <lb></lb>ſed ad trabis pedem orbiculus unus firmiter adnectitur, per <lb></lb>quem funis ductarius juxta trabis longitudinem deſcendens <lb></lb>trajicitur, & ad Ergatæ axem adducitur, ut ex ejus revolutio<lb></lb>ne funis trahatur: orbiculum hunc Græci <foreign lang="grc">ἐπαγόντα</foreign> Latini Ar<lb></lb>temonem vocant, ex Vitruvio lib. 10 cap.5. Hic tamen infi<lb></lb>mus orbiculus cum nihil immutet aut potentiæ velocitatem, <lb></lb>aut ponderis tarditatem, nihil addit momenti ipſi potentiæ ad <lb></lb>onus attollendum, ſed ideò potiſſimùm adhibetur, ut funis <lb></lb>commodiùs Ergatæ circumducatur. </s> </p> <p type="main"> <s id="s.004494">Quare Potentiæ momenta componuntur ex momentis Tro<lb></lb>chlearum & Ergatæ; quæ ſi innoteſcant, & data ſit potentiæ <lb></lb>virtus movendi, manifeſtum erit pondus, quod illa Ergatæ ap<lb></lb>plicata movere poterit. </s> <s id="s.004495">Sic ſi Trochleæ binos habeant orbicu<lb></lb>los, qui dant Rationem quadruplum, Vectis autem Ergatæ ſit <lb></lb>ad ejuſdem Axis ſemidiametrum ut 20 ad 1, Ratio, quæ ex <lb></lb>quadruplâ, & vigecuplâ componitur, eſt octuagecupla; ac <lb></lb>proindè potentia extremo Vecti applicata poterit movere pon<lb></lb>dus octuagecuplum ejus, quod ſinè machinâ movere poteſt. </s> </p> <p type="main"> <s id="s.004496">Hinc ſi potentia movere valeat libras 50, huic machinæ ap<lb></lb>plicata movebit pondus lib. 4000. Illud autem commodi ha<lb></lb>bet Ergata, quod in illâ convolvendâ uti poſſumus jumentis <lb></lb>extremo vecti applicatis: & experimento didicimus trochleis <pb pagenum="604" xlink:href="017/01/620.jpg"></pb>binorum orbiculorum, & Ergatâ attolli à duobus equis pondus <lb></lb>librarum non minùs quàm triginta millium. </s> <s id="s.004497">Cum enim ſint <lb></lb>duo equi, unuſquiſque movet libras 15000; ſed quia Trochleæ <lb></lb>dant Rationem quadruplam, accipe librarum 15000 quadran<lb></lb>tem 3750, & ope trochearum, ſi ſolæ eſſent & ab Ergatâ ſe<lb></lb>junctæ, unicuique equo adhibendus eſſet niſus ſubquadruplus, <lb></lb>videlicet conatus ſufficiens ad <expan abbr="movẽdas">movendas</expan> abſque trochleis libras <lb></lb>3750: quoniam demum Ergatæ Vectis ad ſemidiametrum axis <lb></lb>eſt ex. </s> <s id="s.004498">gr. decuplus, ſinguli equi adhibent conatum adhuc <lb></lb>ſubdecuplum, quo ſcilicer moverent libras 375: eſt nimirum, <lb></lb>ex hypotheſi harum trochlearum, & hujus Ergatæ, motus po<lb></lb>tentiæ ad motum ponderis quadragecuplus; ac proindè poten<lb></lb>tia adhibet conatum, quo moveret abſque machina gravitatem <lb></lb>dati ponderis ſubquadragecuplam. </s> </p> <p type="main"> <s id="s.004499">At verò ſi pondera attollenda ſint omnino ingentia & coloſ<lb></lb>ſicotera, non ſatis fuerit trabem erigere, ſed ex pluribus trabi<lb></lb>bus invicem compactis ſive funibus, ſive ferreis retinaculis, & <lb></lb>clavis quaſi craſſiores columnas erigere, eáſque tranſverſis aliis <lb></lb>trabibus inter ſe colligare, aut etiam obliquis fulcire oportet, <lb></lb>& circa pondus componere validiſſimum caſtellum, quod nul<lb></lb>lam in partem inclinari queat: ut deinde pluribus Trochlea<lb></lb>rum paribus cum ſuis Ergatis ritè collocatis machinator tutò <lb></lb>aggredi poſſit opus. </s> <s id="s.004500">Hìc autem multo commodius accidit plu<lb></lb>res communes Trochleas & Ergatas adhibere, quàm paucio<lb></lb>res trochleas plurimorum orbiculorum conſtruere, quæ lon<lb></lb>giſſimum funem ductarium exigerent, aut ingentes Ergatas ſta<lb></lb>tuere, quarum vectis valde longus non niſi in amplo ſpatio <lb></lb>circumagi poſſet. </s> <s id="s.004501">Illud Machinatoris ſolertiæ relinquitur, <lb></lb>quod Ergatas ſingulas atque Trochleas tam aptè diſponat, ut <lb></lb>ſibi invicem impedimento non ſint. </s> <s id="s.004502">Quod ſi, dum moles ipſa <lb></lb>elevatur, fulcra ſubinde opportuno loco ſubjicias, quibus illa <lb></lb>innitatur, multo certiùs, nec ſine laboris compendio, rem to<lb></lb>tam perficies. </s> </p> <p type="main"> <s id="s.004503">Quod demum ad funes attinet, in ponderum ingentium ele<lb></lb>vatione duplex periculum præcavendum eſt; alterum, quod <lb></lb>plures Trochleæ diverſis ponderis partibus applicantur, ne ſci<lb></lb>licet funes aliquarum Trochlearum vi ponderis plus juſto <lb></lb>diſtendantur, & longiores fiant, quàm par ſit, ut pondus uſque <pb pagenum="605" xlink:href="017/01/621.jpg"></pb>in deſtinatum evehatur locum, & aptè collocetur; una enim <lb></lb>parte jam ferè ſuum in locum deductâ, reliqua pars adhuc diſta<lb></lb>ret, nec potentia trochleis illis applicata ſola ad perficiendum <lb></lb>motum ſufficeret. </s> <s id="s.004504">Alterum eſt, ne ex motu, & vehementi <lb></lb>funium cum orbiculis, aut orbiculorum cum axibus tritu, ni<lb></lb>mis incaleſcant, atque ignem concipiant. </s> <s id="s.004505">Sed utrique pericu<lb></lb>lo occurritur, ſi aquam in promptu habeas, qua funes aut tro<lb></lb>chleæ madefiant; illa enim non ſolum incenſionis periculum <lb></lb>ſubmovet, verùm etiam funes contrahit. </s> </p> <p type="main"> <s id="s.004506">In plano autem horizontali aut inclinato longè facilior eſt <lb></lb>motus, potentia quippe caret labore retinendi onus, quod in<lb></lb>nitur plano; & quamvis hoc ſit inclinatum (non tamen lubri<lb></lb>cum, neque pondus incumbat ſcytalis, ſeu cylindris) ita pon<lb></lb>dus ſuâ gravitate premit ſubjectum planum, ut etiam dimiſſum <lb></lb>non facile prolabatur: ut tamen in hujuſmodi planis faciliùs <lb></lb>trahatur, expedit cylindros ſupponere, aut rotas addere, aut <lb></lb>illud trahæ imponere. </s> <s id="s.004507">Hìc pariter ad trahendum juvari poteſt <lb></lb>potentia, ſi funis ductarij per Trochleas trajecti caput ad Axem <lb></lb>Ergatæ, aut Suculæ, aut Tympani referatur; prout majora aut <lb></lb>mediocria fuerint pondera. </s> <s id="s.004508">In minoribus autem ponderibus <lb></lb>raptandis, etiam ſimplici Vecte, & quidem expeditiſſimè, au<lb></lb>geri poſſunt momenta Trochlearum; ſi nimirum vecti circa <lb></lb>medium alligetur caput funis ductarij, & inclinati vectis caput <lb></lb>ſubinde transferatur. </s> <s id="s.004509">Sit enim ductarij funis extremitas A; <lb></lb>hæc in A religetur vecti BC, qui <lb></lb><figure id="id.017.01.621.1.jpg" xlink:href="017/01/621/1.jpg"></figure><lb></lb>terram premat in C, quod eſt hy<lb></lb>pomochlium, & potentia movens <lb></lb>ſit in B; quæ, manente extremi<lb></lb>tate C, dum promovetur in D, fu<lb></lb>nis caput venit ex A in E: tunc <lb></lb>iterum inclinetur vectis CD, ut <lb></lb>habeat poſitionem FG, & poten<lb></lb>tia ſimiliter circa punctum G ma<lb></lb>nens moveatur ex F versùs D, at<lb></lb>que ulteriùs adducatur funis ca<lb></lb>put E, & ſic deinceps. </s> <s id="s.004510">Quod ſi <lb></lb>progredi nolueris, ſed eodem in loco conſiſtere, ubi vectis po<lb></lb>ſitionem CD nactus fuerit, & A venerit in E, retrahe D ite-<pb pagenum="606" xlink:href="017/01/622.jpg"></pb>rum in B, atque particulam funis AE ita vecti convolve, ut <lb></lb>excurrere nequeat; nam iterato vectis motu trahetur funis, & <lb></lb>cum trochleâ pondus; motúque hujuſmodi continuato deſtina<lb></lb>tumin locum adducetur pondus. </s> <s id="s.004511">Quantum verò ſit hoc compen. </s> <lb></lb> <s id="s.004512">dium, illicò innoteſcet, ſi obſervaveris ut minimum geminari mo<lb></lb>menta potentiæ, ſi videlicet punctum A præcisè medium fuerit <lb></lb>æqualiter ab extremitatibus B & C diſtans: quod ſi AC ſit triens <lb></lb>totius BC, momentum potentiæ triplicatur, & eſt Ratio com<lb></lb>poſita ex Ratione Trochlearum, & Ratione Vectis. </s> </p> <p type="main"> <s id="s.004513">Ut autem manifeſto experimento deprehendas, quàm tenuis <lb></lb>Potentia Trochleis cum Axe in Peritrochio compoſitis non leve <lb></lb>pondus trahat, in extremitate tabulæ non ruditer dolatæ duos <lb></lb>perpendiculares tigillulos erige intervallo digitorum quatuor, <lb></lb>illóſque junge tranſverſario ſimilis craſſitiei, non tamen à plano <lb></lb>abſit niſi quatuor digitos: Deinde ſupra tranſverſarium interval<lb></lb>lo ſaltem digitorum octo ſtatue axem in tigillis facillimè verſati<lb></lb>lem, cujus diameter vix digitalis ſit: alteri autem hujus axis capiti <lb></lb>rotam circumpone, cujus diametrum digiti duodecim <expan abbr="metiãtur">metiantur</expan>: <lb></lb>quæ rota ut levis ſit, bino radiorum ordine conſtet aptè colliga<lb></lb>torum, & circa perimetrum emineant palmulæ, ut in molendino<lb></lb>rum aquaticorum rotis, ex levi materiâ, cujuſmodi eſſet craſſior <lb></lb>charta, aut membrana, aut quid ſimile valens flatum excipere. </s> <lb></lb> <s id="s.004514">Tum parvulæ trochleæ duæ binis orbiculis inſtructæ firmentur, <lb></lb>altera quidem in tranſverſario tigillorum, altera in extremi<lb></lb>tate aſſerculi, cui onus aliquod eſt imponendum, eique aut <lb></lb>rotæ, aut cylindruli ſubjiciantur, ut facilè mobilis ſit; & tro<lb></lb>chleæ mobili adnectatur extremitas funiculi ſerici, qui per or<lb></lb>biculos trochlearum trajectus demum ad Axem referatur. </s> <lb></lb> <s id="s.004515">Nam ſi in rotæ palmulas vehementiùs inſuffles, rota convol<lb></lb>vetur, & cum illâ Axis, atque adeò funiculum involutum ſe<lb></lb>queretur trochlea cum pondere ferè ſexagecuplo ejus, quod <lb></lb>flatu eodem exſufflare poſſes. </s> <s id="s.004516">Aut potius Æolipilam aptè col<lb></lb>loca, ut flatus ex illâ exiens in palmulas incurrat, & ex ponde<lb></lb>ris, quod aſſerculo impoſitum movetur, gravitate cognoſtes <lb></lb>impetum, quo flatus ex Æolipilâ erumpit, ſi Ratio Trochlea<lb></lb>rum, quæ eſt quintupla, componatur cum Ratione diametri ro<lb></lb>tæ ad diametrum axis, quæ, ex conſtructione, eſt duodecupla: <lb></lb>cum enim ſit Ratio, ſexagecupla, fiat ut 60 ad 1, ita gravitas <pb pagenum="607" xlink:href="017/01/623.jpg"></pb>ponderis, quod per hujuſmodi machinulam trahitur, ad pon<lb></lb>dus, quod flatu illo impelli poſſet ſine machinâ. <lb></lb></s> </p> <p type="main"> <s id="s.004517"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004518"><emph type="center"></emph><emph type="italics"></emph>Trochleæ Trochleis additæ plurimùm áugent <lb></lb>momenta Potentiæ.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004519">QUodlibet oblatum pondus datâ Potentiæ virtute movere <lb></lb>adhibitis Trochleis omnes norunt, ſi binas trochleas tot <lb></lb>inſtruant orbiculis, quot exigit Ratio ponderis ad potentiam, <lb></lb>aut plura communium trochlearum paria cum pluribus Ergatis <lb></lb>adhibeant: quo in opere quàm immanes trochleas eſſe oporte<lb></lb>ret, ſi centenos aliquot, aut millenos orbiculos ſingulæ conti<lb></lb>nerent, aut quot Ergatæ, quantóque diſpendio ſtatuendæ eſ<lb></lb>ſent, ex methodo ſuperiori capite traditâ, nemo non videt. </s> <s id="s.004520">Hìc <lb></lb>ergo ſolis Trochleis rem facillimè perfici poſſe me demonſtratu<lb></lb>rum confido, prout in <emph type="italics"></emph>Terrâ machinis motâ<emph.end type="italics"></emph.end> differt. </s> <s id="s.004521">1. indicavi. </s> </p> <p type="main"> <s id="s.004522">Et primò quidem ſimplici orbiculo ſtabili elevari poteſt pon<lb></lb>dus cum incremento momentorum Potentiæ deorſum trahentis <lb></lb>(id quod multo facilius accidit, quàm ſur<lb></lb><figure id="id.017.01.623.1.jpg" xlink:href="017/01/623/1.jpg"></figure><lb></lb>ſum trahere) ſi extremitati funis loco Po<lb></lb>tentiæ adnectatur alius orbiculus, cujus fu<lb></lb>nis in loco inferiore alligatus fuerit. </s> <s id="s.004523">Sit <lb></lb>Pondus P, orbiculo ſtabili A elevandum: <lb></lb>utique Potentia funi ductario in B appli<lb></lb>cata non attollet pondus, niſi ejus gravitas, <lb></lb>aut virtus movendi major fuerit gravitate <lb></lb>ponderis P. </s> <s id="s.004524">Adnectatur in B orbiculus ver<lb></lb>ſatilis E: & paxillo in C firmato alligetur <lb></lb>caput funis per orbiculum E tranſeuntis: <lb></lb>Nam potentia in F funem trahens duplo <lb></lb>velociùs movetur quàm orbiculus E, hoć <lb></lb>eſt B extremitas funis ductarij, quæ cum <lb></lb>pondere P æqualiter movetur: ac proinde <pb pagenum="608" xlink:href="017/01/624.jpg"></pb>potentia, quæ duplo velociùs movetur quàm pondus, ſatis eſt ſi <lb></lb>fuerit paulo major, quàm ſubdupla ponderis. </s> </p> <p type="main"> <s id="s.004525">Ex hoc quaſi rudimento continuò ſe ſe offert methodus com<lb></lb>ponendi trochleas conjugatas; ſi nimirum duabus trochleis aptè <lb></lb>diſpoſitis ad trahendum pondus, funis ductarij extremitati non <lb></lb>applicetur potentia, ſed alia trochlea adnectatur, quaſi ibi eſſet <lb></lb>pondus, & harum trochlearum ſecundò poſitarum funi applice<lb></lb>tur potentia, cujus momenta ex Trochlearum rationibus com<lb></lb><figure id="id.017.01.624.1.jpg" xlink:href="017/01/624/1.jpg"></figure><lb></lb>ponuntur. </s> <s id="s.004526">Sint duæ Trochleæ A & B bi<lb></lb>nis orbiculis inſtructæ; pondus in M ad<lb></lb>nexum ſit trochleæ A; & trochlea B fir<lb></lb>metur in C: funis autem ductarius in D <lb></lb>alligetur trochleæ A mobili: utique Po<lb></lb>tentia in F habet momentum quintuplum, <lb></lb>quia quintuplo velociùs movetur, quàm <lb></lb>pondus in M adnexum. </s> <s id="s.004527">Sed iterum duæ <lb></lb>aliæ Trochleæ H & I binos orbiculos ha<lb></lb>bentes parentur, & trochlea H mobilis <lb></lb>jungatur extremitati funis F, trochlea ve<lb></lb>rò I ſtabilis ſit. </s> <s id="s.004528">Trochlearum H, I funis <lb></lb>ductarius alligetur in G trochleæ mobili; <lb></lb>& potentia in K applicata quinquies velo<lb></lb>ciùs movetur quàm F; ergo & vigeſies <lb></lb>quinquies velociùs quàm pondus adne<lb></lb>xum in M. </s> <s id="s.004529">Illud igitur pondus, quod quin<lb></lb>que homines applicati in F traherent, ab <lb></lb>unico homine in K applicato atque tra<lb></lb>hente adducitur, qui unicus æquivalet vi<lb></lb>gintiquinque hominibus pondus abſque <lb></lb>trochleis trahere conantibus. </s> </p> <p type="main"> <s id="s.004530">Cum itaque communes Trochleæ in promptu ſint, manifeſtum <lb></lb>eſt, quàm facilè multiplicari valeant momenta potentiæ quæ cæ<lb></lb>teroqui in exemplo propoſito duas trochleas ſingulas duodenûm <lb></lb>orbiculorum exigeret, ut unus homo præſtaret idem, quod vi<lb></lb>ginti quinque Quod ſi adhuc duas ſimiles trochleas adhiberes, <lb></lb>& alteram ſimiliter in K adnecteres, unicus homo æquivaleret <lb></lb>hominibus 125 trahentibus: hinc ſi unicus ille homo tanto co<lb></lb>natu trahat, quanto traheret libras 50, omninò ſolus tribus his <lb></lb>trochlearum paribus movebit pondus librarum 6250. </s> </p> <pb pagenum="609" xlink:href="017/01/625.jpg"></pb> <p type="main"> <s id="s.004531">Sed ſi quem admiratio capiat duodecim orbiculis in ſex tro<lb></lb>chleas diſtributis tantum pondus moveri, admiretur adhuc am<lb></lb>pliùs <expan abbr="iiſdẽ">iiſdem</expan> duodecim orbiculis in duodecim ſimplices trochleas <lb></lb>diſtinctis, quæ binæ & binæ conjungentur, longè majus pondus <lb></lb>poſſe trahi. </s> <s id="s.004532">Nam ſi trochleæ mobili, cui alligatur pondus, etiam <lb></lb>funis extremitas adnectatur, jam ſingulæ Trochlearum conjuga<lb></lb>tiones dant rationem triplam; ſunt igitur ſex Rationes triplæ <lb></lb>compoſitæ; ac propterea prima conjugatio dat Rationem 3 ad 1; <lb></lb>ſecunda 9 ad 1; tertia 27 ad 1; quarta 81 ad 1; quinta 243 ad 1; <lb></lb>ſexta 729 ad 1: & hæc eſt Ratio motûs po<lb></lb><figure id="id.017.01.625.1.jpg" xlink:href="017/01/625/1.jpg"></figure><lb></lb>tentiæ ad motum ponderis. </s> <s id="s.004533">Quare ſi po<lb></lb>tentia conetur ut 50, ducatur 729 per 50, <lb></lb>& potentia trahere valebit pondus libra<lb></lb>rum 36450. At verò ſi duodecim illæ ſim<lb></lb>plices trochleæ <expan abbr="nõ">non</expan> fuerint conjugatæ, ſed <lb></lb>ſingulæ ſeorſim ſuos habeant funes, ita ut <lb></lb>primæ adnectatur <expan abbr="põdus">pondus</expan>, & ſecundæ jun<lb></lb>gatur extremitas funis ductarij primæ, at<lb></lb>que ita deinceps, jam multo majus erit <lb></lb>momentum potentiæ; erunt ſcilicet duo<lb></lb>decim Rationes duplæ <expan abbr="cõpoſitæ">compoſitæ</expan>. </s> <s id="s.004534">Sit enim <lb></lb>trochleæ X adnexum pondus, funis illius <lb></lb>alligatus in V, & ejuſdem capiti ad nexa ſit <lb></lb>ſecunda Trochlea T; cujus pariter funis <lb></lb>alligatus in S reliquâ extremitate conjun<lb></lb>gatur cum tertia Trochleâ R, ejuſque fu<lb></lb>nis ſimiliter firmatus in Q veniat ad P, cui <lb></lb>deinceps quarta trochlea adjungatur, & <lb></lb>ſic de cæteris conſequentibus. </s> <s id="s.004535">Certum eſt <lb></lb>T moveri duplo velociùs quàm X, & R <lb></lb>duplo velociùs quàm T, & P duplo velo<lb></lb>ciùs quàm R; ac proinde P moveri octu<lb></lb>plo velociùs quàm pondus in X. </s> <s id="s.004536">Si igitur <lb></lb>duodecim rationes duplæ componantur, <lb></lb>erit demum Ratio 4096 ad 1. Quaprop<lb></lb>ter potentia in extremitate funis trochleæ <lb></lb>duodecimæ ſimiliter conata ut 50, move<lb></lb>bit pondus librarum 204800. </s> </p> <pb pagenum="610" xlink:href="017/01/626.jpg"></pb> <p type="main"> <s id="s.004537">Ex his vides poſterioribus trochleis minùs repugnare pondus <lb></lb>quàm prioribus, atque propterea funes ductarios poſteriorum <lb></lb>trochlearum poſſe exiliores eſſe, quamvis longiores; eóque de<lb></lb>veniri poſſe, ut potentia ſubtiliſſimo funiculo applicetur, & ſe<lb></lb>curè trahat valde magnum pondus. </s> <s id="s.004538">Semper autem tractionis <lb></lb>mentionem feci, non elevationis, quia in illa faciliùs quàm in <lb></lb>hac uti poſſumus hujuſmodi trochlearum complexione: quam<lb></lb>quam etiam in elevatione ad mediocrem altitudinem, diſpoſitis <lb></lb>duabus trochleis, quaſi illas tantum adhibere oporteret, poſſu<lb></lb>mus extremitati funis ductarij adjicere trochleam, cujus com<lb></lb>parem paxillo in terram firmiter depacto alligemus; aut etiam, <lb></lb>ſi altitudo ſuppetat longè major ea, ad quam attollendum eſt <lb></lb>pondus, in ſupremo loco ſtatuere poſſumus trochleam ſtabilem <lb></lb>ſecundæ conjugationis, & mobili trochleæ adnectere extremi<lb></lb>tatem funis ductarij priorum trochlearum, in quibus propterea <lb></lb>caput funis adnectendum eſt trochleæ mobili, cui adhæret pon<lb></lb>dus evehendum. </s> </p> <p type="main"> <s id="s.004539">Non eſt autem diſſimulandum incommodum, quod ex hac <lb></lb>trochlearum diſpoſitione atque complexione oritur, ſcilicet <lb></lb>magnam funium longitudinem requiri, nec non ingens ſpa<lb></lb>tium, in quo diſponantur duo illa Trochlearum pariæ, quibus <lb></lb>vigequintupla fiunt Potentiæ momenta. </s> <s id="s.004540">Quia enim in Tro<lb></lb>chleis adnexam ſarcinam adducentibus ſunt quatuor funis <lb></lb>ductus æquales trochlearum intervallo, utique, ſi eidem tro<lb></lb>chleæ pondus ac funis alligatur, totus explicatur ultra termi<lb></lb>num, cui trochlea ſtabilis adnectitur: quare trochleam mobi<lb></lb>lem ſecundæ conjugationis adnexam extremitati funis priorum <lb></lb>trochlearum conſtituere oportet diſtantem à ſuâ trochleâ ſtabi<lb></lb>li non minùs quàm intervallo quintuplo diſtantiæ priorum: ac <lb></lb>propterea harum poſteriorum funis explicatus excurrit ultra <lb></lb>terminum, cui affigitur compar trochlea ſtabilis ſpatio illius <lb></lb>quintupli intervalli quadruplo, hoc eſt vigecuplo intervalli <lb></lb>priorum trochlearum; cui ſi addatur diſtantia poſteriorum <lb></lb>quintupla diſtantiæ priorum, Potentia trochleæ ſecundæ mo<lb></lb>bili applicata funem trahens movetur vigequintuplo velociùs <lb></lb>quàm pondus, & exigit ſpatium vigequintuplum diſtantiæ prio<lb></lb>rum trochlearum, ſi illa velit progredi, quantum fert longitudo <lb></lb>funis explicati; id quod neceſſe eſt, ſi funis à jumentis trahatur, <pb pagenum="611" xlink:href="017/01/627.jpg"></pb>nec circumducatur Ergatæ; tunc enim non tantum ſpatij re<lb></lb>quiritur, & momentum Ratione Ergatæ augetur. </s> <s id="s.004541">At ſi Poten<lb></lb>tia trahens ſint homines, ſatis eſt ſi propè ſecundam trochleam <lb></lb>ſtabilem conſiſtant. </s> <s id="s.004542">Quare ſi quis voluerit hujuſmodi quatuor <lb></lb>trochlearum complexione uti, ut potentia obtineat momentum <lb></lb>vigequintuplum, requiritur ſpatij longitudo quintupla ſpatij, <lb></lb>per quod deducendum eſt pondus. </s> <s id="s.004543">Quod igitur ad funium <lb></lb>longitudinem ſpectat, longitudo funis priorum trochlearum eſt <lb></lb>quadrupla ſpatij percurrendi à pondere, & longitudo funis <lb></lb>poſteriorum eſt ejuſdem ſpatij vigecupla; hic tamen poſterior <lb></lb>funis poteſt eſſe priore tenuior atque exilior, ut dictum eſt. </s> </p> <p type="main"> <s id="s.004544">Dixerit fortaſſe aliquis, rem minùs attentè conſiderans, poſſe <lb></lb>poſteriores trochleas habere funem non longiorem fune prio<lb></lb>rum; ſed quia, ubi ille totus explicatus fuerit, pondus non eſt <lb></lb>adductum niſi ad quintam partem ſpatij, poſſe trochleas illas <lb></lb>poſteriores ita invicem disjungi, ut ea, quæ eſt mobilis, adjun<lb></lb>gatur funi ductario propè trochleam priorem mobilem; nam <lb></lb>potentia iterum trahens adducet pondus: id quod ſæpius ite<lb></lb>rari poteſt. </s> </p> <p type="main"> <s id="s.004545">Verùm hoc fieri omnino non poſſe deprehendes, ſi obſerva<lb></lb>veris, nunquam hoc pacto adduci pondus niſi per quintam par<lb></lb>tem reliqui ſpatij; quare aliquid ſemper relinquitur, quin ad <lb></lb>deſtinatum locum pondus perveniat. </s> <s id="s.004546">Si placuerit tamen hunc <lb></lb>laborem aſſumere in disjungendis poſterioribus trochleis, prio<lb></lb>res trochleas ita invicem disjunctas initio colloca, ut earum in<lb></lb>tervallum ſit ſaltem ſeſquialterum ſpatij, per quod pondus mo<lb></lb>veri oportet; ſic enim repetito quinquies trahendi labore obti<lb></lb>nebis propoſitum motum: primâ videlicet tractione deducitur <lb></lb>pondus per totius intervalli 1/5; in ſecunda per ejuſdem inter<lb></lb>valli (4/25); in tertiâ per (16/125); in quartâ per (64/625); in quintâ per (256/1125); <lb></lb>quæ partes ſi in ſummam redigantur, dant (2101/3125), hoc eſt paulo <lb></lb>amplius quàm 2/3 propoſiti intervalli, quantum ſatis eſt ad perfi<lb></lb>ciendum deſtinatum ſpatium. </s> <s id="s.004547">Ubi vides; ſi intervallum aſ<lb></lb>ſumptum fuiſſet paulo majus quàm duplum deſtinati ſpatij, ter<lb></lb>tiâ tractione abſolvi propoſitum motum; nam 1/5, (1/25), (16/125) ſi colli<lb></lb>gantur in ſummam, dant (61/125), hoc eſt ferè 1/2. At ſi duobus ſim<lb></lb>plicibus orbiculis utaris, quibus compoſitis potentia habet mo-<pb pagenum="612" xlink:href="017/01/628.jpg"></pb>mentum quadruplum, etiamſi ſecundi orbiculi funem ſtatuas <lb></lb>æqualem funi prioris orbiculi, cui adnectitur pondus, facilli<lb></lb>mum eſt orbiculum ſecundum retrahere ad orbiculum primum, <lb></lb>poſtquam hic primâ tractione abſolvit ſemiſſem ſpatij inter pon<lb></lb>dus & paxillum, cui alligatur funis; & ſecundâ tractione ab<lb></lb>ſolvit quadrantem totius intervalli initio conſtituti: Quare ſa<lb></lb>tis fuerit funem prioris orbiculi æquari intervallo ſeſquitertio <lb></lb>longitudinis ſpatij, per quod deducendum eſt pondus. </s> </p> <p type="main"> <s id="s.004548">Et quoniam hìc mentio incidit orbiculorum ſimplicium, ob<lb></lb>ſerva, quanto faciliùs duobus orbiculis perficiamus id, quod <lb></lb>duabus trochleis binos orbiculos habentibus præſtaremus in <lb></lb>trahendo pondere, quando funis ductarius eſt alligatus tro<lb></lb>chleæ ſtabili; tunc enim potentia ſolùm habet momentum <lb></lb>quadruplum, quod pariter obtinet duobus orbiculis. </s> <s id="s.004549">Sit enim <lb></lb><figure id="id.017.01.628.1.jpg" xlink:href="017/01/628/1.jpg"></figure><lb></lb>AB diſtantia ſeſquitertia ſpatij AI, per quod <lb></lb>trahendum eſt pondus in P adnexum orbicu<lb></lb>lo A: funis in B alligetur, & ejus caput C con<lb></lb>nectatur cum orbiculo E, cujus pariter funis <lb></lb>in B alligetur, atque illius extremitas à Poten<lb></lb>tiâ F trahatur. </s> <s id="s.004550">Quando potentia F adduxerit <lb></lb>orbiculum E prope B, erit orbiculus A in H: <lb></lb>retrahatur orbiculus E ex B, & propè H ad<lb></lb>nectatur funi orbiculi A; factâ enim ſecundâ <lb></lb>tractione, quando orbiculus E fuerit iterum <lb></lb>prope B, orbiculus A erit in I; eſt autem ex <lb></lb>hypotheſi diſtantia AI æqualis ſpatio, per <lb></lb>quod trahendum erat pondus, ſubſeſquitertio <lb></lb>intervalli AB. </s> <s id="s.004551">Ecce igitur Potentia habet <lb></lb>momentum quadruplum, & duorum funium <lb></lb>longitudines ſimul ſumptæ non dant longitu<lb></lb>dinem triplam ſpatij, per quod deducendum <lb></lb>eſt pondus. </s> <s id="s.004552">At ſi eſſent duæ Trochleæ cum <lb></lb>binis orbiculis, exigerent unicum funem qua<lb></lb>druplum longitudinis ſpatij, per quod inſti<lb></lb>tuendus eſt motus. </s> </p> <p type="main"> <s id="s.004553">Sed & illud addendum videtur, quod duobus ſimplicibus or<lb></lb>biculis etiam ad longiora ſpatia adduci poteſt pondus, ita ut <lb></lb>quilibet trahentium habeat momentum quadruplum. </s> <s id="s.004554">Expe-<pb pagenum="613" xlink:href="017/01/629.jpg"></pb>dit autem trahentium numerum geminari, ut alternâ quiete <lb></lb>faciliùs & citiùs onus trahant. </s> <s id="s.004555">Sit orbiculus M adnectendus <lb></lb>ponderi, & ſit <lb></lb><figure id="id.017.01.629.1.jpg" xlink:href="017/01/629/1.jpg"></figure><lb></lb>datus funis du<lb></lb>ctarius SR, cu<lb></lb>jus extremitati<lb></lb>bus R & S re<lb></lb>plicatis quaſi in <lb></lb>laqueum, ſeu <lb></lb>anſam facillimè immitti poſſint & paxillus R, & alterius tro<lb></lb>chleæ uncus S. </s> <s id="s.004556">Funis alius paretur NT prioris duplus extre<lb></lb>mitates ſimiliter replicatas habens, ut in V immitti poſſit tra<lb></lb>hentis manus, & in T paxillus. </s> <s id="s.004557">Quare tantumdem paxillus R <lb></lb>diſtat à Trochleâ M, quantum à paxillo T, & hic tantumdem à <lb></lb>paxillo X. </s> <s id="s.004558">Cum igitur toto fune VNT explicato orbiculus N <lb></lb>fuerit in T, orbiculus M erit in R, & funis extremitas S erit in <lb></lb>T. </s> <s id="s.004559">Itaque ex paxillo T auferatur funis explicatus, & ejus loco <lb></lb>injiciatur extremitas S. </s> <s id="s.004560">Eximatur tunc ex paxillo R extremitas <lb></lb>funis, & adnectatur alteri Trochleæ funem habenti æqualem <lb></lb>funi VNT, cujus extremitas alligata fuerit paxillo X, & ad T <lb></lb>adducetur trochlea M unà cum pondere. </s> <s id="s.004561">Atque ita alternâ ope<lb></lb>rá adducetur pondus ad quancumque diſtantiam; interea enim, <lb></lb>dum orbiculus M ex R trahitur ad T, is qui traxerat funem V, <lb></lb>alligat illum paxillo, ad quem progrediendo pervenitur, & extre<lb></lb>mitatem T exemptam è paxillo trahet, ubi trochleam N eò jam <lb></lb>deductam iterum junxerit extremitati S in T exiſtenti. </s> <s id="s.004562">Sunt <lb></lb>itaque pangendi in terram paxilli æqualibus intervallis. </s> </p> <p type="main"> <s id="s.004563">Monendus eſt autem Lector ad hoc caput non pertinere illam <lb></lb>Trochlearum additionem, quæ non facit rationum Compoſitio<lb></lb>nem; quando ſcilicet plures trochleæ uno loculamento ita in<lb></lb>cluduntur, ut ſingulæ trochleæ tam ſuperior, quam inferior plu<lb></lb>res habeant orbiculorum ordines in latitudinem collocatos, at<lb></lb>que adeo tot funes ductarios, quot ſunt ordines illi orbiculorum, <lb></lb>exigunt; perinde enim eſt atque ſi duæ aut tres trochleæ diver<lb></lb>ſis loculamentis diſtinctæ adhiberentur. </s> <s id="s.004564">Cum autem plures ſint <lb></lb>funes ductarij, qui uno eodémque tempore adducendi ſunt, di<lb></lb>ligenter animum advertere oportet, ut operæ omnes æqualiter <lb></lb>trahant. <pb pagenum="614" xlink:href="017/01/630.jpg"></pb></s> </p> <p type="main"> <s id="s.004565"><emph type="center"></emph>CAPUT VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004566"><emph type="center"></emph><emph type="italics"></emph>Trochlearum ope moveri potest pondus velociter.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004567">HActenus Trochlearum in facilè movendis oneribus vires <lb></lb>expendimus, ubi quò majora momenta ope hujus Faculta<lb></lb>tis adduntur Potentiæ, eò etiam tardior eſt motus ponderis, po<lb></lb>tentiæ autem velocior: quod ſi velociter movendum ſit pondus, <lb></lb>neceſſariò augeri debet potentia. </s> <s id="s.004568">Verùm quia non rarò contin<lb></lb>gere poteſt, ut potentia quidem ipſa per ſe viribus abundet, il<lb></lb>lam tamen tardè moveri oporteat, aut contra in trahendo onere <lb></lb>feſtinato ſit opus, propterea hìc indicandum eſt, qua methodo <lb></lb>uti poſſimus, ut hinc plenior hujus Facultatis notitia habeatur. </s> </p> <p type="main"> <s id="s.004569">Opus ſit in turrim, vel in urbis mœnia commeatum transfer<lb></lb>re velociter: operarum ſuppetat ſatis, at non item temporis. <lb></lb><figure id="id.017.01.630.1.jpg" xlink:href="017/01/630/1.jpg"></figure><lb></lb>Statuatur in ſumma turri, <lb></lb>aut certè in loco opportu<lb></lb>no, Sucula BC cum manu<lb></lb>briis CEF, & BDI, qui<lb></lb>bus plures operæ applicari <lb></lb>poſſint pro gravitate oneris <lb></lb>attollendi; immò etiam ha<lb></lb>beat infixos radios, ut adhuc <lb></lb>plures recipiat, qui illam <lb></lb>verſare poſſint. </s> <s id="s.004570">Circà Axem <lb></lb>involutus ſit funis paulo <lb></lb>longior ſemiſſe altitudinis, <lb></lb>& extremitati ſit adnexus <lb></lb>girgillus G, cui inſertus ſit <lb></lb>funis ductarius. </s> <s id="s.004571">HGL <lb></lb>æqualis altitudini, ad quam <lb></lb>evehendum eſt onus; alte<lb></lb>ri hujus funis extremitati <lb></lb>cohæreat in H validus un<lb></lb>cus, quo onus ſuſpendatur, <lb></lb>alteram verò <expan abbr="extremitatẽ">extremitatem</expan> L <pb pagenum="615" xlink:href="017/01/631.jpg"></pb>firmet clavus, aut quid ſimile ad pedem turris. </s> <s id="s.004572">Nam ſi conver<lb></lb>tatur Sucula, devenient pariter in A tum girgillus G, tum <lb></lb>onus unco H ſuſpenſum; quod ſanè duplo velociùs movetur, <lb></lb>quàm ſi adnexum funi ductario AK traheretur ſurſum ope <lb></lb>ſimplicis ſuculæ. </s> <s id="s.004573">Quare momenta ſuculæ non niſi dimidiata <lb></lb>computanda ſunt, adeò ut ſi duo homines ſuculam BC cir<lb></lb>cumagentes valerent attollere libras 400, eodem conatu, & la<lb></lb>bore poſſint ſolum libras 200 attollere: at quia facilè multipli<lb></lb>cari poſſunt homines ſuculam verſantes, geminetur eorum nu<lb></lb>merus, & attollent libras 400, ſed breviori tempore. </s> <s id="s.004574">In con<lb></lb>trarium autem revoluta ſucula demittet girgillum G, & ſuo pon<lb></lb>dere duplo velociùs deſcendet uncus H. </s> <s id="s.004575">Ex quo habetur quæ<lb></lb>ſitum temporis compendium. </s> </p> <p type="main"> <s id="s.004576">At ſi duabus trochlei, ſimplicibus ſingulos orbiculos haben<lb></lb>tibus res perficienda eſſet, ita ut uni trochleæ adnecteretur Po<lb></lb>tentia, alteri Pondus, funis autem extremitas alicubi clavo re<lb></lb>ligata eſſet, attentè diſpiciendum eſt, utri trochleæ adnecta<lb></lb>tur reliqua funis trochleas jungentis extremitas. </s> <s id="s.004577">Nam ſi tro<lb></lb>chleæ A, quam trahit potentia N, <lb></lb><figure id="id.017.01.631.1.jpg" xlink:href="017/01/631/1.jpg"></figure><lb></lb>adnectatur in C funis per orbicu<lb></lb>los trajectus, trochleæ verò B <lb></lb>pondus M, & funis religatus fue<lb></lb>rit in D, intelligitur motus inci<lb></lb>pere, quando trochleæ adhuc in<lb></lb>vicem abſunt, ità ut in motu tro<lb></lb>chlea ponderis ad trochleam po<lb></lb>tentiæ accedat, ceſſare autem, <lb></lb>cùm illæ proximæ factæ fuerint in <lb></lb>maximâ diſtantiâ à clavo D, ubi <lb></lb>funis extremitas alligatur. </s> <s id="s.004578">Contrà <lb></lb>verò accidit trochleis G & H, ſi <lb></lb>trochleæ H adnectatur pondus B, <lb></lb>atque in I funis ductarij caput: <lb></lb>nam trahente potentia S, quæ ini<lb></lb>tio propiores erant trochleæ, à ſe <lb></lb>invicem recedunt, trochleâ po<lb></lb>tentiæ ſecedente à trochleâ pon<lb></lb>deris; & demum abſolvitur mo-<pb pagenum="616" xlink:href="017/01/632.jpg"></pb>tus, cùm trochlea H ponderis acceſſerit ad R extremitatem <lb></lb>funis religati. </s> <s id="s.004579">Cum itaque in utroque caſu & potentia, & <lb></lb>pondus versùs eandem partem moveantur, in primo tamen <lb></lb>pondus, quod à potentiâ diſtabat, ad illam accedat, & in <lb></lb>ſecundo potentia vicina ponderi ab illo recedat, manifeſto <lb></lb>indicio eſt in primo caſu pondus, in ſecundo potentiam ve<lb></lb>lociùs moveri: quare ibi potentia augenda eſt, ut valeat mo<lb></lb>vere pondus, hìc fieri poteſt additamentum ponderi, ut po<lb></lb>tentiæ virtuti reſpondeat. </s> <s id="s.004580">Eſt autem motuum Ratio ſeſqui<lb></lb>altera, ut palàm faciunt funium ductus, eorúmque explica<lb></lb>tio: Nam in primo caſu maxima trochlearum diſtantia eſt, <lb></lb>quando trochlea A eſt clavo D proxima; igitur potentia <lb></lb>movetur per ſpatium, cujus longitudinem metitur funis ex<lb></lb>plicatus, qui eſt duplus diſtantiæ trochlearum, & pondus <lb></lb>accedens ad potentiam inſuper percurrit ſpatium, quo tro<lb></lb>chleæ diſtabant; igitur motus ponderis eſt ut 3, & poten<lb></lb>tia ut 2. In ſecundo verò caſu, Trochleæ G & H cùm <lb></lb>proximæ ſunt, diſtant à clavo R juxta longitudinem funis <lb></lb>explicati, cùm autem maximè invicem abſunt, & potentia <lb></lb>tranſgreſſa eſt clavum R, totus funis diſtributus eſt in duos <lb></lb>ductus, & trochlearum intervallum eſt medietas longitudi<lb></lb>nis funis; quare ponderis motus eſt ut 1, & motus poten<lb></lb>tiæ ut 1 1/2. </s> </p> <p type="main"> <s id="s.004581">Simili ratione philoſophandum erit, ſi trochleæ inæquales <lb></lb>proponantur, ut ſi altera ſit duorum orbiculorum, altera <lb></lb>unius orbiculi: Utique funis per orbiculos trajectus adnecten<lb></lb>dus eſt ſimplici trochleæ, ejúſque altera extremitas alicubi <lb></lb>firmanda. </s> <s id="s.004582">Non igitur indiſcriminatim ſivè huic, ſivè illi <lb></lb>trochleæ adjungenda eſt potentia, ſed priùs ſtatuendum ti<lb></lb>bi eſt, utrùm velis pondus movere facilè, an velociter; ſi <lb></lb>facilè, tardior ſit ponderis motus, quàm potentiæ; ſi velo<lb></lb>citer, tardior ſit potentia. </s> <s id="s.004583">Facilè movebis pondus, ſi potentia <lb></lb>trahat ſimplicem orbiculum, & pondus cohæreat trochleæ <lb></lb>duorum orbiculorum: Velociter autem movebitur pondus, ſi <lb></lb>illud adnectatur ſimplici orbiculo, potentia verò trahat tro<lb></lb>chleam duorum orbiculorum. </s> <s id="s.004584">Nam in primo caſu funis expli<lb></lb>catus replicatur, & potentia recedit à pondere; in ſecundo fu<lb></lb>nis replićatus explicatur, & pondus accedit ad potentiam. </s> </p> <pb pagenum="617" xlink:href="017/01/633.jpg"></pb> <p type="main"> <s id="s.004585">Sit Trochlea MO, & orbiculus I; huic in L adnectitur fu<lb></lb>nis, cujus altera extremitas religatur in A, <lb></lb><figure id="id.017.01.633.1.jpg" xlink:href="017/01/633/1.jpg"></figure><lb></lb>quò demum devenire poteſt trochlea MO <lb></lb>cum pondere T adjecto. </s> <s id="s.004586">Vice versá Tro<lb></lb>chleæ GC adhibeatur potentia, & pon<lb></lb>dus S adjiciatur orbiculo E: huic in B ad<lb></lb>nectitur funis, qui per orbiculos trajectus <lb></lb>deſinit in F, ubi ille religatur, & trochlea <lb></lb>GO maximè diſtat ab orbiculo E. </s> <s id="s.004587">Inſti<lb></lb>tuto motu, Potentia D ſemper magis re<lb></lb>cedit à pondere T; at pondus S ſemper <lb></lb>magis accedit ad Potentiam H: ibi ergo <lb></lb>potentia celerior eſt pondere, hìc pondus <lb></lb>velocius eſt Potentiâ; motuum autem Ra<lb></lb>tio eſt ſeſquitertia: Nam explicato fune <lb></lb>toto, qui religatur in A, potentia proxi<lb></lb>ma eſt ponderi, & diſtant ab A pro funis <lb></lb>longitudine; potentiâ trahente accedunt <lb></lb>ad A, ſed potentia ulteriùs progreditur, atque abſoluto motu <lb></lb>replicatus eſt funis in tres ductus, & Potentia diſtat à pondere <lb></lb>tertiâ parte ipſius funis, ita ut pondus quidem ſit clavo A proxi<lb></lb>mum, potentia verò tranſgreſſa ſit clavum A intervallo OL: <lb></lb>igitur motus ponderi, quem longitudo funis metitur, eſt ut 1, <lb></lb>potentiæ ut 1 1/3. Ex adverſo Potentia applicata trochleæ GC <lb></lb>proxima eſt clavo F, cum ab illâ pondus maximè abeſt inter<lb></lb>vallo tertiæ partis ipſius funis in tres ductus replicati: inito mo<lb></lb>tu pondus accedit ad Potentiam, cui demum proximum eſt, <lb></lb>quando jam totus funis eſt explicatus; igitur motum potentiæ <lb></lb>metitur funis explicatus, motum autem ponderis adhuc tertia <lb></lb>pars, ſcilicet intervallum BC: adeóque ponderis motus ad mo<lb></lb>tum potentiæ eſt ut 1 1/3 ad 1. </s> </p> <p type="main"> <s id="s.004588">Quæ autem de his trochleis dicta ſunt, ſi attentè conſide<lb></lb>rentur, etiam cæteris trochleis conjugatis, ſed diſpari orbicu<lb></lb>lorum numero inſtructis, conveniunt. </s> <s id="s.004589">Non poſſe verò orbicu<lb></lb>lorum numeros differre niſi unitate, ſatis manifeſtum eſt: nam <lb></lb>ſi differrent binario aut ternario, eorum aliquis aut plures pla<lb></lb>nè otioſi eſſent, quippe qui recipere nequirent funem ducta-<pb pagenum="618" xlink:href="017/01/634.jpg"></pb>rium jam per reliquos orbiculos trajectum. </s> <s id="s.004590">Quare ſi altera tro<lb></lb>chlea minor duos habeat orbiculos, altera major non niſi tres <lb></lb>habere poteſt, aut ſi minor tres habeat, major non niſi quatuor <lb></lb>habere poterit. </s> <s id="s.004591">Attendendum eſt igitur, utri trochlearum tro<lb></lb>chlearum potentia applicetur; ſi enim illa trahendam arripiat <lb></lb>trochleam plures habentem orbiculos, tardiùs movetur, quàm <lb></lb>pondus in Ratione ſubſuperparticulari denominatâ à numero <lb></lb>omnium ſimul <expan abbr="orbiculorũ">orbiculorum</expan>: ut ſi potentia trochleæ trium, pondus <lb></lb>verò trochleæ duorum orbiculorum applicetur, motus potentiæ <lb></lb>eſt ad motum ponderis in Ratione ſubſeſquiquintâ, quia illa mo<lb></lb>vetur ut 5, pondus ut 6: & ſi potentia trochleæ quatuor orbicu<lb></lb>lorum applicetur, pondus autem trochleæ trium, Ratio eſt ſub<lb></lb>ſeſquiſeptima, quia illa movetur ut 7, hoc ut 8. Quare augetur <lb></lb>motus ponderis, & in eadem Ratione difficultas potentiæ. </s> <s id="s.004592">Con<lb></lb>trà autem ſi pondus alligetur majori trochleæ, etiam potentiæ <lb></lb>motus major, eſt motu ponderis in Ratione ſuperparticulari de<lb></lb>nominatâ à numero omnium ſimul orbiculolum: ſic erit Ratio <lb></lb>ſeſquiquinta, ſi potentia duobus, pondus tribus orbiculis allige<lb></lb>tur, nam motus potentiæ eſt ut 6, & motus ponderis ut 5: ſimi<lb></lb>liter erit Ratio ſeſquiſeptima, quando pondus alligatum tro<lb></lb>chleæ quatuor orbiculorum movetur ut 7, dum potentia appli<lb></lb>cata trochleæ trium orbiculorum movetur ut 8. </s> </p> <p type="main"> <s id="s.004593">Quod ſi pari orbiculorum numero conſtet utraque trochlea, <lb></lb>& utraque moveatur, ſimiliter motus erunt in Ratione ſuper<lb></lb>particulari denominatâ à numero omnium orbiculorum ſimul: <lb></lb>hoc tamen erit diſcrimen, quod illud tardius movebitur, quod <lb></lb>applicabitur trochleæ, cui extremitas funis per orbiculos tra<lb></lb>jecti adnectitur. </s> <s id="s.004594">Sic ſi trochleæ ambæ binos habeant orbiculos, <lb></lb>Ratio eſt ſeſquiquarta, ſi ternos ſeſquiſexta: ſi trochleam, cui <lb></lb>funis ductarij extremitas adnectitur, potentia trahat, illa move<lb></lb>tur ut 4, aut ut 6, pondus verò movetur ut 5, aut ut 7: ſed ſi tro<lb></lb>chleæ, cui funis adnectitur, alligetur pondus, potentia movetur <lb></lb>ut 5, aut ut 7, pondus autem ipſum ut 4, aut ut 6. </s> </p> <p type="main"> <s id="s.004595">Ex his itaque duplex trochlearum uſus innoteſcit, alter com<lb></lb>munis quo potentia applicatur extremitati funis ductarij (alterâ <lb></lb>trochlearum manente ſtabili) quem trahens attrahit pariter <lb></lb>pondus, & motus potentiæ eſt in Ratione aliqua multiplici ad <lb></lb>motum ponderis. </s> <s id="s.004596">Alter verò eſt, quando extremitas funis <pb pagenum="619" xlink:href="017/01/635.jpg"></pb>ductarij non trahitur, ſed alicubi firmatur, potentia autem <lb></lb>trahit alteram trochleam, ad cujus motum etiam reliqua tro<lb></lb>chlea cum pondere illorſum movetur, quorſum potentia tendit: <lb></lb>ſi in hoc motu trochleæ disjunguntur, & potentia recedit à <lb></lb>Pondere, Ratio motûs potentiæ ad motum ponderis eſt ſuper<lb></lb>particularis, & potentia conſequitur aliquam movendi facilita<lb></lb>tem: ſin autem pondus ad potentiam accedit, & trochleæ, quæ <lb></lb>disjunctæ erat, fiunt proximæ, Ratio motûs potentiæ ad motum <lb></lb>ponderis eſt ſubſuperparticularis, & potentiam plus adhibere <lb></lb>conatûs oportet, quàm ſi illud abſque trochleis traheret; quia <lb></lb>pondus velociùs movetur quàm potentia. <lb></lb></s> </p> <p type="main"> <s id="s.004597"><emph type="center"></emph>CAPUT VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004598"><emph type="center"></emph><emph type="italics"></emph>Quàm validum eſſe oporteat trochlearum <lb></lb>retinaculum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004599">IN Trochlearum uſu communi alteram ſtabilem eſſe ac fir<lb></lb>mam, alteram mobilem (ſi enim plures eſſent omnino ſtabi<lb></lb>les, quantumvis multæ, non augerent motum potentiæ) illam <lb></lb>autem ab aliquo corpore, cui alligata eſt, retineri, ſatis per ſe <lb></lb>patet; propterea corpus hoc adeò validum eſſe oportet, ut ne<lb></lb>que gravitati ponderis, neque conatui potentiæ cedat, ſed ita <lb></lb>immotum perſiſtat, ut univerſus potentiæ impetus ad vincen<lb></lb>dam ponderis reſiſtentiam referatur. </s> <s id="s.004600">Hinc à veritate non ad<lb></lb>modum receſſiſſe videntur, qui in Mechanicis motionibus qua<lb></lb>ſi duplex munus diſtinguunt, alterum, quo pondus retinetur, ne <lb></lb>vi ſuæ gravitatis labatur, alterum, quo gravitas ipſa ſuperatur, <lb></lb>& cogitur inire motum ſuæ propenſioni adverſantem: poſte<lb></lb>rius hoc ſoli potentiæ tribuendum, prius illud non uni poten<lb></lb>tiæ, ſed etiam corpori, cui machina innititur, adſcribendum <lb></lb>cenſent, & in illud maximam oneris partem rejici aſſerunt. </s> <s id="s.004601">Et <lb></lb>ſanè quid prodeſſet trochleam ſuperiorem aut fune, qui laxari <lb></lb>nequiret, aut ferreo unco, quem revellere nulla gravitas poſ<lb></lb>ſet, connecti cum tigno parieti infixo; ſi timendum eſſet, ne <pb pagenum="620" xlink:href="017/01/636.jpg"></pb>tignum ipſum imbecillum, vímque gravitatis ſuſpenſæ ferre <lb></lb>non valens, frangeretur? </s> <s id="s.004602">Quare ne magnum in diſcrimen res <lb></lb>adducatur, & ad periculum omne ſubmovendum, ne inſtitu<lb></lb>tus motus repentina retinaculi abruptione intercidatur, atque <lb></lb>ut certiùs eligi poſſit, cuinam potiſſimùm corpori (tigno ne pa<lb></lb>rieti infixo? </s> <s id="s.004603">an antennæ erectæ?) concredenda ſit oneris <lb></lb>ſuſtentatio, machinatori attente diſpiciendum eſt, quantam <lb></lb>vim tùm oneris gravitas, tùm potentiæ conatus exerceat adver<lb></lb>sùs hujuſmodi retinaculum. </s> <s id="s.004604">Propterea vim iſtam placuit hoc <lb></lb>capite examinare, ut cætera ſecurè definiri valeant. </s> <s id="s.004605">Ut verò <lb></lb>brevitati & perſpicuitati conſulatur, retinaculum hoc ponamus <lb></lb>eſſe clavum, ex quo trochleæ cum onere ſuſpenſo dependeant; <lb></lb>quæ enim de hujuſmodi clavo dicentur, facilè ad cætera tradu<lb></lb>ci poterunt. </s> </p> <p type="main"> <s id="s.004606">Et primò ſi trochlearum funis per orbiculos ritè trajectus de<lb></lb>mum ſuâ extremitate in nodum colligatur, ne excurrere valeat, <lb></lb>totam atque integram oneris gravitatem (trochleas & funem à <lb></lb>ſuâ inſitâ gravitate nunc quidem mente ſecernamus) à clavo, <lb></lb>ex quo trochleæ ſuſpenduntur, retineri dubium eſſe non poteſt; <lb></lb>nihil aliud quippe adeſt, adversùs quod ponderis gravitas de<lb></lb>orſum ſe ipſa urgens connitatur. </s> <s id="s.004607">Deinde ſi funis ductarij caput, <lb></lb>quod potentia trahere ſolita eſt, alligetur ſolo, aut ingenti ſaxo <lb></lb>longiſſimè graviori, quàm pondus ſuſpenſum, utique neque <lb></lb>ſaxum illud ſubjectæ telluri incumbens, neque tellus ipſa, quip<lb></lb>piam virium exercent adversùs pondus, cui ſolùm ſuâ longè <lb></lb>majori gravitate reſiſtunt Formaliter, non verò Activè; quia ni<lb></lb>mirum nullum efficiunt impetum, quo deſcenſum moliantur; <lb></lb>ac proinde à clavo ſolo pondus trochleis adnexum ſuſtinetur, <lb></lb>& ſolum pondus clavum deorſum trahere conatur. </s> </p> <p type="main"> <s id="s.004608">At verò ſi funis ductarij extremitati adnectatur alia gravitas <lb></lb>pro trochlearum Ratione reſpondens ponderis gravitati, ita ut <lb></lb>æqualibus momentis certantes ambæ ſuſpenſæ conſiſtant, utra<lb></lb>que gravitas collatis viribus clavum trahere conatur, utraque <lb></lb>enim deorſum connititur: & ideò tam validum ſtatui clavum <lb></lb>oportet, ut utriuſque gravitatis conatum ferre valeat. </s> <s id="s.004609">Id quod <lb></lb>multo magis obſervandum eſt, quando gravitas adnexa præpon<lb></lb>derans vim infert oneri, illúdque ſurſum trahit; ipſa ſcilicet gra<lb></lb>vitas plus conatur in motu, quàm in æquilibrio; ac propterea & <pb pagenum="621" xlink:href="017/01/637.jpg"></pb>potentiæ deorſum connitentis in motu impetum, & oneris mo<lb></lb>tui ſurſum repugnantis gravitatem fert clavus utrique reſiſtens <lb></lb>ſuâ ſoliditate. </s> <s id="s.004610">Sicut igitur gravitas inanimata ex trochlearum <lb></lb>fune pendens ſuſpendit pondus, aut attollit; ita potentia vivens <lb></lb>funem retinendo ſuo impetu virtutem ejuſdem gravitatis æquat, <lb></lb>ac ſimilem vim exercet in clavum; funem verò trahendo virtu<lb></lb>tem illam gravitatis ſuperat, atque impreſſo impetu quodam<lb></lb>modo attenuat, ita tamen, ut quod videtur gravitati demptum, <lb></lb>intelligatur additum conatui potentiæ prævalentis. </s> </p> <p type="main"> <s id="s.004611">Mihi autem (quid fruſtra diſſimulem?) non levis injicitur <lb></lb>ſcrupulus & dubitatio, an vis illata clavo, ex quo trochleæ cum <lb></lb>onere dependent, menſuram præcisè recipiat ex abſolutâ gra<lb></lb>vitate oneris, quando abeſt conatus potentiæ illud attollentis <lb></lb>aut ſuſpendentis. </s> <s id="s.004612">Dubitandi anſam offert quædam munerum <lb></lb>commutatio inter Potentiam, Pondus, & Clavum, ſi ad effectio<lb></lb>nes diverſas referantur. </s> <s id="s.004613">Si enim oneris ſuſpenſio aut elevatio vi <lb></lb>potentiæ ex adverſo nitentis conſideretur, Clavus exercet mu<lb></lb>nus Retinaculi: at ſi vim clavo illatam, ejúſque inflexionem, aut <lb></lb>revulſionem intueamur, efficientia vim hujuſmodi inferens tri<lb></lb>buenda eſt aut gravitati oneris, aut impetui potentiæ trahentis: <lb></lb>quapropter ſoliditas clavi inflexionem reſpuentis, aut ejus firma <lb></lb>cohæſio cum pariete aut ligno, cui infixus eſt, vicem ſubit Pon<lb></lb>deris ope trochlearum movendi cum alterâ trochleâ connexi; <lb></lb>Sarcina autem ex reliquâ trochleâ dependens aut retinaculi <lb></lb>munus obtinet, ſi attollatur, aut Potentiæ vices ſubit, ſi deor<lb></lb>ſum moveatur. </s> </p> <p type="main"> <s id="s.004614">Sit clavo A adnexa ſimplex Trochlea B, ejúſque funis <lb></lb>ductarius CDE: adnectatur in C ſa<lb></lb><figure id="id.017.01.637.1.jpg" xlink:href="017/01/637/1.jpg"></figure><lb></lb>xum P, & à Potentia G elevatum ſuſ<lb></lb>pendatur religato funis capite in E. </s> <s id="s.004615">Si <lb></lb>ſaxum P accipiatur, quatenus elevatur, <lb></lb>ipſum eſt Pondus, Clavus A eſt Retina<lb></lb>culum, & Potentia eſt G, ſive illa ſit <lb></lb>inanima ſua majore gravitate contrani<lb></lb>tens, ſive ſit vivens ſuo impetu ſurſum <lb></lb>trahens, & poſtmodum remiſſiore impe<lb></lb>tu, & nervorum contentione impediens, <lb></lb>ne ſaxum elevatum relabatur. </s> <s id="s.004616">At ſi ipſius <pb pagenum="622" xlink:href="017/01/638.jpg"></pb>clavi A preſſio, ſive inflexio conſideretur, jam vis efficien<lb></lb>di preſſionem hanc, ſeu inflexionem, tota tribuenda eſt ſaxo <lb></lb>P, quod propterea inducit rationem Potentiæ, & retinacu<lb></lb>lum eſt paxillus E in terram firmiter depactus, qui nihil agit, <lb></lb>ſed funem duntaxat retinet. </s> <s id="s.004617">Verùm in hac poſitione mo<lb></lb>mentum ſaxi adversùs clavum non eſt ſimplicis gravitatis ab<lb></lb>ſolutè acceptæ, perinde atque ſi funis FG infra orbiculum <lb></lb>reflexus colligeretur in nodum cum fune DC: tunc enim, <lb></lb>collecto in nodum fune, orbiculus eſſet planè otioſus, & ni<lb></lb>hil conferret ad momentorum varietatem, ſed idem accide<lb></lb>ret, ac ſi funis ſimplex proximè & immediatè clavo ad<lb></lb>necteretur ſepoſitâ quacumque trochleâ. </s> <s id="s.004618">Sed fune in E re<lb></lb>ligato (quaſi duplex pondus ex ſimplici fune penderet) ge<lb></lb>minatur ſaxi P momentum adversùs clavum, qui nequit vel <lb></lb>minimum flecti, quin duplo motu ſaxum ipſum moveatur, <lb></lb>neque enim, quod ad geminandum momentum ſpectat, dif<lb></lb>fert ſaxum à potentiâ vivente, quæ utique in C applicata <lb></lb>funi, & trochleam trahens, adversùs pondus trochleæ ad<lb></lb>nexum habet momentum duplum ejus, quod obtineret, ſi <lb></lb>funem ſimplicem traheret: eſt autem trochleæ adnexus <lb></lb>clavus. </s> </p> <p type="main"> <s id="s.004619">Quod ſi in G contra ſaxum P aut gravitas inanimata, aut <lb></lb>potentia vivens nitatur, ſi quidem æqualibus conatibus hinc <lb></lb>& hinc certetur, atque ſuſpenſum conſiſtat ſaxum, aut clavus <lb></lb>ſimiliter premitur atque libræ agina, cùm jugum à duobus <lb></lb>æqualibus ponderibus in æquilibrio retinetur, aut alterutri <lb></lb>munus Potentiæ, & alteri Retinaculi adſcribendum eſt, & <lb></lb>Potentia ſimiliter geminato momento clavum trahit deor<lb></lb>ſum. </s> <s id="s.004620">Sin autem aut ſaxum P, aut virtus movendi in G, <lb></lb>ſuperat, huic Potentiæ ratio tribuatur, oppoſito munus re<lb></lb>tinaculi; ſed Potentiæ abſolutè acceptæ momenta non ge<lb></lb>minantur, quia retinaculum ſtabile non eſt, ſed cedit; <lb></lb>adeóque impetus à Potentia productus duos motus efficit, <lb></lb>alterum trahendo retinaculum, alterum inflectendo clavum, <lb></lb>qui propterea minùs flectitur, quò magis oppoſitum retina<lb></lb>culum movetur. </s> </p> <p type="main"> <s id="s.004621">Neque hæc quicquam habent admirationis: Nam ſi Vectis <lb></lb>ſit CD, habens in C hypomochlium; in medio autem <pb pagenum="623" xlink:href="017/01/639.jpg"></pb>puncto E adnexus ſit funiculus, qui incumbens orbiculo F <lb></lb>verſatili adnexum habeat pondus S <lb></lb><figure id="id.017.01.639.1.jpg" xlink:href="017/01/639/1.jpg"></figure><lb></lb>innixum plano ſubjecto; utique in ex<lb></lb>tremitate D pondus V paulo majus <lb></lb>quàm ſubduplum ponderis S illud ele<lb></lb>vabit, atque præcisè ſubduplum non <lb></lb>elevabit quidem illud, ſed adversùs <lb></lb>orbiculum F conatur momento duplo <lb></lb>ejus, quod obtineret, ſi ex E pende<lb></lb>ret ipſum pondus V, cui reluctaretur <lb></lb>pondus S gravius innixum plano. </s> <s id="s.004622">Ve<lb></lb>rùm ad vectem retinendum in poſitio<lb></lb>ne horizontali CD nihil intereſt. </s> <lb></lb> <s id="s.004623">utrùm in C aliquid ſuperius ſit prohibens, ne illa extremitas <lb></lb>vi ponderis V attollatur, an verò inferiùs funiculo connectatur <lb></lb>cum tellure, aut ex C pendeat onus H (ſed plano ſubjecto in<lb></lb>nixum) vel æquale ipſi V, vel eo majus; ſemper enim pondus <lb></lb>V eadem obtinet momenta. </s> <s id="s.004624">Quare ſi, amoto orbiculo F & pon<lb></lb>dere S, manu retineas funiculum IE, percipies ad ſervandum <lb></lb>vectem horizontalem, quantâ virium acceſſione tibi opus ſit, <lb></lb>ſupra quàm exigeret ſimplex gravitas ponderis V, ſi ex E pen<lb></lb>deret, ubi nulla Vectis ratio intercederet. </s> </p> <p type="main"> <s id="s.004625">Cum itaque hæc in Vecte pariter ratione poſitionis pon<lb></lb>deris contingant, quæ trochleæ accidere diximus ratione <lb></lb>connexionis ponderis vel cum trochleâ, vel cum paxillo <lb></lb>telluri infixo, nil mirum ſi alia atque alia ſint ejuſdem pon<lb></lb>deris momenta adversùs clavum. </s> <s id="s.004626">Sicut autem quando tam ab <lb></lb>hypomochlio quàm à potentiâ ſuſtinetur onus in medio vecte <lb></lb>ſuſpenſum, hypomochlium à pondere non premitur niſi juxta <lb></lb>ſemiſſem gravitatis ponderis; ita quoque cum funis ductarius <lb></lb>alterâ extremitate adnexus eſt clavo, alterâ retinetur à poten<lb></lb>tia, pondus ex trochleâ ſimplici pendens partim à Potentiâ, <lb></lb>partim à clavo ſuſtinetur, adversùs quem minus virium exercet <lb></lb>ejus gravitas, ut conſtabit, ſi clavo orbiculum verſatilem in<lb></lb>figas, & funiculo per orbiculi orbitam excavatam tranſeunti <lb></lb>aliud pondus adnectas, quod ſatis erit, ſi fuerit ſubduplum <lb></lb>ponderis ex trochleâ pendentis; hoc enim ſuſtinebitur à <lb></lb>duplici virtute ſubduplâ gravitatis illius. </s> <s id="s.004627">Non igitur plus <pb pagenum="624" xlink:href="017/01/640.jpg"></pb>reſiſtentiæ requiritur in clavo, quàm in pondere illo ſub<lb></lb>duplo. </s> </p> <p type="main"> <s id="s.004628">His ita in unicâ ſimplici trochleâ conſtitutis, examinandæ <lb></lb>ſunt trochleæ conjugatæ; nec difficile erit ex dictis ſuperiore <lb></lb>capite inveſtigare momenta ponderis adversùs clavum, cui al<lb></lb>tera trochlea adnectitur. </s> <s id="s.004629">Ibi enim alteri trochleæ potentiam <lb></lb>ſurſum trahentem, alteri pondus dependens adnecti poſuimus, <lb></lb>funis verò extremitatem clavo alligari: Hìc loco clavi illius re<lb></lb>tinentis extremitatem funis intelligendum eſt retinaculum, <lb></lb>quodcumque tandem illud ſit, ſive manus hominis, ſive etiam <lb></lb>alius clavus: ſed loco Potentiæ ſuperiorem trochleam ſurſum <lb></lb>trahentis ſit clavus, ex quo trochleæ fune ductario connexæ <lb></lb>unà cum pondere dependent; gravitas autem illa ſuſpenſa ex <lb></lb>inferiore trochleâ exercet munus potentiæ adversùs clavum, <lb></lb>qui ſubit vicem ponderis movendi, quatenus aliquantulum <lb></lb>flectitur, aut inflexioni repugnat. </s> <s id="s.004630">Sicut ergo ibi oſtenſum eſt <lb></lb>in duabus ſimplicibus trochleis ſingulos orbiculos habentibus, <lb></lb>ſi funis ductarij caput alligatum ſit ſuperiori trochleæ, motum <lb></lb>trochleæ ſuperioris ad motum inferioris eſſe ut 2 ad 3; ſi verò <lb></lb>funis caput alligatum fuerit inferiori trochleæ, motum ſuperio<lb></lb>ris ad motum inferioris eſſe ut 3 ad 2: Ita hìc dicendum eſt <lb></lb>(ponamus clavum flecti aliquantulum) in primo caſu motum <lb></lb>flexionis clavi ad motum deſcensûs ponderis eſſe ut 2 ad 3, in <lb></lb>ſecundo autem caſu ut 3 ad 2. Ex quo fit in primo caſu pon<lb></lb>dus habere adversùs clavum majus momentum quàm in ſecun<lb></lb>do caſu; & in primo caſu validiùs deorſum trahere, quàm ſi <lb></lb>ſimplici funiculo dependeret, & motus eſſent æquales; major <lb></lb>ſiquidem eſt Ratio 3 ad 2, quàm 2 ad 2; in ſecundo verò caſu <lb></lb>debiliùs deorſum trahere, quàm ſi nullæ eſſent trochleæ, adeó<lb></lb>que motus æquales eſſent; minor quippe eſt Ratio 2 ad 3, quam <lb></lb>1 ad 1, aut 3 ad 3. </s> </p> <p type="main"> <s id="s.004631">Simili planè methodo philoſophandum eſt in reliquis tro<lb></lb>chleis conjugatis: ſi enim duabus trochleis diſpar inſit orbicu<lb></lb>lorum numerus, ut altera major ſit, altera minor, obſervandum <lb></lb>eſt, an major trochlea alligetur clavo, an verò minor: Si tro<lb></lb>chlea plures habens orbiculos clavo adnectatur, motus flexionis <lb></lb>clavi minor eſt motu deſcensûs ponderis in Ràtione ſubſuper<lb></lb>particulari denominatâ à numero omnium ſimul orbiculorum; <pb pagenum="625" xlink:href="017/01/641.jpg"></pb>ac proinde pondus habet momentum majus, quàm ſi nullæ in<lb></lb>tercederent trochleæ: contra verò ſi clavo adnectatur trochlea <lb></lb>minor, motus flexionis clavi major eſt motu deſcensûs ponde<lb></lb>ris in Ratione ſuperparticulari denominatâ à numero omnium <lb></lb>ſimul orbiculorum; atque ideo pondus adversùs clavum minus <lb></lb>habet momenti, quàm ſi ex illo ſimplici fune penderet. </s> <s id="s.004632">At ſi <lb></lb>utriuſque trochleæ par ſit orbiculorum numerus, & pariter ra<lb></lb>tio ſuperparticularis, aut ſubſuperparticularis denominata à <lb></lb>numero omnium ſimul orbiculorum; & ſi quidem trochleæ ſu<lb></lb>periori adnectatur funis caput, pondus adversùs clavum habet <lb></lb>momentum majus, quàm ſi amotis trochleis ex ſimplici fune <lb></lb>penderet; ſin autem inferiori trochleæ alligetur extremitas fu<lb></lb>nis ductarij, ponderis momentum adversùs clavum minùs <lb></lb>eſt, quàm ſi idem pondus ex eodem clavo ſimplici fune ſuſ<lb></lb>penderetur. </s> </p> <p type="main"> <s id="s.004633">Ex his ſatis apparet clavo eandem vim inferri, ſi pondus de<lb></lb>pendens ex trochleis ſuſpenſum maneat, ſive quia funis extre<lb></lb>mitas religetur paxillo, ſive quia ex eâdem funis extremitate <lb></lb>dependeat onus ſubmultiplex ponderis ex inferiore trochleâ <lb></lb>pendentis, ſecundùm Rationem, quam inferunt ipſi orbiculi. </s> <lb></lb> <s id="s.004634">Sic ex trochleis binos orbiculos habentibus dependeat pondus, <lb></lb>& funis extremitas religetur paxillo: ex dictis, ſuperioris tro<lb></lb>chleæ clavo adnexæ, & funis ductarij caput habentis, motus, <lb></lb>ad motum inferioris trochleæ & ponderis eſt ſubſeſquiquartus; <lb></lb>ac proinde pondus trochleis connexum cum clavo ad vim illi <lb></lb>inferendam perinde ſe habet, atque ſi ex eodem clavo abſque <lb></lb>trochleis ſimplici fune appenderetur pondus aliud dati ponde<lb></lb>ris Seſquiquartum. </s> <s id="s.004635">At ſi extremitati funis adderetur pondus va<lb></lb>lens ſuſpendere onus adnexum trochleæ inferiori, eſſet ex dictis <lb></lb>cap.1. dati oneris ſubquadruplum. </s> <s id="s.004636">Igitur duorum horum pon<lb></lb>derum ſumma ad datum pondus eſſet ut 5 ad 4, cujuſmodi erat <lb></lb>Ratio motuum, ex quibus momentum deſumitur. </s> <s id="s.004637">An non ſi <lb></lb>onus in plano horizontali raptandum ſimplici trochleæ adne<lb></lb>xum proponatur, & duo homines pariter utramque funis ex<lb></lb>tremitatem arripiant, atque trahant, ſinguli medietatem ne<lb></lb>ceſſarij conatûs adhibent? </s> <s id="s.004638">ſi verò alter trahentium deficiat, & <lb></lb>illa funis extremitas alligetur paxillo, nonne qui reliquus eſt <lb></lb>eodem conatu trahens ſolus adducet idem pondus? </s> <s id="s.004639">non niſi <pb pagenum="626" xlink:href="017/01/642.jpg"></pb>quia, cùm ambo trahebant, pondus & potentia æqualiter mo<lb></lb>vebantur; cùm alter tantùm trahit, ille movetur duplo velo<lb></lb>cius, quàm pondus, quod ad ſubduplam velocitatem ſatis ha<lb></lb>bet impetum ſubduplum impetûs neceſſarij ad velocitatem <lb></lb>æqualem. </s> <s id="s.004640">Eadem igitur militat ratio in clavo, cui vis infertur <lb></lb>à duobus ponderibus ſuſpenſis ex trochleis in æquilibrio, quæ <lb></lb>ſimul deorſum trahentia motum habent æqualem cum motu <lb></lb>clavi, qui flectitur, aut revellitur; ſed funis capite religato fir<lb></lb>miter ad paxillum, pondus inferiori trochleæ adnexum motum <lb></lb>habet velociorem comparatum cum ejuſdem clavi motu, ac <lb></lb>propterea majus momentum habet. </s> </p> <p type="main"> <s id="s.004641">Hinc præterea inferendum eſt non ſatis utiliter eos opera<lb></lb>ri, qui pondus ex ſuperiore loco fune ſuſpenſum, ſive orbicu<lb></lb>lus intercedat, ſive non, putant firmius ſuſtineri, ſi funis ca<lb></lb>put in inferiore loco religetur: ſi enim funis excurrere ne<lb></lb>queat, inferiùs hoc retinaculum prorſus inutile accidit, ſin au<lb></lb>tem excurrere valeat, ſuperius illud retinaculum geminatam <lb></lb>vim ſuſcipit, quaſi duplex pondus ab illo ſuſtineretur. <lb></lb></s> </p> <p type="main"> <s id="s.004642"><emph type="center"></emph>CAPUT VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004643"><emph type="center"></emph><emph type="italics"></emph>Aliqui Trochlearum uſus indicantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004644">PRo more in ſuperioribus libris ſervato, hìc pariter indican<lb></lb>di ſunt aliqui Trochlearum uſus, qui facilè ad ſimilia <lb></lb>traduci poterunt, ſpectato motu, qui exhibendus proponitur, <lb></lb>ut ei trochleæ reſpondeant, & aptè collocentur, neque plu<lb></lb>ribus, quàm opus ſit, orbiculis inſtruantur, ne dum poten<lb></lb>tiæ facilitatem conſectaris, nimis tardè moveas pondus, aut <lb></lb>ex adverſo, dum ponderi velocitatem concilias, nimio labore <lb></lb>potentiam opprimas. </s> </p> <pb pagenum="627" xlink:href="017/01/643.jpg"></pb> <p type="main"> <s id="s.004645"><emph type="center"></emph>PROPOSITIO I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004646"><emph type="center"></emph><emph type="italics"></emph>Auram in Conclavi excitare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004647">QUæritur ſæpè æſtivo tempore aliquod ex aëris motu re<lb></lb>frigerium; ſed manuali flabello auram excitare aliquan<lb></lb>do incommodum eſt, ſi aliud agendo diſtinearis: propterea <lb></lb>ventilabrum in conclavis angulo ſtatuere poſſumus, quod ali<lb></lb>quandiu moveatur, aërémque agitet: ideóque illud ad angu<lb></lb>lum ſtatuendum propoſui, ut commotus aër in proximos hinc <lb></lb>atque hinc parietes impactus reflectatur, & faciliùs reliquum <lb></lb>conclavis aërem exagitet. </s> </p> <p type="main"> <s id="s.004648">Excitetur angulo congruens turricula haud abſimilis iis, <lb></lb>quibus horologia reconduntur; in ſupremâ turriculæ parte <lb></lb>ab angulo ad oppoſitum ex diametro angulum Axis horizon<lb></lb>ti parallelus ſtatuatur facilè verſatilis, cujus tamen pars ex<lb></lb>tra turriculam promineat tantæ longitudinis, quanta flabel<lb></lb>lis latitudo deſtinatur. </s> <s id="s.004649">Pars tamen hæc Axis extima nul<lb></lb>lam exigit certam figuram, nihilque refert ſive cylindrica <lb></lb>ſit, ſive quadrata, ſive quæcumque alia; modò ea ſit, ut illi <lb></lb>facilè flabella firmiter infigi, atque eximi pro opportunitate <lb></lb>poſſint, iiſque exemptis aptari valeat manubrium, quo faciliùs <lb></lb>& citiùs ab homine convolvatur Axis. </s> </p> <p type="main"> <s id="s.004650">Parentur duæ trochleæ ternis orbiculis inſtructæ, altera <lb></lb>in ſuperiore turriculæ loco firmetur, altera ad turriculæ pe<lb></lb>dem conſtituatur adnexam habens plumbeam maſſam motui <lb></lb>perficiendo congruentem: huic eidem trochleæ adnectatur <lb></lb>extremitas funis ductarij, qui per omnes trochlearum orbicu<lb></lb>los trajectus demum ad Axem referatur, ibíque alligetur. </s> <s id="s.004651">Tum <lb></lb>appoſito manubrio convolutum Axem circumplectetur funis <lb></lb>ductarius, & plumbea maſſa in ſupremam turriculæ partem <lb></lb>Axi proximam deducetur. </s> <s id="s.004652">Infigantur Axi ventilabra, & amo<lb></lb>to manubrio plumbea maſſa ſibi relicta lentiſſimo motu deſcen<lb></lb>det, convolvénſque axem cum flabellis tandiu aërem commo<lb></lb>vebit, quandiu illa deſcendet. </s> </p> <p type="main"> <s id="s.004653">Hìc commutatas vices inter potentiam & pondus obſervare <lb></lb>quilibet poteſt; potentia ſiquidem movens eſt plumbea maſ-<pb pagenum="628" xlink:href="017/01/644.jpg"></pb>ſa, quæ ſeptuplo tardiùs movetur quàm extremitas funis <lb></lb>ductarij, quem aliàs trahere ſolita eſt potentia. </s> <s id="s.004654">Loco autem <lb></lb>ponderis eſt aër, qui à flabellis impellitur; ac proinde quò <lb></lb>ampliora ſunt flabella, eò major eſt reſiſtentia aëris com<lb></lb>moti, ratione cujus etiam retardatur motus potentiæ. </s> <s id="s.004655">Ubi <lb></lb>quoquè attendenda eſt Ratio longitudinis flabellorum ad ſe<lb></lb>midiametrum axis convoluti: nam ſi hæc Ratio componatur <lb></lb>cum Ratione ſeptuplâ, quam Trochleæ inferunt, habebitur <lb></lb>Ratio motûs extremi flabelli ad motum maſſæ plumbeæ: <lb></lb>quamquam non ita computanda eſt aëris reſiſtentia, quaſi to<lb></lb>ta in flabelli extremitate exerceretur; hæc ſcilicet per univer<lb></lb>ſam