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New Special Instructions
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 30 Jul 2014 15:58:21 +0200 |
| parents | 22d6a63640c6 |
| children |
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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd"> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink"> <info> <author>Casati, Paolo</author> <title>Mechanica</title> <date>1684</date> <place>Lyon</place> <translator/> <lang>la</lang> <cvs_file>casat_mecha_017_la_1684.xml</cvs_file> <cvs_version/> <locator>017.xml</locator> </info> <text> <front> </front> <body> <chap> <pb xlink:href="017/01/001.jpg"/> <p type="head"> <s id="s.000001"><emph type="center"/>R. P. PAULI <lb/>CASATI <lb/>MECHANICA.<emph.end type="center"/></s> </p> <pb xlink:href="017/01/002.jpg"/> <pb xlink:href="017/01/003.jpg"/> <p type="head"> <s id="s.000002"><emph type="center"/>R. P. PAULI <lb/>CASATI <lb/>PLACENTINI <lb/>SOCIET. JESU <lb/>MECHANICORUM <lb/>LIBRI OCTO, <lb/>IN QUIBUS UNO EODEMQUE<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000003"><emph type="center"/>principio Vectis vires Phy&longs;icè explicantur & Geometricè <lb/>demon&longs;trantur,<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000004"><emph type="center"/><emph type="italics"/>Atque Machinarum omnis generis componendarum methodus <lb/>proponitur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <figure id="id.017.01.003.1.jpg" xlink:href="017/01/003/1.jpg"/> <p type="head"> <s id="s.000005"><emph type="center"/>LUGDUNI, <lb/>Apud ANISSONIOS, JOAN, POSUEL <lb/>& CLAUDIUM RIGAUD.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000006"><emph type="center"/><emph type="italics"/>M. </s> <s id="s.000007">D C. LXXXIV.<emph.end type="italics"/><lb/>CUM PRIVILEGIO REGIS.<emph.end type="center"/></s> </p> <pb xlink:href="017/01/004.jpg"/> <figure id="id.017.01.004.1.jpg" xlink:href="017/01/004/1.jpg"/> <pb xlink:href="017/01/005.jpg"/> <p type="head"> <s id="s.000008"><emph type="center"/>CHRISTIANISSIMO <lb/>GALLIARUM <lb/>ET NAVARRÆ REGI <lb/>LUDOVICO <lb/>MAGNO.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.000009"><emph type="italics"/>AD Maje&longs;tatis Tuæ pedes,<emph.end type="italics"/><lb/>REX INVICTISSIME, <lb/><emph type="italics"/>me, meámque hanc de rebus <lb/>Mechanicis lucubrationem, <lb/>ignotus homo, vix forta&longs;&longs;e cre­<lb/>dibili confidentiâ, &longs;i&longs;to: Sed <lb/>quâ Regiâ comitate omnium <lb/>animos concilias, eâdem &longs;u&longs;tentor, ne repul&longs;am <lb/>timeam. </s> <s id="s.000010">In Te Orbis univer&longs;i conjecti &longs;unt oculi, <lb/>quos Tuæ Gloriæ &longs;plendor allicit: à communi feli-<emph.end type="italics"/><pb xlink:href="017/01/006.jpg"/><emph type="italics"/>citate quid me paterer excludi? </s> <s id="s.000011">Amplißima <lb/>Tua in Societatem no&longs;tram merita, quorum nullam <lb/>partem, ne cogitandâ quidem gratiâ, con&longs;equi <lb/>po&longs;&longs;umus, hoc &longs;altem officij ab univer&longs;o Ordine re­<lb/>petunt, ut &longs;inguli, quem cordi penitißimè impre&longs;&longs;um <lb/>ge&longs;tamus non ingrati LVDOVICVM, in <lb/>libris palàm in&longs;criptum velimus. </s> <s id="s.000012">Me verò Natu­<lb/>ræ atque Artis mutuam &longs;ocietatem coëuntium in <lb/>Machinis, ferè dixerim, miracula contemplari <lb/>a&longs;&longs;uetum rapuere ad mirabundum, quæ ip&longs;e patra&longs;ti, <lb/>& bello, & pace, egregia atque præclara facinora <lb/>non modò mirabilia, &longs;ed prodigiis &longs;imilia. </s> <s id="s.000013">Neque <lb/>illa quidem aut ex rerum magnitudine ac difficul­<lb/>tate, aut ex multiplicato numero, aut ex di&longs;simi­<lb/>lium varietate, aut ex &longs;erie non interruptâ, me­<lb/>tienda duxi, quamquam & in his admirabilitatis <lb/>plurimum in&longs;it: Verùm longè omnem admirationem <lb/>multúmque &longs;uperare mihi videtur, quòd paucis <lb/>lu&longs;tris vel &longs;æcula complexus, unus pluribus Regibus <lb/>par, tot, tantáque perficere valui&longs;ti. </s> <s id="s.000014">Ingentis pon­<lb/>deris gravitatem vincit adhibita Machina, &longs;ed <lb/>diuturno impul&longs;u agitanda, ut proficiat aliquid: At <lb/>plurima immen&longs;is munita difficultatibus exiguo tem­<lb/>poris &longs;patio expugnare, atque ad optatum exitum <lb/>perducere, ita Tuum e&longs;t, REX INVICTISSIME, <lb/>ut quemadmodum rerum ge&longs;tarum gloriâ, ac nomi­<lb/>nis celebritate, nemini &longs;uperiorum Regum &longs;ecundus <emph.end type="italics"/><pb xlink:href="017/01/007.jpg"/><emph type="italics"/>prædicaris, &longs;ic Tibi &longs;ecundum, qui Tuis planè in­<lb/>&longs;i&longs;tat ve&longs;tigiis, ventura &longs;æcula &longs;perare vix audeant. </s> <lb/> <s id="s.000015">Patere igitur pro &longs;ummâ, quâ præditus es, huma­<lb/>nitate, qualemcumque hanc rerum Mechanicarum <lb/>tractationem Regio in&longs;igniri Nomine, ut, quos <lb/>meas ha&longs;ce commentationes legere non piguerit, <lb/>vel hinc di&longs;cant, aliud e&longs;&longs;e non imitabile genus <lb/>Facultatis, quâ ingentia citò perficiantur, &longs;i <lb/>LVDOVICI MAGNI mens acce&longs;&longs;erit. </s> <lb/> <s id="s.000016">Incolumem Te diu &longs;ervet DEVS Catholicæ Fi­<lb/>dei incremento, Regníque Tui felicitati; audiát­<lb/>que bonorum omnium Largitor vota, quæ pro Ma­<lb/>je&longs;tate Tuâ &longs;upplex nuncupat<emph.end type="italics"/></s> </p> <p type="head"> <s id="s.000017"><emph type="center"/><emph type="italics"/>MAJESTATIS Tuæ<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000018">Parmæ Kal, Maij 1683. </s> </p> <p type="main"> <s id="s.000019">Humillimus atque Ob&longs;equenti&longs;&longs;imus <lb/>Servus <lb/>PAULUS CASATUS è SOC. JESU. <pb xlink:href="017/01/008.jpg"/><!-- REMOVE S--><emph type="center"/><emph type="italics"/>Facultas R. P. <!-- REMOVE S-->Provincialis Societatis Je&longs;u <lb/>in Provincia Veneta.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000020">EGo Octavius Rubeus Societatis Je&longs;u in Provincia Veneta <lb/>Præpo&longs;itus Provincialis, pote&longs;tate ad id mihi factâ ab <lb/>Adm. </s> <s id="s.000021">R. P. N. <!-- KEEP S--></s> <s id="s.000022">Præpo&longs;ito Generali Jo. <!-- KEEP S--></s> <s id="s.000023">Paulo Oliva, faculta­<lb/>tem facio, ut Opus in&longs;criptum, <emph type="italics"/>Mechanichorum Libri octo, <lb/>Authore P. <!-- REMOVE S-->Paulo Ca&longs;ato Societatis No&longs;træ Sacerdote,<emph.end type="italics"/> eju&longs;dem <lb/>Societatis Doctorum hominum judicio approbatum, typis <lb/>mandetur, &longs;i ita iis, ad quos pertinet, videbitur. </s> <s id="s.000024">Cujus rei <lb/>gratiâ has litteras meâ manu &longs;ub&longs;criptas, & &longs;igillo officij mei <lb/>munitas dedi. </s> <s id="s.000025">Parmæ 23. Februarij 1681. </s> </p> <p type="main"> <s id="s.000026">OCTAVIUS RUBEUS. <lb/></s> </p> <p type="main"> <s id="s.000027"><emph type="center"/><emph type="italics"/>Summa Privilegy à Chri&longs;tiani&longs;&longs;imo Rege conce&longs;&longs;i.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000028">LUDOVICUS MAGNUS Galliarum & Navarræ Rex Chri&longs;tiani&longs;&longs;imus, <lb/>Diplomate &longs;uo &longs;anxit, nequis per univer&longs;os Regnorum &longs;uorum fines <lb/>intra decem proximos annos à die publicationis exemplarium computandos, <lb/>imprimat &longs;eu typis excudendum curet & venale habeat Opus quod in&longs;cribi­<lb/>tur, <emph type="italics"/>Mechanicorum Libri octo, Authore R. P. <!-- REMOVE S-->Paulo Ca&longs;ato Soc. <!-- KEEP S--></s> <s id="s.000029">Ie&longs;u<emph.end type="italics"/>; præter <lb/>Ani&longs;&longs;onios Bibliopolas Lugdunen&longs;es, aut illos quibus ip&longs;imet conce&longs;&longs;erint. </s> <lb/> <s id="s.000030">Prohibuit in&longs;uper eadem auctoritate Regia omnibus &longs;uis &longs;ubditis, idem <lb/>Opus extra Regni &longs;ui limites imprimendum curare, & impre&longs;&longs;um divende­<lb/>re, vel quempiam ubicumque fuerit ad id agendum impellere; ac in&longs;tigare <lb/>&longs;ine con&longs;en&longs;u dictorum ANISSONIORUM; Qui &longs;ecus faxit, confi&longs;ca­<lb/>tione librorum, aliaque gravi pœnâ multabitur, uti latius patet in diplo­<lb/>mate regio. </s> <s id="s.000031">Dabatur Ver&longs;alis die vige&longs;ima prima Januarij anno Dom. <!-- REMOVE S-->1684. </s> </p> <p type="main"> <s id="s.000032"><emph type="italics"/>Ex mandato Regis.<emph.end type="italics"/></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"/> <figure id="id.017.01.009.1.jpg" xlink:href="017/01/009/1.jpg"/> <p type="head"> <s id="s.000035"><emph type="center"/>AD LECTOREM.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.000036">SERO in lucem prodit hæc Me­<lb/>chanicorum tractatio, & vix fide <lb/>me abduco, quam dedi, cùm Di&longs;­<lb/>&longs;ertationes de <emph type="italics"/>Terrâ Machinis motâ <emph.end type="italics"/><lb/>qua&longs;i Prodromum emi&longs;i ante plures <lb/>annos: &longs;cilicet à &longs;tudiis tunc ab&longs;tra­<lb/>ctus, utpote alieni juris, & ad mu­<lb/>nera his non affinia tran&longs;latus, mul­<lb/>tam &longs;alutem & Mathematicis di&longs;ciplinis & Phy&longs;icis dicere <lb/>coactus &longs;um; adeò ut demum tot elap&longs;is annis urgente jam <lb/>&longs;enio cogitationem omnem abjecerim de huju&longs;modi com­<lb/>mentationibus, diffidens me po&longs;&longs;e ad hanc &longs;criptionem <lb/>&longs;atis temporis invenire, quin eam proxima mors interci­<lb/>peret, & &longs;u&longs;ceptum alieni&longs;&longs;imo tempore laborem irritum <lb/>faceret. </s> <s id="s.000037">Adde quòd (pro meâ negligentiâ, quæ calamo <lb/>parcit) temporis diuturnitate deletæ ex animo pleræque <lb/>imagines vix tenue ve&longs;tigium reliquerant, cui novis indu­<lb/>ctis coloribus eas redintegrari po&longs;&longs;e confiderem. </s> <s id="s.000038">Amico­<lb/>rum tamen officio&longs;is &longs;timulis me urgeri pa&longs;&longs;us &longs;um, ut &longs;ub­<lb/>ci&longs;ivis, quæ incurrebant, temporibus tentarem, an de&longs;ti­<lb/>natam animo tractationem, cujus brevem Synop&longs;im au­<lb/>ditoribus meis in Romano Collegio, anno labentis &longs;æculi <lb/>decimi &longs;eptimi quinquage&longs;imo quarto, tradideram, re­<lb/>dordiri, & aliquâ ratione perficere liceret. </s> <s id="s.000039">Licuit autem, <lb/>præter &longs;pem, toties dimi&longs;&longs;um calamum re&longs;umere, ut tan-<pb xlink:href="017/01/010.jpg"/>dem de &longs;ingulis Mechanicis Facultatibus aliquid me &longs;crip­<lb/>&longs;i&longs;&longs;e invenerim, quod Mathematicarum di&longs;ciplinarum can­<lb/>didatis profuturum amici cen&longs;uerunt, &longs;i publici juris fieret. </s> <lb/> <s id="s.000040">Quapropter alienæ utilitati &longs;erviendum potiùs fuit, quàm <lb/>meæ voluntati. </s> </p> <p type="main"> <s id="s.000041">Verùm nete moveat, Amice Lector, quòd Mechanici <lb/>in&longs;cribantur libri, cùm tamen aliqua ad Centrobaryca, ali­<lb/>qua ad Statica pertineant. </s> <s id="s.000042">Cùm enim hæc ad pleniorem <lb/>eorum intelligentiam, quæ de Machinis di&longs;putanda erant, <lb/>referantur, nomen à &longs;copo de&longs;umendum fuit: Nec decrat <lb/>ex Ari&longs;totele (&longs;i tamen ip&longs;i tribuenda &longs;it illa tractatio) &longs;uf­<lb/>fragium, qui Mechanicas Quæ&longs;tiones in&longs;crip&longs;it libellum, <lb/>in quo non de &longs;olis Mechanicis facultatibus agitur. </s> </p> <p type="main"> <s id="s.000043">Methodum ne culpes, quòd non in Theoremata & <lb/>Propo&longs;itiones rem totam dige&longs;&longs;erim, &longs;ed in Capita di&longs;tri­<lb/>buerim, & quidem aliquando longiu&longs;cula: Brevitati nimi­<lb/>rum &longs;tudens non amavi codicem titulis implere, ne fortè, <lb/>ad o&longs;tendendam con&longs;equentium cum præcedentibus con­<lb/>nexionem, cogerer idem &longs;æpiùs inculcare. </s> <s id="s.000044">Facilius au­<lb/>tem duxi ea, quæ conjuncta &longs;unt, uno eodemque ca­<lb/>pite complecti, ut ex ipsâ verborum con&longs;ecutione re­<lb/>rum cognatio innote&longs;cat. </s> <s id="s.000045">Præterquam quod, &longs;i formâ <lb/>illâ Mathematicis familiari u&longs;us fui&longs;&longs;em, animum forta&longs;&longs;e <lb/>induxi&longs;&longs;es, me mihi ineptè blandiri, & qua&longs;i Geometri­<lb/>cas ratiocinationes obtrudere ea, quæ &longs;atis probabili con­<lb/>jecturâ &longs;tabilire conatus &longs;um. <!-- KEEP S--></s> <s id="s.000046">Quamvis enim non pauca <lb/>attulerim, quæ Geometricas demon&longs;trationes recipiunt, <lb/>nec mihi videar p&longs;eudographis &longs;yllogi&longs;mis deceptus; quia <lb/>tamen & apud Phy&longs;icos & apud Mathematicos agenda <lb/>erat cau&longs;a, multa fuere ad Philo&longs;ophicas rationes revocan­<lb/>da; & quidem, quoad ejus fieri potuit, à receptis in &longs;cho-<pb xlink:href="017/01/011.jpg"/>lis opinionibus mihi non erat hìc recedendum, ne quid <lb/>temerè &longs;ine argumentis proferrem, aut ne longiùs ab in­<lb/>&longs;tituto recederem, &longs;i quid novi, quæ&longs;itâ veri &longs;imilitudine, <lb/>molirer. </s> <s id="s.000047">Hoc videlicet mihi poti&longs;&longs;imum curæ fuit, ut Phy­<lb/>&longs;icam admirandorum per Machinas motuum cau&longs;am in­<lb/>ve&longs;tigarem: in Phy&longs;icis autem modum &longs;ciendi Geome­<lb/>tricum inquirens, ne ab Ari&longs;totele redarguerer, timerem. </s> <lb/> <s id="s.000048">Quare alia Geometricè, alia Phy&longs;icè tractata æquo animo <lb/>patere. </s> </p> <p type="main"> <s id="s.000049">Stylum autem quid excu&longs;em? </s> <s id="s.000050">Non e&longs;t, fateor, con­<lb/>&longs;tans & perpetuus, &longs;uíque &longs;imilis: tum quia non eadem <lb/>&longs;emper &longs;ubjecta materia e&longs;t, tum quia, prout tempus fe­<lb/>rebat, animum inæqualiter affectum ad &longs;cribendum at­<lb/>tuli; nec poterat æquabiliter fluere toties interci&longs;a oratio. </s> </p> <p type="main"> <s id="s.000051">Unum e&longs;t inter cætera, quod forta&longs;&longs;e de&longs;ideres, nimi­<lb/>rum illorum, qui de hoc eodem argumento &longs;crip&longs;erunt, <lb/>&longs;ententias explicari, & quæ à me dicuntur, eorum autho­<lb/>ritate muniri. </s> <s id="s.000052">Plurimum &longs;anè mihi lucis afful&longs;i&longs;&longs;et ex do­<lb/>ctorum virorum Commentariis, neque contemnenda or­<lb/>namenti acce&longs;&longs;io hujus meæ lucubrationis tenuitati fieret ex <lb/>diver&longs;is Authorum opinionibus: Verùm ut nunc res&longs;e ha­<lb/>bet, opportunâ librorum &longs;upellectile de&longs;titutus authorum <lb/>mentionem facere plenam non potui, jejunam non debui, <lb/>ne quis per <expan abbr="contemptũ">contemptum</expan> prætermi&longs;&longs;us videretur. </s> <s id="s.000053">Mihi autem <lb/>non ea e&longs;t memoriæ firmitas, quæ, quid aliquando lege­<lb/>rim, aut ubi legerim, &longs;atis explicatâ recordatione &longs;uggerat. </s> <lb/> <s id="s.000054">Quòd &longs;i placui&longs;&longs;et, corrogatis aliunde libris, magnificam <lb/>hanc eruditionis pompam meæ qualicumque commenta­<lb/>tioni adhibere, non &longs;atis otii ad legendum &longs;uppetebat, & <lb/>nimium temporis po&longs;tula&longs;&longs;et &longs;criptio, &longs;i exponendæ pri­<lb/>mùm, dein confirmandæ aut refellendæ fui&longs;&longs;ent aliorum <pb xlink:href="017/01/012.jpg"/>&longs;ententiæ: propterea &longs;atius duxi, quæ animo occurrebant, <lb/>pro meâ con&longs;uetudine breviter &longs;implicitérque &longs;cribere, <lb/>vix aliquando tactâ alicujus Authoris opinione, quam in <lb/>adver&longs;ariis 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/>quæ pro tuâ humanitate mihi condones, plura quæ ama­<lb/>nuen&longs;i, plurima forta&longs;&longs;e quæ Typographo, ubi præ&longs;ertim <lb/>de Numeris, & de Majori aut Minori Ratione &longs;ermo e&longs;t; <lb/>facilis enim contingit o&longs;citanti hallucinatio, ut ab Auto­<lb/>grapho aberret exemplar, & Numerus numero, verbum <lb/>verbo commutetur: Non ægrè tamen ex adjunctis peti <lb/>poterit correctio. </s> <s id="s.000057">In iis verò, in quibus à me per impru­<lb/>dentiam peccatum fuerit, à tuâ Sapientiâ facilè patiar me <lb/>dedoceri. </s> <s id="s.000058">Vale. <!-- KEEP S--></s> </p> <figure id="id.017.01.012.1.jpg" xlink:href="017/01/012/1.jpg"/> <p type="head"> <s id="s.000059">ELENCHUS </s> </p> <pb xlink:href="017/01/013.jpg"/> <figure id="id.017.01.013.1.jpg" xlink:href="017/01/013/1.jpg"/> <p type="head"> <s id="s.000060"><emph type="center"/>ELENCHUS CAPITUM.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="head"> <s id="s.000061"><emph type="center"/>LIBER PRIMUS. </s> <s id="s.000062">De Centro Gravitatis.<emph.end type="center"/><!-- KEEP S--></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>QVid &longs;it Centrum Gravium & Levium.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>An corpora prædita &longs;int gravitate & levitate.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>Quid &longs;it Centrum Gravitatis, & Linea Directionis.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>An gravia centro vicina minùs gravitent.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>Qua ratione Centrum gravitatis corporum inveniatur.<emph.end type="italics"/></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"/>Affertur ratio prædictarum praxeon.<emph.end type="italics"/></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"/>Quomodo gravia &longs;ponte a&longs;cendentia de&longs;cendant.<emph.end type="italics"/></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"/>Cur gravium in plano inclinato de&longs;cendentium alia repant, alia rotentur.<emph.end type="italics"/></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"/>Cur turres inclinatæ non corruant.<emph.end type="italics"/></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"/>An plurium &longs;tructurarum capax &longs;it Mons, quàm &longs;ubjecta planities.<emph.end type="italics"/></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"/>Quomodo animalium motus ordinentur ex centro gravitatis.<emph.end type="italics"/></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"/>An tellus moveatur motu trepidationis.<emph.end type="italics"/></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"/>Qua ratione minuatur gravitatio in plano inclinato.<emph.end type="italics"/></cell> </row> <row> <cell>XIV.</cell> <cell><emph type="italics"/>Qua ratione corpus gravitet in planum inclinatum.<emph.end type="italics"/></cell> </row> <row> <cell>XV.</cell> <cell><emph type="italics"/>Inquiruntur Rationes gravitationis corporum &longs;u&longs;pen&longs;orum.<emph.end type="italics"/></cell> </row> <row> <cell>XVI.</cell> <cell><emph type="italics"/>Tractiones ac elevationes obliquæ expenduntur.<emph.end type="italics"/></cell> </row> </table> <p type="head"> <s id="s.000063"><emph type="center"/>LIBER SECUNDUS. De Cau&longs;is Motûs Machinalis.<emph.end type="center"/></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>QVem ad finem Machinæ in&longs;truantur.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>Impetûs motum proximè efficientis natura explicatur.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>Qua ratione &longs;emel conceptus impetus pereat.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>Qua ratione vis movendi cum impedimentis comparetur.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>In quo Machinarum vires &longs;ita &longs;int.<emph.end type="italics"/></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"/>Quid attendendum &longs;it in Machinæ collocatione, at que materiæ.<emph.end type="italics"/></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"/>Præ&longs;tetne Machinam augere? an componere?<emph.end type="italics"/></cell> </row> <pb xlink:href="017/01/014.jpg"/> <row> <cell>VIII.</cell> <cell><emph type="italics"/>Cur majores rotæ motum juvent præ minoribus.<emph.end type="italics"/></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"/>Quid cylindri & Scytalæ ad faciliorem ponderis motum præ&longs;tent.<emph.end type="italics"/></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"/>Circulorum Concentricorum motus explicatur.<emph.end type="italics"/></cell> </row> </table> <p type="head"> <s id="s.000064"><emph type="center"/>LIBER TERTIUS. De Libra.<emph.end type="center"/></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>LIbræ forma & natura exponitur.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>Libræ inæqualium brachiorum expenditur.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>Quomodo Corporum æquilibria explicentur.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>An, & cur libra ab æquilibrio dimota ad illud redeat.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>An fieri po&longs;&longs;it libra Curva.<emph.end type="italics"/></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"/>Quanam libræ &longs;int omnium exactißimæ.<emph.end type="italics"/></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"/>Libræ dolo&longs;æ vitia reteguntur,<emph.end type="italics"/></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"/>Stateræ Natura & Forma explicatur.<emph.end type="italics"/></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"/>Antiquorum Statera examinatur.<emph.end type="italics"/></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"/>Libræ & Stateræu&longs;us extenditur.<emph.end type="italics"/></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"/>Fundamenta pramittuntur ad explicandum, Cur gravia &longs;u&longs;pen&longs;a modò præponderent, modò æquilibria &longs;int.<emph.end type="italics"/></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"/>Præponderatio & Æquilibritas gravium fune &longs;u&longs;pen&longs;orum con&longs;ideratur.<emph.end type="italics"/></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"/>An aliqua &longs;it Libræ Obliquæ utilitas.<emph.end type="italics"/></cell> </row> </table> <p type="head"> <s id="s.000065"><emph type="center"/>LIBER QUARTUS. De Vecte.<emph.end type="center"/></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>VEctis forma & vires explicantur.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>Quid in hypomochlij collocatione &longs;it ob&longs;ervandum.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>Quaratione &longs;tatuendus &longs;it Ponderi locus in Vecte primi generis.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>Momenta Ponderis in Vecte &longs;eaundi generis con&longs;iderantur.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>Quæ &longs;it Ratio Vectis hypomochlium mobile habentis.<emph.end type="italics"/></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"/>Quanam &longs;int momenta Vectis Pondus fune connexum tra-hentis.<emph.end type="italics"/></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"/>Quid conferat Potentiæ moventis applicatio ad Vectens.<emph.end type="italics"/></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"/>Oneris ex Vecte pendentis momentum inquiritur.<emph.end type="italics"/></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"/>An duo pondus ge&longs;tantes æqualiter premantur.<emph.end type="italics"/></cell> </row> <pb xlink:href="017/01/015.jpg"/> <row> <cell>X.</cell> <cell><emph type="italics"/>An vis Ela&longs;tica ad aliquod Vectis genus pertineat.<emph.end type="italics"/></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"/>Cur longiora corpora faciliùs flectantur, difficiliùs &longs;u&longs;tineantur.<emph.end type="italics"/></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"/>Vnde oriantur forcipum, & forficum vires.<emph.end type="italics"/></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"/>Cur Tollenones juxta puteos con&longs;tituantur.<emph.end type="italics"/></cell> </row> <row> <cell>XIV.</cell> <cell><emph type="italics"/>Remoram vires in agenda navi expenduntur.<emph.end type="italics"/></cell> </row> <row> <cell>XV.</cell> <cell><emph type="italics"/>Quomodo Naves à Gubernaculo moveantur.<emph.end type="italics"/></cell> </row> <row> <cell>XVI.</cell> <cell><emph type="italics"/>An Malus in motu navis habeat Rationem Vectis.<emph.end type="italics"/></cell> </row> <row> <cell>XVII.</cell> <cell><emph type="italics"/>An ex Rationibus Vectis pendeat u&longs;us Anchoræ.<emph.end type="italics"/></cell> </row> <row> <cell>XVIII.</cell> <cell><emph type="italics"/>Plures Vectis u&longs;us exponuntur.<emph.end type="italics"/></cell> </row> </table> <p type="head"> <s id="s.000066"><emph type="center"/>LIBER QUINTUS. De Axe in Peritrochio.<emph.end type="center"/></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>Axis in Peritrochio forma, & vires de&longs;cribuntur.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>Succulæ & Ergata u&longs;us con&longs;ideratur.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>Tympani à calcante circumacti vires expenduntur.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>An Axis in Peritrochio inveniatur etiam &longs;inè tractione.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>Axium in &longs;uis Peritrochiis Compo&longs;itione vires augentur.<emph.end type="italics"/></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"/>Tympanorum dentatorum u&longs;us. & vires exponuntur.<emph.end type="italics"/></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"/>Moletrinarum artificium ex Axe in Peritrochio pendet.<emph.end type="italics"/></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"/>Axis cum Vecte compo&longs;itus auget Potentiæ momenta.<emph.end type="italics"/></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"/>Multiplex Rotarum dentatarum u&longs;us innuitur.<emph.end type="italics"/></cell> </row> </table> <p type="head"> <s id="s.000067"><emph type="center"/>LIBER SEXTUS. De Trochlea.<emph.end type="center"/></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>TRochlearum forma & vires exponuntur.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>An Trochlea ad Vectem revocanda &longs;it.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>An Orbiculi Magnitudo quicquam conferat.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>Qua Ratione Trochlearum vires augeantur.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>Trochlea Trochleis additæ plurimum augent momenta Po-tentiæ.<emph.end type="italics"/></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"/>Trochlearum ope moveri pote&longs;t pondus velociter.<emph.end type="italics"/></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"/>Quàm validum e&longs;&longs;e oporteat Trochlearum retinaculum.<emph.end type="italics"/></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"/>Aliqui Trochlearum u&longs;us indicantur.<emph.end type="italics"/></cell> </row> </table> <pb xlink:href="017/01/016.jpg"/> <p type="head"> <s id="s.000068"><emph type="center"/>LIBER SEPTIMUS. De Cuneo, & Percu&longs;&longs;ionibus.<emph.end type="center"/></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>CVnei farma & vires explicantur.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>Cunei inflexi v&longs;us ad movendum.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>Cuneus Perpetuns circulo excentrico effingitur.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>Ex Cylindro con&longs;trui pote&longs;t Cuneus Perpetuus.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>Cuneum Perpetuum Circulus inclinatus imitatur.<emph.end type="italics"/></cell> </row> <row> <cell>VI.</cell> <cell><emph type="italics"/>Vnde oriatur vis Percu&longs;&longs;ionis.<emph.end type="italics"/></cell> </row> <row> <cell>VII.</cell> <cell><emph type="italics"/>Quàm di&longs;pares ex motûs velocitate &longs;int Percu&longs;&longs;iones.<emph.end type="italics"/></cell> </row> <row> <cell>VIII.</cell> <cell><emph type="italics"/>An validior &longs;it ictus Malles à Situ Verticali ad Horizonta-lem, an verò ab Horizontali ad Verticalem de&longs;cendentis.<emph.end type="italics"/></cell> </row> <row> <cell>IX.</cell> <cell><emph type="italics"/>Quomodo Percu&longs;&longs;iones ex Mele pendeant.<emph.end type="italics"/></cell> </row> <row> <cell>X.</cell> <cell><emph type="italics"/>Quid conferat re&longs;i&longs;tentia corporis percu&longs;&longs;i.<emph.end type="italics"/></cell> </row> <row> <cell>XI.</cell> <cell><emph type="italics"/>Quomodo ex Percu&longs;&longs;ionibus determinentar Reflexiones.<emph.end type="italics"/></cell> </row> <row> <cell>XII.</cell> <cell><emph type="italics"/>Quomodo Impetus in Percu&longs;&longs;ions communicetur.<emph.end type="italics"/></cell> </row> <row> <cell>XIII.</cell> <cell><emph type="italics"/>Cunei u&longs;us promovetur.<emph.end type="italics"/></cell> </row> </table> <p type="head"> <s id="s.000069"><emph type="center"/>LIBER OCTAVUS. De Cochlea.<emph.end type="center"/></s> </p> <table> <row> <cell>CAP.I.</cell> <cell><emph type="italics"/>COchleæ forma & virtus de&longs;cribitur.<emph.end type="italics"/></cell> </row> <row> <cell>II.</cell> <cell><emph type="italics"/>An utilis &longs;it Cochlea duplex contraria.<emph.end type="italics"/></cell> </row> <row> <cell>III.</cell> <cell><emph type="italics"/>Cochlea cum Vecte, atque cum Axe componitur.<emph.end type="italics"/></cell> </row> <row> <cell>IV.</cell> <cell><emph type="italics"/>Cochleæ Infinitæ vires explicantur.<emph.end type="italics"/></cell> </row> <row> <cell>V.</cell> <cell><emph type="italics"/>Cochlea u&longs;us aliqui indicantur.<emph.end type="italics"/></cell> </row> </table> <pb n="1" xlink:href="017/01/017.jpg"/> <figure id="id.017.01.017.1.jpg" xlink:href="017/01/017/1.jpg"/> <p type="head"> <s id="s.000070"><emph type="center"/>MECHANICORUM<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000071"><emph type="center"/>LIBER PRIMUS.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000072"><emph type="center"/><emph type="italics"/>De Centro Gravitatis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000073">MACHINARUM vires, quibus innatæ corporum in <lb/>motum aut quietem propen&longs;ioni ob&longs;i&longs;timus, explo­<lb/>raturus, præterire non po&longs;&longs;um gravitatem ip&longs;am: <lb/>ne &longs;cilicet ignoretur, quid arte vincendum &longs;it. </s> <s id="s.000074">Ideò <lb/>primum hunc Librum Centro gravitatis tribuen­<lb/>dum cen&longs;ui, cùm plura ex illo pendeant examinanda in po&longs;te­<lb/>rioribus. </s> <s id="s.000075">Neque tamen hîc &longs;ubtili&longs;&longs;imam illam &longs;tatices partem <lb/>per&longs;equar, quæ in corporibus &longs;ingulis gravitatis centrum in­<lb/>ve&longs;tigat: id enim, & abundè ab aliis præ&longs;titum, & mihi in hac <lb/>tractatione minimè nece&longs;&longs;arium; quippe cui &longs;atisfuerit cen­<lb/>trum illud phy&longs;icè per&longs;pectum habere, quatenus præcaven­<lb/>dum e&longs;t, ne alienâ ponderis ad machinam applicatione longè <lb/>alia fiat momentorum ratio, quàm oporteat. </s> <s id="s.000076">Ut autem Centri <lb/>gravitatis notitia clarior habeatur, non inutile ducam quæ&longs;tio­<lb/>nes aliquot ad illud enucleatiùs explicandum pertinentes ad­<lb/>dere, ut ip&longs;is etiam tyronibus fiat &longs;atis: quamquam enim illis <lb/>machinalis &longs;cientia carere po&longs;&longs;e alicui forta&longs;&longs;e videatur, rem <lb/>tamen penitiùs intro&longs;piciens eas extrà mechanicæ con&longs;idera­<lb/>tionis fines po&longs;itas non e&longs;&longs;e cogno&longs;cet.</s> </p> <p type="head"> <s id="s.000077"><emph type="center"/>CAPUT I.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000078"><emph type="center"/><emph type="italics"/>Quid &longs;it Centrum gravium, & levium.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000079">QUoniam hæc rerum univer&longs;itas corpora diver&longs;æ inter &longs;e <lb/>rationis complectitur, eorum ordo aliquis nece&longs;&longs;ariu, fuit <pb n="2" xlink:href="017/01/018.jpg"/>ut &longs;uo unumquodque loco di&longs;poneretur; atque adeò æquum <lb/>fuit, ut &longs;ingulis à natura ea tribueretur facultas, quâ & &longs;e &longs;uo <lb/>in loco, hoc e&longs;t, juxta in&longs;itam propen&longs;ionem &longs;ibi debito, con­<lb/>&longs;ervare po&longs;&longs;int, & ad illum &longs;e ip&longs;a promovere, &longs;i fortè indè <lb/>dimota fuerint. </s> <s id="s.000080">Quia verò æqualia non ni&longs;i æqualiter, &longs;imili­<lb/>que ratione di&longs;ponenda erant, nullum autem corpus præter <lb/>&longs;phæram habet perfectam in partium di&longs;po&longs;itione æqualitatem, <lb/>debuerunt corpora omnia orbem unum con&longs;tituere. </s> <s id="s.000081">At in <lb/>&longs;phæra punctum unum e&longs;t, à quo æqualibus radiis extremæ <lb/>&longs;uperficiei partes removentur: igitur ex ordine ad punctum <lb/>hoc, quod Centrum dicitur, comparanda &longs;unt corpora; qua­<lb/>tenus cùm naturâ impellente moventur, ut in loco &longs;ibi debito, <lb/>à quo per vim &longs;ejuncta fuere, demum con&longs;i&longs;tant, vel ad cen­<lb/>trum hoc accedunt, vel ab eo recedunt. </s> </p> <p type="main"> <s id="s.000082">Et quidem &longs;i ad centrum accedant, gravitare dicuntur, &longs;i <lb/>verò recedant, levitare: & quæ propiora centro con&longs;i&longs;tunt, <lb/>graviora, quæ autem remotiora, leviora quoque cen&longs;entur <lb/>&longs;ecundùm &longs;peciem gravitatis, & levitatis: quicquid &longs;it quod <lb/>æqualia e&longs;&longs;e po&longs;&longs;int &longs;ecundùm gravitatem ab&longs;olutam, aut etiam <lb/>&longs;æpè contingat minus habere gravitatis ab&longs;olutæ id, quod e&longs;t <lb/>gravius &longs;ecundùm &longs;peciem. </s> <s id="s.000083">Sic libra plumbi æqualis e&longs;t libræ <lb/>aquæ, immò minor centum libris aquæ; quia tamen plum­<lb/>bum infra aquam de&longs;cendens fit centro vicinius, etiam gra­<lb/>vius e&longs;t &longs;ecundùm &longs;peciem. </s> <s id="s.000084">Quod &longs;i comparare velis duo cor­<lb/>pora &longs;olida, quæ &longs;ibi &longs;ua duritie ita ob&longs;i&longs;tunt, ut neutrum intra <lb/>alterum moveri po&longs;&longs;it tanquam in medio; illud e&longs;&longs;e &longs;ecundùm <lb/>&longs;peciem gravius affirmabis, quod datâ paritate molis cum alio <lb/>corpore, cum quo comparatur, &longs;taterâ expen&longs;um in eodem <lb/>medio, in quo utrumque gravitat puta in aëre, plus habere <lb/>ponderis deprehendes. </s> <s id="s.000085">Sic aurum e&longs;t ferro gravius in &longs;pecie, <lb/>quia ex æqualibus molibus auri & ferri, aurea e&longs;t pondero&longs;ior. </s> </p> <p type="main"> <s id="s.000086">Generatim autem loquendo ea &longs;unt in &longs;pecie graviora, quæ <lb/>&longs;unt den&longs;iora, ea verò in &longs;pecie leviora, quæ rariora: nam & <lb/>inflata ve&longs;ica ob aërem con&longs;tipatum gravior e&longs;t, quàm flaccida; <lb/>& Æolipilam candentem, aëre intus vi caloris raro, leviorem <lb/>primùm, po&longs;teà, ubi refrixerit, graviorem e&longs;&longs;e experimento <lb/>didicimus, aëre a&longs;&longs;umptam raritatem abjiciente. </s> <s id="s.000087">Cùm enim <lb/>radij à &longs;phæræ centro ad &longs;uperficiem ducti longiùs à &longs;e invi- <pb n="3" xlink:href="017/01/019.jpg"/>cem recedant, æquum fuit, ut quæ plus habent materiæ atque <lb/>&longs;ub&longs;tantiæ &longs;ub minori mole, in angu&longs;tiore &longs;patio collocarentur; <lb/>ea verò, quæ &longs;ub majoribus dimen&longs;ionibus continentur, am­<lb/>pliora &longs;patia occuparent, ubi radij magis di&longs;tant: ut videlicet <lb/>hac ratione æqua &longs;ub&longs;tantiæ di&longs;tributio fieret in totâ &longs;phærâ. </s> <lb/> <s id="s.000088">Hinc vides, cur idem corpus, eo ip&longs;o quod rarum fit, a&longs;cendat, <lb/>ut aqua in vaporem re&longs;oluta (ni&longs;i aliunde ad de&longs;cendendum <lb/>determinetur, ut aurum fulminans) quia materies eadem &longs;ub <lb/>majoribus dimen&longs;ionibus petit longiùs abe&longs;&longs;e à centro, ibiquè <lb/>tanti&longs;per conquie&longs;cit, dum con&longs;tipata, atque minorem in mo­<lb/>lem redacta, iterum de&longs;cendat. </s> </p> <p type="main"> <s id="s.000089">Quare centrum hoc, quod motus, vel quies corporum re&longs;pi­<lb/>cit, dicitur <emph type="italics"/>Centrum gravium, & levium<emph.end type="italics"/>; atque idem creditur <lb/>e&longs;&longs;e cum centro univer&longs;i: vel &longs;altem (ne parùm utili nos di&longs;pu­<lb/>tatione torqueamus) centrum eorum, quæ in hac &longs;phærâ ele­<lb/>mentari gravia, aut levia dicuntur, idem e&longs;t cum centro ter­<lb/>raquei hujus globi, ut quotidiana docet experientia: quicquid <lb/>&longs;it, an pars lunaris globi, &longs;i à lunâ &longs;ejungeretur, reditura e&longs;&longs;et <lb/>ad lunam, ut ad centrum &longs;ui motus. </s> <s id="s.000090">Tam itaquè, quæ huju&longs;mo­<lb/>di centro proxima &longs;unt, deor&longs;um po&longs;ita dicuntur, &longs;ur&longs;um verò, <lb/>quæ ab eo longiùs collocata &longs;unt. </s> <s id="s.000091">Hinc telluris &longs;uperficiei in­<lb/>&longs;i&longs;tentes caput &longs;ur&longs;um, pedes deor&longs;um habere dicimur. </s> <s id="s.000092">Ille <lb/>verò, quamvis rectus, & pedes, & caput &longs;ur&longs;um haberet, cu­<lb/>jus umbilicus huic centro univer&longs;i congrueret. </s> <s id="s.000093">Per quod pa­<lb/>riter centrum &longs;i &longs;cala ducta intelligatur, duo po&longs;&longs;ent &longs;ibi non <lb/>occurrere invicem, licet alter a&longs;cenderet, alter de&longs;cenderet; <lb/>hic &longs;iquidem accederet ad centrum, ille inde recederet: per <lb/>eam verò po&longs;&longs;et uterque a&longs;cendere, & tamen licet, æquali mo­<lb/>tu moverentur, &longs;emper invicem di&longs;tarent magis, quò à centro <lb/>ad oppo&longs;itas partes recederent. <lb/></s> </p> <p type="head"> <s id="s.000094"><emph type="center"/>CAPUT II.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000095"><emph type="center"/><emph type="italics"/>An corpora prædita &longs;int gravitate, & levitate.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000096">INter ea, quæ planè homogenea &longs;unt, ordo e&longs;&longs;e non pote&longs;t à <lb/>naturâ in&longs;titutus: hinc &longs;i nulla e&longs;&longs;et corporum di&longs;&longs;imilitudo, <pb n="4" xlink:href="017/01/020.jpg"/>&longs;ed ex omninò &longs;imilibus &longs;ub&longs;tantiæ partibus totus hic orbis <lb/>conflaretur, nulla quoque e&longs;&longs;et aut gravitas, aut levitas. </s> <s id="s.000097">Quid <lb/>enim hæc potiùs pars, nulla naturæ conditione à cæteris di&longs;cre­<lb/>ta, petat abe&longs;&longs;e à centro, illa verò exigat in eo conquie&longs;cere? </s> <s id="s.000098"><lb/>verùm quia multiplici corporum genere coagmentata rerum <lb/>univer&longs;itas inconcinna e&longs;&longs;e non potuit, &longs;uum cuique locum na­<lb/>tura tribuit, in quo &longs;e &longs;i&longs;teret, ut infra hæc quidem de&longs;cende­<lb/>ret, &longs;uprà illa verò a&longs;cenderet, &longs;i quando &longs;ibi invicem con­<lb/>tigua fierent ordine præpo&longs;tero, nec ullus e&longs;&longs;et motui obex. </s> <lb/> <s id="s.000099">Cùm itaque corpora &longs;ingula in&longs;itam habeant propen&longs;ionem <lb/>(ab Ari&longs;totele dicitur <foreign lang="greek">o(rmh/</foreign>) qua petunt certum locum in uni­<lb/>ver&longs;o; con&longs;tat præter de&longs;cendentium gravitatem dari etiam po­<lb/>&longs;itivam levitatem, quâ corpus aliquod &longs;e ip&longs;um promovet ad <lb/>&longs;uperiores partes univer&longs;i à centro magis di&longs;tantes, neque &longs;o­<lb/>lùm admittendam levitatem negativam, quâ corpora minùs <lb/>gravia cen&longs;entur levia, &longs;i eorum cum gravioribus fiat compa­<lb/>ratio. </s> <s id="s.000100">Nam &longs;i ea, quæ levia dicuntur, eatenus dicas a&longs;cendere, <lb/>quatenus à gravioribus in inferiorem locum de&longs;cendentibus <lb/>propelluntur; mihi æquè liberum erit tollere omnem po&longs;iti­<lb/>vam gravitatem, &longs;olâ levitate admi&longs;sâ; & omnia pariter &longs;ol­<lb/>vam dicendo ea gravia cen&longs;eri, quæ minùs levia &longs;unt, atque <lb/>ideò tantùm de&longs;cendere, quòd extrin&longs;ecùs à levioribus a&longs;cen­<lb/>dentibus loco pul&longs;a detrudantur, non quòd ab internâ faculta­<lb/>te deor&longs;um impellantur. </s> <s id="s.000101">Quod &longs;i vel gravitas de medio tollen­<lb/>da &longs;it, vel levitas, &longs;atius e&longs;t levitatem relinquere; naturâ vi­<lb/>delicet ad altiora &longs;emper, & perfectiora a&longs;pirante, nec adeò <lb/>contendente de infimo loco. </s> <s id="s.000102">Quare cùm per gravitatem &longs;olam <lb/>æquè ac per &longs;olam levitatem motus i&longs;ti explicentur, cætero qui <lb/>autem ingenita &longs;it unicuique corpori &longs;ui loci exigentia; utram­<lb/>que admittere rationi maximè con&longs;entaneum fuerit. </s> </p> <p type="main"> <s id="s.000103">Vitreum globum vacuum, qui in tubulum recurvum de&longs;i­<lb/>nat, quoad fieri pote&longs;t, calefactum, ut inclu&longs;us aer rare&longs;cat, <lb/>Hermeticè claude: tum adjiciatur congruens plumbi gravitas, <lb/>quâ infra aquam deprimatur. </s> <s id="s.000104">Sit autem globus, unà cum ad­<lb/>jecto plumbo, connexus cum exqui&longs;itæ libræ brachio, aut lan­<lb/>ce, ejú&longs;que gravitas intrà aquam exploretur: ubi gravitas in­<lb/>notuerit, adhuc &longs;ub aquâ retineatur globus, &longs;ed longiore for­<lb/>cipe extremum tubuli caput occlu&longs;um frangatur: & animad- <pb n="5" xlink:href="017/01/021.jpg"/>vertes globi vitrei cum appen&longs;o plumbo gravitatem augeri; cu­<lb/>jus incrementum indicabitur ab addito in oppo&longs;itâ lance pon­<lb/>dere ad con&longs;tituendum æquilibrium. </s> <s id="s.000105">Cùm itaque idem maneat <lb/>vitrum, idémque plumbum, & nulla facta &longs;it alicujus gravita­<lb/>tis acce&longs;&longs;io, illud unum &longs;upere&longs;t, quòd aör rarus intrà globum <lb/>conclu&longs;us levior, quàm idem aör, aperto tubulo, &longs;ibi re&longs;titu­<lb/>tus, plus elidit gravitatis plumbi & vitri; atque moles compo­<lb/>&longs;ita ex plumbo, vitro, & aëre raro, &longs;ecundùm &longs;peciem levior <lb/>e&longs;t, quàm moles ex plumbo, vitro, & aëre non raro. </s> <s id="s.000106">Aër igi­<lb/>tur intra aquam ita levis e&longs;t, ut aliquid gravitatis imminuat: <lb/>Nam &longs;i globum eundem ex aquâ extractum, omni aëre exclu­<lb/>&longs;o, aquâ repleveris, & iterum eodem plumbo adjecto eju&longs;dem <lb/>gravitatem intrà aquam examinaveris, illam adhuc majorem <lb/>deprehendes; quia &longs;cilicet nulla levitas aöris ade&longs;t, quæ ali­<lb/>quam deterat gravitatem, &longs;ed illa &longs;olùm perire videtur, quam <lb/>infert di&longs;crimen gravitatum &longs;ecundùm &longs;peciem, ut ex Hy­<lb/>dro&longs;taticis con&longs;tat. </s> <s id="s.000107">Neque &longs;u&longs;piceris hæc gravitatum incre­<lb/>menta oriri ex aquâ &longs;ubeunte per apertum tubulum, cùm aër <lb/>a&longs;&longs;umptam ex calore raritatem abjicit, &longs;e in naturalem &longs;uam <lb/>molem re&longs;tituens, &longs;ivè, aëre pror&longs;us exclu&longs;o, ex aquæ globum <lb/>implentis gravitate. </s> <s id="s.000108">Si enim vitrum aliud aut nullius, aut mo­<lb/>dici&longs;&longs;imæ aquæ capax, &longs;ed eju&longs;dem in aëre ponderis cum a&longs;­<lb/>&longs;umpto globo, &longs;imiliter in aquâ expendas, eandem invenies <lb/>gravitatem, &longs;ive multâ, &longs;ive modicâ aquâ repletum fuerit. </s> <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 &longs;ileam, non leviter &longs;uadere pote&longs;t cor­<lb/>pora &longs;uis nutibus non deor&longs;um tantùm, &longs;ed etiam &longs;ur&longs;um co­<lb/>nari, quod mihi haud ita pridem aliud inve&longs;tiganti contigit <lb/>ob&longs;ervare. </s> <s id="s.000111">Cum enim animadverti&longs;&longs;em aliquando, quàm di&longs;­<lb/>par e&longs;&longs;et gravitas aquæ dimidiam &longs;itulam implentis, &longs;i illa in &longs;u­<lb/>perficie horizontali libraret &longs;e&longs;e, ac quandò &longs;uppo&longs;ita ligneo <lb/>globo firmiter cum &longs;uperiore tigillo cohærenti altiùs ad latera <lb/>a&longs;&longs;urgebat locum globo concedens, quem tamen non &longs;u&longs;tine­<lb/>bat; &longs;ubiit animum cupido tentandi, an bubula ve&longs;ica inflata <lb/>tran&longs;ver&longs;is virgulis infra va&longs;is labra depre&longs;&longs;a ita, ut eam aqua <lb/>circumplecteretur, vim haberet pariter augendi momenta gra­<lb/>vitatis; aquam &longs;iquidem cogebat a&longs;&longs;urgere ad altitudinem ma­<lb/>jorem perpendicularem, ac quandò, ve&longs;icâ liberè innatante, <pb n="6" xlink:href="017/01/022.jpg"/>&longs;ub&longs;idebat. </s> <s id="s.000112">Inveni tamen nullum planè ob&longs;ervari po&longs;&longs;e in <lb/>gravitate di&longs;crimen, quamvis tam ampla e&longs;&longs;et ve&longs;ica, ut facilè <lb/>dimidiam va&longs;is capacitatem impleret: in utroque enim ca&longs;u pon­<lb/>dus fuit lib. 44 1/2. </s> <s id="s.000113">Id mihi, fateor, accidit præter opinionem: <lb/> <figure id="id.017.01.022.1.jpg" xlink:href="017/01/022/1.jpg"/><lb/>Nam &longs;i ex pariete extet tigillus, cui adnectatur <lb/>cylindrus P, aut ve&longs;ica ritè firmata, ferè im­<lb/>plens capacitatem va&longs;is AB, va&longs;que illi &longs;up­<lb/>ponatur ita, ut aqua deinde infu&longs;a po&longs;&longs;it libe­<lb/>rè cylindro circumfundi; percipies onus lon­<lb/>gè majus, quàm pro gravitate aquæ infu&longs;æ, <lb/>&longs;i permitteretur &longs;ub&longs;idere: & &longs;i vas ex &longs;taterâ <lb/>pendeat, adducto reductóve &longs;acomate appa­<lb/>rebunt momenta gravitatis longè majora, quàm &longs;i tota illa <lb/>aqua fundum peteret, & cylindri pars, quæ priùs immerge­<lb/>batur, ab&longs;ci&longs;&longs;a, aut ve&longs;ica innataret. </s> <s id="s.000114">Intelligebam id ex majori <lb/>altitudine perpendiculari aquæ &longs;upra eandem ba&longs;im oriri; nam <lb/>depre&longs;&longs;o va&longs;e ita, ut paulatim cylindus emergat, & aqua &longs;ub­<lb/>&longs;idat, &longs;emper minuitur pondus: idem futurum &longs;perabam, &longs;i <lb/>ve&longs;ica intra aquam non ab extrin&longs;eco obice detineretur, &longs;ed à <lb/>virgulis cum va&longs;e ip&longs;o connexis; quandoquidem aqua ad ean­<lb/>dem pariter altitudinem a&longs;&longs;urgebat &longs;uper ba&longs;im eandem: at <lb/>&longs;pem fefellit eventus. </s> <s id="s.000115">Nec alia mihi &longs;e obtulit probabilior ra­<lb/>tio, quàm ut exi&longs;timarem aquam altiorem vehementius qui­<lb/>dem deor&longs;um niti, ve&longs;icam tamen leviorem altiùs depre&longs;&longs;am, <lb/>conantem &longs;ur&longs;um, æqualiter contendere, ut emergeret; cùm <lb/>verò ni&longs;us i&longs;te &longs;ur&longs;um oppo&longs;itas virgulas, atque adeò vas cum <lb/>illis connexum urgeret, elidi adver&longs;um impetum deor&longs;um, <lb/>qui à majore altitudine perpendiculari aquæ oriebatur, & &longs;o­<lb/>lum remanere conatum ex ip&longs;orum corporum &longs;ub&longs;tantiâ pro­<lb/>manantem, quæ &longs;icut eadem &longs;emper erat, &longs;ivè innataret ve&longs;i­<lb/>ca, &longs;ivè per vim immergeretur, ita eadem obtinebat gravita­<lb/>tis momenta. </s> <s id="s.000116">Quo experimento (quamquam non me lateat, <lb/>quid pro &longs;e afferre hîc po&longs;&longs;ent aliter &longs;entientes) vi&longs;us mihi <lb/>&longs;um deprehendere non ob&longs;curum po&longs;itivæ levitatis ve&longs;ti­<lb/>gium. </s> </p> <p type="main"> <s id="s.000117">Ut autem levitatem corporibus adimendam a&longs;&longs;ererent in­<lb/>genio&longs;i Academici, hoc poti&longs;&longs;imum ducti &longs;unt experimento. </s> <s id="s.000118"> <pb n="7" xlink:href="017/01/023.jpg"/>Ligneum <expan abbr="cylindrũ">cylindrum</expan> ABC <lb/> <figure id="id.017.01.023.1.jpg" xlink:href="017/01/023/1.jpg"/><lb/>plano horizontali D, E, <lb/>perpendicularem &longs;tatue­<lb/>runt; & ut cylindri ba­<lb/>&longs;is &longs;ubjecto plano exactè <lb/>congrueret, laminas duas <lb/>accurati&longs;&longs;imè lævigatas, <lb/>tùm cylindri ba&longs;i, tùm <lb/>&longs;ubjecto plano firmiter <lb/>adnexas voluerunt. </s> <s id="s.000119">Tùm <lb/>ne aër facilè inter utrum­<lb/>que &longs;ubiret, erecto &longs;upra <lb/><expan abbr="planũ">planum</expan> in orbem ex cretâ, <lb/>aut cerâ aggerulo, <expan abbr="argen-tũ">argen­<lb/>tum</expan> vivum infuderunt. </s> <s id="s.000120">Cylindrum extremo libræ jugo G, alligâ­<lb/>runt, addito in oppo&longs;itâ libræ extremitate H pondere L cylin­<lb/>dri pondus adæquante; quod utique cylindrum elevare non po­<lb/>te&longs;t. </s> <s id="s.000121">Additum igitur e&longs;t & aliud pondus M u&longs;que eò, dum cy­<lb/>lindrus à &longs;ubjecto plano avelleretur, & fuit librarum circiter <lb/>trium: quam men&longs;uram arguunt e&longs;&longs;e re&longs;i&longs;tentiæ cylindri con­<lb/>tiguo plano adhærentis metu vacui. </s> <s id="s.000122">His peractis concavum <lb/>vas cylindricum NOP, æqualis aut majoris altitudinis parâ­<lb/>runt, laminâ pariter perpolitâ va&longs;is fundo adnexâ, cui impo­<lb/>&longs;itus fuit cylindrus, adeoque adhæ&longs;it, ut, pleno-mercurij <lb/>va&longs;e, omninò non avelleretur, ut innataret; &longs;ed tunc demum <lb/>argento vivo innatavit, cùm per vim à va&longs;is fundo avul&longs;us e&longs;t <lb/>cylindrus: cui, ut iterum fundum peteret, & argento vivo <lb/>immergeretur, imponendum fuit pondus Q librarum circiter <lb/>quinque. </s> <s id="s.000123">Vis ergò levitatis ligni in mercurio (&longs;i qua levitas <lb/>e&longs;&longs;et) æ&longs;timanda e&longs;&longs;et ut quinque, cùm vis adhæ&longs;ionis metu <lb/>vacui &longs;olùm inventa &longs;it ut tria: debui&longs;&longs;et igitur levitas ita præ­<lb/>valere, ut adhæ&longs;ionem vinceret, & cylindrus &longs;ponte elevaretur. </s> <lb/> <s id="s.000124">Non e&longs;t itaque levitas, quæ ligneum cylindrum innatare cogit, <lb/>&longs;ed mercurij gravitas major ip&longs;a e&longs;t, quæ lignum elevat, cum <lb/>primùm locus patet, in quem de&longs;cendat. </s> </p> <p type="main"> <s id="s.000125">Sed antequam experimentum hoc ad examen revocemus, <lb/>ut innote&longs;cat, quid hinc confici po&longs;&longs;it ad levitatem excluden­<lb/>dam, haud ægrè permi&longs;erim, cùm in abeuntis &longs;uâ &longs;ponte cor- <pb n="8" xlink:href="017/01/024.jpg"/>poris locum corpus aliud &longs;uapte vi, & naturâ &longs;uccedit, ab hoc <lb/>illud urgeri po&longs;&longs;e, ut velociùs moveatur: duo &longs;cilicet corpora <lb/>diver&longs;æ &longs;ecundùm &longs;peciem gravitatis &longs;i fuerint perturbatè di&longs;­<lb/>po&longs;ita intrà medium, in quo utrumque gravitat, nil mirum, &longs;i <lb/>à graviore majori ni&longs;u conante extrudatur minùs grave: id <lb/>quod etiam de duobus levibus dicendum perturbatè di&longs;po&longs;itis <lb/>in medio, ubi utrumque levitat: duobus enim &longs;imul currenti­<lb/>bus, ab eo qui ponè &longs;ub&longs;equitur, &longs;i majoribus viribus polleat, <lb/>priorem urgeri atque impelli palam e&longs;t, quamquam motus uni­<lb/>ver&longs;us impul&longs;ioni tribuendus non &longs;it. </s> <s id="s.000126">Ita quoque a&longs;cendentem <lb/>in mercurio ligneum cylindrum à de&longs;cendente mercurio &longs;ur­<lb/>&longs;um urgeri aliquatenus po&longs;&longs;e non diffitebor, &longs;icut & mercu­<lb/>rium ip&longs;um repugnare, ne &longs;ur&longs;um propellatur, atque ab eodem <lb/>lignum innatans prohiberi, ne de&longs;cendat: hinc tamen non <lb/>&longs;equitur ligni a&longs;cendentis motum, aut innatantis quietem, <lb/>prægravis mercurij viribus omnino ad&longs;cribi jure debere, nam, <lb/>& &longs;ua vis a&longs;cendendi, atque con&longs;i&longs;tendi, ligno ip&longs;i tribuen­<lb/>da e&longs;t. </s> </p> <p type="main"> <s id="s.000127">Quid quòd ip&longs;æ innatantis cylindri portiones, altera quidem <lb/>mercurio immer&longs;a, altera verò extans, levitatem ip&longs;i ligno in­<lb/>&longs;itam declarant? </s> <s id="s.000128">Quid enim partis immer&longs;æ ad extantem (&longs;i <lb/>moles &longs;pectetur) ea ratio e&longs;t, quæ &longs;pecificæ gravitatis ligni ad <lb/>differentiam gravitatum ligni, atque mercurij? </s> <s id="s.000129">ni&longs;i quia por­<lb/>tionis mercurio immer&longs;æ levitas, atque extantis in aëre gravi­<lb/>tas, æquilibritatem con&longs;tituent; quemadmodum in <emph type="italics"/>Terra ma­<lb/>chinis mota differt.<emph.end type="italics"/> 5. <emph type="italics"/>n.<emph.end type="italics"/> 105. explicatum e&longs;t. </s> <s id="s.000130">Hanc porrò æqua­<lb/>litatem Algebricè &longs;ic o&longs;tendo. </s> <s id="s.000131">Ratio gravitatis ligni ad gravi­<lb/>tatem mercurij &longs;it ut S. ad R; differentia e&longs;t R—S. Ponatur <lb/>cylindri pars immer&longs;a. A. </s> <s id="s.000132">Quia igitur ut &longs;pecifica gravitas <lb/>corporis innatantis ad differentiam gravitatum, hoc e&longs;t ut <lb/>S ad R — S, ita pars cylindri immer&longs;a A, ad extantem <lb/>(R in A—S in A/S); Si pars extans in aëre in &longs;uam gravitatem S du­<lb/>catur, pars verò immer&longs;a A in differentiam gravitatum R—S, <lb/>hoc e&longs;t in — R + S, quia e&longs;t deficiens, efficitur hinc quantitas <lb/>R in A — S in A, hinc verò — R in A + S in A, quæ &longs;e invi­<lb/>cem elidunt. </s> <s id="s.000133">Æqualia igitur &longs;unt levitatis, & gravitatis mo­<lb/>menta. </s> <s id="s.000134">Sit enim exempli causâ gravitas ligni ad gravitatem <pb n="9" xlink:href="017/01/025.jpg"/>mercurij, ut S. ad 13. differentia e&longs;t 8. </s> <s id="s.000135">E&longs;t igitur cylindri <lb/>pars immer&longs;a eju&longs;dem (5/13), extans verò (8/13): at portio immer&longs;a de­<lb/>ficit à gravitate mercurij &longs;ecundùm &longs;peciem ut 8; igitur (5/13) in - 8 <lb/>dant (40/13): item partis extantis gravitas in aëre e&longs;t S; igitur (8/13) <lb/>in 5 dant (40/13): confligunt itaque inter &longs;e pari conatu levitas (-40/13) & <lb/>gravitas (40/13), adeóque fit con&longs;i&longs;tentia & innatat lignum. </s> </p> <p type="main"> <s id="s.000136">Sed jam ad propo&longs;iti experimenti examen de&longs;cendamus. </s> <s id="s.000137">Aio <lb/>cylindri re&longs;i&longs;tentiam ex adhæ&longs;ione metu vacui non &longs;atis explo­<lb/>ratam fui&longs;&longs;e per libram; hæc enim dum ex pondere M deor&longs;um <lb/>inclinatur, extremitas G &longs;ur&longs;um elevata arcum de&longs;cribit, ac <lb/>proinde cylindri a&longs;cendentis motus non e&longs;t per lineam horizon­<lb/>tali plano perpendiculariter in&longs;i&longs;tentem, &longs;ed per inclinatam: <lb/>Quare cùm A. versùs I libræ centrum trahatur, cylindri ba&longs;is <lb/>non incipit elevari parallela horizonti, &longs;ed cum inclinatione, ita <lb/>ut C priùs elevetur, quàm B: ea autem, quæ &longs;ibi invicem adhæ­<lb/>re&longs;cunt, multò faciliùs divelli manife&longs;tum e&longs;t, &longs;i id cum inclina­<lb/>tione fiat, quàm &longs;i &longs;ervandus &longs;it paralleli&longs;mus. </s> <s id="s.000138">Adde in hac in­<lb/>clinatione faciliùs adhuc divelli cylindrum à &longs;uppo&longs;ito plano, <lb/>quò longior cylindrus fuerit; habet &longs;cilicet rationem vectis, <lb/>cujus potentia e&longs;t in A, hypomochlion in B, re&longs;i&longs;tentia vin­<lb/>cenda in C. </s> <s id="s.000139">Quare pondus M non aptè metitur re&longs;i&longs;tentiam, <lb/>quæ oritur ex corporum adhære&longs;centiâ, metu vacui, &longs;ed hæc <lb/>multò major e&longs;t, &longs;i ad perpendiculum motus fieri debeat; <lb/>quemadmodum & fieri oporteret, &longs;i in va&longs;e NOP mercurij <lb/>pleno cylindrus fundo adhærens rectâ a&longs;cenderet. </s> <s id="s.000140">Quamvis <lb/>igitur pondus Q librarum quinque admitteretur men&longs;ura levi­<lb/>tatis, non continuò argui pote&longs;t hujus exce&longs;&longs;us &longs;upra re&longs;i&longs;ten­<lb/>tiam adhæ&longs;ionis. </s> <s id="s.000141">Quin immo affirmare au&longs;im, &longs;i libræ loco <lb/>adhibita fui&longs;&longs;et amplior trochlea, & ex funiculo ejus orbitam <lb/><expan abbr="cõplectente">complectente</expan> hinc cylindrus A, hinc verò pondus M ad perpen­<lb/>diculum pependi&longs;&longs;ent, non &longs;atis futurum fui&longs;&longs;e <expan abbr="põdus">pondus</expan> librarum <lb/>trium, &longs;ed multò majus adhibendum fui&longs;&longs;e, ut cylindri re&longs;i&longs;ten­<lb/>tiam &longs;uperaret; fui&longs;&longs;et enim avellenda ba&longs;is &longs;ervato paralleli&longs;mo. </s> </p> <p type="main"> <s id="s.000142">Quantum autem virium, ferè &longs;upra fidem, habeat vacui <lb/>horror ad corpora retinenda, &longs;atis apertè declarant gravia, quæ <lb/>&longs;u&longs;penduntur. </s> <s id="s.000143">Ego &longs;anè vidi marmoreum mortarium commu­<lb/>nis magnitudinis &longs;atis vulgari artificio &longs;u&longs;pendi vitreo cyatho: <pb n="10" xlink:href="017/01/026.jpg"/>mortarij &longs;cilicet fundo exteriùs aptata fuerat ma&longs;&longs;a ex farinâ <lb/>ad formandos panes recens macerata, & aquâ ita &longs;ubacta, ut <lb/>illi tenaciter cohæreret: tum vitreo calici injecta &longs;tuppa admo­<lb/>to igne exar&longs;it, applicitu&longs;que calix ma&longs;&longs;æ eam attraxit, &longs;icut & <lb/>medicorum cucurbitulæ carnem attrahunt: quare accepto ca­<lb/>licis vitrei pede facile fuit mortarium elevare, & &longs;u&longs;pendere. </s> <lb/> <s id="s.000144">Quod &longs;i marmoreum mortarium ex metu vacui in aëre pendu­<lb/>lum hæ&longs;it, quid mirum &longs;i & ligneus cylindrus &longs;ubjecto plano <lb/>adhære&longs;cens in mercurio &longs;tetit? </s> </p> <p type="main"> <s id="s.000145">Nondum itaque ex hoc experimento, aut ex &longs;imilibus, ubi <lb/>metu vacui &longs;uos motus moliri corpora non po&longs;&longs;unt, &longs;atis habe­<lb/>mus argumenti, quo levitatem, &longs;olâ gravitate retentâ, expun­<lb/>gamus. </s> <s id="s.000146">Huju&longs;modi e&longs;t illud, ubi in lignei va&longs;is fundo exca­<lb/>vatur &longs;caphium, cui exqui&longs;itè congruat eburneus globus, qui <lb/>&longs;uperinfu&longs;o hydrargyro non a&longs;cendit. </s> <s id="s.000147">Neque enim ideò non <lb/>a&longs;cendit, quia rima nulla patet argento vivo, per quam &longs;ubiens <lb/>extrudat eburneum globum, &longs;ed quia ita &longs;ibi exqui&longs;itè con­<lb/>gruunt ebur, & lignum, ut vis ip&longs;a a&longs;cendendi vincere non va­<lb/>leat vim adhære&longs;centiæ. </s> <s id="s.000148">Nam & eadem vis in aere &longs;u&longs;pendit <lb/>corpora gravia, ne de&longs;cendant. </s> <s id="s.000149">Quamvis autem non totum <lb/>hemi&longs;phærium globi eburnei, &longs;ed &longs;olùm ejus maximus circu­<lb/>lus congrueret excavato ligno, & cavitas ip&longs;a aëre repleretur, <lb/>non propterea tollitur vis adhære&longs;centiæ illius annularis; quia <lb/>&longs;cilicet vis a&longs;cendendi in hydrargyro tanta non e&longs;t, ut valeat <lb/>inclu&longs;um ibi aërem di&longs;trahere, &longs;icut opus e&longs;&longs;et ad incipiendum <lb/>motum citra periculum vacui, & præterea &longs;uperanda e&longs;t re­<lb/>&longs;i&longs;tentia hydrargyri dividendi; corpora enim in motu divi­<lb/>dunt medium, pro cujus cra&longs;&longs;itudine re&longs;i&longs;tentiam experiuntur. </s> <lb/> <s id="s.000150">Adde hemi&longs;phærium inferius in aëre tanquam in loco po&longs;itum <lb/>gravitare non minùs, quàm hemi&longs;phærium &longs;uperius levitet in <lb/>hydrargyro; proinde nil mirum, &longs;i globus non a&longs;cendat. </s> <s id="s.000151">Quod <lb/>&longs;i aëre exclu&longs;o locum illum impleveris hydrargyro, & ebur­<lb/>neum globum ita foramini aptaveris, ut illi exqui&longs;itè congruat; <lb/>&longs;i in &longs;uperinfu&longs;o hydrargyro globus non a&longs;cendat, indicio e&longs;t <lb/>ita globum e&longs;&longs;e foramini infixum, ut neque valeat elevari à &longs;ub­<lb/>jecto hydrargyro in &longs;caphij formam per vim excavato: neque <lb/>enim facilè mihi per&longs;uadebis &longs;pecificarum gravitatum diffe­<lb/>rentiam exigere, ut hemi&longs;phærium integrum præcisè extet: <pb n="11" xlink:href="017/01/027.jpg"/>præter quam quod &longs;i non valebat &longs;ubjectum aërem di&longs;trahere, <lb/>multò minùs id in hydrargyro præ&longs;tare pote&longs;t, ut vacuum <lb/>evitetur. </s> </p> <p type="main"> <s id="s.000152">At, inquis, fi&longs;tulam quadricubitalem &longs;piritu vini plenam <lb/>cum globulo innatante &longs;i clau&longs;eris, & inverteris deor&longs;um, <lb/>a&longs;cendet globulus &longs;patio 200 vibrationum perpendiculi; in eâ­<lb/>dem verò fi&longs;tulâ communis, & &longs;implicis aquæ plenâ a&longs;cendet <lb/>&longs;ubduplo tempore 100 vibrationum. </s> <s id="s.000153">Cur hoc? </s> <s id="s.000154">ni&longs;i quia aqua <lb/>ut pote gravior validiùs extrudit globulum, quàm &longs;piritus vini. </s> <lb/> <s id="s.000155">Nihilominus: &longs;i gravia in levibus magis gravitant, & velociùs <lb/>de&longs;cendunt, quò major e&longs;t &longs;pecificarum gravitatum differen­<lb/>tia; vici&longs;&longs;im levia in gravibus magis levitant, & velociùs <lb/>a&longs;cendunt, quò major e&longs;t &longs;ecundùm &longs;peciem levitatis differen­<lb/>tia: Atqui &longs;piritus vini magis accedit ad &longs;pecificam levitatem <lb/>innatantis globuli, aqua autem magis differt; in aquâ igitur <lb/>globulus magis levitat, & velociùs a&longs;cendit, &longs;icut lapis in aëre <lb/>velociùs de&longs;cendit quàm in aqua, aut in melle. </s> </p> <p type="main"> <s id="s.000156">Addis iterum. </s> <s id="s.000157">Vitreo va&longs;culo, cui longior fi&longs;tula adhæreat, <lb/>fomitem cum filo &longs;ulphurato ope fili ferrei ingere, ut vitrum <lb/>tangat: totum imple hydrargyro, & conver&longs;o deor&longs;um o&longs;culo <lb/>de&longs;cendit hydrargyrus; atque &longs;ub&longs;i&longs;tit in altitudine cubiti, & <lb/>quadrantis: admotâ lucernâ vitrarij vitrum calefiat, ut fomes <lb/>cum filo &longs;ulphurato accendatur: fumus de&longs;cendit, nec ni&longs;i <lb/>aperto &longs;uperiore va&longs;is o&longs;culo a&longs;cendit, aëre videlicet &longs;ubeunte, <lb/>à quo extrudatur &longs;ur&longs;um. </s> <s id="s.000158">Nego fumum ab aëre &longs;ur&longs;um extru­<lb/>di, &longs;ed qui gravior &longs;piritu raro mercurij in illo de&longs;cendebat, <lb/>ubi aërem tangit, ut pote levior in illo a&longs;cendit. </s> </p> <p type="main"> <s id="s.000159">Non au&longs;im tamen in lapide, qui gravitatem in aquâ & aëre, <lb/>levitatem in mercurio, aut plumbo liquente obtinet, duplicem <lb/>&longs;tatuere virtutem, quarum altera &longs;ur&longs;um, altera deor&longs;um con­<lb/>nitatur: Cum enim impetus motum efficiens (ut infrà con&longs;ta­<lb/>bit) eju&longs;dem naturæ &longs;it, in quamcunque demum orbis plagam <lb/>dirigatur motus; &longs;atis video ab uno eodemque principio, pro <lb/>variâ contigui corporis conditione, a&longs;cen&longs;um, de&longs;censúmve <lb/>prodire po&longs;&longs;e. </s> <s id="s.000160">Quandoquidem motus, qui in eadem lineâ per­<lb/>ficitur, &longs;imiles planè includit ubicationes &longs;ucce&longs;&longs;ivè acqui&longs;i­<lb/>tas, &longs;ivè a&longs;cen&longs;us &longs;it, &longs;ivè de&longs;cen&longs;us, ordine tantùm in earum <lb/>adeptione, commutato. </s> <s id="s.000161">Quare cum a&longs;cen&longs;us à de&longs;cen&longs;u hoc <pb n="12" xlink:href="017/01/028.jpg"/>uno differat, quòd quam ubicationem lapis demùm obtineret <lb/>po&longs;t alias propè finem motûs, &longs;i fui&longs;&longs;et centro propior quàm <lb/>mercurius, eam acquirat &longs;ub initium motûs ante alias, &longs;i in <lb/>mercurij locum aër aut aqua &longs;urrogetur centro vicinior quàm <lb/>lapis: ad ordinem hunc permutandum non videtur nece&longs;&longs;aria <lb/>virtutis motricis di&longs;&longs;imilitudo; nihil quippe producitur di&longs;&longs;imi­<lb/>le. </s> <s id="s.000162">Sed &longs;i quis &longs;ufficere dicat conditionum varietatem, nihil <lb/>ab&longs;onum fortè loquatur: debuit enim una virtus activa in &longs;ui <lb/>effectus productione non uni tantùm conditioni alligari, &longs;ed pro <lb/>earum varietate modum quoque operandi mutare po&longs;&longs;e, modò <lb/>præ&longs;titutos fines, quoad &longs;ub&longs;tantiam, non tran&longs;iliret. </s> </p> <p type="main"> <s id="s.000163">Neque arbitror hoc tantùm &longs;en&longs;u negatam ab aliquibus levi­<lb/>tatem po&longs;itivam; potui&longs;&longs;ent enim æquè negare gravitatem, ad­<lb/>mi&longs;&longs;a &longs;olùm potentia motrice. </s> <s id="s.000164">Sed &longs;i vis i&longs;ta &longs;e movendi deor­<lb/>&longs;um gravitas po&longs;itiva dicenda e&longs;t, cùm eadem &longs;it virtus &longs;e mo­<lb/>vendi &longs;ursùm, cur levitas po&longs;itiva non fuerit? </s> <s id="s.000165">Qui enim levita­<lb/>tem à gravitate &longs;ejunctam negat, non illicò levitatem expun­<lb/>git: quemadmodum Angelos intelligentiâ aut voluntate dimi­<lb/>nutos non a&longs;&longs;erunt ij, qui vitalium facultatum di&longs;tinctionem <lb/>non agno&longs;cunt. </s> <s id="s.000166">Nullum igitur corpus &longs;impliciter, & ab&longs;olutè <lb/>grave dicendum e&longs;t, ni&longs;i quod cæteris omnibus ita petat &longs;ube&longs;&longs;e, <lb/>ut nequeat raritatem a&longs;&longs;umere, vi cujus evadat levius corpore <lb/>&longs;imili quidem &longs;ecundùm naturam, di&longs;&longs;imilis tamen raritatis: <lb/>nullum &longs;impliciter, & ab&longs;olutè leve, ni&longs;i quod ita exigat extre­<lb/>mam orbis laciniam occupare, ut nunquam con&longs;tipari po&longs;&longs;it, ac <lb/>fieri gravius proximo corpore rariore. </s> <s id="s.000167">Reliqua omnia non ni&longs;i <lb/>comparatè gravia, aut levia dici po&longs;&longs;unt: &longs;ic plumbum grave e&longs;t <lb/>in aëre, grave in aqua, at pariter leve in mercurio, leve &longs;i cum <lb/>auro conferatur. </s> </p> <p type="main"> <s id="s.000168">Hinc corpus in loco &longs;ibi debito con&longs;titutum, sèque ibi con­<lb/>&longs;ervans (extra tamen &longs;phæræ centrum, nec in extimâ orbis ele­<lb/>mentaris &longs;uperficie) ob idip&longs;um, quia ob&longs;i&longs;tit non tantùm, ne <lb/>infra &longs;ubjectum corpus deprimatur, verùm etiam, ne in locum <lb/>&longs;uperioris attollatur, & levitare &longs;imul dicendum e&longs;t, & gravi­<lb/>tare. </s> <s id="s.000169">At &longs;i in alienum locum transferatur, quia in medio levio­<lb/>re ita repugnat a&longs;cen&longs;ui, ut petat de&longs;cendere, &longs;olùm gravitat; <lb/>quia verò in graviore ita depre&longs;&longs;ioni reluctatur, ut exigat ad <lb/>&longs;uperiora evadere, &longs;olùm levitat. </s> <s id="s.000170">Quod &longs;i corpora huju&longs;modi <pb n="13" xlink:href="017/01/029.jpg"/>in actu &longs;ecundo gravitare aut levitare tunc &longs;olùm dixeris, quan­<lb/>do illa in locum non &longs;uum tran&longs;lata aut de&longs;cendere expetunt, <lb/>aut a&longs;cendere, vel re etiam ipsâ de&longs;cendunt, aut a&longs;cendunt, <lb/>non admodum repugnabo; modò conatum illum, quo &longs;e &longs;uo <lb/>tutantur in loco, gravitationem, & levitationem &longs;altem in actu <lb/>primo, aut pariter a&longs;&longs;eras, aut pariter neges. </s> </p> <p type="main"> <s id="s.000171">Porrò motus omnis gravium, & levium &longs;icut in vacuo exer­<lb/>ceri non pote&longs;t (ut in <emph type="italics"/>Vacuo Pro&longs;cripto cap.<emph.end type="italics"/>2. <emph type="italics"/>num.<emph.end type="italics"/>9. o&longs;tendi) ita <lb/>in medio fit, vel tardiùs, vel citiùs, tùm pro majori vel minori <lb/>ip&longs;ius medij re&longs;i&longs;tentia ad &longs;ci&longs;&longs;ionem partium magis, vel minùs <lb/>connexarum, tùm comparatâ gravitate &longs;eu levitate mobilis <lb/>cum levitate &longs;eu gravitate medij. </s> <s id="s.000172">Hinc e&longs;t gravibus minus <lb/>re&longs;i&longs;tere leviora, magis verò, quæ minùs levia, cæteris pari­<lb/>bus: &longs;ic aër minùs re&longs;i&longs;tit lapidi cadenti, quàm &longs;i idem lapis in­<lb/>ciperet moveri in aquâ, quæ minùs levis e&longs;t, quàm aër. </s> <lb/> <s id="s.000173">Ex oppo&longs;ito autem levibus graviora minùs re&longs;i&longs;tunt, quæ au­<lb/>tem minùs gravia, magis re&longs;i&longs;tunt: &longs;ic exhalatio ex fundo <lb/>aquæ, in vitreâ phialâ ad ignem expo&longs;itâ, per aquam a&longs;cendit <lb/>velociùs, quàm deinde extra aquam po&longs;ita a&longs;cendat in aëre, <lb/>ubi fumeam naturam induerit. </s> <s id="s.000174">Unde patet non adeò &longs;olidum <lb/>ab aliquibus ex hoc experimento &longs;umi argumentum negandi <lb/>po&longs;itivam levitatem. </s> <s id="s.000175">Quæ enim de gravibus ex comparatione <lb/>cum levibus dicuntur, ea de levibus, proportione &longs;ervatâ, di­<lb/>cenda &longs;unt, &longs;i cum gravibus conferantur. </s> <s id="s.000176">Cur autem gravibus <lb/>leviora, levibus graviora minùs re&longs;i&longs;tant, ratio e&longs;t, quia mo­<lb/>bile movetur in medio propter di&longs;&longs;imilitudinem; nam &longs;i corpus <lb/>contiguum e&longs;&longs;et, &longs;imile non moveretur; quando igitur major <lb/>e&longs;t di&longs;&longs;imilitudo, debet velociùs moveri, &longs;egniùs autem, & len­<lb/>tiùs, quò propiùs abe&longs;t à &longs;imilitudine, donec in &longs;imili demum <lb/>quie&longs;cat. </s> </p> <p type="main"> <s id="s.000177">E&longs;t itaque in corporibus gravitas, & levitas, vi cujus motus ali­<lb/>quos juxta naturæ propen&longs;ionem perficiunt, ut certo denique in <lb/>loco con&longs;i&longs;tant, eju&longs;demque vi re&longs;i&longs;tunt, ne oppo&longs;itis motibus <lb/>cieantur, & à &longs;uæ quietis loco avellantur. </s> <s id="s.000178">Quamvis autem <expan abbr="ead&etilde;">eadem</expan> <lb/>maneat gravitas aut levitas, non <expan abbr="id&etilde;">idem</expan> tamen e&longs;t &longs;emper <expan abbr="momentũ">momentum</expan> <lb/>(Græcis <foreign lang="greek"><gap/>ph</foreign>) hoc e&longs;t actualis ad motum inclinatio, dum in actio­<lb/>ne e&longs;t; hæc enim, ut infra patebit, ut plurimum ex po&longs;itione, & <lb/>&longs;itu mutatur, vel comparatè ad <expan abbr="mediũ">medium</expan>, in quo perficitur motus. <pb n="14" xlink:href="017/01/030.jpg"/> </s> </p> <p type="head"> <s id="s.000179"><emph type="center"/>CAPUT III.