flabelli longitudinem diffunditur inæqualiter diſtributa <lb></lb>pro Ratione diſtantiæ à centro motûs, aër quippe pro diversâ <lb></lb>impellentis velocitate inæqualiter reſiſtit. </s> </p> <p type="main"> <s id="s.004656">Quod ſi magis arrideret non continua convolutione flabel<lb></lb>la circumagi, ſed alternâ quadam modò in dextram, modò <lb></lb>in ſiniſtram inflexione agitari; Axi, quem funis ductarius <lb></lb>complectitur, infige rotam dentatam, cujus dentes incurrant <lb></lb>in pinnulas fuſi perpendicularis flabella ſuſtinentes, quemad<lb></lb>modum in Tempore horologij: ſimili enim ratione, ac Tem<lb></lb>pus, ultrò citróque remeabunt flabella, & aërem in oppoſitas <lb></lb>partes commovebunt. </s> <s id="s.004657">Cùm verò antè motum appoſito manu<lb></lb>brio convolvendus erit Axis, ut funem ductarium recipiat, at<lb></lb>que trochlea inferior cum pondere attollatur, ita fuſum pau<lb></lb>liſper elevare oportebit, ut ejus pinnulæ non occurrant denti<lb></lb>bus rotæ, niſi cùm iterum fuſus ſuum in locum reſtituetur. </s> <lb></lb> <s id="s.004658">Hac alternatione diuturnior erit motus. </s> </p> <p type="main"> <s id="s.004659"><emph type="center"></emph>PROPOSITIO II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004660"><emph type="center"></emph><emph type="italics"></emph>Corpus aliquod in gyrum celeriter volvere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004661">IN rebus ſcenicis locum habere non infrequentem poteſt <lb></lb>hæc propoſitio: aliquando ſcilicet ſolis diſcum in ſcenam <lb></lb>producimus, quem licet auro obductum, ac multis facibus <lb></lb>illuſtratum, quas ſpectatorum oculis ex arte ſubducimus, <lb></lb>non tamen radios ejaculantem mentimur, niſi ille circa <pb pagenum="629" xlink:href="017/01/645.jpg"></pb>ſuum centrum velociter circumagatur. </s> <s id="s.004662">Id quod variá qui<lb></lb>dem methodo præſtari poteſt infixum diſci centro cylindrum <lb></lb>convolvendo, ſive ope rotæ dentatæ Vertebram ſtriatam cy<lb></lb>lindro circumpoſitam moventis; ſive fune cylindrum bis aut <lb></lb>ter arctè complexo, & in ſeſe redeunte, ubi majoris alicu<lb></lb>jus tympani orbitam pariter complexus fuerit; ſive pondere <lb></lb>funem cylindro involutum explicante: ſed poſtremus hic <lb></lb>modus non niſi breve temporis ſpatium exigit; duo priores, <lb></lb>ſi paulo longior futurus ſit motus, non niſi à potentiá vi<lb></lb>vente commodè exhiberi poſſunt. </s> <s id="s.004663">Quare ſatius fuerit tro<lb></lb>chleas, ut in ſuperiore propoſitione, diſpoſitas adhibere, at<lb></lb>que loco flabellorum ſolis diſcum Axi adnectere; ſic enim <lb></lb>fiet, ut & celeriter in gyrum agatur, & diu perſeveret <lb></lb>motus. </s> </p> <p type="main"> <s id="s.004664">Similiter ad fingendum mare, & undarum motum vehe<lb></lb>mentiorem, ſtatuuntur horizonti & invicem paralleli ali<lb></lb>quot axes, quos ambiunt ſpiræ profundiùs excavatæ colore <lb></lb>marinam undam imitantes: dum enim hujuſmodi axes con<lb></lb>volvuntur, marini æſtûs curſum ſpectatoribus repræſentant. </s> <lb></lb> <s id="s.004665">Ut autem axes illi citra cujuſquam laborem volvantur tro<lb></lb>chleas duas binis, aut ternis orbiculis inſtructas (prout diu<lb></lb>turnior motus requiritur) compone, & proximas ſtatue, al<lb></lb>teram firmans in ſuperiore loco: Tum funis ductarius per <lb></lb>omnes Trochlearum orbiculos trajectus ſingulorum axium ca<lb></lb>pita ex ordine ambiat unâ ſaltem aut alterâ ſpirâ, & demum <lb></lb>ad peculiarem alium axem deveniat, quem totus plures in <lb></lb>ſpiras complicatus circumplectatur, ita tamen, ut facilè <lb></lb>evolvi queat. </s> <s id="s.004666">Ubi igitur tempus advenerit, inferiori tro<lb></lb>chleæ congruum pondus adnecte; hoc enim licèt lentè de<lb></lb>ſcendat, velociter tamen axes convolvit funem evolvens. </s> <lb></lb> <s id="s.004667">Procellam verò miteſcere aut exaſperari mentieris, factâ <lb></lb>ponderis aliquâ detractione aut acceſſione, id quod difficile <lb></lb>non fuerit. </s> </p> <pb pagenum="630" xlink:href="017/01/646.jpg"></pb> <p type="main"> <s id="s.004668"><emph type="center"></emph>PROPOSITIO III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004669"><emph type="center"></emph><emph type="italics"></emph>Se ipſum ope trochlearum in altum evehere, aut promovere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004670">SElla paretur hinc & hinc habens fulcra, quibus brachia <lb></lb>innituntur, & in hujuſmodi fulcrorum extremitate ante<lb></lb>riore aptetur Sucula manubriata, quam ſedens commodè ver<lb></lb>ſare valeat: ſella autem quatuor funibus in nodum cum an<lb></lb>nulo coëuntibus ſuſpendatur ita, ut inferioris trochleæ uncus <lb></lb>annulo indatur, & funis ductarius per cunctos trochlearum <lb></lb>orbiculos trajectus demum ſuculæ alligetur. </s> <s id="s.004671">Nam in ſellâ ſe<lb></lb>dens, & ſuculæ manubria convertens, funem ductarium <lb></lb>trahit, atque ipſe ſe in altum evehit eâ facilitate, quam in<lb></lb>fert Ratio compoſita ex Rationibus trochlearum, & Suculæ: <lb></lb>eſt ſiquidem Potentia ipſa virtus animalis muſculorum con<lb></lb>tentione verſans manubria, pondus autem eſt inſita corpori <lb></lb>gravitas, quæ eò minor apparet, quo majores ſunt, hoc eſt <lb></lb>pluribus inſtructæ orbiculis, trochleæ, & major eſt Ratio <lb></lb>manubriorum ad ſemidiametrum Axis, qui fune obvolvitur. </s> <lb></lb> <s id="s.004672">Sit enim ex. </s> <s id="s.004673">gr. inferior trochlea, cui pondus movendum ad<lb></lb>nectitur, & funis ductarij extremitas alligatur, orbiculorum <lb></lb>duorum; ſuperior autem trochlea, quæ ſtabilis manet, tres <lb></lb>habeat orbiculos: utique Ratio motûs potentiæ ad motum <lb></lb>ponderis eſt quintupla: manubria autem Suculæ ſint quadru<lb></lb>pla ſemidiametri Axis: Ratio compoſita ex quadruplâ & quin<lb></lb>tuplâ eſt vigecuplâ; igitur conatus Potentiæ manubria verſan<lb></lb>tis ſatis eſt, ſi reſpondeat vigeſimæ parti ponderis. </s> </p> <p type="main"> <s id="s.004674">Similiter ſi cymba adverſo flumine non procul à ripâ dedu<lb></lb>cenda ſit, & qui in eâ ſunt nautæ, ita pauci ſint, ut non va<lb></lb>leant eam adversùs vim profluentis remo agere, aut è ripá fu<lb></lb>ne nautico trahere; ſubſidium ex trochleis petere poterunt; <lb></lb>exſcenſu ſcilicet in terram facto, atque defixo in ripâ paxil<lb></lb>lo alligatur trochlea una, altera adnectitur proræ cymbæ, <lb></lb>in qua nautæ duo funem ductarium trahentes illam adverſo <lb></lb>flumine promovent perinde, atque ſi eſſent octo aut duode<lb></lb>cim homines, ſi trochleæ binos aut ternos habuerint orbicu-<pb pagenum="631" xlink:href="017/01/647.jpg"></pb>los. </s> <s id="s.004675">Quod ſi trochleis illi careant, utantur artificio ſequentis <lb></lb>propoſitionis, unà cum iis, quæ cap. 5. dicta ſunt. </s> </p> <p type="main"> <s id="s.004676"><emph type="center"></emph>PROPOSITIO IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004677"><emph type="center"></emph><emph type="italics"></emph>Trochlearum defectum ſupplere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004678">EX his, quæ cap. 2. hujus libri indicata ſunt, ſatis conſtat <lb></lb>etiam ſinè orbiculis haberi poſſe momentum Trochlea<lb></lb>rum: quare his deficientibus annulos ſufficere facile erit. <lb></lb><figure id="id.017.01.647.1.jpg" xlink:href="017/01/647/1.jpg"></figure><lb></lb>Et primo quidem ſingulis annulis uti poſſumus: nam cùm <lb></lb>cymba communiter adnexum proræ annulum habeat, ut <lb></lb>medio fune, aut catenâ ad ripam religetur, funis ducta<lb></lb>rius unus AB adnectatur paxillo A in ripâ defixo, & per <lb></lb>cymbæ annulum B trajiciatur; illius alteri extremitati C an<lb></lb>nulus alius adnectatur, per quem alter ductarius funis DEF <lb></lb>trajectus & paxillo D alligatus ſi à Potentiâ in F conſtitutâ <lb></lb>trahatur, illa habebit momentum quadruplum, perinde at<lb></lb>que de orbiculi, ſuperiùs dictum eſt cap. 5. At ſi conſiſtentes <lb></lb>in cymbâ trahere illam velint nautæ, adjiciatur paxillo D an<lb></lb>nulus G ſtabilis, per quem productus funis EF tranſeat, & <lb></lb>veniat in H ad nautarum manus in cymbâ; nam illum trahen<lb></lb>do cymbæ prora ex B accedet ad A. </s> </p> <p type="main"> <s id="s.004679">Porrò annuli nomine notatum volo quicquid ejuſmodi eſt, <lb></lb>ut funis per illud trajici poſſit, & liberè excurrere, ſive ſit <lb></lb>ligni fruſtum foramen habens politum & ſatis amplum, ut <lb></lb>per illud funis facilè moveri valeat, ſive etiam ſit flexilis <pb pagenum="632" xlink:href="017/01/648.jpg"></pb>bacilli particula in arcum vel modicè ſinuata; modò illa non <lb></lb>ſit fractioni obnoxia. </s> <s id="s.004680">Illud autem in annullis obſervandum <lb></lb>eſt, quod faciliùs excurrit funis, ſi illi craſſiores fuerint & po<lb></lb>liti, quàm ſi exiles & aſperi. </s> </p> <figure id="id.017.01.648.1.jpg" xlink:href="017/01/648/1.jpg"></figure> <p type="main"> <s id="s.004681">Quod ſi annulis Trochleas propiùs æmulari placuerit, duos <lb></lb>annulos R & S alliga paxillo V, duóſque alios H & G ad<lb></lb>necte in M ponderi trahendo: Tum funem ductarium eidem <lb></lb>paxillo V alligatum trajice primùm per annulum G, deinde <lb></lb>per annulum S, hinc per annulum H, demum per annulum R. </s> <lb></lb> <s id="s.004682">Nam ſi extremitati I potentia trahens applicetur, movebitur <lb></lb>quadruplo velociùs, quàm pondus in M, adeóque etiam ha<lb></lb>bebit momentum quadruplum. </s> <s id="s.004683">Ne autem funes ob nimiam <lb></lb>propinquitatem ſibi invicem impedimento ſint ſe mutuo con<lb></lb>flictu atterentes, annulos tranſverſis bacillis ON, & LP dis<lb></lb>junge. </s> </p> <p type="main"> <s id="s.004684"><emph type="center"></emph>PROPOSITIO V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004685"><emph type="center"></emph><emph type="italics"></emph>Reſiſtentiam ex axium cum orbiculis conflictu in Trochleis <lb></lb>examinare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004686">QUoniam variarum Trochlearum uſum indicavimus, mo<lb></lb>dò trochleâ alterâ manente atque ſtabili, modò utra<lb></lb>que commotâ, placet hìc examinare propoſitas duas tro<lb></lb>chleas, an aliquid impedimenti afferant ex conflictu axium <lb></lb>cum orbiculis, aut etiam trochleas comparare cum annulis <pb pagenum="633" xlink:href="017/01/649.jpg"></pb>earum loco adhibitis, quantum videlicet præ trochleis afferat <lb></lb>impedimenti conflictus funis ductarij cum annulis. </s> </p> <p type="main"> <s id="s.004687">Sit libræ jugum AB æqualium brachiorum aginam cum <lb></lb>examine habens in C; <lb></lb><figure id="id.017.01.649.1.jpg" xlink:href="017/01/649/1.jpg"></figure><lb></lb>adnectatur in A tro<lb></lb>chlea ſuperior, ex qua <lb></lb>cum inferiore trochleâ <lb></lb>pendeat ſaxum F notæ <lb></lb>gravitatis, & funis <lb></lb>ductarij extremitas re<lb></lb>ligetur clavo in E. </s> <s id="s.004688">In<lb></lb>noteſcat autem tùm tro<lb></lb>chlearum ſingularum, <lb></lb>tùm funis ductarij gra<lb></lb>vitas, ut congruum pon<lb></lb>dus parari poſſit in B ap<lb></lb>pendendum. </s> <s id="s.004689">Clavus igi<lb></lb>tur E ſuſtinet inferioris <lb></lb>trochleæ & adnexi ſaxi <lb></lb>gravitatis partem quin<lb></lb>tam, reliquas quatuor <lb></lb>quintas partes, & præ<lb></lb>terea trochleæ ſuperio<lb></lb>ris, atque quatuor duc<lb></lb>tuum funis gravitatem <lb></lb>ſuſtinet brachium libræ <lb></lb>in A. </s> <s id="s.004690">Quare in B tantum ponderis apponendum eſt, quan<lb></lb>tum ſufficiat ad æquilibrium; proinde ſenſim augendum eſt <lb></lb>pondus in D, donec examen in C æqualitatem momentorum <lb></lb>indicet. </s> <s id="s.004691">Hoc peracto adde adhuc ponderi D aliam atque aliam <lb></lb>gravitatem, uſque dum brachium B deorſum inclinetur: hu<lb></lb>juſmodi enim additamentum indicabit reſiſtentiam ortam ex <lb></lb>conflictu axium cum orbiculis. </s> </p> <p type="main"> <s id="s.004692">Jam ſi trochlearum loco annulos ſubſtituas, eadémque me<lb></lb>thodo invento primùm æquilibrio, deinde factâ in D ponderis <lb></lb>acceſſione præponderantiam quæras, deprehendes, quanto <lb></lb>major reſiſtentia ex funis ductarij cum annulis affrictu oriatur, <lb></lb>quàm ex axium cum ſuis orbiculis conflictu in trochleis. </s> </p> <pb pagenum="634" xlink:href="017/01/650.jpg"></pb> <p type="main"> <s id="s.004693">Simili ratione ſi in A ſit clavus, cui ſuperior trochlea ſtabi<lb></lb>lis permanens affigatur; extremitas verò funis ductarij M ad<lb></lb>nectatur brachio libræ ad æquilibrium conſtituendum ſufficit <lb></lb>in N quinta pars gravitatis trochleæ inferioris unà cum ſaxo F, <lb></lb>& ductu funis MI. </s> <s id="s.004694">Facto igitur in H additamento gravitatis, <lb></lb>ut tollatur æquilibrium, indicabitur quanta reſiſtentiæ acceſſio <lb></lb>fiat ponderi F ex axium cum orbiculis conflictu: atque ſimili<lb></lb>ter repoſitis loco trochlearum annulis, poſt æquilibrium aucto <lb></lb>pondere N donec deprimatur, innoteſcet reſiſtentia orta ex fu<lb></lb>nis cum annulis affrictu. </s> </p> <p type="main"> <s id="s.004695">Hinc apparet primò ſatius eſſe hoc poſteriore modo ope<lb></lb>rari, quia longè minus pondus requiritur in N, quàm in D. </s> <lb></lb> <s id="s.004696">Secundò ad tollendum pondus F cum trochlea inferiore, ſi <lb></lb>ſuperior fixa maneat, tantam vim in potentiâ requiri, quan<lb></lb>tâ opus eſſet ad attollendum abſque ullâ machinâ pondus N <lb></lb>præponderans; ad attollendum verò idem ſaxum F cum <lb></lb>utrâque trochleâ, trahendo ſcilicet ſurſum trochleam A, <lb></lb>tantam vim exigi in potentia, quanta requiritur ad attol<lb></lb>lendum pondus D præponderans. </s> <s id="s.004697">Tertiò, retentis iiſdem tro<lb></lb>chleis, ſed mutato pondere F, examinari poſſe, an, & quan<lb></lb>to major reſiſtentia oriatur ex majore preſſione axium, <lb></lb>quando pondus eſt majus. </s> <s id="s.004698">Quartò. </s> <s id="s.004699">mutatis trochleis, & pon<lb></lb>dere eodem retento, diſparitatem aliquam inveniri, quia non <lb></lb>omnium trochlearum axes ſunt æquè teretes, ac politi, <lb></lb>& ſuorum orbiculorum foramini congruentes. </s> <s id="s.004700">Quod ſi, exa<lb></lb>mine hujuſmodi ſemel inſtituto, orbiculos manu pauliſper <lb></lb>convertas, & iterum idem examen inſtituas, neque æqua<lb></lb>lis inveniatur reſiſtentia, indicium erit foramen orbiculi, <lb></lb>aut fortaſſe etiam axem, non eſſe, exquiſitè rotundum. </s> <lb></lb> <s id="s.004701">Quintò. </s> <s id="s.004702">ſimili examine in annulis inito deprehendi poſſe, <lb></lb>an faciliùs ſuccedat tractio fune craſſiore, an verò te<lb></lb>nuiore. </s> </p> <p type="main"> <s id="s.004703">Non ita tamen neceſſe eſt indicatâ methodo uti, ut, ſi <lb></lb>non placeat jugum libræ æqualium brachiorum adhibere, <lb></lb>nequeas loco libræ ſtateram applicare ut trochleæ A, aut <lb></lb>funis extremitati M: primùm enim indicabitur æquili<lb></lb>brium: deinde longiùs reducto ſacomate, uſque dum appa<lb></lb>rere incipiat præponderatio, innoteſcet quantitas impedimen-<pb pagenum="635" xlink:href="017/01/651.jpg"></pb>ti, quin opus ſit gravitatem aliam atque aliam addere, ut <lb></lb>in librâ. </s> </p> <p type="main"> <s id="s.004704"><emph type="center"></emph>PROPOSITIO VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004705"><emph type="center"></emph><emph type="italics"></emph>Vim Retinaculi Trochlearum augere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004706">SÆpè contingit infixo parieti tigillo alligari ſuperiorem <lb></lb>trochleam; & niſi paries valdè firmus ac ſolidus fuerit, <lb></lb>cujuſmodi ſunt antiqui parietes, non leve periculum immi<lb></lb>net, ne ponderis vi labefactetur ipſe paries, maximè ſi recens <lb></lb>fuerit, & tigillus non admodum procul à ſummitate infigatur; <lb></lb>ut ſi recentis parietis BC <lb></lb><figure id="id.017.01.651.1.jpg" xlink:href="017/01/651/1.jpg"></figure><lb></lb>foramini immittatur tigillus <lb></lb>brevior AH, ex quo in A <lb></lb>dependet trochlea, & ex illâ <lb></lb>pondus cum reliquâ tro<lb></lb>chleâ: fieri enim poteſt, ut <lb></lb>tigillus ipſe quaſi Vectis à <lb></lb>pondere adnexo depreſſus at<lb></lb>tollat lateres impoſitos, & <lb></lb>ſuperioris parietis compa<lb></lb>gem diſſolvat. </s> <s id="s.004707">Foramen igi<lb></lb>tur ita fiat, ut paries pervius <lb></lb>ſit, illúmque pervadat lon<lb></lb>gior tigillus AF, cujus ca<lb></lb>put F fune FI connectatur <lb></lb>cum annulo in I parieti in<lb></lb>fixo: ſic enim ponderis gra<lb></lb>vitas nullam inferre poterit <lb></lb>parieti labem quamvis re<lb></lb>centi; & quò longior fuerit <lb></lb>tigilli pars HF ſupra partem HA, eò validius retinebitur tro<lb></lb>chlea in A, tigillo rationem Vectis habente. </s> </p> <pb pagenum="636" xlink:href="017/01/652.jpg"></pb> <p type="main"> <s id="s.004708"><emph type="center"></emph>PROPOSITIO VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004709"><emph type="center"></emph><emph type="italics"></emph>Trochleis vim Vectis augere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004710">QUamvis ex dictis obvium ſit Trochleas cum aliis Faculta<lb></lb>tibus componere, placet tamen hìc eas cum Vecte com<lb></lb>ponendas indicare. </s> <s id="s.004711">Sit prælum in torculari, ſed fortè diſſipa<lb></lb>ta fuerit Cochlea, qua illud deorſum trahebatur, aut certè ad <lb></lb>ſubitum uſum properato prælo utendum ſit: trabem ſtatue <lb></lb>tranſverſam, quæ altero capite retineatur objecto repagulo, ne <lb></lb>ſurſum attollatur. </s> <s id="s.004712">Vectis eſt ſecundi generis in medio habens <lb></lb>pondus premendum. </s> <s id="s.004713">Alteri trabis extremitati adnectatur tro<lb></lb>chlea, ejúſque compar in inferiore loco firmetur: nam funem <lb></lb>ductarium trahentes momentum habebunt, quod ex Ratione <lb></lb>Vectis, & ex Ratione Trochlearum componitur. </s> </p> <p type="main"> <s id="s.004714">Simili methodo utendum eſt, ſi Vecte ſecundi generis at<lb></lb>tollendum ſit pondus: loco enim potentiæ deſtinato, hoc eſt <lb></lb>Vectis extremitati attollendæ, adnectatur Trochlea, ejúſque <lb></lb>compar in ſuperiore loco firmetur: hìc enim pariter Vectis at<lb></lb>que Trochlearum Rationes componuntur. </s> <s id="s.004715">Quòd ſi funem Su<lb></lb>culâ traxeris, aut Ergatâ, tres erunt Rationes compoſitæ; dua<lb></lb>bus quippe illis addenda eſt Ratio Suculæ aut Ergatæ. <lb></lb><figure id="id.017.01.652.1.jpg" xlink:href="017/01/652/1.jpg"></figure></s> </p> <pb pagenum="637" xlink:href="017/01/653.jpg"></pb> <figure id="id.017.01.653.1.jpg" xlink:href="017/01/653/1.jpg"></figure> <p type="main"> <s id="s.004716"><emph type="center"></emph>MECHANICORUM <emph.end type="center"></emph.end><emph type="center"></emph>LIBER SEPTIMUS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004717"><emph type="center"></emph><emph type="italics"></emph>De Cuneo & Percuſsionibus.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004718">MECHANICARUM Facultatum Quarta ſpecies, <lb></lb>Cuneus, præſentem diſputationem exigit: neque <lb></lb>enim ſolam corporum gravitatem, ob id ipſum <lb></lb>quia gravitas eſt, vincere oportet, ac loco dimo<lb></lb>vere, quemadmodum Vecte, Axe, & Trochleis, <lb></lb>ſed etiam ſæpè conjunctas corporum partes, aut cohærentia <lb></lb>proximè corpora ſejungere atque divellere: id quod Cuneo <lb></lb>potiſſimùm perficimus, & iis, quæ ad Cunei rationem ſpecta<lb></lb>re videntur. </s> <s id="s.004719">Quoniam verò in iis, in quibus præcipuè Cu<lb></lb>nei vis elucet, percuſſione utimur, quæ ſanè in paulò lon<lb></lb>giorem ſermonem nos vocat, non erit abs re aliquanto latiùs <lb></lb>Percuſſionis naturam explicare, ut potentiæ Cuneo applicatæ <lb></lb>virtus manifeſta fiat. </s> <s id="s.004720">Quamquam non ſemper percuſſione <lb></lb>indigeat Cuneus, ſed non rarò impulſione contentus ſit, ne<lb></lb>que ſemper ad divellenda ea, quæ conjuncta ſunt, illo uta<lb></lb>mur, ſed aliquando etiam ad deprimendum, aut attollendum <lb></lb>corpus aliquod, ut ex ſequentibus patebit. </s> <s id="s.004721">Ea verò, quæ <lb></lb>Cunei figuram imitantur, quia in apicem deſinunt, ut trian<lb></lb>gula, ideóque Cunei nomine ſunt indicata ſæpiùs à Vitruvio <lb></lb>in Architecturæ libris, ad præſentem diſputationem non atti<lb></lb>nent; quemadmodum neque ſubſcudes, ſeu ſecuriclæ, quibus <lb></lb>arctè duæ tabulæ compinguntur; licèt enim cunei formam <lb></lb>imitentur, non tamen ſimilem, ſed cuneo oppoſitam effectio<lb></lb>nem habent, & inter retinacula connumerandæ ſunt. <pb pagenum="638" xlink:href="017/01/654.jpg"></pb></s> </p> <p type="main"> <s id="s.004722"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004723"><emph type="center"></emph><emph type="italics"></emph>Cunei forma, & vires explicantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004724">CUneus, quo ad ligna findenda communiter utimur, ſatis <lb></lb>notum eſt inſtrumentum, quod ex ampliori baſe faſtigia<lb></lb>tur in acumen deſinens; duas ſcilicet facies quadrangulas in li<lb></lb>neam coëuntes, & duobus triangulis hinc atque hinc connexas <lb></lb>ſuper quadrilateram baſim erigit, adeò ut ſcindendo corpori <lb></lb>acies ipſa applicetur, percuſſionem excipiat baſis. </s> <s id="s.004725">Nihil tamen <lb></lb>prohibet aliam cuneo figuram tribui: nam cum aliquando com<lb></lb>planandus eſſet colliculus, cujus creta arenæ permixta in lapi<lb></lb>deam quandam materiem concreverat, & ita obduruerat, ut li<lb></lb>gonibus ægerrimè cederet, parari juſſi longiuſculos ferreos cy<lb></lb>lindrulos digitorum duûm craſſitudine extremitate alterâ capi<lb></lb>tatos ad excipiendum mallei ictum, altera in planas duas ſuper<lb></lb>ficies compreſſos atque exacutos, qui juxta lapidis intervenia ap<lb></lb>plicati, ac tudite adacti lapidem penetrabant; qui demum rimas <lb></lb>agens diſſiliebat in fruſta ſatis conſpicua non pœnitendo labore. </s> <lb></lb> <s id="s.004726">Sed quæcumque demum figura cuneo ſtatuatur, illud omnibus <lb></lb>commune eſt, quod ex minori latitudine in majorem procedant, <lb></lb>ut quibus corporibus Cuneus inſeritur, magis atque magis alte<lb></lb>rum ab altero ſecedat, dum ille penitiùs adigitur. </s> </p> <p type="main"> <s id="s.004727">Nunc verò ſciſſionem tantiſper ſeponamus, & ſolum motum <lb></lb>corporis gravis ex cunei impulſione conſideremus, ſicuti ſi duro <lb></lb>plano incumbentem marmoreum cubum, cui vectis ſubjici ne<lb></lb>quiret, ut attolleretur, addacto cuneo disjungeremus à ſub<lb></lb>jecto plano: id quod faciliùs acutiore cuneo præſtatur, ut omnes <lb></lb><figure id="id.017.01.654.1.jpg" xlink:href="017/01/654/1.jpg"></figure><lb></lb>nôrunt, quàm ſi ille in mi<lb></lb>nus acutum angulum de<lb></lb>ſineret. </s> <s id="s.004728">Sit enim cubus H <lb></lb>marmoreus plano AB in<lb></lb>cumbens; & applicatus <lb></lb>cuneus DE, atque tudite <lb></lb>in D validè percuſſus ita <lb></lb>cubo ſubjiciatur, ut hic <pb pagenum="639" xlink:href="017/01/655.jpg"></pb>elevetur ad altitudinem FI. </s> <s id="s.004729">Cùm itaque citrà omnem dubi<lb></lb>tationem is, qui tuditem movet, eóque cuneum percutit, illo <lb></lb>eodem conatu nequeat cubum attollere ſinè cuneo, quæritur, <lb></lb>unde vis tanta illi accedat, ſpectatâ præcisè cunei figurâ, nihil <lb></lb>interim ad percuſſionem reſpiciendo. </s> </p> <p type="main"> <s id="s.004730">Qui Ariſtoteli Mechan. quæſt. </s> <s id="s.004731">17. in lateribus cunei ligno <lb></lb>ſcindendo interjecti duplicem Vectem agnoſcenti adhærent, <lb></lb>illicò ad Vectis Rationes confugient, quas in latere DE re<lb></lb>cognoſcere conabuntur. </s> <s id="s.004732">Verùm quærenti, primi ne? </s> <s id="s.004733">an ſe<lb></lb>cundi generis Vectis ſit DE? vix ſuppetet, quid reſpondeant. </s> <lb></lb> <s id="s.004734">Si primi generis, ut voluiſſe videtur Ariſtoteles; cum tria ſint <lb></lb>puncta D, & I, & E, atque primum D procul dubio potentiæ <lb></lb>aſcribatur; reliquum eſt medium punctum I eſſe hypomochlij, <lb></lb>extremum E ponderis. </s> <s id="s.004735">Atqui nihil prorſus apparet, quod cu<lb></lb>bum H applicet puncto E, à quo neque ſuſtinetur, neque tan<lb></lb>gitur: igitur vectis non eſt primi generis. </s> <s id="s.004736">Quod ſi in E pon<lb></lb>dus eſſe ultrò conceſſerim, illud certè ex hoc efficitur, atque <lb></lb>conſequens eſt, quod cuneo novâ percuſſione ulteriùs adacto, <lb></lb>& Potentia ad hypomochlium, ſcilicet D ad I, accedat, & pon<lb></lb>dus ab hypomochlio, ſcilicet E ab I, remotius fiat; igitur & <lb></lb>potentiæ momenta decreſcerent, & ponderis momenta auge<lb></lb>rentur; ac proinde hanc momentorum deceſſionem, & acceſ<lb></lb>ſionem, major movendi difficultas, & quidem notabilis atque <lb></lb>conſpicua, conſequeretur; quæ tamen cum experimentis non <lb></lb>conſentit. </s> <s id="s.004737">Adde in Vecte primi generis potentiam & pondus <lb></lb>oppoſitis motibus circa hypomochlium, quaſi circà centrum, <lb></lb>circulariter moveri; at hìc neque ullus intercedit circa <lb></lb>punctum I motus circularis, neque potentia deſcendit pondere <lb></lb>aſcendente. </s> </p> <p type="main"> <s id="s.004738">At fortaſſe, quod aliis magis placet, vectem ais eſſe ſecundi <lb></lb>generis; pondus quippe in I ſuſtinetur, & eſt potentiæ in D <lb></lb>exiſtenti proximum; quare hypomochlio relinquitur extre<lb></lb>mum punctum E. </s> <s id="s.004739">Id quidem aliquantò magis appoſitè dictum <lb></lb>videretur, ſi, quæ vectis Rationibus conveniunt, hìc quoquè <lb></lb>in Cuneo locum habere poſſent; Vectis ſiquidem quò longior <lb></lb>eſt, & potentia magis abeſt ab hypomochlio, cæteris paribus, <lb></lb>plus momenti tribuit potentiæ: at Cunei DE longitudo ſi au<lb></lb>geatur, manente eodem angulo IEF, eádemque diſtantiâ IE, <pb pagenum="640" xlink:href="017/01/656.jpg"></pb>nullam facit momentorum acceſſionem: nulla igitur ibi Vectis <lb></lb>Ratio intercedit. </s> <s id="s.004740">Contrà verò manente eadem cunei DE lon<lb></lb>gitudine, eademque diſtantiâ IE, diminuto autem, ſive aucto <lb></lb>angulo ad E eædem perſeverarent vectis Rationes; ergo & ea<lb></lb>dem momenta movendi: id tamen longiſſimè à vero abeſſe <lb></lb>manifeſtis docemur experimentis, nam major angulus ad E <lb></lb>movendi difficultatem auget, minor minuit. </s> <s id="s.004741">Cùm verò Cu<lb></lb>neus, ex hypotheſi, ſemper ſubjecto plano incumbat, utique <lb></lb>potentia in D ab eo ſemper æqualiter diſtat, & nunquam altiùs <lb></lb>aſſurgit, pondere tamen altiùs ſublato, quo magis illi cuneus <lb></lb>ſubjicitur: in Vecte autem ſecundi generis potentia aſcendit, ſi <lb></lb>pondus attollitur. </s> <s id="s.004742">Non igitur cuneus habet rationem vectis ſe<lb></lb>cundi generis; ſi maximè nullum hìc haberi circa hypomo<lb></lb>chlium, tanquam circa centrum, motum circularem potentiæ <lb></lb>animadvertas. </s> </p> <p type="main"> <s id="s.004743">Cùm itaque à Cuneo abſint Rationes Vectis, illius vires pe<lb></lb>tendæ ſunt ex eo, quod olim conſtitutum eſt, Facultatibus om<lb></lb>nibus Mechanicis communi principio: videlicet, quia Cunei <lb></lb>forma ea eſt, ut majore motu moveatur Potentia Cuneum im<lb></lb>pellens, quàm pondus à Cuneo repulſum ad latus; hoc minùs <lb></lb>reſiſtit, quàm ſi æquali motu cum potentiâ moveretur; atque <lb></lb>adeò impetus motum in potentiâ efficiens tantæ velocitatis, & <lb></lb>valens pari velocitate movere certum pondus, cui ineſſet æquè <lb></lb>intenſus ac in potentiâ, poterit in majori pondere entitativè <lb></lb>æqualis, ſed minùs intenſus efficere motum tardiorem pro Ra<lb></lb>tione minoris intenſionis, ita tamen, ut, quæ Ratio eſſet inten<lb></lb>ſionis majoris in minori pondere, ad intenſionem minorem in <lb></lb>majori pondere, ea pariter ſit Ratio majoris gravitatis ad mi<lb></lb>norem gravitatem; ſic enim contingit æqualem eſſe entitativè <lb></lb>motum tardiorem majoris ponderis, atque motum velociorem <lb></lb>minoris ponderis; quemadmodum aliàs in Vecte & in Tro<lb></lb>chleâ explicatum eſt. </s> <s id="s.004744">Quoniam igitur cuneus, vi impetûs im<lb></lb>preſſi à potentia, dum promovetur ſub pondus juxta lineam EF, <lb></lb>repellit pondus juxta lineam FI, linea EF motum potentiæ me<lb></lb>titur, linea autem FI motum ponderis. </s> <s id="s.004745">Atqui Cunei confor<lb></lb>matio hoc habet, ut in triangulo EFI minimus angulus ſit ad <lb></lb>apicem E; igitur per 19. lib. 1 minimum latus eſt FI, atque <lb></lb>proinde minùs movetur pondus per FI, quàm potentia per EF. <pb pagenum="641" xlink:href="017/01/657.jpg"></pb>Quanto igitur major eſt EF quàm FI, tanto majus eſſe poteſt <lb></lb>gravitatis momentum in I vincendum, quàm eſſet momentum <lb></lb>gravitatis propellendæ in E juxta directionem FE motûs po<lb></lb>tentiæ. </s> </p> <p type="main"> <s id="s.004746">Hinc planiſſimè conſtat, cur acutiores cunei majora habeant <lb></lb>movendi momenta, cæteris paribus. </s> <s id="s.004747">Fac enim angulum E eſſe <lb></lb>adhuc minorem, utique oppoſitum latus minus erit quàm FI; <lb></lb>eadem igitur linea EF ad lineam breviorem, quàm FI, habet <lb></lb>majorem Rationem, quàm ad eandem lineam FI; ac propterea <lb></lb>pondus adhuc multo tardiùs movetur quàm potentia, & pote<lb></lb>rit eſſe majus; vel, ſi majus non fuerit, potentia indigebit mino<lb></lb>re conatu, & faciliùs movebit. </s> </p> <p type="main"> <s id="s.004748">Obſerva autem (quantum quidem ex Cunei Ratione eſt) ut <lb></lb>ſe initium dederit, eandem ſemper eſſe facilitatem in proceſſu <lb></lb>motûs, quia eadem permanet Ratio motuum potentiæ cuneo <lb></lb>applicatæ, & ponderis: nam ex 4. lib. 6. ut EF ad FI, ita EG <lb></lb>ad CO, & IS, hoc eſt FC, ad SO, propter triangulorum ſi<lb></lb>militudinem, cùm ſit IS parallela ipſi EC. Quantum, inquam, <lb></lb>eſt ex Cunei Ratione; quandoquidem cubi H, dum manente <lb></lb>extremitate A elevatur ex I, momenta ſubinde variari, ſuo lo<lb></lb>co, ſuperiùs indicatum eſt. </s> <s id="s.004749">In ſcindendis autem corporibus, <lb></lb>prout variè contingit ſciſſio, aliquando peculiaris intercedere <lb></lb>poteſt cauſa faciliorem vel difficiliorem in proceſſu ſciſſionem <lb></lb>reddens. </s> </p> <p type="main"> <s id="s.004750">Et quidem in ſciſſione corporum vi cunei faciendâ non eſt <lb></lb>ita proclivè Geometricas leges perſequi, ad explicandam eo<lb></lb>rum reſiſtentiam: neque enim ſicut gravitas loco dimovenda <lb></lb>facilè innoteſcit, certámque ſub menſuram cadit, ita corpo<lb></lb>rum reſiſtentia, ne findantur, fieri poteſt manifeſta: Eſt ſiqui<lb></lb>dem ſciſſio partium conjunctarum ſeparatio; earum autem con<lb></lb>junctionem adeò variam eſſe contingit, ut certam legem ſubi<lb></lb>re nequeat. </s> <s id="s.004751">Nam quemadmodum inter lapides, ut monet Vi<lb></lb>truvius lib. 2. cap. 7. alij ita molles ſunt, ut etiam ſerra dentatâ, <lb></lb>quaſi ligna, ſecentur, immò ſecundùm oras maritimas ab ſal<lb></lb>ſugine exeſa diffluant, & in locis patentibus atque apertis, prui<lb></lb>ná & gelu frientur, ac diſſolvantur; alij duriores, ſed qui inter<lb></lb>veniorum vacuitates habeant, quapropter ab igne tuti non ſint, <lb></lb>quin rareſcente aëre vacuitatibus illis interjecto diſſiliant & diſ-<pb pagenum="642" xlink:href="017/01/658.jpg"></pb>ſipentur; alii ita ſpiſſis compactionibus ſolidati, ut neque ab <lb></lb>tempeſtatibus, neque ab ignis vehementiâ timeant: Ita pariter <lb></lb>inter ligna alia aliis ſolidiora ſunt, & unum præ alio faciliùs eſt <lb></lb>fiſſile, prout particulæ componentes craſſiores, aut tenuiores, <lb></lb>ſunt magis aut minus exquiſite permiſtæ, atque nimio, ſive <lb></lb>modico, ſive temperato humore concretæ, & prout juxta ſta<lb></lb>minum ductum, aut illa oblique ſecando, inſtituitur ſciſſio. </s> <lb></lb> <s id="s.004752">Sunt autem corpora illa (quantum quidem ad præſentem <lb></lb>tractationem ſpectat) partium ſeparationi difficiliùs obnoxia, <lb></lb>quorum materia ita probè ſubacta eſt, ut eorum elementa in <lb></lb>minutiſſimas particulas conciſa, & quaſi individua corpuſcula <lb></lb>in unam naturam inobſervabili permiſtione temperata coalue<lb></lb>rint eâ tantùm humoris copiâ, quæ ſatis fuerit ad illa firmiter <lb></lb>agglutinanda. </s> <s id="s.004753">Ex quo fit, ut hujuſmodi corpora ſolidiora ſint, <lb></lb>minúſque conſpicuas inanitates admittant, atque proinde, ſi <lb></lb>expoliantur, ſuperficiem induant lævem & undique æquabi<lb></lb>lem: id quod cæteris non accidit, quorum particulæ frequenti<lb></lb>bus hiatibus interciſæ, cùm aliæ emineant, aliæ ſuperentur, <lb></lb>ſemper aliquid habeant aſperitatis; quemadmodum animadver<lb></lb>tere poterit, quiſquis lapides cum marmoribus comparaverit. </s> </p> <p type="main"> <s id="s.004754">Si igitur ex ſolido corpore avellenda eſt particula aliqua, hæc <lb></lb>iſtáque disjungenda eſt à circumſtantibus particulis, quibus <lb></lb>conjungitur, neque fieri poteſt, ut illa moveatur, quin proxi<lb></lb>marum particularum aliæ motui oppoſitæ impellantur, aliæ <lb></lb>diſtrahantur; omnes autem ægrè à ſtatu ſibi ſecundùm natu<lb></lb>ram debito recedentes repugnant: quò verò plures particulæ <lb></lb>vim ſubire coguntur, eò major eſt reſiſtentia plurium quaſi col<lb></lb>latis viribus ſimul repugnantium. </s> <s id="s.004755">Hinc ſi duriora ligna ſecan<lb></lb>da offerantur, potior eſt uſus ſubtilioris ſerræ minutos denticu<lb></lb>los habentis; quia videlicet, quò exilior atque ſubtilior eſt den<lb></lb>ticulus, minorem particulam obviam habet, quam impellat, & <lb></lb>pauciores particulæ, à quibus ſeparetur, illam circunſtant; <lb></lb>ideóque ab iis faciliùs avellitur, quàm particula major, quæ à <lb></lb>pluribus disjungenda eſſet. </s> <s id="s.004756">Contra verò quorum particulæ le<lb></lb>vi impulſu divelluntur, quia non adeò dura ſunt, craſſiore ſer<lb></lb>râ facilè ſecantur, quæ in durioribus majorem reſiſtentiam in<lb></lb>veniens parùm utilis accideret. </s> <s id="s.004757">Hæc autem in limá pariter ob<lb></lb>ſervari poſſunt; quàm enim diſſipari aſperitate opus eſt in limâ, <pb pagenum="643" xlink:href="017/01/659.jpg"></pb>qua chalybs, aut qua lignum terendo expolitur? </s> <s id="s.004758">Sed & in mar<lb></lb>morum ſectione mirantur aliqui ſerras adhiberi nullis dentibus, <lb></lb>ſaltem conſpicuis, aſperas, non ſatis animum advertentes ad <lb></lb>arenas aquâ aſperſas, quæ hujuſmodi in opere interveniunt; ut <lb></lb>enim loquitur Plinius lib. 36. cap. 6. <emph type="italics"></emph>arená hoc fit, & ferro vide<lb></lb>tur fieri, ſerrá in prætenui lineâ premente arenas, verſandóque tractu <lb></lb>ipſo ſecante.<emph.end type="italics"></emph.end></s> <s id="s.004759"> Expedit autem ſubtiliore arenâ uti, <emph type="italics"></emph>craſſior enim are<lb></lb>na laxioribus ſegmentis terit, & plus erodit marmoris, majúſque <lb></lb>opus ſcabritiâ polituræ relinquit:<emph.end type="italics"></emph.end> Sunt ſcilicet arenæ granula tam <lb></lb>quæ premuntur, quàm quæ ſerræ adhærent, quaſi denticuli <lb></lb>mobiles mordacis limæ eodem ductu tùm cruſtarum faciem le<lb></lb>viter expolientis, tùm ſubjectum marmor ſecantis. </s> </p> <p type="main"> <s id="s.004760">Eſt autem manifeſtum non eâdem vi, qua ſuper lignum ſer<lb></lb>ram adducentes & reducentes ſciſſionem inchoamus, idem pa<lb></lb>riter obtineri, ſi cuneo lignum premamus citrà percuſſionem: <lb></lb>quia nimirum cuneo prementes urgemus in directum ſubjectas <lb></lb>ligni partes, quæ conjunctim reſiſtunt, ne comprimantur, at<lb></lb>que à lateribus cohærentes particulæ repugnant, ne diſtrahan<lb></lb>tur: ſerram verò ducentes obliquè urgemus ligni particulas <lb></lb>dentibus reſpondentes, ac proinde pauculæ illæ tantùm, quæ <lb></lb>urgentur, reſiſtunt compreſſioni, & illis attiguæ diſtractioni. </s> <lb></lb> <s id="s.004761">Hinc fit cultro faciliùs aliquid ſcindi, ſi illius aciem quamvis <lb></lb>hebetem & obtuſam adversùs corpus ſciſſile urgeas ſimul, atque <lb></lb>tranſverſam agas; quia particulæ à cultro preſſæ minorem in<lb></lb>veniunt reſiſtentiam in tranſverſo motu, ubi anteriores eâdem <lb></lb>cum poſterioribus directione moventur, nec ſibi adverſantur, <lb></lb>quàm ſi ſolo preſſu extimæ urgerent interiores, quæ comprimi <lb></lb>renuunt. </s> <s id="s.004762">Huc pariter referenda eſt cauſa, cur adeò valida con<lb></lb>tingat ſciſſio, ſi Harpe ictus infligatur: ſic hujuſmodi genere <lb></lb>enſis in ſummitate falcati, & in exteriore latere exacuti uſum <lb></lb>Perſeum in amputando Meduſæ capite, & Mercurium in occi<lb></lb>dendo Argo centoculo refert Ovidius lib. 5. Metam. <emph type="italics"></emph>Vertit in <lb></lb>hunc Harpem madefactam cæde Meduſæ:<emph.end type="italics"></emph.end> id enim non ex ſolâ <lb></lb>Harpes gravitate, ſed ex ipſo potiſſimùm flexu oritur, qui ef<lb></lb>ficit, ut dum vi impetùs deſcendit, acies etiam tranſverſo mo<lb></lb>tu ducatur ſupra partes corporis ſciſſilis; ex quo & facilior ſciſ<lb></lb>ſio. </s> <s id="s.004763">Sic quidam vulgari enſe, quo equitantes viatores non in <lb></lb>ſpeciem, ſed ad uſum, præcingi ſolent, vituli caput uno ictu <pb pagenum="644" xlink:href="017/01/660.jpg"></pb>amputabat; averſo ſcilicet ictu percutiens gladius circulum <lb></lb>deſcribebat, adeóque non ſolùm premendo ſecabat, verùm <lb></lb>etiam motu tranſverſo: quo in negotio dexteritate potiùs, <lb></lb>quàm viribus opus eſt. </s> <s id="s.004764">Quòd autem alij reſimum Harpes la<lb></lb>tus in canaliculum excavant, eíque aliquid argenti vivi in<lb></lb>dunt, quod à capulo ad cuſpidem excurrat, id faciunt ad per<lb></lb>cuſſionem augendam, quia tranſlata ad cuſpidem gravitate <lb></lb>Mercurij, etiam percuſſionis centrum transfertur longiùs à ca<lb></lb>pulo, ideóque ictus fit validior, accedente præſertim impetu, <lb></lb>quem Mercurius deſcendens concipit. </s> </p> <p type="main"> <s id="s.004765">Non eſt itaque comparandus gladius motu tranſverſo ſcin<lb></lb>dens cum ſerrâ ſecante; hæc enim obvias particulas in ſcabem <lb></lb>abeuntes ſenſim à ligno divellens illud demum ſublatis omni<lb></lb>bus intermediis particulis bifariam diviſum relinquit: ille ve<lb></lb>rò ſuâ acie premens atque penetrans, ſed nihil abradens, inter<lb></lb>ponitur partibus, quæ invicem ſeparantur. </s> <s id="s.004766">Motus ſerræ motui <lb></lb>particularum abſciſſarum planè æqualis eſt; dens quippe parti<lb></lb>culam, in quam incurrit, tangens impellit, ſuóque impulſu vin<lb></lb>cens nexum, quo particula ſibi cohærentibus jungebatur, illam <lb></lb>avellit. </s> <s id="s.004767">Tranſverſus verò gladij motus ſi comparetur cum mo<lb></lb>tu particularum compreſſarum, atque invicem divulſarum, <lb></lb>multò major eſt illo; nam culter manûs moventis motui obſe<lb></lb>cundat; & particulæ compreſſæ in latus recedunt; ac proinde <lb></lb>multo velociùs movetur potentia gladium adducens aut redu<lb></lb>cens, quàm id, cui hoc motu vis infertur: atque idcircò gladij <lb></lb>vis ſcindendi hoc motu refertur ad cuneum. </s> <s id="s.004768">Quòd ſi gladius <lb></lb>non motu tranſverſo ducatur ſuper id, quod ſcinditur, ſed om<lb></lb>nino motu recto preſſionis, ſit autem ferri craſſities ſenſim exte<lb></lb>nuata in aciem, quemadmodum cuneis omnibus commune eſt, <lb></lb>idem planè dicendum erit, quod de vulgari cuneo, cui nomen <lb></lb>hoc præcipuè inditum eſt. </s> </p> <p type="main"> <s id="s.004769">Quare Cuneus in corpus ſcindendum adactus conſiderandus <lb></lb>eſt ratione habitâ ipſius corporis, quod tenerum ac molle eſſe <lb></lb>poteſt, atque ita flexibile, ut ſequatur quocunque torqueas, aut <lb></lb>etiam durum, & minimè tractabile. </s> <s id="s.004770">Si molle illud ſit, immiſ<lb></lb>ſum cuneum recipit, séque illi accommodat, & comprimun<lb></lb>tur tùm ſubjectæ particulæ cunei aciem tangentes, tùm quæ à <lb></lb>lateribus cunei faciei congruunt: illæ omnino æqualiter pro-<pb pagenum="645" xlink:href="017/01/661.jpg"></pb>moventur, ac adigitur cuneus, neque motus earum à cuneo <lb></lb>pendet, quâ cuneus eſt, ſed quâ corpus eſt ſuâ mole objectam <lb></lb>molem trudens: hæ verò ad latus ſecedentes magis & magis, <lb></lb>prout cunei craſſitudo excreſcit, minori motu moventur, quàm <lb></lb>promoveatur immiſſus Cuneus; quandoquidem major eſt cunei <lb></lb>longitudo ejuſdem motum metiens, quàm craſſities dextras at<lb></lb>que ſiniſtras partes impellens. </s> <s id="s.004771">Sin autem durum ſit corpus, cui <lb></lb>cuneus inſeritur, illud quidem vix excogitari poteſt ex adeò <lb></lb>conſtipatis partibus inter ſe quàm aptiſſimè cohærentibus con<lb></lb>ſtare, ut nullius prorſus compreſſionis ſit capax, neque vel te<lb></lb>nuiſſimum cunei apicem admittat. </s> <s id="s.004772">Comprimuntur igitur ini<lb></lb>tio partes proximè cuneo ſubjectæ multò magis, quàm quæ la<lb></lb>teri adjacent; ſed propter duritiem certum compreſſionis mo<lb></lb>dum natura finivit, extra quem ſubjectas partes diffindi potiùs <lb></lb>patiatur, atque à ſe mutuò divelli. </s> <s id="s.004773">Qua in fiſſione ipſæ etiam <lb></lb>ſuperiores partes majori compreſſioni ſemper validiùs re<lb></lb>pugnantes, quò penitiùs adigitur cuneus, plurimum habent <lb></lb>momenti, ut cunei vis, quâ cuneus eſt, exerceatur: Quia vide<lb></lb>licet ſuperiores partes cum inferioribus connexæ, neque flexi<lb></lb>biles, dum ad latera cuneo urgente ſecedunt, cogunt pariter in<lb></lb>feriores ad dexteram & ad ſiniſtram recedere; atque propterea <lb></lb>quæ adhuc connexæ erant, diſtrahuntur ita, ut demum di<lb></lb>vellantur. </s> </p> <p type="main"> <s id="s.004774">Sit cuneus HI in ſubjectum lignum immiſſus inter B & C; <lb></lb>dum percuſſione urgetur <lb></lb><figure id="id.017.01.661.1.jpg" xlink:href="017/01/661/1.jpg"></figure><lb></lb>introrſum, partes B versùs <lb></lb>A, & partes C verſus D re<lb></lb>cedunt, & cum illis pariter <lb></lb>inferiores BM, atque CM: <lb></lb>ex quo fit partes in M con<lb></lb>nexas diſtrahi atque invi<lb></lb>cem divelli, & ſciſſionem <lb></lb>longiùs promoveri. </s> <s id="s.004775">Cum <lb></lb>igitur hoc ſit propoſitum cuneo ſcindere ſubjectum corpus, non <lb></lb>attendendus eſt ſimpliciter motus in B & C, ſed etiam qui in M <lb></lb>efficitur, ibi quippe ſciſſio contingit, ſemperque longiùs diſtat <lb></lb>à punctis B, & C, locus ſciſſionis, quò magis introrſum urge<lb></lb>tur cuneus. </s> <s id="s.004776">Ex quo fit attentè diſtinguendam eſſe facilitatem <pb pagenum="646" xlink:href="017/01/662.jpg"></pb>adigendi cunei, à facilitate ſcindendi: quandiu enim partes <lb></lb>faciem cunei tangentes non admodum repugnant compreſſio<lb></lb>ni, facilè cedunt adacto cuneo; ubi verò ulteriùs comprimi <lb></lb>renuunt, tota vis exercenda eſt in diſtractione partium ſejun<lb></lb>gendarum; quam diſtractionem quò majorem eſſe contingit, <lb></lb>augetur ſanè & cunei adigendi, & ſcindendi difficultas. </s> <s id="s.004777">Hæc <lb></lb>tamen alio ex capite minuitur, quia, quò magis punctum M, in <lb></lb>quo diſtrahendæ ſunt partes conjunctæ, abeſt à punctis B & C, <lb></lb>faciliùs conſequitur ſciſſio, nam, cæteris paribus, motus particu<lb></lb>larum, quæ ſejunguntur, minorem habet Rationem ad motum <lb></lb>punctorum B & C. </s> <s id="s.004778">Sic fuſtem craſſiorem ab alterâ extremitate <lb></lb>fiſſum juxta notabilem longitudinem ulteriùs findimus etiam <lb></lb>ſolis manibus eò faciliùs, quò longior fuerit prior ſciſſio: plu<lb></lb>rimum ſiquidem intereſt in lignis, quorum textura certum <lb></lb>quendam & rectum ſtaminum ordinem habet, utrùm juxta eo<lb></lb>rumdem ſtaminum ductum inſtituatur ſciſſio, an hæc obliquè <lb></lb>ſecentur; quemadmodum & in lapidibus præſtat cuneum in<lb></lb>terveniis applicare, ut faciliùs ſcindantur: propterea nodoſis <lb></lb>arborum partibus applicatus Cuneus ægrè illas findit, quia no<lb></lb>dorum ſtamina non recto tramite, ſed per anfractus & tortuosè <lb></lb>procedunt. </s> <s id="s.004779">In hac autem ligni ſciſſione ſi placeat tum particu<lb></lb>las, quæ in M diſtrahuntur, tum illis ſubjectas atque adhuc im<lb></lb>motas, puta in N conſiderare, atque veſtigium aliquod dupli<lb></lb>cis Vectis ſecundi generis recognoſcere, itaut commune hypo<lb></lb>mochlium ſit in N, longitudines vectium BN, & CN, poten<lb></lb>tia medio cuneo applicata in B & C, atque reſiſtentia vincen<lb></lb>da in M; non me difficilem præbebo: ſed & illud ſtatim ad<lb></lb>dam, non eſſe hunc duplicem illum Vectem, quem alij in Cu<lb></lb>neo quærunt; cum potiùs ſint duo vectes in diverſa impulſi ab <lb></lb>interjecto cuneo. </s> </p> <p type="main"> <s id="s.004780">Quapropter ex his; quæ latiùs explicare placuit, illud confi<lb></lb>citur, quod Cuneo aliquando reſiſtit gravitas, ut cùm ille cor<lb></lb>pori gravi elevando ſupponitur, aut cùm disjunctorum quidem <lb></lb>corporum gravium, ſed proximorum, ſaltem alterum remove<lb></lb>tur; aliquando reſiſtit partium nexus in M, qui niſi ſolvatur, <lb></lb>propelli nequeunt partes B & C cunei faciem tangentes: vis ſi<lb></lb>quidem cunei proximè exercetur adversùs B & C, & propter <lb></lb>partium connexionem etiam adversùs M, quamvis hoc poſtre-<pb pagenum="647" xlink:href="017/01/663.jpg"></pb>mum ſit ſcopus ſcindentis: quò autem validiùs partes in M <lb></lb>conjunguntur, etiam difficiliùs urgentur partes B & C: ligna <lb></lb>verò adhuc viridia, & lento humore plena, quia particulæ ma<lb></lb>jorem diſtractionem ferunt, nec facilè diſſiliunt, difficiliùs <lb></lb>ſcinduntur, quàm ligna arida. </s> </p> <p type="main"> <s id="s.004781">Quæ itaque de Cuneo vulgari dicta ſunt, facilè innoteſcit <lb></lb>ea pariter convenire forficibus, cultris enſibus, novaculis, ſcal<lb></lb>pris, dentibus hominis anterioribus, & ſimilibus, quibus ad <lb></lb>ſcindendum utimur; ſunt enim cunei diverſimodè juxta varios <lb></lb>uſus conformati; neque egent percuſſione, quia res ſcindendæ <lb></lb>non admodum reſiſtunt, & ſolus impulſus ſæpè ſufficit. </s> <s id="s.004782">Forfi<lb></lb>ces autem ſunt quidem cunei, ſed vectem conjunctum haben<lb></lb>tes, adeò ut potentia momenti augmentum acquirat ex Ratio<lb></lb>nibus vectis juxta diſtantias tùm potentiæ, tùm corporis ſcin<lb></lb>dendi, à clavo, ubi decuſſantur. </s> <s id="s.004783">An verò etiam ſcalpra, qui<lb></lb>bus Marmorarij aſſulas ex operibus dejiciunt; Cuneorum ratio<lb></lb>nem habeant, non admodum curo; videntur ſiquidem non in<lb></lb>cidere marmor, ſed partes ſuperfluas decutere: quod ſi per<lb></lb>cuſſi ſcalpri mucro penetrat, & dividit marmor, cuneus perin<lb></lb>de eſt atque ſcalpra, quibus lignum cælatur. </s> </p> <p type="main"> <s id="s.004784">Similiter acus, ſubulæ, aculei, clavi ad cuneum referun<lb></lb>tur: & quidem hujuſmodi corpora quò ſubtiliora ſunt, eò fa<lb></lb>ciliùs penetrant, quia immiſſa, & juxta ſuam longitudinem <lb></lb>progredientia valdè moventur interea, dum corporis perforan<lb></lb>di particulæ diſtrahendæ atque comprimendæ exiguo motu in <lb></lb>latera ſecedunt. </s> <s id="s.004785">Hinc conſtat, cur terebellâ paulo minore ape<lb></lb>riendum ſit foramen, cui immittatur clavus craſſiuſculus, ſi <lb></lb>præſertim lignum tenue ſit; ne videlicet immiſſo clavo tot par<lb></lb>tes adeò invicem comprimantur, ut ulteriorem compreſſionem <lb></lb>recuſantes cogant alias diſtrahi, ac demum rimâ factâ lignum <lb></lb>diſſiliat: ſublatis autem terebrâ particulis aliquot, reliquæ com<lb></lb>primendæ ut clavum arctè complectantur, cum pauciores ſint, <lb></lb>faciliùs compreſſionem ferunt citrà periculum fractionis aut <lb></lb>ſciſſionis ligni. </s> </p> <p type="main"> <s id="s.004786">Demum ſecuris, & gladius cæſim feriens, cuneus eſt, cui quo<lb></lb>dammodo junctus eſt tudes; illo ſiquidem percutimus: quid enim <lb></lb>intereſt, quod cuneum manentem tudite percutiamus, ſive cu<lb></lb>neo velociter moto percutiatur corpus ſcindendum? <pb pagenum="648" xlink:href="017/01/664.jpg"></pb></s> </p> <p type="main"> <s id="s.004787"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004788"><emph type="center"></emph><emph type="italics"></emph>Cunei inflexi uſus ad movendum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004789">PRæter vulgarem Cunei formam, quæ planis extremitatibus <lb></lb>circumſcribitur, ſi non ad ſcindendum, ſed ad movendum <lb></lb>adhibeatur, utilis eſſe poteſt Cuneus inflexus, ita ut, qua ſal<lb></lb>tem parte movendo corpori applicatur, faciem habeat non pla<lb></lb>nam, ſed inflexam, moveatur autem non circa ejuſdem circu<lb></lb>laris curvitatis centrum. </s> <s id="s.004790">In craſſiore tabulâ aſſumpto A puncto <lb></lb><figure id="id.017.01.664.1.jpg" xlink:href="017/01/664/1.jpg"></figure><lb></lb>tanquam centro deſcribatur pla<lb></lb>cito intervallo AB pars arcûs <lb></lb>circuli BC, ſive Quadrans, ſive <lb></lb>Quadrante minor, ſive major <lb></lb>fuerit. </s> <s id="s.004791">Tum centro alio aſſump<lb></lb>to, & majore aliquo intervallo, <lb></lb>alia circuli pars deſcribatur DC <lb></lb>occurrens priori arcui BC in <lb></lb>puncto C, ſive ibi ſe contingant, <lb></lb>ſive ſecent arcus, prout tibi <lb></lb>commodius acciderit. </s> <s id="s.004792">Reſecatis igitur ſupervacuis tabulæ par<lb></lb>tibus, & retentâ parte curvilineâ, habetur cuneus BCD in<lb></lb>flexus, qui ſuam vim exerceat non motu recto, ut cæteri cunei, <lb></lb>ſed curvo: propterea illi ad firmitatem addantur tranſverſaria <lb></lb>ab extremitatibus cunei exeuntia, & in unum punctum coëun<lb></lb>tia, circa quod, tanquam centrum, moveri poſſit cuneus. </s> <s id="s.004793">Ad <lb></lb>firmitatem, inquam, quia, ad motum, ſatis eſſet craſſiori ex<lb></lb>tremitati BD addere appendicem BM, in qua aſſumi poſſit <lb></lb>punctum, circa quod moveatur, quodcumque illud ſit, modò <lb></lb>non ſit centrum arcûs DC, ſi arcus ille impellat corpus moven<lb></lb>dum, neque punctum D minùs diſtet ab hujuſmodi centro mo<lb></lb>tûs, quàm punctum aliud extremum C. </s> <s id="s.004794">Contra verò ſi cunei <lb></lb>conatus exercendus ſit trahendo, & corpori applicetur arcus <lb></lb>BC, oportet centrum motûs minùs abeſſe ab extremitate B, <pb pagenum="649" xlink:href="017/01/665.jpg"></pb>quàm ab apice cunei C. </s> <s id="s.004795">Hùc ſcilicet ſpectare videntur ferrei <lb></lb>uncini, quibus duo corpora fibulantur, aut refibulantur, ut <lb></lb>cum armaria, feneſtræ, aut capſulæ clauduntur & recludun<lb></lb>tur. </s> <s id="s.004796">Quare intellectâ rectâ DM tanquam parte diametri cir<lb></lb>culi, cujus arcus exerceat vim cunei, ſi hic ſit arcus DC, <lb></lb>oportet ejus centrum inter extremitatem D, & punctum M <lb></lb>centrum motûs, interjacere; ſin autem ſit arcus BC, inter A <lb></lb>centrum circuli, & extremitatem B, oportet interjici centrum <lb></lb>motûs H: ab illo quippe centro M in primo caſu removeri <lb></lb>oportet corpus movendum, & ad hoc centrum H accedere <lb></lb>oportet corpus trahendum. </s> <s id="s.004797">Quod ſi recta DM non fuerit pars <lb></lb>diametri tranſeuntis per centra arcûs & motûs, ſaltem oportet <lb></lb>illam hujuſmodi diametro propiorem eſſe, quàm ſit recta ex <lb></lb>centro motûs ad C ducta; id quod ex 7.lib.3. manifeſtum eſt. </s> </p> <p type="main"> <s id="s.004798">Firmato itaque pro loci opportunitate centro motûs, & appli<lb></lb>catum ad impellendum pondus cuneum BCD urgens poten<lb></lb>tia in D, deſcribit circa M centrum motûs arcum circularem <lb></lb>DE, ſed pondus non propellitur niſi juxta differentiam linea<lb></lb>rum à punctis arcûs DC ad M centrum motûs ductarum. </s> <s id="s.004799">Ex <lb></lb>quo fit movendi facilitatem <lb></lb>ſubinde augeri, cæteris pa<lb></lb><figure id="id.017.01.665.1.jpg" xlink:href="017/01/665/1.jpg"></figure><lb></lb>ribus. </s> <s id="s.004800">Hoc autem ut pla<lb></lb>niùs explicetur, ſit arcus <lb></lb>LQ in quinque æquales <lb></lb>partes diviſus, & ſingulæ <lb></lb>ſint gr. 3. linea HL tran<lb></lb>ſeat per centrum I, & ex H <lb></lb>ducantur rectæ HM, HN, <lb></lb>HO &c; Certum eſt lineas <lb></lb>haſce omnes ex Q ad L <lb></lb>ſemper majores eſſe, & ma<lb></lb>ximam eſſe HL ex 7. lib. 3. atque aſſumptâ HF æquali ipſi <lb></lb>HQ, differentiam totam eſſe FL. </s> <s id="s.004801">Quare ſi LQ ſit latus cu<lb></lb>nei inflexi, potentia deſcribens arcum æqualem arcui LQ ha<lb></lb>beret motum, qui ad motum ponderis eſſet ut arcus LQ ad <lb></lb>rectam FL, ſeu QS. </s> <s id="s.004802">Sed quoniam centrum motûs eſt H, po<lb></lb>tentia circa illud deſcribit arcum LS, cujus quantitas inno<lb></lb>teſcit, ſi dato IL, hoc eſt IQ, Radio, atque diſtantiâ IH, <pb pagenum="650" xlink:href="017/01/666.jpg"></pb>cùm ex hypotheſi notus ſit angulus LIQ, inveſtigetur angu<lb></lb>lus IHQ, quem metitur arcus LS; hujus autem quantitas <lb></lb>prodit ex datis partibus Radij HL. </s> <s id="s.004803">Quare angulus LIQ ſit <lb></lb>gr. 15. IL partium 10000, IH partium 5000. Igitur in trian<lb></lb>gulo HIQ angulus IHQ eſt gr. 10. 0′. </s> <s id="s.004804">45″: atque ſi ex Cy<lb></lb>clometricis ineatur Ratio menſuræ arcûs LS, invenietur in <lb></lb>partibus Radij IL 10000 ferè par arcui <expan abbr="Lq;">LQ</expan> hic eſt partium <lb></lb>2618, ille 2621. Quapropter in tam exiguâ circuli portione <lb></lb>perinde eſt arcum LQ, atque arcum LS conſiderare. </s> <s id="s.004805">Singu<lb></lb>læ itaque partes quintæ arcûs deſcripti ſunt particularum 524. </s> </p> <p type="main"> <s id="s.004806">Jam verò per Trigonometriam, ex datis lateribus HI 5000, <lb></lb>& IM 10000, atque angulo comprehenſo HIM, ut pote ſup<lb></lb>plemento ad duos rectos noti anguli LIM ex hypotheſi gr. 3. <lb></lb>(ſimiliter in reliquis triangulis eadem ſunt latera, & angulus <lb></lb>comprehenſus ſenſim per gr. 3. minuitur) inveniatur linearum <lb></lb>longitudo; & eſt in iiſdem Radij IL partibus 10000 linea <lb></lb>HQ 14885, HP 14927, HO 14959, HN 14981, HM 14995, <lb></lb>atque demum HL 15000. Sunt igitur linearum ex Q ad L in<lb></lb>crementa inæqualia, videlicet 42, 32, 22, 14, 5, quibus reſpon<lb></lb>det motus ponderis cuneo propulſi, qui ſemper decreſcit, dum <lb></lb>potentia motus æquales perficit. </s> <s id="s.004807">Ratio proinde motûs potentiæ <lb></lb>ad motum ponderis initio, dum cunei pars QP ſubinde ponde<lb></lb>ri applicatur, atque Potentia venit ex L in M, eſt ut 524 ad 42, <lb></lb>deinde in PO ut 524 ad 32, in ON ut 524 ad 22, in NM ut <lb></lb>524 ad 14, in ML ut 524 ad 5. Cum itaque ſemper major fiat <lb></lb>Ratio motuum, augetur movendi facilitas; atque perinde fit, <lb></lb>ac ſi acutior ſemper atque acutior cuneus adhiberetur. </s> </p> <p type="main"> <s id="s.004808">Hìc tamen obſervandum eſt ita temperandum eſſe movendi <lb></lb>facilitatem cum ipſo ponderis motu, ut illam conſectando hoc <lb></lb>minùs moveri non contingat, quàm par fuerit: quò enim <lb></lb>punctum, quod eſt centrum motûs, minùs abeſt ab I centro <lb></lb>arcûs LQ, eò quidem faciliùs movetur pondus, quia ad hujus <lb></lb>motum potentiæ motus majorem habet Rationem, ſed à pon<lb></lb>dere minus ſpatium percurritur. </s> <s id="s.004809">Nam ſi centrum motûs ſit G, <lb></lb>& IG partium 2500, quarum IQ eſt 10000, linea GQ eſt <lb></lb>12432, GP 12456, GO 12475, GN 12489, GM 12497, <lb></lb>GL 12500: atque adeò linearum incrementa ſunt 24, 19, 14, 8, 3; <lb></lb>cum tamen quinta pars arcûs intervallo GL deſcripti à poten-<pb pagenum="651" xlink:href="017/01/667.jpg"></pb>tiâ ſit proximè 524; Major eſt autem Ratio 524 ad ſingula hæc <lb></lb>linearum incrementa, quàm cum motûs centrum eſt H. </s> <s id="s.004810">Cum <lb></lb>hac tamen movendi facilitate connectitur exiguus ponderis mo<lb></lb>tus; nam inter 12432 & 12500, quæ ſunt extremæ lineæ GQ <lb></lb>& GL, differentia 68 minor eſt quàm differentia 115 inter HQ <lb></lb>14885 & HL 15000, quæ differentia inter extremas lineas me<lb></lb>titur ponderis motum: eſt ſiquidem differentia inter aggrega<lb></lb>tum laterum & baſim trianguli HIQ, aut GIQ, menſura, juxta <lb></lb>quam pondus promovetur impulſu cunei. </s> <s id="s.004811">Cum verò IQ & IL, <lb></lb>utpote ſemidiametri, æquales ſint, baſis autem HQ major ſit baſi <lb></lb>GQ (nam in triangulo HGQ amblygonio baſis HQ opponi<lb></lb>tur majori angulo) fieri non poteſt, ut eadem ſit motûs ponderis <lb></lb><expan abbr="mẽſura">menſura</expan> æqualis ipſis FL aut QS, niſi ab aſſumpto motûs centro <lb></lb>deſcriptus arcus (intervallo uſque ad Q punctum illi centro pro<lb></lb>ximum) tranſeat per extremitates eaſdem Q & F, per quas tran<lb></lb>ſiret arcus ex H intervallo HQ deſcriptus. </s> <s id="s.004812">Quare duo circuli <lb></lb>ſe in duobus punctis ſecantes communem haberent rectam li<lb></lb>neam QF, ad quam bifariam ſectam in V perpendicularis VH <lb></lb>tranſiret per utriuſque circuli centrum, ex 3. lib. 3. ac proinde, <lb></lb>cum ex. </s> <s id="s.004813">5. lib.3. non habeant idem centrum H, alterius circuli <lb></lb>centrum eſſet extra rectam HI, puta in T. </s> </p> <p type="main"> <s id="s.004814">Utrùm autem dato eodem motûs centro, & datâ pari arcûs <lb></lb>portione, præſtet arcum eſſe majoris, an minoris, circuli partem, <lb></lb>vix eſt dubitandi locus. </s> <s id="s.004815">Quando enim duo circuli idem planum <lb></lb>in eodem puncto contingunt, peripheria majoris interjicitur in<lb></lb>ter planum datum & peripheriam minoris circuli; atque adeò <lb></lb>ab eodem motûs centro lineæ ad illam majoris circuli periphe<lb></lb>riam ductæ omnes ſecant peripheriam minoris, ac propterea, ut<lb></lb>pote longiores, majorem efficiunt preſſionem, longiúſque pro<lb></lb>pellunt corpus, quod impellitur. </s> <s id="s.004816">Hinc ſi viribus potentia abun<lb></lb>det, & ad majus ſpatium protrudere oporteat pondus, adhiben<lb></lb>da eſt peripheria majoris circuli; contra verò minore utendum <lb></lb>eſt, ſi parum movendum ſit, & potentia imbecillior. </s> </p> <p type="main"> <s id="s.004817">Porrò hìc exerceri cunei vires, quis ambigat? </s> <s id="s.004818">neque enim <lb></lb>admodum intereſt, plana-ne? </s> <s id="s.004819">an inflexa? </s> <s id="s.004820">ſit ejus facies, modò <lb></lb>ex ejus interjectu duo disjuncta corpora magis invicem remo<lb></lb>veantur, ſive utrumque ſimul in diverſas partes abeant, ſive al<lb></lb>tero manente, alterum tantummodo moveatur. </s> <s id="s.004821">Hìc autem im-<pb pagenum="652" xlink:href="017/01/668.jpg"></pb>motum manet centrum, circa quod vertitur portio circuli ex<lb></lb>centrici, quæ ſive ſimplici impulſione, ſive etiam percuſſione, <lb></lb>adacta urget corpus, quod contingit, neque aliter quàm ſi inter <lb></lb>validum ſtipitem humi defixum, atque pondus interjiceretur <lb></lb>vulgaris cuneus planus. </s> </p> <p type="main"> <s id="s.004822">Verùm quamvis hactenus potentiam in ipſa cunei inflexi ex<lb></lb>tremitate poſuerimus ad explicandum ejus motum, nihil tamen <lb></lb>refert: nam ſi etiam circa medium cuneum fuerit anſa, qua ar<lb></lb>reptâ ille valeat circumduci, perinde eſt; motus ſiquidem po<lb></lb>tentiæ ad ponderis motum eandem ſervat Rationem. </s> <s id="s.004823">Ex quo <lb></lb>manifeſtò deprehenditur nullam eſſe in Cuneo Vectis umbram; <lb></lb>in Vecte ſiquidem certus eſt potentiæ locus, quo mutato etiam <lb></lb>momenta variantur: at in hujuſmodi Cuneo non contingit mo<lb></lb>mentorum mutatio, cujuſcumque tandem cunei parti applice<lb></lb>tur potentia, dummodo ea ſit diſpoſitio, ut vires ſuas æquè exer<lb></lb>cere valeat, ſive in hac, ſive in illâ arcûs extremitate, hoc eſt ad <lb></lb>L aut Q, ſive circa medium ad O, aut N, conſtituatur. </s> <s id="s.004824">Cave ta<lb></lb>men putes æquè liberum eſſe in majore aut minore diſtantiâ à <lb></lb>centro motûs H aut G potentiam collocare: id enim ſanè per<lb></lb>peram fieret; pro Ratione ſiquidem diſtantiæ à centro motûs <lb></lb>majorem aut minorem arcum potentia ſuo motu deſcriberet: <lb></lb>eſto nihil interſit, cuinam parti applicetur, ſervatâ eâdem à præ<lb></lb>dicto motûs centro diſtantiâ. </s> <s id="s.004825">Propterea ſi ad Q applicetur po<lb></lb>tentia cuneum trahens, anſa ejuſmodi apponenda eſt ſurſum re<lb></lb>curva, cui applicata potentia non minorem arcum deſcribat, <lb></lb>quàm ſi illa applicaretur puncto L impellens cuneum: non eſt <lb></lb>ſcilicet par potentiæ motus, qui fit intervallo HQ, ac interval<lb></lb>lo HL. </s> <s id="s.004826">Similiter autem Cuneo plano uti licebit, cujus latera ſi <lb></lb>ferreo paxillo hinc atque hinc extante trajeceris, ut arreptis <lb></lb>utrâque manu paxilli extremitatibus cuneum adducere valëas, <lb></lb>aut impellere, duo corpora, quibus cuneus interjicitur, disjun<lb></lb>ges: immò ſi fiſſili ligno bicubitali juxta ſtaminum ductum cu<lb></lb>neum eumdem ita per vim immiſeris, ut cuneum elevatum ſe<lb></lb>quatur pariter & lignum, tùm ligni calce ſaxum percuſſeris, <lb></lb>cuneus ſciſſionem promovebit, quocumque tandem in loco ſive <lb></lb>juxta ipſius cunei apicem, ſive juxta baſim immiſſus fuerit pa<lb></lb>xillus ille, cui potentia applicatur. <pb pagenum="653" xlink:href="017/01/669.jpg"></pb></s> </p> <p type="main"> <s id="s.004827"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004828"><emph type="center"></emph><emph type="italics"></emph>Cuneus perpetuus circulo excentrico effingitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004829">CUneum Perpetuum voco non eum, qui perpetuâ, hoc eſt <lb></lb>majore ſemper atque majore impulſione corpus propellat <lb></lb>longiùs atque longiùs; id enim, ut ſatis clarum eſt, infinitam <lb></lb>exigeret cunei longitudinem: ſed eatenus dico <emph type="italics"></emph>perpetuum,<emph.end type="italics"></emph.end> qua<lb></lb>tenus potentia illi ſemel applicata inſtitutum motum juxta ean<lb></lb>dem directionem perpetuare poteſt: id quod neutiquam con<lb></lb>tingit ſecundùm rectam lineam, quæ ſpatium requireret infini<lb></lb>tum perpetuo illo motu percurrendum; ſed potentia in orbem <lb></lb>progrediens, & cuneum contorquens, motus alternos perficit. </s> </p> <p type="main"> <s id="s.004830">Et primùm quidem id fieri poteſt circulo, cujus motûs cen<lb></lb>trum abſit ab ejuſdem circuli centro; vis enim cunei erit ad <lb></lb>propellendum corpus intervallo duplo intervalli centrorum; <lb></lb>momentum verò potentiæ deſumetur ex ſemiperipheriâ circuli, <lb></lb>cujus Radius æqualis ſit dati circuli ſemidiametro auctæ cen<lb></lb>trorum illorum intervallo; ſi tamen extremitati à centro motûs <lb></lb>maximè diſtanti ipſa potentia applicetur. </s> </p> <p type="main"> <s id="s.004831">Sit datus circulus BED, cujus centrum A: fiat motûs cen<lb></lb>trum C, circa quod in gyrum <lb></lb><figure id="id.017.01.669.1.jpg" xlink:href="017/01/669/1.jpg"></figure><lb></lb>agatur conſtitutus circulus. </s> <lb></lb> <s id="s.004832">In hoc motu duo circuli con<lb></lb>centrici deſcribuntur; alter <lb></lb>quidem à puncto D, Radio <lb></lb>CD, alter verò à puncto B, <lb></lb>Radio CB. </s> <s id="s.004833">Quare dum po<lb></lb>tentia ex B per G venit in H, <lb></lb>punctum D per K venit in L, <lb></lb>& corpus, quod puncto D ap<lb></lb>plicitum erat, à peripheriâ <lb></lb>circuli BED ſenſim propel<lb></lb>litur, donec veniat ex D in H. </s> <lb></lb> <s id="s.004834">Eſt autem DH æqualis ipſi <lb></lb>BL, quia ex æqualibus CH & CB auferuntur æquales CD <pb pagenum="654" xlink:href="017/01/670.jpg"></pb>& CL: inter diametros verò BD & LD differentia eſt BL: <lb></lb>igitur quia diametrorum differentia dupla eſt differentiæ ſemi<lb></lb>diametrorum, BL eſt dupla ipſius AC differentiæ ſemidiame<lb></lb>trorum AD & CD. </s> <s id="s.004835">Quapropter etiam DH ſpatium, quod à <lb></lb>pondere propulſo percurritur, duplum eſt intervalli centrorum <lb></lb>AC Demùm potentia in B, ex hypotheſi, applicata momentum <lb></lb>habet juxta Rationem ſemiperipheriæ BGH ad ſpatium DH <lb></lb>duplum intervalli centrorum AC: hæc ſiquidem eſt Ratio mo<lb></lb>tuum potentiæ & ponderis. </s> <s id="s.004836">Hinc ſi ponatur dati circuli Ra<lb></lb>dius AB 100, & centrorum diſtantia AC 13, erit DH 26: At <lb></lb>ſemiperipheria BGH ad ſuum Radium BC eſt ut 355 ad 113; <lb></lb>igitur BGH ad DH eſt ut 355 ad 26. Quare, cæteris paribus, <lb></lb>quò majus eſt centrorum intervallum, eò majores requiruntur <lb></lb>in potentiâ vires; quia hujus intervalli duplum eſt ſpatium, per <lb></lb>quod impellitur pondus, manente eodem potentiæ motu. </s> </p> <p type="main"> <s id="s.004837">Eſt <expan abbr="autẽ">autem</expan> attentè <expan abbr="conſiderandũ">conſiderandum</expan>, utrùm præſtet, cæteris paribus, <lb></lb>majore circulo uti: Cæteris, inquam, paribus, ut ſcilicet idem ſit <lb></lb>centrorum intervallum, & eadem potentiæ à <expan abbr="cẽtro">centro</expan> motûs diſtan<lb></lb>tia. </s> <s id="s.004838">Et primò <expan abbr="obſervandũ">obſervandum</expan> eſt cuneum eſſe LBED, cujus vertex <lb></lb>eſt angulus <expan abbr="cõtingentiæ">contingentiæ</expan> factus à peripheriâ dati circuli, & à pe<lb></lb>ripheriâ circuli, quem circa centrum C in motu deſcribit extre<lb></lb>mitas D. </s> <s id="s.004839">Deinde ſemiperipheria BGH à potentiâ deſcripta in <lb></lb>motu (& eſt ex hypotheſi <expan abbr="partiũ">partium</expan> 355, <expan abbr="quarũ">quarum</expan> Radius CB eſt 113) <lb></lb>dividatur in partes æquales duodecim, ita ut ſingulæ reſpon<lb></lb>deant gradibus 15, & ſingulis competant partes 29 1/2. Similiter <lb></lb>ſemiperipheria DOL in 12 æquales partes ſingulas gr.15. divi<lb></lb>datur: adeò ut cùm linea BD circa punctum C circumacta an<lb></lb>gulum gr. 15 deſcripſerit, potentia ſit progreſſa per partes 29 1/2. <lb></lb>Examinandum eſt, quanto ſpatio interim propellatur pondus, <lb></lb>quod erat in D, versùs H. </s> </p> <p type="main"> <s id="s.004840">Sit angulus DCO, hoc eſt arcus DO, gr. 15: ducta intelli<lb></lb>gatur recta CO uſque in N peripheriam dati circuli; eſt igitur <lb></lb>ON motus ponderis ex D verſus H. </s> <s id="s.004841">Quapropter inveſtiganda <lb></lb>eſt ipſius CN longitudo, ut appareat ejuſdem exceſſus ſupra <lb></lb>CD. </s> <s id="s.004842">Ducatur dati circuli Radius AN notus partium 100; da<lb></lb>tur item intervallum AC partium 13; notus eſt angulus ACN <lb></lb>gr. 165: ergo per Trigonometriam innoteſcit primò angu<lb></lb>lus CNA gr.1.55′. 41″; atque ex eo reliquus angulus NAC <pb pagenum="655" xlink:href="017/01/671.jpg"></pb>gr.13.4′. 19″, hoc eſt arcus ND: deinde habetur longitudo <lb></lb>CN partium (87 38/100), quarum CO, hoc eſt CD eſt 87: igitur <lb></lb>ON eſt (38/100). Quod ſi arcus DO ponatur gr. 30, in triangulo <lb></lb>ACN dantur <expan abbr="eadẽ">eadem</expan> latera AN 100, & AC 13, & angulus ACN <lb></lb>gr. 150: igitur invenitur CNA gr.3. 43′. </s> <s id="s.004843">37″; atque angulus <lb></lb>NAC, hoc eſt arcus ND gr. 26. 16′. </s> <s id="s.004844">23″, & linea CN par<lb></lb>tium (88 52/100): igitur ON eſt part. (1 52/100). Atarcus DO ſit gr. 45; <lb></lb>eſt angulus ACN gr. 135: datis iiſdem lateribus AN 100, & <lb></lb>AC 13, invenitur angulus CNA gr. 5. 16′. </s> <s id="s.004845">27″, atque angu<lb></lb>lus NAC, hoc eſt arcus ND gr. 39. 43′. </s> <s id="s.004846">33″. & linea CN <lb></lb>part. (90 38/100): igitur ON part. (3 38/100). Similiter ſi DO ſit gr. 60: <lb></lb>invenitur ON part. (5 86/100); ſi verò fuerit gr. 75, eſt ON (8 84/100): ſi <lb></lb>gr. 90, eſt ON (12 15/100). Mutetur jam hypotheſis, & circuli dati <lb></lb>Radius ſit duplex, ſcilicet AD, hoc eſt AN, partium 200, <lb></lb>quarum AC eſt 13. Sit arcus DO gr. 15: invenitur angulus <lb></lb>CNA gr. 0. 57′. </s> <s id="s.004847">51″: atque angulus NAC gr. 14. 2′. </s> <s id="s.004848">9″: ac <lb></lb>proinde linea GN partium (187 40/100), quarum CD, hoc eſt CO, <lb></lb>eſt 187; quare ON eſt (40/100). Sit deinde arcus DO gr. 30: in<lb></lb>venitur angulus CNA gr. 1.5 1′.45″, & angulus NAC, hoc eſt <lb></lb>arcus DN, gr. 28. 8′. </s> <s id="s.004849">15″, atque demum linea CN part. (188 63/100): <lb></lb>igitur ON part. (1 63/100). Denique arcus DO ſit gr. 45: deprehen<lb></lb>ditur angulus CNA gr. 2. 38′. </s> <s id="s.004850">4″. angulus NAC, hoc eſt ar<lb></lb>cus DN, gr. 42, 21′. </s> <s id="s.004851">56″; & linea CN part. (190 59/100); atque <lb></lb>adeò ON part. (3 59/100). Si DO ſit gr. 60, ON eſt part. (6 18/100); ſi <lb></lb>DO ſit gr. 75, ON eſt (9 35/100). Si ſit gr. 90, ON eſt (12 57/100). </s> </p> <p type="main"> <s id="s.004852">Ex his manifeſtò conſtat initio motûs in primo quadrante à <lb></lb>circulo majore paulò ampliùs propelli pondus ex D versùs H, <lb></lb>quàm à circulo minore, datâ angulorum motûs ad centrum C <lb></lb>paritate. </s> <s id="s.004853">Verùm in circulo majoris diametri non ſolùm pari <lb></lb>graduum numero reſpondet longior arcus pro Ratione diame<lb></lb>trorum, ſed etiam, ut ex ſuperioribus calculis conſtat, major <lb></lb>circulus plures gradus ponderi coaptat, quàm minor. </s> <s id="s.004854">Sic in <lb></lb>motu ad centrum C gr. 15, circulo minori, cujus Radius 100, <lb></lb>competunt gr. 13. 4′. </s> <s id="s.004855">19″; at circulo majori, cujus Radius 200, <lb></lb>competunt gr. 14. 2′. </s> <s id="s.004856">9″. </s> <s id="s.004857">Quare præterquam quod duplex eſt <pb pagenum="656" xlink:href="017/01/672.jpg"></pb>longitudo arcus majoris, quia duplex eſt Radius, adhuc ſu<lb></lb>pereſt longitudo gr. 0. 57′. </s> <s id="s.004858">50″: cum tamen motus ponderis in <lb></lb>minore ſit (38/100), in majore (40/100); quod diſcrimen (2/100) longè minus eſt <lb></lb>illo exceſſu arcûs. </s> </p> <p type="main"> <s id="s.004859">Quapropter ſi Potentia peripheriæ dati circuli partibus ſub<lb></lb>inde applicetur (ut ſi extarent ad orbitam paxilli perpendicula<lb></lb>res) patet in majore circulo haberi majora momenta; multo <lb></lb>magis, ſi applicetur juxta maximam à motûs centro diſtantiam; <lb></lb>id quod fieri expedit, ſi nihil obſit: At ſi Potentia à centro mo<lb></lb>tûs æquè abſit in majore atque in minore circulo, non habetur <lb></lb>hoc momentorum compendium, quod ex diſtantia facilè obti<lb></lb>neri poſſet. </s> </p> <p type="main"> <s id="s.004860">Si itaque corpus ex D in H impulſum aut vi elaſticâ reſti<lb></lb>tuere ſe poſſit ex H in D, aut illud ſublevatum (ſi circulus <lb></lb>fuerit in plano Verticali) ſuâ gravitate deſcendere valeat ex H <lb></lb>in D, paulatim in priorem locum redibit, cum potentia tranſ<lb></lb>greſſa punctum H per I ſe reſtituet in B: potentia igitur perpe<lb></lb>tuò in gyrum circumactâ, circulum ſimiliter verſando, corpus <lb></lb>illud in motu reciprocando ſervabit conſtantiam. </s> <s id="s.004861">Quod ſi vir<lb></lb>tute elaſticâ præditum ſit corpus impulſum ex D in H, illa pa<lb></lb>riter, præter inſitam corpori gravitatem, movendi difficulta<lb></lb>tem augebit, quippe cui vis inferenda eſt, quam deinde excu<lb></lb>tere valeat. </s> <s id="s.004862">Quare ſatius fuerit omnem virtutem elaſticam amo<lb></lb>vere (ſi id quidem fieri poſſit) ut ſola gravitatis reſiſtentia ſu<lb></lb>peranda ſit. </s> </p> <p type="main"> <s id="s.004863">Ut autem corpus ultro citróque remeare poſſit ex D in H, & <lb></lb>viciſſim ex H in D, regula ſtatuatur in Verticali plano erecta, <lb></lb>ſed verſatilis circa axem, aut annulum alteri ejuſdem regulæ <lb></lb>extremitati infixum, & reliqua regulæ extremitas mobilis oc<lb></lb>currat circulo in B: Tum funiculo longitudine diametrum BD <lb></lb>æquante connectatur regula cum pondere movendo; ſic enim <lb></lb>fiet ut regulæ extremitas devenerit in L, quando pondus fue<lb></lb>rit in H; atque propellendo regulam ex L in B, pondus ex H <lb></lb>trahetur in D, & perpetua viciſſitudine tum regula, tùm pon<lb></lb>dus à circumacto cuneo impellentur: ſemper verò reſiſtentia <lb></lb>orietur ex ponderis modò impulſi, modò attracti gravitate; re<lb></lb>gula ſiquidem per ſe nihil obſiſtit, ſed quatenus cum pondere <lb></lb>trahendo conjungitur. </s> <s id="s.004864">Verùm qua poſitione collocandus ſit fu-<pb pagenum="657" xlink:href="017/01/673.jpg"></pb>niculus circuli diametro BD reſpondens, an ſupra, an infra cir<lb></lb>culum BED, quid opus eſt explicare? </s> <s id="s.004865">ſatis enim cuique mani<lb></lb>feſtum eſt attendendum eſſe, qua ratione ipſi circulo applicetur <lb></lb>potentia movens; nam ſi illum Potentia agitet paxillo in ſupe<lb></lb>riori aut inferiori facie extremæ orbitæ infixo, patet funiculum <lb></lb>adverſæ faciei reſpondere, ne in illum paxillus incurrat. </s> <s id="s.004866">Sin <lb></lb>autem potentia circulo non proximè adhæreat, nec illum tan<lb></lb>gat, quia ex centro C exeunti axi additum eſt manubrium cir<lb></lb>culo parallelum, aut Vectis, aut circulus alius eidem parallelus, <lb></lb>liberum erit funiculum alterutri circuli faciei reſpondentem <lb></lb>collocare, ex neutrâ ſcilicet parte impedimento eſſe poteſt <lb></lb>potentiæ ſe in gyrum contorquenti. </s> </p> <p type="main"> <s id="s.004867">Ne me verò carpendum puta, quòd integrum circulum <lb></lb>FBED propoſuerim, cum ſatis eſſe poſſit ſegmentum paulo <lb></lb>majus ſemicirculo BED (ut ſcilicet ſit locus axi infigendo in C <lb></lb>motûs centro) cui tota vis impellendi ſive pondus, ſive regu<lb></lb>lam, tribuenda eſt. </s> <s id="s.004868">Eo conſilio integrum circulum FBED <lb></lb>propoſui, ut liberum potentiæ ſit ſive in dexteram, ſive in ſi<lb></lb>niſtram motum inſtituere, atque promiſcuè uti modò cuneo <lb></lb>inflexo BED, modò BFD, prout commodius acciderit. </s> <s id="s.004869">Dein<lb></lb>de ſi ſemicirculo tantùm BED utamur, & vis elaſtica interve<lb></lb>niat, aut gravitas ſublevata recidat, ubi potentia venerit in H, <lb></lb>& ſemicirculi BED ſit facta poſitio HML, fieri non poteſt, ut <lb></lb>potentia versùs I procedat, quin illicò & quaſi momento pon<lb></lb>dus redeat ad D; hujuſmodi verò motus adeò velox vix contin<lb></lb>gere ſæpiùs poteſt citra aliquod detrimentrum; cui periculo <lb></lb>occurritur, ſi integer fuerit circulus FBED, ſenſim enim fit <lb></lb>regreſſus ex H in D. <lb></lb></s> </p> <p type="main"> <s id="s.004870"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004871"><emph type="center"></emph><emph type="italics"></emph>Ex Cylindro construi potest Cuneus perpetuus.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004872">ALtera ſpecies Cunei perpetui deſumi poterit ex Cylindro <lb></lb>Recto obliquè ſecto, erit ſiquidem ſectio Ellipſis; & potiſ-<pb pagenum="658" xlink:href="017/01/674.jpg"></pb>ſimùm inſerviet ad deprimendum, ſi cylindrus fuerit horizon<lb></lb>ti perpendicularis, atque addito vecte circumagatur Ergatæ in <lb></lb>morem: Sin autem cylindrus ſit horizonti parallelus, inſerviet <lb></lb>ad propellendum pondus, & Radios admittet quemadmodum <lb></lb>Sucula. </s> <s id="s.004873">Sit cylindrus BC Rectus, <lb></lb><figure id="id.017.01.674.1.jpg" xlink:href="017/01/674/1.jpg"></figure><lb></lb>cujus ſcilicet Axis eſt ad baſim <lb></lb>perpendicularis, & obliquè ſece<lb></lb>tur plano per DC; fit enim ſectio <lb></lb>DECF ellipſis. </s> <s id="s.004874">Quod ſi hujuſ<lb></lb>modi Ellipſis planities officere poſ<lb></lb>ſit motui, interiores partes aliquan<lb></lb>tulùm excavari oportebit, quando<lb></lb>quidem ſufficit limbus perimetri, <lb></lb>modò ſit ſolidus, & ſatis validus; <lb></lb>quem & ferreâ laminâ non ruditer <lb></lb>politâ munire operæ pretium fue<lb></lb>rit. </s> <s id="s.004875">Ut autem hujuſmodi ſectionis obliquitas major aut minor <lb></lb>opportunè fiat, ſtatuenda eſt primùm menſura depreſſionis aut <lb></lb>impulſionis, qua movendum eſt pondus, & ſit ex. </s> <s id="s.004876">gr. CA, cui <lb></lb>æqualis ſumatur ID: tùm ex D in C fiat ſectio, & erit conſti<lb></lb>tutus cuneus ADFC, cujus latus unum eſt cylindri ſemipe<lb></lb>ripheria AD, aliud DFC ſemiperimeter Ellipſis, cujus partes <lb></lb>ponderi in D conſtituto ſubinde applicantur ex convolutione <lb></lb>cylindri, & quatenus ab AD recedunt, pondus deprimunt, aut <lb></lb>impellunt, donec demum à puncto C attingatur pondus propul<lb></lb>ſum ex D in I. </s> </p> <p type="main"> <s id="s.004877">Quoniam verò Ellipticum limbum ferreâ laminâ muniendum <lb></lb>dixi, non omninò abs re fuerit indicare, qua methodo illius <lb></lb>perimetrum indagare poſſimus, ut laminæ in ellipticam figu<lb></lb>ram inflectendæ longitudo innoteſcat. </s> <s id="s.004878">In circulo quidem nota <lb></lb>eſt aliqua Ratio diametri ad peripheriam, ſive ut 7 ad 22, ſi<lb></lb>ve ut 71 ad 223, ſive ut 113 ad 355, ſive quæcumque alia ma<lb></lb>gis arrideat majoribus numeris explicata: at in Ellipſi nulla hu<lb></lb>juſmodi Ratio perimetri ad alterutrum Axem (quod quidem <lb></lb>ſciam) deprehenſa adhuc eſt: quapropter ad inveſtigandam <lb></lb>ejus perimetrum ex datis Axibus variæ tentatæ ſunt viæ, quas <lb></lb>inire nunc non eſt operæ pretium. </s> <s id="s.004879">Mihi hanc, utpote perbre<lb></lb>vem, nec à veritatis formâ, quantum res Phyſica patitur, re-<pb pagenum="659" xlink:href="017/01/675.jpg"></pb>cedentem ineundam ſuſcepi. </s> <s id="s.004880">Illud verò tanquam certum & <lb></lb>demonſtratum pono, quod Ellipſis, quæ ex Coni ſectione, ab <lb></lb>ea quæ ex Cylindri ſectione oritur, non differt, ut quidam mi<lb></lb>nùs attenti perperam exiſtimârunt; proinde quælibet oblata <lb></lb>Ellipſis ad aliquem Cylindrum ſpectare poteſt. </s> <s id="s.004881">Ad quem au<lb></lb>tem cylindrum illa pertineat, facile eſt determinare ex ipſius <lb></lb>Ellipſis Axe minori, qui eſt æqualis diametro Cylindri. </s> <s id="s.004882">Datis <lb></lb>igitur, Axibus, Majore, & Minore, invenire oportet, quan<lb></lb>ta ſit in hujuſmodi Cylindro obliquitas ſectionis Ellipſim <lb></lb>conſtituens: id quod obtinetur, ſi ex quadrato Axis Majoris <lb></lb>auferatur quadratum Axis Minoris; reſidui enim Radix qua<lb></lb>drata dabit in Cylindri latere longitudinem, qua diſtant inter <lb></lb>ſe duo plana baſi parallela, inter quæ intercipitur ſectio <lb></lb>obliqua. </s> </p> <p type="main"> <s id="s.004883">Sit data Ellipſis DECF, cujus major Axis DC 25, Minor <lb></lb>FE 20. Axi FE æqualis eſt cylindri diameter CI. </s> <s id="s.004884">Igitur pla<lb></lb>num per axem cylindri ductum habet cum plano obliquè ſe<lb></lb>cante communem ſectionem DC Axem Ellipſis, & cum cy<lb></lb>lindri baſe ſectionem facit CI, atque in ſuperficie dat latus DI. </s> <lb></lb> <s id="s.004885">Quare eſt triangulum rectangulum CID, cujus datur hypo<lb></lb>thenuſa DC 25, & baſis CI 20: ex ipſius DC quadrato 625 <lb></lb>ablatum quadratum ex CI 400, relinquit 225, quadratum per<lb></lb>pendiculi DI, quod propterea eſt 15. Plana igitur CI, & AD <lb></lb>parallela diſtant intervallo CI 15. Cum itaque ex baſis dia<lb></lb>metro CI 20 innoteſcat ejuſdem cylindricæ baſis periphe<lb></lb>ria (62 84/100) proximè, ſuperficies cylindrica ACID manifeſta eſt <lb></lb>(942 60/100), cujus ſemiſſem (471 30/100) dividit bifariam ſemiperimeter <lb></lb>Ellipſis DEC. </s> <s id="s.004886">Eſt igitur triangulum rectangulum, cujus late<lb></lb>ra circa rectum ſunt latus DI 15, & cylindricæ baſis ſemiperi<lb></lb>pheria CHI (31 42/100): horum quadrata 225, & (987 2164/10000) in ſum<lb></lb>mam colligantur, & quadrati (1212 2164/10000) Radix (34 82/100) ferè, eſt <lb></lb>Ellipſis ſemiperimeter DEC, integra verò DECF erit (69 64/100). </s> </p> <p type="main"> <s id="s.004887">Quapropter cum plana per AD & CI ex hypotheſi ſint pa<lb></lb>rallela, etiam DI, & AC æqualia ſunt latera. </s> <s id="s.004888">Igitur cum cy<lb></lb>lindri dati nota ſit diameter 20, atque adeò ſemiperipheria <lb></lb>AD (31 42/100), ſit data obliquitas, quam metitur AC 15, ex ſum<lb></lb>mâ quadratorum rectæ AC, & ſemiperipheriæ AD, cruatur <pb pagenum="660" xlink:href="017/01/676.jpg"></pb>Radix quadrata, & dabit ſemiperimetrum Ellipſis DFC proxt<lb></lb>mam veræ, quæ ex jam datis eſt illa eadem, quam paulo antè <lb></lb>invenimus (34 82/100). </s> </p> <p type="main"> <s id="s.004889">Hinc igitur innoteſcunt momenta hujuſmodi cunei, com<lb></lb>paratis inter ſe lineis, quæ definiunt motum ponderis atque <lb></lb>potentiæ; pondus enim movetur juxta lineam AC, potentia <lb></lb>autem juxta ſemiperipheriam cylindri, quatenus videlicet præ<lb></lb>cisè atque ſimpliciter ratione ipſius cunei motus illi convenit. </s> <lb></lb> <s id="s.004890">Verùm quia non facilè potentia applicatur proximè ſuperficiei <lb></lb>cylindri, & ſæpius expedit potentiæ momenta augere; propterea <lb></lb>Cylindro infigitur Vectis MN, cujus longitudo deſumitur à <lb></lb>puncto, ubi ille concurrit cum Axe Cylindri, uſque ad extre<lb></lb>mitatem N, cui potentia applicatur. </s> <s id="s.004891">Hæc autem longitudo, <lb></lb>manente eodem cuneo, varia omnino eſſe poteſt, atque adeò <lb></lb>potentiæ momenta repræſentabit ſemiperipheria circuli ab ex<lb></lb>tremitate N deſcripti, quæ comparanda erit cum motu ipſius <lb></lb>ponderis ab obliquitate ſectionis definito, ut dictum eſt. </s> </p> <p type="main"> <s id="s.004892">At ſubdubitare contingit, utrùm craſſiore, an graciliore cy<lb></lb>lindro uti expediat, manente eâdem obliquitatis menſurâ, at<lb></lb>que eâdem vectis longitudine; manet ſiquidem eadem mo<lb></lb>tuum Ratio; ſed augeri videtur corporum conflictus ex mutuo <lb></lb>tritu, nam in craſſiore cylindro major eſt elliptica ſemiperime<lb></lb>ter, quàm in tenuiore, ut manifeſtum eſt, ſi methodo paulò <lb></lb>antè indicatâ res ad calculos revocetur: quamobrem ex majore <lb></lb>hoc tritu augeri videtur difficultas movendi, cum maneat ea<lb></lb>dem corporis gravitas, eadem potentiæ virtus, eadem motuum <lb></lb>Ratio. </s> <s id="s.004893">Longè tamen aliter ſe res habet; quandoquidem duo<lb></lb>rum corporum ſe ſe invicem in motu contingentium conflictus, <lb></lb>qui ex ſuperficiei aſperitate oritur (hìc corporis unius conatum <lb></lb>adversùs aliud vi ſuæ gravitatis mente ſecernimus à conatu, <lb></lb>quo illud repellit præcisè vi ſuæ molis tanquam objectum im<lb></lb>pedimentum, etiamſi adversùs illud non gravitet) conſide<lb></lb>randus eſt, quatenus corpus impulſum adverſatur directioni <lb></lb>motûs corporis impellentis. </s> <s id="s.004894">Hinc eſt minimum eſſe conflictum, <lb></lb>ſi ambæ facies ſe in plano Verticali contingant, & alterutrum <lb></lb>corpus in eodem plano Verticali moveatur; nam reliquum cor<lb></lb>pus non repellitur, quia in illud non incurrit linea directionis <lb></lb>motûs alterius corporis, ſed ſolùm prominulæ utriuſque corpo-<pb pagenum="661" xlink:href="017/01/677.jpg"></pb>ris particulæ, quatenus aliæ in alias incurrunt, impediunt mo<lb></lb>tum pro earum magnitudine & numero: quoad impedimentum <lb></lb>maximâ ex parte tollitur, ſi pingui aliquo humore delibutæ fa<lb></lb>cies lubricæ fiant; replentur ſcilicet inanitates inter prominulas <lb></lb>particulas interjectæ, quas intercapedines ſubire non tam faci<lb></lb>lè poſſunt corporis proximi particulæ. </s> <s id="s.004895">Sic ſi integer eſſet cylin<lb></lb>drus, ſuâ baſi aut limbo CHI contingens ſubjectum corpus, <lb></lb>minimo tritu cum illo confligeret in motu circà ſuum Axem, <lb></lb>quia hujuſmodi motui non opponitur corpus illud in I poſitum. </s> <lb></lb> <s id="s.004896">At verò major eſt conflictus, quando directioni motûs illud ad<lb></lb>verſatur, ut cùm prope D eſſe intelligitur aliquâ ſui parte ſub<lb></lb>jectum cylindro, qui obliquè ſectus circumagi non poteſt, quin <lb></lb>urgeat illud ex D versùs I. </s> <s id="s.004897">Quò autem majore angulo planum <lb></lb>Ellipticum CEDF inclinatur ad baſis planum CHI, eò magis <lb></lb>converſioni cylindri adverſatur objectum corpus, adeóque ma<lb></lb>jor invenitur difficultas. </s> </p> <p type="main"> <s id="s.004898">Cùm itaque in majore cylindro, datâ æquali obliquitatis <lb></lb>menſurâ AC (æqualem obliquitatem non dico) planum obli<lb></lb>què ſecans minorem angulum cum plano baſis cylindri conſti<lb></lb>tuat, magiſque ad ipſam baſim accedat, minus habet reſiſten<lb></lb>tiæ ab objecto corpore, ſi particulæ ſingulæ conſiderentur, <lb></lb>quamvis cunctæ reſiſtentiæ ſimul collectæ demùm in æqualem <lb></lb>ſummam à menſura AC definitam coëant. </s> <s id="s.004899">Sit enim minoris cy<lb></lb>lindri ſemiperipheria RS, men<lb></lb>ſura obliquitatis RO, ſemiperi<lb></lb><figure id="id.017.01.677.1.jpg" xlink:href="017/01/677/1.jpg"></figure><lb></lb>meter Ellipſis SO: Manente au<lb></lb>tem eâdem RO, ſit majoris cy<lb></lb>lindri ſemiperipheria RT, & el<lb></lb>lipſis ſemiperimeter TO; utique <lb></lb>angulus RSO, utpote externus, <lb></lb>major eſt interno oppoſito RTO; ac proinde alternus SOX <lb></lb>major eſt alterno TOX, quibus angulis repræſentatur plani <lb></lb>obliquè ſecantis inclinatio ad baſim cylindri. </s> <s id="s.004900">Quamvis igitur <lb></lb>ex cylindri convolutione ſemiperimeter Ellipſis SO impellat <lb></lb>pondus juxta menſuram RO, & juxta eandem menſuram RO <lb></lb>impellatur pondus à ſemiperimetro Ellipſis TO; item ab illius <lb></lb>quadrante SP, atque hujus quadrante TQ, æqualiter impella<lb></lb>tur juxta menſuram MP, & NQ, quæ æquales ſunt (utraque <pb pagenum="662" xlink:href="017/01/678.jpg"></pb>ſcilicet eſt quadrans ipſius RO, ſiquidem propter triangulo<lb></lb>rum ſimilitudinem, ut OS ad PS, ita RO ad MP, & ut OT <lb></lb>ad QT, ita RO ad NQ; ſed ex hypotheſi OS eſt quadrupla <lb></lb>ipſius PS, ſicut OT eſt quadrupla ipſius QT; igitur RO eſt <lb></lb>quadrupla ipſius MP, & ipſius NQ, quæ propterea ſunt æqua<lb></lb>les) quia tamen QT major eſt quàm PS, qua Ratione RT ma<lb></lb>jor eſt quàm RS, & OT major quàm OS; idcircò eadem re<lb></lb>ſiſtentia diſtributa per plures particulas longioris QT, ſeu OT, <lb></lb>minor eſt in ſingulis particulis, quàm cùm diſtribuitur per pau<lb></lb>ciores particulas brevioris PS, ſeu OS. </s> <s id="s.004901">Minùs igitur TO ſuis <lb></lb>particulis ſubinde contingens corpus, quod impellit, cum eo <lb></lb>confligit, quàm confligat SO pertinens ad minorem cy<lb></lb>lindrum. </s> </p> <p type="main"> <s id="s.004902">Hujuſmodi Cuneum inflexum ex cylindro obliquè ſecto per<lb></lb><figure id="id.017.01.678.1.jpg" xlink:href="017/01/678/1.jpg"></figure><lb></lb>petuum eſſe in reciprocando motu, ſatis <lb></lb>manifeſtum eſt, ſi poſtquam pondus ex <lb></lb>D propulſum eſt in I, iterum redeat ad <lb></lb>D; abſolutâ enim cylindri converſione <lb></lb>iterum pondus ex D ad I propellitur. </s> <s id="s.004903">Id <lb></lb>autem ut fiat, ſtatuatur jugum KL cir<lb></lb>ca axem in V verſatile, ita tamen ut V <lb></lb>reſpondeat axi cylindri: tùm in L ad<lb></lb>nectatur pondus, quod ex D impelletur <lb></lb>in I, dum Cylindri dimidia revolutio ex <lb></lb>D per F in C perficietur: quia autem in reliquâ dimidiâ cylin<lb></lb>dri revolutione jugi extremitas K jam repulſa ſurſum versùs A, <lb></lb>impelletur iterum ad C, pondus reſtituetur ex I in D, atque ita <lb></lb>deinceps reciprocando impulſionem tum ponderis adnexi in L, <lb></lb>tùm extremitatis K. </s> <s id="s.004904">Hinc ſi ex laqueari pendeat ſtatua ventum <lb></lb>referens, & in ſpeciem volantis ingentes das expandens junctas <lb></lb>jugo KL, atque in ſuperiore conclavi adſit qui cylindrum cir<lb></lb>cumagat, alis reciprocantibus commovebitur aër, & aura exci<lb></lb>tabitur ad refrigerandum. </s> </p> <p type="main"> <s id="s.004905">Pro variis demum uſibus ſtatuetur cylindrus modò horizon<lb></lb>ti, tanquam Ergata, perpendicularis, modò velut Sucula, paral<lb></lb>lelus: Erítque expeditiſſima ejus converſio, ſi centrum Ellipſis <lb></lb>nulli polo innitatur, ſed cylindrus ipſe congruo loculamento <lb></lb>ita inſeratur, ut in eo ſit verſatilis, &, quam primò dederis, po-<pb pagenum="663" xlink:href="017/01/679.jpg"></pb>ſitionem deinceps ſervet, ac neutram in partem nutet. Id qui<lb></lb>dem paulò longiorem cylindrum exigit; non tamen eſt neceſ<lb></lb>ſe unius perpetuæ craſſitiei eſſe cylindrum; ſæpè enim nimis <lb></lb>craſſum atque incommodum eſſe contingeret; ſed fruſto cy<lb></lb>lindrico craſſiori obliquè ſecto firmiter inſeri poterit gracilior <lb></lb>cylindrus, ita ut axis axi conveniat, & rectam lineam conſti<lb></lb>tuant, atque hic in foramen immiſſus, in quo verſatilis eſt, dum <lb></lb>contorquetur, craſſiorem cylindrum pariter convolvit. <lb></lb> </s> </p> <p type="main"> <s id="s.004906"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004907"><emph type="center"></emph><emph type="italics"></emph>Cuneum perpetuum Circulus inclinatus imitatur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004908">TErtiam hanc cunei perpetui ſpeciem duabus ſuperioribus <lb></lb>adjicere non inutile fuerit, ut legenti hujus libri cap. ult. </s> <lb></lb> <s id="s.004909">prop. 6. conſtabit, quanquam fortaſſis alicui à ſuperiore parùm <lb></lb>diſtare videatur; ibi enim Ellipſim ex cylindri recti ſectione <lb></lb>obliquâ, hìc circulum ſuo quidem centro inſiſtentem, ſed in<lb></lb>clinatum proponimus. </s> <s id="s.004910">Ut autem res <lb></lb>clariùs exponatur, concipiamus circu<lb></lb><figure id="id.017.01.679.1.jpg" xlink:href="017/01/679/1.jpg"></figure><lb></lb>lum à plano horizontali RS ſectum <lb></lb>per centrum A, ita ut eorum commu<lb></lb>nis ſectio ſit diameter BC, ſemicircu<lb></lb>lus autem ſuperior ad horizontem in<lb></lb>clinatus ſit BDC; qui per AD bifa<lb></lb>riam ſecetur plano Verticali ad ſub<lb></lb>jectum planum RS horizontale recto; <lb></lb>sítque horum planorum communis <lb></lb>ſectio recta AE. </s> <s id="s.004911">Tum ex D Quadran<lb></lb>tis extremitate demittatur per 11.lib.11. perpendicularis ad ſub<lb></lb>jectum planum recta DE; quæ propterea ex defin. </s> <s id="s.004912">3. lib.11. fa<lb></lb>cit cum rectà AE angulum rectum. </s> <s id="s.004913">Accipiatur arcus DF, & <lb></lb>per F in circuli plano ductâ FG parallelâ ipſi AD, per eam <lb></lb>ductum intelligatur planum parallelum plano tranſeunti per <lb></lb>AD: Igitur planum horizontale plana illa parallela ſecans, per <lb></lb>16. lib. 11. facit ſectiones AE & GH parallelas, ac proinde, <lb></lb>cum duæ rectæ AD & AE duabus rectis GF & GH ſint pa-<pb pagenum="664" xlink:href="017/01/680.jpg"></pb>rallelæ, etiam per 10. lib. 11 anguli DAE, & FGH (cum illæ <lb></lb>ſint ſimiliter poſitæ) ſunt æquales. </s> <s id="s.004914">Jam ex F demittatur in ſub<lb></lb>jectum planum perpendicularis FH, quæ cum rectâ GH <lb></lb>conſtituit angulum rectum. </s> <s id="s.004915">Cum itaque duo triangula AED <lb></lb>& GHF rectangula habeant angulum DAE angulo FGH <lb></lb>æqualem, & reliquus reliquo æqualis eſt, atque ſimilia ſunt <lb></lb>triangula: Quapropter ut AD ad DE, ita GF ad FH, & per<lb></lb>mutando ut AD ad GF, ita DE ad FH. </s> <s id="s.004916">Eadem erit ratioci<lb></lb>natio in triangulo KLI ſimiliter facto, quod erit reliquis ſimi<lb></lb>le, & ut AD ad KI, ita DE ad IL: atque ita deinceps de cæ<lb></lb>teris omnibus triangulis, quæ efformari poſſunt à lineis paral<lb></lb>lelis Radio AD, tanquam hypothenuſis, & à perpendiculari<lb></lb>bus cadentibus in ſubjectum planum ex peripheriâ circuli in<lb></lb>clinati, & à rectis, quæ jungunt punctum, in quod cadit per<lb></lb>pendiculum, cum extremitate altera hypothenuſæ. </s> <s id="s.004917">Quoniam <lb></lb>verò omnes lineæ parallelæ Radio AD ſunt Sinus arcuum à <lb></lb>puncto C incipientium (ſic IK eſt Sinus arcûs IC, FG eſt Si<lb></lb>nus arcûs FC) omnium illarum Ratio manifeſta eſt ex Canone <lb></lb>Sinuum, ſi arcuum quantitas in gradibus data ſit, vel nota; <lb></lb>quare etiam nota eſt Ratio perpendiculorum DE, FH, IL. </s> <lb></lb> <s id="s.004918">Hæc autem, quæ de Quadrante DAC dicta ſunt, etiam de <lb></lb>reliquo Quadrante DAB intelliguntur; & quæ de hoc ſupe<lb></lb>riore ſemicirculo demonſtrata ſunt, etiam de inferiore ſemicir<lb></lb>culo vera ſunt, quatenus ille ad hoc idem planum horizontale <lb></lb>RS refertur, à quo circulus bifariam ſecatur. </s> </p> <p type="main"> <s id="s.004919">Jam verò integer circulus cum alio plano horizontali non <lb></lb>Secante, ſed Tangente circulum inclinatum in puncto infimo, <lb></lb>comparetur: ſunt autem duo hæc plana horizontalia invicem <lb></lb>parallela; & perpendiculum à puncto D cadens in planum ho<lb></lb>rizontale Tangens, eſt duplum perpendiculi DE, quemadmo<lb></lb>dum totius circuli diameter eſt dupla Radij AD. </s> <s id="s.004920">Quapropter <lb></lb>cum nota ſit dati circuli diameter ſecundùm certam menſuram, <lb></lb>& data ſit circuli inclinatio, ſive perpendiculi longitudo, qui <lb></lb>eſt Sinus anguli inclinationis, facile eſt invenire ſingularum <lb></lb>perpendicularium quantitatem. </s> <s id="s.004921">Nam ſi datur angulus inclina<lb></lb>tionis circuli ad planum Tangens (cùm hoc ſit parallelum pla<lb></lb>no Secanti) angulus ille æqualis eſt angulo DAE: quare ſicut <lb></lb>in triangulo DAE rectangulo datur hypothenuſa AD Radius <pb pagenum="665" xlink:href="017/01/681.jpg"></pb>circuli, & angulus acutus adjacens DAE, ex quibus inveni<lb></lb>tur latus DE, ita manifeſtum fit perpendiculum à ſummo cir<lb></lb>culi inclinati puncto D in planum horizontale Tangens, quod <lb></lb>eſt duplum lateris DE inventi. </s> </p> <p type="main"> <s id="s.004922">Sivè igitur detur puncti D à plano horizontali Tangente <lb></lb>diſtantia, ſivè inveniatur, diſtantia hæc bipartitò dividatur, <lb></lb>ejúſque medietas tribuatur perpendiculari DE. </s> <s id="s.004923">Tum Qua<lb></lb>drans DC in quotlibet partes æquales diviſus intelligatur, puta <lb></lb>in decem, & ex Canone accipiantur ſingulorum arcuum Sinus <lb></lb>gr. 81.72. 63.54. 45.36.27.18.9: deinde fiat ut Radius ad ſingu<lb></lb>los Sinus, ita notum perpendiculum DE ad aliud, & proveniet <lb></lb>ſingulorum perpendiculorum in planum Secans cadentium <lb></lb>menſura: quibus ſingillatim addenda eſt quantitas ipſius DE, <lb></lb>hoc eſt dimidia altitudo ſumma, ut habeatur ſingulorum alti<lb></lb>tudo ſuprà planum horizontale Tangens. </s> <s id="s.004924">Ponatur ſumma cir<lb></lb>culi elevatio à poſitione horizontali, palmi unius: igitur DE <lb></lb>eſt ſemipalmus, qui intelligatur diſtinctus in particulas 100.000, <lb></lb>adeóque totus palmus in part. </s> <s id="s.004925">200.000. Ideò in ſuperiori <lb></lb>Quadrante ſingulis perpendiculis addito ſemipalmo perpen<lb></lb>diculorum altitudo ea eſt, quam adjecta tabella exhibet. </s> </p> <p type="table"> <s id="s.004926">TABELLE WAR HIER</s> </p> <p type="main"> <s id="s.004927">Pro inferiori autem Quadrante ponendo gr. 90. in puncto con<lb></lb>tactûs circuli cum plano Tangente, lineæ perpendiculares ad <pb pagenum="666" xlink:href="017/01/682.jpg"></pb>planum Secans auferendæ ſunt à ſemiſſe datæ elevationis, hoc <lb></lb>eſt ex ſemipalmo part. </s> <s id="s.004928">100000, & reſiduum eſt diſtantia per<lb></lb>pendicularis à ſubjecto plano Tangente ſingulis partium <lb></lb>punctis reſpondens; quemadmodum adjectæ tabellæ pars alte<lb></lb>ra oſtendit. </s> </p> <p type="main"> <s id="s.004929">His fundamentis poſitis innititur ſpecies hæc cunei petita ex <lb></lb>circulo inclinato, qui eandem ſervans inclinationem circa <lb></lb>ſuum centrum convertitur. </s> <s id="s.004930">Sit enim circu<lb></lb><figure id="id.017.01.682.1.jpg" xlink:href="017/01/682/1.jpg"></figure><lb></lb>lus BFED, cujus centrum C, ad horizon<lb></lb>tem, ſive ad planum Verticale inclinatus, & <lb></lb>ſit in D pondus impellendum. </s> <s id="s.004931">Potentia in B <lb></lb>exiſtens, circulumque retinens in eâdem in<lb></lb>clinatione, ſi illum circa ſuum centrum C <lb></lb>circumagat, paulatim pondus impellit, prout <lb></lb>illud tangitur modò à puncto E, modò à <lb></lb>puncto F, donec demum à puncto B infimo <lb></lb>ad extremum motûs terminum deducatur. </s> <s id="s.004932">Quare potentiæ <lb></lb>motus definitur à ſemicirculi peripheriâ BFED, motus verò <lb></lb>ponderis à rectâ DH. </s> </p> <p type="main"> <s id="s.004933">Ut autem circulus in converſione eandem ſemper inclinatio<lb></lb>nem ſervet, fruſtum ligni G obliquè in parte ſuperiori ſectum, <lb></lb>inferiùs affigatur circulo (aut contrà in parte inferiori ſectum <lb></lb>ſuperiùs affigatur, prout commodius acciderit) atque in ligno <lb></lb>foramen fiat reſpondens circuli centro C, per quod foramen <lb></lb>tranſeat polus, cui centrum inſiſtit. </s> <s id="s.004934">Cum enim foramen illud <lb></lb>perpendiculare maneat ad horizontem, circulus eandem reti<lb></lb>net in converſione inclinationem. </s> </p> <p type="main"> <s id="s.004935">Verùm quia priùs ſtatuendum eſt ſpatium DH, per quod <lb></lb>ponderi commeandum eſt, quàm circuli amplitudo definiatur, <lb></lb>non ſolùm ut Potentiæ motus ad ponderis motum Rationem <lb></lb>habeat majorem pro circuli ſemiperipheriæ longitudine, ſed <lb></lb>etiam ut quàm minimum fieri poſſit, inclinatio ipſa recedat à <lb></lb>paralleliſmo cum plano, ad quod inclinari dicitur, ſive illud <lb></lb>horizontale ſit, ſive Verticale, quò enim minùs directioni mo<lb></lb>tûs potentiæ opponitur pondus, eò minùs reſiſtit: Propterea <lb></lb>data linea DH ſtatuatur ut Sinus anguli inclinationis, & Ra<lb></lb>dio reſpondebit diameter circuli opportuni. </s> <s id="s.004936">Sic ſi recta DH ſit <lb></lb>linea palmaris, & angulus inclinationis ponatur gr. 10: fiat ut <pb pagenum="667" xlink:href="017/01/683.jpg"></pb>gr.10 Sinus 17365 ad Radium 100000, ita 1 palmus ad pal<lb></lb>mos (5 76/100) ferè, quæ eſſet diameter circuli eam inclinationem <lb></lb>habentis; atque adeò motus potentiæ cum circulo in gyrum <lb></lb>actæ eſſet ad motum ponderis ſaltem noncuplus. </s> <s id="s.004937">Ex quibus ſa<lb></lb>tis apertum eſt ampliorem circulum præ minore utiliorem eſſe, <lb></lb>cæteris paribus. </s> </p> <p type="main"> <s id="s.004938">At circulum ipſum convolvere aut non placet, aut non licet, <lb></lb>quia fortaſſe pondus illius peripheriæ adnexum eſt, atque id<lb></lb>circò non niſi in gyrum pariter cum circulo ageretur. </s> <s id="s.004939">Idem <lb></lb>planè aſſequemur, ſi circulum horizonti, aut plano Verticali, <lb></lb>conſtitutum parallelum potentia urgeat in B; tùm puncto D <lb></lb>applicetur pondus; deinde potentia pergens in F & E percurrat <lb></lb>circuli ambitum illum deprimendo & inclinando; quandoqui<lb></lb>dem utroque modo mutatur diſtantia ponderis ab infimo <lb></lb>puncto circuli. </s> <s id="s.004940">Ponatur enim punctum F æquè diſtans à puncto <lb></lb>B, atque punctum E diſtat à puncto D. </s> <s id="s.004941">Si potentia manens ap<lb></lb>plicata eidem puncto B convertat circulum ita, ut ipſa poten<lb></lb>tia diſtet à pondere arcu BE, pondus non adnexum circulo <lb></lb>impellitur pro Ratione, quam arcus ille exigit: at verò ſi ma<lb></lb>nente pondere applicato ad punctum D, cui adnectitur, poten<lb></lb>tia pergat ex B in F, ſimiliter diſtat à pondere arcu FD, qui <lb></lb>eſt æqualis arcui BE; atque proinde æqualiter deprimitur, pro<lb></lb>ut idem arcus exigit, juxta ſuperius explicata, & in tabellâ ex<lb></lb>poſita; atque ita deinceps, donec potentia veniat in D: ſingulas <lb></lb>autem impulſiones metitur differentia perpendicularium. </s> </p> <p type="main"> <s id="s.004942">Hìc igitur ubi pondus circulo adnexum ponitur, manifeſta <lb></lb>eſt motûs reciprocatio: potentia ſiquidem ubi per F & E vene<lb></lb>rit in circuli punctum D, & impulerit pondus uſque in H, <lb></lb>percurrendo reliquum ſemicirculum DIB iterum retrahit pon<lb></lb>dus ex H in D. </s> <s id="s.004943">At quando pondus non connectitur cum circu<lb></lb>lo, & circulus ipſe convertitur, tunc opus eſt aliquo artificio, <lb></lb>ut pondus ex H remeet in D, quemadmodum indicatum eſt <lb></lb>capite ſuperiori. </s> </p> <p type="main"> <s id="s.004944">Porrò circulus iſte non convolutus, ſed à potentiâ ejus am<lb></lb>bitum percurrente ſecundùm alias atque alias partes inclinatus, <lb></lb>non eſt à Ratione Cunei excludendus; quandoquidem parùm <lb></lb>intereſt utrùm ſimili motu potentia atque organum moveantur, <lb></lb>an verò diſſimili motu. </s> <s id="s.004945">Quando potentia in eodem puncto B <pb pagenum="668" xlink:href="017/01/684.jpg"></pb>ſemper applicata in gyrum pergit, ſimili motu cum circulo in <lb></lb>gyrum acto movetur: quando verò potentia quidem circulari<lb></lb>ter movetur, ſed non ſecum rapit circulum, quem ſolummodò <lb></lb>inclinat, eſt quidem diverſus potentiæ motus à motu organi, <lb></lb>ſed ponderis motus idem planè efficitur in utroque caſu, & <lb></lb>æqualis eſt potentiæ ipſius motus determinatus à Rationibus <lb></lb>Cunei, quamvis hic non promoveatur, ſed ſolùm impellatur. <lb></lb></s> </p> <p type="main"> <s id="s.004946"><emph type="center"></emph>CAPUT VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004947"><emph type="center"></emph><emph type="italics"></emph>Vnde oriatur vis Percuſsionis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.004948">CUnei vires, quatenus ex ejus formâ proveniunt, hactenus <lb></lb>conſideravimus; nunc ad id, quod potiſſimum in hac <lb></lb>tractatione videtur, tranſeundum eſt, videlicet ad percuſſio<lb></lb>nem, qua dum adigitur Cuneus, multo faciliùs conſequitur mo<lb></lb>tus (ſive ſciſſio ſit, ſive ſimplex impulſio, citrà corporis diviſio<lb></lb>nem) quàm ſi onere impoſito prægravaretur, aut Vecte ſeu aliâ <lb></lb>qualibet Facultate augerentur Potentiæ momenta. </s> <s id="s.004949">Certè <lb></lb>Ariſtoteles Mechan. quæſt. </s> <s id="s.004950">19. quærit, <emph type="italics"></emph>Cur ſi quis ſuper lignum <lb></lb>magnam imponat ſecurim, deſupérque illi magnum adjiciat pondus, <lb></lb>ligni quippiam, quod curandum ſit, non dividit: Si verò ſecurim ex<lb></lb>tollens percutiat, illud ſcindit; cum alioquin multo minus habeas <lb></lb>ponderis id, quod percutit, quàm id quod ſuperjacet, & premit?<emph.end type="italics"></emph.end> Id <lb></lb>quod in cæteris quoquè percuſſionibus, ubi nulla intervenit <lb></lb>Cunei Ratio, manifeſtum eſt; quemadmodum in ſimplici com<lb></lb>preſſione, ut cùm lamella aurea in ſubtiliſſimam bracteolam di<lb></lb>ducitur repetitâ mallei percuſſione; quod enim, licèt immen<lb></lb>ſum, pondus vi ſuæ gravitatis tantumdem præſtare poſſet? </s> <s id="s.004951">Per<lb></lb>cuſſionis igitur natura inveſtiganda eſt, ut ejus vires in Cuneo <lb></lb>innoteſcant. </s> </p> <p type="main"> <s id="s.004952">Certum autem eſſe debet, & extra omnem controverſiam <lb></lb>poſitum nihil eſſe in hac rerum univerſitate, quod vacet cor<lb></lb>pore, ſed corporibus omnem obſideri locum, nullúmque eſſe <lb></lb>inane, in quod ſe recipere valeant, ac propterea corpora om-<pb pagenum="669" xlink:href="017/01/685.jpg"></pb>nia ita ſibi viciſſim ſuâ mole obſiſtere, ut nullum moveri va<lb></lb>leat, quin alterius in locum ſuccedat; quod proinde loco pelli <lb></lb>neceſſe eſt, quantum ſatis fuerit, ut ſubeunti corpori ſpatium <lb></lb>concedat; ſive id contingat, quia obſiſtens corpus inter an<lb></lb>guſtias deprehenſum ſe comprimi patiatur, ſivè quia diviſum <lb></lb>in latera ſecedat, ſive quia circumfuſa corpora circumpellat, <lb></lb>quæ abeuntis veſtigia ſequantur. </s> <s id="s.004953">Cum itaque nec omnia pla<lb></lb>nè corpora perpetuò quieſcant, nec omnia æquali prorſus agi<lb></lb>tatione commoveantur, fieri non poteſt, quin aliquibus vis <lb></lb>aliqua ſaltem aliquando inferatur, ſeu quia non licet diu juxta <lb></lb>naturæ inſtitutum quieta conſiſtere, ſeu quia externo pulſu ad <lb></lb>velociorem motum incitantur. </s> <s id="s.004954">Quare nullius corporis ex loco <lb></lb>in locum migratio excogitari poteſt, cui nullum aliud corpus <lb></lb>adverſetur & repugnet, vel ut ſuo ſe tutetur in loco juxta præ<lb></lb>ſcriptum à natura ordinem, vel ut partium nexum, & natura<lb></lb>lem earum poſitionem ſervet citrà diviſionem, aut compreſſio<lb></lb>nem, aut diſtractionem. </s> <s id="s.004955">Ex quo & illud conſequens eſt, quod <lb></lb>nullum reipſa (quicquid animo finxeris) quieſcit corpus ad <lb></lb>omnem omninò motum adeò indifferens, ut nihil prorſus re<lb></lb>tundat impetûs ab alio corpore commoto ſponte concepti, aut <lb></lb>extrinſecùs impreſſi: nullum quippe eſt, quod neque quicquam <lb></lb>habeat proni, neque ſurſum ſubvolare contendat, ſi diſparis ſe<lb></lb>cundùm ſpeciem gravitatis corpori permeabili proximum <lb></lb>conſiſtat, ubi fortè ordinem perturbari contigerit: ac propterea <lb></lb>ad motum indifferens cenſendum non eſt, niſi ut exquiſitam <lb></lb>circuli peripheriam circa centrum gravium percurrat externâ <lb></lb>vi impellente: id quod animo fingere facile eſt, opere exequi, <lb></lb>ut mitiſſimè loquar, difficillimum; certè ſemper incertum. </s> </p> <p type="main"> <s id="s.004956">Hoc verò diſcrimen eſt inter corpora (quantum quidem ad <lb></lb>præſentem diſputationem attinet) quod aliqua ita liberè fluunt, <lb></lb>ut nuſquam adhæreſcere videantur, quemadmodum aër, & ex<lb></lb>tenuatus vapor: Alia liquida & fuſa manant, atque labuntur, <lb></lb>ut aqua cæteríque humores, per quos tranſire & permeare li<lb></lb>cet, dirempti enim iterùm coëunt. </s> <s id="s.004957">Alia partibus conſtant, quæ <lb></lb>junctione aliquâ tenentur, & ſub certâ quidem conformatione, <lb></lb>atque figurâ conſiſtunt, quandiu nullo impellente urgentur; <lb></lb>quia tamen facilè comprimi queunt, in aliam figuram transfe<lb></lb>runtur; cujuſmodi ſunt lutum, cera, & reliqua mollia ac tene-<pb pagenum="670" xlink:href="017/01/686.jpg"></pb>ra, quæ aut ita tractabilia ſunt, ut quamcumque in formam <lb></lb>fingantur, aut ita flexibilia, ut ſequantur quocumque tor<lb></lb>queas: Alia demum ſolida & dura ſunt, quæ figuræ terminos, <lb></lb>quibus circumſcribuntur, non facilè mutant, & ſi fortè ſe ali<lb></lb>quatenus comprimi patiantur, priſtinam formam ſibi reparant: <lb></lb>Ex hiſce quatuor corporum generibus priora rationem medij <lb></lb>ſubire poſſunt, in quo reliquorum corporum motus exercean<lb></lb>tur, ut ex alio in alium locum commigrent; poſteriora, ſi unum <lb></lb>in aliud incurrat, aut ſi ſibi invicem occurrant, ea ſunt, per <lb></lb>quæ tranſitus non pateat, ſed aliorum corporum motui tantiſ<lb></lb>per mole ſuâ unumquodque obluctatur, dum pulſu externo re<lb></lb>moveatur. </s> </p> <p type="main"> <s id="s.004958">Porrò Impulſionem à Percuſſione diſtinguere opus eſt, niſi <lb></lb>vocabulis abuti velimus; quamvis enim utraque objecti corpo<lb></lb>ris reſiſtentiam inveniat, nemo tamen dixerit idem eſſe, ap<lb></lb>prehenſum manu Vectem impellendo, atque illum percutien<lb></lb>do deprimere, innatans aquæ lignum conto propellere, atque <lb></lb>inflicto ictu illud à ripâ longiùs abſtrahere, etiamſi æquè & <lb></lb>Vectis deprimatur, & lignum promoveatur. </s> <s id="s.004959">Simplex nimi<lb></lb>rum Impulſio nullum per ſe antecedentem corporis impellentis <lb></lb>motum exigit: at Percuſſio ob idipſum, quia Percuſſio eſt, cor<lb></lb>poris percutientis motum requirit, qui ipſorum corporum col<lb></lb>liſionem præcedat. </s> <s id="s.004960">Quare in Percuſſione intervenit inſtituti <lb></lb>jam & inchoati motûs interruptio ex novâ objecti corporis re<lb></lb>ſiſtentiâ. </s> <s id="s.004961">Hinc Ariſtoteles lib. 4. Meteor. ſumma 3. cap. 2. ait, <lb></lb><emph type="italics"></emph>Eſt autem Pulſio, motus à movente, qui fit à tactu; Percuſſio autem; <lb></lb>cum à latione.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.004962">Hæc omnia conjunctim Percuſſio poſtulat: Primò inchoa<lb></lb>tum eſſe jam & inſtitutum motum oportet: quamvis etenim <lb></lb>impulſio omnis vincat corporis urgendi aut ſcindendi reſiſten<lb></lb>tiam etiam primo motûs momento; quia tamen præcedens cor<lb></lb>poris impellentis motus, quo antè acceſſit ad corporis impulſi <lb></lb>contactum, quàm illud urgere incipiat, omnino præter Impul<lb></lb>ſionis naturam accidit (hæc ſi quidem eadem ſequetur, etiam <lb></lb>ſi priùs in mutuo contactu diutiſſimè quieſcant) propterea non <lb></lb>ſatis eſt reſiſtentiam invenire, ſed hanc inſtituto jam motui in<lb></lb>tervenire neceſſe eſt, ut ſit Percuſſio. </s> <s id="s.004963">Deinde, licèt præceſſe<lb></lb>rit motus, atque adhuc continuatus novam inveniat reſiſten-<pb pagenum="671" xlink:href="017/01/687.jpg"></pb>tiam, quia alias medij ejuſdem ſcindendi partes offendit; ſi ta<lb></lb>men æquabilis perſeveret priſtinam velocitatem aut tarditatem <lb></lb>nulla ex parte imminutam continenter ſervans, non eſt cenſen<lb></lb>da nova reſiſtentia; ſed quemadmodum continuus eſt idem <lb></lb>motus aliis atque aliis partibus ſibi ſuccedentibus, ita conti<lb></lb>nuatur eadem reſiſtentia, nec poſteriores medij partes percuti <lb></lb>dicuntur, ſed, ut priùs, præcisè impelli, aut ſcindi: quia vide<lb></lb>licet præcedens motus nihil confert ad novam hanc impulſio<lb></lb>nem prioribus omnino ſimilem. </s> <s id="s.004964">Quod ſi corpus in motu ob id<lb></lb>ipſum quia movetur, majorem atque majorem adhiberet cele<lb></lb>ritatem, adeò ut in medio, hoc eſt aëre ipſo, reſiſtentiæ mo<lb></lb>dus augeretur, facilè acquieſcam contendenti aërem verberari <lb></lb>& percuti: ſed quia nimis facilem ſe præbet aër ad hoc, ut <lb></lb>ſcindatur, non de hujuſmodi percuſſione medij, in quo fit mo<lb></lb>tus, mihi hìc eſt ſermo, ſed potiùs de percuſſione corporis, ad <lb></lb>quod per medium accedit corpus percutiens. </s> <s id="s.004965">Hinc aquam per<lb></lb>cuti non negaverim, quando enſis bonitatem examinaturi, <lb></lb>utrùm ſcilicet ritè & æquabiliter in chalybem temperatum ſit <lb></lb>ferrum, horizontalem aquæ ſtagnantis ſuperficiem plano gla<lb></lb>dio vehementer percutimus; per aërem ſcilicet, tanquam per <lb></lb>medium antecedentis motûs, ad aquam devenit gladius; quic<lb></lb>quid ſit, quod & ipſa aqua ad ulteriorem motum, quo enſis <lb></lb>profundiùs immergatur, medij rationem habere poſſit. </s> <s id="s.004966">Simi<lb></lb>liter aërem ipſum percuti à corpore, quod ex aquâ emergit, <lb></lb>haud ægrè conceſſerim, ſi id quidem ex vi præcedentis motûs <lb></lb>contingat: eſto, minùs obſiſtat aër, quàm aqua, obſiſtit tamen, <lb></lb>ſi à quiete dimoveatur, aut velociùs moveri cogatur, quàm mo<lb></lb>veretur, ſi hujuſmodi nova impulſio vi præcedentis motûs non <lb></lb>accideret: Neque aër, cum primùm emergens corpus in eum <lb></lb>incurrit, habet rationem medij, ſed perinde ſe habet primo il<lb></lb>lo momento, atque ſi tabella ſuſpenſa horizonti parallela fa<lb></lb>ciem aquæ proximè contingeret, & in eam ex aqua emergens <lb></lb>corpus incurreret; quanquam ab hac majorem, quàm ab aere, <lb></lb>reſiſtentiam ſubiret, atque adeo validiorem huic ictum infli<lb></lb>geret. </s> </p> <p type="main"> <s id="s.004967">Sed quid vocabula in quæſtionem fruſtra vocamus? </s> <s id="s.004968">De his <lb></lb>loquere, ut libet: per me ſanè licebit, quando corpus ab uno <lb></lb>fluido, per quod inchoatus eſt motus, ad aliud fluidum tranſit <pb pagenum="672" xlink:href="017/01/688.jpg"></pb>(ſive hoc magis, ſive minùs craſſum atque concretum fuerit) <lb></lb>hujus poſterioris fluidi primum contactum cum impulſione <lb></lb>Percuſſionem æquè appellare, atque ſi non fluidum eſſet, ſed <lb></lb>durum; validiùs ſcilicet impetitur vi antecedentis motûs, quàm <lb></lb>ſi corpus incurrens tunc primùm à quiete recederet. </s> <s id="s.004969">Hìc ſoli<lb></lb>dorum atque conſiſtentium corporum percuſſiones perſequi<lb></lb>mur, quarum vim inquirimus, & modum recipiunt à reſiſten<lb></lb>tiâ corporis percuſſi, quæ quò major eſt, validior quoquè cæte<lb></lb>ris paribus efficitur percuſſio. </s> <s id="s.004970">Sic ſi quis velit alteri alapam in<lb></lb>fligere, nullus erit ictus, ſi æquè velociter ad eaſdem partes <lb></lb>moveantur tum percutientis manus, tum is, cui deſtinata eſt <lb></lb>alapa; quia nulla eſt reſiſtentia motum manûs impediens, aut <lb></lb>retardans: erit verò ictus genere ipſo validiſſimus, ſi ſibi oc<lb></lb>currant, & quò majore impetu atque velocitate occurrent, eò <lb></lb>validior; quia nullum eſt majus reſiſtentiæ genus, quàm ſi duo <lb></lb>oppoſiti motus ſe invicem retundant. </s> <s id="s.004971">Quod ſi demum percu<lb></lb>tientis manus moveatur velociùs, quàm is, qui percutitur, <lb></lb>quamvis ad eaſdem partes moveantur, ictus infligetur validus <lb></lb>pro Ratione exceſsûs velocitatis, cui motus tardior reſiſtit, qua<lb></lb>tenus corpus tardum tandem à velociore deprehenditur, atque <lb></lb>urgetur: antè ictum verò ſi corpus percuſſum quieſcat, quo ve<lb></lb>locior erit percutientis motus, validior quoquè erit ictus; ad <lb></lb>eandem enim reſiſtentiam major motus habet majorem Ratio<lb></lb>nem, quàm minor. </s> </p> <p type="main"> <s id="s.004972">Vim igitur percuſſionis ex antecedenti motu originem duce<lb></lb>re manifeſtum videtur; non quidem quâ motus eſt ex loco in <lb></lb>locum tranſitus, hic enim ante corporum contactum ictum <lb></lb>nullum infligere poteſt, in ictu autem ipſo motus omnis præce<lb></lb>dens evanuit, nec jam extinctus quicquam efficere poteſt, <lb></lb>etiamſi motui præſenti vis aliqua efficiendi tribueretur. </s> <s id="s.004973">Sed <lb></lb>quia cum motu illo antecedente acquiſitus eſt impetus, qui <lb></lb>adhuc durans ipſo percuſſionis momento longè plus habet vi<lb></lb>rium, quàm ſi tunc omnino inciperet motus cum impulſione; <lb></lb>augetur ſiquidem in motu impetus ab eâdem cauſa movente <lb></lb>productus ſingulis momentis, ſemper enim ad agendum cauſa <lb></lb>neceſſaria applicata eſt, atque, ſi maneat, utilitate non caret <lb></lb>impetus, quem ſubſequi poteſt motus. </s> <s id="s.004974">Quid nimirum cauſæ <lb></lb>eſt, quare ligneus globus leniter aquæ impoſitus innataret, ſi <pb pagenum="673" xlink:href="017/01/689.jpg"></pb>verò ex editâ turri in ſubjectam foſſam dimittatur, aquam al<lb></lb>tiùs penetrat? </s> <s id="s.004975">niſi quia impetum in motu globus acquiſivit, <lb></lb>quo perſeverante terminos ſuæ gravitati à Naturâ præſcriptos <lb></lb>tranſilit, eóque demum langueſcente, aut illum aqua ſurſum <lb></lb>extrudet, aut vi ſuæ levitatis ſponte aſcendet. </s> <s id="s.004976">Sic citrà nota<lb></lb>bilem doloris ſenſum ſuſtinemus capiti impoſitum lapidem for<lb></lb>tè bipedalem, at non item ſcrupuli duorum digitorum ex alti<lb></lb>tudine centum cubitorum decidentis ictum ferre poſſumus ci<lb></lb>trà incommodum non ſanè leve: id quod ex acquiſito impetu <lb></lb>contingere palàm eſt, nulla quippe alia præter impetum in <lb></lb>promptu eſt cauſa, cui vis hæc efficiendi commodè, atque pro<lb></lb>babili conjecturâ, tribuenda ſit. </s> </p> <p type="main"> <s id="s.004977">Hunc impetum in motu acquiſitum <emph type="italics"></emph>Gravitatis<emph.end type="italics"></emph.end> nomine indi<lb></lb>gitare placuit Ariſtoteli, cùm propoſitæ quæſtioni 19. ſatisfa<lb></lb>cere contendens ait, <emph type="italics"></emph>An quia omnia cum motu fiunt, & grave <lb></lb>ipſum gravitatis magis aſſumit motum, dum movetur, quàm dum <lb></lb>quieſcit? </s> <s id="s.004978">Incumbens igitur connatam gravi motionem non movetur; <lb></lb>motum verò & ſecundùm hanc movetur, & ſecundùm eam, quæ eſt <lb></lb>percutientis.<emph.end type="italics"></emph.end></s> <s id="s.004979"> Neque enim adeò in rebus Phyſicis Ariſtotelem, <lb></lb>ejúſque peritiores aſſeclas cæcutiiſſe dixerim, ut gravitatem <lb></lb>corpori inſitam, quæ prima radix atque origo eſt, cui motus <lb></lb>debeatur, in ipſo motu revera augeri exiſtimaverint (quamvis <lb></lb>nullâ factâ naturæ, ſaltem conſtipatis partibus, mutatione) <lb></lb>haud ſecus, ac calori calor addatur. </s> <s id="s.004980">Sed idcirco plus gravitatis <lb></lb>aſſumi dicitur à corpore gravi dum movetur, quàm dum quieſ<lb></lb>cit, quia in motu vi ac poteſtate ſe movendi æquiparat corpora <lb></lb>graviora, atque adeò plus habet gravitatis non Formaliter, ſed <lb></lb>Virtualiter & Æquivalenter, ut ipſorum Peripateticorum vo<lb></lb>cabulis utar. </s> <s id="s.004981">Cæterùm <emph type="italics"></emph>gravitatis<emph.end type="italics"></emph.end> nomine non ipſum pondus <lb></lb>intelligi ab Ariſtotele ſuadet ipſa loquendi formula, qua gravi<lb></lb>tatem aſſumptam dum movetur, confert cum gravitate aſſump<lb></lb>tâ dum quieſcit, ut hæc illâ minor cenſeatur: videtur enim <lb></lb>Ariſtoteles in corpore gravi ad motum prono agnoſcere aſ<lb></lb>ſumptum impetum, quo fieret connata ipſi corpori gravi mo<lb></lb>tio, niſi impediretur, dum quieſcit, & præter hunc impetum, <lb></lb>alium in motu acquiſitum, adeò ut demum utroque impetu <lb></lb>moveatur, ac proinde dicatur in motu plus aſſumere gravitatis. </s> </p> <p type="main"> <s id="s.004982">Quòd ſi hæc philoſophandi ratio placeat, qua corpori gravi <pb pagenum="674" xlink:href="017/01/690.jpg"></pb>idcircò ſaltem ad ſpeciem quieſcenti, quòd removere non va<lb></lb>leat ea, quæ obſtant, & motum, qui ſub ſenſum cadat, impe<lb></lb>diunt, conceditur impetus Innatus, qui ſit ipſa actualis gravi<lb></lb>tatio inſitæ gravitati addita, reipſa connitens aut adversùs ſub<lb></lb>jectum corpus, aut contra vim ſuſpendentem: Cùm gravitas <lb></lb>in motu alium atque alium adhibeat novum conatum ad <lb></lb>deſcendendum, perinde videtur contingere, ac ſi toties mul<lb></lb>tiplicata fuiſſet eadem gravitas, quoties multiplicatus fuit co<lb></lb>natus priori illi æqualis. </s> <s id="s.004983">Hac autem ratione non ineptè dixit <lb></lb>Ariſtoteles grave ipſum aſſumere plus gravitatis in motu, quia <lb></lb>ſublato motûs impedimento & impetus Innatus ſuas omnes vi<lb></lb>res exerit, & augetur Acquiſito: ac propterea minor gravitas <lb></lb>ſic æquivalenter multiplicata longè plus efficit, quàm ſi major <lb></lb>gravitas incumberet, cujus impetus Innatus impediretur, ne <lb></lb>motum efficeret ullum neque compreſſionis corporis ſubjecti, <lb></lb>neque diſtentionis corporis ſuſpendentis. </s> <s id="s.004984">Quando autem nul<lb></lb>lus omnino motus, ſivè qui ſub ſenſum cadat, ſive qui aciem <lb></lb>omnem fugiat, tribuitur corpori gravi, nullus quoquè ſuperad<lb></lb>ditus Impetus ipſi connatæ gravitati reſpondens concedendus <lb></lb>eſt; neque enim deorſum reipſa connititur; quamvis in ſe ha<lb></lb>beat principium & originem gravitandi, ſi impedimentum ſal<lb></lb>tem ex parte removeatur. </s> </p> <p type="main"> <s id="s.004985">Cùm itaque omne id, quod percuſſionis ictum conſequitur, <lb></lb>ab impetu oriatur, neque impetum gravitas, aut ulla moven<lb></lb>di facultas, concipere valeat, quin aliquo ſaltem motu ipſa mo<lb></lb>veatur; nil mirum, ſi ingens moles prohibita, ne prorsùs mo<lb></lb>veatur, nullam labem inferat ſubjecto lapidi, quem minor gra<lb></lb>vitas cadens, atque percutiens in fruſta comminuit: minor ſci<lb></lb>licet gravitas liberè deſcendens multum concipit impetum, <lb></lb>quem lapidi percuſſo communicans cogit eum in fruſta diſſili<lb></lb>re, ſi vis impetûs ſuperet partium nexum, aut ſaltem cum con<lb></lb>cutit. </s> <s id="s.004986">Nullum autem effectum impetûs ab ingenti mole prorsùs <lb></lb>quieſcente expectare poſſumus, quippe quæ nullum imprime<lb></lb>re poteſt impetum ſubjecto corpori. </s> </p> <p type="main"> <s id="s.004987">Hinc mirari ceſſent, qui plumbeum globulum primo mallei <lb></lb>ictu certam compreſſionem pati obſervant, ſecundo verò ictu <lb></lb>priori omninò æquali adhuc magis comprimi, quamvis minore <lb></lb>compreſſione, quia particulæ jam per vim conſtipatæ validiùs <pb pagenum="675" xlink:href="017/01/691.jpg"></pb>rejiciunt majorem violentiam. </s> <s id="s.004988">At ſi globulum ſimilem ſubji<lb></lb>ciant ponderi, quod illum æquè comprimat, ac prior ictus mal<lb></lb>lei, addito adhuc æquali pondere non ſequitur compreſſio glo<lb></lb>buli tanta, quæ reſpondeat ſecundo ictui mallei: ex quo ſatis <lb></lb>conſtat duplicis percuſſionis vires non æquari à duplici gravita<lb></lb>te. </s> <s id="s.004989">Si enim animum attentè advertant, videbunt mallei mo<lb></lb>tus tam in primâ quàm in ſecunda percuſſione planè æquales <lb></lb>eſſe, tùm ratione velocitatis, tùm ratione ſpatij, ac proinde <lb></lb>æquali impetu malleum percutere: At factâ jam primâ ſubjecti <lb></lb>globuli compreſſione, in qua gravitas incumbens motum ha<lb></lb>buit illi compreſſioni reſpondentem, manifeſtum eſt, propter <lb></lb>majorem globuli jam compreſſi reſiſtentiam, non poſſe ſecun<lb></lb>dam gravitatem priori æqualem additam æquali motu, nec <lb></lb>æquali velocitate moveri, atque propterea neque poſſe æqua<lb></lb>lem impetum concipere, quo poſſit effectum ſecundo mallei <lb></lb>ictui ſimilem producere. </s> <s id="s.004990">Adde quod ſecundum pondus addi<lb></lb>tum priori, atque illi impoſitum, ſuum habet gravitatis cen<lb></lb>trum, & commune totius ponderis globulo incumbentis cen<lb></lb>trum gravitatis transfertur in aliud molis compoſitæ punctum; <lb></lb>ideóque linea directionis non ſimiliter incurrit in ſubjectum <lb></lb>globulum, adversùm quem ſimiles exhibeat vires. </s> <s id="s.004991">Neque mihi <lb></lb>facilè perſuadebis tam accuratè ſecundum pondus adjectum <lb></lb>priori, ut poſterius gravitatis centrum in eadem ſit lineâ di<lb></lb>rectionis, nec ab illâ quicquam deflectat. </s> <s id="s.004992">Quare vel poſterior <lb></lb>hæc gravitas addita priori, jam quieſcenti, ubi facta eſt vis <lb></lb>comprimendi par virtuti reſiſtendi, omni prorſus motu caret, & <lb></lb>nihil impetûs poteſt concipere, aut imprimere; vel ſolùm te<lb></lb>nuiſſimum, & qui vix poſt multum tempus conſpicuus fiat, mo<lb></lb>tum habet, & non niſi levem impetum imprimit, quo ſubjectus <lb></lb>globulus demum aliquantulo compreſſior appareat: ideò, ut <lb></lb>compreſſio ſimilis illi, quæ fit à ſecundo mallei ictu, habeatur, <lb></lb>neceſſe eſt gravitatem additam eſſe adhuc majorem, ut gravi<lb></lb>tas tota compoſita impetum efficere valeat, quem conſequa<lb></lb>tur motus æqualis ſecundæ illi compreſſioni à malleo factæ. </s> </p> <p type="main"> <s id="s.004993">Cur itaque ſecuris ligno incumbens, quamvis ingenti præ<lb></lb>gravata pondere, vix levem fiſſionem inferat ſubjecto ligno, <lb></lb>quod tamen altiùs penetratur ab eâdem ſecuri cadente & per<lb></lb>cutiente, in promptu cauſa eſt: quia videlicet compreſſio, quæ <pb pagenum="676" xlink:href="017/01/692.jpg"></pb>& impulſio eſt, cum motu quidem fit, ſed ipſo ſtatim initio & <lb></lb>in progreſſu adeſt reſiſtentia, ne producatur totus impetus, <lb></lb>quem vis motiva poſſet efficere, & motus non eſt, niſi quan<lb></lb>tum impellitur objectum corpus; ideóque ſecuris vi ponderis <lb></lb>incumbentis non valet in hujuſmodi motu alium impetum con<lb></lb>cipere præter illum, quem fert præſens motus, qui valde exi<lb></lb>guus eſt: At percuſſio ea eſt, ut cùm primùm ſecuris cadens <lb></lb>applicatur ligno, jam multum habeat concepti impetus in aëre <lb></lb>libero, & nihil adhuc reſiſtente ligno, ac propterea poſſit ve<lb></lb>lociùs moveri comprimendo & dividendo ſubjectum lignum. </s> <lb></lb> <s id="s.004994">Ex quo fit onus ſecuri impoſitum tantæ gravitatis eſſe oportere, <lb></lb>ut quæ Ratio eſt ſpatij à ſecuri cadente decurſi ad ſpatium, quo <lb></lb>illa penetrat lignum, ea ſaltem ſit Ratio gravitatis conflatæ ex <lb></lb>ſecuri & addito pondere ad <expan abbr="gravitatẽ">gravitatem</expan> ſimplicis ſecuris, ut fieret <lb></lb>æqualis ſciſſio ab eâdem ſecuri: ut videlicet tantumdem im<lb></lb>petûs concipiatur à magnâ gravitate in exiguo motu præſente <lb></lb>reſiſtentiâ, quantum impetûs concipitur à ſecuri in anteceden<lb></lb>te motu longiore abſque reſiſtentiâ ullâ, præterquam medij. </s> </p> <p type="main"> <s id="s.004995">Similiter nullum adhiberi poſſe pondus, quo aureæ lamellæ <lb></lb>impoſito hæc diduci poſſit in ſubtiliſſimam bracteolam, quem<lb></lb>admodum vi mallei percutientis, ex iiſdem principiis conſtat. </s> <lb></lb> <s id="s.004996">Attende enim, quanto motu moveri poſſit illud pondus com<lb></lb>primens; utique non niſi quantum eſt altitudinis diſcrimen in<lb></lb>ter lamellam & bracteolam: at tantillum ſpatium, in quo exer<lb></lb>cendus eſſet motus, quam Rationem habet ad toties multipli<lb></lb>catum ſpatium, in quo iteratis ſæpiùs ictibus liberè movetur <lb></lb>malleus? </s> <s id="s.004997">Cùm itaque minimus motus, aut etiam fortaſsè nul<lb></lb>lus, poſt tenuiſſimam auri compreſſionem ingenti illi oneri con<lb></lb>veniat, nil mirum ſi exiguo impetu ferè nihil efficiat, cùm ta<lb></lb>men malleus novo ſemper impetu ſingulis ictibus concepto ali<lb></lb>quam, licèt ſemper minorem atque minorem, compreſſionem <lb></lb>efficiat. </s> </p> <p type="main"> <s id="s.004998">Ut autem res hæc pleniùs innoteſcat, obſerva impulſionem, <lb></lb>qua corpus urgetur, opponi tractioni, & compreſſionem par<lb></lb>tium diſtractioni, atque ſicut corporis, quod urgetur, particu<lb></lb>læ aliquando comprimuntur, ita corporis, quod trahitur, par<lb></lb>ticulas aliquando diſtrahi, aut divelli, neque diſſimilem eſſe <lb></lb>reſiſtentiam corporum vi ſuæ gravitatis, ne impellantur, aut <pb pagenum="677" xlink:href="017/01/693.jpg"></pb>ratione poſitionis partium, ne comprimantur, ac ne trahantur, <lb></lb>aut particularum nexus diſſolvatur. </s> <s id="s.004999">Quapropter ubi primùm <lb></lb>incipit impulſio aut tractio, ſive compreſſio aut diſtractio, in<lb></lb>cipit etiam reſiſtentia, quæ eò major evadit, quò majorem vio<lb></lb>lentiam ſubit corpus. </s> <s id="s.005000">Hinc eſt potentiam impellentem aut tra<lb></lb>hentem ſemper minore impetu ferri, quàm ſi liberè moveretur, <lb></lb>dum nulla adeſſet reſiſtentia. </s> <s id="s.005001">Sic ſi quis funiculum, quem re<lb></lb>tinet clavus parieti infixus, arripiat, atque jam extentum <lb></lb>trahat, illum quidem multo niſu intendit, ſed nec illum diſ<lb></lb>rumpere valet, nec clavum revellere: ſed ſi eodem conatu fu<lb></lb>niculum languidum nec dum extentum trahat, celeriter mo<lb></lb>vetur manus, antequam funiculus extendatur, & facilè aut hic <lb></lb>abrumpitur, aut ille revellitur. </s> <s id="s.005002">Quia nimirum extenti jam fu<lb></lb>niculi reſiſtentia, ne intendatur, impedit, ne potentia pro Ra<lb></lb>tione ſui conatûs moveatur, multo impetu abſumpto in vincen<lb></lb>dâ illâ reſiſtentia; neque movetur potentia niſi cunctabunda, <lb></lb>& per breviſſimum ſpatium, quantum vi intenſionis funiculus <lb></lb>magis extenditur: At ubi languidus eſt funiculus, potentia <lb></lb>abſque ullo retinente per aliquantum ſpatij liberè movetur, & <lb></lb>totum impetum ſuo conatui reſpondentem in efficiendo celeri <lb></lb>motu impendit, quem jam notabiliter auctum invenit funicu<lb></lb>lus, cum primùm eſt extentus, & adhuc magis augetur perſe<lb></lb>verante eodem conatu. </s> <s id="s.005003">Quare cùm multò major ſit impetus, <lb></lb>ſatis eſſe poteſt non ſolùm ad intendendum funiculum, verùm <lb></lb>etiam ad illum diſrumpendum, aut, ſi, hujus particulæ validio<lb></lb>re nexu jungantur, ad revellendum clavum. </s> </p> <p type="main"> <s id="s.005004">Ex his habes, quid reſpondeat doctiſſimis viris vim percuſ<lb></lb>ſionis inveſtigantibus. </s> <s id="s.005005">Ut apparet, quantâ vi plumbeus globu<lb></lb>lus unciarum duarum ex cubitali altitudine cadens percuteret <lb></lb>ſubjectum corpus, exiſtimârunt ſatis innoteſcere, ſi globulus <lb></lb>ille funiculo cubitali adnecteretur chordæ arcûs medio loco in<lb></lb>ter extremitates. </s> <s id="s.005006">Tum ſublatus globulus uſque ad chordam <lb></lb>ipſam, dimiſſus eſt, atque obſervatum eſt punctum, ad quod <lb></lb>adducta eſt chorda: proclive enim erat arguere, globulum <lb></lb>tantâ vi percuſſurum ſubjectum corpus, quantâ vi inflectebat <lb></lb>baliſtæ arcum. </s> <s id="s.005007">Quare tentando varia pondera addiderunt <lb></lb>chordæ arcûs, donec demum pondus decem librarum chordam <lb></lb>ad idem punctum adduxit, ad quod adducta fuerat à globo ca-<pb pagenum="678" xlink:href="017/01/694.jpg"></pb>dente, atque in eodem flexionis ſtatu chordam & arcum deti<lb></lb>nuit. </s> <s id="s.005008">Arguebant igitur percuſſionem globi plumbei duarum <lb></lb>unciarum ex cubitali altitudine cadentis æquiparari preſſioni <lb></lb>decem librarum. </s> <s id="s.005009">Ulteriùs autem progrediendo, adhibita eſt ba<lb></lb>liſta alia validior, cujus arcus ob duriorem ferri temperationem <lb></lb>minùs erat flexibilis: quapropter cum ejuſdem potentiæ eadem <lb></lb>ſit vis, ejuſdem globuli ex eâdem altitudine ſimiliter cadentis <lb></lb>non niſi eædem eſſe poterant vires ad vincendam æqualem re<lb></lb>ſiſtentiam: atque adeò durioris arcûs minor flexio æquè re<lb></lb>ſiſtens, ac major flexio arcûs mollioris, breviore termino de<lb></lb>finivit deſcenſum globuli plumbei, & ad propius punctum ad<lb></lb>ducta eſt chorda. </s> <s id="s.005010">Verùm, ut in eodem flexionis ſtatu fortior <lb></lb>hic arcus retineretur, non ſatis fuit decem libras appendere, <lb></lb>ſed viginti librarum pondere opus fuit. </s> <s id="s.005011">Hinc inferebant ean<lb></lb>dem ejuſdem globuli duarum unciarum percuſſionem æquare <lb></lb>non ſolùm vires librarum decem, ſed & viginti: atque uſque <lb></lb>eò argumentationem deducebant, ut aſſumpto robuſtiore ali<lb></lb>quo arcu concluderent, ne pondus quidem librarum mille ſatis <lb></lb>eſſe ad arcum illum in eâ poſitione retinendum, ad quam fuiſ<lb></lb>ſet adductus à globo duarum unciarum cadente: id quod vim <lb></lb>quandam percuſſionis infinitam indicare videbatur. </s> </p> <p type="main"> <s id="s.005012">Verùm quamvis hos ingenioſorum hominum conatus non <lb></lb>modò non improbem, ſed multâ commendatione dignos exiſti<lb></lb>mem, liceat tamen mihi argumentationis infirmitatem expo<lb></lb>nere; tam enim non eſt vis percuſſionis duarum unciarum in<lb></lb>finita, quàm infinita non eſt vis preſſionis decem librarum. </s> <lb></lb> <s id="s.005013">Quando enim vi ponderis adnexi flectitur arcus, utique pon<lb></lb>dus deſcendit, & ſuâ gravitate ſuperat rigidi chalybis vires, <lb></lb>donec demum æqualitas quædam intercedat inter vim arcûs <lb></lb>elaſticam, & gravitatis conatum ad deſcendendum; tunc ſcili<lb></lb>cet fit conſiſtentia. </s> <s id="s.005014">Prout igitur robuſtiores ſunt arcus, minùs <lb></lb>permittunt deſcendere pondus chordæ appenſum, ſi omnia ſint <lb></lb>paria: Nam ſi brevior ſit arcus mollis & languidus, longior ve<lb></lb>rò arcus durioris temperationis, fieri poteſt, ut idem pondus <lb></lb>æqualiter adducat longiorem chordam atque breviorem, ſimi<lb></lb>li planè ratione ac de ponderibus fune ſuſpenſis præponderan<lb></lb>tibus atque æquilibribus dictum eſt lib.3. cap.12: ideò ponendi <lb></lb>ſunt arcus ita ſimiles & æquales, ut ſolâ ferri temperatione diſ-<pb pagenum="679" xlink:href="017/01/695.jpg"></pb>crepent. </s> <s id="s.005015">Si igitur validioris arcus repugnantia, ut flectatur ad <lb></lb>duos digitos, tanta eſt, quanta repugnantia mollioris arcûs, ut <lb></lb>flectatur ad ſex digitos, patet non eſſe eumdem impetum de<lb></lb>cem librarum deſcendentium ſolùm per duos priores digitos, <lb></lb>atque per ſex: ac proinde cùm decem libræ applicatæ arcui va<lb></lb>lidiori ſolùm poſſunt per duos digitos (& quidem lentiùs <lb></lb>propter majorem reſiſtentiam) moveri, minus poſſent, quàm <lb></lb>per impetum conceptum in motu ſex digitorum; & propterea <lb></lb>neque poſſent illius robuſtioris arcûs chordam adducere ad <lb></lb>duos digitos; ſed neque adductam ab aliâ potentiâ poſſent reti<lb></lb>nere in eo ſtatu ac poſitione: quia etiam ſi vis elaſtica arcûs ro<lb></lb>buſtioris inflexi ad duos digitos par eſſet virtuti elaſticæ arcûs <lb></lb>imbecillioris inflexi ad ſex digitos, cui reluctantur decem li<lb></lb>bræ; hæ minùs repugnant, ne ad duos digitos, quàm ne ad ſex <lb></lb>attollantur; igitur decem libræ minùs reſiſtunt virtuti elaſticæ <lb></lb>arcûs fortioris, adeóque nec poſſunt in eo flexionis ſtatu reti<lb></lb>nere arcum fortiorem & chordam: ſi enim pares ſunt vires <lb></lb>elaſticæ arcûs inflexi ad duos digitos, & arcûs inflexi ad ſex di<lb></lb>gitos, pari impetu ſe reſtituunt, ut parem violentiam excu<lb></lb>tiant; at pondus par utrique chordæ adnexum non pari veloci<lb></lb>tate movetur, ſi ad duos ac ſi ad ſex digitos attollatur; igitur <lb></lb>minùs reſiſtunt decem libræ motui duorum, quàm motui ſex <lb></lb>digitorum. </s> </p> <p type="main"> <s id="s.005016">Porrò vis globi cadentis non eſt comparanda cum pondere <lb></lb>quatenus retinente chordam in eadem flexione, ſed quatenus <lb></lb>illam adducente & flectente, ut motus cum motu, non verò <lb></lb>motus cum quiete comparetur. </s> <s id="s.005017">In eo autem motu ponderis ad<lb></lb>ducentis chordam, & arcum inflectentis, quò major eſt re<lb></lb>ſiſtentia, eò minor eſt impetus & velocitas, qua pondus illud <lb></lb>movetur: igitur idem pondus non parem vim habere poteſt, <lb></lb>ubi diſpari impetu & velocitate movetur. </s> <s id="s.005018">At globus cadens <lb></lb>antequam incipiat trahere chordam, nullum prorſus habet im<lb></lb>pedimentum, ſed ſivè fortior, ſivè mollior ſit arcus, eodem im<lb></lb>petu & velocitate movetur; ubi verò reſiſtentiam invenit, ſo<lb></lb>lùm deſcendit ulteriùs pro ratione repugnantiæ; & factä de<lb></lb>mum æqualitate inter vim deſcendendi à globulo acquiſitam, <lb></lb>& vim elaſticam in arcu, ceſſat deſcenſus, atque extincto im<lb></lb>petu acquiſito, vi elaſticâ vincente globuli gravitatem, hic ſur-<pb pagenum="680" xlink:href="017/01/696.jpg"></pb>ſum trahitur. </s> <s id="s.005019">Cùm itaque quicquid vi extrinſecùs aſſumptâ <lb></lb>movetur, moveatur juxta exceſſum virtutis motivæ ſupra re<lb></lb>ſiſtentiam; ſi æqualis reſiſtentiæ menſura, quæ ex diſſimilium <lb></lb>arcuum majori aut minori flexione deſumitur, eumdem exceſ<lb></lb>ſum virtutis motivæ exigat, ut vincatur, & hunc exceſſum ha<lb></lb>beat globulus cadens, nil mirum, ſi idem globulus cadens id <lb></lb>præſtare poſſit, quod ſuperat vires alicujus ponderis, cujus vis <lb></lb>movendi non eumdem ſemper exceſſum habet ſupra illam re<lb></lb>ſiſtentiam priori reſiſtentiæ æqualem; quia videlicet non æqua<lb></lb>li impetûs intenſione aggreditur motum, ubi ipſo ſtatim initio <lb></lb>major invenitur difficultas, & tardior eſt motus. </s> <s id="s.005020">Non eſt igitur <lb></lb>vis infinita globuli duarum unciarum nullo impedimento prohi<lb></lb>biti, quin ad trahendam cujuſcumque arcûs chordam ſemper <lb></lb>afferat, exempli gratiâ, centum gradus impetûs in motu acqui<lb></lb>ſitos, quando pondera majora & majora tractionem incipientia <lb></lb>à quiete non parem habent impetûs exceſſum, ſed minorem & <lb></lb>minorem pro duriore arcûs temperatione. </s> <s id="s.005021">An infinitam dixe<lb></lb>ris equi virtutem, qui ſolus in liberâ planitie currum trahat, ad <lb></lb>quem trahendum in eâdem planitie altioribus atque altioribus <lb></lb>nivibus obſitâ requiruntur plures & plures equi? </s> <s id="s.005022">igitur nec in<lb></lb>finita eſt vis decem librarum, qua flectitur arcus mollis, quia <lb></lb>ad flectendos arcus fortiores majus & majus pondus requiritur: <lb></lb>huic autem virtuti decem librarum æqualis eſt vis globuli ca<lb></lb>dentis; hæc igitur & ipſa finita eſt. </s> <s id="s.005023">Nimirum aucta reſiſtentia <lb></lb>quodammodo imminuit virtutem agendi; ac propterea non ſa<lb></lb>tis aptè comparantur decem libræ cum viginti libris perinde, <lb></lb>atque ſi utræque eſſent omnino liberæ; ſed unumquodque pon<lb></lb>dus componi debet cum ſuâ reſiſtentiâ, ut demum habeatur <lb></lb>exceſſus virtutis motivæ ſupra reſiſtentiam. </s> </p> <p type="main"> <s id="s.005024">At, inquis, arcus fortior retinetur à libris viginti, & infir<lb></lb>mior à libris decem. </s> <s id="s.005025">Ita planè eſt: ſed hìc pondera propriè non <lb></lb>habent rationem efficientis, ſed potiùs reſiſtentis, quatenus <lb></lb>impediunt arcuum vim elaſticam, ne ſe reſtituant: cùm verò <lb></lb>virtutes elaſticæ ex genere ſuo propter diſparem temperatio<lb></lb>nem inæquales ſint, nil mirum, ſi ab inæqualibus reſiſtentiis <lb></lb>impediendæ ſint, ne agant. </s> <s id="s.005026">Hinc autem non eſt deſumenda <lb></lb>ulla comparatio cum virtute globuli cadentis, quippe qui ac<lb></lb>quiſitum impetum amittens non habet vim retinendi arcum in <pb pagenum="681" xlink:href="017/01/697.jpg"></pb>eo ſtatu; ad quem illum adduxit: at ponderis adnexi gravitas <lb></lb>manet, & ibi retinet arcum, quò eum adduxit; niſi fortè ali<lb></lb>quem impetum acquiſierit in deſcenſu, quo pereunte, aliquan<lb></lb>tulum præpolleat vis elaſtica, & ſurſum retrahat appenſum <lb></lb>pondus. </s> <s id="s.005027">Licet igitur globulo cadenti æqualiter reſiſtere dican<lb></lb>tur arcus fortior qui minùs flectitur, & mollior qui magis flecti<lb></lb>tur; poſtquam tamen jam per vim inflexi ſunt arcus, naturali<lb></lb>ter partes minùs flexibiles validiùs conantur ſe reſtituere, quàm <lb></lb>flexibiliores: quemadmodum gravitas ut quatuor, & gravitas <lb></lb>ut duo, ſi moveantur per vim motu reciprocè ſubduplo, æqua<lb></lb>liter reſiſtunt moventi; ſed ſi utraque ſuſpendatur, inæquali<lb></lb>ter conantur ſuos motus naturales. <lb></lb></s> </p> <p type="main"> <s id="s.005028"><emph type="center"></emph>CAPUT VII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005029"><emph type="center"></emph><emph type="italics"></emph>Quàm diſpares ex motûs velocitate ſint <lb></lb>percuſsiones.