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000180"><emph type="center"/><emph type="italics"/>Quid &longs;it centrum gravitatis, & linea directionis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000181">QUamvis non minùs levitate, quàm gravitate prædita &longs;int <lb/>corpora, quia tamen frequentiùs gravitatem vincere co­<lb/>namur, quàm levitatem; ideò illa poti&longs;&longs;imùm cadit &longs;ub con­<lb/>templationem &longs;cie&ngrave;tiæ Machinalis: vix enim aliquando con­<lb/>tingere poterit, ut opus &longs;it infra aquam corpus aliquod leve <lb/>per vim deprimere. </s> <s id="s.000182">Hinc factum e&longs;t, ut de &longs;olo gravitatis cen­<lb/>tro &longs;ermo communiter &longs;it, levitatis autem centrum &longs;ilentio <lb/>obvolvatur: quia nimirùm quæ de gravitate de&longs;cendente ex­<lb/>plicantur, ea de levitate a&longs;cendente, pro rata portione, dicta <lb/>facile intelliguntur. </s> </p> <p type="main"> <s id="s.000183">Ad centrum terræ (quod & centrum gravium ac levium <lb/>dicimus) properant corpora quæcumque gravia in medio le­<lb/>viore con&longs;tituta &longs;ibi redduntur, ut motus &longs;uos perficiant. </s> <s id="s.000184">Quo­<lb/>niam verò natura finem propo&longs;itum per media, quæ pote&longs;t, bre­<lb/>vi&longs;&longs;ima pro&longs;equitur, ambages, & diverticula fugiens; mo­<lb/>ventur per lineam rectam, ut pote brevi&longs;&longs;imam, ni&longs;i externo <lb/>aliquo impedimento cogantur à rectitudine deflectere: Hæc <lb/>autem recta linea intelligi debet ex terræ centro ducta ad cor­<lb/>pus ip&longs;um, quod movetur; ac proinde tùm in &longs;phæricam &longs;u­<lb/> <figure id="id.017.01.030.1.jpg" xlink:href="017/01/030/1.jpg"/><lb/>perficiem, tùm in planum Horizon­<lb/>tis ad perpendiculum cadit. </s> <s id="s.000185">Sed quia <lb/>corpus, quod deor&longs;um contendit, <lb/>plures habet partes, quibus con&longs;tat, <lb/>&longs;ingulas &longs;uâ gravitate præditas, lineæ <lb/>verò à &longs;ingulis hi&longs;ce partibus exeun­<lb/>tes in terræ centro concurrunt; fieri <lb/>non pote&longs;t, ut &longs;ervatâ corporis figu­<lb/>râ, atque continuo partium nexu non <lb/>di&longs;&longs;oluto, per rectam &longs;uam lineam ad <lb/>centrum ductam unaquæque pars <lb/>de&longs;cendat. </s> <s id="s.000186">Si enim parallelepipe­<lb/>dum AB in aëre dimittatur, ut &longs;pon- <pb n="15" xlink:href="017/01/031.jpg"/>te &longs;ua de&longs;cendat, fieri non pote&longs;t, ut A rectam AC percur­<lb/>rat, quin oppo&longs;itum extremum B à recta BC longi&longs;&longs;ime rece­<lb/>dat, & contra: utramque verò extremitatem &longs;imul A & B <lb/>rectâ in centrum C tendere non po&longs;&longs;e e&longs;t manife&longs;tum: Quare <lb/>cum &longs;ibi invicem ob&longs;i&longs;tant æqualiter, ob gravitatis æqualita­<lb/>tem, eas ex perpendicularibus AC, BC æqualiter &longs;ecedere <lb/>oportet ad latera, atque parallelas BE, AF de&longs;cendendo de&longs;­<lb/>cribere. </s> <s id="s.000187">Eadem e&longs;t ratio de cæteris partibus æquali intervallo <lb/>&longs;ejunctis à medio D; omnes enim à &longs;uis perpendiculis rece­<lb/>dunt, præter punctum medium D, cujus perpendicularis <lb/>DC parallela e&longs;t lineis à reliquis partibus in motu de&longs;criptis. </s> <lb/> <s id="s.000188">Ex omnibus itaque particulis datum grave componentibus, eæ <lb/>&longs;olùm, quæ puncto D imminent, per rectam DC in centrum <lb/>moventur; quæ tàm plano horizontis in C, quàm &longs;uperficiei <lb/>&longs;phæricæ in H perpendicularis e&longs;t; cæteræ verò parallelæ BE, <lb/>AF perpendiculares quidem in horizontem cadunt, &longs;ed &longs;phæ­<lb/>ricam &longs;uperficiem obliquè &longs;ecant. </s> </p> <p type="main"> <s id="s.000189">Jam verò &longs;i eju&longs;dem parallelepipedi aliud planum AO hori­<lb/>zonti parallelum moveri versùs C intelligas, erit in eo &longs;imiliter <lb/>aliud punctum unicum, quod rectam DC percurrat; & intra <lb/>corporis &longs;oliditatem unica linea puncto illi imminens viâ eâdem <lb/>in centrum perget non declinans à perpendiculo: cæteræ partes, <lb/>tam quæ ad <expan abbr="dextrã">dextram</expan>, quàm quæ ad <expan abbr="levã">levam</expan>, tam quæ antè, quàm quæ <lb/>ponè, &longs;ibi mutuò adver&longs;antes à recto in <expan abbr="centrũ">centrum</expan> itinere deflectent <lb/>æqualiter. </s> <s id="s.000190">Cum itaque, in priori po&longs;itione, linea puncto D <lb/>imminens, e&longs;&longs;et in communi &longs;ectione planorum, quorum alte­<lb/>rum partes dextras à &longs;ini&longs;tris, alterum anteriores à po&longs;terioribus <lb/>æqualiter &longs;ecernebat; in &longs;ecundâ autem po&longs;itione linea à per­<lb/>pendiculo non recedens &longs;it quoquè in duorum planorum com­<lb/>muni &longs;ectione, quibus pariter corporis gravitas in æquas tribui­<lb/>tur partes; unum verò ex planis &longs;ecantibus &longs;it utrique po&longs;itioni <lb/>commune; unicum e&longs;t punctum tribus planis commune, in quo <lb/>binorum planorum &longs;ectiones &longs;e invicem &longs;ecant, & &longs;it ex. gr. <lb/>punctum I; quod unicum rectâ pergit in centrum C, quemcum­<lb/>que tandem &longs;itum in motu obtineat corpus datum AB, ip&longs;um <lb/>enim e&longs;t duabus illis lineis commune, quæ in &longs;ingulis po&longs;itioni­<lb/>bus ad &longs;ui perpendiculi latera non recedunt: cætera illarum li­<lb/>nearum puncta, mutatâ po&longs;itione corporis, lineam quoque mo­<lb/>tûs mutant. </s> </p> <pb n="16" xlink:href="017/01/032.jpg"/> <p type="main"> <s id="s.000191">Illud itaquè punctum in quocumque corpore gravi, quod <lb/>&longs;emper in motu de&longs;cribit lineam rectà in terræ centrum <lb/>ductam, dicitur <emph type="italics"/>Centrum Gravitatis<emph.end type="italics"/>; & linea, quæ centrum <lb/>gravitatis conjungit cum terræ centro, <emph type="italics"/>Linea directionis<emph.end type="italics"/> dicitur; <lb/>&longs;ecundùm quam videlicet dirigitur motus, & dimentienda e&longs;t <lb/>corporis à centro terræ di&longs;tantia, &longs;i quatenus grave con&longs;idere­<lb/>tur. </s> <s id="s.000192">Porrò punctum I centrum gravitatis dicitur, quia centri <lb/>nomen tribuitur puncto, quod e&longs;t medium: & quemadmodum <lb/>magnitudinis alicujus centrum vocatur punctum illud, quod <lb/>æquales magnitudines circun&longs;tant, &longs;i partes, quæ ex adver&longs;o <lb/>&longs;unt, accipiantur; ita in gravibus centrum gravitatis dicitur, <lb/>quod æquales gravitates, vel æqualia gravitatum momenta cir­<lb/>cun&longs;tant. </s> <s id="s.000193">Quod &longs;i punctum I non haberet hinc, & hinc æqua­<lb/>les gravitatum vires, ab alterutrâ parte præ&longs;tante viribus pro­<lb/>pelleretur in latus extra lineam directionis, à quâ nunquam re­<lb/>cedit, &longs;i liberè moveatur. </s> <s id="s.000194">Cave tamen, ne partium æqualita­<lb/>tem dimetiaris linearum longitudine à céntro gravitatis exeun­<lb/>tium, ita ut &longs;ingulas lineas æqualiter dividendas putes; &longs;ed to­<lb/>tum corpus debet intelligi divi&longs;um bifariam à plano per cen­<lb/>trum gravitatis ip&longs;ius corporis, & per centrum gravium ac le­<lb/>vium tran&longs;eunte, ita ut &longs;i planum à dextrâ in &longs;ini&longs;tram ductum <lb/>&longs;ecernat partes anteriores à po&longs;terioribus, æqualia &longs;int gravita­<lb/>tum momenta antè, & ponè; &longs;i aliud planum per eandem di­<lb/>rectionis lineam ductum partes dextras à &longs;ini&longs;tris di&longs;tinguat pa­<lb/>ria &longs;imiliter hinc & hinc gravitatum momenta relinquat. </s> </p> <p type="main"> <s id="s.000195">Gravitatum, inquam, momenta, non gravitates; ne locus <lb/>pateat æquivocationi; neque enim quoties æqualia &longs;unt mo­<lb/>menta, toties æquales &longs;unt gravitates hinc & hinc centrum gra­<lb/>vitatis complectentes, ut patebit ex iis, quæ de æquilibrio dice­<lb/>mus. </s> <s id="s.000196">Unde fit in iis tantùm corporibus, quæ partibus unius eju&longs;­<lb/>demque naturæ, ac ductu perpetuo &longs;imiliter con&longs;titutis, <expan abbr="con&longs;tãt">con&longs;tant</expan>, <lb/> <figure id="id.017.01.032.1.jpg" xlink:href="017/01/032/1.jpg"/><lb/>idem e&longs;&longs;e centrum gravitatis atque magni­<lb/>tudinis; reliqua certis regulis non circum­<lb/>&longs;cripta, aut ex variis naturis compo&longs;ita, in <lb/>alio puncto, molis centrum habere, in alio, <lb/>gravitatis. </s> <s id="s.000197">Si enim duo &longs;olida VT, cujus <lb/>centrum gravitatis, & magnitudinis R, <lb/>& MN, cujus centrum S, æqualia &longs;ecun- <pb n="17" xlink:href="017/01/033.jpg"/>dùm gravitatem coagmententur, non erit centrum gravitatis <lb/>totius molis compo&longs;itæ in I, ubi planum tran&longs;iens per VN &longs;e­<lb/>cat lineam RS jungentem centra &longs;ingularum gravitatum æqua­<lb/>lium, &longs;ed erit in L, ubi recta RS bifariam dividitur: planum <lb/>autem per centrum terræ, & punctum L ductum non ita &longs;ecat <lb/>hanc molem, ut &longs;int æquales hinc, & hinc gravitates, quamvis <lb/>æqualia &longs;int gravitatum inæqualium momenta, quæ ex figuræ <lb/>po&longs;itione poti&longs;&longs;imùm pendent. </s> <s id="s.000198">Quod &longs;i corporis VT gravitas <lb/>ad corporis MN gravitatem, eam haberet rationem, quam SI <lb/>ad IR, e&longs;&longs;et I gravitatis centrum molis compo&longs;itæ, quæ à plano <lb/>per terræ centrum, & punctum I ducto non in gravitates æqua­<lb/>les, &longs;ed in momenta æqualia divideretur; ut in loco inferiùs ex­<lb/>plicabitur. </s> </p> <p type="main"> <s id="s.000199">Ob&longs;erva autem non &longs;emper centrum gravitatis e&longs;&longs;e in ip&longs;o <lb/>corpore gravi, ut patet in corporibus annularibus, aut angulos <lb/>cavos habentibus, in quibus nullum e&longs;t punctum per quod tran­<lb/>&longs;euntia plana quæcunque dividant in æquas partes momenta <lb/>gravitatum: ita tamen e&longs;t extra corporis cavi &longs;oliditatem, ut &longs;it <lb/>intra ip&longs;am cavitatem punctum, ex quo &longs;i intelligatur annulus, <lb/>vel fru&longs;tum annulare &longs;u&longs;pendi, manet po&longs;itionem habens hori­<lb/>zonti parallelam, cum habeat æqualia hinc, & hinc gravita­<lb/>tum momenta. </s> <s id="s.000200">Quod &longs;i corpus in cavos angulos &longs;inuatum ha­<lb/>beat particulam aliquam procurrentem, pote&longs;t contingere, ut <lb/>in illius particulæ extremo &longs;it totius molis centrum gravitatis: <lb/>&longs;ic brevioris alicujus bacilli extremitati alteri &longs;i duos cultros in­<lb/>fixeris, ut &longs;inguli cum bacillo hinc, & hinc angulum acutum <lb/>ad ea&longs;dem partes con&longs;tituant, ita inclinari po&longs;&longs;unt, ut extremo <lb/>ungue &longs;uppo&longs;ito reliquæ bacilli extremitati tota illa moles &longs;u&longs;ti­<lb/>neatur citrà periculum cadendi, cùm gravitatis centrum in illa <lb/>extremitate, intrà cavitatem, quam inclinati cultri faciunt, <lb/>æqualia habeat ex omni parte gravitatum momenta, &longs;i planum <lb/>&longs;ecans per illud tran&longs;eat. <pb n="18" xlink:href="017/01/034.jpg"/></s> </p> <p type="head"> <s id="s.000201"><emph type="center"/>CAPUT IV.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000202"><emph type="center"/><emph type="italics"/>An gravia centro vicina minùs gravitent.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000203">COrpora non intelliguntur gravitare ni&longs;i in alieno loco; <lb/>quando &longs;cilicet corpus contiguum inter illa & centrum <lb/>terræ interjectum, quod medii rationem habere pote&longs;t, levius <lb/>e&longs;t; petit enim infra illud e&longs;&longs;e: ni&longs;um autem hunc deorsùm <lb/><emph type="italics"/>Gravitationem<emph.end type="italics"/> dicimus. </s> <s id="s.000204">Sed quoniam ni&longs;us i&longs;te videtur idcircò <lb/>à naturâ in&longs;titutus, ut perturbatus corporum ordo re&longs;tituatur; <lb/>&longs;i ex fine ratio petenda &longs;it, &longs;atis apparet corpora gravia ćentro <lb/>terræ vicina minùs gravitare. </s> <s id="s.000205">Quemadmodum enim quotie&longs;­<lb/>cunque aliquis à propo&longs;ito fine magis di&longs;tat, eò magis anxius <lb/>e&longs;t, atque &longs;olicitus de mediis ad illum a&longs;&longs;equendum nece&longs;&longs;ariis, <lb/>& animo æquiore toleratur modica, quàm multa violentia; ita <lb/>natura minorem ordinis debiti perturbationem &longs;entiens, &longs;i gra­<lb/>ve parùm ab&longs;it, quàm &longs;i longè abe&longs;&longs;et, à loco, ubi juxta inge­<lb/>nitam propen&longs;ionem exigit con&longs;i&longs;tere, minùs &longs;olicita e&longs;&longs;e debet <lb/>de illo re&longs;tituendo, nec adeò vehementi conatu, hoc e&longs;t gravi­<lb/>tatione, illud urgere debet in locum &longs;uum. </s> </p> <p type="main"> <s id="s.000206">Ad hæc omnibus aperti&longs;&longs;imè liquet eò majore naturæ impe­<lb/>tu corpora deorsùm niti, quò levius e&longs;t corpus, in quo tan­<lb/>quam in medio perficiendus e&longs;t motus, &longs;i dimittantur. </s> <s id="s.000207">Sic à <lb/>&longs;axo in aëre pendente manum deorsùm validiùs trahi &longs;enti­<lb/>mus, quàm ab eodem aquæ immer&longs;o trahatur, & multò lan­<lb/>guidiùs conatur deor&longs;um lapis in melle de&longs;cendens, quàm in <lb/>aqua; quia videlicet aqua levior e&longs;t melle, & aër levior aquâ. </s> <lb/> <s id="s.000208">Hinc e&longs;t quod, &longs;i medij partes fuerint diversâ gravitate prædi­<lb/>tæ, pars centro terræ propior etiam erit gravior; atque ideò <lb/>corpus in parte medij graviore minùs gravitabit propè centrum <lb/>terræ, quàm procul. </s> <s id="s.000209">E&longs;&longs;e autem eju&longs;dem medij non commoti <lb/>partes graviores in imo, omnium ferè hominum &longs;en&longs;us e&longs;t: <lb/>quotus enim qui&longs;que e&longs;t, qui ne&longs;ciat mellis optimam partem <lb/>e&longs;&longs;e, quæ in va&longs;is fundo, vini quæ in medio, olei quæ in &longs;um­<lb/>mo? </s> <s id="s.000210">id autem verum non e&longs;&longs;et, ni&longs;i liquoris eju&longs;dem partes <lb/>e&longs;&longs;ent diversâ gravitate delatæ in loca à terræ centro di&longs;pari- <pb n="19" xlink:href="017/01/035.jpg"/>bus intervallis remota: Quia enim oleum eò perfectius e&longs;t, <lb/>quò propiùs aëris levitatem &longs;pirituum &longs;ubtilitate æmulatur, <lb/>ideò quod in &longs;ummo va&longs;e innatat, optimum e&longs;t: At vini &longs;ua­<lb/>vitas in exqui&longs;itâ &longs;ui tartari &longs;ufficienti humore diluti cum &longs;pi­<lb/>ritibus permi&longs;tione con&longs;i&longs;tens medium locum in va&longs;e exigit, <lb/>&longs;icut media e&longs;t illius gravitas inter vagantium &longs;pirituum levita­<lb/>tem, & fæculenti tartari gravitatem: Mellis demùm dulcedo <lb/>ex &longs;ui &longs;alis, &longs;eu &longs;acchari, copiâ proveniens iis partibus poti&longs;&longs;i­<lb/>mum ine&longs;t, quæ multo &longs;ale refertæ graviores quoquè &longs;unt, & <lb/>in fundo &longs;ub&longs;idunt. </s> <s id="s.000211">Nec e&longs;t iis abroganda fides, qui in alti&longs;&longs;i­<lb/>mo mari adeò gravem aquam à &longs;e deprehen&longs;am alicubi te&longs;tan­<lb/>tur, ut &longs;upta reliquum maris fundum ambulantes ad alti&longs;&longs;i­<lb/>mam fo&longs;&longs;am venerint, in quam penetrare &longs;æpiùs irrito conatu <lb/>tentârint: his enim non ægrè fidem habeo, qui aërem in imis <lb/>vallibus cra&longs;&longs;iorem atquè graviorem, in &longs;ummis verò montibus <lb/>puriorem atque leviorem ab omnibus admitti video. </s> <s id="s.000212">Cum ita­<lb/>que (&longs;i ex notis ad minùs nota progredi philo&longs;ophando liceat) <lb/>propè centrum gravium ac levium medij partes graviores &longs;int, <lb/>quam procul ab illo; minor e&longs;t gravitatio corporum, &longs;i centro <lb/>propiora fiant, ac quando longè ab illo remota detinebantur. </s> <lb/> <s id="s.000213">Hinc autem re&longs;ponderi pote&longs;t quærentibus, cur in fodinis lon­<lb/>gè faciliùs crudi metalli ma&longs;&longs;a moveatur, quàm in &longs;uperficie <lb/>terræ: aër &longs;cilicet profundis illis cuniculis inclu&longs;us gravior mul­<lb/>tò ac cra&longs;&longs;ior e&longs;t aëre i&longs;to, quem in&longs;piramus, atque adeò ibi <lb/>metallum minùs gravitat. </s> </p> <p type="main"> <s id="s.000214">Quòd &longs;i libeat minorem hanc gravitationem experimento <lb/>deprehendere, &longs;ume vitream fi&longs;tulam &longs;upernè clau&longs;am longio­<lb/>rem pedibus tribus Romanis, eam imple argento vivo, digito­<lb/>que o&longs;culum accuratè claudens inverte, ac argento vivo &longs;ub­<lb/>jecti va&longs;is immerge; tùm amoto digito de&longs;cendet mercurius in <lb/>fi&longs;tulâ, iterúmque a&longs;cendet, & in certâ demum altitudine per­<lb/>pendiculari quie&longs;cet. </s> <s id="s.000215">Ob&longs;ervatâ igitur altitudine perpendicu­<lb/>lari, quam mercurius obtinet, &longs;i in imâ valle experimentum <lb/>in&longs;tituatur, eâque comparatâ cum altitudine perpendiculari, <lb/>in qua con&longs;i&longs;tit, cùm in &longs;ummo montis alti&longs;&longs;imi vertice expe­<lb/>rimentum idem &longs;umitur, animadvertes altitudinem mercurij <lb/>per vim in fi&longs;tulâ &longs;u&longs;pen&longs;i minorem e&longs;&longs;e in &longs;ummo monte, quàm <pb n="20" xlink:href="017/01/036.jpg"/>in valle; Quia nimirum mercurius intra fi&longs;tulam detentus tan­<lb/>quàm in va&longs;e, e&longs;t in aëre fi&longs;tulam ambiente tanquam in loco; <lb/>in aëre autem leviori cùm magis gravitet, in minori etiam al­<lb/>titudine perpendiculari con&longs;i&longs;tit. </s> <s id="s.000216">Experimentum hoc in valle, <lb/>& in monte &longs;umere mihi otium non fuit, quamvis in eo &longs;æ­<lb/>piùs me exercuerim: &longs;ed de illius veritate ambigere non &longs;inunt <lb/>te&longs;tes in Galliâ luculenti&longs;&longs;imi, qui di&longs;crimen hoc in mercuri; <lb/>altitudine ob&longs;ervârunt in altioribus montibus. </s> </p> <p type="main"> <s id="s.000217">Verùm, ex alio præteteà capite imminui debet gravitatio <lb/>corporum in minori à centro remotione, habitâ &longs;olùm ratione <lb/>&longs;itûs. </s> <s id="s.000218">Cùm enim totius corporis gravitatio conflata &longs;it ex &longs;in­<lb/>gularum partium impetu, quo deor&longs;um nituntur, manife&longs;tum <lb/>e&longs;t &longs;ingulis partibus languidiùs deor&longs;um conantibus, totius cor­<lb/>poris gravitationem e&longs;&longs;e pariter languidiorem. </s> <s id="s.000219">Quoniam verò <lb/>quicquid in motu cogitur à recto &longs;ecundùm naturam tramite <lb/>deflectere, lentiùs atque remi&longs;&longs;iùs pergit ad præ&longs;titutum mo­<lb/>tûs terminum; particulæ autem corporis &longs;olidi gravis, propio­<lb/>res centro factæ, magis à &longs;uo perpendiculo, &longs;ibi invicem ad­<lb/>ver&longs;antes, declinant; &longs;atis con&longs;tat &longs;ingulas fractis quodammo­<lb/>do viribus languentes plurimum de conatu remittere. </s> <s id="s.000220">Si enim <lb/> <figure id="id.017.01.036.1.jpg" xlink:href="017/01/036/1.jpg"/><lb/>&longs;olidum AB fiat centro vicinius ita, <lb/>ut A &longs;it in K, & B in L, lineæ di­<lb/>rectionis partium extremarum &longs;unt <lb/>KC, LC: at coguntur per lineas <lb/>KF, LE parallelas de&longs;cendere, <lb/>fiuntque anguli CKF, CLE ex­<lb/>terni majores internis CAK, CBL <lb/>per 16. l. 1. magis igitur in K & L <lb/>recedunt à perpendiculo, quàm re­<lb/>cederent in A & B. </s> <s id="s.000221">Quia itaque <lb/>pars in K exi&longs;tens magis impeditur <lb/>ab oppo&longs;itâ extremitate, quæ in L, <lb/>ne per KC de&longs;cendat (ni&longs;i enim <lb/>pars, quæ in L, urgeret oppo&longs;itam <lb/>tentans per LC de&longs;cendere, non cogeretur pars in K exi&longs;tens <lb/>adeò recedere à &longs;uâ directionis lineâ) minori etiam impetu <lb/>deor&longs;um fertur. </s> <s id="s.000222">E&longs;t autem eadem de reliquis partibus ratio, <pb n="21" xlink:href="017/01/037.jpg"/>præter eas, quæ in eâdem directionis lineâ &longs;unt cum centro <lb/>gravitatis; &longs;ingulæ enim ad centrum terræ accedentes magis à <lb/>&longs;uo perpendiculo recedunt, minú&longs;que deor&longs;um gravitant. </s> <s id="s.000223">Quî <lb/>igitur fieri po&longs;&longs;it, ut debilitato &longs;ingularum particularum cona­<lb/>tu, atque impetu deor&longs;um, non minuatur pariter totius cor­<lb/>poris gravitatio, &longs;i fiat centro vicinius? </s> </p> <p type="main"> <s id="s.000224">Illud tamen non diffiteor, quod &longs;i medij levitates, aut angu­<lb/>lorum CLE, CBL inclinationes eo tantùm di&longs;crimine &longs;ecer­<lb/>nantur, quod omnem &longs;en&longs;um fugiat, vel &longs;altem ex medij gra­<lb/>vitate, & anguli magnitudine conjunctim &longs;umptis oriri non <lb/>po&longs;&longs;it varietas, quæ &longs;ub &longs;en&longs;um cadat; neque percipietur gra­<lb/>vitationis differentia in majori vicinitate. </s> <s id="s.000225">Sed hoc non facit, <lb/>quin inter gravitationes di&longs;crimen intercedat; neque enim <lb/>continuò, &longs;i quid &longs;en&longs;um latet, id omninò non e&longs;&longs;e dicendum <lb/>e&longs;t: contingere &longs;i quidem pote&longs;t motum aliquem ita &longs;en&longs;im, & <lb/>&longs;ine &longs;en&longs;u fieri, ut non ni&longs;i elap&longs;o temporis &longs;patio demùm inno­<lb/>te&longs;cat. </s> <s id="s.000226">Sic &longs;i vinum, cujus gravitas vix minor &longs;it gravitate aquæ <lb/>arte &longs;atis notâ affuderis aquæ ita, ut innatet, & &longs;upremam va­<lb/>&longs;is partem occupet, aliudque vas &longs;imili vino plenum, &longs;ed paulò <lb/>altius, habeas, tum ex libra centrum motûs habente in cen­<lb/>tro gravitatis jugi pendeant æqualia pondera intrà vinum <lb/>utriu&longs;que va&longs;is; fiet utique ponderum æquilibrium, & con­<lb/>&longs;i&longs;tent eo in &longs;itu, quem illis dederis: at &longs;i alterum libræ extre­<lb/>mum ita deprimas, ut pondus, quod ex eo pendet, ex vino ad <lb/>aquam vix graviorem tran&longs;eat, reliquo pondere intra vinum <lb/>manente; initio quidem non apparebit motus libræ &longs;e re&longs;ti­<lb/>tuentis, quia pondus in vino non excedit gravitationem pon­<lb/>deris æqualis in aquâ ni&longs;i eo exce&longs;&longs;u, quo gravitas aquæ &longs;upe­<lb/>rat gravitatem vini; hic autem exce&longs;&longs;us cum minimus &longs;it, mo­<lb/>tum quoque efficiet, quem ægrè à quiete di&longs;cernas, ni&longs;i ubi <lb/>po&longs;t aliquod tempus deprehenderis pondus altius de&longs;cendi&longs;&longs;e, <lb/>depre&longs;&longs;ius autem a&longs;cendi&longs;&longs;e. </s> <s id="s.000227">Haud &longs;ecus philo&longs;ophandum e&longs;t <lb/>de majore, aut minore corporum gravitatione, &longs;i di&longs;paribus in­<lb/>tervallis à terræ centro removeantur, diutiùs enim propè cen­<lb/>trum incumbere poterunt &longs;u&longs;tinenti, quàm procul: id quod <lb/>&longs;atis erit ad minorem gravitationem patefaciendam, quæ non <lb/>&longs;tatim innote&longs;cat. </s> </p> <pb n="22" xlink:href="017/01/038.jpg"/> <p type="main"> <s id="s.000228">Hæc autem non leviter confirmari videntur ex iis, quæ quo­<lb/>tidiè ferè videmus; nam &longs;i circinus, quo circulos de&longs;cribere <lb/>&longs;olemus, cadat, &longs;emper nodus prævertit cu&longs;pides, & prior ter­<lb/>ram ferit; ni&longs;i fortè nodus ad perpendiculum immineat cru­<lb/>ribus: & omnia ferè corpora, quæ centrum gravitatis ex una <lb/>parte habent, &longs;i ex modicâ altitudine dimittantur, videntur <lb/>quidem cadere parallela; &longs;ed ex majori altitudine &longs;i de&longs;cen­<lb/>dant, pars gravior prior terram attingit. </s> <s id="s.000229">Sit enim corpus ES, <lb/> <figure id="id.017.01.038.1.jpg" xlink:href="017/01/038/1.jpg"/><lb/>cujus gravitatis centrum H, linea <lb/>directionis HA; &longs;i horizonti paral­<lb/>lelum de&longs;cenderet, per rectas EI, <lb/>SR parallelas lineæ directionis mo­<lb/>veretur; id quod in modicâ tantùm <lb/>altitudine contingere videtur, quia <lb/>nondum facta e&longs;t ea gravitationis <lb/>imminutio in extremitate S, quæ <lb/>percipi po&longs;&longs;it. </s> <s id="s.000230">Si enim E per EI <lb/>de&longs;cenderet, S verò per SR, an­<lb/>gulus IEA æqualis alterno EAH <lb/>per 29. lib. 1. minor e&longs;&longs;et angulo <lb/>RSA, qui æqualis e&longs;t alterno <lb/>HAS; nam ex hypothe&longs;i minùs <lb/>di&longs;tat E, quàm S, à centro gravi­<lb/>tatis H, & e&longs;t angulus EAH minor angulo HAS; pars igi­<lb/>tur S magis deflecteret à &longs;uo perpendiculo SA, quàm E de­<lb/>flecteret ab EA; cùm itaque S magis in latus propelleretur, <lb/>plus etiam de conatu deor&longs;um remitteret, quàm E; atque adeò <lb/>non po&longs;&longs;et æqualiter de&longs;cendere ac moveri, contra hypothe&longs;im <lb/>paralleli&longs;mi. </s> <s id="s.000231">Dicendum e&longs;t igitur non per parallelas EI, SR <lb/>fieri motum, &longs;ed intra illas paulatim partem E graviorem præ­<lb/>currere: quia &longs;cilicet partes omnes extra lineam directionis <lb/>AH con&longs;titutæ dum removentur à &longs;uo perpendiculo, aliquid <lb/>amittunt de impetu, quo deor&longs;um nituntur, propiores quidem <lb/>minus, remotiores autem plus; pars &longs;i quidem G in principio <lb/>motûs de&longs;cendens parallela lineæ directionis per GM facit an­<lb/>gulum AGM internum per 16.lib.1. minorem externo GMS, <lb/>qui per 29. 1. e&longs;t æqualis alterno MSR. </s> <s id="s.000232">Quia ergo AGM <pb n="23" xlink:href="017/01/039.jpg"/>minor e&longs;t angulo ASR, pars G minus de &longs;uo impetu deor&longs;um <lb/>amittit, quàm pars S; & quamvis initio di&longs;crimen hoc non <lb/>percipiatur, demum fit, ut additis pluribus differentiis mani­<lb/>fe&longs;tè appareat partem S minùs gravitare, quia tardiùs deor­<lb/>&longs;um movetur; & tandem ip&longs;a &longs;equitur partem E præcur­<lb/>rentem, po&longs;tquam minori illâ gravitatione permi&longs;it parti E, <lb/>ut propiùs accederet ad lineam directionis, fieretquè quæ­<lb/>dam virtualis conver&longs;io circa centrum gravitatis H, in qua <lb/>extremitas E occuparet infimum locum, S autem &longs;upre­<lb/>mum. </s> <s id="s.000233">Quare cùm nos doceat experientia partem HS <lb/>æquiponderantem parti HE, &longs;i &longs;u&longs;pendantur ex H, in mo­<lb/>tu tamen minùs gravitare, quàm oppo&longs;itam, ideóque fieri <lb/>illam conver&longs;ionem, ut pars E fiat inferior; neque aptior <lb/>a&longs;&longs;ignari po&longs;&longs;it ratio, quàm quæ petitur ex rece&longs;&longs;u partium <lb/>majori à &longs;uo perpendiculo: &longs;atis liquet, quantum momenti <lb/>habeat hæc declinatio à perpendiculo ad minuendam gra­<lb/>vitationem. </s> <s id="s.000234">Ex majori igitur declinatione à lineâ perpen­<lb/>diculari, quæ con&longs;equitur corpus con&longs;titutum non adeò <lb/>procul à centro terræ ut priùs, non ineptè arguitur minor <lb/>corporis gravitatio in eo &longs;itu, &longs;i cætera &longs;int paria: neque <lb/>enim comparo corpus, quod per motum de&longs;cendit, per&longs;e­<lb/>verans in &longs;uo motu, cum corpore in loco altiori tran&longs;eun­<lb/>te à quiete ad motum; nam tunc ex impetu per motum <lb/>concepto major e&longs;t gravitatio in loco inferiore, quàm in &longs;u­<lb/>periore: &longs;ed tantùm corpora invicem comparo, vel pariter <lb/>quie&longs;centia, vel æquali tempore mota, illudque, quod ter­<lb/>ræ vicinius e&longs;t, a&longs;&longs;ero, vel minori ni&longs;u conari à quiete in <lb/>loco alieno tran&longs;ire ad motum, vel æquali tempore, quo præ­<lb/>ce&longs;&longs;it motus, minus impetus acqui&longs;ii&longs;&longs;e ac minoribus viribus <lb/>motum continuare. </s> </p> <p type="main"> <s id="s.000235">Ex his quæ de gravibus hactenus di&longs;putata &longs;unt, aliquis <lb/>forta&longs;sè inferat levia à centro remotiora minùs levitare, &longs;i­<lb/>cut gravia centro propiora minùs gravitant. </s> <s id="s.000236">Verùm res e&longs;t <lb/>pen&longs;iculatiùs examinanda, nec &longs;impliciter ex oppo&longs;itis gra­<lb/>vium, ac levium naturis definienda, qua&longs;i ob id ip&longs;um, <lb/>quia &longs;ibi gravitas atque levitas adver&longs;antur, contraria ha­<lb/>berent omnia con&longs;equentia. </s> <s id="s.000237">Et quidem quod &longs;pectat ad <pb n="24" xlink:href="017/01/040.jpg"/>&longs;olam corporis levioris po&longs;itionem, non minuitur levitatio, <lb/>&longs;ed potiùs augetur in majoribus à terræ centro intervallis; <lb/>ubi minùs à &longs;uo perpendiculo declinant partes centrum le­<lb/>vitatis circun&longs;tantes, & idcirco minùs de conatu remit­<lb/>tunt, quò nituntur ad &longs;upe­<lb/> <figure id="id.017.01.040.1.jpg" xlink:href="017/01/040/1.jpg"/><lb/>riora evadere. </s> <s id="s.000238">Sit namque <lb/>Globus HG, cujus centrum <lb/>levitatis M, & linea di&longs;cretio­<lb/>nis OMN; cui parallelæ <lb/>&longs;unt HD & GF, quas de&longs;­<lb/>cribunt a&longs;cendendo extremi­<lb/>tates H & G, & motum eum­<lb/>dem continuabunt, &longs;i globus <lb/>in N tran&longs;latus intelligatur. </s> <lb/> <s id="s.000239">Quando igitur globus e&longs;t in M, <lb/>extremitas H recedit à per­<lb/>pendiculo OI, & cum eo <lb/>facit angulum IHT; quan­<lb/>do autem e&longs;t in N, extremi­<lb/>tas T a&longs;cendens per TD fa­<lb/>cit cum perpendiculo OR an­<lb/>gulum RTD, qui per 15.lib.1. <lb/>æqualis e&longs;t angulo HTO ad <lb/>verticem, hic autem, inter­<lb/>nus cum &longs;it, per 16. 1. minor <lb/>e&longs;t externo IHT. </s> <s id="s.000240">E&longs;t ergo <lb/>RTD minor angulo IHT, <lb/>atque ideò plus habet mo­<lb/>menti &longs;ur&longs;um, ubi minus à <lb/>recto &longs;ecundum naturam tra­<lb/>mite deflectit. </s> </p> <p type="main"> <s id="s.000241">Di&longs;crimen hoc momentorum ab angulorum inæqualitate <lb/>proveniens optimè intelligit natura, quæ ita motum perfi­<lb/>cit, ut, &longs;i duo inæqualiter levia coagmentata fuerint, le­<lb/>vius præcurrat. </s> <s id="s.000242">Sic &longs;i A cortex &longs;uberis coagmentetur ligno <lb/>fagino B, & intra aquam mediocriter profundam horizon<lb/>taliter collocetur &longs;olidum DC, ita per lineam directio- <pb n="25" xlink:href="017/01/041.jpg"/>nis TO a&longs;cendit centrum <lb/> <figure id="id.017.01.041.1.jpg" xlink:href="017/01/041/1.jpg"/><lb/>levitatis, ut demum A in <lb/>loco &longs;uperiore, B autem in <lb/>inferiore con&longs;tituatur, ex­<lb/>tremo D per rectam DO <lb/>a&longs;cendente: Quo in motu <lb/>natura magnum invenit <lb/>compendium. </s> <s id="s.000243">Quia enim <lb/>partes centro levitatis vi­<lb/>ciniores magis levitant, <lb/>quòd linea parallela lineæ <lb/>directionis faciat minorem <lb/>angulum cum earum per­<lb/>pendiculo (&longs;ic &longs;i linea di­<lb/>rectionis &longs;it FL, eique pa­<lb/>rallelæ NG, RX, angu­<lb/>lus NGX internus per <lb/>29. 1. e&longs;t æqualis externo <lb/>RXY, at PGX externus per 16. 1. major e&longs;t interno GXF, <lb/>hoc e&longs;t VXY ad verticem, ergo PGX major e&longs;t angulo <lb/>VXY, & &longs;i uterque auferatur ex æqualibus NGX, RXY, <lb/>remanet NGP minor angulo RXV, ideoque G magis levi­<lb/>tat, quam X) ex majore impedimento, quod initio motûs ha­<lb/>betur ob anguli HDI magnitudinem, dum pars D minùs le­<lb/>vitat, centrum levitatis per SO a&longs;cendens inclinat corpus DC, <lb/>& extremitas D in recta DO con&longs;tituitur, in qua longê ci­<lb/>tiùs minuuntur impedimenta, quàm &longs;i per parallelam DI <lb/>a&longs;cenderet: vix enim a&longs;cendit in E, cum impedimenta &longs;unt <lb/>æquè diminuta, ac &longs;i a&longs;cendi&longs;&longs;et in I; quandoquidem angu­<lb/>lus KEI per 29. 1. e&longs;t æqualis alterno EID, atque adeò <lb/>etiam angulo, quem in I faceret parallela DI cum perpendi­<lb/>culo; e&longs;t igitur angulus KEI minor quocunque alio angulo, <lb/>qui fieret in punctis intermediis lineæ DI; &longs;ed quoniam cen­<lb/>trum levitatis a&longs;cendendo acqui&longs;ivit majorem impetum, quàm <lb/>extremitas in E exi&longs;tens, per vim illam rapit extra paralle­<lb/>lam EK, trahitque per lineam EO, & perpendiculum facit <lb/>angulum &longs;emper minorem cum lineâ directionis; unde fit <lb/>partem inferiorem &longs;emper faciliùs trahi, quo minùs in diver&longs;a <pb n="26" xlink:href="017/01/042.jpg"/>abit ejus perpendiculum, cum quo &longs;emper minorem, & mi­<lb/>norem angulum facit linea motûs DO; donec demùm to­<lb/>tum &longs;olidum obtineat &longs;itum perpendicularem; quod initio erat <lb/>in æquilibrio. </s> </p> <p type="main"> <s id="s.000244">Cæterum, quamvis habitâ ratione &longs;itûs, levia altiora magis <lb/>levitent, &longs;ivè parallela horizonti jaceant extrema, &longs;ivè incli­<lb/>nata, ratione tamen medij, quod in &longs;uperioribus e&longs;t levius, <lb/>quàm in inferioribus, minùs levitant: experientia enim o&longs;ten­<lb/>dit ea lentiùs a&longs;cendere, quæ propiùs accedunt ad medij na­<lb/>turam &longs;ecundùm levitatem: nam ex tribus globulis &longs;phæricis, <lb/>quorum diameter unc. 2 1/5 pedis Romani, cereus erat ponderis <lb/>drachmarum 24, faginus drachm. 22, vitraëreus drachm. 7. <lb/>in aëre expen&longs;i, &longs;ed eorum motus in aquâ ad altitudinem pe­<lb/>dum 14, valdè inæqualis fuit, numeratis vibrationibus eju&longs;­<lb/>dem perpendiculi; cereus &longs;iquidem a&longs;cendit lenti&longs;&longs;imè vibra­<lb/>tionibus 88, faginus vibrationibus 37, vitraëreus vibrationi­<lb/>bus 33: unde patet cereum, qui minimùm ab aquâ differt in <lb/>pondere (aquæ etenim molis æqualis e&longs;t drachm. 25 3/5) minùs <lb/>in eâ levitare. </s> <s id="s.000245">Sicut igitur diver&longs;a levia in eodem medio inæ­<lb/>qualiter levitant, &longs;ic idem leve in medio di&longs;&longs;imili inæqualiter <lb/>levitabit pro majore aut minore levitatum di&longs;&longs;imilitudine. </s> <lb/> <s id="s.000246">Conveniunt itaque gravia, & levia, quod hæc procul à cen­<lb/>tro offendentia medium levius minùs levitant, illa propè cen­<lb/>trum habentia medium gravius minùs gravitant. </s> <s id="s.000247">Differunt au­<lb/>tem ratione po&longs;itionis, quia, in loco remotiore à centro, per­<lb/>pendicula omnia concurrunt ad angulos magis acutos, minú&longs;­<lb/>que differunt à lineâ rectâ, ideo qua&longs;i collatis viribus magis <lb/>gravitant, & magis levitant; at prope centrum cum perpendi­<lb/>cula magis in diver&longs;a abeant, & levia minùs levitant, & gravia <lb/>minùs gravitant. </s> <s id="s.000248">Porrò hanc &longs;imilitudinem gravitationis gra­<lb/>vium, & levitationis levium in eodem loco, à me vocari di&longs;cri­<lb/>men, & differentiam, quia habita ratione oppo&longs;itorum videba­<lb/>tur leve remotius debere minùs levitare, &longs;icut grave propius <lb/>minùs gravitat, ne te moveat; litem de verbo non faciam. <pb n="27" xlink:href="017/01/043.jpg"/></s> </p> <p type="head"> <s id="s.000249"><emph type="center"/>CAPUT V.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000250"><emph type="center"/><emph type="italics"/>Quâ ratione centrum gravitatis corporum <lb/>inveniatur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000251">OPus mechanicum plerunque non indiget puncto illo, <lb/>quod intra corporum &longs;oliditatem latet, ac centrum gra­<lb/>vitatis definivimus; &longs;ed &longs;atis e&longs;t &longs;i in extimâ corporis &longs;uperfi­<lb/>cie innote&longs;cat punctum, aut linea imminens ip&longs;i gravitatis <lb/>centro, pro ratione &longs;itûs, in quo corpus grave con&longs;i&longs;tere cu­<lb/>pimus. </s> <s id="s.000252">Ideo geometricum laborem inveniendi punctum illud <lb/>intimum Centrobarycæ relinquens, mechanica tantùm inqui­<lb/>&longs;itione, & qua&longs;i tentans, perve&longs;tigo punctum illud, aut li­<lb/>neam in corporis &longs;uperficie, cui re&longs;pondet planum per lineam <lb/>directionis ductum, & &longs;ecans corpus in certo &longs;itu con&longs;titu­<lb/>tum. </s> <s id="s.000253">Et quidem &longs;i corpus &longs;phæricum fuerit ex partibus eju&longs;­<lb/>dem naturæ conflatum, aut &longs;altem ex partibus heterogeneis <lb/>quidem, &longs;ed circa &longs;phæræ centrum &longs;imiliter di&longs;po&longs;itis ita, ut <lb/>intima &longs;phærula folliculis quibu&longs;dam obvolvatur; quia idem <lb/>e&longs;t molis atque gravitatis centrum, punctum quodcumque in <lb/>&longs;phærica &longs;uperficie a&longs;&longs;umatur, aptum erit; &longs;ingula enim &longs;i­<lb/>milem habent po&longs;itionem. </s> <s id="s.000254">Sin autem aut &longs;phæræ &longs;egmentum, <lb/>aut &longs;phæra ex partibus heterogeneis inæqualiter di&longs;po&longs;itis fue­<lb/>rit; imponatur plano horizontali accuratè levi, & maximè æqua­<lb/>bili; & quod punctum tangetur à &longs;uppo&longs;ito plano, ubi motus <lb/>omnis ce&longs;&longs;averit, illud e&longs;t, quod poti&longs;&longs;imùm quæritur, ac <lb/>punctum &longs;uperius, quod huic è regione e&longs;t, erit pariter aptum <lb/>ad propo&longs;itum finem. </s> </p> <p type="main"> <s id="s.000255">Quod &longs;i cylindricum fuerit oblatum corpus, aut pri&longs;ma quod­<lb/>cunque continuo, & &longs;imili ductu productum; &longs;ecetur bifariam <lb/>longitudo, & punctum habebitur cylindri centro gravitatis <lb/>re&longs;pondens: pri&longs;matis autem &longs;ingula plana parallelogramma &longs;i <lb/>dividantur in æquas tum longitudinis, tum latitudinis partes, <lb/>planum per inventa puncta ductum tran&longs;ibit per centrum <lb/>gravitatis pri&longs;matis, dividet enim in partes æquales, & &longs;imi­<lb/>liter po&longs;itas, unde oritur momentorum gravitatis æqualitas. </s> <pb n="28" xlink:href="017/01/044.jpg"/> <figure id="id.017.01.044.1.jpg" xlink:href="017/01/044/1.jpg"/> <lb/> <s id="s.000256">Ut &longs;i parallelepipedi BC plana ita <lb/>dividantur, ut habeant puncta me­<lb/>dia I, & O, & per ea agatur pla­<lb/>num, con&longs;tat æqualia e&longs;&longs;e momenta <lb/>gravitatis partium IB, & IC, cùm <lb/>nullo ex capite po&longs;&longs;it oriri momento­<lb/>rum inæqualitas. </s> <s id="s.000257">At &longs;i non facies parallelogrammæ pri&longs;matis <lb/>dividendæ &longs;int, &longs;ed potius ba&longs;is, quæ &longs;æpè varia e&longs;t, & irre­<lb/>gularis, tunc inveniendum e&longs;t in ea punctum, in quo &longs;ibi oc­<lb/>currunt &longs;ectiones planorum &longs;ecantium datum corpus in mo­<lb/>menta æqualia, illudque re&longs;pondet centro gravitatis intra &longs;o­<lb/>liditatem exi&longs;tenti. </s> </p> <figure id="id.017.01.044.2.jpg" xlink:href="017/01/044/2.jpg"/> <p type="main"> <s id="s.000258">Sit autem primò ba&longs;is pri&longs;matis <lb/>trigona AHI; dividatur unum <lb/>ex lateribus ex. gr. HI bifariam <lb/>in G, planum enim tran&longs;iens per <lb/>A & G, atque bifariam &longs;ecans pa­<lb/>rallelogrammum HV tran&longs;ibit per <lb/>centrum gravitatis pri&longs;matis trigo­<lb/>ni. </s> <s id="s.000259">Nam &longs;i datum pri&longs;ma &longs;ecetur <lb/>pluribus planis parallelis plano <lb/>HV facientibus &longs;ectiones ML, <lb/>BO, NS, CE, & ex harum <lb/>&longs;ectionum extremis exeant alia <lb/>plana &longs;ecantia parallela plano AG; <lb/>ab&longs;cinduntur ex pri&longs;mate dato pa­<lb/>rallelepipeda LF, OK &c. quæ à plano AG dividuntur in <lb/>partes GL, GM æquales ac &longs;imiliter po&longs;itas; item DO, DB, &c. </s> <lb/> <s id="s.000260">Igitur &longs;ingula in eodem plano AG habent gravitatis centrum, <lb/>ac proinde tota moles ex iis parallelepipedis compo&longs;ita in eo­<lb/>dem plano habet centrum gravitatis. </s> <s id="s.000261">Quoniam verò, &longs;i adhuc <lb/>plana &longs;ecantia frequentiora &longs;int, plura fiunt parallelepipeda, <lb/>quorum omnium moles compo&longs;ita adhuc minus differt à mole <lb/>totius pri&longs;matis dati, ita ut toties multiplicari po&longs;&longs;it bi&longs;ectio, <lb/>ut demum relinquatur differentia minor quacunque minimâ <lb/>mole excogitabili; hinc fit molem compo&longs;itam ex parallelepi­<lb/>pedis illis infinitis (&longs;ic loqui liceat, quia non e&longs;t certus eorum <lb/>numerus explicabilis) habere centrum gravitatis in plano AG; <pb n="29" xlink:href="017/01/045.jpg"/>ac proinde etiam pri&longs;ma trigonum ex iis conflatum parallelepi­<lb/>pedis habere in eodem plano AG centrum &longs;uæ gravitatis, <lb/>quandoquidem non differt ab illis ni&longs;i differentiâ minore qua­<lb/>cumque minimâ excogitabili. </s> <s id="s.000262">Sunt igitur partium AGH, <lb/>AGI momenta æqualia; quia &longs;i inæqualia e&longs;&longs;ent haberent <lb/>differentiam, qua po&longs;&longs;et dari minor (neque enim e&longs;&longs;et indivi­<lb/>dua) hæc autem differentia &longs;i e&longs;&longs;et, alia non e&longs;&longs;et, quàm quæ <lb/>intercedit inter pri&longs;ma datum, & omnia parallelepipeda, cu­<lb/>jus differentiæ inæquales partes e&longs;&longs;ent in AGH, & AGI: <lb/>igitur differentia partium AGH, AGI e&longs;&longs;et minor diffe­<lb/>rentiâ pri&longs;matis, & omnium parallelepipedorum; nam e&longs;&longs;e non <lb/>pote&longs;t major, vel illi æqualis: &longs;ed jam ex hypothe&longs;i differentia <lb/>inter molem compo&longs;itam ex omnibus parallelepipedis, & pri&longs;­<lb/>ma, e&longs;t minor quacumque minimâ datâ, ergo &longs;i e&longs;&longs;ent inæ­<lb/>qualia momenta partium AGH, AGI haberent differen­<lb/>tiam minorem, & non minorem eâdem differentiâ inter pri&longs;­<lb/>ma & omnia parallelepipeda. </s> <s id="s.000263">Non &longs;unt igitur inæqualia. </s> <s id="s.000264">Res <lb/>autem forta&longs;sè &longs;ic breviùs explicabitur; &longs;i partes AGH, AGI <lb/>non &longs;unt æquales, &longs;it AGH minor quàm AGI, differentiâ Y. </s> <lb/> <s id="s.000265">Tot autem fiant bi&longs;ectiones, ut parallelepipeda relinquant <lb/>differentiam minorem quàm Y. </s> <s id="s.000266">Quia ergo parallelepipeda <lb/>in AGI habent differentiam minorem quàm Y, à parte pri&longs;­<lb/>matis AGI, illa &longs;unt majora quàm pars pri&longs;matis AGH, <lb/>quæ deficit à parte AGI differentiâ Y. </s> <s id="s.000267">Atqui parallelepepida <lb/>in AGH &longs;unt æqualia parallelepipedis in AGI, ergo etiam <lb/>parallelepipeda in AGH majora &longs;unt, quàm tota pars AGH, <lb/>quod e&longs;t manife&longs;tè fal&longs;um. </s> <s id="s.000268">Non e&longs;t igitur altera pars major, <lb/>altera minor. </s> <s id="s.000269">Porrò ex continua bi&longs;ectione laterum AC, <lb/>& CN &c. relinqui &longs;emper minorem differentiam, hoc e&longs;t &longs;e­<lb/>mi&longs;&longs;em præcedentis differentiæ, con&longs;tat, quia &longs;i AC &longs;ecetur <lb/>in P, & ducantur plana parallela planis AG, & HV, dividi­<lb/>tur CT bifariam in Q, & e&longs;t TP parallelepipedum ablatum <lb/>duplum pri&longs;matis trigoni CPQ, cui æquale e&longs;t pri&longs;ma APX; <lb/>adeóque duobus hi&longs;ce pri&longs;matis æquale e&longs;t ablatum parallele­<lb/>pipedum TP, quod e&longs;t &longs;emi&longs;&longs;is differentiæ ATC, quæ priùs <lb/>relinquebatur: & eadem e&longs;t de cæteris ratio. </s> <s id="s.000270">Quare &longs;i ex datâ <lb/>quantitate auferatur &longs;emi&longs;&longs;is, & iterum &longs;emi&longs;&longs;is re&longs;idui, & &longs;ic <lb/>in infinitum, nece&longs;&longs;e e&longs;t aliquando eò devenire, ut re&longs;idua <pb n="30" xlink:href="017/01/046.jpg"/>quantitas minor &longs;it quacunque datâ quantitate, ut colligitur <lb/>ex prop. 1. lib. 10. Eucl. </s> <s id="s.000271">Ideo fieri non pote&longs;t, ut pri&longs;mate di­<lb/>vi&longs;o à plano AG, altera pars excedat momenta alterius quan­<lb/>titate Y, quia tot po&longs;&longs;unt ab&longs;cindi purallelepipeda, ut relin­<lb/>quatur differentia illorum à pri&longs;mate minor, quàm &longs;it Y: pla­<lb/>num autem AG æqualiter dividit momenta parallelepipedo­<lb/>rum, igitur cum tota re&longs;idua differentia minor &longs;it quam Y, <lb/>e&longs;&longs;e omnino non pote&longs;t, ut altera pars habeat exce&longs;&longs;um quan­<lb/>titati Y re&longs;pondentem &longs;i enim quantitates illæ differrent, po&longs;­<lb/>&longs;et dari quantitas minor illarum differentiâ; &longs;ed non pote&longs;t hu­<lb/>ju&longs;modi minor quantitas dari, nam quælibet data e&longs;t major, <lb/>igitur non differunt, &longs;ed &longs;unt æquales. </s> </p> <p type="main"> <s id="s.000272">His ita con&longs;titutis facilè definitur punctum centro gravitatis <lb/>imminens in ba&longs;i pri&longs;matis: quia enim o&longs;ten&longs;um e&longs;t planum <lb/>ab angulo per medium latus oppo&longs;itum ductum tran&longs;ire per <lb/>centrum gravitatis, & dividere in momenta æqualia totum <lb/>pri&longs;ma, centrum gravitatis erit non &longs;olùm in plano AG, &longs;ed <lb/>etiam in plano IN propter eandem rationem. </s> <s id="s.000273">Punctum igi­<lb/> <figure id="id.017.01.046.1.jpg" xlink:href="017/01/046/1.jpg"/><lb/>tur D, in quo occurrunt &longs;ibi communes <lb/>&longs;ectiones planorum &longs;ecantium, & ba&longs;is, e&longs;t <lb/>punctum, quod quæritur, imminens centro <lb/>gravitatis. </s> <s id="s.000274">Punctum D autem &longs;ecare rectam <lb/>NI ita, ut ND ad DI &longs;it ut 1 ad 2, &longs;ic <lb/>o&longs;tenditur. </s> <s id="s.000275">Ducatur recta NG, quæ per 2. lib. 6. e&longs;t paral­<lb/>lela ip&longs;i AI; ergo ut HG ad HI, ita NG ad AI per 4. lib. 6. <lb/>ergo NG ad AI e&longs;t ut 1 ad 2: ergo triangula NGA, AGI <lb/>&longs;unt ut 1 ad 2, per 1. lib. 6. </s> <s id="s.000276">Cum autem ut ND ad DI, <lb/>ita NDA ad DIA, & NDG ad DIG per 1. 6. erit <lb/>etiam, ex 12. lib. 5. ut ND ad DI, ita NGA ad AGI, <lb/>hoc e&longs;t 1 ad 2. </s> <s id="s.000277">Eadem ratione o&longs;tenditur GD ad DA e&longs;&longs;e, <lb/>ut 1 ad 2. </s> <s id="s.000278">Vel etiam breviùs: Quia enim NG, AI &longs;unt pa­<lb/>rallelæ, triangula NDG, ADI &longs;unt &longs;imilia propter angulo­<lb/>rum æqualitatem; ergo ut NG ad AI, hoc e&longs;t ut 1 ad 2, <lb/>ita GD ad DA, & ND ad DI. </s> <s id="s.000279">Quare &longs;atis erit latus unum <lb/>trianguli bifariam &longs;ecare, & ab oppo&longs;ito angulo rectam duco­<lb/>re; cujus tertia pars ver&longs;us ba&longs;im divi&longs;am dabit centrum gravi­<lb/>tatis trianguli. </s> </p> <p type="main"> <s id="s.000280">Jam verò &longs;i ba&longs;is pri&longs;matis quadrangula fuerit parallelogram- <pb n="31" xlink:href="017/01/047.jpg"/>ma, ductis diametris apparebit quæ&longs;itum punctum, per quod <lb/>tran&longs;eunt omnia plana dividentia æqualiter corporis dati mo­<lb/>menta, cum &longs;int partes utrinque æquales, & &longs;imiliter po&longs;itæ. </s> <lb/> <s id="s.000281">Et ob eandem rationem &longs;i ba&longs;is pri&longs;matis fuerit aliqua ex figu­<lb/>ris ordinatis, &longs;eu æquilateris; centrum figuræ e&longs;t punctum im­<lb/>minens centro gravitatis; planum &longs;i <expan abbr="quid&etilde;">quidem</expan> per illud tran&longs;iens, & <lb/>per <expan abbr="unũ">unum</expan> angulorum, dividit <expan abbr="totũ">totum</expan> pri&longs;ma in partes æquales &longs;imi­<lb/>literque po&longs;itas; atque adeò momenta hinc, & hinc &longs;unt æqualia. </s> </p> <p type="main"> <s id="s.000282">At &longs;i ba&longs;is trapezia fuerit, duc utramque <lb/> <figure id="id.017.01.047.1.jpg" xlink:href="017/01/047/1.jpg"/><lb/>diametrum EC, & BD: tum in ba&longs;i trigo­<lb/>nâ BCD pri&longs;matis partialis inveniatur <lb/>punctum centro gravitatis re&longs;pondens (pun­<lb/>ctum hoc deinceps, brevitatis gratiâ, dice­<lb/>tur centrum gravitatis, quamvis per abu&longs;ionem) & &longs;it H; & in <lb/>oppo&longs;ita ba&longs;i trigona reliqui pri&longs;matis BDE pariter invenia­<lb/>tur punctum F; & per utrumque punctum tran&longs;eat planum <lb/>FH; nam in hoc eodem plano e&longs;t centrum gravitatis totius <lb/>pri&longs;matis trapezij, quod dividitur in momenta æqualia: hoc &longs;i­<lb/>quidem planum tran&longs;iens per H gravitatis momenta æqualia <lb/>habet hinc, & hinc in pri&longs;mate trigono BDC; &longs;imiliter cum <lb/>tran&longs;eat per F, habet hinc, & hinc momenta æqualia gravitatis <lb/>pri&longs;matis trigoni BED: &longs;i igitur æqualia æqualibus jungantur, <lb/>planum idem æqualiter partitur momenta gravitatis pri&longs;matis <lb/>trapezij EDCB, & in eo e&longs;t centrum gravitatis illius. </s> <s id="s.000283">Eadem <lb/>ratione in ba&longs;i trigona EBC inveniatur punctum G, & in ba&longs;i <lb/>EDC punctum S, per quæ &longs;i agatur planum GS, in eo pariter <lb/>erit <expan abbr="centrũ">centrum</expan> gravitatis totius pri&longs;matis trapezij. </s> </p> <p type="main"> <s id="s.000284">E&longs;t igitur centrum gravitatis in communi <lb/> <figure id="id.017.01.047.2.jpg" xlink:href="017/01/047/2.jpg"/><lb/>&longs;ectione planorum FH, & GS; ac proinde <lb/>punctum I illud e&longs;t, quod quæritur. </s> <s id="s.000285">Aliter <lb/>etiam, & facillimè in ba&longs;i trapezia ABCD <lb/>invenitur centrum gravitatis: ductis enim <lb/>diametris AC, BD, altera diameter ex. gr. <lb/>AC bifariam &longs;ecetur in E, ducanturque rectæ <lb/>DE, BE; trianguli ADC centrum gravi­<lb/>tatis e&longs;t in recta DE, & quidem in F, ita ut EF <lb/>&longs;it tertia pars totius ED, ut con&longs;tat ex paulò <lb/>ante demon&longs;tratis. </s> <s id="s.000286">Ducatur igitur FG pa- <pb n="32" xlink:href="017/01/048.jpg"/>rallela alteri diametro BD, & erit &longs;imiliter G centrum gravita­<lb/>tis trianguli ABC, quia per 2. lib. 6. ut EF ad FD, ita EG <lb/>ad GB; Quia ergo diameter AC &longs;ecatur in H, &longs;umatur FO <lb/>æqualis ip&longs;i GH, & e&longs;t O centrum gravitatis trapezij, e&longs;t enim <lb/>triangulum ABC ad triangulum ADC, ut FO ad OG, hoc <lb/>e&longs;t ut HG ad HF. </s> <s id="s.000287">E&longs;t autem HG ad HF ut BI ad ID pro­<lb/>pter paralleli&longs;mum linearum GF, BD. </s> <s id="s.000288">Porrò con&longs;tat triangu­<lb/>lum ABC ad triangulum ADC e&longs;&longs;e ut BI ad ID, nam trian­<lb/>gula ABI, ADI &longs;unt ut ba&longs;es BI, DI, item BCI, DCI &longs;unt <lb/>ut eædem ba&longs;es BI, DI per 1. lib. 6; igitur, & totum triangu­<lb/>lum ABC ad totum ADC e&longs;t ut BI ad DI: igitur, & trian­<lb/>gulum ABC ad triangulum ADC e&longs;t ut FO ad OG. </s> </p> <figure id="id.017.01.048.1.jpg" xlink:href="017/01/048/1.jpg"/> <p type="main"> <s id="s.000289">Hinc facilis patet via ad inve&longs;ti­<lb/>gandum idem punctum in ba&longs;i pri&longs;­<lb/>matis pentagoni BDEAC. </s> <s id="s.000290">Pri­<lb/>mùm enim ducto plano per BE, in­<lb/>veniatur in ba&longs;i trigonâ BDE <lb/>punctum R, & in ba&longs;i BEAC qua­<lb/>drangulâ punctum P; & ducto plano <lb/>per RP, in eo erit centrum gravi­<lb/>tatis pri&longs;matis pentagoni, cum in eo­<lb/>dem &longs;int centra gravitatis partium. </s> <lb/> <s id="s.000291">Deinde ducto per D & A plano, inveniatur in ba&longs;i trigona <lb/>DEA punctum L centrum gravitatis, & in ba&longs;i quadrangu­<lb/>lâ ACBD punctum M centrum gravitatis: in plano pariter <lb/>ducto per ML e&longs;t centrum gravitatis totius pri&longs;matis pentago­<lb/>ni, quod proinde e&longs;t in communi planorum per PR, & LM <lb/>ductorum &longs;ectione; atque adeò punctum, quod quæritur, e&longs;t O. </s> <lb/> <s id="s.000292">Eadem e&longs;t methodus in pri&longs;mate hexagono; ducto enim plano <lb/>dividente in duo pri&longs;mata, quorum alterum e&longs;t trigonum, al­<lb/>terum pentagonum, inveniatur utriu&longs;que centrum gravitatis, <lb/>& per inventa puncta agatur planum. </s> <s id="s.000293">Deinde iterum alio pla­<lb/>no &longs;ecetur in duo pri&longs;mata, quorum alterum pariter &longs;it trigo­<lb/>num, alterum pentagonum, & per inventa &longs;ingularia gravi­<lb/>tatum centra agatur planum: duo &longs;iquidem plana ducta per <lb/>centra gravitatis partium, tran&longs;eunt pariter per centrum gra­<lb/>vitatis totius, quod e&longs;t in communi eorum &longs;ectione. </s> <s id="s.000294">Eademque <lb/>de reliquis pri&longs;matis e&longs;t ratio. </s> </p> <pb n="33" xlink:href="017/01/049.jpg"/> <p type="main"> <s id="s.000295">Sed hæc indica&longs;&longs;e &longs;ufficiat, quæ operi Mechanico &longs;atis e&longs;&longs;e <lb/>po&longs;&longs;unt in omnibus ferè pri&longs;matis: Si enim ba&longs;is non fuerit <lb/>planè rectilinea, in&longs;cripto polygono rectilineo, quod mini­<lb/>mùm differat à plano ba&longs;is, quæres ejus centrum gravitatis, <lb/>methodo jam tradita; illoque u&longs;urpato tanquam vero dati pri&longs;­<lb/>matis centro quæ&longs;ito, minimùm aberrabis; aliquando tamen <lb/>aberrabis, aliquando continget, ut inventum cum quæ&longs;ito <lb/>conveniat. </s> <s id="s.000296">Quod &longs;i accuratiori inve&longs;tigatione opus fuerit: <lb/>quemadmodum in cæteris corporibus, quæ continuum ductum <lb/>non habent, &longs;ed inæquali cra&longs;&longs;itudine cre&longs;cunt, aut decre&longs;­<lb/>cunt, ut in obeli&longs;cis, aut pyramidibus truncatis, reliqui&longs;què <lb/>planè inordinatis molibus; tunc ad geometricam Centrobary­<lb/>ces methodum confugiendum e&longs;t; quam hic ego non per&longs;e­<lb/>quor. </s> <s id="s.000297">Praxes igitur aliquæ proponendæ &longs;unt, quibus centrum <lb/>gravitatis phy&longs;icè per&longs;pectum habere po&longs;&longs;imus in corporibus, <lb/>quorum frequentior, vulgari&longs;que u&longs;us e&longs;&longs;e pote&longs;t. </s> </p> <p type="main"> <s id="s.000298">Prima praxis &longs;it ad inveniendum gra­<lb/> <figure id="id.017.01.049.1.jpg" xlink:href="017/01/049/1.jpg"/><lb/>vitatis centrum in cingulis, quæ laminis <lb/>quoque communis e&longs;&longs;e pote&longs;t. </s> <s id="s.000299">Sit datum <lb/>cingulum AH, quod primùm &longs;u&longs;penda­<lb/>tur ex H, & inde pendens perpendicu­<lb/>lum &longs;ecet oppo&longs;itum latus IA in C; note­<lb/>tur igitur punctum C. </s> <s id="s.000300">Deinde iterum <lb/>&longs;u&longs;pendatur ex R, & perpendiculum ca­<lb/>dat in punctum F, quod notetur. </s> <s id="s.000301">His cognitis ducatur filum <lb/>ex R in F, ibique intentum alligetur; aliud filum &longs;imiliter <lb/>ex H in C ducatur, & &longs;ecans in S filum RF, dabit punctum S <lb/>quæ&longs;itum centrum gravitatis: ex quo &longs;i &longs;u&longs;penderetur datum <lb/>cingulum, maneret horizonti parallelum. </s> <s id="s.000302">Quod &longs;i e&longs;&longs;et corpus <lb/>talis figuræ, ut &longs;patium non clauderet, &longs;ed haberet angulum <lb/>cavum, aut e&longs;&longs;et fru&longs;tum annulare, eadem e&longs;t methodus factâ <lb/>&longs;u&longs;pen&longs;ione illius ex duobus punctis, ex quibus perpendiculum <lb/>cadere po&longs;&longs;it intrà corporis &longs;uperficiem; in qua &longs;i notentur <lb/>puncta, per quæ tran&longs;it, & ducantur fila, ut priùs, eorum com­<lb/>munis &longs;ectio dabit quæ&longs;itum centrum gravitatis. </s> <s id="s.000303">Hinc &longs;i vel la­<lb/>mina e&longs;&longs;et perforanda, ut axi infigeretur, vel cingulum e&longs;&longs;et <lb/>axi imponendum, in utrâque &longs;uperficie oppo&longs;ita quærere opor­<lb/>teret punctum S, ut axis per centrum gravitatis tran&longs;iret, eique <pb n="34" xlink:href="017/01/050.jpg"/>uterque polus re&longs;ponderet: in cingulis autem præterea haben­<lb/>da e&longs;&longs;et ratio tran&longs;ver&longs;ariorum, per quæ axis infigendus e&longs;&longs;et, <lb/>ea enim po&longs;&longs;unt centrum gravitatis compo&longs;itæ in alio puncto <lb/>con&longs;tituere. </s> </p> <p type="main"> <s id="s.000304">Secunda praxis laminis poti&longs;&longs;imùm accommodata, in quibus <lb/>punctum medium &longs;atis accuratè inquiritur, ut &longs;i lamina metal­<lb/>lica e&longs;&longs;et in calicem excavanda, hæc e&longs;&longs;e pote&longs;t. </s> <s id="s.000305">Impone lami­<lb/>nam acutæ cu&longs;pidi cultri, aut &longs;tyli, eamque ultrò citróque <lb/>tanti&longs;per move, dum con&longs;i&longs;tat citrà periculum cadendi: <lb/>punctum enim, quod à cultri aut &longs;tyli cu&longs;pide notatur, cen­<lb/>trum e&longs;t quæ&longs;itum. </s> </p> <p type="main"> <s id="s.000306">Tertia praxis &longs;it iis corporibus conveniens, quæ præ&longs;tant <lb/>longitudine, qualia &longs;unt p&longs;eudocylindrica, conica, pyrami­<lb/>des &c. quæ &longs;i non prædita &longs;int multâ gravitate, imponantur <lb/>funiculo brevi horizontaliter exten&longs;o, at &longs;i graviora fuerint, vel <lb/>cylindrulo vel aciei pri&longs;matis trigoni imponantur, & u&longs;que <lb/>dum in æquilibrio con&longs;i&longs;tant, promoveantur: ubi enim quie­<lb/>verit corpus impo&longs;itum, ex loco contactûs innote&longs;cet vel <lb/>punctum, &longs;i in puncto &longs;e contingant, vel linea, &longs;i in lineâ, per <lb/>quam &longs;i ducatur planum à centro terræ, di&longs;tinguetur impo&longs;i­<lb/>tum corpus in momenta gravitatis æqualia. </s> <s id="s.000307">Inventâ autem hu­<lb/>ju&longs;modi lineâ facilè prodet &longs;e quæ&longs;itum punctum. </s> </p> <p type="main"> <s id="s.000308">Quarta praxis non multùm di&longs;tat à &longs;uperiore: &longs;i nimirum <lb/>oblatum corpus impo&longs;ueris plano alicui horizontali, quod ta­<lb/>men à pavimento ab&longs;it mediocri aliquo intervallo, habeat au­<lb/>tem extremum marginem exactè rectum: extra &longs;uppo&longs;iti pla­<lb/>ni marginem illud paulatim promove, donec eò venerit, ut &longs;i <lb/>vel minimum ulteriùs promoveretur, &longs;ponte caderet; ibíque <lb/>&longs;ecundùm rectitudinem marginis plani duc &longs;tylo lineam in cor­<lb/>pore impo&longs;ito. </s> <s id="s.000309">Deinde &longs;uperficie eâdem planum tangente, &longs;i <lb/>corpus, præter longitudinem, non modicam præterea habeat <lb/>latitudinem, convertatur aliquantulum, & &longs;imili methodo in­<lb/>venietur linea alia &longs;ecans priorem in puncto quæ&longs;ito, quod &longs;ci­<lb/>licet re&longs;pondet centro gravitatis intra corporis &longs;oliditatem de­<lb/>lite&longs;centi. </s> </p> <p type="main"> <s id="s.000310">Hæc &longs;unt quæ Mechanices in&longs;tituto &longs;ufficere po&longs;&longs;int ad cen­<lb/>trum gravitatis inveniendum; in molibus enim majoribus, quæ <lb/>plerumque vix differunt à pri&longs;matis, non indigemus commu- <pb n="35" xlink:href="017/01/051.jpg"/>niter Geometricâ &longs;ubtilitate. </s> <s id="s.000311">Illud re&longs;tat, ut earum, quas at­<lb/>tuli praxes, ratio, & cau&longs;æ explicentur, ex quibus clarion ha­<lb/>beatur notitia eorum, quæ ad centrum gravitatis pertinent. <lb/></s> </p> <p type="head"> <s id="s.000312"><emph type="center"/>CAPUT VI.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000313"><emph type="center"/><emph type="italics"/>Affertur ratio prædictarum praxeon.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000314">UT palam fiat praxibus capite &longs;uperiore allatis inveniri <lb/>punctum re&longs;pondens centro gravitatis, quod inquiritur, <lb/>indicandi &longs;unt fontes, ex quibus illæ deducuntur. </s> <s id="s.000315">Earum ita­<lb/>que ratio petenda e&longs;t ex gravium naturâ, quæ extra locum &longs;ibi <lb/>debitum con&longs;tituta, in medio videlicet leviore, conantur de­<lb/>or&longs;um pro viribus, ni&longs;i impediantur: quod &longs;i interpellentur <lb/>quidem, non tamen pror&longs;us de&longs;cen&longs;u prohibeantur, de&longs;cen­<lb/>dunt, prout fert ob&longs;tantium impedimentorum conditio. </s> <s id="s.000316">Sic <lb/>lapis &longs;phæricus in montis clivo po&longs;itus cùm non valeat rectâ; <lb/>&longs;icut in aëre libero, deor&longs;um ferri, per planum illud inclina­<lb/>tum de&longs;cendit: Sic plumbum, quod filo adnectitur laqueari, à <lb/>perpendiculo remotum de&longs;cendit circulariter. </s> <s id="s.000317">Porrò quæ de <lb/>toto ip&longs;o corpore vera e&longs;&longs;e intelligimus, ejus quoque partibus <lb/>&longs;ingulis conveniunt; cùm enim &longs;ingulæ &longs;uam habeant gravita­<lb/>tem, ni&longs;i quid ob&longs;tet, de&longs;cendunt. </s> <s id="s.000318">Jam verò &longs;i contingat ita <lb/>corpus grave oppo&longs;ito extrin&longs;ecùs obice impediri, ut cunctæ <lb/>&longs;imul partes, qua&longs;i moles unà de&longs;cendere nequeant; &longs;ublato <lb/>partium nexu de&longs;cendunt, quæcunque carent impedimento: <lb/>ut &longs;i ceream candelam, aut glaciem, quam manu &longs;u&longs;tines, igni <lb/>admoveas; haud dubium, quin partes extremæ igni proximæ <lb/>lique&longs;centes, &longs;olutâ unione cum cæteris, &longs;uis nutibus deor&longs;um <lb/>latæ liberè de&longs;cendant. </s> <s id="s.000319">At &longs;i partes omnes colligatæ invicem <lb/>permaneant, eandemque figuram &longs;ervent; corpore illo &longs;u&longs;pen­<lb/>&longs;o aut &longs;u&longs;tentato, fieri non pote&longs;t, ut partes aliquæ de&longs;cendant, <lb/>quin aliæ, ouæ è regione &longs;unt trans &longs;u&longs;pen&longs;ionis, aut &longs;u&longs;tenta­<lb/>tionis punctum, a&longs;cendant; id autem harum gravitati re­<lb/>pugnat: non igitur a&longs;cendere po&longs;&longs;unt, ni&longs;i de&longs;cendentes op­<lb/>po&longs;itæ viribus ac momentis præ&longs;tent ita, ut harum gravitati <pb n="36" xlink:href="017/01/052.jpg"/>vim inferre valeant. </s> <s id="s.000320">Quare &longs;i fiat corporis &longs;u&longs;pen&longs;i, aut &longs;u&longs;ten­<lb/>tati con&longs;i&longs;tentia, argumentum e&longs;t æqualitatis momentorum <lb/>punctum &longs;u&longs;pen&longs;ionis, aut &longs;u&longs;tentationis hinc, & hinc u&longs;que­ <lb/>quaque circun&longs;tantium; &longs;i qua enim e&longs;&longs;et inæqualitas, alterutra <lb/>pars præponderaret, & ad motum incitaretur. </s> </p> <figure id="id.017.01.052.1.jpg" xlink:href="017/01/052/1.jpg"/> <p type="main"> <s id="s.000321">Sit corpus grave AB, cujus <lb/>centrum gravitatis H, linea di­<lb/>rectionis HT in centrum uni­<lb/>ver&longs;i producta. </s> <s id="s.000322">Si &longs;u&longs;pendatur <lb/>ex puncto C, quod e&longs;t in eadem <lb/>lineâ directionis, nece&longs;&longs;ariò con­<lb/>&longs;i&longs;tit corpus horizonti paralle­<lb/>lum, quia rectâ de&longs;cendere non <lb/>pote&longs;t per HT, cum in C reti­<lb/>neatur; neque alterutra pars pote&longs;t de&longs;cendere, quia momen­<lb/>ta partis HB, quibus deor&longs;um nititur, æqualia &longs;unt momen­<lb/>tis, quibus pars HA re&longs;i&longs;tit, no elevetur; & vici&longs;&longs;im viribus <lb/>gravitatis- AH cætero qui de&longs;cen&longs;uræ reluctatur gravitas HB <lb/>pari ni&longs;u repugnans, ne attollatur; totum ergo con&longs;i&longs;tit. </s> <s id="s.000323">At &longs;i <lb/>ex M puncto &longs;u&longs;pendatur, non pote&longs;t quidem per MT per­<lb/>pendicularem de&longs;cendere versùs terræ centrum, &longs;ed neque <lb/>con&longs;i&longs;tet horizonti parallelum; quia &longs;i planum intelligatur ex <lb/>terræ centro per rectam MT ductum, non dividitur corpus in <lb/>momenta æqualia, cum non tran&longs;eat per H centrum gravita­<lb/>tis; igitur cum majora &longs;int momenta partis MB, quàm par­<lb/>tis MA, illa præponderabit, atque de&longs;cendens circa <lb/>punctum M permanens convertetur, donec centrum gra­<lb/>vitatis H &longs;it in perpendiculari MT, cui congruat recta <lb/>MO: tunc autem demum con&longs;i&longs;tet, quia planum tran&longs;iens <lb/>per MHO æqualiter di&longs;pertit momenta gravitatis; neutrâ <lb/>autem parte præponderante, utraque quie&longs;cit. </s> <s id="s.000324">Idem dicen­<lb/>dum, &longs;i corpus ex I puncto &longs;u&longs;penderetur; tunc enim &longs;o­<lb/>lùm fieret con&longs;i&longs;tentia, ubi in eadem directionis lineâ <lb/>e&longs;&longs;et punctum I atque H centrum gravitatis. </s> <s id="s.000325">Quod &longs;i du­<lb/>plici funiculo &longs;u&longs;pendatur pondus, & illi paralleli non &longs;int, <lb/>quia neque horizonti perpendiculares, illi &longs;i producantur, <lb/>concurrent in punctum aliquod lineæ directionis, &longs;ivè &longs;upra <lb/>pondus, &longs;ivè infra, pro ratione angulorum, quos con&longs;tituunt. <pb n="37" xlink:href="017/01/053.jpg"/>Sit enim corpus AB, cujus cen­<lb/> <figure id="id.017.01.053.1.jpg" xlink:href="017/01/053/1.jpg"/><lb/>trum gravitatis O, linea directio­<lb/>nis IOC, &longs;i ex I &longs;u&longs;pendatur <lb/>per O, in co &longs;itu manebit; ergo <lb/>etiam, &longs;i funiculi &longs;int IH, IL, <lb/>manebit: ergo etiam, &longs;i &longs;int PH, <lb/>SL, funiculorum enim longitudo <lb/>nihil facit; Idem etiam dicendum <lb/>cum funiculi &longs;unt DH, FL; pro­<lb/>ducti enim concurrunt cum linea <lb/>directionis in C, &longs;emper &longs;cilicet <lb/>perinde &longs;e habet atque, &longs;i ex I &longs;u&longs;penderetur. </s> </p> <p type="main"> <s id="s.000326">Quæ verò de &longs;u&longs;pen&longs;ione dicta &longs;unt, ea, analogiâ &longs;ervatâ, de <lb/>&longs;u&longs;tentatione quoque dicta intelligantur; tunc &longs;olùm videlicet <lb/>corpus con&longs;i&longs;tere, cùm ex centro gravitatis ducta directionis <lb/>linea tran&longs;it per punctum &longs;u&longs;tentationis, quia tunc &longs;olùm æqua­<lb/>lia hinc, & hinc &longs;unt momenta virtutis ad de&longs;cendendum, at­<lb/>que re&longs;i&longs;tentiæ ad a&longs;cendendum: ut quando corpus aliquod <lb/>imponitur cono, vel pri&longs;ma &longs;phæræ, vel &longs;egmentum &longs;phæri­<lb/>cum, plano, vel cylindrus aciei pri&longs;matis trigoni in tran&longs;ver­<lb/>&longs;um; cadet enim in alterutram partem impo&longs;itum corpus, ni&longs;i <lb/>in eadem linea fuerint centrum terræ, punctum contactus, & <lb/>centrum gravitatis. </s> <s id="s.000327">Quod &longs;i corpus &longs;u&longs;tentans, atque &longs;u&longs;tenta­<lb/>tum &longs;e tangant in linea, opus e&longs;t lineam illam e&longs;&longs;e in plano per <lb/>lineam directionis ducto, ut fiat æqualium momentorum con­<lb/>&longs;i&longs;tentia. </s> <s id="s.000328">Quare &longs;i impo&longs;itum corpus con&longs;i&longs;tat, certi&longs;&longs;imo ar­<lb/>gumento con&longs;tabit punctum, &longs;eu lineam, contactûs re&longs;pon­<lb/>dere centro gravitatis. </s> <s id="s.000329">Hinc patet ratio &longs;ecundæ, & tertiæ <lb/>praxis. </s> </p> <p type="main"> <s id="s.000330">In prima praxi quia facies extima, &longs;upra quam perpendicu­<lb/>lum liberè movetur, e&longs;t in plano verticali, perpendiculum HC <lb/>e&longs;t parallelum lineæ directionis corporis gravis, quæ tran&longs;it <lb/>etiam per punctum &longs;u&longs;pen&longs;ionis H: planum igitur tran&longs;iens per <lb/>punctum &longs;u&longs;pen&longs;ionis H, & per perpendiculum HC, tran&longs;it <lb/>quoque per centrum gravitatis corporis. </s> <s id="s.000331">Cum verò idem pror­<lb/>&longs;us dicendum &longs;it de plano tran&longs;eunte per punctum &longs;u&longs;pen&longs;io­<lb/>nis R, & perpendiculum RF, illud &longs;cilicet tran&longs;ire per cen­<lb/>trum gravitatis corporis; apertum e&longs;t centrum gravitatis e&longs;&longs;e in <pb n="38" xlink:href="017/01/054.jpg"/>communi illorum planorum &longs;ectione, eique re&longs;pondere <lb/>punctum S inventum. </s> </p> <p type="main"> <s id="s.000332">Quia demum, &longs;i corpus quod &longs;u&longs;tinet, & id, quod &longs;u&longs;tine­<lb/>tur, in &longs;uperficie &longs;e tangant, corpus impo&longs;itum in alterutram <lb/>partem cadere non pote&longs;t (ni&longs;i fortè &longs;uppo&longs;itum planum fuerit <lb/>inclinatum) quin planum per lineam directionis ductum ita &longs;it <lb/>extra &longs;uperficiem, in qua fit contactus, ut neque illam con­<lb/>tingat; con&longs;tat ratio quartæ praxis. </s> <s id="s.000333">Si namque planum ex ter­<lb/> <figure id="id.017.01.054.1.jpg" xlink:href="017/01/054/1.jpg"/><lb/>ræ centro ductum per C cen­<lb/>trum gravitatis dati corporis <lb/>OS, &longs;ecet &longs;ubjectum planum, <lb/>pars corporis extra marginem <lb/>FE in aëre extans minora ha­<lb/>bet momenta gravitatis, quàm <lb/>reliqua pars; hæc igitur gra­<lb/>vior non pote&longs;t ab illa elevari: <lb/>ubi verò promotum corpus eò <lb/>venerit, ut planum per cen­<lb/>trum gravitatis C ductum tangat extremum marginem &longs;ub­<lb/>jecti plani ita, ut in eodem plano, in quo e&longs;t centrum gravi­<lb/>tatis C, &longs;it etiam FE, æqualia &longs;unt gravitatis momenta par­<lb/>tis CS in aëre extantis, ac CO partis plano incumbentis; & <lb/>&longs;i vel minimum ulteriùs promoveretur, pars extra planum &longs;ub­<lb/>jectum extans gravior e&longs;&longs;et, adeóque de&longs;cenderet. </s> <s id="s.000334">Quare &longs;i in <lb/>corporis OS &longs;uperficie infimâ lineam de&longs;crip&longs;eris &longs;ecundùm <lb/>marginem FE, ea erit in plano tran&longs;eunte per centrum gravi­<lb/>tatis. </s> <s id="s.000335">Quia verò idem contingit, &longs;i ii&longs;dem &longs;uperficiebus &longs;e con­<lb/>tingentibus alium &longs;itum corpori dederis, pariterque eò u&longs;que <lb/>promoveris, ut citrà cadendi periculum promoveri ulteriùs <lb/>non po&longs;&longs;it; alia linea &longs;ecundùm marginem FE ducta erit pari­<lb/>ter in plano per gravitatis centrum tran&longs;eunte, &longs;ecabitque <lb/>priorem lineam, punctum mutuæ linearum &longs;ectionis illud e&longs;&longs;e, <lb/>quod quæritur, &longs;atis liquet. </s> <s id="s.000336">Hæc e&longs;t di&longs;par philo&longs;ophandi ra­<lb/>tio, &longs;i pars CO adeò longa e&longs;&longs;et, ut etiam extaret extra an­<lb/>gu&longs;tias &longs;ubjecti plani; &longs;emper enim con&longs;i&longs;tit impo&longs;itum corpus, <lb/>quandiù planum per lineam directionis tran&longs;iens, aut tangit, <lb/>aut &longs;ecat &longs;ubjectum planum. </s> <s id="s.000337">Quandocunque enim linea di­<lb/>rectionis non tran&longs;it per punctum, vel lineam, vel &longs;uperficiem, <pb n="39" xlink:href="017/01/055.jpg"/>in quibus corpus grave tangitur à &longs;u&longs;tentante (idem dic de <lb/>&longs;u&longs;pen&longs;ione) &longs;emper in alterutram partem grave inclinatur, in <lb/>eam &longs;cilicet, in qua reperitur centrum gravitatis, cùm plura <lb/>&longs;int ex ea parte momenta gravitatis. <lb/></s> </p> <p type="head"> <s id="s.000338"><emph type="center"/>CAPUT VII.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000339"><emph type="center"/><emph type="italics"/>Quomodo gravia &longs;pontè a&longs;cendentia de&longs;cendant.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000340">EX his, quæ proximè dicta &longs;unt, grave &longs;u&longs;tentatum in eam <lb/>partem inclinari, in qua e&longs;t gravitatis centrum, oritur ali­<lb/>quando a&longs;cen&longs;us gravium, qui rerum naturalium ignaros in <lb/>admirationem adducit non mediocrem, &longs;i maximè tunc cor­<lb/>pus de&longs;cendere intelligant, quando illud cernunt altiùs ab ho­<lb/>rizonte a&longs;cendere. </s> <s id="s.000341">Sit <lb/> <figure id="id.017.01.055.1.jpg" xlink:href="017/01/055/1.jpg"/><lb/>enim &longs;uper planum in­<lb/>clinatum RN rota tantæ <lb/>latitudinis, ut po&longs;&longs;it in <lb/>plano verticali erecta <lb/>permanere, dum conver­<lb/>titur; habeat autem ad <lb/>PO adnexam laminam <lb/>plumbeam cra&longs;&longs;iorem, <lb/>adeò ut totius rotæ cen­<lb/>trum gravitatis &longs;it S. </s> <s id="s.000342">Jam <lb/>verò ea &longs;it plani &longs;ubjecti <lb/>inclinatio, ut rotâ illud <lb/>tangente puncto H, li­<lb/>nea à terræ centro per H punctum contactûs tran&longs;iens non <lb/>tran&longs;eat per S centrum gravitatis (&longs;eu ut veriùs dicam, quia <lb/>extima &longs;uperficies rotæ cylindrica tangit planum in lineâ, pla­<lb/>num ex centro terræ per lineam contactûs in H ductum non <lb/>tran&longs;eat per S) &longs;ed illud relinquat versùs &longs;uperiorem plani par­<lb/>tem N; planum per rectam HO perpendicularem ductum <lb/>di&longs;tinguit rotam in momenta gravitatis inæqualia: non pote&longs;t <lb/>igitur rota in H con&longs;i&longs;tere, &longs;ed convertitur, ita ut tangat pla­<lb/>num in I primùm, deinde in E, demùm in P, ubi con&longs;i&longs;tet, <pb n="40" xlink:href="017/01/056.jpg"/>cùm linea directionis ex gravitatis centro S ducta in terræ cen­<lb/>trum tran&longs;ibit per P locum contactús. </s> <s id="s.000343">In hac autem conver­<lb/>&longs;ione dum rotæ partes inter H & P deinceps aptantur &longs;ubjecto <lb/>plano, centrum quidem molis a&longs;cendit, &longs;ed centrum gravita­<lb/>tis S de&longs;cendit. </s> <s id="s.000344">Lineam porrò SP minorem e&longs;&longs;e lineá SE, & <lb/>hanc minorem lineâ SI, & hanc lineâ SH, con&longs;tat ex prop.7. <lb/>lib.3. Eucl. &longs;i nimirum per S, & C centrum agatur diameter. </s> <lb/> <s id="s.000345">Non e&longs;t tamen cen&longs;endum quamlibet ponderis additionem <lb/>in OP &longs;atis e&longs;&longs;e, ut in quolibet plano inclinato rota a&longs;cendat; <lb/>&longs;i enim di&longs;tantia centri gravitatis à centro rotæ minor fuerit, <lb/>quàm Sinus inclinationis plani, &longs;emper de&longs;cendet; &longs;i eidem <lb/>Sinui æqualis, non a&longs;cendet; &longs;i demum eo &longs;inu major, poterit <lb/>a&longs;cendere. </s> </p> <figure id="id.017.01.056.1.jpg" xlink:href="017/01/056/1.jpg"/> <p type="main"> <s id="s.000346">Sit planum inclinatum <lb/>AB, quod in H contingat <lb/>circulum (hunc &longs;umo cir­<lb/>culum, qui tran&longs;eat per <lb/>centrum tum molis tum <lb/>gravitatis rotæ) cujus cen­<lb/>trum C, & ducatur recta <lb/>CH, quæ cum perpendi­<lb/>culari HO faciat angu­<lb/>lum CHO. </s> <s id="s.000347">Quia enim <lb/>OH producta cadit in ho­<lb/>rizontem AD perpendicularis, & angulus OHA per 32.lib.1. <lb/>æqualis e&longs;t duobus internis HFA, FAH, e&longs;t autem AHC ad <lb/>contingentem factus à &longs;emidiametro rectus per 18.lib. 3. &longs;icut <lb/>& HFA e&longs;t rectus; reliquus CHO æqualis e&longs;t angulo HAF <lb/>inclinationis plani. </s> <s id="s.000348">Certum e&longs;t igitur, quòd in eam partem ro­<lb/>ra convertetur, in qua fuerit centrum gravitatis. </s> <s id="s.000349">Quoniam <lb/>verò CI e&longs;t Sinus anguli CHI, po&longs;ito radio CH, e&longs;t au­<lb/>tem CI minima omnium, quæ ex C puncto cadant in rectam <lb/>HO, manife&longs;tum e&longs;t, quòd, &longs;i centrum gravitatis fuerit cen­<lb/>tro rotæ vicinius, ut in R, rota &longs;emper de&longs;cendet, quia cen­<lb/>trum gravitatis re&longs;picit declivitatem plani: at, &longs;i fuerit <lb/>in I, a&longs;cendere non pote&longs;t, quia pars re&longs;piciens acclivita­<lb/>tem plani non præponderat: &longs;i demum longiùs à centro <lb/>di&longs;titerit, ut in S, a&longs;cendere poterit, u&longs;que dum punctum S <pb n="41" xlink:href="017/01/057.jpg"/>fuerit in lineâ perpendiculari ad horizontem tran&longs;eunte per <lb/>punctum contactûs. </s> </p> <p type="main"> <s id="s.000350">Ex his apertè con&longs;tat futurum, ut rota de&longs;cendat, &longs;i angulus, <lb/>quem in puncto contactûs faciunt lineæ ductæ ex centris mo­<lb/>lis, & gravitatis (&longs;uppono molis centrum idem e&longs;&longs;e cum centro <lb/>rotæ, quâ rota e&longs;t) minor fuerit angulo inclinationis plani, <lb/>tunc enim centrum gravitatis re&longs;picit declivitatem plani; fu­<lb/>turum autem, ut rota a&longs;cendat, &longs;i angulus ille major fuerit co­<lb/>dem angulo inclinationis, quia centrum gravitatis re&longs;picit ac­<lb/>clivitatem plani; futurum demùm, ut con&longs;i&longs;tat, &longs;i angulus il­<lb/>le fuerit æqualis eidem angulo inclinationis plani, quia nimi­<lb/>rum planum perpendiculare dividit æqualiter momenta gravi­<lb/>tatis, cum tran&longs;eat per centrum gravitatis exi&longs;tens in lineâ <lb/>perpendiculari. </s> </p> <p type="main"> <s id="s.000351">Hinc patet &longs;emper de&longs;cen&longs;uram rotam, &longs;i habeat centrum <lb/>gravitatis R, quia &longs;emper facit angulum, de quo dictum e&longs;t, <lb/>minorem angulo inclinationis, hoc e&longs;t angulo CHI, nam &longs;i <lb/>ducatur ad CR perpendicularis RE, & ex centro ducatur <lb/>recta CE, angulus CER e&longs;t maximus omnium, quos faciunt <lb/>lineæ ex punctis C, & R ductæ ad idem punctum circumfe­<lb/>rentiæ, ut mox o&longs;tendam; atqui CER minor e&longs;t angulo CHI, <lb/>(quia ob lineas RE, IH parallelas, angulus IHC internus <lb/>per 29.lib.1. e&longs;t æqualis externo RLC, & RLC externus per <lb/>16. lib. 1. major e&longs;t interno CER, ac proinde IHC major <lb/>quàm CER) igitur quicunque angulus con&longs;titutus à rectis <lb/>exeuntibus ex C, & R minor e&longs;t angulo inclinationis; atque <lb/>adeò &longs;emper de&longs;cendet. </s> </p> <p type="main"> <s id="s.000352">At &longs;i centrum gravitatis fuerit S, ductâ ad CS perpendicu­<lb/>lari SM, angulus omnium maximus e&longs;t CMS: hic autem e&longs;t <lb/>æqualis externo CKI, cum IK, & SM parallelæ &longs;int con&longs;ti­<lb/>tutæ; angulus verò CKI externus major e&longs;t interno CHI, <lb/>igitur angulus CMS major e&longs;t angulo CHI, hoc e&longs;t angulo <lb/>inclinationis. </s> <s id="s.000353">A&longs;cendere igitur poterit rota, quando angulus <lb/>ad contractum factus à lineis ex C, & S exeuntibus major e&longs;t <lb/>angulo inclinationis; &longs;in autem contactus fiat in co puncto, ad <lb/>quod fit angulus æqualis, con&longs;i&longs;tet; &longs;i in iis punctis, ad quæ fit <lb/>angulus minor, de&longs;cendet. </s> </p> <p type="main"> <s id="s.000354">Porrò quamvis iis, qui in A&longs;tronomicarum Pro&longs;taphære&longs;eon <pb n="42" xlink:href="017/01/058.jpg"/>doctrinâ ver&longs;ati &longs;unt, &longs;upervacaneum &longs;it o&longs;tendere angulum <lb/>ad peripheriam factum à Radio circuli, & à linea perpendicu­<lb/>lari in diametrum, e&longs;&longs;e maximum omnium, qui fieri po&longs;&longs;int à <lb/>Radio, & à lineâ ductâ ex eodem diametri puncto, in quod <lb/>cadebat perpendicularis; ut omnibus tamen fiat &longs;atis, non pi­<lb/> <figure id="id.017.01.058.1.jpg" xlink:href="017/01/058/1.jpg"/><lb/>gebit hîc demon&longs;trare. </s> <s id="s.000355">Sit in diametro <lb/>circuli punctum R extra centrum C, & <lb/>ad CR ducatur perpendicularis HR, <lb/>quæ producta in G, bifariam dividitur <lb/>in R: & ductis ex centro rectis CH, <lb/>CG æqualibus, &longs;unt anguli CHR, <lb/>CGR æquales, per 5. vel 8. lib.1. </s> <s id="s.000356">Fiat <lb/>angulus CER, ductis ex C & R rectis <lb/>lineis ad idem punctum E peripheriæ. </s> <lb/> <s id="s.000357">Dico angulum CER minorem e&longs;&longs;e an­<lb/>gulo CHR. </s> <s id="s.000358">Ducatur enim recta EG; & erunt in I&longs;o&longs;cele <lb/>CEG æquales anguli CEG, CGE. </s> <s id="s.000359">Quia verò, per 7.lib.3. <lb/>RE major e&longs;t quàm RG, angulus RGE major e&longs;t angulo <lb/>REG, per 18.lib. 1. & ablatis æqualibus remanet REC mi­<lb/>nor angulo RGC, hoc e&longs;t RHC. </s> <s id="s.000360">Similiter o&longs;tendetur angu­<lb/>lum RIC minorem e&longs;&longs;e angulo RHC: ductâ enim IG, angu­<lb/>li CIG, CGI &longs;unt æquales: & quoniam per 7.lib.3. RG ma­<lb/>jor e&longs;t quàm RI, angulus RIG major e&longs;t angulo RGI, per <lb/>18.lib.1. &longs;i igitur ex æqualibus auferantur inæquales anguli, re­<lb/>manet RIC minor, quàm RGC, hoc e&longs;t quam RHC. </s> <s id="s.000361">Ea­<lb/>dem erit methodus demon&longs;trandi angulos ad puncta periphe­<lb/>riæ propiora puncto H e&longs;&longs;e majores angulo CER. </s> <s id="s.000362">Ductâ enim <lb/>RD æquali ip&longs;i RE, ad punctum &longs;cilicet D æqualiter di&longs;tans à <lb/>diametro, ac di&longs;tet punctum E, & ducto radio CD, e&longs;t angu­<lb/>lus CDR æqualis angulo CER. </s> <s id="s.000363">Sit autem puncto H vicinior <lb/>angulus COR, quem dico e&longs;&longs;e majorem angulo CER per <lb/>7.lib.3. & 8.lib.1. </s> <s id="s.000364">Ducta lineâ OD, anguli COD, CDO <lb/>&longs;unt æquales, quia latera CO, CD æqualia &longs;unt: at per 7.lib.3. <lb/>RO minor e&longs;t, quàm RE, hoc e&longs;t RD, igitur angulus ROD <lb/>per 18.lib.1. major e&longs;t angulo RDO, & ablatis æqualibus re­<lb/>manet ROC major quam RDC, hoc e&longs;t quàm REC. </s> <s id="s.000365">Angu­<lb/>li itáque recedentes à puncto H &longs;emper fiunt minores, acce­<lb/>dentes verò fiunt majores. </s> </p> <pb n="43" xlink:href="017/01/059.jpg"/> <p type="main"> <s id="s.000366">Hoc probato con&longs;equens e&longs;t illud, quod in rotæ peripheriâ <lb/>duo &longs;unt puncta, inter quæ quodlibet punctum contingat pla­<lb/>num <expan abbr="inclinatũ">inclinatum</expan>, rota a&longs;cendit, &longs;i angulus maximus factus à lineis <lb/>ductis ex centro rotæ, & ex centro gravitatis &longs;it major angulo <lb/>inclinationis; quia nimirum anguli à puncto H recedentes ad <lb/>utramque partem &longs;emper fiunt minores; ergo ad utramque e&longs;t <lb/>angulus unus æqualis angulo inclinationis, & &longs;patium inter <lb/>huju&longs;modi angulos e&longs;t quantitas peripheriæ, quæ a&longs;cendens <lb/>pote&longs;t coaptari plano inclinato: ac proinde ex horum puncto­<lb/>rum di&longs;tantia definietur &longs;patium, quod pote&longs;t rota a&longs;cendens <lb/>percurrere. </s> </p> <p type="main"> <s id="s.000367">Sit igitur rota, cujus centrum C, & <lb/> <figure id="id.017.01.059.1.jpg" xlink:href="017/01/059/1.jpg"/><lb/>centrum gravitatis S: &longs;it autem CS par­<lb/>tium 11, quarum CH Radius e&longs;t 16: <lb/>e&longs;t igitur CS æqualis Sinui gr. 43. 