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005030">PErcuſſionem ex ea parte, quatenus à ſimplici Impulſione <lb></lb>diſtinguitur, motum exigere antecedentem, quo impetus <lb></lb>acquiratur, ſuperiori capite definitum eſt. </s> <s id="s.005031">Nunc verò, quia pro <lb></lb>motuum velocitate diversâ diſpares ſunt percuſſionum vires, <lb></lb>quærendum eſt, unde diſſimilitudo iſta procreetur, & quænam <lb></lb>ſervari Ratio videatur, ſivè inæquales ejuſdem corporis, ſive <lb></lb>diverſorum corporum percuſſiones inter ſe comparentur. </s> <s id="s.005032">Eſt <lb></lb>autem conſiderandum in Impetu, qui eſt proximè efficiens mo<lb></lb>tum, aliud eſſe ejus quantitatem, ſivè entitatem accipere, aliud <lb></lb>in ejuſdem Intenſione conſiſtere: intenſionem conſequitur ve<lb></lb>locitas motûs, at ex entitate ipsâ magis extensâ, quamvis mi<lb></lb>nùs intensâ, ac proinde ex motu tardiore, oriri poteſt validior <lb></lb>ictus, de quo in ſequentibus. </s> <s id="s.005033">Ex motûs autem velocitate, quæ <lb></lb>corpori vi præcedentis motûs congrueret, ſi nihil obſtaret, per<lb></lb>cuſſionem fieri majorem tam certis experimentis conſtat, ut vel <lb></lb>cæci, ſi quando præfidentes concitatiùs ambulando caput ad <lb></lb>objectum parietem allidunt, id abundè teſtari valeant; corpus <pb pagenum="682" xlink:href="017/01/698.jpg"></pb>ſiquidem, quod motui reſiſtit, majori & velociori motui ma<lb></lb>gis reſiſtit, quare & percuſſio fit validior. </s> </p> <p type="main"> <s id="s.005034">Ubi verò de motûs velocitate ſermo eſt, non videtur diſſimu<lb></lb>landa medij ſcindendi reſiſtentia; hoc quippe tardiori motui <lb></lb>minùs, velociori magis obſtat. </s> <s id="s.005035">Si enim ex lento & flexili vir<lb></lb>gulto abſtractam virgam per aërem molli brachio huc, illuc, <lb></lb>ſurſum, deorſum duxeris, hæc, aëre tenuiſſimam aut ferè nul<lb></lb>lam compreſſionem ſubeunte, vix, aut ne vix quidem, tantu<lb></lb>lum à directâ ſuarum partium poſitione deflectet: at ſi eam ve<lb></lb>hementius agitaveris, aëre tantam particularum compreſſio<lb></lb>nem renuente, manifeſtè inflexam videbis, & illatam ſibi vim <lb></lb>aër acuto ſibilo prodet. </s> <s id="s.005036">Sic baculo aquam ſenſim ac leniter di<lb></lb>videns non admodum repugnantem experiris; at velociùs con<lb></lb>citanti illa validè reſiſtit, eóque validiùs, quò craſſior fuerit <lb></lb>baculus. </s> <s id="s.005037">Ex his liquidò conficitur de percuſſione philoſophan<lb></lb>tem fruſtra medij reſiſtentiam mente abſtrahere: Nam ſi nulla <lb></lb>eſt ſine motu percuſſio, nullus motus niſi per medium, neque <lb></lb>ſinè certa velocitatis aut tarditatis menſurâ, cui medium inæ<lb></lb>qualiter reſiſtit; utique & motum à percuſſione ita mente pari<lb></lb>ter ſejungere poteris, ut nihil prorſus de motu cogites, ſi nul<lb></lb>lam cum medio rationem habendam exiſtimas: At motum, ejúſ<lb></lb>que velocitatem attendendam eſſe in percuſſione nemo negat; <lb></lb>igitur neque medij reſiſtentiam, quæ velocitati modum ali<lb></lb>quem ſtatuit, omnino contemnere oportet. </s> </p> <p type="main"> <s id="s.005038">Hinc duæ ferè ex diametro oppoſitæ ſententiæ cavendæ <lb></lb>ſunt, quarum altera gravium inæqualium motum ſtatuit ipſo<lb></lb>rum gravitatibus analogum, ut decuplò velociùs moveatur il<lb></lb>lud, quod eſt decuplò gravius: altera æqualem omnibus velo<lb></lb>citatem tribuit. </s> <s id="s.005039">Utramque manifeſta experimenta falſitatis re<lb></lb>darguunt, ſi ex congruâ altitudine inſtituantur: Si enim ex val<lb></lb>dè editâ turri inæqualia corpora, aut ejuſdem, aut diverſæ ſe<lb></lb>cundùm ſpeciem gravitatis dimittas, illud, quod gravius eſt, <lb></lb>terram citiùs attingere obſervabis, ſed tam brevi momentorum <lb></lb>diſcrimine, ut nulla ſubeſſe poſſit ſuſpicio ſervatæ velocitatum <lb></lb>cum gravitatibus analogiæ, neque tamen de velocitatum inæ<lb></lb>qualitati dubitari queat. </s> <s id="s.005040">Meis ſcilicet auribus & oculis fidem <lb></lb>abrogare nequeo, quicquid obtrudant aliqui in contrarium ſua <lb></lb>aut aliorum experimenta afferentes ex nimis brevi altitudine. <pb pagenum="683" xlink:href="017/01/699.jpg"></pb>Nam & ſæpiùs in profundiſſimum puteum inæquales lapides <lb></lb>dimiſi ſimul, & ictuum ſonitum alium alio priorem ſemper au<lb></lb>divi; id quod ſatis eſt ad illam velocitatum omnimodam æqua<lb></lb>litatem rejiciendam, quamvis uter prior aquam attigerit, certò <lb></lb>dignoſcere non valeret auris; quòd ſi alter in ſubjectam peluim <lb></lb>æneam, alter in vas ligneum decidiſſet, potuiſſet auris dijudi<lb></lb>care ex ſonitu: Et ex altiſſimâ turri Bononienſi dimiſſa pondera <lb></lb>inæqualia obſervavi initio quaſi æqualiter deſcendere ita, ut <lb></lb>oculus nullam velocitatum diſſimilitudinem adhuc dignoſce<lb></lb>ret; deinde procedente deſcenſu paulatim gravitas major præ<lb></lb>currere notabiliter incipiebat, ſempérque magis augebatur ve<lb></lb>locitas, adeò ut aliquando gravitas major terram attigerit, <lb></lb>quando minor adhuc aberat intervallo pedum quadraginta, <lb></lb>quemadmodum ex notâ in turris latere dimetiri licuit. </s> <s id="s.005041">Verum <lb></lb>quidem eſt breviſſimâ temporis menſurâ & hanc minorem in <lb></lb>terram decidiſſe. </s> <s id="s.005042">Id quod fortaſſe fucum fecit non animadver<lb></lb>tentibus magnæ velocitati multum reſpondere ſpatij, quod <lb></lb>quaſi momento percurritur; ac propterea æqualitatem veloci<lb></lb>tatum utrique gravitati tribuendam cenſuerunt, quia exiguum <lb></lb>erat temporum diſcrimen. </s> <s id="s.005043">An vellus, quantum pugno com<lb></lb>prehenditur, æquè velociter ac prægrande ſaxum deſcenſurum <lb></lb>exiſtimas? </s> <s id="s.005044">Figuræ dices tribuendum plurimum, non enim ab <lb></lb>omnibus corporibus æquè facilè dividitur aër nunquam non <lb></lb>fluctuans. </s> <s id="s.005045">Ita ſane: igitur ſi aër inæqualiter reſiſtit, inæqua<lb></lb>liter moveri poſſunt corpora cadentia. </s> <s id="s.005046">Adde non diſſimiliter <lb></lb>gravia & levia ad ſuos motus à naturâ incitari; atque adeò ſi, <lb></lb>ubi plus eſt levitatis, velociorem motum ſurſum obſervamus, <lb></lb>etiam, ubi plus eſt gravitatis concitatiorem motum deorſum <lb></lb>arguere debemus. </s> <s id="s.005047">Aquam in longiore fiſtulâ vitreá aliquan<lb></lb>diu agita, ut aër fiſtulæ incluſus aquæ admiſceatur: ubi ab agi<lb></lb>tatione ceſſatum fuerit, majores aëris particulas citò aſcenden<lb></lb>tes videbis, dum minores cunctabundæ paulatim moventur: id <lb></lb>quod clariùs conſtabit, ſi aquæ loco hydrargyrum in fiſtulam <lb></lb>admiſeris. </s> <s id="s.005048">Quidni igitur gravia pariter deorſum diſpari velo<lb></lb>citate moveantur, ſi inæqualia fuerint? </s> </p> <p type="main"> <s id="s.005049">Non tamen ſervandam eſſe gravitatum analogiam hinc <lb></lb>apertè conſtat, quod in corporibus ejuſdem ſpeciei Ratio gra<lb></lb>vitatum eadem eſt ac magnitudinum: magnitudines autem <pb pagenum="684" xlink:href="017/01/700.jpg"></pb>ſunt in triplicatâ Ratione homologorum laterum: At impedi<lb></lb>mentum, quod ex medio ſcindendo oritur, & velocitati mo<lb></lb>dum ſtatuit, non eſt ipſis magnitudinibus analogum, ſed ad <lb></lb>ſummum ea eſſe poteſt Ratio, quæ inter corporum ſuperficies <lb></lb>intercedit; hæ autem tantùm ſunt in duplicatâ Ratione late<lb></lb>rum homologorum. </s> <s id="s.005050">Non igitur velocitates, quatenus ab im<lb></lb>pedimento temperantur, ſunt directè gravitatibus analogæ. </s> <lb></lb> <s id="s.005051">Ubi autem corpora non ejuſdem ſecundùm ſpeciem gravitatis, <lb></lb>habuerint gravitates magnitudinibus reciprocè analogas, atque <lb></lb>adeò æquali gravitate abſolutâ, ſeu pondere, prædita fuerint, <lb></lb>adhuc inæquales eſſe aëris reſiſtentias, ſi figuræ ſimiles ſint, <lb></lb>ſatis probabiliter concedimus, plus ſiquidem majori repugnat, <lb></lb>quàm minori: ſi verò & diſſimiles figuræ, & inæquales gravi<lb></lb>tates ponantur, ex omnibus ſimul compoſitis quodammodo <lb></lb>conflari reſiſtentiarum Rationem facile eſt opinari; ſed Ratio<lb></lb>nis terminos temerè definire non auſim. </s> </p> <p type="main"> <s id="s.005052">Porrò certam legem, qua in reſiſtendo aër contineatur, om<lb></lb>ninò afferre non poſſumus, ſi quemadmodum ille reſiſtat, per<lb></lb>pendamus. </s> <s id="s.005053">Non eadem eſt aquæ & aëris reſiſtendi Ratio, ne <lb></lb>dividatur; aqua enim nondum in vaporem extenuata, & fuſa, <lb></lb>conſtipari ſe, & in anguſtiora ſpatia coarctari non ſinit; ſed ubi <lb></lb>locum deſcendenti ex aëre corpori concedere cogitur, ſuper<lb></lb>ficiem intrà vas, quo continetur, attollens tantumdem aëri, <lb></lb>quem impellit, ſurripit ſpatij, quantum immerſo corpori per<lb></lb>mittit. </s> <s id="s.005054">Aut ſi non deſcendat corpus, ſed obliquè feratur (ut ſi <lb></lb>baculum partim aquæ immiſſum, partim extantem tranſverſum <lb></lb>agas, aut navis prora illam findat) tunc quæ motui opponitur, <lb></lb>aqua criſpatur, & baculus ſivè navis foveam ponè relinquit, in <lb></lb>quam deinde aqua refluat; quò autem craſſior baculus aut am<lb></lb>plior prora, & vehementior atque concitatior fuerit motus, <lb></lb>aqua impulſa altiùs aſſurgit, & magis depreſſa fovea apparet. </s> <lb></lb> <s id="s.005055">Ex quo ſatis apertè conſtat à corpore, quod movetur, proximas <lb></lb>aquæ oppoſitæ particulas impelli, & per has interjectas etiam <lb></lb>reliquas in aëris locum protrudi. </s> </p> <p type="main"> <s id="s.005056">At verò aër, quem facilè comprimi & dilatari tam multis ex<lb></lb>perimentis novimus, dum locum corpori commoto concedit, <lb></lb>neque opus eſt, ut ſupremi ætheris regionem invadat locum <lb></lb>ſibi quærens, neque in foveam excavatus vel ad momentum <pb pagenum="685" xlink:href="017/01/701.jpg"></pb>hiat; ſed tantiſper dum ejus particulæ circumpulſæ in abeun<lb></lb>tis corporis relictum ſpatium ſuccedant, quæ antè ſunt, com<lb></lb>primuntur, quæ ponè, dilatantur; compreſſæ autem ſe expli<lb></lb>cantes aërem lateribus adhærentem repellunt, quem dilatatæ <lb></lb>attrahunt ſe contrahentes. </s> <s id="s.005057">Si tardus ſit motus, exiguâ aëris <lb></lb>conſtipatione aut diſtractione opus eſt; at ſi velocior, oppoſitæ <lb></lb>aëris particulæ magis comprimuntur, ſequentes magis dilatan<lb></lb>tur; quæ proinde ſe reſtituere vehementiùs conantes, etiam <lb></lb>velociorem efficiunt reliquarum particularum circumpulſio<lb></lb>nem. </s> <s id="s.005058">Verùm quia ſapientiſſimo Naturæ inſtituto ita compara<lb></lb>tum eſt, ut quàm minimum ejus ordo perturbetur, & mini<lb></lb>mam, quoad fieri poſſit, corpora ſingula patiantur violentiam, <lb></lb>hanc pluribus potiùs diſpertiendam cenſuit, quàm uni ſubeun<lb></lb>dam: propterea ſi digitale ſpatium multo aëri ſurripiendum eſt, <lb></lb>exigua contingit ſingulis particulis naturalis ſpatij jactura, <lb></lb>quam diſſimulanter ferunt, nec admodum repugnant; contra <lb></lb>verò ſi modicus ſit aër, & tantumdem de ejus ſpatio demendum <lb></lb>ſit, reluctatur acriùs, ut pro viribus naturæ jura tueatur. </s> <s id="s.005059">Hinc <lb></lb>ſi corpus, quod movetur, brevi intervallo abſit à corpore ſoli<lb></lb>do & duro, quod ejus motum obſiſtendo compeſcet, atque <lb></lb>adeò etiam aërem impulſum remoratur, hunc inter anguſtias <lb></lb>deprehenſum magis conſtipari neceſſe eſt, magíſque reſiſtere. </s> <lb></lb> <s id="s.005060">Non eſt tamen aëri denegandum, quod cæteris corporibus ul<lb></lb>tro concedimus; nam & ipſe jam commotus ex concepto per <lb></lb>impulſionem externam, aut ex vi ſuâ elaſticâ, impetu faciliùs <lb></lb>pergit inſtitutum iter conficere, quàm ſi tunc primùm à quie<lb></lb>te recederet: Ex quo fit in motu corporis accelerato, licèt ra<lb></lb>tione habitâ velocitatis augenda eſſet reſiſtentia aëris, hanc ta<lb></lb>men non augeri niſi pro exceſſu velocitatis illius ſupra motum, <lb></lb>quo aër moveretur ad eaſdem partes, niſi acriùs ab ipſo corpo<lb></lb>re urgeretur. </s> </p> <p type="main"> <s id="s.005061">Hanc aëris reſiſtentiam paulò explicatiùs commemorare pla<lb></lb>cuit eo conſilio, ut mihi ipſe perſuadeam non modò ipſum <lb></lb>nihil omnino non officere motui, verùm etiam tam fieri non <lb></lb>poſſe, ut percuſſionibus certiſſimam legem ſtatuamus, quàm <lb></lb>evidens eſt adeò inconſtantem & variam eſſe aëris reſiſten<lb></lb>tiam, ut ad calculos ſubtiliter & exquiſitè revocari nequeat, <lb></lb>quippe quæ ex tam variis cauſis pendet: quemadmodum enim <pb pagenum="686" xlink:href="017/01/702.jpg"></pb>aquæ, cæterorúmque liquorum diſſimilium reſiſtentia inæqua<lb></lb>lis conſpicua eſt, ita purum ac tenuem aërem non æquè reſiſte<lb></lb>re atque craſſum & concretum ratio ſuadet: quis autem ſynce<lb></lb>rum aërem ab aëre cum terræ expirationibus permiſto diſcer<lb></lb>nat? </s> <s id="s.005062">Quid ſi corpus in motu aërem aliò directum, aut in con<lb></lb>trarias partes reflexum, aut turbine aliquo perverſum atque <lb></lb>adhuc agitatum offendat? </s> <s id="s.005063">an non aliquis velocitatis gradus im<lb></lb>minuitur? </s> <s id="s.005064">Sed quis certum habeat, utrùm quieſcat aër, neque <lb></lb>corporis impetum frangat, aut reprimat aliena impreſſione ad<lb></lb>verſus, an verò ad eaſdem partes delatus motui obſecundet, & <lb></lb>velocitati faveat? </s> <s id="s.005065">Quare laudandi quidem quicumque percuſ<lb></lb>ſionum naturam veſtigantes, & ea, quibus in ejus notitiam <lb></lb>deduci poſſent, conjecturá proſpicientes, in inſtituendi expe<lb></lb>rimentis ſedulò ſe exercuerunt; parùm tamen mihi de veritate <lb></lb>blandiri me poſſe arbitrarer, ſi hæc quaſi Apodixes temerè reci<lb></lb>perem; ſed neque ore tam duro fuerim, ut ea prorsùs rejiciam. </s> <lb></lb> <s id="s.005066">Confirmatis igitur experimentis me duci ſinam, quatenus ad <lb></lb>veritatis ſimilitudinem me proximè acceſſurum ſpero. </s> </p> <p type="main"> <s id="s.005067">Percuſſio itaque, ſi motum naturalem ex gravitate ortum ſub<lb></lb>ſequatur, certam aliquam Rationem ob idipſum ſortiri videtur, <lb></lb>quia cum velocitate conſentit impetus: velocitas autem ex ſpa<lb></lb>tio deprehenditur æqualibus temporibus reſpondente; ſpatia <lb></lb>verò cum temporibus comparata certis Rationibus definita vi<lb></lb>deri, iterata experimenta docuerunt, quæ vix quiſquam ſanus <lb></lb>neget; in iis ſiquidem tot doctiſſimi viri poſt Galilæum verſati <lb></lb>ſunt pari exitu, & ſummo conſenſu, ut in his omnibus inſit <lb></lb>quidam, ſine ullo fuco veritatis color. </s> <s id="s.005068">Hujus rei ſpecimen <lb></lb>exhibeamus in globo argillaceo unciarum octo, qui ſpatio unius <lb></lb>ſcrupuli ſecundi (quantus ferè eſt pulſus arteriæ hominis ſani) <lb></lb>obſervatus eſt percurrere pedes Romanos 15; duplo autem <lb></lb>tempore incipiendo à quiete, hoc eſt ſcrupulis ſecundis duo<lb></lb>bus, pedes 60: quare ſi priori ſcrupulo ſecundo reſpondent pe<lb></lb>des 15, poſteriori tribuendi ſunt pedes 45: igitur motus eſt ce<lb></lb>lerior, cùm majus ſpatium pari tempore confecerit. </s> <s id="s.005069">Plura hu<lb></lb>juſmodi experimenta (ſi te à tentando abſterreat labor) ſuppe<lb></lb>ditabit Ricciolius tom.1. Almag. lib.9. ſect.4. cap.16. ex quibus <lb></lb>demum infertur velocitatis incrementa fieri juxta incremen<lb></lb>tum progreſſionis Arithmeticæ numerorum imparium ab unita-<pb pagenum="687" xlink:href="017/01/703.jpg"></pb>te incipientis 1.3.5.7.9.11.13, &c. </s> <s id="s.005070">adeò ut, ſi quod ſpatium pri<lb></lb>mo momento percurritur, ſtatuatur ut 1, triplo velociùs movea<lb></lb>tur corpus grave deſcendens in ſecundo momento, quintuplo <lb></lb>velocius in tertio, ſeptuplo velociùs in quarto, atque ita dein<lb></lb>ceps. </s> <s id="s.005071">Quoniam verò numerorum imparium ſeries ab unitate <lb></lb>incipiens hoc habet, quòd, ſi colligantur in ſummam, nume<lb></lb>ros quadratos conſtituant; hinc eſt, quòd collectis in ſummam <lb></lb>omnibus incrementis velocitatis (hoc eſt omnibus ſpatiis, ex <lb></lb>his quippe dignoſcitur velocitas) habeatur numerus quadratus <lb></lb>temporis, quod duravit motus. </s> <s id="s.005072">Collatis igitur invicem duobus <lb></lb>motibus naturalibus ejuſdem corporis gravis, ſed non iſochro<lb></lb>nis, erunt ut quadrata temporum ita & ſpatia, atque è conver<lb></lb>ſo ut ſpatia inter ſe, ita & temporum quadrata. </s> </p> <p type="main"> <s id="s.005073">Hinc cognito ſpatio, quod à dato corpore gravi percurritur <lb></lb>dato tempore, ſtatim innoteſcet, quantum ſpatij conficere va<lb></lb>leat alio tempore dato, vel quanto tempore aliud datum ſpa<lb></lb>tium. </s> <s id="s.005074">Quæratur enim, quantum ſpatium deſcendendo percur<lb></lb>ret uno horæ quadrante globus idem argillaceus, qui uno mi<lb></lb>nuto ſecundo Romanos pedes 15 percurrit? </s> <s id="s.005075">Datum tempus, <lb></lb>ſcilicet horæ quadran, ſcrupula Secunda 900 continet, cujus <lb></lb>numeri quadratum eſt 810000. Fiat igitur ut 1 ad 810000, ita <lb></lb>pedes 15 ad 12150000: qui pedum numerus in milliaria Itali<lb></lb>ca reſolutus dat milliaria 2430, quæ uno horæ quadrante con<lb></lb>ficeret. </s> <s id="s.005076">Viciſſim quæratur quantum temporis idem globus in<lb></lb>ſumeret in primo milliari percurrendo, hoc eſt ped. 5000. Fiat <lb></lb>ut 15 ad 5000, ita 1 quadratum dati temporis, ſcilicet unius <lb></lb>ſcrupuli ſecundi, ad 333 1/3 quadratum quæſiti temporis; cujus <lb></lb>quadrati Radix inveſtiganda eſt, & demum invenitur Scrup. <lb></lb></s> <s id="s.005077">ſec. 18 1/4 & paulo ampliûs; nam huic tempori præcisè reſpon<lb></lb>dent ſolùm pedes (4995 15/16). </s> </p> <p type="main"> <s id="s.005078">Incrementa hæc velocitatis ex concepti impetûs incremento <lb></lb>deſumenda eſſe nullus dubito; ſed operoſum videri poſſet au<lb></lb>geſcentis impetûs cauſam exponere. </s> <s id="s.005079">Cùm junior Ariſtotelem <lb></lb>interpretarer, & primas curas hujuſmodi rerum contemplatio<lb></lb>ni impenderem, hanc excogitavi hypotheſim; videlicet impe<lb></lb>tûs producti diuturnitatem maximam duobus tantùm momen<lb></lb>tis circumſcribebam, ita ut primo momento oriretur, ſecundo <pb pagenum="688" xlink:href="017/01/704.jpg"></pb>æqualem ſibi impetum gigneret, in quo ſuperſtes eſſet, tertio <lb></lb>periret: gravitati autem ſingulis momentis vim producendi <lb></lb>certum impetûs gradum ſibi congruentem tribuebam. </s> <s id="s.005080">Hinc <lb></lb>corpus deſcendens primo momento primum habebat impetûs <lb></lb>gradum à gravitate productum; ſecundo momento primus ille <lb></lb>gradus alium gradum gignebat præter eum, qui à gravitate <lb></lb>tunc oriebatur; quare duo novi gradus cum uno antiquo tres <lb></lb>gradus conſtituebant. </s> <s id="s.005081">Tertio momento primus gradus peribat, <lb></lb>duo ſecundi gradus duos pariter producebant, & gravitas ſuum <lb></lb>tertium gradum; quare quinque gradus erant. </s> <s id="s.005082">Quarto mo<lb></lb>mento duobus ſecundis gradibus pereuntibus, tres gradus ter<lb></lb>tio momento producti reliqui erant, & ſibi tres alios gradus <lb></lb>addebant, quos producebant, atque gravitas ſuum quartum <lb></lb>gradum efficiebat, ut in univerſum eſſent ſeptem gradus. </s> <s id="s.005083">Ex <lb></lb>his ſeptem quinto momento peribant tres tertij gradus; qua<lb></lb>tuor reliqui item alios quatuor adjiciebant quinto gradui à gra<lb></lb>vitate proficiſcenti, & erant novem. </s> <s id="s.005084">Atque ita deinceps, tot <lb></lb>pereuntibus gradibus, quotum erat momentum uno interjecto <lb></lb>præcedens, & tot productis, quotum erat ipſius motûs momen<lb></lb>tum. </s> <s id="s.005085">Sic momento vigeſimo nono peribant gradus 27 pro<lb></lb>ducti momento vigeſimo ſeptimo, remanentibus gradibus 28 <lb></lb>productis momento vigeſimo octavo, à quibus totidem produ<lb></lb>cebantur unâ cum gradu proprio gravitatis, hoc eſt gradus 29, <lb></lb>& tunc erat impetûs intenſio graduum 57 momento vigeſimo <lb></lb>nono. </s> <s id="s.005086">Quemlibet verò terminum in ſerie numerorum impa<lb></lb>rium facilè invenies, ſi illi duplicato demas unitatem: ſic quæ<lb></lb>rens octavum terminum ex denominatore termini duplicato, <lb></lb>ſcilicet bis 8, hoc eſt 16, deme unitatem, & 15 eſt octavus ter<lb></lb>minus: ſic terminus ſeptuageſimus habetur demptâ unitate ex <lb></lb>140, & eſt 139. </s> </p> <p type="main"> <s id="s.005087">Huic hypotheſi cum Phenomeno optimè conveniebat, & ea <lb></lb>ſtatuebatur impetûs intenſio, quæ velocitati efficiendæ par eſ<lb></lb>ſet, ſervatâ incrementorum Ratione, quæ ex iteratis experimen<lb></lb>tis innotuerat. </s> <s id="s.005088">Verùm commentitia, & fabulæ proxima vide<lb></lb>batur tàm brevis impetûs vita, quam non niſi duo momenta <lb></lb>metirentur: in iis ſanè, quæ vi externâ moventur, & longiùs <lb></lb>projiciuntur, aut in gyrum aguntur, licet extinctâ effectrice <lb></lb>causâ impreſſus impetus diutiùs permanet; quidni & impetus <pb pagenum="689" xlink:href="017/01/705.jpg"></pb>ſponte ſuâ conceptus, ſuæque origini cohærens aliquandiu per<lb></lb>ſeveret? </s> <s id="s.005089">quippe qui aut ejuſdem, aut ſaltem non deterioris na<lb></lb>turæ cenſendus eſt. </s> <s id="s.005090">Adde nimis incertum eſſe, an impetus im<lb></lb>petum producere valeat in eodem corpore, cui ineſt, quamvis <lb></lb>impetum in alienis corporibus percuſſis efficiendi vis illi conce<lb></lb>datur: nam & calor, & cæteræ qualitates effectrices, quas de<lb></lb>perditas ſibi forma ſubſtantialis reparare incipit, non alios ſimi<lb></lb>les gradus ſibi addunt, licèt eos in proximo corpore efficere va<lb></lb>leant. </s> <s id="s.005091">Præterquam quod, cur illo ipſo momento, quò primùm <lb></lb>exiſtit impetus, ſimilem gradum non producit? </s> <s id="s.005092">nihil ſcilicet il<lb></lb>li deeſt, nullo impedimento prohibetur, neque cauſam ætate <lb></lb>præcedere effectum, ſed origine, neceſſe eſt. </s> <s id="s.005093">Si autem primo <lb></lb>momento & oritur impetus, & impetum efficit, hic pariter ſuam <lb></lb>vim primo eodem momento exerens alium impetum producit, <lb></lb>& infinita graduum impetûs æqualium multitudo conſurgit; <lb></lb>cujus ne veſtigium quidem apparere poteſt, cùm in cauſarum <lb></lb>& effectuum ſerie ſemper ab infinitate natura diſcedat. </s> </p> <p type="main"> <s id="s.005094">Quare impetum à gravitate deſcendente productum, ex tam <lb></lb>expedito interitu vendicandum, & virtute ſe novo impetu au<lb></lb>gendi ſpoliandum, longè probabiliore conjecturâ cenſui. </s> <s id="s.005095">Im<lb></lb>petum igitur certâ quadam menſurâ gravitati corporis con<lb></lb>gruente, ſtatim ac in motum erumpere poteſt, produci exiſtimo, <lb></lb>& quandiu motus perſeverat, permanere; eadem enim gravi<lb></lb>tas, quæ primo momento illum effecit, reliquis conſequentibus <lb></lb>momentis conſervare valet; finis, quò refertur, & cujus causâ <lb></lb>productus eſt, adhuc obtineri poteſt, videlicet motus; liberè <lb></lb>deſcendenti corpori nullum objicitur impedimentum; nihil <lb></lb>adeſt, quod ipſius concepti impetûs interitum exigat: ergo im<lb></lb>petum à gravibus deſcendentibus conceptum non perire in mo<lb></lb>tu ſi dixerimus, ſimilitudinem veri nos conſecutos arbitror. </s> <lb></lb> <s id="s.005096">Quoniam verò gravitas inter eas cauſas enumeratur, in quibus <lb></lb>ineſt efficiendi neceſſitas, & quandiu opus eſt juxta naturæ pro<lb></lb>poſitum, quantum poſſunt, efficiunt; ſingulis momentis, qui<lb></lb>bus poteſt deſcendere, ſingulos impetûs gradus æquales priori<lb></lb>bus adjicit, adeò ut, quot momenta motum metiuntur, tot gra<lb></lb>dus impetûs poſtremo momento intenſionem conſtituant, cui <lb></lb>motûs velocitas reſpondeat. </s> <s id="s.005097">Velocitatum igitur incrementa <lb></lb>fiunt juxta naturalem numerorum progreſſionem 1.2.3.4.5, &c. <pb pagenum="690" xlink:href="017/01/706.jpg"></pb>nam juxta hanc eandem ſeriem impetûs, velocitatis cauſa, au<lb></lb>getur, ſingulis gradibus in ſingula momenta additis. </s> </p> <p type="main"> <s id="s.005098">Ab omni tamen infinitatis ſuſpicione recedendum eſt hìc, <lb></lb>ubi momentorum vocabulum uſurpo, quaſi infinita puncta <lb></lb>temporis agnoſcerem, & ad vim percuſſionis infinitam adſtruen<lb></lb>dam, infinitis momentis ſingulis impetûs gradum tribuerem. </s> <lb></lb> <s id="s.005099">Quemadmodum enim corpora punctis prorsùs individuis non <lb></lb>conſtare ſuadetur multiplici argumento præſertim ex Aſympto<lb></lb>tis lineis deſumpto, ita motui atque tempori puncta omnibus <lb></lb>omnino partibus carentia nunquam concedenda cenſui. </s> <s id="s.005100">Sed <lb></lb>huic verbo, cum momentum dico, ſubjecta notio eſt, minima <lb></lb>temporis particula Phyſica, quæ licèt particulas alias adhuc mi<lb></lb>nores contineat ſibi ordine ſuccedentes, ex quibus illa conſti<lb></lb>tuitur, tota tamen ad primi impetûs effectionem ita requiritur, <lb></lb>ut juxta naturæ leges nihil effici poſſet motûs, niſi integra illa <lb></lb>temporis particula ſuppeteret. </s> <s id="s.005101">Hujuſmodi autem non indivi<lb></lb>duas particulas minimas certas atque æquales in tempore, aut <lb></lb>motu, finito non eſſe niſi certo numero definitas manifeſtum <lb></lb>eſt: Quapropter ſicut momentorum, ita & graduum impetûs <lb></lb>æqualium multitudo finita eſt. </s> </p> <p type="main"> <s id="s.005102">Verùm nemo temerè hanc momentorum multitudinem ad <lb></lb>calculos revocare inſtituat; res enim planè incerta eſt. </s> <s id="s.005103">Utique <lb></lb>tardiſſimos reperiri & languidiſſimos motus aliquos novimus, <lb></lb>qui diu latent, nec niſi poſt tempus benè conſpicuum demum <lb></lb>innoteſcunt: Ex quo deprehendimus in tempore aut motu, qui <lb></lb>ſenſibus percipi poſſit, multas numerari hujuſmodi momento<lb></lb>rum myriadas: ſi enim in uno aliquo motu exiguo particulæ il<lb></lb>lius ſibi ex ordine ſuccedentes reſpondent motui longiſſimè ma<lb></lb>jori (cujuſmodi eſt cælorum motus) ex quo definitur tempus, <lb></lb>& in hoc plurimæ partes notabiles, & ſub Phyſicam menſuram <lb></lb>cadentes numerantur, utique & in illo plurimæ particulæ om<lb></lb>nem ſenſus aciem fugientes inveniuntur. </s> <s id="s.005104">Et quidem ſi cum il<lb></lb>lis <expan abbr="Aſtronõmis">Aſtronomis</expan> philoſophemur, qui cæleſtium graduum minu<lb></lb>ta uſque eò in ſexageſimas partiuntur, ut demum in ſcrupulis <lb></lb>Decimis conſiſtant, cùm Æquatoris gradus quindecim in Pri<lb></lb>mo mòbili uni horæ reſpondeant, ſatis conſtat, quantus ſit hu<lb></lb>juſmodi ſcrupulorum Decimorum numerus, in quorum fluxum <lb></lb>unica hora reſolvatur: ac proinde in uno horæ minuto Secun-<pb pagenum="691" xlink:href="017/01/707.jpg"></pb>do, hoc eſt in pulſu arteriæ, continentur pluſquam decies mil<lb></lb>lies millena millia myriadum hujuſmodi ſcrupulorum Decimo<lb></lb>rum, quæ momenta appellari poſſunt. </s> <s id="s.005105">Quapropter illud uni<lb></lb>cum generatim ſtatuere poſſumus, in quolibet tempore Phyſi<lb></lb>cè notabili plurima eſſe momenta, quamvis eorum certum nu<lb></lb>merum explicare nequeamus: ideóque cùm incrementa velo<lb></lb>citatum menſuram deſumant ex momentorum numero, qui <lb></lb>ſemper unitatis additione augetur, intenſio autem impetûs ha<lb></lb>beat graduum numerum parem numero momentorum; quàm <lb></lb>difficiles explicatus habet momentorum multitudo, tam obſcu<lb></lb>ra eſt impetûs intenſio; ſi minutam ſubtilitatem perſequamur. </s> <lb></lb> <s id="s.005106">Sed ſi datum tempus in aliquot particulas noſtro arbitratu <lb></lb>diſtinguamus, quo plures fuerint hujuſmodi particulæ, eò pro<lb></lb>piùs accedemus ad id, quod experimentis deprehenſum eſt; vi<lb></lb>delicet, etiam ſi ſpatia in motu decurſa juxta ſeriem naturalem <lb></lb>numerorum augeantur in motu, demum eorum collectiones in<lb></lb>cipiendo à quiete habere inter ſe duplicatam Rationem tempo<lb></lb>rum inveniemus. </s> </p> <p type="main"> <s id="s.005107">Comparatis igitur invicem motibus alicujus corporis gravis <lb></lb>deſcendentis, cujus motus unus jam innotuerit, quantum ſcili<lb></lb>cet ſpatij dato tempore confecerit, innoteſcet, quantus futurus <lb></lb>ſit alio tempore motus, ſi fiant ut quadrata datorum temporum, <lb></lb>ita & ſpatia; vel quanto tempore percurrendum ſit ſpatium de<lb></lb>finitum, ſi fiant ut Radices quadratæ datorum ſpatiorum, ita & <lb></lb>tempora. </s> <s id="s.005108">Quia nimirum poſita illa incrementa impetûs, & ve<lb></lb>locitatum, atque ſpatiorum juxta ſeriem naturalem numero<lb></lb>rum ab unitate incipientem conſtituunt collectiones habentes <lb></lb>inter ſe proximè Rationem duplicatam temporum. </s> <s id="s.005109">Habemus <lb></lb>experimento globum, argillaceum unciarum octo percurrere <lb></lb>uno minuto Secundo horæ pedes 15, & duobus Secundis pe<lb></lb>des 60, hoc eſt ſpatium quadruplum, & quia tempora ſunt ut 1 <lb></lb>ad 2, ſpatia ſunt ut quadrata, ſcilicet ut 1 ad 4. Ponamus in uno <lb></lb>Secundo eſſe momenta 10000; ſunt igitur ultimo momento <lb></lb>10000 gradus velocitatis ſimiles & æquales primo gradui primi <lb></lb>momenti, & ſpatium ultimo hoc momento decurſum, ad ſpa<lb></lb>tium primi momenti eſt ut 10000 ad 1. Coge igitur in ſum<lb></lb>mam omnia ſpatia incipiendo ab unitate uſque ad 10000, vi<lb></lb>delicet ultimi termini dimidiato quadrato adde ejuſdem ultimi <pb pagenum="692" xlink:href="017/01/708.jpg"></pb>termini ſemiſſem, & prodibit omnium ſpatiorum ſumma. </s> <s id="s.005110">Ultimi <lb></lb>termini 10000 quadratum eſt 100000000, cui adde ipſum ul<lb></lb>timum terminum; & hujus ſummæ medietas 50005000 eſt <lb></lb>ſumma minimorum ſpatiorum, quibus conflantur pedes 15. In <lb></lb>duobus Secundis erunt momenta 20000, & ſimiliter invenitur <lb></lb>ſumma 200010000 minimorum ſpatiorum, quibus conſtant <lb></lb>pedes 60. Non ſunt quidem duæ hujuſmodi ſummæ hìc <lb></lb>inventæ 50005000 & 200010000, omnino ut 1 ad 4, ſed ut 1 <lb></lb>ad (3 49995/50005): verùm tantula differentia (10/50005) quid officit allato ex<lb></lb>perimento? </s> <s id="s.005111">an potuit obſervari? </s> <s id="s.005112">Si 4. ſunt pedes 60, quid <lb></lb>ſunt (3 49995/50005)? utique pedes (59 49855/50005); deeſt igitur pedis particula <lb></lb>(150/50005), hoc eſt unciæ quaſi pars vigeſima ſeptima. </s> <s id="s.005113">Quis autem <lb></lb>tam minutæ ſubtilitati locus ſit in obſervando motu? </s> </p> <p type="main"> <s id="s.005114">Ut autem perſpicuè appareat hanc hypotheſim incrementi <lb></lb>juxta ſeriem naturalem numerorum conſentire cum experi<lb></lb>mentis, & ſpatia ſe habere ut quadrata temporum, ſtatuamus <lb></lb>eadem ſpatia, ut primum ſit ad ſecundum in Ratione 50005000 <lb></lb>ad 200010000. Radix primi ſpatij eſt (7071 5959/14142), Radix autem <lb></lb>ſecundi ſpatij eſt (14142 6918/14142); quæ ſunt ut 1 ad 2, ſi fractiones <lb></lb>contemnantur; nec repugnat experimentum; nam tantula <lb></lb>differentia temporum, ne ſit Ratio præcisè dupla, diſcerni <lb></lb>non potuit: quarum enim partium (7071 5959/14142) eſt unum minu<lb></lb>tum Secundum horæ, deeſt unius partis (5000/14142), ut ſint duo mi<lb></lb>nuta Secunda, hoc eſt unius pulsûs alteriæ pars una vicies mil<lb></lb>leſima deſideratur, ut ſint planè duo Secunda. </s> <s id="s.005115">Quære argu<lb></lb>menta, ſi qua potes; an experimento revinces eſſe planiſſimè <lb></lb>duo minuta Secunda, nec vel unicum momentum defuiſſe? </s> </p> <p type="main"> <s id="s.005116">Hoc idem, quod exempli causâ in Ratione duplâ temporum <lb></lb>& quadruplâ ſpatiorum explicatum eſt, in cæteris pariter de<lb></lb>prehendes. </s> <s id="s.005117">Fac enim eſſe tempus quadruplum, hoc eſt Secun<lb></lb>dorum 4, hoc eſt minimorum temporis 40000. Tota collectio <lb></lb>ſpatiorum erit 800020000. Quare 50005000 ad 800020000 <lb></lb>eſt ut 1 ad (15 49925/50005), quaſi ut 1 ad 16; eſt autem defectus (60/50005). <lb></lb>Spatium igitur uno Secundo decurſum cum ſit ped. 15, qua<lb></lb>tuor Secundis erit ped. 240 minùs una ferè ſexageſima nona <lb></lb>particulâ unciæ. </s> <s id="s.005118">Viciſſim ut tempora invenias in ſubduplicatâ <pb pagenum="693" xlink:href="017/01/709.jpg"></pb>Ratione ſpatiorum, quære illorum tanquam quadratorum Ra<lb></lb>dices; & primi quidem Radix eſt, ut priùs fuit inventa <lb></lb>(7071 5959/14142), ſecundi Radix eſt (28284 8836/14142), quarum Ratio eſt <lb></lb>quadrupla, ſi fractiones ſpernantur; at aliquid deeſt, ut ſint <lb></lb>integra quatuor Secunda minuta horæ; qui defectus demum <lb></lb>vix major eſt quàm (1/6667) unius pulsûs arteriæ. </s> </p> <p type="main"> <s id="s.005119">Cùm itaque conſtituta hypotheſis incrementi ſpatiorum, <lb></lb>velocitatis, atque impetûs juxta ſeriem naturalem numerorum <lb></lb>ſit naturæ conſentanea, nequè Phyſicè repugnet experimentis, <lb></lb>non debemus eſſe ſolliciti, ut aliam quæramus hypotheſim ad <lb></lb>ſtatuenda incrementa exquiſitè juxta numeros impares; cùm <lb></lb>maximè in aquâ ob majorem reſiſtentiam, quam in illâ divi<lb></lb>denda inveniunt corpora gravia deſcendentia, non exactè ſer<lb></lb>vari eandem Rationem incrementorum, quæ in aëre apparet, <lb></lb>experimenta iterata declarent, quamvis ad illam Rationem <lb></lb>proximè accedant, ut apud Ricciolium tom. </s> <s id="s.005120">1. Almag. lib. 9. <lb></lb>ſect.4. cap. 16. n. </s> <s id="s.005121">16. varia experimenta afferentem legi poteſt. </s> <lb></lb> <s id="s.005122">Præterquam quod ſi tam in aëre quàm in aquâ adhibeantur in <lb></lb>experimentum corpora ſecundùm gravitatem ſpecificam ab il<lb></lb>lis minimum diſcrepantia, ſtatim apparebit non ſervari illam <lb></lb>temporum atque ſpatiorum Analogiam: id quod pariter obſer<lb></lb>vabitur, ſi in diverſis liquoribus eadem gravia corpora dimit<lb></lb>tantur: varia ſcilicet eſt reſiſtentia; hæc autem latet, quando <lb></lb>grave dimiſſum valde differt à gravitate, aut levitate medij. </s> </p> <p type="main"> <s id="s.005123">His ita conſtitutis, percuſſionis vires, quatenus ex velocita<lb></lb>te oriuntur, proximè definire poterimus, ſi innoteſcant ſpatia, <lb></lb>per quæ idem corpus grave liberè deſcendit; nam hinc inno<lb></lb>teſcet Ratio intenſionum impetûs ultimo deſcensûs momento, <lb></lb>quo contingit percuſſio. </s> <s id="s.005124">Eſt ſiquidem numerus graduum im<lb></lb>petus in motu naturali libero concepti par numero momento<lb></lb>rum motûs; at momenta, quibus conſtant tempora motuum <lb></lb>inæqualium, ſunt in Ratione ſubduplicatâ Rationis ſpatiorum: <lb></lb>igitur ſicut Radices quadratæ ſpatiorum <expan abbr="indicãt">indicant</expan> Rationem tem<lb></lb>porum, ita pariter eædem indicant Rationem intenſionum im<lb></lb>petûs. </s> <s id="s.005125">Quare ſi alicujus gravis ex datâ altitudine cadentis per<lb></lb>cuſſio manifeſta fuerit, facilè inferemus, quanta proximè ſit <lb></lb>futura ejuſdem percuſſio ex majori, aut minori altitudine, ſi fiat <pb pagenum="694" xlink:href="017/01/710.jpg"></pb>ut Radix datæ altitudinis prioris ad Radicem poſterioris altitu<lb></lb>dinis, ita nota percuſſio ad quæſitam percuſſionem. </s> </p> <p type="main"> <s id="s.005126">Neque hìc conabor dicta confirmare experimentis tùm à <lb></lb>Ricciolio loc. </s> <s id="s.005127">cit. </s> <s id="s.005128">cap.16. n.12, tùm à Merſenno tom.3. in Re<lb></lb>flexionibus Phyſico-Mathemat. cap. 8. allatis, ex quibus proxi. </s> <lb></lb> <s id="s.005129">mé infertur hæc Ratio ſubduplicata ſpatiorum. </s> <s id="s.005130">Nam cùm adhi<lb></lb>bita ſit libra, ut in alteram lancem cadens pondus ex diverſis <lb></lb>altitudinibus dimiſſum elevaret pondera inæqualia oppoſitæ <lb></lb>lanci impoſita, res eſt anceps & incetta. </s> <s id="s.005131">Quandoquidem, ut <lb></lb>obſervavi jam tum ab anno 54 labentis ſæculi ſcriptis Romæ de <lb></lb>hoc eodem argumento publicè traditis, & funiculi, ex quibus <lb></lb>lanx percuſſa pendet, diſtrahuntur, & libræ jugum flectitur, <lb></lb>immò & lanx ipſa ictum cadentis ponderis excipiens flexilis <lb></lb>eſt; ac propterea impetûs vim retundunt, cùm maximè vi <lb></lb>elaſticâ ſe reſtituentes conantur ſurſum. </s> <s id="s.005132">Præterquam quod, ſi <lb></lb>pondus cadens non exactè incidat in lancis centrum reſpon<lb></lb>dens extremitati jugi, plurimum intereſt ad varianda momen<lb></lb>ta, prout libræ brachium aut decurtatum aut productum intel<lb></lb>ligitur. </s> <s id="s.005133">Quò autem majus eſt pondus in oppoſitâ lance attollen<lb></lb>dum ex vi depreſſionis lancis percuſſæ, magis reſiſtit, ac proin<lb></lb>de locus eſt flexioni majori ipſius libræ, aut funiculorum <lb></lb>diſtractioni, præſertim ſi lanx fuerit concava, & cadens pon<lb></lb>dus illam tangens prolabatur in depreſſiorem lancis locum. </s> <s id="s.005134">Ex <lb></lb>quo accidit, ut docuit experientia, aucto pondere elevando non <lb></lb>ſatis eſſe dimittere pondus cadens ex altitudine, quæ ſit ad prio<lb></lb>rem altitudinem ut quadratum ponderis majoris ad quadratum <lb></lb>ponderis minoris initio elevati; percuſſio enim contingit in lan<lb></lb>ce, cujus reſiſtentiam ad hoc, ut vi ponderis cadentis deprima<lb></lb>tur, metitur reſiſtentia ponderis elevandi in oppoſitâ lance: at <lb></lb>hæc ſi fuerit major quàm reſiſtentia jugi, aut lancis, ne flecta<lb></lb>tur, aut funiculorum ne diſtrahantur, in hac flexione aut <lb></lb>diſtractione inſumitur vis percuſſionis, quin oppoſita lanx attol<lb></lb>latur. </s> <s id="s.005135">Quapropter ex altitudine adhuc majori dimittendum eſt <lb></lb>pondus cadens; nam adhuc majore impetu concepto tam validè <lb></lb>percutiet lancem, ut reſiſtentia jugi & lancis ad flexionem ul<lb></lb>teriorem, atque funiculorum ad longiorem diſtractionem, ma<lb></lb>jor ſit quàm reſiſtentia ponderis oppoſitæ lancis: atque adeò <lb></lb>lanx percuſſa non deprimetur ſolùm, quantum funiculorum <pb pagenum="695" xlink:href="017/01/711.jpg"></pb>diſtractio & ipſa flexio permittit, ſed adhuc ulteriùs; atque op<lb></lb>poſita lanx attolletur, quatenus impetûs vires excedunt oppo<lb></lb>ſitæ gravitatis reſiſtentiam. </s> </p> <p type="main"> <s id="s.005136">Ex his, quæ de Percuſſionibus corporum gravium naturali<lb></lb>ter deſcendentium hactenus dicta ſunt, conjectura ſumenda eſt <lb></lb>de reliquis percuſſionibus, quæ motûs originem non à ſolâ gra<lb></lb>vitate ducunt: nam in his pariter ex motûs velocitate, qua in<lb></lb>dicatur intenſio impetûs, oritur validior percuſſio: nam ſive <lb></lb>conceptus à potentiâ vivente impetus, ſive extrinſecùs im<lb></lb>preſſus (niſi novam offendat reſiſtentiam, quæ illum retundat <lb></lb>ac minuat) augetur novo impetu, quem potentia ſimiliter ap<lb></lb>plicata, iiſdémque viribus prædita, nec obſtaculo ullo aut reti<lb></lb>naculo impedita, ſequentibus momentis efficere poteſt: ideó<lb></lb>que quò major eſt potentiæ percutientis motus quoad ſpatium, <lb></lb>cæteris paribus, validiùs percutit. </s> <s id="s.005137">Cæteris, inquam, paribus; <lb></lb>nam ſi poſterioribus momentis motûs, minor impetus addatur à <lb></lb>potentiâ movente, quàm deperdatur ex priore impetu impreſ<lb></lb>ſo, quem natura repugnans excutit, langueſcit motus, & mi<lb></lb>nuitur impetus: aut ſi tantumdem acquiratur impetûs, quan<lb></lb>tum deperditur, motus eſt æquabilis, nec ad validiorem per<lb></lb>cuſſionem quicquam confert diuturna potentiæ moventis appli<lb></lb>catio, cum eadem ſit impetûs intenſio in fine horæ, atque in <lb></lb>primo momento. </s> <s id="s.005138">Quia tamen frequentiùs (etiam ſi fortè ali<lb></lb>quid antiquioris impetûs ex novâ reſiſtentiâ deteratur) plus ad<lb></lb>ditur, velocitas incrementum ſumit, ſi potentia maneat diutiùs <lb></lb>applicata. </s> </p> <p type="main"> <s id="s.005139">Hinc patet, cur, cum quis hoſtem ſariſsâ confodere tentat, <lb></lb>aut poſtes craſſiore fuſte arietare, brachium retrahat quantum <lb></lb>poteſt; ut nimirum potentia diutiùs applicata maneat in motu, <lb></lb>ſempérque novum impetum gignens ſariſſæ aut fuſti imprimat. </s> <lb></lb> <s id="s.005140">Sic duobus digladiantibus, ſi alter alteri ſiniſtrum latus obver<lb></lb>tat, & tam longo enſe utatur, ut protecto corpore poſſit manum <lb></lb>valde retrahere, hic validiſſimum ictum infliget extento dex<lb></lb>tro brachio, tùm quia ex celerrimâ corporis totius converſione <lb></lb>impetus aliquis brachio communicatur præter impetum, quem <lb></lb>conferunt muſculi movendo brachio deſtinati, tùm quia diu<lb></lb>tiùs movetur gladius à manu per majus ſpatium. </s> <s id="s.005141">Quod ſi eo <lb></lb>ictu, quem Itali <emph type="italics"></emph>Quartam<emph.end type="italics"></emph.end> vocamus, hoſtem impetat, adhuc va-<pb pagenum="696" xlink:href="017/01/712.jpg"></pb>dior erit ictus, quia longior motus; nam in converſione corpo<lb></lb>ris obvertitur hoſti dextrum latus præcisè ita, ut brachium to<lb></lb>tum extendi queat; & præterea pars exterior manûs deorſum <lb></lb>convertitur, ex quo propter conformationem juncturarum cubi<lb></lb>ti & manus ictus evadit duos ferè digitos longior, quàm ſi non <lb></lb>fieret hujuſmodi manûs converſio: demum cùm ſiniſtrum bra<lb></lb>chium in poſteriora projiciatur extentum, fit, ut corpori major <lb></lb>impetus in anteriora poſſit imprimi citrà periculum cadendi; <lb></lb>brachium ſiquidem eo pacto in poſteriora projectum, tranſlato <lb></lb>gravitatis, ut gravitationis, centro ſervat totius corporis æqui<lb></lb>librium. </s> </p> <p type="main"> <s id="s.005142">Nec diſſimili ratione manifeſtum fit, cur pugnum validiùs <lb></lb>impingant, qui brachium magis contrahunt, ideóque validio<lb></lb>res ictus ab iis proveniant, qui longiora habentes brachia ea <lb></lb>magis contrahunt; ſicut calce fortiùs impetunt, qui longiora <lb></lb>habent crura; quia videlicet diutiùs moventur, magiſque im<lb></lb>petum augent & velocitatem, antè ictum. </s> <s id="s.005143">Sic vides ab equis <lb></lb>calcitronibus pedem in anteriora retrahi, & ab irato tauro col<lb></lb>lum depreſſum in poſteriora inflecti, corpore pariter curvato, <lb></lb>& quaſi in poſteriora retracto, ut longiore motu validiùs impe<lb></lb>tant: hinc qui propior eſt equo calcitranti, minùs læditur, quia <lb></lb>nondum tantum impetum concepit, quantum longiore motu <lb></lb>concepiſſet. </s> </p> <p type="main"> <s id="s.005144">Hujuſmodi percuſſionibus à potentiâ vivente, quæ ſuos mo<lb></lb>tus ex arbitrio temperat, provenientibus certam legem ſtatui <lb></lb>non poſſe nemo non videt, ſive corpus percutiens impetu ex<lb></lb>trinſecùs aſſumpto feratur naturâ omnino repugnante, ſive im<lb></lb>petus impreſſus cum impetu vi interiore acquiſito in eumdem <lb></lb>motum conſpirent. </s> <s id="s.005145">Hoc unum tanquam manifeſtò comper<lb></lb>tum atque deprehenſum tenemus, quod in longiore motu factâ <lb></lb>novi impetûs acceſſione velocitas augetur, & vis percuſſionis <lb></lb>eſt major. </s> <s id="s.005146">Hinc ſicut quando fiſtucâ cadente pali in terram <lb></lb>adiguntur, initio illa modicum attollitur, quia exigua ſuperan<lb></lb>da eſt reſiſtentia, hac autem creſcente quò altiùs adacti fuerint, <lb></lb>magis illa attollitur; ſic lignarios fabros clavum in tabulam in<lb></lb>figentes, initio quidem breviore mallei motu uti videmus, quem <lb></lb>deinceps augent, donec demum totâ brachij extenſione con<lb></lb>nitantur, prout reſiſtentiæ incrementa validiore percuſſione <pb pagenum="697" xlink:href="017/01/713.jpg"></pb>vinci oportet. </s> <s id="s.005147">Sic ruſticos ligna findentes altiùs elevare ſecu<lb></lb>rim, aut tuditem, quo cuneum percutiant, obſervamus, quò <lb></lb>major eſt ſcindendi difficultas, ut auctus impetus velociorem <lb></lb>motum efficiat, quem percuſſio conſequitur. </s> </p> <p type="main"> <s id="s.005148">Cum itaque duæ velocitates inter ſe comparatæ conferri poſ<lb></lb>ſint vel ratione temporis, vel ratione ſpatij, ita ut vel æqualia <lb></lb>ſpatia inæqualibus temporibus, vel inæqualia ſpatia tempore <lb></lb>eodem conficiant; illa utique erit major velocitas, quando in <lb></lb>motu temporis brevitas, & ſpatij amplitudo conſenſerint. </s> <s id="s.005149">Hinc <lb></lb>percuſſio contingit validior à corpore, quod multum ſpatij bre<lb></lb>vi tempore decurrat, quàm à corpore conficiente minus ſpatij <lb></lb>longiore tempore. </s> <s id="s.005150">Propterea quæ de velocitate dicta ſunt, & <lb></lb>percuſſionum viribus, ſpectatâ diuturnitate motûs, ita intelli<lb></lb>genda ſunt, ut corpus percutiens vel eodem tempore plus ſpa<lb></lb>tij, vel breviore tempore æquale ſpatium, vel breviore tempore <lb></lb>plus ſpatij decurrat: hoc enim ex majore impetûs intenſione <lb></lb>oritur. <lb></lb></s> </p> <p type="main"> <s id="s.005151"><emph type="center"></emph>CAPUT VIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005152"><emph type="center"></emph><emph type="italics"></emph>An validior ſit ictus Mallei à ſitu Verticali ad <lb></lb>Horizontalem, an verò ab Horizontali ad <lb></lb>Verticalem deſcendentis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005153">CErtum eſt percuſſiones fieri validiores, quando cum impe<lb></lb>tu ab extraneâ potentiâ impreſſo conſentit vis intrinſeca <lb></lb>ipſius corporis percutientis motu naturali deſcendentis; ipſum <lb></lb>enim ſuum pariter impetum concipit, quem addit impreſſo: ſic <lb></lb>ſaxum, quod ex editâ turri deorſum rectâ projicis, validiùs per<lb></lb>cutit, quàm ſi illud dimitteres ſponte ſua caſurum. </s> <s id="s.005154">Hinc qui <lb></lb>malleo deorſum percutit aliquod corpus, ad motum eundem, <lb></lb>cum percutientis impulſu conſpirantem invenit mallei gravita<lb></lb>tem, & citrà omnem controverſiam majorem ictum infligit, <lb></lb>quàm ſi ſurſum, aut in latus urgeret malleum, gravitate aut re<lb></lb>pugnante, aut ſaltem nihil juvante. </s> </p> <pb pagenum="698" xlink:href="017/01/714.jpg"></pb> <p type="main"> <s id="s.005155">Porrò cùm circa juncturam brachij cum humero, tanquam <lb></lb>circa centrum, deſcribatur ſemicirculus deſcendens, in quo <lb></lb>ſunt duo Quadrantes, alter cùm brachium ſummè elevatum in <lb></lb>perpendiculo exiſtens deſcendit, ut fiat horizonti parallelum, <lb></lb>alterùm brachium à poſitione horizonti parallelâ ad imum de<lb></lb>primitur, ut iterum in perpendiculo ſtatuatur; dubitari poteſt, <lb></lb>an mallei ictus juxta duas haſce poſitiones ſint omnino æquales, <lb></lb>an verò inæquales; & hoc quidem non ex vi impulſionis exter<lb></lb>næ, quam æquabilem ponimus, homine æqualiter connitente, <lb></lb>brachio æqualiter extento, & datâ eâdem manubrij longitudi<lb></lb>ne, ſed ratione ipſius gravitatis mallei naturaliter deſcendentis. </s> <lb></lb> <s id="s.005156">Prioris motus, quo in circulo Verticali malleus deprimitur uſ<lb></lb>que ad planum horizonti parallelum, in quo fit percuſſio, exem<lb></lb>plum præbent tùm fabri ferrarij incudem tundentes, tùm ligna<lb></lb>rij clavos aſſeribus in plano horizontali conſtitutis infigentes. </s> <lb></lb> <s id="s.005157">Poſterioris autem motûs, quo malleum horizonti parallelum <lb></lb>uſque eò deprimimus, ut fiat perpendicularis, ſpecimen habe<lb></lb>mus, cum corpus in pavimento jacens, aut non procul ab illo, <lb></lb>ita percutimus, ut percuſſum moveatur horizonti ferè paralle<lb></lb>lum; cujuſmodi eſſet, quando cuneum jacenti ſaxo ſubjicere <lb></lb>conamur, aut ligneam pilam ludentes malleo excutimus. </s> </p> <p type="main"> <s id="s.005158">Ut autem dilucidè propoſita quæſtio exponatur, ſecernendus <lb></lb><figure id="id.017.01.714.1.jpg" xlink:href="017/01/714/1.jpg"></figure><lb></lb>eſt mallei motus naturalis ab <lb></lb>ea parte, quam externa im<lb></lb>pulſio addit; & perinde con<lb></lb>ſiderandus eſt malleus, atque <lb></lb>ſi manubrij extremitas axi in<lb></lb>fixa eſſet circa eum verſatilis, <lb></lb>adeò ut ſibi relictus malleus <lb></lb>arcum deſcendendo deſcri<lb></lb>beret. </s> <s id="s.005159">Sit malleus AB; & <lb></lb>manubrij extremitas A ſit <lb></lb>circa axem in A verſatilis; <lb></lb>centrum autem gravitatis <lb></lb>mallei intelligatur in B: quod <lb></lb>quandiu in perpendiculo im<lb></lb>minet axi A, totam ſuam vim <lb></lb>in illum exerens ſuſtinetur, <pb pagenum="699" xlink:href="017/01/715.jpg"></pb>nec motum inchoat, niſi à perpendiculo BA removeatur; hoc <lb></lb>verò ubi tranſgreſſus fuerit malleus, deſcenſum molitur: ſed <lb></lb>quia rigido manubrio connectitur cum Axe A, cogitur in latus <lb></lb>ſecedere, & deſcribere arcum BC, cui motui reſpondet ſolùm <lb></lb>deſcenſus BD Sinus Verſus anguli BAC; & deſcripto integro <lb></lb>Quadrante BE, deſcenſum metitur Radius BA. </s> </p> <p type="main"> <s id="s.005160">Verùm quamvis motus hujuſmodi per arcum BE ſit conſen<lb></lb>taneus propenſioni gravitatis, quæ proinde ſingulis momentis <lb></lb>novam impetûs particulam concipiens motum, quantum poteſt, <lb></lb>accelerat, non tamen illa Ratio ſervatur, de qua ſuperiori Capi<lb></lb>te dictum eſt, ut Ratio ſpatiorum ſit in duplicatâ Ratione tem<lb></lb>porum; neque enim hìc liberè deſcendit malleus, ſed in ſingu<lb></lb>lis punctis Quadrantis alia atque alia habet momenta deſcen<lb></lb>dendi, ſingula minora momento, quod haberet idem malleus <lb></lb>nullo impedimento prohibitus; quod momentum integrum ille <lb></lb>obtinet tantummodo in Quadrantis extremitate E, ubi nullâ <lb></lb>ratione ſuſtinetur aut retinetur ab axe A manubrium ſuſtinen<lb></lb>te, aut retinente. </s> <s id="s.005161">Eſt autem momentum gravitatis in unoquo<lb></lb>que arcûs puncto, quaſi illa eſſet in plano ibi circulum tangen<lb></lb>te, ac propterea inclinato: idcirco, ut lib.1. cap.13. dictum eſt, <lb></lb>ejuſdem gravitatis momentum in plano inclinato, ad momen<lb></lb>tum in lineâ perpendiculari eam Rationem habet, quam in <lb></lb>triangulo rectangulo, cujus angulus Verticalis ſit æqualis an<lb></lb>gulo inclinationis plani, habet Perpendiculum ad Hypothe<lb></lb>nuſam. </s> <s id="s.005162">Quare in puncto C malleus habet momentum, ac ſi eſ<lb></lb>ſet in plano inclinato FC, & ad momentum liberum ita ſe ha<lb></lb>bet, ut DF ad FC, hoc eſt per 8.lib.6. ut DC ad CA, & ſi<lb></lb>militer in puncto G ut IG ad GA. </s> <s id="s.005163">Ex quo patet momentorum <lb></lb>incrementa analoga eſſe incrementis Sinuum Rectorum arcu<lb></lb>bus ſubinde majoribus convenientium. </s> </p> <p type="main"> <s id="s.005164">At hìc, ubi de gravitatis momentis ſermo inſtituitur, caven<lb></lb>dum eſt, ne quem fortè in errorem inducat ambiguitas nomi<lb></lb>nis. </s> <s id="s.005165">Nam quando in C momentum dicimus eſſe ut DC, & in <lb></lb>G momentum eſſe ut IG, hoc intelligendum eſt præcisè ra<lb></lb>tione poſitionis, quatenus in hoc aut illo puncto conſtituta gra<lb></lb>vitas concipitur, nullâ habitâ ratione antecedentis motûs aut <lb></lb>quietis: & ſub voce momenti Gravitatis hæc ſubjecta eſt ſen<lb></lb>tentia, ut gravitas mallei, quæ non impedita ſingulis punctis <pb pagenum="700" xlink:href="017/01/716.jpg"></pb>temporis conciperet novum impetum ut AC, AG &c. </s> <s id="s.005166">quia à <lb></lb>rigido manubrio modò magis, modò minùs ſuſtinetur, quando <lb></lb>eſt in C, impeditur, ne concipiat impetum niſi ut DC, & in G <lb></lb>ut IG, atque ita de cæteris, donec in E concipiat impetum ut <lb></lb>AE. </s> <s id="s.005167">Cæterùm quia in præcedentibus temporis punctis acqui<lb></lb>ſitæ ſunt particulæ impetûs reſpondentes. </s> <s id="s.005168">Sinubus Rectis præ<lb></lb>cedentium arcuum, ea fit impetûs intenſio, ac proinde motûs <lb></lb>velocitas, quæ omnium illorum Sinuum aggregato ferè reſpon<lb></lb>deat: Et in fine Quadrantis in E vis eſt complectens omnes <lb></lb>impetus, quibus additio facta eſt ſemper non tamen æqualis pri<lb></lb>mo impetui, qui valdè languidus fuit, ſed ſemper major atque <lb></lb>major, prout Sinus Recti excreverunt. </s> </p> <p type="main"> <s id="s.005169">Et hæc quidem de Superiore Quadrante. </s> <s id="s.005170">Jam inferior Qua<lb></lb>drans conſiderandus eſt, in quo Axis A retinet malleum, ne ex <lb></lb>E recto tramite deſcendat, ſed eum cogit deflectere, & arcum <lb></lb>ES deſcribere, in cujus ſingulis punctis momentum perinde eſt <lb></lb>atque in plano inclinato. </s> <s id="s.005171">Quare in L intelligitur deſcendens <lb></lb>in plano inclinato KL, & ibi ejus momentum ad momentum <lb></lb>liberum eſt ut RK ad KL, hoc eſt ut ML, ad LK, hoc eſt per <lb></lb>8. lib.6. ut MA ad AL, hoc eſt ut Sinus Complementi arcûs <lb></lb>EL ad Radium. </s> <s id="s.005172">Similiter in N eſt ut PA ad AN; & ſic de <lb></lb>cæteris. </s> <s id="s.005173">Eſt autem manifeſtum hujuſmodi Sinus Complemen<lb></lb>torum eoſdem planè eſſe cum Sinubus Rectis Superioris Qua<lb></lb>drantis, ſed ordine præpoſtero acceptis, atque adeò horum ag<lb></lb>gregatum eſſe illorum ſummæ æquale. </s> </p> <p type="main"> <s id="s.005174">Non tamen hinc ſtatim conficitur eandem eſſe in S vim mal<lb></lb>lei deſcendentis ex E, atque eſt in E vis ejuſdem deſcendentis <lb></lb>ex B. </s> <s id="s.005175">Non inficior æqualem in utroque Quadrante produci <lb></lb>impetûs entitatem, ſi in ſummam referantur omnes impetûs <lb></lb>particulæ, quæ momentis ſingulis efficiuntur; ſed an æqualem <lb></lb>demum conflent intenſionem, ex qua vis percuſſionis oritur, <lb></lb>non omninò temerè, ut mihi quidem videor, ſubdubito. </s> <s id="s.005176">Cùm <lb></lb>enim acquiſitus impetus interveniente reſiſtentiâ imminuatur, <lb></lb>ac debilitetur, & quidem eò magis, ſi à rectâ ſecundùm natu<lb></lb>ram lineâ magis declinare cogitur; utique inſtitutâ Superioris <lb></lb>cum Inferiori Quadrante comparatio oſtendit in illo quidem <lb></lb>reſiſtentiam ſemper decreſcere, in hoc ſemper augeri, ac proin<lb></lb>de impetum prioribus momentis acquiſitum, licèt aliquid in <pb pagenum="701" xlink:href="017/01/717.jpg"></pb>conſequentibus amittat, tamen hujus decrementi menſurâ ma<lb></lb>gis ac magis extenuatâ, non adeò in Superiore Quadrante lan<lb></lb>gueſcere, ſicut in inferiore, ubi reſiſtentia ſemper augetur, & <lb></lb>impetus magis ac magis deteritur. </s> <s id="s.005177">Adde impetum de novo <lb></lb>productum in poſterioribus momentis, in ſuperiore quidem <lb></lb>Quadrante eſſe majorem, in inferiore verò minorem. </s> <s id="s.005178">Quare <lb></lb>cùm in poſtremis motûs momentis in ſuperiore Quadrante <lb></lb>multus producatur impetus, & ferè nulla ſit reſiſtentia, in in<lb></lb>feriore autem Quadrante multa inveniatur reſiſtentia, & valdè <lb></lb>exiguus impetus producatur, ſatis probabili conjecturâ ali<lb></lb>quam ſtatuemus ictuum inæqualitatem, ita ut aliquanto vali<lb></lb>dior ſit ex B in E, quàm ex E in S. </s> </p> <p type="main"> <s id="s.005179">Neque obſtat, quod in E maximus producatur impetus, qui <lb></lb>deinde in L & N, atque conſequentibus punctis additamen<lb></lb>tum accipiat, licèt ſemper minus ac minus, quo ita augetur, ut <lb></lb>ſemper incitetur motus. </s> <s id="s.005180">Hoc enim non facit, quin impetus in E <lb></lb>conceptus majoribus ſemper decrementis imminuatur uſque <lb></lb>in S, & impetus in L conceptus ſimiliter magis langueſcat, at<lb></lb>que ita de cæteris. </s> <s id="s.005181">Finge ſcilicet nullum impetum novum con<lb></lb>cipi in L, aut nullum in N, adhuc impetus in E conceptus deor<lb></lb>ſum tenderet, ſed retinaculo illo debilitatus languidiùs adduce<lb></lb>ret malleum in S: idem dic de quolibet impetu ſingulis mo<lb></lb>mentis concepto, qui creſcente reſiſtentiâ majoribus decre<lb></lb>mentis imminueretur, & languidè veniret in S. </s> <s id="s.005182">At quoniam <lb></lb>plurima ſunt momenta in breviſſimi temporis particulâ, tot <lb></lb>ſunt reliqui impetus, ut ſimul conſtituant notabilem inten<lb></lb>ſionem. </s> </p> <p type="main"> <s id="s.005183">Quod autem reſiſtentia in ſuperiore Quadrante minuatur, <lb></lb>argumento non eſt opus; manifeſtum quippe eſt gravitatem in <lb></lb>arcu BE deſcendentem ſubinde transferri à plano magis incli<lb></lb>nato in minùs inclinatum, & magis accedens ad perpendicu<lb></lb>lare: quis autem neget deſcendenti gravitati eò minùs obſtare <lb></lb>planum ſubjectum, quò fuerit minùs inclinatum? </s> <s id="s.005184">Atqui an<lb></lb>gulus CAF minor eſt angulo GAH, anguli autem ad C & G <lb></lb>ſunt recti; igitur angulus AFC major eſt angulo AHG; at<lb></lb>que propterea planum FC eſt magis inclinatum, quàm pla<lb></lb>num HG, & cætera plana conſequentia uſque ad planum per<lb></lb>pendiculare in E. </s> <s id="s.005185">Contra verò in inferiore Quadrante reſiſten-<pb pagenum="702" xlink:href="017/01/718.jpg"></pb>tiam ſemper augeri ex eo conſtat, quòd à plano perpendicula<lb></lb>ri ad inclinatum, immò ad ſemper magis atque magis inclina<lb></lb>tum, fit tranſitus, donec demum deſcendens gravitas veniat ad <lb></lb>planum horizontale. </s> <s id="s.005186">Angulus videlicet inclinationis plani eſt <lb></lb>æqualis angulo, quem denotat arcus in inferiore Quadrante <lb></lb>decurſus. </s> <s id="s.005187">Nam in triangulo ALK, cujus in baſim AK cadit <lb></lb>perpendicularis LM, ex 8.lib.6. angulo KAL æqualis eſt an<lb></lb>gulus KLM, & propter paralleliſmum linearum ML & KR, <lb></lb>angulo KLM æqualis eſt alternus LKR angulus inclinationis <lb></lb>plani KL; igitur hic angulus inclinationis eſt æqualis angulo, <lb></lb>quem denotat arcus EL. </s> <s id="s.005188">Idem dic de angulo EAN & cæte<lb></lb>ris, qui ſemper majores indicant gravitatem tranſire ad plana <lb></lb>magis & magis inclinata. <lb></lb></s> </p> <p type="main"> <s id="s.005189"><emph type="center"></emph>CAPUT IX.