26′. <lb/>qui erit maximus angulus CIS ad peri­<lb/>pheriam factus à Radio, & à lineâ IS <lb/>perpendiculari ad SC. </s> <s id="s.000368">Quare in quoli­<lb/>bet plano habente minorem inclinatio­<lb/>nem poterit a&longs;cendere. </s> <s id="s.000369">Ponatur plani <lb/>inclinatio gr. 15, cui æqualis &longs;it angulus CHS. </s> <s id="s.000370">Fiat igitur <lb/>ut CS 11 ad CH 16, ita Sinus anguli CHS 25882 ad <lb/>37646 Sinum Anguli CSH gr. 22. 7′; eritque angulus <lb/>SCH gr. 142. 53′. </s> <s id="s.000371">Cre&longs;cet ergo &longs;upra angulum H angulus <lb/>ad peripheriam, &longs;i ultra punctum H fiat contactus rotæ <lb/>in alio puncto viciniore puncto I, ex quo ad SC perpendi­<lb/>cularis cadit; & ex I decre&longs;cit u&longs;que dum in P fiat angu­<lb/>lus SPC grad. 15 æqualis angulo inclinationis. </s> <s id="s.000372">In triangu­<lb/>lo itaque SPC invenitur ex ii&longs;dem datis angulus PSC <lb/>gr. 157. 53′. & angulus SCP gr. 7. 7′. qui ex angulo SCH <lb/>gr. 142. 53′ ablatus relinquit PCH gr. 135. 46′. quæ e&longs;t quan­<lb/>titas arcûs HIP, quæ plano coaptatur in a&longs;cen&longs;u. </s> <s id="s.000373">Quoniam <lb/>verò quarum partium CG Radius e&longs;t 16, peripheria e&longs;t 100 1/2 <lb/>earum parirer e&longs;t arcus HP ferè 38, &longs;i Radius rotæ fuerit un­<lb/>ciarum pedis 16, rota a&longs;cendet in plano percurrens &longs;patium <lb/>pedum 3, & eo ampliùs. </s> <s id="s.000374">Hinc poteris aut rotæ diametrum au­<lb/>gere, aut plani inclinationem minuere, &longs;i volveris rotam lon­<lb/>giore &longs;patio moveri: auctâ enim rotæ diametro augetur peri- <pb n="44" xlink:href="017/01/060.jpg"/>pheria, &longs;ervatâ ratione eadem di&longs;tantiæ centri gravitatis. </s> <s id="s.000375">At &longs;i <lb/>data fuerit rota (oportet non ignorari di&longs;tantiam centri gravi­<lb/>tatis à centro rotæ, poterit autem primâ praxi cap.5. inve&longs;tiga­<lb/>ri) certum e&longs;t illam non po&longs;&longs;e a&longs;cendere ni&longs;i per &longs;patium mi­<lb/>nus longitudine &longs;emiperipheriæ; con&longs;tituto autem &longs;patio inve­<lb/>nietur inclinatio plani nece&longs;&longs;aria, hac methodo. </s> <s id="s.000376">Data &longs;patij <lb/>longitudo PH reducatur ad denominationem graduum, & erit <lb/>notus angulus PCH: & quoniam anguli ad H & ad P debent <lb/>e&longs;&longs;e æquales, anguli verò in R ad verticem &longs;unt æquales, erunt <lb/>pariter æquales PCH, & PSH, qui proinde notus e&longs;t. </s> <s id="s.000377">Hujus <lb/>&longs;emi&longs;&longs;is auferatur ex recto CSI, & innote&longs;cet angulus CSH, <lb/>cum quo & duobus lateribus CS, CH invenietur per Trigo­<lb/>nometriam angulus CHS æqualis angulo inclinationis plani <lb/>nece&longs;&longs;ariæ. </s> <s id="s.000378">Quod autem angulus HSI &longs;it &longs;emi&longs;&longs;is totius HSP, <lb/>hoc e&longs;t dati PCH, &longs;ic o&longs;tendo. </s> <s id="s.000379">Quia in duobus triangulis <lb/>CSP, CHS idem latus CS opponitur angulis æqualibus ad H, <lb/>& ad P, æqualia autem latera CH, & CP opponuntur angulis <lb/>quæ&longs;itis CSH, & CSP, con&longs;tat horum duorum angulorum <lb/>e&longs;&longs;e unum eundemque &longs;inum; ergo &longs;imul &longs;umpti &longs;unt æquales <lb/>duobus rectis; auferatur ex eorum &longs;ummâ unus rectus, rema­<lb/>nebunt duo anguli &longs;imul CSH, ISP æquales uni recto, hoc <lb/>e&longs;t angulo ISC: auferatur communis CSH, remanebit HSI <lb/>æqualis angulo ISP: id quod oportuit demon&longs;trare. </s> </p> <p type="main"> <s id="s.000380">Colligere po&longs;&longs;umus ex his, quæ hactenus explicata &longs;unt, fie­<lb/>ri quidem po&longs;&longs;e, ut, &longs;i rota in plano inclinato primùm con&longs;ti­<lb/>tuta exactè tangat in H, pror&longs;us con&longs;i&longs;tat; id tamen vix po&longs;&longs;e <lb/>&longs;perari, quia &longs;i in alio puncto remotiore ab I tangat, cadet, &longs;i <lb/>in puncto viciniore, a&longs;cendet. </s> <s id="s.000381">At ubi venerit in P, &longs;i ex con­<lb/>cepto impetu pergat adhuc aliquantulum a&longs;cendere; centro <lb/>gravitatis S tran&longs;lato versùs plani declivitatem, & diminuto <lb/>angulo, de&longs;cendet; & ubi tran&longs;ilierit punctum P, iterùm aucto <lb/>angulo a&longs;cendet, donec omninò in P con&longs;i&longs;tat. </s> <s id="s.000382">Ubi licet <lb/>animadvertere non idem e&longs;&longs;e punctum contactus, in quo <lb/>quie&longs;ceret in plano horizontali, ac inclinato; in plano enim <lb/>horizontali quie&longs;ceret in O, ubi linea à centro rotæ C perpen­<lb/>dicularis horizonti, ac tran&longs;iens per S centrum gravitatis, ter­<lb/>minatur: in eo autem puncto O con&longs;i&longs;tere non po&longs;&longs;e &longs;upra pla­<lb/>num inclinatum &longs;atis patet ex dictis. </s> <s id="s.000383">Porrò hæc, quæ de rotâ <pb n="45" xlink:href="017/01/061.jpg"/>con&longs;i&longs;tente, aut cadente di&longs;putata &longs;unt, dicenda e&longs;&longs;e de &longs;phæ­<lb/>râ quie&longs;cente in plano inclinato, clarius e&longs;t, quàm ut oporteat <lb/>pluribus explicare. </s> </p> <p type="main"> <s id="s.000384">Unum &longs;upere&longs;&longs;e videtur o&longs;tendendum, quî verum &longs;it cen­<lb/>trum gravitatis de&longs;cendere ita, ut fiat horizonti vicinius, dum <lb/>rota a&longs;cendit, & fit remotior. </s> <s id="s.000385">Id ut manife&longs;tum fiat, primò in­<lb/>veniatur HS: & &longs;it ut Sinus anguli CHS gr. 15. ad &longs;inum an­<lb/>guli SCH gr. 14.2. 53′. hoc e&longs;t ut 25882 ad 60344, ita CS <lb/>partium 11 ad HS 25 2/3: quæ e&longs;t altitudo centri gravitatis ante <lb/>motum. </s> <s id="s.000386">Deinde inveniatur SP; & &longs;it ut Sinus SPC gr. 15 ad <lb/>Sinum SCP gr.7. 7′ hoc e&longs;t, ut 25882 ad 12389, ita CS par­<lb/>tium 11 ad SP 5 1/4, quæ in fine motus erit altitudo centri gravi­<lb/>tatis &longs;upra planum inclinatum; huic autem addenda e&longs;t altitu­<lb/>do, quam &longs;upra horizontem habet punctum illud plani inclinati, <lb/>in quo tanget P. </s> <s id="s.000387">Quia ergo inclinatio plani e&longs;t gr. 15, & HP <lb/>e&longs;t partium 38, tantum e&longs;t &longs;patium, quod in plano percurritur <lb/>à rota a&longs;cendente, fiat ut Radius 100000 ad 25882 Sinum an­<lb/>guli inclinationis, ita 38 ad 9 4/5 altitudinem &longs;upra horizontem, <lb/>cui &longs;i addas SP 5 1/4, erit in fine motûs altitudo centri gravitatis <lb/>&longs;upra horizontem partium 15, cùm initio di&longs;taret partibus 25 2/3. </s> <lb/> <s id="s.000388">Centrum igitur gravitatis &longs;impliciter, & ab&longs;olutè de&longs;cendit, <lb/>dum rota in plano inclinato a&longs;cendit. </s> </p> <p type="main"> <s id="s.000389">Po&longs;&longs;em hîc afferre aquam vi &longs;uæ gravitatis a&longs;cendentem in <lb/>cochleâ Archimedis, dum cylindrus, quem cochlea ambit, <lb/>convertitur: ab&longs;tineo tamen, quia non vacat hîc examinare, <lb/>an motus ille compo&longs;itus &longs;it ex conver&longs;ione, quâ pul&longs;u externo <lb/>agitata aqua attollatur, & ex naturali de&longs;cen&longs;u, quo per tubum <lb/>in &longs;piras &longs;inuatum de&longs;cendat; an verò quemadmodum &longs;uppo&longs;i­<lb/>to cuneo reluctans pondus elevatur, vel etiam cochleâ trahitur <lb/>in plano horizontali, ita dicendum &longs;it aquam vi &longs;uæ gravitatis <lb/>in imo per&longs;i&longs;tentem à cochleâ &longs;en&longs;im &longs;ubeunte elevari &longs;imul, & <lb/>trahi, quin illa &longs;ponte &longs;ua a&longs;cendat: nam aquæ facilè tribuitur <lb/>aliquando motus, qui &longs;ubjecto corpori, cui illa in&longs;idet, conve­<lb/>nit; ut liquet &longs;i ampliorem peluim ex fune &longs;u&longs;penderis, vel lu­<lb/>brico in plano horizontali collocaveris, in qua &longs;it non multa <lb/>aqua in depre&longs;&longs;iore fundi parte quie&longs;cens; va&longs;e &longs;iquidem ex <lb/>improvi&longs;o vehementiùs impul&longs;o videtur aqua in oppo&longs;itam par- <pb n="46" xlink:href="017/01/062.jpg"/>tem refluere, cum tamen vas ip&longs;um potiùs infra aquam mo­<lb/>veatur, quàm aqua in va&longs;e: quanquam ratione adhæ&longs;ionis aquæ <lb/>ad peluim etiam ip&longs;a motum concipiat. </s> <s id="s.000390">Quare in cen&longs;u &longs;ponte <lb/>a&longs;cendentium numeranda non videtur aqua tubo &longs;peciali cy­<lb/>lindrum circumplexo elevata. </s> </p> <p type="main"> <s id="s.000391">Videatur forta&longs;&longs;e aqua &longs;ponte a&longs;cen&longs;ura in tubo non æquabi­<lb/>li &longs;ed conico, in plano verticali rotæ &longs;piraliter circumducto: <lb/>dum enim aqua æquilibrium &longs;uperficiei faciens in parte tubi <lb/>ampliore præponderat, convertitur rota, & illa iterum æqua­<lb/>liter &longs;e librans totius molis compo&longs;itæ centrum gravitatis trans­<lb/>fert extra lineam perpendicularem: &longs;i tamen ea cautio adhi­<lb/>beatur, ut tanta &longs;it aquæ quantitas, quæ non planam obtineat <lb/>&longs;uperficiem &longs;ed tubi inflexione conformetur; neque ita &longs;it <lb/>&longs;piræ a&longs;cendentis ardua altitudo, ut aqua po&longs;t &longs;uperficiei libra­<lb/>tionem ex ea parte ob &longs;ui paucitatem non præponderet; & præ­<lb/>terea ejus figuræ &longs;it tubus, ut aqua in parte angu&longs;tiore remo­<lb/>tior à perpendiculari, non ita ratione &longs;itûs augeat momenta &longs;ui <lb/>conatus deor&longs;um, ut repugnare valeat aquæ ampliorem tubi <lb/>partem occupanti. </s> <s id="s.000392">Si hæc, inquam, ob&longs;erventur (an autem <lb/>ita facile &longs;it ea ob&longs;ervare, ut quidam autumant, hic non de­<lb/>finio) & centrum gravitatis transferatur extra perpendicula­<lb/>rem versùs ampliorem tubi &longs;piralis partem, futurum quidem <lb/>e&longs;t, ut aqua a&longs;cendat; id tamen non e&longs;t opus centri gravitatis, <lb/>&longs;ed potius virtutis illius, qua humor &longs;e æquabiliter librat. <lb/> </s> </p> <p type="head"> <s id="s.000393"><emph type="center"/>CAPUT VIII.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000394"><emph type="center"/><emph type="italics"/>Cur gravium in plano inclinato de&longs;cendentium <lb/>alia repant, alia rotentur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000395">QUæ capite &longs;uperiori dixi de globi aut rotæ &longs;uper planum <lb/>inclinatum con&longs;i&longs;tentiâ in puncto, in quo linea à centro <lb/>globi, aut rotæ ducta cum eâ, quæ ex centro gravitatis duci­<lb/>tur, facit angulum æqualem angulo inclinationis plani, non ita <lb/>intelligi velim, qua&longs;i motus omnis deor&longs;um adimatur rotæ aut <lb/>globo cuju&longs;libet gravitatis, & in quovis plano inclinato: ibi <lb/>enim con&longs;i&longs;tentiæ, aut quietis nomine &longs;olam conver&longs;ionem <pb n="47" xlink:href="017/01/063.jpg"/>excipio, non lap&longs;um nego. </s> <s id="s.000396">Fieri &longs;i quidem pote&longs;t, ut adeò con­<lb/>tinuo lævore lubricum &longs;it planum, exactéque rotundatus globus, <lb/>ut nullam ex eminulis particulis moram recipiens deor&longs;um la­<lb/>batur, volubilitate ipsâ motum nihil juvante, &longs;ed &longs;olo pondere <lb/>urgente, cum in lineâ ad horizontem perpendiculari &longs;emper <lb/>maneat centrum gravitatis, & punctum contactûs. </s> </p> <p type="main"> <s id="s.000397">Neque e&longs;&longs;et diver&longs;a ratio &longs;phæræ centrum gravitatis haben­<lb/>tis extra centrum molis, ac cæterorum corporum non &longs;phæri­<lb/>corum: Nam gravia quæcunque in plano inclinato con&longs;tituta <lb/>tantum habent ad de&longs;cendendum momenti, ut a&longs;peritatis re­<lb/>&longs;i&longs;tentiam vincant, repunt quidem, &longs;i linea directionis ab eo­<lb/>rum gravitatis centro in terræ centrum ducta tran&longs;eat per con<lb/>tactum &longs;ubjecti plani, & impo&longs;iti gravis; rotantur verò, &longs;i di­<lb/>rectionis linea in plani declivitatem cadat extra contactum: <lb/>&longs;ivè demùm in puncto, &longs;ivè in lineâ, &longs;ivè in &longs;uperficie con­<lb/>tactus fiat. </s> <s id="s.000398">E&longs;t autem animadvertendum non e&longs;&longs;e opus, ut una <lb/>continua &longs;uperficies &longs;it, aut linea, &longs;ecundùm quam &longs;e tangant; <lb/>&longs;ed pro &longs;uperficie aut linea contactûs accipitur totum illud &longs;pa­<lb/>tium, quod inter extrema contingentia rectis lineis conjuncta <lb/>intercipitur. </s> </p> <p type="main"> <s id="s.000399">Sit planum inclinatum AB, <lb/> <figure id="id.017.01.063.1.jpg" xlink:href="017/01/063/1.jpg"/><lb/>cui globus C incumbit con­<lb/>tingens in puncto D. </s> <s id="s.000400">Ex cen­<lb/>tro gravitatis C, quod & cen­<lb/>trum molis e&longs;t ex hypothe&longs;i, <lb/>cadat linea directionis CE <lb/>perpendicularis in horizon­<lb/>tem FB; quæ nece&longs;&longs;ariò ca­<lb/>dit extra punctum contactûs <lb/>D; alioquin eadem linea CE <lb/>caderet ad angulos rectos &longs;u­<lb/>pra planum inclinatum, & &longs;upra horizontale, id quod fieri <lb/>non pote&longs;t, cum huju&longs;modi plana non &longs;int invicem parallela. </s> <lb/> <s id="s.000401">Per D igitur punctum &longs;u&longs;tentationis ductâ GH parallelâ lineæ <lb/>directionis, &longs;i per utramque plana parallela ducantur, planum <lb/>per GH &longs;ecat &longs;phæram in partes inæqualiter graves; & idcir­<lb/>co pars præponderans, in qua e&longs;t centrum gravitatis globi, mo­<lb/>vetur circa punctum &longs;u&longs;tentationis D, atque adeò in gyrum <pb n="48" xlink:href="017/01/064.jpg"/>conver&longs;a circa centrum C de&longs;cendit, ac rotatur. </s> <s id="s.000402">Quod &longs;i inæ­<lb/>qualis fuerit &longs;phæræ &longs;ub&longs;tantia, & centrum gravitatis I in per­<lb/>pendiculari GH, non de&longs;cendet &longs;phæra in gyrum acta, &longs;ed <lb/>tantùm repet, cum neutra pars præponderet. </s> </p> <p type="main"> <s id="s.000403">Simili ratione parallelepipedum KL, cujus centrum gravi­<lb/>tatis M, non repit; quia, cùm linea directionis MN cadat ex­<lb/>tra ba&longs;im KO, quæ contingit &longs;ubjectum planum, &longs;i per extre­<lb/>mam lineam KP tran&longs;eat planum PQ horizonti perpendicu­<lb/>lare, dividitur parallelepipedum in duo pri&longs;mata inæqualia, & <lb/>non æquiponderantia: cum verò pri&longs;ma trapezium QLKP <lb/>præponderet pri&longs;mati trigono KOQ, quod &longs;u&longs;tinetur à ba&longs;i, <lb/>illud nece&longs;&longs;ariò de&longs;cendit, & circa lineam KP convertitur. </s> <lb/> <s id="s.000404">Contrà autem quando intra ba&longs;im contactûs, ut in cubo PR, <lb/>cujus centrum S, cadit linea directionis ST, tunc repit, & non <lb/>rotatur cubus; quia &longs;cilicet ab extrema &longs;u&longs;tentationis lineâ KP <lb/>ductum planum horizonti perpendiculare dividit cubum in <lb/>partes inæquales ita, ut pars illa, in qua e&longs;t centrum gravitatis, <lb/>& quæ à &longs;ubjecto plano tota &longs;u&longs;tinetur, præponderet, nec po&longs;­<lb/>&longs;it à reliquâ parte elevari, ut circa KP convertatur. </s> </p> <p type="main"> <s id="s.000405">Hinc apparet ad quantam altitudinem pertinere po&longs;&longs;it paral­<lb/>lelepipedum, ut in dato plano inclinato non rotetur, &longs;ed repat: <lb/>nam ab extremâ &longs;u&longs;tentationis lineâ KP excitatum planum <lb/>horizonti perpendiculare PQ, quod bifariam in partes æqui­<lb/>ponderantes dividit parallelepipedum KQ, determinat altitu­<lb/>dinem maximam <expan abbr="Xq;">Xque</expan> in omni quippe majori altitudine non <lb/>repit, &longs;ed rotatur, quia linea directionis cadit extra ba&longs;im <lb/>&longs;u&longs;tentationis: in omni verò minori altitudine non rotatur, &longs;ed <lb/>repit, quia linea directionis cadit intra ba&longs;im &longs;u&longs;tentationis. </s> <lb/> <s id="s.000406">Hoc idem in corporibus cæteris, quamvis non parallelepipe­<lb/>dis, ob&longs;ervandum e&longs;t, an &longs;cilicet linea directionis cadat extra <lb/>ba&longs;im &longs;u&longs;tentationis, nec ne. </s> </p> <p type="main"> <s id="s.000407">Quæ tamen de cubo repente dicta &longs;unt, intelligi velim &longs;pecta­<lb/>tâ per &longs;e gravium figurâ: quia per accidens fieri pote&longs;t, ut cor­<lb/>pus non repat, &longs;ed rotetur, quamvis linea directionis cadat in­<lb/>tra ba&longs;im, quæ planum inclinatum contingit. </s> <s id="s.000408">Nam &longs;i in motu <lb/>occurrat &longs;uper plano inclinato offendiculum aliquod, cui de­<lb/>&longs;cendens corpus illidatur, fieri pote&longs;t, ut impetus ex motu con­<lb/>ceptus ita promoveat centrum gravitatis in anteriora, ut linea <pb n="49" xlink:href="017/01/065.jpg"/>directionis cadat extra ba&longs;im ultrà punctum illud, quod proxí­<lb/>mum e&longs;t offendiculo, ac proinde circa illud convertatur. </s> <s id="s.000409">Hæc <lb/>autem poti&longs;&longs;imùm e&longs;t ratio, cur ex clivis de&longs;cendentes lapides, <lb/>quamquam nec orbiculares, nec admodum alti, rotentur ta­<lb/>men; quia &longs;cilicet multa offendicula in clivo occurrunt, & ab <lb/>impetu per motum concepto partes &longs;uperiores promoventur <lb/>ulteriùs, inferioribus retardatis. </s> <s id="s.000410">Sic &longs;æpè ce&longs;pitantes cadimus, <lb/>quia ab offendiculo retinentur pedes, cum interim corpus re­<lb/>liquum ex concepto impetu ulteriùs promoveatur, ita ut linea <lb/>directionis cadat extra ba&longs;im &longs;u&longs;tentationis. <lb/></s> </p> <p type="head"> <s id="s.000411"><emph type="center"/>CAPUT IX.<emph.end type="center"/></s> </p> <p type="head"> <s id="s.000412"><emph type="center"/><emph type="italics"/>Cur turres inclinatæ non corruant.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000413">OB&longs;ervandum e&longs;t, ait Vitruvius lib.6. cap. 11, uti omnes <lb/>&longs;tructuræ perpendiculo re&longs;pondeant, neque habeant in <lb/>ulla parte proclinationes. </s> <s id="s.000414">Nemo e&longs;t qui non intelligat præ­<lb/>ceptum hoc ad ædificiorum con&longs;i&longs;tentiam pertinere; &longs;ed neque <lb/>defuerunt, qui rem &longs;ubtiliùs, quàm par &longs;it, perpendentes ina­<lb/>ni timore &longs;e torquebant, ne fortè aliquando domus corrueret, <lb/>cujus parietes inter &longs;e paralleli fuerant con&longs;tituti; cùm enim <lb/>perpendicula &longs;ibi demum in terræ centro occurrant, fieri non <lb/>po&longs;&longs;e putabant, ut &longs;imul paralleli e&longs;&longs;ent parietes. </s> <s id="s.000415">Id quod Geo­<lb/>metricè quidem verum e&longs;t; Phy&longs;icè tamen paralleli&longs;mus cum <lb/>perpendiculis con&longs;entit: nam &longs;i funiculos duos longitudinis <lb/>ped. 100. clavo affixos ita extendas, ut extrema eorum palmi <lb/>intervallo di&longs;tent, angulum facient acuti&longs;&longs;imum; & &longs;i lineas <lb/>duas bipedales duxeris eorum extremitatibus congruentes, vix <lb/>different à parallelis, cum intervalla jungentia utro&longs;que linea­<lb/>rum terminos differant inter &longs;e &longs;olum palmi parte quinquage­<lb/>&longs;ima. </s> <s id="s.000416">Longè autem majorem rationem terræ &longs;emidiameter ha­<lb/>bet ad quamlibet ædificiorum altitudinem; ut proinde à paral­<lb/>leli&longs;mo multo minùs recedant parietes, etiam&longs;i fuerint turrium <lb/>in&longs;tar alti&longs;&longs;imi. </s> <s id="s.000417">Ponantur enim parietes duo, aut potiùs turres, <lb/>di&longs;tare inter &longs;e pa&longs;&longs;.300; &longs;it autem parietum, vel turrium alti­<lb/>tudo pa&longs;&longs;. 60, hoc e&longs;t ped.300. </s> <s id="s.000418">Con&longs;tat mihi, ut aliàs o&longs;tendi, <lb/>terrenam &longs;emidiametrum non e&longs;&longs;e minorem pa&longs;&longs;ibus Rom. <pb n="50" xlink:href="017/01/066.jpg"/><expan abbr="antiq.">antique</expan> 4128635: quarè &longs;i fiat ut terræ &longs;emidiameter 4128635 <lb/>ad altitudinem 60, ita di&longs;tantia parietum, aut turrium in imo <lb/>300, ad aliud, proveniet differentia, qua di&longs;tantia turrium in <lb/>&longs;ummo vertice &longs;uperat earum di&longs;tantiam in imo pede, & erit <lb/>partium (4359/1000000) unius pa&longs;&longs;us, quæ e&longs;t minor quàm 2/5 digiti: quis <lb/>autem parallelas non dixerit turres, quæ vix uno aut altero <lb/>hordei grano di&longs;tant à paralleli&longs;mo? </s> <s id="s.000419">Quod &longs;i in tanta altitudine <lb/>atque di&longs;tantiâ di&longs;crimen hoc adeò exiguum e&longs;t, &longs;atis patet, <lb/>quid de columnarum paralleli&longs;mo dicendum &longs;it. </s> <s id="s.000420">Con&longs;tat autem <lb/>ex his ædificia in alti&longs;&longs;imis montibus con&longs;tituta habere parie­<lb/>tes minùs à paralleli&longs;mo recedentes, &longs;i fuerint ad perpendicu­<lb/>lum ædificati, quàm in locis depre&longs;&longs;ioribus: atque adeò, &longs;i duæ <lb/>columnæ eandem inter &longs;e po&longs;itionem &longs;ervantes de&longs;cenderent <lb/>cum &longs;ubjecto plano, ita ut alterutra columnarum illarum ad <lb/>perpendiculum de&longs;cenderet, reliqua demùm adeò inclinare­<lb/>tur, ut caderet. </s> </p> <p type="main"> <s id="s.000421">Sed quàm inanem &longs;ibi &longs;truant &longs;olicitudinem, qui nimis exi­<lb/>guè, & exiliter ad calculos revocant &longs;tructurarum perpendicu­<lb/>la, &longs;atis indicant turres inclinatæ, quæ po&longs;t aliquot &longs;ecula con­<lb/>&longs;i&longs;tunt citrà ullum ruinæ periculum, quamvis illam timeant <lb/>imperiti. </s> <s id="s.000422">Duas habemus in Italiâ turres ob in&longs;ignem inclina­<lb/>tionem con&longs;picuas; altera e&longs;t Bononiæ quadrata opere lateri­<lb/>tio, altera Pi&longs;is rotunda ex albo marmore affabrè expolito, & <lb/>columnis 284 rite di&longs;po&longs;itis ornata. </s> <s id="s.000423">Ædificari cœpit anno <lb/>1173 Germano quodam architecto, quem ab aliis Guillel­<lb/>mum, ab aliis Joannem OEnipontanum dici reperio. </s> <s id="s.000424">Rotunda <lb/>e&longs;t forma duplici muro concludente &longs;calas cochleæ in modum <lb/>ab imo ad &longs;ummum ductas: parietis cra&longs;&longs;ities e&longs;t cubitorum <lb/>6 1/3, turris altitudo cubitorum 78, ambitus in imo pede cubi­<lb/>torum 80; unde colligitur diameter cubitorum ferè 25 1/2; incli­<lb/>natio, &longs;eu intervallum inter ba&longs;im, & perpendiculum e&longs;t cu­<lb/>bitorum 7 1/3, ut ex literis ad me inde datis habeo; quamvis <lb/>apud aliquos legerim tantùm cubitos 7, apud alios 6 1/2. </s> <s id="s.000425">Factâ <lb/>ne fuerit illa inclinatio de indu&longs;triâ, an verò &longs;ub&longs;identibus fun­<lb/>damentis, incertum e&longs;t. </s> <s id="s.000426">Ego non facilè eo in illorum &longs;enten­<lb/>tiam, qui id &longs;cribunt contigi&longs;&longs;e ex artificis imperitia, cui non <lb/>&longs;atis per&longs;pecta e&longs;&longs;et &longs;oli natura; tum quia fundamenta altitudi- <pb n="51" xlink:href="017/01/067.jpg"/>nem habent, atque amplitudinem ingentem, quibus con­<lb/>&longs;truendis annus &longs;olidus &longs;atis non fuit; tum quia nullam unquam <lb/>egit rimam, id quod &longs;ub&longs;idente &longs;olo rari&longs;&longs;imum e&longs;t; tum quia <lb/>potuit architectus excitari ad artis &longs;pecimen exhibendum à tur­<lb/>ri Bononien&longs;i Gari&longs;endâ excitatâ anno 1110. </s> </p> <p type="main"> <s id="s.000427">Turris Bononien&longs;is altitudinem habet pedum Bonon. 130; <lb/>exteriùs inclinatur ped. 9, interiùs verò ped. 1, & paulo am­<lb/>plius: muri cra&longs;&longs;ities in parte infimâ e&longs;t pedum 6 1/2, in &longs;upre­<lb/>ma ped. 4; cava turris ped. 7. quare lateris longitudo e&longs;t ped. <lb/>20, & ambitus, quoniam quadrata e&longs;t, ped. 80. Ex his men­<lb/>&longs;uris, quas in <emph type="italics"/>Bononïá Perlu&longs;tratâ<emph.end type="italics"/> anno 1650 typis evulgatâ at­<lb/>tulit Antonius Pauli Ma&longs;ini, turris &longs;pe­<lb/> <figure id="id.017.01.067.1.jpg" xlink:href="017/01/067/1.jpg"/><lb/>ciem exhibeo, & e&longs;t AB latus unum <lb/>ped. 20, BD inclinationis men&longs;ura <lb/>ped. 9. DC altitudo perpendicularis <lb/>ped.130; EB & AF ped. 6 1/2 cra&longs;&longs;ities <lb/>imi parietis, & CH ped. 4. cra&longs;&longs;ities <lb/>eju&longs;dem parietis EC exteriùs inclinati. </s> <lb/> <s id="s.000428">At quoniam inclinatio interior FI dici­<lb/>tur e&longs;&longs;e ped.1, & paulo ampliùs, erit ID <lb/>paulo major ped.21; erecta autem ex I <lb/>perpendicularis dabit punctum G termi­<lb/>num cra&longs;&longs;itiei muri AG in parte &longs;upre­<lb/>mâ, & erit CG major ped. 21, cum &longs;it <lb/>æqualis ip&longs;i ID. </s> <s id="s.000429">Quare fieri non pote&longs;t, <lb/>ut KG &longs;it ped. 4; quemadmodum HC; <lb/>alioquin e&longs;&longs;et CK &longs;altem ped.25, cum <lb/>ba&longs;is AB &longs;it tantum ped.20. </s> <s id="s.000430">Hinc &longs;i li­<lb/>ceat conjecturas per&longs;equi (quandoqui­<lb/>dem veritatem a&longs;&longs;equi non potui, cum <lb/>non careat periculo a&longs;cen&longs;us per &longs;calas <lb/>ligneas à pluviis maximam partem cor­<lb/>ruptas) exi&longs;timo AF majorem e&longs;&longs;e quàm <lb/>EB, hoc e&longs;t majorem pedibus 6 1/2, KG <lb/>verò minorem quam HC, ut turri &longs;ua <lb/>con&longs;tet Eurithmia; id quod obtineretur, &longs;i ID uno, aut alte­<lb/>ro pede minor e&longs;&longs;et quàm AB, differentia enim inter ID, <lb/>& AB e&longs;&longs;et cra&longs;&longs;ities KG. </s> <s id="s.000431">Et &longs;anè memini aliquando me au- <pb n="52" xlink:href="017/01/068.jpg"/>divi&longs;&longs;e &longs;upremam cra&longs;&longs;itiem muri oppo&longs;iti parti inclinatæ <lb/>non excedere integrum pedem. </s> <s id="s.000432">Id autem valde opportu­<lb/>num accidebat, ut longè faciliùs paries AFGK &longs;uâ mole <lb/>&longs;taret: neque enim ca&longs;u inclinatam fui&longs;&longs;e turrim dicere po­<lb/>teris, quam con&longs;tat prope A&longs;inellam recti&longs;&longs;imam ideò fui&longs;&longs;e <lb/>conditam, ut multo clariùs appareret inclinatio: præterquam <lb/>quod inclinatio interior minor externâ &longs;atis o&longs;tendit muros <lb/>nunquam fui&longs;&longs;e parallolos. </s> </p> <p type="main"> <s id="s.000433">Porrò ut con&longs;tet ex huju&longs;modi inclinatione non magis <lb/>e&longs;&longs;e de ruinâ timendum, quàm &longs;i exactè perpendicularis e&longs;­<lb/>&longs;et, examinemus, &longs;i placet, centrum gravitatis in turri Bo­<lb/>nonien&longs;i; hinc enim facilis erit conjectura de cæteris. </s> <s id="s.000434">Et <lb/> <figure id="id.017.01.068.1.jpg" xlink:href="017/01/068/1.jpg"/><lb/>primò parietis maximè inclinati &longs;ectio <lb/>verticalis illum bifariam &longs;ecans ac tran­<lb/>&longs;iens per centrum gravitatis &longs;it HCBE: <lb/>cujus latera parallela HC, EB bifariam <lb/>&longs;ecta in V & R jungantur rectâ VR, cu­<lb/>jus longitudo inve&longs;tiganda e&longs;t, ut in eâ <lb/>definiatur punctum S centrum gravitatis, <lb/>ac innote&longs;cat utrum perpendicularis SX, <lb/>&longs;cilicet linea directionis cadat intra ba­<lb/>&longs;im EB &longs;u&longs;tentantem. </s> <s id="s.000435">Et ut à fractioni­<lb/>bus minus incommodi &longs;ubeamus, liceat <lb/>a&longs;&longs;umere pedem in partes cente&longs;imas di­<lb/>vi&longs;um. </s> <s id="s.000436">Cum autem EB &longs;it ped. 6 1/2, &longs;emi&longs;­<lb/>&longs;is RB e&longs;t ped. 3. 25″; & quia HC e&longs;t <lb/>ped. 4, VC e&longs;t ped. 200″. </s> <s id="s.000437">Et ducatur <lb/>recta BV. </s> </p> <p type="main"> <s id="s.000438">In triangulo BDC rectangulo datis BD, <lb/>inclinatione ped. 90′0′, & altitudine per­<lb/>pendiculari CD ped. 130′0′, additis late­<lb/>rum quadratis fit quadratum hypothenu­<lb/>&longs;æ BC, quæ e&longs;t ped. 13031″. </s> <s id="s.000439">Ex datis autem lateribus BD, <lb/>& DC invenitur angulus CBD gr. 88. 33′, cui æqualis e&longs;t <lb/>inter parallelas VC, BD alternus VCB: angulus verò CBR <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/>ped. 130. 31″, & angulo verticali VCB gr. 88. 33′, reperitur <pb n="53" xlink:href="017/01/069.jpg"/>CVB gr. 90. 34′. 14″, & VBC gr. 0. 52′. 46″.. </s> <s id="s.000441">Ex his autem <lb/>inve&longs;tigatur VB ped. 130. 26″. </s> </p> <p type="main"> <s id="s.000442">Quoniam autem angulus CBR notus erat gr. 91. 27′, &longs;i de­<lb/>matur ex illo angulus VBC gr. 0. 52′. 46″. remanet VBR <lb/>gr. 90. 34′, 14″, æqualis angulo CVB alterno inter parallelas; <lb/>& nota &longs;unt latera illum con&longs;tituentia BR ped 3. 25″. & BV <lb/>ped. 130. 26″. </s> <s id="s.000443">Ex quibus datis invenitur angulus BRV gr. 88. <lb/>0′. 2″, BVR gr. 1. 25′. 44″ & ba&longs;is VR ped. 130. 326‴. </s> </p> <p type="main"> <s id="s.000444">Jam verò, ex prop. 15 lib.1. Æquipond. Archimedis, divi­<lb/>datur VR in S eâ ratione, ut &longs;it VS ad SR, ut duplum EB <lb/>majoris parallelarum unâ cum minore HC, ad duplum HC <lb/>unâ cum majore EB, hoc e&longs;t (quia EB e&longs;t ped. 6 1/2) & HC <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. 326‴, <lb/>ad SR ped. 59. 99″. </s> <s id="s.000446">Demum ex S ducta perpendiculari SX, <lb/>quia in triangulo RXS rectangulo datur angulus SRX gr.88. <lb/>0′.. 2″. atque adeò ejus complementum RSX gr.1. 59′. 58″. & <lb/>latus SR ped. 59. 99″. invenitur latus RX ped. 209″. </s> <s id="s.000447">E&longs;t igi­<lb/>tur RX linea minor, quàm RB po&longs;ita ped. 3. 25″; & idcirco <lb/>perpendicularis linea directionis SX cadit intrà ba&longs;im parie­<lb/>tis EBCH. </s> </p> <p type="main"> <s id="s.000448">Sed quia facturum me puto rem aliquibus gratam, &longs;i quas <lb/>inij rationes hîc exhibeam, calculi totius progre&longs;&longs;um per lo­<lb/>garithmos hîc addo, ut illum po&longs;&longs;is, &longs;i placeat examinare. <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/> —— l</cell> <cell>4,1038;,79160</cell> </row> <row> <cell>CBD gr.88.33. m</cell> <cell>1,15970,08429</cell> <cell>Semi&longs;umma ang.</cell> <cell>gr.44.42′.53″,-m</cell> <cell>9,99567.51920</cell> </row> <row> <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"/> <p type="main"> <s id="s.000449">Quod &longs;i paries exteriùs inclinatus etiam &longs;olitarius con&longs;i&longs;tere <lb/>po&longs;&longs;et, modò ea e&longs;&longs;et partium connexio, ut unum quid &longs;oli­<lb/>dum conflarent, quia directionis linea intra ba&longs;im &longs;u&longs;tentan­<lb/>tem cadit, & planum per extremam ba&longs;is lineam, & terræ cen­<lb/>trum tran&longs;iens relinquit interiorem parietis partem præponde­<lb/>rantem exteriori: quis po&longs;&longs;it de turris ruinâ dubitare, &longs;i eâdem <lb/>methodo deprehendat oppo&longs;iti parietis AG centrum gravita­<lb/>tis e&longs;&longs;e in O, ac proinde comparatis reliquorum duorum pa­<lb/>rietum centris gravitatum, totius turris centrum gravitatis e&longs;&longs;e <lb/>in intimis turris partibus? </s> <s id="s.000450">Quò igitur firmiùs &longs;ibi cohærebunt <lb/>partes turris, eò major erit inclinatio, quam obtinere pote&longs;t ci­<lb/>tra cadendi periculum. </s> <s id="s.000451">Id quod pueris ip&longs;is noti&longs;&longs;imum e&longs;t, <lb/>qui turriculas inclinatas architectantur ex buxeis orbiculis, <lb/>quibus in alveolo ludunt. </s> </p> <p type="main"> <s id="s.000452">Et ut res i&longs;ta plani&longs;&longs;imè o&longs;tendatur, <lb/><figure id="id.017.01.070.1.jpg" xlink:href="017/01/070/1.jpg"/><lb/>&longs;it &longs;upra planum inclinatum AB, pa­<lb/>rallelepipedum ligneum ID ita, ut <lb/>recta CE ad horizontem perpendicu­<lb/>laris tran&longs;eat per centrum gravitatis: <lb/>con&longs;tat ex dictis cap. 8. futurum e&longs;&longs;e, <lb/>ut grave ID repat, non autem rote­<lb/>tur, quia pars CED non præponderat parti CEI, &longs;iqui­<lb/>dem po&longs;&longs;it de&longs;cendere per planum inclinatum; quod &longs;i à lap­<lb/>&longs;u impediatur, &longs;ub&longs;i&longs;tet. </s> <s id="s.000453">Jam verò intellige per C planum <lb/>FH horizontale, & adnecti pri&longs;ma trigonum CIK pa­<lb/>rallelepipedo ID; utique pars CEK præponderat parti <lb/>CED, multóque minùs dubitandum erit de &longs;olidi KD rui­<lb/>nâ ver&longs;us H. <!-- KEEP S--></s> <s id="s.000454">Quid autem aliud e&longs;t &longs;olidum KD, quam tur­<lb/>ris inclinata? </s> </p> <p type="main"> <s id="s.000455">Scrip&longs;eram hæc jam tum ab anno labentis &longs;æculi quinquage­<lb/>&longs;imo &longs;exto; cum animum &longs;ubiit &longs;u&longs;picari, an &longs;uperiùs allatæ ex <lb/>Ma&longs;ino turris Bononien&longs;is men&longs;uræ omninò veritati re&longs;ponde­<lb/>rent. </s> <s id="s.000456">Quare litteris ad P. <!-- REMOVE S-->Franci&longs;cum Mariam Grimaldum da­<lb/>tis rogavi, ut pro eâ, quam ad res omnes conferre &longs;olebat, di­<lb/>ligentiâ, accuratè men&longs;uras illas inquireret: hæc igitur ex ejus <lb/>re&longs;pon&longs;ione habui, quibus &longs;uperiùs dicta corrigenda &longs;unt; quæ <lb/>tamen expungere nolui, ut &longs;i lubeat, vulgarem opinionem &longs;e­<lb/>qui valeas. </s> </p> <pb pagenum="55" xlink:href="017/01/071.jpg"/> <p type="main"> <s id="s.000457">Extimus turris ambitus tam in imâ, quam in &longs;upremâ parte <lb/>æqualis e&longs;t, adeò ut oppo&longs;itæ facies parallelæ excurrant: &longs;in­<lb/>gulorum autem laterum ad ba&longs;im latitudo e&longs;t ped. <!-- REMOVE S-->Bonon. <!-- REMOVE S-->17. <lb/>unc. </s> <s id="s.000458">8. murorum cra&longs;&longs;ities in imo æqualis e&longs;t; eo tantum di&longs;­<lb/>crimine, quod murus, qua parte o&longs;tium patet, cra&longs;&longs;us e&longs;t ped.5. <lb/>unc.11. qui verò Septentrionem &longs;pectat, propiùs accedit ad pe­<lb/>des 6. Porrò in &longs;ummâ turri murorum cra&longs;&longs;ities pariter æqualis <lb/>e&longs;t, & vix deficit à pedibus 5, quantum quidem ex a&longs;pectu à <lb/>&longs;uperiori proximæ turris A&longs;inellæ podio conjicere potuit &longs;ingu­<lb/>lorum murorum lateres numerans. </s> <s id="s.000459">Areæ demum vacuæ ad ba­<lb/>&longs;im latus unum e&longs;t ped. <!-- REMOVE S-->6. alterum ped.6. unc.1. </s> </p> <p type="main"> <s id="s.000460">Cum autem pluviæ per hiantem, & patulum turris verticem <lb/>deciduæ &longs;calas corruperint, nec eò veniri po&longs;&longs;it, ut demi&longs;&longs;o <lb/>perpendiculo altitudo turris inve&longs;tigetur, &longs;ub&longs;idium peten­<lb/>dum fuit ex Trigonometriâ, & ex proximâ turri A&longs;inellâ, cu­<lb/>jus men&longs;uræ multiplici ob&longs;ervatione innotuerant. </s> <s id="s.000461">Sit itaque <lb/>turris inclinata DC, &longs;uperioris autem podij <lb/><figure id="id.017.01.071.1.jpg" xlink:href="017/01/071/1.jpg"/><lb/>A&longs;inellæ altitudo EB ped.234 1/2, unde ob&longs;er­<lb/>vatus e&longs;t angulus CEB gr. <!-- REMOVE S-->18. 40′. </s> <s id="s.000462">Item in <lb/>eadem turri A&longs;inellâ patet fene&longs;tra in F, adeò <lb/>ut di&longs;tantia EF &longs;it ped.141: ibi pariter ob&longs;er­<lb/>vatus e&longs;t angulus EFC gr. <!-- REMOVE S-->51. 51′. </s> <s id="s.000463">Quare in <lb/>triangulo CEF, notum e&longs;t latus EF, & duo <lb/>anguli adjacentes, ex quibus datis colligi­<lb/>tur EC di&longs;tantia ped. (117 7/12). Jam verò intelli­<lb/>gantur ex C cadere duæ perpendiculares, al­<lb/>tera quidem CH in planum horizontale, alte­<lb/>ra verò CG in turrim A&longs;inellam; erit enim al­<lb/>titudo CH æqualis altitudini GB, nam CG <lb/>e&longs;t parallela horizonti, cui turris EB perpen­<lb/>dicularis in&longs;i&longs;tit. </s> <s id="s.000464">Ut igitur innote&longs;cat quæ&longs;i­<lb/>ta altitudo, inveniatur in triangulo rectangu­<lb/>lo CGE, ex datis latere CE ped. (117 7/12) & <lb/>angulo ob&longs;ervato CEG, gr.18.40′, latus EG <lb/>ped. (111 5/12). Jam verò &longs;i EG ped.(111 5/12) dematur <lb/>ex EB ped. <!-- REMOVE S-->234 1/2, remanet altitudo GB, hoc e&longs;t CH, <lb/>ped. (123 1/12). </s> </p> <pb pagenum="56" xlink:href="017/01/072.jpg"/> <p type="main"> <s id="s.000465">Demum ad inve&longs;tigandam turris inclinationem, applicito <lb/>ad punctum I perpendiculo ob&longs;ervatus e&longs;t angulus DIL <lb/>gr. <!-- REMOVE S-->3. 10′.: cùm autem IL parallela &longs;it perpendiculari CH, erit <lb/>pariter angulus DCH gr.3.10′. <!-- KEEP S--></s> <s id="s.000466">Igitur in triangulo DCH <lb/>rectangulo ad H notum e&longs;t latus CH ped.(123 1/12), & angulus <lb/>DCH gr.3.10′, ergo & innote&longs;cit latus DH ped.6. (10/12), quæ e&longs;t <lb/>men&longs;ura inclinationis quæ&longs;itæ. </s> </p> <p type="main"> <s id="s.000467">Ex his accuratioribus men&longs;uris indagemus, &longs;i placet, in <lb/>orientali pariete inclinato centrum gravitatis, & lineam di­<lb/>rectionis methodo eâdem, qua &longs;uperiùs u&longs;i &longs;umus; eademque <lb/>figura &longs;ectionis verticalis re&longs;umatur. </s> <s id="s.000468">E&longs;t igitur EB ped. <!-- REMOVE S-->6. ac <lb/>propterea RB ped. <!-- REMOVE S-->300″; & quia HC e&longs;t ped. <!-- REMOVE S-->5, VC e&longs;t <lb/>ped.2. 50″. <!-- KEEP S--></s> <s id="s.000469">BD autem e&longs;t ped. <!-- REMOVE S-->6. unc.10, hoc e&longs;t ped.(6 10/12). </s> </p> <figure id="id.017.01.072.1.jpg" xlink:href="017/01/072/1.jpg"/> <p type="main"> <s id="s.000470">In Triangulo BDC rectangulo datis BD <lb/>ped. <!-- REMOVE S-->6. (10/12), & altitudine perpendiculari CD <lb/>ped. (123 1/12), additis laterum quadratis fit qua­<lb/>dratum hypothenu&longs;æ BC, quæ e&longs;t ped.123.27″. <!-- KEEP S--></s> <lb/> <s id="s.000471">Fiat igitur ut CB ped. <!-- REMOVE S-->123. 27″, ad BD <lb/>ped. <!-- REMOVE S-->6. 83″. <!-- KEEP S--></s> <s id="s.000472">ita Radius ad &longs;inum anguli BCD <lb/>gr. <!-- REMOVE S-->3. 10′ 34″. <!-- KEEP S--></s> <s id="s.000473">Quare angulus reliquus CBD <lb/>gr. <!-- REMOVE S-->86. 49′. </s> <s id="s.000474">26″, cui æqualis e&longs;t alternus VCB <lb/>inter parallelas VC, RD; angulus autem, <lb/>qui e&longs;t deinceps, CBR gr. <!-- REMOVE S-->93. 10′. </s> <s id="s.000475">34′. </s> <s id="s.000476">In <lb/>triangulo VCB datis lateribus VC ped.2-50″, <lb/>CB ped. <!-- REMOVE S-->123. 27″, & angulo verticali VCB <lb/>gr. <!-- REMOVE S-->86. 49′. </s> <s id="s.000477">26″, reperitur CVB gr. <!-- REMOVE S-->92. 0′. </s> <s id="s.000478">36″, <lb/>& VBC. gr. <!-- REMOVE S-->1. 9′, 58″. <!-- KEEP S--></s> <s id="s.000479">Ex his verò invenitur <lb/>VB ped. <!-- REMOVE S-->122. 76″. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000480">Jam verò in Triangulo VBR, notus e&longs;t <lb/>angulus RBV æqualis alterno CVB gr.92. <lb/>0′. </s> <s id="s.000481">36′. </s> <s id="s.000482">& nota &longs;unt latera RB ped. <!-- REMOVE S-->300″, & <lb/>VB ped. <!-- REMOVE S-->122. 76″. <!-- KEEP S--></s> <s id="s.000483">Quare invenitur angulus <lb/>VRB gr. <!-- REMOVE S-->86. 35′ 43″. <!-- REMOVE S-->BVR gr. <!-- REMOVE S-->1. 23′. </s> <s id="s.000484">41″, & ba&longs;is VR <lb/>ped. <!-- REMOVE S-->123. 17″. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000485">Tum fiat ut 17 ad 16; hoc e&longs;t duplum majoris EB cum mi­<lb/>nore HC, ad duplum minoris HC cum majore EB, ita VS <lb/>ad SR, & erit SR ped.59.72″. <!-- KEEP S--></s> <s id="s.000486">Ductâ igitur ex S centro gra-<pb pagenum="57" xlink:href="017/01/073.jpg"/>vitatis perpendiculari lineâ directionis SX, ex datis latere SR <lb/>ped. <!-- REMOVE S-->59. 72″, & angulo VRX gr. <!-- REMOVE S-->86, 35′, 43″, innote&longs;cit RX <lb/>ped. <!-- REMOVE S-->3. 54″. <!-- KEEP S--></s> <s id="s.000487">Quare RX major e&longs;t quàm RB: & &longs;i paries ille <lb/>&longs;olitarius e&longs;&longs;et, non utique con&longs;i&longs;teret; &longs;ed quoniam reliqui <lb/>tres parietes adjecti &longs;unt, con&longs;tat ita totius molis centrum gra­<lb/>vitatis e&longs;&longs;e in intima turris parte, ut linea directionis cadat in­<lb/>trà turris ba&longs;im &longs;u&longs;tentantem. </s> </p> <p type="main"> <s id="s.000488">Ex his di&longs;cuties timorem eorum, qui &longs;oliciti &longs;unt de obeli&longs;­<lb/>corum con&longs;i&longs;tentiâ, ex inclinatione aliquâ verticis ruinam <lb/>proximam præ&longs;agientes: cum enim in huju&longs;modi molibus cen­<lb/>trum gravitatis vicinius &longs;it ba&longs;i quàm vertici, &longs;i centrum incli­<lb/>netur in alterutram partem &longs;patio tantùm digitali, vertex in­<lb/>&longs;ignem acquiret inclinationem, con&longs;i&longs;tet tamen, quandiu linea <lb/>directionis tran&longs;ibit per ba&longs;im &longs;u&longs;tentationis. </s> <s id="s.000489">Inclinatio enim <lb/>non e&longs;t &longs;patium illud, quod inter ba&longs;im, & perpendiculum à <lb/>turris, vel obeli&longs;ci vertice demi&longs;&longs;um intercipitur (quamvis hoc <lb/>vocabulo hactenus abuti placuerit, ne à vulgo di&longs;creparem) <lb/>&longs;ed e&longs;t angulus, quem turris facit cum plano; & manente ea­<lb/>dem inclinatione, intervallum illud mutari pote&longs;t pro majore, <lb/>aut minore turris longitudine. </s> <s id="s.000490">Quare quò longior e&longs;t moles in­<lb/>clinata, cæteris paribus, minùs e&longs;t timendum, quia minor e&longs;t <lb/>declinatio à perpendiculari: &longs;i enim KE &longs;it pedum 100, KC <lb/>verò ped.1. angulus KEC æqualis declinationi à perpendiculo <lb/>e&longs;t gr. <!-- REMOVE S-->0. 34. 22″. <!-- REMOVE S-->at &longs;i KE &longs;it ped. <!-- REMOVE S-->50, & KC iterum ped. <!-- REMOVE S-->1. <lb/>angulus KEC e&longs;t grad. <!-- REMOVE S-->11. 32′. </s> <s id="s.000491">13″. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000492">Hîc autem qua&longs;i præteriens &longs;atisfaciam quærenti, cur lon­<lb/>giores ha&longs;tas faciliùs, quàm breviores virgas digiti extremitate <lb/>&longs;u&longs;tineamus, quin cadant. </s> <s id="s.000493">Quia nimirum minimus angulus <lb/>declinationis à perpendiculo &longs;tatim &longs;e prodit ha&longs;tæ vertice ad <lb/>partem unam &longs;ecedente, cui &longs;tatim occurrimus ha&longs;tæ calcem <lb/>manu transferentes, ac &longs;ub vertice collocantes: verùm quia fa­<lb/>cilior ha&longs;tæ con&longs;i&longs;tentia innote&longs;cit etiam, quando à &longs;uppo&longs;itâ <lb/>manu calx ejus non movetur (nam &longs;i militarem &longs;ari&longs;&longs;am terræ <lb/>perpendiculariter in&longs;i&longs;tentem con&longs;titueris, potes te &longs;emel in gy­<lb/>rum contorquere, & illam qua&longs;i perpendicularem recipere, id <lb/>quod in breviore ha&longs;tâ non obtinebis) alia e&longs;t ratio petenda <lb/>primùm ex dictis, quia &longs;cilicet longior ha&longs;ta, cæteris paribus, <lb/>minùs declinat à perpendiculo, ideóque difficiliùs de&longs;cendit; <pb pagenum="58" xlink:href="017/01/074.jpg"/>deinde quemadmodum longiorem ha&longs;tam &longs;i in aquá agitaveris <lb/>majorem percipies re&longs;i&longs;tentiam, quàm &longs;i breviorem virgam in­<lb/>citares; ita aërem variis &longs;emper motibus turbatum plus etiam <lb/>impedire de&longs;cen&longs;um longioris ha&longs;tæ cen&longs;endum e&longs;t, præ&longs;ertim <lb/>&longs;i in &longs;uperiore parte aër versùs unam, in inferiore autem versùs <lb/>aliam partem moveatur: id quod in breviore virgâ non accidit, <lb/>quam modicus aër contingit, nec pote&longs;t aut adeò re&longs;i&longs;tere di­<lb/>vi&longs;ioni, aut adeò diver&longs;is motibus cieri. </s> <s id="s.000494">Hinc a&longs;ta longior <lb/>tardiùs de&longs;cen&longs;um molitur, & faciliùs &longs;u&longs;tinetur, quia major <lb/>aëris dividendi quantitas, ac motus varius, magis re&longs;i&longs;tit, & <lb/>datâ æqualitate motûs minùs declinat à perpendiculo. <lb/></s> </p> <p type="head"> <s id="s.000495"><emph type="center"/>CAPUT X.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="head"> <s id="s.000496"><emph type="center"/><emph type="italics"/>An plurium &longs;tructurarum capax &longs;it mons, quàm <lb/>&longs;ubjecta planities.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000497">POte&longs;t mons cum &longs;ubjectâ planitie, cui in&longs;i&longs;tit, dupliciter <lb/>comparari; primùm conferendo &longs;olam planitiem in ver­<lb/>tice montis exi&longs;tentem cum parte &longs;ubjecti plani &longs;ibi re&longs;­<lb/>pondente; deinde clivum montis comparando cum plano <lb/>horizontali. </s> <s id="s.000498">Et &longs;anè &longs;i planities in &longs;ummo montis jugo con­<lb/>&longs;ideretur, certum e&longs;t illam e&longs;&longs;e plurium &longs;tructurarum ca­<lb/>pacem, quàm &longs;ubjectum planum in &longs;uperficie globi ter­<lb/>re&longs;tris: Quemadmodum enim &longs;uperficies &longs;phæræ majoris <lb/>plura capit ædificia, quàm minor, ita etiam &longs;phærarum <lb/>inæqualium partes &longs;imiles inæqualis &longs;unt capacitatis: Con&longs;tat <lb/>autem planitiem in &longs;ummo monte pertinere ad &longs;phæram <lb/>majorem, quàm pertineat &longs;imilis planities illi &longs;ubjecta; ac <lb/>proinde & amplior e&longs;t, & magis capax. </s> <s id="s.000499">Harum verò pla­<lb/>nitierum differentia ea erit, quæ e&longs;t quadratorum di&longs;tan­<lb/>tiarum à centro terræ: quòd &longs;i quadratorum huju&longs;modi <lb/>differentia exigua &longs;it & contemnenda, eo quod ad illam <lb/>quadratum &longs;emidiametri terræ habeat nimis magnam ratio­<lb/>nem; planitierum pariter differentia fugiet omnem &longs;en&longs;um. <pb pagenum="59" xlink:href="017/01/075.jpg"/>Sit terræ &longs;emidiameter CS, altitudo au­<lb/><figure id="id.017.01.075.1.jpg" xlink:href="017/01/075/1.jpg"/><lb/>tem montis SR, in cujus vertice &longs;it pla­<lb/>nities RH, cui &longs;imilis e&longs;t in &longs;uperficie <lb/>globi terreni planities SO illi parallela: <lb/>hæ autem planities &longs;imiles habent, per <lb/>20. lib. 6. duplicatam Rationem laterum <lb/>RI, SL, hoc e&longs;t, per 4. lib. 6. duplica­<lb/>tam Rationis, quam habet CR ad CS. <!-- KEEP S--></s> <lb/> <s id="s.000500">E&longs;t igitur ut quadratum di&longs;tantiæ CR. <lb/>ad quadratum di&longs;tantiæ CS, ita plani­<lb/>ties RH ad planitiem SO. <!-- KEEP S--></s> <s id="s.000501">Plura itaque <lb/>ædificia perpendiculariter in&longs;i&longs;tentia <lb/>po&longs;&longs;unt in planitie RH majori excitari <lb/>in montis vertice, quàm in &longs;ubjectâ <lb/>plani tie. </s> </p> <p type="main"> <s id="s.000502">At &longs;i montis clivus RMOL comparetur cum &longs;ubjectâ pla­<lb/>nitie SO, certum e&longs;t illum e&longs;&longs;e majorem, &longs;icuti latus RL op­<lb/>po&longs;itum angulo RSL, qui non e&longs;t minor recto, majus e&longs;t la­<lb/>tere SL in triangulo RSL, & RM ad SF e&longs;t ut RC ad SC: <lb/>&longs;uperficies igitur LM comprehen&longs;a &longs;ub majoribus lateribus, <lb/>& angulis non minoribus, quàm &longs;uperficies SO, major erit, <lb/>&longs;i illa per &longs;e con&longs;ideretur. </s> <s id="s.000503">Non tamen continuò major dicenda <lb/>e&longs;t capacitas, quæ plura aut ampliora recipiat ædificia; ni&longs;i <lb/>mons ad ingentem altitudinem a&longs;cendat; tunc enim perpendi­<lb/>cula non &longs;unt inter &longs;e parallela, propter in&longs;ignem eorum <lb/>di&longs;tantiam. </s> <s id="s.000504">Nam &longs;i &longs;uper clivo AB <lb/>&longs;it &longs;tructura AL, cujus parietes per­<lb/><figure id="id.017.01.075.2.jpg" xlink:href="017/01/075/2.jpg"/><lb/>pendiculares, &longs;int etiam paralleli <lb/>LB, DA, illi non magis inter &longs;e <lb/>di&longs;tant, quàm &longs;i &longs;uper plano hori­<lb/>zontali NB fui&longs;&longs;ent excitati: quic­<lb/>quid &longs;it, quod, &longs;icut linea AB ma­<lb/>jor e&longs;t quàm NB, ita planum incli­<lb/>natum majus &longs;it plano horizontali. </s> <lb/> <s id="s.000505">Non igitur plures aut ampliores &longs;tructuras recipit clivus collis, <lb/>quàm &longs;ubjectum planum horizontale. </s> <s id="s.000506">Quod verò de &longs;tructuris <lb/>dicitur, de cæteris quoque intelligendum e&longs;t, quæ perpendi­<lb/>cularia in&longs;i&longs;tunt, & &longs;patium implent; at &longs;i ita &longs;e habeant, ut <pb pagenum="60" xlink:href="017/01/076.jpg"/>perpendicularia non in&longs;i&longs;tant, certum e&longs;t plures aut longiores <lb/>homines jacere po&longs;&longs;e in clivo AB, quos non capit planum NB: <lb/>vel &longs;i in clivo &longs;e minùs invicem impediant, tunc plura huju&longs;­<lb/>modi corpora in colle e&longs;&longs;e po&longs;&longs;unt quàm in planitie: &longs;i enim ra­<lb/>mi arboris inferioris re&longs;pondeant trunco &longs;uperioris, certum e&longs;t <lb/>quod multò viciniores e&longs;&longs;e po&longs;&longs;unt arbores, quàm in planitie, <lb/>ubi rami &longs;e vici&longs;&longs;im impedientes majorem po&longs;tulant truncorum <lb/>di&longs;tantiam; ac proinde etiam multo plures arbores intra ea&longs;­<lb/>dem parallelas erunt. </s> <s id="s.000507">Sic plures homines e&longs;&longs;e po&longs;&longs;unt in gradi­<lb/>bus amphitheatri, quàm in &longs;ubjecto plano, quia graciliores, <lb/>partes &longs;uperiorum re&longs;pondent cra&longs;&longs;ioribus inferiorum, & &longs;e <lb/>minùs invicem impedientes minus relinquunt &longs;patij vacui: <lb/>quod &longs;i non homines, &longs;ed parallelepipeda, &longs;tatueres in gradi­<lb/>bus, non plura &longs;tatui in iis po&longs;&longs;ent, quàm in planâ areâ gradi­<lb/>bus &longs;ubjectâ. </s> </p> <p type="main"> <s id="s.000508">Hæc autem ædificiorum æqualitas in clivo & in plani­<lb/>tie, locum non habet ni&longs;i intra illud &longs;patium, quod inter­<lb/>cipitur à perpendiculis Phy&longs;icè parallelis; &longs;tatim enim ac à <lb/>paralleli&longs;mo recedunt perpendicula, &longs;i ea fuerit altitudo, ad <lb/>quam clivus a&longs;cendens venit, ut planities parallela plano <lb/>horizontali in eâ altitudine major &longs;it, quàm &longs;imilis plani­<lb/>ties depre&longs;&longs;ior, etiam plura ædificia recipiet clivus, quàm <lb/>unica planities horizontalis &longs;ubjecta. </s> <s id="s.000509">Ponamus enim per­<lb/>pendicula GC, & OC jam non e&longs;&longs;e parallela, eamque e&longs;&longs;e <lb/>altitudinem KG, ut planum per G tran&longs;iens horizonti <lb/>parallelum majus &longs;it plano per O intra eadem perpendicu­<lb/>la intercepto, erit quidem capacitas plani inclinati GOLF <lb/>æqualis capacitati &longs;ubjecti plani EKOL: at ulteriùs a&longs;cen­<lb/>dendo capacitas FGMR non erit æqualis capacitati plani <lb/>SK continuati cum priore plano EO, &longs;ederit major, quip­<lb/>pe quæ æqualis e&longs;t capacitati plani VG; e&longs;t autem pla­<lb/>num VG ad planum &longs;imile SK, ut quadratum GC ad <lb/>quadratum KC: major igitur e&longs;t totius clivi ML capacitas, <lb/>quàm planitiei SO. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000510">Et ut res apertius con&longs;tet, quandoquidem clivi alti&longs;­<lb/>&longs;imorum montium, &longs;i eandem &longs;ervent inclinationem, non <lb/>&longs;unt ab imo pede ad &longs;ummum jugum æquabili, & conti­<lb/>nuo ductu exten&longs;i, Sit terræ centrum H, & &longs;uperficies <pb pagenum="61" xlink:href="017/01/077.jpg"/>AD; cujus arcus dividatur in par­<lb/><figure id="id.017.01.077.1.jpg" xlink:href="017/01/077/1.jpg"/><lb/>tes AB, BC, CD æquales, ita ut <lb/>&longs;inguli arcus pro rectâ lineâ, & &longs;u­<lb/>perficies pro plano horizontali <lb/>Phy&longs;icè u&longs;urpari po&longs;&longs;int; & tunc <lb/>&longs;olùm intelligatur mutari horizon, <lb/>quando ex A jam venerit in B, <lb/>deinde in C &c. </s> <s id="s.000511">Si igitur &longs;it pla­<lb/>num inclinatum AE, ubi venerit <lb/>in E punctum perpendiculi HB <lb/>producti, non pote&longs;t rectâ progre­<lb/>di, quin mutet inclinationem &longs;upra horizontem novum, ad <lb/>quem venit; quare ut &longs;ervetur &longs;imilis inclinatio, deflectit in EF, <lb/>& e&longs;t angulus HEF æqualis angulo HAE cui demum ubi ve­<lb/>nerit in F, debet fieri æqualis angulus HEG. </s> <s id="s.000512">Centro autem H, <lb/>intervallis HE & HF de&longs;cribantur arcus EI, & FK. </s> <s id="s.000513">Certum <lb/>e&longs;t duarum linearum angulum con&longs;tituentium partem aliquam <lb/>extremam e&longs;&longs;e, &longs;ecundùm quam lineæ illæ non differunt, &longs;en&longs;u <lb/>judice, à parallelis; at &longs;i major pars accipiatur, jam perit paral­<lb/>leli&longs;mus: Sic RA, & EB pro parallelis u&longs;urpari &longs;i po&longs;&longs;int, non <lb/>poterunt &longs;imiliter pro parallelis accipi RA, & LB: Sic LE, & <lb/>FI &longs;umuntur tanquam parallelæ citrà errorem, at non item LB, <lb/>& MC. <!-- KEEP S--></s> <s id="s.000514">Quare perpendicula non &longs;olùm recedunt à paralleli&longs;­<lb/>mo &longs;en&longs;ibili, quia majorem angulum in centro H con&longs;tituunt, <lb/>&longs;ed etiam quia major eorum pars a&longs;&longs;umitur, in qua jam apparet <lb/>convergentia, quæ in parte minore latebat. </s> </p> <p type="main"> <s id="s.000515">Cum itaque &longs;tructuræ perpendiculares in plano inclinato <lb/>occupent &longs;patium eodem modo, ac &longs;i e&longs;&longs;ent in plano horizon­<lb/>tali intra ea&longs;dem parallelas, jam con&longs;tat clivi partem EF com­<lb/>parandam e&longs;&longs;e cum plano EI, non autem cum plano BC; quia <lb/>in E, & I terminatur paralleli&longs;mus linearum LE, FI. <!-- KEEP S--></s> <s id="s.000516">E&longs;t igi­<lb/>tur capacitas clivi EF æqualis capacitati EI; at capacitas EI <lb/>major e&longs;t quàm capacitas BC, ergo capacitas clivi AF major <lb/>e&longs;t, quàm capacitas planitiei AC. <!-- KEEP S--></s> <s id="s.000517">Eademque e&longs;to de cæteris <lb/>ratio. </s> <s id="s.000518">Hinc manife&longs;tum e&longs;t non omninò in univer&longs;um vera e&longs;&longs;e, <lb/>quæ pa&longs;&longs;im dicuntur de æquali capacitate collium, & planitiei <lb/>&longs;ubjectæ, ni&longs;i hæc certis limitibus circum&longs;cribantur; videlicet <lb/>&longs;i &longs;ermo &longs;it de iis quæ tantùm perpendiculariter in&longs;i&longs;tunt, & <pb pagenum="62" xlink:href="017/01/078.jpg"/>intrà illud &longs;patium, ac in eá altitudine, ubi perpendiculorum <lb/>convergentia adeò exigua e&longs;t, ut evane&longs;cat. </s> <s id="s.000519">Cæterùm &longs;atis <lb/>mihi videor o&longs;tendi&longs;&longs;e fieri po&longs;&longs;e, ut clivus aliquis plures <lb/>&longs;tructuras recipere po&longs;&longs;it, quàm &longs;uperficies &longs;phærica globi illi <lb/>re&longs;pondens. </s> <s id="s.000520">Si enim eadem e&longs;t &longs;emper, ut &longs;upponitur, plani <lb/>inclinatio, etiam latera turrium, vel domorum parietes æquè <lb/>invicem remoti intercipient æquales partes plani inclinati: Si <lb/>ergo &longs;tructura intercipiens &longs;emi&longs;&longs;em plani AE transferatur in <lb/>EF, æqualem partem intercipiet; at hæc minor e&longs;t &longs;emi&longs;&longs;e <lb/>ip&longs;ius EF, igitur duæ &longs;tructuræ occupantes totum planum AE, <lb/>tran&longs;latæ in EF æquale &longs;patium occupabunt, & relinquent <lb/>adhuc partem &longs;patij inanem. </s> <s id="s.000521">E&longs;&longs;e autem EF lineam majorem <lb/>linea AE patet; quia triangula AHE, EHF æquiangula <lb/>&longs;unt, & latera habent proportionalia, adeóque ut AH ad HE, <lb/>ita AE ad EF; atqui HE excedit lineam HA; igitur & EF <lb/>major e&longs;t quàm AE: ergo multo major erit &longs;uperficies ip&longs;ius <lb/>EF, quàm &longs;uperficies &longs;imilis ip&longs;ius AE. <!-- KEEP S--></s> <s id="s.000522">In &longs;patio igitur, quo <lb/>&longs;uperficies EF excedit &longs;uperficiem AE, poterit alia præterea <lb/>&longs;tructura excitari. <lb/></s> </p> <p type="head"> <s id="s.000523"><emph type="center"/>CAPUT XI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="head"> <s id="s.000524"><emph type="center"/><emph type="italics"/>Quomodo animalium motus ordinentur ex centro <lb/>gravitatis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="head"> <s id="s.000525">DEi &longs;apientiam nunquam &longs;atis admirari po&longs;&longs;umus, quæ in <lb/>ordinandis naturæ motibus elucet; animalia enim &longs;olo <lb/>naturæ ductu adeò accuratè &longs;e ip&longs;a &longs;i&longs;tunt in lineâ directionis, <lb/>ut nemo mathematicus Geometriæ apices per&longs;crutatus po&longs;&longs;it <lb/>tam &longs;ubtiliter deprehendere, ac brevi&longs;&longs;imo temporis momento, <lb/>centrum gravitatis. </s> <s id="s.000526">Quandoquidem &longs;ive con&longs;i&longs;tentium quie­<lb/>tem, &longs;ivè gradientium motum, &longs;ivè reclinantium &longs;e &longs;e inflexio­<lb/>nem con&longs;ideres, miram naturæ artem intelliges, quâ præcavit, <lb/>ne corpus ingenitâ gravitate delatum præceps caderet. </s> <s id="s.000527">Id au­<lb/>tem a&longs;&longs;ecuta e&longs;t motus ita di&longs;ponendo, ut linea directionis nun-<pb pagenum="63" xlink:href="017/01/079.jpg"/>quam caderet extrà ba&longs;im &longs;u&longs;tentationis, ni&longs;i fortè in cur&longs;u, in <lb/>quo tamen &longs;atis con&longs;ultum e&longs;t animalis incolumitati, dum ab <lb/>anteriore pede, ubi terram attigerit, retinetur, ne ulteriùs <lb/>de&longs;cendat. </s> </p> <p type="main"> <s id="s.000528">Ba&longs;is autem &longs;u&longs;tentationis non &longs;unt &longs;oli pedes, &longs;ed totum <lb/>illud &longs;patium interceptum à lineis pedum extremitates jun­<lb/>gentibus; &longs;ic in quadrupedibus linea directionis debet cadere <lb/>intrà &longs;patium comprehen&longs;um lineis, quæ jungunt extrema <lb/>pedum terram contingentium, ut po&longs;&longs;it animal con&longs;i&longs;tere. </s> <lb/> <s id="s.000529">Hinc equus in po&longs;teriores pedes &longs;e erigens flexis poplitibus <lb/>reclinat &longs;e &longs;e in po&longs;teriora, & tanti&longs;per in eo &longs;itu con&longs;i&longs;tit, <lb/>dum centrum gravitatis imminet &longs;patio, quod à pedibus oc­<lb/>cupatur, & ab illis intercipitur; & &longs;i extra illud &longs;patium ca­<lb/>dat linea directionis, vel aver&longs;us cadit, vel iterum quatuor <lb/>pedibus in&longs;i&longs;tit. </s> <s id="s.000530">Ubi tamen ob&longs;ervandum e&longs;t ex equo & equi­<lb/>te fieri unam molem compo&longs;itam unum habentem commune <lb/>centrum gravitatis: unde fit equum magis defatigari, &longs;i eques <lb/>non rectus in&longs;ideat; &longs;ed inclinatus in alterutram partem, cen­<lb/>tro enim gravitatis tran&longs;lato motûs facilitas mutatur; & equite <lb/>in anteriora inclinato ac premente caput equi in po&longs;teriores <lb/>pedes erecti, centrum gravitatis in anteriora transfertur, & <lb/>occurritur periculo, ne equus aver&longs;us cadat. </s> </p> <p type="main"> <s id="s.000531">Porrò dum &longs;patium à pedibus occupatum voco ba&longs;im &longs;u&longs;ten­<lb/>tationis, non &longs;emper &longs;atis e&longs;t lineam directionis cadere non <lb/>extrà pedes; quia &longs;i pedes ip&longs;i &longs;olùm ex parte tangant &longs;ub­<lb/>jectum corpus, ut contingit in funambulis, debet linea di­<lb/>rectionis cadere in funem, cui in&longs;i&longs;tunt pedes, & &longs;i extra il­<lb/>lum cadat, certa e&longs;t ruina, quia latitudo pedum non juvat. </s> <lb/> <s id="s.000532">Cum autem difficillimum &longs;it diutiùs con&longs;i&longs;tere ita, ut centrum <lb/>gravitatis &longs;emper immineat funi, ideò funambuli, vel ha&longs;tam <lb/>plumbeis laminis gravem in extremitatibus manu tenent, vel <lb/>brachiis expan&longs;is &longs;e librant, ut ha&longs;tam vel brachia extenden­<lb/>tes in partem oppo&longs;itam ei, in quam gravitas inclinat, cen­<lb/>trum gravitatis con&longs;tituatur in puncto, quod immineat funi <lb/>&longs;ui tentanti. </s> <s id="s.000533">Hinc oritur difficultas con&longs;i&longs;tendi, quam expe­<lb/>riuntur grallatores; cum enim grallæ exiguâ &longs;ui parte tangant <lb/>terram, e&longs;t qua&longs;i linea, in qua fit &longs;u&longs;tentatio, extra quam fa­<lb/>cilè cadit linea directionis: ideò tertium ge&longs;tant baculum, cui <pb pagenum="64" xlink:href="017/01/080.jpg"/>innitantur, quoties quie&longs;cere voluerint, lineâ directionis ca­<lb/>dente intrà &longs;patium triangulare comprehen&longs;um à grallis, & <lb/>baculo. </s> </p> <p type="main"> <s id="s.000534">Hîc autem maximè &longs;e prodit naturæ providentia in tam va­<lb/>riâ pedum conformatione, ut ad &longs;u&longs;tentandum idonei e&longs;&longs;ent: <lb/>quadrupedibus &longs;iquidem non adeò amplos pedes tribuit, quia <lb/>ex eorum inter &longs;e di&longs;tantiâ plurimum &longs;patium intercipitur, cui <lb/>immineat centrum gravitatis: bipedibus verò latiores tribuit <lb/>pedes, quâ parte timeri potuit ca&longs;us: &longs;ic quia ex duorum cru­<lb/>rum modicâ divaricatione non facilè periculum erat cadendi <lb/>in alterutrum latus, ideò humanis pedibus minorem dedit la­<lb/>titudinem, quàm longitudinem; hanc verò non in æquas <lb/>di&longs;tribuit partes, &longs;ed minimam calci (præterquam in Scauris, <lb/>quos pravis fultos male talis appellat Horatius, talis &longs;cilicet <lb/>extantioribus) maximam anteriori parti conce&longs;&longs;it, ne impetu <lb/>per motum concepto tran&longs;latum centrum gravitatis in anterio­<lb/>ra tran&longs;iliret ba&longs;im &longs;u&longs;tentationis. </s> <s id="s.000535">Aliquam tamen mediocrem <lb/>latitudinem pedibus conce&longs;&longs;it, ut po&longs;&longs;et homo, &longs;i res ferret, uni <lb/>tantùm pedi in&longs;i&longs;tere, & e&longs;&longs;et aliqua &longs;patij amplitudo, intrà <lb/>quam quodlibet punctum opportunum e&longs;&longs;et con&longs;i&longs;tentiæ cen­<lb/>tri gravitatis. </s> <s id="s.000536">Sic aves illæ, quæ uni pedi in&longs;i&longs;tunt, cuju&longs;modi <lb/>&longs;unt grues, & ciconiæ, digitos habens longiores, quos valdè <lb/>explicant qua&longs;i in gyrum, ut amplior &longs;it ba&longs;is &longs;u&longs;tentationis; in­<lb/>trà quam ut cadat linea directionis, altero pede elevato inclina­<lb/>tur corpus in oppo&longs;itam partem, ut centrum gravitatis immineat <lb/>pedi &longs;u&longs;tentanti. </s> <s id="s.000537">Eandem ob cau&longs;am an&longs;eres, & anates, quæ <lb/>multâ carne abundant, & amplo &longs;unt pectore, alternâ qua­<lb/>dam in dextrum, & &longs;ini&longs;trum latus inclinatione gradiuntur, <lb/>ideóque ampliores habent palmas, ut citrà cadendi periculum <lb/>centrum gravitatis faciliùs vel immineat pedi &longs;u&longs;tentanti, vel <lb/>minimùm ab eo declinet, ne majore, quàm par &longs;it, impetu <lb/>de&longs;cendens corpus & anteriori pedi incumbens, tibiæ mu&longs;cu­<lb/>los, & tendines lædat. </s> <s id="s.000538">Aves verò, quæ &longs;ubtilioribus ramu&longs;cu­<lb/>lis in&longs;ident non palmipedes &longs;unt, &longs;ed digitatæ (palmæ enim <lb/>avibus amphibiis ad natandum poti&longs;&longs;imum datæ videntur) ut <lb/>ramis tenaciùs inhæreant; quæ præterquàm quod exiguæ &longs;unt <lb/>gravitatis, facilè &longs;e &longs;i&longs;tunt in lineâ directionis, quæ cadat in <lb/>ramu&longs;culum, cui in&longs;i&longs;tunt, majore, vel minore angulo, quem <pb pagenum="65" xlink:href="017/01/081.jpg"/>faciunt tibiæ cum coxâ; ideò ubi ramum arripuerint, &longs;ub&longs;ul­<lb/>tantes &longs;e librant, ramumque arctè apprehentes prohibent, ne <lb/>repentino ca&longs;u circumagantur à centro gravitatis nondum im­<lb/>minente ba&longs;i &longs;u&longs;tentationis. </s> </p> <p type="main"> <s id="s.000539">Verùm quoniam ad aves delap&longs;us &longs;um, prætereundus non <lb/>e&longs;t u&longs;us centri gravitatis involatu; quia enim avis dum alis <lb/>aërem verberans in volatu &longs;e librat atque &longs;u&longs;pendit, ita alas <lb/>debet extendere, ut centrum gravitatis exi&longs;tat intra illud <lb/>alarum &longs;patium, in quo exercetur &longs;u&longs;tentatio; ideò &longs;i vo­<lb/>luerit ad &longs;uperiora volatum dirigere, alas in anteriora ver­<lb/>&longs;us caput extendit, ut centro gravitatis in po&longs;terioribus re­<lb/>licto, ac deor&longs;um præponderante, caput &longs;ur&longs;um dirigatur: <lb/>contra verò, ut motum deor&longs;um dirigat, alas retrahit, ut <lb/>caput præponderet, ac deor&longs;um feratur. </s> <s id="s.000540">Hinc &longs;atis patet, <lb/>cur ubi Pavo caudæ pompam explicuerit, erecto pectore & <lb/>capite in&longs;i&longs;tat pedibus, quibus immineat centrum gravita­<lb/>tis: at &longs;i caput ad anteriora inclinare voluerit, & pectus <lb/>inflectere, cogitur explicatam caudam demittere, ut &longs;yrma­<lb/>te illo æquilibrium &longs;tatuat corpori, ne proruat, ut verè pro­<lb/>cumberet, &longs;i pectore inclinato expan&longs;a cauda retineretur in <lb/>po&longs;itione eâdem. </s> </p> <p type="main"> <s id="s.000541">Infinitum e&longs;&longs;et &longs;ingulos animalium motus per&longs;equi, in qui­<lb/>bus centri gravitatis ratio habetur; &longs;atis fuerit ob&longs;erva&longs;&longs;e nos <lb/>ex declivi loco de&longs;cendentes non in&longs;i&longs;tere plantis pedum ad <lb/>angulos rectos; &longs;ed paululum in po&longs;teriora inclinari; contra <lb/>verò a&longs;cendentes jugum acclive curvari in anteriora; ut nimi­<lb/>rum linea directionis cadat intrà &longs;patium, cui pedes in&longs;i&longs;tunt; <lb/>extra quod illa &longs;i caderet, nec alteri fulcro inniteremur, quod <lb/>unà cum pedibus includeret ba&longs;im &longs;u&longs;tentationis, nece&longs;&longs;ariò <lb/>nobis cadendum e&longs;&longs;et. </s> <s id="s.000542">Quòd &longs;i quis onus habens dor&longs;o impo­<lb/>&longs;itum in montosâ regione iter habeat, multò magis curvari de­<lb/>bet, cum a&longs;cendit, ut pedibus immineat centrum gravitatis <lb/>compo&longs;itæ ex corpore, & ex onere: quare &longs;apienti&longs;&longs;imè ru&longs;tici <lb/>aliqui in Alpibus, quæ Germaniam ab Italiá di&longs;terminant, ar­<lb/>culam ex levibus a&longs;&longs;erculis, & virgulis compactam habent, cui <lb/>onera immittunt, ba&longs;is autem arculæ, quæ ge&longs;tantis corpori <lb/>adhæret, imitatur Re&longs;c Hebraicum, ita ut pars quidem dor­<lb/>&longs;o, pars autem capiti incumbat: unde fit, ut centrum gravita-<pb pagenum="66" xlink:href="017/01/082.jpg"/>tis compo&longs;itæ minùs recedat à medio humani corporis, adeó­<lb/>que faciliùs etiam motus perficiatur, quin opus &longs;it tantâ corpo­<lb/>ris inflexione. </s> <s id="s.000543">Simile quid experimur, &longs;i quis à &longs;ede &longs;urgat; <lb/>caput enim cum thorace in anteriora reclinat; pedes verò in <lb/>po&longs;teriora versùs &longs;edem retrahit, ut nimirum pedes &longs;upponan­<lb/>tur centro gravitatis, quod primùm imminet parti digitis proxi­<lb/>mæ, deinde corpore erecto linea directionis versùs talos rece­<lb/>dit. </s> <s id="s.000544">Hinc etiam patet cur homo &longs;upinus jacens &longs;urgere non <lb/>po&longs;&longs;it, ni&longs;i retractis &longs;ub &longs;e pedibus, & thorace in anteriora pro­<lb/>pul&longs;o per impetum &longs;ibi impre&longs;&longs;um. </s> <s id="s.000545">Vidi tamen non &longs;emel ho­<lb/>minem, qui cum &longs;upinus jaceret, non retractis &longs;ub &longs;e pedibus <lb/>&longs;urgebat planè rectus &longs;icut &longs;tipes; ad caput autem appone­<lb/>bat, vel globum tormentarium majorem, vel &longs;axum non <lb/>modicæ gravitatis; quod manu utrâque apprehen&longs;um attol­<lb/>lebat, & velociter in anteriora movebat, &longs;ibique impetum <lb/>imprimebat: impetus enim impre&longs;&longs;us promovens ad ante­<lb/>riora &longs;axum, & corpus ip&longs;um vincebat gravitatem corpo­<lb/>ris cæteroqui ca&longs;uri; ex brachiis autem exten&longs;is &longs;axum à <lb/>corpore remotum tenentibus oriebatur, ut centrum gravi­<lb/>tatis molis compo&longs;itæ longè citiùs immineret pedibus, à <lb/>quibus &longs;u&longs;tentabatur, etiam antequam planta terram at­<lb/>tingeret, &longs;ed cum adhuc &longs;oli calci inniteretur. </s> <s id="s.000546">Quantum <lb/>verò impetus valeat ad vincendam oppo&longs;itam gravitatem <lb/>corporis, patet in ce&longs;pitantibus, qui naturæ ductu illico bra­<lb/>chia extendunt, & in contrariam partem projiciunt, ut &longs;ci­<lb/>licet impetus in oppo&longs;itam partem exæquet exce&longs;&longs;um gravita­<lb/>tis, quæ ad eam partem reperitur, in quam ex ce&longs;pitatione <lb/>facta e&longs;t inclinatio. </s> </p> <p type="main"> <s id="s.000547">Ex his quid in &longs;ingulis motibus dicendum &longs;it, intelli­<lb/>ges; neque enim otium e&longs;t ire per &longs;ingula. </s> <s id="s.000548">Caput hoc <lb/>claudo explicatione quæ&longs;tionis, qua quæritur, quantò ma­<lb/>jus &longs;patium percurrat caput quàm pedes; certum &longs;iquidem <lb/>e&longs;t hominem in lineâ directionis imminere &longs;emper terræ <lb/>centro; ac proinde &longs;i pedes ex B venerunt in C, caput ex <lb/>F in E tran&longs;latum e&longs;t per arcum FE majorem arcu BC. <!-- KEEP S--></s> <lb/> <s id="s.000549">Cum enim uterque arcus BC, FE &longs;ubtendatur eidem an­<lb/>gulo ad centrum, &longs;unt &longs;imiles, & ut arcus BC ad totam <lb/>&longs;uam peripheriam, ita arcus FE ad &longs;uam peripheriam; &longs;unt <pb pagenum="67" xlink:href="017/01/083.jpg"/>autem peripheriæ inter &longs;e ut &longs;emi­<lb/><figure id="id.017.01.083.1.jpg" xlink:href="017/01/083/1.jpg"/><lb/>diametri, igitur BC ad FE, ut TB, <lb/>ad TF; atqui TF major e&longs;t quàm <lb/>TB, igitur & FE arcus major arcu <lb/>BC: ab&longs;cindatur FI, quæ ex hypo­<lb/>the&longs;i intelligatur æqualis ip&longs;i BC; <lb/>e&longs;t igitur ut TB ad TF, ita FI ad <lb/>FE, & dividendo ut TB ad BF <lb/>ita FI, hoc e&longs;t BC, ad IE. </s> <s id="s.000550">Fiat ita­<lb/>que ut TB &longs;emidiameter terræ mil­<lb/>liar. </s> <s id="s.000551">Rom. <!-- REMOVE S-->ant.4128.pa&longs;&longs;.635. ad BF <lb/>altitudinem hominis ex. </s> <s id="s.000552">gr. <!-- REMOVE S-->ped. <!-- REMOVE S-->Rom. <!-- REMOVE S-->ant. </s> <s id="s.000553">6. ita BC iter pe­<lb/>dum mill. <!-- REMOVE S-->500, ad IE exce&longs;&longs;um itineris capitis qui e&longs;t (726632/1000000) <lb/>unius pedis. </s> <s id="s.000554">Quòd &longs;i fiat ut terræ &longs;emidiameter ad hominis al­<lb/>titudinem, ita circulus terræ maximus mill. <!-- REMOVE S-->25941 ad exce&longs;­<lb/>&longs;um itineris capitis &longs;upra iter pedum terræ ambitum percurren­<lb/>tium, proveniet exce&longs;&longs;us ped. <!