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005190"><emph type="center"></emph><emph type="italics"></emph>Quomodo percuſsiones ex mole pendeant.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005191">QUamvis ad validiorem ictum infligendum corporis percu<lb></lb>tientis velocitas, impetûs intenſionem indicans, pluri<lb></lb>mum conferat, ut dictum eſt; non ad hanc tamen velut ad uni<lb></lb>cam cauſam referenda eſt vis percuſſionis; ſed & corporis ejuſ<lb></lb>dem percutientis moles attendenda eſt: videmus ſcilicet ex <lb></lb>mole ipsâ percuſſiones augeri, cæteris paribus; perinde enim <lb></lb>eſt, atque ſi tot corpora percutientia eſſent, quàm multiplex <lb></lb>eſt moles major collata cum minore. </s> <s id="s.005192">Nam ſi nota eſt vis per<lb></lb>cutiendi, quæ ineſt globulo duarum unciarum cadenti ex cer<lb></lb>tâ quadam altitudine utique probabilis conjectura & ratio ſua<lb></lb>det ſextuplam eſſe vim globi ex ſimili materiâ unciarum duo<lb></lb>decim ex altitudine eâdem cadentis: gravitas ſiquidem ſexies <lb></lb>multiplicata etiam impetum efficere poteſt ſextuplum, non qui<lb></lb>dem intenſivè, ſed entitativè; neque enim pro Ratione molis <lb></lb>augetur velocitas, quippe quæ requireret ſextuplam intenſio<lb></lb>nem. </s> <s id="s.005193">Hanc tamen majorem vim Ratione molis ita intelligi <lb></lb>velim, ut medij reſiſtentia diſſimuletur: nam eo ipſo, quod mo-<pb pagenum="703" xlink:href="017/01/719.jpg"></pb>les ejuſdem ſecundùm ſpeciem gravitatis augetur, etiam ſuper<lb></lb>ficiem augeri neceſſe eſt, quæ non eâdem facilitate medium di<lb></lb>vidit. </s> <s id="s.005194">Sed quoniam ita augeri poteſt moles, ut non ſimilem <lb></lb>ſervet figuram, ſed alias atque alias induat figuras manente <lb></lb>æquali gravitate, ac propterea valde incerta eſt reſiſtentiæ <lb></lb>menſura, quæ ex medij diviſione oritur, prout hanc aut illam <lb></lb>faciem medio dividendo obvertit ipſa moles; hinc eſt, quod il<lb></lb>lam reſiſtentiam tantiſper diſſimulare licet, dum reliquas per<lb></lb>cuſſionis cauſas veſtigamus. </s> <s id="s.005195">Cæterùm illius quoque ratio eſt <lb></lb>habenda, ut aliquid virium detractum intelligatur percuſſioni, <lb></lb>ubi & moles & ſpatium conſideratur; quatenus velocitas, auctâ <lb></lb>mole, hoc eſt aucto ex medij reſiſtentia impedimento, aliquan<lb></lb>tulum imminuitur, ita tamen ut pariter plures majoris molis <lb></lb>partes, collatis viribus medium urgentes, aliquid afferant faci<lb></lb>litatis in dividendo medio, adeóque etiam velocitatis. </s> </p> <p type="main"> <s id="s.005196">Ex his habetur percuſſionis vires componi ex mole & ex ve<lb></lb>locitate corporis percutientis: Moles ſiquidem determinat en<lb></lb>titativè menſuram impetûs ſingulis momentis producti, veloci<lb></lb>tas indicat intenſionem, hoc eſt ſummam impetuum in motu <lb></lb>acquiſitorum, hoc eſt ejuſdem impetûs gravitati, aut potentiæ <lb></lb>virtuti, primo momento reſpondentis multiplicationem. </s> <s id="s.005197">Qua<lb></lb>re ſi duorum corporum ictus comparentur, percuſſionum Ratio <lb></lb>erit compoſita ex Rationibus velocitatum, & gravitatum, ſeu <lb></lb>potentiarum, quibus vis movendi tribuitur. </s> <s id="s.005198">Velocitatem au<lb></lb>tem illam intelligo, quæ ratione impetus acquiſiti conveniret <lb></lb>corpori eo momento, quo percutit, niſi inveniret reſiſtentiam: <lb></lb>Hujuſmodi verò impetûs eo momento intenſionem indicat ſpa<lb></lb>tium in antecedenti motu decurſum, ex quo, prout dictum eſt <lb></lb>cap.7. cognoſcitur Ratio temporum, quibus eſt analoga inten<lb></lb>ſio impetûs. </s> <s id="s.005199">Hinc ſi cadat ex altitudine 100 palmorum globus <lb></lb>unciarum duarum, deinde ex altitudine decem palmorum glo<lb></lb>bus unciarum 12, ſunt duæ Rationes, altera velocitatum, hoc eſt <lb></lb>intenſionum impetûs in ſubduplicatâ Ratione altitudinum, vi<lb></lb>delicet ut 10 ad (3 16/100), altera gravitatum ut 1 ad 6; quæ compoſi<lb></lb>tæ dant Rationem ut 10 ad (18 96/100); ac proinde percuſſio globi <lb></lb>minoris eſtimari poterit proximè ut 10, majoris ut (18 96/100): Nam <lb></lb>duæ unciæ majoris deſcendendo per decem palmos haberent <pb pagenum="704" xlink:href="017/01/720.jpg"></pb>impetum ut (3 16/100): ergo ſexies duæ unciæ habent entitativè im<lb></lb>petum ut (18 96/100). Quare ſi gravitates fuerint reciprocè ut velo<lb></lb>citates, hoc eſt impetûs intenſiones, erunt æquales percuſſio<lb></lb>nes, ut eſt manifeſtum. </s> </p> <p type="main"> <s id="s.005200">Hoc autem, quod de gravitate motu naturali deſcendente <lb></lb>dictum eſt, de potentiis pariter, ſervatâ analogiâ, eſt intelligen<lb></lb>dum (aſſumendo ſcilicet loco gravitatis vim ipſam potentiæ, <lb></lb>quæ ſimiliter conata perſeveret in motu antecedente percuſſio<lb></lb>nem) ſi innoteſcat, quantum potentiæ inæqualiter conentur, <lb></lb>& per inæqualia ſpatia moveant idem corpus, quo ictum infli<lb></lb>gunt; nam ex Rationibus conatuum, & velocitatum componi<lb></lb>tur Ratio percuſſionum. </s> <s id="s.005201">At ſi potentiæ duæ inæqualiter co<lb></lb>nantes per inæqualia ſpatia moveant inæqualia corpora, qui<lb></lb>bus alteri corpori ictus infligatur, attendendum eſt, an ſolùm <lb></lb>tam diuturnus ſit motus ictum præcedens, ut nihil impreſſi im<lb></lb>petûs deteratur: nam ſi alternis quibuſdam incrementis & de<lb></lb>crementis modò augeatur, modò minuatur, non eſt habenda <lb></lb>ratio totius temporis, aut ſpatij, in quo factus eſt motus: quis <lb></lb>enim exiſtimet aptè computari poſſe, utrùm navis percurrerit <lb></lb>ſex, aut octo milliaria, ut ejus ictus, quo cymbam percutit, <lb></lb>cognoſcatur? </s> <s id="s.005202">Quare ſatius erit in hujuſmodi motibus ab impe<lb></lb>tu extrinſecùs impreſſo provenientibus, qui ut plurimum re<lb></lb>pugnantem habet ipſius corporis naturam, nec totus permanet <lb></lb>quemadmodum impetus acquiſitus, ipſam velocitatem conſi<lb></lb>derare, quatenus apparet non multo tempore antè ictum: tunc <lb></lb>enim, quia potentia movens eum impetum imprimit, qui ſatis <lb></lb>ſit ad molem illam movendam tantâ velocitate, ut impetus hu<lb></lb>juſmodi innoteſcat, & molis & velocitatis ratio habenda eſt; <lb></lb>atque idcircò ad comparandas invicem percuſſiones compo<lb></lb>nenda eſt Ratio ex Rationibus molium, & velocitatum. </s> <s id="s.005203">Hinc <lb></lb>ſi navis oneraria lentè moveatur velocitate ut duo, & navis alia <lb></lb>ſextuplo minor moveatur velocitate ut decem (quia videlicet <lb></lb>paulò antè ictum obſervatum eſt, quo tempore illa procedebat <lb></lb>duos paſſus, hanc percurriſſe decem paſſus) ictus majoris ad <lb></lb>ictum minoris erit ut 12 ad 10, compoſitis ſcilicet Ratione mo<lb></lb>lium 6 ad 1, & Ratione velocitatum 2 ad 10. </s> </p> <p type="main"> <s id="s.005204">Hìc autem ubi Molis nomen uſurpamus, cavendus eſt in <pb pagenum="705" xlink:href="017/01/721.jpg"></pb>vocabuli ambiguitate lapſus: neque enim corporis tantummo<lb></lb>do amplitudinem, quatenus ſub Geometricam dimenſionem <lb></lb>cadens ſpatium occupat, intelligere oportet, verùm etiam na<lb></lb>turam ipſam atque ſubſtantiam: ea ſcilicet, quæ minore ſecun<lb></lb>dùm ſpeciem gravitate prædita ſunt, ſub magnis dimenſionibus <lb></lb>parum habent ſubſtantiæ atque materiæ, ideóque & tenuem <lb></lb>movendi ac impetum producendi virtutem; neque propterea <lb></lb>quod molem magnam præſeferant, validiora in percutiendo <lb></lb>cenſenda ſunt, quaſi à globo ligneo librarum duarum, quia fe<lb></lb>rè decuplo major eſt globo plumbeo ejuſdem ponderis, expecta<lb></lb>ri poſſet validior ictus, ſi ex eâdem altitudine dimittantur: nam <lb></lb>vis producendi impetum connata eſt ſubſtantiæ, quâ ſubſtantia <lb></lb>talis eſt, non quantitati, prout extenſio eſt. </s> <s id="s.005205">Quare ubi molis <lb></lb>habendam eſſe rationem diximus, ut fiat Rationum Compoſi<lb></lb>tio, ex qua percuſſionum incrementa aut decrementa inno<lb></lb>teſcant, ipſam potiſſimùm ſubſtantiam intelligimus, quam, non <lb></lb>niſi intra idem genus corporis, extenſio major aut minor conſe<lb></lb>qui ſolet: propterea ſi ferrei cylindri ictus, atque lignei, confer<lb></lb>re invicem volueris, non ipſos cylindros, quatenus cylindri ſunt <lb></lb>ſub tantâ baſi & altitudine, dimetiri oportet, ſed potiùs eorum <lb></lb>gravitatem, ut quanta ſit moles virtutis ipſi naturæ atque ſub<lb></lb>ſtantiæ reſpondens, ex gravitate inferatur. </s> </p> <p type="main"> <s id="s.005206">Non tamen idcircò extenſionis atque figuræ animadverſio <lb></lb>otioſa eſt, aut contemnenda, in percuſſionibus; quinimmò non <lb></lb>oſcitanter conſideranda, ut deprehendatur, qua ſui corporis <lb></lb>parte validiſſimum ictum infligat inſtrumentum percutiens. </s> <lb></lb> <s id="s.005207">Hoc autem tripliciter potiſſimùm movetur, videlicet, primò ad <lb></lb>perpendiculum deſcendendo motu naturali; deinde horizonta<lb></lb>liter, ſeu obliquè, cùm à dextrâ in ſiniſtram, aut viciſſim à lævâ <lb></lb>in dexteram, aut in anteriora extrinſecùs motu recto impelli<lb></lb>tur; demum in orbem, circuli arcum deſcribendo. </s> </p> <p type="main"> <s id="s.005208">Et quidem corpus ſponte ſuâ deſcendens, quodcumque tan<lb></lb>dem illud ſit, ſuam habet Directionis lineam, per quam in mo<lb></lb>tu Centrum gravitatis progreditur. </s> <s id="s.005209">In infimâ igitur corporis <lb></lb>parte ipſa Directionis linea definit punctum, in quo ſi fiat cor<lb></lb>poris percutientis contactus, ille erit validiſſimus ictus, quem <lb></lb>hujuſmodi corpus ex datâ altitudine deſcendens infligere po<lb></lb>teſt, ibi quippe maximam reperit reſiſtentiam, cum æquales <pb pagenum="706" xlink:href="017/01/722.jpg"></pb>vires hinc atque hinc conſiſtentes ibi conſpirent, & obicem <lb></lb>motui directè oppoſitum offendant, adeò ut neque in hanc, <lb></lb>neque in illam partem Centrum gravitatis dirigatur. </s> <s id="s.005210">Quod ſi <lb></lb>punctum contactûs corporum colliſorum non ſit in lineâ Di<lb></lb>rectionis corporis cadentis, ſed à latere; eò validior erit ictus, <lb></lb>quo minore intervallo punctum contactûs ab hujuſmodi lineâ <lb></lb>Directionis aberit; magis videlicet opponitur motui directo, <lb></lb>quàm ſi ab eâ longiùs abeſſet: quando enim contactus procul <lb></lb>eſt à lineâ Directionis, ab hac minùs deflectere cogitur cen<lb></lb>trum gravitatis, quod multò magis repellendum eſſet à contactu <lb></lb><figure id="id.017.01.722.1.jpg" xlink:href="017/01/722/1.jpg"></figure><lb></lb>propiore. </s> <s id="s.005211">Sic globus, cujus centrum gra<lb></lb>vitatis ſit C, deſcendens per lineam Di<lb></lb>rectionis CD, ſi percutiat puncto D, om<lb></lb>nium validiſſimum ictum infligit, quia <lb></lb>corpus percuſſum omninò opponitur mo<lb></lb>tui CD, nec centro C relinquit locum <lb></lb>ſaltem obliquè deſcendendi: at verò ſi <lb></lb>contactus fiat in E, impeditur quidem <lb></lb>deſcenſus globi per rectam CD ulteriùs <lb></lb>productam, poteſt tamen centrum gravitatis deſcendere deſcri<lb></lb>bendo circa punctum E manens arcum CF; quapropter in E <lb></lb>minorem invenit reſiſtentiam quàm in D, ubi nihil deſcende<lb></lb>re poteſt, ſi ſubjectum corpus loco non cedat. </s> <s id="s.005212">Similiter ſi con<lb></lb>tactus fiat in G, adhuc impeditur motus directus per CD, atta<lb></lb>men centrum gravitatis C poteſt obliquè deſcendere deſcri<lb></lb>bendo arcum CH. </s> <s id="s.005213">Sed quoniam per arcum CF magis decli<lb></lb>nat à perpendiculo, & minùs deſcendit, quàm per arcum CH <lb></lb>(quamvis arcus illi æquales ponantur paribus ℞adiis EC, & <lb></lb>GC deſcripti) propterea magis impeditur motus in contactu E <lb></lb>propiori lineæ Directionis, quàm in G remotiori. </s> <s id="s.005214">Cum itaque <lb></lb>eò validiorem ictum infligant corpora percutientia, quò ma<lb></lb>jorem inchoato motui reſiſtentiam offendunt, manifeſtum eſt <lb></lb>in corporibus naturali motu deſcendentibus validiſſimum eſſe <lb></lb>ictum in puncto, quod lineæ directionis motûs reſpondet, ſem<lb></lb>pérque imbecilliores eſſe ictus, quò magis puncta contactûs ab<lb></lb>ſunt à lineâ Directionis. </s> </p> <p type="main"> <s id="s.005215">Hoc idem, quod de lineâ Directionis gravium ſponte ſuâ <lb></lb>deſcendentium dictum eſt, analogiâ ſervatâ, traducendum eſt <pb pagenum="707" xlink:href="017/01/723.jpg"></pb>ad ea corpora, quæ externo impulſu agitata motu recto ſivè <lb></lb>Horizonti parallelo, ſivè ad Horizontem aut obliquè, aut ad <lb></lb>perpendiculum, inclinato moventur. </s> <s id="s.005216">Cum enim, ex hypo<lb></lb>theſi, partes omnes hujuſmodi corporis impulſi æquali veloci<lb></lb>tate per æqualia ſpatia moveantur, æqualem impetum ſingulæ <lb></lb>recipiunt, à movente impreſſum. </s> <s id="s.005217">Similiter igitur in corpore <lb></lb>illo concipiendum eſt punctum, quod <emph type="italics"></emph>Centrum Impetûs<emph.end type="italics"></emph.end> vocari <lb></lb>poteſt, quia illud æquales hinc & hinc Impetus circumſtant, <lb></lb>quemadmodum Centrum Gravitatis dicitur, circa quod æqua<lb></lb>lia gravitatis momenta diſpoſita intelliguntur. </s> <s id="s.005218">Hinc ſi corpo<lb></lb>ris particulæ fuerint omnino homogeneæ, adeóque æquè capa<lb></lb>ces impetûs recipiendi, illud idem erit Centrum Impetûs, quod <lb></lb>eſt centrum molis, ſeu magnitudinis; nam eadem plana, quæ <lb></lb>molem æqualiter dividunt, etiam æqualiter dividunt Impetum <lb></lb>per ſingulas particulas æquabiliter diffuſum. </s> <s id="s.005219">At ſi non ejuſdem <lb></lb>generis fuerint partes corpus illud componentes, ſed raræ aliæ, <lb></lb>aliæ denſæ, hoc eſt ex materiâ partim tenui, partim conſtipatâ, <lb></lb>ſicut non eſſet idem Centrum Gravitatis, atque Centrum <lb></lb>Magnitudinis, ita neque idem eſt cum Molis centro Centrum <lb></lb>Impetûs impreſſi; quia, ut ex Projectis conſtat, ea quæ ſecun<lb></lb>dùm ſpeciem leviora ſunt, cæteris paribus, minorem impetum <lb></lb>concipiunt (& globuli ex argillâ efficti, quos baliſtæ evibrant, <lb></lb>majorem ictum infligunt, quàm pares globuli lignei, qui ſunt <lb></lb>argillâ leviores) ac proinde Centrum Impetûs impreſſi aſſumi <lb></lb>poteſt idem, ac punctum illud, quod in motu naturali eſſet <lb></lb>Centrum gravitatis ipſi corpori inexiſtens. </s> </p> <p type="main"> <s id="s.005220">Quare in motibus corporum externâ vi impulſorum atten<lb></lb>denda eſt pariter linea, ſecundùm quam dirigitur motus hujuſ<lb></lb>modi Centri Impetûs: & punctum illud in corporis percutien<lb></lb>tis ſuperficie, quod linea directionis motûs à Centro Impetûs <lb></lb>ducta deſignat, ipſum eſt, in quo corpus percutiens vim ſuam <lb></lb>validiſſimè exercet. </s> <s id="s.005221">Cum enim omnia plana per hanc Di<lb></lb>rectionis motûs lineam tranſeuntia (quorum illa eſt communis <lb></lb>ſectio) dividant univerſum Impetum in partes hinc & hinc <lb></lb>æquales, quippe quæ etiam per Centrum Impetûs tranſeunt, <lb></lb>ita ex percuſſione in puncto illo impeditur motus, ut neque ad <lb></lb>hanc, neque ad illam partem deflectere poſſit corpus impactum <lb></lb>in obicem, qui reſiſtit. </s> <s id="s.005222">Quod ſi punctum contactûs fuerit ex-<pb pagenum="708" xlink:href="017/01/724.jpg"></pb>tra lineam Directionis motûs, inæquales ſunt impetus, & majo<lb></lb>re præpollente, corpus pergit in motu, quamvis ad latus in<lb></lb>flectatur ſivè magis, ſivè minùs, prout majus aut minus fuerit <lb></lb>intervallum inter punctum contactûs, & lineam Directionis <lb></lb>motûs. </s> </p> <p type="main"> <s id="s.005223">Sit corpus AB, quod tranſlatum à potentiâ impellente ha<lb></lb>beat Centrum Impetûs C, & linea, per quam dirigitur motus, <lb></lb><figure id="id.017.01.724.1.jpg" xlink:href="017/01/724/1.jpg"></figure><lb></lb>ſit CD, cui parallelæ <lb></lb>ſunt lineæ à ſingulis par<lb></lb>tibus in motu deſcriptæ. </s> <lb></lb> <s id="s.005224">Si ergo in obicem incur<lb></lb>rat punctum D, ita im<lb></lb>peditur motus, ut ulte<lb></lb>riùs promoveri nequeat <lb></lb>corpus, niſi obex loco cedat; quia nimirum impetus in DA <lb></lb>æqualis eſt impetui in DB, ideò neutra pars æquali impetu af<lb></lb>fecta promoveri poteſt: eſt igitur maxima reſiſtentia, & ictus <lb></lb>validiſſimus. </s> <s id="s.005225">Sin autem non puncto. </s> <s id="s.005226">D, ſed puncto E fiat per<lb></lb>cuſſio ſecundùm eandem directionem GE, jam impetus ſunt <lb></lb>inæquales, & minor impetus eſt in EA, quàm in EB; proin<lb></lb>de pars EB validior pergens in motu inflectitur circa obicem <lb></lb>in puncto E, tanquam circa centrum, & reſiſtentia eſt minor, <lb></lb>quàm ad punctum D. </s> <s id="s.005227">Simile quid contingit, fi fiat percuſſo <lb></lb>in puncto F, multo enim major impetuum inæqualitas interce<lb></lb>dit inter FA, & FB, quàm inter EA, & EB, atque faciliùs <lb></lb>fit converſio & inflexio motûs circa obicem in puncto F, quàm <lb></lb>in puncto E: arcus ſiquidem majore Radio FD deſcriptus mi<lb></lb>nùs deſciſcit à rectitudine lineæ, per quam dirigitur motus, <lb></lb>quàm arcus minore Radio ED deſcriptus. </s> <s id="s.005228">Quò igitur magis <lb></lb>punctum contactûs in percuſſione abeſt à puncto D, eò infir<lb></lb>mior eſt ictus, minorem quippe invenit reſiſtentiam. </s> </p> <p type="main"> <s id="s.005229">At ſi corpus idem AB ita impellatur, ut linea directionis <lb></lb>motûs ducta ex C centro impetûs ſit CA, ſimiliter conſtat va<lb></lb>lidiſſimum ictum fieri in A, imbecilliorem verò in extremis an<lb></lb>gulis ejuſdem ſuperficiei. </s> <s id="s.005230">Hinc vides, cur ex vetere diſciplinâ <lb></lb>Poliorceticâ ad murorum, aut poſtium expugnationem, arie<lb></lb>tes, quibus concutiebantur, non planâ facie, ſed convexâ com<lb></lb>muniter, aut acutâ conſtruerentur: quia ſcilicet trabem ferro <pb pagenum="709" xlink:href="017/01/725.jpg"></pb>in capite armatam funibus ſuſpenſam (ne ſuſtinendi laborem <lb></lb>ſubirent, ſed vires omnes in motu impenderent) retro ducen<lb></lb>tes, ac deinde propellentes, non planè horizontaliter, ſed qua<lb></lb>ſi circulariter movebant; planum autem ſi fuiſſet trabis caput, <lb></lb>ictus inflictus fuiſſet ab extremo illius ſuperficiei latere, non ve<lb></lb>rò à partibus circa medium exiſtentibus, à quibus multò vali<lb></lb>dior ictus expectari potuiſſet; quemadmodum certiùs contingit <lb></lb>facie convexâ, aut in apicem deſinente. </s> </p> <p type="main"> <s id="s.005231">Si demum linea directionis motûs eſſet CF, utique in F eſ<lb></lb>ſet validiſſimus ictus, quia planum FH bifariam divideret <lb></lb>æqualiter univerſum impetum, & impetus FAH æqualis eſſet <lb></lb>impetui FBH. </s> <s id="s.005232">Eſſet autem infirmior ictus, quem infligeret <lb></lb>punctum D, cujus directio DI parallela directioni Centri FH <lb></lb>inæqualiter divideret impetum, & pars impetûs DBI minor <lb></lb>eſſet parte DAHI; quapropter hæc circa obicem in D mo<lb></lb>veri poſſet, & minorem inveniret reſiſtentiam quàm in F. </s> </p> <p type="main"> <s id="s.005233">Ubi obſervandum eſt non aptè quæri, quonam in puncto va<lb></lb>lidiſſimus fiat ictus, niſi pariter ſtatuatur, quænam ſit linea Di<lb></lb>rectionis motûs: Nam in eodem puncto D validiſſimus eſt ictus, <lb></lb>ſi directio fuerit CD, quia tunc eſt maxima reſiſtentia; nullus <lb></lb>eſt ictus in directione CA, quia nihil illi opponitur; imbecillis <lb></lb>eſt ictus in Directione CF, quia mediocrem offendit reſiſten<lb></lb>tiam. </s> <s id="s.005234">Præterea comparatis invicem Directionibus CD & CA, <lb></lb>validior eſt ictus in A quàm in D; plures ſiquidem partes in <lb></lb>eandem longitudinis lineam directè conſpirantes plus obtinent <lb></lb>virium, quàm pauciores in lineâ latitudinis: præterquam quod <lb></lb>partium ad latera adjacentium lineæ, quæ propiores ſunt lineæ <lb></lb>Directionis Centri Impetûs, quaſi in unam Phyſicè coaleſcunt; <lb></lb>id quod non contingit partibus notabili intervallo disjunctis ab <lb></lb>illâ Directionis lineâ: quæ eatenus ſolùm in percuſſionem con<lb></lb>ſentiunt, quatenus cum intermediis conjunctæ nexu non facilè <lb></lb>diſſolubili eas pariter juvant; nam ſi eſſet corpus percutiens in <lb></lb>plures partes, ceu virgulas, diſſectum, iis, quæ obicem con<lb></lb>tingerent, manentibus, reliquæ ſinè ictu excurrerent: propterea <lb></lb>etiam conjunctæ faciliùs à directâ poſitione deflectentes corpo<lb></lb>ris longitudinem inflectunt: ut in tenui & gracili ligno accide<lb></lb>re poteſt extremis partibus A & B, quæ ex impulſu, quo pro<lb></lb>moventur, poſſunt circa punctum D inflecti; ex quo infirmior <pb pagenum="710" xlink:href="017/01/726.jpg"></pb>percuſſio, quàm cùm tanta eſt corporis craſſities, ut pro longi<lb></lb>tudine breviore non valeat flecti. </s> <s id="s.005235">At ſi cum his directionibus <lb></lb>CD & CA, ad perpendiculum incidentibus in faciem corpo<lb></lb>ris percutientis, comparetur Directionis linea CF obliquè in<lb></lb>cidens, licèt longior ſit linea CF, quàm CD, non idcirco va<lb></lb>lidior eſt in F ictus Directionis CF, quàm in D ictus Directio<lb></lb>nis CD: id quod oritur ex minore reſiſtentiâ ratione obliqui<lb></lb>tatis; faciliùs quippe poteſt ulteriùs excurrere corpus obliquè <lb></lb>percutiens, quàm ſi directè percuteret. </s> </p> <p type="main"> <s id="s.005236">Porrò non levis error obreperet minùs accuratè perpenden<lb></lb>tibus ea, quæ hactenus de Centro Impetûs diſputata ſunt, ſi hoc <lb></lb>Centrum abſolutè in eo inſtrumento, quo ad percutiendum <lb></lb>utimur, quærendum eſſe exiſtimarent: non enim rarò etiam <lb></lb>ejus, à quo inſtrumentum impellitur, conſiderandus eſt impe<lb></lb>tus & motus. </s> <s id="s.005237">Sic quando duo lanceis concurrunt, non eſt æſti<lb></lb>manda percuſſio ex ſolo impetu lanceæ impreſſo, verùm etiam <lb></lb>ex eo, quem militis corpori imprimit equus, cui currenti inſi<lb></lb>det, immò & ipſius equi impetus, quem virtute ſuâ animali <lb></lb>concipit: univerſum quippe hunc impetum retundi oportet <lb></lb>ab eo, qui ictum recipit: hinc ſi miles minùs robuſtus fuerit, <lb></lb>infirmior eſt ictus, quia ipſo ictûs momento ille cedit, & perin<lb></lb>de eſt, atque ſi lancea ipſa cederet, aut flecteretur. </s> <s id="s.005238">Non eſt igi<lb></lb>tur Centrum impetûs in lanceâ ipsâ, ſed potiùs in corpore mili<lb></lb>tis non procul ab equo; ac proinde inclinata lancea, ut, quam <lb></lb>minimùm fieri poſſit, recedat à poſitione parallelâ lineæ Di<lb></lb>rectionis motûs, & ab hac lineâ non longè abſit, validiſſimum <lb></lb>ictum infliget: hoc autem quia faciliùs obtinetur longiore lan<lb></lb>ceâ, quàm breviore, ideò, cæteris paribus, præſtat longiore <lb></lb>lanceâ uti. </s> </p> <p type="main"> <s id="s.005239">Res autem aliter ſe habet, quando percuſſio contingit inſtru<lb></lb>mento non ampliùs cohærente ipſi cauſæ, à qua impetum re<lb></lb>cipit, ſed jam ab eâ disjuncto; in eo enim præcisè eſt Cen<lb></lb>trum Impetûs, & attendenda eſt linea Directionis motûs ab <lb></lb>hujuſmodi centro ducta, ut vis percuſſionis maxima innoteſcat. </s> </p> <p type="main"> <s id="s.005240">At hìc quæris; ſi haſtam manu ſtringentes impetum illi im<lb></lb>primimus, & brachium pariter impetum concipit, atque ex <lb></lb>utroque impetu æſtimandus eſt ictus; cur validiùs haſtam ean<lb></lb>dem intorquemus jaculantes, quàm manu tenentes? </s> <s id="s.005241">Sic anti-<pb pagenum="711" xlink:href="017/01/727.jpg"></pb>quis, ad acriùs feriendum, placuit haſtis amentatis uti, ut poſt <lb></lb>ejaculationem haſtam loris ligatam retraherent, iterúmque <lb></lb>evibrarent: Eſſe autem validiorem ictum hinc cognoſces, quòd <lb></lb>haſtæ evibratæ mucro altiùs infigitur objectæ tabulæ, quàm <lb></lb>cùm illam manu retinentes ſimiliter tabulam cuſpide percuti<lb></lb>mus. </s> <s id="s.005242">Ex multiplici causâ id petendum videtur. </s> <s id="s.005243">Et primò qui<lb></lb>dem, quia cùm haſtam manu ſtringimus, caro, quæ eſt in volâ <lb></lb>manûs, illicò cedit, ac obicem haſtæ reſiſtentem offendit, ex <lb></lb>qua ceſſione minuitur impetûs; qui & multo magis debilitatur, <lb></lb>ſi brachium pariter in poſteriora modicè revocemus timentes, <lb></lb>ne ex præconcepto impetu, & corporis percuſſi reſiſtentiâ <lb></lb>oriatur nimia aliqua partium convulſio, aut dolor: hoc autem <lb></lb>incommodum vitatur in haſtâ jam emiſsâ. </s> <s id="s.005244">Deinde quando ali<lb></lb>quid jaculamur, ultimo momento, quo illud tenemus, bra<lb></lb>chium validiſſimo conatu in anteriora movemus, ſtatímque re<lb></lb>trahimus dimittentes miſſile, cui propterea plurimus impetus <lb></lb>imprimitur: conſtat autem non poſſe à nobis haſtam retinenti<lb></lb>bus (alia ſcilicet eſt muſculorum contentio & motio) moveri <lb></lb>brachium motu adeò concitato. </s> <s id="s.005245">Demum impetus breviſſimo <lb></lb>illo motu tantâ vi productus in miſſili ſuam retinet directionem <lb></lb>(quicquid ſit, an gravitas inſita aliquid officiat) quæ in lon<lb></lb>giore motu brachij ſi non dimittatur, aliquantulum labefacta<lb></lb>tur, eo quod plures motus circa diverſa centra, videlicet circa <lb></lb>os humeri, & os cubiti, miſceantur; atque ex diverſa illâ di<lb></lb>rectione vis impetûs minuitur. </s> <s id="s.005246">Cùm itaque ex omnibus hiſce <lb></lb>cauſis major inveniatur impetus in haſtâ evibratâ, quo momen<lb></lb>to illa percutit, majorem quoquè ictum ab eâ infligi conſe<lb></lb>quens eſt. </s> </p> <p type="main"> <s id="s.005247">Ad hoc percuſſionum horizontalium genus ſpectat illa per<lb></lb>cuſſio, qua in ludo minoris tudiculæ globus unus tudiculâ im<lb></lb>pellente emiſſus alium globulum percutit. </s> <s id="s.005248">Si enim in eâdem <lb></lb>directionis motûs lineâ reperiantur centra utriuſque globi, per<lb></lb>cutientis ſcilicet & percuſſi, maximus ictus infligitur, quia <lb></lb>maximam invenit reſiſtentiam, cum totus globulus percuſſus <lb></lb>toti percutienti opponatur, cujus ſingularum partium lineæ di<lb></lb>rectionis motûs ſi producantur, occurrunt globulo percuſſo <lb></lb>(æquales ſunt globuli ex hypotheſi) quamvis ſola linea directio<lb></lb>nis Centri illum contingat. </s> <s id="s.005249">Sin autem globus emiſſus ita alium <pb pagenum="712" xlink:href="017/01/728.jpg"></pb>quieſcentem tangat, ut recta linea per contactûs punctum ducta <lb></lb>ſit utriuſque globi Tangens; & lineæ directionis motûs paralle<lb></lb>la, nullus eſt ictus, quia nullum motui impedimentum infertur. </s> <lb></lb> <s id="s.005250">Demum ſi linea, per quam dirigitur centrum globuli emiſſi, non <lb></lb>occurrat centro globi percuſſi, ita tamen ſe habeat, ut lineæ <lb></lb>utrumque globulum Tangenti occurrat extra punctum con<lb></lb>tactûs, tunc major aut minor erit ictus pro ratione impedimen<lb></lb>ti & reſiſtentiæ, prout majori aut minori parti globuli percu<lb></lb>tientis opponitur globulus percuſſus: Ex quo fit eò majorem <lb></lb>eſſe reſiſtentiam, quò linea directionis motûs Centri percu<lb></lb>tientis propiùs ad punctum contactûs occurrit lineæ Tangenti. </s> </p> <p type="main"> <s id="s.005251">Sit globus B emiſſus adversùs globum A quieſcentem, & <lb></lb><figure id="id.017.01.728.1.jpg" xlink:href="017/01/728/1.jpg"></figure><lb></lb>linea, per quam dirigitur motus <lb></lb>centri B, ſit BC occurrens <lb></lb>puncto contactûs C, atque adeò, <lb></lb>ut colligitur ex 12. lib. 3. etiam <lb></lb>centro A, ſi producta intelliga<lb></lb>tur. </s> <s id="s.005252">Hìc invenit maximam re<lb></lb>ſiſtentiam globus percutiens, ne<lb></lb>que ad hanc, neque ad illam <lb></lb>partem deflectere poteſt à priori <lb></lb>directione; linea ſiquidem Di<lb></lb>rectionis BC ad angulos rectos <lb></lb>incidit in Tangentem DE, & omnes ſingularum partium di<lb></lb>rectiones occurrerent globulo A, ſi productæ intelligantur ex<lb></lb>tra globulum B. </s> <s id="s.005253">Quòd ſi linea directionis motûs fuiſſet BF pa<lb></lb>rallela Tangenti DE, nullum planè inferretur impedimentum <lb></lb>motui à globo quieſcente A; nam partium globi impulſi Di<lb></lb>rectiones parallelæ lineæ BF, eæ eſſent, ut earum nulla incur<lb></lb>reret in globum A, ideóque nullus eſſet ictus, ubi nulla eſt re<lb></lb>ſiſtentia. </s> <s id="s.005254">At ſi globus B habeat Directionem BE, aut BG, & <lb></lb>quieſcentem globum tangeret in C, etiamſi neque BE, neque <lb></lb>BG directiones centri incurrerent in globum A, tamen par<lb></lb>tium aliquarum ejuſdem globi B Directiones parallelæ Directio<lb></lb>ni BE, aut BG, ſi productæ intelligantur, incurrerent in glo<lb></lb>bum A, atque invenirent ex eo impedimentum. </s> <s id="s.005255">Sit enim Di<lb></lb>rectio BE, & in globo B linea LO ipſi BE paralella, quæ pro<lb></lb>ducta contingeret in K globum A: utique omnes partes ſeg-<pb pagenum="713" xlink:href="017/01/729.jpg"></pb>menti OLS habentes Directionem parallelam Directioni BE, <lb></lb>quæ eſt directio Centri, inveniunt reſiſtentiam, cum earum di<lb></lb>rectiones incurrant in oppoſitum globum A. </s> <s id="s.005256">Similiter ſi Di<lb></lb>rectio Centri ſit BG, parallela Directio HI producta tangit <lb></lb>in P globum A, qui proinde opponitur Directionibus omnium <lb></lb>partium ſegmenti IHS. </s> <s id="s.005257">Cùm autem ſegmentum IHS majus <lb></lb>ſit ſegmento OLS, etiam major eſt ictus, quando Directio BG <lb></lb>ea eſt, ut Tangenti lineæ DE occurrat in puncto G non ita <lb></lb>procul à contactu C, quàm ſi directio BE ea eſſet, quæ in <lb></lb>puncto E remotiore à contactu C occurreret eidem Tangenti <lb></lb>DE: Maximam ſiquidem habet veritatis ſpeciem, in hujuſmo<lb></lb>di ictibus impetum in globo percuſſo eâ intenſione imprimi, <lb></lb>quæ proportione reſpondeat partibus, quæ directè ſecundùm <lb></lb>illam Directionis lineam impediuntur. </s> <s id="s.005258">Quoniam verò impe<lb></lb>tus globo percuſſo impreſſus illum afficit æquabiliter, hinc eſt, <lb></lb>quod ille non poteſt à percuſſione determinari, niſi ut moveatur <lb></lb>per lineam Directionis, quæ conjungat punctum contactûs C <lb></lb>cum centro A: quandoquidem æquales ſunt globi partes circa <lb></lb>centrum, adeóque & æquales impetus. </s> </p> <p type="main"> <s id="s.005259">Obſerva hìc à me aſſumptos eſſe circulos pro globis, & lineas <lb></lb>vice planorum ſecantium globos, ut res faciliùs explicaretur: <lb></lb>cæterùm quæ de lineis dicta ſunt, ſi de planis per lineas illas <lb></lb>tranſeuntibus intelligantur, rem ſimiliter ob oculos ponent, ſi <lb></lb>ipſa parallela fuerint, aut inclinata, prout de lineis conſtituta <lb></lb>eſt hypotheſis. </s> <s id="s.005260">Supereſt tertius percutientis motus, videlicet <lb></lb>in arcûs circularis ſpeciem ductus, quando, altera extremitate <lb></lb>manente, corpus in gyrum movetur. </s> <s id="s.005261">Experimentis autem do<lb></lb>cemur validiſſimum ictum non ſemper fieri ab extremitate, <lb></lb>quamvis hæc velociſſimè moveatur præ cæteris punctis alte<lb></lb>ri extremitati manenti propioribus. </s> <s id="s.005262">Ut igitur inveniatur <lb></lb>punctum, in quo corpus percutiens maximam habeat re<lb></lb>ſiſtentiam, ponendum eſt illud eſſe æquabiliter ductum, & <lb></lb>ex materiâ homogeneâ æqualiter capaci impetûs; atque <lb></lb>ibi ſanè maxima erit reſiſtentia, ubi impetûs momenta <lb></lb>æqualiter dividuntur: id quod contingit in puncto ita re<lb></lb>moto à motûs centro, ut cadat inter beſſem, & dodrantem <lb></lb>totius longitudinis, quæ habet rationem Radij, quo arcus <lb></lb>deſcribitur. </s> </p> <pb pagenum="714" xlink:href="017/01/730.jpg"></pb> <p type="main"> <s id="s.005263">Sit corporis percutientis longitudo AB. </s> <s id="s.005264">Si motu naturali <lb></lb><figure id="id.017.01.730.1.jpg" xlink:href="017/01/730/1.jpg"></figure><lb></lb>ſponte ſuâ deſcenderet, & <lb></lb>in motu poſitionem hori<lb></lb>zonti parallelam ſervaret, <lb></lb>utique validiſſimus eſſet <lb></lb>ictus in D puncto, quod <lb></lb>reſpondet centro gravita<lb></lb>tis C; eſſent enim hinc atque hinc æquales gravitates, & æqua<lb></lb>lia impetûs momenta, ut ſuperiùs dictum eſt. </s> <s id="s.005265">At manente ex<lb></lb>tremitate A, tanquam centro motûs, & corpore ipſo vi ſuæ gra<lb></lb>vitatis deſcendente, licèt ſingulæ particulæ, utpote naturæ <lb></lb>ejuſdem, paribus viribus ſint præditæ, non tamen æquali mo<lb></lb>mento feruntur; ſed cum in A retineantur, quæ puncto A pro<lb></lb>piores ſunt, magis detorquentur à directione naturalis gravita<lb></lb>tis, adeóque plus momenti habent partes inter DB, quàm in<lb></lb>ter AD conſtitutæ. </s> <s id="s.005266">Porrò momenta ſunt in Ratione Diſtan<lb></lb>tiarum: Momentum ſiquidem eſt Exceſſus virtutis moventis <lb></lb>ſupra reſiſtentiam, qua impedimentum prohibet, ne ſequatur <lb></lb>motus juxta naturalem propenſionem: quare ſingularum par<lb></lb>tium momenta ex earum motu dignoſcuntur: moventur autem <lb></lb>per circulorum arcus ſimiles, quorum etiam ſimiles ſunt Sinus <lb></lb>deſcenſum metientes, qui ſunt in Ratione Radiorum, hoc eſt <lb></lb>diſtantiarum ab A communi centro. </s> <s id="s.005267">Sic momentum puncti <lb></lb>D eſt ut AD, puncti G ut AG, puncti H ut AH, atque ita <lb></lb>de cæteris. </s> <s id="s.005268">Hinc eſt omnium momentorum ſummam conflari <lb></lb>ex illorum aggregato, quàſi ex aggregato arcuum quos deſcri<lb></lb>bunt, aut Sinuum arcubus ſimilibus reſpondentium, quorum <lb></lb>Ratio eadem eſt cum aggregato Radiorum, ex quibus deſcri<lb></lb>buntur arcus. </s> <s id="s.005269">Cum autem univerſa longitudo AB in particu<lb></lb>las æquales diviſa intelligatur, manifeſtum eſt diſtantias à cen<lb></lb>tro A conſtituere Progreſſionem Arithmeticam juxta ſeriem <lb></lb>naturalem numerorum, ac proinde punctum, quod vocari po<lb></lb>teſt <emph type="italics"></emph>Centrum Momentorum Impetûs,<emph.end type="italics"></emph.end> illud eſſe, in quo momenta <lb></lb>illa bifariam æqualiter dividuntur. </s> </p> <p type="main"> <s id="s.005270">Hoc verò punctum eſſe ultra beſſem totius longitudinis hinc <lb></lb>apparet, quòd, ſi longitudo AB in tres æquales partes diſtincta <lb></lb>intelligatur, prima centro A proxima habet momentum ut 1, ſe<lb></lb>cunda ut 2, tertia ut 3: igitur poſt finem ſecundæ, hoc eſt in G, <pb pagenum="715" xlink:href="017/01/731.jpg"></pb>videtur eſſe æqualitas momentorum; nam AG habet ut 3, <lb></lb>& GB item ut 3. Sed non eſſe in G momentorum æqualitatem <lb></lb>conſtat, ſi adhuc plures in partes AB diſtincta intelligatur, & <lb></lb>eadem ſit Ratio dupla AG ad GB. </s> <s id="s.005271">Sit AB partium 6; AG <lb></lb>eſt 4, GB 2: Momenta AG ſunt 10, GB 11; ſunt ſcilicet <lb></lb>ipſius AG momenta quatuor diſtantiarum 1. 2. 3. 4. hoc eſt 10, <lb></lb>GB verò momenta quintæ & ſextæ diſtantiæ 5 & 6, hoc eſt 11. <lb></lb>Quod ſi ponatur AB partium 9, & AG 6, GB 3; momenta <lb></lb>AG ſunt 21, GB ſunt 24. Similiter ſtatuamus longitudinem <lb></lb>AB partium 12, ſcilicet AG 8, GB 4, momenta AG ſunt 36, <lb></lb>GB 42. Item AB ſit partium 15; AG 10, GB 5: momenta <lb></lb>AG ſunt 55, GB 65. Demum AB ſit partium 18; AG 12, <lb></lb>GB 6: momenta AG ſunt 78, GB 93. Non igitur momen<lb></lb>torum æqualitas eſt præcisè in G ad beſſem longitudinis AB, <lb></lb>ſed eſt ultra G versùs B. </s> </p> <p type="main"> <s id="s.005272">Verùm, ſi AH ſit dodrans longitudinis AB, non in H, ſed <lb></lb>citrà H, inter G & H eſt quæſitum punctum, in quo momen<lb></lb>torum æqualitas invenitur. </s> <s id="s.005273">Nam ſi AB ſit partium 4, atque <lb></lb>AH ſit 3, HB verò ſit 1; momenta AH ſunt 6, & HB 4: <lb></lb>Si AB ſit part. </s> <s id="s.005274">8: & AH 6, HB 2; momenta AH ſunt 21, <lb></lb>HB 15; & ſic de cæteris ſervatâ eádem Ratione triplâ AH <lb></lb>ad HB. </s> <s id="s.005275">Cum itaque momenta AG minora ſint quàm momen<lb></lb>ta GB, contrà verò momenta AH majora ſint momentis HB, <lb></lb>conſtat æqualitatem momentorum eſſe inter G & H, hoc eſt <lb></lb>inter beſſem & dodrantem. </s> <s id="s.005276">Ubi autem proximè ſit hujuſmodi <lb></lb>punctum, deprehendes, ſi totam AB ſtatus partium 576, <lb></lb>& AI partium 407: Cum enim momenta omnia totius AB <lb></lb>ſint 166176, & momenta AI ſint 83028, remanent momen<lb></lb>ta IB 83148, proximè æqualia momentis AI. </s> <s id="s.005277">Eſt autem Ra<lb></lb>tio 407 ad 576 minor Ratione 3 ad 4: & Ratio AI ad IB mi<lb></lb>nor eſt Ratione AH ad HB, hoc eſt minor Ratione Dodran<lb></lb>tis ad Aſſem: item Ratio 407 ad 576 major eſt Ratione 2 ad 3, <lb></lb>hoc eſt Ratione Beſſis ad Aſſem. </s> <s id="s.005278">Quæ de lineâ AB hactenus <lb></lb>dicta ſunt, in reliquis pariter illi parallelis vera eſſe deprehen<lb></lb>duntur ſimili ratiocinatione; ac propterea à linea IL omnes in <lb></lb>eâdem Ratione ſecantur. </s> <s id="s.005279">Maxima igitur percuſſio à corpore <lb></lb>AE circa lineam AF in gyrum acto fiet in lineâ IL; in qua <lb></lb>medium punctum K denotat locum validiſſimi ictûs; in eo ſci-<pb pagenum="716" xlink:href="017/01/732.jpg"></pb>licet omnia momenta impetûs æqualiter dividuntur tam jux<lb></lb>ta longitudinem, quàm juxta latitudinem. </s> </p> <p type="main"> <s id="s.005280">Sed quoniam rarò contingit corpus, quo percutimus, ita æqua<lb></lb>bili ductu partes omnes diſpoſitas habere, ſicut hactenus hypo<lb></lb>theſim in cylindro aut priſmate conſtituimus, idcircò frequen<lb></lb>tiſſimè centrum hoc momentorum Impetûs, ex quo ictus vehe<lb></lb>mentia pendet, aut magis ad centrum motûs accedit, aut ab <lb></lb>hoc magis recedit, prout ad hanc aut illam extremitatem plu<lb></lb>res ſunt partes majoris impetûs, aut majorum momentorum ca<lb></lb>paces: fieri ſiquidem poteſt, ut plures partes centro motûs pro<lb></lb>ximæ tenuioribus momentis ſint præditæ, pauciores autem par<lb></lb>tes ab hujuſmodi motûs centro remotæ majora obtineant mo<lb></lb>menta, adeò ut inæqualitas partium reciprocâ quadam inæqua<lb></lb>litate momentorum compenſetur; & fieri poteſt, ut plures par<lb></lb>tes cum majore diſtantiâ componantur, adeò ut centrum mo<lb></lb>mentorum impetûs proximum ſit extremitati, quæ velociſſimè <lb></lb>movetur. </s> <s id="s.005281">Hinc quia malleo & ſecuri infligendus eſt ictus, in <lb></lb>illorum manubriis ſtatuendis cavendum eſt, ne nimis gravia <lb></lb>ſint, ne fortè extra malleum aut ſecurim, quibus fit percuſſio, ſit <lb></lb>centrum momentorum impetûs. </s> <s id="s.005282">Centrum hoc momentorum ſi <lb></lb>appellare libeat <emph type="italics"></emph>Centrum Percuſſionis,<emph.end type="italics"></emph.end> per me licet; neque enim <lb></lb>hæreo in vocabulis. </s> </p> <p type="main"> <s id="s.005283">Ut autem oblato quocumque corpore ad percutiendum apto, <lb></lb>quo utendum ſit motu circulari, cujuſmodi eſt malleus, clava, <lb></lb>ſecuris, & ſimilia, ejus centrum Momentorum impetûs Phyſicè <lb></lb>& Mechanicè habeamus, hæc methodus fortaſſe non inutilis <lb></lb>accidat. </s> <s id="s.005284">Extremam illam partem, quæ manu apprehendi ſolet, <lb></lb>ex clavo immobili ita ſuſpende, ut circa illum liberè moveri <lb></lb>valeat: tum ſuſpenſam clavam à perpendiculo remove, & in <lb></lb>hanc atque illam partem vibrari permitte. </s> <s id="s.005285">Interim ex ſubtiliſſi<lb></lb>mo filo æreus, aut plumbeus, globulus pendeat, qui pariter vi<lb></lb>bretur: & hujus perpendiculi vibrationes cum clavæ ſuſpenſæ <lb></lb>vibrationibus compara, an videlicet ſingulæ ſingulis iſochronæ <lb></lb>ſint, hoc eſt æqualis durationis, an verò inæqualis; ſi una per<lb></lb>pendiculi vibratio diuturnior ſit, quàm una clavæ vibratio, de<lb></lb>curtandum eſt filum, ſi brevior, producendum uſque eò, dum <lb></lb>perpendiculi vibrationes ſingulæ ſingulis clavæ vibrationibus <lb></lb>iſochronæ fuerint. </s> <s id="s.005286">Hoc ubi conſecutus fueris, haud temerè <pb pagenum="717" xlink:href="017/01/733.jpg"></pb>pronunciabis quæſitum Centrum momentorum Impetûs clavæ <lb></lb>tanto intervallo abeſſe à puncto ſuſpenſionis, quanta eſt per<lb></lb>pendiculi longitudo, non quidem exactiſſimè & Geometricè, <lb></lb>ſed quantum ſatis eſt ad Phyſicum opus. </s> <s id="s.005287">Cur ita argumentari <lb></lb>liceat, ſi rationem repoſcas, hæc ſatis probabilis afferri poteſt; <lb></lb>quia ſcilicet Centrum momentorum Impetûs eſt punctum illud, <lb></lb>in quo cum ſit æqualitas momentorum, omnes ipſius clavæ par<lb></lb>tes ſuam vim exercent ad motum illius oſcillationis; quemad<lb></lb>modum in centro globuli ex filo pendentis (filum ex hypothe<lb></lb>ſi nullum habet notabile momentum ad motum adnexi globuli <lb></lb>variandum) eſt æqualitas momentorum ejuſdem deſcendentis, <lb></lb>& ad poſitionem perpendicularem ſe reſtituentis. </s> <s id="s.005288">E ſt igitur cla<lb></lb>va quaſi perpendiculum tantæ longitudinis, quantum eſt inter<lb></lb>vallum inter Centrum motûs atque Centrum momentorum Im<lb></lb>petûs. </s> <s id="s.005289">At perpendicula æqualis longitudinis ſunt iſochrona: <lb></lb>igitur invento perpendiculo iſochrono cum oſcillationibus cla<lb></lb>væ, nota erit ex hujus longitudine etiam longitudo rigidi illius <lb></lb>perpendiculi, quod concipitur in clavâ, videlicet diſtantia Cen<lb></lb>tri momentorum Impetûs à Centro motûs. </s> </p> <p type="main"> <s id="s.005290">Hìc tamen animadvertas velim ex hac methodo non haberi <lb></lb>exactè punctum Centri momentorum in clavâ, ſed illud adhuc <lb></lb>paulò longiùs abeſſe; quia nimirum perpendiculorum omnino <lb></lb>æqualium, præterquam in gravitate ponderis appenſi, illud, <lb></lb>quod gravius eſt, plures vibrationes eodem tempore perficit; <lb></lb>atque perpendiculorum omnino æqualium, præterquam in lon<lb></lb>gitudine, illud, quod longius eſt, paucioribus vibrationibus <lb></lb>eodem tempore agitatur. </s> <s id="s.005291">Cum autem clava ſit perpendiculum <lb></lb>gravius globulo, qui ex filo pendet, poſitâ æquali longitudine, <lb></lb>clava velociùs moveretur: Si igitur motus clavæ eſt iſochronus <lb></lb>cum motu globuli ex filo ſuſpenſi, neceſſe eſt longitudine gra<lb></lb>vioris inferente vibrationum raritatem compenſari ejus gravita<lb></lb>tem, quæ crebriores efficeret vibrationes. </s> <s id="s.005292">Quare hoc certum <lb></lb>habebis, quæſitum Centrum Momentorum eſſe ultra punctum <lb></lb>illud inventum ex longitudine perpendiculi adhibiti. </s> </p> <p type="main"> <s id="s.005293">Sed & illud præterea obſervandum eſt, motus iſtos circula<lb></lb>res corporum percutientium communiter non habere pro ſui <lb></lb>motûs Centro alteram extremitatem, niſi fortè, quando ad ſolius <lb></lb>manûs motum moventur, cubito ac brachio immotis: cæterùm <pb pagenum="718" xlink:href="017/01/734.jpg"></pb>pro centro motûs habent aut cubiti, aut humeri juncturam, <lb></lb>prout cubitus, vel totum brachium movetur: & tunc Centrum <lb></lb>momentorum transferri contingit, nec opus eſt adeò longa eſſe <lb></lb>manubria; ut vides malleos, quibus ad contundendos libros <lb></lb>utuntur bibliopægi, brevioris eſſe manubrij, quia extento bra<lb></lb>chio percutiunt, quod fungitur vice valde longi manubrij: <lb></lb>contra verò fabrorum ferrariorum mallei, bipennes, & cætera <lb></lb>inſtrumenta, quæ utrâque manu apprehenſa tractamus, lon<lb></lb>giora habent manubria, tunc enim brachium adeò extendere <lb></lb>nequimus. <lb></lb></s> </p> <p type="main"> <s id="s.005294"><emph type="center"></emph>CAPUT X.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005295"><emph type="center"></emph><emph type="italics"></emph>Quid conferat reſistentia corporis percuſsi.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005296">EAtenus ictum corpori percuſſo infligi, quatenus hoc motui <lb></lb>corporis percutientis opponitur, illíque obſiſtit, dictum eſt <lb></lb>cap.6. eóque vehementiorem eſſe percuſſionem, quò major eſt <lb></lb>reſiſtentia. </s> <s id="s.005297">Hæc autem reſiſtentia originem ducit ex ipsâ cor<lb></lb>porum naturâ; omne ſiquidem corpus, quâ corpus eſt, nulli <lb></lb>corpori penetrabile eſt, neque fieri poteſt, citrà Divinam vir<lb></lb>tutem longiùs, quàm naturæ termini poſtulant, excurrentem, <lb></lb>ut uno eodémque in ſpatio duo corpora collocentur, quemad<lb></lb>modum duæ ſubſtantiæ ab omni prorſus ſenſu disjunctæ, ſed <lb></lb>quæ ſolâ ratione, & intelligentiâ comprehenduntur, ſe viciſſim <lb></lb>eodem in loco facilè patiuntur. </s> <s id="s.005298">Corpus igitur percuſſum tùm <lb></lb>ex ſuâ conſtitutione, & temperie, tum ex recedendi difficulta<lb></lb>te, tùm ex poſitione, ſecundùm quam ictum excipit, habet, ut <lb></lb>modum percuſſioni ſtatuat; ex triplici enim hoc capite reſiſten<lb></lb>tiarum varietas petenda eſt, qua corpus percuſſum reluctatur, <lb></lb>ne loco cedat. </s> </p> <p type="main"> <s id="s.005299">Et ad primum quidem quod attinet, corpora dura magis re<lb></lb>ſiſtere, quàm mollia, manifeſtum eſt ex ipsá Duri & Mollis <lb></lb>notione. <emph type="italics"></emph>Eſt autem Durum,<emph.end type="italics"></emph.end> ut ait Ariſtoteles lib. 4. Meteor. <lb></lb>ſumma 2. cap. 1. <emph type="italics"></emph>quod non cedit in ſeipſum ſecundùm ſuperficiem:<emph.end type="italics"></emph.end><pb pagenum="719" xlink:href="017/01/735.jpg"></pb><emph type="italics"></emph>Molle autem, quod cedit, non circumobſiſtendo; aqua enim non Mol<lb></lb>lis, non enim cedit compreſſione ſuperficies in profundum, ſed circum<lb></lb>obſiſtit:<emph.end type="italics"></emph.end> aqua ſcilicet, & humores urgenti quidem cedunt, at non <lb></lb>induendo ſuperficiem, quæ maneat, (ut ceræ ac luto accidit) <lb></lb>ſed ita ſecedendo, ut, urgente remoto, ad ſuperficiem antiquæ <lb></lb>ſuperficiei ſimilem confluant particulæ, quæ ſeceſſerant. </s> <s id="s.005300">Qua<lb></lb>propter ad mollium genus, in rem præſentem ſpectare cenſen<lb></lb>da ſunt, quæcumque ſe premi patiuntur, hoc eſt, externo im<lb></lb>pulſu in ſe ipſa coëunt, cùm in profundum ſuperficies permu<lb></lb>tatur, nec dividitur; ſivè imprimi & formari poſſint, pulſu tan<lb></lb>tùm, ut cera & argilla, aut percuſſione, ut plumbum; ſivè im<lb></lb>preſſionem & formam rejiciant, ut lana, & ſpongiæ. </s> <s id="s.005301">Ea autem, <lb></lb>quæ dura ſunt, ſed ductilia, quia <emph type="italics"></emph>eâdem percußione poſſunt ſimulin <lb></lb>latus, & in profundum ſecundùm ſuperficiem transferri ſecundùm <lb></lb>partem,<emph.end type="italics"></emph.end> ut ferro candenti, aliíſque metallis ſub fabri malleo <lb></lb>contingit, aliquatenus ad mollia pertinere videntur, ſaltem <lb></lb>comparatè, quia videlicet cedunt percutienti, quod propterea <lb></lb>durius cenſetur. </s> <s id="s.005302">Sic in arce Antuerpienſi memini me vidiſſe <lb></lb>ænea aliquot ingentia tormenta bellica, olim ex Sckenckianâ <lb></lb>munitione, cum in Hiſpanorum poteſtatem venit, aſportata, <lb></lb>in quorum tubis non mediocres contuſionum notæ ab hoſtili<lb></lb>bus globis impreſſæ apparebant. </s> <s id="s.005303">Quæ verò corpora dura ſunt, <lb></lb>neque ſe ita comprimi patiuntur, ut ſuperficies depreſſa craſſi<lb></lb>tiem minuat, ſed ſolùm, ſervatâ longitudine, flexibilia ſunt eâ <lb></lb>ratione, ut à rectitudine ad curvitatem, aut viciſſim à curvitate <lb></lb>ad rectitudinem torqueantur, cedunt quidem, ſed inter mollia, <lb></lb>ex hoc quidem capite, recenſenda non ſunt. </s> <s id="s.005304">Quod ſi vehe<lb></lb>mentiore percuſſione non ſolùm flectantur, ſed etiam frangan<lb></lb>tur (quemadmodum contingit craſſiuſculo baculo, cujus ex<lb></lb>tremitates in acumen deſinentes innituntur duobus vitreis cya<lb></lb>this; qui circa medium valido fuſte percuſſus flectitur, & in<lb></lb>flexione declinans vitra, iis integris frangitur) in magnas par<lb></lb>tes dividuntur, & ſeparantur: at ſi in partes plures diſſiliant ex <lb></lb>unicâ percuſſione, friantur, ut vitrum, lapis, fictile; id quod ex <lb></lb>duritie oritur. </s> </p> <p type="main"> <s id="s.005305">Hæc eadem corporis habitudo, quæ particularum compo<lb></lb>nentium complexionem reſpicit, æquè in percutiente, ac in <lb></lb>percuſſo attendenda eſt; quandoquidem ſi diſpar fuerit eorum <pb pagenum="720" xlink:href="017/01/736.jpg"></pb>durities, fieri poteſt, ut ex ictu labefactetur potiùs percutiens, <lb></lb>quàm percuſſum: ſic globus plumbeus ex editâ turri decidens <lb></lb>in ſubjectum ſilicem ex ictu contunditur, rotundâ ſuperficie in <lb></lb>planam mutatâ, qua parte fuit contactus; & vitrum ad ſaxa <lb></lb>alliſum friatur; & follis luſorius in parietem impactus compri<lb></lb>mitur. </s> <s id="s.005306">Hinc tamen non fit, quò minus corpus illud, in quod <lb></lb>plumbeus globus, aut vitrum, aut follis incurrit, percuſſum di<lb></lb>catur; ex ictu enim ſaltem concutitur, & contremiſcit. </s> <s id="s.005307">Neque <lb></lb>tremorem hujuſmodi temerè confictum ſuſpicabitur, quiſquis <lb></lb>longiſſimæ trabis extremitati aurem admoverit, ut alterâ extre<lb></lb>mitate quamvis leviſſimè digito percuſsâ ſonitum audiat, aut <lb></lb>noctu ſcrobiculo in terrâ facto aurem immiſerit, ut adventan<lb></lb>tis alicujus adhuc procul poſiti paſſus percipiat: nullum verò <lb></lb>ſonum fieri ſine tremore & motu, extra controverſiam poſuit <lb></lb>experientia. </s> </p> <p type="main"> <s id="s.005308">Porrò ſpectatâ corporum temperie, percuſſionum vehemen<lb></lb>tiâ æſtimatur ex iis, quæ conſequuntur reſiſtentiam ortam ex <lb></lb>corporum colliſorum duritie ſeu mollitudine majori au minori, <lb></lb>tùm abſolutè, tùm comparatè. </s> <s id="s.005309">Cum <emph type="italics"></emph>Abſolutè<emph.end type="italics"></emph.end> dico, alterutrius <lb></lb>ſolùm duritiem ſeu mollitudinem conſidero ita, ut aut corpora <lb></lb>percuſſa inter ſe, aut corpora percutientia ſimiliter inter ſe <lb></lb>conferantur: <emph type="italics"></emph>Comparatè<emph.end type="italics"></emph.end> autem, quando percutiens cum per<lb></lb>cuſſo comparatur, prout duritie ſe excedunt. </s> <s id="s.005310">Si corpus percu<lb></lb>tiens valdè durum ponatur, & corpus percuſſum molle fuerit, <lb></lb>hoc cedendo retundit ictum; ex levi enim illâ reſiſtentiâ tan<lb></lb>diu durante, quandiu fit partium compreſſio, minuitur in per<lb></lb>cutiente impetus, & quod corpori molli ſubjectum eſt corpus, <lb></lb>leviſſimam impreſſionem ex ictu recipit. </s> <s id="s.005311">Sic apud Sinas, ut in <lb></lb>Atlante Sinico pag. </s> <s id="s.005312">127. <emph type="italics"></emph>In flumine, per quod ad Ienping navi<lb></lb>gatur, Catadupæ aquarum multæ ſunt, & periculoſiſſima Syrtibus <lb></lb>loca, duo præſertim propè Cinglieu, unus Kieulung, alter Changcung <lb></lb>dictus. </s> <s id="s.005313">Cum naves tranſeunt, ne cum aquâ decidentes fractionis in<lb></lb>currant periculum, ſcitè præmittunt aliquot ſtraminis faſces, ad quos <lb></lb>navis leviùs impingat, ac tranſeat.<emph.end type="italics"></emph.end></s> <s id="s.005314"> Sic ferreis tormentorum glo<lb></lb>bis objecti ſacci lanâ aut terrâ repleti illorum vim elidunt, ne <lb></lb>diruant muros hujuſmodi ſaccis protecto: ſic farti goſſypio <lb></lb>thoraces non levi munimento ſunt digladiantibus. </s> <s id="s.005315">Quò autem <lb></lb>mollius fuerit corpus percuſſum, quia magis cedit, minùs læ-<pb pagenum="721" xlink:href="017/01/737.jpg"></pb>ditur à percutiente; & viciſſim quò durius illud fuerit, magis <lb></lb>ab eodem percutiente læditur, cujus impulſum excipit. </s> </p> <p type="main"> <s id="s.005316">Hinc vides cur ferreos militum thoraces, & galeas noſtro <lb></lb>hoc ævo aliter temperare oporteat, ac antiquis temporibus, <lb></lb>quando gladiorum, haſtarum, ſagittarum ictus tantummodo <lb></lb>repellere opus erat; tunc enim durâ temperatione ſolidandum <lb></lb>erat ferrum, ne prorſus cederet, hujuſmodi armorum mucro<lb></lb>nem admittendo: nunc verò ut innoxiè excipiantur ictus glo<lb></lb>borum à Sclopis emiſſorum, ferrum molle eſſe expedit, ut con<lb></lb>tuſum flectatur, & aliquantulum cedens ita imminuat globi <lb></lb>ejaculati vires, ut penetrare ulteriùs non valeat. </s> <s id="s.005317">Quod ſi in <lb></lb>chalybem temperatus eſſet thorax militaris, nec admodum <lb></lb>craſſus eſſet, ne gravitate nimiâ incommodus, aut inutilis ac<lb></lb>cideret, facilè chalybs ex globi ictu diſſiliret, & vulneri locum <lb></lb>aperiret. </s> <s id="s.005318">Sed quia globi plumbei ſunt, & ſe comprimi patiun<lb></lb>tur, ex hujuſmodi percuſſione compreſſio quaſi diſtribuitur in<lb></lb>ter plumbeum globum exploſum, atque ferreum thoracem, qui <lb></lb>multo magis contunderetur (aut fortè etiam perforaretur à glo<lb></lb>bulo ferreo; globulus autem plumbeus, ſi thorax aut ipſa galea <lb></lb>nihil cederet, magis comprimeretur, quemadmodum cùm in <lb></lb>marmor exploditur. </s> </p> <p type="main"> <s id="s.005319">At ubi corpus percuſſum non cedit in ſeipſum ſecundùm ſu<lb></lb>perficiem, flectitur tamen, adhuc minùs reſiſtit, quàm corpo<lb></lb>ra rigida, nec flexioni notabili obnoxia. </s> <s id="s.005320">Notabili, inquam, ne <lb></lb>in quæſtionem vocemus, utrùm flecti dicenda ſint illa corpora, <lb></lb>quæ ex ictu tremorem concipiunt, ut æri campano, cum pulſa<lb></lb>tur, accidit: nam vix excogitari poteſt corpus aliquod, cui ex <lb></lb>vi percuſſionis accidere nequeat tremor; cum & terram ipſam <lb></lb>licèt altiùs defoſſam in cuniculis concuti & contremiſcere <lb></lb>oſtendant lapilli, & fabæ in tympani militaris planâ facie ſubſi<lb></lb>lientes ex profundo illo ligonis ictu. </s> <s id="s.005321">Certè, ſi Atlanti Sinico <lb></lb>pag.57. credimus, ubi in IV Provinciâ Xantung mentionem <lb></lb>facit de monte, cui nomen Mingxe, hoc eſt Sonorum lapis; <lb></lb><emph type="italics"></emph>in hujus montis vertice cippus erectus ſtat centum altus perticas <emph.end type="italics"></emph.end><lb></lb>(Pertica apud Sinas eſt decem cubitorum) <emph type="italics"></emph>qui vel leviter digito <lb></lb>percuſſus ad tympani modum ſonum edere dicitur, à quo monti nomen<emph.end type="italics"></emph.end>; <lb></lb>nullus autem ſonus abſque corporis ſonori tremore efficitur. </s> <lb></lb> <s id="s.005322">Quod ſi non niſi leviſſimè flecti queat corpus percuſſum, ſed ci-<pb pagenum="722" xlink:href="017/01/738.jpg"></pb>tra tremorem frangatur, aut frietur, indicium eſt majoris re<lb></lb>ſiſtentiæ, ac proinde, cæteris paribus, vehementiorem futu<lb></lb>ram percuſſionem, quàm ſi conſpicuam flexionem admitteret. </s> <lb></lb> <s id="s.005323">Cæterùm reſiſtentia ferè maxima eorum corporum eſt, quæ & <lb></lb>partes nexu ægrè diſſolubili copulatas habent, & congruá craſ<lb></lb>ſitudine prædita non niſi creberrimo & minutiſſimo tremore <lb></lb>concuti poſſunt, ſi percutiantur. </s> <s id="s.005324">Nam omnium reſiſtentiarum <lb></lb>abſolutè maxima eſt, cùm prorſus immotum à percuſſione ma<lb></lb>net corpus. </s> </p> <p type="main"> <s id="s.005325">Hinc ſi percuſſi corporis durities major fuerit, quàm percu<lb></lb>tientis, fieri poteſt, ut impetus qui corpori percuſſo imprimi <lb></lb>non poteſt, disjiciat ipſius percutientis partes, aut in latus im<lb></lb>pellat ita, ut vel contundatur, vel frangatur, vel frietur, sítque <lb></lb>percutientis conditio deterior, quàm percuſſi. </s> <s id="s.005326">Hujuſmodi eſſet <lb></lb>apud nos conditio gladij, quo marmor percuteremus; neque <lb></lb>enim noſtrates enſes comparandi ſunt cum illo, de quo Atlas <lb></lb>Sinicus pag.159.in XV Provincia Junnam ad urbem Chinkiang, <lb></lb>ubi hæc habet. <emph type="italics"></emph>Ad urbis Borealem partem ad hæc uſque tempora in<lb></lb>gens conſpicitur lapis, ubi Mung Rex Sinulo alterius Regis legatos <lb></lb>excipiens, cum illi minimè ſatisfacerent, extracto gladio rapidens <lb></lb>ita percuſſit, ut ictus ad tres cubitos penetraret, verbis inſuper mi<lb></lb>nacibus legatos alloquens; Ite, & Regi veſtro renunciare, quales apud <lb></lb>me gladij ſint.<emph.end type="italics"></emph.end></s> </p> <p type="main"> <s id="s.005327">Altera reſiſtentiæ origo habetur ex difficultate recedendi; <lb></lb>quando videlicet corpus percuſſum ſivè ratione figuræ, ſivè ra<lb></lb>tione molis & gravitatis, ſivè ratione obſtaculi alicujus, aut <lb></lb>retinaculi, ſivè ratione motûs oppoſiti, nequit obſecundare <lb></lb>motui percutientis, ſed potiùs illum aut cohibet, aut retardat, <lb></lb>aut reflectit; hæc enim tria accidere poſſunt motui percutientis <lb></lb>ex percuſſi reſiſtentiâ. </s> <s id="s.005328">Primum ſiquidem ſi corpus percuſſum <lb></lb>volubile non fuerit, & in orbem incitari nequeat, ſed planâ fa<lb></lb>cie incumbat ſolo, præſertim ſalebroſo, quò ampliori facie fit <lb></lb>contactus, eò difficiliùs impelli poteſt. </s> <s id="s.005329">Deinde etiamſi rotun<lb></lb>dum fuerit corpus, & facilis motionis principium habeat ſpecta<lb></lb>tâ figurâ, ſi tamen ingens fuerit globus marmoreus, aut æreus, <lb></lb>tanta eſſe poteſt gravitas, ut vix, aut ne vix quidem, loco dimo<lb></lb>veri queat. </s> <s id="s.005330">Non tamen ſemper faciliùs moventur ex percuſſio<lb></lb>ne, quæ leviora ſunt; nam ſi quis ex cupreſſu galbulum, aut <pb pagenum="723" xlink:href="017/01/739.jpg"></pb>ex quercu gallam decerpat, & malleo percutiat, ob nimiam <lb></lb>galbuli, & gallæ levitatem multo infirmior erit ictus, quàm ſi <lb></lb>æqualem globulum eburneum percuteret. </s> <s id="s.005331">Ad hæc, forma <lb></lb>quidem apta eſſe poteſt, nec gravitas aut moles nimia, ſed quia <lb></lb>corpus percuſſum nequit percutienti cedere, niſi corpus aliud <lb></lb>proximum repellatur, propterea augeri poteſt reſiſtentia: ſic <lb></lb>ſublicas acuminatas in terram adigimus fiſtucâ ſivè directas ad <lb></lb>perpendiculum, ſivè pronas; ſed ea poteſt eſſe telluris denſitas <lb></lb>compreſſionem reſpuens, ut ſæpiùs cadens fiſtuca parum profi<lb></lb>ciat. </s> <s id="s.005332">Demum ſi corpus percuſſum antè ictum non quieſcat, ſed <lb></lb>oppoſito motu occurrat percutienti, quò velociùs movetur, & <lb></lb>directione magis oppoſitâ, etiam magis reſiſtit, & utrumque <lb></lb>viciſſim eſt percutiens & percuſſum; niſi quod percutientis vo<lb></lb>cabulum validiori conceditur. </s> <s id="s.005333">Contra verò languidior accidit <lb></lb>percuſſio, ſi corpus percutiens aſſequatur aliud, quod ad eaſ<lb></lb>dem partes tardiùs movetur; eóque minor eſt reſiſtentia, quò <lb></lb>minor eſt in velocitate motuum differentia. </s> <s id="s.005334">Sic decidentis ex <lb></lb>altitudine non modicâ lapidis ictum manu citrà læſionem exci<lb></lb>pimus, ſi illius motui, ubi manum attigerit, exiguo minoris <lb></lb>velocitatis diſcrimine obſecundemus: hæc ſiquidem exigua re<lb></lb>ſiſtentia modicum quid impetûs deterit, & quia aliquot mo<lb></lb>menta durat, ita ſenſim extenuatur impetus, ut demum qui re<lb></lb>liquus eſt nocere non valeat. </s> </p> <p type="main"> <s id="s.005335">Hoc artificio procul dubio utebatur quidam, qui ante ali<lb></lb>quot annos, ut ex viro fide digno tanquam rem notiſſimam ac<lb></lb>cepi, Mutinæ enſem eâ dexteritate in altum projiciebat, ut per<lb></lb>pendicularis recideret mucrone deorſum converſo, quem ca<lb></lb>dentem nuda manûs vola innoxiè excipiebat; ſed cum aliquan<lb></lb>do invitus cogeretur, ut id noctu experiretur in conclavi mul<lb></lb>tis facibus illuſtrato, cùm (deficiente conſtanti & clariſſimâ <lb></lb>diurnâ luce, quam æmulari non poteſt tremula & inconſtans <lb></lb>facium, quamvis multarum, flamma) non ita exactè aſſeque<lb></lb>retur deſcendentis gladij motum & velocitatem, finem fecit lu<lb></lb>do manum trajectam referens. </s> </p> <p type="main"> <s id="s.005336">Poſtremum caput, ex quo reſiſtentiæ modus deſumitur in <lb></lb>percuſſionibus, eſt ipſa poſitio corporis percuſſi, prout directè, <lb></lb>aut obliquè, ictum excipit, hoc eſt quatenus linea directionis <lb></lb>motûs, quo fertur corpus percutiens, incurrit in corporis per-<pb pagenum="724" xlink:href="017/01/740.jpg"></pb>cuſſi ſuperficiem ad angulos æquales, aut inæquales. </s> <s id="s.005337">Si enim <lb></lb>ad angulos æquales opponatur Directioni motûs, cum ad neu<lb></lb>tram partem corpus percutiens declinare poſſit, tota vis ictus <lb></lb>excipitur à corpore percuſſo. </s> <s id="s.005338">Sin autem obliquè, & ad angu<lb></lb>los inæquales, à corpore percuſſo excipiatur percutientis ictus, <lb></lb>quò major erit angulorum inæqualitas, eò languidior erit per<lb></lb>cuſſio, minùs quippe motûs Directioni opponitur ſuperficies <lb></lb>percuſſa, quò fuerit angulus Incidentiæ magis acutus. </s> <s id="s.005339">Ut au<lb></lb>tem horum ictuum Ratio aliqua innoteſcat, nulla mihi con<lb></lb>gruentior methodus occurrit, quàm ſi philoſophemur ſimili <lb></lb>planè ratiocinatione, ac cùm lib. 1. cap. 14. expendimus gravi<lb></lb>tationem corporis in planum inclinatum; ſicut enim ibi gravi<lb></lb>tatem cum ſuâ directione deorſum ad centrum gravium conſi<lb></lb>deravimus, ita hìc in percuſſione impetum corporis percutien<lb></lb>tis, & ejus directionem accipere oportet: & quemadmodum in <lb></lb>plano inclinato gravia obtinent momenta deſcendendi majora, <lb></lb>aut minora, prout angulus inclinationis plani cum perpendi<lb></lb>culo minor eſt, aut major; ſimiliter in percuſſione momentum <lb></lb>progrediendi juxta conceptam aut impreſſam directionem mo<lb></lb>tûs iiſdem tenetur legibus, juxta plani percuſſi obliquitatem; <lb></lb>ac proinde minor invenitur reſiſtentia, ubi majus eſt progre<lb></lb>diendi momentum. </s> <s id="s.005340">Quare hìc ſatis erit recolere, quæ dicta <lb></lb>ſunt lib. 1. cap. 13. & 14. de gravitatione in plano inclinato, & <lb></lb>in planum inclinatum, eáque percuſſionibus ſervatâ analogiâ <lb></lb>applicare. </s> </p> <p type="main"> <s id="s.005341">Cum itaque in percutiente conſideranda ſit & moles, & mo<lb></lb>tûs velocitas, & Directio motûs, & durities; in corpore autem <lb></lb>percuſſo & naturæ temperatio, & recedendi difficultas, & po<lb></lb>ſitio, ſecundùm quam excipitur ictus, ſpectanda ſit; mani<lb></lb>feſtum eſt ex his omnibus ictuum vim temperari; atque adeò ſi <lb></lb>duo ictus comparandi ſint, aſſumendæ ſunt in corporibus per<lb></lb>cutientibus invicem comparatis Rationes omnes & molis ad <lb></lb>molem (hoc eſt gravitatis ad gravitatem, aut virtutis moventis <lb></lb>ad virtutem moventem) & velocitatis ad velocitatem, & di<lb></lb>rectionis ad directionem, & duritiei ad duritiem; & ſimiliter <lb></lb>in corporibus percuſſis Rationes eorum, quæ in illis conſide<lb></lb>rantur: atque demum facta Rationum compoſitio indicabit Ra<lb></lb>tionem ictuum. </s> <s id="s.005342">Hinc vides quàm multæ fieri poſſint hujuſmodi <pb pagenum="725" xlink:href="017/01/741.jpg"></pb>Rationum complexiones; quas ſi juxta earum varietatem in <lb></lb>Propoſitiones digerere otium eſſet, in molem non exiguam hæc <lb></lb>ſcriptio excreſceret, ſed non majore fructu, quàm ſi tu ipſe Ra<lb></lb>tiones, ut indicatum eſt, componas. <lb></lb></s> </p> <p type="main"> <s id="s.005343"><emph type="center"></emph>CAPUT XI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005344"><emph type="center"></emph><emph type="italics"></emph>Quomodo ex Percuſsionibus determinentur <lb></lb>Reflexiones.