-- REMOVE S-->37. unc.8. hoc e&longs;t pa&longs;&longs;.7. & pau­<lb/>lò ampliùs: Quare vides in &longs;ingulis milliariis motum capitis non <lb/>habere exce&longs;&longs;um ni&longs;i partium (17429/1000000) unciæ pedis Romani anti­<lb/>qui; quæ differentia &longs;en&longs;um omnem fugit. </s> </p> <p type="main"> <s id="s.000555">Liceat hic ex morâ, quam in hoc Tractatu perficiendo duxi, <lb/>id utilitatis capere, quod po&longs;&longs;im pro me ip&longs;e brevi Apologiâ <lb/>re&longs;pondere, ne videar in Ageometriam lap&longs;us, cui nulla ni&longs;i ex <lb/>o&longs;citantiâ &longs;uppeteret excu&longs;atio (nam & quandoque bonus dor­<lb/>mitat Homerus) & quidem tunc, cùm Mathematicas di&longs;cipli­<lb/>nas in Collegio Romano publicè pro&longs;itentem maximè ocula­<lb/>tum fui&longs;&longs;e oportuerat. </s> <s id="s.000556">Incidi in Magiam Naturalem P. <!-- REMOVE S-->Ga&longs;paris <lb/>Schotti part.3.lib.1. pag. </s> <s id="s.000557">71, ubi mihi tribuit &longs;ententiam maxi­<lb/>mè ab&longs;urdam, qua&longs;i in mechanicâ meâ manu&longs;criptâ (quam <lb/>&longs;cilicet anno 1653. Romæ auditoribus meis tradidi) docuerim <lb/>exce&longs;&longs;um motûs capitis &longs;upra motum pedum <emph type="italics"/>e&longs;&longs;e valde modi­<lb/>cum, nimirum &longs;olum pedum &longs;ex cum dimidio, adeò ut in milliaribus<emph.end type="italics"/><lb/>500 <emph type="italics"/>tantum reperiatur exce&longs;&longs;us<emph.end type="italics"/> (15/17) <emph type="italics"/>unius pedis, po&longs;itá hominis altitu­<lb/>dine pedum &longs;ex, & terræ ambitu milliariorum<emph.end type="italics"/> 21600. Hæ&longs;i pri­<lb/>mùm attonitus, meamque o&longs;citantiam admiratus illicò anti­<lb/>quàs illas meas &longs;chedulas per&longs;crutari cœpi; & nihil minus in­<lb/>veniens errorem Typographo, qui pro pa&longs;&longs;ibus pedes &longs;uppo-<pb pagenum="68" xlink:href="017/01/084.jpg"/>&longs;uerit, tribuendum cen&longs;ui&longs;&longs;em, ni&longs;i Author ip&longs;e modicum il­<lb/>lum exce&longs;&longs;um pedum &longs;ex cum dimidio redargueret. </s> <s id="s.000558">Quare <lb/>contingere facile potuit, ut ille, qui tunc Romæ degebat, ex <lb/>aliquo manu&longs;cripto codice meam &longs;ententiam re&longs;cribens, ubi <lb/>men&longs;uram hanc pedibus definiebam, brevitatis ergo ad pa&longs;­<lb/>&longs;us revocaverit, quam litera P notatam demùm pro pedibus &longs;it <lb/>interpretatus. </s> <s id="s.000559">Cæterùm prudens, & attentus lector me facilli­<lb/>mè ab hoc errore vindicabit, &longs;i terræ ambitum mill. 21600. di­<lb/>vidat per mill.500; & quotientem 43 multiplicet per (15/17) unius <lb/>pedis; deprehendet enim totum exce&longs;&longs;um pedum ferè 38, qui <lb/>excedunt pa&longs;&longs;us &longs;eptem cum dimidio. </s> <s id="s.000560">Quod &longs;i ex diametro pe­<lb/>dum 34400000, & ex diametro pedum 34400012, quas ibi Au­<lb/>thor ponit congruentes peripheriæ juxta Rationem 7 ad 22 con­<lb/>&longs;iderentur, erit differentia circulorum pedum 38 eadem plane <lb/>cum no&longs;trâ; &longs;ed longi&longs;&longs;imè minor eâ, quam ille ibi &longs;tatuit. </s> </p> <p type="main"> <s id="s.000561">Cæterùm quantus &longs;it peripheriæ majoris exce&longs;&longs;us &longs;upra mi­<lb/>norem, habebitur facillimè, &longs;i majoris Radij TF, exce&longs;&longs;um <lb/>BF, &longs;tatuas tanquam circuli Radium; hujus namque circuli <lb/>peripheria e&longs;t æqualis exce&longs;&longs;ui illi. </s> <s id="s.000562">Quia enim ut minor Ra­<lb/>dius TB ad majorem Radium TF, ita minor peripheria ad <lb/>majorem peripheriam, etiam convertendo & dividendo, ut <lb/>TB ad BF, ita minor peripheria ad exce&longs;&longs;um peripheriæ ma­<lb/>joris, & vici&longs;&longs;im permutando ut Radius TB minor ad &longs;uam <lb/>minorem peripheriam, ita BF exce&longs;&longs;us Radij majoris ad exce&longs;­<lb/>&longs;um majoris peripheriæ. </s> <s id="s.000563">Atqui exce&longs;&longs;us hic BF a&longs;&longs;umptus ut <lb/>Radius circuli habet ad &longs;uam peripheriam eandem Rationem, <lb/>quam TB Radius minor ad &longs;uam peripheriam; igitur e&longs;t ea­<lb/>dem Ratio BF exce&longs;sûs Radij, ad exce&longs;&longs;um peripheriæ majo­<lb/>ris, quæ e&longs;t eju&longs;dem BF ut Radij ad &longs;uam peripheriam: ergo <lb/>per 9. lib. 5. hæc peripheria æqualis e&longs;t illi exce&longs;&longs;ui periphe­<lb/>riæ majoris. </s> <s id="s.000564">Cum itaque Ratio diametri ad peripheriam &longs;it ut <lb/>7 ad 22, &longs;eu ut 113 ad 355, fiat ut Radius 7 ad peripheriam <lb/>44, &longs;eu ut 113 ad 710, ita BF altitudo ped. <!-- REMOVE S-->6. ad ped. <!-- REMOVE S-->37. <lb/>unc. 8: qui numerus con&longs;entit cùm &longs;uperiore. <pb pagenum="69" xlink:href="017/01/085.jpg"/></s> </p> <p type="head"> <s id="s.000565"><emph type="center"/>CAPUT XII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="head"> <s id="s.000566"><emph type="center"/><emph type="italics"/>An tellus moveatur motu trepidationis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000567">QUoniam centrum gravitatis e&longs;t in quolibet corpore <lb/>punctum illud, quod æquales gravitates circum&longs;tant, <lb/>manife&longs;tum e&longs;t non permanere idem gravitatis centrum, &longs;i <lb/>aliqua corpori additio fiat, aut detractio; neque enim manet <lb/>eadem momentorum gravitatis æqualitas circa illud punctum; <lb/>&longs;ed aliud e&longs;t punctum, per quod ducta plana dividunt totius <lb/>corporis gravitatem in momenta æqualia, & e&longs;t novum cen­<lb/>trum gravitatis. </s> <s id="s.000568">Hinc patet in telluris globo, qui plurimas <lb/>mutationes &longs;ubit, corporibus gravibus ex alio in alium locum <lb/>tran&longs;latis, tolli æqualitatem partium &longs;altem in actu primo gra­<lb/>vitantium, cum hæc quidem, quæ oppo&longs;itæ parti ante erat <lb/>æqualis, &longs;ubtractione nunc fiat minor, illa verò, quæ pariter <lb/>&longs;ibi oppo&longs;itæ parti proximè fuit æqualis, additione evadat ma­<lb/>jor. </s> <s id="s.000569">Ex quo nece&longs;&longs;ariò colligitur mutatio centri gravitatis. </s> </p> <p type="main"> <s id="s.000570">Sed quia, ut tellus &longs;uis librata ponderibus in loco &longs;ibi debi­<lb/>to con&longs;i&longs;teret, debuit initio ejus centrum gravitatis congrue­<lb/>re centro univer&longs;i, circa quod gravia & levia di&longs;ponuntur; id­<lb/>circò dubitari pote&longs;t, utrùm mutato gravitatis centro terra mo­<lb/>veri debeat, ut novum gravitatis centrum collocetur in centro <lb/>univer&longs;i. </s> <s id="s.000571">Quoniam verò huc illuc pa&longs;&longs;im tran&longs;latis corpori­<lb/>bus, terra nunc in hanc, nunc in illam partem moveretur, ut <lb/>proinde qua&longs;i trepidaret; hinc factus e&longs;t quæ&longs;tioni locus, an <lb/>tellus moveatur motu trepidationis; quicquid &longs;it an motus i&longs;te <lb/>&longs;ub &longs;en&longs;um cadat, nec ne. </s> </p> <p type="main"> <s id="s.000572">Terram univer&longs;am & &longs;ingulas ejus partes &longs;uâ gravitate re­<lb/>pugnare, ne &longs;ur&longs;um moveantur, certum e&longs;t; at univer&longs;i cen­<lb/>trum occupare, toti quidem elemento gravi&longs;&longs;imo convenit, &longs;ed <lb/>non partibus &longs;ingulis: neque enim gravitas e&longs;t appetitus &longs;ub­<lb/>&longs;i&longs;tendi in centro, quem natura non &longs;atis aptè gravibus &longs;ingu­<lb/>lis indidi&longs;&longs;et; cui nimirùm fieri &longs;atis non pote&longs;t, ni&longs;i corpora <lb/>&longs;e invicem penetrent; unum autem grave in centro exi&longs;tens <pb pagenum="70" xlink:href="017/01/086.jpg"/>cætera omnia inde excludit. </s> <s id="s.000573">Re&longs;tituunt &longs;e gravia in locum <lb/>&longs;uum versùs centrum pergendo, non ut ad centrum veniant; <lb/>&longs;ed ut nihil levius infra &longs;e habeant; quemadmodum & levia <lb/>versùs cælum a&longs;cendunt, non ut cælum petant, ibíque demum <lb/>quie&longs;cant, &longs;ed ne quid gravius &longs;upra &longs;e patiantur. </s> <s id="s.000574">Cæterùm <lb/>hoc ip&longs;o, quòd natura, & vacuitatem omnem eliminavit, & <lb/>corporum penetrationem pro&longs;crip&longs;it, & vim &longs;e &longs;uis locis di&longs;po­<lb/>nendi corporibus indidit, &longs;atis univer&longs;i con&longs;i&longs;tentiæ & ordini <lb/>con&longs;ultum e&longs;t. </s> <s id="s.000575">Quare corpori nihil levius infra &longs;e habenti nul­<lb/>lam præterea gravitationem tribuendam cen&longs;eo, præter re­<lb/>&longs;i&longs;tentiam, ne &longs;ur&longs;um moveatur. </s> <s id="s.000576">Gravitas &longs;iquidem non ni&longs;i <lb/>comparatè dicitur, habitâ ratione proximi corporis, in quo <lb/>tanquam in loco exi&longs;tit id, quod grave dicitur; nam &longs;i orbis <lb/>univer&longs;us con&longs;taret unico corpore homogeneo, nihil e&longs;&longs;et aut <lb/>grave aut leve, cum nihil e&longs;&longs;et, quòd præ aliis expo&longs;ceret pro­<lb/>piùs admoveri centro univer&longs;i. </s> <s id="s.000577">Cum itaque terra ad hoc uni­<lb/>ver&longs;i centrum perinde &longs;e habeat, atque &longs;i corporibus levioribus <lb/>non circumfunderetur, his namque &longs;ublatis illa nec propiùs ad <lb/>univer&longs;i centrum accederet, nec longiùs ab eo recederet; ideò <lb/>pars terræ quæcumque cum reliquis comparata (ponatur hîc <lb/>tellus tota homogenea) nec gravis e&longs;t nec levis; ac proinde, <lb/>cùm nulla pars centro propior e&longs;&longs;e exigat, quàm alia, nulla <lb/>quoque e&longs;t, quæ aliam urgeat, aut premat propriè, &longs;ed omnes, <lb/>& &longs;ingulæ tantummodò repugnant, ne &longs;ur&longs;um in medium leve <lb/>transferantur. </s> </p> <p type="main"> <s id="s.000578">Hinc e&longs;t quod terræ con&longs;i&longs;tentiam in loco &longs;uo, non propriè <lb/>ex libræ rationibus explicandam cen&longs;eo; quia in librâ utraque <lb/>lanx non repugnat &longs;olùm, ne attollatur, verùm etiam in aere <lb/>con&longs;tituta deor&longs;um nititur; terræ autem partes &longs;uperiores nil <lb/>infrà &longs;e levius habentes non conantur deor&longs;um. </s> <s id="s.000579">Et quemad­<lb/>modum &longs;i libræ lanx utraque &longs;ubjecto plano incumberet, ea­<lb/>rum con&longs;i&longs;tentia non e&longs;&longs;et æquilibrio tribuenda, quamvis <lb/>æquilibres &longs;int, &longs;ed idcircò &longs;olùm con&longs;i&longs;terent, quia infrà &longs;e <lb/>haberent corpus, quod permeare vel non exigit, vel non po­<lb/>te&longs;t earum gravitas: ita terræ partes licèt adeò æqualiter &longs;int <lb/>di&longs;po&longs;itæ circa &longs;uum commune gravitatis centrum (in quo vi­<lb/>res &longs;uas exererent tellure totâ in aeris locum tran&longs;latâ) ut ex illo <lb/>&longs;u&longs;pensâ tellure in æquilibrio con&longs;i&longs;terent; re tamen ipsâ non <pb pagenum="71" xlink:href="017/01/087.jpg"/>con&longs;i&longs;tunt propter æquilibrium; &longs;ed quia nulla pars habet in­<lb/>fra &longs;e aliquid, &longs;ub quo petat exi&longs;tere, atque adeò nulla e&longs;t, <lb/>quæ deor&longs;um nitatur. </s> <s id="s.000580">Quare Poëticè &longs;olùm, non verò Philo­<lb/>&longs;ophicè dictum e&longs;t. <lb/><emph type="italics"/>Terra pilæ &longs;imilis, nullo fulcimine nixa, <lb/>Aëre &longs;ubjecto tam grave pendet onus.<emph.end type="italics"/><lb/>Aer &longs;i quidem non e&longs;t &longs;ubjectus terræ, &longs;ed circumfu&longs;us; ea <lb/>namque &longs;ubjecta &longs;unt, quæ inferiora; inferiora autem, quæ <lb/>centro propiora. </s> <s id="s.000581">Terræ itaque globus nihil habet, in quod <lb/>gravitatis vires exerceat deor&longs;um conando. </s> </p> <p type="main"> <s id="s.000582">Quæ cum ita &longs;int, nulla unquam continget in terrâ mutatio <lb/>atque gravium tran&longs;latio, quæ efficiat motum trepidationis. </s> <lb/> <s id="s.000583">Sit enim terræ globus AB, cujus cen­<lb/><figure id="id.017.01.087.1.jpg" xlink:href="017/01/087/1.jpg"/><lb/>trum C &longs;it pariter centrum gravitatis: <lb/>ducto per C plano IL, hemi&longs;phærium <lb/>IAL e&longs;t æquale hemi&longs;phærio IBL; <lb/>ex quo ab&longs;ci&longs;&longs;a intelligatur portio <lb/>&longs;phærica DEB, in cujus locum &longs;uc­<lb/>cedat aër. </s> <s id="s.000584">Si qua igitur pars deberet <lb/>deor&longs;um versùs C niti, non alia uti­<lb/>que e&longs;&longs;et præter D & E, quæ longiùs <lb/>à centro ab&longs;unt, quàm contiguus aër <lb/>DE. <!-- KEEP S--></s> <s id="s.000585">At portio IDEL prævalere non <lb/>pote&longs;t hemi&longs;phærio IAL, quod deberet &longs;ur&longs;um propelli; ergo <lb/>non pote&longs;t centrum C moveri versùs A, ut punctum aliquod <lb/>inter C & K congruat centro univer&longs;i. </s> <s id="s.000586">Sed neque hemi&longs;phæ­<lb/>rium IAL debet de&longs;cendere, quia nullum habet corpus leve <lb/>&longs;ibi contiguum, quod univer&longs;i centro vicinius &longs;it; non ergo <lb/>debet propellere oppo&longs;itum &longs;egmentum IDEL; cujus omnes <lb/>partes non &longs;olùm reluctantur motui, quo recedant ab univer&longs;i <lb/>centro C, &longs;ed etiam illarum aliquæ &longs;e ip&longs;æ urgent, & conan­<lb/>tur versùs C. <!-- KEEP S--></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/>po&longs;itam partem, & addatur hemi&longs;phærio &longs;uperiori etiam mons <lb/>FHG æqualis ab&longs;ci&longs;&longs;æ portioni &longs;phæricæ. </s> <s id="s.000589">Aio ne dum factam <lb/>e&longs;&longs;e mutationem, quæ ad motum telluri conciliandum &longs;ufficiat. </s> <lb/> <s id="s.000590">Quamvis enim mons ille FHG, quippe quem ambit aër le-<pb pagenum="72" xlink:href="017/01/088.jpg"/>vior vicinior centro, conetur deor&longs;um; certum e&longs;t illum de­<lb/>&longs;cendere non po&longs;&longs;e, quin totam reliquam terram impellat, eju&longs;­<lb/>que re&longs;i&longs;tentiam &longs;uperet; re&longs;i&longs;tit autem primò &longs;egmentum <lb/>IDEL, cujus omnes partes magis à centro removerentur; ni­<lb/>&longs;i igitur mons FHG major &longs;it &longs;egmento &longs;phærico IDEL <lb/>(vel &longs;altem non multò minor, &longs;i quidem ob majorem à centro <lb/>di&longs;tantiam augerentur momenta gravitatis, ex dictis cap. 4.) <lb/>non poterit &longs;ubjectam terram loco dimovere. </s> <s id="s.000591">Præterea etiam <lb/>hemi&longs;phærium IAL repugnat de&longs;cen&longs;ui montis FHG, quia <lb/>fieri non pote&longs;t hic motus, ni&longs;i hemi&longs;phærij partes tran&longs;iliant <lb/>planum IL, atque magis à centro recedant. </s> <s id="s.000592">Quanta igitur <lb/>gravitate præditum e&longs;&longs;e montem oporteret, qui tantam re­<lb/>&longs;i&longs;tentiam &longs;uperare valeret? </s> <s id="s.000593">At nunquam fieri tantam partium <lb/>permutationem, ut id quod transfertur, &longs;it non minus &longs;emi&longs;&longs;e <lb/>hemi&longs;phærij, ut &longs;altem ratione habitâ di&longs;tantiæ à centro po&longs;­<lb/>&longs;it prævalere, ita omnibus e&longs;t manife&longs;tum, ut probatione non <lb/>indigeat. </s> <s id="s.000594">Quare neque hanc gravium tran&longs;lationem motus ul­<lb/>lus con&longs;equitur, quo tellus trepidare dicatur. </s> </p> <p type="main"> <s id="s.000595">At, inquis, &longs;i in utrâque libræ lance &longs;int unciæ 100, & al­<lb/>terutri uncia una addatur, lanx illa deprimitur, & oppo&longs;ita <lb/>elevatur; ergo exiguum pondus vim habet movendi ingens <lb/>pondus; ergo pariter mons FHG producere pote&longs;t impetum, <lb/>qui ad movendum &longs;egmentum IDEL, quantumvis gravius, <lb/>abundè &longs;ufficiat. </s> <s id="s.000596">Ego vero nego con&longs;equentiam; quia non ab <lb/>unciâ illâ additâ &longs;olâ elevatur oppo&longs;itum pondus, &longs;ed omnes <lb/>unciæ &longs;imul in medio leviore &longs;u&longs;pen&longs;æ collatis viribus deor&longs;um <lb/>conantur, atque præponderantes oppo&longs;itæ lancis pondus at­<lb/>tollunt. </s> <s id="s.000597">Hoc autem nil in rem no&longs;tram facit, ubi neque mons <lb/>FHG &longs;olitariè &longs;umptus pote&longs;t &longs;ursùm propellere molem <lb/>IDEL majorem &longs;e, neque juvari pote&longs;t ab hemi&longs;phærio IAL, <lb/>quod cum nihil infrà &longs;e habeat, quod & levius &longs;it, & inter <lb/>ip&longs;um ac univer&longs;i centrum intercipiatur, neque pote&longs;t &longs;e ip&longs;um <lb/>versùs centrum urgere &longs;ecundùm aliquas &longs;ui partes ab eo remo­<lb/>tiores, cum maximè partes centro proximæ valde reluctentur, <lb/>ne ab illo removeantur. </s> <s id="s.000598">Id quod in libræ lance, cui uncia fue­<lb/>rit addita, reperire non poteris; totum &longs;iquidem lancis pon­<lb/>dus deor&longs;um nititur. </s> </p> <p type="main"> <s id="s.000599">Quod &longs;i ex librâ &longs;imilitudinem ducere placeat, petenda po-<pb pagenum="73" xlink:href="017/01/089.jpg"/>tiùs e&longs;t ex librâ, cujus lanx altera &longs;ubjecto plano incumbat, al­<lb/>tera in aëre libera pendeat; &longs;i enim utraque lanx plena æquali­<lb/>bus ponderibus con&longs;i&longs;tat in æquilibrio, & incumbenti lanci ad­<lb/>datur ponderis pars, quæ à pendulâ lance detrahatur, lances <lb/>non moventur, nec inter &longs;e mutuò confligunt ponderum gra­<lb/>vitates, ni&longs;i quatenùs lanx gravior &longs;emper magis re&longs;i&longs;tit leviori, <lb/>ne ab illâ elevetur: cæterùm gravior lanx non movet leviorem, <lb/>ni&longs;i ubi demum tanto pondere prægravata fuerit, ut &longs;ubjecti <lb/>plani re&longs;i&longs;tentiam vincens illud aut frangat, aut &longs;altem depri­<lb/>mat. </s> <s id="s.000600">Sic hemi&longs;phærium IAL habet rationem lancis non tan­<lb/>tùm &longs;ubjecto plano incumbentis, &longs;ed, quod potius e&longs;t, &longs;uo in <lb/>loco quie&longs;centis; cui quò plus addideris ponderis, auges qui­<lb/>dem re&longs;i&longs;tentiam ne &longs;ursùm versùs H propellatur, ip&longs;um verò <lb/>non conatur deor&longs;um versùs C; &longs;ed totus conatus impo&longs;ito & <lb/>adjecto monti tribuendus e&longs;&longs;et, vel (ut &longs;im maximè liberalis) <lb/>etiam exce&longs;&longs;ui illi, quo hemi&longs;phærium IAL &longs;uperat &longs;egmen­<lb/>tum &longs;phæricum IDEL, qui exce&longs;&longs;us e&longs;t æqualis ip&longs;i monti, <lb/>hoc e&longs;t &longs;egmento DEB. </s> <s id="s.000601">Quare &longs;i fuerit ab&longs;ci&longs;&longs;a tertia pars <lb/>hemi&longs;phærij unius, & addatur alteri hemi&longs;phærio è regione &longs;e­<lb/>cundùm diametrum, tunc ad &longs;ummum æqualis erit pars terræ <lb/>deor&longs;um nitens FMGH parti oppo&longs;itæ repugnanti IDEL; & <lb/>&longs;i velis partem FMGH remotiorem à centro magis gravitare <lb/>ita, ut ratio hujus exce&longs;sûs in gravitando po&longs;&longs;it vincere non &longs;o­<lb/>lùm re&longs;i&longs;tentiam &longs;egmenti IDEL, ne &longs;ur&longs;um propellatur, &longs;ed <lb/>etiam &longs;egmenti FILG, ne &longs;ecundùm partes IL centro proxi­<lb/>mas ab eo removeatur; non admodum repugnabo. </s> <s id="s.000602">Sed cum <lb/>nunquam mille&longs;ima, ne dum &longs;exta, pars terreni globi ex alio <lb/>in alium locum ex diametro oppo&longs;itum transferatur, nulla un­<lb/>quam fit gravium permutatio, vi cujus tellus trepidet. </s> </p> <p type="main"> <s id="s.000603">Sed unum adhuc &longs;upere&longs;t, quod per di&longs;&longs;imulantiam præ­<lb/>tereundum non videtur. </s> <s id="s.000604">E&longs;to inquis, nulla fiat in tellure gra­<lb/>vium tran&longs;latio, quæ tanta &longs;it, ut novum gravitatis centrum in <lb/>univer&longs;i centro con&longs;tituere valeat, ac proinde nulla &longs;it centri <lb/>terræ trepidatio: circa centrum &longs;altem nutabit tellus motu <lb/>conver&longs;ionis, validâ ventorum vi &longs;ummos montes impellente, <lb/>orbemque totum, pro variâ ip&longs;orum incur&longs;ione, modò hanc, <lb/>modò illam partem ver&longs;ante: unde forta&longs;&longs;e ortam acû magne­<lb/>ticæ eodem in loco po&longs;t aliquot annos variationem &longs;u&longs;picari <pb pagenum="74" xlink:href="017/01/090.jpg"/>quis po&longs;&longs;it. </s> <s id="s.000605">Cum enim tellus æqualibus circà centrum nutibus <lb/>librata permaneat, multo faciliùs omnem in partem converti <lb/>po&longs;&longs;e videtur, quàm rota ingens &longs;uo in axe &longs;u&longs;pen&longs;a: Rota &longs;ci­<lb/>licet &longs;uo pondere axem premens illum, dum convertitur, te­<lb/>rit; hancque affrictûs difficultatem vincat nece&longs;&longs;e e&longs;t, quod <lb/>una ex parte additur pondus, vel quæ applicatur Potentia, ut <lb/>conver&longs;ionem efficiat: tellus verò in orbem diffu&longs;a nec cen­<lb/>trum premit, nec axem, cum quo ullus fiat affrictus; ac <lb/>proptereà faciliorem præbet conver&longs;ionis an&longs;am Potentiæ unam <lb/>aliquam in partem urgenti. </s> <s id="s.000606">Huju&longs;modi autem Potentia ventus <lb/>e&longs;t, non ad perpendiculum in terram incidens, &longs;ed obliquè in <lb/>præaltos &longs;altem montes incurrens; cujus viribus nihil ob&longs;tare <lb/>videtur, quin telluris globum &longs;ibi ob&longs;ecundantem inclinet; <lb/>quemadmodum, & ingentes naves, vela implens, impellit. </s> </p> <p type="main"> <s id="s.000607">Huic difficultati ut me &longs;ubducam, non me in abditos magne­<lb/>ti&longs;mi rece&longs;&longs;us recipio, a&longs;&longs;erendo tellurem ita arcanis nodis cæ­<lb/>lo connexam, ut à &longs;ummo axium polorumque cæle&longs;tium atque <lb/>terre&longs;trium con&longs;en&longs;u divelli ac di&longs;trahi prorsùs nequeat: ne­<lb/>que enim hi&longs;ce magneti&longs;mi latebris me &longs;atis protectum exi&longs;ti­<lb/>marem; demptâ quippe &longs;olis Au&longs;tralibus atque Borealibus ven­<lb/>tis hâc facultate tellurem convertendi, ne &longs;cilicet terre&longs;tres <lb/>poli à cæle&longs;tibus di&longs;crepent, quid prohibeat reliquos ad Orti­<lb/>vum, aut Occiduum limitem pertinentes, quin &longs;uo flatu or­<lb/>bem hunc volvant, adhuc &longs;upere&longs;&longs;et explicandum. </s> <s id="s.000608">Hoc qui­<lb/>dem &longs;atis e&longs;&longs;e videretur ad &longs;ubmovendam &longs;u&longs;picionem illam de <lb/>acûs magneticæ variatione ob telluris conver&longs;ionem; manente <lb/>nimirum axe terre&longs;tri ita, ut cum cæle&longs;ti conveniat, aut illi <lb/>&longs;altem parallelus exi&longs;tat, nihil e&longs;t quod, etiam tellure circa <lb/>axem conversâ, magneticam declinationem commutare queat: <lb/>nam quod ad &longs;yderum a&longs;pectus &longs;pectat, parum intere&longs;t, tellus­<lb/>ne? </s> <s id="s.000609">an cælum volvatur; &longs;i igitur diurna cæli conver&longs;io magne­<lb/>tis declinationem non mutat, neque ad illam mutandam &longs;uffi­<lb/>ceret telluris circa &longs;uum axem conver&longs;io, vi cujus alia atque <lb/>alia &longs;ydera re&longs;piceret: Præterquam quod non id temporum lap­<lb/>&longs;u accideret; &longs;ed ubi ventorum impetus elangui&longs;&longs;et, illicò va­<lb/>riatio illa declinationis magneticæ deprehenderetur: id quod <lb/>ab omni experimento longè abe&longs;t. </s> <s id="s.000610">Verùm adeò à no&longs;tris &longs;en­<lb/>&longs;ibus &longs;ejunctæ &longs;unt magneticorum &longs;ymptomatum cau&longs;æ, ut ad <pb pagenum="75" xlink:href="017/01/091.jpg"/>aliarum difficultatum &longs;olutionem non facilè advocandus &longs;it in <lb/>Philo&longs;ophicam &longs;cenam magneti&longs;mus. </s> </p> <p type="main"> <s id="s.000611">Illud potius hìc attendendum videtur, quod montis altitu­<lb/>do, atque magnitudo ad totius telluris molem Rationem habet <lb/>&longs;atis exiguam. </s> <s id="s.000612">Cum enim terræ ambitus probabiliter &longs;tatuatur, <lb/>ut aliàs o&longs;tendi, milliarium Rom. <!-- REMOVE S--><expan abbr="antiq.">antique</expan> 30598, eju&longs;que <lb/>propterea diameter &longs;it proximè mill. (9738 4/51), tota &longs;uperficies <lb/>&longs;phærica (ut pote quadrupla maximi circuli ex demon&longs;tratis <lb/>ab Archimede) e&longs;t mill. <!-- REMOVE S-->quadratorum 297. 987800 proximè. </s> <lb/> <s id="s.000613">Mons &longs;tatuatur altitudinis perpendicularis milliarium quin­<lb/>que; hæc e&longs;t ad terre&longs;trem diametrum ut 1 ad 1947: ba&longs;is <lb/>montis occupet milliaria quadrata 500; hæc e&longs;t ad &longs;phæricam <lb/>totius globi &longs;uperficiem, ut 1 ad 595975. Finge jam pro mon­<lb/>te granum hordei, quod promineat &longs;ecundùm &longs;uam latitudi­<lb/>nem ex &longs;phærâ habente diametrum granorum 1947, hoc e&longs;t <lb/>pa&longs;&longs;uum geometricorum &longs;ex, &longs;eu pedum Rom. <!-- REMOVE S--><expan abbr="antiq.">antique</expan> 30. cir­<lb/>culi maximi ambitus erit pedum 94 1/4: quare hujus &longs;phæræ &longs;u­<lb/>perficies habet pedes quadratos 2827, hoc e&longs;t quadratas lati­<lb/>tudines grani hordei paulò plures quàm 11. 579000. Igitur <lb/>grani hordei jacentis altitudo ad hujus &longs;phæræ diametrum <lb/>eandem ex hypothe&longs;i habet rationem, quam prædicti montis <lb/>altitudo ad telluris diametrum: & &longs;i decem grana &longs;ibi invicem <lb/>attigua di&longs;ponantur, ut montis ba&longs;im æmulentur, eadem erit <lb/>ratio ad &longs;uperficiem. </s> <s id="s.000614">Quamvis itaque &longs;phæra illa intelligatur <lb/>planè inanis ac levi&longs;&longs;ima &longs;olam habens &longs;uperficiem papyra­<lb/>ceam, ex qua granum ordei agglutinatum promineat, an pu­<lb/>tas à flatu quantumvis valido per fi&longs;tulam emi&longs;&longs;o in granum il­<lb/>lud hordei incurrente convertendum e&longs;&longs;e globum papyra­<lb/>ceum? </s> <s id="s.000615">Id &longs;anè ex cæteris experimentis conjicere non licet; <lb/>perinde enim e&longs;t atque &longs;i nihil promineret; neque vel mini­<lb/>mùm obe&longs;t Phy&longs;icæ rotunditati. </s> <s id="s.000616">Quare neque montis altitu­<lb/>do con&longs;tituta quicquam detrahet orbicularis figuræ, quod &longs;ub <lb/>Phy&longs;icam con&longs;iderationem cadat; ac proptereà nihil virium ad <lb/>tellurem convertendam obtinet ventus in montem incurrens. </s> </p> <p type="main"> <s id="s.000617">Et quidem conver&longs;ionem hanc re ipsâ non fieri manife&longs;tum <lb/>e&longs;t; &longs;i quidem cum nulla vincenda e&longs;&longs;et gravitas, quæ longiùs <lb/>à centro gravium recederet, vel quæ axem tereret, facillima <lb/>videretur e&longs;&longs;e globi totius conver&longs;io circa centrum, non &longs;olùm <pb pagenum="76" xlink:href="017/01/092.jpg"/>validioribus atque incitatioribus, &longs;ed temperatis etiam atque <lb/>mediocribus ventis flantibus. </s> <s id="s.000618">Hi autem aliquando diuturni <lb/>&longs;unt; cuju&longs;modi poti&longs;&longs;imum &longs;unt Ete&longs;iæ, quibus maritimi cur­<lb/>&longs;us celeres, & certi diriguntur. </s> <s id="s.000619">Tot igitur dierum &longs;patio, ven­<lb/>to oppo&longs;itos montes vehementiùs urgente, non modica fieret <lb/>terreni globi inclinatio; ac propterea non eadem demum per­<lb/>maneret eodem in loco Poli &longs;uprà Horizontem altitudo, quo­<lb/>ties ab alterutro cardine Au&longs;trali Boreali ve, aut à &longs;ol&longs;titiali <lb/>Brumali-ve limite tam ortivo quàm occiduo ventus &longs;piraret, at­<lb/>que multarum ædium facies non eandem ampliùs re&longs;picerent <lb/>cæli plagam; quare & &longs;cietherica Horologia quantumvis ac­<lb/>curatè &longs;emel de&longs;cripta po&longs;t non adeò multas temporum inclina­<lb/>tiones toto ferè cælo di&longs;creparent; aliis enim, atque aliis &longs;ub­<lb/>inde flantibus ventis, varia oriretur orbis conver&longs;io, atque alia <lb/>planorum cum circulis horariis &longs;ectio, quæ de&longs;criptis lineis non <lb/>congrueret. </s> <s id="s.000620">Hujus autem mutationis nullum in toto terra­<lb/>rum orbe ve&longs;tigium apparet, ni&longs;i fortè fabulas liceat com­<lb/>mini&longs;ci. </s> </p> <p type="main"> <s id="s.000621">Quòd &longs;i conver&longs;ionem hanc non omninò circa centrum <lb/>quamcumque in partem fieri, &longs;ed tantummodo circa axem, <lb/>dixeris, ut argumenti vim effugias; Quid illud e&longs;t, quod ita <lb/>terre&longs;trem axem cum cæle&longs;ti colligatum velit, ut tamen ter­<lb/>re&longs;tres meridianos à primâ mundi molitione con&longs;titutos tem­<lb/>poris lap&longs;u cum cæle&longs;tibus meridianis non convenire permit­<lb/>tat? </s> <s id="s.000622">Sed & aliud profectò, nec illud quidem leve, incommo­<lb/>dum &longs;ubeas nece&longs;&longs;e e&longs;t; dum enim conver&longs;ionem ad&longs;truis ab <lb/>ortu in occa&longs;um, & vici&longs;&longs;im ab occa&longs;u in ortum, fieri poterit, <lb/>ut po&longs;t aliquot annos non planè &longs;pernenda conver&longs;io facta fue­<lb/>rit, ac proinde temporum numeratio cælo non re&longs;pondeat. </s> <lb/> <s id="s.000623">Nam &longs;i ab ortu in occa&longs;um ex. </s> <s id="s.000624">gr. <!-- REMOVE S-->proce&longs;&longs;erit tellus, minus tem­<lb/>poris numerabitur quàm pro ratione cæle&longs;tium motuum; ut <lb/>contigi&longs;&longs;e fertur navi cui à Victoriâ nomen inditum e&longs;t, in ex­<lb/>peditione Magellanicâ; cum &longs;cilicet po&longs;t totius orbis ambitum <lb/>redux in Hi&longs;palen&longs;em portum, ex quo ante tres annos &longs;olve­<lb/>rat, intraret, tunc primùm ob&longs;ervarunt &longs;e à rectâ temporis nu­<lb/>meratione defeci&longs;&longs;e die uno; quippe qui cum juxta diurnam <lb/>cæli conver&longs;ionem ab ortu in occa&longs;um iter in&longs;titui&longs;&longs;ent, ju&longs;to <lb/>tardiùs &longs;emper &longs;ol illis occiderat, exiguo quidem &longs;ingulis die-<pb pagenum="77" xlink:href="017/01/093.jpg"/>bus, quibus procedebant, di&longs;crimine, &longs;ed quod demùm modi­<lb/>cis illis acce&longs;&longs;ionibus in integrum diem excreverat. </s> <s id="s.000625">Contra ve­<lb/>rò accideret, &longs;i ab occa&longs;u in ortum &longs;emper navigaretur; ju&longs;to <lb/>enim breviores e&longs;&longs;ent dies, ac propterea eorum numerus ac­<lb/>cre&longs;ceret. </s> <s id="s.000626">Hæc autem in temporum numeratione incon&longs;tan­<lb/>tia, &longs;i ventorum impetu tellus modò in ortum, modò in occa­<lb/>&longs;um converteretur, quantam perturbationem inveheret in <lb/>A&longs;tronomiam? </s> <s id="s.000627">Neque tibi quicquam &longs;uffragari exi&longs;times, &longs;i <lb/>ex varia ventorum oppo&longs;itas in plagas &longs;ivè &longs;imul, &longs;ivè &longs;ubinde, <lb/>&longs;pirantium commutatione conver&longs;iones illas compen&longs;ari dixe­<lb/>ris: id enim ad incertum revocat omnes A&longs;tronomorum calcu­<lb/>los, ubi meridianorum circulorum &longs;ectiones &longs;tabiles non perma­<lb/>neant; cum ad orbem totum inclinandum, ut tu quidem au­<lb/>tumas, &longs;atis &longs;it, &longs;i unâ aliquâ in regione ventus montes impel­<lb/>lat; quî verò certus &longs;im factam ab Arge&longs;te telluris conver&longs;io­<lb/>nem in ortum, æquatam demum fui&longs;&longs;e à Vulturno, aut ab <lb/>Euro-Au&longs;tro? </s> </p> <p type="main"> <s id="s.000628">Verùm quàm infirmæ &longs;int validi&longs;&longs;imorum ventorum vires ad <lb/>globum hunc terraqueum inclinandum, expendamus, etiam&longs;i <lb/>montium perpendicula non quinque tantùm milliaribus defini­<lb/>ta velis, &longs;ed multò altiora. </s> <s id="s.000629">Statue in ingenti lacu compo&longs;itam <lb/>ex trabibus aliquot ratem, quam in littore &longs;tans facilè funiculo <lb/>modereris: Tùm ratem aliam paris quidem latitudinis, &longs;ed cen­<lb/>tuplò longiorem, compone: Poteris-ne hanc funiculo eodem, <lb/>ac labore non majori, trahere perinde atque priorem? </s> <s id="s.000630">Negabis <lb/>utique, quamvis enim utraque lacui &longs;tagnanti innatet, nec <lb/>vincenda &longs;it alterutrius gravitas, ut à centro gravium magis re­<lb/>cedat; licet utraque parem in motu ab aquâ dividendâ re&longs;i&longs;ten­<lb/>tiam inveniat (eju&longs;dem quippe &longs;unt latitudinis &longs;olâ di&longs;crepan­<lb/>tes longitudine, & æqualis e&longs;t utriu&longs;que immer&longs;io propter ean­<lb/>dem &longs;ingularum trabium molem, atque &longs;pecificam gravitatem) <lb/>quia tamen di&longs;par e&longs;t ratium magnitudo, & impetu extrin&longs;e­<lb/>cùs accepto utraque eget, ut moveatur, palàm e&longs;t majore im­<lb/>petu opus e&longs;&longs;e, ut ratis major trahatur, ac propterea po&longs;&longs;e hanc <lb/>adeò augeri, ut impetus ad illam movendam nece&longs;&longs;arius exce­<lb/>dat vires Potentiæ ratem minorem funiculo moderantis. </s> <s id="s.000631">Ita <lb/>planè e&longs;t. </s> <s id="s.000632">Sed jam animum transfer ad in&longs;titutam di&longs;putatio­<lb/>nem, ut di&longs;picias, undè irrep&longs;erit dubitatio hæc de telluris <pb pagenum="78" xlink:href="017/01/094.jpg"/>conver&longs;ione ex ventorum impul&longs;u, & quàm facilè fucum fece­<lb/>rit rota &longs;uo in axe &longs;u&longs;pen&longs;a, quæ levi negotio, nec valido im­<lb/>pul&longs;u, volvitur. </s> <s id="s.000633">Rota &longs;iquidem tota deor&longs;um gravitat, ac <lb/>proptereà axem premit; quia autem in axe &longs;u&longs;penditur, fieri <lb/>non pote&longs;t, ut pars altera de&longs;cendat, quin oppo&longs;ita a&longs;cendat. </s> <lb/> <s id="s.000634">Quandiu conatus ad de&longs;cendendum æqualis e&longs;t re&longs;i&longs;tentiæ ad <lb/>a&longs;cendendum, rota quie&longs;cit; nec volvitur, ni&longs;i alterutri parti <lb/>fiat acce&longs;&longs;io Potentiæ, quæ pariter de&longs;cen&longs;um juvet, vel quia <lb/>ip&longs;a quoquè deor&longs;um conatur cum parte de&longs;cendente, vel quia <lb/>&longs;ur&longs;um nitens partem alteram elevat, oppo&longs;itamque deprimet <lb/>&longs;uapte naturâ de&longs;cendentem. </s> <s id="s.000635">Non tamen huju&longs;modi rotæ &longs;u&longs;­<lb/>pen&longs;æ conver&longs;io tribuenda e&longs;t &longs;oli Potentiæ; &longs;ed pars rotæ de­<lb/>&longs;cendens atque Potentia collatis viribus elevant partem rotæ <lb/>a&longs;cendentem, eíque impetum imprimunt. </s> <s id="s.000636">At in telluris circa <lb/>&longs;uum centrum, vel axem, conver&longs;ione nihil ade&longs;&longs;et, quod Po<lb/>tentiam juvaret; quia nulla e&longs;t pars, quæ deor&longs;um conetur, <lb/>aut &longs;ur&longs;um, ut po&longs;&longs;it oppo&longs;itæ parti impetum aliquem impri­<lb/>mere; nulla etenim pars in huju&longs;modi conver&longs;ione ad centrum <lb/>gravium accederet, aut ab illo recederet. </s> <s id="s.000637">Totus igitur impe­<lb/>tus à vento imprimendus e&longs;&longs;et toti telluris globo, ut à &longs;uâ, quæ <lb/>&longs;ecundùm naturam e&longs;t, quiete dimoveretur. </s> <s id="s.000638">Atqui globi ter­<lb/>raquei ea e&longs;t moles, ut contineat milliaria cubica proximè <lb/>48670. 200000 (omnis nimirum &longs;phæra æqualis e&longs;t cono, cu­<lb/>jus altitudo par e&longs;t Radio &longs;phæræ, ba&longs;is autem æqualis &longs;uperfi­<lb/>ciei &longs;phæræ, ex dictis verò paulò &longs;uperiùs, & &longs;uperficies & Ra­<lb/>dius globi hujus innote&longs;cit) nullus igitur adeò vehemens e&longs;t <lb/>ventus, qui tantæ moli impetum imprimere valeat; nullus &longs;i­<lb/>quidem excogitari pote&longs;t ventus, qui globum marmoreum, aut <lb/>etiam ex argillâ, in planitie æqui&longs;&longs;imâ con&longs;titutum, &longs;i mille <lb/>pa&longs;&longs;us Geometricos in diametro numeret, convolvere valeat. </s> <lb/> <s id="s.000639">Adde in telluris conver&longs;ione, &longs;i illa fieret, quò vehementior <lb/>e&longs;&longs;et ventus in montem incurrens, validior e&longs;&longs;et re&longs;i&longs;tentia aëris <lb/>à reliquis montibus dividendi; &longs;ed & multorum ingentium <lb/>fluminum contrariam in partem labentium impetus ob&longs;i&longs;teret, <lb/>ne tellus vento flanti ob&longs;ecundaret. </s> <s id="s.000640">Quod &longs;i hæc levis e&longs;&longs;e mo­<lb/>menti dixeris ad ob&longs;i&longs;tendum, levis pariter momenti e&longs;&longs;e ven­<lb/>torum impetum, nece&longs;&longs;e e&longs;t, fatearis: neque hic arduum e&longs;&longs;et <lb/>ventorum atque fluminum vires invicem conferre, aquarum-<pb pagenum="79" xlink:href="017/01/095.jpg"/>que impetum multò validiorem o&longs;tendere; &longs;ed ad alia prope­<lb/>randum e&longs;t: &longs;atisfuerit monui&longs;&longs;e non mediocrem intercedere <lb/>analogiam inter aquarum guttas in rivulos primùm, deinde in <lb/>majores rivos, ac demum in torrentem concurrentes, atque <lb/>terræ expirationes in ventum congregatas, quæ multum vi­<lb/>rium obtinent, &longs;i plurimæ in unum coëant, quemadmodum <lb/>& aquis contingit. <lb/></s> </p> <p type="head"> <s id="s.000641"><emph type="center"/>CAPUT XIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="head"> <s id="s.000642"><emph type="center"/><emph type="italics"/>Quâ ratione minuatur gravitatio in plano <lb/>inclinato.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000643">PLanum inclinatum dicitur planum quodcumque non tran­<lb/>&longs;it per centrum gravium & levium, hoc e&longs;t per centrum <lb/>univer&longs;i; huju&longs;modi &longs;iquidem planum non cadit ad angulos <lb/>æquales in &longs;phæricam terræ &longs;uperficiem. </s> <s id="s.000644">Hinc etiam planum <lb/>horizonti parallelum reipsâ e&longs;t inclinatum, ni&longs;i adeò exiguum <lb/>&longs;it ac breve, ut puncti vicem obtineat, &longs;i cum terreni globi &longs;u­<lb/><figure id="id.017.01.095.1.jpg" xlink:href="017/01/095/1.jpg"/><lb/>perficie conferatur. </s> <s id="s.000645">Sit univer&longs;i <lb/>centrum A, plana BA, & CA &longs;unt <lb/>verticalia & perpendicularia, qui­<lb/>bus &longs;i corpus aliquod grave appli­<lb/>cueris, illud non impedietur, quin <lb/>per &longs;uam directionis lineam de&longs;cen­<lb/>dat. </s> <s id="s.000646">At verò tam planum BC, quam <lb/>planum CD inclinata &longs;unt, nec cor­<lb/>pus grave illis impo&longs;itum pote&longs;t <lb/>rectâ &longs;ecundùm directionis lineam <lb/>de&longs;cendere, &longs;ed ab illâ declinare co­<lb/>gitur plano ob&longs;i&longs;tente. </s> <s id="s.000647">Sunt autem anguli inclinationis ABC, <lb/>ACD. <!-- KEEP S--></s> <s id="s.000648">Quod &longs;i planum parallelum horizonti ita exiguum &longs;it, <lb/>ut à &longs;phæricâ &longs;uperficie, quam tangit, non recedat; tunc in <lb/>quacumque ejus parte con&longs;tituatur corpus grave, perinde e&longs;t, <lb/>atque &longs;i in puncto D collocatum concipiatur. </s> <s id="s.000649">Sin autem ita à <pb pagenum="80" xlink:href="017/01/096.jpg"/>puncto D di&longs;titerit, ut à &longs;phæricâ &longs;uperficie recedat, quemad­<lb/>modum &longs;i e&longs;&longs;et planum DF, illud e&longs;t inclinatum, & fit angulus <lb/>DFA inclinationis. </s> <s id="s.000650">Ubi ob&longs;ervandum e&longs;t non eandem e&longs;&longs;e <lb/>&longs;ingularum plani partium inclinationem; angulus enim in­<lb/>clinationis AEC major e&longs;t inclinatione ABC, per 16. <lb/>lib. 1. & &longs;imiliter AFD maior e&longs;t angulo ACD. <!-- KEEP S--></s> <s id="s.000651">Quare <lb/>&longs;tatim atque ea e&longs;t puncti E à puncto B di&longs;tantia, ut an­<lb/>gulus à perpendiculis in centro A factus contemni non po&longs;­<lb/>&longs;it, alia e&longs;t etiam phy&longs;icè inclinatio, & corporis eju&longs;dem <lb/>gravitatio mutatur. </s> </p> <p type="main"> <s id="s.000652">Quoniam verò corpus grave plano inclinato impo&longs;itum ita <lb/>aëre circumfunditur, ut petat infrà illum de&longs;cendere, & re­<lb/>&longs;i&longs;tat, ne &longs;ur&longs;um moveatur; ideò gravitare dicitur. </s> </p> <p type="main"> <s id="s.000653">Sed cavendum e&longs;t, ne ex vocabulorum &longs;imilitudine er­<lb/>ror &longs;ubrepat: quandoquidem aliud e&longs;t <emph type="italics"/>gravitare in plano <lb/>inclinato,<emph.end type="italics"/> aliud <emph type="italics"/>gravitare in planum inclinatum:<emph.end type="italics"/> nam intrà <lb/>aërem corpus grave, putà, lapis, gravitat in quocunque <lb/>plano etiam perpendiculari, non tamen gravitat in pla­<lb/>num perpendiculare, nulla&longs;que vires &longs;uæ gravitatis con­<lb/>tra illud exercet, quamvis in eo exi&longs;tens, & re&longs;i&longs;tat &longs;ur­<lb/>&longs;um trahenti, & conetur, ut vincat vires retinentis, ac <lb/>quicquid moram infert, & impedimentum motui. </s> <s id="s.000654">In pla­<lb/>no itaque inclinato exi&longs;tens corpus grave (&longs;ubjectum pla­<lb/>num &longs;upponitur optimè lævigatum, nec motui officiens <lb/>partium prominularum a&longs;peritate) gravitat quidem, &longs;ed mi­<lb/>nùs quàm in plano perpendiculari, & pro variâ planorum <lb/>inclinatione, varia pariter e&longs;t gravitatio, ut quotidiana nos <lb/>docet experientia. </s> <s id="s.000655">Quâ igitur ratione gravitatio minuatur, <lb/>hîc e&longs;t examinandum; capite &longs;equenti gravitatio in Planum <lb/>inclinatum explicabitur. </s> </p> <p type="main"> <s id="s.000656">Cogno&longs;citur autem gravitatio ex re&longs;i&longs;tentiâ, quâ corpus <lb/>repugnat contra vires illud retinentis, ne deor&longs;um feratur, <lb/>aut &longs;ur&longs;um trahentis; neque enim alio ni&longs;u gravia gravi­<lb/>tant, quàm quo re&longs;i&longs;tunt impedienti motum gravitati <lb/>convenientem. </s> <s id="s.000657">Et quidem experimento aliquo pote&longs;t gra­<lb/>vitationis varietas inve&longs;tigari; &longs;i nimirum planum BO ex <lb/>ligno, aut marmore accuratè lævigetur, & extremitati B <lb/>adnectatur orbiculus D facillimè circa axem ver&longs;atilis, pon-<pb pagenum="81" xlink:href="017/01/097.jpg"/>deri autem A &longs;ubjiciantur <lb/><figure id="id.017.01.097.1.jpg" xlink:href="017/01/097/1.jpg"/><lb/>rotulæ, & adnectatur funi­<lb/>culus per D tran&longs;iens, ex <lb/>cujus extremo pendeat lanx <lb/>E, cui pondera immitti po&longs;­<lb/>&longs;int: pro variâ enim plani <lb/>BO inclinatione etiam pon­<lb/>dera in lance mutare opor­<lb/>tebit, ut pondus A &longs;u&longs;ti­<lb/>neatur, & plura erunt, quò magis ad perpendiculare accedet <lb/>planum BO. <!-- KEEP S--></s> <s id="s.000658">Verùm quia nunquam carere poteris &longs;u&longs;picione, <lb/>an corporum affrictus aliquid afferat impedimenti; ideò &longs;eclu­<lb/>&longs;is omnibus, quæ extrin&longs;ecùs accidere po&longs;&longs;unt, re&longs;i&longs;tentiam ex <lb/>&longs;olâ gravitate ortam opus e&longs;t con&longs;iderare. </s> </p> <p type="main"> <s id="s.000659">Re&longs;i&longs;tentia verò omnis re&longs;pondet violentiæ, quam patitur <lb/>id quod re&longs;i&longs;tit; minori etenim conatu minorem vim illatam <lb/>propul&longs;are &longs;tudet natura, quæ validiùs ob&longs;i&longs;tit majori violen­<lb/>tiæ: id quod ita rationi e&longs;t con&longs;onum, & obviis experimentis <lb/>manife&longs;tum, ut in hoc demon&longs;trando &longs;upervacaneum &longs;it im­<lb/>morari. </s> <s id="s.000660">Con&longs;tituantur itaque duo <lb/><figure id="id.017.01.097.2.jpg" xlink:href="017/01/097/2.jpg"/><lb/>æqualis ponderis corpora in D & <lb/>in C; &longs;ingulis alligetur funiculus, <lb/>qui per B tran&longs;eat, & &longs;ur&longs;um tra­<lb/>hantur &longs;imul ita, ut æqualiter mo­<lb/>veantur. </s> <s id="s.000661">Ab&longs;olutâ motûs particu­<lb/>lâ, corpus alterum ex D a&longs;cendit <lb/>in H in plano perpendiculari; al­<lb/>terum in plano inclinato ex C ve­<lb/>nit in E, & CE linea æqualis e&longs;t <lb/>lineæ motûs DH. <!-- KEEP S--></s> <s id="s.000662">Non eandem <lb/>tamen utrumque grave &longs;ubiit vio­<lb/>lentiam; nam motus DH fuit &longs;impliciter, & ab&longs;olutè violen­<lb/>tus; at motus CE eatenus &longs;olùm gravitati adver&longs;atur, quate­<lb/>nus a&longs;cendit; a&longs;cen&longs;um autem metitur linea DG, quam ab­<lb/>&longs;cindit EG horizonti parallela. </s> <s id="s.000663">Hîc &longs;cilicet planum DC in­<lb/>tellige horizontale nihil à &longs;phæricá &longs;uperficie di&longs;crepans, ut <lb/>communiter contingit: quòd &longs;i non ita &longs;e haberet; &longs;ed e&longs;&longs;et <lb/>ampli&longs;&longs;imum planum, men&longs;ura violentiæ illatæ ponderi in C <pb pagenum="82" xlink:href="017/01/098.jpg"/>con&longs;tituto, in E elevato de&longs;umenda e&longs;&longs;et ex differentiâ inter <lb/>KC & OE. </s> <s id="s.000664">E&longs;t itaque gravitatio in plano perpendiculari ad <lb/>gravitationem in plano inclinato, ut re&longs;i&longs;tentia ad a&longs;cenden­<lb/>dum in uno ad re&longs;i&longs;tentiam ad a&longs;cendendum in alio; re&longs;i&longs;tentiæ <lb/>autem &longs;unt, ut violentia, quam corpora &longs;ubeunt in motu; vio­<lb/>lentia demum e&longs;t ut HD ad GD, hoc e&longs;t per 7. lib. 5. ut CE <lb/>ad DG. <!-- KEEP S--></s> <s id="s.000665">Sed ut CE ad DG, ita EB ad GB, per 2. lib. 6. & <lb/>ut BE, ad BG ita BC ad BD, per 4. lib. 6. igitur gravitatio <lb/>in perpendiculari ad gravitationem in inclinato e&longs;t ut BC ad <lb/>BD, hoc e&longs;t ut Secans anguli inclinationis ad Radium. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000666">Quæ autem de totis DH, & CE lineis dicta &longs;unt, de &longs;ingu­<lb/>lis earum particulis æqualibus dicta intelligantur; ductis quip­<lb/>pe parallelis horizonti, eadem e&longs;t omnium Ratio: hîc namque <lb/>&longs;upponimus planum BC non adeò magnum e&longs;&longs;e, ut &longs;ingula <lb/>ejus puncta cum diver&longs;is horizontibus comparanda &longs;int, omnes <lb/>&longs;iquidem perpendiculares lineæ directionis non qua&longs;i conver­<lb/>gentes, &longs;ed phy&longs;icè parallelæ accipiuntur. </s> <s id="s.000667">Quòd &longs;i tam lon­<lb/>gum e&longs;&longs;et planum, ut phy&longs;icè mutatus intelligeretur angulus <lb/>inclinationis, non eadem e&longs;&longs;et Ratio gravitationis in toto, ac in <lb/>partibus: &longs;ed mutato angulo inclinationis mutaretur utique <lb/>ejus Secans; ac proinde inæqualium Secantium Ratio ad eum­<lb/>dem Radium inæqualis, gravitationum pariter inæqualem ra­<lb/>tionem o&longs;tenderet. </s> </p> <p type="main"> <s id="s.000668">Quod &longs;i a&longs;cendentium per vim extrin&longs;ecùs illatam corporum <lb/>re&longs;i&longs;tentiam atque gravitationem metimur ex violentiâ, quam <lb/>pro planorum varietate &longs;ubeunt; eorum pariter in de&longs;cendendo <lb/>efficacitatem ex ip&longs;o de&longs;cen&longs;u argui æquum e&longs;&longs;et, datâ motûs <lb/>in diver&longs;is planis æqualitate. </s> <s id="s.000669">Sed quia de&longs;cen&longs;us naturæ pro­<lb/>pen&longs;ioni congruit, fieri non pote&longs;t, ut in alio atque alio plano <lb/>æquales &longs;int motus i&longs;ochroni; tardior enim e&longs;t, qui in plano in­<lb/>clinato perficitur, neque, &longs;i æqualis ponderis corpora de&longs;cen­<lb/>dant ex H & E, quando illud ad D pervenit, hoc pote&longs;t attin­<lb/>gere punctum C: ideò non ex de&longs;cen&longs;u gravitationem metiri <lb/>oportet, cum motus æquales non habeantur: ni&longs;i fortè ea&longs;dem <lb/>movendi vires tribuas gravitati non impeditæ in perpendicula­<lb/>ri, ac impeditæ in plano inclinato. </s> <s id="s.000670">Qua propter gravitationis <lb/>momenta ad de&longs;cendendum non aliunde meliùs æ&longs;timantur, <lb/>quàm ex repugnantiâ ad a&longs;cendendum: &longs;ic enim vulgari argu-<pb pagenum="83" xlink:href="017/01/099.jpg"/>mento &longs;ingulorum corporum gravitates librâ expendimus, tan­<lb/>tumque iis ad de&longs;cendendum virium tribuimus, quantum re­<lb/>&longs;i&longs;tunt, ne ab oppo&longs;itâ libræ lance deor&longs;um conante eleventur. </s> <lb/> <s id="s.000671">Eadem igitur e&longs;t gravitationis Ratio, &longs;eu propen&longs;ionis ad de­<lb/>&longs;cendendum, quæ e&longs;t re&longs;i&longs;tentiæ ad a&longs;cendendum: Cum verò <lb/>re&longs;i&longs;tentiam in plano inclinato ad re&longs;i&longs;tentiam in perpendicu­<lb/>lari o&longs;ten&longs;um &longs;it e&longs;&longs;e, ut Radius ad Secantem anguli inclinatio­<lb/>nis, hoc e&longs;t ut BD ad BC, erit pariter vis de&longs;cendendi in <lb/>plano BC ad vim de&longs;cendendi in plano BD, reciprocè ut BD <lb/>ad BC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000672">Eadem ratione in plano CD &longs;uperficiem globi tangente, <lb/>gravitatio in CD ad gravitationem in perpendiculari CA e&longs;t <lb/>ut CD ad CA; e&longs;t enim CA Secans anguli inclinationis <lb/>DCA. </s> <s id="s.000673">Si enim ducatur KF Tangens, triangula CKF, <lb/>CDA &longs;unt &longs;imilia, angulus enim ad C communis e&longs;t, & am­<lb/>bo rectangula ad D & K; quare ut CK ad CF, ita CD ad <lb/>CA; &longs;ed gravitatio in CF ad gravitationem in CK e&longs;t reci­<lb/>procè ut CK ad CF: igitur gravitatio in plano inclinato CD <lb/>globum tangente, ad gravitationem in perpendiculari CA, e&longs;t <lb/>ut CD ad CA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000674">Hinc e&longs;t quod in planis horizontalibus, quæ ut plurimum <lb/>habemus, corpora non de&longs;cendant, aut moveantur: quia ni­<lb/>mirum à puncto, in quo grave &longs;tatuitur, ex. </s> <s id="s.000675">gr. <!-- REMOVE S-->F, ductæ li­<lb/>neæ FA perpendicularis & FD Tangens faciunt angulum <lb/>DFA inclinationis adeò magnum, ut Radius ad ejus &longs;ecan­<lb/>tem penè infinitam non habeat &longs;en&longs;u perceptibilem Rationem, <lb/>vel &longs;altem non tantam, ut gravitatio, quæ ratione inclinatio­<lb/>nis plani congruit corpori, non elidatur à re&longs;i&longs;tentiâ, quæ ori­<lb/>tur ex corporum a&longs;peritate. </s> <s id="s.000676">Quare &longs;ublatâ, aut potiùs impeditâ, <lb/>gravitatione corpus quie&longs;cit in plano horizontali. </s> </p> <p type="main"> <s id="s.000677">Et hæc e&longs;t ratio, cur violentiam determinans, quam grave <lb/>a&longs;cendens patitur, a&longs;&longs;ump&longs;erim in perpendiculari BA par­<lb/>tem GD, quam ab&longs;cindit parallela horizonti; hæc enim <lb/>men&longs;ura phy&longs;icè non di&longs;crepat à verâ men&longs;urâ, quæ a&longs;&longs;umen­<lb/>da e&longs;&longs;et, &longs;i mente concipias rectam lineam DC tangere circu­<lb/>lum, cujus &longs;emidiameter &longs;it millecuplo major. </s> <s id="s.000678">Men&longs;ura &longs;i qui­<lb/>dem a&longs;censûs petenda e&longs;t ex exce&longs;&longs;u, quo perpendicularis EA <lb/>&longs;uperat perpendicularem AC; illo enim intervallo, quo magis <lb/>rece&longs;&longs;it à centro, a&longs;cendit. </s> </p> <pb pagenum="84" xlink:href="017/01/100.jpg"/> <p type="main"> <s id="s.000679">Ex quo fit quod, &longs;i planum inclinatum BC cum perpendi­<lb/>culari CA faceret angulum acutum ACB, corpus ex C u&longs;que <lb/>in L (in quod punctum cadit perpendicularis AL) de&longs;cende­<lb/>ret, quia &longs;emper magis ad centrum accederet: ex L autem in E <lb/>a&longs;cenderet, & a&longs;cen&longs;um metiretur exce&longs;&longs;us perpendiculi EA <lb/>&longs;uprà perpendiculum LA. <!-- KEEP S--></s> <s id="s.000680">Quare ut ex C a&longs;cenderet, debe­<lb/>ret e&longs;&longs;e planum inclinatum IC, quod cum CA faceret angu­<lb/>lum ICA &longs;altem rectum. </s> <s id="s.000681">Ubi ex occa&longs;ione licet ob&longs;ervare <lb/>po&longs;&longs;e dari duos montes, qui cum valle intermediâ planitiem <lb/>unam con&longs;tituant; &longs;i nimirum montium vertices e&longs;&longs;ent E, & C, <lb/>ex quibus in imam vallem L de&longs;cenderetur: & aqua per mon­<lb/>tium venas de&longs;cendens in L po&longs;&longs;et fontem aut lacum creare. </s> </p> <p type="main"> <s id="s.000682">Re autem ipsâ &longs;emper contingit angulum BCA e&longs;&longs;e obtu&longs;um <lb/>vel non minorem recto. </s> <s id="s.000683">Ponatur enim terræ &longs;emidiameter DA <lb/>1000, & planum DC: (e&longs;&longs;et autem planum DC longius <lb/>milliar.4.) erit angulus DAC, gr. <!-- REMOVE S-->0. 3′. </s> <s id="s.000684">26′; atque adeò DCA <lb/>gr. <!-- REMOVE S-->89. 56′. </s> <s id="s.000685">34″. <!-- KEEP S--></s> <s id="s.000686">Jam verò &longs;it CD ad DB ut 100 ad 87; erit <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/> <s id="s.000689">Nunc &longs;i libeat comparare perpendiculum EA cum perpendi­<lb/>culo GA, &longs;tatue GD &longs;emi&longs;&longs;em totius BD; e&longs;t igitur & GE <lb/>&longs;emi&longs;&longs;is ip&longs;ius DC: Quare GE e&longs;t partium 50, quarum GA e&longs;t <lb/>100043 1/2: addantur quadrata GE 2500 & GA 10008701892 1/4, <lb/>& &longs;ummæ radix quadrata (100043 102543/200086) major verâ e&longs;t EA, quæ <lb/>non excedit perpendicularem GA 100043 1/2 ni&longs;i particulis (2500/400172). <lb/>Quoniam autem DAC angulus inventus e&longs;t grad. <!-- REMOVE S-->0. 3′. </s> <s id="s.000690">26′; <lb/>eju&longs;que Secans AC e&longs;t partium (100000 5017/100000), quarum AD <lb/>po&longs;ita e&longs;t 100000; di&longs;crimen inter AC, & AE &longs;uperiùs in­<lb/>ventam, e&longs;t partium (43 46227/100000), quæ e&longs;t proximè eadem men&longs;u­<lb/>ra, ac DG po&longs;ita partium 43 1/2. Quod &longs;i in plani inclinati lon­<lb/>gitudine <expan abbr="tantã">tantam</expan> Rationem habente ad terræ <expan abbr="&longs;emidiametrũ">&longs;emidiametrum</expan>, quan­<lb/>ta con&longs;tit ita e&longs;t, pote&longs;t citrà errorem a&longs;&longs;umi tanquam men&longs;ura <lb/>a&longs;censûs pars perpendiculi BA intecepta ab horizontali DC, <lb/>& parallelâ EG, &longs;atis patet id multò magis licere in planorum <lb/>longitudinibus minorem Rationem habentibus ad eandem ter­<lb/>ræ &longs;emidiametrum. </s> <s id="s.000691">Manet itaque con&longs;tituta regula gravitatio­<lb/>nis, videlicet gravitationem in plano inclinato ad gravitationem <lb/>in perpendiculari e&longs;&longs;e, ut e&longs;t Radius ad &longs;ecantem anguli incli­<lb/>nationis. </s> </p> <pb pagenum="85" xlink:href="017/01/101.jpg"/> <p type="main"> <s id="s.000692">Quamvis verò in partibus inferioribus plani inclinati &longs;it &longs;em­<lb/>per major angulus inclinationis, quàm in &longs;uperioribus, & pro­<lb/>inde minor &longs;it Ratio, quam habet Radius ad &longs;ecantem anguli <lb/>majoris, ac ea, quam idem Radius habet ad &longs;ecantem anguli <lb/>minoris: non tamen ea e&longs;t gravitationis differentia, cujus ratio <lb/>habenda &longs;it; cum enim adeò exiguus &longs;it angulus BAC, ejus <lb/>quantitas di&longs;tribuitur per omnes inclinationis angulos, qui <lb/>fiunt in punctis intermediis inter B & C; atque adeò contem­<lb/>nendum e&longs;t in praxi di&longs;crimen illud, quod oritur ex alio atque <lb/>alio inclinationis angulo in codem plano. </s> <s id="s.000693">Quod &longs;i in&longs;ignis e&longs;&longs;et <lb/>Rationum varietas, notabilis quoque e&longs;&longs;et gravitationis diver­<lb/>&longs;itas idem enim contingeret, ac &longs;i non idem e&longs;&longs;et planum. </s> <s id="s.000694">Sed <lb/>hoc communiter non accidit. </s> </p> <p type="main"> <s id="s.000695">Ex his illud manife&longs;tâ con&longs;ecutione conficitur, quod &longs;i duo <lb/>plana inclinata inter &longs;e comparentur, eju&longs;dem corporis gravita­<lb/>tiones in illis &longs;unt reciproce ut Secantes angulorum inclinatio­<lb/>nis: hoc e&longs;t, &longs;i fuerint duo plana inclinata BS, BC, gravitatio <lb/>in BS ad gravitationem in BC e&longs;t ut BC ad BS. <!-- KEEP S--></s> <s id="s.000696">Quia enim <lb/>gravitatio in BC ad gravitationem in BD e&longs;t ut BD ad BC; <lb/>& gravitatio in BD ad gravitationem in BS e&longs;t ut BS ad BD, <lb/>igitur ex æqualitate, per 23. lib.5. gravitatio in BC ad gravi­<lb/>tationem in BS e&longs;t ut BS ad BC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000697">Hinc prætereà fit, ut, &longs;i gravia in planis con&longs;tituta habeant <lb/>Rationem eandem, quam &longs;ecantes angulorum inclinationis ha­<lb/>bent inter &longs;e vel ad Radium, eorum gravitatione, &longs;int æquales. </s> <lb/> <s id="s.000698">Sit ad horizontalem, SC per­<lb/><figure id="id.017.01.101.1.jpg" xlink:href="017/01/101/1.jpg"/><lb/>pendicularis BD, & inclina­<lb/>tæ BS, BC, per quas lineas <lb/>ducta intelligantur plana, & <lb/>in planis gravia diver&longs;a, & ut <lb/>BD ad BC ita pondus O ad <lb/>pondus M, & ut BD ad BS <lb/>ita pondus O ad pondus N. <!-- KEEP S--></s> <lb/> <s id="s.000699">Dico ponderum M, O, N, gravitationes in &longs;uis planis e&longs;&longs;e <lb/>æquales. </s> <s id="s.000700"><expan abbr="Quoniã">Quoniam</expan> enim duorum gravium gravitationes in eadem <lb/>perpendiculari BD &longs;unt ut <expan abbr="ip&longs;orũ">ip&longs;orum</expan> pondera, gravitatio M in per­<lb/>pendiculari BD, ad gravitationem O in eadem perpendiculari, <lb/>e&longs;t ut M ad O, hoc e&longs;t ut BC ad BD; &longs;ed gravitatio M in per-<pb pagenum="86" xlink:href="017/01/102.jpg"/>pendiculari BD, ad gravitationem eju&longs;dem M in inclinatâ <lb/>BC, e&longs;t pariter ut BC ad BD; igitur per 11. lib. 5. gravita­<lb/>tio M in perpendiculari ad gravitationem O in perpendiculari <lb/>e&longs;t, ut gravitatio M in perpendiculari BD ad gravitationem <lb/>M in inclinatâ BC; igitur per 14. lib. 5. gravitatio O in per­<lb/>pendiculari BD æqualis e&longs;t gravitationi M in inclinatâ BC. <!-- KEEP S--></s> <lb/> <s id="s.000701">Eâdem methodo o&longs;tenditur æqualem e&longs;&longs;e gravitationem N in <lb/>inclinatâ BS, gravitationi O in perpendiculari BD. <!-- KEEP S--></s> <s id="s.000702">Quare <lb/>gravitationes M & N æquales inter &longs;e &longs;unt, cum æquales &longs;int <lb/>gravitationi O. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000703">Con&longs;tat itaque ii&longs;dem viribus retineri po&longs;&longs;e, aut &longs;ur&longs;um trahi, <lb/>majus pondus in plano inclinato, quàm in perpendiculari, ea­<lb/>dem enim e&longs;t illorum gravitatio, ut o&longs;tendi; vires autem reti­<lb/>nentis aut trahentis debent gravitationi corporis proportione <lb/>re&longs;pondere. </s> <s id="s.000704">Quare datis viribus, quæ po&longs;&longs;int datum pondus O <lb/>&longs;u&longs;tinere in perpendiculari BD, cogno&longs;ci pote&longs;t gravitas pon­<lb/>deris quod eædem vires &longs;u&longs;tinere valebunt in dato plano BC in­<lb/>clinato: &longs;i nimirùm fiat ut Radius ad &longs;ecantem anguli datæ in­<lb/>clinationis, ita datum pondus O ad pondus M quæ&longs;itum. </s> <s id="s.000705">De­<lb/>tur O lib. 15. & angulus DBC gr. <!-- REMOVE S-->36. Fiat ut radius 10000000 <lb/>ad &longs;ecantem 12360680, ita lib. 15. ad lib. 18 1/2; quod e&longs;t pon­<lb/>dus M æquè gravitans in plano BC cum pondere O in per­<lb/>pendiculari. </s> <s id="s.000706">Contra verò dato pondere M &longs;u&longs;tinendo ii&longs;dem <lb/>viribus, quibus &longs;u&longs;tinetur O in perpendiculari, invenietur in­<lb/>clinatio plani: &longs;i fiat ut pondus O lib. 15. ad pondus M datum <lb/>lib. 50, ita Radius 10000000 ad 333.33333.&longs;ecantem anguli in­<lb/>clinationis DBC gr. <!-- REMOVE S-->72. 32′. </s> <s id="s.000707">32″. <!-- KEEP S--></s> <s id="s.000708">Demum dato pondere & pla­<lb/>ni inclinatione nota fiet potentia, &longs;i ut Secans datæ inclinatio­<lb/>nis ad Radium, ita fiat datum pondus ad aliud pondus, quod <lb/>potentia valet &longs;u&longs;tinere in perpendiculari. </s> <s id="s.000709">Sit enim DBC <lb/>gr. <!-- REMOVE S-->36, & M lib. 50. Erit ut Secans 12360680 ad Radium <lb/>10000000, ita M lib. 50 ad pondus O ferè lib.40 1/2, quod po&longs;&longs;it <lb/>à potentia in aere libero &longs;u&longs;tineri. </s> <s id="s.000710">Quare potentia &longs;u&longs;tinens <lb/>pondus in plano inclinato e&longs;t ad pondus, ut Radius ad Secan­<lb/>tem anguli inclinationis; & potentia potens movere cum &longs;it ma­<lb/>jor potentiâ &longs;u&longs;tinente, etiam majorem habet Rationem quàm <lb/>habeat Radius ad Secantem. </s> <s id="s.000711">Id quod intelligitur ex vi præcisè <lb/>gravitationis; quicquid inferat di&longs;criminis partium conflictus. <pb pagenum="87" xlink:href="017/01/103.jpg"/></s> </p> <p type="head"> <s id="s.000712"><emph type="center"/>CAPUT XIV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="head"> <s id="s.000713"><emph type="center"/><emph type="italics"/>Quâ ratione corpus gravitet in planum inclinatum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000714">COn&longs;tituta Ratione gravitationis in plano inclinato, deter­<lb/>minatis &longs;cilicet momentis, quæ ad de&longs;cendendum obtinet <lb/>corpus grave exi&longs;tens in plano inclinato, &longs;upere&longs;t explicanda <lb/>gravitatio, quam idem corpus exercet in planum inclinatum <lb/>illud urgendo, atque deor&longs;um premendo. </s> <s id="s.000715">Certum e&longs;t autem <lb/>planum verticale &longs;eu perpendiculare nullo pacto urgeri à cor­<lb/>pore gravi, quod liberè de&longs;cendere pote&longs;t per &longs;uam directionis <lb/>lineam, quæ cum non occurrat plano verticali, nullum ab eo <lb/>recipit impedimentum. </s> <s id="s.000716">Quare corporis gravitas vires totas <lb/>exercet, aut de&longs;cendendo, aut repugnando contra retinentem, <lb/>qui non plus adhibere debet conatûs in retinendo, etiam &longs;i pla­<lb/>num verticale amoveatur: atque adeò nihil omninò gravitat in <lb/>planum verticale. </s> <s id="s.000717">Contra verò in planum horizontale, quam <lb/>maximè gravitant corpora; eò quod directionis lineâ in illud <lb/>incurrente ad angulos rectos, motus omnis impeditur, & <lb/>cunctas gravitatis vires deor&longs;um contendentes ita &longs;ubjectum <lb/>planum excipit, ut nihil reliquum &longs;it virium, quas vel minimo <lb/>motu exerceat. </s> <s id="s.000718">Hinc &longs;i corporis in plano horizontali jacentis <lb/>an&longs;am teneas, nihil tibi pror&longs;us e&longs;t laborandum, nec quicquam <lb/>percipis ponderis; at &longs;ubmoto plano lacertis omnibus e&longs;t con­<lb/>tendendum, ut illud retineas; tota enim gravitatio cum reti­<lb/>nente luctatur, quæ planum &longs;u&longs;tinens urgebat. </s> <s id="s.000719">In hoc itaque <lb/>planum verticale cum horizontali comparatur, quod cum ver­<lb/>ticale nihil impediat motum, corpus in plano verticali omninò <lb/>gravitat, &longs;ed in illud non gravitat: cum autem horizontale <lb/>pror&longs;us impediat motum, corpus in plano horizontali nihil gra­<lb/>vitat, &longs;ed in illud totam &longs;uam gravitationem exercet. </s> <s id="s.000720">Eædem <lb/>igitur vires, quæ ad de&longs;cendendum in plano verticali impen­<lb/>derentur, in urgendo plano horizontali in&longs;umuntur. </s> </p> <p type="main"> <s id="s.000721">Quæ cum ita &longs;int, &longs;atis con&longs;tat corpora gravia ita in pla­<lb/>no inclinato gravitare, & obtinere momenta ad de&longs;cenden-<pb pagenum="88" xlink:href="017/01/104.jpg"/>dum, ut etiam in illud, à quo impediuntur, gravitent, il­<lb/>ludque urgeant. </s> </p> <p type="main"> <s id="s.000722">Id verò fieri non pote&longs;t ni&longs;i pro ratione impedimenti & mo­<lb/>ræ, quam &longs;ubjectum planum motui infert &longs;u&longs;tinendo corpora <lb/>gravia; quæ proinde &longs;ibi relicta à directionis lineâ declinant, <lb/>motúmque deflectunt. </s> <s id="s.000723">Porrò in plano inclinato quantum &longs;ub­<lb/>&longs;it impedimenti, &longs;tatim apparet, ac innote&longs;cit, quantum reli­<lb/>quum &longs;it virium ad de&longs;cendendum; vires enim, quæ reliquæ <lb/>&longs;unt, &longs;i adjiciantur viribus impeditis, totam virium omnium <lb/>&longs;ummam conflare debent. </s> <s id="s.000724">Atqui ex &longs;uperiori capite notæ &longs;unt <lb/>vires, quibus corpus gravitat in plano inclinato; igitur quæ e&longs;t <lb/>differentia gravitationis in plano inclinato, à gravitatione in <lb/>plano verticali, quod & perpendiculare, ea e&longs;t men&longs;ura im­<lb/>pedimenti, quod à &longs;ubjecto plano infertur motui; atque <lb/><figure id="id.017.01.104.1.jpg" xlink:href="017/01/104/1.jpg"/><lb/>adeò gravitationis corporis in planum. </s> </p> <p type="main"> <s id="s.000725">Cum itaque o&longs;ten&longs;um fuerit <expan abbr="gravitation&etilde;">gravitationem</expan> <lb/>in plano BS ad gravitationem in plano BD <lb/>e&longs;&longs;e reciprocè ut BD ad BS, hoc e&longs;t, ut Ra­<lb/>dius ad &longs;ecantem anguli inclinationis cum <lb/>verticali, hoc e&longs;t ut BV ad BS, patet vires <lb/>non impeditas ad vires impeditas e&longs;&longs;e ut <lb/>BV ad VS, quandoquidem totas gravita­<lb/>tis vires refert BS. <!-- KEEP S--></s> <s id="s.000726">In planum igitur inclinatum BS gravitatio <lb/>e&longs;t ut VS, quæ in planum horizontale e&longs;&longs;et &longs;ecundùm totas <lb/>vires ut BS. <!-- KEEP S--></s> <s id="s.000727">Quare gravitatio in planum horizontale ad gra­<lb/>vitationem in planum inclinatum e&longs;t ut Secans BS ad exce&longs;­<lb/>&longs;um Secantis &longs;upra Radium, VS; &longs;eu, quod in idem recidit, &longs;i <lb/>gravitatio in plano inclinato ad gravitationem in verticali po­<lb/>natur ut Sinus complementi anguli inclinationis ad Radium, <lb/>ita BR Radius ad DR Sinum ver&longs;um anguli inclinationis. </s> <s id="s.000728">Id <lb/>autem, quod de plano BS dictum e&longs;t, de plano quoque BC, & <lb/>cæteris quibu&longs;cunque dictum intelligatur; cum enim gravita­<lb/>tio in plano inclinato BC ad gravitationem in perpendiculari <lb/>&longs;it ut BD, hoc e&longs;t BX, ad BC, erit gravitatio in planum ho­<lb/>rizontale ad gravitationem in inclinatum ut BC ad XC, hoc <lb/>e&longs;t ut BT ad DT. </s> <s id="s.000729">Quare gravitatio in planum BS ad gravi­<lb/>tationem in planum BC, e&longs;t ut DR Sinus ver&longs;us inclinationis <lb/>DBS, ad DT Sinum ver&longs;um inclinationis DBC; a&longs;&longs;umptis <pb pagenum="89" xlink:href="017/01/105.jpg"/>&longs;cilicet numeris tabulatis ad eundem Radium relatis; nam &longs;i li­<lb/>neæ &longs;pectentur, non e&longs;t Ratio ut DR ad DT, &longs;ed ut OT <lb/>ad DT; neque enim idem e&longs;t Radius BS & BC; ac proinde <lb/>OT major e&longs;t, quàm DR, &longs;icuti Radius BI major e&longs;t Radio <lb/>BS; vel a&longs;&longs;umpto eodem Radio BD, Ratio e&longs;t ut VS ad XC, <lb/>exce&longs;&longs;us &longs;ecantium &longs;upra Radium. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000730">Id verò ex dictis &longs;ub finem capitis &longs;uperioris videtur mani­<lb/>fe&longs;tum: nam &longs;i in plano BC retinetur pondus lib. 50. ii&longs;dem <lb/>viribus, quibus in perpendiculari &longs;u&longs;penderentur lib. 40 1/2, pa­<lb/>tet à plano &longs;u&longs;tineri lib.9 1/2; ac proinde grave, quod habet gra­<lb/>vitatem totam ut 100, in plano BC gravitabit ut 81, & urge­<lb/>bit ut 19 &longs;ubjectum planum. </s> </p> <p type="main"> <s id="s.000731">Ex his fieri pote&longs;t &longs;atis quæ­<lb/><figure id="id.017.01.105.1.jpg" xlink:href="017/01/105/1.jpg"/><lb/>renti, cur &longs;u&longs;tinens columnam <lb/>OR plus gravitatis percipiat, <lb/>quàm qui &longs;u&longs;tinet columnam <lb/>SR: quia nimirum, qui &longs;u&longs;tinet, <lb/>e&longs;t pars plani inclinati, in quo ja­<lb/>cens concipitur columna: quan­<lb/>do igitur e&longs;t pars plani habentis <lb/>inclinationem LOR, gravitas, <lb/>quæ &longs;u&longs;tinetur à &longs;ubjecto plano, &longs;e habet ad totam gravitatem <lb/>ut Sinus Ver&longs;us anguli LOR ad Radium; Quando autem e&longs;t <lb/>pars plani habentis inclinationem VSR, gravitatio in &longs;ub­<lb/>jectum planum &longs;u&longs;tinens e&longs;t ad totam gravitationem ut Sinus <lb/>Ver&longs;us anguli VSR ad eumdem Radium. <!-- KEEP S--></s> <s id="s.000732">Atqui Sinus Ver&longs;us <lb/>anguli VSR minoris minor e&longs;t Sinu Ver&longs;o anguli LOR ma­<lb/>joris; igitur minor e&longs;t gravitatio SR, quam OR. </s> <s id="s.000733">Verum qui­<lb/>dem e&longs;t illud, quod &longs;i in R aliquo obice prohibeatur, ne de­<lb/>&longs;cendat; variatâ inclinatione, quo fit minor &longs;u&longs;tinentis labor, <lb/>eò augetur magis conatus potentiæ in R detinentis columnam, <lb/>ne juxta plani inclinationem de&longs;cendat. </s> <s id="s.000734">Hinc &longs;i duo &longs;int co­<lb/>lumnam inclinatam deferentes, qui illam in R &longs;u&longs;tinet, plus <lb/>&longs;ubit laboris, quàm qui in O, aut S: quia præter gravitatio­<lb/>nem, quam percipit tanquam pars plani inclinati SR aut OR, <lb/>debet præterea retinere columnam proclivem ad de&longs;cen&longs;um <lb/>propter plani inclinationem; ideò cùm &longs;calas, aut montis cli­<lb/>vum con&longs;cendunt, qui in &longs;uperiore loco e&longs;t, minimum &longs;ubit <pb pagenum="90" xlink:href="017/01/106.jpg"/>laboris. </s> <s id="s.000735">Huc etiam revocari po&longs;&longs;e videtur ratio, ob quam in <lb/>elevando pontes illos ver&longs;atiles, qui arcium portis opponuntur, <lb/>initio major percipiatur difficultas, &longs;ed demùm facillimè ele­<lb/>ventur. </s> <s id="s.000736">Verùm id ex dicendis inferiùs clariùs con&longs;tabit; neque <lb/>enim omnium gravium, quocunque &longs;e tandem modo habeant, <lb/>eadem e&longs;t ratio; cum animum diligenter advertere oporteat, ut <lb/>innote&longs;cat planum inclinatum, in quo &longs;uam gravitationem <lb/>exercent, & habent vires ad de&longs;cendendum. </s> </p> <p type="main"> <s id="s.000737">Non e&longs;t autem per di&longs;&longs;imulantiam prætereunda difficultas, <lb/>quæ face&longs;&longs;ere po&longs;&longs;et aliquid negotij, & gravitationis Rationem <lb/>con&longs;titutam convellere videretur. </s> <s id="s.000738">E&longs;t &longs;iquidem certum apud <lb/>omnes mechanicos, tam ubi de libra, quàm ubi de vecte &longs;ermo <lb/>e&longs;t, aliam &longs;ervari Rationem quàm Sinuum Ver&longs;orum in mo­<lb/>mento potentiæ, aut ponderis determinando. </s> <s id="s.000739">Sit vectis, aut <lb/><figure id="id.017.01.106.1.jpg" xlink:href="017/01/106/1.jpg"/><lb/>libræ brachium EC, hypomochlion <lb/>&longs;eu centrum C; attollatur in H, aut <lb/>in D; omnes con&longs;entiunt momentum <lb/>potentiæ aut ponderis in E ad mo­<lb/>mentum in H, e&longs;&longs;e ut HC ad IC, <lb/>ad momentum verò in D e&longs;&longs;e ut DC <lb/>ad FC. </s> <s id="s.000740">E&longs;t igitur, inquis, gravitatio <lb/>in planum DC ad gravitationem in <lb/>planum horizontale EC, ut FC ad DC; in planum verò HC, <lb/>ut IC ad HC, hoc e&longs;t ut Sinus Rectus anguli inclinationis ad <lb/>Radium. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000741">Priùs verò, quàm me ab hac difficultate expediam, o&longs;tendo <lb/>non &longs;atis aptè gravitationem in planum inclinatum de&longs;umi po&longs;­<lb/>&longs;e ex Sinu Recto anguli inclinationis. </s> <s id="s.000742">Quandoquidem vis de­<lb/>&longs;cendendi in plano DC ad <expan abbr="totã">totam</expan> corporis liberi <expan abbr="gravitation&etilde;">gravitationem</expan> e&longs;t <lb/>ut DF ad DC, igitur &longs;i gravitatio in <expan abbr="planũ">planum</expan> DC ad totam <expan abbr="gravi-tation&etilde;">gravi­<lb/>tationem</expan> e&longs;t ut FC ad DC, tota virium &longs;umma e&longs;t DF plus FC, <lb/>ac tota vis gravitandi, ubi nullum e&longs;t impedimentum, e&longs;t DC; <lb/>igitur DC, & DF plus FC, æquales &longs;unt, contra 20.lib.1.Eucl. <!-- KEEP S--></s> <lb/> <s id="s.000743">Neque hic liceat ad æqualitatem potentiarum confugere, ut <lb/>&longs;icut per 47. lib. 1. Eucl. <!-- REMOVE S-->linea DC pote&longs;t quadrata linearum <lb/>DF, FC, ita vis totius gravitatis æqualis gravitationibus in <lb/>plano inclinato & in planum inclinatum eandem &longs;ervet pro­<lb/>portionem laterum trianguli DFC, adeò ut totam gravitatem <pb pagenum="91" xlink:href="017/01/107.jpg"/>Secans anguli inclinationis exprimat, gravitationem in plano <lb/>inclinato Radius, Tangens verò gravitationem in planum in­<lb/>clinatum. </s> <s id="s.000744">Si enim Quadratum DC æquale e&longs;t quadratis DF, <lb/>& FC &longs;imul &longs;umptis, non tamen linea DC æqualis e&longs;t aggre­<lb/>gato linearum DF & FC: neque eadem e&longs;t inter lineas DF <lb/>& DC Ratio, quæ inter earum quadrata; &longs;ed e&longs;t &longs;ub duplica­<lb/>tâ quadratorum: Quare cum gravitatio in plano inclinato DC <lb/>ad gravitationem in perpendiculari, non &longs;it ut quadratum DF <lb/>ad quadratum DC; &longs;ed ut linea DF ad lineam DC, fru&longs;trà ad <lb/>quadrata confugimus, quorum nulla hîc habetur ratio. </s> </p> <p type="main"> <!-- paragraph type checked til this point --> <s id="s.000745">In eo itaque æquivocatio con&longs;i&longs;tit, quod pondus in D con&longs;ti­<lb/>tutum, & applicatum brachio DC concipitur e&longs;&longs;e in plano in­<lb/>clinato DC, contra quàm res e&longs;t: in eo &longs;iquidem plano intel­<lb/>ligendum e&longs;t, in quo ad motum determinatur; illud autem e&longs;t <lb/>planum DG, quod tangit circulum ED; & &longs;ic deinceps, pro <lb/>ut diver&longs;a circuli puncta à diver&longs;is planis contingi po&longs;&longs;unt. </s> <lb/> <s id="s.000746">Quare in D momentum ad de&longs;cendendum per DG ad totam <lb/>gravitationem e&longs;t ut DF ad DG, hoc e&longs;t ut FC ad CD, per <lb/>8. lib.6. hoc e&longs;t ut FC ad EC. <!-- KEEP S--></s> <s id="s.000747">E&longs;t igitur brachium libræ &longs;eu <lb/>vectis CD, &longs;u&longs;tinens pondus &longs;eu potentiam D, quæ cum ha­<lb/>beat vires univer&longs;as ut EC, gravitationis autem momenta ha­<lb/>beat &longs;olùm ut FC, impeditur à &longs;u&longs;tinente ut FE; e&longs;t autem <lb/>EF Sinus Ver&longs;us anguli FCD, hoc e&longs;t anguli inclinationis <lb/>FDG. </s> <s id="s.000748">Quare gravitatio ponderis contrà &longs;ubjectum corpus, <lb/>quod impedit motum perpendicularem, ad totam gravitatio­<lb/>nem e&longs;t, ut Sinus Ver&longs;us anguli inclinationis plani, per quod <lb/>fieri pote&longs;t motus, ad Radium. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000749">Hinc vides valdè di&longs;parem e&longs;&longs;e rationem gravitationis in <lb/>&longs;u&longs;tinendo corpore inclinato, &longs;i illud liberè moveri po&longs;&longs;it, ac &longs;i <lb/>circa centrum perfici debeat motus. </s> <s id="s.000750">Nam &longs;i DC &longs;it columna, <lb/>aut pons ver&longs;atilis, retineaturque in C, jam punctum C vicem <lb/>obtinens &longs;ubjecti plani, illiu&longs;que munere fungens, &longs;u&longs;tinet <lb/>ponderis partem EF, reliqua FC, quæ e&longs;t men&longs;ura momenti <lb/>ad de&longs;cendendum, debet &longs;u&longs;tineri à potentia motum impe­<lb/>diente per DG. <!-- KEEP S--></s> <s id="s.000751">Sin autem per DC planum columna moveri <lb/>po&longs;&longs;it rectâ & de&longs;cendere, vis de&longs;cendendi ad totam gravitatio­<lb/>nem e&longs;t ut DF ad DC, gravitatio autem contra &longs;u&longs;tinentem <lb/>e&longs;t ad totam gravitationem ut Sinus Ver&longs;us anguli inclinationis <pb pagenum="92" xlink:href="017/01/108.jpg"/>FDC ad Radium; qui enim &longs;u&longs;tinet grave, dum de&longs;cendit in­<lb/>clinatum, habet rationem plani inclinati. </s> <s id="s.000752">Neque id mirum vi­<lb/>deri debet, quandoquidem plurimum refert, an per planum <lb/>DG an verò per DC &longs;it determinatio ad motum, & quâ ra­<lb/>tione &longs;u&longs;tinens opponatur virtuti motivæ: quare cùm diversâ <lb/>ratione opponatur motui circa centrum C, ac motui per pla­<lb/>num DC, etiam di&longs;par erit in &longs;u&longs;tinendo difficultas. </s> </p> <p type="main"> <s id="s.000753">Ex his, quæ tùm hoc, tùm &longs;uperiori capite di&longs;putata &longs;unt, <lb/>habes quid funambulis re&longs;pondeas volatum mentiri meditanti­<lb/>bus, cum pectore in&longs;i&longs;tentes intento funi, diductis cruribus & <lb/>exten&longs;is brachiis, corpus æqualibus momentis librant, séque <lb/>ex editâ turri in depre&longs;&longs;iorem locum præcipites dant; &longs;i fortè, <lb/>ut noverint, quàm &longs;olidus e&longs;&longs;e debeat ac validus funis, quo iis <lb/>utendum e&longs;t, quærant, quantis momentis corpus urgeat &longs;ub­<lb/><figure id="id.017.01.108.1.jpg" xlink:href="017/01/108/1.jpg"/><lb/>jectum funem. </s> <s id="s.000754">Datâ enim turris altitudi­<lb/>ne BD, & depre&longs;&longs;ioris loci, in quem de­<lb/>&longs;cendendum e&longs;t, di&longs;tantiâ DC, collectí&longs;­<lb/>que in &longs;ummam harum quadratis, Radix <lb/>&longs;ummæ dabit BC funis longitudinem; ex <lb/>quâ &longs;i auferatur BX turris altitudini BD <lb/>æqualis, erit BC divi&longs;a in X juxtà Ratio­<lb/>nem momentorum, quæ corporis gravitas <lb/>exercet in plano inclinato, & in planum <lb/>inclinatum. </s> <s id="s.000755">Sic po&longs;itâ BD ped. <!-- REMOVE S-->150, & DC ped. <!-- REMOVE S-->200, BC e&longs;t <lb/>ped. <!-- REMOVE S-->250: ex quâ &longs;i auferatur BD, erit BX 150, & XC 100. <lb/>Statue autem totius gravitatis corporis funambuli momenta <lb/>220; hæc dividantur in duas partes, quarum major &longs;it &longs;e&longs;qui­<lb/>altera minoris, &longs;icut BX inventa e&longs;t ip&longs;ius XC &longs;e&longs;quialtera, & <lb/>erunt momenta quidem ad de&longs;cendendum in plano inclinato <lb/>132, momenta verò gravitationis in planum inclinatum, hoc <lb/>e&longs;t in &longs;ubjectum funem, 88. Hæc tamen intelligenda &longs;unt eâ <lb/>factâ hypothe&longs;i, quòd funis rectâ intentus permaneret: cæte­<lb/>rùm cum & &longs;uopte pondere, & &longs;ub impo&longs;iti corporis mole &longs;ub­<lb/>&longs;idat, atque inflectatur, præ&longs;ertim circà medium, &longs;atis apparet <lb/>adhuc majorem &longs;ubjecti plani inclinationem æ&longs;timandam e&longs;&longs;e, <lb/>quàm quæ ex altitudine DB & di&longs;tantiâ DC inferatur, quin <lb/>& illam pro diversâ ab extremitatibus di&longs;tantiâ &longs;ubinde muta­<lb/>ri, ac proinde validiori fune opus e&longs;&longs;e. <pb pagenum="93" xlink:href="017/01/109.jpg"/></s> </p> <p type="main"> <s id="s.000756"><emph type="center"/>CAPUT XV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.000757"><emph type="center"/><emph type="italics"/>Inquiruntur Rationes gravitationis corporum <lb/>&longs;uspen&longs;orum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000758">COn&longs;ideratâ corporum gravitatione tùm in plano inclinato, <lb/>tùm in planum inclinatum, con&longs;equens e&longs;t, ut ad eorum­<lb/>dem gravitationem, &longs;i ex fune &longs;u&longs;pendantur, gradum facia­<lb/>mus; hæc enim illi valdè affinis e&longs;t &longs;peculatio: id quod facilè <lb/>intelligat, qui&longs;quis animum advertere voluerit, remque totam <lb/>penitiùs intro&longs;picere. </s> <s id="s.000759">Ex his &longs;i quidem, quæ hactenus di&longs;puta­<lb/>ta &longs;unt, lux, opinor, non modica ad hanc, quam examinan­<lb/>dam &longs;u&longs;cipimus quæ&longs;tionem, derivabitur. </s> </p> <p type="main"> <s id="s.000760">Pendeat ex clavo C ad perpen­<lb/><figure id="id.017.01.109.1.jpg" xlink:href="017/01/109/1.jpg"/><lb/>diculum globus ferreus A, quem <lb/>&longs;uppo&longs;itum planum horizontale <lb/>BD ita exactè contingat, ut nihil <lb/>de funiculi CA intentione remit­<lb/>tatur. </s> <s id="s.000761">Satis apparet &longs;ubjecto pla­<lb/>no BD non incumbere globum A, <lb/>&longs;ed omnia &longs;uæ gravitationis, qua <lb/>deor&longs;um nititur, momenta exer­<lb/>cere contrà clavum C, ex quo &longs;u&longs;pen&longs;us ad perpendiculum <lb/>pendet. </s> <s id="s.000762">Quod &longs;i aut clavus C, nemine funem retinente, revel­<lb/>leretur, aut funis CA præcideretur, jam tota vis de&longs;cendendi, <lb/>quæ corpori A ine&longs;t, urgeret &longs;ubjectum planum BD; nec ta­<lb/>men in motum erumperet globus, quia planum BD; pari u&longs;que­<lb/>quaque ad perpendiculum inclinatione libratur, atque adeò <lb/>motui pror&longs;us ob&longs;i&longs;tit. </s> </p> <p type="main"> <s id="s.000763">Jam verò &longs;i globum A pariter ex perpendiculo CA penden­<lb/>tem contingat planum aliud non quidem horizontale, &longs;ed in­<lb/>clinatum EF, manife&longs;tum e&longs;t totam pariter gravitationem <lb/>exerceri contra clavum C retinentem, planumque contingens <pb pagenum="94" xlink:href="017/01/110.jpg"/>omninò non urgeri, ni&longs;i præci&longs;o funiculo &longs;ibi relinquatur glo­<lb/>bus, ut in inclinato plano EF ad de&longs;cen&longs;um pronus contra &longs;ub­<lb/>jectum planum nitatur, à quo cogitur, ut in motu à recto, quod <lb/>ad univer&longs;i centrum e&longs;t, itinere deflectat. </s> </p> <p type="main"> <s id="s.000764">Quod &longs;i planum inclinatum EF ita &longs;u&longs;pen&longs;o globo A &longs;ubji­<lb/>ciatur, ut recta linea centrum gravitatis A, & punctum &longs;u&longs;­<lb/>pen&longs;ionis H conjungens parallela &longs;it lineæ EF, quam in plano <lb/>inclinato de&longs;cendens globus percurreret; momenta quidem <lb/>gravitationis, quæ in eo plano obtineret globus ad de&longs;cenden­<lb/>dum, exercebit adversùs clavum retinentem in H, &longs;ubjectum <lb/>verò planum EF perinde urgebitur, atque &longs;i nullo retinente li­<lb/>bera e&longs;&longs;et globo de&longs;cendendi facultas: vis enim, quâ prohibe­<lb/>tur globus, ne moveatur &longs;ecundùm rectam lineam, ut con&longs;tat, <lb/>opponitur de&longs;cen&longs;ui in plano inclinato; ejus autem directio <lb/>AH non opponitur nitenti in planum, cui parallela e&longs;t. </s> </p> <p type="main"> <s id="s.000765">Contra verò &longs;i globus in plano inclinato con&longs;titutus retinea­<lb/>tur &longs;ecundùm rectam lineam, quæ ad perpendiculum cadit in <lb/>&longs;ubjectum planum EF, nimirum &longs;ecundùm lineam LO, im­<lb/>peditur quidem, ne contra planum nitatur; &longs;ed vis i&longs;ta &longs;ic reti­<lb/>nens nullâ ratione adver&longs;atur motui in plano inclinato, quin <lb/>ii&longs;dem gravitatis momentis de&longs;cendat globus in eo plano; &longs;i <lb/>quidem retinentis directio LO maneat &longs;emper adversùs illud <lb/>planum perpendicularis. </s> <s id="s.000766">Nam &longs;i potentia retinens &longs;ecundùm <lb/>eam directionem agat, ut neque congruat perpendiculari LO, <lb/>neque parallelæ HA, ob&longs;i&longs;tet gravitationi corporis &longs;ivè in pla­<lb/>no inclinato, &longs;ivè in planum inclinatum pro ratione anguli, <lb/>quem retinentis directio inter perpendicularem LO, & paral­<lb/>lelam HA interjecta, con&longs;tituet cum plano inclinato. </s> <s id="s.000767">Quæ <lb/>enim inter LO & CA fuerit, elidet omnem corporis conatum <lb/>adversùs planum, à quo illud avellit; non autem omnem eum, <lb/>qui in plano inclinato deor&longs;um rapit. </s> <s id="s.000768">Quæ verò fuerit inter <lb/>CA & HA, tollet quidem de&longs;cen&longs;um in plano EF inclinato; <lb/>&longs;ed non omninò prohibebit, quin &longs;ubjectum planum, cui aliqua­<lb/>tenus nititur, urgeat. </s> <s id="s.000769">Id quod facilè intelligas, &longs;i plana &longs;ubjecta <lb/>BD horizontale, & EF inclinatum ex maximè flexili mate­<lb/>ria, puta, papyro, concipias; in quâlibet enim &longs;u&longs;pen&longs;ione <lb/>inter C, & L, planum BD horizontale flectetur ex pondere, <lb/>non autem inclinatum EF: contrà verò in omni &longs;u&longs;pen&longs;ione <pb pagenum="95" xlink:href="017/01/111.jpg"/>inter C & H, planum inclinatum EF flectetur; at non item ho­<lb/>rizontale BD, quia nimirum inclinatum EF prohibet, ne recta <lb/>HA ad perpendiculum accedens verticalis fiat. </s> </p> <p type="main"> <s id="s.000770">Unum hic præterea con&longs;iderandum venit, quod &longs;uperiori <lb/>capite &longs;ubindicatum fuit; &longs;i videlicet non ex flexili fune deor­<lb/>&longs;um pendeat globus, &longs;ed rigido bacillo circà axem inferiùs po­<lb/>&longs;itum ver&longs;atili adnectatur &longs;upe­<lb/><figure id="id.017.01.111.1.jpg" xlink:href="017/01/111/1.jpg"/><lb/>riùs. </s> <s id="s.000771">Sic rectus bacillus AB, cujus <lb/>extremitas altera adnexum ha­<lb/>beat globum B, altera &longs;it circà <lb/>axem A ver&longs;atilis. </s> <s id="s.000772">Satis aperta <lb/>conjectura e&longs;t bacillum AB vi­<lb/>cem &longs;ubire plam, cui innitatur <lb/>globus in B, qui proinde prohi­<lb/>betur, tùm ne ad perpendiculum <lb/>cadat per BD, tùm ne per BA <lb/>delabatur: linea igitur plani, per quod moliri motum poterit <lb/>globus B, nulla alia congruentiùs a&longs;&longs;ignari queat præter BC, <lb/>quæ cum bacillo BA rectum angulum con&longs;tituit. </s> <s id="s.000773">Perindè igi­<lb/>tur in motum incitabitur, atque &longs;i in plano e&longs;&longs;et, cujus inclina­<lb/>tio angulum efficeret æqualem angulo elevationis bacilli &longs;uprà <lb/>planum horizontale GA. <!-- KEEP S--></s> <s id="s.000774">Cum enim recta BD producta ca­<lb/>dens in planum horizontale, angulum BSA Rectum efficiat, <lb/>reliqui duo &longs;imul SAB, ABS, Recto ABC æquales &longs;unt; & <lb/>communi ABS dempto, &longs;upere&longs;t SAB elevationis angulus <lb/>æqualis angulo SBC inclinationis plani. </s> <s id="s.000775">Quare ductâ Tan­<lb/>gente DE, erit BE Secans anguli inclinationis, BD verò Ra­<lb/>dius: ac proptereà ad de&longs;cendendum in huju&longs;modi plano BC <lb/>momenta, ad totam gravitatem in perpendiculo BD, erunt ut <lb/>Radius BD ad Secantem BE, juxta ea, quæ cap. 13. hujus lib. <!-- REMOVE S--><lb/>demon&longs;travimus. </s> </p> <p type="main"> <s id="s.000776">Quia tamen in motu globus ex bacilli conver&longs;ione circà <lb/>axem A non pote&longs;t percurrere rectam BC, &longs;ed ita retinetur à <lb/>bacillo, cui adnectitur, ut de&longs;cendat in F, jam in alio plano <lb/>minorem inclinationem habente con&longs;titutus intelligitur, nimi­<lb/>rùm in plano FG, quod cum perpendiculo FL efficit angulum <lb/>inclinationis GFL æqualem angulo LAF elevationis: id quod <lb/>eâdem planè methodo, ac &longs;uperiùs factum e&longs;t, demon&longs;tratur. <pb pagenum="96" xlink:href="017/01/112.jpg"/>Ex quo fit, quemadmodum in huju&longs;modi conver&longs;ione globus <lb/>in alio atque alio plano inclinato con&longs;tituitur, ita alia atque alia <lb/>obtinere gravitatis momenta: in B &longs;iquidem gravitat ut BD ad <lb/>BE, in F verò ut HF ad FI. <!-- KEEP S--></s> <s id="s.000777">Cum igitur Radius utrobique ex <lb/>hypothe&longs;i æqualis &longs;it, videlicet DB, & HF, major autem &longs;it <lb/>BE Secans majoris anguli DBE, quàm FI Secans minoris an­<lb/>guli HFI, con&longs;tat ex 8. lib. 5. majorem Rationem e&longs;&longs;e HF ad <lb/>FI minorem, quàm DB ad BE majorem, atque adeò globum <lb/>magis in F quàm in B gravitare, ut deor&longs;um moveatur, atque <lb/>adeò minùs etiam conniti contrà planum, in quo e&longs;t, videlicet <lb/>adversùs bacillum FA, magis verò adversùs bacillum BA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000778">Ex his attentè perpen&longs;is facilis e&longs;t tran&longs;itus ad &longs;u&longs;pen&longs;orum <lb/>corporum gravitationem inve&longs;tigandam. </s> <s id="s.000779">Sit enim jam non in­<lb/><figure id="id.017.01.112.1.jpg" xlink:href="017/01/112/1.jpg"/><lb/>feriùs, &longs;ed &longs;uperiùs po&longs;itus <lb/>Axis A, circa quem ver&longs;a­<lb/>tilis e&longs;t funiculus AB, cui <lb/>globus B adnectitur. </s> <s id="s.000780">Con­<lb/>&longs;tat &longs;anè non ad perpendi­<lb/>culum BD cadere po&longs;&longs;e <lb/>globum B; &longs;ed à recto deor­<lb/>&longs;um tramite deflectere, fu­<lb/>niculo &longs;cilicet AB eum re­<lb/>tinente, quemadmodum ri­<lb/>gidus bacillus OB eum ali­<lb/>quatenùs &longs;u&longs;tineret. </s> <s id="s.000781">Quia autem bacillo OB &longs;u&longs;tinente, vis <lb/>de&longs;cendendi ea e&longs;&longs;et, quæ per planum inclinatum BC, eadem <lb/>pariter e&longs;t funiculo retinente; videlicet per planum BC, in <lb/>quod recta AB ad rectos angulos incidit. </s> <s id="s.000782">Momenta igitur gra­<lb/>vitatis in eo plano inclinato, ad gravitatis momenta &longs;i corpus <lb/>liberè de&longs;cenderet, in eâ &longs;unt Ratione, quæ e&longs;t DB ad BE; <lb/>hoc e&longs;t DO ad OB per 8. lib.6. hoc e&longs;t KB ad BA per 4.lib.6. <lb/>Haud di&longs;pari methodo ratiocinantes o&longs;tendemus globi in F <lb/>con&longs;tituti momenta ad gravitandum e&longs;&longs;e perinde, atque &longs;i e&longs;­<lb/>&longs;et in plano inclinato FI, in quod ad rectos angulos cadit fu­<lb/>niculus AF; ac proinde gravitatio in F, &longs;i de&longs;cendendi vis <lb/>præcisè &longs;pectetur, ad gravitationem globi liberi, e&longs;t ut HF <lb/>ad FI, hoc e&longs;t, ut GF ad FA. <!-- KEEP S--></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"/>immorari oporteat, alia &longs;ubinde atque alia e&longs;&longs;e momenta gra­<lb/>vitatis corporis &longs;u&longs;pen&longs;i, pro ut major aut minor e&longs;t angulus <lb/>declinationis à perpendiculo AG, haud aliter quàm &longs;i in aliis <lb/>atque aliis planis inclinatis con&longs;titueretur; quo enim minor e&longs;t <lb/>declinationis angulus GAF, eò major e&longs;t angulus inclinatio­<lb/>nis plani, quippe qui e&longs;t illius complementum. </s> <s id="s.000784">Con&longs;tat &longs;i qui­<lb/>dem angulos GAF, GFA &longs;imul, e&longs;&longs;e æquales tùm Recto <lb/>AFI, tùm Recto GFH; ac proinde dempto communi GFI, <lb/>remanet HFI angulus inclinationis plani æqualis angulo <lb/>GFA, qui e&longs;t complementum anguli declinationis GAF. </s> <lb/> <s id="s.000785">Quare quò declinationis angulus major e&longs;t, eò minus e&longs;t <lb/>complementum, ac propterea e&longs;t minor angulus inclinationis <lb/>plani: in plano autem minùs inclinato majora &longs;unt gravitatis <lb/>momenta. </s> <s id="s.000786">Quò igitur corpus &longs;u&longs;pen&longs;um magis à perpendiculo <lb/>removetur, eò majora percipiuntur gravitatis momenta, ma­<lb/>jorque vis requiritur in eo, qui motum prohibere voluerit, ut <lb/>& ip&longs;a experientia unicuique facilè demon&longs;trat, & ratio evin­<lb/>cit; cum enim AB & AF æquales &longs;int, major e&longs;t Ratio KB <lb/>ad BA, quàm GF ad FA per 8. lib. 5. e&longs;t nimirum KB major, <lb/>& GF minor. </s> </p> <p type="main"> <s id="s.000787">Quoniam verò quò major e&longs;t gravitatio in plano inclinato, <lb/>minor e&longs;t in planum inclinatum; hoc ip&longs;o, quod facto declina­<lb/>tionis angulo GAB majore, quàm GAF, major e&longs;t ad de&longs;cen­<lb/>dendum propen&longs;io, minor e&longs;t conatus adversùs axem A reti­<lb/>nentem. </s> <s id="s.000788">Id quod manife&longs;to etiam experimento deprehen­<lb/>des, &longs;i ob&longs;ervaveris minùs intentum e&longs;&longs;e funiculum AB, <lb/>quàm AF. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000789">Hinc & illud &longs;atis dilucidè apparet, quod longitudinis <lb/>funiculi non exigua ratio habenda e&longs;t; ex eâ &longs;cilicet pen­<lb/>det, quod in plano magis aut minùs inclinato con&longs;titutum <lb/>cen&longs;eatur corpus grave &longs;u&longs;pen&longs;um. </s> <s id="s.000790">Si enim globus F ex fu­<lb/>niculo AF pendeat, declinationis angulus e&longs;t GAF: at <lb/>verò &longs;i funiculus, quo &longs;u&longs;penditur, &longs;it MF, angulum de­<lb/>clinationis facit GMF, qui cum externus &longs;it, major e&longs;t <lb/>interno MAF per 16. lib. 1. ac propterea minor e&longs;t incli­<lb/>natio plani FN facientis cum rectâ MF angulum Rectum, <lb/>quàm &longs;it inclinatio plani FI, cui perpendicularis e&longs;t recta <lb/>AF. <!-- KEEP S--></s> <s id="s.000791">Plus igitur momenti ad gravitandum habet glo-<pb pagenum="98" xlink:href="017/01/114.jpg"/>bus F, &longs;i ex breviore funiculo MF pendeat, quàm &longs;i ex <lb/>longiore AF. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000792">Quæ cum ita &longs;int, haud &longs;anè incongrua &longs;e nobis offert me­<lb/>thodus pondus ex depre&longs;&longs;iore in altiorem locum transferendi; <lb/>&longs;i videlicet id curemus, ut ex &longs;atis valido & longiore fune &longs;u&longs;­<lb/>pendatur; &longs;ublato etenim partium attritu, qui fieret, &longs;i per pla­<lb/>num raptaretur pondus, minore virium jacturâ trahi pote&longs;t. </s> </p> <p type="main"> <s id="s.000793">Sit corpus grave ubi A, quod at­<lb/><figure id="id.017.01.114.1.jpg" xlink:href="017/01/114/1.jpg"/><lb/>tollere oporteat, & in &longs;uperiorem <lb/>locum RS transferre. </s> <s id="s.000794">Si ex C brevio­<lb/>ri fune &longs;u&longs;pendatur, trahere illud po­<lb/>terit u&longs;que in R, quicumque facto de­<lb/>clinationis angulo ACR pote&longs;t illud <lb/>cum aliquo virium exce&longs;&longs;u retinere, <lb/>& ob&longs;i&longs;tere gravitatis momentis, quæ <lb/>obtinet in R. <!-- KEEP S--></s> <s id="s.000795">At &longs;i ex longiore fune <lb/>DA pendeat, idem corpus A trahi <lb/>poterit, & retineri in S, ne deor&longs;um labatur, & quidem mino­<lb/>re conatu; facto enim declinationis angulo ADS minore, <lb/>quàm ACR, in S pariter minùs gravitat quàm in R. <!-- KEEP S--></s> <s id="s.000796">Angu­<lb/>lum autem ADS minorem e&longs;&longs;e angulo ACR con&longs;tat, &longs;i rectæ <lb/>AR, AS ducantur: nam CA, CR æqualia &longs;unt latera ex hy­<lb/>pothe&longs;i, item DA, DS æqualia; e&longs;t &longs;cilicet idem funiculus, <lb/>qui primum perpendicularis cadit, deinde à perpendiculo re­<lb/>movetur: in Triangulo I&longs;o&longs;cele CAR anguli ad ba&longs;im AR <lb/>æquales &longs;unt per 5. lib. 1. item in triangulo I&longs;o&longs;cele DAS an­<lb/>guli ad ba&longs;im AS æquales inter &longs;e &longs;unt. </s> <s id="s.000797">Porrò angulus DAS <lb/>major e&longs;t angulo CAR; ergo & reliquus DSA major reliquo <lb/>CRA. </s> <s id="s.000798">Cum itaque tres anguli utriu&longs;que trianguli &longs;int æquales <lb/>duobus Rectis per 32. lib. 1. &longs;i ex &longs;ummâ duorum Rectorum au­<lb/>ferantur duo majores anguli DAS, DSA, relinquitur ADS <lb/>minor, quàm &longs;i ex eâdem duorum Rectorum &longs;ummâ auferan­<lb/>tur duo minores CAR, CRA, hoc e&longs;t minor quàm ACR. </s> <lb/> <s id="s.000799">Ut autem clariùs innote&longs;cat, quænam &longs;it gravitationum Ratio <lb/>pro funiculi longitudine, &longs;it corpus grave in R: & primùm <lb/>quidem ex C pendeat funiculo breviore CR, deinde ex D lon­<lb/>giore funiculo DR: qui&longs;quis retineat corpus in R con&longs;titu­<lb/>tum, atque de&longs;cen&longs;u prohibeat, faciliùs retinebit, cum ex D, <pb pagenum="99" xlink:href="017/01/115.jpg"/>quàm cùm ex C, pendebit; quia declinationis angulus XCR <lb/>major e&longs;t angulo XDR per 16. lib. 1. Verùm qua Ratione, in­<lb/>quis, vires, quas in utroque ca&longs;u retinens exerit, di&longs;criminan­<lb/>tur? </s> <s id="s.000800">utique &longs;ecundùm Reciprocam funiculorum Rationem co­<lb/>natur ob&longs;i&longs;tens corporis propen&longs;ioni ad de&longs;cen&longs;um; quæ enim <lb/>Ratio gravitationum corporis, ea e&longs;t virium gravitationibus <lb/>repugnantium: comparatà autem corporis in R con&longs;tituti gra­<lb/>vitatione, &longs;i ex C pendeat, cum eju&longs;dem ibidem po&longs;iti gravita­<lb/>tione, &longs;i pendeat ex D, e&longs;t reciprocè ut DR ad CR; igitur <lb/>& vires retinentis corpus ex C pendens &longs;unt ut DR, retinen­<lb/>tis verò idem corpus ex D pendens &longs;unt ut CR. </s> <s id="s.000801">Id quod hinc <lb/>conficitur, quia corpus in &longs;u&longs;pen&longs;ione, po&longs;itionem habens CR, <lb/>gravitat ut XR ad RC, po&longs;itionem verò habens DR gravitat <lb/>ut XR ad RD; duæ autem Rationes XR ad RC, & XR ad <lb/>RD &longs;unt reciprocè ut RD ad RC. </s> <s id="s.000802">Quotie&longs;cumque enim duæ <lb/>&longs;unt Rationes, quarum idem e&longs;t Antecedens terminus, & di­<lb/>ver&longs;us Con&longs;equens, eæ &longs;unt reciprocè ut con&longs;equentes. </s> </p> <p type="main"> <s id="s.000803">Quòd &longs;i quis Rationes inter &longs;e comparare non a&longs;&longs;uetus de <lb/>hoc ambigeret, an Rationes eumdem vel æqualem anteceden­<lb/>tem terminum habentes &longs;int reciprocè ut Con&longs;equentes, facilè <lb/>intelliget, &longs;i animadvertat Rationes eumdem Con&longs;equentem <lb/>terminum habentes e&longs;&longs;e inter &longs;e directè, ut antecedentes. </s> <lb/> <s id="s.000804">Quemcumque enim interrogaveris, quæ &longs;it Ratio 2/7 ad 6/7 illicò <lb/>re&longs;pondebit e&longs;&longs;e &longs;ubtriplam, &longs;ecunda &longs;cilicet ter continet pri­<lb/>mam, ut con&longs;tat &longs;i ter po&longs;itam Rationem 2/7 in &longs;ummam colligas; <lb/>neque enim <expan abbr="id&etilde;">idem</expan> e&longs;t Rationem Rationis e&longs;&longs;e &longs;ubtriplam, ac &longs;ub­<lb/>triplicatam; Ratio &longs;iquidem 2/7 e&longs;t &longs;ubtriplicata Rationis (8/343). Si <lb/>igitur pariter quæras, quænam &longs;it Ratio 7/2 ad 7/6 rectè re&longs;ponde­<lb/>bit eam e&longs;&longs;e triplam, hoc e&longs;t reciprocè ut 6 ad 2: id quod ma­<lb/>nife&longs;tè apparebit, &longs;i illas ad denominationem eandem, hoc e&longs;t <lb/>ad eumdem Con&longs;equentem terminum reduxeris, &longs;unt nimirum <lb/>ut (42/12) ad (14/12), hoc e&longs;t ut 6 ad 2. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000805">Ex quibus obiter patet methodus exponendi per lineas pro­<lb/>portionem duarum Rationum etiam numeris non explicabi­<lb/>lium; &longs;i videlicet fiat ut Antecedens &longs;ecundæ Rationis ad &longs;uum <lb/>Con&longs;equentem, ita Antecedens datus primæ Rationis ad alium <lb/>novum Con&longs;equentem; erit enim prima Ratio data ad &longs;ecun-<pb pagenum="100" xlink:href="017/01/116.jpg"/>dam rationem datam reciprocè ut novus Con&longs;equens terminus <lb/>ad datum Con&longs;equentem primæ Rationis: aut etiam &longs;i fiat ut <lb/>Con&longs;equens &longs;ecundæ Rationis ad &longs;uum Antecedentem, ita con­<lb/>&longs;equens primæ Rationis ad alium novum Antecedentem; erit <lb/>enim prima ratio data ad &longs;ecundam Rationem datam, directè <lb/>ut datus Antecedens primæ Rationis ad novum Antecedentem. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000806">Con&longs;ideratâ hactenus unicâ & &longs;implici corporis gravis &longs;u&longs;­<lb/>pen&longs;ione, gradum facere oportet ad gravitationis rationes in­<lb/>ve&longs;tigandas, &longs;i duplex fuerit &longs;u&longs;pen&longs;io. </s> <s id="s.000807">Sit enim globus A tùm <lb/><figure id="id.017.01.116.1.jpg" xlink:href="017/01/116/1.jpg"/><lb/>ex B, tùm ex C &longs;u&longs;pen&longs;us fu­<lb/>niculis BA & CA. <!-- KEEP S--></s> <s id="s.000808">Haud du­<lb/>bium quin tota corporis gravi­<lb/>tas ex B & C pendeat; &longs;ed quâ <lb/>Ratione &longs;ingulæ vires eidem <lb/>gravitati ob&longs;i&longs;tant, de hoc po­<lb/>te&longs;t ambigi. </s> <s id="s.000809">Verùm ni&longs;i mea <lb/>mihi nimiùm blanditur opi­<lb/>nio, ex dictis facilis videtur <lb/>explicatio. </s> <s id="s.000810">Corpus &longs;iquidem <lb/>ex duplici fune &longs;u&longs;pen&longs;um ita <lb/>con&longs;titutum e&longs;t, ut alterutro <lb/>fune præci&longs;o ex reliquo pen­<lb/>deat, & de&longs;cendens moveatur <lb/>circà punctum, cui alligatur <lb/>funis. </s> <s id="s.000811">Quare unu&longs;qui&longs;que ob&longs;i&longs;tit momentis, quibus ex altero <lb/>gravitat; nimirum funiculus CA retinens globum, ne de&longs;cen­<lb/>dat, repugnat momentis gravitatis, quibus globus A &longs;e ip&longs;e <lb/>deor&longs;um urget circa punctum B ex fune BA: Contrà verò fu­<lb/>niculus BA eundem globum retinet, ne circa punctum C ex <lb/>funiculo CA moveatur de&longs;cendens, atque adcò ob&longs;i&longs;tit, mo­<lb/>mentis gravitatis ad de&longs;cendendum circà idem punctum C. <!-- KEEP S--></s> <s id="s.000812">At­<lb/>qui momenta de&longs;cendendi ex fune BA ad gravitatem in per­<lb/>pendiculo &longs;unt ut DA ad AB, & ex fune CA &longs;unt ut EA ad <lb/>AC, ex his, quæ &longs;uperiùs di&longs;putata &longs;unt. </s> <s id="s.000813">Sunt igitur duæ Ra­<lb/>tiones DA ad AB, & EA ad AC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000814">Quare fiat angulus DAF æqualis angulo EAC, & e&longs;t trian­<lb/>gulum DAF ob angulorum æqualitatem &longs;imile triangulo <lb/>EAC; ac propterea per 4. lib. 6. ut EA ad AC, ita DA ad <pb pagenum="101" xlink:href="017/01/117.jpg"/>AF. <!-- KEEP S--></s> <s id="s.000815">Ergo vis de&longs;cendendi ex CA e&longs;t ut DA ad AF, & vis <lb/>de&longs;cendendi ex BA e&longs;t ut DA ad AB: igitur duæ hæ Ratio­<lb/>nes &longs;unt reciprocè ut BA ad AF; atque adeò B quidem reti­<lb/>nens, ne de&longs;cendat ex CA, exerit vires ut BA; C verò reti­<lb/>nens, ne de&longs;cendat ex BA, adhibet conatum ut FA; & quæ <lb/>componitur ex BA, AF, totum gravitatis momentum, quod <lb/>corpori &longs;u&longs;pen&longs;o ine&longs;t, repræ&longs;entat. </s> <s id="s.000816">Momentum, inquam, <lb/>gravitatis potiùs, quàm gravitatem totam; totius &longs;i quidem <lb/>gravitatis nomine vires ip&longs;as de&longs;cendendi intelligimus, quas <lb/>corpus grave obtinet &longs;ibi prorsùs relictum &longs;eclu&longs;o quolibet im­<lb/>pedimento, à quo certam de&longs;cendendi regulam accipiat: Mo­<lb/>menti autem vocabulo ip&longs;as de&longs;cendendi vires &longs;ignificamus <lb/>non per &longs;e & &longs;olitariè acceptas; &longs;ed quatenus ex corporis po&longs;i­<lb/>tione, cæterorumque quæ circum&longs;tant, ad majorem aut mino­<lb/>rem motùs velocitatem determinatur. </s> <s id="s.000817">Con&longs;iderato itaque ni&longs;u <lb/>corporis A ad de&longs;cendendum & cùm perpendicularis e&longs;t funi­<lb/>culus BD, & cum declinat BA, Ratio momentorum e&longs;t ut <lb/>BA ad AD. <!-- KEEP S--></s> <s id="s.000818">Similiter momentum ex perpendiculari CE ad <lb/>momentum ex declinante CA e&longs;t ut CA ad AE, hoc e&longs;t ut <lb/>FA ad AD: e&longs;t igitur corporis A ex duplici funiculo BA, CA <lb/>pendentis totum gravitandi momentum, quod ex lineis BA, <lb/>AF componitur. </s> </p> <p type="main"> <s id="s.000819">Hic autem hæ&longs;itantem videre mihi videor non neminem ex <lb/>iis, quæ dicebantur, colligentem corpus A primùm ex decli­<lb/>nante BA æquè ac ex perpendiculari BD gravitare; deinde <lb/>plus ad de&longs;cendendum momenti obtinere, &longs;i ex duobus funi­<lb/>culis, quàm &longs;i ex unico pendeat. </s> <s id="s.000820">Si enim angulus declinatio­<lb/>nis DBA &longs;it gr. <!-- REMOVE S-->22. 12′; e&longs;t DA &longs;inus dati anguli ad radium <lb/>BA ut 37784 ad 100000: & &longs;i angulus declinationis ECA <lb/>&longs;it gr. <!-- REMOVE S-->54. 35, e&longs;t EA &longs;inus dati anguli ad Radium CA ut <lb/>81496 ad 100000. At ex con&longs;tructione triangulum DAF &longs;i­<lb/>mile e&longs;t triangulo EAC; igitur DA ad AF e&longs;t ut 81496 ad <lb/>100000. E&longs;t autem DA in particulis Radij BA partium 37784; <lb/>igitur &longs;i fiat ut 81496 ad 100000, ita 37784, ad aliud, erit AF <lb/>earumdem particularum 46363, quarum BA e&longs;t 100000. Qua­<lb/>re compo&longs;ita BA, AF momenta &longs;unt 146363, cum tamen <lb/>momentum in perpendiculari AD &longs;it tantum 100000. Cum <lb/>verò dictum &longs;it B clavum re&longs;i&longs;tere ponderi A ut BA, C autem <pb pagenum="102" xlink:href="017/01/118.jpg"/>ut FA, manife&longs;tum e&longs;t B clavum retinere ut 100000 quando <lb/>declinat BA à perpendiculo: Atqui etiam in perpendiculo BD <lb/>retinet ut 100000, igitur idem e&longs;t ponderis tùm ex BD, tùm <lb/>ex BA momentum; id quod e&longs;t ab&longs;urdum. </s> </p> <p type="main"> <s id="s.000821">Sed & illud prætereà ex dictis con&longs;equi videtur, quod eju&longs;­<lb/>dem corporis majus &longs;it momentum, &longs;i ex duobus funiculis, quàm <lb/>&longs;i ex unico pendeat. </s> <s id="s.000822">Fiat enim angulus DBH æqualis angulo <lb/>declinationis ECA, & a&longs;&longs;umptâ BH æquali ip&longs;i BA, ducatur <lb/>ad BD perpendicularis HI: erit utique triangulum BHI &longs;imi­<lb/>le triangulo CAE, ac propterea ut EA ad AC, ita IH ad <lb/>HB, hoc e&longs;t ad AB. <!-- KEEP S--></s> <s id="s.000823">Sunt igitur duæ Rationes eundem Con­<lb/>&longs;equentem terminum habentes, atque adeò inter &longs;e in ratione <lb/>Antecedentium, ac proinde cùm vis de&longs;cendendi ex BA &longs;it ut <lb/>DA ad AB, & vis de&longs;cendendi ex CA &longs;it ut IH ad AB, vires <lb/>de&longs;cendendi invicem comparatæ &longs;unt ut DA ad IH, totum­<lb/>que momentum componitur ex DA 37784, & IH 81496. <lb/>Quare momentum quod in perpendiculari, &longs;i unico funiculo <lb/>penderet ex BD, e&longs;&longs;et 100000, pendente corpore A ex duo­<lb/>bus funiculis BA, CA, fit majus, &longs;cilicet 119280. ut quid igi­<lb/>tur ex pluribus funiculis illud &longs;u&longs;pendere oportuit? </s> </p> <p type="main"> <s id="s.000824">Quibus difficultatibus ut fiat &longs;atis, & id, quod inquirimus, <lb/>enucleatiùs explicetur, illud ob&longs;ervo, quod funiculus BA &longs;i <lb/>præcisè &longs;pectetur, quatenus ex eo corpus grave pendet, retinet <lb/>globum A, ne rectâ de&longs;cendat per lineam ip&longs;i BD parallelam, <lb/>&longs;ed cogit illum deflectere in motu: quare adversùs clavum B, <lb/>globus A exercet ea momenta, quæ exerceret in planum incli­<lb/>natum, cui BA ad rectos angulos in&longs;i&longs;teret. </s> <s id="s.000825">At &longs;i globus ex alio <lb/>prætereà funiculo CA pendeat, idem funiculus BA re&longs;i&longs;tit <lb/>etiam momentis illis, quibus globus A de&longs;cenderet in plano in­<lb/>clinato, cui CA ad rectos angulos in&longs;i&longs;teret, quæ momenta (ut <lb/>&longs;ummum) &longs;unt ad BA radium ut 81496. Momenta verò qui­<lb/>bus urgeret planum inclinatum perpendiculare ad BA, &longs;unt, ex <lb/>dictis &longs;uperiori capite, ut Sinus Ver&longs;us anguli inclinationis pla­<lb/>ni; inclinatio autem plani, ut paulò &longs;uperiùs hoc eodem capite <lb/>demon&longs;travimus, e&longs;t complementum anguli declinationis <lb/>DBA. </s> <s id="s.000826">Quare differentia inter DA 37784 &longs;inum rectum an­<lb/>guli declinationis, & radium BA 100000, cum &longs;it Sinus Ver­<lb/>&longs;us anguli inclinationis plani, &longs;unt momenta 62216 addenda <pb pagenum="103" xlink:href="017/01/119.jpg"/>prioribus 81496; adeò ut &longs;umma &longs;it 143712 momentorum, qui­<lb/>bus funiculus BA repugnat, &longs;i pondus pendeat etiam ex CA; <lb/>cum tamen &longs;i ex ip&longs;o tantùm funiculo BA penderet, & aliquis <lb/>e&longs;&longs;et præcisè obluctans viribus ad de&longs;cendendum, idem funicu­<lb/>lus BA re&longs;i&longs;teret &longs;olùm momentis 62216. </s> </p> <p type="main"> <s id="s.000827">Eâdem methodo deprehendes funiculum CA, &longs;i ex eo &longs;olo <lb/>globus pendeat, retinere momenta 18504: at &longs;i etiam ex BA <lb/>globus pendeat, additis momentis 37784, tota momentorum <lb/>&longs;umma e&longs;t 56288. Jam &longs;ummam hanc priori 143712 adde, & <lb/>erit tota momentorum &longs;umma 200000: perinde atque &longs;i corpo­<lb/>ris gravitas fui&longs;&longs;et duplicata. </s> <s id="s.000828">Id quod deprehendes, quo&longs;cum­<lb/>que demùm declinationis angulos &longs;tatueris &longs;ivè majores, &longs;ivè <lb/>minores; &longs;emper enim eandem &longs;ummam momentorum om­<lb/>nium invenies 200000: & funiculus minoris declinationis plus <lb/>momentorum &longs;u&longs;tinebit, tùm quia Sinus Ver&longs;us majoris incli­<lb/>nationis plani major e&longs;t, tum quia Sinus Rectus alterius anguli <lb/>declinationis majoris item major e&longs;t. </s> </p> <p type="main"> <s id="s.000829">Hæc tamen ut veritati congruant, ita &longs;olùm accipienda &longs;unt, <lb/>ut momenta &longs;ingula ex utrâque funiculorum declinatione orta <lb/>particulatim &longs;umantur: pondus &longs;cilicet ex utroque &longs;u&longs;pen&longs;um <lb/>perinde hactenus con&longs;ideratum e&longs;t, ac &longs;i momenta ip&longs;a de&longs;cen­<lb/>dendi in diver&longs;as partes abeuntia momentum quoddam ex <lb/>utri&longs;que temperatum non con&longs;tituerent; re autem ipsa quod ex <lb/>iis componitur momentum, non ex ip&longs;orum momentorum ad­<lb/>ditione conflatur, &longs;ed ex ip&longs;is temperatur. </s> <s id="s.000830">Si enim mobile &longs;it <lb/>ubi A, impetum verò cum tali <lb/><figure id="id.017.01.119.1.jpg" xlink:href="017/01/119/1.jpg"/><lb/>directione habeat, quâ deferri <lb/>po&longs;&longs;it æquabiliter per rectam <lb/>AB, alio autem impetu feratur <lb/>æquabiliter directum in C, no­<lb/>tum omnibus e&longs;t motum, qui ex <lb/>AB & AC componitur, non fieri ex earum additione, &longs;ed tem­<lb/>perari in lineam AD, quæ dimetiens e&longs;t parallelogrammi, quod <lb/>ex earumdem linearum AB, AC longitudine, ac mutuâ incli­<lb/>natione formam de&longs;umit. </s> <s id="s.000831">Quâ in re plurimum intere&longs;t, quam <lb/>invicem habeant inclinationem directiones motuum in diver&longs;a <lb/>abeuntium; quò enim acutiorem angulum con&longs;tituunt, eò lon­<lb/>giùs provehitur mobile, ut AB, AC in acutum angulum <pb pagenum="104" xlink:href="017/01/120.jpg"/>coëuntibus mobile ex A in D venit: quò verò obtu&longs;ior fuerit <lb/>angulus, eò etiam brevius e&longs;t iter ip&longs;ius mobilis, ut contingit, <lb/>&longs;i ex B directum per rectas BA, BD ad obtu&longs;um angulum <lb/>con&longs;titutas moveatur, &longs;i&longs;titur enim in C, & brevior e&longs;t diame­<lb/>ter BC quàm AD, ut ex 24. lib. 1. &longs;atis manife&longs;tum e&longs;t geo­<lb/>metris, & ip&longs;a motuum natura po&longs;tulat; qui nimirum &longs;ibi in­<lb/>vicem magis adver&longs;antur, magi&longs;que in diver&longs;a abeunt, &longs;e ma­<lb/>gis elidunt, id quod fit ex angulo obtu&longs;o DBA; qui verò mi­<lb/>nùs in diver&longs;a abeunt, id quod fit ex angulo acuto CAB, &longs;e pa­<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/>dus &longs;u&longs;penditur: ducatur ad BA perpendicularis AR, & e&longs;t <lb/>planum inclinatum, in quo de&longs;cendendi momentum e&longs;t ut <lb/>DA; &longs;imiliter ad CA perpendicularis AG ducatur referens <lb/>planum inclinatum, in quo de&longs;cendendi momentum e&longs;t AE. <!-- KEEP S--></s> <lb/> <s id="s.000833">Sumatur igitur AR quidem ip&longs;i AD æqualis, AG verò ip&longs;i <lb/>AE pariter æqualis, &longs;i funiculi BA, & CA æquales fuerint; <lb/>&longs;in autem inæquales &longs;int, fiat angulus DBH æqualis angulo <lb/>declinationis ECA, & &longs;umptâ BH æquali ip&longs;i BA, duca­<lb/>tur ad BD perpendicularis HI, eritque ut EA ad AC, <lb/>ita IH ad HB, hoc e&longs;t ad AB; ac propterea ip&longs;i IH, quæ <lb/>refert momentum AE, &longs;umatur AG æqualis. </s> <s id="s.000834">Ex quo fit cor­<lb/>pus A &longs;u&longs;pen&longs;um hâc ratione momenta de&longs;cendendi habe­<lb/>re in diver&longs;as partes abeuntia AR, AG: perfecto igitur paral­<lb/>lelogrammo ARNG, ex duobus illis momentis temperatur <lb/>momentum AN. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000835">Ip&longs;ius autem AN longitudinem inve&longs;tigare non e&longs;t diffici­<lb/>le; cum enim noti &longs;upponantur anguli declinationum DBA, <lb/>ECA, angulus RAG conflatur ex eorum complementis, <lb/>quippe qui æqualis e&longs;t duobus angulis inclinationis planorum <lb/>AR, & AG. <!-- KEEP S--></s> <s id="s.000836">Porrò ex hypothe&longs;i &longs;unt angulus DBA gr. <!-- REMOVE S-->22. <lb/>12′, & angulus ECA gr. <!-- REMOVE S-->54. 35′: jungantur &longs;imul, & eorum <lb/>&longs;umma gr. <!-- REMOVE S-->76. 47′ auferatur ex gr. <!-- REMOVE S-->180, ut re&longs;iduum gr. <!-- REMOVE S-->103. <lb/>13′ &longs;it angulus RAG, cui æqualis e&longs;t oppo&longs;itus RNG; ac <lb/>proinde notus e&longs;t angulus G, qui e&longs;t &longs;uo oppo&longs;ito R æqualis, <lb/>uterque &longs;cilicet gr. <!-- REMOVE S-->76. 47′ quæ e&longs;t &longs;umma angulorum decli­<lb/>nationis. </s> <s id="s.000837">Sunt igitur in triangulo AGN nota latera AG, <lb/>GN (e&longs;t enim ex 34. lib. 1. GN oppo&longs;ito lateri AR æquale) <pb pagenum="105" xlink:href="017/01/121.jpg"/>unâ cum angulo G comprehen&longs;o, & ex Trigonometriâ inno­<lb/>te&longs;cit tertium latus AN. <!-- KEEP S--></s> <s id="s.000838">Quare cum latus AG &longs;it ex &longs;upe­<lb/>riùs con&longs;titutis 81496, & GN, hoc e&longs;t AR, 37784, fiat ut <lb/>laterum AG, GN &longs;umma 119280 ad eorumdem differen­<lb/>tiam 43712, ita &longs;emi&longs;ummæ angulorum ad ba&longs;im, hoc e&longs;t <lb/>gr. <!-- REMOVE S-->51. 36 1/2 Tangens 126205 ad 46249 Tangentem gr. <!-- REMOVE S-->24. 49′ 2/5 <lb/>differentiæ infra, vel &longs;upra eandem &longs;emi&longs;ummam. </s> <s id="s.000839">E&longs;t igitur <lb/>angulus GAN gr. <!-- REMOVE S-->26. 47′ (3/10). In triangulo itaque AGN noti <lb/>&longs;unt duo anguli A, & G, ac latus GN angulo A oppo&longs;i­<lb/>tum; igitur ut anguli A gr. <!-- REMOVE S-->26. 47′ (3/10) Sinus 45070 ad anguli G <lb/>gr. <!-- REMOVE S-->76. 47′ Sinum 97351, ita latus GN 37784 ad latus AN <lb/>81613. </s> </p> <p type="main"> <s id="s.000840">Ex quibus apparet de&longs;cendendi momentum, quod compo­<lb/>nitur ex momentis in planis inclinatis, non e&longs;&longs;e 119280 ex eo­<lb/>rum &longs;ummâ, &longs;ed ita temperari, ut longè minus &longs;it, videlicet &longs;o­<lb/>lùm 81613. </s> </p> <p type="main"> <s id="s.000841">Methodo eâdem operantes deprehendemus ponderis in H <lb/>con&longs;tituti, ac ex funiculis BH, CH &longs;u&longs;pen&longs;i momentum ita <lb/>componi ex momento HI bis &longs;umpto (&longs;i quidem anguli decli­<lb/>nationum DBH, ECH & funiculi æquales &longs;int) ut in unum <lb/>ex utroque nimirum HI & HO temperatum HS coale&longs;cat. </s> <lb/> <s id="s.000842">Unde con&longs;tabit quò majores fuérint declinationum anguli, eò <lb/>longiorem futuram lineam HS, atque adeò etiam majus mo­<lb/>mentum de&longs;cendendi; plana &longs;iquidem inclinata acutiorem <lb/>angulum con&longs;tituunt. </s> <s id="s.000843">Quam momentorum varietatem pau­<lb/>lò inferiùs manife&longs;to experimento comprobabimus: ubi con&longs;ta­<lb/>bit pondus hâc ratione &longs;u&longs;pen&longs;um ex duobus funiculis plus ha­<lb/>bere aliquando momenti ad de&longs;cendendum, quàm in perpen­<lb/>diculari &longs;u&longs;pen&longs;ione. </s> </p> <p type="main"> <s id="s.000844">Quemadmodum verò de momentis de&longs;cendendi in planis <lb/>inclinatis ratiocinati &longs;umus, ita pariter in unum coale&longs;cere di­<lb/>cenda &longs;unt momenta, quibus funiculi pondus retinentes ip&longs;um <lb/>quodammodo avellere conantur à plano inclinato, ne illud ur­<lb/>geat; hæc enim pariter momenta in diver&longs;a abeunt &longs;ecun­<lb/>dùm ip&longs;am funiculorum directionem. </s> <s id="s.000845">Sunt autem momenta <lb/>illa Sinus Ver&longs;i angulorum inclinationis planorum; qui haben­<lb/>tur, &longs;i Sinus Recti complementorum, hoc e&longs;t angulorum de-<pb pagenum="106" xlink:href="017/01/122.jpg"/><figure id="id.017.01.122.1.jpg" xlink:href="017/01/122/1.jpg"/><lb/>clinationis funiculorum, de­<lb/>mantur ex Radio. <!-- KEEP S--></s> <s id="s.000846">Itaque ex <lb/>BA auferatur BF ip&longs;i DA <lb/>æqualis, & e&longs;t FA Sinus Ver­<lb/>&longs;us anguli inclinationis: po&longs;ita <lb/>e&longs;t autem declinatio DBA <lb/>gr.22. 12′, igitur FA e&longs;t parti­<lb/>cularum 62216; & declinatio <lb/>ECA gr. <!-- REMOVE S-->54. 35′; igitur factâ <lb/>CG æquali ip&longs;i AE, remanet <lb/>GA particularum 18504, quarum CA e&longs;t 100000. Quare ut <lb/>habeantur particulæ eju&longs;dem rationis cum particulis AF, fiat <lb/>ut CA ad AG, ita BA ad AH, & e&longs;t AH particularum 18504 <lb/>homologarum particulis AF. <!-- KEEP S--></s> <s id="s.000847">Perficiatur parallelogrammum <lb/>AHIF; & quia funiculus CA retrahit à plano inclinato juxta <lb/>momentum ac directionem HA, funiculus verò BA retrahit à <lb/>plano inclinato &longs;ecundùm momentum ac directionem FA, di­<lb/>rectionibus in diver&longs;a abeuntibus, temperatur ex his momentis <lb/>momentum AI diameter parallelogrammi. </s> </p> <p type="main"> <s id="s.000848">Porrò in diametri AI inve&longs;tigatione methodus e&longs;t eadem, <lb/>quâ paulò antè utebamur: Cum enim tres anguli BAD, BAC, <lb/>CAE &longs;int duobus Rectis æquales, anguli verò BAD, CAE <lb/>noti &longs;int, quippe complementa angulorum declinationis DBA, <lb/>ECA, innote&longs;cit reliquus FAH, qui æqualis e&longs;t &longs;ummæ an­<lb/>gulorum declinationis. </s> <s id="s.000849">E&longs;t igitur FAH gr.76.47′, ac proinde <lb/>angulus AFI gr.103.13′ notus e&longs;t, unâ cum lateribus FA 62216 <lb/>& FI 18504. Fiat igitur ut laterum &longs;umma 80720 ad eorum­<lb/>dem differentiam 43712, ita angulorum ad ba&longs;im AI &longs;emi&longs;um­<lb/>mæ gr. <!-- REMOVE S-->38. 23′1/2. Tangens 79235 ad 42907 Tangentem dif­<lb/>ferentiæ infra vel &longs;upra eandem &longs;emi&longs;ummam, hoc e&longs;t gr. <!-- REMOVE S-->23. <lb/>13′.1/2 dempta igitur hæc differentia ex &longs;emi&longs;&longs;ummâ gr.38.23′ 1/2, <lb/>reliquum facit angulum FAI gr.15.10′. <!-- KEEP S--></s> <s id="s.000850">Fiat demùm ut anguli <lb/>FAI gr.15.10′. <!-- KEEP S--></s> <s id="s.000851">Sinus 26163 ad anguli AFI gr. <!-- REMOVE S-->103. 13′. </s> <s id="s.000852">hoc e&longs;t <lb/>ad &longs;upplementi gr.76.47′. <!-- REMOVE S-->Sinum 97351, ita latus FI 18504 <lb/>ad ba&longs;im AI 68852. </s> </p> <p type="main"> <s id="s.000853">Inventa itaque momenta compo&longs;ita tùm in planis inclinatis, <lb/>tùm in plana inclinata, dividantur juxta Rationem momento-<pb pagenum="107" xlink:href="017/01/123.jpg"/>rum &longs;implicium, ut innote&longs;cat, quid demum cuique funicolo <lb/>tribuendum &longs;it in pondere retinendo. </s> <s id="s.000854">Momentum de&longs;cenden­<lb/>di compo&longs;itum inventum e&longs;t &longs;u&longs;periùs 81613, &longs;implicia &longs;unt <lb/>81496, & 37784. Fiat ut igitur ut &longs;implicium momentorum <lb/>&longs;umma 119280 ad eorum alterutrum, puta ad 37784, ita mo­<lb/>mentum compo&longs;itum 81613 ad aliud, & provenit 25852 pars <lb/>illius momenti pertinens ad funiculum CA, qui retinet pon­<lb/>dus; cujus vis de&longs;cendendi e&longs;t DA 37784. Reliqua autem mo­<lb/>menti 81613 pars 55761 pertinet ad funiculum BA retinentem <lb/>pondus, cujus vis de&longs;cendendi e&longs;t EA 81496. Pari ratione fiat <lb/>ut Sinuum Ver&longs;orum angulorum inclinationis &longs;implicium <lb/>62216, atque 18504 &longs;umma 80720 ad eorum alterutrum, pu­<lb/>ta ad 18504, ita momentum compo&longs;itum inventum 68852 ad <lb/>aliud, & provenit pro minori 15783, pro majori verò 53069. <lb/>Quare funiculus BA minorem habens declinationem, & plus <lb/>&longs;u&longs;tinet in &longs;uo plano magis inclinato, cui perpendicularis e&longs;t, <lb/>nimirum ut 53069, & plus retinet in plano reliquo minùs in­<lb/>clinato, nimirum ut 55761: contra verò funiculus CA, & mi­<lb/>nus &longs;u&longs;tinet, &longs;cilicet ut 15783, & minus retinet &longs;cilicet ut <lb/>25852. Funiculus itaque BA exercet vires ut 108830, & fu­<lb/>niculus CA ut 41635, & totum corporis &longs;u&longs;pen&longs;i momentum <lb/>e&longs;t 150465. </s> </p> <p type="main"> <s id="s.000855">Non &longs;ola autem momenta de&longs;cendendi in planis inclinatis <lb/>con&longs;iderari oportere, &longs;ed & ea, quæ e&longs;&longs;ent adversùs plana <lb/>ip&longs;a inclinata, uti dictum e&longs;t, ex eo apertè conficitur, quòd <lb/>ubi funiculi concurrerent ad acuti&longs;&longs;imum angulum, vix quic­<lb/>quam virium in retinendo pondere exercere opus e&longs;&longs;et; te­<lb/>nui&longs;&longs;imum quippe, e&longs;&longs;et momentum, quod ex parvis mo­<lb/>mentis per acuti&longs;&longs;imorum angulorum Sinus Rectos definitis <lb/>componeretur: &longs;i verò nihil præterea momenti addendum e&longs;­<lb/>&longs;et; à magnâ gravitatione, quæ in perpendiculari e&longs;t, ad ferè <lb/>nullam tran&longs;itus e&longs;&longs;et, facta vel modicâ à perpendiculo decli­<lb/>natione; atque adeò vix intenti e&longs;&longs;e deberent funiculi: id quod <lb/>manife&longs;to experimento adver&longs;atur. </s> </p> <p type="main"> <s id="s.000856">Illud po&longs;tremò hîc o&longs;tendendum &longs;upere&longs;t, plus &longs;cilicet in­<lb/>e&longs;&longs;e po&longs;&longs;e momenti ad de&longs;cendendum corpori ex duobus funi­<lb/>culis invicem inclinatis &longs;u&longs;pen&longs;o, quàm &longs;i ex unico ad per­<lb/>pendiculum pendeat. </s> <s id="s.000857">Orbiculo circà &longs;uum axem C ver&longs;atili, <pb pagenum="108" xlink:href="017/01/124.jpg"/><figure id="id.017.01.124.1.jpg" xlink:href="017/01/124/1.jpg"/><lb/>ac &longs;ecundùm extremam <lb/>oram excavato, in&longs;eratur <lb/>funiculus AFB, ex quo <lb/>æqualia hinc, & hinc <lb/>pondera A, & B pen­<lb/>deant: nullus planè &longs;e­<lb/>quitur motus, quia utrum­<lb/>que ex perpendiculo pen­<lb/>det, & quantâ vi alterum conatur deor&longs;um, pari nu&longs;u alterum <lb/>repugnat, ne elevetur. </s> <s id="s.000858">Quærenti igitur, quantum momenti <lb/>pondus B habeat ad de&longs;cendendum, utique re&longs;pondebis omni­<lb/>nò par e&longs;&longs;e momento ponderis A. <!-- KEEP S--></s> <s id="s.000859">Jam verò &longs;it funiculus AFD, <lb/>qui in D religetur, & ponderi A &longs;umatur æquale pondus E, <lb/>vel potiùs ip&longs;um B transferatur in E, & funiculo AFD ad­<lb/>nectatur in H; ut &longs;int qua&longs;i duo funiculi DH, FH. <!-- KEEP S--></s> <s id="s.000860">Quæro <lb/>quantum ad de&longs;cendendum momenti habeat pondus E, hoc e&longs;t <lb/>pondus B in H tran&longs;latum, quod e&longs;t æquale ponderi A: &longs;i tan­<lb/>tumdem habet momenti, quantum pondus A, planè manebit <lb/>immotum, intento funiculo FD; at &longs;i E de&longs;cendens cogat <lb/>a&longs;cendere pondus A, utique plus momenti habet quàm A, hoc <lb/>e&longs;t, plu&longs;quàm B perpendiculariter pendens. </s> <s id="s.000861">Id quod re ipsâ <lb/>contingit; & quidem tàm certo experimento, ut non &longs;olùm <lb/>pondus E prævaleat ponderi A, &longs;i &longs;it ei æquale, verùm etiam &longs;i <lb/>minus &longs;it eodem pondere A. <!-- KEEP S--></s> <s id="s.000862">Non igitur hoc ab&longs;urdum e&longs;t, <lb/>quod con&longs;titutam à nobis momentorum hypothe&longs;im con&longs;equa­<lb/>tur, &longs;ed potiùs ip&longs;i naturæ no&longs;tra con&longs;entit hypothe&longs;is, cui ro­<lb/>bur adjicit experientia; nec ex eo capite perperam philo&longs;opha­<lb/>ti videmur, quòd in perpendiculo minus momenti, quàm ex <lb/>duplici funiculo &longs;u&longs;pen&longs;um pondus habere dicendum &longs;it. </s> </p> <p type="main"> <s id="s.000863">Ex his, quæ de corpore ex binis funiculis &longs;u&longs;pen&longs;o hactenus <lb/>di&longs;putata &longs;unt, non difficilis erit conjectura eorum, quæ dicen­<lb/>da &longs;int, &longs;i ex tribus aut quatuor &longs;u&longs;pendatur, &longs;ivè illi immedia­<lb/>tè adnectantur ip&longs;i ponderi, &longs;ivè funiculus unus demum in plu­<lb/>ra capita dividatur, ex quibus fiat &longs;u&longs;pen&longs;io: neque enim his <lb/>diutiùs ad nau&longs;eam immorandum cen&longs;eo. <pb pagenum="109" xlink:href="017/01/125.jpg"/></s> </p> <p type="main"> <s id="s.000864"><emph type="center"/>CAPUT XVI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.000865"><emph type="center"/><emph type="italics"/>Tractiones ac elevationes obliquæ expenduntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000866">PRoxima e&longs;t iis, quæ hactenus di&longs;putata &longs;unt, præ&longs;ens in­<lb/>ve&longs;tigatio gravitationis corporum, &longs;ive nisûs, quo motui <lb/>re&longs;i&longs;tunt, cùm obliquè in plano aliquo trahuntur, aut elevan­<lb/>tur: &longs;icut enim toto conatu repugnant elevanti ad perpendicu­<lb/>lum, & ab&longs;trahenti à plano, cui in&longs;ident, ita pro majori, aut <lb/>minori obliquitate tractionis aut elevationis magis etiam, aut <lb/>minùs, ob&longs;i&longs;tere experimur. </s> <s id="s.000867">Et primùm quidem &longs;uper plano <lb/><figure id="id.017.01.125.1.jpg" xlink:href="017/01/125/1.jpg"/><lb/>inclinato AB duo pondera <lb/>pror&longs;us æqualia, & &longs;imilia <lb/>intelligantur po&longs;ita in B <lb/>& C, atque linea CE &longs;it <lb/>horizonti BE perpendicu­<lb/>laris, ac pondus C filo DC <lb/>ad perpendiculum &longs;u&longs;pen­<lb/>datur, ita tamen, ut con­<lb/>tingat planum in C, & &longs;it <lb/>recta DE. <!-- KEEP S--></s> <s id="s.000868">Item ex D <lb/>puncto ducatur filum DB, <lb/>ut &longs;ur&longs;um trahatur B pon­<lb/>dus incumbens plano in­<lb/>clinato, dum pariter pon­<lb/>dus C &longs;ur&longs;um rectâ trahi­<lb/>tur, & à plano avellitur: horum autem funiculorum trahatur <lb/>ex D pars æqualis. </s> <s id="s.000869">Quando igitur C venerit in V, æquali men­<lb/>&longs;urâ BP multatum intelligitur filum DB, & remanet longi­<lb/>tudo DP, hoc e&longs;t DO; pondus enim, cum filum in D trahe­<lb/>retur, ex B venit in O. <!-- KEEP S--></s> <s id="s.000870">Ductâ itaque lineâ ON horizonti pa­<lb/>rallelâ, erit EN altitudo perpendicularis, ad quam a&longs;cendit <lb/>pondus B in plano inclinato interea, dum pondus C venit in V, <lb/>aut E venit in M, e&longs;t enim EM a&longs;&longs;umpta ip&longs;i CV æqualis. </s> <lb/> <s id="s.000871">Quare cum pondus B obliquè trahitur &longs;uper planum inclina-<pb pagenum="110" xlink:href="017/01/126.jpg"/>tum, minorem &longs;ubit violentiam, quàm cum ab illo perpendi­<lb/>culari elevatione avellitur. </s> </p> <p type="main"> <s id="s.000872">Hoc tamen ita intelligendum e&longs;t, ut ob&longs;ervetur alia e&longs;&longs;e <lb/>momenta, cùm tractionis linea parallela e&longs;t ip&longs;i plano inclina­<lb/>to, ac cùm in planum inclinatum cadit obliqua, ut hîc li­<lb/>nea DB. <!-- KEEP S--></s> <s id="s.000873">Si enim in plano inclinato &longs;umatur BR æqualis <lb/>perpendiculari EM, gravitatio per rectam BC, &longs;eu per li­<lb/>neam eidem parallelam, ad gravitationem in perpendiculo <lb/>CE e&longs;t reciprocè ut EC ad BC, &longs;eu ut ES ad BR aut EM, <lb/>ex &longs;uperiùs dictis cap.13. At verò cum tractio obliqua e&longs;t, <lb/>gravitatio e&longs;t ut EN ad EM, &longs;ivè ut BO ad BX: punctum <lb/>autem O altius e&longs;t puncto R, ac proptereà in huju&longs;modi <lb/>obliquâ tractione plus violentiæ infertur ponderi, quàm in <lb/>tractione parallelâ, plus enim a&longs;cendit. </s> <s id="s.000874">Porrò lineam BO <lb/>longiorem e&longs;&longs;e lineâ BR e&longs;t manife&longs;tum; &longs;iquidem duo la­<lb/>tera DO, OB per 20. lib.1. majora &longs;unt reliquo DB: e&longs;t <lb/>autem ex hypothe&longs;i DP ip&longs;i DO æqualis, ergo reliqua <lb/>BP minor e&longs;t, quàm BO: &longs;ed & ip&longs;i BP, hoc e&longs;t ip&longs;i <lb/>EM, æqualis a&longs;&longs;umpta e&longs;t BR; igitur BR minor e&longs;t quàm <lb/>BO. <!-- KEEP S--></s> <s id="s.000875">Id quod etiam hinc con&longs;tat, quia in triangulo I&longs;o­<lb/>&longs;cele DOP angulus OPB infra ba&longs;im major e&longs;t recto, <lb/>cum &longs;it deinceps angulo DPO ad ba&longs;im acuto; ergo per <lb/>19.lib.1. latus BO majus e&longs;t latere BP, hoc e&longs;t BR; igi­<lb/>tur etiam EN major e&longs;t quàm ES, & plus difficultatis <lb/>percipitur in obliquâ hâc tractione, quàm in tractione pa­<lb/>rallelà. </s> </p> <p type="main"> <s id="s.000876">Similiter intelligatur pondus C elevatum fui&longs;&longs;e ex D <lb/>(quod punctum D concipiatur multò altius, quàm in præ­<lb/>&longs;enti &longs;chemate) ad perpendiculum altitudine æquali ip&longs;i ET, <lb/>pondus verò B æquali tractione funiculi veni&longs;&longs;e ex B in G, <lb/>demptâ &longs;cilicet longitudine BF ip&longs;i ET æquali, atque <lb/>adeò DF, DG æquales &longs;unt: ip&longs;i autem ET æqualis &longs;u­<lb/>matur BI; quæ &longs;imili ratione demon&longs;tratur brevior, quàm <lb/>BG: ex quo pariter &longs;it hîc etiam ad majorem altitudi­<lb/>nem perpendicularem EH elevari, quàm &longs;i tractio pa­<lb/>rallela fui&longs;&longs;et plano inclinato, & elevatio ad altitudi­<lb/>nem EL. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000877">Ex his manife&longs;tum e&longs;t plus virium requiri ad trahendum <pb pagenum="111" xlink:href="017/01/127.jpg"/>pondus idem per lineam DB, aut DO, aut DG obli­<lb/>quas, quàm per lineam plani inclinati BC, aut illi paral­<lb/>lelam: dum enim per obliquas illas lineas fit tractio, pon­<lb/>dus quidem non omninò ab&longs;trahitur à plano, &longs;icut in tractio­<lb/>ne perpendiculari, &longs;ed nec omninò incumbit plano, &longs;i­<lb/>cut in tractione parallelâ ip&longs;i plano; ac propterea, quò ma­<lb/>gis tractio ad perpendicularem accedit, eò majorem inve­<lb/>nit in pondere re&longs;i&longs;tentiam. </s> <s id="s.000878">Patet autem altitudinum per­<lb/>pendicularium EH, EL differentiam HL majorem e&longs;&longs;e, <lb/>quàm &longs;it altitudinum perpendicularium EN, ES differen­<lb/>tia NS. </s> <s id="s.000879">Comparatis enim triangulis i&longs;o&longs;celibus DPO, <lb/>DFG, anguli ad ba&longs;im PO majores &longs;unt angulis ad ba&longs;im <lb/>FG, quia angulus PDO minor e&longs;t angulo FDG: ergo <lb/>angulus BPO, qui e&longs;t infra ba&longs;im, minor e&longs;t angulo <lb/>BFG infra ba&longs;im. </s> <s id="s.000880">Fiat igitur ip&longs;i BPO æqualis angulus <lb/>BFK, ac proinde K cadit inter puncta I & G. <!-- KEEP S--></s> <s id="s.000881">Sunt ergo <lb/>triangula BPO, BFK habentia angulum ad B communem <lb/>æquiangula, & &longs;imilia, ac per 4. lib.6. ut PB, hoc e&longs;t BR, <lb/>ad BO, ita FB, hoc e&longs;t BI, ad BK; & invertendo, ac <lb/>dividendo, & iterùm invertendo ut BR ad RO, ita BI <lb/>ad IK. <!-- KEEP S--></s> <s id="s.000882">Atqui IG major e&longs;t quam IK, ergo per 8.lib.5. <lb/>Ratio BI ad IG minor e&longs;t Ratione BI ad IK, hoc e&longs;t BR <lb/>ad RO. <!-- KEEP S--></s> <s id="s.000883">Cum itaque per 2. lib.6. ut BR ad RO, ita ES <lb/>ad SN; & ut BI ad IG, ita EL ad LH, major e&longs;t Ra­<lb/>tio ES ad SN, quàm EL ad LH, & permutando major <lb/>e&longs;t Ratio ES ad EL, quàm SN ad LH; e&longs;t autem ES <lb/>minor quàm EL, ergo etiam SN multò minor e&longs;t quàm <lb/>LH; ac proinde quo magis à perpendiculari recedet obli­<lb/>qua tractio, momentum ponderis magis accedit ad momen­<lb/>tum eju&longs;dem in plano inclinato per tractionem parallelam, <lb/>hoc e&longs;t, minore differentiâ hoc excedit. </s> <s id="s.000884">Momentum igitur <lb/>perpendicularis tractionis ad momentum obliquæ tractionis <lb/>minorem Rationem habet, quàm ad momentum tractionis pa­<lb/>rallelæ plano inclinato. </s> </p> <p type="main"> <s id="s.000885">Ex his ob&longs;ervare e&longs;t aliquod paradoxum, pondus &longs;cilicet obli­<lb/>quâ hâc elevatione tractum plus moveri, quàm potentiam tra­<lb/>hentem; hæc enim movetur &longs;ecundùm men&longs;uram funiculi <lb/>tracti, hoc e&longs;t BP &longs;eu BR illi æqualis, o&longs;ten&longs;um e&longs;t autem <pb pagenum="112" xlink:href="017/01/128.jpg"/>BR minorem e&longs;&longs;e quàm BO. <!-- KEEP S--></s> <s id="s.000886">Id quod etiam manife&longs;tum e&longs;t, <lb/>&longs;i tractio obliqua non ab&longs;trahat pondus à plano, &longs;ed qua&longs;i il­<lb/><figure id="id.017.01.128.1.jpg" xlink:href="017/01/128/1.jpg"/><lb/>lud adversùs planum trahat. </s> <lb/> <s id="s.000887">Sit enim planum AB, &longs;uper <lb/>quo globus C, & funiculus <lb/>obliquus DC; ex D autem <lb/>pendeat ad perpendiculum <lb/>æquale pondus E. <!-- KEEP S--></s> <s id="s.000888">Uterque fu­<lb/>niculus pariter trahatur, & <lb/>cum E venerit in F, æqualis <lb/>pars CG decedit funiculo <lb/>DC; remanet autem longitu­<lb/>do DG æqualis longitudini <lb/>DH, & centrum globi C ve­<lb/>nit in H. <!-- KEEP S--></s> <s id="s.000889">Dico CH motum <lb/>globi majorem e&longs;&longs;e &longs;upra CG <lb/>motum potentiæ trahentis. </s> <lb/> <s id="s.000890">Ducatur enim recta GH; e&longs;t <lb/>I&longs;o&longs;celes DGH, ergo angulus HGC infra ba&longs;im major e&longs;t <lb/>recto; ergo CH per 19.lib.1. major e&longs;t quàm CG. <!-- KEEP S--></s> <s id="s.000891">Ip&longs;i autem <lb/>CH æqualem e&longs;&longs;e di&longs;tantiam contactuum RS manife&longs;tum <lb/>e&longs;t, quia ex centris H & C rectæ cadunt in S & R ad angu­<lb/>los rectos, atque adeò &longs;unt parallelæ: &longs;unt æquales CR & HS, <lb/>ut pote Radij eju&longs;dem globi; igitur per 33.lib.1. CH, & RS <lb/>æquales &longs;unt & parallelæ. </s> <s id="s.000892">Quare &longs;ivè centrum &longs;pectetur, &longs;ivè <lb/>puncta contactuum, perinde e&longs;t; &longs;emper enim major e&longs;t glo­<lb/>bi motus motu potentiæ trahentis; & quia RS major e&longs;t quàm <lb/>CG, hoc e&longs;t quàm motus, qui fieret in ip&longs;o plano inclinato <lb/>tractione parallelâ, hinc e&longs;t quod huju&longs;modi obliquâ tractio­<lb/>ne ad majorem altitudinem perpendicularem pari tempore tra­<lb/>hitur, majorémque proptereà violentiam &longs;ubiens majoribus <lb/>indiget viribus, quàm &longs;i tractione parallelâ elevaretur. </s> </p> <p type="main"> <s id="s.000893">Sed jam trahatur iterum funiculus ita, ut ip&longs;i CG primæ <lb/>tractioni æqualis &longs;it &longs;ecunda tractio HL; & crit centrum globi <lb/>in M, & æquales DM, DL. </s> <s id="s.000894">Anguli MDH, HDC &longs;i di­<lb/>cantur æquales, etiam per 3.lib.6. ut MD ad DC ita MH <lb/>ad HC: e&longs;t igitur MH minor quàm HC, major tamen quàm <lb/>HL, quia &longs;ubten&longs;a e&longs;t angulo MLH obtu&longs;o, ut pote infra ba-<pb pagenum="113" xlink:href="017/01/129.jpg"/>&longs;im I&longs;o&longs;celis MDL. </s> <s id="s.000895">Atqui ex hypothe&longs;i anguli MDL, HDG <lb/>&longs;unt æquales; ergo I&longs;o&longs;celium anguli infra ba&longs;es, hoc e&longs;t MLH, <lb/>HGC &longs;unt æquales: angulus autem externus MHL major e&longs;t <lb/>interno HCD, hoc e&longs;t HCG, per 16.lib.1. igitur reliquus <lb/>HML minor e&longs;t reliquo CHG. </s> <s id="s.000896">Itaque in duobus triangulis, <lb/>angulis CGH, HLM ex hypothe&longs;i o&longs;ten&longs;is æqualibus &longs;ub­<lb/>tenditur illi quidem majus latus CH, huic verò minus HM, <lb/>& angulis inæqualibus CHG majori, HML minori æquale <lb/>latus CG, HL: id quod omninò ab&longs;urdum e&longs;&longs;e con&longs;tat ex <lb/>doctrinâ & Canone Sinuum; &longs;ubten&longs;æ &longs;iquidem inæquales an­<lb/>gulorum æqualium &longs;unt in circulis inæqualibus, major in majori <lb/>circulo, minor in minori, in quibus utique fieri non pote&longs;t, ut <lb/>angulorum inæqualium &longs;ubten&longs;æ &longs;int æquales. </s> <s id="s.000897">Non igitur fieri <lb/>pote&longs;t ut factá &longs;ecunda tractione HL æquali priori CG, angu­<lb/>lus MDH æqualis &longs;it angulo HDC; alioquin triangulum <lb/>HLM (cujus ba&longs;is HM ex hypothe&longs;i arguitur minor ba&longs;e <lb/>CH, quæ tamen &longs;unt angulis ad G & L æqualibus &longs;ubten&longs;æ) <lb/>e&longs;&longs;et in circulo minore, quàm &longs;it circulus, in quo e&longs;&longs;et triangu­<lb/>lum CGH; in circulo autem minore, angulo minori HML <lb/>&longs;ubten&longs;a HL e&longs;&longs;et æqualis ip&longs;i CG &longs;ubten&longs;æ angulo majori <lb/>CHG in circulo majore. </s> </p> <p type="main"> <s id="s.000898">Quod &longs;i dicatur angulus MDH minor, quàm HDC, ergo <lb/>angulus MLH infra ba&longs;im minor e&longs;t angulo HGC infra ba­<lb/>&longs;im: atqui angulus MHL externus major e&longs;t <expan abbr="interño">interno</expan> HCG; <lb/>igitur reliquus angulus LMH vel e&longs;t æqualis angulo GHC, <lb/>vel illo minor, vel illo major. </s> <s id="s.000899">Sit æqualis: quoniam æqualibus <lb/>lineis CG, HL &longs;ubtenduntur, &longs;unt in circulis æqualibus; ergo <lb/>cùm angulus MHL major &longs;it angulo HCG, etiam oppo&longs;itum <lb/>latus ML majus e&longs;t quàm HG: ergo I&longs;o&longs;celes MDL habens <lb/>angulum minorem &longs;ub brevioribu lateribus habet majorem <lb/>ba&longs;im, & I&longs;o&longs;celes HDG habens angulum majorem &longs;ub late­<lb/>ribus <expan abbr="lõgioribus">longioribus</expan> habet <expan abbr="brevior&etilde;">breviorem</expan> ba&longs;im; id quod e&longs;t manife&longs;tè <expan abbr="ab-&longs;urdũ">ab­<lb/>&longs;urdum</expan>, ut patet ex 24. & 25.lib.1.Fieri igitur non pote&longs;t, ut anguli <lb/>LMH, GHC &longs;int æquales, &longs;i MDH minor e&longs;t quàm HDC. </s> </p> <p type="main"> <s id="s.000900">Quandoquidem igitur LMH, GHC non &longs;unt æquales, dica­<lb/>tur angulus LMH minor quàm GHC, & quia æqualibus li­<lb/>neis HL, CG &longs;ubtenduntur, triangulum HLM e&longs;t in circulo <lb/>majore, triangulum verò CHG in minore. </s> <s id="s.000901">Cum autem angu-<pb pagenum="114" xlink:href="017/01/130.jpg"/>lus MHL, ex &longs;æpiùs dictis, &longs;it major quàm HCG, etiam &longs;ub­<lb/>ten&longs;a illius, ut potè in circulo majori, &longs;cilicet ML major e&longs;t <lb/>quàm HG &longs;ubten&longs;a anguli minoris in circulo minori: atque <lb/>hinc idem quod priùs, &longs;equitur ab&longs;urdum angulum verticalem <lb/>MDL, ex hypothe&longs;i minorem, & brevioribus lateribus com­<lb/>prehen&longs;um ba&longs;im habere majorem, quàm &longs;it ba&longs;is anguli verti­<lb/>calis HDG majoris &longs;ub lateribus longioribus. </s> </p> <p type="main"> <s id="s.000902">Sed neque dici pote&longs;t angulus HML major quàm CHG; <lb/>quia, &longs;i MDL minor e&longs;t quàm HDG, angulus DML ad ba­<lb/>&longs;im I&longs;o&longs;celis major e&longs;t quàm DHG pariter ad ba&longs;im; ergo &longs;i <lb/>DML majori addatur major HML, & DHG minori adda­<lb/>tur minor CHG, erit totus DMH major toto angulo DHC, <lb/>internus &longs;cilicet major externo, contra 16.lib.1. Si igitur an­<lb/>gulus HML comparatus cum angulo CHG non pote&longs;t e&longs;&longs;e <lb/>æqualis, neque minor, neque major, factâ hypothe&longs;i anguli <lb/>MDL minoris quàm HDC, nece&longs;&longs;ariâ con&longs;ecutione confici­<lb/>tur angulum MDL non e&longs;&longs;e minorem angulo HDG. </s> </p> <p type="main"> <s id="s.000903">Cum itaque angulus MDL neque æqualis, neque minor &longs;it <lb/>angulo HDG, &longs;equitur quod &longs;it major: igitur & angulus in­<lb/>fra ba&longs;im MLH major e&longs;t angulo HGC; item angulus MHL <lb/>major e&longs;t quàm HCG; ergo HML reliquus minor e&longs;t reliquo <lb/>CHG: at i&longs;tis æquales lineæ HL, CG &longs;ubtenduntur, igitur <lb/>triangulum HML e&longs;t in majore circulo, ac proinde angulo <lb/>MLH majori, quàm CGH, etiam majus latus &longs;ubtenditur: <lb/>quapropter MH, hoc e&longs;t SN, illi parallela & æqualis, major <lb/>e&longs;t quàm CH, hoc e&longs;t RS: atque adeò ad majorem altitudi­<lb/>nem elevatur per SN, quàm per RS factâ æquali tractione, &longs;eu <lb/>æquali motu potentiæ trahentis. </s> <s id="s.000904">Ex quo & manife&longs;tum e&longs;t pro <lb/>majori obliquitate & rece&longs;&longs;u tractionis à paralleli&longs;mo cum pla­<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/>altioribus rotis in&longs;truantur birota (antiquis Ci&longs;ia dicebantur) <lb/>adeò ut unicus equus temoni applicitus, illumque &longs;ubjecto pla­<lb/>no proximè parallelum &longs;ervans, dum clivum a&longs;cendit, ingentia <lb/>pondera trahat, quibus &longs;anè par non e&longs;&longs;et, &longs;i rotarum axis mi­<lb/>nùs à &longs;ubjecto plano di&longs;taret, & equitractio e&longs;&longs;et obliqua &longs;ur­<lb/>&longs;um: quamvis, ut aliàs &longs;uo loco explicabitur, ip&longs;a rotarum am­<lb/>plitudo plurimum conferat. </s> <s id="s.000906">Similiter in navium tractione, quæ <pb pagenum="115" xlink:href="017/01/131.jpg"/>adver&longs;o flumine deducuntur fune ab&longs;idi mali conjuncto, ali­<lb/>quid juvare funis longitudinem, ut &longs;cilicet minùs obliqua &longs;it <lb/>tractio, ex dictis confirmatur: quamvis enim tractiones in plano <lb/>inclinato con&longs;ideraverimus, ut gravium elevationem expende­<lb/>remus, aliquid etiam facit obliquitas tractionis in plano horizon­<lb/>tali, cuju&longs;modi e&longs;t aqua, cui navis innatat; pars &longs;iquidem de­<lb/>mer&longs;a ob&longs;tantem undam repellere debet; nec planè inutile e&longs;t, <lb/>&longs;ecundùm quam lineam dirigatur motus potentiæ trahentis, vi <lb/>cujus impedimentum &longs;uperandum e&longs;t. </s> </p> <p type="main"> <s id="s.000907">Hactenus nobis de tractione &longs;ermo fuit, quæ motum inferens <lb/>non ni&longs;i &longs;patiis, per quæ motus e&longs;t, determinari potuit. </s> <s id="s.000908">Quo­<lb/>niam verò in obliquis tractionibus non eandem &longs;emper analo­<lb/>giam &longs;ervari, quæ in parallelâ tractione eadem perpetuò e&longs;t, de­<lb/>prehendimus, inquirendum &longs;upere&longs;t, quæ demum Ratio mo­<lb/>mentorum &longs;it pro &longs;ingulis obliquitatibus, ut con&longs;tet, quibus vi­<lb/>ribus retineri po&longs;&longs;it, ne in proclive labatur pondus, etiam&longs;i vires <lb/>ad illud ulteriùs elevandum non &longs;uppetant. </s> <s id="s.000909">Quamquam autem <lb/>pondera qua&longs;i molis expertia unico puncto expre&longs;&longs;imus in plano <lb/>ip&longs;o inclinato, ut in 1.fig.hujus cap. re tamen verâ centrum gra­<lb/>vitatis attendendum e&longs;t, ut in 2. &longs;chemate, quod utique di&longs;tat à <lb/>plano, cui corpus grave incumbit: hujus verò di&longs;tantiam nulla <lb/>certior men&longs;ura definit, quàm linea ex eo cadens in &longs;ubjectum <lb/>planum ad angulos rectos, hæc quippe omnium brevi&longs;&longs;ima e&longs;t. <lb/><figure id="id.017.01.131.1.jpg" xlink:href="017/01/131/1.jpg"/><lb/>Sit igitur planum inclinatum AB, <lb/>cui impo&longs;itus globus centrum ha­<lb/>bet gravitatis C, & contingit pla­<lb/>num in D; ac propterea etiam, quæ <lb/>à centro ad contactum ducitur <lb/>recta CD, di&longs;tantiam determinat, <lb/>cum &longs;it plano perpendicularis ex <lb/>18.lib.3. Jam recta CE parallela <lb/>plano ducatur, & &longs;it linea &longs;u&longs;pen­<lb/>&longs;ionis, quam claritatis gratiâ paral­<lb/>lelam vocemus: & per D punctum, <lb/>in quod cadit linea di&longs;tantiæ cen­<lb/>tri gravitatis tran&longs;eat perpendicu­<lb/>laris horizonti linea FD quæ in G <lb/>&longs;ecat lineam CE. <!-- KEEP S--></s> <s id="s.000910">Con&longs;tat trian-<pb pagenum="116" xlink:href="017/01/132.jpg"/>gulum DGC &longs;imile e&longs;&longs;e triangulo BAS: quia enim GD pa­<lb/>rallela e&longs;t lineæ AS pariter perpendiculari ad horizontem, an­<lb/>guli SAB, ADG alterni æquales &longs;unt per 27.lib.1. Et quo­<lb/>niam angulus CDA ex con&longs;tructione e&longs;t rectus, complemen­<lb/>tum CDG æquale e&longs;t angulo complementi ABS; anguli verò <lb/>DCG, BSA &longs;unt recti, hic quidem ex hypothe&longs;i, ille autem <lb/>propter linearum CE, DA paralleli&longs;mum: igitur reliquus <lb/>CGD reliquo BAS æqualis e&longs;t; ac proptereà per 4. lib. 6. ut <lb/>BA ad AS, ita DG ad GC. <!-- KEEP S--></s> <s id="s.000911">Quoniam itaque, &longs;i pondus in <lb/>plano inclinato ad pondus in perpendiculari &longs;it ut inclinata BA <lb/>ad perpendicularem AS, eorum momenta æqualia &longs;unt, & <lb/>æquiponderant, etiam globus æqualia ad de&longs;cendendum habet <lb/>momenta, ac potentia habeat vires ad retinendum in parallelâ <lb/>EC, &longs;i globi gravitas ad potentiam retinendum &longs;it ut DG ad <lb/>GC. <!-- KEEP S--></s> <s id="s.000912">Verum quidem e&longs;t globum non per lineam FD, &longs;ed per <lb/>CT à centro gravitatis perpendicularem horizonti deor&longs;um ni­<lb/>ti: Sed quia CT ip&longs;i FD parallela e&longs;t, triangulum CTD <lb/>triangulo DGC &longs;imile e&longs;t & æquale; atque adeò parùm in­<lb/>tere&longs;t, utrùm lineis DG, GC, an verò lineis CT, TD eadem <lb/>Ratio exponatur. </s> </p> <p type="main"> <s id="s.000913">Sed jam retineatur globus per rectam CH; utique perinde &longs;e­<lb/>cundùm eam directionem &longs;e habet, atque &longs;i e&longs;&longs;et planum HCK; <lb/>globus enim &longs;u&longs;tinetur per lineam DC, & retinetur ex H, ac <lb/>proinde &longs;ecundùm <expan abbr="rectã">rectam</expan> HCK conatur deor&longs;um co &longs;itu: quam­<lb/>quam &longs;ubjecti plani inclinatio ob&longs;taret, ne &longs;ecundùm rectam <lb/>HCK procederet, &longs;i &longs;ibi dimitteretur, & alia atque alia plana <lb/>con&longs;tituerentur. </s> <s id="s.000914">Planum itaque illud HC declinat à perpen­<lb/>diculari, cum quâ con&longs;tituit angulum CID æqualem externo <lb/>KCT propter paralleli&longs;mum perpendicularium FD, CT per <lb/>27. lib. 1. qui utique CID minor e&longs;t externo CGD per 16. <lb/>lib. 1. & quidem differentia anguli ICG per 32.lib.1. Fiat <lb/>ergo angulus BAP æqualis angulo CIG; quia BAS o&longs;ten&longs;us <lb/>e&longs;t æqualis ip&longs;i CGD, remanet PAS æqualis angulo ICG. </s> <lb/> <s id="s.000915">Quare BPA externus æqualis e&longs;t duobus internis, &longs;cilicet recto <lb/>PSA, & acuto SAP, per 32.lib.1. igitur idem angulus BPA <lb/>æqualis e&longs;t toti angulo DCI. </s> <s id="s.000916">Sunt itaque æquiangula & &longs;imi­<lb/>lia duo triangula BAP & DIC, atque per 4.lib.6. ut BA ad <lb/>AP, ita DI ad IC. </s> <s id="s.000917">Atqui pondera &longs;uper BA & AP, quæ &longs;int <pb pagenum="117" xlink:href="017/01/133.jpg"/>ut BA ad AP, æquiponderant ex dictis cap. 13. ergo etiam <lb/>æqualium momentorum e&longs;t globus, & potentia retinens per <lb/>HC, &longs;i globus ad potentiam &longs;it ut DI ad IC, hoc e&longs;t ut CN <lb/>ad ND, &longs;i ex D intelligatur exire DN parallela ip&longs;i HC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000918">Eâdem ratione &longs;i linea obliqua, per quam globus retinetur, <lb/>&longs;it infra parallelam CE, ut &longs;i &longs;it CX, o&longs;tendetur globi gravita­<lb/>tem ad potentiam retinentem e&longs;&longs;e ut DQ ad QC, e&longs;t enim <lb/>qua&longs;i planum inclinatum faciens cum perpendiculari angulum <lb/>DQC majorem interno DGC, hoc e&longs;t majorem angulo BAS <lb/>illi æquali. </s> <s id="s.000919">Fiat igitur angulo DQC æqualis angulus BAY: <lb/>& quia ABY æqualis e&longs;t angulo CDQ, ut &longs;uperiùs dictum <lb/>e&longs;t, triangula BAY, DQC &longs;unt æquiangula & &longs;imilia, ac per <lb/>4.lib.6. ut BA ad AX, ita DQ ad QC: ergo quia pondera &longs;u­<lb/>per BA, & AY, quæ &longs;int in Ratione BA ad AY, æquiponde­<lb/>rant, etiam globi & potentiæ retinentis momenta æqualia &longs;unt, <lb/>&longs;i fuerint ut DQ ad QC. </s> </p> <p type="main"> <s id="s.000920">Hic autem tria ob&longs;ervanda occurrunt. </s> <s id="s.000921">Primum e&longs;t, quòd <lb/>Rationes prædictæ momentorum potentiæ retinentis compara­<lb/>tæ ad pondus idem, quamvis pro diversâ obliquitate aliis atque <lb/>aliis lineis explicentur DQ ad QC, & DG ad GC, DI ad <lb/>IC, omnes tamen exponuntur comparatè ad eandem BA in <lb/>triangulo BAY; in quo ip&longs;æ quoque inter &longs;e invicem compara­<lb/>ri po&longs;&longs;unt. </s> <s id="s.000922">Secundum e&longs;t, quòd &longs;i obliquitas tàm &longs;upra, quàm <lb/>infra parallelam CE æqualis &longs;it, hoc e&longs;t angulus ICG æqualis <lb/>&longs;it angulo GCQ, momenta potentiæ retinentis in H & X <lb/>æqualia &longs;unt; inter &longs;e &longs;iquidem &longs;unt ut AP, & AY, quæ lineæ <lb/>æquales &longs;unt; nam anguli PAS, YAS æquales &longs;unt ex hypo­<lb/>the&longs;i, & con&longs;tructione, anguli autem ad S &longs;unt recti & latus <lb/>AS e&longs;t utrique triangulo commune; ergo etiam per 26.lib.1.la­<lb/>tera AP & AY æqualia &longs;unt. </s> <s id="s.000923">Tertium e&longs;t, quòd in lineá CE <lb/>parallelâ minus virium exigitur ad retinendum globum, quàm <lb/>in cæteris: nam & linea AS vires potentiæ repræ&longs;entans om­<lb/>nium minima e&longs;t, utpote perpendicularis. </s> </p> <p type="main"> <s id="s.000924">Ex his & illud colligitur, quod &longs;i linea, &longs;ecundùm quam <lb/>pondus retinetur in plano inclinato, &longs;it parallela horizonti, <lb/>eadem e&longs;t philo&longs;ophandi methodus. </s> <s id="s.000925">Si enim &longs;uper plano in­<lb/>clinato AB &longs;it pondus tangens in C, cujus gravitatis centrum <lb/>&longs;it D, & linea retentionis DE horizonti parallela, ducatur <pb pagenum="118" xlink:href="017/01/134.jpg"/><figure id="id.017.01.134.1.jpg" xlink:href="017/01/134/1.jpg"/><lb/>CF perpendicularis horizonti; & Ratio <lb/>ponderis ad vires retinentes erunt ut CF <lb/>ad FD. <!-- KEEP S--></s> <s id="s.000926">Fiat enim angulus BAH æqua­<lb/>lis angulo CFD, qui utique e&longs;t rectus, <lb/>cum DE ex hypothe&longs;i &longs;it horizonti pa­<lb/>rallela, FC verò perpendicularis: ergo <lb/>&longs;uper AB, AH æquiponderant pondera, <lb/>quæ &longs;int ut AB ad AH; paria igitur &longs;unt <lb/>momenta, &longs;i pondus ad vires potentiæ re­<lb/>tinentis in eâdem Ratione &longs;it ut AB ad AH, hoc e&longs;t ut CF ad <lb/>FD. <!