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005345">UT Percuſſionis natura plenè perfectéque innoteſcat, diſ<lb></lb>piciendum ſupereſt, quomodo ex illâ determinetur Re<lb></lb>flexio. </s> <s id="s.005346">Motus ſiquidem, qui propriè eſt reflexus, percuſſionem <lb></lb>conſequitur, quatenus id, quod motu directo ferebatur, inve<lb></lb>nit obicem, ne ulteriùs juxta eandem Directionem progredia<lb></lb>tur: ſed quia adhuc acquiſitus, ſeu impreſſus, impetus ſupereſt, <lb></lb>aliam inire viam cogitur; eóque magis reflectitur, quò majo<lb></lb>rem invenit reſiſtentiam ortam ex utriuſque corporis impene<lb></lb>trabilitate, atque duritie. </s> <s id="s.005347">Quòd ſi utrique corpori, percutien<lb></lb>ti videlicet atque percuſſo, ſumma durities ineſſe poneretur, <lb></lb>ita ut in neutro ex vi percuſſionis ulla ſequeretur partium com<lb></lb>preſſio, aut depreſſio, aut attritio ſeu diviſio, perfecta quoquè <lb></lb>intelligeretur reflexio, in qua corpus percutiens non niſi in <lb></lb>tranſitu, citrà omnem vel breviſſimam morulam, contingeret <lb></lb>corpus, à quo reflectitur; & nulla fieret impetûs acquiſiti, ſive <lb></lb>impreſſi, diminutio præter eam, quam ſecum trahit nova re<lb></lb>flectentis determinatio oppoſita lineæ directionis, ſecundùm <lb></lb>quam priùs movebatur. </s> <s id="s.005348">Nemini autem dubium eſſe debet, an <lb></lb>corpus reflexum pergat moveri ex vi impetûs adhuc reſidui <lb></lb>poſt motum directum: nam corpus reflectens prorſus immo<lb></lb>tum & quieſcens non poteſt impetum illi communicare; cum <lb></lb>perpetuis experimentis doceamur nihil moveri ab alio <lb></lb>quieſcente. </s> </p> <p type="main"> <s id="s.005349">Sæpiùs tamen contingit (ſi quis dixerit <emph type="italics"></emph>ſemper,<emph.end type="italics"></emph.end> quibus argu<lb></lb>mentis eum coarguerem manifeſtæ falſitatis, me non habere <pb pagenum="726" xlink:href="017/01/742.jpg"></pb>candidè profiteor) non fieri puram reflexionem ex merâ re<lb></lb>ſiſtentiâ; ſed in alterutro ſaltem corporum colliſorum, ex per<lb></lb>cuſſione ſequitur aliqua partium violenta compreſſio, aut <lb></lb>diſtractio; hanc autem naturæ repugnantem partium poſitio<lb></lb>nem excutere dum nititur, séque in priſtinum ſtatum reſtitue<lb></lb>re, novum impetum concipit, quem & poteſt reflectens re<lb></lb>flexo imprimere, atque in eo diminuti ex reſiſtentiâ impetûs <lb></lb>jacturam, aliquâ ſaltem ex parte, reſarcire. </s> <s id="s.005350">Hinc, in rem præ<lb></lb>ſentem diſtinguere oportet, quid inter Compreſſionem & De<lb></lb>preſſionem interſit: quæ enim deprimuntur, ut plumbum, cera, <lb></lb>argilla, non reſiliunt ſecundùm ſuperficiem, ut priſtinam figu<lb></lb>ram induant; ideóque quando hujuſmodi corpora in aliud im<lb></lb>pinguntur, vel aliud in illa impingitur, valdè debilitatur re<lb></lb>flexio, ſi modò aliqua contingere poteſt. </s> <s id="s.005351">Quæ verò compri<lb></lb>muntur, externâ vi deficiente ſe in priſtinam figuram abſque <lb></lb>cunctatione reſtituunt concepto novo impetu. </s> <s id="s.005352">Exemplum ex <lb></lb>folle pugillatorio peti poteſt, ut res in apertum deducatur. </s> <s id="s.005353">Ca<lb></lb>dens in ſubjectum pavimentum follis luſorius, ritè inflatus, im<lb></lb>peditur, ne ulteriùs procedat; ſed quia incluſi aëris particulæ <lb></lb>eæ ſunt, quæ per vim conſtipari ampliùs poſſint, ideò ex illis <lb></lb>anteriores hinc urgentur à poſterioribus, quæ vi acquiſiti im<lb></lb>petûs inchoatum iter proſequuntur, hinc tellure reſiſtente, <lb></lb>hinc alutâ continente, inter anguſtias deprehenſæ comprimun<lb></lb>tur: id quod cùm motum exigat, certam aliquam breviſſimi <lb></lb>temporis menſuram requirit, quo fluente, terra à folle tangi<lb></lb>tur, motúſque aliquatenus impeditur (nunquam tamen ita, ut <lb></lb>ceſſet omnino motus illarum ſaltem partium, à quibus anterio<lb></lb>res urgentur ac premuntur) & quò diutiùs hujuſmodi com<lb></lb>preſſio durat, eò magis impeditur motus totius follis, atque adeò <lb></lb>plus impetûs deperditur. </s> <s id="s.005354">Sed quoniam ſtatus ille majoris com<lb></lb>preſſionis aëri intrà follem conſtipato contra naturam accidit, <lb></lb>ubi primùm, debilitato impetu urgente, reſtituere ſe poteſt <lb></lb>aër, impetum ſibi imprimit, quo moveatur ad ampliorem lo<lb></lb>cum occupandum, ſi facta fuerit condenſatio, vel certè ad par<lb></lb>tes in priſtino & naturali ſtatu conſtituendas (quemadmodum <lb></lb>alutæ contingit, cujus partes aliæ compreſſæ, aliæ diſtractæ ſe<lb></lb>ſe reſtituunt) cúmque id præſtare nequeat motu ad terram di<lb></lb>recto, quippe quæ reſiſtit, in oppoſitam partem motum dirigit; <pb pagenum="727" xlink:href="017/01/743.jpg"></pb>novóque hoc impetu ſi non æquatur, qui reſiſtentiâ diminutus <lb></lb>fuerat, ſaltem incrementi alicujus compenſatione lenitur in<lb></lb>commodum detrimenti, & major fit motus, quàm pro Ratione <lb></lb>reſidui impetûs ante percuſſionem & compreſſionem concepti. </s> </p> <p type="main"> <s id="s.005355">Hæc eadem proportione dicenda ſunt, quando non corpus <lb></lb>percutiens, ut follis in terram decidens, ſed percuſſum com<lb></lb>primitur aut diſtrahitur, & virtute elaſticâ ſe reſtituit; impe<lb></lb>tum enim concipiens, quo amiſſam figuram recuperet, etiam <lb></lb>percutienti impetum imprimit, quo repellitur. </s> <s id="s.005356">Sic in ſphæ<lb></lb>riſterio immiſſæ pilæ ſi reticulum ex contortis animalium in<lb></lb>teſtinis in plagas diſtinctum objeceris, validiùs reflectitur pila, <lb></lb>quàm objecto batillo ligneo, intenti enim nervi illi, ex impetu <lb></lb>pilæ inflexi, validiſſimè ſe reſtituunt, id quod ligno non con<lb></lb>tingit, quippe quod vix elaſticam hanc virtutem exercet, ſi ta<lb></lb>men à pilâ impacta quicquam inflexionis recipit, quæ compreſ<lb></lb>ſio ſit potiùs, quàm depreſſio. </s> <s id="s.005357">Quæ ſcilicet corpora eam par<lb></lb>tium texturam habent, ut minùs ferant ſe à priore poſitione & <lb></lb>figurâ dimoveri, illa ſeſe majore impetu reſtituunt. </s> </p> <p type="main"> <s id="s.005358">Ex his conſtat, cur partium depreſſio officiat reflexioni cor<lb></lb>poris percutientis: quia nimirum à poſterioribus illius partibus <lb></lb>urgentur anteriores contactui proximæ, quæ interim vel quieſ<lb></lb>cunt, vel multò tardiùs moventur, vel ad latus ſecedunt, & <lb></lb>idcircò vel totum, vel ferè totum, ſuum impetum deperdunt: <lb></lb>poſteriores verò dum urgent ac premunt, moventur quidem, <lb></lb>ſed reperiunt reſiſtentiam ſubſidentium partium anteriorum, <lb></lb>atque adeò in illis pariter minuitur impetus; ſæpiúſque tanta fit <lb></lb>impetûs diminutio, ut, depreſſione abſolutâ, partes illæ poſte<lb></lb>riores reliquum non habeant tantum impetûs, qui vincere va<lb></lb>leat gravitatem, & reflexionem efficere; neque enim aliquid <lb></lb>amiſſi impetûs compenſatur ab impetu novo partium ſe reſti<lb></lb>tuentium, quemadmodum fieri diximus in compreſſione. </s> <s id="s.005359">Quan<lb></lb>do autem depreſſio partium accidit corpori, ad quod alliditur <lb></lb>corpus percutiens, ut cùm in arenam ſiccam ac pulverulentam, <lb></lb>aut in limoſam terram decidit globus, tunc multum impetûs <lb></lb>deperditur, ut dictum eſt ſuperiùs de ictu, qui eò infirmior eſt, <lb></lb>quò mollius eſt corpus percuſſum; reflexio autem eò major eſt, <lb></lb>quò validiore ictu percutitur corpus reflectens. </s> <s id="s.005360">Quòd ſi <lb></lb>utrumque corpus, tam percutiens, quàm percuſſum, patiatur <pb pagenum="728" xlink:href="017/01/744.jpg"></pb>compreſſionem aut depreſſionem, aut partium attritum, tunc <lb></lb>multò minor eſt reflexio, quia dum invicem cedunt, aliquo <lb></lb>tempore durat reſiſtentia, multóque magis minuitur impetus. </s> <lb></lb> <s id="s.005361">id quod adhuc magis contingit, ſi ſe invicem conterant, & <lb></lb>particulæ aliquæ majores reſiliant. </s> </p> <p type="main"> <s id="s.005362">Quapropter cum incerta ſemper, & varia ſit complexio hu<lb></lb>juſmodi reſiſtentiarum & ceſſionum, juxta variam corporum <lb></lb>temperationem; ut reflexionis certæ regulæ ſtatuantur, ſemo<lb></lb>tis iis, quæ percuſſionis accidunt, conſideranda & aſſumenda <lb></lb>eſt reſiſtentia abſque ullâ ceſſione, perinde atque ſi duriſſimo<lb></lb>rum corporum colliſio fieret. </s> </p> <p type="main"> <s id="s.005363">Cum itaque in reflexione, corporis duri in aliud decidentis, <lb></lb>aut impacti, motus ad novam lineam dirigatur, nova hæc di<lb></lb>rectio oritur ex lineâ directionis prioris motûs quatenus compa<lb></lb>ratâ cum plano reflectente, videlicet quatenus ad illud incli<lb></lb>natur, & cum eo angulum conſtituit in puncto contactûs. </s> <lb></lb> <s id="s.005364">Quando autem ſuperficies corporis reflectentis eo loco, ubi <lb></lb>percutitur, plana non eſt, ſed convexa (ſimile quid dicendum, <lb></lb>ſi cava fuerit) ſivè ſphærica ſit, ſivè Elliptica, ſivè Conica, re<lb></lb>verâ nullum ibi eſt planum reflectens (niſi fortè hujuſmodi <lb></lb>convexas ſuperficies ex plurimis planis minimis conſtitui fin<lb></lb>gas, quemadmodum circuli peripheriam ex infinitis lineolis <lb></lb>rectis, quarum rectitudo ſenſum omnem fugiat, componi opi<lb></lb>nantur aliqui) ſed communiter mente concipiunt planum, <lb></lb>quod in puncto percuſſionis tangeret ſuperficiem con<lb></lb>vexam; & ex illo angulos tùm Incidentiæ, tùm Reflexionis <lb></lb>definiunt. </s> </p> <p type="main"> <s id="s.005365">Porrò planum reflectens (quod quidem ſpectat ad novam <lb></lb>directionem motûs ſtatuendam corpori percutienti, quem po<lb></lb>namus eſſe globum) ita ſe habere videtur, ac ſi in globum <lb></lb>quieſcentem motu parallelo impingeretur ipſum planum tanto <lb></lb>impetu, quanto impetu fertur globus adversùs planum: ſi enim <lb></lb>ex duobus colliſis alterum quieſcit, alterum movetur, ad ra<lb></lb>tionem ictûs nil refert, utrum illorum quieſcat, aut moveatur, <lb></lb>modò cætera omnia paria fuerint; ad rationem verò reflexio<lb></lb>nis, quà reflexio eſt, attenditur potiſſimum ordinatio novæ li<lb></lb>neæ motûs, quæ ex obſtaculi poſitione deſumitur, adeò ut no<lb></lb>va linea directionis, quatenus à plano reflectente pendet, & à <pb pagenum="729" xlink:href="017/01/745.jpg"></pb>centro gravitatis deſcendentis, aut à centro Impetûs corpo<lb></lb>ris impacti, certâ Ratione reſpiciat priorem lineam directio<lb></lb>nis. </s> <s id="s.005366">Eo igitur ipſo quod concipimus planum reflectens mo<lb></lb>veri motu parallelo, hoc eſt ſervatâ poſitione priori poſi<lb></lb>tioni parallelâ, adversùs globum quieſcentem, manifeſtum <lb></lb>eſt novam determinationem ex illo ortam eſſe versùs lineam <lb></lb>plano perpendicularem, ex puncto contactûs erectam: nam <lb></lb>impetus, qui ex illo plani motu imprimeretur globo <lb></lb>quieſcenti, hunc deferret per lineam jungentem punctum <lb></lb>contactûs cum centro globi: hæc autem linea ex centro <lb></lb>ſphæræ ducta ad punctum contactûs plani eſt ipſi plano per<lb></lb>pendicularis, ut ex Sphæricis conſtat. </s> <s id="s.005367">Quamvis igitur res <lb></lb>contrario modo ſe habeat, ſcilicet planum quieſcat, & glo<lb></lb>bus moveatur, directio tamen, quatenus orta ex reſiſtentiâ <lb></lb>plani, eodem modo ſe habet, & eſt versùs perpendicula<lb></lb>rem ex puncto contactûs. </s> <s id="s.005368">Sed quia cum impetu globi ut <lb></lb>plurimùm manet adhuc prior directio, ex his duabus mo<lb></lb>tuum ordinationibus oritur tertia mixta; ita ut neque ad <lb></lb>perpendiculum reflectatur, niſi incidentiæ linea perpendi<lb></lb>cularis fuerit, neque recta inſtitutum iter proſequatur. </s> </p> <p type="main"> <s id="s.005369">Quoniam igitur ex puncto contactûs innumeræ lineæ exi<lb></lb>re poſſunt cùm variâ inclinatione ad planum reflectens, <lb></lb>nec ulla peculiaris eſt cauſa, cur ad hos potiùs, quàm ad <lb></lb>illos angulos, reflectatur corpus percutiens, qui majores <lb></lb>ſint aut minores angulo incidentiæ, quem linea directio<lb></lb>nis motûs conſtituit cum eodem plano reflectente; reli<lb></lb>quum eſt, ut angulo incidentiæ æqualis ſit angulus refle<lb></lb>xionis; hæc ſiquidem linea ad angulum priori æqualem re<lb></lb>flexa unica eſt, quæ inter innumeras alias lineas magis aut <lb></lb>minùs inclinatas potiori quodam jure exigitur à naturâ <lb></lb>prioris directionis leges, quoad fieri poteſt, retinente. </s> <lb></lb> <s id="s.005370">Non eſt autem neceſſe tyronem monere, duas lineas, di<lb></lb>rectam & reflexam in puncto reflexionis concurrentes eſſe <lb></lb>in uno & eodem plano, ut conſtat ex. </s> <s id="s.005371">2. lib. 11. ab hoc <lb></lb>autem plano ſecari planum reflectens, ac proinde ad lineam, <lb></lb>quæ eſt duorum planorum communis ſectio, referendam eſſe <lb></lb>linearum illarum inclinationem. </s> </p> <p type="main"> <s id="s.005372">Quare ſit plani reflectentis, & plani, in quo fit motus, <pb pagenum="730" xlink:href="017/01/746.jpg"></pb>communis ſectio linea AB, & ſuper planum ad rectos angulos <lb></lb><figure id="id.017.01.746.1.jpg" xlink:href="017/01/746/1.jpg"></figure><lb></lb>cadat linea directionis prioris DC, <lb></lb>per quam movetur globus tanto <lb></lb>impetu, ut niſi planum obſtaret, <lb></lb>ulteriùs procederet rectà versùs E: <lb></lb>Verùm quoniam à plano obſiſten<lb></lb>te repellitur per lineam perpendi<lb></lb>cularem CD, nova hæc determi<lb></lb>natio ad motum eſt omninò & <lb></lb>adæquatè oppoſita priori directioni DC, ideóque ictus eſt va<lb></lb>lidiſſimus propter maximam reſiſtentiam. </s> <s id="s.005373">Hinc quia ex re<lb></lb>ſiſtentiâ oritur reflexio, maxima eſt reflexio, quæ fit per li<lb></lb>neam perpendicularem, nihil enim remanet de priori directio<lb></lb>ne: in hoc quippe comparantur invicem reflexiones, ut illa <lb></lb>major dicatur, in qua nova motûs ordinatio magis minuit prio<lb></lb>rem directionem, ut ſcilicet minùs pergat ad eam partem, ad <lb></lb>quam ferebatur motu directo corpus percutiens. </s> <s id="s.005374">In reflexione <lb></lb>autem perpendiculari ita tollitur prior directio, ut nullo pacto <lb></lb>globus, qui ex D per DC movebatur, ampliùs versùs E ten<lb></lb>dat. </s> <s id="s.005375">Cùm ergo nova ordinatio ſit per perpendicularem CD <lb></lb>ad angulos rectos, manifeſtò conſtat, angulum reflexionis eſſe <lb></lb>æqualem angulo incidentiæ; nam omnes anguli recti ſunt <lb></lb>æquales. </s> </p> <p type="main"> <s id="s.005376">At moveatur corpus per lineam FC, & fiat incidentiæ an<lb></lb>gulus FCB acutus: niſi planum reſiſteret, progrederetur cor<lb></lb>pus juxta eandem directionem ultra C in G; quo motu rece<lb></lb>dens à puncto C partim tenderet à C versùs A, partim à C ver<lb></lb>sùs E, ita ut à lineâ CA diſtaret intervallo AG, à lineâ au<lb></lb>tem CE intervallo EG; eſſet enim directio CG æquivalens <lb></lb>directioni mixtæ ex CA, & CE. </s> <s id="s.005377">Verùm nova motûs ordina<lb></lb>tio à plano reflectente, quatenus opponitur ulteriori motui, eſt <lb></lb>per lineam perpendicularem CD; hæc autem priori directio<lb></lb>ni FCG adverſatur ſolùm, prout æquivalet Directioni CE <lb></lb>(nam quatenus æquivalet directioni CA, non illi opponitur; <lb></lb>globo ſcilicet, qui per CA moveretur, planum non reſiſteret, <lb></lb>nec illum reflecteret) ac propterea dat oppoſitam directionem <lb></lb>CD, cujus longitudinem ponamus æqualem ipſi CE. </s> <s id="s.005378">Manen<lb></lb>te igitur directione per CA, & directione CE mutatâ in CD, <pb pagenum="731" xlink:href="017/01/747.jpg"></pb>eſt ex utrâque mixta directio CH, ſecundùm quam movetur <lb></lb>corpus reflexum. </s> <s id="s.005379">Quoniam itaque directionum ſingularum <lb></lb>menſuræ ſunt CA, & CE, per A ducatur parallela ipſi DE; & <lb></lb>per E, atque per D, ducantur EG & DH ipſi CA parallelæ. </s> <lb></lb> <s id="s.005380">Eſt ergo rectangulum HE; & quia CD aſſumpta eſt æqualis <lb></lb>ipſi CE, etiam AH & AG ſunt illis æquales. </s> <s id="s.005381">Quapropter cum in <lb></lb>triangulis CAH, CAG rectangulis, latera AC & AG æqualia <lb></lb>ſint lateribus AC & AH, atque angulus comprehenſus ad A <lb></lb>ſit rectus, per 4. lib. 1. angulus ACH (qui eſt angulus Re<lb></lb>flexionis) eſt æqualis angulo ACG: at angulo ACG æqualis <lb></lb>eſt ad verticem angulus incidentiæ FCB, per 15. lib. 1: ergo <lb></lb>angulo FCB incidentiæ æqualis eſt ACH angulus re<lb></lb>flexionis. </s> </p> <p type="main"> <s id="s.005382">Eâdem methodo, ſi angulus incidentiæ fuerit ICB, often<lb></lb>demus angulum reflexionis KCA eſſe illi æqualem; quando<lb></lb>quidem directio CM mutatur in CN, & manet directio CA; ac <lb></lb>propterea directio mixta ex CN, & CA, eſt CK. </s> <s id="s.005383">Atque ita de <lb></lb>cæteris. </s> </p> <p type="main"> <s id="s.005384">Ex quibus obſervabis, quò acutior fuerit angulus inciden<lb></lb>tiæ, in reflexione ita miſceri novam directionem cum anti<lb></lb>quâ, ut magis prævaleat antiqua; nova ſiquidem ad anti<lb></lb>quam, ſecundùm id, quod de illâ remanet, ſe habet ut <lb></lb>Sinus Rectus anguli incidentiæ ad Sinum Complementi. </s> <lb></lb> <s id="s.005385">Nam ſi incidentiæ angulus ſit FCB, & illi æqualis <lb></lb>HCA, nova directio CD, hoc eſt AH ad antiquæ <lb></lb>reſiduum CA, ſe habet ut HA ad AC: Sin autem in<lb></lb>cidentiæ angulus fuerit ICB, hoc eſt illi æqualis angu<lb></lb>lus reflexionis KCA, nova directio ad id, quod de anti<lb></lb>quâ remanet, eſt ut KA ad CA. </s> <s id="s.005386">Eſt autem major Ratio <lb></lb>AC ad AK minorem, quàm ejuſdem AC ad AH majo<lb></lb>rem per 8. lib. 5. Quare quandiu angulus incidentiæ mi<lb></lb>nor eſt ſemirecto, majus eſt reſiduum antiquæ directionis <lb></lb>(attentè obſerva me de ſolâ directione loqui) quâm nova <lb></lb>ordinatio: ubi fuerit angulus ſemirectus, ſunt æquales; <lb></lb>ſi angulus incidentiæ fuerit ſemirecto major, nova ordina<lb></lb>tio major eſt eo, quod remanet de antiquâ directione: ubi <lb></lb>demum fuerit angulus rectus in perpendiculari inciden<lb></lb>tiâ, nova directio ad priorem ſe habet ut Radius ad <pb pagenum="732" xlink:href="017/01/748.jpg"></pb>nihil, motus enim reflexus nihil retinet de priori di<lb></lb>rectione. </s> </p> <p type="main"> <s id="s.005387">Antè tamen quàm in hac diſputatione procedamus, mentis <lb></lb><figure id="id.017.01.748.1.jpg" xlink:href="017/01/748/1.jpg"></figure><lb></lb>oculos tantiſper in globum <lb></lb>C, à quo percutitur planum <lb></lb>AB, convertamus; hacte<lb></lb>nus enim univerſa contem<lb></lb>platio in meris lineis verſa<lb></lb>ta eſt. </s> <s id="s.005388">Et quidem ſi directio<lb></lb>nis linea ſit RS perpendi<lb></lb>cularis tranſiens per globi <lb></lb>centrum C, & punctum <lb></lb>contactûs S, nulla eſſe po<lb></lb>teſt difficultas, quin per <lb></lb>eandem lineam SR reſiliat <lb></lb>ad angulos rectos. </s> <s id="s.005389">Sed ſi in <lb></lb>planum obliquè incidat li<lb></lb>nea directionis globi per <lb></lb>centrum C deducta, & ſit MN; certum eſt in plano AB <lb></lb>punctum N, in quod directionis linea MC producta incurrit, <lb></lb>non eſſe punctum contactûs; alioquin linea à globi centro <lb></lb>ducta ad punctum contactûs, caderet ad angulos inæquales ex <lb></lb>hypotheſi, cum tamen angulos rectos conſtituere demonſtretur <lb></lb>in Sphæricis. </s> <s id="s.005390">Eſt igitur contactus in puncto S, extra lineam <lb></lb>directionis centri, ideóque angulus reflexionis non eſt BNQ <lb></lb>æqualis angulo ANM. </s> <s id="s.005391">Propterea in globo attendendum eſt <lb></lb>punctum S, à quo reipsâ percutitur planum; & ſicuti in circu<lb></lb>lo globum bifariam dividente punctum I delatum eſt per li<lb></lb>neam MI, ita punctum S per lineam OS ipſi MI parallelam <lb></lb>(pono hìc globum non rotari dum movetur, ſed recto itinere <lb></lb>deduci) venit ad contactum & percuſſionem plani. </s> <s id="s.005392">Cum igi<lb></lb>tur OS & MN ſint parallelæ, anguli OSA, & MNA ſunt <lb></lb>æquales: & ſicuti ſi punctum I ſolitarium eſſet, atque juxta <lb></lb>ſuam directionem veniret in N, reflecteretur per NQ, ut re<lb></lb>flexionis angulus QNB eſſet æqualis angulo Incidentiæ <lb></lb>MNA; ita punctum S globi reflectitur per SP, & angulus <lb></lb>reflexionis PSB æqualis eſt incidentiæ angulo OSA; ac pro<lb></lb>inde anguli PSB, & QNB ſunt æquales inter ſe. </s> <s id="s.005393">Centrum <pb pagenum="733" xlink:href="017/01/749.jpg"></pb>igitur C cùm in directione MN haberet directionem mixtam <lb></lb>ex directione CS versùs planum, & directione SB, eum im<lb></lb>pediatur à globi ſoliditate ne ad planum ulteriùs accedat, mu<lb></lb>tatâ directione CS in CR, atque retentâ priore directione SB, <lb></lb>habet directionem mixtam CH omnino ſimilem directioni <lb></lb>puncti S; atque propterea CH eſt parallela ipſi SP. </s> <s id="s.005394">Quod ſi <lb></lb>globus rotari intelligatur, loco linearum, de quibus hactenus <lb></lb>fuit ſermo, concipe plana, in quibus puncta illa ſuas periodos <lb></lb>deſcriberent in motu rotationis, & plana illa eſſent ad planum <lb></lb>reflectens ſimiliter inclinata, ut de lineis dictum eſt. </s> </p> <p type="main"> <s id="s.005395">In cæteris verò corporibus non rotundis idem de eorum re<lb></lb>flexione dicendum eſt, ſervatâ analogiâ, quantum ferre poteſt <lb></lb>anomala eorum figura, & diſpar partium poſitio circa centrum <lb></lb>gravitatis aut magnitudinis: in multis enim hujuſmodi æqua<lb></lb>litas angulorum incidentiæ & reflexionis non exactè ſervatur. </s> <lb></lb> <s id="s.005396">Sic haſtam ſi obliquè contorqueas in rupem, non modò inæ<lb></lb>qualitatem angulorum deprehendes, ſed vix reflexionem fieri <lb></lb>admittes; quia videlicet extremo haſtæ calce rupem tangente, <lb></lb>reliquæ partes habentes circa centrum gravitatis inæqualia mo<lb></lb>menta, valdè turbant motum: ſolùm autem quando haſta in <lb></lb>planum impingitur, aut cadit, ad perpendiculum, ſervata re<lb></lb>flexionis regulâ ad angulos rectos reſilit; quia tunc partes om<lb></lb>nes circa centrum gravitatis paria habent momenta. </s> <s id="s.005397">Hæc au<lb></lb>tem momentorum diverſitas in globo non reperitur, niſi fortè <lb></lb>aut deficiat à perfectâ rotunditate, aut centrum magnitudinis <lb></lb>non ſit idem cum centro gravitatis, adeò ut linea punctum <lb></lb>contactûs cum centro gravitatis, jungens non ſit plano re<lb></lb>flectenti perpendicularis; tunc enim perturbaretur globi <lb></lb>reflexio. </s> </p> <p type="main"> <s id="s.005398">Ex dictis ſatis apertè conſtat reflexionem non ex impetu de<lb></lb>ſumendam eſſe, ſed ex directione motus, cui opponitur corpus <lb></lb>reflectens, juxta hu us poſitionem perpendicularem aut obli<lb></lb>quam: multus enim impetus aliquando officere poteſt æquali<lb></lb>tati angulorum, ſi ex colliſione corporis impacti cum corpore <lb></lb>reflectente, aut alterutrum, aut utrumque notabiliter cedat, <lb></lb>adeò ut non contingat ſincera reflexio. </s> <s id="s.005399">Cæterùm cum ſemper <lb></lb>in reflexione ſit nova directio priori directioni oppoſita, ali<lb></lb>quid impetûs perit pro Ratione oppoſitionis. </s> <s id="s.005400">Ex quo fit in re-<pb pagenum="734" xlink:href="017/01/750.jpg"></pb>flexione ad angulos magis acutos impetum minori decremento <lb></lb>minui, quia nova directio minùs opponitur antiquæ, & minùs <lb></lb>impeditur motus; idcirco globus ad angulum valde acutum re<lb></lb>flexus, ſi offendat in motu reflexo aliquem obicem, multò va<lb></lb>lidiùs illum percutit, quàm ſi ad angulum minùs acutum re<lb></lb>flecteretur, quia, cæteris paribus, majore impetu ſuperſtite <lb></lb>ictum infligit. </s> </p> <p type="main"> <s id="s.005401">Quapropter obvium eſt cuique rationem reddere omnium, <lb></lb>quæ in pilæ ludo contingunt circa ſaltus in pavimento, in quod <lb></lb>pila emiſſa decidit, & reflexiones ad parietem, in quem illa <lb></lb>impingitur. </s> <s id="s.005402">Duo tamen potiſſimùm obſervare placet. </s> <s id="s.005403">Primùm, <lb></lb>quando pila cadit obliqua in pavimentum non procul à pariete, <lb></lb>ſæpè fit duplex reflexio, altera ſcilicet à pavimento, altera à <lb></lb>pariete: ex quo fit, ut pila aliquando longè altiorem ſaltum <lb></lb>edat, ſi multum habeat impetus; quia videlicet à pavimento <lb></lb>reſiliens, ſi in parietem non incurreret, lineam curvam in re<lb></lb>flexione deſcribens vi ſuæ gravitatis impetum extrinſecùs im<lb></lb>preſſum temperantis, citiùs deprimeretur, & magis à recto tra<lb></lb>mite deorſum deflecteret: at quia proximus ponitur eſſe paries, <lb></lb>linea primò reflexa nondum differt notabiliter à lineâ rectâ; <lb></lb>atque proinde in ſecundâ reflexione altiùs pila aſſurgit, quàm <lb></lb>à pavimento diſtaret apex lineæ curvæ, quæ ex primâ reflexio<lb></lb>ne deſcriberetur; nam directio illa ſecunda magis elevata ſupra <lb></lb>horizontem minùs permittit pilam à rectâ lineâ declinare; ut <lb></lb>in baliſtarum & bombardarum globis cum majori elevatione <lb></lb>emiſſis conſtat. </s> <s id="s.005404">Deinde quando reticulis luditur, non rarò re<lb></lb>ticulum movetur in plano aliquo horizontali, aut valde incli<lb></lb>nato (nos Itali dicimus <emph type="italics"></emph>Tagliare, ò Trinciare una palla<emph.end type="italics"></emph.end>) ita ut, <lb></lb>dum pilam rectâ expellit, illi etiam motum quendam imprimat, <lb></lb>quo ipſa circa ſuum centrum movetur: unde fit, ut, niſi pilam <lb></lb>excipias, repelláſque antè, quàm pavimentum attingat, fruſtra <lb></lb>deinde ſaltum illius expectes juxta regulas reflexionis, quia ni<lb></lb>mirum pila terram tangens, dum pergit moveri circa ſuum <lb></lb>centrum motu orbiculari, nequit à plano impediente recipere <lb></lb>directionem illam, cujus eſſet capax, ſi ſolùm ſimplici motu <lb></lb>centri mota fuiſſet; motus enim peripheriæ globi contrarius eſt <lb></lb>motui centri. </s> <s id="s.005405">Idem accidit quando pila leviore affrictu funem <lb></lb>perſtringit; tunc ſcilicet concipit motum circularem, adeóque <pb pagenum="735" xlink:href="017/01/751.jpg"></pb>ſaltus fallit. </s> <s id="s.005406">Quantum autem in motu valeat directiones com<lb></lb>miſcere, alteram centri rectam, alteram peripheriæ circularem <lb></lb>ſed oppoſitam, ſatis norûnt, qui minoribus orbiculis ludentes <lb></lb>globum quaſi pendentem ex manu tenent, dúmque illum pro<lb></lb>jiciunt, manu ei motum circularem communicant; unde oritur, <lb></lb>quod, ubi terram globus attigerit, vel ſiſtit ſe, ſi directio peri<lb></lb>pheriæ ad motum circularem eſt æqualis directioni centri ad <lb></lb>motum rectum; vel tardiùs promovetur, quàm ſi ſolam centri <lb></lb>directionem haberet, prout directio centri major eſt directione <lb></lb>peripheriæ, quæ cum primùm terram attingit, apta eſt ſuá con<lb></lb>verſione retrahere centrum versùs projicientem. </s> </p> <p type="main"> <s id="s.005407">Quandoquidem verò in ludicris philoſophamur, liceat hìc <lb></lb>vulgarem errorem retegere; quando ſcilicet rotundum verticil<lb></lb>lum impreſſus impetus in gyrum agit, ſi verticillus corruat, mo<lb></lb>vetur, quoad impetus extinguatur, ſed ita ut videatur omnino <lb></lb>in contrarias partes agi, ac priùs: id quod communiter tribuunt <lb></lb>reflexioni, quia in pavimentum recidit. </s> <s id="s.005408">Nullam hìc reflexio<lb></lb>nem intercedere, & eandem permanere motûs directionem, <lb></lb>memini me aliquando aſſerentem viſum fuiſſe pluribus, qui <lb></lb>aderant, paradoxum loqui: ſed ubi inter nos convenit eundem <lb></lb>motum eſſe, quandiu circà axem ita fit convolutio, ut quæ par<lb></lb>tes peripheriæ verticilli præcedebant axe inſiſtente, eædem axe <lb></lb>inclinato & procumbente præcedant; juſſi aliquas notas peri<lb></lb>pheriæ imprimi, ut priores à poſterioribus diſcerni poſſent; de<lb></lb>inde yerticillo corruente, & procumbente obſervatum eſt par<lb></lb>tes, quæ priores erant in circuitione, eaſdem ſubinde altiùs at<lb></lb>tolli à pavimento, atque circa axem eandem fieri converſio<lb></lb>nem: & quia verticilli pes quaſi centrum retinet inclinatam <lb></lb>peripheriam, illa eadem converſio circa axem facit, ut peri<lb></lb>pheria ſecundùm poſteriores partes ſubinde attingat ſubjectum <lb></lb>alveolum, adeóque ratione habitâ alveoli videatur in contrarias <lb></lb>partes ferri ac priùs. </s> <s id="s.005409">Quare cùm nulla ſit nova motûs directio <lb></lb>ex plani oppoſitione, nulla quoque eſt reflexio. </s> </p> <p type="main"> <s id="s.005410">At ſi duo corpora ſibi invicem occurrant, ſibi mutuo ob<lb></lb>ſiſtunt, & diminuto ex reſiſtentiâ impetu, ſi quid adhuc reſi<lb></lb>duum fuerit impetûs, qui excedat inſitam repugnantiam ex <lb></lb>gravitate ortam, fit reflexio, aut alterius tantùm, ſi in reliquo <lb></lb>impetus obtundatur, aut utriuſque, ſi fuerint ſibi invicem per-<pb pagenum="736" xlink:href="017/01/752.jpg"></pb>cutiens & percuſſum: ut cùm duo globi ſibi in motu occur<lb></lb>runt aut æquali, aut non immodicè inæquali impetu acti. <lb></lb><emph type="italics"></emph>Æquali<emph.end type="italics"></emph.end> inquam, <emph type="italics"></emph>impetu,<emph.end type="italics"></emph.end> non <emph type="italics"></emph>æquali velocitate<emph.end type="italics"></emph.end>; ſi enim inæqua<lb></lb>les fuerint globi, fieri poteſt, ut eorum velocitates ſint in Re<lb></lb>ciprocâ Ratione gravitatum; tunc ſcilicet impetus æquales ſunt; <lb></lb>contingere ſiquidem poteſt majorem globum tardè quidem <lb></lb>moveri, ſed multo impetu reſpondente ejus moli, adeò ut ex<lb></lb>cedat minoris globi impetum, qui proptereà non præcisè re<lb></lb>flectatur, ſed à majore globo & impetum recipiat, & directio<lb></lb>nem non ex ſolâ reſiſtentiâ definitam, ſed etiam ex ipſius ma<lb></lb>joris globi motu. </s> <s id="s.005411">Id quod ſi contingat, minor quidem reflecti<lb></lb>tur, ſed qui majore impetu ferebatur, modicam inveniens re<lb></lb>ſiſtentiam non reflectitur, quia dum minor globus cedit, plu<lb></lb>rimum impetûs deperditur à majore; & ubi reſiſtentia minor <lb></lb>eſt ceſſione, eſſe nequit reflexio. </s> </p> <p type="main"> <s id="s.005412">Ponamus itaque globos duos tanto impetu actos, ut poſſit <lb></lb>uterque reflecti. </s> <s id="s.005413">Non placet inter illos ad punctum contactûs <lb></lb>interjicere planum, ut ex angulis determinetur reflexio; hoc <lb></lb>enim planum cogitatione nobis ipſi fingimus; ſed, licèt in <lb></lb>idem res recidat, tamen ad veritatem ſinceriùs me acceſſurum <lb></lb>ſpero, ſi rem ex ipſis motuum directionibus & reſiſtentiis deſi<lb></lb><figure id="id.017.01.752.1.jpg" xlink:href="017/01/752/1.jpg"></figure><lb></lb>niero. </s> <s id="s.005414">Quare occurrant ſibi <lb></lb>globi in puncto A; & illo<lb></lb>rum directiones primò eæ <lb></lb>ſint, quæ ſibi maximè ad<lb></lb>verſantes in rectam lineam <lb></lb>BC coëant: haud dubium <lb></lb>quin globus V per rectam <lb></lb>AB, & globus R per rectam <lb></lb>AC reſiliat, uterque per li<lb></lb>neam, quâ venit, jungentem <lb></lb>cum puncto contactûs Centrum impetûs: ſimplici enim di<lb></lb>rectione alter adversùs alterum fertur, & ſibi toto conatu re<lb></lb>pugnant. </s> <s id="s.005415">Deinde obliquus ſit morus, & globi R directio ſit DE: <lb></lb>quapropter RE directio quædam eſt mixta ex lineâ maximæ re<lb></lb>ſiſtentiæ RA, & ex lineâ nullius reſiſtentiæ RI, ita ut partim <lb></lb>versùs A, partim versùs I tendat: priorem directionem versùs A <lb></lb>metitur linea RF, poſteriorem versùs I metitur linea FE. </s> <s id="s.005416">Cum <pb pagenum="737" xlink:href="017/01/753.jpg"></pb>igitur globus V ſolùm priori directioni RF opponatur, hæc <lb></lb>mutatur in oppoſitam RG, manet autem directio versùs I æqua<lb></lb>lis ipſi FE, & eſt GH: propterea motus centri R eſt RH <lb></lb>parallelus motui puncti A, quod per lineam KA incidens in <lb></lb>Tangentem MN, reflecteretur ad angulos æquales KAN & <lb></lb>LAM percurrendo lineam AL. </s> <s id="s.005417">Simili ratione, ſi globi V di<lb></lb>rectio ſit PO, à globo R occurrente in A reflectitur centrum <lb></lb>per rectam VS. <lb></lb></s> </p> <p type="main"> <s id="s.005418"><emph type="center"></emph>CAPUT XII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005419"><emph type="center"></emph><emph type="italics"></emph>Quomodo impetus in percuſsione communicetur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005420">ANtè ſatisfaciendum eſt Phyſicis, quàm percuſſionum con<lb></lb>templationem dimittamus. </s> <s id="s.005421">Quoniam percuſſio omnis mo<lb></lb>tum antecedentem exigit; motus non habetur abſque impetu <lb></lb>concepto aut impreſſo; ex impetu pendet ictus, quo corporis <lb></lb>percuſſi reſiſtentia aliqua vincitur, ſivè illud totum impellatur, <lb></lb>ſivè expellatur, ſivè concutiatur, ſivè flectatur, ſivè compri<lb></lb>matur, ſivè deprimatur, ſivè diſſiliat in partes earum unione <lb></lb>ſolutâ, ſivè quamcumque aliam vim ſubeat; corporis percuſſi <lb></lb>partes, vel omnes, vel aliquæ ſaltem, moveantur, & impetum <lb></lb>recipiant neceſſe eſt, à quo motus ipſe efficiatur impreſſi impe<lb></lb>tûs intenſioni reſpondens. </s> <s id="s.005422">Quærat autem Phyſicus, cuinam <lb></lb>tribuenda ſit virtus efficiendi impetum corpori percuſſo im<lb></lb>preſſum. </s> </p> <p type="main"> <s id="s.005423">Exiſtimabit fortaſſe non nemo à virtute eâdem, quæ in cor<lb></lb>pore percutiente inſidet, ut ſeipſum moveat, effici novum im<lb></lb>petum, quo corpus percuſſum impellatur, aut agitetur. </s> <s id="s.005424">Sed <lb></lb>quid? </s> <s id="s.005425">ſi percutiens neque animans ſit, cujus in poteſtate poſita <lb></lb>ſit motio, neque juxta inſitæ gravitatis directionem ſeipſum <lb></lb>agat. </s> <s id="s.005426">Huic certè inhærens facultas ſe movendi planè otioſa eſt, <lb></lb>quippe quæ prorſus immota conſiſteret, niſi impetum extra<lb></lb>neum reciperet. </s> <s id="s.005427">Aliunde igitur quàm ex hac ſe movendi facul<lb></lb>tate originem ducit impetus corpori percuſſo impreſſus. </s> <s id="s.005428">Dein-<pb pagenum="738" xlink:href="017/01/754.jpg"></pb>de certum eſt corporis percutientis naturam non priùs imprime<lb></lb>re poſſe percuſſo impetum; quàm illud attingat: at in ipſo per<lb></lb>cutientis appulſu ea eſt percuſſi reſiſtentia, ut ejuſdem percu<lb></lb>tientis motum ex ipsâ naturâ provenientem imminuat: cùm <lb></lb>igitur natura percutientis vix ſeipſa movere valeat, quàm te<lb></lb>nues habet vires ad vincendam obicis reſiſtentiam? </s> <s id="s.005429">Præterea, <lb></lb>niſi facta fuerit notabilis in longiore motu naturali acquiſiti im<lb></lb>petûs acceſſio, manifeſtò apparet valdè languida & enervata <lb></lb>percuſſio; &, quamvis ſivè longior, ſivè exiguus motus præ<lb></lb>ceſſerit, eadem manens virtus movendi, nec ſibi diſſimilis, va<lb></lb>rietatem in ſe habet nullam: cum tamen ex diſparibus incre<lb></lb>mentis impetûs in motu acquiſiti diſſimiles fiant percuſſiones: <lb></lb>Non igitur à ſolâ inſitâ vi movendi producitur in percuſſo im<lb></lb>petus. </s> </p> <p type="main"> <s id="s.005430">Propterea, ut una atque eadem in percuſſionibus omnibus <lb></lb>aſſignetur producti impetûs cauſa, ſivè percutiens ſponte ſuâ, <lb></lb>ſivè per vim ſibi illatam moveatur, percutientis impetum plu<lb></lb>res cenſent dicendum eſſe principium & cauſam effectricem <lb></lb>impetûs percuſſo impreſſi; ab illo enim, prout major fuerit, aut <lb></lb>minor, hujus menſuram pendere ſatis innotuiſſe videtur ex <lb></lb>quotidianis experimentis. </s> </p> <p type="main"> <s id="s.005431">Verùm, ne raptim in hanc ſententiam pedarius Philoſophus <lb></lb>curram, illud me remoratur, quod, ſicuti eam eſſe conſtat im<lb></lb>petûs naturam, ut illico prorſus pereat, ac motus ceſſat omni<lb></lb>no illius corporis, in quo priùs inerat motum efficiens, ita pari<lb></lb>ter eodem momento impetum minui neceſſe eſt, eáque Ratio<lb></lb>ne, quo momento, & qua Ratione illius ejuſdem corporis mo<lb></lb>tus ex parte impeditur. </s> <s id="s.005432">Quò igitur magis impeditur percutien<lb></lb>tis motus, eò magis ejuſdem impetum minui conſequens eſt: <lb></lb>propterea, quo momento à percutiente attingitur corpus per<lb></lb>cuſſum, extenuatur in illo impetus, quia tunc illius motus im<lb></lb>peditur; eóque minor evadit in percutiente impetus, quò ma<lb></lb>jus invenit impedimentum motûs. </s> <s id="s.005433">Cùm autem effectui tenui<lb></lb>tatem importet cauſæ imbecillitas, exiguum utique impetum <lb></lb>in corpore percuſſo efficere valeret attenuatus percutientis im<lb></lb>petus, quo momento accidit appulſus atque alliſio; eóque mi<lb></lb>norem impetum reciperet corpus percuſſum, quò magis re<lb></lb>ſiſtens plus inferret impedimenti motui percutientis, quippe <pb pagenum="739" xlink:href="017/01/755.jpg"></pb>cujus impetus fieret languidior; neque enim quicquam juvat <lb></lb>antiqua virtus, ſi nunc eſt effœta. </s> <s id="s.005434">Quò igitur magis reſiſtit <lb></lb>corpus percuſſum, languidiorem ictum exciperet, cum levior <lb></lb>infirmiórque impetus in eo efficeretur à tenuiore & languidio<lb></lb>re percutientis impetu. </s> <s id="s.005435">Sed cum manifeſta refragetur expe<lb></lb>rientia validiores ictus à majore reſiſtentia ortos demonſtrans, <lb></lb>quæſo à Philoſophis, ut in hac causâ mihi dent hanc veniam, <lb></lb>ut patiantur me ab eorum placitis aliquantulum diſcedere, nec <lb></lb>percutientis impetui tribuere facultatem effectricem impetûs <lb></lb>in corpore percuſſo, lyceo quamvis reclamante; cui ſilentium <lb></lb>ſi tantiſper indicere poſſem, dum me audiret poſtulantem id, <lb></lb>quod æquiſſimum eſt, ut ne quid huc præjudicati afferat, meam <lb></lb>fortaſſe in ſententiam volens deduceretur. </s> </p> <p type="main"> <s id="s.005436">Cùm itaque nec à virtute movendi, quæ corpori percutien<lb></lb>ti inhæret, nec ab impetu ejuſdem percutientis effici novum <lb></lb>impetum in corpore percuſſo, ſatis probabili conjecturâ dicen<lb></lb>dum videatur, quænam demum erit cauſa impetûs, & eorum, <lb></lb>quæ impetum conſequuntur, in corpore percuſſo? </s> <s id="s.005437">Ut quæſtio<lb></lb>nibus ſatisfiat, quas percuſſiones excitant, nihil ſe mihi offert <lb></lb>vero propius, quàm ſi dicamus ex percutiente in corpus per<lb></lb>cuſſum migrare impetum, aut totum, aut ex parte, prout alicu<lb></lb>jus motûs capax fuerit corpus, quod motui percutientis reſiſtit. </s> <lb></lb> <s id="s.005438">Si totus impetus à percutiente recedat, hoc neque reflectitur <lb></lb>ab obice percuſſo, neque quicquam procedit in motu: Si quid <lb></lb>impetûs in percutiente remaneat, hoc aut juxta inſtitutam di<lb></lb>rectionem pergit moveri unà cum corpore percuſſo, ſive lentiùs <lb></lb>illud ſequitur, aut aliò reflectitur, pro reſidui impetûs inten<lb></lb>ſione, aut vibratur, & concutitur. </s> </p> <p type="main"> <s id="s.005439">Hinc quia graviſſima ſimul & duriſſima corpora tantum im<lb></lb>petûs obtinere à percutiente nequeunt, quanto opus eſſet, ut <lb></lb>motum aliquem conſpicuum ex percuſſione reciperent, pro<lb></lb>pterea validiſſimè reſiſtunt, & reflectunt, cùm univerſus ferè <lb></lb>impetus in percutiente remaneat: in corpus enim percuſſum <lb></lb>non migrat niſi impetus, qui reſpondeat motui, cujus illud <lb></lb>tunc eſt capax. </s> <s id="s.005440">Contra verò à corporibus, quæ leviter re<lb></lb>ſiſtunt, & facilè moventur aliquo motu, aut nihil, aut langui<lb></lb>dè reflectitur percutiens; quia illa plurimum impetûs reci<lb></lb>piunt, & exiguus impetus in percutiente reliquus eſt. </s> <s id="s.005441">Hinc <pb pagenum="740" xlink:href="017/01/756.jpg"></pb>pariter globus æqualem in mole & gravitate globum percu<lb></lb>tiens eâ directione, quæ per utriuſque globi centra tranſeat, <lb></lb>conſiſtit in loco, ubi percutit, & percuſſum globum vehemen<lb></lb>ter excutit; quia videlicet globus æqualis ſatis reſiſtit, & capax <lb></lb>eſt totius impetûs eum æquali intenſione afficientis, hic deſti<lb></lb>tuens globum percutientem æquè velocem motum percuſſo <lb></lb>conciliat, & percutiens omni deſtitutus impetu conſiſtit. </s> <s id="s.005442">Sin <lb></lb>autem percutiatur globus major & gravior; hic quidem (niſi <lb></lb>nimia ſit gravitatis aut molis differentia) loco cedit; ſed quia ad <lb></lb>motum æquè velocem plus requirit impetus, quàm illi impri<lb></lb>mere valeat globus minor, propterea minore intenſione af<lb></lb>fectus tardiùs movetur, & minorem globum aliquando reflectit. </s> <lb></lb> <s id="s.005443">Si demum globus major minorem & leviorem percutiat, hic <lb></lb>languidiùs reſiſtens impetum recipit velociori motui congruum; <lb></lb>& quia in globo majore adhuc aliquid ſupereſt impetûs, ille <lb></lb>pariter pergit moveri, ſed tardiùs. </s> </p> <p type="main"> <s id="s.005444">At, inquis, impetus ex eo genere eſt, quod Accidentia tan<lb></lb>quam partes complectitur: Accidentia autem ex ſubjecto in <lb></lb>ſubjectum non tranſire, ipſi ſcholarum parietes clamant. </s> <s id="s.005445">Mul<lb></lb>ta iſtiuſmodi, non diffiteor, dicuntur in ſcholis: verùm an ſatis <lb></lb>examinata, momentóque ſuo ponderata fuerint, ignoro: non <lb></lb>pauca quippe habemus de manu, ut aiunt, in manum tradita, <lb></lb>non ad aurificis ſtateram revocata, ſed populari trutinæ per<lb></lb>miſſa. </s> <s id="s.005446">In illis certè Accidentium generibus, quæ poſtremis no<lb></lb>vem Categoriis comprehenduntur, ſi ſex demas, Relationem, <lb></lb>Actionem, Paſſionem, Ubi, Quando, Situm, quos alij (libera<lb></lb>liter-ne? </s> <s id="s.005447">dicam, an prodigè?) <emph type="italics"></emph>Modos<emph.end type="italics"></emph.end> certæ naturæ, à qua avel<lb></lb>li nequeunt, affixos appellant, alij minimo contenti, & parciùs <lb></lb>philoſophantes, nihil eſſe præter mera nomina, aut abſtractas <lb></lb>à rebus inter ſe comparatis intelligentias exiſtimant; vix tria <lb></lb>reliqua genera Quantitas, Qualitas, Habitus conſtituere con<lb></lb>troverſiam poſſunt. </s> <s id="s.005448">Et quidem de Habitu nullus videtur re<lb></lb>lictus ambigendi locus; quis enim neget potuiſſe Therſitem eâ<lb></lb>dem Achillis galeâ, eodémque thorace armari, & regiâ chla<lb></lb>myde ſervum indui? </s> <s id="s.005449">mutatâ ſcilicet armorum aut indumento<lb></lb>rum Ubicatione comparatâ cum hominis, qui armatus dicitur, <lb></lb>aut veſtitus, Ubicatione & poſitione. </s> <s id="s.005450">Quantitatem verò, qua <lb></lb>locus obſidetur (nam de Numero, qui præter individua cogita-<pb pagenum="741" xlink:href="017/01/757.jpg"></pb>tioni ea complectenti ſubjecta nihil eſt, non attinet dicere) <lb></lb>quam multi à materiâ non dividunt? </s> <s id="s.005451">quot Philoſophi ſuam ſin<lb></lb>gulis corporeis rebus tribuunt quantitatem? </s> <s id="s.005452">De ſolâ igitur <lb></lb>Qualitate oriri poteſt quæſtio: cujus tamen ſpecies aliæ me<lb></lb>ram membrorum aut terminorum corporis collocationem & <lb></lb>conformationem dicunt, ut Forma & Figura; aliæ particula<lb></lb>rum in extimâ ſuperficie poſitionem, ut aſperitas & lævor; aliæ <lb></lb>earumdem toto corpore diffuſarum complexionem, ut molli<lb></lb>tudo & durities, raritas & denſitas; aliæ non niſi intelligentiâ <lb></lb>ſecretæ accidere dicuntur Naturæ, cujuſmodi non paucæ Na<lb></lb>turales Potentiæ & Impotentiæ; aliæ Patibiles Qualitates aut <lb></lb>Paſſiones immiſſione corpuſculorum effluentium communican<lb></lb>tur, quemadmodum Odores & Sapores, & fortè etiam quas Pri<lb></lb>mas Qualitates vocant. </s> </p> <p type="main"> <s id="s.005453">Sed quicquid tandem de hujuſmodi Accidentibus aſſerere <lb></lb>placeat (neque enim hìc de iis philoſophandi eſt locus) ultro <lb></lb>demus ea eſſe, quæ licèt à ſubſtantiâ diſtinguantur, per ſe ta<lb></lb>men ſtare nequeant, & neceſſariò ſubjectam aliquam naturam <lb></lb>afficiant, in qua inhæreant: verùm Qualitates omnes (niſi ex <lb></lb>earum genere ſint, quos Modos appellant, quia Actuales De<lb></lb>terminationes, cujuſmodi ſunt cogitationes, appetitiones, & <lb></lb>motus, quibus actio vitæ continetur) quid prohibet nunc huic, <lb></lb>mox illi ſubjecto inhærere, quemadmodum in locum pereun<lb></lb>tis Cauſæ Effectricis, cujus virtute hactenus conſervabantur, <lb></lb>aliam ſubſtitui cauſam, cujus vi adhuc permaneant, omnes fa<lb></lb>temur? </s> <s id="s.005454">Nonne causâ effectrice magis indigent Accidentia, <lb></lb>quàm Materiali & Subjectivâ? </s> <s id="s.005455">Divinâ ſiquidem vi accidentia <lb></lb>à Subjecto avulſa permanere poſſe docemur ex Myſteriis Eu<lb></lb>chariſticis; at ſinè ullâ causâ effectrice conſiſtere nullatenus <lb></lb>poſſunt: hanc ſubinde permutant citrà Naturæ incommodum; <lb></lb>quidni & ſubjectum? </s> <s id="s.005456">Nihil igitur extra modum abſonum & ab<lb></lb>ſurdum loquatur, qui impetum migrantem ex percutiente in <lb></lb>percuſſum ita ſubjectum mutare dixerit, quemadmodum om<lb></lb>nes novum impetum à percutiente malleo produci in percuſſo <lb></lb>& excuſſo globo opinantes, aliam ejuſdem impetûs, quandiu <lb></lb>durat, cauſam, à qua conſervetur, ultro admittunt. </s> </p> <p type="main"> <s id="s.005457">Quamvis autem hoc cæteris qualitatibus ratum ac firmum <lb></lb>eſſet, quod ita ſubjecto, cui ſemel inhæſerint, affigantur, ut <pb pagenum="742" xlink:href="017/01/758.jpg"></pb>aut in illo inſidere neceſſe ſit, aut interire; impetui tamen <lb></lb>privatam legem à Naturâ irrogatam fuiſſe non eſt incon<lb></lb>gruum, quippe qui motui efficiendo, & locorum commutatio<lb></lb>ni, tamquam proxima cauſa, deſtinatus eſt; ſi enim illi corpo<lb></lb>rum tranſlatio tribuenda eſt, quidni & ipſe à corpore, quod <lb></lb>jam commovere nequit ob reſiſtentiam, in aliud corpus proxi<lb></lb>mum faciliùs mobile tranſmittat, ut ſubmoveatur impedimen<lb></lb>tum? </s> <s id="s.005458">Neque mihi videor temerè in hanc ſententiam diſceſſiſſe: <lb></lb>obſervavi ſcilicet quàm multum interſit in vehementi brachij <lb></lb>projectione, ſi verè lapidem manu longiùs excutias, ac ſi tan<lb></lb>tummodo, eâdem quidem contentione, ſed manu vacuâ, te la<lb></lb>pidem jactare mentiaris: hoc enim poſtremum ſinè dolore non <lb></lb>accidit, quia impetus à brachio in lapidem jactandum transfe<lb></lb>rendus ſi in brachio permaneat, hoc ſecum rapit, & nexum <lb></lb>diſtrahit, quo tenetur cum humero colligatum. </s> <s id="s.005459">At ne fortè <lb></lb>me potiùs opinionis commento, quàm re ductum ſuſpiceris <lb></lb>(quamquam & alij hunc eundem brachij dolorem experientes <lb></lb>non ſemel probârunt) baliſtæ arcum chalybeum intento nervo <lb></lb>inflecte, ac ſæpiùs, nullo adjecto globo aut telo, quod explodat <lb></lb>& ejiciat, ſubmoto nervi adducti retinaculo dimitte: an diutiùs <lb></lb>inani hoc ludo uti licebit? </s> <s id="s.005460">ſexcenties utique & millies baliſtâ <lb></lb>hac globos argillaceos ejaculaberis citrà arcûs detrimentum; <lb></lb>ſed non item ſine incommodo ſæpiùs vacuum nervum dimittes, <lb></lb>quin arcus ipſe in periculum ac diſcrimen vocetur, ne facilè <lb></lb>diſrumpatur: impetus ſiquidem, quem miſſili imprimere opor<lb></lb>tuit, in arcu, dum ſeſe vi elaſticâ reſtituit, permanens illum va<lb></lb>lidiùs concutit, ac ſæpiùs labefactans demum diffindit. </s> </p> <p type="main"> <s id="s.005461">Quapropter, cùm ex projectionibus ſatis habeamus argumen<lb></lb>ti, poſſe impetum ex projiciente migrare in projectum, quo mo<lb></lb>mento projicitur; cur non item poterit impetus ex percutiente <lb></lb>in percuſſum tranſire, quo momento percutitur, prout hoc mo<lb></lb>tum aliquem concipere poteſt pro impetûs Ratione? </s> </p> <p type="main"> <s id="s.005462">Neque ut percuſſi impetum à percutientis virtute tunc pri<lb></lb>mò productum adſtruas, conferenda eſt Percuſſio cum Impul<lb></lb>ſione; non enim par eſt in Percuſſione aut Projectione, atque in <lb></lb>Simplici Impulſione aut Tractione philoſophandi ratio: Poten<lb></lb>tia enim corpori impulſo aut raptato applicata quandiu cum <lb></lb>illo nectitur, & ſe, & illud movet quaſi corpus unum ex utro-<pb pagenum="743" xlink:href="017/01/759.jpg"></pb>que conflatum: propterea ſicut muſculi in animante oſſa ſibi <lb></lb>cohærentia attollentes & ſe movent, & oſſa; ita potentia Vecti <lb></lb>applicata & ſe movet, & vectem, & pondus, atque equi cur<lb></lb>rui adjuncti non modò ſeipſi, ſed & currum, trahentes movent. </s> <lb></lb> <s id="s.005463">At Percuſſio ſæpè corpus percuſſum procul à percutiente ejicit, <lb></lb>quemadmodum & Projectio. </s> <s id="s.005464">Quod ſi cum Percuſſione junga<lb></lb>tur Impulſio (quæ ſemper Projectionem præcedit) impetus in <lb></lb>Impulſione producitur à potentiâ impellente; ſed ſicut momen<lb></lb>to Projectionis qui erat in projiciente impetus, migrat in pro<lb></lb>jectum, quod diſcedit; ita in Percuſſione primo Percuſſionis mo<lb></lb>mento tranſit impetus in corpus percuſſum pro ejus capacitate: <lb></lb>quod ſi præterea impellatur à corpore percutiente, cujus motus <lb></lb>juxta ſuam directionem procedat, & urgeat partes corporis per<lb></lb>cuſſi (ut in iis, quæ deprimuntur, aut comprimuntur contin<lb></lb>git & cùm ſublicas, dum panguntur, fiſtuca ex caſu non reſi<lb></lb>liens impellit) impetum aliquem habet ab impellente pro<lb></lb>ductum præter impetum ab eodem tamquam percutiente, ipſo <lb></lb>percuſſionis momento communicatum: ſed qui ab impellente <lb></lb>efficitur, non admodum multus eſt, ſi cum eo componatur, qui <lb></lb>ex percuſſione habetur. </s> </p> <p type="main"> <s id="s.005465">Simile quid Impulſioni, quæ Percuſſionem ſequitur, habe<lb></lb>tur in Tractione, quam Excurſus præceſſit, in quo acquiſitus <lb></lb>eſt impetus: quo enim momento Excurſus ceſſat, & incipit <lb></lb>Tractio, tranſit impetus, & minuitur in trahente; ut ſi lapis in <lb></lb>pavimento jacens fune jungatur alteri lapidi paulò minori, fu<lb></lb>nis autem orbiculo verſatili inſideat, & lapis ille minor cadens, <lb></lb>donec funem intendat, impetum ex motu acquirat; ſtatim ac <lb></lb>intentus eſt funis, & lapis jacens deſcendentis lapidis motui re<lb></lb>ſiſtit impetus acquiſitus migrat ad vincendam jacentis lapidis <lb></lb>reſiſtentiam, atque acceptâ à trahentis motu directione cogi<lb></lb>tur aſcendere, quandiu alter deſcendit, & hunc aliquantulum <lb></lb>trahit; ſed impetu impreſſo langueſcente in lapide graviore hic <lb></lb>deſcendit, & ſurſum viciſſim rapit eum, à quo vim paſſus fue<lb></lb>rat. </s> <s id="s.005466">Sic potentia velociter languidum funem intendens mul<lb></lb>tum concipit impetum, quem ponderi adnexo imprimit, <lb></lb>dum illo deſtituitur, cum primùm reſiſtentiam patitur, ſed & <lb></lb>aliam impetûs particulam trahendo producit atque efficit in <lb></lb>pondere. </s> </p> <pb pagenum="744" xlink:href="017/01/760.jpg"></pb> <p type="main"> <s id="s.005467">Cum igitur duplex ſit in motu ſubmovendorum impedimen<lb></lb>torum genus, alia, videlicet, quæ inchoatum motum ab<lb></lb>rumpunt, alia quæ obſiſtunt, ne fiat motus; illa tollenda <lb></lb>ſunt per impetum, quo motus continuandus fuiſſet, niſi im<lb></lb>pedimentum occurriſſet; hæc verò ſuperat pars impetûs pro<lb></lb>ducta à potentiâ, quæ ſe tardiùs movet, quia vires dividit, <lb></lb>partem impetûs ſibi reſervans, partem impertiens obſtaculo, <lb></lb>quod removet impellendo aut trahendo. </s> <s id="s.005468">Quare nil mirum, ſi <lb></lb>impetus, qui periturus eſſet in percutiente, cujus motus im<lb></lb>peditur, tranſeat in obicem percuſſum, quem ſubmovendo <lb></lb>locum relinquit ulteriori motui, ſi facultas ſe movendi ſuppe<lb></lb>tat corpori percutienti. <lb></lb></s> </p> <p type="main"> <s id="s.005469"><emph type="center"></emph>CAPUT XIII.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005470"><emph type="center"></emph><emph type="italics"></emph>Cunei uſus promovetur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005471">NE quis fortè Cuneum ſolis ruſticis ad findenda ligna uſui <lb></lb>eſſe ſibi perſuadeat, fontes aliquos indicare placet ex <lb></lb>quibus non levis utilitas derivatur. </s> <s id="s.005472">Ad Machinarum ſcilicet <lb></lb>Rationem pertinet potiſſimùm motus corporis, cujus reſiſten<lb></lb>tia ſuperatur, ſivè illa demum ex gravitate oriatur, ſivè ex <lb></lb>nexu, quo colligatur cum proximo corpore; id quod iis con<lb></lb>tingit, quæ in corpus unum coaleſcunt, & fiſſione ſejun<lb></lb>guntur. </s> </p> <p type="main"> <s id="s.005473"><emph type="center"></emph>PROPOSITIO I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005474"><emph type="center"></emph><emph type="italics"></emph>Vectis vires Cuneo augere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005475">COntingit aliquando potentiam incommodè applicari <lb></lb>vecti, ut cum hominem valde curvari oportet ad <lb></lb>vectem ſecundi generis ferè in ſolo jacentem attollendum; <lb></lb>tunc ſubſidium à Cuneo non incongruè peti poteſt. </s> <s id="s.005476">Sit <lb></lb>Vectis AB ſubjectus foribus DC ſuis è cardinibus avellen-<pb pagenum="745" xlink:href="017/01/761.jpg"></pb>dis, ut reficiantur: hypomochlium eſt in A, & pondus in C. </s> <lb></lb> <s id="s.005477">At ſi potentiam adeò in<lb></lb><figure id="id.017.01.761.1.jpg" xlink:href="017/01/761/1.jpg"></figure><lb></lb>clinari atque curvari opor<lb></lb>teat, ut arripiat extremum <lb></lb>vectem B, ſatis manifeſtum <lb></lb>eſt, quanto id incommodo <lb></lb>fiat. </s> <s id="s.005478">Subjiciatur vecti in B <lb></lb>(ſiquidem ſolum æquo <lb></lb>mollius fuerit) aſſeris aut <lb></lb>lapidis pars, quæ compreſſioni reſiſtat, atque inter illam & <lb></lb>vectem apex Cunei E immittatur. </s> <s id="s.005479">Nam ſi tudite cuneum per<lb></lb>cutias, vectem facilè attollet, ac proinde etiam valvas in C <lb></lb>incumbentes. </s> </p> <p type="main"> <s id="s.005480">Quod ſi vectis ſecundi generis FG habens hypomochlium <lb></lb>in G ita fuerit altiùs col<lb></lb><figure id="id.017.01.761.2.jpg" xlink:href="017/01/761/2.jpg"></figure><lb></lb>locatus, ut ægrè brachio<lb></lb>rum contentione attolle<lb></lb>re valeas pondus in K ad<lb></lb>nexum, utere cuneo in<lb></lb>flexo FH, quem ſolo in I <lb></lb>incumbentem, & vecti in <lb></lb>F ſubjectum, ſi propellas <lb></lb>lateri HI, arrepto manu<lb></lb>brio LM, applicatus, prout commodiùs acciderit, vectem cum <lb></lb>pondere eatenus elevabis, quoad latus IH longius, ſolo ad <lb></lb>pendiculum inſiſtat. </s> <s id="s.005481">Vectem autem, qua parte cuneum hujuſ<lb></lb>modi contingit, ita extenuatum eſſe oportere, ut cunei orbitæ ex<lb></lb>cavatæ congruat, ne elabatur, res per ſe ipſa loquitur. </s> </p> <p type="main"> <s id="s.005482">Maximè verò opportunum duxerim hujuſmodi cuneo in<lb></lb>flexo uti, ubi tertij generis Vectis <lb></lb><figure id="id.017.01.761.3.jpg" xlink:href="017/01/761/3.jpg"></figure><lb></lb>adhibendus fuerit RS, & pondus <lb></lb>adnexum ex S in V ſuſtollendum: <lb></lb>Nam ſi loco Potentiæ deſtinato <lb></lb>in T ſubjicias cuneum inflexum <lb></lb>TP, ſolo in X incumbentem, & <lb></lb>hunc urgeas ex latere XP, aut <lb></lb>trahas ex latere TX, ubi P venerit <lb></lb>in Q, pondus ex S erit in V. </s> </p> <pb pagenum="746" xlink:href="017/01/762.jpg"></pb> <p type="main"> <s id="s.005483"><emph type="center"></emph>PROPOSITIO II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005484"><emph type="center"></emph><emph type="italics"></emph>Vecte, aut Trochleâ, aut Succulâ, Cunei inflexi <lb></lb>vires augere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005485">INſtrumenta preſſoria varij generis excogitantur; ſed ad ſubi<lb></lb>tum uſum, & non ſinè compendio, uti aliquando poſſumus <lb></lb>Cuneo inflexo, cui, ſi validiùs premendum ſit, vectem adjice<lb></lb><figure id="id.017.01.762.1.jpg" xlink:href="017/01/762/1.jpg"></figure><lb></lb>re licebit. </s> <s id="s.005486">Sit craſſior tabula AB, <lb></lb>cui ſupponatur id, quod premen<lb></lb>dum proponitur. </s> <s id="s.005487">Paretur cuneus <lb></lb>inflexus DE, & pro illius motûs <lb></lb>centro ſtatuatur punctum C, cui <lb></lb>axis infigatur: Nam urgendo lon<lb></lb>gius latus CE, aut trahendo bre<lb></lb>vius CD, ſubjectam tabulam AB <lb></lb>premes. </s> <s id="s.005488">Quod ſi validiore preſſu <lb></lb>opus fuerit, lateri longiori CE <lb></lb>Vectem FG adhibe: potentia ſi<lb></lb>quidem in G majorem arcum deſ<lb></lb>cribens circa centrum motûs C, <lb></lb>majora obtinebit momenta, quàm <lb></lb>ſi proximè illa applicaretur Cuneo: illa tamen momenta poten<lb></lb>tiæ in G ſenſim minuuntur, prout cunei partes tabulam con<lb></lb>tingentes propiores ſunt extremo puncto E. </s> <s id="s.005489">Ne verò ipſa ea<lb></lb>dem tabula AB impedimento ſit, ſi lateri CE proximè vectis <lb></lb>adhæreret, affigatur cuneo unum aut alterum Chelonion HF, <lb></lb>IK, in quæ conjectus vectis FG diſtet à Cuneo citrà pericu<lb></lb>lum incurrendi in ſubjectam tabulam AB. </s> </p> <p type="main"> <s id="s.005490">Vel ſi Vectem cuneo affigere non placuerit, ipſius vectis ca<lb></lb>put hypomochlio reſpondens ita collocetur, ut vectis horizonti <lb></lb>ferè paralleli longitudo tranſverſa cadat in latus cunei DE, <lb></lb>séque non procul ab E decuſſent: hac enim ratione vecti ſua <lb></lb>conſtabunt momenta, quibus momenta cunei augeantur. </s> </p> <p type="main"> <s id="s.005491">At ſi fortè loci diſpoſitio non ferat, ut vectis adhibeatur im<lb></lb>pellendo cuneo, trahatur ille ex D, ubi aut annulus infigatur, <pb pagenum="747" xlink:href="017/01/763.jpg"></pb>aut foramini inſeratur funis, cui deinde trochlea ſive ſimplex, <lb></lb>ſive multiplex adnectatur, prout opus fuerit. </s> <s id="s.005492">Immò & Succula <lb></lb>addi poterit, ad quam funis caput religetur: erúntque momen<lb></lb>ta potentiæ, quæ componuntur ex Rationibus Succulæ, Tro<lb></lb>chleæ, & Cunei. </s> </p> <p type="main"> <s id="s.005493"><emph type="center"></emph>PROPOSITIO III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005494"><emph type="center"></emph><emph type="italics"></emph>Cuneum inflexum validiſsimum conſtruere ad <lb></lb>Vectem tùm trahendum, tùm repellendum.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005495">ASſumatur planum aliquod circulare circa axem per cen<lb></lb>trum ductum verſatile, ita craſſum & validum, ut in eo <lb></lb>inſculpi poſſit profundiùs ſpira, quæ Vectis caput ferreo clavo <lb></lb>capitato, & in globum rotundato, armatum contineat, ne ela<lb></lb>batur. </s> <s id="s.005496">Hinc enim fiet, ut in ſpiræ cavitatem immiſſum Vectis <lb></lb>caput aut propellatur, aut attrahatur, prout plani illius motus <lb></lb>in hanc aut illam partem dirigitur: tantus ſcilicet erit Vectis <lb></lb>motus, quanta erit Radiorum à centro ad ſpiræ ambitum ducto<lb></lb>rum differentia. </s> <s id="s.005497">Ex quo orietur tractio, aut impulſio; Radiis <lb></lb>enim decreſcentibus trahitur Vectis ad centrum, illis creſcen<lb></lb>tibus propellitur à centro. </s> <s id="s.005498">Potentia igitur certæ plani illius cir<lb></lb>cularis parti applicata integrum circulum deſcribit, dum vectis <lb></lb>caput per unum ſpiræ flexum excurrit, & tot circulos potentia <lb></lb>deſcribit, quot ſpiræ flexus vectis caput ſubinde complectun<lb></lb>tur. </s> <s id="s.005499">Quare comparanda eſt diſtantia à centro plani circumacti, <lb></lb>quam in motu caput Vectis mutavit, cum univerſis circulis, <lb></lb>quos potentia interim deſcripſit, & ſtatim innoteſcet Ratio mo<lb></lb>mentorum. </s> <s id="s.005500">Hinc ſi plano hujuſmodi, in quo excavata eſt ſpi<lb></lb>ra, addideris circa extremam orbitam Radios, quemadmodum <lb></lb>Axi in Peritrochio, motus potentiæ ſatis amplos circulos <lb></lb>deſcribet. </s> </p> <p type="main"> <s id="s.005501">Statuamus, exempli gratiâ, plani circularis aſſumpti diame<lb></lb>trum cubitalem, hoc eſt ſeſquipedalem, ſeu digitorum 24; ſit <lb></lb>autem inter ſpiræ excavatæ flexum & flexum intercapedo digi<lb></lb>torum duûm, adeò ut, peractâ circulatione unâ, vectis caput <lb></lb>per ſpiram excurrens digitos duos à primâ ſuâ ſede dimotum <pb pagenum="748" xlink:href="017/01/764.jpg"></pb>fuerit: at potentia in extremâ orbitâ plani circularis conſtituta <lb></lb>ſuo motu integram peripheriam circuli, cujus diameter digi<lb></lb>torum 24, deſcripſerit, hoc eſt digitorum 75. Motus igitur <lb></lb>potentiæ ad motum capitis vectis eſt ut 75 ad 2: cui ſi addatur <lb></lb>Ratio ad ipſum Vectem ſpectans, quatenus cum pondere com<lb></lb>paratur, fiet Ratio Compoſita indicans Rationem motûs poten<lb></lb>tiæ ad motum ponderis. </s> </p> <p type="main"> <s id="s.005502">Quapropter etiamſi ad conciliandum ponderi motum paulò <lb></lb>velociorem, uteremur Vecte primi generis ſed inverſo, ita ut <lb></lb>ab hypomochlio plus diſtaret pondus, quàm potentia capiti <lb></lb>vectis applicata, adhuc haberetur non modicum momentorum <lb></lb>compendium. </s> <s id="s.005503">Sit enim diſtantia potentiæ in capite vectis ab <lb></lb>hypomochlio ut 1, ponderis verò ut 5; atque adeò, dum Vectis <lb></lb>caput deprimitur digitos duos, pondus attollatur digitos de<lb></lb>cem: Componantur duæ Rationes 75 ad 2, & 1 ad 5; erit Ra<lb></lb>tio 15 ad 2, & potentia ſpiræ applicata movebit pondus hujuſ<lb></lb>modi vecti adnexum lib.150, quo conatu abſque ullâ machinâ <lb></lb>moveret pondus lib. 20. </s> </p> <p type="main"> <s id="s.005504">Hanc propoſitionem hìc potiùs afferre placuit, quàm in ſe<lb></lb>quentem librum de Cochleâ reſervare, quia hìc caput vectis <lb></lb>excurrit per ipſam ſpiram, & proximè pertinere videtur hìc <lb></lb>motus ad motum ſuper faciem Cunei inflexi: in Cochleâ verò, <lb></lb>prout communiter illa uſurpatur, pondus movetur ad motum <lb></lb>cylindri, cui inſculpta eſt Cochlea. </s> <s id="s.005505">Dixi, prout communiter <lb></lb>uſurpatur, quia aliquid ſimile contingit Cochleæ infinitæ, ut <lb></lb>videbimus. </s> </p> <p type="main"> <s id="s.005506"><emph type="center"></emph>PROPOSITIO IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005507"><emph type="center"></emph><emph type="italics"></emph>Flatum vehementem non interruptum excitare <lb></lb>follibus adhibitis.