-- KEEP S--></s> <s id="s.000927">Quia enim BAH angulus e&longs;t rectus per 8.lib.6. e&longs;t ut <lb/>BA ad AH, ita BG ad GA; e&longs;t autem BG ad GA ut CF ad <lb/>FD; quia nimirum FC perpendicularis horizonti e&longs;t paralle­<lb/>la ip&longs;i AG, & anguli BAG, FCA alterni &longs;unt æquales per <lb/>27.lib.1. DCA verò e&longs;t rectus ex hypothe&longs;i; igitur & DCF <lb/>complementum recti æquale e&longs;t angulo ABG: utrumque <lb/>triangulum e&longs;t rectangulum; ergo ut BG ad GA, ita CF <lb/>ad FD. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000928">Hinc apparet fieri po&longs;&longs;e, ut ad retinendum pondus in tali &longs;i­<lb/>tu aliquando plus virium requiratur, quàm ad &longs;u&longs;tinendum il­<lb/>lud in perpendiculari; quando videlicet ex inclinatione plani <lb/>AB con&longs;equitur lineam CF minorem e&longs;&longs;e quàm FD: immò <lb/>cre&longs;cit retinendi difficultas, &longs;i adhuc retentio fiat per lineam <lb/>inferiorem horizontali DE, quæ cum perpendiculari CF con­<lb/>&longs;tituat angulum DIC obtu&longs;um; cum enim cre&longs;ceret linea DI <lb/>&longs;upra DF, & IC decre&longs;ceret infra FC, e&longs;&longs;et minor Ratio pon­<lb/>deris in perpendiculo ad potentiam obliquè retinentem, <lb/>quæ proinde major e&longs;&longs;e deberet, ut fieret momentorum æqua­<lb/>litas. </s> </p> <p type="main"> <s id="s.000929">Concipe autem &longs;ublatum triangulum totum BAH, & DC <lb/>e&longs;&longs;e columnam, quæ in eodem &longs;itu inclinata retineri debeat: <lb/>jam &longs;atis con&longs;tat ex dictis, quâ ratione di&longs;poni oporteat funes, <lb/>ut qui funium extremitates tenent, minus laboris impendant. </s> <lb/> <s id="s.000930">Non e&longs;t tamen eadem funis retinentis, & fulcri &longs;u&longs;tentantis <lb/>ratio: in &longs;upponendis enim fulcris illud poti&longs;&longs;imùm attenditur, <lb/>quòd fulcrum ip&longs;um integrum permaneat, citrà &longs;ci&longs;&longs;ionis aut <lb/>fractionis periculum; id quod habetur, quò magis perpendicu­<lb/>lari ad horizontem &longs;itui proximum collocatur; parùm &longs;cilicet <pb pagenum="119" xlink:href="017/01/135.jpg"/>intere&longs;t, quanto conatu &longs;ubjectam tellurem urgeat modò certi <lb/>&longs;imus de fulcri ip&longs;ius firmitate. </s> <s id="s.000931">Cæterùm &longs;i tu ip&longs;e fu&longs;tem <lb/>manu tenens cogaris inclinatam columnam &longs;u&longs;tinere, punctum <lb/>autem &longs;u&longs;tentationis, cui fulcrum applicatur, magis à &longs;ub­<lb/>jecto plano di&longs;tet, vel &longs;altem non minùs, quàm centrum gra­<lb/>vitatis columnæ, experieris minori conatu opus e&longs;&longs;e, &longs;i ful­<lb/>crum axi columnæ perpendiculare &longs;it, qui &longs;itus re&longs;pondet re­<lb/>tentioni parallelæ plano inclinato, majorem verò adhiben­<lb/>dum e&longs;&longs;e conatum, &longs;i fulcrum cum eodem axe acutum aut ob­<lb/>tu&longs;um angulum con&longs;tituat; id quod obliquis elevationibus <lb/>re&longs;pondet. </s> </p> <p type="main"> <s id="s.000932">Quòd &longs;i infra centrum gravitatis applicetur fulcrum, jam <lb/>con&longs;tat hoc ita e&longs;&longs;e collocandum, ut ei idem centrum im­<lb/>mineat, alioquin aut columna corruet, aut multis viri­<lb/>bus tibi contendendum erit, ut illam &longs;u&longs;tentes à lap&longs;u; &longs;i <lb/>tamen ea &longs;it complexio tùm inclinationis, tùm obicis co­<lb/>lumnæ pedem retinentis, ne excurrat, aut elevetur, tùm po­<lb/>&longs;itionis fulcri, ut aliquatenus &longs;u&longs;tineri columna po&longs;&longs;it, ne pror­<lb/>sùs ruat. </s> </p> <p type="main"> <s id="s.000933">Sed quoniam hîc columnæ mentio incidit, præ&longs;tat ele­<lb/>vationes corporum, quæ non tota elevantur, &longs;ed eorum <lb/>altera extremitas &longs;ubjecto alicui fulcro aut plano innititur, <lb/>altera elevatur aut &longs;u&longs;penditur, con&longs;iderare: neque enim hîc <lb/>reputanda &longs;unt momenta gravitatis perinde, ac &longs;i totum cor­<lb/>pus elevaretur aut &longs;u&longs;penderetur, quemadmodum paulò an­<lb/>te dicebatur; immò verè longè minora &longs;unt pro ratione <lb/>di&longs;tantiæ à centro gravitatis, ut ex inferiùs dicendis, ubi de <lb/>æquilibrio, atque de vecte &longs;ermo erit, con&longs;tabit. </s> <s id="s.000934">Cavendum <lb/>autem plurimum e&longs;t ab æquivocationibus, quæ obrepere <lb/>po&longs;&longs;unt, ni&longs;i animum advertas ad gravitatem, &longs;ivè per totam <lb/>longitudinem, quæ movetur, aut ad motum incitari pote&longs;t, <lb/>diffu&longs;am, &longs;ivè qua&longs;i in unum punctum ibi collectam, ubi ele­<lb/>vans applicatur, ut in vecte, aut librâ; hinc enim non mo­<lb/>dica momentorum inæqualitas oritur. </s> <s id="s.000935">Nam &longs;i puncto appli­<lb/>cationis re&longs;pondeat centrum gravitatis, multò majores ad <lb/>elevandum, aut &longs;u&longs;pendendum corpus requiruntur vires, <lb/>quàm &longs;i centrum gravitatis à puncto applicationis aliquo in­<lb/>tervallo &longs;ejungatur. </s> </p> <pb pagenum="120" xlink:href="017/01/136.jpg"/> <figure id="id.017.01.136.1.jpg" xlink:href="017/01/136/1.jpg"/> <p type="main"> <s id="s.000936">Hinc &longs;i &longs;it pri&longs;ma AB ho­<lb/>rizontaliter collocatum, eju&longs;­<lb/>que extremitas A innitatur <lb/>apici pyramidis, altera verò <lb/>extremitas B &longs;u&longs;pendatur per­<lb/>pendiculari funiculo CB, vel <lb/>&longs;u&longs;tentetur &longs;uppo&longs;ito ad <expan abbr="per-pendiculũ">per­<lb/>pendiculum</expan> fulcro DB, æqua­<lb/>liter res &longs;e habet, & pares requiruntur vires tam in &longs;u&longs;penden­<lb/>te CB, quàm in &longs;u&longs;tentante DB: hæ tamen vires non pares <lb/>e&longs;&longs;e debent toti ponderi pri&longs;matis; &longs;ed quia centrum gravita­<lb/>tis E ab utroque extremo æqualiter di&longs;tare &longs;upponitur, &longs;e­<lb/>mi&longs;&longs;is tantùm gravitatis percipitur in B. </s> <s id="s.000937">Quod &longs;i in codem <lb/>horizontali &longs;itu retineatur pri&longs;ma &longs;ivè à &longs;u&longs;pendente obliquo <lb/>IB, &longs;ivè ab obliquo &longs;u&longs;tentante OB, utique retinentis, aut <lb/>&longs;u&longs;tentantis vires æquipollere debent viribus retinentis aut <lb/>&longs;u&longs;tentantis ad perpendiculum CB aut DB. <!-- KEEP S--></s> <s id="s.000938">Quemadmo­<lb/>dum igitur pondera illa &longs;uper BO & BD æquiponderant, <lb/>quæ &longs;unt ut BO ad BD, ita vires, quæ &longs;ecundùm ea&longs;dem <lb/>lineas ac directiones æqualem effectum præ&longs;tare debent; in <lb/>eâdem Ratione BO ad BD e&longs;&longs;e oportet: Vires ergo retinen­<lb/>tis BI obliqui ad vires retinentis CB ad perpendiculum &longs;unt <lb/>ut BO ad BD, hoc e&longs;t, ductâ parallelâ CI, ut IB ad CB, <lb/>propter triangulorum OBD, CBI &longs;imilitudinem. </s> </p> <p type="main"> <s id="s.000939">Ut autem non hîc perperam nos philo&longs;ophari innote&longs;cat, <lb/>finge &longs;ublatam ex A pyramidem, & con&longs;titutam in G ita, <lb/>ut ex B ad perpendiculum dependeat pondus aliquod æqui­<lb/>librium efficiens cum pri&longs;mate: quo perpendiculari pondere <lb/>&longs;ublato, ut pri&longs;ma horizontale permaneat, certum e&longs;t &longs;uper <lb/>plano inclinato BO requiri pondus, quod ad pondus per­<lb/>pendiculare ex BD &longs;it ut BO ad BD: igitur &longs;i loco pon­<lb/>deris applicentur &longs;ecundùm eandem rectam lineam BO vires <lb/>alicujus viventis, à quo retineatur pri&longs;ma in eodem &longs;itu ho­<lb/>rizontali, &longs;atis apparet conatum debere e&longs;&longs;e ut BO ad cona­<lb/>tum, qui &longs;ecundùm perpendicularem requireretur ut BD. <!-- KEEP S--></s> <lb/> <s id="s.000940">Sicut itaque conatus deor&longs;um trahens, cum fulcrum e&longs;t in <lb/>G citrà centrum gravitatis E, ex inclinatione lineæ, &longs;ecun­<lb/>dùm quam fit, de&longs;umitur, ita etiam conatus &longs;u&longs;pendens IB, <pb pagenum="121" xlink:href="017/01/137.jpg"/>aut &longs;ur&longs;um urgens OB, cum fulcrum e&longs;t in A ultrà centrum <lb/>gravitatis E, de&longs;umendus e&longs;t pariter ex inclinatione lineæ, &longs;e­<lb/>cundùm quam applicatur pri&longs;mati, comparatè ad conatum per­<lb/>pendicularem CB, vel DB, habita &longs;emper ratione di&longs;tantiæ <lb/>fulcri à centro gravitatis. </s> </p> <p type="main"> <s id="s.000941">Ne quid verò dubitationis <lb/><figure id="id.017.01.137.1.jpg" xlink:href="017/01/137/1.jpg"/><lb/>&longs;uper&longs;it, utrum OB deor&longs;um, <lb/>& IB &longs;ur&longs;um trahentium pa­<lb/>res &longs;int vires &longs;ecundùm ean­<lb/>dem rectam lineam OI, &longs;int <lb/>rotulæ duæ H & F circa &longs;uum <lb/>axem ver&longs;atiles infixæ extre­<lb/>mitatibus regulæ, aut tigilli, <lb/>& ex funiculo rotularum ca­<lb/>vitatibus in&longs;erto dependeant <lb/>æqualia pondera L & G. <!-- KEEP S--></s> <s id="s.000942">Hæc <lb/>pondera &longs;ibi vici&longs;&longs;im æquipon­<lb/>derare manife&longs;tum e&longs;t, quem­<lb/>cumque tandem &longs;itum &longs;ivè <lb/>perpendicularem, &longs;ivè incli­<lb/>natum, habeat regula, aut ti­<lb/>gillus, cui rotulæ infixæ &longs;unt. </s> <s id="s.000943">Sit libræ jugum AB æqualiter <lb/>in E divi&longs;um, circa quod punctum &longs;tabile moveri queat, & <lb/>in A adnectatur funiculo HF: ex B autem dependeat pondus <lb/>D æquale ponderi G, &longs;ed ita obliquè di&longs;po&longs;itum, ut linea BO <lb/>parallela &longs;it lineæ AF. <!-- KEEP S--></s> <s id="s.000944">Submove pondus L, remanent G <lb/>& D, quorum neutrum prævalere pote&longs;t; &longs;unt enim æqualia <lb/>inter &longs;e, & per lineas &longs;imiliter inclinatas AF, BO agunt. </s> <s id="s.000945">Re­<lb/>pone pondus L, & amove pondus G, item removeatur pon­<lb/>dus D, & &longs;ur&longs;um ponatur æquale C; aio libræ jugum AB <lb/>adhuc retinere eumdem &longs;itum; quia &longs;cilicet pondera C & D <lb/>vici&longs;&longs;im æquiponderabant, &longs;icut etiam G & L: igitur quantum <lb/>virium habebat pondus D ad æquiponderandum ip&longs;i G, tan­<lb/>tumdem virium habet pondus C ad æquiponderandum ponde­<lb/>ri L, hoc e&longs;t eidem ponderi G. <!-- KEEP S--></s> <s id="s.000946">Sivè igitur in &longs;uperiori &longs;che­<lb/>mate con&longs;iderentur vires deor&longs;um trahentes aut &longs;u&longs;tentantes <lb/>OB, &longs;ive retinentes IB, perinde e&longs;t, & æqualium momento­<lb/>rum cen&longs;endæ &longs;unt. </s> </p> <pb pagenum="122" xlink:href="017/01/138.jpg"/> <figure id="id.017.01.138.1.jpg" xlink:href="017/01/138/1.jpg"/> <p type="main"> <s id="s.000947">Non jam horizontale &longs;it <lb/>pri&longs;ma AB, &longs;ed inclinatum, <lb/>& puncto A &longs;tabili innixum: <lb/>momenta ad de&longs;cendendum, <lb/>ac proinde repugnantia ad <lb/>a&longs;cendendum, ut &longs;uperiùs in­<lb/>nuimus cap.14; æ&longs;timanda <lb/>&longs;unt in plano DC inclinato, <lb/>quod cum AB angulos facit <lb/>rectos, & cum horizonte AE <lb/>concurrit in puncto E. <!-- KEEP S--></s> <s id="s.000948">Ducatur per B perpendicularis ad ho­<lb/>rizontem FH, & ex H ad BE perpendicularis HO. <!-- KEEP S--></s> <s id="s.000949">Momen­<lb/>ta gravitatis pri&longs;matis in perpendiculari ad momenta eju&longs;dem <lb/>in inclinatà &longs;unt reciprocè ut inclinata EB ad perpendicula­<lb/>rem BH, hoc e&longs;t per 8.lib.6. ut HB ad BO, &longs;ive (ductâ ex D <lb/>&longs;uper DB inclinatam perpendiculari DG &longs;ecante rectam HF <lb/>in F) ut BF ad BD, propter &longs;imilitudinem triangulorum OBH, <lb/>DBF. <!-- KEEP S--></s> <s id="s.000950">Vires ergo retinentes in D ad vires retinentes in F &longs;unt <lb/>ut DB ad BF. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000951">Retineatur pri&longs;ma &longs;ecundùm obliquam GB, quæ producta <lb/>u&longs;que ad Horizontalem concurrat in L. <!-- KEEP S--></s> <s id="s.000952">Iterum ex L ad DE <lb/>cadat ad angulos rectos LC, quæ perpendicularem FH &longs;ecabit <lb/>in I: e&longs;t autem IC parallela ip&longs;i HO; ac propterea per 4.lib.6. <lb/>ut HB ad BO, ita IB ad BC, & per 11.lib.5. ut IB ad BC, <lb/>ita BF ad BD. <!-- KEEP S--></s> <s id="s.000953">Ad retinendum igitur pri&longs;ma in eodem &longs;itu in­<lb/>clinationis BAE per obliquam GB, vires æquipollentes viri­<lb/>bus retinentibus in perpendiculari FB e&longs;&longs;e oportet ut BL ad <lb/>BI, quemadmodum retinentes per rectam DB &longs;unt ut BC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000954">Quare datâ corporis inclinatione, cujus gravitas retinenda e&longs;t <lb/>in eodem &longs;itu, &longs;umatur eju&longs;dem axis tran&longs;iens per gravitatis <lb/>centrum, & ad axis extremitatem mobilem ducatur ip&longs;i axi per­<lb/>pendicularis DB, in quâ a&longs;&longs;umpto quolibet puncto D, ducatur <lb/>prædicto axi parallela DG, quæ &longs;ecans lineas qua&longs;libet obli­<lb/>quas, & perpendicularem ad Horizontem, dabit omnium obli­<lb/>quarum &longs;u&longs;pen&longs;ionum Rationem: Sic recta DG &longs;ecans perpen­<lb/>dicularem FB & obliquam GB determinat Rationem virium in <lb/>utrâque &longs;u&longs;pen&longs;ione, ut &longs;cilicet &longs;int in Ratione BF ad BG, & <lb/>&longs;ic de reliquis. </s> </p> <pb pagenum="123" xlink:href="017/01/139.jpg"/> <p type="main"> <s id="s.000955">Quòd &longs;i in gradibus data &longs;it inclinatio pri&longs;matis, & funiculi <lb/>oblique &longs;u&longs;pendenti declinatio a perpendiculo, &longs;tatim ex tabu­<lb/>lis Sinuum, aut etiam Secantium, apparebit Ratio quæ&longs;ita li­<lb/>nearum: angulus enim, quem perpendicularis ad axem facit <lb/>cum perpendiculari ad Horizontem, æqualis e&longs;t angulo incli­<lb/>nationis pri&longs;inatis; angulo &longs;iquidem BAE inclinationis pri&longs;ma­<lb/>tis, æqualis e&longs;t angulus EBH per 8.lib.6. ac proptereà etiam <lb/>ex 15.lib.1. qui illi e&longs;t ad verticem DBF. <!-- KEEP S--></s> <s id="s.000956">Hinc &longs;i inclinatio­<lb/>nis angulus &longs;it gr. <!-- REMOVE S-->36. DB ad BF erit ut Radius ad Secantem <lb/>gr. <!-- REMOVE S-->36. vel ut Sinus gr.54. complementi gr.36. ad Radium. <!-- KEEP S--></s> <s id="s.000957">At <lb/>angulus, quem facit linea obliquæ &longs;u&longs;pen&longs;ionis cum perpendi­<lb/>culari ad horizontem tran&longs;eunte per pri&longs;matis punctum; in quo <lb/>&longs;u&longs;penditur, e&longs;t æqualis angulo, quem eadem &longs;u&longs;pen&longs;ionis li­<lb/>nea facit cum perpendiculo tran&longs;eunte per aliud extremum <lb/>eju&longs;dem lineæ &longs;u&longs;pen&longs;ionis, cui applicatur potentia retinens: <lb/>duæ enim perpendiculares prædictæ &longs;unt inter &longs;e parallelæ, & <lb/>linea &longs;u&longs;pen&longs;ionis in eas incidens alternos angulos facit æquales <lb/>per 27.lib.1. Si igitur GB à &longs;uo perpendiculo, quod ex G in <lb/>horizontem cadat, declinat gr.25. etiam FBG e&longs;t gr.25. To­<lb/>tus igitur angulus DBG e&longs;t aggregatum anguli inclinationis <lb/>pri&longs;matis, & anguli declinationis funiculi &longs;u&longs;pendentis: igitur <lb/>DBG e&longs;t gr.61, & po&longs;itâ DB ut Radio, erit BG Secans gr.61. <lb/>Vel &longs;i comparanda &longs;it BG cum BF, qui angulus GFB ex­<lb/>ternus per 32.lib.1. æqualis e&longs;t duobus internis oppo&longs;itis tran­<lb/>guli DBF, erit GFB gr.126; at FBG e&longs;t gr.25, igitur FGB <lb/>e&longs;t gr.29. Quare BF ad BG e&longs;t ut Sinus gr. <!-- REMOVE S-->29. ad Sinum <lb/>gr.126, hoc e&longs;t &longs;upplementi gr.54. <!-- REMOVE S-->Apparet ex his primò minimas vires exerceri, &longs;i linea reten­<lb/>tionis cadat ad perpendiculum in axem corporis elevati cum in­<lb/>clinatione; quia &longs;cilicet cum in D &longs;it angulus rectus, recta BD <lb/>e&longs;t omnium linearum ex B puncto excuntium, & in rectam <lb/>DG cadentium minima: quò autem major fuerit obliquitas, <lb/>eò etiam majores vires requiri, quia longiores &longs;unt Secantes <lb/>angulorum majorum in B po&longs;ito Radio BD. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.000958">Secundò fieri pote&longs;t, ut pare: vires requirantur, &longs;i linea re­<lb/>tentionis faciat cùm axe corporis elevati angulum acutum, ac <lb/>&longs;i faciat cùm eodem angulum obtu&longs;um, ut &longs;i fuerit recta MB; <lb/>ip&longs;a enim pariter opponitur angulo recto BDM, ac proinde <pb pagenum="124" xlink:href="017/01/140.jpg"/>eò major e&longs;t quàm recta BD, quò fuerit major angulus MBD, <lb/>qui pote&longs;t e&longs;&longs;e æqualis angulo DBF, vel DBG; quo ca&longs;u <lb/>etiam ip&longs;a BM æqualis erit ip&longs;i BF aut BG. <!-- KEEP S--></s> <s id="s.000959">Ex quo ulteriàs <lb/>&longs;equitur, &longs;i à retinente obliquè fiat tractio elevando magis ac <lb/>magis pri&longs;ma &longs;ic inclinatum, mutari &longs;ubinde momenta: hoc ta­<lb/>men intercedit di&longs;crimen, quod trahentis linea initio applicata, <lb/>ut angulum faciat acutum cum axe pri&longs;matis, in ipsâ tractione <lb/>&longs;emper majorem facit cum ip&longs;o axe angulum, donec veniat ad <lb/>angulum rectum con&longs;tituendum, ut &longs;i MB traheretur, donec <lb/>coincidat cùm DB, quæ pariter moveri intelligatur: contrà <lb/>verò trahentis linea applicata, ut cum axe faciat angulum ob­<lb/>tu&longs;um, in ipsâ tractione magis adhuc obtu&longs;um angulum con&longs;ti­<lb/>tuit, donec tractionis linea (&longs;i tamen fieri id po&longs;&longs;it) in unam <lb/>rectam lineam cum axe pri&longs;matis conveniat. </s> <s id="s.000960">Quare in primâ <lb/>illâ tractione minuitur conatus, in hac &longs;ecunda augetur. <lb/><figure id="id.017.01.140.1.jpg" xlink:href="017/01/140/1.jpg"/></s> </p> <pb pagenum="125" xlink:href="017/01/141.jpg"/> <figure id="id.017.01.141.1.jpg" xlink:href="017/01/141/1.jpg"/> <p type="main"> <s id="s.000961"><emph type="center"/>MECHANICORUM<emph.end type="center"/><!-- REMOVE S--><emph type="center"/>LIBER SECUNDUS.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.000962"><emph type="center"/><emph type="italics"/>De cau&longs;is motus Machinalis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000963">INNOTUIT, opinor, quantum ad præ&longs;ens in&longs;titu­<lb/>tum &longs;atis e&longs;&longs;e po&longs;&longs;it, centrum gravitatis ex iis, quæ <lb/>libro &longs;uperiore dicta &longs;unt: nunc propiùs ad ip&longs;am <lb/>machinalem &longs;cientiam accedendum, quam Mecha­<lb/>nicam dicimus. </s> <s id="s.000964">Hæc Geometriæ &longs;ubjicitur; neque <lb/>enim, ut illa, puram corporum quantitatem moli&longs;que exten­<lb/>&longs;ionem ab&longs;tractè con&longs;iderat, &longs;ed quatenus gravitati illigatam <lb/>aut levitati; nihil tamen &longs;olicita de ipsâ corporum materie, au­<lb/>reáne &longs;it, an lapidea. </s> <s id="s.000965">Quamvis autem ea quoque Statices pars, <lb/>quam Hydro&longs;taticen indigitamus, &longs;e pariter in corporum gra­<lb/>vitate con&longs;iderandâ exerceat, aliam tamen &longs;ibi contemplatio­<lb/>nem a&longs;&longs;umit; motum &longs;iquidem corporum &longs;ingulorum naturæ <lb/>congruentem, pro humorum, in quos incurrunt, diver&longs;itate, <lb/>poti&longs;&longs;imùm &longs;peculatur: Mechanice verò eatenus &longs;olùm ingeni­<lb/>tam corporibus propen&longs;ionem in motum aut quietem explorat, <lb/>ut earum facultati per&longs;pectæ vim po&longs;&longs;it opportunâ in&longs;trumento­<lb/>rum machinatione inferre. </s> <s id="s.000966">Quapropter ut certâ methodo ma­<lb/>chinas oneribus movendis pares con&longs;truere valeamus, motus <lb/>machinalis cau&longs;as antè cognitas habere nece&longs;&longs;e e&longs;t, quàm ma­<lb/>chinas ip&longs;as aggrediamur. </s> <s id="s.000967">His porrò jactis fundamentis ope­<lb/>ro&longs;um non erit inædificare, & machinarum &longs;ingularum vires, <lb/>&longs;ivè &longs;implices illæ &longs;int, &longs;ivè compo&longs;itæ, exponere: adeò ut iis <lb/>ritè intellectis, quæ hoc &longs;ecundo libro di&longs;putabuntur, vix quie­<lb/>quam in reliquo opere &longs;uper&longs;it difficultatis. <pb pagenum="126" xlink:href="017/01/142.jpg"/></s> </p> <p type="main"> <s id="s.000968"><emph type="center"/>CAPUT I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.000969"><emph type="center"/><emph type="italics"/>Quem ad finem Machinæ in&longs;truantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.000970">FInis, quò demum unaquæque actio refertur, primus animo <lb/>concipitur, præ&longs;tituiturque, & idonea ad agendum &longs;ub&longs;i­<lb/>dia, quæ deligenda &longs;unt, moderatur. </s> <s id="s.000971">Hinc ille primus nobis <lb/>in hâc contemplatione occurrit; quem &longs;cilicet ad finem ma­<lb/>chinæ in&longs;tituantur, in&longs;truantúrque, con&longs;iderandum; ut ad <lb/>hanc qua&longs;i regulam cæteræ cau&longs;æ dirigantur, & formentur. </s> <lb/> <s id="s.000972">Fortè dixerit qui&longs;piam magnificè, eo con&longs;ilio machinas à no­<lb/>bis excogitatas, ut naturam arte vincamus; quemadmodum <lb/>enim &longs;cribit Antipho Poeta apud Ari&longs;totelem in quæ&longs;t.Mechan. <lb/><!-- REMOVE S-->&longs;ub initium, <foreign lang="greek">te/xnh| kratou=men, w)_n fu/s<gap/> nikw/meqa. </foreign></s> <s id="s.000973">Sed hic plani&longs;­<lb/>&longs;imè philo&longs;ophandi locus e&longs;t, non gloriandi in&longs;olentiùs. </s> <s id="s.000974">Quare <lb/>fatendum e&longs;t apertè, adhiberi machinas in &longs;ub&longs;idium infirmi­<lb/>tatis; ut quod virium imbecillitas onus loco movere, aut omni­<lb/>nò, aut ni&longs;i ægerrimè &longs;ola nequiret, illud demum facilè, quò <lb/>libuerit, aut trahat, aut impellat, aut etiam expellat quantum­<lb/>vis reluctans, &longs;i machina accedat. </s> </p> <p type="main"> <s id="s.000975">Dupliciter autem in&longs;ita corporibus gravitas ob&longs;i&longs;tit moventi, <lb/>&longs;i ab alio in alium locum transferenda fuerit: di&longs;paribus enim <lb/>momentis mora infertur motui, &longs;i hic fluido in corpore ac &longs;e­<lb/>quaci, puta in aëre aut aquâ, perficiatur, ac &longs;i &longs;uprà &longs;olidam <lb/>con&longs;i&longs;tentemque planitiem raptetur moles, &longs;ive Horizonti pa­<lb/>rallela jaceat planities, &longs;ive molli aut arduâ inclinatione eriga­<lb/>tur in clivum. </s> <s id="s.000976">Et quidem &longs;i &longs;olidum in corpus non incumbat <lb/>onus, &longs;ed in aëre &longs;u&longs;pen&longs;um pendeat, ac &longs;ur&longs;um trahere opor­<lb/>teat, certos ad calculos revocari gravitatis momenta poterunt, <lb/>quibus machina proportione re&longs;pondeat: nam quamvis aër aëri <lb/>præ&longs;tet tenuitate, non ea tamen e&longs;t in levitatibus differentia, ut <lb/>hinc in gravium corporum momentis di&longs;&longs;imilitudo notabilis <lb/>oriatur. </s> <s id="s.000977">Quare &longs;icut laberetur turpiter, qui machinam &longs;axo ab <lb/>imo mari ad &longs;ummam &longs;uperficiem elevando parem in&longs;trueret, &longs;i <lb/>nullâ factâ virium acce&longs;&longs;ione illud in aërem extrahi po&longs;&longs;e &longs;ibi <pb pagenum="127" xlink:href="017/01/143.jpg"/>per&longs;uaderet; ita nimis exiguè & exiliter ad calculos revocaret <lb/>aërem, qui pro di&longs;pari ejus levitate modum machinæ &longs;tatueret; <lb/>in materiâ etenim, ex quâ machina componitur, nullus e&longs;t <lb/>huic minutæ &longs;ubtilitati locus, quæ aciem omnem fugit, ni&longs;i <lb/>cum veritas in di&longs;putatione limatur. </s> <s id="s.000978">Id quod de eâ pariter <lb/>gravitationis inæqualitate dictum velim, quæ ex inæquali à cen­<lb/>tro gravium di&longs;tantiâ ortum habet, ut lib.1. cap. 4. di&longs;putatum <lb/>e&longs;t: Quia in tantulo Spatio, in quo nos labor no&longs;ter exercet, <lb/>illa momentorum exuperantia &longs;ub &longs;en&longs;um non cadit. </s> <s id="s.000979">Quo cir­<lb/>ca &longs;atis &longs;upérque habemus, quòd moventis vires ac molis mo­<lb/>vendæ pondus reputantes ita inter &longs;e conferamus, ut virium <lb/>imbecillitas adhibitâ machinâ convale&longs;cat, & repugnanti one­<lb/>ris gravitati non re&longs;i&longs;tat modò, &longs;ed & præ&longs;tare po&longs;&longs;it, nullâ aut <lb/>loci aut aëris habitâ ratione. </s> </p> <p type="main"> <s id="s.000980">Verùm quàm facile e&longs;t corporis gravitatem cùm ex mate­<lb/>riæ &longs;pecie, tùm ex molis magnitudine inve&longs;tigare; tàm mul­<lb/>tis difficultatibus impedita res e&longs;t, &longs;i examinandum &longs;it, <lb/>quantùm ex mutuo corporum &longs;e contingentium tritu retardetur <lb/>motus: non enim qui&longs;quis pendulum in aere majoris campanæ <lb/>malleum pote&longs;t à perpendiculo dimovere, earum e&longs;t virium, ut <lb/>illum pariter in terrâ jacentem propellere valeat: & decennis <lb/>puer arrepto fune illigatam cymbam, modicè fluctuante &longs;alo, <lb/>ad &longs;e trahit; quam vix, aut ne vix quidem, robu&longs;tioris lacerti <lb/>vir dimoveat, ubi areno&longs;o vado in&longs;ederit: cum tamen eadem aut <lb/>ligneæ cymbæ aut ferreo malleo gravitas innata permaneat. </s> <lb/> <s id="s.000981">E&longs;t autem tùm &longs;ubjecti corporis con&longs;i&longs;tentis, tùm impo&longs;iti one­<lb/>ris movendi &longs;uperficies &longs;pectanda, quatenus &longs;e contingunt: <lb/>Nam &longs;i lapideum globum pondo 100 in planitie con&longs;titutum <lb/>non rotare modo, &longs;ed & rectâ urgere po&longs;&longs;is, non itidem cubum <lb/>pondere parem & materiâ &longs;imilem æquali facilitate urgebis; <lb/>quia &longs;cilicet globus tenui&longs;&longs;imâ &longs;ui parte &longs;uppo&longs;itam planitiem <lb/>contingens minus invenit impedimenti ex proximè &longs;ubjecti <lb/>corporis a&longs;peritate, quæ prominulas impo&longs;iti globi particulas re­<lb/>moretur; at cubus longè pluribus &longs;ui partibus plano adhæret, at­<lb/>que adeò multiplicatá partium hujus in illius partes incurren­<lb/>tium re&longs;i&longs;tentiâ, augeri quoque movendi <expan abbr="difficultat&etilde;">difficultatem</expan> nece&longs;&longs;e e&longs;t. </s> </p> <p type="main"> <s id="s.000982">Quoniam verò obtineri nequit, ut corporum &longs;e contingen­<lb/>tium &longs;uperficies &longs;int continuo lævore lubricæ, earum autem <pb pagenum="128" xlink:href="017/01/144.jpg"/>a&longs;peritates anomalæ &longs;unt ac multiformes, re&longs;i&longs;tentia indè pro­<lb/>veniens &longs;ub certam legem non cadit; &longs;ed quantum conjectura <lb/>a&longs;&longs;equi valemus, illa potius ex antiquis experimentis æ&longs;timanda <lb/>videtur, quàm mathematicis ratiocinationibus indaganda. </s> <s id="s.000983">In <lb/>hoc uno nimirùm facem præferre pote&longs;t Geometria, ut &longs;i reli­<lb/>qua pror&longs;us paria &longs;int, nec alia &longs;it quàm molis aut figuræ di&longs;&longs;i­<lb/>militudo, quantum ex hoc capite movendi difficultas augea­<lb/>tur, minuaturve, innnote&longs;cat: cæterùm plenè atque perfectè <lb/>explicare, quantum re&longs;i&longs;tentiæ ex a&longs;perarum &longs;uperficierum <lb/>conflictione oriatur, quis ni&longs;i temerè conetur? </s> </p> <p type="main"> <s id="s.000984">Po&longs;teriori huic malo, quod &longs;uperficierum aliqua a&longs;peritas <lb/>creat, occurritur, &longs;i pingui &longs;equacíque materiâ oblitæ lubri­<lb/>cæ fiant: Sic Automatis, rotarum &longs;e &longs;e mutuá collabellatione <lb/>mordentium conver&longs;ione, horas indicantibus velocitas conci­<lb/>liatur, &longs;i quis denticulos oleo leviter perungat: &longs;ic plau&longs;trorum <lb/>tarditatem, equorumque laborem, ut imminuant aurigæ, axes <lb/>rotarúmque modiolos axungiâ illinunt; & cæmentarij majora <lb/>&longs;axa attollentes, trochleæ orbiculis &longs;apone perfricatis, quærunt <lb/>laboris compendium. </s> <s id="s.000985">Hinc Am&longs;terodami pa&longs;&longs;im ob&longs;ervatur <lb/>lubricas fieri trahas cerui&longs;iæ doliis, &longs;imilíve pondere, onu&longs;tas; <lb/>cum enim equus non procul abe&longs;t à ponte, in quem a&longs;cenden­<lb/>dum e&longs;t, is, qui equum agit, centonem unguine delibutum <lb/>currenti trahæ &longs;ub&longs;ternit, ut expre&longs;&longs;us ex centone pinguis hu­<lb/>mor inficiat duo illa longiora tigna, quibus traha in&longs;i&longs;tit, ac <lb/>proinde lubrica machina faciliùs raptetur per vias lateribus <lb/>&longs;tratas. </s> <s id="s.000986">Sic Dio lib.50. de Augu&longs;to loquens. <emph type="italics"/>Audivi eum trire­<lb/>mes ex mari exteriore per murum in &longs;inum tran&longs;tuli&longs;&longs;e, & loco Pa­<lb/>langum, per quos ducerentur, tergoribus animalium recens cæ&longs;orum <lb/>loco inunctis u&longs;um,<emph.end type="italics"/> Et Silius Ital. <!-- KEEP S--></s> <s id="s.000987">lib.13.v.444. <lb/><emph type="italics"/>Lubrica roboreis aderant &longs;ub&longs;tramina plau&longs;tris, <lb/>Atque recens cæ&longs;i tergo prolap&longs;a juvenci, <lb/>Æquorcem rota ducebat per gramina puppim.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.000988">Verùm nec frequens e&longs;&longs;e pote&longs;t, nec commodum, remedium <lb/>hoc ex pingui liquore petitum; illud certius erit ad imminuen­<lb/>dam moram ex tritu corporum ortam, quod ea &longs;e invicem <lb/>quàm minimùm contingant. </s> <s id="s.000989">Quoniam verò deducendi one­<lb/>ris &longs;uperficiem amplam mutare &longs;æpè nequimus, aut illud rap­<lb/>tandum trahæ imponimus, quæ non ni&longs;i tigillis duobus læviga-<pb pagenum="129" xlink:href="017/01/145.jpg"/>tis &longs;ubjectam planitiem tangit; aut in plau&longs;trum injicimus, cu­<lb/>jus rotæ &longs;olum calcantes dum convertuntur, axem tantum­<lb/>modo terunt, compendio &longs;anè mirabili; nam dum rotæ modio­<lb/>lus axem &longs;emel terit, pedes circiter viginti provehitur onus, aut <lb/>demum &longs;ublato corporum mutuo tritu cylindros, vel &longs;cytalas <lb/>illi &longs;ubjicimus, ut nihil noceat &longs;oli a&longs;peritas, ni&longs;i quatenus hæc <lb/>cylindrorum vel &longs;cytalarum conver&longs;ionem remoratur. </s> </p> <p type="main"> <s id="s.000990">Huc &longs;pectat id, quod non &longs;ine voluptate ob&longs;ervare aliouan­<lb/>do contigit Bononiæ. </s> <s id="s.000991">Tres erant viri nec admodum robu&longs;ti, <lb/>qui ut aliquot ingentes &longs;accos farinâ plenos in domum infer­<lb/>rent, paratum habuerunt axem binis rotulis circiter &longs;e&longs;quipal­<lb/>maribus in&longs;tructum; axi jungebatur cra&longs;&longs;iu&longs;culus temo &longs;acco­<lb/>rum longitudinem vix &longs;uperans. </s> <s id="s.000992">Erecto &longs;acco machinulam ap­<lb/>plicabant, tùm &longs;accum pariter cum temone reclinabant, & ne <lb/>temoni incumbens juxtà longitudinem &longs;accus in alterutram <lb/>partem inclinaretur, duo hinc & hinc retinebant pariter, ac <lb/>propellebant, ut tertium arrepto temone trahentem labore le­<lb/>varent: Hâc ratione alium atque alium &longs;accum tenui&longs;&longs;imo la­<lb/>bore in domum importarunt; erectoque iterum temone delap­<lb/>&longs;us e&longs;t ex machinulâ &longs;accus, &longs;tetitque erectus. </s> </p> <p type="main"> <s id="s.000993">Ex his itaque con&longs;tat in machinâ in&longs;truendâ non &longs;olùm in­<lb/>genitæ corpori movendo gravitatis rationem habendam e&longs;&longs;e; <lb/>&longs;ed & plani, &longs;uper quo illud deducendum e&longs;t, jacens-ne &longs;it? </s> <lb/> <s id="s.000994">an erectum? </s> <s id="s.000995">læve, an a&longs;perum? </s> <s id="s.000996">amplâ, an tenui &longs;uperficie <lb/>contingat? </s> <s id="s.000997">hinc &longs;i quidem varia re&longs;i&longs;tentiæ momenta exur­<lb/>gunt. </s> <s id="s.000998">Illud tamen plerumque contingit, quod &longs;i attollendo ad <lb/>perpendiculum oneri par fuerit machina, illa pariter &longs;ufficiat <lb/>ad onus idem &longs;uper plano horizontali, aut inclinato deducen­<lb/>dum: vix enim fieri pote&longs;t (ni&longs;i &longs;umma &longs;it &longs;uperficierum &longs;e <lb/>contingentium a&longs;peritas) ut quantum re&longs;i&longs;tentiæ demitur à <lb/>plano &longs;u&longs;tinente, tantumdem addatur ex mutuo prominentium <lb/>particularum conflictu. </s> </p> <p type="main"> <s id="s.000999">Quamquam & ip&longs;a a&longs;peritas facit aliquod laboris compen­<lb/>dium: nam licèt continens ac perpetuus non &longs;it motus, &longs;ed al­<lb/>ternâ quiete interruptus &longs;uper arduo clivo, modico tamen co­<lb/>natu prohibetur moles, ne prolap&longs;a &longs;i&longs;ipheum crect laborem; <lb/>quia a&longs;pera &longs;uper&longs;icies motui ob&longs;i&longs;tens efficit ne corporis gravi­<lb/>tas deor&longs;um conetur pro plani inclinatione. </s> <s id="s.001000">Satis igitur fuerit <pb pagenum="130" xlink:href="017/01/146.jpg"/>ab&longs;olutæ oneris gravitati machinam ita re&longs;pondere, ut illi ad <lb/>perpendiculum &longs;u&longs;tollendo cæteroqui impares vires &longs;ufficiant: <lb/>qui enim valuerit, adhibitâ machinâ, molem attollere, poterit <lb/>illam pariter, eju&longs;dem machinæ ope, in plano quocunque tra­<lb/>here aut propellere; &longs;i maximè cylindri aut rotæ ei &longs;ubji­<lb/>ciantur. </s> </p> <p type="main"> <s id="s.001001">Hîc autem fortè nec à præ&longs;enti in&longs;tituto alienum, nec lecto­<lb/>ri injucundum accidat, &longs;i quæ, aliquando commini&longs;ci placuit, <lb/>&longs;ubjiciam, cum narrantem quendam audirem de campaná in­<lb/>gentis ponderis facillimè agitatâ &longs;ubjectis æneis rotulis, quæ <lb/>demum longo ævo confectæ di&longs;&longs;ipatæ fuere; &longs;ed quonam artifi­<lb/>cio, quóve ordine di&longs;po&longs;itæ fui&longs;&longs;ent, enarrare omninò non <lb/>poterat. </s> <s id="s.001002">Quare mecum ip&longs;e reputans, quî fieri id potui&longs;&longs;et, in <lb/>eam incidi &longs;ententiam, ut exi&longs;timarem gravi&longs;&longs;imam campanam <lb/>potui&longs;&longs;e facilè pul&longs;ari, imminutâ re&longs;i&longs;tentiâ, quæ oritur ex mu­<lb/><figure id="id.017.01.146.1.jpg" xlink:href="017/01/146/1.jpg"/><lb/>tuo fulcri, & axis tritu. </s> <s id="s.001003">Sint <lb/>enim binæ rotulæ B & C ex <lb/>ære &longs;olido, quarum diameter <lb/>&longs;it in aliquâ Ratione multiplici <lb/>ad diametrum axis, cui cam­<lb/>pana innititur. </s> <s id="s.001004">Axis autem &longs;e­<lb/>midiameter &longs;it AE, rotulæ ve­<lb/>rò BE in ratione duplâ; ergo <lb/>& periphæriæ &longs;unt in eâdem Ratione: dum igitur punctum I <lb/>in H perficit quadrantem, convertit pariter rotulam; cujus pe­<lb/>ripheriæ &longs;emiquadranti coæquatur. </s> <s id="s.001005">Quare &longs;i rotula infixa e&longs;&longs;et <lb/>axi, cujus &longs;emidiameter BG e&longs;&longs;et æqualis &longs;emidiametro AE, <lb/>fieret affrictus cum octante peripheriæ axis rotulæ B; &longs;ed quia <lb/>etiam in rotulâ C fieret æqualis affrictus cum eju&longs;dem axe, jam <lb/>nihil ferè emolumenti haberetur, quia totus affrictus æquè e&longs;­<lb/>&longs;et, ac &longs;i quadrans EO in fulcro &longs;tabili & cavo converteretur: <lb/>& potiùs laboris in agitandâ campanâ compendium e&longs;&longs;et, &longs;i ro­<lb/>tulæ fixæ hærerent, axis &longs;i quidem cylindricus cum &longs;it, &longs;ubjectas <lb/>rotulas in lineâ tangeret modico &longs;cilicet tritu; rotularum autem <lb/>axes concavis earum partibus congruunt in &longs;uperficie, quæ te­<lb/>ritur, dum rotulæ convertuntur: ni&longs;i fortè cylindrica axis <lb/>BG &longs;uperficies convexa paulò minor e&longs;&longs;et concavâ rotulæ <lb/>&longs;uperficie, eæque propterea &longs;ecundùm lineam &longs;e continge-<pb pagenum="131" xlink:href="017/01/147.jpg"/>rent, ut ex 13. lib.3. facilè e&longs;t demon&longs;trare; id quod nec rarò <lb/>contingit. </s> </p> <p type="main"> <s id="s.001006">Verum non e&longs;t nece&longs;&longs;e rotulis B & C tàm &longs;olidos axes dare; <lb/>nam &longs;i axis AE toti campanæ oneri ferendo par e&longs;t, bini æqua­<lb/>les axes duplici ponderi re&longs;i&longs;tunt: &longs;atis igitur e&longs;&longs;et, &longs;i axes &longs;in­<lb/>guli B & C, oneris &longs;emi&longs;&longs;em &longs;u&longs;tinerent. </s> <s id="s.001007">Cum verò cylindro­<lb/>rum re&longs;i&longs;tentiæ, ne frangantur, &longs;int in triplicatâ Ratione &longs;ua­<lb/>rum diametrorum, &longs;ufficeret inter &longs;emidiametrum AE, & ejus <lb/>&longs;emi&longs;&longs;em duas medias proportione continuâ reperire, quæ enim <lb/>proxime minor e&longs;&longs;et ipsá AE, e&longs;&longs;et &longs;ufficiens &longs;emidiameter cy­<lb/>lindri &longs;ubduplam habentis &longs;oliditatem ac re&longs;i&longs;tentiam. </s> <s id="s.001008">Sed <lb/>adhuc minor requiritur &longs;emidiameter, quia onus axes rotula­<lb/>rum B & C obliquè premit; ex quo fit campanæ gravitationem <lb/>in axes illos e&longs;&longs;e &longs;ecundùm lineas AB, AC, non autem juxtà <lb/>perpendiculum AD: igitur ut AD ad AB, ita reciprocè gra­<lb/>vitatio &longs;uper AB ad gravitationem &longs;uper AD: atqui gravita­<lb/>tio in alterutrum axium, ut &longs;ummum &longs;ubdupla e&longs;t totius gra­<lb/>vitationis; ergo gravitatio &longs;uper BA minor e&longs;t &longs;ubduplâ. </s> <s id="s.001009">Quâ <lb/>autem Ratione minor &longs;it con&longs;tat. </s> <s id="s.001010">Cum enim detur tùm &longs;emi­<lb/>diameter AE, tùm etiam BE, nota e&longs;t tota BA, & BD, pari­<lb/>ter, ip&longs;i BE æqualis, nota e&longs;t; igitur ex 47 lib. 1. etiam AD <lb/>innote&longs;cit, cujus &longs;cilicet quadratum habetur, &longs;i ex BA quadra­<lb/>to dematur quadraturm BD. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001011">Cum itaque, ex hypothe&longs;i, BA &longs;it 3, cujus quadratum 9, & <lb/>BD 2, cujus quadratum 4, remanet quadratum 5, eju&longs;que Ra. </s> <lb/> <s id="s.001012">dix 2. 23″. <!-- REMOVE S-->e&longs;t recta DA: gravitatio igitur &longs;uper BA ad totam <lb/>campanæ &longs;uper utrumque axem B, & C, gravitationem e&longs;t <lb/>223 ad 600′. </s> <s id="s.001013">Quoniam verò &longs;olidorum &longs;imilium re&longs;i&longs;tentia <lb/>e&longs;t in triplicatâ Ratione laterum homologorum (in cylindris <lb/>autem diametrorum ratio habetur) quærantur duo medij pro­<lb/>portionales numeri inter 600″ & 223″. <!-- KEEP S--></s> <s id="s.001014">Id quod a&longs;&longs;equeris, &longs;i <lb/>cuju&longs;libet extremi quadratum ducas in alium extremum, pro­<lb/>ducti enim Radix cubica e&longs;t terminus proximus illi numero, <lb/>cujus quadratum a&longs;&longs;ump&longs;i&longs;ti. </s> <s id="s.001015">Primi igitur 600 quadratum <lb/>360000 duc in 223, & producti 80280000, Radix cubica e&longs;t <lb/>431 1/3 proximè: alterius verò extremi 223 quadratum 49729 <lb/>ductum in 600 dat 29837400, cujus Radix cubica 310 proxi­<lb/>mè e&longs;t alter medius. </s> <s id="s.001016">Sunt igitur quatuor numeri 600. 431 1/3. <pb pagenum="132" xlink:href="017/01/148.jpg"/>310. </s> <s id="s.001017">223 continuè proportionales proximè, &longs;pretis fractiuncu­<lb/>lis. </s> <s id="s.001018">Quare &longs;i &longs;iat ut 600′ ad 431′, ita &longs;emidiameter AE ad BN, <lb/>erit hæc &longs;emidiameter quæ&longs;ita &longs;ufficienter re&longs;i&longs;tens. </s> </p> <p type="main"> <s id="s.001019">Quoniam itaque BE dupla e&longs;t ip&longs;ius AE, & AE ad BN <lb/>facta e&longs;t ut 600 ad 431, erit BE ad BN ut 1200 ad 431; & &longs;e­<lb/>cundùm hane eandem Rationem &longs;e habebunt &longs;emiquadrantes <lb/>ab illis de&longs;eripti. </s> <s id="s.001020">Atqui octans peripheriæ ex Radio BE æqua­<lb/>lis e&longs;t quadranti ex Radio. <!-- KEEP S--></s> <s id="s.001021">AE; igitur quadrans EO ad &longs;emi­<lb/>quadrantem ex Radio BN e&longs;t pariter ut 1200 ad 431: Qui igi­<lb/>tur affrictus axis campanæ cum fulcro &longs;tabili & cavo e&longs;&longs;et 1200, <lb/>rotulæ B cum &longs;uo axe e&longs;t 431, cui æqualis e&longs;t alterius rotulæ C <lb/>affictus cum &longs;uo axe; ac proinde &longs;ubjectis rotulis, quarum dia­<lb/>meter &longs;it tantum dupla diametri axis. </s> <s id="s.001022">campanæ, affrictus e&longs;t ut <lb/>862, ad affrictum qui e&longs;&longs;et ut 1200. Si itaque rotularum dia­<lb/>meter ad campanæ axem. </s> <s id="s.001023">non tantùm dupla, &longs;ed vel tripla, <lb/>vel quadrupla &longs;it, multò minor erit affrictus, majorque in agi­<lb/>tanda campanâ facilitas. </s> </p> <p type="main"> <s id="s.001024">Quamvis autem i&longs;tâ con&longs;imilivè diligentiâ indu&longs;triâque plu­<lb/>rimum imminui po&longs;&longs;it particularum conflictus, quæ &longs;e vici&longs;&longs;im <lb/>terentes moram atque impedimentum motui inferrent; non illa <lb/>tamen ex eo propriè veréque dicitur motio machinalis, quòd <lb/>in&longs;trumento atque apparatu aliquo perficiatur, ni&longs;i, &longs;pectatâ <lb/>dumtaxat oneris gravitate, potentia illi movendo cæteroqui im­<lb/>par, &longs;ub&longs;idium &longs;ibi comparet ex machinâ. </s> <s id="s.001025">Machina autem non <lb/>idem e&longs;t, &longs;i plenè atque perfectè interpretari velis, ac in&longs;tru­<lb/>mentum; licet enim machina omnis in&longs;trumentum &longs;it, non ta­<lb/>men in&longs;trumentum quodlibet machinæ vocabulum continuò <lb/>&longs;ortitur, &longs;i motionem aliquatenùs juvet; &longs;ed illud prætereà ef­<lb/>ficiat nece&longs;&longs;e e&longs;t, quod ejus ope naturalem ac in&longs;itam vim cor­<lb/>poris loco dimovendi &longs;uperet vis minor extrin&longs;ecùs adhibita. </s> <lb/> <s id="s.001026">Cum ergò onus hærere in &longs;alebrâ, non ex in&longs;itâ vi, &longs;ed ex proxi­<lb/>mi etiam atque continentis corporis a&longs;peritate proveniat, & <lb/>in&longs;trumenta, quibus hoc tantummodo impedimentum tollitur, <lb/>idem planè efficiant, quod pinguis humor lubricum parans iter; <lb/>neque hæc machinæ magis dici po&longs;&longs;unt, quàm centones ungui­<lb/>ne delibuti, &longs;i ritè &longs;ub&longs;ternantur, neque motus propterea inter <lb/>machinales numerandus videtur, quorum hîc cau&longs;as ye&longs;tigare <lb/>nobis propo&longs;itum e&longs;t. </s> <s id="s.001027">Quamquam negandum non &longs;it hæc pari-<pb pagenum="133" xlink:href="017/01/149.jpg"/>ter ad mechanicam contemplationem pertinere; quippe quæ <lb/>machinis, præcipuo nimirum mechanices &longs;copo. </s> <s id="s.001028">affinia &longs;unt; <lb/>etiam&longs;i ad illas non velut &longs;ubjectæ partes ad genus revocentur: <lb/>& in&longs;trumentis huju&longs;modi &longs;i machinæ appellationem tribuere <lb/>placuerit, non admodum de nomine di&longs;putabo; res enim hîc <lb/>&longs;pectatur, non verba penduntur. </s> </p> <p type="main"> <s id="s.001029">Sed neque hîc di&longs;putare velim, utrùm in motuum machina­<lb/>lium cen&longs;um irrepant, an verò iis ritè annumerandi &longs;int motus <lb/>illi, quos &longs;ur&longs;um deor&longs;um, ultrò citróque perficiendos eatenus <lb/>expeditè, nec exiguo laboris compendio, molimur, quatenus <lb/>eos intervallis ita di&longs;tinguimus, ut nos quidem corpus deprima­<lb/>mus, ut adducamus, ab alio verò extollatur, aut reducatur: in <lb/>his &longs;iquidem &longs;æpè nihil e&longs;t, quod no&longs;tram imminuat operam, <lb/>&longs;i motiones &longs;ingulæ attendantur; quamquam motui univer&longs;o <lb/>adjumentum importat continens illa conatûs no&longs;tri, alienique <lb/>&longs;ub&longs;idij, vici&longs;&longs;itudo. </s> <s id="s.001030">Hinc &longs;i quis <lb/><figure id="id.017.01.149.1.jpg" xlink:href="017/01/149/1.jpg"/><lb/>ad contundendam in æneo morta­<lb/>rio A contumacem aliquam mate­<lb/>riam graviore pi&longs;tillo ferreo opus <lb/>habeat, haud dubium quin ei mul­<lb/>tâ lacertorum vi contendendum <lb/>&longs;it, ut illum extollat; cumque ope­<lb/>ro&longs;ius multo &longs;it inflexum corpus <lb/>erigere, quàm erectum inclinare, <lb/>multóque mole&longs;tius brachia tanto <lb/>pondere pre gravata attollere, quàm <lb/>eorum gravitati ob&longs;ecundando de­<lb/>primere, &longs;atis con&longs;tat, quantum &longs;i­<lb/>bi laboris detractum eat, &longs;i &longs;uperio­<lb/>re in loco tran&longs;ver&longs;um tigillum <lb/>CD circa axem E ver&longs;atilem &longs;tatuat, paribú&longs;que intervallis <lb/>hinc ex C pendeat fune &longs;u&longs;pen&longs;us pi&longs;tillus B, hinc verò in D <lb/>plumbea ma&longs;&longs;a adnectatur, quâ ita pi&longs;tillus præponderetur, ut, <lb/>nemine hunc retinente aut deprimente, illa aliquanto gravior <lb/>in &longs;ubjectum prodeuntis è pariete tigni caput G recidens &longs;pon­<lb/>te &longs;ub&longs;idat. </s> <s id="s.001031">Omnis &longs;cilicet extollendi pi&longs;tilli labore &longs;ublato, <lb/>vel &longs;olum brachiorum pondus pi&longs;tillo additum &longs;atis e&longs;&longs;e ali­<lb/>quando poterit ad leviu&longs;culè tundendam materiam, licebitque <pb pagenum="134" xlink:href="017/01/150.jpg"/>modò contento, modò remi&longs;&longs;o conatu opus urgere. </s> <s id="s.001032">Id quod <lb/>pariter continget, &longs;i operâ unâ opus duplex efficere placuerit; <lb/>nam &longs;i ex D plumbeæ ma&longs;&longs;æ loco alius pendeat æque, ac plum­<lb/>bum, gravis pi&longs;tillus, pondere præpollens elevabit pi&longs;tillum B, <lb/>aliámque vici&longs;&longs;im in altero &longs;ubjecto mortario conteret mate­<lb/>riam &longs;ponte &longs;uâ cadens: cumque pi&longs;tillorum gravitates non ad­<lb/>modum inter &longs;e di&longs;pares &longs;int, neque multum laboris eum &longs;ubi­<lb/>re nece&longs;&longs;e erit, cui pi&longs;tillum B deprimendi munus incumbit. </s> </p> <p type="main"> <s id="s.001033">Quâ in re, &longs;i motus univer&longs;us ita tribuatur in partes, ut tun­<lb/>dentis quidem motiones &longs;ingulæ &longs;eor&longs;im &longs;pectentur, non ille <lb/>profectò &longs;e juvari &longs;entit, quippe quem, præter vires ad commi­<lb/>nuendam materiam nece&longs;&longs;arias, conatum quoque adhibere <lb/>oportet ad vincendam præponderantis plumbi, aut pi&longs;tilli gra­<lb/>vitatem. </s> <s id="s.001034">Cæterùm &longs;i totius motûs, qui Ar&longs;i pariter con&longs;tat ac <lb/>The&longs;i, habeatur ratio, in&longs;iciari nemo poterit, minus multo la­<lb/>boris impendi, quàm &longs;i hæc omnia &longs;ublata intelligantur. </s> <s id="s.001035">Qua­<lb/>re nec incongruum pror&longs;us videatur motûs machinalis voca­<lb/>bulum, cum ver&longs;atilis tigillus CD ad libræ Rationes manife&longs;tò <lb/>revocetur, quam certè ex machinarum albo nemo expungit, ni­<lb/>&longs;i qui &longs;olas quinque facultates, & quæ ex his componuntur, ma­<lb/>chinas indigitare voluerit, & libram ad vectem referri po&longs;&longs;e <lb/>pernegarit. </s> </p> <p type="main"> <s id="s.001036">Nec di&longs;&longs;imilis ineunda videtur dicendi ratio, &longs;i quid alternis <lb/>ciendum motibus &longs;ic di&longs;ponitur, ut, cum primùm quidem mo­<lb/>vetur, corpus aliud vi flectatur, quod po&longs;tmodum facultate <lb/>ela&longs;ticâ, &longs;e re&longs;tituens illud vici&longs;&longs;im moveat; quemadmodum <lb/>pa&longs;&longs;im in eorum officinis videre e&longs;t, qui rudes arborum, aut <lb/>elephantini dentis particulas in toreumata elaborant: primùm <lb/>enim artifex pede &longs;ubjectum vectem premens, toreuma in gy­<lb/>rum ducit, ha&longs;tulámque &longs;uperiore in loco po&longs;itam pariter in­<lb/>flectit; quæ &longs;ibi mox &longs;uam reparans rectitudinem, funiculum­<lb/>que cylindrulo ver&longs;atili circumplicatum retrahens, illud iterum <lb/>&longs;ua per ve&longs;tigia ver&longs;at, ut accuratè exqui&longs;itéque tornetur. </s> <s id="s.001037">Sic <lb/>aliquid &longs;ubtiliter ac delicatè &longs;ecturus, ut &longs;errulam rectâ addu­<lb/>cas, reducá&longs;que, operæ tantùm &longs;emi&longs;&longs;em tibi re&longs;ervans, arcum <lb/>intentum ex adver&longs;o &longs;tatuito, ac medio nervo &longs;errulam alliga­<lb/>to; hac enim adductâ magis flectetur arcus, qui &longs;e &longs;e mox re&longs;ti­<lb/>tuens illam vici&longs;&longs;im reducet. </s> </p> <pb pagenum="135" xlink:href="017/01/151.jpg"/> <p type="main"> <s id="s.001038">Hæc &longs;anè laboris in movendo compendia ex ela&longs;mate, vel ex <lb/>anti&longs;acomate petita, quemadmodum & ea, quæ mutuum cor­<lb/>porum tritum atque conflictum minuunt, ut pote Mechanico <lb/>artificio con&longs;tituta, eumdemque in finem ac machinæ, quibus <lb/>hoc nomen præcipuè tribuitur, videlicet in infirmæ potentiæ <lb/>&longs;ub&longs;idium excogitata, e&longs;to illis primas deferant, non tamen <lb/>omninò rejicerem, &longs;i in machinarum cen&longs;u prodirent, ii&longs;que <lb/>&longs;e peterent ad&longs;cribi. </s> <s id="s.001039">Triplicem enim in &longs;peciem tribui po&longs;&longs;e vi­<lb/>detur univer&longs;um machinarum genus: Prima eas complectitur <lb/>facultates, quarum ope motui facilitas conciliatur, quocum­<lb/>que tandem ex capite &longs;ivè tantummodo ex in&longs;itâ in corporibus <lb/>gravitate, &longs;ivè non ex eâ dumtaxat, &longs;ed ex partium a&longs;peritate <lb/>movendi difficultas con&longs;urgat. </s> <s id="s.001040">Altera e&longs;t, quæ mutuam qui­<lb/>dem corporum &longs;e contingentium conflictionem minuit, &longs;ed ad <lb/>vincendam oneris gravitatem ip&longs;i potentiæ momenta non addit. </s> <lb/> <s id="s.001041">Tertia demùm eatenus per &longs;e, quia talis e&longs;t, moventem juvat, <lb/>quatenus ejus operam alternam efficit, cum tamen neque gra­<lb/>vitatem vincat, neque quod ex partium triru impedimentum <lb/>oritur, extenuet, ni&longs;i cum alterutra, aut utraque &longs;uperiori &longs;pe­<lb/>cie, amico fœdere copuletur. </s> <s id="s.001042">Alternam autem operam appel­<lb/>lo, cum in motu ex duplici motione compo&longs;ito alterutram effi­<lb/>cit potentia, &longs;ivè illæ &longs;ibi invicem adver&longs;antes &longs;uccedant, ut <lb/>Ar&longs;is ac The&longs;is, Adductio atque Reductio, &longs;ivè in unam tem­<lb/>perentur, ut cum premere &longs;imul oportet ac agitare: &longs;ic plana <lb/>vitra expolientes in &longs;pecula, inter ip&longs;a, & lacunar bacillum in­<lb/>flectunt, qui &longs;e re&longs;tituere tentans vi ela&longs;ticâ, &longs;peculum validè, <lb/>quantum opus e&longs;t, admovet atque applicat ad &longs;ubjectum pla­<lb/>num, adeò ut ad artificem à pre&longs;&longs;u immunem nil aliud &longs;pectet, <lb/>quàm &longs;peculum urgere, retrahere, contorquere. </s> <s id="s.001043">Verùm ta­<lb/>met&longs;i de his omnibus in hac tractione pa&longs;&longs;im &longs;e offeret dicendi <lb/>locus, primus tamen di&longs;putationis no&longs;træ &longs;copus erit prima illa <lb/>&longs;pecies, ip&longs;æ nimirum facultates, quarum poti&longs;&longs;imum momen­<lb/>ta expendimus, cum motûs machinalis cau&longs;as inquirimus. <pb pagenum="136" xlink:href="017/01/152.jpg"/></s> </p> <p type="main"> <s id="s.001044"><emph type="center"/>CAPUT II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001045"><emph type="center"/><emph type="italics"/>Impetùs motum proximè efficientis natura <lb/>explicatur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001046">QUicquid movetur, qualecumque e&longs;t, cau&longs;am habeat mo­<lb/>ventem nece&longs;&longs;e e&longs;t, ut hoc quidem &longs;ponte &longs;uâ, illud ve­<lb/>rò alienâ vi ex alio in alium locum migret. </s> <s id="s.001047">Suopte ingenio mo­<lb/>ventur tùm corpora gravia aut levia, ut &longs;i extrà præ&longs;criptum <lb/>&longs;ibi à naturâ locum con&longs;tituta fuerint, &longs;uo quæque ordine di&longs;­<lb/>ponantur; tùm rara aut den&longs;a, ut &longs;i per vim hæc extenuata fue­<lb/>rint, illa concreverint, naturæ &longs;tatum &longs;ibi reparent; tùm ani­<lb/>mantia, quibus cum à naturâ tributum &longs;it, ut &longs;e, vitam, cor­<lb/>pu&longs;que tueantur, &longs;timulos admovet appetitus, ut ea declinent, <lb/>quæ nocitura videantur, omniaque, quæ &longs;int ad vivendum ne­<lb/>ce&longs;&longs;aria, acquirant, & parent. </s> <s id="s.001048">Vi extrin&longs;ecus impre&longs;sâ locum <lb/>mutant, quæcumque in motu non &longs;erviunt naturæ, &longs;ed alieno <lb/>reguntur arbitrio; ut iis contingit, quæ raptantur, pelluntur, in <lb/>gyrum ducuntur, projiciuntur, & hujus generis motibus <lb/>cientur. </s> </p> <p type="main"> <s id="s.001049">Quoniam verò gravium, & levium celeritatem naturâ ur­<lb/>gente incitari, jaculorum autem, ac mi&longs;&longs;ilium, motum u&longs;que <lb/>eò &longs;en&longs;im langue&longs;cere, ut planè deficiat, ob&longs;ervamus; etiam&longs;i <lb/>moventi naturæ, quæ ex Philo&longs;ophi decretis &longs;ub&longs;tantia e&longs;t, mo­<lb/>tûs originem ultimam tribuamus, jure tamen optimo aliquid <lb/>naturæ ip&longs;i ac motui, interjectum agno&longs;cimus (Impetum no­<lb/>minamus) cujus intentionem ac remi&longs;&longs;ionem velocitas ac tar­<lb/>ditas con&longs;equatur. </s> <s id="s.001050">Cum enim eadem de&longs;cendentis lapidis na­<lb/>tura per&longs;everet, nec illa in &longs;uâ pote&longs;tate &longs;it, aut optione delatâ, <lb/>ut eligat utrum velit, motum arbitrio &longs;uo incitare, aut remit­<lb/>tere valeat; qui fieri po&longs;&longs;it, ut de&longs;cendens velocitatem augeat, <lb/>ni&longs;i ei, quem primùm produxit, alium atque alium momentis <lb/>&longs;ingulis impetum adjiciat? </s> <s id="s.001051">Illud certè extrà omnem controver­<lb/>&longs;iam po&longs;itum videtur, naturam gravem &longs;ponte &longs;uâ non a&longs;cen-<pb pagenum="137" xlink:href="017/01/153.jpg"/>dere: quid ergo illud e&longs;t, quod eburneum globulum in &longs;ub­<lb/>jectam rupem delap&longs;um re&longs;ilire cogit, aut &longs;ibi relictum plum­<lb/>bum ex fune &longs;u&longs;pen&longs;um ultrà perpendiculum, naturá repugnan­<lb/>te, &longs;ur&longs;um provehit, & eò quidem altiùs, quò ex altiore loco <lb/>globulus aut plumbum deciderunt? </s> <s id="s.001052">ni&longs;i quia conceptus naturâ <lb/>procurante impetus pergit motum efficere, ipsâ etiam naturâ <lb/>quantum pote&longs;t, ob&longs;i&longs;tente. </s> <s id="s.001053">Quòd &longs;i corpus alienâ vi longiùs <lb/>emi&longs;&longs;um moveatur, extrin&longs;ecùs impetum imprimi nece&longs;&longs;e e&longs;t: <lb/>quem &longs;anè non concipit, ubi primùm à projiciente &longs;ejunctum <lb/>fuerit; nihil enim prode&longs;&longs;et ad longiorem lapidis jactum fun­<lb/>dam iterum ac tertiò circumducere, ni&longs;i alium atque alium im­<lb/>petum lapis conciperet, quandiù funditori adhærens unâ cum <lb/>ip&longs;o movetur. </s> </p> <p type="main"> <s id="s.001054">Quæcumque igitur moventur, impetum habent, quo ferun­<lb/>tur; cui &longs;atis probabili conjectura, proxima vis motum efficien­<lb/>di tribuenda videtur. </s> <s id="s.001055">Id quod in projectis quidem, ii&longs;que om­<lb/>nibus, quæ naturâ repugnante moventur, ita manife&longs;tum e&longs;t, <lb/>ut id pluribus demon&longs;trare non oporteat; nulla &longs;iquidem ade&longs;t <lb/>in&longs;ita motûs cau&longs;a; ab impetu igitur illo extrin&longs;ecùs impre&longs;&longs;o <lb/>motum effici nece&longs;&longs;e e&longs;t. </s> <s id="s.001056">At in cæteris, quibus &longs;e movendi <lb/>principium ine&longs;t, neme jure negaverit aut in motu impetum <lb/>acquiri, aut velocitatis incrementum ex impetus acce&longs;&longs;ione ori­<lb/>ri: quî enim fieret, ut excurrentes objectam fo&longs;&longs;am ampliore <lb/>&longs;altu tran&longs;ilirent faciliùs, quàm nullo præcedente cur&longs;u, &longs;i in <lb/>cur&longs;u ip&longs;o conceptus impetus non augeretur? </s> <s id="s.001057">Jam verò &longs;i &longs;e­<lb/>cundo temporis momento incitatur magis motus, quàm primo, <lb/>urgente &longs;cilicet etiam impetu, quem corpus priore motu acqui­<lb/>&longs;ivit; hic utique impetus, quem nunc gignere non pote&longs;t <lb/>prior motus, cum perierit, extitit pariter cum priore motu: <lb/>natura igitur movens priore momento & motum effecit & im­<lb/>petum. </s> <s id="s.001058">Atqui impetum ex eorum &longs;altem genere e&longs;&longs;e, quæ mo­<lb/>tum efficiant, con&longs;tat ex velociore motu po&longs;terioribus momen­<lb/>tis, naturâ pror&longs;us immutatâ, factoque impetûs incremento: <lb/>contrà verò motu, quâ motus e&longs;t, impetum non augeri &longs;atis <lb/>indicant mi&longs;&longs;ilia, quorum velocitas, dum moventur, &longs;en&longs;im <lb/>elangue&longs;cit. </s> <s id="s.001059">Igitur & priore illo temporis momento non mo­<lb/>tus impetum; &longs;ed impetus motum proximè effecit; impetum <lb/>autem procreavit innata movendi vis; cui id circo motio tri-<pb pagenum="138" xlink:href="017/01/154.jpg"/>buitur, quia id illa gignit, quod proximè motus con&longs;equitur, <lb/>& ad motum efficiendum natura de&longs;tinavit. </s> <s id="s.001060">Quid? <!-- KEEP S--></s> <s id="s.001061">quòd mo­<lb/>tui per &longs;e, quia ex alio in alium locum continuata migratio e&longs;t, <lb/>efficientiam ægrè tribuere po&longs;&longs;umus: quippe qui, cum in <lb/>fluxione con&longs;i&longs;tat, ita ut locus loco, &longs;eu potius, ut &longs;cholæ lo­<lb/>quuntur, Ubicatio Ubicationi, priori &longs;cilicet pereunti &longs;uccedat <lb/>po&longs;terior æquè fugax, inferioris notæ cen&longs;endus e&longs;t quàm im­<lb/>petus naturâ &longs;uâ aliquandiù permanens: labentia enim &longs;tanti­<lb/>bus deteriora e&longs;&longs;e, cæteris paribus, quis neget? </s> <s id="s.001062">effectum au­<lb/>tem causâ præ&longs;tabiliorem e&longs;&longs;e non po&longs;&longs;e ip&longs;a originis notio &longs;ua­<lb/>det, ne quid effectus habeat, quod non acceperit, aut aliquid <lb/>cau&longs;a dederit, quo ip&longs;a careret. </s> <s id="s.001063">Non igitur impetum motus, <lb/>&longs;ed motum impetus efficit. </s> </p> <p type="main"> <s id="s.001064">Porrò cum definitas ad agendum vires unaquæque cau&longs;a ob­<lb/>tineat, certa e&longs;t impetûs men&longs;ura, quæ cum innatâ movendi <lb/>facultate ita adæquatur, ut eo qua&longs;i termino circum&longs;cripta cen­<lb/>&longs;enda &longs;it potentia movens, nec unquam validiore conatu po&longs;&longs;it <lb/>&longs;e ip&longs;a urgere; &longs;i tamen omnem impetum antecedente motu a&longs;­<lb/>&longs;umptum mente &longs;ecernas. </s> <s id="s.001065">Et quidem omne animal (quippe <lb/>cui ine&longs;t appetitio & declinatio naturalis ejus, quod naturæ ac­<lb/>commodatum e&longs;t, aut infen&longs;um) non &longs;emper univer&longs;am illam <lb/>impetûs men&longs;uram exequitur, &longs;ed ut vult, ita utitur motu &longs;ui <lb/>corporis, quem aucto aut diminuto impetu modò intendit, mo­<lb/>dò remittit, pro ut interiore motu, rerumque appetitu &longs;imula­<lb/>tur. </s> <s id="s.001066">Contrà verò inanimum non &longs;uo arbitrio motûs intentio­<lb/>nem moderatur, &longs;ed naturæ juribus ob&longs;equens nihil prætermit­<lb/>tit impetûs, & quantum eniti pote&longs;t, opportunum in locum, &longs;i­<lb/>bique à naturâ con&longs;titutum, contendit. </s> <s id="s.001067">Cave tamen exi&longs;times <lb/>parem e&longs;&longs;e lapidis eju&longs;dem, & in aëre, & in aquâ de&longs;cendentis <lb/>impetum: natura &longs;cilicet ex medio dividendo, in quo perficien­<lb/>dus e&longs;t motus, metitur impetûs modum. </s> </p> <p type="main"> <s id="s.001068">Sed quoniam non pauca &longs;unt, quæ motui &longs;æpè adver&longs;antur, <lb/>hinc e&longs;t non &longs;emper eandem e&longs;&longs;e corporis &longs;e moventis velocita­<lb/>tem, quamvis pari impetu producto connitatur: deteritur nimi­<lb/>rum tantum impetus, quantum &longs;atis e&longs;t ad impedimentum &longs;ub­<lb/>movendum. </s> <s id="s.001069">Sivè enim objectum corpus propellendum &longs;it, &longs;ivè <lb/>medij particulæ locum ægrè dantes divellendæ aut compri­<lb/>mendæ &longs;int, &longs;ivè connexam molem pariter rapi oporteat, &longs;ivè <pb pagenum="139" xlink:href="017/01/155.jpg"/>quid aliud huju&longs;modi ad&longs;it, cui ni&longs;i vis inferatur, ut ex alio <lb/>in alium locum migret præter naturam, irritus reddatur corpo­<lb/>ris in motum propen&longs;i conatus; &longs;atis con&longs;tat illud motu agitan­<lb/>dum e&longs;&longs;e exteriùs: atque adeò quantum impetus illi imprimi­<lb/>tur oppo&longs;itæ propen&longs;ioni æquale, motui tantumdem &longs;ub­<lb/>trahitur. </s> </p> <p type="main"> <s id="s.001070">In iis &longs;anè, quæ alienâ vi extrin&longs;ecùs moventur, quia infi­<lb/>nitè progredi non licet, aliqua demum origo deprehenditur, <lb/>cui naturalis &longs;it motus: natura &longs;iquidem vis e&longs;t ciens motus in <lb/>corporibus nece&longs;&longs;arios; ita tamen certis tenetur legibus uni­<lb/>ver&longs;itatis rerum concinnitatem &longs;pectantibus, ut ne ab iis di&longs;ce­<lb/>dat, &longs;ingularibus corporibus vim aliquam inferri permittat, ubi <lb/>adver&longs;is propen&longs;ionibus inter &longs;e confligentibus validior præ&longs;tat <lb/>imbecilliori. </s> <s id="s.001071">Sic quia nefas e&longs;t aut corpora inanitatibus inter­<lb/>jectis conci&longs;a hiare, aut unum in proximi corporis locum, ni&longs;i <lb/>eo recedente, penetrare, aut diverticula flexione&longs;que in motu <lb/>&longs;ponte quærere; ideò & liquor in longiore &longs;iphonis, aut &longs;piri­<lb/>talis diabetis, crure de&longs;cendens continuum liquorem in brevio­<lb/>re crure a&longs;cendere cogit, totumque ex va&longs;e demum exhaurit; & <lb/>rapidè lap&longs;us torrens &longs;axa rapit, objecta&longs;que moles disjicit; & <lb/>ad perpendiculum cadens lapis &longs;ubjectum vitrum comminuit, <lb/>&longs;uique ve&longs;tigium in terrâ validiùs pre&longs;sâ relinquit. </s> <s id="s.001072">Verùm il­<lb/>lud firmum ac perpetuum e&longs;t, quòd ubi plus violentiæ opus e&longs;t, <lb/>parem conatum languidior motus con&longs;equitur. </s> <s id="s.001073">Id quod in <lb/><figure id="id.017.01.155.1.jpg" xlink:href="017/01/155/1.jpg"/><lb/>&longs;iphone ABC ob&longs;ervare in promptu e&longs;t, ex <lb/>cujus o&longs;culo C inæqualis aquæ copia de­<lb/>fluit paribus temporis intervallis: quò enim <lb/>magis aquæ &longs;uperficies in va&longs;e deprimitur, <lb/>eò lentiùs aqua ex &longs;iphone dilabitur: <lb/>quamvis &longs;cilicet aquæ crus BC implentis <lb/>pares &longs;int &longs;emper ad de&longs;cendendum vires, &longs;i <lb/>nihil, aut &longs;altem non inæqualiter, repugnet, <lb/>aquæ tamen crus BD brevius, & BI longius, & BA adhuc <lb/>longius implentis di&longs;par e&longs;t in a&longs;cen&longs;u repugnantia; ac pro­<lb/>pterea cum earumdem virium BC minor &longs;it Ratio ad majorem <lb/>re&longs;i&longs;tentiam BI, quàm ad minorem BD, languidior quoque <lb/>motus e&longs;t de&longs;cendentis aquæ ex BC, cùm graviorem aquam <lb/>BI, quàm cùm minùs gravem BD &longs;ursùm trahere oportet. </s> <s id="s.001074">At <pb pagenum="140" xlink:href="017/01/156.jpg"/>&longs;i externum &longs;iphonis crus ità decurtatum &longs;it in E, ut o&longs;culum E <lb/>& aquæ in va&longs;e &longs;uperficies I paribus ab&longs;int ab Horizonte inter­<lb/>vallis, aquam ideò hærere, nec amplius ex E fluere con&longs;tat, <lb/>quia aquæ BE ad de&longs;cendendum propen&longs;ionem, par aquæ BI <lb/>repugnantia, ne a&longs;cendat, elidit. </s> <s id="s.001075">Quòd &longs;i demum aquam in <lb/>va&longs;e imminuas, ut ejus &longs;uperficies paulò infra I, atque adeò <lb/>infra E o&longs;culum deprimatur, non jam aqua hæret in E, &longs;ed &longs;ua <lb/>per ve&longs;tigia in EB remeare cogitur, præponderatâ nimirum <lb/>majore gravitate aquæ implentis crus paulo longiùs quàm BI, <lb/>atque adeò quàm BE, quod illi ex hypothe&longs;i con&longs;tituimus <lb/>æquale; tantóque velociùs ab aquâ interioris cruris raperetur <lb/>exterior, quantò depre&longs;&longs;ior facta fui&longs;&longs;et in va&longs;e aquæ &longs;uper­<lb/>ficies. </s> </p> <p type="main"> <s id="s.001076">Hinc itaque fit, ut pro variâ corporis motui ob&longs;i&longs;tentis re­<lb/>pugnantiâ modò plus, modò minus impetûs reliquum &longs;it, quo <lb/>motû, celeritas aut tarditas perficiatur. </s> <s id="s.001077">Et &longs;i tanta &longs;it eorum <lb/>omnium, quæ motui moram inferunt, ob&longs;i&longs;tentia, ut ad eam <lb/>vincendam plus impetûs nece&longs;&longs;e &longs;it, quàm pro potentiæ facul­<lb/>tate, tunc nullus efficitur motus, quo corpus ex loco in locum <lb/>transferatur, &longs;ed aliqua ex peregrino impetu fit partium com­<lb/>pre&longs;&longs;io, aut di&longs;tractio; neque enim omnes corporis particulæ <lb/>homogeneæ &longs;unt, aut ita compactæ citrà omnes poros, ut nul­<lb/>la tenuiorum particularum compre&longs;&longs;io aut di&longs;tractio con&longs;equi <lb/>po&longs;&longs;it. </s> <s id="s.001078">Quod &longs;i ea &longs;it corporis per vim movendi natura aut po&longs;i­<lb/>tio, ut nullum planè &longs;ivè lationis, &longs;ivè rotationis, &longs;ivè vibratio­<lb/>nis, &longs;ivè con&longs;tipationis, &longs;ivè dilatationis motum concipere po&longs;­<lb/>&longs;it, aut violento in &longs;tatu permanere languido illo impetu, quem <lb/>vis extrin&longs;eca efficere valeret, nullum quoque impetum reci­<lb/>pit; quippe qui idcircò imprimeretur, ut motum præter natu­<lb/>ram efficeret, aut ut naturalem motum retunderet, aut etiam <lb/>pror&longs;us impediret. </s> <s id="s.001079">Quemadmodum enim &longs;i corporis alicujus <lb/>&longs;pecificam gravitatem in aquâ mutari non po&longs;&longs;e con&longs;tet, infer­<lb/>re continuò licet, corpus idem neque raritatem neque den&longs;ita­<lb/>tem in aquâ a&longs;&longs;ùmere po&longs;&longs;e; ex his &longs;iquidem &longs;pecificæ gravita­<lb/>tis mutatio oriretur: ita pariter ubi nihil haberi pote&longs;t eorum, <lb/>quæ impetum extrin&longs;ecùs impre&longs;&longs;um nece&longs;&longs;ariò con&longs;equuntur, <lb/>impetum quoque abe&longs;&longs;e non immeritò conjectamus. </s> </p> <p type="main"> <s id="s.001080">Si quis tamen animum diligentiùs adverrat, manife&longs;tò de-<pb pagenum="141" xlink:href="017/01/157.jpg"/>prehendet corpus idem magis repugnare motui, &longs;i celeriùs mo­<lb/>vendum &longs;it, minùs verò, &longs;i tardiùs: &longs;ic ferreæ an&longs;æ cubiculi <lb/>o&longs;tio infixæ magnetem armatum applicui, & &longs;iquidem paulò <lb/>velociùs magnetem traherem, disjungebatur ab ansâ; at len­<lb/>tiùs trahentem &longs;ub&longs;equebatur o&longs;tium, magnetis &longs;cilicet vim <lb/>non &longs;uperans, ubi lentè res peragebatur. </s> </p> <p type="main"> <s id="s.001081">An non oneri, quod potentia præ &longs;ui tenuitate propellere <lb/>non po&longs;&longs;e videtur, motus, qui momentis &longs;ingulis &longs;en&longs;um om­<lb/>nem fugiat, conciliari pote&longs;t, adeò ut, &longs;i illa quidem con&longs;tan­<lb/>ter urgeat, elap&longs;o demùm longo temporis intervallo appareat? </s> <lb/> <s id="s.001082">Sic incumbentem glebam tenerrimus na&longs;centis frugis caulicu­<lb/>lus tandem di&longs;cutit; duri&longs;&longs;ima marmora &longs;cindens caprificus lo­<lb/>co movet; & ædificia &longs;ub&longs;edi&longs;&longs;e, ac inæquabile &longs;olum pre&longs;&longs;i&longs;&longs;e, <lb/>rimæ demùm loquuntur. </s> <s id="s.001083">Tota igitur corporis, quod præter <lb/>naturam movendum e&longs;t, repugnantia metienda e&longs;t, quâ ex <lb/>principio ip&longs;o motum detrectante, quâ ex motûs celeritate, aut <lb/>tarditate: adeò ut pro variâ horum connexione di&longs;par movendi <lb/>difficultas oriatur. </s> </p> <p type="main"> <s id="s.001084">Ex quo fit impetu eodem moveri celeriùs po&longs;&longs;e corpus, quod <lb/>minorem &longs;ubit violentiam, tardiùs verò, cui vis major infer­<lb/>tur, &, &longs;i eadem &longs;it reciprocè Ratio tarditatis ad velocitatem, <lb/>quæ e&longs;t minoris violentiæ ad majorem violentiam, parem fore <lb/>utrobique movendi difficultatem, cùm par &longs;it repugnantia, quæ <lb/>ex motûs tùm &longs;pecie, tùm intentione componitur. </s> <s id="s.001085">Si enim mo­<lb/>les aliquâ tantâ vi raptetur, ut, quo tempore decies arteria pul­<lb/>&longs;um edit, pa&longs;&longs;um unum conficiat; quantum virium adhiberi <lb/>oporteat, ut paribus temporis momentis ad tres pa&longs;&longs;us eadem <lb/>moles promoveatur? </s> <s id="s.001086">utique, &longs;i cætera omnia paria &longs;int, triplo <lb/>majorem conatum adhibendum concedes, inten&longs;ione exten­<lb/>&longs;ionem compen&longs;ante: nam quemadmodum iterùm ac tertiò re­<lb/>petendus fui&longs;&longs;et prior ille conatus ad æquale &longs;emper &longs;patium pa­<lb/>ri tarditate percurrendum; ita quamvis conatui conatus non <lb/>&longs;uccedat, triplici tamen conatu opus erit, ut tempore eodem <lb/>motus ille triplo major perficiatur. </s> <s id="s.001087">Nonnè & agricolæ terram <lb/>&longs;ubigentes fo&longs;&longs;ione glebarum, tam multiplices adhibent operas, <lb/>quàm breviori tempore opus ab&longs;olvere meditantur? </s> <s id="s.001088">Eò igitur <lb/>magis re&longs;i&longs;tit corpus motui, quò celeriùs agitandum e&longs;t; con­<lb/>trà verò minùs repugnat, quò tardiùs. </s> </p> <pb pagenum="142" xlink:href="017/01/158.jpg"/> <p type="main"> <s id="s.001089">Quare &longs;i duo &longs;int corpora, quorum alterum alteri præ&longs;tet <lb/>triplo majori gravitate, atque hæc pari celeritate attollenda &longs;int, <lb/>di&longs;parem exigunt conatum pro gravitatis Ratione: &longs;i par &longs;it eo­<lb/>rum gravitas, motus autem alterius reliquo triplo velocior e&longs;&longs;e <lb/>debeat, inæqualem pariter exigunt conatum, &longs;ed pro ratione <lb/>velocitatis: &longs;i demùm & di&longs;par &longs;it gravitas, & inæqualis velo­<lb/>citas, eam e&longs;&longs;e con&longs;tat repugnantiam, quæ tùm ex gravitate, <lb/>tùm ex velocitate componitur; atque adeò &longs;i corpus alterum <lb/>triplo gravius triplo etiam velociùs movendum e&longs;&longs;et, noncuplex <lb/>e&longs;&longs;et ejus repugnantia; &longs;in autem triplo levius triplo majori <lb/>velocitate quàm corpus triplo gravius, moveretur, par e&longs;&longs;et eo­<lb/>rum ob&longs;i&longs;tentia, paremque conatum exigerent. </s> </p> <p type="main"> <s id="s.001090">Hinc &longs;atis apertè con&longs;tat, datâ tum re&longs;i&longs;tentiarum, tum velo­<lb/>citatum Ratione, &longs;i gravitas altera nota &longs;it, reliquam facilè inno­<lb/>te&longs;cere: &longs;i nimirùm nota gravitas per &longs;uam velocitatem ducatur, <lb/>& in datâ Ratione re&longs;i&longs;tentiarum reperiatur huic producto ter­<lb/>minus homologus; quo per ignotæ gravitatis velocitatem da­<lb/>tam divi&longs;o, prodibit Quotiens index quæ&longs;itæ gravitatis. </s> <s id="s.001091">Sint <lb/>duo corpora inæqualia, & ad ea movenda requiratur conatus <lb/>in Ratione &longs;e&longs;quialterâ, motus autem eorum &longs;int ut 7 ad 8, & <lb/>illud quod minùs re&longs;i&longs;tit, moveturque velocitate ut 7, numeret <lb/>gravitatis libras 4. Reliqui corporis validiùs re&longs;i&longs;tentis, cujus <lb/>velocitas e&longs;t ut 8, gravitas &longs;ic invenietur. </s> </p> <p type="main"> <s id="s.001092">Libræ 4 ducantur per numerum &longs;uæ velocitatis 7, & fit 28. <lb/>Quia igitur re&longs;i&longs;tentiæ &longs;unt, ut 2 ad 3 ex hypothe&longs;i, & unius <lb/>corporis re&longs;i&longs;tentiâ, quæ ex gravitate & motûs velocitate com­<lb/>ponitur, e&longs;t 28, fiat ut 2 ad 3, ita 28 ad aliud, & erit 42 re­<lb/>&longs;i&longs;tentia alterius corporis compo&longs;ita ex ejus velocitate & gravi­<lb/>tate. </s> <s id="s.001093">Atqui velocitas nota e&longs;t 8; igitur divisâ totâ re&longs;i&longs;tentiâ <lb/>42 per 8; prodibit quotiens 5 1/4 index quæ&longs;itæ gravitatis. </s> <s id="s.001094">Quare <lb/>ad movendas libras 5 1/4 velocitate ut 8, requiritur conatus &longs;e&longs;­<lb/>quialter conatûs nece&longs;&longs;arij ad movendas libras 4 velocitate <lb/>ut 7. Eadem e&longs;to de reliquis ac &longs;imilibus conjectura. </s> </p> <p type="main"> <s id="s.001095">Ex his præterea manife&longs;tum e&longs;t corporis per vim dimovendi <lb/>re&longs;i&longs;tentiam ex &longs;olâ naturâ, & principio in&longs;ito, quod motui re­<lb/>pugnat, ab&longs;olutè definiri non po&longs;&longs;e; motum &longs;i quidem ab omni <lb/>prorsùs celeritatis aut tarditatis men&longs;urâ &longs;ejungere non po&longs;&longs;u­<lb/>mus; idcircò non ni&longs;i habitâ ratione celeritatis, aut tarditatis, <pb pagenum="143" xlink:href="017/01/159.jpg"/>ex quibus re&longs;i&longs;tentia componitur, re&longs;i&longs;tentia ip&longs;a innote&longs;cere <lb/>poterit. </s> <s id="s.001096">Quare & impetus à facultate movendi principium ha­<lb/>bente productus major &longs;it nece&longs;&longs;e e&longs;t, quàm dimoti corporis <lb/>repugnantia; quæ varia prorsùs cùm &longs;it, nunc quidem majo­<lb/>rem, nunc verò minorem impetum exigit, ut ab eo vincatur; <lb/>nam &longs;i pares confligerent vires, à neutrâ parte &longs;taret victoria. </s> </p> <p type="main"> <s id="s.001097">Quod autem ad ip&longs;am motûs originem &longs;pectat, ea, quæ vi­<lb/>vunt, ab iis, quæ vitâ omnino carent, &longs;ecernenda &longs;unt: hæc <lb/>enim (&longs;cilicet non viventia) propterea motum expetunt, ut <lb/>violentiam, quam &longs;ubeunt, excutiant, nec unquam à loco, &longs;eu <lb/>&longs;tatu, &longs;ecundùm naturam opportuno &longs;ponte recedunt; quem­<lb/>admodum eunti per &longs;ingula con&longs;tabit. </s> <s id="s.001098">Sic gravibus & levibus <lb/>&longs;uis in locis quietem natura indixit, non motum; nec deor­<lb/>&longs;um conantur aut &longs;ur&longs;um, ni&longs;i alieno in loco, hoc e&longs;t, in me­<lb/>dio di&longs;pari gravitate aut levitate prædito con&longs;titutâ: &longs;ic quæ­<lb/>cumque ela&longs;ticâ facultate pollent, motum non moliuntur, ni&longs;i <lb/>cum &longs;ibi naturalem partium figuram, &longs;itumque reparare opor­<lb/>tet. </s> <s id="s.001099">At motum, cujus origo vita e&longs;t, natura perficit, etiam&longs;i <lb/>nulla præce&longs;&longs;erit violentia: &longs;ic &longs;tirpes dum augentur, & cre&longs;­<lb/>cunt, earum particulæ locum mutant; &longs;ic vitali facultate in­<lb/>fluentibus per nervos in <expan abbr="animaliũ">animalium</expan> mu&longs;culos &longs;piritibus, quos ani­<lb/>males vocant, intenduntur mu&longs;culi, motu&longs;que membrorum con­<lb/>&longs;equitur: quamvis ante motum nec &longs;tirpis particulæ, nec anima­<lb/>lis membra vim <expan abbr="ullã">ullam</expan> &longs;ubierint in loco minimè congruo retenta. </s> </p> <p type="main"> <s id="s.001100">Quæcunque igitur ob id ip&longs;um in motum prona &longs;unt, quia <lb/>vim patiuntur, impetum illicò concipiunt, ac vis iis illata e&longs;t, <lb/>quo naturalem locum, &longs;eu &longs;tatum, recipere valeant, licèt &longs;æpè <lb/>irrito conatu, ni&longs;i quatenùs adver&longs;o hoc impetu illatam ab ob­<lb/>&longs;i&longs;tente violentiam retundunt, vim aliquam illi vici&longs;&longs;im infe­<lb/>rentes. </s> <s id="s.001101">Sic onera bajulorum humeros, quibus &longs;u&longs;tinentur, <lb/>premunt, aut penduli brachij; ex quo &longs;u&longs;penduntur, mu&longs;cu­<lb/>los ac ligamenta fatigant: id quod pariter in corpore inanimo <lb/>cernere licet; quemadmodum enim ex diuturnâ prementis <lb/>deor&longs;um ponderis, ac mu&longs;culorum &longs;ursùm urgentium luctâ, <lb/>di&longs;&longs;ipatis &longs;piritibus, la&longs;&longs;itudo in animali oritur, ita pariter &longs;ub­<lb/>jectum a&longs;&longs;erem longâ temporis morâ pondus curvat, aut etiam <lb/>demùm frangit, & funem, ex quo pendet, non intendit &longs;olùm, &longs;ed <lb/>etiam tandem aliquando corrupto particularum nexu disjicit. </s> </p> <pb pagenum="144" xlink:href="017/01/160.jpg"/> <p type="main"> <s id="s.001102">Quo id autem pacto contingat, explicare opero&longs;um non fue­<lb/>rit funiculi texturam con&longs;ideranti; ex tenui&longs;&longs;imis &longs;cilicet linei <lb/>aut cannabini corticis longâ maceratione, & plurimâ tun&longs;ione <lb/>extenuati particulis in &longs;piram contortis filum cohæret; ex filis <lb/>autem plu&longs;culis in &longs;piram pariter contortis funiculus, & pluri­<lb/>bus funiculis cra&longs;&longs;iores rudentes conflantur: quod &longs;i di&longs;&longs;olvatur <lb/>omnis &longs;pira, non cohærent funiculi aut fili partes. </s> <s id="s.001103">Spira di&longs;­<lb/>&longs;olvitur factâ in contrarium revolutione; quò autem laxioribus <lb/>gyris flectitur, eò faciliùs villi &longs;inguli ex cæteris, quibus im­<lb/>plicantur, extrahuntur; & uno ab aliorum communione &longs;e­<lb/>juncto, amplitudo &longs;patij faciliorem exitum proximis relinquit: <lb/>ex quo fit faciliùs &longs;emper ac faciliùs po&longs;&longs;e funiculum frangi; <lb/>filo enim uno rupto, aut extracto, facilior e&longs;t in contrarium re­<lb/>volutio, & &longs;pira fit amplior, ac reliqua fila faciliùs extrahun­<lb/>tur. </s> <s id="s.001104">Ob&longs;ervamus autem non rarò appen&longs;um ex funiculo pon­<lb/>dus aliquandiu in gyrum contorqueri; dum &longs;cilicet &longs;uâ gravi­<lb/>tate deor&longs;um connitens intendit funiculum, contorta fila in <lb/>contrarium revolvuntur. </s> <s id="s.001105">Sed &, quamvis nulla fieret in con­<lb/>trarium revolutio, &longs;atis con&longs;tat ex illâ inten&longs;ione funiculum <lb/>di&longs;trahi, ac produci; atque adeò &longs;piram laxiorem fieri, paula­<lb/>timque unum aut alterum villum educi, locumque fieri vapo­<lb/>ribus, qui proximum villum corrumpentes faciliori &longs;ci&longs;&longs;ioni pa­<lb/>rant, atque adeò, &longs;erpente lue, demùm non tot integri &longs;uper­<lb/>&longs;unt villi, qui po&longs;&longs;int ponderis gravitati ob&longs;i&longs;tere, quin dif­<lb/>fringantur. </s> <s id="s.001106">Ex quo &longs;atis apparet &longs;u&longs;pen&longs;um pondus, licèt non <lb/>omninò de&longs;cendat, impetum tamen concipere, quo retinenti <lb/>repugnat, & vim aliquam vici&longs;&longs;im infert. </s> </p> <p type="main"> <s id="s.001107">Nec ab&longs;imili ratione in reliquis vim patientibus contingere <lb/>ob&longs;ervabimus, ea &longs;cilicet moliri illicò naturalis &longs;tatûs repara­<lb/>tionem, aliquidque efficere, licèt tenui&longs;&longs;imum, quod demum <lb/>appareat, ubi temporis morâ augmentum ceperit. </s> <s id="s.001108">Sic ha&longs;tam <lb/>per vim inflexam &longs;i continuò dimittas, illa &longs;e&longs;e re&longs;tituit, facul­<lb/>tate ela&longs;ticâ; at &longs;i dies aliquot, aut etiam diutiù per vim &longs;i­<lb/>nuata perman&longs;erit, &longs;ibi dimi&longs;&longs;a antiquam rectitudinem non re­<lb/>parat; elanguit nimirùm facultas ela&longs;tica, quæ ex violentâ par­<lb/>ticularum compre&longs;&longs;ione aut di&longs;tractione oriebatur. </s> <s id="s.001109">Cùm enim <lb/>primùm ha&longs;ta flectitur, particulæ concavam curvaturæ partem <lb/>re&longs;picientes comprimuntur, contra verò, quæ convexam re&longs;pi-<pb pagenum="145" xlink:href="017/01/161.jpg"/>ciunt, di&longs;trahuntur; quare tùm quæ, raræ, tùm quæ den&longs;æ factæ <lb/>&longs;unt, dum vim illicò prorsùs excutere conantur, con&longs;pirant, ut <lb/>pri&longs;tinam ha&longs;tæ rectitudinem moliantur: Quod &longs;i id non li­<lb/>cuerit, hæ quidem aliam ex angu&longs;tiis evadendi, quâ facilior <lb/>patet via, rationem tentant, ita ut demùm &longs;ubtili&longs;&longs;imas in ru­<lb/>gas cri&longs;pentur, illæ verò &longs;e&longs;e ad angu&longs;tiora &longs;patia &longs;en&longs;im reci­<lb/>pientes mutuum nexum &longs;olvunt, tenui&longs;&longs;imo&longs;que poros relin­<lb/>quunt, aut &longs;i qui priùs interjecti fuerint, ampliùs hiare per­<lb/>mittunt. </s> <s id="s.001110">Id quod ubi jam contigerit, fru&longs;trà &longs;ubmoves, quæ <lb/>admoveras impedimenta; & &longs;pontè curvaturam ha&longs;ta &longs;ervat, <lb/>ni&longs;i fortè particulis omnibus adhuc per tempus non licuerit <lb/>vim totam excutere; tunc enim &longs;e &longs;e languidiùs re&longs;tituunt, pro <lb/>ratione reliquæ violentiæ. </s> <s id="s.001111">Hinc patet arcum, quò fuerit con­<lb/>tentus atque adductus vehementiùs, remitti aliquando, & ma­<lb/>nualium tormentorum rotas interdum laxari oportere, ne vis <lb/>ela&longs;tica languidior facta minùs utilis fiat. </s> </p> <p type="main"> <s id="s.001112">Ex his igitur paulò enucleatiùs explicatis, in quibus longio­<lb/>re temporis fluxu motum aliquem tardi&longs;&longs;imum contigi&longs;&longs;e, at­<lb/>que adeò etiam impetum jam tum ab initio &longs;tatim fui&longs;&longs;e pro­<lb/>ductum con&longs;tat, conjecturam in reliquis capio, & ab iis impe­<lb/>tum concipi &longs;tatuo, quæ aut loco naturali dimota, aut incon­<lb/>gruam partium po&longs;itionem nacta id repetunt, quod natura exi­<lb/>git. </s> <s id="s.001113">Motus autem non pro impetûs tantum, &longs;ed & pro re­<lb/>&longs;i&longs;tentiæ modo con&longs;equitur. <lb/></s> </p> <p type="main"> <s id="s.001114"><emph type="center"/>CAPUT III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001115"><emph type="center"/><emph type="italics"/>Quâ ratione &longs;emel conceptus impetus pereat.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001116">UT impetûs natura, quam inquirimus, explicatiùs atque <lb/>di&longs;tinctiùs innote&longs;cat, ex quo pariter, quæ corpora, quâ­<lb/>ve ratione, impetum re&longs;puant, intelligamus, hîc nobis e&longs;t <lb/>ve&longs;tigandum, quâ ratione conceptum &longs;emel impetum abji­<lb/>ciant: hinc nimirum in uberiorem ip&longs;ius re&longs;i&longs;tentiæ notitiam <lb/>venientes ad explicandam motûs machinalis cau&longs;am propiùs <lb/>accedemus. </s> </p> <pb pagenum="146" xlink:href="017/01/162.jpg"/> <p type="main"> <s id="s.001117">Et &longs;anè conceptum impetum, naturâ &longs;uâ, nec flabilem &longs;em­<lb/>per permanere, nec ad unicum temporis punctum durare, &longs;a­<lb/>tis con&longs;tat: &longs;ivè enim &longs;pontè profluat ex naturâ debitum &longs;ibi <lb/>locum quærente, &longs;ivè alienâ vi impre&longs;&longs;us &longs;uo loco corpus ex­<lb/>trudat, perpetuus e&longs;&longs;e nequit; omnis &longs;cilicet motus terminum <lb/>habeat nece&longs;&longs;e e&longs;t; nam &longs;i violentus quidem e&longs;t, perennis uti­<lb/>que non e&longs;t; &longs;in autem naturalis, quem violentus præce&longs;&longs;erit, <lb/>certis definitur terminis; à loco enim, in quo quietem natura <lb/>indixit, corpus infinito intervallo non abe&longs;t, ac proinde ubi <lb/>eum attigerit, demùm conquie&longs;cet, nec impetu perpetuo opus <lb/>erit, cùm motum ce&longs;&longs;are oporteat. </s> <s id="s.001118">Sed neque temporis mo­<lb/>mento circum&longs;cribi impetum &longs;ivè in naturali motu acqui&longs;itum, <lb/>&longs;ive in violento impre&longs;&longs;um, plura &longs;unt, quæ palam faciunt: ut <lb/>enim reliqua &longs;ileam nullæ e&longs;&longs;ent funependulorum o&longs;cillatio­<lb/>nes, nullus emi&longs;&longs;æ &longs;agittæ motus, &longs;i conceptus impetus illicò <lb/>periret. </s> </p> <p type="main"> <s id="s.001119">In duo autem veluti genera tribuendus e&longs;t Impetus ex natu­<lb/>râ dimanans; alius Innatus, &longs;eu qua&longs;i in&longs;itus, alius Acqui&longs;itus <lb/>dicitur, Innatum, &longs;eu qua&longs;i in&longs;itum, voco, non quem corpus <lb/>jugiter obtineat, &longs;ive &longs;uo in loco, &longs;ive in alieno quie&longs;cat; &longs;ed <lb/>eum, qui facultati &longs;e movendi præcisè re&longs;pondet, nullo facto <lb/>per continuam adjectionem incremento: quandiù enim corpus <lb/>ita &longs;imili &longs;ecundùm gravitatem corpore circumfunditur, ut na­<lb/>turali in loco con&longs;i&longs;tere dicendum &longs;it, quare conetur motum: <lb/>conatum autem hîc ab impetu non di&longs;tinguo: &longs;atis igitur citrà <lb/>quemlibet impetum &longs;uo &longs;e tutatur in loco per hoc, quod eá fa­<lb/>cultate &longs;it præditum, quæ in contrariam partem conniti valeat <lb/>illicò, ac vis inferri cæperit. </s> <s id="s.001120">Hinc nullum aquæ impetum tri­<lb/>buo intrà aquam con&longs;i&longs;tenti; &longs;ed tunc &longs;olùm cùm &longs;itula plena <lb/>è lacu extrahitur, ea aquæ pars impetum habet, quæ &longs;uprà &longs;ub­<lb/>jectam lacûs &longs;uperficiem aere circumfuia motum expetit, quo <lb/>&longs;uum repetat locum repugnans &longs;u&longs;tinenti. </s> <s id="s.001121">Impetum hunc, qui <lb/>naturali &longs;e movendi facultati re&longs;pondet, & e&longs;t ip&longs;a gravitatio, <lb/>&longs;eu naturalis ad de&longs;cen&longs;um propen&longs;io, Innatum voco, & is e&longs;t, <lb/>cui extrin&longs;eca cau&longs;a repugnat motum impediens. </s> <s id="s.001122">Quòd &longs;i &longs;u&longs;­<lb/>pen&longs;um corpus &longs;ibi relinquatur, ita &longs;uum in locum contendit, <lb/>ut vis naturalis æquè &longs;emper ad agendum applicata, nec impe­<lb/>dita, momentis &longs;ingulis novum impetum acquirat, qui propterea <pb pagenum="147" xlink:href="017/01/163.jpg"/>Acqui&longs;itus dicitur, & po&longs;terior priori additus inten&longs;ionem ef­<lb/>ficit: &longs;apienti &longs;anè naturæ in&longs;tituto; nam &longs;i corpora per &longs;e ip&longs;a <lb/>ac &longs;uâ &longs;ponte mota non accelerarent; &longs;ed naturalis motus pla­<lb/>ne æquabilis e&longs;&longs;et, tardè nimis locum &longs;uum con&longs;equerentur; <lb/>atque adeò augendus continuò fuit impetus, ut & motus in­<lb/>crementum acciperet: at &longs;i innatus impetus valdè <expan abbr="int&etilde;&longs;us">inten&longs;us</expan> e&longs;&longs;et, <lb/>corpora nonni&longs;i ægerrimè aliò transferri, aut alieno in loco re­<lb/>tineri pro animalium, & hominis utilitate po&longs;&longs;ent; finge &longs;cili­<lb/>cet animo tibiam tanto impetu innato repugnare, ne attollatur, <lb/>quanto impetu in aëre ex 200 pa&longs;&longs;uum altitudine de&longs;cenderet; <lb/>quanto id tibi e&longs;&longs;et incommodo? </s> <s id="s.001123">Quare peropportunum acci­<lb/>dit, ut vehemens non e&longs;&longs;et &longs;ingularum particularum impetus <lb/>innatus, qui tamen ubi motum efficeret, novâ acce&longs;&longs;ione po&longs;­<lb/>&longs;et augeri. </s> </p> <p type="main"> <s id="s.001124">Quod ad impetum Innatum &longs;pectat, quem à gravitatione <lb/>ipsá & proxima motus exigentia non &longs;ejungo, utique fru&longs;trà <lb/>e&longs;&longs;et, &longs;i omni pror&longs;us effectu careret; impetus autem motum <lb/>aut efficit, aut &longs;altem exigit: propterea illum &longs;tatim perire au­<lb/>tumo, ac fuerit corpus in loco &longs;uo: Id quod hoc deprehendes <lb/>experimento. </s> <s id="s.001125">Scrobem defo&longs;sâ humo altè excavato; &longs;itulam <lb/>aquæ pienam, & noti ponderis, intrà illam &longs;u&longs;pendito; tùm <lb/>aquam in &longs;crobem tantâ copiâ derivato; ut &longs;itulan u&longs;quequa­<lb/>que circumplectatur: illicò evane&longs;cet totius aquæ priùs in &longs;itu­<lb/>lâ gravitantis pondus, quin & &longs;itula ip&longs;a pro gravitatum &longs;ecun­<lb/>dùm &longs;peciem di&longs;&longs;imilitudine levior apparebit, ut ex Hydro&longs;ta­<lb/>ticis con&longs;tat. </s> <s id="s.001126">Periit ergo innatus impetus, quo aqua &longs;itulam <lb/>replens de&longs;cen&longs;um moliebatur. </s> </p> <p type="main"> <s id="s.001127">At impetum Acqui&longs;itum non continuò perire, ac eò ventum <lb/>fuerit, ubi quie&longs;cendum e&longs;&longs;et, hinc &longs;altem di&longs;ces, quod <lb/>ligneum globum aquæ cæteroqui innataturum &longs;i in &longs;ublime at­<lb/>tollas, & ex illâ altitudine cadere permittas, infrà aquæ &longs;uper­<lb/>ficiem de&longs;cendere, ac penitùs immergi videbis; quamquam <lb/>po&longs;tea emergat, & ubi aliquoties &longs;ub&longs;ultaverit, demùm pro <lb/>gravitatum aquæ, & ligni di&longs;paritate emer&longs;us quie&longs;cat. </s> <s id="s.001128">Quæ <lb/>&longs;anè immer&longs;io, ni&longs;i Acqui&longs;itus impetus adhuc duraret, omninò <lb/>non contingeret. </s> <s id="s.001129">Verùm nihil rem per &longs;e &longs;atis ab&longs;tru&longs;am æquè <lb/>in lucem evocat, ac funependulorum motus; plumbum enim <lb/>ex filo &longs;u&longs;pen&longs;um, & à perpendiculo dimotum, ita de&longs;cendens <pb pagenum="148" xlink:href="017/01/164.jpg"/>arcum de&longs;cribit, ut ferè parem arcum, & vix (aut fortè ne vix <lb/>quidem) minori tempore a&longs;cendens de&longs;cribat. </s> <s id="s.001130">Cui autem, re­<lb/>pugnante plumbi gravitate à naturá in&longs;itâ, tribuatur a&longs;cen&longs;us, <lb/>ni&longs;i impetui acqui&longs;ito dum de&longs;cenderet, adhuc po&longs;t de&longs;cen&longs;um <lb/>duranti? </s> <s id="s.001131">Quemadmodum verò in de&longs;cen&longs;u po&longs;teriores motûs <lb/>partes prioribus velociores &longs;unt, factâ nimirum novi impetûs <lb/>acce&longs;&longs;ione, ita ex oppo&longs;ito a&longs;cen&longs;us ex celeritate in tarditatem <lb/>de&longs;init, factâ acqui&longs;iti impetûs dece&longs;&longs;ione continuâ, donec ita <lb/>elanguerit, ut gravitas ip&longs;a &longs;uperet, & iterum de&longs;cendens al­<lb/>ternas vibrationes efficiat. </s> <s id="s.001132">Perit igitur Acqui&longs;itus impetus non <lb/>totus &longs;imul; &longs;ed &longs;en&longs;im extenuatur; idque non aliâ ratione, <lb/>quàm quâ proportione impeditur motus, quocumque tandem <lb/>ex capite impedimenta oriantur. </s> <s id="s.001133">Cum enim impetus contra­<lb/>rium impetum non habeat, &longs;i præci&longs;a quidem impetûs natura <lb/>&longs;pectetur (quippe qui unus & idem contrariorum motuum ori­<lb/>go e&longs;t, ut ex funependulis ultrò citróque &longs;ponte vibratis & ex <lb/>pilâ lu&longs;oriâ deor&longs;um cadente, ac vi concepti impetûs &longs;ur&longs;um <lb/>re&longs;iliente, con&longs;tat) reliquum e&longs;t, ut pereat pro ratione eorum, <lb/>quæ aut motui corporis ob&longs;i&longs;tunt, aut illud aliò quoquomodo <lb/>dirigunt. </s> </p> <p type="main"> <s id="s.001134">Præ&longs;tat autem hîc funependuli <lb/><figure id="id.017.01.164.1.jpg" xlink:href="017/01/164/1.jpg"/><lb/>motum paulò attentiùs con&longs;iderare. </s> <lb/> <s id="s.001135">Sit plumbeus globulus B filo AB <lb/>connexus clavo in A. <!-- KEEP S--></s> <s id="s.001136">Si globulo li­<lb/>ceret, quâ impetus innatus urget viâ, <lb/>de&longs;cendere, utique rectam BC per­<lb/>curreret; &longs;ed funiculo retinente co­<lb/>gitur arcum BK de&longs;cribere, adeò ut <lb/>&longs;emper in alio & alio plano inclinato <lb/>con&longs;titutus, alia, & alia habeat gra­<lb/>vitatis momenta, ut lib. 1. cap. 15 explicatum e&longs;t; hæc autem <lb/>&longs;unt pro Ratione Sinuum angulorum declinationis à perpendi­<lb/>culo AK. <!-- KEEP S--></s> <s id="s.001137">Quare totum momentum, quod in B e&longs;&longs;et ut AB, <lb/>&longs;ingulis momentis in de&longs;cen&longs;u libero per rectam BC paribus <lb/>&longs;altem incrementis augeretur (Quicquid &longs;it an etiam pro Ra­<lb/>tione duplicatâ temporum, de quo alias di&longs;putabimus) &longs;ed <lb/>cum à rectitudine deflectat, cum venerit in D, non additur <lb/>momentum ut EF, &longs;ed ut ED; &longs;imiliter in G momentum non <pb pagenum="149" xlink:href="017/01/165.jpg"/>e&longs;t ut HI, &longs;ed ut HG. </s> <s id="s.001138">Augetur igitur impetus in de&longs;cen&longs;u <lb/>BK non omninò pro Ratione <expan abbr="momentorũ">momentorum</expan> temporis, quo motus <lb/>durat, &longs;ed pro Ratione momentorum gravitatis, quæ &longs;ubinde <lb/>obtinet minora & minora; pars <expan abbr="&longs;iquid&etilde;">&longs;iquidem</expan> impetûs ab in&longs;itâ globuli <lb/>gravitate producti deteritur in intendendo filo, quo retinetur. </s> <lb/> <s id="s.001139">Quapropter ubi in K venerit per arcum BK, non tantum ha­<lb/>bet impetûs, quantum &longs;i per lineam perpendicularem arcui <lb/>BK æqualem de&longs;cendi&longs;&longs;et; in motu enim ad perpendiculum <lb/>cum nihil retineat aut impediat, totus impetus ad de&longs;cen&longs;um <lb/>urget velociùs, quàm ubi repugnat aliquid. </s> <s id="s.001140">Ex quo fit quod, <lb/>cùm arcus BK ad Radium AB, hoc e&longs;t ad BC æqualem <lb/>proximè ut 11 ad 7, ex Cyclometricis, multò plus temporis in <lb/>percurrendo arcu BK, quàm in rectâ BC, in&longs;umitur; tardiùs <lb/>&longs;cilicet movetur quàm in perpendiculari, quæ ad BC e&longs;&longs;et ut <lb/>11 ad 7. manente itaque, quamdiu corpus naturâ urgente mo-<lb/>vetur, impetu acqui&longs;ito, qui re&longs;i&longs;tentiam excedit, in fine <lb/>de&longs;censûs in K totus impetus e&longs;t ut aggregatum omnium Si­<lb/>nuum Quadrantis: at in perpendiculari BC in fine de&longs;censùs <lb/>in C e&longs;&longs;et ut aggregatum omnium parallelarum ip&longs;i AB in <lb/>Quadrato AC; ac propterea (in re Phy&longs;icâ &longs;i liceat cum geo­<lb/>metrizantibus per Indivi&longs;ibilia ratiocinari) erit impetus per ar­<lb/>cum BK acqui&longs;itus ad impetum per rectam BC acqui&longs;tum ut <lb/>Quadrans ABK ad Quadratum AC, hoc e&longs;t ut 11 ad 14, ex <lb/>iis quæ in Cyclometriâ demon&longs;trantur. </s> </p> <p type="main"> <s id="s.001141">Quoniam verò ubi ad perpendiculum AK globulus de&longs;cen­<lb/>dens venerit, nihil objicitur, quod motum pror&longs;ùs impediar, <lb/>quin ad ea&longs;dem partes pergat ferri ex præconcepti impetûs di­<lb/>rectione, non &longs;i&longs;tit in perpendiculo; &longs;ed ulteriùs pergens a&longs;cen­<lb/>dit, nec ni&longs;i per arcum circà centrum A, funiculo &longs;cilicet reti­<lb/>nente. </s> <s id="s.001142">Sed jam repugnat a&longs;cen&longs;ui gravitas plumbi, non qui­<lb/>dem quantum in perpendiculo KA, verùm pro ratione Sinuum <lb/>angulorum declinationis; qui cum &longs;emper a&longs;cendendo cre&longs;­<lb/>cant, major e&longs;t etiam momentorum gravitatis Ratio nitentium <lb/>contrà impetum de&longs;cendendo acqui&longs;itum. </s> <s id="s.001143">Quare tantum abe&longs;t, <lb/>ut novus &longs;ingulis temporis punctis impetus &longs;ur&longs;um directus pro­<lb/>ducatur, ut potius ex eo tantumdem dematur, quanta e&longs;t <lb/>a&longs;cendentis plumbi repugnantia. </s> <s id="s.001144">Hinc e&longs;t a&longs;cen&longs;um initio ve­<lb/>lociorem e&longs;&longs;e, quia adhuc multus e&longs;t impetus acqui&longs;itus, & pro <pb pagenum="150" xlink:href="017/01/166.jpg"/>Sinuum declinationis brevitate, exigua illius pars deteritur, <lb/>atque adeò motus efficitur celerior: quia verò diminuto &longs;en&longs;im <lb/>impetu, & auctis <expan abbr="cõtrariæ">contrariæ</expan> gravitatis <expan abbr="mom&etilde;tis">momentis</expan> pro Sinuum decli­<lb/>nationis <expan abbr="increm&etilde;to">incremento</expan>, minor fit ip&longs;ius impetûs ad <expan abbr="contrariũ">contrarium</expan> ni&longs;um <lb/>Ratio, tardior &longs;equitur motus, & plus acqui&longs;iti impetûs perit, do­<lb/>nec demùm pror&longs;us evanuerit, & &longs;uperante gravitate globulus <lb/>iterum de&longs;cendat. </s> <s id="s.001145">Quamvis autem &longs;i po&longs;itio &longs;ola &longs;pectetur, ii&longs;­<lb/>dem Reciproce gradibus minui videatur impetus, quibus fuit <lb/>auctus, totidemque momentis temporis, ita ut quantum po&longs;tre­<lb/>mo temporis puncto acce&longs;&longs;it, tantumdem primo decedat, adhuc <lb/>tamen aliqua e&longs;t ob&longs;i&longs;tentiæ appendicula ex aëre dividendo, ac <lb/>propterea paulo ampliùs extenuatur impetus acqui&longs;itus, quàm <lb/>pro Ratione incrementi Sinuum declinationis: quò autem ve­<lb/>locior e&longs;t motus, magis etiam aër dividendus comprimitur, <lb/>den&longs;atú&longs;que plus ob&longs;i&longs;tit quàm rarus; quòd &longs;i medium non fue­<lb/>rit compre&longs;&longs;ionis capax, &longs;altem æquali tempore plures medij <lb/>partes &longs;cinduntur, quàm in motu tardiori, ac propterea etiam <lb/>multiplex e&longs;t medij re&longs;i&longs;tentia: Ex quo fit arcum a&longs;censûs pau­<lb/>lò minorem &longs;emper e&longs;&longs;e arcu de&longs;censûs, &, cum vici&longs;&longs;im glo­<lb/>bus remaneat ex humiliore loco ac priùs de&longs;cendens, brevio­<lb/>rem pariter &longs;ecundi a&longs;censûs arcum perfici, atque ita deinceps, <lb/>ut &longs;ervatâ eâ in motu &longs;emper minori reciprocando con&longs;tantiâ <lb/>demum quie&longs;cat in perpendiculo. </s> </p> <p type="main"> <s id="s.001146">At, inquis, dura magis ob&longs;i&longs;tunt corpori, ejú&longs;que motum <lb/>validiùs impediunt, quàm mollia, quæ dum &longs;e comprimi pa­<lb/>tiuntur, & loco pauli&longs;per cedunt, motui aliquantulùm & ex <lb/>parte ob&longs;ecundant: &longs;i igitur pro Ratione impedimenti debili­<lb/>tatur acqui&longs;itus impetus, minus detrahitur impetûs corpori, <lb/>quod ex alto decidens à &longs;ub&longs;tratis paleis excipitur, quàm &longs;i ad <lb/>&longs;axum allideretur; vehementiùs igitur à luto quàm à &longs;àxo re­<lb/>flecteretur, contrà quàm docet experientia. </s> </p> <p type="main"> <s id="s.001147">Fateor eburneum globum &longs;egniùs re&longs;ilire delap&longs;um in gle­<lb/>bam humore perfu&longs;am, quàm in marmor; non tamen his con­<lb/>&longs;equens e&longs;t, ut impetûs acqui&longs;iti diminutioni alius &longs;tatuendus <lb/>&longs;it modus, quàm ex impedimento: ubi enim globus cadens ex<lb/>timam &longs;ubjecti corporis &longs;uperficiem attigerit, non quie&longs;cit, &longs;ed <lb/>pergit moveri, aut deor&longs;um comprimendo corpus molle, aut <lb/>illicò &longs;ursùm reflexum à duro. </s> <s id="s.001148">Ita autem à corpore molli ex-<pb pagenum="151" xlink:href="017/01/167.jpg"/>cipitur, ut licèt hoc cedat, impediat tamen & remoretur mo­<lb/>tum; ac proinde quò magis cedit &longs;ubjectum corpus, eò diutiùs <lb/>movetur globus cum ip&longs;o, vel intrà ip&longs;um; atque interea plus <lb/>impetûs perit: quid igitur mirum, &longs;i languidiùs po&longs;tea re&longs;iliat, <lb/>cum exigua impetûs portio reliqua &longs;it? </s> <s id="s.001149">Quòd &longs;i <expan abbr="durũ">durum</expan> e&longs;&longs;et &longs;ub­<lb/>jectum corpus, impetu nondum debilitato reflecteretur vali­<lb/>diùs. </s> <s id="s.001150">Hinc fieri pote&longs;t adeò molle e&longs;&longs;e &longs;ubjectum corpus, ut <lb/>dum illud penetrat decidens globus, tantum impetûs deper­<lb/>dat, ut, quod reliquum fit, non &longs;atis &longs;it ad vincendam in&longs;itam <lb/>globo gravitatem, qui propterea neque re&longs;ilire valeat. </s> <s id="s.001151">Quam­<lb/>vis itaque corpus molle minùs ob&longs;i&longs;tat quàm durum, diutiùs <lb/>tamen re&longs;i&longs;tit; & per aliquot momenta aliquoties diminutus <lb/>impetus minore men&longs;urâ, eò decrementi venire pote&longs;t, ut ma­<lb/>gis imminutus demum fuerit, quàm &longs;i unico momento magis <lb/>ob&longs;titi&longs;&longs;et corpus durum. </s> <s id="s.001152">Cæterùm paribus momentis plus pe­<lb/>rit impetûs ex alli&longs;ione ad corpus durum, quàm ad molle, quip­<lb/>pe quod magis opponitur motui. </s> <s id="s.001153">Porrò huic rei explicandæ <lb/>&longs;imilitudo aliqua peti po&longs;&longs;et ex luce, cui &longs;anè &longs;i contingat per <lb/>medium diaphanum quidem, &longs;ed den&longs;um, pergere, languidiùs <lb/>multò reflectitur à &longs;peculo, in quod incurrit, &longs;i den&longs;ioris me­<lb/>dij longior fuerit tractus, quàm &longs;i brevior, perinde atque eò <lb/>minùs reflectitur corpus, quò molliori magi&longs;que &longs;ub&longs;identi cor­<lb/>pori occurrit. </s> <s id="s.001154">&longs;ed quoniam quæ de luce dicenda e&longs;&longs;ent, fortè ob­<lb/>&longs;curiora acciderent, ab huju&longs;modi &longs;imilitudine <expan abbr="prud&etilde;">prudem</expan>, ab&longs;tineo. </s> </p> <p type="main"> <s id="s.001155">Sed ex illud e&longs;t in durorum corporum colli&longs;ione ob&longs;ervan­<lb/>dum, quod aliqua particularum compre&longs;&longs;io aliquando contin­<lb/>git &longs;ivè in alterutro, &longs;ivè in utróque, quæ &longs;e facultate ela&longs;ticâ <lb/>re&longs;tituentes motum reflexum juvant: id autem manife&longs;to ex­<lb/>perimento con&longs;tat in pilâ ex gummi, ut vocant, Indico, quæ <lb/>ad terram eli&longs;a frequenti&longs;&longs;imè &longs;ub&longs;ultat; at ubi in corpus molle <lb/>incidit, neque hujus neque illius partes violentam compre&longs;&longs;io­<lb/>nem &longs;ubeunt, quam &longs;e&longs;e re&longs;tituentes excutere debeant. </s> <s id="s.001156">Sic & <lb/>pilá in &longs;phæri&longs;terio ludentes &longs;atis nôrunt eam validiùs reflecti <lb/>objecto recticulo, quàm ligneo batillo; intenti &longs;cilicet nervi ex <lb/>contortis &longs;iccati&longs;que animalium inte&longs;tinis reticulum con&longs;tituen­<lb/>tes cùm pilæ ictum excipiunt, flectuntur quidem aliquantu­<lb/>lum; &longs;ed illicò &longs;ibi pri&longs;tinam rectitudinem reparantes pilam ex­<lb/>cutiunt (id quod ligneo ba&longs;tillo non contingit) novoque hoc <pb pagenum="152" xlink:href="017/01/168.jpg"/>impetu auctus reliquus pilæ impetus motum quoquè efficit <lb/>majorem: quòd &longs;i in reticulo flaccidi, & remi&longs;&longs;i &longs;int nervi, lan­<lb/>guidè pila reflectitur. </s> </p> <p type="main"> <s id="s.001157">Ad quandam autem reflexionis &longs;peciem pertinere cen&longs;enda <lb/>e&longs;t concu&longs;&longs;io, &longs;ive vibratio, aliquarum &longs;altem corporis partium, <lb/>ubi totum ex reliquo impetu re&longs;ilire nequit: &longs;ic corpus ita at­<lb/>tollens, ut &longs;ummis pedibus innitaris, po&longs;tmodum recidens in <lb/>talos, eò validiorem partium concu&longs;&longs;ionem percipies, quò ve­<lb/>lociùs recides. </s> <s id="s.001158">Simile quid etiam in inanimis contingere ratio <lb/>&longs;uadet, neque enim ita &longs;emper &longs;olida aut pror&longs;us homogenea <lb/>tota moles e&longs;t, ut nullæ omninò partes concuti valeant: quin <lb/>etiam alli&longs;i corporis partes, &longs;i non adeò tenaci vinculo inter &longs;e <lb/>cohæreant, ex reliquo impetu aliæ aliò di&longs;tractæ de&longs;iliunt. </s> </p> <p type="main"> <s id="s.001159">Hinc, docente naturâ, ex alto de&longs;ilientes ubi terram pedi­<lb/>bus attigerint, genua antror&longs;um inflectunt, qua&longs;i calcaneis in­<lb/>&longs;e&longs;&longs;uri, ne conceptus ex &longs;altu impetus &longs;uperiorem corporis par­<lb/>tem deor&longs;um validiùs urgens &longs;ubjectas tibias, & genua ita pre­<lb/>mat, ut inde divi&longs;io aliqua membrorum, aut o&longs;&longs;ium luxatio, aut <lb/>nervorum &longs;eu tendinum nimia di&longs;ten&longs;io dolorem gignat: hoc <lb/>autem valet illa genuum inflexio ad extenuandum impetum, <lb/>quod & flexili mollitiâ &longs;ub&longs;idens terra uligino&longs;a, &longs;i quando la­<lb/>pis in eam ex alto deciderit. </s> <s id="s.001160">Sic Atlas Sinicus pag. </s> <s id="s.001161">123. in XI. <!-- KEEP S--></s> <lb/> <s id="s.001162">Provinciâ Fokion, ubi &longs;ermo e&longs;t de flumine Min, quod vio­<lb/>lento cur&longs;u per &longs;axa volvitur, ait naves, quibus ibi navigatur, <lb/>ex diverbio vocari <emph type="italics"/>Papyraceas, eo quòd tenuibus ac minime re­<lb/>&longs;i&longs;tentibus con&longs;tent a&longs;&longs;eribus, imò ne clavis quidem compaginatis; <lb/>&longs;ed vimine quodam lenti&longs;&longs;imo; unde tamet&longs;i in &longs;axa impingat na­<lb/>vis, &longs;apè tamen minimè rumpitur, quia vix re&longs;i&longs;tit.<emph.end type="italics"/></s> <s id="s.001163"> Et pag.127. <lb/>de catadupis aquarum in flumine per quod ad Jenping naviga­<lb/>tur loquens ait. <emph type="italics"/>Cum naves tran&longs;eunt, ne cum aquâ decidentes <lb/>f actionis incurrant periculum, &longs;citè præmittunt nautæ aliquot &longs;tra­<lb/>minis fa&longs;ces, ad quos navis leviùs impingat, ac tran&longs;eat.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.001164">Jam verò ad impetum extrin&longs;ecùs impre&longs;&longs;um mentem ocu­<lb/>lo&longs;que intendente, non illum &longs;emper momento perire animad­<lb/>vertimus, aut illicò, ac externus agitator ce&longs;&longs;at. </s> </p> <p type="main"> <s id="s.001165">Unde enim tit, ut concitato navigio, cùm vela nautæ con­<lb/>traxerunt, aut remiges inhibuerunt, retineat tamen ip&longs;a navis <lb/>motum & cur&longs;um &longs;uum, intermi&longs;&longs;o ventorum incur&longs;u, pulsúve <pb pagenum="153" xlink:href="017/01/169.jpg"/>remorum? </s> <s id="s.001166">ni&longs;i quia navis, etiam nullo impellente, vi impre&longs;sâ <lb/>urgetur. </s> <s id="s.001167">Quid rhedam cur&longs;u procedente faciliùs quàm initiò <lb/>promovet, equis licet languidius connitentibus? </s> <s id="s.001168">curve onus <lb/>aliquod ingens protrudentes, aut trahentes hoc maximè ca­<lb/>vent, ne contentionem illam quies interrumpat, experientiâ <lb/>&longs;atis edocti incitatum &longs;emel minori labore propelli, quàm com­<lb/>moveri quie&longs;cens? </s> <s id="s.001169">ni&longs;i quia reliquus ex priore motu impetus <lb/>adhuc per&longs;everans po&longs;teriorem motum juvat. </s> <s id="s.001170">Hoc tamen tria <lb/>hæc differunt, quòd onus, ce&longs;&longs;antibus iis, qui protrudebant, <lb/>con&longs;i&longs;tit illicò (ni&longs;i fortè volubilitatem habens, aut &longs;ubjectis <lb/>cylindris innixum, adhuc modicum quid volvi aut progredi <lb/>pergat) rheda currentes equo, &longs;ubita funium abruptione dis­<lb/>junctos &longs;equitur ad pa&longs;&longs;us aliquot non adeò multos pro viæ <lb/>æquabilitate præcedenti&longs;que velocitatis ratione; navigium verò <lb/>&longs;ubmi&longs;&longs;is antennis, remi&longs;que ce&longs;&longs;atione torpentibus aliquandiu, <lb/>intervallo non &longs;anè contemnendo, provehitur. </s> <s id="s.001171">Oneris &longs;cilicet <lb/>motui, cui volubilitatem neque ars, neque natura dederit, im­<lb/>pedimento e&longs;t ip&longs;a extremitas a&longs;pera &longs;ubjectam planitiem &longs;ale­<lb/>bris quandóque non carentem contingens, gravita&longs;que ita va­<lb/>lidè premens, ut major futurus e&longs;&longs;et partium tritus, quàm pro <lb/>impetûs modo, qui reliquus e&longs;&longs;et, &longs;uperari po&longs;&longs;et: Id quod cur­<lb/>renti rhedæ idcircò non contingere planum e&longs;t, quia licèt <lb/>nihilo levior &longs;it quàm onus protru&longs;um, minùs tamen rotarum <lb/>modioli leniter cum axibus confligentes motum retardant. </s> <s id="s.001172">At <lb/>navis &longs;ponte &longs;uâ innatans, ventorum incur&longs;ione, remorúmve <lb/>pul&longs;u diutiùs acta, vix, aut fortè ne vix quidem, mole &longs;uâ re­<lb/>luctatur, ni&longs;i quatenus diffindenda e&longs;t aqua; nec &longs;inè multo fa­<lb/>cilitatis compendio, prior &longs;iquidem unda, quam prora impel­<lb/>lens excitat, aliam ante &longs;e urget ad ea&longs;dem partes: propterea <lb/>impre&longs;&longs;us navi impetus modicum nactus impedimentum diù <lb/>durat, illámque promovet. </s> <s id="s.001173">Quare idem de impetu extrin&longs;ecùs <lb/>a&longs;&longs;umpto dicendum e&longs;t, quod de acqui&longs;ito; nimirùm minui pro <lb/>Ratione eorum, quæ in&longs;tituto motui ob&longs;i&longs;tunt, aut etiam pror­<lb/>sùs perire. </s> </p> <p type="main"> <s id="s.001174">Præter ea autem quæ utrique motui tùm naturali, tùm vio­<lb/>lento æquè opponuntur, (cuju&longs;modi e&longs;t medium dividendum, <lb/>objecti corporis occur&longs;us, aut contingentis tritus atque con­<lb/>flictus, retinaculum, quod certo limite motum definiat, & alia <pb pagenum="154" xlink:href="017/01/170.jpg"/>id genus) illa e&longs;t externo impul&longs;ui peculiaris repugnantia, <lb/>quæ ex inhærente corpori gravitate oritur, &longs;ive illi innatus im­<lb/>petus, &longs;ive acqui&longs;itus modum &longs;tatuat. </s> <s id="s.001175">Neque id &longs;impliciter <lb/>tantùm, &longs;ed comparatè con&longs;iderandum e&longs;t, quam &longs;cilicet in <lb/>plagam impul&longs;us motum dirigat, & quatenus gravitatis pro­<lb/>pen&longs;ioni opponatur. </s> <s id="s.001176">Quemadmodum enim qui in pilâ aroma­<lb/>ta pin&longs;unt, nihil repugnantem, quin & impul&longs;ui ob&longs;ecundan­<lb/>tem, experiuntur pi&longs;tilli gravitatem deprimentes; contrà verò <lb/>attollentes fatigat eadem gravitas directò deor&longs;um urgens; me­<lb/>dium autem quiddam tenet in ob&longs;i&longs;tendo, &longs;i motio tran&longs;ver&longs;a <lb/>contingat; &longs;icut experiri licet, &longs;i ex funiculo pendens idem <lb/>pi&longs;tillus à perpendiculo dimoveatur; minore enim conatu opus <lb/>e&longs;t: ita quò minùs in oppo&longs;itam gravitati plagam dirigitur im­<lb/>pul&longs;us, eò etiam diutiùs per&longs;everat minus habens impedimenti. </s> <lb/> <s id="s.001177">Hinc e&longs;t quod gravitas æquabiliter toto corpore fu&longs;a &longs;i aut ex <lb/>centro &longs;u&longs;pendatur, aut coni apici in&longs;i&longs;tat, levi negotio, ac &longs;a­<lb/>tis diù, in gyrum convertitur; innatum videlicet gravitatis im­<lb/>petum vis ip&longs;a &longs;u&longs;pendens aut &longs;u&longs;tentans elidit; nihil verò im­<lb/>pul&longs;um remoratur præter aut funiculi &longs;u&longs;pendentis &longs;piras paulò <lb/>&longs;pi&longs;&longs;iores, aut tritum cum &longs;ubjecto cono, aëri&longs;que dividendi <lb/>re&longs;i&longs;tentiam; quæ tamen &longs;i tollatur in corpore orbiculari circà <lb/>centrum commoto, etiam longior fit conver&longs;io. </s> <s id="s.001178">Sic ferream <lb/>&longs;agittam palmarem cra&longs;&longs;iu&longs;culam in&longs;tar acûs magneticæ in <lb/>æquilibrio con&longs;titutam levi&longs;&longs;imo impul&longs;u ac diuti&longs;&longs;imè in gy­<lb/>rum agi ob&longs;ervavi; vix enim acuti&longs;&longs;imum verticem, cui innite­<lb/>batur, terebat, & aëris intrà eumdem gyrum circumducti mo­<lb/>dica erat re&longs;i&longs;tentia. </s> <s id="s.001179">Id autem multo luculentiùs apparet in <lb/>verticillo, cujus axem perpolito alveolo in&longs;i&longs;tentem extremo <lb/>pollice ac indice leviter comprimens, ac paulò celeriùs vertens, <lb/>eò diuturniori vertigine contorqueri videbis, quò pauciores <lb/>minore&longs;que offenderit in &longs;ubjectâ tabulâ a&longs;peritates, ad quas al­<lb/>li&longs;us paululùm inclinetur, aut aliò reflectatur. </s> </p> <p type="main"> <s id="s.001180">Quòd &longs;i magnetis polo ritè armato chalybeum axiculum <lb/>congruo verticulo in&longs;tructum admoveris, ut planè à magnete <lb/>&longs;u&longs;pendatur, tùm &longs;ummis digitis opportunè axem terentibus <lb/>vertiginem ei delicatè ac molliter conciliaveris, miraculi loco <lb/>tibi erit tàm diuturna conver&longs;io; quippe cui non &longs;ubjectialveoli <lb/>a&longs;peritates &longs;altitare cogentes, non gravitas ip&longs;a premens, tritum-<pb pagenum="155" xlink:href="017/01/171.jpg"/>que augens, non &longs;u&longs;pendentis funiculi violenta contortio ob­<lb/>&longs;i&longs;tunt, motúmve aliquatenus impedientes impre&longs;&longs;um impe­<lb/>tam imminuunt; &longs;ed magnetico radio &longs;u&longs;pen&longs;us intra &longs;e perpe­<lb/>tuò volvitur lævi&longs;&longs;imum chalybem magnetis polo adhærentem <lb/>leni&longs;&longs;imè terens. </s> </p> <p type="main"> <s id="s.001181">Illud etiam in motu, qui ab extrin&longs;eco provenit, con&longs;ide­<lb/>randum e&longs;t, quòd contingere pote&longs;t duos ade&longs;&longs;e motores, qui <lb/>corporis motum in diver&longs;as partes dirigant: quare alter alteri <lb/>ob&longs;i&longs;tit, & motus ex duplici directione compo&longs;itus is e&longs;t, qui <lb/>non re&longs;pondeat men&longs;uræ duplicis illius impetûs, &longs;i &longs;inguli in­<lb/>tegrè accipiantur. </s> <s id="s.001182">Con&longs;tat enim, &longs;i æquabili & æquali cona­<lb/>tu urgeant corpus, moveri aut per diametrum Quadrati, &longs;i di­<lb/>rectiones &longs;int ad angulum rectum con&longs;titutæ; aut per Diago­<lb/>nalem lineam Rhombi, &longs;i directiones obliquæ &longs;int: &longs;i verò <lb/>æquabiles quidem &longs;int, &longs;ed inæquales conatus, per diametrum <lb/>Rectanguli aut Rhomboidis moveri, pro ut ad rectum aut obli­<lb/>quum angulum directiones &longs;ibi invicem re&longs;pondent. </s> <s id="s.001183">Semper <lb/>autem minor e&longs;t motus quàm pro duorum illorum impul&longs;uum <lb/>ratione; diameter &longs;iquidem brevior e&longs;t aggregato duorum <lb/>adjacentium laterum. </s> <s id="s.001184">Quòd &longs;i æquabiles non &longs;int impetus, <lb/>vel &longs;altem alter æquabilis &longs;it, alter acceleratus aut retardatus, <lb/>linea curva de&longs;cribitur; quæ pariter minor e&longs;t duabus rectis, <lb/>quæ vi &longs;ingulorum impetuum de&longs;criberentur; ab illis &longs;i qui­<lb/>dem continetur. </s> </p> <p type="main"> <s id="s.001185">Hîc tamen advertendus animus e&longs;t, & ob&longs;ervare oportet <lb/>æquabilem impul&longs;um (&longs;i continuus &longs;it, nec morulis inter­<lb/>ruptus) e&longs;&longs;e non po&longs;&longs;e, ni&longs;i ab animali &longs;emper æqualiter conan­<lb/>te efficiatur; quia gravium de&longs;cen&longs;us naturaliter acceleratur; <lb/>ela&longs;mata verò dum &longs;e re&longs;tituunt, &longs;emper languidiùs &longs;ingulis <lb/>momentis conantur, &longs;i quidem virtus ela&longs;tica con&longs;ideretur: <lb/>quamquàm po&longs;teriore momento quod e&longs;t reliquum prioris im­<lb/>petûs, inten&longs;ionem efficit additum po&longs;teriori licèt remi&longs;&longs;o. </s> <lb/> <s id="s.001186">Vix igitur contingere pote&longs;t motum unum à duplici impetu <lb/>extrin&longs;ecùs impre&longs;&longs;o fieri per lineam rectam ni&longs;i corpus à du­<lb/>plici motore æquabiliter urgeatur. </s> </p> <p type="main"> <s id="s.001187">Cum itaque impetus acqui&longs;itus, aut aliundè impre&longs;&longs;us, &longs;it <lb/>qualitas propter motum in&longs;tituta, quæ non ni&longs;i in motu pro­<lb/>ducitur, ita pariter ni&longs;i in motu, & cum motu non con&longs;erva-<pb pagenum="156" xlink:href="017/01/172.jpg"/>tur. </s> <s id="s.001188">Quare &longs;i corpus eò deveniat, ut nullo pror&longs;us pacto agi­<lb/>tari queat, aut interiore motu cieri, quo momento impeditur <lb/>motus, ne &longs;it, eo momento impetus perit, ce&longs;&longs;ante videlicet <lb/>causa effectiva ab ejus con&longs;ervatione eo ip&longs;o quod ce&longs;&longs;at finis, <lb/>propter quem impetus e&longs;t. </s> <s id="s.001189">Quod &longs;i impedimentum occurrat <lb/>non prorsùs motum tollens (ut &longs;i globus in plano horizontali <lb/>rotatus veniat ad planum inclinatum, per quod ex concepto <lb/>impetu a&longs;cendat) tunc pro ratione impedimenti extenuatur <lb/>impetus, donec tandem pereat. <lb/> </s> </p> <p type="main"> <s id="s.001190"><emph type="center"/>CAPUT IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001191"><emph type="center"/><emph type="italics"/>Quâ ratione vis movendi cum impedimentis <lb/>comparetur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001192">MOtus omnis nec in oppo&longs;itas, nec in diver&longs;as plagas, &longs;ed <lb/>per certam lineam dirigitur; unico quippe in loco, non <lb/>in pluribus, eodem temporis puncto e&longs;&longs;e pote&longs;t corpus. </s> </p> <p type="main"> <s id="s.001193">Nihil igitur motui moram & impedimentum inferre pote&longs;t, <lb/>ni&longs;i directò aut obliquè illi &longs;ecundùm eam lineam, per quam <lb/>in&longs;tituendus e&longs;&longs;et, antè, ponè, ad dextram, ad lævam, &longs;ur&longs;um, <lb/>deor&longs;um opponatur. </s> <s id="s.001194">Si enim duo corpora eádem pergerent viâ, <lb/>& maximâ velocitatis, aut tarditatis con&longs;piratione con&longs;entirent, <lb/>tunc neque po&longs;terius ab eo quod antè e&longs;t, traheretur, neque <lb/>prius à po&longs;teriore urgeretur, neque alterum alteri impedimen­<lb/>to e&longs;&longs;et. </s> <s id="s.001195">Hinc manife&longs;tum e&longs;t non po&longs;&longs;e impedimentum &longs;upe­<lb/>rari, quin ei vis aliqua inferatur. </s> </p> <p type="main"> <s id="s.001196">Rem porrò univer&longs;am duas in partes tribuere po&longs;&longs;umus, ut <lb/>duplex Re&longs;i&longs;tentiæ genus &longs;tatuatur; Formalem alteram, alte­<lb/>ram Activam &longs;cholæ vocarent. </s> <s id="s.001197">Corpus enim, quod ob&longs;tat, aut <lb/>retinet, &longs;i motum prorsùs nullum conetur in&longs;tituto aut de&longs;ti­<lb/>nato motui adver&longs;antem, re&longs;i&longs;tit quidem, &longs;ed Formaliter; nihil <lb/>&longs;cilicet efficit, quo repugnet, &longs;ed &longs;uo tantùm &longs;e tutatur in loco: <lb/>Sin autem & contrà nitatur, aut retrahat, jam non ob&longs;i&longs;tit &longs;o­<lb/>lùm, ne loco per vim dimoveatur; &longs;ed etiam impetum in con-<pb pagenum="157" xlink:href="017/01/173.jpg"/>trariam plagam directum efficit, cujus vi motum impedit, ac <lb/>proptereà Activè re&longs;i&longs;tit. </s> <s id="s.001198">Huic autem verbo, cum <emph type="italics"/>Re&longs;i&longs;tere<emph.end type="italics"/> di­<lb/>cimus, &longs;ubjecta notio e&longs;t, in causá e&longs;&longs;e ne motus fiat, aut &longs;al­<lb/>tem non ea velocitate, quæ virtuti movendi non impeditæ cæ­<lb/>teroqui re&longs;ponderet. </s> <s id="s.001199">Sic paries, in quem incurris, tibi re&longs;i&longs;tit <lb/>Formaliter, ne procedas, & aqua &longs;tagnans, cui collo tenus im­<lb/>mergeris, progredienti re&longs;i&longs;tit Formaliter, ne velociter, &longs;icut <lb/>intra aërem movearis pro ratione impetus, quo conaris progre­<lb/>di: qui verò occurrens te repellit, ut &longs;i coneris contra ictum <lb/>fluvij, non Formaliter tantùm, &longs;ed etiam Activè re&longs;i&longs;tit; non <lb/>&longs;olùm enim ob&longs;tat, quia ejus in locum &longs;uccedere non potes, <lb/>ni&longs;i cum loco dimoveas, &longs;ed etiam tibi adver&longs;um impetum im­<lb/>primit, ut te loco extrudat. </s> </p> <p type="main"> <s id="s.001200">Cum itaque impedimenta motûs externo impetu &longs;ubmoven­<lb/>da &longs;int, virtus autem movendi certa &longs;it ac definita, con&longs;tat vi­<lb/>res omnes, quæ in corpore promovendo, &longs;i nihil ob&longs;taret, exer­<lb/>cerentur, duas in partes di&longs;trahi, ad movendum &longs;cilicet cor­<lb/>pus, & ad tollenda impedimenta, Concipit igitur impetum, <lb/>qui motum efficiat, & ob&longs;tanti corpori impetum imprimit, ut <lb/>loco cedat. </s> <s id="s.001201">Quid igitur mirum, &longs;i di&longs;tractis viribus languidior <lb/>&longs;equatur metus? </s> <s id="s.001202">Quia verò quò majori velocitate corpus <lb/>ob&longs;tans propellendum e&longs;t, aut trahendum, majori quoque im­<lb/>petu impre&longs;&longs;o opus habet, palàm e&longs;t majorem quoque in pro­<lb/>pellente, aut &longs;ecum rapiente, impetum requiri, ut majorem re­<lb/>&longs;i&longs;tentiam vincens &longs;e ip&longs;um pariter moveat. </s> </p> <p type="main"> <s id="s.001203">Hic autem quid monui&longs;&longs;e oporteat vim re&longs;i&longs;tendi &longs;uperan­<lb/>dam e&longs;&longs;e à virtute movendi? </s> <s id="s.001204">quis enim ambigat, an, &longs;i pares <lb/>illæ fuerint, nullus futurus &longs;it motus? </s> <s id="s.001205">Quòd &longs;i impedimentum <lb/>prorsùs immotum adversùs conantem per&longs;tat, nullum pariter <lb/>recipit impetum; qui &longs;cilicet, etiam &longs;i priùs fui&longs;&longs;et, motu ce&longs;­<lb/>&longs;ante periret. </s> <s id="s.001206">Hinc in animali defatigatio membrorum oritur, <lb/>quando prorsùs in irritum conatus cadit; impetus enim, quem <lb/>concipit, ut æqualem motum imprimeret impedimento, &longs;i hoc <lb/>&longs;uperari po&longs;&longs;et, in animali ip&longs;o motum aliquem efficit, &longs;ed quia <lb/>progredi vetatur ab o&longs;tante aut retinente impedimento, impe­<lb/>tus ille non totius animalis motum ulteriùs promovet; &longs;ed mem­<lb/>brorum partes alias comprimit, alias di&longs;tendit, unde & dolor <lb/>aliquis, & la&longs;&longs;itudo provenit. </s> <s id="s.001207">At &longs;i corpus, cui motus debetur, <pb pagenum="158" xlink:href="017/01/174.jpg"/>cùm inanimum &longs;it, nequeat impetum, quemadmodum animan­<lb/>tes, ex arbitrio temperare, & quia &longs;olidum e&longs;t ac durum, nul­<lb/>lam pati compre&longs;&longs;ionem aut di&longs;tentionem partium po&longs;&longs;it, &longs;icut <lb/>& corpus ob&longs;tans aut retinens compre&longs;&longs;ionem omnem aut <lb/>di&longs;tentionem re&longs;puit; tunc nullum concipit aut imprimit im­<lb/>petum præter innatam gravitationem, aut levitationem, cùm <lb/>per vim in loco non debito detineatur. </s> <s id="s.001208">Ex hoc conjecturam ca­<lb/>pere licet de eo, quod contingit, quando virtute movendi re­<lb/>&longs;i&longs;tentiam vincente impedimentum &longs;ubmovetur; impediri vi­<lb/>delicet, ne producatur motus, juxta re&longs;i&longs;tentiæ modum atque <lb/>men&longs;uram; quæ &longs;icuti non quâlibet minimâ vi &longs;uperari pote&longs;t, <lb/>ita majori cedit. </s> </p> <p type="main"> <s id="s.001209">Verùm quonam id pacto contingat, ut explicare conemur, <lb/>illud ob&longs;erva, quòd &longs;i corpus idem quadruplo velociùs moveri <lb/>debeat, ac moveretur priùs certâ impetûs men&longs;urâ, utique qua­<lb/>druplo majorem impetum exigit, ut pro impetûs inten&longs;ione <lb/>aut remi&longs;&longs;ione velocior aut tardior &longs;equatur motus. </s> <s id="s.001210">At &longs;i cor­<lb/>pus aliud movendum quadruplo gravius exhibeatur, in hoc im­<lb/>petus ille quadruplex &longs;ubquadruplam efficiet inten&longs;ionem, ac <lb/>propterea etiam motum habebit tardiorem, &longs;i cætera &longs;int paria, <lb/>pro impetûs inten&longs;ione. </s> <s id="s.001211">Si cætera, inquam, &longs;int paria; &longs;æpè <lb/>enim aër, aut aqua plus velociori motui re&longs;i&longs;tunt, quàm tardio­<lb/>ri, & moles major efficit, ut non omninò velocitas inten&longs;ioni <lb/>impetûs re&longs;pondeat. </s> <s id="s.001212">Hæc tamen nunc mente &longs;ecernamus, per­<lb/>inde atque &longs;i nihil officerent motui. </s> </p> <p type="main"> <s id="s.001213">Quoniam igitur motus ab omni velocitatis aut tarditatis men­<lb/>&longs;urâ &longs;ejungi nequit, finge corpus per vim movendum huju&longs;­<lb/>modi e&longs;&longs;e, ut &longs;pectatâ mole &longs;eu materiâ, ac &longs;pecificâ gravitate, <lb/>ad percurrendum &longs;patium pa&longs;&longs;uum 100 unius horæ quadrante, <lb/>indigeret impetu, cujus inten&longs;io e&longs;&longs;et particularum 4 in &longs;ingu­<lb/>lis corporis movendi partibus: molem autem, exempli gratiâ, <lb/>di&longs;tinctam concipe in particulas 100 minimas. </s> <s id="s.001214">Quare &longs;pectatâ <lb/>tùm exten&longs;ione tùm inten&longs;ione impetûs, nece&longs;&longs;e e&longs;t illi à mo­<lb/>tore imprimi impetûs particulas 400. Quòd &longs;i corporis per vim <lb/>movendi moles ac materia e&longs;&longs;et quadruplex alterius, &longs;i nimi­<lb/>rum ratione materiæ exten&longs;ionis particulas haberet 400, jam <lb/>impetus idem &longs;ubquadruplam efficeret inten&longs;ionem, & &longs;ingulæ <lb/>impetûs particulæ &longs;ingulis corporis particulis ine&longs;&longs;ent; atque <pb pagenum="159" xlink:href="017/01/175.jpg"/>adeò etiam hujus velocitas e&longs;&longs;et &longs;ubquadrupla prioris velocita­<lb/>tis: partamen utrobique e&longs;&longs;et, illud quidem velociùs, hoc tar­<lb/>diùs movendi difficultas, cum in utroque particulas 400 impe­<lb/>tûs produci oporteret; utriu&longs;que enim impetûs exten&longs;iones & <lb/>inten&longs;iones e&longs;&longs;ent Reciprocè in eadem Ratione. <!-- KEEP S--></s> <s id="s.001215">In corpore <lb/>itaque, ex quo motus originem ducit, tanta vis movendi ine&longs;&longs;e <lb/>debet, ut & corpori impedienti, quod &longs;ubmovetur, congruen­<lb/>tem motui impetum imprimat, hoc e&longs;t particulas 400, & ip&longs;um <lb/>&longs;e pariter promoveat: nihil enim accepto extrin&longs;ecùs impetu <lb/>agitatur à motore prorsùs immoto, ut eunti per &longs;ingula patebit. </s> </p> <p type="main"> <s id="s.001216">Jam verò quoniam idem corpus modò remi&longs;&longs;iùs, modò con­<lb/>citatiùs moveri pro impetûs inten&longs;ione videmus, probabilis <lb/>conjectura e&longs;t in iis, quæ non &longs;uo arbitrio, &longs;ed naturæ reguntur <lb/>imperio, totum impetum produci, qui virtuti efficiendi re&longs;pon­<lb/>det: hæc autem in impedimento, cujus re&longs;i&longs;tentia vincitur, <lb/>impetum eâ inten&longs;ionis men&longs;urâ imprimit, quæ illi motûs ve­<lb/>locitatem conciliet ip&longs;ius corporis moventis velocitati con­<lb/>gruentem, adeò ut movendi facultas totas &longs;uas vires exerat <lb/>partim impetum imprimens &longs;ubmovendo impedimento, partim <lb/>motum efficiens in ip&longs;o corpore: ex quo fit quod eò remi&longs;&longs;iorem <lb/>motum in &longs;e motor efficiat, quò major &longs;ecundùm inten&longs;ionem <lb/>impetus impeditur ab impedimento. </s> <s id="s.001217">Sic plumbeus globus bili­<lb/>bri, &longs;i, funiculo excavatæ volubilis orbiculi curvaturæ in&longs;erto, <lb/>connectatur cum globulo &longs;ubduplæ gravitatis, non eá veloci­<lb/>tate de&longs;cendit, qua de&longs;cenderet &longs;ibi relictus ab&longs;que ullâ appen­<lb/>dice; velociùs tamen movetur, quàm &longs;i e&longs;&longs;et globuli adjuncti <lb/>tantùm &longs;e&longs;quialter; quia &longs;cilicet ut ad æqualem velocitatem <lb/>temperentur motus tùm impedimenti &longs;ur&longs;um, tùm corporis mo­<lb/>ventis deor&longs;um, minor inten&longs;ivè impetus impediendus e&longs;t à glo­<lb/>bulo &longs;ubduplo quàm à &longs;ub&longs;e&longs;quialtero; ac propterea major e&longs;t <lb/>&longs;ecundùm inten&longs;ionem reliquus impetus motum efficiens con­<lb/>citatiorem. </s> </p> <p type="main"> <s id="s.001218">Quòd autem à globo de&longs;cendente imprimatur impetus glo­<lb/>bulo, quem &longs;ur&longs;um trahit, hinc con&longs;tat, quod &longs;i globulus ille <lb/>non &longs;it admodum gravis, tùm demum &longs;ub&longs;ilit, ubi globus ve­<lb/>lociter de&longs;cendens &longs;ubjectum planum attigerit: quid enim il­<lb/>lum &longs;ub&longs;ilire cogeret quie&longs;cente jam globo, à quo trahebatur, <lb/>ni&longs;i adhuc aliquid impre&longs;&longs;i impetûs remaneret? </s> <s id="s.001219">At quòd im-<pb pagenum="160" xlink:href="017/01/176.jpg"/>pre&longs;&longs;us hîc impetus non ab ip&longs;o motore, &longs;ed ab impetu, quem <lb/>ille concepit, proximè efficiatur, hinc &longs;ibi &longs;uadent plures, quia <lb/>ex alterâ parte impetum ab impetu produci po&longs;&longs;e manife&longs;tum <lb/>videtur ex percu&longs;&longs;ionibus projectorum, ut cùm globus pro­<lb/>jectus in quie&longs;centem globum impactus illum trudit; ex alterâ <lb/>cau&longs;am proximam effectui homogeneam congruenter naturæ <lb/>&longs;tatuimus; &longs;ic enim & calorem in nobis à calore potiùs quàm à <lb/>&longs;ub&longs;tantiâ ignis proximè produci exi&longs;timamus. </s> <s id="s.001220">Sed quid de <lb/>percu&longs;&longs;ionum impetu dicendum &longs;it, &longs;uo loco con&longs;tabit inferiùs. </s> </p> <p type="main"> <s id="s.001221">Motoris demùm velocitatem inten&longs;ioni impetûs concepti <lb/>non re&longs;pondere experimur, cum valdè conantes ut onus rapte­<lb/>mus; parùm progredimur; at &longs;i funis ex improvi&longs;o abrumpa­<lb/>tur, illicò corruimus, impetu &longs;cilicet concepto motum validiùs <lb/>efficiente, ubi de&longs;ierit impetum oneri, quod raptabatur, im­<lb/>primere. </s> </p> <p type="main"> <s id="s.001222">Hinc fit quòd, &longs;i ea fuerit corporum di&longs;po&longs;itio, ut impedi­<lb/>mentum tardè &longs;ubmovendum &longs;it, ac proinde remi&longs;&longs;iore impetu <lb/>opus habeat, qui &longs;ibi imprimatur; corpus verò, cui motus <lb/>omnis tribuitur, non æquali tarditate cum impedimento ferri <lb/>nece&longs;&longs;e &longs;it, &longs;ed velociùs præ illo moveri po&longs;&longs;it, hoc &longs;anè eò mi­<lb/>nùs habet re&longs;i&longs;tentiæ, quò minorem in intentione impetûs men­<lb/>&longs;uram impedimento eidem imprimere debet, ut illud &longs;ubmo­<lb/>veatur. </s> <s id="s.001223">Contrà verò &longs;i ita fuerint di&longs;po&longs;ita, ut impedimentum <lb/>velociùs præ ip&longs;o motore moveri oporteat, multò magis re&longs;i&longs;tit, <lb/>quàm &longs;i pariter moverentur, plus enim impetûs imprimendum <lb/>e&longs;t, ut motus con&longs;equatur. </s> </p> <p type="main"> <s id="s.001224">Hactenùs re&longs;i&longs;tentiam poti&longs;&longs;imùm Formalem, impedimento <lb/>nihil in adver&longs;um conante, contemplati &longs;umus; jam ad Acti­<lb/>vam tran&longs;eamus, cum &longs;cilicet duo corpora invicem aut omni­<lb/>nò, aut ex parte repugnant, quia motum in diver&longs;as aut oppo­<lb/>&longs;itas plagas directum moliuntur. </s> <s id="s.001225">In medio va&longs;e aquâ pleno &longs;ta­<lb/>tuatur lignea tabella cra&longs;&longs;iu&longs;cula, eique lapis imponatur: dum <lb/>illa conatur a&longs;cendere, hic de&longs;cendere, &longs;e invicem urgent; &longs;ed <lb/>cum &longs;e vici&longs;&longs;im permeare nequeant, &longs;i paribus quidem viribus <lb/>confligant, &longs;ine motu con&longs;i&longs;tunt; &longs;in autem imparibus, aut <lb/>ambo a&longs;cendunt, aut ambo de&longs;cendunt, pro ut &longs;ive tabellæ le­<lb/>vitas, &longs;ive lapidis gravitas oppo&longs;itam vicerit. </s> <s id="s.001226">Quod &longs;i lapis ta­<lb/>bellæ non impo&longs;itus, &longs;ed &longs;uppo&longs;itus, arctè tamen connexus <pb pagenum="161" xlink:href="017/01/177.jpg"/>fuerit, adhue contrarios motus conantur, non &longs;e tamen invi­<lb/>cem urgent, &longs;ed vici&longs;&longs;im retrahunt, quandiù vinculum non <lb/>revellatur, aut rumpatur. </s> <s id="s.001227">Hic verò &longs;ubdubitet qui&longs;piam, <lb/>utrùm corpora, quæ contrario ni&longs;u reluctantur, &longs;ibi vici&longs;­<lb/>&longs;im impetum imprimant, nec ne, aut æqualem, &longs;i pares fue­<lb/>rint vires, aut, &longs;i impares, inæqualem: Quando enim ob vi­<lb/>rium æqualitatem utrumque corpus con&longs;i&longs;tit, codem pacto <lb/>quies &longs;equitur, &longs;i unumquodque &longs;uam gravitationem aut levi­<lb/>tationem &longs;ervans nihil alteri imprimat, ac &longs;i lignea tabella levi­<lb/>tans partem impetûs &longs;ur&longs;um directi conferat impo&longs;ito lapidi, à <lb/>quo gravitante vici&longs;&longs;im recipiat tantumdem impetús deor&longs;um <lb/>directi; ex quo fiat, ut lapis habens concepti ac innati impe­<lb/>tûs deor&longs;um directi vires æquales viribus impetûs &longs;ur&longs;um di­<lb/>recti con&longs;i&longs;tat, idemque in ligneâ tabellâ contingat. </s> <s id="s.001228">Cùm ve­<lb/>rò inæquales fuerint vires, id quod validius e&longs;t, eodem modo <lb/>&longs;uperat, &longs;ive nihil contrarij impetûs ab infirmiore oppo&longs;ito re­<lb/>cipiat, &longs;ed minorem motum vi &longs;ui impetûs producat pro ratio­<lb/>ne virium, quibus &longs;uperat; &longs;ivè partem impetûs contrarij reci­<lb/>piat, quæ proprij impetûs vires attenuet. </s> </p> <p type="main"> <s id="s.001229">Quotidianum e&longs;t hujus æqualitatis aut inæqualitatis experi­<lb/>mentum in iis, quæ innatant humori; hæc enim humori im­<lb/>po&longs;ita, quia in aëre gravitant, de&longs;cendunt; pars verò immer&longs;a <lb/>levitat in humore; prægravata tamen à reliquâ parte extante <lb/>deor&longs;um adhuc urgetur, donec inter partem immer&longs;am & ex­<lb/>tantem fiat æquilibrium, & tantumdem pars immer&longs;a levitet in <lb/>humore, ac extans gravitat in aëre. </s> <s id="s.001230">Sic ma&longs;&longs;a plumbea argento <lb/>vivo impo&longs;ita de&longs;cendit, donec molis plumbeæ pars (2/13) extet; e&longs;t <lb/>enim &longs;pecifica plumbi gravitas ad &longs;pecificam mercurij gravita­<lb/>tem ut 11 ad 13. levitat itaque plumbum in mercurio ut 2, gra­<lb/>vitat in aëre ut 11; igitur plumbeæ ma&longs;&longs;æ partes 11 levitantes <lb/>&longs;ingulæ ut 2 parem habent conatum &longs;ur&longs;um, ac partes 2 gra­<lb/>vitantes &longs;ingulæ ut 11 conantur deor&longs;um. </s> <s id="s.001231">Quòd &longs;i ita depri­<lb/>meretur plumbum, ut ejus partes 12 immergerentur, & una <lb/>extaret; jam unica pars gravitans ut 11 vinceretur à partibus <lb/>12 levitantibus &longs;ingulis ut 2, ac propterea adhuc pars una <lb/>emergeret: quemad modum &longs;i quatuor partes extarent, & no­<lb/>vem immergerentur, harum levitas 18 ab illarum gravitate 44 <lb/>vinceretur, ideóque adhuc duæ immergerentur. </s> </p> <pb pagenum="162" xlink:href="017/01/178.jpg"/> <p type="main"> <s id="s.001232">Jam &longs;i dixeris à partis immer&longs;æ levitantis momentis 18 impe­<lb/>diri momenta 18 partis extantis gravitantis, adeò ut &longs;uper&longs;int <lb/>tantùm vires juxtà exce&longs;&longs;um gravitatis, &longs;cilicet momentorum <lb/>26, juxta quem exce&longs;&longs;um impetum imprimat parti immer&longs;æ, ut <lb/>deprimatur, tunc autem cum paria &longs;uerint levitatis atque gra­<lb/>vitatis momenta, jam non invicem agere, &longs;ed &longs;e vici&longs;&longs;im impe­<lb/>dire, probabilior forta&longs;&longs;e videatur alicui philo&longs;ophandi ratio <lb/>hîc, ubi directè &longs;ibi invicem adver&longs;antur directiones; alteruter <lb/>enim aut neuter impetus movet oppo&longs;itum corpus. </s> <s id="s.001233">Verùm <lb/>quoniam ubi lineæ directionum motûs non &longs;unt in directum <lb/>po&longs;itæ; &longs;ed inclinationem habent, motus mixtus, qui &longs;equitur, <lb/>ex utroque impetu unum motum temperari indicat, in eam fe­<lb/>ror &longs;ententiam, ut exi&longs;timem duo corpora obliquè &longs;ibi invicem <lb/>repugnantia vici&longs;&longs;im imprimere, & recipere impetum in diver­<lb/>&longs;as plaga directum pro modo virtutis uniu&longs;cuju&longs;que, adeò ut <lb/>&longs;i paria &longs;int momenta, medius planè inter utramque directio­<lb/>nem &longs;equatur motus, &longs;i di&longs;paria, &longs;equatur pro modo exce&longs;sûs. </s> </p> <p type="main"> <s id="s.001234">Fieri autem hane mutuam impetûs communicationem hinc <lb/>apparet, quòd &longs;i duo corpora, quorum virtus movendi ut AB <lb/><figure id="id.017.01.178.1.jpg" xlink:href="017/01/178/1.jpg"/><lb/>& AC, inloco, ubi A, con&longs;ti­<lb/>tuta moveri cœperint, alterum <lb/>quidem, quod ad dexteram e&longs;t, <lb/>cum directione AB, alterum <lb/>verò, quod ad &longs;ini&longs;tram, cum di­<lb/>rectione AC, ita &longs;e impediunt, <lb/>ut quod ad lævam e&longs;t, urgeat reliquum, ne per rectam AB proce­<lb/>dat; hoc verò quod ad <expan abbr="dexterã">dexteram</expan> e&longs;t, illud impediat, ne per rectam <lb/>AC incedat; &longs;ed propellat ita, ut ambo habeant directionem <lb/>mixtam AD. <!-- KEEP S--></s> <s id="s.001235">Hæc autem lineæ AD cum major &longs;it &longs;ingulis <lb/>lateribus AB, AC in rectangulo, aut rhomboide, ut quadra­<lb/>to, aut rhombo, cavè nè putes &longs;ingulis corporibus &longs;upra pro­<lb/>prium impetûs modum factam e&longs;&longs;e aliquam ab externo impetu <lb/>virium acce&longs;&longs;ionem: quî enim fieri po&longs;&longs;it, ut corpus nullo re­<lb/>pugnante po&longs;&longs;it certo tempore percurrere lineam AB, dimi­<lb/>nutis verò impetûs viribus ex re&longs;i&longs;tentià, pari tempore longio­<lb/>rem lineam AD percurrat? </s> <s id="s.001236">An quia recipiat à corpore re­<lb/>pugnante impetum, cujus acce&longs;&longs;ione augeatur proprius impe­<lb/>tus, qui reliquus e&longs;t? </s> <s id="s.001237">At &longs;i propter virium æqualitatem percur-<pb pagenum="163" xlink:href="017/01/179.jpg"/>rant Quadrati diametrum, utique tantumdem alterum ab alte­<lb/>ro recipit impetús, quantum tribuit: igitur non e&longs;t major vis <lb/>impetus, quàm &longs;i nihil repugnaret: ex quo fit neque motum ve­<lb/>lociorem e&longs;&longs;e po&longs;&longs;e, ut pari tempore diametrum percurrant, <lb/>quo &longs;ingula de&longs;criberent latus Quadrati. </s> </p> <p type="main"> <s id="s.001238">Non igitur ex illà mutuá impetus in diversâ directi commu­<lb/>nicatione fit in &longs;ingulis corporibus impetûs inten&longs;io major (&longs;i <lb/>propriè loquendum &longs;it, habent enim impetus illi, conceptus <lb/>&longs;cilicet, & impre&longs;&longs;us, directionem diver&longs;am) quàm ferat pro­<lb/>pria &longs;ingulorum virtus: id autem poti&longs;&longs;imùm con&longs;tat, quando <lb/><expan abbr="&longs;ingulorũ">&longs;ingulorum</expan> directiones valdè obtu&longs;um <expan abbr="angulũ">angulum</expan> con&longs;tituunt; cor­<lb/>pora enim in motu breviorem Rhombi aut Rhomboidis <expan abbr="diame-trũ">diame­<lb/>trum</expan> de&longs;cribunt, quæ linea aliquando minor e&longs;t &longs;ingulis lateribus. </s> </p> <p type="main"> <s id="s.001239">Finge itaque corpus, quod percurreret AB, nullo impedi­<lb/>mento prohiberi, quin moveatur eádem velocitate per AD; <lb/>utique &longs;olùm æquale &longs;patium AI decurreret, impediret tamen, <lb/>ne aliud corpus habens directionem AC, illique perpetuò <lb/>adhærens, decurreret juxta &longs;uam directionem &longs;patium æquale <lb/>ip&longs;i AC; &longs;ed tantùm EI, hoc e&longs;t Sinum anguli BAD loco <lb/>Tangentis eju&longs;dem anguli, po&longs;ito Radio AI. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001240">Finge iterum alterum corpus habens directionem AC eâ­<lb/>dem velocitate moveri per AD; utique non ni&longs;i &longs;patium AF, <lb/>ip&longs;i AC æquale, motu dimetiretur, prohiberetque, ne reli­<lb/>quum corpus habens directionem AB, illique perpetuò adhæ­<lb/>rens, progrederetur ni&longs;i in F, hoc e&longs;t &longs;patio æquali ip&longs;i BD; <lb/>&longs;ed versùs B non procederet ni&longs;i juxta men&longs;uram AG mino­<lb/>rem ipsâ AC. <!-- KEEP S--></s> <s id="s.001241">Atqui utrumque &longs;uam habet directionem, & <lb/>non per AD, &longs;eque vici&longs;&longs;im impediunt; igitur dum &longs;imul mo­<lb/>ventur, neque &longs;ub&longs;i&longs;tunt in F, neque veniunt in I; &longs;ed medio <lb/>loco con&longs;i&longs;tunt, puta in O. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001242">Dixeris forta&longs;&longs;e AO æqualem ip&longs;i AE ita, ut &longs;it &longs;icut DB <lb/>ad BA, ita IE ad EA, hoc e&longs;t ad AO, aut AO e&longs;&longs;e medio <lb/>loco proportionalem inter AF & AI, hoc e&longs;t inter AC & AB <lb/>men&longs;uras virium impetûs &longs;ingulorum corporum. </s> <s id="s.001243">Hoc tamen <lb/>&longs;ecundo loco propo&longs;itum non facilè admi&longs;erim, quia ubi æqua­<lb/>les &longs;unt virtutes movendi, medio loco proportionalis e&longs;t æqua­<lb/>lis &longs;ingulis extremis, ac propterea utrumque corpus impeditum <lb/>æque velociter moveretur, ac non impeditum. </s> <s id="s.001244">Primum verò, <pb pagenum="164" xlink:href="017/01/180.jpg"/>quod &longs;cilicet AO æqualis &longs;it ip&longs;i AE, gratis a&longs;&longs;eritur; neque <lb/>enim potior ulla apparet ratio, cur ad in&longs;tituendam analogiam <lb/>a&longs;&longs;umatur potiùs IE, quàm quælibet alia minor linea cadens <lb/>inter G & E. <!-- KEEP S--></s> <s id="s.001245">Ego autem libentiùs profiteor me ne&longs;cire, quà <lb/>Ratione analogia hæc in&longs;tituatur, quam aliquid certi divinan­<lb/>do &longs;tatuere. </s> </p> <p type="main"> <s id="s.001246">Verùm quamvis non utrumque corpus velociùs moveatur <lb/>quàm pro &longs;uâ virtute, alterum tamen quod urgetur, &longs;eu rapitur <lb/>à validiori, pote&longs;t, factâ impetûs acce&longs;&longs;ione, plus &longs;patij percur­<lb/>rere, quàm pro &longs;uis viribus: impeditur &longs;iquidem motus non ab­<lb/>&longs;olutè, &longs;ed juxtà eam directionem. </s> <s id="s.001247">Hinc fit corpus habens di­<lb/>rectionem & velocitatem AC minorem velocitate AB promo­<lb/>veri ultrà punctum F in linea mixti motûs AD. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001248">At inquis: an &longs;i nautæ remis incumbant, veli&longs;que obliquis <lb/>ventum excipiant, tardior erit motus, quàm &longs;i navis vel à &longs;olis <lb/>remigibus, vel à &longs;olo vento impelleretur? </s> <s id="s.001249">contrarium &longs;anè vi­<lb/>detur experientia evincere. </s> <s id="s.001250">Verùm &longs;i rem attentiùs con&longs;ideres, <lb/>aliam planè e&longs;&longs;e rationem deprehendes, cum duo corpora &longs;e <lb/>moventia vici&longs;&longs;im &longs;e impediunt, aliam cùm unum à duplici ex­<lb/>trin&longs;eco impetu in diver&longs;a directo impellitur: de illis hactenùs <lb/>&longs;ermo fuit, neque ulla ratio &longs;uadere pote&longs;t velocius à tardiore <lb/>incitari, quamquam tardius à velociore urgeatur, ut dictum e&longs;t. </s> </p> <p type="main"> <s id="s.001251">At &longs;i unum corpus à duobus æqualis aut inæqualis virtutis <lb/>impetum recipiat, utique magis inten&longs;us, vel &longs;i inten&longs;ionem <lb/>propriè dictam neges, certè major e&longs;t impetus, quàm &longs;i ab al­<lb/>terutro tantùm reciperet impetum: quare nil mirum, &longs;i ea mo­<lb/>tûs velocitas con&longs;equatur, quæ utrumque impetum &longs;ingillatim <lb/>&longs;umptum vincat, quamvis utroque &longs;imul &longs;umpto minor &longs;it, quia <lb/>habent directiones oppo&longs;itas, ut alibi explicabitur. </s> <s id="s.001252">Hinc e&longs;t <lb/>navim velociùs agi velis remi&longs;que, quàm &longs;i aut &longs;olâ ventorum <lb/>vi, aut &longs;olâ remigum ope propelleretur, & cymbam, dum &longs;e­<lb/>cundo flumine rapitur, &longs;imulque remis ad alteram ripam im­<lb/>pellitur, velociùs moveri, quàm aut in &longs;tagno eâdem remigum <lb/>operâ, aut à flumine ce&longs;&longs;antibus remis ageretur. </s> <s id="s.001253">Quemadmo­<lb/>dum enim neque ventus remos impellit, neque ab his ventus <lb/>impellitur, ita neque &longs;e vici&longs;&longs;im immediatè impediunt, aut &longs;ibi <lb/>mutuò repugnant; atque adeò non e&longs;t hîc eadem philo&longs;ophan­<lb/>di ratio, ac cum duo corpora &longs;ibi invicem immediatè re&longs;i&longs;tunt, <pb pagenum="165" xlink:href="017/01/181.jpg"/>& alterum alterius vires extenuat impediens, ne juxtà propriæ <lb/>virtutis men&longs;uram motum concipiat. </s> </p> <p type="main"> <s id="s.001254">Ex his quæ hactenùs dicta &longs;unt, illud &longs;atis con&longs;tare videtur, <lb/>quòd animal eatenùs in motu difficultatem ac re&longs;i&longs;tentiam per­<lb/>cipit, quatenùs multum impetûs concipere debet, ex quo mu&longs;­<lb/>culorum contentio oritur, neque tamen ea &longs;equitur motûs ve­<lb/>locitas, quæ tanto impetui re&longs;ponderet, dum &longs;ubmovendo im­<lb/>pedimento maximam virium partem impendit impetum impri­<lb/>mens: unde fit plurimum influentis &longs;piritûs animalis ab&longs;umi in <lb/>tàm diuturnâ, vel tàm validâ mu&longs;culorum contentione, ac <lb/>proinde la&longs;&longs;itudinem &longs;equi, atque aliquando etiam contento­<lb/>rum mu&longs;culorum dolorem, cum id non contingat &longs;ine aliquâ <lb/>partium compre&longs;&longs;ione aut di&longs;tentione. </s> <s id="s.001255">Quò igitur velociùs <lb/>moveri pote&longs;t animal pro ratione concepti impetûs, eò mino­<lb/>rem percipit in &longs;ubmovendo impedimento difficultatem; & <lb/>quidem maximè &longs;i alternâ contentionis ac remi&longs;&longs;ionis mu&longs;cu­<lb/>lorum vici&longs;&longs;itudine labor mite&longs;cat. </s> </p> <p type="main"> <s id="s.001256">Curio&longs;iùs autem inquirenti, quam Rationem habeat motoris <lb/>impetus ad impetum corpori, quod movetur, quatenus move­<lb/>tur, impre&longs;&longs;um, ut aliquatenus &longs;atisfaciam, a&longs;&longs;ero ut minimum <lb/>duplam e&longs;&longs;e, non quidem inten&longs;ivè, aut exten&longs;ivè &longs;ed enti­<lb/>tativè. </s> <s id="s.001257">Quatenùs, inquam, movetur, hoc e&longs;t quatenus vinci­<lb/>tur ejus re&longs;i&longs;tentia: cæterùm potentia movens in &longs;e producit, & <lb/>in mobili æqualem impetum; &longs;ed quemadmodum ubi calor fri­<lb/>gori permi&longs;cetur illud vincens, non percipitur ni&longs;i quatenus <lb/>excedit vim frigoris, ita impetus oneri impre&longs;&longs;us eatenus mo­<lb/>vet, quatenùs eju&longs;dem re&longs;i&longs;tentiam &longs;uperat: Hunc autem ex­<lb/>ce&longs;&longs;um &longs;ubduplum impetûs motoris &longs;atis probabili conjecturâ <lb/>affirmo. </s> <s id="s.001258">Illud enim hoc mihi &longs;uadet, quòd motoris virtutem <lb/>metitur exce&longs;&longs;us impetûs, quem ille habet &longs;uprà impedimenti <lb/>re&longs;i&longs;tentiam: re&longs;i&longs;tentiæ autem modus, ut &longs;æpiùs dictum e&longs;t, <lb/>ex velocitate motûs, quæ concilianda e&longs;t gravitati corporis &longs;ub­<lb/>movendi, de&longs;umitur; hoc enim ideò re&longs;i&longs;tit partibus ex gr.100 <lb/>impetûs, quia &longs;i &longs;olùm fuerint 100 partes impetûs, fieri non po­<lb/>te&longs;t ut moveatur tantâ velocitate, &longs;ed pluribus impetûs parti­<lb/>bus indiget: exce&longs;&longs;us igitur virtutis motoris æqualis e&longs;t ut mi­<lb/>nimum re&longs;i&longs;tentiæ mobilis; atque adeò tota virtus motoris, hoc <lb/>e&longs;t impetus ab eo conceptus, æquivalet tùm re&longs;i&longs;tentiæ mobi-<pb pagenum="166" xlink:href="017/01/182.jpg"/>lis juxta men&longs;uram requi&longs;itam ad motum, qui &longs;equitur, tùm <lb/>principio motûs eju&longs;dem mobilis: atqui motus hic æqualis e&longs;t <lb/>motui, cui illud re&longs;i&longs;tit, totus igitur impetus motoris duplus e&longs;t <lb/>impetùs, qui motum efficit in mobili, quatenus movetur. </s> </p> <p type="main"> <s id="s.001259">Hinc e&longs;t eodem conatu motoris di&longs;parem effici motum, &longs;i <lb/>potentia æqualiter moveatur cum mobili, ut con&longs;tat: quia ni­<lb/>mirum impetus mobili impre&longs;&longs;us inæqualem habet inten&longs;io­<lb/>nem, quamvis entitativè æqualis &longs;it. </s> <s id="s.001260">Si enim tota motoris vir­<lb/>tus &longs;it 20, & decem impetûs particulas re&longs;i&longs;tentiam &longs;uperantes <lb/>mobili imprimat, in quo inten&longs;io fiat ut 1, in mobili gravitatis <lb/>&longs;e&longs;quialteræ, particulæ eædem decem impetûs inten&longs;ionem ef­<lb/>ficiunt ut 2/3; quare & hujus motus erit &longs;ub&longs;e&longs;qui alter, ac pro­<lb/>inde motor, qui æqualiter cum mobili movetur, etiam tardio­<lb/>rem habet motum, quàm cùm motum priori mobili conci­<lb/>liabat. </s> </p> <p type="main"> <s id="s.001261">Patet igitur ex his nunquam fieri po&longs;&longs;e, ut corpus grave mi­<lb/>noris aut æqualis virtutis alterum moveat ita, ut planè in velo­<lb/>citate con&longs;entiant; illud enim corpus minùs aut æquè grave <lb/>concipere non pote&longs;t impetum, qui & &longs;ibi ad motum &longs;ufficiat, <lb/>& alteri impetum imprimat: finge &longs;cilicet animo fui&longs;&longs;e impe­<lb/>tum impre&longs;&longs;um corpori æquè vel magis gravi; hîc utique cum <lb/>non excedat re&longs;i&longs;tentiam mobilis, nullum efficere pote&longs;t mo­<lb/>tum; igitur neque impre&longs;&longs;us fuit impetus, ne &longs;it omninò inuti­<lb/>lis. </s> <s id="s.001262">Quòd &longs;i eâ ratione di&longs;ponantur ut motor velociùs moveri <lb/>po&longs;&longs;it quàm mobile, jam fieri pote&longs;t, ut à minore majus movea­<lb/>tur: nam &longs;i motor certâ quâdam velocitate movere po&longs;&longs;it pon­<lb/>dus unius libræ motu &longs;ibi æquali, eodem conatu & eádem ve­<lb/>locitate &longs;e movens movebit pondus centum librarum, &longs;i hoc ita <lb/>&longs;it di&longs;po&longs;itum, ut centuplo tardiùs moveatur: quia nimirum <lb/>idem entitativè impetus in hoc pondere centuplo remi&longs;&longs;ior, <lb/>quàm in pondere unius libræ, &longs;ufficit ad motum centuplo tar­<lb/>diorem. </s> <s id="s.001263">Motus &longs;iquidem centum librarum &longs;ubcentuplus in ve­<lb/>locitate, æqualis e&longs;t motui unius libræ centuplo in velocitate; <lb/>&longs;i enim libra percurrit centum &longs;patij digitos &longs;ibi &longs;uccedentes in <lb/>longitudine, pari tempore centum libræ percurrunt quidem <lb/>unicum digitum longitudinis &longs;patij, centum tamen &longs;patia digi­<lb/>talia percurrunt, &longs;ingulæ &longs;cilicet libræ digitum. <pb pagenum="167" xlink:href="017/01/183.jpg"/> </s> </p> <p type="main"> <s id="s.001264"><emph type="center"/>CAPUT V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001265"><emph type="center"/><emph type="italics"/>In quo Machinarum vires &longs;itæ &longs;int.