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005508">GLebam metallicam ex fodinis erutam valido igne exco<lb></lb>quere oportet, ut metallum fluat, atque id, quod utile eſt, <lb></lb>ab inutili ſecernatur. </s> <s id="s.005509">Ignis autem ut ex carbonibus excitetur <lb></lb>eâ vehementiâ, qua opus eſt, etiam vehementem flatum, qui <lb></lb>ex follibus exprimatur adhibendum manifeſtum eſt omnibus: <pb pagenum="749" xlink:href="017/01/765.jpg"></pb>neque enim ubique commodum reperiri poteſt conclave hy<lb></lb>pogæum, in quod præceps delapſa aqua aërem vapori miſtum <lb></lb>per tubum in camini focum impellat. </s> <s id="s.005510">Quare præter vulgarem <lb></lb>& notiſſimam methodum folles alterno motu agitandi, ex his, <lb></lb>quæ hujus lib. cap. 3. dicta ſunt, rationem aliquam mire poſ<lb></lb>ſumus, qua plurimum flatus in carbones accenſos immit<lb></lb>tamus. </s> </p> <p type="main"> <s id="s.005511">Quod ad folles ipſos ſpectat, non illos ſimplices vellem, ſed <lb></lb>ſingulos duplices, ita videlicet conformatos ut ſinguli ex binis <lb></lb>aſſeribus conſtent invicem ſecundùm alteram extremitatem in<lb></lb>clinatis, quaſi in angulum coituri eſſent, qui omnino ſtabiles <lb></lb>permaneant. </s> <s id="s.005512">In ipſo autem tigillo, cui firmiter infixa manent <lb></lb>aſſerum illorum capita, excavatus ſit congruè ductus, per quem <lb></lb>flatus exprimatur in tubum adnexum, quo ad focum defertur: <lb></lb>atque opportuno loco in ſingulis aſſeribus, ut moris eſt, fora<lb></lb>men excipiendo aëri deſtinatum aſſario muniatur. </s> <s id="s.005513">Hos inter <lb></lb>immotos, planum aliud ſimile, ad extremitatem fibulâ verſatili <lb></lb>connexum cum tigillo illo communi, adjiciatur, & cum extre<lb></lb>mis aſſeribus corio plicatili jungatur, adeò ut duo ſint conjuncti <lb></lb>folles, quorum alter clauditur, alter recluditur, cùm ex medio <lb></lb>hoc plano mobili exiens anſa adducitur & reducitur: hoc enim <lb></lb>mobile planum eſt diaphragma ſejungens folles, ne ex altero in <lb></lb>alterum compreſſus aër effugiat, ſed per infimum ductum in <lb></lb>tubum erumpat, per quem ad focum devehatur. </s> <s id="s.005514">Quatuor pa<lb></lb>rentur hujuſmodi folles duplices, quorum bini ſibi ex diametro <lb></lb>oppoſiti ita ſtatuantur, ut inter illos diſcus circularis congruæ <lb></lb>magnitudinis interjectus eorum anſas ſubinde propellere va<lb></lb>leat: bini autem oppoſiti funiculo jungantur aut loro, aut ca<lb></lb>tenulâ, anſas connectente longitudinis æqualis diametro cir<lb></lb>culi: Ex quo fiet, ut operâ eâdem follis unius anſa propellatur, <lb></lb>oppoſiti verò anſa trahatur. </s> </p> <p type="main"> <s id="s.005515">Porrò attendendum eſt, quantum ſpatij percurrat ſingulo<lb></lb>rum follium anſa ultro citróque remeando, quo loco illa tangi<lb></lb>tur à circulo: hujus enim ſpatij ſemiſſe definietur intervallum, <lb></lb>quo circuli centrum abeſſe oportet à centro, quod ſtatuendum <lb></lb>eſt, ut circa illud fiat ejuſdem circuli convolutio. </s> <s id="s.005516">Huic motûs <lb></lb>centro infigendus eſt firmiter axis, ſivè ille ſit communis exte<lb></lb>riori rotæ ab aquâ fluente convolutæ, ſivè cui vectis opportu-<pb pagenum="750" xlink:href="017/01/766.jpg"></pb>næ longitudinis adjiciatur, ut ab homine, aut à jumento con<lb></lb><figure id="id.017.01.766.1.jpg" xlink:href="017/01/766/1.jpg"></figure><lb></lb>torqueatur. </s> <s id="s.005517">Sic anſæ motus <lb></lb>univerſus. </s> <s id="s.005518">Sit ex. </s> <s id="s.005519">gr. AB <lb></lb>circuli centrum ſit C: acci<lb></lb>piatur intervallum CD ſub<lb></lb>duplum ipſius AB; & erit <lb></lb>in D infigendus axis, ex cu<lb></lb>jus convolutione circulus pa<lb></lb>riter circumagatur, & fol<lb></lb>lium anſas in quatuor oppo<lb></lb>ſitis punctis A, E, G, F ſubinde tangat, eáſque viciſſim propel<lb></lb>lat, & trahat: Cùm ſcilicet incipit propelli follis anſa, quæ eſt <lb></lb>in E, propellitur pariter ea, quæ eſt in G (ſi quidem conver<lb></lb>ſio fiat ex E in F) atque ex adverſo tantumdem trahitur quæ <lb></lb>eſt in A, quantùm propellitur quæ eſt in E; atque ſimiliter <lb></lb>tractio ejus, quæ eſt in F, eſt æqualis impulſioni anſæ, quæ <lb></lb>eſt in G. </s> </p> <p type="main"> <s id="s.005520">Si igitur circulus non ſit in plano Verticali, & axem in D in<lb></lb>fixum non habeat communem cum rotâ, quæ ab aquâ volva<lb></lb>tur, ſed ſit in plano horizontali, axi infixo in D addatur vectis <lb></lb>DH, ut potentia in H vectem impellens aut trahens circum<lb></lb>agat circulum. </s> <s id="s.005521">Quo autem loco ſtatuendus ſit vectis, pendet <lb></lb>ex loci poſitione, prout vel in ſuperiori, vel in inferiori, vel in <lb></lb>eodem conclavi folles collocantur; axis ſiquidem certam non <lb></lb>exigit longitudinem, ſed ea illi tribuenda eſt, quæ commodior <lb></lb>acciderit. </s> <s id="s.005522">Vectis tamen longitudinem ita temperare oportet, <lb></lb>ut, dum potentiæ movendi facilitatem affectas, nimiam tardi<lb></lb>tatem compreſſionis follium effugias. </s> </p> <p type="main"> <s id="s.005523">Ex his ſatis apparet, quantùm aëris impellatur in prunas à <lb></lb>quatuor follibus, qui clauduntur, dum quatuor reliqui reclu<lb></lb>duntur, perpetuúſque eſt flatus nunquam interruptus. </s> <s id="s.005524">Quòd <lb></lb>ſi potentia movens viribus abundet, & circulus fieri poſſit am<lb></lb>plior ita, ut non quatuor ſolùm follibus duplicibus, ſed etiam <lb></lb>ſex aut octo ſimilibus in gyrum diſponendis commodus locus <lb></lb>ſuppetat, ſatis vides, quantus excitari poſſit flatus. </s> </p> <p type="main"> <s id="s.005525">Quoniam verò ex dictis infertur folles eſſe erigendos, ut in <lb></lb>eorum anſas circulus horizonti parallelus incurrat, obſerva poſ<lb></lb>ſe illos etiam jacentes (modò aſſarium ſuperioris aſſeris foramen <pb pagenum="751" xlink:href="017/01/767.jpg"></pb><expan abbr="claudẽs">claudens</expan> exactè fungi poſſit ſuo munere) uſui eſſe poſſe, ſi <expan abbr="artificiũ">artificium</expan> <lb></lb>aliquod adhibeatur, quo anſa attollatur, atque deprimatur. </s> <s id="s.005526">Fiat <lb></lb>inflexus vectis, ſeu quaſi vec<lb></lb><figure id="id.017.01.767.1.jpg" xlink:href="017/01/767/1.jpg"></figure><lb></lb>tis RSG, cujus angulo S <lb></lb>addatur axis, circa quem fa<lb></lb>cilè converti poſſit, & ad ex<lb></lb>tremitatem G ſit regula GH <lb></lb>follis anſæ adnexa; & ſimilia <lb></lb>omnia ex adverſo parentur, <lb></lb>atque funiculo MP conjun<lb></lb>gantur. </s> <s id="s.005527">Nam circulus im<lb></lb>pellens in R attollet extre<lb></lb>mitatem G, ac proinde anſam illi adnexam, trahendo autem <lb></lb>funiculum MP deprimet extremitatem O, & cum illâ anſam <lb></lb>oppoſiti follis: atque ita viciſſim impellendo N, attolletur O, <lb></lb>& deprimetur G. </s> <s id="s.005528">Hinc colliges hoc eodem artificio, ſi haſtulæ <lb></lb>GH non follis anſam, ſed antliæ embolum adjeceris, fieri poſ<lb></lb>ſe machinam, qua, multiplicatis antiis, plurimum aquæ ſur<lb></lb>ſum impellere valeas. </s> <s id="s.005529">An autem diſci circularis exteriorem or<lb></lb>bitam ferreo annulo polito & lævi munire, atque ferream lami<lb></lb>nam pariter politàm anſæ follis, aut vecti inflexo, apponere <lb></lb>præſtet, ut quàm minimo tritu inter ſe confligant, non eſt opus <lb></lb>monere, ſi fuerit operæ pretium machinæ diuturnitati conſu<lb></lb>lere, & faciliorem motum exhibere. </s> </p> <p type="main"> <s id="s.005530">Quòd demum ſpectat ad ipſius circuli collocationem, quam<lb></lb>quam cylindrus illi infixus poſſit inniti polis, circa quos verſe<lb></lb>tur; ut tamen ſubter circulum liberrimè trahi poſſint funiculi, <lb></lb>placeret potiùs illum omnino ſuſpenſum pendere ex craſſiore <lb></lb>(ſive ſimplici, ſive ex duobus compacto) tigno, cujus forami<lb></lb>ni inſeratur cylindrus, ferreo annulo munitus tam in ſuperiori, <lb></lb>quàm in inferiori parte, qua foramini reſpondet, ita ut neque <lb></lb>ſurſum agi, neque deorſum deſcendere valeat, ſed intrà fo<lb></lb>ramen illud convertatur, quod pariter utrinque circulis fer<lb></lb>reis muniatur reſpondentibus ſuperiori & inferiori annulo <lb></lb>cylindri. </s> <s id="s.005531">Sit circularis diſcus AB, in quo motûs centrum <lb></lb>ſit C, cui infigatur cylindrus CD convertendus à vecte, <lb></lb>ſivè in I, ſivè in H immittendo. </s> <s id="s.005532">Horizonti parallelum <pb pagenum="752" xlink:href="017/01/768.jpg"></pb>tignum KL ſecundùm ſuas extremitates in pariete, aut ali<lb></lb><figure id="id.017.01.768.1.jpg" xlink:href="017/01/768/1.jpg"></figure><lb></lb>ter, firmetur, in eóque ſit <lb></lb>foramen F capax cylindri: <lb></lb>foramen ferreo circulo, <lb></lb>quoad fieri poſſit, lævi at<lb></lb>que polito muniatur, cui <lb></lb>æqualis annulus cylindrum <lb></lb>veſtiens, illíque affixus, <lb></lb>reſpondeat. </s> <s id="s.005533">Manebit ex F <lb></lb>ſuſpenſus cylindrus DC uná <lb></lb>cum adjuncto circulari diſ<lb></lb>co AB. </s> <s id="s.005534">In inferiore tigni <lb></lb>facie ſimiliter ſit circulus <lb></lb>ferreus, & annulus, ne ſur<lb></lb>ſum excurrere queat cylin<lb></lb>drus. </s> <s id="s.005535">Si tignum quidem <lb></lb>valde diſtet à circulo AB, <lb></lb>immitti poterit vectis in H; <lb></lb>at ſi exiguum fuerit inter<lb></lb>valium inter tignum & cir<lb></lb>culum AB, atque tignum exiſtat infra planum, in quo ho<lb></lb>mo aut jumentum vectem impellens aut trahens movetur, <lb></lb>vectis in I immittatur. </s> </p> <p type="main"> <s id="s.005536"><emph type="center"></emph>PROPOSITIO V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005537"><emph type="center"></emph><emph type="italics"></emph>Plures antlias duplices perpetuo ductu agitare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005538">UBi jumentorum operâ uti oportet ad agitandas antlias, <lb></lb>quibus aqua in ſuperiorem locum aut attrahitur, aut im<lb></lb>pellitur, illa in gyrum agere neceſſe eſt; id quod multo <lb></lb>tempore eget; neque enim circuitus illos currendo efficere <lb></lb>poſſunt: quapropter rotarum dentatarum complexionem <lb></lb>conſtruere ſolemus, ut, dum ſemel jumentum ſuam con<lb></lb>verſionem abſolvit, ſæpiùs antliæ agitentur. </s> <s id="s.005539">Verùm minore <lb></lb>impendio abſque rotis idem fortaſſe aſſequemur, ſi potiſſimùm <lb></lb>aqua in altum propellenda ſit. </s> </p> <pb pagenum="753" xlink:href="017/01/769.jpg"></pb> <p type="main"> <s id="s.005540">Ad perpendiculum erigatur tignum, quod imo puteo in<lb></lb>nitatur, ſive ſolidiori tigno <lb></lb><figure id="id.017.01.769.1.jpg" xlink:href="017/01/769/1.jpg"></figure><lb></lb>AB putei lateribus infixo <lb></lb>inſiſtat brevius tignum <lb></lb>CD, ita tamen obliquè <lb></lb>conſtitutum, ut hujus an<lb></lb>guli latera illius reſpiciant, <lb></lb>quatenus haſtulæ ex hoc <lb></lb>exeuntes, medio jugo, <lb></lb>nullum recipiant à ſub<lb></lb>jecto tigno AB impedi<lb></lb>mentum. </s> <s id="s.005541">Suprema pars <lb></lb>tigni CD ita ſecetur, ut <lb></lb>circa axem in E infixum <lb></lb>liberè verſati poſſit ju<lb></lb>gum, cujus extremitatibus <lb></lb>adnexæ ſunt haſtulæ an<lb></lb>tliarum embolos attollen<lb></lb>tes atque deprimentes. </s> <lb></lb> <s id="s.005542">Paulò infra axem E ape<lb></lb>riatur foramen F, cui pa<lb></lb>riter immitti queat jugum <lb></lb>alterum verſatile circa <lb></lb>axem H infixum paulò <lb></lb>infrà crenam ſuperiori ju<lb></lb>go ſubſervientem. </s> </p> <p type="main"> <s id="s.005543">Porrò utriuſque jugi non eadem eſt forma: Nam ſupe<lb></lb>rius jugum axi E infixum rectum eſt GI, additamento <lb></lb>ad K auctum, ut paulo depreſſius ſit foramen K ad reci<lb></lb>piendum axem, quàm ſint axes ad G & I, quibus jun<lb></lb>guntur cum jugo haſtulæ ad embolorum motum perficien<lb></lb>dum deſtinatæ. </s> <s id="s.005544">At verò jugum inferius non niſi extremi<lb></lb>tates LM & NO rectas habet, cætera inflexum eſt, & <lb></lb>ad mediam curvaturam habet in P foramen, quo innita<lb></lb>tur axi in H infixo. </s> <s id="s.005545">Quantam autem eſſe oporteat hu<lb></lb>juſmodi inflexionem MPN, ex hoc definies, quod ubi <lb></lb>jugum GI in ſuo axe conſiſtens horizonti parallelum fue<lb></lb>rit, etiam inferioris jugi in ſuo axe H conſiſtentis extre-<pb pagenum="754" xlink:href="017/01/770.jpg"></pb>mitates LM & NO in eodem horizontali plano cum GI <lb></lb>conveniant. </s> </p> <p type="main"> <s id="s.005546">His paratis fruſtum cylindricum diametri (ſi id quidem <lb></lb>commodè fieri poſſit) non multo minoris, quàm ſit jugi GI <lb></lb>intercapedo inter haſtularum axes, conſtruatur. </s> <s id="s.005547">Quod ſi <lb></lb>tantæ craſſitudinis lignum præſto non fuerit, plura aptè <lb></lb>compinge, atque ferreo circulo conſtringe, ne diſſilire va<lb></lb>leant. </s> <s id="s.005548">Tum deſtinatæ embolorum depreſſioni atque eleva<lb></lb>tioni æqualis ſaltem pars QS toreutæ operâ rotundetur; <lb></lb>deinde ſerrâ obliquè ſecetur, ut fiat ellipſis QT: cujus <lb></lb>limbus ferreâ lamellâ exactè planâ & politâ muniatur tùm <lb></lb>ad perpetuitatem, ne lignum atteratur, tùm ad faciliorem <lb></lb>motum, ut minor ſit cum ſubjectis ligneis jugis conflictus: <lb></lb>interiores autem ellipſis partes ſcalpro eximi poſſunt, ut factâ <lb></lb>cavitate nullum motui impedimentum afferant tigni CD ſu<lb></lb>premi anguli. </s> </p> <p type="main"> <s id="s.005549">Fruſto huic cylindrico ad axem firmiter inſeratur minor <lb></lb>cylindrus VR, cui immitti poſſit vectis XY à jumento in X <lb></lb>circumducendus. </s> <s id="s.005550">Minor hujuſmodi cylindrus methodo ſu<lb></lb>periùs indicatâ ſub finem prop. 4. ſuſpendatur eâ lege, ut <lb></lb>jugo GI maximè inclinato conveniat ellipſis diameter QT, <lb></lb>minori autem ellipſis Axi conveniant extremitates LM & <lb></lb>NO inferioris jugi, quæ erunt horizonti parallelæ. </s> <s id="s.005551">Hinc <lb></lb>fiet, ut converſo cylindro ſubinde deveniant ad maximam <lb></lb>depreſſionem, atque viciſſim ad maximam elevationem, ſin<lb></lb>guli antliarum emboli. </s> </p> <p type="main"> <s id="s.005552">Quod ſi liceret in puteo, ubi aqua ſcaturit, aut in vaſe, in <lb></lb>quod aqua influit, in altum elevanda vi antliæ propellentis, <lb></lb>non quadratum tantùm, ſed hexagonum aut octogonum priſ<lb></lb>ma erigere ad perpendiculum, & in oppoſitis faciebus fora<lb></lb>mina excavare, quibus immitterentur inflexa juga, eâ in<lb></lb>flexione, quæ ſatis eſſet, ut demum omnium extremitates in <lb></lb>eodem horizontali plano convenirent, ſatis manifeſtum eſt <lb></lb>tribus aut quatuor jugis poſſe deinceps ſex aut octo antlias <lb></lb>agitari. </s> </p> <pb pagenum="755" xlink:href="017/01/771.jpg"></pb> <p type="main"> <s id="s.005553"><emph type="center"></emph>PROPOSITIO VI.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005554"><emph type="center"></emph><emph type="italics"></emph>Alia ratione plures antlias componere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005555">EX iis, quæ hujus libri cap. 5. dicta ſunt, genus aliud ad <lb></lb>Cuneum pertinens excogitare poſſumus, quo ſimul plu<lb></lb>res antlias agitare poſſit potentia, cui maximè virium copia <lb></lb>ſuppetat, & valde ſimplex machina conſtruenda proponatur. </s> <lb></lb> <s id="s.005556">Ex ſolidis aſſeribus <lb></lb><figure id="id.017.01.771.1.jpg" xlink:href="017/01/771/1.jpg"></figure><lb></lb>compingatur cir<lb></lb>culus: hic in octo <lb></lb>partes diſtribuatur, <lb></lb>& duæ proximæ <lb></lb>confixum habeant <lb></lb>tigillum AB, cu<lb></lb>jus extremitates ita <lb></lb>extra circulum pro<lb></lb>mineant, ut inciſis <lb></lb>crenis haſtulæ em<lb></lb>bolo adnexæ circa <lb></lb>ſuum axem verſa<lb></lb>tiles ſinè impedi<lb></lb>mento moveri que<lb></lb>ant. </s> <s id="s.005557">Tres ſimiles <lb></lb>tigilli tranſverſarij <lb></lb>affigantur CD, EF, GH extremitatibus ſimiliter prominenti<lb></lb>bus extra circuli ambitum, & excavatis in crenas haſtularum <lb></lb>capaces. </s> <s id="s.005558">Haſtularum verò formam ſuaderem, quæ prope em<lb></lb>bolum eſſent plicatiles in dextram atque ſiniſtram, quemad<lb></lb>modum in ſupremâ parte, ubi tranſverſariis cohærent, ſunt <lb></lb>circa axem flexiles in anteriorem atque in poſteriorem partem: <lb></lb>ex hac enim flexibilitate in omnem partem facilior oritur mo<lb></lb>tus. </s> <s id="s.005559">Duos autem tigillos CD & EF exiſtimo apponendos eſſe <lb></lb>tranſverſarios, ad majorem circuli firmitatem: quamquam ſuf<lb></lb>ficeret ad propoſitum finem breviores apponere ad CE & DF, <lb></lb>omnino ſimiles & æquales ipſis AB & GH. </s> </p> <p type="main"> <s id="s.005560">His paratis alius æqualis circulus ſuperponatur, firmitérque <pb pagenum="756" xlink:href="017/01/772.jpg"></pb>cum inferiore cohæreat. </s> <s id="s.005561">Tum validus ſtylus ferreus RT figu<lb></lb>ræ primùm cylindricæ, deinde ad S ſphæricæ, demum in T <lb></lb>deſinens in conum conſtruatur, & columnæ, cui univerſa ma<lb></lb>china inniti debet, ad perpendiculum infigatur. </s> <s id="s.005562">Ad centrum <lb></lb>verò circuli inferioris foramen fiat, per quod facilè globus S <lb></lb>immitti poſſit, & in centro circuli ſuperioris aliud pariter fo<lb></lb>ramen aperiatur, ſed tantùm capax coni ST, adeò ut machi<lb></lb>na ſuſtineatur à globo S, & in quancumque partem facilè in<lb></lb>clinari queat: id quod etiam faciliùs continget, ſi foramen il<lb></lb>lud ſuperioris circuli, qua parte globum S contingit, annulo, <lb></lb>ſeu limbo ferreo muniatur. </s> <s id="s.005563">Quod ſi circulus ille ſuperior craſ<lb></lb>ſior fuerit, quàm ut facilè inclinari poſſit, ne ſuperior ora fo<lb></lb>raminis incurrat in conum, abradi poterit, quantum ſatis fue<lb></lb>rit, in calathoidem, ut magis pateat, atque liberam inclina<lb></lb>tionem permittat. </s> </p> <p type="main"> <s id="s.005564">Circulari hac compage impoſitâ ſtylo RS, haſtulæ embolo<lb></lb>rum ſuis axibus adnectantur extremitatibus tigillorum promi<lb></lb><figure id="id.017.01.772.1.jpg" xlink:href="017/01/772/1.jpg"></figure><lb></lb>nentibus. </s> <s id="s.005565">Tum ad conciliandum <lb></lb>motum machinæ, cylindrus IK <lb></lb>ſuo centro K innitatur apici ſty<lb></lb>li T, & in ſuperiore loco, axis I <lb></lb>congruo foramini immiſſus ier<lb></lb>vet cylindri poſitionem perpen<lb></lb>dicularem. </s> <s id="s.005566">Sit autem in cylin<lb></lb>dri latere profundiùs excavata <lb></lb>crena, cui inſeri poſſit triangu<lb></lb>lum OPN obtuſangulum ad P, <lb></lb>quod validum ſit, & cum cylin<lb></lb>dro firmiſſimè cohæreat: ſic enim fiet, ut trianguli extremi<lb></lb>tas O tangens circulum, illum à poſitione horizonti parallelâ <lb></lb>removeat, & in eam partem inclinet, atque ex adversâ elevet. </s> <lb></lb> <s id="s.005567">Potentia verò vecti VX applicata, & cylindrum volvens, alam <lb></lb>pariter NOP circumducet; quæ aliis atque aliis ſubjecti circu<lb></lb>li partibus ſubinde applicata illas deprimet, & ex diametro op<lb></lb>poſitas elevabit: intermediæ autem aliæ deprimentur, ad quas <lb></lb>ſcilicet extremitas O accedit, aliæ elevabuntur, à quibus ea<lb></lb>dem extremitas O recedit. </s> </p> <p type="main"> <s id="s.005568">Quantum autem extremitas O infra baſim cylindri deſcen-<pb pagenum="757" xlink:href="017/01/773.jpg"></pb>dere oporteat, definiendum eſt primò ex motu, quem embolus <lb></lb>elevatus atque depreſſus perficit; cujus motús medietas acci<lb></lb>pienda eſt: deinde attendenda eſt diſtantia baſis cylindri à pla<lb></lb>no circuli, ſi hoc conſtitueretur horizonti parallelum; hæc ve<lb></lb>rò diſtantia addenda eſt ſemiſſi motûs emboli, ut innoteſcat, <lb></lb>quantum oporteat extremitatem O deprimi infra baſim cylin<lb></lb>dri. </s> <s id="s.005569">Neque cuiquam dubium eſſe poteſt; an ſic definienda ſit <lb></lb>hujuſmodi depreſſio extremitatis O; ſiquidem inclinato circu<lb></lb>lo tantum extremitas altera diametri deprimitur infra planum <lb></lb>horizontale, quantum altera attollitur; hæc autem duplicata <lb></lb>differentia dat univerſum motum emboli; igitur hujus motús <lb></lb>ſemiſſe definitur circuli depreſſio & inclinatio. </s> <s id="s.005570">Quia autem ad <lb></lb>faciliorem motum, tùm ne cylindri craſſities plano circuli in<lb></lb>clinato occurrat, tùm ne latus PO circulum tangat præter<lb></lb>quam extremitate O, ad vitandum tritum atque conflictum <lb></lb>partium, præſtat cylindrum non proximè adhærere circulo; <lb></lb>propterea diſtantia baſis cylindri à centro ſubjecti circuli com<lb></lb>putanda eſt. </s> </p> <p type="main"> <s id="s.005571">Porrò expedire extremitatem O munitam ferreâ laminâ <lb></lb>percurrere in ſubjecto circulo laminam pariter ferream ex<lb></lb>quiſitè politam, non opus eſt monere: ſatis quippe per ſe <lb></lb>patet. </s> <s id="s.005572">Illud cavendum eſt, ut modum ſerves in alæ NOP <lb></lb>amplitudine; nam ſi nimis exigua ſit, paulò difficiliùs mo<lb></lb>vet, quia nimis diſtat ab haſtulis embolorum: ſin autem <lb></lb>æquo amplior fuerit, cùm maximam reſiſtentiæ partem illa <lb></lb>ſuſtineat, ſubit periculum luxationis. </s> <s id="s.005573">Cæterùm hoc pende<lb></lb>bit ex circuli amplitudine, cujus diametrum conſtituen<lb></lb>dam eſſe habitâ ratione motûs embolo antliæ communican<lb></lb>di, nemo ignorat; quemadmodum & in ſimplici antliá ex <lb></lb>hoc eodem definitur diſtantia haſtulæ à centro motûs. </s> <s id="s.005574">Quo<lb></lb>niam enim motus ille depreſſionis & elevationis emboli con<lb></lb>nectitur cum motu circulari ſemidiametri circuli, cui haſtu<lb></lb>læ adnectuntur, eum Radium circulo tribuere oportet, ut <lb></lb>arcus ab extremo puncto deſcriptus quàm minimùm differat <lb></lb>à lineâ rectâ; ſic enim faciliùs movetur embolus. </s> <s id="s.005575">Quare ar<lb></lb>cus ejuſmodi deſcribendus eſt, ut illius medietas Sinum <lb></lb>Verſum habeat, quoad fieri poterit, minimum. </s> <s id="s.005576">Ponamus <lb></lb>univerſum emboli motum eſſe unciarum 4, ejus ſemiſſem <pb pagenum="758" xlink:href="017/01/774.jpg"></pb>unciarum 2: Sit circuli Radius BD unciarum 8. Inveniatur <lb></lb><figure id="id.017.01.774.1.jpg" xlink:href="017/01/774/1.jpg"></figure><lb></lb>in Canone Sinuum arcus, cujus Si<lb></lb>nus ad Radium ſit ut 2 ad 8, & eſt <lb></lb>proximè gr. 14. 28′ 40″. Eſt igitur <lb></lb>arcus ab extremâ ſemidiametro D <lb></lb>deſcribendus CE gr. 28. 57′. </s> <s id="s.005577">20″: <lb></lb>quo bifariam diviſo in D eſt arcus <lb></lb>CD gr.14. 28′. </s> <s id="s.005578">40″; cujus Sinus CI; <lb></lb>& Sinus Verſus ID eſt totius Radij BD (3/100), hoc eſt unius <lb></lb>unciæ 4/25; quæ deflexio arcûs CE à rectitudine non admodum <lb></lb>nocet. </s> <s id="s.005579">Satis igitur fuerit, ſi circuli diameter ſit unc.14. & ti<lb></lb>gilli hinc atque hinc aliquantulum præter unam unciam pro<lb></lb>mineant, ubi illis haſtulæ embolorum adnectuntur; ſic enim <lb></lb>fiet, ut haſtulæ ſatis commodè moveantur, maximè ſi lon<lb></lb>giores fuerint. </s> </p> <p type="main"> <s id="s.005580">Quod ſi ligneis tigillis uti nolueris, ſed potius ferreis priſ<lb></lb>matibus inter utrumque ligneum circulum aptè conſerendis, <lb></lb>adeò ut circuli plana ſibi viciſſim adhæreant, non dubium, <lb></lb>quin multò firmior futura ſit machina: hoc te monitum volo, <lb></lb>quod circulos craſſiuſculos eſſe oportet, ut in illis opportunum <lb></lb>foramen excavetur, quo commodè machina inſiſtat ſtyli glo<lb></lb>bulo, &, prout oportet, inclinetur. <lb></lb><figure id="id.017.01.774.2.jpg" xlink:href="017/01/774/2.jpg"></figure></s> </p> <pb pagenum="759" xlink:href="017/01/775.jpg"></pb> <figure id="id.017.01.775.1.jpg" xlink:href="017/01/775/1.jpg"></figure> <p type="main"> <s id="s.005581"><emph type="center"></emph>MECHANICORUM <emph.end type="center"></emph.end><emph type="center"></emph>LIBER OCTAVUS.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005582"><emph type="center"></emph><emph type="italics"></emph>De Cochlea.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005583">POSTREMO loco inter Mechanicas Facultates <lb></lb>numeratur Cochlea, non tamen poſtremo loco <lb></lb>habenda, ſi ejus vires perpendantur; immò ſi <lb></lb>cum cæteris Facultatibus comparetur, omnium <lb></lb>efficaciſſima cenſenda erit, cæteris paribus, ut <lb></lb>ex iis, quæ hoc libro diſputabuntur, mani<lb></lb>feſtum fiet. </s> <s id="s.005584">Cur de Cochleâ poſtremus habeatur ſermo, ſi <lb></lb>quis inquirat, non pauci ex iis, qui inter Mechanicas faculta<lb></lb>tes cognationis nexus quoſdam perveſtigant, ideò poſt Cu<lb></lb>neum numerari Cochleam autumabunt, quia Cochlea longior <lb></lb>quidam Cuneus cylindro convolutus cenſeri poteſt, cujus <lb></lb>propterea vires ad Cuneum revocare contendunt. </s> <s id="s.005585">Mihi tamen, <lb></lb>qui Facultates ſingulas ita à reliquis abſolutas agnoſco, ut nul<lb></lb>lo alio vinculo invicem copulentur, niſi quatenus omnes ab <lb></lb>uno eodémque principio ortum ducunt, ea tantummodo eſſe <lb></lb>videtur cauſa, quod reliquæ Facultates ſimplices ſint, ac faci<lb></lb>liùs parabiles, quàm Cochlea, atque hæc ſi ſolitaria adhibea<lb></lb>tur, nec cum ullâ reliquarum Facultatum componatur, licèt <lb></lb>validè urgeat, aut trahat, eâ tamen communiter non utamur <lb></lb>ad majores motus efficiendos, quos unâ aliquâ reliquarum Fa<lb></lb>cultatum, minore operâ, conſequimur. </s> </p> <p type="main"> <s id="s.005586">Hùc autem non ſpectat Archimedea Cochlea ad aquam in <lb></lb>altum evehendam inſtituta: eſt enim tubus in ſpiram convolu<lb></lb>tus circa ſuperficiem conicam aut cylindricam, ſeu in cono ipſo <lb></lb>aut cylindro ita excavatus, ut aquam continere valeat, quam <lb></lb>extremum tubi oſculum ex ſubjectâ profluente hauſit: dum ſci-<pb pagenum="760" xlink:href="017/01/776.jpg"></pb>licet circa ſuum axem Conus aut Cylinder ad horizontem in<lb></lb>clinatus convertitur, quæ ingreſſa fuerat aqua, per ſpiras aſcen<lb></lb>dens ad alteram tubi extremitatem ſuperiorem demum effundi<lb></lb>tur; atque hac ratione ad tantam altitudinem illa attollitur, <lb></lb>quantus eſt Sinus anguli, quo ad horizontem inclinatur axis <lb></lb>eoni aut cylindri, poſito codem axe tanquam Radio. </s> <s id="s.005587">Hic, in<lb></lb>quam, motus aquæ in tubo hujuſmodi ſpirali aſcendentis, non <lb></lb>eſt præſentis diſputationis, aqua ſiquidem non trahitur ſurſum, <lb></lb>ſed ſemel ingreſſa in tubo ſpirali convoluto ſponte deſcendit, <lb></lb>donec ad ſupremum oſculum provehatur; haud ſecus ac plum<lb></lb>beus globulus in eundem tubum immiſſus, ſi volvatur cylin<lb></lb>der, non valens conſiſtere in eâ ſpiræ parte, quæ priùs infima <lb></lb>& horizonti proxima, modò in converſione removetur ab ho<lb></lb>rizonte & attollitur, ſuâ autem gravitate repugnans aſcenſui, <lb></lb>ſponte deſcendit per tubum tanquam per planum inclinatum, <lb></lb>atque ita deinceps, quoad ex ſupremo tubi oſculo erumpat. </s> <lb></lb> <s id="s.005588">Idem planè contingit aquæ in hujuſmodi tubo ſpirali vi ſuæ <lb></lb>gravitatis ſubinde fluenti ac deſcendenti in ſingulis ſpiris ſta<lb></lb>tim, ac modicum quid elevata eſt in converſione. </s> </p> <p type="main"> <s id="s.005589">Cochlea igitur, de qua hìc diſputabitur, ea eſt, quæ ad vim <lb></lb>gravitati inferendam, ſi repugnet, inſtituta eſt, adeò ut corpo<lb></lb>ris vim paſſi motus impulſui à Potentiâ per Cochleam commu<lb></lb>nicato adæquatè tribuendus ſit; & ſi quid gravitas ipſa confe<lb></lb>rat, id planè contingens reputetur. </s> <s id="s.005590">Nomen autem Cochleæ <lb></lb>inditum eſt ex ſimili quadam convolutione in teſtâ limacis, <lb></lb>quæ in ſpiras contorquetur, ſicut & Cochlides dicuntur ſcalæ, <lb></lb>per quas in gyrum aſcenditur. <lb></lb></s> </p> <p type="main"> <s id="s.005591"><emph type="center"></emph>CAPUT I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005592"><emph type="center"></emph><emph type="italics"></emph>Cochleæ forma & virtus deſcribitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005593">COchlea, quam explicandam ſuſcipimus, ex limacis teſtâ <lb></lb>eatenus ſolùm ſimilitudinem ducit, quatenus in ſpiras du<lb></lb>citur, cæterùm animalis illius ſpiræ inæquales ſunt, & major <pb pagenum="761" xlink:href="017/01/777.jpg"></pb>ſpira minorem quaſi complectitur, non quemadmodum helix <lb></lb>in plano deſcripta, ſed ferè ſicut ſpira in coni aut globi ſuper<lb></lb>ficie deformata. </s> <s id="s.005594">Spira autem conicè ducta, aut ſphæricè, pa<lb></lb>rum utilis accideret Machinatoris inſtituto; cum enim, ut fir<lb></lb>metur, inſerenda ſit foramini ſimiliter in ſpiram excavato, ma<lb></lb>jores coni, aut globi, ſpiræ non congruerent minoribus ſpiris <lb></lb>foraminis conici aut ſphærici in modum ſcaphij, nec per eas <lb></lb>promoveri poſſent; atque minores coni, aut globi, ſpiræ in am<lb></lb>plioribus ſpiris foraminis firmari nequirent. </s> <s id="s.005595">Oportet igitur ſpi<lb></lb>ram omnino ſimilibus ductibus, atque æqualibus conſtare; id <lb></lb>quod non niſi in cylindro obtinetur. </s> <s id="s.005596">Quapropter Cochlea, de <lb></lb>qua hìc agimus, eſt ſolida ſpira in ſuperficie excavati cylindri <lb></lb>efformata; quæ vitium Capreolos arboris ramum complexos <lb></lb>imitata vulgari vocabulo <emph type="italics"></emph>Vitis<emph.end type="italics"></emph.end> (& fortaſſe aptiùs) nominatur. </s> <lb></lb> <s id="s.005597">Receptaculum verò concavum, cui cylindrus in helicem de<lb></lb>formatus immittitur, habétque ſpirales cavitates ſolidæ cylin<lb></lb>dri ſpiræ congruentes, <emph type="italics"></emph>Matrix<emph.end type="italics"></emph.end> dicitur, alij <emph type="italics"></emph>Tylum, Cochlidium <emph.end type="italics"></emph.end><lb></lb>alij, vocabulo ad hanc ſignificationem detorto, vulgus <emph type="italics"></emph>Matrem <lb></lb>Vitis<emph.end type="italics"></emph.end> nuncupat. </s> </p> <p type="main"> <s id="s.005598">Ut autem ſpiram cylindro æqualibus atque ſimilibus gyris <lb></lb>circumductam intelligas, concipe triangulum rectangulum, <lb></lb>cujus perpendiculum æquale ſit dato lateri aut Axi cylindri <lb></lb>Recti, baſis verò trianguli toties contineat perimetrum baſis <lb></lb>cylindri, quoties ſpira cylindrum ipſum complecti debet; nam <lb></lb>hujuſmodi trianguli hypothenuſa lineam ſpiralem omnino ſi<lb></lb>militer ductam in cylindri ſuperficie deſcribet, ſi triangulum <lb></lb>cylindro circumvolvatur. </s> </p> <p type="main"> <s id="s.005599">Sit cylindri altitudo AB, ejúſque baſis circulari peripheriæ <lb></lb>ſit æqualis recta BC ad rectum an<lb></lb><figure id="id.017.01.777.1.jpg" xlink:href="017/01/777/1.jpg"></figure><lb></lb>gulum CBA conſtituta. </s> <s id="s.005600">Oporteat <lb></lb>autem ſpiram quatuor gyris com<lb></lb>plecti cylindrum; idcirco recta BC <lb></lb>producatur, ut tota BF ſit ipſius BG <lb></lb>quadrupla: ducta enim hypothenu<lb></lb>ſa FA, ſi triangulum cylindro cir<lb></lb>cumplicetur quadruplici convolutio<lb></lb>ne, deſignabit in cylindri ſuperficie quatuor ſpiras omnino ſi<lb></lb>miles & æquales. </s> <s id="s.005601">Spirarum æqualitatem & ſimilitudinem fa-<pb pagenum="762" xlink:href="017/01/778.jpg"></pb>cilè demonſtrabis, ſi trianguli baſim BF, & altitudinem BA, <lb></lb>utramque in quatuor æquales partes diſtinxeris; deinde ex ſin<lb></lb>gulis diviſionum punctis rectas CM, DL, EK altitudini BA <lb></lb>parallelas, & rectas GK, HL, IM parallelas baſi BF excita<lb></lb>veris; ſibi enim occurrentes in punctis K, L, M, divident hy<lb></lb>pothenuſam in quatuor æquales partes, ut patet ex 2. lib. 6: Ni<lb></lb>mirum ut FE ad ED, ita FK ad KL; & ut FD ad DC, ita FL <lb></lb>ad LM; & ut FC ad CB, ita FM ad MA: ſunt autem FE & <lb></lb>ED ex hypotheſi æquales, igitur etiam FK & KL æquales: <lb></lb>FD poſita eſt ipſius DC dupla, ergo FL ipſius LM dupla; <lb></lb>ergo LM æqualis eſt ipſi KL, aut FK: Demum FC ex con<lb></lb>ſtructione eſt ipſius CB tripla; igitur etiam FM eſt tripla <lb></lb>ipſius MA; quare MA æqualis eſt ſingulis reliquis partibus <lb></lb>FK, KL, LM; & tota hypothenuſa divifa eſt in quatuor <lb></lb>æquales partes. </s> <s id="s.005602">Item in parallelogrammo KD, per 34. lib. 1. <lb></lb>æqualia ſunt oppoſita latera KN & ED, atque in parallelo<lb></lb>grammo LC æqualia ſunt LO & DC, quemadmodum & in <lb></lb>parallelogrammo MB æqualia ſunt MI & CB: Sicut igitur <lb></lb>rectæ FE, ED, DC, CB ex hypotheſi ſunt æquales, etiam <lb></lb>FE, KN, LO, MI ſunt inter ſe æquales. </s> <s id="s.005603">Similiter oſtendes <lb></lb>ſicut æquales ſunt ex conſtructione BG, GH, HI, IA, ita <lb></lb>æquales inter ſe eſſe EK, NL, OM, IA. </s> <s id="s.005604">Cum itaque trian<lb></lb>gula FEK, KNL, LOM, MIA habeant tria latera ſingula <lb></lb>ſingulis æqualia, & ſimiliter poſita, ipſa ſunt quoque æquian<lb></lb>gula, ac proinde ſimiliter inclinatæ ſunt ſingulæ ſpiræ FK, KL, <lb></lb>LM, MA, quæ pariter demonſtratæ ſunt æquales. </s> <s id="s.005605">Quam ſi<lb></lb>milem inclinationem oſtendit æqualitas angulorum ad F, K, <lb></lb>L, M, propter linearum paralleliſmum. </s> <s id="s.005606">Triangulum igitur <lb></lb>ABF ſuâ hypothenusâ FA deſignat in cylindri ſuperficie qua<lb></lb>tuor ſimiles & æquales ſpiras. </s> </p> <p type="main"> <s id="s.005607">Verùm quid juvaret in exteriore cylindri ſuperficie ſpiralem <lb></lb>lineam exquiſitè deſcripſiſſe, niſi corpus ipſum cylindricum in <lb></lb>ſolidam ſpiram deformaretur? </s> <s id="s.005608">Quapropter neceſſariò cylin<lb></lb>drum circumplectuntur duæ ſpiræ, cava altera & depreſſa, al<lb></lb>tera convexa & prominens, quibus ſimiliter atque æqualiter <lb></lb>depreſſæ & prominentes duæ ſpiræ in receptaculi ſeu Matricis <lb></lb>foramine cylindricè excavato requiruntur ita illis reſponden<lb></lb>tes, ut depreſſam receptaculi ſpiram ſubeat prominens cylindri <pb pagenum="763" xlink:href="017/01/779.jpg"></pb>ſpira, & viciſſim prominentem receptaculi ſpiram excipiat de<lb></lb>preſſa cylindri ſpira. </s> <s id="s.005609">Ex quo fit, ut convolutus circa ſuum <lb></lb>axem cylindrus attollatur aut deprimatur, adducatur aut redu<lb></lb>catur, prout opus fuerit, atque cum eo corpus baſi illius proxi<lb></lb>mum, ſeu adnexum urgeatur, aut trahatur, elevetur, aut pie<lb></lb>matur. </s> </p> <p type="main"> <s id="s.005610">Vulgatiſſimus autem & frequentiſſimus eſt hujus Facultatis <lb></lb>uſus, ubi potiſſimùm opus eſt validâ preſſione, ut in prælis vi<lb></lb>nariis ad exprimendum ex uvæ jam preſſæ reliquiis tortivum <lb></lb>muſtum, apud typographos ad imprimendos ſubjectæ chartæ <lb></lb>ex typis characteres, apud bibliopægos ad comprimendos li<lb></lb>bros, jam compactos, apud fabros ferrarios ad firmandas fer<lb></lb>reas laminas limâ expoliendas, atque apud alios artifices. </s> <lb></lb> <s id="s.005611">Quamquam & ſæpiſſimè clavorum loco, quibus ligna, aut me<lb></lb>tallicæ laminæ configuntur citrà mallei percuſſionem, cochleis <lb></lb>utimur, & quidem ad validiorem atque perennem firmitatem; <lb></lb>neque enim revelli poteſt cochlea, aut excuti, quemadmodum <lb></lb>clavus. </s> <s id="s.005612">Sed tunc hujuſmodi cochleæ non exercent vim facul<lb></lb>tatis Mechanicæ; eatenus ſcilicet validiùs, quàm clavi, duo <lb></lb>corpora, quæ compinguntur, connectunt, quatenus multipli<lb></lb>ces in cylindruli facie ſolidarum ſpirarum ductus pluribus cavis <lb></lb>foraminum ſpiris implicantur ex cylindruli convolutione; qui <lb></lb>propterea eximi non poteſt, niſi in contrarium revolvatur; <lb></lb>quandiu quidem incorruptum permanet lignum neque ex hu<lb></lb>more putreſcens, neque vermiculo erodente carioſum, neque <lb></lb>calore nimio ita diſcedens atque dehiſcens, ut laxato foramine <lb></lb>jam non ampliùs ſolida cylindruli ſpira congruentibus ſtriis <lb></lb>coërceatur. </s> </p> <p type="main"> <s id="s.005613">Hinc eſt in ſuſtentando pondere ex cochleâ ſuſpenſo pro<lb></lb>priè non exerceri vim Mechanicam; nihil enim ampliùs co<lb></lb>nante Potentiâ (quemadmodum in Vecte, aut Axe in Peritro<lb></lb>chio, aut fune Trochlearum retinendo opus eſt, quæ pondus <lb></lb>elevavit convoluto cylindro in cochleam deformato, ſola ſpira<lb></lb>rum cavæ atque convexæ complexio efficit, ut cylindrus cum <lb></lb>adnexo pondere retineatur, ne recidat, quatenus à ſubjectâ lo<lb></lb>culamenti ſpirâ ſolidâ ſuſtinetur: quemadmodum & ſubſcudi<lb></lb>bus compagem cohibentibus accidit, quatenus ſecuricla ex mi<lb></lb>nore in majorem amplitudinem explicata decreſcentis recepta-<pb pagenum="764" xlink:href="017/01/780.jpg"></pb>culi anguſtiis coërcetur, ne excurrat; adeóque confixum hu<lb></lb>juſimodi ſubſcude corpus grave inferius retinetur, ne à ſuperio<lb></lb>re disjungatur, & cadat. </s> </p> <p type="main"> <s id="s.005614">Tota igitur vis Machinalis à Cochleâ exercetur in motu, <lb></lb>quem à potentiâ illam circumagente recipit. </s> <s id="s.005615">Et ſanè ſi poten<lb></lb>tiæ cylindrum verſantis motum comparemus cum motu ponde<lb></lb>ris, quod à cochleâ urgetur, aut trahitur; ſtatim apparebit po<lb></lb>tentiam quidem circulum deſcribere circa convoluti cylindri <lb></lb>axem, pondus verò recta moveri, prout promovetur, aut retra<lb></lb>hitur cylindrus. </s> <s id="s.005616">Cum itaque in ſingulis cylindri converſioni<lb></lb>bus ejus motum definiat ſpiræ à ſpirâ intervallum; ſi hoc cum <lb></lb>circulari peripheriâ conferatur, innoteſcet motuum Ratio, & <lb></lb>Potentiæ momentum, quæ eò minorem in pondere reſiſten<lb></lb>tiam invenit, quò tardiùs hoc movetur. </s> <s id="s.005617">Hinc ſi cylindri alti<lb></lb>tudo ad ejuſdem diametrum ſit ut 20 ad 1, numeratáſque ſpiras <lb></lb>cylindrum complectentes inveneris eſſe 35, rectè definies con<lb></lb>volutionibus 35 reſpondere totum cylindri motum, atque adeò <lb></lb>ſpiræ à ſpirà intervallum eſſe ad cylindri diametrum ut 4. ad 7: <lb></lb>ex quo infertur circuli peripheriam ad ſpirarum diſtantiam, <lb></lb>hoc eſt potentiæ motum ad motum ponderis, eſſe proximè <lb></lb>ut 22 ad 4, atque potentiæ conatum ut 4. vincere poſſe quam<lb></lb>libet reſiſtentiam minorem quàm ut 22, ſpectatâ Ratione, quam <lb></lb>infert cylindri craſſities, & ſpirarum obliquitas. </s> </p> <p type="main"> <s id="s.005618">Verùm quia non niſi parvulis cochleis, aut ubi levis conatus <lb></lb>requiritur, ita applicatur potentia, ut cylindri ſuperficiei ap<lb></lb>plicata intelligatur, complanatâ ſcilicet ejuſdem cylindri extre<lb></lb>mitate, quam ſummis digitis apprehendere valeas, communi<lb></lb>ter adhuc majus eſt momentum Potentiæ, quàm ut ex circuli <lb></lb>peripheriâ baſim cylindri ambiente circumſcribatur; additur <lb></lb>enim aut Radius cylindri Capiti quadrato infixus, aut aliquid <lb></lb>manubrij rationem habens, adeò ut potentia longè majorem <lb></lb>circulum deſcribat, quàm ſit cylindri in ſpiram deformati baſis: <lb></lb>ac proinde non ex cylindri craſſitie, ſed ex diſtantiâ potentiæ <lb></lb>ab axe cylindri definiendus eſt ejuſdem potentiæ circulum per<lb></lb>ficientis motus, atque cum ſpirarum intervallo motum ponde<lb></lb>ris metiente comparandus. </s> </p> <p type="main"> <s id="s.005619">Hinc ad imprimendas metallicæ laminæ ex argento, aut <lb></lb>auro, aut cupro imagines citrà percuſſionem, ſuper ſolido pla-<pb pagenum="765" xlink:href="017/01/781.jpg"></pb>no erectis atque infixis ad perpendiculum duobus ferreis pedi<lb></lb>bus ferreo pariter tranſverſario firmatis, in quo excavata co<lb></lb>chleæ congruens Matrix, typus inter laminam & cylindrum <lb></lb>interjectus validè urgetur ex cylindri convolutione, & imagi<lb></lb>nem exprimit: quia videlicet ſuperiori cylindri Capiti quadra<lb></lb>to inſeritur longior ferreus vectis hinc atque hinc productus, <lb></lb>ut duplici ejus extremitati duplex potentia, ſi opus fuerit, ap<lb></lb>plicetur. </s> <s id="s.005620">Quapropter ab axe cylindri ad vectis hujuſmodi ex<lb></lb>tremitatem ducta linea eſt Radius circuli potentiæ motum de<lb></lb>terminantis; atque ſi hujuſmodi Radij longitudo ad ſpirarum <lb></lb>intervallum fuerit ut 50 ad 1, circuli diameter eſt 100, ejuſque <lb></lb>peripheria major quàm 314; & potentiæ motus ad motum typi <lb></lb>laminam prementis eſt ut 314 ad 1: idcirco ſi in vectis extremi<lb></lb>tatibus ſint ſinguli homines perinde conantes, ac ſi libras 50 <lb></lb>ſinguli moverent, premitur typus vi hujus cochleæ quaſi à <lb></lb>pondere librarum 31400. </s> </p> <p type="main"> <s id="s.005621">Quod autem de preſſione dicitur, ſimili ratione intelligen<lb></lb>dum eſt de ponderis elevatione, ſi fortè aut inferiori cylindri <lb></lb>baſi adnexum fuerit, aut ejus capiti impoſitum; ſicut enim cor<lb></lb>pus prementi reſiſtit ratione particularum conſtipatarum, ita <lb></lb>elevanti repugnat ratione ſuæ gravitatis: utrobique igitur ſimi<lb></lb>lem virtutem habet potentia ad vincendam reſiſtentiam, quan<lb></lb>do utrobique eadem invenitur Ratio motuum atque momento<lb></lb>rum. </s> <s id="s.005622">Propterea in hujuſmodi cochleis, quæ infixo Radio con<lb></lb>volvuntur, non eſt admodum anxiè procuranda cylindri craſ<lb></lb>ſities, modò ſatis ſolidus ſit, nec fragilis: eadem quippe eſt cir<lb></lb>culi à potentiæ motu deſcripti peripheria, ſivè major ſit, ſivè <lb></lb>minor cylindri craſſitudo, quando eadem eſt potentiæ diſtan<lb></lb>tia à cylindri axe, ac proinde idem eſt momentum. </s> </p> <p type="main"> <s id="s.005623">Illud quidem ad rem facit maximè, quam obliquè inclinatus <lb></lb>ſit ſpirarum ductus; hinc enim oritur intervalli Ratio inter pro<lb></lb>ximos ſpirarum circuitus, qui frequentiſſimi ſunt, ac brevi in<lb></lb>tervallo disjuncti, ſi linea ſpiralis ſit maximè inclinata, rari au<lb></lb>tem atque notabiliter ſejuncti, ſi illa fuerit ad majorem angu<lb></lb>lum (acutum tamen) erecta: Eſt nimirum hujuſmodi interval<lb></lb>lum æquale Tangenti anguli inclinationis, poſito Radio am<lb></lb>bitu baſis cylindri; ipſa autem ſpira eſt ejuſdem anguli Secans. </s> <lb></lb> <s id="s.005624">Quare datâ cylindri diametro, invenitur peripheria baſis; & <pb pagenum="766" xlink:href="017/01/782.jpg"></pb>dato ſpirarum intervallo, invenitur angulus huic intervallo tan<lb></lb>quam Tangenti oppoſitus, ſcilicet inclinatio ſpiræ, & hypo<lb></lb>thenuſa tanquam ejuſdem anguli Secans dat ipſius lineæ ſpira<lb></lb>lis longitudinem. </s> </p> <p type="main"> <s id="s.005625">Quod ſi totius lineæ ſpiralis univerſum cylindrum com<lb></lb>plectentis lineam deſideras, toties peripheriam baſis multipli<lb></lb>ca, quot ſunt ſpirarum circuitus, & habebis Radium; cylindri <lb></lb>altitudo dabit Tangentem, cui reſpondens Secans indicabit to<lb></lb>tius ſpiræ integram longitudinem. </s> <s id="s.005626">Sit ex. </s> <s id="s.005627">gr. cylindri altitudo <lb></lb>ped. 3. hoc eſt unciarum 36. ejus diameter unciarum 7; ergo <lb></lb>baſis perimeter unc. </s> <s id="s.005628">22: Spirarum circuitus ſint 25: igitur <lb></lb>ductâ perimetro 22 in 25, habetur 550 tanquam Radius, & 36 <lb></lb>tanquam Tangens: igitur ut unciæ 550 ad uncias 36, ita Ra<lb></lb>dius 100000 ad 6545 Tangentem gr. 3. m. </s> <s id="s.005629">45. cui reſpondet <lb></lb>Secans 100214: Quare ut 100000 ad 100214, ita unciæ 550 <lb></lb>ad uncias (551 177/1000) longitudinem totius lineæ ſpiralis; quam, ſi <lb></lb>careas Canone Trigonometrico, etiam habebis ex 47. lib. 1. <lb></lb>addendo quadrata numerorum 550 & 36, erit enim horum <lb></lb>ſumma quadratum, cujus Radix dabit eandem quæſitam ſpiræ <lb></lb>longitudinem. </s> </p> <p type="main"> <s id="s.005630">Cum autem hæc ſpiræ longitudo, ſive univerſa, ſive parti<lb></lb>culatim aſſumatur, ſemper longior ſit, ſivè multiplici, ſivè ſin<lb></lb>gulari perimetro circuli, qui eſt baſis cylindri, utique motus <lb></lb>potentiæ, ejúſque momentum, non ex hac ſpirali lineâ deſu<lb></lb>mendum eſt; neque enim ipſa eſſe poteſt menſura motûs po<lb></lb>tentiæ cylindro applicatæ ad ejus diametri extremitatem. </s> <s id="s.005631">Hinc <lb></lb>eſt mihi non arridere eorum ſententiam, qui cochleæ vires re<lb></lb>ferunt ad planum inclinatum, quod ab ipsâ lineâ ſpirali repræ<lb></lb>ſentetur. </s> <s id="s.005632">In plano ſiquidem inclinato momentum gravitatis, <lb></lb>ad ejuſdem gravitatis momentum in perpendiculo, ſe habet re<lb></lb>ciprocè ut perpendiculum ad ipſam lineam inclinatam; ac pro<lb></lb>pterea eandem Rationem ſervant conatus Potentiæ moventis <lb></lb>pondus aut in perpendiculo, aut in plano inclinato. </s> <s id="s.005633">At hìc po<lb></lb>tentia non movetur juxta lineæ inclinatæ longitudinem, ſed <lb></lb>breviore motu juxta baſim trianguli rectanguli, cujus hypothe<lb></lb>nuſa eſt ipſa linea inclinata: Igitur potentiæ momentum ali<lb></lb>quanto minus cenſendum eſt, quam pro Ratione plani inclina<lb></lb>ti. </s> <s id="s.005634">Adde momenta gravitatis ponderis alicujus tunc ſolùm <pb pagenum="767" xlink:href="017/01/783.jpg"></pb>fieri minora in plano inclinato, quando illi inſiſtit, & deorſum <lb></lb>nititur premendo ipſum planum: at ſi pondus idem incumbat <lb></lb>plano horizontali, verum quidem eſt planum verticale, quod <lb></lb>adversùs pondus moveatur non ſecundùm directionem, quæ <lb></lb>recta occurrat centro gravitatis ejuſdem ponderis, ſed obliquè, <lb></lb>minus invenire reſiſtentiæ: ſed pondus illud propriè non mo<lb></lb>vetur ſuper plano, licèt ab eo obliquè repellatur; & potiùs pla<lb></lb>num movetur juxta pondus: hìc verò ſi pondus in plano hori<lb></lb>zontali jacens ſit adnexum cochleæ trahenti, aut oppoſitum <lb></lb>cochleæ repellenti & urgenti, non movetur obliquè, ſed motu <lb></lb>directo: non igitur movetur ſuper planum inclinatum. </s> </p> <p type="main"> <s id="s.005635">Porrò unum eſt in Cochleâ quodammodo ſingulare, quod <lb></lb>in nullam aliam Facultatem æquè convenire deprehenditur: <lb></lb>Cum enim requiratur & cylindrus in helicem inflexus, & Ma<lb></lb>trix illi congruens, ita ut alteri quies, alteri motus debeatur; <lb></lb>perinde eſt ſi matrice immotâ cylindrus convertatur, atque ſi <lb></lb>manente cylindro matrix ipſa convolvatur, modò Potentia <lb></lb>æquali Radio utatur, ſivè cylindri capiti, ſivè Matrici infixo: <lb></lb>eadem ſiquidem ſunt potentiæ momenta, & æqualis motus pon<lb></lb>deris; æqualiter enim promovetur matrix in cylindro ſtabili, at<lb></lb>que cylindrus in Matrice immotâ. </s> <s id="s.005636">Id quod maxime locum <lb></lb>habet, ubi opus eſt compreſſione, & vulgatiſſimus eſt apud va<lb></lb>rios artifices uſus. </s> </p> <p type="main"> <s id="s.005637">Jam verò quod ad diuturnitatem ſpectat, diffitendum non eſt <lb></lb>cochleam frequenti uſu atteri, eſt enim perpetuus illius cum <lb></lb>ſuâ Matrice conflictus, quamvis plurimùm juvet, ſi ſmegmate, <lb></lb>aut pingui aliquo humore inungatur, quo lubrica fiat, ut faci<lb></lb>liùs convolvatur, minúſque atteratur. </s> <s id="s.005638">Deinde quamvis unica <lb></lb>ſpira matrici ſufficiat, ut vel ipſa, vel cylindrus promoveatur, <lb></lb>aut retrahatur, nihilominus faciliùs labem patitur, quàm ſi plu<lb></lb>res in ſpiras fuerit excavata: cum enim aut à gravitate ponde<lb></lb>ris ſuſtollendi, aut à partium conſtipatione repugnantium com<lb></lb>preſſioni, ipſa unica reſiſtentiam inveniat, utique vel ponderis <lb></lb>gravitas ipſi uni innititur, vel potentiæ conatus, reluctante cor<lb></lb>pore comprimendo aut trahendo, in illam ſolam effunditur. </s> <lb></lb> <s id="s.005639">Propterea expedit alteram ſaltem, aut tertiam ſpiram addere, <lb></lb>ut diviſo in plures conatu firmitati conſulatur. </s> </p> <p type="main"> <s id="s.005640">Eandem ob cauſam aliquando cylindrum complectitur non <pb pagenum="768" xlink:href="017/01/784.jpg"></pb>unica ſpirarum ſeries, ſed & alia illi parallela additur (nec quic<lb></lb>quam prohibet, quin & plures duabus ſint hujuſmodi paralle<lb></lb>larum ſpirarum ſeries) ut multo validior ac firmior ſit cochlea, <lb></lb>ne facilè ſpira aliqua diſſipetur, aut ſi qua labefactetur, nullum <lb></lb>ſequatur incommodum, alterâ ſpirâ parallelâ ejus vices ſup<lb></lb><figure id="id.017.01.784.1.jpg" xlink:href="017/01/784/1.jpg"></figure><lb></lb>plente. </s> <s id="s.005641">Sic ſpiræ ABCDEF paral<lb></lb>lela ſtatuitur altera ab I incipiens, & <lb></lb>per KLMNOP ſimili lapſu ſerpens. </s> <lb></lb> <s id="s.005642">Ex hac tamen multiplici ſpirâ non au<lb></lb>getur momentum potentiæ applicatæ <lb></lb>Radio VS; neque enim circulus à po<lb></lb>tentiâ in S applicatâ deſcriptus com<lb></lb>parandus eſt cum AK, ſed cum AC; <lb></lb>unicâ ſiquidem cylindri convolutione <lb></lb>promovetur cylindrus non ex A in K, <lb></lb>ſed ex A in C. </s> <s id="s.005643">Propterea oblato cy<lb></lb>lyndro in Cochleam deformato dili<lb></lb>genter attendendum eſt, utrum plures ſint ſpirarum ſeries, an <lb></lb>unica; ne fortè ex brevi inter proximas ſpiras intervallo perpe<lb></lb>ram conjicias lineam ſpiralem magis inclinatam, quàm reipſa <lb></lb>ſit; prius enim dijudicandum eſt, an illæ proximæ ſpiræ ad ean<lb></lb>dem Helicem ſpectent; attenditur ſcilicet intervallum ſpirarum <lb></lb>ad eandem ſeriem continuo ductu pertinentium. <lb></lb></s> </p> <p type="main"> <s id="s.005644"><emph type="center"></emph>CAPUT II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005645"><emph type="center"></emph><emph type="italics"></emph>An utilis ſit Cochlea duplex contraria.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005646">QUamvis ad ſuperandam modico labore reſiſtentiam non <lb></lb>modicam corporis, quod Cochlea urget, aut trahit, hu<lb></lb>juſmodi Facultas ſit potiſſimùm excogitata, ſæpiſſimè tamen <lb></lb>cochleam adhibemus non ad vincendam reſiſtentiam, quæ ali<lb></lb>quando tenuiſſima eſt, ſed unicè ad motum ita temperandum, <lb></lb>ut pro opportunitate exiguus ſit; neque enim muſculorum mo<lb></lb>tum ita attenuare pro arbitrio poteſt homo, ut ſemper quàm <pb pagenum="769" xlink:href="017/01/785.jpg"></pb>minimus contingat: propterea deformato in cochleam cylin<lb></lb>dro utimur, ut majori potentiæ motui minor motus in corpore <lb></lb>movendo reſpondeat. </s> <s id="s.005647">Sic ad excipiendas objectorum corpo<lb></lb>rum ſpecies opticas aut lumen, cum non eadem ſemper tu<lb></lb>boſpecilli longitudo opportuna ſit pro variâ tum objecti diſtan<lb></lb>tiâ, tum oculi conformatione, prudenter ab aliquibus extre<lb></lb>mus tubulus, cui lens ocularis inſeritur, in ſpiram contorque<lb></lb>tur, ut faciliùs & citiùs juſtam longitudinem aſſequantur: id <lb></lb>quod ægrè obtinerent, ſi rectà tubulum illum adducerent, aut <lb></lb>reducerent, ut ſatis experientiâ conſtat. </s> </p> <p type="main"> <s id="s.005648">Hinc aliquando contingit oppoſitos motus conciliandos eſſe <lb></lb>duobus corporibus ita, ut aut ad ſe mutuò accedant, aut ma<lb></lb>gis invicem disjungantur, ſive illa motui valde repugnent, ſive <lb></lb>ſola motûs tarditas requiratur. </s> <s id="s.005649">Propterea ejuſdem cylindri lon<lb></lb>gitudo in duas helices diſtinguitur, quæ ſimili quidem ductu <lb></lb>cylindrum circumplectuntur, ſed illis in diverſa abeuntibus, <lb></lb>unius ſpiræ non ſunt alterius ſpiris parallelæ; quæ eatenus con<lb></lb>trariæ vocari poſſunt, quatenus oppoſitos motus efficiunt, ne<lb></lb>que eæ ſunt, quæ in unam continuam ſpiram coaleſcere queant. <lb></lb><figure id="id.017.01.785.1.jpg" xlink:href="017/01/785/1.jpg"></figure><lb></lb>Ex cylindri AB medio puncto C exeant duæ ſpiræ ad eaſdem <lb></lb>partes inclinatæ, hinc CD versùs extremitatem A procedens, <lb></lb>hinc verò CE versùs extremitatem B; utraque enim ſuam ma<lb></lb>tricem habens, cui inſeratur, dum convolvitur cylindrus, ma<lb></lb>tricem longiùs à medio promovet, aut ad medium attrahit; at<lb></lb>que cum matrice adnexa corpora ſimili & æquali motu moven<lb></lb>tur. </s> <s id="s.005650">Hinc ſi utraque matrix proxima ſit medio puncto C, ex <lb></lb>primâ convolutione cylindri altera per CD removetur uſque <lb></lb>in F, altera per CE in G; atque adeò ſicut matrices moven<lb></lb>tur per CF, & CG, ita eadem menſura corporum adnexorum <pb pagenum="470" xlink:href="017/01/786.jpg"></pb>motum metitur, quæ invicem removentur intervallo FG; & <lb></lb>ita deinceps in cæteris cylindri convolutionibus. </s> </p> <p type="main"> <s id="s.005651">Quod ſi movendorum in oppoſitas partes corporum reſiſten<lb></lb>tia exigua ſit, ſatis fuerit extremitatibus cylindri anſulas appo<lb></lb>nere, quibus circumactis cylindrus ipſe in cochleas deforma<lb></lb>tus convertatur. </s> <s id="s.005652">Sic antè annos ferè quadraginta (cum non ar<lb></lb>rideret vulgaris tunc apud artifices circinorum forma, qui in<lb></lb>terjecto elatere crura divaricant; ſed inflexam in arcum co<lb></lb>chleam alteri crurum infixam, & per alterum trajectam, de<lb></lb>currente matrice exteriùs appoſitâ, dilatationem moderatur ar<lb></lb>tifex) juſſi mihi parari abſque ullo elatere circinum, quem ipſe <lb></lb>dilatarem atque contraherem pro arbitrio, cochleam hujuſmo<lb></lb>di duplicem in hanc atque in illam partem convertens. </s> <s id="s.005653">Ad <lb></lb>trientem totius longitudinis à nodo, ſingula crura cylindricum <lb></lb>foramen habent, ut ſingulis inſerantur cylindruli congruentes <lb></lb>exquiſitè politi, quorum ſuperiori extremitati ſunt adnexæ co<lb></lb>chlearum matrices, inferior extremitas extra circini ſoliditatem <lb></lb>exiens in helicem deſinit, ut appoſitâ matrice cylindrulus in<lb></lb>tra foramen contineatur. </s> <s id="s.005654">Hujuſmodi eſt cylindrulus IS craſ<lb></lb>ſitiei circini reſpondens, ſuperior pars eſt matrix R, infima ex<lb></lb>tra circini ſoliditatem in helicem deformata eſt SV, cui addita <lb></lb>matrix X continet cylindrulum intrà foramen, cui inditus eſt, <lb></lb>ita tamen, ut cylindruli ipſius opportunam convolutionem non <lb></lb>impediat. </s> <s id="s.005655">Duæ igitur matrices, cujuſmodi eſt R, coaptantur <lb></lb>duplici cochleæ cujus deinde extremitatibus, ad facilem <lb></lb>converſionem, anſulæ adduntur, adeò ut illæ matrices non <lb></lb>ſint exemptiles. </s> <s id="s.005656">Quare utrique crurum circini foramini, <lb></lb>utriuſque matricis cylindrulus IS inſeratur, & inferius ma<lb></lb>trice X firmetur: Nam convertendo cylindrum in dupli<lb></lb>cem cochleam deformatum, circini crura divaricabis, aut <lb></lb>adduces, ut libuerit. </s> <s id="s.005657">Neque quicquam officiet cylindri <lb></lb>rectitudo, quia matricum cylindruli IS pro opportunitate <lb></lb>volvuntur. </s> <s id="s.005658">Hinc circino eodem uti poteris abſque cochleâ, <lb></lb>deduci enim hæc poteſt, exemptis matricibus è foramine, <lb></lb>cui inſeruntur. </s> </p> <p type="main"> <s id="s.005659">At verò ſi validiore conatu opus fuerit, ad medium cy<lb></lb>lindrum, ubi cochlearum ſpiræ incipiunt, oportebit forami<lb></lb>na excavare, quibus immitti queat vectis, ut potentiæ mo-<pb pagenum="771" xlink:href="017/01/787.jpg"></pb>tus ad ponderum motum habeat majorem Rationem. </s> <s id="s.005660">Sic <lb></lb><figure id="id.017.01.787.1.jpg" xlink:href="017/01/787/1.jpg"></figure><lb></lb>duplici cochleâ cylindro circumductâ ad medium E ſint fora<lb></lb>mina, quibus vectis BC ſubinde inferri poſſit: duo autem <lb></lb>membra FD, & MN ex materiâ ſatis ſolidâ, qua extremita<lb></lb>te reſpiciunt vectem, matricem habeant cochleæ congruen<lb></lb>tem, ut ex vectis & cochleæ converſione aut ad ſe invicem <lb></lb>accedant, aut ſejungantur: reliqua extremitas exterior D <lb></lb>& N cava ſit, ut corpus repellendum comprehendatur, re<lb></lb>flectatur verò quaſi in uncos K & R, ut ſi duo corpora attra<lb></lb>henda fuerint, iis apprehendantur, ſivè proximè & immedia<lb></lb>tè, ſivè funibus adnexa. </s> </p> <p type="main"> <s id="s.005661">Quanta ſit hujus inſtrumenti vis, etiam ad frangenda aut <lb></lb>dilatanda ferrea clathra, hinc patet, quòd, longiore vecte ad<lb></lb>dito, potentiæ momenta notabiliter augentur; quia potentia <lb></lb>percurrit peripheriam circuli, cujus Radius à cylindri centro <lb></lb>ad vectis extremitatem producitur, pondera verò non niſi pro <lb></lb>ſpirarum intervallo moventur. </s> <s id="s.005662">Ubi tamen advertendum eſt, <lb></lb>utrum cylindrus ita ſit alicui loculamento inſertus, ut ejus <lb></lb>medium DE nec ad dexteram, nec ad ſiniſtram declinare <lb></lb>queat, an verò liber omninó ſit. </s> <s id="s.005663">Si enim interjectum mo<lb></lb>vendis corporibus inſtrumentum omnino liberum ſit, pon<lb></lb>dera verò movenda inæqualiter reſiſtant comparatis, aut eo<lb></lb>rum gravitatibus, aut momentis ratione planorum non uno <lb></lb>modo inclinatorum, aut ex diſparili ſuperficierum aſperita<lb></lb>te, non ſequitur æqualis eorum motus, ſed qua parte ma<lb></lb>jor invenitur reſiſtentia, minor quoque eſt motus; quamvis <lb></lb>utrumque æqualiter diſter à medio cylindri, quod repellitur <lb></lb>quodammodo ad eam partem, ubi levior eſt reſiſtentia. </s> <s id="s.005664">Si <lb></lb>enim ad N ſit aliquid obſtans motui, ut paries, aut firmi-<pb pagenum="472" xlink:href="017/01/788.jpg"></pb>ter infixus paxillus, ad D verò corpus aliquod repellendum; <lb></lb>utique ex vectis BC converſione etiam cochlea convolvitur, <lb></lb>& corpus in D poſitum tantumdem promovetur, quanto in<lb></lb>tervallo abſunt F & M ex cochleæ converſione; nam propter <lb></lb>impedimentum in N exiſtens, nullo pacto ipſum M move<lb></lb>tur. </s> <s id="s.005665">Sin autem corpus in N non omnino reluctetur motui, <lb></lb>ſed tamen reſiſtat magis, quàm corpus in D, illud quidem <lb></lb>aliquantulum repellitur, ſed multò magis repellitur corpus, <lb></lb>quod eſt in D; & in hoc motu medium cylindri punctum E <lb></lb>ad eas partes accedit, ad quas movetur corpus in D repul<lb></lb>ſum. </s> <s id="s.005666">Quod ſi medium E ita eſſet loculamento aliquo con<lb></lb>cluſum, ut poſitionem mutare nequeat, ſed ſolùm convolvi <lb></lb>poſſit, tunc utrumque corpus æqualiter repellitur, quia ſpi<lb></lb>rarum intervalla in utráque cochleâ æqualia ſunt. </s> <s id="s.005667">Hinc pa<lb></lb>tet poſſe fieri motus inæquales, ſi ſpirarum inclinationes non <lb></lb>fuerint æquales; minùs enim movetur illud, quod ſpiris ſpiſ<lb></lb>ſioribus urgetur. </s> </p> <p type="main"> <s id="s.005668">Quamvis autem hujuſmodi duplex cochlea ad duo corpo<lb></lb>ra disjungenda aut attrahenda ſæpè utilis ſit, ubi tamen exi<lb></lb>guus motus requiritur, ſivè ad augenda potentiæ momenta, <lb></lb>ſive ad affectandam tarditatem, præſtabit ſimplicem co<lb></lb>chleam adhibere. </s> <s id="s.005669">Nam in duplicis cochleæ converſione <lb></lb>disjunguntur, aut ad ſe invicem accedunt matrices (ac pro<lb></lb>inde & corpora, quæ moventur) quantum eſt duplex inter<lb></lb>vallum, quo ſpira abeſt à ſpira; ſingulis nimirum cochleis <lb></lb>ſuum reſpondet intervallum: at in ſimplicis cochleæ con<lb></lb>verſione motus reſpondet ſimplici duarum proximarum ſpi<lb></lb>rarum intervallo; ad quod idem potentiæ motus majorem <lb></lb>habet Rationem, quàm ad duplex intervallum. </s> <s id="s.005670">Quare fa<lb></lb>tis eſt, ſi alterutrum membrum cochleam includens longius <lb></lb>ſit, & matricem habeat; reliquum membrum, ut MN, <lb></lb>brevius eſſe poteſt, quantum opus fuerit ad recipiendum <lb></lb>cylindri caput extenuatum in minorem cylindrum, intrà <lb></lb>foramen cylindricum exquiſitè politum, ut facillimè con<lb></lb>verti poſſit: ita verò caput illud muniatur, ut ex locula<lb></lb>mento extrahi nequeat, quando utendum fuerit inſtru<lb></lb>mento ad corpora attrahenda: nam ad illa disjungenda cùm <lb></lb>adhibetur, ſatis reluctatur major diameter cylindri in co-<pb pagenum="773" xlink:href="017/01/789.jpg"></pb>chleam deformati, ne intrà foramen ulteriùs excurrat. </s> </p> <p type="main"> <s id="s.005671">Simili planè ratione ad divaricanda aut contrahenda cir<lb></lb>cini crura, uti <lb></lb><figure id="id.017.01.789.1.jpg" xlink:href="017/01/789/1.jpg"></figure><lb></lb>poteram unicâ & <lb></lb>ſimplici cochleâ <lb></lb>RS: cylindri il<lb></lb>lius extremitas <lb></lb>extenuetur in mi<lb></lb>norem cylindrum <lb></lb>exquiſitè lævigatum, qui prominentis capitis O foramini <lb></lb>cylindrico pariter polito inſeratur, & exteriore anſula V <lb></lb>converti pro arbitrio poſſit. </s> <s id="s.005672">Alterius clavi TX caput T <lb></lb>matricem habeat cochleæ congruentem: nam converſa an<lb></lb>ſula V adducet clavum T, & cum eo crus circini, ad O, <lb></lb>aut ab hoc illum removebit, & crura divaricabit: & qui<lb></lb>dem faciliùs licebit minutam in accipiendis punctorum di<lb></lb>ſtantiis ſubtilitatem perſequi; quandoquidem uni integræ <lb></lb>converſioni cylindri reſpondet unicum ſpirarum intervallum, <lb></lb>non autem duo intervalla hujuſmodi, quemadmodum cùm <lb></lb>duplex eſt cochlea. </s> </p> <p type="main"> <s id="s.005673">Quæ verò hì dicta ſunt, in pluribus aliis locum habe<lb></lb>re poſſunt, in quibus pro opportunitate modò ſimplicem, <lb></lb>modò duplicem cochleam prudens Machinator adhibebit: <lb></lb>Et quidem ſi duplex futura ſit cochlea, neque æquali mo<lb></lb>tu movenda ſint in oppoſitas partes corpora, cochleas ipſas <lb></lb>non ſimili, ſed inæquali, ſpirarum inclinatione formari ju<lb></lb>bebit. </s> <s id="s.005674">Cochleam autem ipſam opportuno loco ſtatuet: & <lb></lb>ſi fortè corporum ipſorum motus paulò velocior aut major <lb></lb>requiratur, quàm ferat ipſa cochleæ convolutio, duobus <lb></lb>vectibus decuſſatis, & circa axem in decuſſatione verſati<lb></lb>libus uti poterit, atque cochleam cum ſuis clavis & matri<lb></lb>cibus (ut ſuperiùs de circino dictum eſt) non procul à de<lb></lb>cuſſatione collocabit; nam modica illius convolutio non <lb></lb>exiguum motum tribuet corporibus in vectium illorum ex<lb></lb>tremitate poſitis, quippe quæ à decuſſatione magis diſtant, <lb></lb>quàm cochlea. </s> </p> <p type="main"> <s id="s.005675">At, inquies, hoc idem Ergatâ præſtari poterit: ſi enim <lb></lb>funes breviorum brachiorum extremitatibus C & E adnexi <pb pagenum="774" xlink:href="017/01/790.jpg"></pb>connectantur cum Ergatæ cylindro, ex hujus converſione ac<lb></lb><figure id="id.017.01.790.1.jpg" xlink:href="017/01/790/1.jpg"></figure><lb></lb>cedent ad ſe invicem ex<lb></lb>tremitates C & E, ac pro<lb></lb>pterea etiam velociùs cor<lb></lb>pora in F & D movebun<lb></lb>tur. </s> <s id="s.005676">Ita planè: non diffi<lb></lb>teor: ſed ſi extremitates C <lb></lb>& E proximas disjungere <lb></lb>oporteat, adeóne promptus <lb></lb>erit Ergatæ uſus, quin alio artificio opus ſit, ut hujus ope <lb></lb>disjungantur? </s> <s id="s.005677">Præterquam, quod ſæpè multum ſpatij ad col<lb></lb>locandam Ergatam requiritur; ſi maximè opus ſit illi pegma <lb></lb>conſtruere. </s> <s id="s.005678">Quid verò ſi vectes ipſi promovendi ſint, non <lb></lb>retrahendi, ut in rebus ſcenicis contingere poteſt? </s> <s id="s.005679">Quid ſi in <lb></lb>ſublimiore loco res perficienda ſit? </s> <s id="s.005680">quàm incommodè opportu<lb></lb>na Ergata ſatis longo vecte inſtructa ibi parabitur? </s> <s id="s.005681">Sed illud <lb></lb>potiſſimum attendendum eſt, quod vis cochleæ longè major <lb></lb>eſt; nam in Ergatâ Ratio motûs potentiæ ad motum ponderis <lb></lb>eſt eadem cum Ratione peripheriæ ab extremitate vectis de<lb></lb>ſcriptæ ad cylindri ambitum, hoc eſt longitudinis vectis uſque <lb></lb>ad axem cylindri, ad ipſius cylindri ſemidiametrum: at in co<lb></lb>chleâ peripheria ab extremo vecte deſcripta non comparatur <lb></lb>cum ipſius cylindri perimetro, ſed cum proximarum ſpirarum <lb></lb>intervallo, quod ſæpiſſimè minus eſt (ſaltem poteſt eſſe minus, <lb></lb>ſi magis inclinatæ & ſpiſſæ ſint ſpiræ) quàm cylindri diameter, <lb></lb>aut ejus perimeter: ac proinde major eſt Ratio motûs potentiæ <lb></lb>ad motum ponderis. <lb></lb></s> </p> <p type="main"> <s id="s.005682"><emph type="center"></emph>CAPUT III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005683"><emph type="center"></emph><emph type="italics"></emph>Cochlea cum Vecte, atque cum Axe componitur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005684">COntingere poteſt aliquando onus Vecte elevandum eſſe, <lb></lb>tantam verò illius gravitatem deprehendi, ut ſufficiens <lb></lb>Vectis longitudo non ſuppetat pro ratione operarum, quas <pb pagenum="775" xlink:href="017/01/791.jpg"></pb>adhibere poſſumus, aut ſaltem eæ ſint loci anguſtiæ, ut neque <lb></lb>hujuſmodi Vectis longitudinem, neque operarum multitudi<lb></lb>nem capiat: plures autem Vectes componere omnino incom<lb></lb>modum ſit, quia, ut in loco dictum eſt, minimus fieret oneris <lb></lb>elevandi motus. </s> <s id="s.005685">Præſtabit igitur Cochleam Vecti addere, ubi <lb></lb>maximè frequens futura ſit hujuſmodi ponderum elevatio, <lb></lb>quemadmodum propè telonia, ubi ingentes mercium ſarcinæ <lb></lb>attollendæ ſunt, ut plauſtris avehendæ imponantur, aut ad<lb></lb>vectæ ex iis deponantur. </s> </p> <p type="main"> <s id="s.005686">Erigatur tignum AB ad perpendiculum firmiter infixum <lb></lb>plano ſubjecto, tanta verò ſit craſſities tigni, ut in eo excavari <lb></lb><figure id="id.017.01.791.1.jpg" xlink:href="017/01/791/1.jpg"></figure><lb></lb>poſſit crena CD, per quam tignum aliud EF immitti facilè <lb></lb>valeat adeò ſolidum, ut Vectis munere fungatur, ubi extremo <lb></lb>unco G funibus adnexum fuerit onus. </s> <s id="s.005687">Ut igitur facillimè one<lb></lb>ris elevatio perficiatur, cochlea HI priſmati HL infixa (ſa<lb></lb>tiùs fuerit, ſi ejuſdem ligni pars in priſma, pars in cochleam <lb></lb>deformetur) ad perpendiculum erigatur inſerta matrici NM: <lb></lb>ſit autem ita ſolida matrix, ut illi adnecti queat tignum EF <lb></lb>clavo E, circa quem facilè verſari poſſit tignum ipſum, quan<lb></lb>do attollitur aut deprimitur. </s> <s id="s.005688">Demum ſubjecto priſmati HL <pb pagenum="776" xlink:href="017/01/792.jpg"></pb>non ſolùm in quatuor faciebus inſint foramina, quibus immit<lb></lb>tatur Radius KO, verùm etiam in infimâ baſis parte, qua <lb></lb>reſpondet axi cylindri in cochleam deformati, ſit polus, circa <lb></lb>quem convolvi poſſit: Hic tamen (ut ſatis manifeſtum eſt) in<lb></lb>trà ferream laminam ritè applumbatam lapidi in terrâ firmiſſi<lb></lb>mè defixo, aut certè adeò gravi, ut longè omnem elevandarum <lb></lb>ſarcinarum gravitatem vincat, ita contineri debet, ut indè nul<lb></lb>lo pacto eximi valeat, neque à priſmate avelli. </s> </p> <p type="main"> <s id="s.005689">Quod ſi non fuerit inter pavimentum & laqueare interval<lb></lb>lum enorme, faciliùs erit congruam trabis partem in cochleam <lb></lb>deformare, illámque baſi imponere (cujus altitudo commodam <lb></lb>Radij KO converſionem præſtet) atque circa duos polos, al<lb></lb>terum eidem ſubjectæ baſi, alterum lacunari infixum, convol<lb></lb>vere. </s> <s id="s.005690">Aut ſaltem proximo parieti infigatur tignum horizonta<lb></lb>le, quod prominens excipiat ſuperiorem polum, atque prohi<lb></lb>beat, ne vi ponderis ex unco G dependentis, in altum abripia<lb></lb>tur cochlea. </s> </p> <p type="main"> <s id="s.005691">Hìc vides compoſitam cum Vecte Cochleam, quæ Vectis vi<lb></lb>res notabili incremento auget. </s> <s id="s.005692">Eſt autem vectis hypomochlium <lb></lb>in eâ inciſæ aut inſculptæ crenæ parte, quæ cochleam reſpicit, <lb></lb>quando pondus attollitur, & vectis caput F ſupra lineam hori<lb></lb>zontalem elevatur; ſecùs verò, quando deprimitur vectis infra <lb></lb>lineam horizontalem, tunc enim hypomochlium eſt in D. </s> <s id="s.005693">Sit <lb></lb>igitur hypomochlium ad totius longitudinis EF beſſem; ac <lb></lb>proinde Ratio motûs potentiæ in E, ad motum ponderis in F, <lb></lb>eſt dupla. </s> <s id="s.005694">Ponamus Radij KO extremitatem O ab axe cylin<lb></lb>dri diſtare intervallo ſaltem decuplo intervalli ſpirarum co<lb></lb>chleæ HV: ergo potentia deſcribens peripheriam circuli, cu<lb></lb>jus diameter eſt 20, habet motum, qui eſt, ut minimum, ut 62 <lb></lb>ad motum HV, hoc eſt ad motum extremitatis E: hujus autem <lb></lb>motus eſt duplus motûs ponderis in F: igitur motus potentiæ eſt <lb></lb>ad motum ponderis ut 124 ad 1. Quare ut major ſit motus pon<lb></lb>deris, poterit hypomochlium minùs abeſſe ab extremitate E; <lb></lb>vix enim tales ſunt ſarcinæ, quæ ut moveantur, citrà laborem <lb></lb>ſuſtentandi, indigeant 60 hominibus. </s> </p> <p type="main"> <s id="s.005695">At ſi loci ratio ferat, ſuaderem potiùs Vectem ſecundi gene<lb></lb>ris, ita ut altera vectis extremitas inſiſteret tigno AB, & pon<lb></lb>dus inter tignum & cochleam interciperetur: ſic enim quò <pb pagenum="777" xlink:href="017/01/793.jpg"></pb>pondus eſſet propius cochleæ, ad majorem altitudinem attolle<lb></lb>retur, licèt majore conatu; ſed cochleæ vis abundare videtur, <lb></lb>& Vectis ſecundi generis ſemper auget momenta. </s> </p> <p type="main"> <s id="s.005696">Nec diſſimili ratione Vectem manu tractabilem ita cochleâ <lb></lb>inſtruere poſſumus, ut ad ingentia pondera movenda ſatis ſit. </s> <lb></lb> <s id="s.005697">Finge ſiquidem re<lb></lb><figure id="id.017.01.793.1.jpg" xlink:href="017/01/793/1.jpg"></figure><lb></lb>vellendas ſuis è car<lb></lb>dinibus ingentes a<lb></lb>licujus baſilicæ val<lb></lb>vas, ut reficiantur; <lb></lb>ferreus vectis AB <lb></lb>paretur extremita<lb></lb>te A ſubjiciendus <lb></lb>ponderi, & altera <lb></lb>extremitas B in ma<lb></lb>tricem cochleæ ex<lb></lb>cavetur: in hanc <lb></lb>immittatur cochlea CD, manubrium habens CE: atque ut <lb></lb>cochlea faciliùs convertatur, neque pavimentum atterat, pa<lb></lb>ratam habeto ferream laminam H, quæ illi ſubjiciatur. </s> <s id="s.005698">Nam <lb></lb>ſi ponderi ſupponatur vectis ex. </s> <s id="s.005699">gr. in F ad totius longitudinis <lb></lb>ſextantem, manubrij autem longitudo CE ſit ſaltem decupla <lb></lb>intervalli ſpirarum cochleæ, utique peripheria deſcripta à po<lb></lb>tentiâ in E, ad elevationem extremitatis B, eſt ſaltem ut 62 <lb></lb>ad 1: elevatio autem ipſius B, ad elevationem ponderis in F, <lb></lb>eſt ut 6 ad 1: igitur motus potentiæ ad motum ponderis eſt ut <lb></lb>372 ad 1. Quapropter, etſi initio parùm attollantur fores, & <lb></lb>ſubjecto cuneo, ne recidant, atque revolutâ cochleâ deprimen<lb></lb>dus, ac promovendus ſit vectis, ut pondus ſit in I, puta ad to<lb></lb>tius longitudinis quadrantem aut trientem, adhuc potentiæ <lb></lb>momenta erunt ut 248, aut 186 ad 1: quæ ſanè exigua non ſunt <lb></lb>pro ſimplici hujuſmodi machinulâ. </s> </p> <p type="main"> <s id="s.005700">Huc pariter ſpectat præli genus in meâ patriâ vulgare (in lo<lb></lb>cis potiſſimùm montanis, ubi faciliùs ingentes lapides non pro<lb></lb>cul advehendi ſuppetunt) quo ex uvæ jam calcatæ reliquiis <lb></lb>tortivum muſtum exprimitur. </s> <s id="s.005701">Roboris, quoad fieri poteſt, lon<lb></lb>giſſimum truncum unâ cum imo caudice aſſumunt, & ita ramis <lb></lb>omnibus ſpoliant, ut tamen bifurcum relinquant, quatenus <pb pagenum="778" xlink:href="017/01/794.jpg"></pb>bifidæ illi extremitati inniti, atque connecti valeat matrix co<lb></lb>chleæ, quæ convertatur circa polum ingenti ſubjecto lapidi in<lb></lb>ſiſtentem, ſed eâ ratione, ut demum etiam lapis attolli queat. </s> <lb></lb> <s id="s.005702">Truncum verò illum, qui præli munere fungi debet, præter <lb></lb>extremum caudicem craſſiſſimum, dedolant, ut inter bina tigna <lb></lb>hinc atque hinc in alvei lateribus ad perpendiculum erecta in<lb></lb>terjectum prælum attolli ac deprimi poſſit citrà impedimentum, <lb></lb>quod alioqui ipſa rudis aſperitas pareret. </s> <s id="s.005703">Porrò tigna illa bina <lb></lb>erecta, aut ex adverſo rotundis aliquot foraminibus perforant, <lb></lb>quibus immitti poſſit craſſiuſculus cylindrus, ſeu ferreus vectis, <lb></lb>aut illa incidunt patente crenâ, cui inſeri valeat repagulum; eo <lb></lb>conſilio, ut alterutra præli extremitas pro opportunitate prohi<lb></lb>beatur, ne aſcendat, aut deſcendat. </s> <s id="s.005704">Quare convoluta cochlea <lb></lb>attollit matricem, & oppoſita præli extremitas amotis omnibus <lb></lb>ſubjectis repagulis ſenſim deſcendit: ubi autem eò venerit, ut <lb></lb>non abſit ab altitudine eorum, quæ in torculari calcanda ſunt, <lb></lb>immittitur ſuperiùs repagulum, ne amplius attolli valeat; Tum <lb></lb>revoluta in contrarium cochleâ matricem cum præli extremita<lb></lb>te deorſum trahit: & quoniam reliqua extremitas attolli nequit <lb></lb>obſtante repagulo, premuntur uvæ, & in Lacum defluit <lb></lb>muſtum. </s> <s id="s.005705">Ubi demum adeò compreſſa fuerint vinacea, ut fa<lb></lb>cilius ſit lapidem cochleæ adnexum attollere, quàm illa magis <lb></lb>comprimere, ex cochleæ converſione attollitur lapis; quem ad <lb></lb>mediocrem altitudinem elevatum pendere diutiùs permittunt, <lb></lb>ut, lapidis gravitate deorſum conante, à prælo exprimatur, <lb></lb>quantulumcumque muſti adhuc vinaceis ineſt. </s> <s id="s.005706">Duplex igitur <lb></lb>hìc conſideranda eſt preſſio: altera quidem vi potentiæ co<lb></lb>chleam volventis; & hìc cochlea cum vecte ſecundi generis <lb></lb>componitur; eſt enim prælum vectis, cujus hypomochlium eſt <lb></lb>in eâ extremitate, quæ repagulo prohibetur, ne attollatur; po<lb></lb>tentiæ autem vectem hujuſmodi deprimentis vices obit cochlea <lb></lb>claviculatim ſtriata; quæ tamen motûs originem non habens <lb></lb>ſibi inſitam, potentiæ munus & nomen relinquit vectiariis il<lb></lb>lam verſantibus. </s> <s id="s.005707">Altera preſſio fit, ceſſante convolutione co<lb></lb>chleæ, vi gravitatis lapidis ſuſpenſi; & tunc non niſi Ratio <lb></lb>vectis intervenit, atque Potentia eſt ipſa gravitas. </s> </p> <p type="main"> <s id="s.005708">Sed quoniam non ubique reperiuntur aut tam ingentes lapi<lb></lb>des, aut tam longæ arbores, communiter univerſus premendi <pb pagenum="779" xlink:href="017/01/795.jpg"></pb>labor vectiariis incumbit cochleam unam, aut alteram verſan<lb></lb>tibus. </s> <s id="s.005709">Si duæ ſint cochleæ ad oppoſita torcularis latera conſti<lb></lb>tutæ, matricem habent in ipſo prælo excavatam, quod ſuá con<lb></lb>verſione deorſum trahunt, ut ex ſubjectis vinaceis exprimatur <lb></lb>muſtum: & tunc nihil eſt, quod Vectis momenta exerceat, <lb></lb>ſed ſola vis Cochleæ habetur. </s> <s id="s.005710">At ſi unica fuerit cochlea <lb></lb>(quemadmodum & in typographorum torculis) præli non eſt <lb></lb>uſus; ſed tranſverſæ trabi ſuperiori immotæ inſeritur per exca<lb></lb>vatas congruentes ſtrias cochlea, quæ in converſione depreſſa <lb></lb>calcat impoſitum vinaceis planum ex ſolidis aſſeribus. </s> <s id="s.005711">Verùm <lb></lb>contingere poteſt ut non ſit vectiario ſpatium expeditum, <lb></lb>quando, poſt modicam & faciliorem compreſſionem breviore <lb></lb>vecte peractam, adhuc longiore Radio utendum eſſet ad con<lb></lb>torquendam cochleam. </s> <s id="s.005712">Propterea ab Axe in Peritrochio ſub<lb></lb>ſidium facile peti poteſt; ſi videlicet extra torcularis alveum <lb></lb>ligneus cylindrus ad perpendiculum erigatur circa ſuos polos, <lb></lb>alterum ſubjecto plano, alterum exporrectæ è proximo pariete <lb></lb>trabi, infixos verſatilis: huic funem adnecte, qui extremo un<lb></lb>co apprehendat annulum Radij, quo cochlea verſatur: cylin<lb></lb>dro enim infixus Radius dum illum volvit, & funem illi cir<lb></lb>cumducit, cochleæ vectem ad ſe rapit, & vehementiùs premi<lb></lb>tur ſubjectum cochleæ planum, quàm ſi eadem cochlea duplo <lb></lb>longiore vecte convolveretur. </s> </p> <p type="main"> <s id="s.005713">Unum tamen hìc obſervandum, videlicet in hujuſmodi con<lb></lb>verſione non eadem eſſe momenta, quandoquidem funis exten<lb></lb>tus non eundem ſemper cum cochleæ vecte angulum conſti<lb></lb>tuit; eò autem minora ſunt momenta, quò magis hic ab an<lb></lb>gulo recto recedit, ut ex iis conſtat, quæ lib.4. cap.7. dicta ſunt. </s> <lb></lb> <s id="s.005714">Quapropter expedit cylindrum illum verſatilem non longiùs à <lb></lb>torculari abeſſe, ut funis minùs acutum angulum cum vecte <lb></lb>conſtituat, quando vectis extremitas incipit ſuæ peripheriæ ar<lb></lb>cum deſcribere: atque adeò ita ſtatuendus videtur cylindrus, <lb></lb>ut quando funis angulum rectum conſtituet cum cochleæ vecte, <lb></lb>hic jam percurrerit arcum non majorem ſemirecto angulo, reſ<lb></lb>pondentem gradibus 45: ſic enim fiet, non nimis acutum eſſe <lb></lb>angulum initio tractionis, & progrediendo augeri, donec fiat <lb></lb>rectus: deinde, licèt momenta decreſcant angulo in obtuſum <lb></lb>tranſeunte, ubi nimis obliquus factus fuerit angulus, poterit in <pb pagenum="780" xlink:href="017/01/796.jpg"></pb>aliud foramen immitti Radius, laxato priùs fune ex cylindri <lb></lb>revolutione. </s> <s id="s.005715">Quapropter ut potentiæ cylindrum volventis mo<lb></lb>menta ad calculos revoces, maximum momentum eſt fune ad <lb></lb>cochleæ Radium perpendiculari: quamvis autem non ſemper <lb></lb>progrediente motu conſtituat angulum rectum, tunc tamen <lb></lb>perinde computandum eſt momentum, atque ſi eandem poſi<lb></lb>tionem ad angulum rectum ſervatura eſſet potentia per funem <lb></lb>Radio applicata; præterita ſiquidem atque futura applicatio <lb></lb>nihil minuit præſentis applicationis virtutem. </s> <s id="s.005716">In eâ verò appli<lb></lb>catione perpendiculari, momenti Ratio deſumenda eſt ex circu<lb></lb>li à Radio deſcripti peripheriâ, atque ex intervallo ſpirarum co<lb></lb>chleæ: quæ Ratio componenda eſt cum Ratione Radij cylin<lb></lb>drum volventis ad ejuſdem cylindri ſemidiametrum. </s> </p> <p type="main"> <s id="s.005717">Sed ut habeantur momenta aliarum poſitionum, inquiren<lb></lb>dus eſt angulus applicationis funis ad eundem cochleæ vectem. <lb></lb><figure id="id.017.01.796.1.jpg" xlink:href="017/01/796/1.jpg"></figure><lb></lb>Et primò qui<lb></lb>dem datur Ra<lb></lb>dij cochleæ in<lb></lb>fixi longitudo <lb></lb>AB, ſemidia<lb></lb>meter cylindri <lb></lb>CD, & diſtan<lb></lb>tia AD. Inqui<lb></lb>ratur, in quo <lb></lb>puncto accidat <lb></lb>poſitio funis <lb></lb>perpendicularis <lb></lb>ad Radium co<lb></lb>chleæ: hæc uti<lb></lb>que non fit, niſi <lb></lb>fune tangente <lb></lb>utramque peri<lb></lb>pheriam tum <lb></lb>circuli à Radio <lb></lb>AB deſcripti, <lb></lb>tum cylindri; & erit BC. </s> <s id="s.005718">Quia igitur linea BC utrumque <lb></lb>circulum tangit, ductis ſemidiametris AB & CD, anguli ABC, <lb></lb>& DCB ſunt recti, ex 18. lib.3: igitur ex 27. lib.1. lineæ AB <pb pagenum="781" xlink:href="017/01/797.jpg"></pb>& DC ſunt parallelæ, & per 29. lib.1. anguli alterni BAD, <lb></lb>& CDA ſunt æquales: ſed & anguli ad verticem E ſunt æqua<lb></lb>les: ergo triangula BAE, CDE ſunt ſimilia; & per 4. lib. 6. <lb></lb>ut BA ad CD, ita AE ad DE; & componendo ut AB plus <lb></lb>CD ad CD, ita AD ad DE: innoteſcit itaque DE. </s> <s id="s.005719">Quare <lb></lb>ex quadrato ipſius DE auferatur quadratum lateris DC, & re<lb></lb>ſidui Radix erit recta CE. </s> <s id="s.005720">Fiat ergo ut DC ad CE, ita AB <lb></lb>ad BE; atque additis BE & CE nota eſt tota BC. </s> </p> <p type="main"> <s id="s.005721">Deinde aſſumatur poſitio Radij AF, ita ut arcus FB non ſit <lb></lb>major gradibus 45. Dato igitur arcu illo, hoc eſt angulo FAB, <lb></lb>noti ſunt anguli AFB, ABF trianguli iſoſcelis ad baſim BF, <lb></lb>ſinguli enim habent ſemiſſem reſidui ad duos rectos: atque <lb></lb>adeò, ſi recto CBA addatur notus ABF, innoteſcit anguli ob<lb></lb>tuſi CBF quantitas: Inventum jam eſt latus CB, & latus BF <lb></lb>ſubtenſa dati arcûs ex Canone Sinuum innoteſcit in partibus <lb></lb>Radij AB; quapropter ex Trigonometria inveniri poteſt baſis <lb></lb>CF, & angulus BFC; qui ſi auferatur ex noto angulo BFA, <lb></lb>remanet quæſitus angulus CFA applicationis funis CF ad <lb></lb>vectem AF. </s> <s id="s.005722">Habetur itaque ex hujuſmodi applicatione ad <lb></lb>vectem per angulum acutum CFA Ratio momenti comparati <lb></lb>cum momento applicationis ad angulum rectum CBA: eſt <lb></lb>enim ex dictis lib. 4. cap.7. ut Sinus anguli acuti ad Radium. </s> <lb></lb> <s id="s.005723">Fiat igitur ut Radius ad Sinum anguli AFC, ita peripheria <lb></lb>deſcripta à vecte AB ad aliud; & hoc inventum comparandum <lb></lb>eſt cum intervallo ſpirarum cochleæ, ut habeatur Ratio mo<lb></lb>menti potentiæ in C conſtitutæ, & applicatæ ad vectem AF <lb></lb>cum directione CF. </s> <s id="s.005724">Componenda deinde eſt hæc Ratio cum <lb></lb>Ratione Radij DH cylindrum volventis, ad ejuſdem cylindri <lb></lb>ſemidiametrum DC, & habebitur adæquata Ratio momenti <lb></lb>potentiæ in H. </s> </p> <p type="main"> <s id="s.005725">Tertiò. </s> <s id="s.005726">Ex puncto C ad A ducatur recta CA: & cum in <lb></lb>triangulo ABC rectangulo nota jam ſint latera AB & BC cir<lb></lb>ca rectum, invenitur hypothenuſa AC, & angulus BAC. </s> <lb></lb> <s id="s.005727">Tum ex 33. lib.3. ſuper rectâ AC deſcripto circuli ſegmento <lb></lb>capiente angulum obtuſum æqualem Supplemento anguli <lb></lb>AFC ad duos rectos, in puncto I, ubi hujus ſegmenti arcus <lb></lb>ſecat peripheriam à Cochleæ vecte deſcriptam, concurrant duæ <lb></lb>rectæ AI & CI. </s> <s id="s.005728">Eſt igitur triangulum CAI, in quo data ſunt <pb pagenum="782" xlink:href="017/01/798.jpg"></pb>latera CA & AI unâ cum angulo AIC noto, utpote ex con<lb></lb>ſtructione æquali ſupplemento ad duos rectos anguli jam noti <lb></lb>AFC. </s> <s id="s.005729">Quapropter inveniatur angulus IAC, qui demptus ex <lb></lb>jam invento angulo BAC, relinquit angulum BAI; ac <lb></lb>propterea notus eſt arcus BI, qui additus arcui BF dabit totum <lb></lb>arcum FI, in quo primùm momenta creſcunt ex F in B, dein<lb></lb>de decreſcunt ex B in I, ubi angulus obtuſus tantumdem exce<lb></lb>dit rectum, quantum à recto deficit acutus AFC; atque adeò <lb></lb>in F & I æqualia ſunt momenta. </s> </p> <p type="main"> <s id="s.005730">Antequam verò praxim hanc exemplo illuſtrem, ut ſub<lb></lb>ductis calculis noverit Machinator, quonam pacto omnia diſ<lb></lb>ponenda ſint, monendus eſt lector à me ideò ſemper idem <lb></lb>punctum C aſſumptum fuiſſe, quia in re Phyſicâ nullus ſubre<lb></lb>pere poteſt error notabilis. </s> <s id="s.005731">Cæterùm ſi funem extentum con<lb></lb>ſideremus ſemper quaſi lineam tangentem cylindri periphe<lb></lb>riam, ſatis manifeſtum eſt, ſi linea BC eſt tangens in puncto <lb></lb>C, & angulus BCD eſt rectus, non poſſe lineam à puncto F <lb></lb>productam ad contactum cadere in punctum C, ſed ultrà illud, <lb></lb>ita ut demum veniat punctum contactûs in C, quando poſitio <lb></lb>vectis fuerit AB, & iterum punctum contactûs recedat à C, <lb></lb>quando poſitio vectis fiat AI. </s> <s id="s.005732">Verùm quia exiguum eſt hujuſ<lb></lb>modi diſcrimen, propterea unum idémque punctum C aſ<lb></lb>ſumptum eſt, cum non ſequatur phyſicè ullum incommodum <lb></lb>ex hoc Geometricæ accurationis contemptu. </s> </p> <p type="main"> <s id="s.005733">Sit igitur ex. </s> <s id="s.005734">gr. ſpirarum cochleæ intervallum unc. </s> <s id="s.005735">2; & <lb></lb>vectis longitudo AB cubitorum 3, hoc eſt unc. </s> <s id="s.005736">36: quare inte<lb></lb>gra peripheria hoc Radio eſt unc. </s> <s id="s.005737">226; atque ideò in B, ubi <lb></lb>applicatio eſt ad angulum rectum, Ratio motuum ſeu momen<lb></lb>torum eſt ut 226 ad 2, hoc eſt 113 ad 1. Sit cylindri ſemidia<lb></lb>meter DC unc. </s> <s id="s.005738">3, atque DH ſimiliter unc. </s> <s id="s.005739">36: eſt igitur Ra<lb></lb>tio motûs ſeu momenti potentiæ in H, ad motum ſeu momen<lb></lb>tum in C ut 12 ad 1. Ratio itaque compoſita ex Rationibus 113 <lb></lb>ad 1, & 12 ad 1 eſt Ratio 1356 ad 1: quæ longê major eſt, <lb></lb>quàm ſi cochleæ adhiberi potuiſſet Radius cubitorum 6, non <lb></lb>addito Axe in Peritrochio. </s> <s id="s.005740">Ut inveniatur longitudo BC, pri<lb></lb>mùm fiat ut AB plus DC ad DC, hoc eſt ut unc.39. ad unc.3. <lb></lb>ita diſtantia AD data unc. </s> <s id="s.005741">65, ad ED unc. </s> <s id="s.005742">5. Igitur in trian<lb></lb>gulo ECD rectangulo, cujus hypothenuſa ED unc. </s> <s id="s.005743">5. la-<pb pagenum="783" xlink:href="017/01/799.jpg"></pb>tus DC unc. </s> <s id="s.005744">3. eſt latus EC unc. </s> <s id="s.005745">4: atque adeò ut DC 3 ad <lb></lb>CE 4, ita AB 36 ad BE 48; cui addita CE 4 dat totam per<lb></lb>pendicularem BC unc. </s> <s id="s.005746">52. </s> </p> <p type="main"> <s id="s.005747">Ponatur arcus FB gr.45; ergo ejus ſubtenſa 76536 partium, <lb></lb>quarum Radius AB unc.36 eſt 100000, erit unc. </s> <s id="s.005748">27 1/2: anguli <lb></lb>verò AFB, ABF ſunt ſinguli gr. 67. 30′. </s> <s id="s.005749">Quare in triangu<lb></lb>lo FCB datur angulus CBF gr. 157. 30′. </s> <s id="s.005750">comprehenſus à la<lb></lb>teribus CB unc. </s> <s id="s.005751">52, & BF unc. </s> <s id="s.005752">27 1/2. Invenitur ergo angulus <lb></lb>BFC gr. 14. 45′, qui ex angulo BFA gr. 67. 30. demptus relin<lb></lb>quit angulum AFC gr. 52. 45′; cujus Sinus eſt particularum <lb></lb>79600. Igitur ut Radius 100000 ad 79600, ita momentum <lb></lb>Applicationis per angulum rectum, quod erat ut 113, ad 90 <lb></lb>proximè, momentum Applicationis per hunc angulum AFC <lb></lb>acutum. </s> <s id="s.005753">Compoſitis itaque Rationibus 90 ad 1, & 12 ad 1, mo<lb></lb>mentum potentiæ in H erit 1080. </s> </p> <p type="main"> <s id="s.005754">Demum in triangulo ABC rectangulo ex lateribus AB 36, <lb></lb>& BC 52, reperitur hypothenuſa AC 63 1/4, & angulus BAC <lb></lb>gr.55. 18′. </s> <s id="s.005755">Quapropter in triangulo AIC datur latus AC 63 1/4, <lb></lb>& angulus illi oppoſitus AIC gr. 127. 15′. </s> <s id="s.005756">& præterea latus AI <lb></lb>36: ex quibus invenitur huic oppoſitus angulus ICA gr.26.56. <lb></lb>Igitur tertius angulus CAI eſt gr. 25. 49′: qui ſi auferatur ex <lb></lb>angulo BAC gr. 55. 18, reliquus eſt angulus BAI gr. 29. 29′. </s> <lb></lb> <s id="s.005757">Igitur totus arcus FI eſt gr.74. 29′. </s> </p> <p type="main"> <s id="s.005758">Ut autem appareat, quid conferat amplitudo arcûs BF, ſta<lb></lb>tuatur hic gr. 60. & huic æquales ſunt anguli ad baſim BF; <lb></lb>quæ recta BF eſt ipſi AB æqualis, hoc eſt unc.36. Quare in <lb></lb>triangulo FBC datur latus FB unc.36. & latus BC unc. </s> <s id="s.005759">52, & <lb></lb>angulus ab iis comprehenſus gr. 150: invenitur ergo angulus <lb></lb>BFC gr.17.47′; qui ablatus ex BFA gr. 60. relinquit CFA <lb></lb>gr.42, 13′: cujus Sinus eſt partium 67193. Igitur ut Radius <lb></lb>100000 ad 67193, ita 113 ad 76, quod eſt momentum Applica<lb></lb>tionis per hunc angulum acutum: atque compoſitis Rationibus <lb></lb>76 ad 1, & 12 ad 1, momentum potentiæ in H eſt ut 912. In trian<lb></lb>gulo verò AIC dantur latera AI unc. </s> <s id="s.005760">36, & AC unc. </s> <s id="s.005761">63 1/4 & <lb></lb>& Supplementum anguli AFC ad duos rectos eſt angulus <lb></lb>AIC gr. 137, 47″: ergo invenitur angulus ACI gr. 22. 29′ <lb></lb>Eſt igitur angulus IAC gr. 19. 44′: qui demptus ex angu-<pb pagenum="784" xlink:href="017/01/800.jpg"></pb>lo BAC gr.55. 18′. </s> <s id="s.005762">ſuperiùs invento, relinquit angulum IAB, <lb></lb>hoc eſt arcum IB gr. 35. 34′. </s> <s id="s.005763">Quare totus arcus FI eſſet <lb></lb>gr. 95. 34′. </s> <s id="s.005764">Ex quo vides intra eoſdem terminos æqualium mo<lb></lb>mentorum, minora eſſe extrema momenta in F & I, ſed per <lb></lb>majorem arcum, ſi incipias motum in majore diſtantiá à puncto <lb></lb>Applicationis per angulum rectum: propterea ſatius videtur <lb></lb>majora obtinere momenta, & minorem arcum deſcribere: ideò <lb></lb>dixi aſſumendum eſſe arcum BF non majorem gradibus 45. </s> </p> <p type="main"> <s id="s.005765">His ſimilia de Succulâ dicenda ſunt, quæ de Axe perpendi<lb></lb>culari diximus, ſi ſucculâ potiùs utendum loci & motûs quæ<lb></lb>ſiti opportunitas ſuadeat: id quod ita per ſe clarum eſt, ut in <lb></lb>his diutiùs immorari non ſit opus. <lb></lb></s> </p> <p type="main"> <s id="s.005766"><emph type="center"></emph>CAPUT IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005767"><emph type="center"></emph><emph type="italics"></emph>Cochleæ Infinitæ vires explicantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005768">VAlidiſſimam omnium Facultatum Cochleam eſſe ex ſupe<lb></lb>rioribus manifeſtum eſt: ſed illud accidit incommodum, <lb></lb>quod nimis brevibus terminis coërcetur; quos nimirum ejus <lb></lb>longitudo definit; ſivè illa circa ſuum axem convoluta intrà <lb></lb>Matricem immotam moveatur, ſivè illa poſitionem non mu<lb></lb>tans ex convolutione attrahat aut repellat Matricem & pondus <lb></lb>ei adnexum. </s> <s id="s.005769">Propterea alius cochleæ uſus excogitatus eſt citrà <lb></lb>ullam Matricem, cui inſeratur, atque ejuſmodi, ut cochleæ <lb></lb>converſioni nullus ſtatuatur finis, eaſdémque ſemper exerceat <lb></lb>vires. </s> <s id="s.005770">Hinc Cochleæ Infinitæ, aut Viti Perpetuæ nomen in<lb></lb>ditum eſt. </s> </p> <p type="main"> <s id="s.005771">Cylindrus circa ſuum axem, appoſito manubrio, verſatilis <lb></lb>in brevem cochleam deformatur unâ aut alterâ ſpirâ conten<lb></lb>tus: ita autem ad tympani dentes accommodatur, ut eorum in<lb></lb>tervallum ſit ſpirarum intervallo congruens; hoc eſt initium <lb></lb>ſpiræ apprehendat unum tympani dentem; dúmque ex Co<lb></lb>chleæ convolutione dens primus tántum promovetur, quantum <lb></lb>exigit ſpirarum diſtantia, unâ converſione abſolutâ iterum ini-<pb pagenum="785" xlink:href="017/01/801.jpg"></pb>tium ſpiræ apprehendat ſecundum tympani dentem proximè <lb></lb>conſequentem, ex tympani convolutione jam conſtitutum in <lb></lb>codem loco, in quo erat primus dens initio motûs: atque ita <lb></lb>deinceps omnes ſubinde dentes apprehenduntur à cochleâ; <lb></lb>ſemelque revoluto tympano, iterum à primo dente incipit ſe<lb></lb>cunda illius convolutio. </s> <s id="s.005772">Hinc quia cochleâ hujuſmodi, quate<lb></lb>nus ad ſe pertinet, nullum ſtatuit convolutionibus terminum, <lb></lb>etiamſi definitum habet ſpirarum numerum, immò unicam ha<lb></lb>beat ſpiram, Infinita dicitur, nam & tympanum orbitam ha<lb></lb>bens in ſeſe redeuntem plurimis ſine fine convolutionibus cir<lb></lb>cumagi poteſt. </s> <s id="s.005773">At ſi tympani loco rectam appoſueris laminam <lb></lb>denticulatam, quæ ex Cochleæ hujuſmodi converſione alium <lb></lb>atque ſubinde alium dentem apprehendentis adduceretur, aut <lb></lb>repelleretur; an illa appellanda eſſet Cochlea Infinita, quia <lb></lb>longiorem atque longiorem ſinè fine laminam ſimiliter movere <lb></lb>poſſet; iis examinanda relinquatur quæſtio, quibus de vocabu<lb></lb>lo diſputandi otium eſt. </s> </p> <p type="main"> <s id="s.005774">Tympano autem infixus eſt Axis, ſive ille ſimplex ſit, cui <lb></lb>ductarius funis circumvolvatur, ſivè ſtriatus fuerit, qui aliud <lb></lb>tympanum convertat, prout ſuo loco, ubi de Axe in Peritro<lb></lb>chio diſputatum eſt. </s> <s id="s.005775">Quapropter vis Cochleæ componitur cum <lb></lb>vi tympani, quod ab illà convertitur: idcirco huic Machinæ <lb></lb>Cochleæ Compoſitæ aliqui nomen fecerunt. </s> <s id="s.005776">Cùm itaque ſin<lb></lb>gulis cochleæ converſionibus ſinguli dentes tympani promo<lb></lb>veantur, toties convertitur cochlea, quot in tympani orbitâ <lb></lb>numerantur dentes. </s> <s id="s.005777">Potentiæ igitur motus, quo illa manubrium <lb></lb>verſans deſcribit circuli peripheriam, ducendus eſt per den<lb></lb>tium numerum, ut habeatur Ratio motûs Potentiæ, ad motum <lb></lb>orbitæ tympani. </s> <s id="s.005778">Cum verò data ſit Ratio tympani ad ſuum <lb></lb>Axem, data eſt Ratio motûs orbitæ tympani ad motum ponde<lb></lb>ris fune ductario attracti. </s> <s id="s.005779">Hæ duæ Rationes componantur, & <lb></lb>nota erit Ratio motû potentiæ ad motum ponderis. </s> <s id="s.005780">Sit co<lb></lb>chleæ manubrium digitorum 7; igitur peripheria circuli à po<lb></lb>tentiâ manubrio applicatâ deſcripti eſt ferè digit. </s> <s id="s.005781">44: tympani <lb></lb>ſemidiameter ad ſui Axis ſemidiametrum ſit ut 4 ad 1: Sit au<lb></lb>tem tympani orbita in dentes 24. diſtincta; ac propterea dum ſe<lb></lb>mel tympanum cum ſuo Axe volvitur, motus Potentiæ eſt digi<lb></lb>torum ferè 44 vicies & quater ſumptorum hoc eſt digit. </s> <s id="s.005782">1056. <pb pagenum="786" xlink:href="017/01/802.jpg"></pb>Si igitur tympani ſemidiameter ſit digit. </s> <s id="s.005783">4, & Axis ſemidiame<lb></lb>ter dig.1, illius peripheria eſt ſaltem digit. </s> <s id="s.005784">25, hujus verò pe<lb></lb>ripheria ſaltem digit. </s> <s id="s.005785">6 1/4, quantus eſt ex unâ tympani conver<lb></lb>ſione motus ponderis. </s> <s id="s.005786">Itaque motus Potentiæ ad motum pon<lb></lb>deris eſt ut 1056 ad 6 1/4, hoc eſt proximè ut 169 ad 1. </s> </p> <p type="main"> <s id="s.005787">Hinc ſi plura fuerint Compoſita Tympana, eorum Ratio, <lb></lb>quæ ex Rationibus diametrorum tympanorum ad ſuorum <lb></lb>Axium diametros componitur, aſſumenda eſt, atque attenden<lb></lb>dum quoties volvatur cochlea, ut primum tympanum cochleæ <lb></lb>proximum circumagatur: deinde per numerum dentium primi <lb></lb>tympani ducendus eſt motus potentiæ manubrio cochleæ appli<lb></lb>catæ; & ex Ratione tympanorum, atque ex Ratione Cochleæ, <lb></lb>componenda eſt Ratio. </s> <s id="s.005788">Sit cochlea eadem, quæ priùs, eodém<lb></lb>que manubrio inſtructa, adeò ut potentia ſemel cochleam ver<lb></lb>ſans deſcribat circuli peripheriam digitorum ferè 44; & pri<lb></lb>mum tympanum habens peripheriam dig.25 in dentes 24 diſtri<lb></lb>butam, dum ſemel volvitur, potentia vicies & quater periphe<lb></lb>riam dig.44 deſcribens percurrit digitos 1056. Sit idem pri<lb></lb>mum tympanum ad ſuum Axem ſtriatum ut 4 ad 1, ſecundum <lb></lb>tympanum ad ſuum Axem fune ductario involutum ſit ut 3 ad 2: <lb></lb>Ratio compoſita horum duorum tympanorum eſt ut 6 ad 1. <lb></lb>Cum verò motus potentiæ manubrio applicatæ ad integrum <lb></lb>motum peripheriæ tympani primi ſit ut 1056 ad 25 (nam ſingu<lb></lb>læ converſiones manubrij cochleæ ad motum unius dentis ſunt <lb></lb>ut 44 ad 25/24) componatur hæc Ratio cum Ratione 6 ad 1, & erit <lb></lb>motus potentiæ ad motum ponderis axi ſecundi tympani per fu<lb></lb>nem ductarium applicati, ut 6336 ad 25, hoc eſt ferè ut 253 1/2 <lb></lb>ad 1: atque adeò quo conatu potentia moveret libras decem; <lb></lb>hac machinâ movebit libras 2535. </s> </p> <p type="main"> <s id="s.005789">Verùm adhuc augeri poſſunt vires Cochleæ Infinitæ non <lb></lb>multiplicatis tympanis dentatis, ſed cum illo unico, quod à <lb></lb>Cochleâ movetur, componendo Trochleas: ſi videlicet alteri <lb></lb>Trochleæ adnectatur pondus, altera Trochlea alicubi firmetur: <lb></lb>tum funis ductarius, qui à Potentiâ arripiendus eſſet atque <lb></lb>trahendus, axi tympani alligetur. </s> <s id="s.005790">Nam ſi Ratio, quam Tro<lb></lb>chleæ inferunt, componatur cum Ratione Axis in Peritrochio, <lb></lb>atque Ratione Cochleæ, fit Ratio ex tribus Rationibus trium <pb pagenum="787" xlink:href="017/01/803.jpg"></pb>Facultatum compoſita. </s> <s id="s.005791">Sic Ratio Cochleæ ſit, ut priùs, 44 ad 25/24, <lb></lb>Ratio Tympani ad Axem ſit 4 ad 1, Ratio Trochlearum, capite <lb></lb>funis ad trochleam ponderis alligato (ſint autem Trochleæ bi<lb></lb>norum orbiculorum) ſit 5 ad 1: tres hæ Rationes Compoſitæ <lb></lb>conſtituunt Rationem 845 ad 1. Quare quo conatu moveres <lb></lb>libras decem, movebis libras 8450 tam facili & parabili ma<lb></lb>chinâ. </s> </p> <p type="main"> <s id="s.005792">Obſervanda ſunt autem tam commoda, quàm incommoda, <lb></lb>quæ hujus machinæ, ſcilicet Cochleæ Infinitæ uſum comitan<lb></lb>tur. </s> <s id="s.005793">Neque in poſtremis illud numerandum eſt, quod tantula <lb></lb>machinula facillimè transferri poteſt, ad pondera ſatis magna <lb></lb>dimovenda; maximè ſi in plano raptanda ſint ſuppoſitis ſcytalis <lb></lb>& trochleæ adhibeantur, quas non adeò craſſo fune connecti <lb></lb>oportet, quemadmodum ſi in ſublime attollendum eſſet pon<lb></lb>dus, & fune ipſo retinendum, ne relabatur. </s> </p> <p type="main"> <s id="s.005794">Adde non requiri ampliora ſpatia, ut cochlea hujuſmodi infi<lb></lb>nita circumagatur, & vel ſedentem hominem ſolâ, neque mul<lb></lb>tâ, lacertorum manubrium verſantium contentione poſſe mo<lb></lb>tum quæſitum perficere: atque ſi pondus attollatur, licet poten<lb></lb>tiæ, quandocumque libitum fuerit, ceſſare à motu, quin pon <lb></lb>dus ſuſpenſum recidat, etiamſi neque illi fulcrum ſubjiciatur <lb></lb>neque cochleæ manubrium retinaculo aliquo firmetur. </s> <s id="s.005795">Verùm <lb></lb>in attollendis ingentibus oneribus non expedit hac machinâ <lb></lb>uti, niſi tympanum dentatum ſatis magnum fuerit, ut Axem <lb></lb>craſſiorem atque validiorem admittat, cui ductarius funis cir <lb></lb>cumduci queat; hic autem funis cum <expan abbr="teñuis">tenuis</expan> eſſe non poſſit, ne<lb></lb>que exilem Axem exigit. </s> <s id="s.005796">Præterea diſſimulandum non eſt per <lb></lb>culum, ne cochlea inutilis fiat; ſi videlicet vel unicus tympano <lb></lb>dens excutiatur: ubi enim in converſione ad cam lacunam <lb></lb>ventum fuerit, illico ceſſat tympani converſio, cum nullus ejus <lb></lb>dens occurrat cochleæ. </s> <s id="s.005797">Propterea rem prudenter adminiſtrare <lb></lb>oportet, ut congrua machina eligatur. </s> </p> <p type="main"> <s id="s.005798">Porrò non contemnenda utilitas ex Cochleâ hac infinitâ per<lb></lb>cipi poteſt ad augendas communis Cochleæ vires ſivè preme<lb></lb>tis, ſivè etiam attrahentis. </s> <s id="s.005799">Eo videlicet loco, ubi aptandus eſ<lb></lb>ſet Radius ad Cochleæ converſionem, tympanum dentatum ad<lb></lb>jiciatur, ex cujus centro exeat cylindrus in cochleam deforma<lb></lb>tus, & Matrici inſertus: tympani verò dentes congruâ cochleæ <pb pagenum="788" xlink:href="017/01/804.jpg"></pb>infinitæ ſpirâ excipiantur: Manubrio enim verſato cochlea in<lb></lb>finita convertitur, & ſingulis converſionibus ſingulos tympani <lb></lb>dentes, alios ſubinde atque alios promovens, tympani covolu<lb></lb>tionem efficit, atque cum eo pariter infixa cochlea verſatur. </s> <lb></lb> <s id="s.005800">Prudenti autem Machinatori non deerit methodus, qua hujuſ<lb></lb>modi Cochlea infinita applicetur, & ſimul cum tympano den<lb></lb>tato deprimatur aut attollatur, ſi opus fuerit. </s> <s id="s.005801">Quapropter Ra<lb></lb>tio peripheriæ tympani ad intervallum ſpirarum ſuæ cochleæ, <lb></lb>componenda eſt cum Ratione peripheriæ à manubrio deſcriptæ <lb></lb>ad intervallum ſpirarum cochleæ infinitæ: ex hoc ſiquidem in<lb></lb>tervallo pendet motus peripheriæ tympani, cujus dentes ap<lb></lb>prehenduntur; quo enim preſſior eſt cochleæ infinitæ ſpira, eò <lb></lb>tenuiores & frequentiores inſunt tympano dentes. </s> <s id="s.005802">Sit ex. </s> <s id="s.005803">gr. <lb></lb>ſpirarum cochleæ prementis intervallum ſubtriplum ſemidia<lb></lb>metri tympani, cui illa infixa eſt: igitur Ratio perimetri tym<lb></lb>pani ad intervallum ſpirarum eſt ut 18 84/100 ad 1. At Cochleæ in<lb></lb>finitæ manubrium ad ejuſdem ſpirarum diſtantiam ſit ut 10 ad 1: <lb></lb>Motus igitur potentiæ manubrium verſantis eſt ut peripheria <lb></lb>deſcripta 62 83/100 ad motum unius dentis tympani ut 1. Ratio <lb></lb>itaque ex his duabus Rationibus Compoſita eſt 1183 7/10 ad 1. <lb></lb>Ex quo ſatis innoteſcit, quanto virium incremento addatur co<lb></lb>chleæ vulgari cochlea hæc infinita tam brevi manubrio in<lb></lb>ſtructa, loco vectis admodum longi, quem ſpatij anguſtiæ non <lb></lb>caperent. </s> </p> <p type="main"> <s id="s.005804">Verùm non ad augendas tantummodo vires, ſeu, ut veriùs <lb></lb>dicam, ad momentorum potentiæ incrementum, adhiberi po<lb></lb>teſt cochlea infinita, ſed ad motum quantumvis exiguum: <lb></lb>ſæpè enim motum extenuare opus eſt. </s> <s id="s.005805">Sic in automatis horas <lb></lb>indicantibus vi laminæ elaſticæ longioris in ſpiram convolutæ, <lb></lb>ad rotarum celeritatem aut tarditatem moderandam oportet <lb></lb>ipſum elaterem modò intendere, modò remittere: quia verò <lb></lb>in vulgaribus horologiis id perficitur convolutione rotæ denta<lb></lb>tæ (cujus axi intimum ſpiræ elaſticæ caput adnectitur, atque <lb></lb>ne lamina per vim complicata ſe in laxiorem ſpiram reſtituat, <lb></lb>axem ipſum & rotam dentatam revolvendo, obliquis rotæ ejuſ<lb></lb>dem dentibus, qua parte recti ſunt, objicitur virgula elaſtica) <lb></lb>ut minimum dentem unum promovere aut retrahere neceſſe <pb pagenum="789" xlink:href="017/01/805.jpg"></pb>eſt. </s> <s id="s.005806">At ſæpè contingere poteſt, ut elaſticam laminam jam val<lb></lb>de intentam amplius intendere, quantum fert integra dentis <lb></lb>unius converſio, celeriorem motum inferat, quam temporis ra<lb></lb>tio poſtularet; propterea ſcientiſſimi artifices, rejecta virgulâ <lb></lb>illâ elaſticâ, ita rotæ illius dentes conformant, ut cochleolæ <lb></lb>infinitæ congruant; hæc enim convoluta valde minutis pro<lb></lb>greſſionibus laminam elaſticam intendit, aut remittit, & ubi<lb></lb>cunque placuerit, ſiſtitur. </s> </p> <p type="main"> <s id="s.005807">Illud quoque non leve commodum (ut paulò ſuperius indi<lb></lb>catum eſt) in attollendis ponderibus animadverſione dignum <lb></lb>eſt, quod ſublato pondere atque ſuſpenſo, ceſſare poteſt po<lb></lb>tentia; & quamvis nec ab illâ, nec ab alio quolibet retinaculo <lb></lb>manubrium cochleæ infinitæ retineatur, neque pendenti oneri <lb></lb>fulcrum ullum ſubjiciatur, ipſa per ſe cochlea tympanum ſiſtit, <lb></lb>& ſuſpenſum pondus impeditur, ne ſuâ vi recidat. </s> <s id="s.005808">Id quod in <lb></lb>tympanis dentatis, neque in Succulis, neque in Trochleis, ne<lb></lb>que in Vecte obtinetur: quas Facultates ſi potentia dimiſerit, <lb></lb>inchoato jam motu, neque illas aliquo retinaculo coërceat, <lb></lb>priorem laborem irritum facit gravitas ſibi dimiſſa, ut ſatis aper<lb></lb>tè conſtat. </s> </p> <p type="main"> <s id="s.005809">Poſtremò Cocheas infinitas cochleis pariter infinitis coag<lb></lb>mentare ſi quis voluerit, is profectò momentis potentiæ immen<lb></lb>ſam quandam acceſſionem fecerit. </s> <s id="s.005810">Si enim primi tympani den<lb></lb>tati Axem deformaveris in cochleam, quæ aliud tympanum <lb></lb>pariter dentatum moveat, & ſecundi hujus tympani Axem item <lb></lb>in ſpiralem ſtriam excavaveris, quæ tertium tympanum con<lb></lb>vertat unà cum Axe, cui ductarius funis circumducitur; ecce <lb></lb>quot Rationibus componitur Ratio motuum potentiæ & pon<lb></lb>deris. </s> <s id="s.005811">Prima Ratio eſt Peripheriæ à manubrio deſcriptæ ad di<lb></lb>ſtantiam ſpirarum primæ cochleæ. </s> <s id="s.005812">Secunda Ratio eſt periphe<lb></lb>riæ primi tympani ad intervallum ſpirarum ſecundæ cochleæ. </s> <lb></lb> <s id="s.005813">Secunda Ratio eſt peripheriæ primi tympani ad intervallum <lb></lb>ſpirarum ſecundæ cochleæ. </s> <s id="s.005814">Tertia Ratio eſt peripheriæ ſecun<lb></lb>di tympani ad intervallum ſpirarum tertiæ cochleæ. </s> <s id="s.005815">Quarta <lb></lb>demum eſt Ratio peripheriæ tertij tympani ad ambitum ſui <lb></lb>Axis. </s> <s id="s.005816">Ponamus ſingulas peripherias ad ſuæ cochleæ ſpirarum <lb></lb>intervallum eſſe ut 30 ad 1, & tertij tympani orbitam ad ſui <lb></lb>Axis ambitum eſſe ut 5 ad 1; componendæ ſunt tres Rationes <pb pagenum="790" xlink:href="017/01/806.jpg"></pb>trigecuplæ cum unâ quintuplâ, & exurgit Ratio motûs poten<lb></lb>tiæ manubrio applicatæ, ad motum ponderis ut 135000 ad 1. <lb></lb>Quo igitur conatu potentia moveret libras decem, hac trium <lb></lb>cochlearum infinitarum complexione movebit millies mille tre<lb></lb>centas quinquaginta libras, ſeu, ut vulgari vocabulo utar, mil<lb></lb>lionem & trecenta quinquaginta millia librarum. </s> <s id="s.005817">Quid autem, <lb></lb>ſi plura tympana cochleas infinitas habentia addantur? </s> <s id="s.005818">utique <lb></lb>ſi primæ cochleæ manubrio agitatæ quatuor conſequentia tym<lb></lb>pana cum ſuis cochleis addantur, eandem Rationem trigecu<lb></lb>plam habentia, & quintum tympanum cum ſuo Axe Rationem <lb></lb>quintuplam habeat, demum potentia momentum obtinebit <lb></lb>ut 121. 500000: &, ſi abſque machinâ moveret libras decem, <lb></lb>hac machinâ ex quinque cochleis cum ſibi congruentibus <lb></lb>tympanis movere poterit mille ducentos quindecim milliones <lb></lb>librarum. </s> </p> <p type="main"> <s id="s.005819">Neque ſibi quiſquam perſuadeat opus eſſe ingentibus tym<lb></lb>panis, ut validiſſimis cochleis reſpondeant: Experimento enim <lb></lb>didicimus valde exiguas cochleas ſatis eſſe ad ingentia pondera <lb></lb>attollenda, modò axis funi ductario deſtinatus ſatis firmus ſit ac <lb></lb>validus, & ferendo oneri par. </s> <s id="s.005820">Hic autem Axis (quemadmo<lb></lb>dum & in Ergatâ) ſi plurimum funem excipere debeat, ne in <lb></lb>nimiam longitudinem protendatur, conformari poteſt in Cy<lb></lb>lindroides Hyperbolicum: nam ductarius funis illum aliquo<lb></lb>ties complexus (quantum ſatis fuerit, ne excurrat) colligi po<lb></lb>terit, & in convolutione ſe ad apicem Hyperbolæ continebit. </s> </p> <p type="main"> <s id="s.005821">At, inquis, hujuſmodi motus ponderis nimis longa temporis <lb></lb>ſpatia exigit. </s> <s id="s.005822">Ita planè: neque aliter contingere poteſt, ſi qui<lb></lb>dem tam ingens pondus movere volueris: an non præſtat tan<lb></lb>tam molem demum loco ceſſiſſe, quam omnino immotam cui<lb></lb>cumque conatui reluctari? </s> <s id="s.005823">Sed quid, ſi opportuniſſimum ſe <lb></lb>præbeat proximus rivulus perennis? </s> <s id="s.005824">primæ cochleæ apponatur <lb></lb>loco manubrij rota cum pinnis, in quas aqua incurrat; illa enim <lb></lb>circumacta cochleam & conſequentia tympana verſabit, ac de<lb></lb>mum vel dormientibus operis moles ab exiguâ aquâ dimo<lb></lb>vebitur. </s> </p> <p type="main"> <s id="s.005825">Quod ſi ex pluribus cochleis infinitis compoſitam machinam <lb></lb>tibi conſtruere volueris, ita tamen, ut modò majoribus, modò <lb></lb>minoribus ponderibus movendis ſit idonea citrà temporis diſ-<pb pagenum="791" xlink:href="017/01/807.jpg"></pb>pendium, ubi ſatis virium habetur in potentiâ; eâ ratione in <lb></lb>loculamento diſpone ſingulos axes in cochleam deformatos, ut <lb></lb>eorum poli ex loculamento promineant, atque pro re natâ pro<lb></lb>pelli ſeu retrahi aliquantiſper valeat hic aut ille axis, ne ejus <lb></lb>ſtria occurrat ſubjecti tympani dentibus. </s> <s id="s.005826">Nam ſi alterius ſal<lb></lb>tem poli extremitas in quadratam figuram deſinat, quæ inſeri <lb></lb>poſſit manubrio, hoc poterit huic aut illi axi aptari, quin ſupe<lb></lb>riores cochleæ hujus tympani convolutionem impediant. </s> <s id="s.005827">Quod <lb></lb>ſi majora adhuc requirantur potentiæ momenta, proximè ſu<lb></lb>perior axis ſuum in locum reſtituatur, ut cochleæ ſtria in ſub<lb></lb>jecti tympani dentes incurrat. </s> <s id="s.005828">Quapropter ad minora pondera <lb></lb>movenda adhibeantur inferiores cochleæ, ad majora ſuperiores. <lb></lb></s> </p> <p type="main"> <s id="s.005829"><emph type="center"></emph>CAPUT V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005830"><emph type="center"></emph><emph type="italics"></emph>Cochleæ uſus aliqui indicantur.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005831">ADeò frequens eſt & vulgatus apud pleroſque artifices co<lb></lb>chleæ uſus, ut ex tam variâ ejus cum cæteris complexio<lb></lb>ne unuſquiſque facilè colligere poſſit, quid facto ſit opus, ubi <lb></lb>eâ utendum neceſſitas aut utilitas ſuaſerit. </s> <s id="s.005832">Ne tamen ab initâ <lb></lb>in antecedentibus libris conſuetudine in hujus operis calce re<lb></lb>cedam, pauca quædam indicare placuit, quæ in reliquis non <lb></lb>admodum diſſimilibus facem præferant. </s> </p> <p type="main"> <s id="s.005833"><emph type="center"></emph>PROPOSITIO I.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005834"><emph type="center"></emph><emph type="italics"></emph>Aërem validè comprimere, aut dilatare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005835">FOllibus luſoriis aërem pyulco ingerentes majorem ſubinde <lb></lb>atque majorem difficultatem percipiunt; quo enim magis <lb></lb>aër concluſus à naturali raritate recedere cogitur, etiam majo<lb></lb>re niſu reſiſtit, neque ſolùm magis denſari renuit, ſed & ſe la<lb></lb>tiùs explicare molitur. </s> <s id="s.005836">Hinc didicimus & pneumaticos fontes <lb></lb>conſtruere, qui Spiritu interno urgente aquam in altum evi-<pb pagenum="792" xlink:href="017/01/808.jpg"></pb>brant, & plumbeas glandes fiſtulis ejaculari, non pulvere ni<lb></lb>trato ignem concipiente, ſed aëre per vim denſato ad antiquas <lb></lb>dimenſiones recuperandas erumpente. </s> <s id="s.005837">Quoniam verò ingeſta <lb></lb>jam in conceptaculum non exigua aëris copia difficiliùs com<lb></lb>primitur nová aëris acceſſione, quàm ut manus valeat truſillum <lb></lb>rectâ impellere; idcirco truſilli haſtulam deformatam in heli<lb></lb>cem, & ſuæ matrici inſertam, adhibere operæ pretium erit: <lb></lb>dum enim manubrio agitante contorquetur cochlea, ſenſim de<lb></lb>primitur embolus, aërémque ingerit. </s> <s id="s.005838">Ne autem morâ longio<lb></lb>re opus ſit perpetuâ verſatione manubrij, ita cochleæ matrix <lb></lb>externam vaſis faciem contingat, ut illi adnecti, atque ab eo <lb></lb>disjungi valeat: initio enim, quando adhuc levis eſt aeris mo<lb></lb>dicè compreſſi reſiſtentia, lamella illa ſuo foramine interiùs <lb></lb>claviculatim ſtriato cohærens haſtulæ emboli, ſi à vaſe dis<lb></lb>juncta fuerit, unà cum haſtulâ movebitur: deinde verò, quan<lb></lb>do jam truſillus ægrè impellitur, lamella illa cum vaſe con<lb></lb>nectatur, & non niſi verſato manubrio adduci atque reduci em<lb></lb>bolus poterit; id quod ſatis lentè perficietur. </s> <s id="s.005839">Rem claritatis <lb></lb>gratia in fonte pneumatico explicemus. </s> </p> <p type="main"> <s id="s.005840">Sit vas AB ex materiâ metallicâ, in cujus ſuperiore parte la<lb></lb><figure id="id.017.01.808.1.jpg" xlink:href="017/01/808/1.jpg"></figure><lb></lb>brum, ex quo per fo<lb></lb>ramen A immittatur <lb></lb>in vas aqua, ita ta<lb></lb>men, ut non implea<lb></lb>tur; aqua enim in vas <lb></lb>modicè inclinatum <lb></lb>deſcendens aërem ex<lb></lb>pellet per tubulum <lb></lb>CD. </s> <s id="s.005841">Ubi ſatis aquæ <lb></lb>immiſſum fuerit, oc<lb></lb>cludatur foramen A <lb></lb>diligentiſſimè co<lb></lb>chleolâ congruente, <lb></lb>& convoluto epiſto<lb></lb>mio E, tubus DC ſit <lb></lb>aëri impervius ad vaſis latus ſtatuatur modiolus cum embolo <lb></lb>congruente HI, & emboli haſtula ſit connexa cum mobili va<lb></lb>ſis ansâ HO. </s> </p> <pb pagenum="793" xlink:href="017/01/809.jpg"></pb> <p type="main"> <s id="s.005842">Porrò haſtula HK perforata ſit, & continuo ductu uſque <lb></lb>ad emboli KS fundum pateat aëri ingredienti via HS; ſed fo<lb></lb>ramini S adjecta ſit valvula, quæ aëri regreſſum obſtruat. </s> <s id="s.005843">Simi<lb></lb>liter modioli fundo in I valvula exteriùs appoſita aperiatur in<lb></lb>geſto aëri tranſitum præbens, ſed aëri intra vas compreſſo cum <lb></lb>nuſquam exitus pateat, valvula ipſa modioli foramen I occlu<lb></lb>dit. </s> <s id="s.005844">Haſtulæ verò HK exterior facies ſit in helicem ſtriata, & <lb></lb>lamellæ MN tanquam matrici congruat, quæ in M & N co<lb></lb>chleolis adnecti queat exteriùs vaſi, quaſi eſſet anſæ fulcrum. </s> </p> <p type="main"> <s id="s.005845">Ubi immiſſum fuerit quantum ſatis eſt aquæ, cochleolis M <lb></lb>& N revolutis disjungatur matrix à vaſe: tum attractâ ansâ <lb></lb>HO, unà cum lamellâ MN attrahitur embolus KS, & per <lb></lb>apertum ductum HS ingreditur aër modiolum implens. </s> <s id="s.005846">Im<lb></lb>pulſo deinde embolo, valvula ad S clauditur, & aër ex modio<lb></lb>lo per patentem valvulam I ingeritur in vas; ex quo nequit exi<lb></lb>re, neque aquam propellere, clauſo ſcilicet epiſtomio E, & fo<lb></lb>ramine A: quapropter comprimitur, & denſatur; ideóque at<lb></lb>tracto denuo embolo KS incluſus vaſi aër ſe latiùs explicare <lb></lb>connitens valvulam I valide applicat foramini modioli, ſibíque <lb></lb>exitum obſtruit. </s> <s id="s.005847">Toties adducitur atque reducitur embolus, & <lb></lb>aër ingeritur, quoad magna premendi difficultas percipiatur; <lb></lb>ubi eò ventum fuerit, tunc lamella MN iterum vaſi adnecta<lb></lb>tur ſuis cochleolis; nec jam embolus rectâ adduci poteſt; ſed <lb></lb>arreptum in O manubrium verſatur, & embolus intrà modio<lb></lb>lum circumactus ſenſim attollitur, qui deinde revoluto in con<lb></lb>trarium manubrio deprimitur, & multâ vi aër in vaſe compri<lb></lb>mitur. </s> <s id="s.005848">Laxato demum Epiſtomio E, compreſſus in vaſe aër <lb></lb>aquam exprimit per tubum CD, primùm quidem vehemen<lb></lb>tiùs, ſubinde remiſſius, prout aëris vis elaſtica ſenſim lan<lb></lb>gueſcit. </s> </p> <p type="main"> <s id="s.005849">Hoc idem quod de aëre intra vas comprimendo ad aquam <lb></lb>evibrandam comminiſci placuit, ſervatâ analogiâ dicendum <lb></lb>eſt de aëre, tùm conatu manûs rectâ truſillum impellentis, tum <lb></lb>ope cochleæ ſimiliter conformatæ, intrà conceptaculum com<lb></lb>primendo, ut ex fiſtulâ deinde multâ vi emittatur plumbea <lb></lb>glans, ubi reſeratus aëri exitus illum ſubitò dilatari permiſerit. </s> <lb></lb> <s id="s.005850">Quin & pneumatica hujuſmodi tormenta citrà conceptaculum <lb></lb>aëris compreſſi conſtruere non inutile accidat, ſi, quemadmo-<pb pagenum="794" xlink:href="017/01/810.jpg"></pb>dum noſtrates pueri ſurculos ſambuceos fungosâ medullâ <lb></lb>exhauriunt, & utráque tubuli extremitate papyraceis globulis <lb></lb>obſtructâ, alterum globulum congruo cylindro propellunt, at<lb></lb>que incluſum aërem denſant, quoad aëris vim elaſticam, & im<lb></lb>pellentis manûs conatum non ferens extremus alter globulus <lb></lb>edito ſcloppo expellatur; ita ferream fiſtulam longiorem para<lb></lb>veris, cujus alteri extremitati immittatur plumbea glans ob<lb></lb>ducta papyro, aut ſimili materiâ, ut exquiſitè tubi oſculum <lb></lb>implens demum univerſam aëris vim excipiat, alteram extre<lb></lb>mitatem aliquot ſpiris ambiat cava cochlea, quam impleat cy<lb></lb>lindrus ferreus in congruentem cochleam deformatus: Si enim <lb></lb>hujuſmodi cylindrus vix brevior fuerit, quàm fiſtula, & apto <lb></lb>manubrio convolutus in fiſtulam ſenſim immittatur, totum aë<lb></lb>rem, quo fiſtula replebatur, ad exiguas ſpatij anguſtias adiget, <lb></lb>ex quibus magnâ vi demum, quâ data porta, erumpens ejacu<lb></lb>labitur plumbeum globulum. </s> </p> <p type="main"> <s id="s.005851">Quod ſi aërem non comprimere, ſed diſtrahere atque dila<lb></lb>tare libitum fuerit, eâdem ratione parandus eſt modiolus cum <lb></lb>embolo, ac haſtulâ in helicem ſtriatâ, atque perforatâ, & co<lb></lb>chleæ matrici inſerta, niſi quod valvulæ contrariam poſitionem <lb></lb>exigunt; nam modioli valvula I intrà ipſum modiolum ſtatuen<lb></lb>da eſt, ut adducto embolo aperiatur, & ex vaſe aër in modio<lb></lb>lum attrahatur: Emboli verò valvula non ad S, ſed in H ap<lb></lb>ponenda eſt, ut reducto embolo, aër in modiolum admiſſus ex<lb></lb>primatur per tubulum SH, ſivè manu urgeatur truſillus, ſive <lb></lb>cochlea convolvatur. </s> <s id="s.005852">Aërem autem, licèt valde compreſſum, <lb></lb>magis etiam convolutâ cochleâ denſari, aut valde rarum ma<lb></lb>gis adhuc dilatari manifeſtum eſt; id quod rectâ manûs impul<lb></lb>ſione aut attractione nequaquam fieri poſſet. </s> </p> <p type="main"> <s id="s.005853"><emph type="center"></emph>PROPOSITIO II.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005854"><emph type="center"></emph><emph type="italics"></emph>Forcipum vires cochleâ augere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005855">DUplicem exerceri à forcipibus vim conſtat; altera eſt con<lb></lb>ſtringendo id, quod illis apprehenditur, & earum vis <lb></lb>major aut minor ex eo æſtimatur, quod brachia longè à nodo, <lb></lb>aut prope illum, arripiantur: altera vis eſt in extrahendo ali-<pb pagenum="795" xlink:href="017/01/811.jpg"></pb>quid, ut clavum tabulæ aut parieti infixum; cum enim curva <lb></lb>ſit forceps, qua parte clavum apprehendit, adnexum in ipſo <lb></lb>flexu habet hypomochlium, & brachia inclinando, pro eorum <lb></lb>longitudine, vis extrahendi exercetur quaſi per vectem. </s> <s id="s.005856">At <lb></lb>aliquando opus eſt majore conatu, quàm ut ſolis forcipibus va<lb></lb>leat potentia infixum clavum extrahere; momentum ſiquidem <lb></lb>potentiæ pendet ex Ratione, quam habet diſtantia potentiæ ad <lb></lb>diſtantiam clavi ab ipſo flexu, qui fungitur munere hypomo<lb></lb>chlij. </s> <s id="s.005857">Quare vis extrahendi major communicari poteſt ope co<lb></lb>chleæ, ita tamen, ut forceps non exerceat munus vectis. </s> </p> <p type="main"> <s id="s.005858">Paretur itaque valida & ſatis craſſa lamina chalybea AB, <lb></lb>matricem cochleæ habens in C, & ſit <lb></lb><figure id="id.017.01.811.1.jpg" xlink:href="017/01/811/1.jpg"></figure><lb></lb>cochlea FE, manubrium habens ED. </s> <lb></lb> <s id="s.005859">Cochleæ verò extremitas in cylindrum <lb></lb>deſinat, qui craſſioris laminæ HI fora<lb></lb>mini exquiſitè polito inſeratur, & in eo <lb></lb>facillimè convolvi valeat. </s> <s id="s.005860">Cylindri ex<lb></lb>tremitas infra laminam HI ita dilatetur, <lb></lb>ut eandem laminam HI ſuſtineat, non <lb></lb>tamen convolutionem impediat. </s> <s id="s.005861">Porrò <lb></lb>laminæ HI adnexi ſint duo annuli ita <lb></lb>conformati, ut forcipis brachia exci<lb></lb>piant: nam ſi brachia in hujuſmodi an<lb></lb>nulos immittantur, ut hi proximi ſint nodo forcipis maximè di<lb></lb>latatæ, antequam apprehendat clavum extrahendum, poſtmo<lb></lb>dum conſtrictâ forcipe & clavum apprehendente, elevata la<lb></lb>mina HI annulos ſecum rapiet, qui per forcipis brachia diva<lb></lb>ricata excurrentes demum validè illa conſtringent, nec ulte<lb></lb>riùs excurrere poterunt. </s> <s id="s.005862">His paratis utrique extremitati AB <lb></lb>ſubjiciantur fulcra (ſivè ſint tigillorum fruſta, ſivè quæcum<lb></lb>que alia) inter ipſam laminam & planum, ex quo educendus <lb></lb>eſt clavus, interjecta: Nam manubrio DE convoluta cochlea <lb></lb>ita matricem AB applicabit fulcris, ut firmiſſimè cohæreant <lb></lb>cum ſubjecto plano. </s> <s id="s.005863">Jam ſi pergas cochleam contorquere, hæc <lb></lb>ſecum rapiet laminam HI, & adjectos annulos cum forcipe, & <lb></lb>clavo, quem revellit. </s> </p> <p type="main"> <s id="s.005864">Quod ſi fortè placuerit forcipem habere peculiarem huic <lb></lb>inſtrumento aptandam, habeat in brachiorum extremitatibus <pb pagenum="796" xlink:href="017/01/812.jpg"></pb>uncos aut annulos annulis H & I inſerendos aut connectendos, <lb></lb>eâ tamen ratione diſpoſitos, ut dum lamina HI vi cochleæ tra<lb></lb>hitur, brachia ipſa ad ſe invicem accedendo forcipem con<lb></lb>ſtringant. </s> </p> <p type="main"> <s id="s.005865">Unum præterea addendum, quod non levis eſt momenti, & <lb></lb>aliàs quoque obſervari poterit. </s> <s id="s.005866">Contingere poteſt, ut omnibus <lb></lb>modo dicto paratis, potentia ſe infirmiorem ſentiat, quàm ut <lb></lb>valeat circumducto manubrio DE cochleam contorquere. </s> <s id="s.005867">Hoc <lb></lb>igitur tibi remedium compara: longiorem vectem validis funi<lb></lb>culis colliga cum manubrio DE, & vecte illo quaſi manubrio <lb></lb>utens experieris pro Ratione longitudinis aucta momenta; am<lb></lb>plior ſiquidem peripheria, quæ tunc à potentiâ deſcribitur, ad <lb></lb>ſpirarum cochleæ intervallum habet Majorem Rationem. </s> </p> <p type="main"> <s id="s.005868"><emph type="center"></emph>PROPOSITIO III.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005869"><emph type="center"></emph><emph type="italics"></emph>Numerum paſſuum aut rotæ converſionem metiri.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005870">HOc idem problema lib.5. cap.9. prop.2. propoſitum eſt, & <lb></lb>per rotulas dentatas ſingulis prioris rotæ converſionibus <lb></lb>excipientes impulſionem ſingulorum dentium, in quos promi<lb></lb>nens paxillus incurrat, perfici poſſe indicatum eſt. </s> <s id="s.005871">Nunc <lb></lb>aliam methodum indicare placet ex iis, quæ ſuperiore capite <lb></lb>ſunt dicta de Cochleâ Infinitâ. </s> <s id="s.005872">Primam quidem rotulam, cui <lb></lb>motus origo ineſt ex funiculi tractione, prout ibi dictum eſt, <lb></lb>eandem ſtatue, & illius axis extremitas appoſito indice tot paſ<lb></lb>ſus, aut tot rotæ converſiones indicabit, quot in dentes ipſa <lb></lb>prima rotula diſtributa intelligitur. </s> <s id="s.005873">Hujus rotulæ axis in co<lb></lb>chleam infinitam deformetur, cui ſua rotula dentata congruat; <lb></lb>& ſingulis primæ rotulæ converſionibus ſinguli dentes ſecundæ <lb></lb>promoventur: atque adeò quot dentes ſecundæ huic rotulæ in<lb></lb>ſunt, ut hæc integram converſionem perficiat, tot requiruntur <lb></lb>prioris rotulæ converſiones. </s> <s id="s.005874">Similiter ſecundæ rotulæ axis in <lb></lb>cochleam infinitam deformetur, & tertiam rotulam dentatam <lb></lb>convertat, cujus axis pariter tertiam cochleam infinitam conſti<lb></lb>tuere poteſt, & quartam rotulam cum ſuo axe & indice convol<lb></lb>vere. </s> <s id="s.005875">Singulorum axium extremitates in facie loculamenti ad<lb></lb>jecto indice ob oculos ponunt numerum revolutionum proxi-<pb pagenum="797" xlink:href="017/01/813.jpg"></pb>mè antecedentis rotulæ. </s> <s id="s.005876">Quapropter numerus à poſtremâ ro<lb></lb>tulâ indicatus multiplicandus eſt per numerum omnium den<lb></lb>tium penultimæ rotulæ, & productus per numerum dentium <lb></lb>antepenultimæ ducendus; atque iterum hunc productum per <lb></lb>numerum omnium dentium antecedentis rotulæ multiplicare <lb></lb>oportet, ut omnium paſſuum, aut converſionum rotæ currûs, nu<lb></lb>merus innoteſcat. </s> <s id="s.005877">Quare artificis induſtria in hoc requiritur, <lb></lb>ut rotularum dentibus eos numeros ſtatuat, quorum rationem <lb></lb>inire non ſit nimis operoſum. </s> </p> <p type="main"> <s id="s.005878">Illud autem, commodum-ne dixeris? </s> <s id="s.005879">an incommodum? </s> <s id="s.005880">in <lb></lb>cochlearum infinitarum complexione contingit neceſſariò, <lb></lb>quod axes ſunt in planis invicem rectis, ac proinde indices non <lb></lb>in eâdem loculamenti facie conſtitui poſſunt: cum enim unuſ<lb></lb>quiſque axis ad planum ſui tympani dentati, cui infigitur, ſit <lb></lb>rectus, ipſum verò tympanum ſit in eodem plano, in quo eſt <lb></lb>cochlea infinita, à qua convertitur, manifeſtum eſt plana ipſa, <lb></lb>in quibus ſunt axes, eſſe invicem recta, atque idcirco non ad <lb></lb>eandem loculamenti faciem pertinere eorum indices. </s> </p> <p type="main"> <s id="s.005881"><emph type="center"></emph>PROPOSITIO IV.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005882"><emph type="center"></emph><emph type="italics"></emph>Lunæ motum & phaſes in automato indicare.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005883">QUæ communiter parantur automata horas indicantia, in<lb></lb>dicem habent horis duodecim perficientem integrum cir<lb></lb>cuitum: quapropter lunæ motum, ejuſque ætatem ob oculos <lb></lb>ponere cupiens, ſatis erit, ſi axem, cui horarum index inſeri<lb></lb>tur, in cochleam infinitam deformaveris, quæ convertat tym<lb></lb>panum in dentes 59 diſtributum; axis enim tympani indicem <lb></lb>convertens ætatem lunæ commonſtrabit in dexterâ, aut in ſi<lb></lb>niſtrâ facie loculamenti, cui automatum includitur. </s> <s id="s.005884">Cum <lb></lb>enim lunaris menſis Synodicus complectatur dies 29 1/2, index <lb></lb>autem horarum ſemiſſem diei perficiat, erunt indicis hujus <lb></lb>converſiones 59, dum ſemel index lunæ ſuam converſionem <lb></lb>abſolvit. </s> <s id="s.005885">Si igitur index lunæ ſit lamina rotundum habens fo<lb></lb>ramen propè indicis lingulam, per quod appareat pictus in ſub<lb></lb>jectâ facie circulus centrum habens extra indicis centrum, adeò <lb></lb>ut primâ die lunæ nihil illius circuli appareat, & die decima-<pb pagenum="798" xlink:href="017/01/814.jpg"></pb>quintâ foramen integrum exhibeat ejuſdem circuli colorem, <lb></lb>lunæ Phaſes à foramine, & ejus ætas à lingulâ indicabuntur. </s> </p> <p type="main"> <s id="s.005886">Quod ſi placuerit in eádem facie, in qua deſcriptæ ſunt ho<lb></lb>ræ, etiam lunæ phaſes & motum apparerere, oportebit axi in<lb></lb>dicis horarum aptatam rotulam denticulos habere ad perpendi<lb></lb>culum infixos, qui curriculum, ſeu Vertebram ſtriatam con<lb></lb>vertant, ita ut vertebræ hujuſmodi una converſio planè iſo<lb></lb>chrona ſit uni converſioni indicis horarum. </s> <s id="s.005887">Curriculi autem <lb></lb>axis in cochleam infinitam deformatus convertat tympanum in <lb></lb>dentes 59 diſtinctum, quod collocetur faciei loculamenti pa<lb></lb>rallelum; hujus ſiquidem converſio in eâdem loculamenti fa<lb></lb>cie, in qua & horæ indicantur, repræſentabit lunæ phaſes. </s> </p> <p type="main"> <s id="s.005888">At ſi fortaſſe volueris in eâdem Automati facie ita apparere <lb></lb>horas & lunæ ætatem, ut proximè ſaltem indicetur, quotâ ho<lb></lb>râ accidat Novilunium aut Plenilunium, poſtquam ſemel jux<lb></lb>ta Ephemerides conciliaveris indices horarum & lunæ; non ſa<lb></lb>tis erit in dentes 59 diſtinxiſſe tympanum, cujus ſinguli dentes <lb></lb>horis 12 promoveantur; ſiquidem menſis lunaris Synodicus <lb></lb>complectitur dies 29, horas 12, minuta 44, hoc eſt ferè tres <lb></lb>horæ quadrantes; atque adeò poſt duos menſes index lunæ in<lb></lb>dicaret Novilunium ſeſquihorâ citiùs, quàm par fuerit, & poſt <lb></lb>annum index anteverteret verum Novilunium novem horis. </s> <lb></lb> <s id="s.005889">Quare axi horas indicanti non eſſet copulandus axis cochleæ <lb></lb>infinitæ, cujus tympanum aliam exigeret dentium multitudi<lb></lb>nem; ſed peculiaris axis ſtatuendus eſſet, cujus converſio ita <lb></lb>temperaretur, ut horis undecim cum quadrante abſolveretur; <lb></lb>tympanum verò, ex cujus converſione convolveretur index lu<lb></lb>næ, diſtribuendum eſſet in dentes 63; hujus enim unica con<lb></lb>verſio reſponderet converſionibus 63 axis, cujus ſingulæ con<lb></lb>verſiones perficerentur horis 11 1/4: quapropter index lunæ <lb></lb>ſuam converſionem abſolveret horis 708 3/4, hoc eſt diebus 29, <lb></lb>horis 12, minutis 45. Eſſet igitur in ſingulis lunationibus pau<lb></lb>lò tardior non niſi uno minuto; ſed demum abſolutis duode<lb></lb>cim lunationibus exiguum eſſet diſcrimen. </s> <s id="s.005890">Quod ſi rotulæ ho<lb></lb>ras indicantis faciem interiorem in partes 16 diſtinxeris, & <lb></lb>denticulos ad perpendiculum erexeris, qui Curriculum con<lb></lb>vertant, ita tamen, ut curriculus unâ converſione excipiat ſo<lb></lb>lùm quindecim denticulos, utique una curriculi converſio <pb pagenum="799" xlink:href="017/01/815.jpg"></pb>perficietur horis 11 1/4, hoc eſt (15/16) horarum duodecim, ſeu hora<lb></lb>rum quadrantibus 45; qui per 63 multiplicati dant horæ qua<lb></lb>drantes 2835, quot una lunatio complectitur. </s> </p> <p type="main"> <s id="s.005891"><emph type="center"></emph>PROPOSITIO V.<emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005892"><emph type="center"></emph><emph type="italics"></emph>Pancratium ad onera Vecte attollenda opportunum conſtruere.<emph.end type="italics"></emph.end><emph.end type="center"></emph.end></s> </p> <p type="main"> <s id="s.005893">SÆpè contingit Vecte ſecundi generis attollendum eſſe ali<lb></lb>quod onus, cui impar ſit potentia: idcirco præſtò eſſe po<lb></lb>teſt inſtrumentum (cui Pancratio nomen fieri poſſe oſtendit <lb></lb>vis ſatis magna) plures in alios uſus accommodatum, quod & <lb></lb>facillimè quocumque in loco collocari valet, & quocumque <lb></lb>transferri. </s> <s id="s.005894">Cochlea infinita cum ſuo tympano dentato con<lb></lb>gruente paretur: tympani axis ſit excavatus in tres aut quatuor <lb></lb>ſtrias convenientes dentibus laminæ rectæ chalybeæ dentatæ <lb></lb>ſatis ſolidæ, cujuſmodi illa eſt, quam lib. 5. cap. 6. exhibui. </s> <lb></lb> <s id="s.005895">Nam ſi hæc includantur capſulæ paulò longiori, quàm ſit la<lb></lb>mina illa dentata, & cochleæ axis extra loculamentum promi<lb></lb>neat, ut ei aptari poſſit manubrium; ex Cochleæ converſione <lb></lb>volvitur tympanum, & unà cum illo ejuſdem axis ſtriatus, qui <lb></lb>dentes laminæ chalybeæ ſubiens illam elevat. </s> <s id="s.005896">Et quoniam hu<lb></lb>jus laminæ caput ſinuatum ſubjicitur vecti, etiam vectis attol<lb></lb>litur, & cum eo pondus. </s> <s id="s.005897">Quanta ſit cochleæ infinitæ cum ſuo <lb></lb>tympano & axe vis ad elevandam laminam, conſtat ex dictis: <lb></lb>Componenda eſt autem hæc Ratio cum Ratione Vectis, ut ha<lb></lb>beatur momentum Potentiæ manubrio applicatæ comparatæ <lb></lb>cum onere. </s> </p> <p type="main"> <s id="s.005898"><emph type="center"></emph>FINIS.<emph.end type="center"></emph.end></s> </p> </chap> </body> <back></back> </text> </archimedes>