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001266">POtentiam oneri movendo cæteroqui imparem præ&longs;tare po&longs;­<lb/>&longs;e, &longs;i machina adhibeatur, quotidiano experimento di&longs;ci­<lb/>mus; adeò ut ip&longs;a unica pluribus potentiis machinâ de&longs;titutis <lb/>virtute æqualis &longs;it, & quæ pondus &longs;olitarium ac &longs;implex loco <lb/>pror&longs;us movere non poterat, ubi &longs;e ad machinam applicuerit, <lb/>jam non ponderi tantùm, &longs;ed & machinæ motum conciliet. </s> <lb/> <s id="s.001267">Quid ergo illud &longs;it, ex quo huju&longs;modi virium incrementum <lb/>oritur, hîc perve&longs;tigandum e&longs;t; & ad illud cau&longs;æ genus revo­<lb/>catur, quam Scholæ Formalem appellant; e&longs;t &longs;cilicet ratio, per <lb/>quam fit, ut &longs;it, atque dicatur Machina: hoc autem incremen­<lb/>tum virium, ut ex dicendis con&longs;tabit, ex machinæ figurâ pen­<lb/>det &longs;ecundùm quam potentiæ, & ponderis motus &longs;ibi invicem <lb/>pro ratâ portione re&longs;pondent. </s> </p> <p type="main"> <s id="s.001268">A machinâ quâ machina e&longs;t, potentiæ moventis vires non <lb/>augeri certum e&longs;t; nihil enim illi interioris virtutis impertitur, <lb/>& quâ machina e&longs;t, ab omni innatâ gravitate &longs;ejuncta intelli­<lb/>gitur: vectis &longs;iquidem, ferreus &longs;it, &longs;ive ligneus, machinæ ra­<lb/>tionem non immutat, &longs;i &longs;ola intercedat materiæ gravioris aut <lb/>levioris di&longs;paritas. </s> </p> <p type="main"> <s id="s.001269">Quòd &longs;i faciliùs ferreo vecte tricubitali deor&longs;um premens at­<lb/>tollas &longs;axum, quàm &longs;i ligneo vecte pariter tricubitali utaris <lb/>(quia nimirum ferreus vectis habet &longs;ibi adnexam ex gravi ma­<lb/>teriâ, quâ con&longs;tat, potentiam, quæ deor&longs;um urgendo te juvat, <lb/>ut &longs;axum attollatur,) id planè e&longs;&longs;e extra vectis naturam, quâ <lb/>vectis e&longs;t, manife&longs;tum erit, &longs;i non deor&longs;um, &longs;ed &longs;ur&longs;um, aut à <lb/>lævâ in dextram connitendum &longs;it, ut duo connexa disjungas; <lb/>tunc enim ferrei vectis gravitas &longs;u&longs;tentanda laborem potiùs <lb/>creabit, quàm ut præ &longs;imili ligneo vecte motum hunc facilio­<lb/>rem reddat. </s> <s id="s.001270">Quare præter Mechanicæ facultatis in&longs;titutum <lb/>machinis accidit, ut gravitate &longs;uâ potentiæ moventis vires ad­<lb/>augeant, non quidem illam immutando, facto interiore virtu-<pb pagenum="168" xlink:href="017/01/184.jpg"/>tis additamento; &longs;ed aliam potentiam, quæ conjunctis cum illi <lb/>viribus agat, con&longs;ociando. </s> </p> <p type="main"> <s id="s.001271">Sed & illud animadvertendum e&longs;t, vix unquam fieri po&longs;&longs;e, <lb/>ut potentia movens nihil pror&longs;us impedimenti à machina reci­<lb/>piat: &longs;ivè enim machinæ ip&longs;ius pars aliqua gravis elevanda e&longs;t; <lb/>&longs;ivè membrorum, in quæ machina di&longs;tribuitur, invicem con­<lb/>fligentium, &longs;eque vici&longs;&longs;im terentium a&longs;peritas ob&longs;i&longs;tit; &longs;ive mo­<lb/>tus (ut machinæ ip&longs;i, cui applicatur potentia, ob&longs;ecundet) à <lb/>&longs;uâ directione inflectitur; &longs;ivè quid huju&longs;modi intercedit, quod <lb/>aliquid de motûs velocitate imminuat, quæ cæteroqui concep­<lb/>tum potentiæ ab omni machinâ ab&longs;olutæ impetum con&longs;equere­<lb/>tur. </s> <s id="s.001272">Ex his tamen aliqua &longs;unt, quæ ita motui potentiæ offi­<lb/>ciunt, ut ad retinendum onus juvent; hujus &longs;iquidem gravitas <lb/>minùs adversùs potentiam valet, &longs;i & ip&longs;um, quia machinæ il­<lb/>ligatum à recto in centrum gravium tramite deflectere, vel <lb/>mutuum partium &longs;e terentium conflictum vincere cogatur, ut <lb/>vim potentiæ inferat. </s> <s id="s.001273">Verùm hæc, quamvis, ubi res ad praxim <lb/>deducitur, per incuriam di&longs;&longs;imulanda non &longs;int, &longs;ub &longs;taticam <lb/>con&longs;iderationem hîc non cadunt, ubi machinarum vires ex­<lb/>penduntur; harum enim figura perindè attenditur, atque &longs;i <lb/>nihil adjumenti, nihil detrimenti ex materiâ accederet. </s> </p> <p type="main"> <s id="s.001274">Ad rem itaque propiùs accedentibus recolenda &longs;unt ea, quæ <lb/>in &longs;uperioribus hujus libri capitibus di&longs;putata &longs;unt, proximam <lb/>videlicet motûs effectricem cau&longs;am impetum e&longs;&longs;e &longs;ive ab inte­<lb/>riore virtute manantem in iis, quæ &longs;ponte &longs;uâ moventur, &longs;ivè <lb/>extrin&longs;ecùs aliunde impre&longs;&longs;um iis, quæ naturâ repugnante per <lb/>vim cientur: ex cujus impetûs inten&longs;ione, quatenùs omnem <lb/>re&longs;i&longs;tentiam &longs;uperat, motuum velocitas oritur: nunquam autem <lb/>à velocitate aut tarditate motum &longs;ejungi po&longs;&longs;e certum e&longs;t, quip­<lb/>pe qui nec &longs;inè &longs;patio per quod decurratur, nec &longs;inè partium <lb/>&longs;ibi certâ lege &longs;uccedentium continuatione ac &longs;erie intelli­<lb/>gi pote&longs;t. </s> <s id="s.001275">Quare & re&longs;i&longs;tentiæ momenta tùm ex corporis <lb/>movendi gravitate, tùm ex velocitate componi &longs;æpiùs innui­<lb/>mus, ut hinc innote&longs;cat fieri facilè po&longs;&longs;e, ut, &longs;icut eju&longs;dem <lb/>gravitatis re&longs;i&longs;tentia inæqualis e&longs;t, &longs;i velocitate inæquali mo­<lb/>venda &longs;it, & gravitatum inæqualium di&longs;paria &longs;unt re&longs;i&longs;tentiæ <lb/>momenta, &longs;i Ratio, quæ ex gravitatum & velocitatum Ratio­<lb/>nibus componitur, &longs;it Ratio Inæqualitatis, quia gravior velo-<pb pagenum="169" xlink:href="017/01/185.jpg"/>ciùs, minùs gravis tardiùs movetur; ita gravitatum inæqualium <lb/>par &longs;it re&longs;i&longs;tentia, &longs;i quæ inter gravitates intercedit Ratio, ea­<lb/>dem reciprocè inter velocitates inveniatur. </s> <s id="s.001276">Quemadmodum <lb/>enim quæcumque calori adver&longs;antur, vehementiorem quidem <lb/>validi&longs;&longs;imè re&longs;puunt, tenui&longs;&longs;imum verò facillimè admittunt; <lb/>haud di&longs;pari ratione pondera, &longs;i velociùs incitare velis, im­<lb/>pensiùs reluctantur, minimo ac tardi&longs;&longs;imo motui levi&longs;&longs;imè ob­<lb/>&longs;i&longs;tunt. </s> </p> <p type="main"> <s id="s.001277">Quoniam igitur naturâ definitum e&longs;t, quantam gravitatem, <lb/>quantáque velocitate, pro certâ impre&longs;&longs;i impetûs men&longs;urâ, mo­<lb/>vere po&longs;&longs;it Potentia concepto impetu, qui pro ratâ portione <lb/>re&longs;pondeat impetui quem illa oneri imprimit, ut Potentia, & <lb/>onus æquali velocitate moveantur; &longs;atis con&longs;tat eandem impe­<lb/>tús men&longs;uram parem e&longs;&longs;e movendo oneri graviori, &longs;i quá Ra­<lb/>tione po&longs;terior hæc gravitas priorem gravitatem vincit, eâdem <lb/>Reciprocè Ratione prioris velocitas po&longs;terioris tarditatem &longs;u­<lb/>peret; utrobique &longs;cilicet par e&longs;t re&longs;i&longs;tentia, ac proinde ab eâ­<lb/>dem potentiâ vinci pote&longs;t. </s> <s id="s.001278">Cùm enim ea, quæ &longs;imul æqualiter <lb/>moventur, æquali impetu ferantur; &longs;i Potentia tàm tardè mo­<lb/>veretur ac pondus per machinam, indigeret impetu ex. </s> <s id="s.001279">gr. <!-- REMOVE S-->&longs;ub­<lb/>quintuplo ejus quo illa movetur quintuplo velociùs ac ip&longs;um <lb/>Pondus. <!-- KEEP S--></s> <s id="s.001280">Verùm impetus hîc &longs;ubquintuplus ineptus e&longs;&longs;et ad <lb/>oneris re&longs;i&longs;tentiam quintuplo ferè majorem vincendam; &longs;ed &longs;o­<lb/>lum &longs;uperare po&longs;&longs;et ac movere 1/5 ponderis. </s> <s id="s.001281">Quinque igitur im­<lb/>petus huic æquales po&longs;&longs;unt totam re&longs;i&longs;tentiam &longs;uperare. </s> <s id="s.001282">Cum <lb/>itaque in motu quintuplo velociori Potentiæ &longs;it verè impetus <lb/>quintuplus, poterit etiam elevare pondus, quod e&longs;t quintuplo <lb/>majus, quàm &longs;it 1/5 ip&longs;ius. </s> <s id="s.001283">Verùm híc ubi de motûs velocitate <lb/>&longs;ermo e&longs;t, non is quidem ab&longs;olutè accipiendus e&longs;t; &longs;ed quâ <lb/>parte gravium naturæ repugnat: &longs;i enim plumbeus globus <lb/>A ex C dependeat funiculo CA, & circà ver&longs;atilem or­<lb/>biculum B &longs;tabili axi infixum ducatur filum connectens <lb/>globos A & D, cottum quidem e&longs;t globum A, &longs;i u&longs;que ad <lb/>B perveniat, tantumdem &longs;patij in arcu AB percurrere, <lb/>non tamen tantumdem a&longs;cendere, quantum globus D &longs;e­<lb/>cundùm rectam BD de&longs;cendit; &longs;ed a&longs;cen&longs;um metitur AE, <lb/>nimirum Sinus Ver&longs;us arcûs AB, qui minor e&longs;t codem <lb/>arcu (arcus &longs;iquidem major e&longs;t rectâ AB lineâ ip&longs;um &longs;ub-<pb pagenum="170" xlink:href="017/01/186.jpg"/><figure id="id.017.01.186.1.jpg" xlink:href="017/01/186/1.jpg"/><lb/>tendente, quæ oppo&longs;ita <lb/>recto angulo E major e&longs;t <lb/>quàm trianguli ba&longs;is AE) <lb/>ac propterea re&longs;i&longs;tentiæ <lb/>momenta non ea &longs;unt, quæ <lb/>ex velocitate motûs AB, <lb/>&longs;ed AE, & ipsâ globi A <lb/>gravitate componuntur. </s> <s id="s.001284">Ex <lb/>quo fit globum D quam­<lb/>vis minorem po&longs;&longs;e globo A <lb/>graviori præ&longs;tare, ac illum <lb/>ad certam altitudinem ele­<lb/>vare, ut cuilibet experiri <lb/>licet, cum tamen illi a&longs;cen­<lb/>&longs;um &longs;uo de&longs;cen&longs;ui æqualem <lb/>nullatenùs conciliare po&longs;&longs;it. </s> <lb/> <s id="s.001285">Quòd &longs;i idem globus A ex breviore funiculo HA dependeat, <lb/>experimento con&longs;tat opus e&longs;&longs;e globo D gravitatem addere, ut <lb/>valeat illum per arcum AF elevare ad eandem altitudinem <lb/>AE: magis quippè laborio&longs;um e&longs;t breviore motu AF, quàm <lb/>longiore motu AB ad eandem altitudinem a&longs;cendere; atque <lb/>adeò plus virium in D requiritur, ut globo A majorem impetum <lb/>imprimat, ex cujus inten&longs;ione plus &longs;ingulis temporis momentis <lb/>a&longs;cendat in hoc po&longs;teriore motu, quàm in priore. </s> <s id="s.001286">Ne tamen <lb/>motui globi D tribue men&longs;uram arcûs AB &longs;ed rectæ AB. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001287">Sicut autem ubi potentiæ & oneris æquales e&longs;&longs;e debent mo­<lb/>tus, potentiæ vires gravitate oneris majores e&longs;&longs;e oportet, ut vim <lb/>illi inferant; ita pariter ubi potentia & onus in motuum velo­<lb/>citate di&longs;&longs;entiunt, & illa quidem velociùs, hoc tardiùs move­<lb/>tur, nece&longs;&longs;e e&longs;t majorem e&longs;&longs;e Rationem Potentiæ ad Onus (licet <lb/>illa minor &longs;it onere) quàm &longs;it Ratio tarditatis hujus ad illius <lb/>velocitatem; ut &longs;cilicet ratio Potentiæ ad onus, quæ ex mo­<lb/>tuum, & virium Rationibus componitur, &longs;it Ratio majoris inæ­<lb/>qualitatis. </s> <s id="s.001288">Sit ex. </s> <s id="s.001289">gr. <!-- REMOVE S-->Ratio motûs Potentiæ ad motum Oneris <lb/>ut 3 ad 2; &longs;i Ratio virium potentiæ ab&longs;olutè &longs;umptæ ad gravi­<lb/>tatem oneris &longs;it Reciprocè ut 2 ad 3, Ratio ex his Rationibus <lb/>compo&longs;ita e&longs;t Æqualitatis, &longs;cilicet 1 ad 1, & motus nullus &longs;e­<lb/>quitur; multò minùs &longs;i fuerit Ratio minor quàm 2 ad 3; prove-<pb pagenum="171" xlink:href="017/01/187.jpg"/>niret enim Ratio minoris Inæqualitatis: debet ergo e&longs;&longs;e major <lb/>Ratione 2 ad 3. Sit ex hypothe&longs;i Ratio 4 ad 5; jam Ratio com­<lb/>po&longs;ita ex Rationibus 3 ad 2, & 4 ad 5, e&longs;t Ratio 6 ad 5 majoris <lb/>Inæqualitatis. </s> </p> <p type="main"> <s id="s.001290">Neque hoc ita dictum intelligas, qua&longs;i motus ip&longs;e Potentiæ, <lb/>eju&longs;que velocitas, efficiendi vim haberet; &longs;ed ex ipsá majore <lb/>potentiæ velocitate innote&longs;cit impetum, qui radix e&longs;t motûs, <lb/>minus invenire impedimenti ex onere, quod minùs re&longs;i&longs;tit, eo <lb/>quòd tardiùs movendum e&longs;t, quàm &longs;i æqualem velocitatis gra­<lb/>dum cum potentiâ &longs;ortiri deberet. </s> <s id="s.001291">Quare licèt potentia minor <lb/>&longs;it, ac pauciores entitativè particulas impetús producere valeat, <lb/>quàm potentia major, &longs;atis in aperto e&longs;t fieri po&longs;&longs;e, ut potentia <lb/>major majorem inveniens re&longs;i&longs;tentiam nequeat impetum im­<lb/>primere, ac movere onus, quod movebitur à minore potentiâ, <lb/>&longs;i onus idem minùs re&longs;i&longs;tat, cum &longs;it tardiùs movendum: impe­<lb/>tus enim à minore potentiâ oneri impre&longs;&longs;us &longs;atis e&longs;t ad vincen­<lb/>dam minorem hanc re&longs;i&longs;tentiam; cum tamen potentia major <lb/>non &longs;atis habeat virtutis, ut eam impetús men&longs;uram oneri im­<lb/>primat, quæ majorem illius re&longs;i&longs;tentiam &longs;uperaret. </s> </p> <p type="main"> <s id="s.001292">In eo igitur totum Mechanices artificium con&longs;i&longs;tit, ut &longs;ua <lb/>in&longs;trumenta ita di&longs;ponat, loci&longs;que congruis ita Potentiam, & <lb/>Onus collocet, ut Potentiæ motus velocior &longs;it præ motu Oneris: <lb/>tùm horum motuum Ratione attentè per&longs;pectâ definies, quæ­<lb/>nam Potentia datum Onus movere, vel quodnam Onus à datâ <lb/>Potentiâ moveri queat; &longs;i nimirum Potentiæ vires ad oneris <lb/>gravitatem majorem habeant Rationem, quàm &longs;it Ratio motùs <lb/>Oneris ad motum Potentiæ. <!-- KEEP S--></s> <s id="s.001293">Neque enim Machina aut Poten­<lb/>tiæ vires auget, aut oneris gravitatem minuit, &longs;ed Ponderis re­<lb/>&longs;i&longs;tentiam ad Potentiæ virtutem accommodat. </s> </p> <p type="main"> <s id="s.001294">Phy&longs;ica autem cau&longs;a hæc e&longs;t, quia impetus à Potentiâ pro­<lb/>ductus, qui in onere minori movendo æque velociter cum po­<lb/>tentiâ <expan abbr="major&etilde;">majorem</expan> haberet inten&longs;ionem, in onere majore &longs;ed tardiùs <lb/>movendo minorem quidem habet inten&longs;ionem, &longs;ed quæ &longs;atis e&longs;t <lb/>pro minore <expan abbr="re&longs;i&longs;t&etilde;tia">re&longs;i&longs;tentia</expan>. </s> <s id="s.001295">Fac enim oneris particulas graves e&longs;&longs;e 20, <lb/>illique à <expan abbr="Pot&etilde;tiâ">Potentiâ</expan> <expan abbr="aliquãto">aliquanto</expan> graviore imprimi particulas 100 impe­<lb/>tûs, quibus vincitur Oneris re&longs;i&longs;tentia: inten&longs;io in &longs;ingulis par­<lb/>ticulis gravitatis e&longs;t particularum impetûs 5, juxtà quam inten­<lb/>&longs;ionis men&longs;uram &longs;equitur motus æque velox Potentiæ & oneris, <pb pagenum="172" xlink:href="017/01/188.jpg"/>hujus quidem per vim &longs;ursùm; illius verò juxtà naturam deor­<lb/>&longs;um. </s> <s id="s.001296">Sit adhuc eadem Potentia; &longs;ed offeratur Onus, cujus <lb/>particulæ gravitatis &longs;int non jam 20; &longs;ed 50: Potentiæ virtus e&longs;t <lb/>eadem; quapropter non ni&longs;i re&longs;i&longs;tentiam vincere pote&longs;t, cui <lb/>vincendæ &longs;ufficiant particulæ 100 impetus; hæ autem in One­<lb/>re graviore ut 50 efficerent &longs;olùm inten&longs;ionem ut 2: Non igitur <lb/>Potentia & onus æquè veloci motu, qui re&longs;pondeat inten&longs;ioni <lb/>ut quinque, &longs;icuti priùs, moveri poterunt; &longs;ed ut onus moveri <lb/>po&longs;&longs;it, impetúmque à potentiâ recipere, opus e&longs;t ita illud col­<lb/>locare, ut quò magis Ratione gravitati re&longs;i&longs;tit; eò minùs ra­<lb/>tione tarditatis motûs re&longs;i&longs;tat, &longs;eque eâ ratione temperent duæ <lb/>hæ re&longs;i&longs;tentiæ, ut una confletur re&longs;i&longs;tentia non major illâ, quæ <lb/>oriebatur ex onere gravi ut 20 æqualiter movendo: id quod <lb/>fiet, &longs;i motus Potentiæ, quatenùs machinæ applicatur, ad mo­<lb/>tum oneris &longs;it ut 5 ad 2 in Reciprocâ Ratione inten&longs;ionum im­<lb/>petûs producti. </s> <s id="s.001297">Quare motus Potentiæ ad motum oneris e&longs;t <lb/>duplus &longs;e&longs;quialter, quemadmodum po&longs;terior hæc oneris gravi­<lb/>tas ut 50 e&longs;t prioris gravitatis ut 20 dupla &longs;e&longs;quialtera: atque <lb/>hinc manife&longs;tum e&longs;t particulas gravitatis 50 re&longs;i&longs;tentes ut 2 ra­<lb/>tione motûs comparati cum motu potentiæ, requirere particu­<lb/>las 100 impetûs, quemadmodum particulæ gravitatis 20 re­<lb/>&longs;i&longs;tentes ut 5 ratione motûs comparati cum motu eju&longs;dem Po­<lb/>tentiæ requirunt particulas 100 impetûs. </s> <s id="s.001298">Quid igitur mirum, &longs;i <lb/>potentia eadem eodem conatu movet onus ut 50 velocitate ut 2, <lb/>quo conatu movet onus ut 20 velocitate ut 5? </s> </p> <p type="main"> <s id="s.001299">Servatur itaque perpetua quædam ju&longs;titia inter potentiæ vi­<lb/>res, oneris gravitatem, &longs;patia motuum, ac tempora; quò enim <lb/>decre&longs;cunt potentiæ vires, aut oneris gravitas augetur, eò bre­<lb/>viora &longs;unt &longs;patia, & longiora tempora motuum ip&longs;ius oneris; <lb/>&longs;ed ampliora &longs;patia motuum potentiæ debilioris, quæ præ one­<lb/>re velociùs movetur. </s> <s id="s.001300">Hinc dato onere graviori &longs;ubmovendo, <lb/>aut potentiam augeri, aut, &longs;i illa immutata permaneat, oneris <lb/>motum imminui, &longs;eu potentiæ motum augeri nece&longs;&longs;e e&longs;t: Te­<lb/>nui enim potentiâ ingens pondus citò moveri non pote&longs;t. </s> </p> <p type="main"> <s id="s.001301">Formalem igitur Machinæ Rationem, quâ Machina e&longs;t, in eo <lb/>&longs;itam e&longs;&longs;e deprehendimus, quòd ea figura &longs;it, quæ potentiæ, <lb/>& oneris motibus legem ita &longs;tatuat, ut Potentia velociter, Pon­<lb/>dus lentè moveatur; &longs;ic enim fit, ut minor oneris re&longs;i&longs;tentia vir-<pb pagenum="173" xlink:href="017/01/189.jpg"/>tuti vim movendi, etiam&longs;i minorem, habenti pro ratâ portio­<lb/>ne re&longs;pondeat. </s> <s id="s.001302">Satis igitur erit, ubi &longs;ingularum machinarum <lb/>vires expendendæ erunt motuum inire rationes, qui ex machi­<lb/>næ agitatione oriuntur: nam &longs;i Potentia præ Onere velociùs <lb/>moveatur, operæ pretium faciet Machinator; modò non adeò <lb/>tenuis &longs;it motuum Ratio, ut quicquid utilitatis ex machinæ fi­<lb/>gurà accedit, deferatur ex partium &longs;e terentium conflictu; nam <lb/>perinde e&longs;&longs;et, ac &longs;i oneri gravitas adderetur. </s> </p> <p type="main"> <s id="s.001303">Ex his liquet à non paucis plus operæ labori&longs;que con&longs;ump­<lb/>tum, quàm par e&longs;&longs;et, ut Ari&longs;toteli adhærerent in referendis <lb/>machinarum viribus in circuli naturam planè admirandam: <lb/><emph type="italics"/>Quapropter<emph.end type="italics"/> inquit initio <expan abbr="qq.">qque</expan> Mechan. <emph type="italics"/>non e&longs;t inconveniens ip&longs;um <lb/>miraculorum omnium e&longs;&longs;e principium. </s> <s id="s.001304">Ea igitur quæ circà libram fiunt, <lb/>ad circulum referuntur, quæ verò circa vectem, ad ip&longs;am libram; <lb/>alia autem ferè omnia, quæ circa mechanicas &longs;unt motiones, ad <lb/>vectem.<emph.end type="italics"/></s> <s id="s.001305"> Ni&longs;i enim fucum veritati faciamus, quæ demum mi­<lb/>racula ita circulum à reliquo figurarum vulgo &longs;ecernunt, ut in <lb/>cum admiratio omnis corrivata confluat, nec ni&longs;i hinc in cæte­<lb/>ras derivetur? </s> <s id="s.001306">An quòd linea eadem, quâ circuli ambitus de­<lb/>finitur, omnis latitudinis expers, cava pariter atque convexa <lb/>amico fœdere copulat, quæ &longs;ibi invicem repugnant? </s> <s id="s.001307">Cavum <lb/>&longs;i quidem à convexo, quæ recto interjecto di&longs;criminantur, per­<lb/>inde di&longs;&longs;idere cen&longs;emus, atque minus à majori, inter quæ &longs;ibi <lb/>adver&longs;antia id, quod æquale e&longs;t, intercedit. </s> <s id="s.001308">At hæc ita vulga­<lb/>ria &longs;unt, ut non Hyperbolæ &longs;olùm, ac Parabolæ, aut Nicome­<lb/>dis Conchoidi, aut Archimedis Spiralibus, aut Dino&longs;trati <lb/>Quadratici, cæteri&longs;que omnibus extrà Geometricas leges cur­<lb/>vis lineis communia &longs;int; verùm etiam in angulo quocumque <lb/>rectilineo facilè ab omnibus ob&longs;erventur; cum lineæ rectæ, qui­<lb/>bus inclinatis angulus con&longs;tituitur, hinc quidem &longs;ibi mutuis <lb/>nutibus annuere, hinc verò abnuere videantur; quibus oppo­<lb/>&longs;itis nutibus media pariter interjacet directa po&longs;itio, omni in­<lb/>clinatione &longs;ubmotâ. </s> </p> <p type="main"> <s id="s.001309">An ipsâ na&longs;centis Circuli exordia admiratione non carent, <lb/>quòd æquè ex Radij eju&longs;dem in centro &longs;ub&longs;i&longs;tentis quiete, ac <lb/>circumlati motu oriatur? </s> <s id="s.001310">Sed quid hæc in circulo potiùs &longs;u&longs;­<lb/>piciamus, quàm in Helice, cui gene&longs;is haud di&longs;par contingit? </s> <lb/> <s id="s.001311">Quòd &longs;i circulo primas ideò deferendas exi&longs;timemus, quòd <pb pagenum="174" xlink:href="017/01/190.jpg"/>in &longs;e recurrens peripheria ibi &longs;ui motûs terminum inveniat, <lb/>unde &longs;ump&longs;it exordium; & circumacta, quæ ex adver&longs;o <lb/>&longs;unt, partes oppo&longs;itis cieat motibus, ita ut progredientibus <lb/>&longs;upremis infimæ regrediantur, & in ima detrudantur &longs;i­<lb/>ni&longs;træ, dextris in altiora provectis: Quid Ellip&longs;im præjudi­<lb/>cio repellimus? </s> <s id="s.001312">cum & hæc unico limite cavo pariter atque <lb/>convexo in &longs;e&longs;e redeunte circum&longs;cripta in contrarias partes <lb/>incitetur; nec à rectâ tantummodo lineâ alternis auctà cre­<lb/>mentis, imminutáque decrementis altero terminorum quie&longs;­<lb/>cente, &longs;ed etiam (quod verè miraculo proximum e&longs;t) <lb/>utroque extremo flexilis lineæ in binis Ellip&longs;eos umbilicis <lb/>defixo ab illâ in alios, atque alios angulos &longs;inuata de&longs;­<lb/>cribatur. </s> </p> <p type="main"> <s id="s.001313">At, inquis, in circulo &longs;emidiametri partes codem im­<lb/>pellente circà centrum agitatæ ita di&longs;pari velocitate ferun­<lb/>tur, ut earum tarditas aut concitatio intervallo, quo &longs;in­<lb/>gulæ à centro ab&longs;unt, &longs;it analoga. </s> <s id="s.001314">Verùm & hoc Ellip&longs;i, <lb/>ac plano Helicoidi aliquatenùs pro &longs;uo modulo commune <lb/>e&longs;t; &longs;emidiametri enim circumactæ puncta à centro remo­<lb/>tiora velociùs feruntur. </s> <s id="s.001315">Partes autem quie&longs;centi centro pro­<lb/>piores cunctabundas moveri, naturæ pro viribus oppo&longs;ita <lb/>di&longs;terminantis in&longs;tituto con&longs;entaneum e&longs;&longs;e nemo non videt, <lb/>qui tarditatem interjici videt quietem inter, ac motûs ve­<lb/>locitatem. </s> <s id="s.001316">Quare &longs;apienti&longs;&longs;imo con&longs;ilio factum, ut eorum, <lb/>quæ firmo nexu invicem &longs;olidata &longs;ub&longs;i&longs;tunt, vel particu­<lb/>læ omnes æquis pa&longs;&longs;ibus moveantur, vel &longs;i qua moræ di&longs;­<lb/>pendium &longs;ubeat, finitimarum velocitas, &longs;ervatâ aliquâ vi­<lb/>cinitatis analogiâ minuatur: ne &longs;cilicet &longs;olutâ compage di&longs;­<lb/>&longs;iliant. </s> </p> <p type="main"> <s id="s.001317">Quæ verò ad explicandum, cur ea, quæ centro propiora <lb/>&longs;unt, tardiùs in gyrum contorqueantur, Author illius libri <lb/>Quæ&longs;t. mechan. </s> <s id="s.001318">commini&longs;citur de duplici motu, naturali vi­<lb/>delicet, ac præter naturam, quibus feratur ea, quæ circu­<lb/>lum de&longs;cribit linea (qua&longs;i breviorem lineam vis major à tra­<lb/>hente centro illata magis à naturali motu, qui &longs;ecundùm <lb/>Tangentem e&longs;t, deflecteret) ea &longs;unt, quæ facillimè cor­<lb/>ruant, & minimè cum Ari&longs;totelis doctriná cohæreant, qui <lb/>lib. 1. de Cælo. <!-- KEEP S--></s> <s id="s.001319">&longs;umma 4. circularem motum & &longs;implicem, & <pb pagenum="175" xlink:href="017/01/191.jpg"/>naturalem, & priorem recto di&longs;erti&longs;&longs;imè pronunciat; <emph type="italics"/>Perfectum <lb/>enim,<emph.end type="italics"/> inquit text. </s> <s id="s.001320">12; <emph type="italics"/>prius naturâ e&longs;t imper&longs;ecto; circulus autem <lb/>perfectorum e&longs;t, recta verò linea nulla.<emph.end type="italics"/></s> <s id="s.001321"> Quis ergo in circulo <lb/>motus præter naturam? <emph type="italics"/>nece&longs;&longs;arium e&longs;t,<emph.end type="italics"/> ait text. </s> <s id="s.001322">8. <emph type="italics"/>e&longs;&longs;e ali­<lb/>quod corpus &longs;implex, quod natum e&longs;t ferri circulari motu &longs;ecun­<lb/>dùm &longs;uam ip&longs;ius naturam.<emph.end type="italics"/></s> <s id="s.001323"> Ea certè quibus in&longs;ita e&longs;t in mo­<lb/>tum propen&longs;io, in gyrum aguntur, ut &longs;ydera; aut &longs;altem mo­<lb/>tu in &longs;e recurrente circulum æmulantur, ut ex cerebri & cor­<lb/>dis &longs;y&longs;tole ac dia&longs;tole &longs;pirituum ac &longs;anguinis circuitio oritur; <lb/>aut plurium circularium motuum commixtione unum tempe­<lb/>rant motum, ut animalia cum progrediuntur; o&longs;&longs;a &longs;iquidem, <lb/>quibus membra &longs;ub&longs;i&longs;tunt, ita à mu&longs;culis commoventur, ut <lb/>unum quod que &longs;ui motus centrum con&longs;tituat in eâ finitimi o&longs;&longs;is <lb/>parte, cui &longs;ivè <foreign lang="greek">*kaq´<gap/>e)na/rsqrwsin</foreign>, &longs;ive <foreign lang="greek">kata/ dia)rqrwsin</foreign> flexili com­<lb/>page in&longs;eritur. </s> <s id="s.001324">At motu recto, ut potè brevi&longs;&longs;imo, nihil fertur, <lb/>ni&longs;i cui ex naturæ in&longs;tituto cedit quies certo in loco, à quo <lb/>ab&longs;tractum fuerit, eóque &longs;ibi redditum &longs;pontè remigrat. </s> <s id="s.001325">Nihil <lb/>igitur præter naturam in circuli motu deprehendi pote&longs;t, ex <lb/>quo di&longs;par illa intimarum atque extimarum partium velocitas <lb/>petenda &longs;it; cum vix alium natura per &longs;e expetat &longs;implicem <lb/>motum præter circularem. </s> <s id="s.001326">Cur autem qui &longs;ecundùm rectam <lb/>extremæ &longs;emidiametro ad perpendiculum in&longs;i&longs;tentem lineam <lb/>fit motus, naturalis cen&longs;eatur? </s> <s id="s.001327">An quia gravia &longs;uis nutibus ad <lb/>terræ centrum rectâ feruntur? </s> <s id="s.001328">Semidiametro igitur, ni&longs;i in <lb/>verticali plano con&longs;tituatur horizonti parallela, motus qui &longs;e­<lb/>cundùm lineam circuli Tangentem e&longs;t, præter naturam con­<lb/>tinget, quippe qui à rectâ, quæ gravia in centrum dirigit, de­<lb/>flectat: & in circulo horizonti parallelo circumacta &longs;emidiame­<lb/>ter nullo naturali motu agitabitur; nulla enim recta linea cir­<lb/>culi Tangens in eo plano e&longs;t, quæ lineæ directionis gravium <lb/>congruat: & tamen quemcumque demum &longs;itum circulus eju&longs;­<lb/>que &longs;emidiameter obtineat, eandem &longs;emper motuum analo­<lb/>giam &longs;ervant partes pro ratione intervalli à centro, citrà ullam <lb/>motuum naturalis, & præter naturam, commi&longs;tionem. </s> </p> <p type="main"> <s id="s.001329">Verùm mirifica &longs;it circuli natura; quid hæc ad explicandam <lb/>Mechanicarum motionum cau&longs;am? </s> <s id="s.001330">an ut hanc ignotam fatea­<lb/>mur, quia admirandam prædicamus? </s> <s id="s.001331">&longs;ed unico argumento, <lb/>commenta huju&longs;modi disjiciamus. </s> <s id="s.001332">Si minor potentia majori <pb pagenum="176" xlink:href="017/01/192.jpg"/>ponderi prævaleat, nullú&longs;que intercedat circularis motus, <expan abbr="certũ">certum</expan> <lb/>e&longs;t hoc virtutis <expan abbr="increm&etilde;tum">incrementum</expan> neque in Vectem, neque in libram <lb/>neque in Circulum referri po&longs;&longs;e: adeóque principium aliud e&longs;&longs;e <lb/>magis latè patens, à circulo ab&longs;olutum: Atqui citrà omnem cir­<lb/>cularem <expan abbr="motũ">motum</expan> minor potentia præpollet graviori ponderi: Mani­<lb/>fe&longs;tum e&longs;t igitur fru&longs;trà ex circulo peti Mechanicarum motio­<lb/>num principium; &longs;ed illud e&longs;&longs;e, quod à nobis indicatum e&longs;t, <lb/>quippe quod, ubicumque reperitur, hoc efficit, ut minor po­<lb/>tentia majori ponderi motum conciliet, nec is unquam &longs;ine illo <lb/>contingit. </s> </p> <p type="main"> <s id="s.001333">A&longs;&longs;umptionis veritas ut innote&longs;cat, ingen&longs;que pondus tardè <lb/>movendum à tenui virtute &longs;ine circulari motu propelli po&longs;&longs;e <lb/>confirmem, non ego te in &longs;uburbanum campum deducam, ut <lb/>tenerrimo germini &longs;uppullulanti incumbentes glebas demùm <lb/>loco ce&longs;&longs;i&longs;&longs;e ob&longs;erves, aut marmora Me&longs;&longs;alæ &longs;cindentem capri­<lb/>ficum obtrudam, turre&longs;que longâ annorum &longs;erie labefactatas <lb/>enatis fruticibus atque virgultis; ne mihi fortè herbe&longs;centes <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"/> <p type="main"> <s id="s.001334">Sed age raptandus &longs;it in plano horizontali, aut inclinato, aut <lb/>etiam elevandus &longs;it ad perpendiculum cylindrus A. <!-- KEEP S--></s> <s id="s.001335">Experire <lb/>primùm quanto labore id præ&longs;tes illum trahens illigato fune <lb/>in C, & arreptâ extremitate funis B. </s> <s id="s.001336">Tùm in B infixo firmi­<lb/>ter paxillo ductarius funis alligetur; hic porrò in&longs;eratur annu­<lb/>lo C optimè ferruminato, & quoad ejus fieri poterit exqui&longs;itè <lb/>polito, arreptáque alterâ funis extremitate D iterum trahe cy­<lb/>lindrum, & quantò minori labore id perficias, tu te ip&longs;e doce­<lb/>bis. </s> <s id="s.001337">At hîc nulla circuli vides miracula; hîc libra nulla; nullus <lb/>hîc vecti locus: motus enim tùm potentiæ trahentis, tùm cy­<lb/>lindri, rectus e&longs;t. </s> <s id="s.001338">Facilitatis autem di&longs;crimen non ex ullo cir­<lb/>culari motu, qui nu&longs;quam apparet, &longs;ed ex eo oritur, quòd pri­<lb/>mùm potentia & onus æqualiter moventur; po&longs;teà verò cylin-<pb pagenum="177" xlink:href="017/01/193.jpg"/>dri velocitas &longs;ubdupla e&longs;t velocitatis potentiæ; quia cum ex C <lb/>cylindrus venit in B funis ultrà B extenditur juxtà longitudi­<lb/>nem CB u&longs;que in E; ac propterea motus potentiæ duplus e&longs;t, <lb/>&longs;cilicet CE. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001339">Statue item in pariete puncta duo A & B (quo autem majo­<lb/>re intervallo disjuncta fuerint, res meliùs &longs;uccedet) ibique <lb/>clavos rotundos nihil ha­<lb/><figure id="id.017.01.193.1.jpg" xlink:href="017/01/193/1.jpg"/><lb/>bentes a&longs;peritatis infige. </s> <lb/> <s id="s.001340">Tùm pondera duo H & <lb/>G æqualia a&longs;&longs;ume, eáque <lb/>funiculo nullis nodis a&longs;pe­<lb/>ro, &longs;ive &longs;erico crudo, &longs;ive <lb/>crinibus equinis connexa <lb/>impone claviculis A & B, <lb/>ut liberè ex iis depen­<lb/>deant: &longs;uâ autem gravitate <lb/>funiculum AB intentum Horizonti parallelum &longs;ervabunt, & <lb/>neutro prævalente ob gravitatis æqualitatem prorsùs immota <lb/>con&longs;i&longs;tent. </s> <s id="s.001341">Elige jam pondus tertium I, quod alteri datorum <lb/>H & G æquale &longs;it, aut etiam &longs;ingulis aliquantò minus; illud­<lb/>que in E extento funiculo AB adnecte: &longs;tatim pondus I &longs;ecun­<lb/>dùm rectam EF de&longs;cendens videbis; pondera autem H & G <lb/>per rectas HA, & GB a&longs;cendentia, quâ men&longs;urâ funiculi in­<lb/>flexi partes AF, BF &longs;imul &longs;umptæ excedunt rectam AB. <!-- KEEP S--></s> <s id="s.001342">Nul­<lb/>lus igitur motus circularis hîc e&longs;t; &longs;ed omnes recti ad perpendi­<lb/>culum, & tamen potentia I minor commovet majus pondus, <lb/>quod ex H & G conflatur. </s> </p> <p type="main"> <s id="s.001343">Id autem ideò contingere, quia motus EF de&longs;cendentis I <lb/>major e&longs;t motu a&longs;cendentium H & G, hinc manife&longs;tum e&longs;t, <lb/>quòd pondus I u&longs;que ad certum terminum de&longs;cendit, ibique <lb/>&longs;ub&longs;i&longs;tit: quòd &longs;i illud manu apprehen&longs;um adhuc deor&longs;um <lb/>trahens eleves pondera H & G, ubi manum indè ab&longs;traxeris, <lb/>pondera H & G prævalent, ac de&longs;cendentia elevant pondus I <lb/>ad certum illum terminum, ubi &longs;ponte &longs;ub&longs;titerat: quia nimi­<lb/>rum ultrà illum terminum non jam major e&longs;t Ratio ponderis I <lb/>ad pondera HG, quàm &longs;it Ratio motuum H & G ad motum I. <!-- KEEP S--></s> <lb/> <s id="s.001344">Hæc autem inferiùs, ubi de librâ & Æquilibrio &longs;ermo erit, <lb/>paulò fu&longs;iùs & dilucidiùs explicabuntur; nunc enim &longs;atis e&longs;t <pb pagenum="178" xlink:href="017/01/194.jpg"/>pro in&longs;titutâ di&longs;putatione o&longs;tendi&longs;&longs;e minorem gravitatem præ­<lb/>pollere citrà omnem motum circularem. </s> </p> <p type="main"> <s id="s.001345">Ratum itaque e&longs;to ad nullum certum machinæ genus cætera <lb/>e&longs;&longs;e revocanda; &longs;ed omnibus commune e&longs;&longs;e principium, ex quo <lb/>vires de&longs;umunt; impetûs &longs;cilicet à potentiâ producti proportio <lb/>ad ponderis re&longs;i&longs;tentiam (quæ eò minor e&longs;t, quò tardiùs mo­<lb/>veri debet) ea e&longs;t, quæ motûs facilitatem conciliat; nullus <lb/>quippe adeò tenuis impetus reperitur, cui lenti&longs;&longs;imus aliquis <lb/>motus non re&longs;pondeat, &longs;i intereà à velociori motu potentia non <lb/>prohibeatur. </s> <s id="s.001346">Ubi autem de potentiæ velocitate &longs;ermo e&longs;t, non <lb/>ea intelligatur, quæ e&longs;&longs;et, ubi præter &longs;e nihil ip&longs;a moveret, ab­<lb/>&longs;oluta ab omni re&longs;i&longs;tentiâ; &longs;ed eam velocitatem intellige, quæ <lb/>comparatè dicitur, ubi ejus motus cum oneris motu confertur. </s> <lb/> <s id="s.001347">Semper tamen impetus, qui in Potentiâ reperitur quatenùs ex­<lb/>cedit re&longs;i&longs;tentiam ponderis, majorem in eâ intentionem ha­<lb/>bet, quàm in pondere, quamvis pares entitativè &longs;int impetus <lb/>Potentiæ, & oneris. </s> <s id="s.001348">Hæc autem clariùs patebunt lib.4. cap.1. <lb/> </s> </p> <p type="main"> <s id="s.001349"><emph type="center"/>CAPUT VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001350"><emph type="center"/><emph type="italics"/>Quid attendendum &longs;it in Machinæ collocatione, <lb/>atque materie.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001351">QUamvis in&longs;tructarum Machinarum vires ad calculos revo­<lb/>centur in&longs;pectâ earum figurâ, ut Potentiæ atque oneris <lb/>motus invicem comparentur; quo tamen loco & &longs;itu Machina <lb/>ip&longs;a collocetur, di&longs;piciendum e&longs;t, ut innote&longs;cat, quanta illi vis <lb/>inferatur tùm ab oneris gravitate, tùm à potentiæ conatu: ex <lb/>hoc &longs;iquidem decernendum erit, quàm &longs;olidam con&longs;trui opor­<lb/>teat Machinam. </s> <s id="s.001352">Quotus enim qui&longs;que e&longs;t, qui ignoret longè <lb/>&longs;olidiorem requiri machinam, &longs;i ex illa dependeat, aut illi in­<lb/>cumbat onus, quàm &longs;i non machinæ; &longs;ed &longs;ubjecto plano, inni­<lb/>tatur idem pondus, aut aliunde dependeat? </s> <s id="s.001353">alia &longs;cilicet &longs;unt <lb/>gravitatis momenta contrà virtutem &longs;u&longs;tinentem etiam citrà <lb/>motum, alia verò momenta, quatenus motui adver&longs;atur. <pb pagenum="179" xlink:href="017/01/195.jpg"/>Hinc operæ pretium fuerit non contemnendum, &longs;i res ita à <lb/>Machinatore di&longs;ponantur, ut pondus, quàm minimum fieri <lb/>po&longs;&longs;it, à machinâ &longs;u&longs;tineatur: hâc enim ratione fiet, ut lon­<lb/>giùs avertatur periculum luxationis aut fractionis membrorum, <lb/>quibus machina di&longs;tinguitur, etiam&longs;i exilior illa fuerit; & ma­<lb/>chinæ gravitas aliqua &longs;ubtrahetur, dum moles ip&longs;a minuitur, <lb/>atque proinde movendi oneris difficultas non augebitur ex ma­<lb/>chiná; quæ etiam minore impendio parabitur. </s> </p> <p type="main"> <s id="s.001354">Sit exempli gratiâ pondus A, quod &longs;it trochleâ attollendum <lb/>in D. <!-- KEEP S--></s> <s id="s.001355">Poterit id duplici ratione fieri; primùm raptando illud in <lb/>plano Horizontali ita, ut ex B <lb/><figure id="id.017.01.195.1.jpg" xlink:href="017/01/195/1.jpg"/><lb/>veniat in C, tùm alligatâ tro­<lb/>cleâ in I illud attollendo ad <lb/>perpendiculum u&longs;que in D: <lb/>cum raptatur, totum incumbit <lb/>pondus &longs;ubjecto plano; cum at­<lb/>tollitur, totum ex trochleâ de­<lb/>pendet. </s> <s id="s.001356">At &longs;i trochleâ utaris, <lb/>de cujus firmitate &longs;ubdubites, <lb/>& loci di&longs;po&longs;itio ferat, ut po&longs;­<lb/>&longs;it ex E & H onus &longs;u&longs;pendere, <lb/>res faciliùs perficietur. </s> <s id="s.001357">Ponde­<lb/>ri enim A adnecte funem OE, <lb/>ex quo pendere po&longs;&longs;it in E, ac <lb/>prætereà tantumdem funis OS liberè vagantis; trochleam au­<lb/>tem alliga in F: ubi verò ope trochleæ adduxeris pondus ex O <lb/>in G, tùm funem OS liberè vagantem eleva, ac benè inten­<lb/>tum adnecte in H, ut jam pondus ex H dependeat ad perpen­<lb/>diculum: Ex hoc fiet, ut re&longs;oluto fune OE, liberéque vagan­<lb/>te, ope trochleæ in F alligatæ adducas pondus ex G in D mul­<lb/>tò minori labore, quàm &longs;i ex B in C illud raptâ&longs;&longs;es, & ex C <lb/>in D &longs;u&longs;tuli&longs;&longs;es. </s> <s id="s.001358">Con&longs;tat autem pondus idem minùs conniti <lb/>adversùs lineas FG aut FD, quàm adversùs perpendiculares <lb/>HG aut ID, ex iis quæ di&longs;putata &longs;unt lib. 1. cap. 15, ac <lb/>propterea etiam minùs dubitari pote&longs;t de trochleæ firmitate. </s> </p> <p type="main"> <s id="s.001359">Hoc autem compendium elevandi pondera perinde, atque <lb/>&longs;i per planum inclinatum attollerentur, ea &longs;cilicet &longs;u&longs;pendendo <lb/>atque obliquè trahendo, ubi in praxim ritè deduxeris, appa-<pb pagenum="180" xlink:href="017/01/196.jpg"/>rebit quanto labori, & quàm magnis &longs;umptibus parcatur: cum <lb/>neque vincendus &longs;it partium tritus atque conflictus inter pon­<lb/>dus, ac &longs;ubjectum planum, neque &longs;ternendum &longs;it multo robo­<lb/>re planum ip&longs;um, quod oneri &longs;u&longs;tinendo non impar &longs;it. </s> <s id="s.001360">At ubi <lb/>funem EO, quoad ejus fieri poterit, intenderis, aquá largiter <lb/>imbuito; hoc enim fiet, ut &longs;e&longs;e contrahens etiam paulò inten­<lb/>tior, atque ad de&longs;tinatum opus evadat aptior. </s> </p> <p type="main"> <s id="s.001361">Quæ cum ita &longs;int, alia &longs;e offert methodus elevandi pondera <lb/>non levi laboris compendio, &longs;i nimirùm duplex adhibeatur <lb/><figure id="id.017.01.196.1.jpg" xlink:href="017/01/196/1.jpg"/><lb/>trochlea, altera quidem in A imminens pon­<lb/>deri ad perpendiculum, altera verò in B. </s> <lb/> <s id="s.001362">Adhibita igitur trochlea B elevabit pondus <lb/>ex C in D, ibique totum ex B pendebit: <lb/>tùm vici&longs;&longs;im trochleâ A utere, & ex D in E <lb/>a&longs;cendet pondus, quod ibi totum ex A pen­<lb/>debit: iterum igitur adhibe trochleam B, ut <lb/>ex E in F a&longs;cendat; atque vici&longs;&longs;im, adhibitâ <lb/>trochleâ A a&longs;cendet ex F in G; & &longs;ic de­<lb/>inceps. </s> </p> <p type="main"> <s id="s.001363">Ubi vides motum ponderis a&longs;cendentis per arcus CDEFG <lb/>majorem e&longs;&longs;e quàm &longs;i rectâ ad perpendiculum elevatum fui&longs;&longs;et <lb/>ex C in G. <!-- KEEP S--></s> <s id="s.001364">Quia verò altitudines perpendiculares &longs;ingulis ar­<lb/>cubus re&longs;pondentes &longs;ubinde majores fiunt, propterea plus vi­<lb/>rium à potentia movente adhibendum e&longs;t in progre&longs;&longs;u. </s> <s id="s.001365">Quâ <lb/>autem Ratione altitudines illæ perpendiculares cre&longs;cant, faci­<lb/>lè innote&longs;cet, &longs;i arcuum &longs;ingulorum Sinus ver&longs;os &longs;uis Radiis <lb/>re&longs;pondentes ad calculos revocaveris; arcus enim &longs;uperiores & <lb/>plurium e&longs;&longs;e graduum, & ex Radio minori, manife&longs;tum e&longs;t: <lb/>di&longs;tantia autem parallelarum AC, BD perpendicularium ea­<lb/>dem &longs;emper e&longs;t; quapropter & æquales lineæ &longs;unt Sinus Recti <lb/>arcuum inæqualium in circulis inæqualibus, videlicet arcuum <lb/>majorum in circulis minoribus. </s> <s id="s.001366">Quamquam nec omninò ne­<lb/>ce&longs;&longs;e e&longs;t ità &longs;ingulis tractionibus pondus attollere, ut ad per­<lb/>pendiculum dependeat, &longs;i maximè trochleæ invicem non mo­<lb/>dicum di&longs;tarent; &longs;ed &longs;ufficeret alternis operis trochleas agita­<lb/>re, ut a&longs;cendens pondus modò ad hoc, modò ad illud perpen­<lb/>diculum accederet, ita tamen ut ultró citróque tran&longs;grediatur <lb/>perpendiculum, quod medium cadit inter extremas AC & BD; <pb pagenum="181" xlink:href="017/01/197.jpg"/>alioquin par non e&longs;&longs;et utriu&longs;que trahentis labor. </s> <s id="s.001367">Cæterùm <lb/>&longs;atius e&longs;t A & B parùm di&longs;tare. </s> </p> <p type="main"> <s id="s.001368">Ut autem exemplo aliquo res manife&longs;ta fiat, &longs;tatuamus alti­<lb/>tudinem AC e&longs;&longs;e pedum 70, di&longs;tantiam verò AB pedum 30, <lb/>cui æqualis e&longs;t ea, quæ ex D cadit perpendicularis in AC, &longs;ci­<lb/>licet DS. </s> <s id="s.001369">Quare in triangulo ASD rectangulo nota e&longs;t Hy­<lb/>pothenu&longs;a AD, quæ æqualis e&longs;t ip&longs;i AC, & nota e&longs;t Ba&longs;is <lb/>SD. <!-- KEEP S--></s> <s id="s.001370">Atqui con&longs;tat Perpendiculum AS e&longs;&longs;e medio loco pro­<lb/>portionale inter &longs;ummam atque differentiam Hypothenu&longs;æ ac <lb/>ba&longs;is, &longs;cilicet inter 100 & 40; igitur ducta prima in tertiam, <lb/>videlicet ducta &longs;umma in differentiam dabit 4000 Quadratum <lb/>Mediæ (hoc e&longs;t perpendiculi AS) cujus Radix ped. <!-- REMOVE S-->63 1/4 ferè <lb/>e&longs;t Perpendiculum AS. </s> <s id="s.001371">Igitur elevatio CS e&longs;t ped. <!-- REMOVE S-->6 3/4. </s> </p> <p type="main"> <s id="s.001372">Cum itaque BD æqualis &longs;it ip&longs;i AS (jungunt enim paral­<lb/>lelas æquales AB & SD) iterum in triangulo BVE rectangu­<lb/>lo nota e&longs;t Hypothenu&longs;a BE ped. <!-- REMOVE S-->63 1/4, & Ba&longs;is EV e&longs;t ped. <!-- REMOVE S-->30: <lb/>Quare inter &longs;ummam ped. <!-- REMOVE S-->93 1/4, ac differentiam ped. <!-- REMOVE S-->33 1/4 media <lb/>proportionalis ped. <!-- REMOVE S-->55. 67″. <!-- KEEP S--></s> <s id="s.001373">e&longs;t Perpendiculum BV; atque <lb/>adeò elevatio DV e&longs;t ped. <!-- REMOVE S-->7. 58″. <!-- REMOVE S-->major quàm CS. <!-- KEEP S--></s> <s id="s.001374">Et &longs;ic de <lb/>reliquis. </s> </p> <p type="main"> <s id="s.001375">At &longs;tatue di&longs;tantiam AB &longs;olùm ped. <!-- REMOVE S-->20: reperies perpendi­<lb/>culum AS vix excedere ped. <!-- REMOVE S-->67; quare elevatio CS erit ped. <!-- REMOVE S-->3 <lb/>ferè; ac propterea etiam Perpendiculum BV erit paulò majus <lb/>ped. <!-- REMOVE S-->63. 94″; & elevatio DV ped. <!-- REMOVE S-->3. 06″; & &longs;ic de cæteris. </s> </p> <p type="main"> <s id="s.001376">Potentiæ verò elevantis motum metitur differentia, quæ <lb/>inter lineas BC & BD intercedit: quando autem di&longs;tantia <lb/>AB e&longs;t ped. <!-- REMOVE S-->30, linea BC e&longs;t ped. <!-- REMOVE S-->76. 15″; at cum e&longs;t ped. <!-- REMOVE S-->20, <lb/>BC e&longs;t ped. <!-- REMOVE S-->72 4/5. Cum igitur in primo ca&longs;u BD &longs;it ped. <!-- REMOVE S-->63 1/4, <lb/>motus potentiæ e&longs;t ped. (12 9/10); in &longs;ecundo autem ca&longs;u cum BD <lb/>&longs;it ped. <!-- REMOVE S-->67; linea autem BC &longs;it ped. <!-- REMOVE S-->72 4/5, motus potentiæ e&longs;t <lb/>ped. <!-- REMOVE S-->5 4/5. Quare in primo Ratio motûs Potentiæ ad motum <lb/>ponderis e&longs;t (12 9/10) ad 6 3/4, in &longs;ecundo Ratio e&longs;t 5 4/5 ad 3: & factâ <lb/>reductione ad alias denominationes, prima Ratio e&longs;t 86 ad 45, <lb/>&longs;ecunda Ratio e&longs;t 29 ad 15, quæ &longs;i ad eumdem denominato­<lb/>rem 45 reducatur, erit 87 ad 45. Con&longs;tat autem majorem e&longs;&longs;e <lb/>Rationem 87 ad 45, quàm 86 ad 45. per 8. l. <!-- KEEP S--></s> <s id="s.001377">5. Majorem igi­<lb/>tur Rationem habet motus Potentiæ ad motum ponderis, quan-<pb pagenum="182" xlink:href="017/01/198.jpg"/>do A & B minùs di&longs;tant, quàm cum &longs;eparantur intervallo ma­<lb/>jore; atque adeò major e&longs;t etiam movendi facilitas. </s> </p> <p type="main"> <s id="s.001378">Quòd &longs;i rei hujus minimè dubium experimentum &longs;umere <lb/>placeat, ip&longs;i&longs;que oculis rem totam &longs;ubjicere citrà omnem de­<lb/><figure id="id.017.01.198.1.jpg" xlink:href="017/01/198/1.jpg"/><lb/>ludentis phanta&longs;iæ &longs;u&longs;picio­<lb/>nem, firmetur in A orbiculus <lb/>circà &longs;uum axem ver&longs;atilis, & <lb/>ex eo æqualia pondera D & E <lb/>funiculo connexa dependeant <lb/>ad perpendiculum; quæ prop­<lb/>ter gravitatis æqualitatem im­<lb/>mota permanent. </s> <s id="s.001379">Tùm in B <lb/>firmetur orbiculus circà &longs;uum <lb/>axem pariter ver&longs;atilis, & a&longs;­<lb/>&longs;umatur pondus C ponderi E <lb/>æquale, cui adnectatur funi­<lb/>culo EBC. </s> <s id="s.001380">Si manu retineas <lb/>pondus C, ne gravitet, per­<lb/>&longs;i&longs;tit pondus E in &longs;uo perpendiculo: jam manu retine <lb/>pondus D, ne pror&longs;us moveatur, ac dimitte pondus C, vi­<lb/>debis hoc quidem de&longs;cendere, pondus verò E a&longs;cendere, <lb/>donec ex B dependeat, & in æquilibrio cum pondere C <lb/>&longs;ub&longs;i&longs;tat. </s> <s id="s.001381">Iterum retine pondus C, & dimitte pondus D, <lb/>pariterque pondus D de&longs;cendens videbis, E verò adhuc <lb/>a&longs;cendens; & &longs;ic deinceps u&longs;que eò, dum pondus E uni­<lb/>cum ambobus D & C æquipolleat, ut &longs;uperiori capite in­<lb/>dicatum e&longs;t. </s> <s id="s.001382">Id igitur quod à ponderibus D & C præ&longs;tatur, <lb/>à quâlibet potentiâ æquali in D & C con&longs;titutâ præ&longs;tari po&longs;&longs;e <lb/>manife&longs;tum e&longs;t. </s> <s id="s.001383">Si itaque &longs;implicibus orbiculis fit, ut pondus <lb/>æquale po&longs;&longs;it prævalere, multò magis id fiet, &longs;i trochleæ adhi­<lb/>beantur. </s> </p> <p type="main"> <s id="s.001384">Ex his apparet, quid & in cæteris machinarum generibus, <lb/>analogiâ &longs;ervatâ, dicendum &longs;it, ex quarum opportunâ col­<lb/>locatione facilitas movendi augentur. </s> <s id="s.001385">Si enim, exempli gra­<lb/>tiâ, cubus A marmoreus elevandus fuerit vecte BC, mul­<lb/>tò faciliùs id fiet, &longs;i ille &longs;upponatur cubo, quàm &longs;i ex I ad <lb/>perpendiculum elevaretur eodem vecte &longs;u&longs;pen&longs;um: ex I &longs;ci­<lb/>licet totus cubus à vecte &longs;u&longs;tineretur; at &longs;ubjectus vectis <pb pagenum="183" xlink:href="017/01/199.jpg"/>BC ita cubum &longs;u&longs;tentat, ut <lb/><figure id="id.017.01.199.1.jpg" xlink:href="017/01/199/1.jpg"/><lb/>etiam reliquo latere cubus <lb/>idem &longs;ubjecto plano incumbat. </s> </p> <p type="main"> <s id="s.001386">Quemadmodum autem non <lb/>quemlibet vectem cuilibet <lb/>oneri <expan abbr="elevãdo">elevando</expan> parem e&longs;&longs;e om­<lb/>nes intelligunt; &longs;ed habita ra­<lb/>tione materiæ, ex quâ con&longs;tat, <lb/>congrua &longs;oliditas ei tribuenda <lb/>e&longs;t; ita pariter in cæteris omnibus, quæ hùc &longs;pectant (&longs;ive <lb/>&longs;int machinarum membra, &longs;ive paxilli &longs;int aut tigilli, quibus <lb/>machinæ adnectuntur) materiæ &longs;oliditatem attendendam e&longs;&longs;e <lb/>manife&longs;tum e&longs;t, ne frangantur. </s> <s id="s.001387">Et quidem quod ad materiam <lb/>attinet, non omnium &longs;olidorum partes pari nexu cohærent, <lb/>&longs;ed alia aliis fragiliora &longs;unt: &longs;ic lignum quernum difficiliùs <lb/>frangitur, quàm fraxineum aut populeum: neque enim in <lb/>omni ligno æque opero&longs;a &longs;imili&longs;que &longs;taminum textura repe­<lb/>ritur; cum etiam lignum idem quaqua ver&longs;um findi non po&longs;­<lb/>&longs;it pari facilitate; permagni quippe intere&longs;t, recta ne juxtà <lb/>venarum ductum? </s> <s id="s.001388">an obliquè? </s> <s id="s.001389">&longs;ectio facienda &longs;it. </s> <s id="s.001390">Id quod <lb/>in ip&longs;is quoque lapidibus, atque marmoribus ob&longs;ervare quan­<lb/>doque nece&longs;&longs;e e&longs;t, ubi non æquè per omnes partes compacta <lb/>materia venas habet &longs;ci&longs;&longs;ioni maximè obnoxias. </s> <s id="s.001391">In metallis <lb/>pariter eorum natura con&longs;ideranda e&longs;t, molli&longs;ne illa &longs;it, ac <lb/>flexibilis? </s> <s id="s.001392">an verò dura? </s> <s id="s.001393">ut eam, quam &longs;emel induit figu­<lb/>ram, con&longs;tanter retineat. </s> <s id="s.001394">Ex quo fit, ut pro materiæ di&longs;&longs;i­<lb/>militudine di&longs;par etiam cra&longs;&longs;ities requiratur: quis enim ne&longs;ciat, <lb/>quantum ligneum inter ac ferreum eju&longs;dem molis vectem in­<lb/>ter&longs;it? </s> </p> <p type="main"> <s id="s.001395">Verùm illud potiùs con&longs;iderandum videtur, quod ad &longs;oli­<lb/>ditatem ip&longs;am &longs;pectat, etiam&longs;i materies diver&longs;a non &longs;it; pro <lb/>variâ enim cra&longs;&longs;itudine mutatur frangendi difficultas; & quia <lb/>in mole majori plures in&longs;unt partes divi&longs;ioni re&longs;i&longs;tentes, fran­<lb/>gendi pariter difficultas augetur pro Ratione multitudinis par­<lb/>tium, &longs;i cætera paria &longs;int. </s> <s id="s.001396">Dubitare videlicet nemo pote&longs;t à <lb/>duplici partium dividendarum numero duplicem oriri re&longs;i&longs;ten­<lb/>tiam. </s> <s id="s.001397">Si cætera, inquam, &longs;int paria; nam &longs;i filum &longs;ericum ut <lb/>rumpatur, requirit vim ut unum, & decem fila &longs;erica paris <pb pagenum="184" xlink:href="017/01/200.jpg"/>cra&longs;&longs;itiei ac longitudinis parallela &longs;imul po&longs;ita requirant vim <lb/>decuplam; &longs;i in unum funiculum decem illa fila ritè contor­<lb/>queantur, multò majorem vim quàm decuplam requiri, ut fu­<lb/>niculus frangatur, manife&longs;tum e&longs;t: quemadmodum & ligneus <lb/>tigillus multo validiùs re&longs;i&longs;tit fractioni, quàm virgarum fa&longs;ci­<lb/>culus eidem tigillo æqualis; major e&longs;t enim particularum unio, <lb/>ubi in unum corpus coale&longs;cant, quàm ubi plura minora corpo­<lb/>ra con&longs;tituantur. </s> </p> <p type="main"> <s id="s.001398">Hinc &longs;i fuerint duo parallelepipeda quadrata A & B, quorum <lb/>latera &longs;int in Ratione quadruplâ, altitudines verò AC, & BD <lb/><figure id="id.017.01.200.1.jpg" xlink:href="017/01/200/1.jpg"/><lb/>æquales; con&longs;tat ex 32. <lb/>l. <!-- REMOVE S-->11 ea e&longs;&longs;e inter &longs;e ut ba­<lb/>&longs;es; ba&longs;es autem &longs;unt qua­<lb/>drata laterum; igitur pa­<lb/>rallelepipedum B e&longs;t &longs;ede­<lb/>cuplum parallelepipedi A. <!-- KEEP S--></s> <lb/> <s id="s.001399">Finge &longs;exdecim parallele­<lb/>pipeda ip&longs;i A æqualia in <lb/>fa&longs;ciculum colligata, & <lb/>&longs;ci&longs;&longs;ionem faciendam jux­<lb/>ta lineam OS vi oneris in <lb/>O po&longs;iti: certum e&longs;t faci­<lb/>liùs frangi po&longs;&longs;e &longs;exdecim <lb/>illa parallelepipeda, quàm <lb/>parallelepipedum B illis <lb/>omnibus æquale; ut enim &longs;cindatur, curvari oportet vi oneris <lb/>incumbentis; illa autem &longs;exdecim faciliùs curvantur quàm <lb/>ip&longs;um B. </s> <s id="s.001400">Id quod manife&longs;tum fiat, &longs;i virgam ex &longs;alicto <lb/>decerpens, eamque leniter inflectens ob&longs;erves, quâ quidem <lb/>parte virga curvata e&longs;t, tenerum corticem in rugas a&longs;&longs;urge­<lb/>re atque cri&longs;pari, quâ verò parte convexa e&longs;t, corticem <lb/>di&longs;trahi atque di&longs;tendi. </s> <s id="s.001401">Ex quo facilè arguimus, quid durio­<lb/>ribus corporibus contingat, quæ non adeò manife&longs;tè corru­<lb/>gari po&longs;&longs;unt; flecti &longs;cilicet nequeunt, quin aliqua fiat inte­<lb/>riorum partium compre&longs;&longs;io, & exteriorum di&longs;tractio. </s> <s id="s.001402">Hinc <lb/>in parallelepipedo B, quod flecti intelligitur, ut &longs;cindatur, <lb/>partes, quæ circa O, comprimuntur; quæ verò circà S, <lb/>di&longs;trahuntur: huic autem motioni repugnant omnes particu-<pb pagenum="185" xlink:href="017/01/201.jpg"/>læ vi nexûs, quo unaquæque cum &longs;ibi proximè cohærentibus <lb/>particulis colligatur. </s> <s id="s.001403">Cum autem &longs;exdecim illa parallelepipe­<lb/>da minora non &longs;int invicem connexa, quemadmodum particu­<lb/>læ omnes parallelepipedi B in unam molem coaluerunt, con&longs;tat <lb/>pauciores nexus faciliùs, quàm plures, di&longs;&longs;olvi. </s> </p> <p type="main"> <s id="s.001404">Hoc verò ut pleniùs atque apertiùs explicetur, intellige &longs;o­<lb/>lidum longiu&longs;culum RS in plures tenues laminas plano RI <lb/>parallelas divi&longs;um, &longs;ibi­<lb/><figure id="id.017.01.201.1.jpg" xlink:href="017/01/201/1.jpg"/><lb/>que ita vici&longs;&longs;im con­<lb/>gruentes, ut earum ex­<lb/>tremitates con&longs;tituant <lb/>planum HI. <!-- KEEP S--></s> <s id="s.001405">Omnes <lb/>ha&longs;ce laminas &longs;ecun­<lb/>dùm extremitates ful­<lb/>cris impo&longs;itas pondus <lb/>&longs;uper DC con&longs;titutum <lb/>adeò premat, ut cur­<lb/>vari aliquantulum cogantur. </s> <s id="s.001406">Ob&longs;ervabis illicò extremitates <lb/>illas non jam ampliùs in eandem planitiem HI exæquari; &longs;ed <lb/>eas quidem laminas, quæ cavitatem &longs;pectant, magis curvari; <lb/>minùs verò eas, quæ convexitati re&longs;pondent, ac proptereà ex­<lb/>timæ laminæ extremitatem ab extremitate intimæ laminæ, quæ <lb/>ponderi impo&longs;ito cohæret, magis recedere, quàm interme­<lb/>diarum extremitates. </s> <s id="s.001407">Con&longs;tat itaque in hoc motu &longs;ingula­<lb/>rum laminarum particulas, dum curvantur, non iis re&longs;pon­<lb/>dere adhærentis laminæ particulis, quas priùs contingebant, <lb/>cùm omnis curvitatis expertes erant, atque faciliùs potui&longs;&longs;e <lb/>&longs;ingulas laminas moveri, quia nullo nexu invicem copulan­<lb/>tur. </s> <s id="s.001408">Quòd &longs;i ex iis unum &longs;olidum RS planè integrum coa­<lb/>le&longs;cat, manife&longs;tum e&longs;t planitiem HI permanere, ac propterea, <lb/>dum curvatur, nece&longs;&longs;e e&longs;t, ut interiores particulæ invicem <lb/>connexæ di&longs;trahantur, cum nequeant aliæ ab aliis &longs;ecedere, <lb/>quemadmodum in laminis contingere ob&longs;ervavimus. </s> <s id="s.001409">Hinc <lb/>oritur major &longs;olidi, quàm laminarum, re&longs;i&longs;tentia, ne fran­<lb/>gatur. </s> <s id="s.001410">Non negarim tamen aliquando &longs;atius e&longs;&longs;e duobus me­<lb/>diocribus tigillis uti, quàm cra&longs;&longs;iore tigno illis æquali; quia <lb/>nimirum alterutro labem patiente rima&longs;vè agente, alter faci­<lb/>liùs integer per&longs;everat; in cra&longs;&longs;iore autem tigno, &longs;i rimam du-<pb pagenum="186" xlink:href="017/01/202.jpg"/>cere occœperit, periculum e&longs;t, ne malum &longs;erpat juxta vena­<lb/>rum aut fibrarum ductum. </s> <s id="s.001411">Cæterum &longs;ublato huju&longs;modi peri­<lb/>culo, ubi reliqua paria &longs;int, cra&longs;&longs;iora corpora difficiliùs fran­<lb/>guntur. </s> </p> <p type="main"> <s id="s.001412">Quare &longs;olidorum re&longs;i&longs;tentia, ne frangantur, major e&longs;t <lb/>quam pro Ratione &longs;ectionum; hæc &longs;iquidem Ratio &longs;ectionum <lb/>&longs;ervari quidem intelligitur, &longs;i limâ aut &longs;errâ &longs;ecari corpora <lb/>oporteat; illæ enim tantummodo particulæ re&longs;i&longs;tunt., quæ <lb/>&longs;ectionem admittunt; at ubi de fractione agitur, quæ præter <lb/>motum particularum, quæ dividuntur, motum etiam aliquem <lb/>exigit aliarum, quas comprimi aut di&longs;trahi opus e&longs;t, plus, <lb/>minùs, pro Ratione vicinitatis, longè alia e&longs;t Ratio, pro ut <lb/>compre&longs;&longs;io illa atque di&longs;tractio particularum faciliùs aut dif­<lb/>ficiliùs perfici poterit. </s> <s id="s.001413">Hoc autem ex ipsâ figurâ poti&longs;&longs;imùm <lb/>pendet: Solidi enim RS &longs;ectio CDE eadem quidem e&longs;t, &longs;i­<lb/>vè illud circà DE longiorem lineam, &longs;ivè circa CD brevio­<lb/>rem, curvari debeat, ut frangatur; &longs;ed non eadem e&longs;t in <lb/>fractione CD ac in fractione DE frangendi difficultas; nam <lb/>cum propiores &longs;int puncto D partes, quæ ad C, quàm quæ <lb/>ad E &longs;itæ &longs;unt, con&longs;tat has quidem magis cum circà lineam <lb/>CD curvatur &longs;olidum, illas verò, cùm circà lineam DE <lb/>curvatur, minùs di&longs;trahi oportere, ut fractio &longs;equatur. </s> <s id="s.001414">Quò <lb/>autem magis di&longs;trahi debent particulæ, quæ ex D ver ûs E <lb/>recedunt, magis interim comprimi nece&longs;&longs;e e&longs;t eas, quæ ad D <lb/>accedunt &longs;ecundùm lineam RO in plano RI. </s> <s id="s.001415">Major igi­<lb/>tur e&longs;t difficultas, &longs;i circà breviorem lineam CD curve­<lb/>tur, & fractio &longs;ecundùm longiorem lineam DE &longs;equatur, <lb/>quàm &longs;i contrà curvetur circà longiorem DE, & fractio &longs;it <lb/>juxtà breviorem CD. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001416">Jam igitur &longs;i duo &longs;olida invicem comparentur, quæ eju&longs;­<lb/>dem &longs;int materiæ eju&longs;demque longitudinis, & in pari ab ex­<lb/>tremitatibus di&longs;tantiâ frangi oporteat, &longs;tatuatur in utroque <lb/>&longs;olido punctum fractionis, per quod intelligatur planum &longs;e­<lb/>cans &longs;imiliter inclinatum, facien&longs;que in utroque &longs;olido &longs;uper­<lb/>ficies, quas vocemus <emph type="italics"/>Ba&longs;es.<emph.end type="italics"/></s> <s id="s.001417"> Item planum per quod movetur <lb/>Potentia vim frangendi habens, ita productum intelligatur, ut <lb/>Ba&longs;ibus prædictis &longs;imili inclinatione occurrens de&longs;cribat &longs;ectio­<lb/>num lineas, quas vocemus Cra&longs;&longs;ities. </s> <s id="s.001418">Ut &longs;i fuerint duo &longs;oli-<pb pagenum="187" xlink:href="017/01/203.jpg"/>da CD & EF æqualis longitu­<lb/><figure id="id.017.01.203.1.jpg" xlink:href="017/01/203/1.jpg"/><lb/>dinis, parieti infixa &longs;ecundùm <lb/>æquales partes CI & EH, ut <lb/>in punctis I & H fiat fractio, <lb/>ex hypothe&longs;i. </s> <s id="s.001419">Si per ea puncta <lb/>agantur plana &longs;imiliter inclina­<lb/>ta, erunt &longs;uperficies IL & <lb/>HM, quas vocamus hîc <emph type="italics"/>Ba&longs;es.<emph.end type="italics"/><lb/>Jam in extremitatibus D & F <lb/>æquè remotis à punctis I & H <lb/>&longs;int Potentiæ vim frangendi habentes, & per lineam motûs <lb/>huju&longs;modi Potentiarum intelligantur plana cum &longs;imili inclina­<lb/>tione occurrentia ba&longs;ibus IL & HM, ponamu&longs;que communes <lb/>horum planorum &longs;ectiones e&longs;&longs;e lineas parallelas, & æquales li­<lb/>neis IN & HO; quas &longs;ectiones vocamus <emph type="italics"/>Cra&longs;&longs;ities<emph.end type="italics"/> &longs;olidorum, <lb/>atque pro earum men&longs;urâ u&longs;urpamus lineas IN & HO. <!-- KEEP S--></s> <s id="s.001420">Cum <lb/>itaque frangendi difficultas oriatur tùm ex numero partium, <lb/>quæ &longs;eparandæ &longs;unt, has autem ip&longs;æ Ba&longs;es IL & HM defi­<lb/>niunt, tùm ex violento motu di&longs;tractionis partium, qui ex ipsâ <lb/>&longs;olidorum cra&longs;&longs;itie IN, & HO digno&longs;citur; illud con&longs;equens <lb/>e&longs;t, quòd Re&longs;i&longs;tentiæ &longs;olidorum Ratio ea &longs;it, quæ ex Ratione <lb/>Ba&longs;ium, & Ratione Cra&longs;&longs;itierum componitur. </s> <s id="s.001421">Hinc e&longs;t quòd <lb/>&longs;i Ba&longs;es fuerint &longs;imiles, & quæ e&longs;t Ratio laterum homologo­<lb/>rum, ea etiam &longs;it Cra&longs;&longs;itierum Ratio, re&longs;i&longs;tentiæ ad fractionem <lb/>invicem comparatæ erunt in Ratione triplicatâ laterum homo­<lb/>logorum; ac propterea cylindrorum re&longs;i&longs;tentia ad fractionem <lb/>erit in Ratione triplicatâ Diametrorum, &longs;eu Cra&longs;&longs;itierum. </s> </p> <p type="main"> <s id="s.001422">Hanc, de quâ hactenus nobis &longs;ermo fuit, <emph type="italics"/>Re&longs;i&longs;tentiam ab&longs;olu­<lb/>tam<emph.end type="italics"/> dicimus, quam &longs;olidum habet, ne dividatur: quò enim <lb/>plures partes debent præter naturam comprimi, aut di&longs;trahi, <lb/>plures &longs;unt re&longs;i&longs;tentiæ; & quò magis hoc motu debent mo­<lb/>mento eodem præter naturam moveri, eò etiam magis re­<lb/>&longs;i&longs;tunt: quâ igitur ratione plures &longs;unt re&longs;i&longs;tentes, & quâ Ra­<lb/>tione magis re&longs;i&longs;tunt, tota re&longs;i&longs;tentiæ ratio componitur; quæ <lb/>ex ipsâ corporis &longs;oliditate pendet, nullâ habitâ ratione longi­<lb/>tudinis ip&longs;ius &longs;olidi: Propterea <emph type="italics"/>Ab&longs;oluta<emph.end type="italics"/> dicitur. </s> <s id="s.001423">Nam &longs;i lon­<lb/>gitudines frangendorum corporum comparemus, quæ &longs;uâ va­<lb/>rietate mutant frangendi difficultatem, aut facilitatem, re-<pb pagenum="188" xlink:href="017/01/204.jpg"/>&longs;i&longs;tentia hæc dicenda erit <emph type="italics"/>Re&longs;pectiva<emph.end type="italics"/>; quæ aliquando ea e&longs;&longs;e <lb/>pote&longs;t, ut corpus majore re&longs;i&longs;tentiâ ab&longs;olutâ præditum redda­<lb/>tur magis obnoxium fractioni; longitudo &longs;iquidem auget fran­<lb/>gendi facilitatem: ideo autem <emph type="italics"/>Re&longs;pectivam<emph.end type="italics"/> dicimus, quia com­<lb/>paratè ad momenta potentiæ &longs;umitur; hæc verò momenta ex <lb/>variâ longitudine, &longs;eu di&longs;tantia à puncto fractionis pendere <lb/><figure id="id.017.01.204.1.jpg" xlink:href="017/01/204/1.jpg"/><lb/>manife&longs;tum e&longs;t. </s> <s id="s.001424">Sit enim <lb/>&longs;olidum AB, quod ita <lb/>flectatur, ut fiat fractio <lb/>CD: Potentia movens in <lb/>B con&longs;tituta dum perficit <lb/>&longs;patium BE, di&longs;tractio par­<lb/>ticularum &longs;olidi fit &longs;olùm <lb/>per &longs;patium CD (aut ve­<lb/>riùs per CHD, nam etiam partes inter C & H di&longs;trahuntur; <lb/>Sed hîc claritatis gratiâ &longs;olùm extremæ CD con&longs;iderantur) <lb/>quod e&longs;t multo minus &longs;patio BE &longs;ecundùm Rationem HD ad <lb/>HE. <!-- KEEP S--></s> <s id="s.001425">At &longs;i &longs;olidum frangendum &longs;it AF, aut &longs;i &longs;it totum AB, <lb/>tamen Potentia movens &longs;it &longs;olùm applicata in F, Potentia perfi­<lb/>ciens &longs;patium FG (quod e&longs;t minus quàm BE in Ratione HF <lb/>ad HB) major e&longs;&longs;e debet quàm Potentia in B &longs;ecundùm Ratio­<lb/>nem Reciprocam motuum BE & FG, ut &longs;equatur idem motus <lb/>di&longs;tractionis partium CD; nam ex 8. l. <!-- REMOVE S-->5. minor e&longs;t Ratio FG <lb/>ad CD, quàm &longs;it Ratio BE ad eandem CD. <!-- KEEP S--></s> <s id="s.001426">Con&longs;tat igitur <lb/>à longitudine augeri facilitatem frangendi, ac proinde Re­<lb/>&longs;i&longs;tentiam hanc Re&longs;pectivam e&longs;&longs;e &longs;ecundùm Reciprocam Ra­<lb/>tionem longitudinum. </s> </p> <p type="main"> <s id="s.001427">Ex quo obiter apparet, cur &longs;olida Horizonti perpendicularia <lb/>magis re&longs;i&longs;tant fractioni, &longs;i potentiæ motus, &longs;eu conatus, &longs;it ad <lb/>perpendiculum Horizonti: quia videlicet in huju&longs;modi motu <lb/>ad perpendiculum æqualiter moveri oportet Potentiam cum <lb/>&longs;olidi particulis, quæ di&longs;trahi aut comprimi debent: ut autem <lb/>Potentia &longs;uperet vim re&longs;titivam, aut major e&longs;&longs;e debet Ratio <lb/>motûs potentiæ ad motum corporis re&longs;i&longs;tentis, quàm &longs;it Ratio <lb/>virium re&longs;i&longs;tendi ad virtutem movendi, aut virtus movendi ab­<lb/>&longs;olutè major e&longs;&longs;e debet vi re&longs;i&longs;tendi: Cum itaque in motu per­<lb/>pendiculari intercedere non po&longs;&longs;it motuum inæqualitas, ne­<lb/>ce&longs;&longs;e e&longs;t virtutem movendi vehementer augeri, ut &longs;uperet vim, <pb pagenum="189" xlink:href="017/01/205.jpg"/>quâ particulæ &longs;olidi invicem connexæ repugnant, ne di&longs;tra­<lb/>hantur, aut comprimantur. </s> </p> <p type="main"> <s id="s.001428">Hinc ex ha&longs;tâ ad perpendiculum &longs;u&longs;pensâ pendebit ingens <lb/>&longs;axum, & tigillum perpendiculariter terræ in&longs;i&longs;tentem pre­<lb/>met moles, penè dixerim, immen&longs;a, citrà ha&longs;tæ aut ti­<lb/>gilli fractionem: quia omnes ha&longs;tæ atque tigilli partes & <lb/>æqualiter cum onere &longs;u&longs;pen&longs;o aut incumbente moveri de­<lb/>berent, & omnes æqualiter re&longs;i&longs;tunt di&longs;tractioni aut com­<lb/>pre&longs;&longs;ioni: At &longs;i ad horizontem inclinata aut parallela fue­<lb/>rint huju&longs;modi &longs;olida (ha&longs;ta videlicet atque tigillus) non <lb/>e&longs;t æqualis omnium partium di&longs;tractio aut compre&longs;&longs;io, mi­<lb/>nùs enim di&longs;trahuntur, quæ puncto H proximæ &longs;unt, quam <lb/>quæ ad D accedunt (concipe H in media cra&longs;&longs;itie) con­<lb/>trà verò illæ magis, hæ minùs comprimuntur; quemad­<lb/>modum neque motui di&longs;tractionis aut compre&longs;&longs;ionis e&longs;&longs;et <lb/>æqualis motus oneris deorsùm urgentis in ha&longs;tæ, vel tigil­<lb/>li non perpendicularium extremitate con&longs;tituti, &longs;ed multò <lb/>major e&longs;&longs;et hîc oneris motus. </s> <s id="s.001429">Quoniam verò rerum natu­<lb/>ra magis repugnat corporum penetrationi, ad quam quodam­<lb/>modo accedere videtur compre&longs;&longs;io, quàm corporum unito­<lb/>rum divi&longs;ioni, ubi vacui metus ab&longs;it; hinc e&longs;t majorem <lb/>molem faciliùs &longs;u&longs;tineri à fulcro ad perpendiculum &longs;ubjecto, <lb/>quàm &longs;u&longs;pendi ex &longs;olido perpendiculari citrà fractionis pe­<lb/>riculum. </s> <s id="s.001430">Quamvis negandum non &longs;it ad huju&longs;modi facili­<lb/>tatem, quam experimur in &longs;u&longs;tinendo potiùs, quàm in re­<lb/>tinendo onere, conferre plurimum, quòd tellus, cui ful­<lb/>crum infigitur, demùm non &longs;ub&longs;idit; at laqueare &longs;eu for­<lb/>nix ex quo &longs;olidum pendet onere prægravatum, tantam <lb/>gravitatem non ita facilè ferre pote&longs;t. </s> <s id="s.001431">Quare ad tollenda <lb/>in &longs;uperiores ædificiorum partes ingentia &longs;axa multo cau­<lb/>tiùs atque tutiùs ij operantur, qui longam trabem, aut plu­<lb/>ra tigna ritè connexa, qua&longs;i navis malum rudentibus u&longs;<lb/>quequaque firmatum, ne à perpendiculo deflectat, &longs;ta­<lb/>tuunt, cui &longs;uperiorem trochleam adnectant; quàm qui tra­<lb/>bem Horizonti parallelam parieti infigunt ad idem munus <lb/>præ&longs;tandum; hæc &longs;iquidem horizonti parallela magis fractio­<lb/>ni obnoxia e&longs;t, quàm perpendicularis; præterquam quod <lb/>parietem aliquatenus labefactare pote&longs;t, cum habeat ratio-<pb pagenum="190" xlink:href="017/01/206.jpg"/>nem vectis in &longs;uperiora propellentis &longs;axo deor&longs;um urgente; <lb/>ni&longs;i huic periculo ex arte obviam eatur. </s> </p> <p type="main"> <s id="s.001432">Comparatis itaque invicem &longs;olidorum frangendorum lon­<lb/>gitudinibus, hoc e&longs;t intervallis inter fractionum puncta & <lb/>locum, ubi potentia vim frangendi habens con&longs;tituta intel­<lb/>ligitur, quò major e&longs;t longitudo, eò minor e&longs;t re&longs;i&longs;tentia <lb/>&longs;olidi, ne frangatur. </s> <s id="s.001433">Qua propter ubi duo data &longs;olida con­<lb/>ferantur, quæcumque demùm illa &longs;int, non &longs;olùm eorum <lb/>Re&longs;i&longs;tentia Ab&longs;oluta, quæ ex Rationibus Ba&longs;ium, & Cra&longs;­<lb/>&longs;itierum componitur, attendenda e&longs;t, &longs;ed etiam Re&longs;i&longs;tentia <lb/>Re&longs;pectiva, quæ ex longitudinibus pendet: atque adeò <lb/>adæquata Ratio re&longs;i&longs;tentiæ, ne frangantur, ea e&longs;t, quæ <lb/>componitur ex Rationibus Ba&longs;ium & Cra&longs;&longs;itierum atque ex <lb/>Ratione longitudinum Reciprocè &longs;umptarum: cùm enim <lb/>longitudini majori re&longs;pondeat minor re&longs;i&longs;tentia, manife&longs;tum <lb/>e&longs;t longitudinum Rationem e&longs;&longs;e Reciprocè &longs;umendam, ut <lb/>re&longs;i&longs;tentiæ, quæ ex illis oritur, Ratio habeatur. </s> <s id="s.001434">Hinc e&longs;t <lb/>fieri aliquando po&longs;&longs;e, ut &longs;olidum cra&longs;&longs;ius minùs re&longs;i&longs;tat <lb/>fractioni, quàm &longs;ubtilius, &longs;i hoc breve &longs;it, illud verò valdè <lb/>longum, &longs;i videlicet longitudo cra&longs;&longs;ioris ad longitudinem <lb/>&longs;ubtilioris Rationem habeat majorem, quàm &longs;it ea, quæ ex <lb/>Rationibus Ba&longs;ium, & Cra&longs;&longs;itierum componitur. </s> <s id="s.001435">Sic &longs;i duo <lb/>fuerint cylindri, & alter triplo cra&longs;&longs;ior fuerit reliquo, &longs;ed <lb/>etiam trigecuplo longior fuerit illo, minùs etiam fractioni <lb/>re&longs;i&longs;tet; quia re&longs;i&longs;tentia ab&longs;oluta majoris cylindri ad mino­<lb/>rem e&longs;t ut 27 ad 1, &longs;ed re&longs;i&longs;tentia Re&longs;pectiva eju&longs;dem ma­<lb/>joris ad minoris re&longs;i&longs;tentiam pariter re&longs;pectivam e&longs;t ut 1 ad <lb/>30: Ratio ergo ex his Rationibus 27 ad 1, & 1 ad 30 <lb/>Compo&longs;ita, e&longs;t Ratio 27 ad 30, hoc e&longs;t 9 ad 10, ac propterea <lb/>major cylindrus re&longs;i&longs;tit fractioni ut 9, minor verò fractioni <lb/>re&longs;i&longs;tit ut 10. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001436">De&longs;ine jam mirari, &longs;i quando paxillum maximis viribus <lb/>re&longs;i&longs;tere videris; quia nimirùm potentia, quæ motum co­<lb/>natur, proximè applicata e&longs;t parieti aut plano, cui paxil­<lb/>lus infigitur: quòd &longs;i remotior illa fuerit, etiam minùs hic <lb/>re&longs;i&longs;tet. </s> <s id="s.001437">Sic defixo in terram paxillo AB, cui funis AC al­<lb/>ligatur, experientia docet paxillum eò re&longs;i&longs;tere validiùs, quò <lb/>propiùs ad A alligatur funis, debiliùs autem re&longs;i&longs;tere, quò <pb pagenum="191" xlink:href="017/01/207.jpg"/>magis ad B accedit; <lb/><figure id="id.017.01.207.1.jpg" xlink:href="017/01/207/1.jpg"/><lb/>in A nimirùm motus <lb/>potentiæ trahentis vix <lb/>excederet motum pa­<lb/>xilli, qui ibi flectere­<lb/>tur ex hypothe&longs;i; at <lb/>fune in B po&longs;ito, po­<lb/>tentia ibi con&longs;tituta, <lb/>& per funem applica­<lb/>ta multò velociùs mo­<lb/>veretur, quàm paxilli <lb/>partes propè A, quæ <lb/>ibi flecterentur. </s> </p> <p type="main"> <s id="s.001438">Quòd &longs;i loci conditio, aut ip&longs;a oneris movendi con&longs;titutio <lb/>id exigat, ut funis propè B alligetur, & de paxilli AB firmi­<lb/>tate dubitetur, paxillum alterum DE paulò remotiorem com­<lb/>modo loco depange ita, ut funis primùm in D firmetur, de­<lb/>inde circa B convolutus extendatur, pro ut operis faciendi ra­<lb/>tio fieret. </s> </p> <p type="main"> <s id="s.001439">Eâdem ratione &longs;i tigillus, ex quo onus dependere debet, pa­<lb/>rieti &longs;it infixus, & &longs;it GH, fractioni magis erit obnoxius, quò <lb/>propiùs accedet pondus ad H: <lb/><figure id="id.017.01.207.2.jpg" xlink:href="017/01/207/2.jpg"/><lb/>propterea aut ei &longs;ubjicitur brevior <lb/>tigillus IR omninò contiguus, <lb/>aut &longs;upponitur fulcrum OS in­<lb/>clinatum; quod fractionem eò va­<lb/>lidiùs impediet, quò minùs di&longs;ta­<lb/>bunt H & S, & quò acutior fue­<lb/>rit angulus, quem fulcrum SO <lb/>cum pariete con&longs;tituit, &longs;eu, quod <lb/>eôdem recidit, quò magis ad <lb/>recti anguli quantitatem acce­<lb/>det angulus GSO. </s> <s id="s.001440">Quæ omnia <lb/>ita ex dictis aperta &longs;unt, ut ulte­<lb/>riori explicatione non egeant. </s> </p> <p type="main"> <s id="s.001441">Sed & illud hîc, ubi de Re&longs;i&longs;tentiâ Re&longs;pectivâ &longs;ermo e&longs;t, <lb/>adjiciendum videtur, quòd ex &longs;olâ majori longitudine hæc non <lb/>minuitur, ni&longs;i cùm longitudo &longs;olidi ad perpendiculum in&longs;i&longs;tit <pb pagenum="192" xlink:href="017/01/208.jpg"/>Horizonti; tunc enim gravitas ip&longs;a &longs;olidi tota incumbit <lb/>&longs;ubjecto plano; & tantùm Potentia oblique atque in tran&longs;­<lb/>ver&longs;um trahens applicata extremitati longioris &longs;olidi plus ha­<lb/>bet momenti, quàm applicata extremitate brevioris, quin <lb/>velociùs, & faciliùs movetur &longs;ecundùm Rationem longitu­<lb/>dinum illarum. </s> <s id="s.001442">At quando &longs;olida &longs;unt horizonti parallela, <lb/>aut ad illum ita inclinata, ut centrum gravitatis partis illius, <lb/>quæ erumpit ex corpore, cui &longs;olidum infigitur, non immi­<lb/>neat ba&longs;i &longs;u&longs;tentationis, non &longs;ola longitudo attendenda e&longs;t, <lb/>&longs;ed & ip&longs;a gravitas, quæ etiam nullo addito extrin&longs;eco mo­<lb/>tore &longs;ua habet momenta, quibus deor&longs;um connititur. </s> <s id="s.001443">Ex <lb/>quo fit pro majori gravitate etiam frangendi facilitatem au­<lb/>geri, ip&longs;a nimirum gravitas e&longs;t potentia conjuncta, quæ au­<lb/>getur pro ratione materiæ; materia autem augetur pro ra­<lb/>tione longitudinis (cætera &longs;iquidem paria e&longs;&longs;e hîc claritatis <lb/>gratiâ, ponamus) ac propterea longius pri&longs;ma comparatum <lb/>cum breviori pri&longs;mate, eo quòd majorem habeat gravita­<lb/>tem, minùs re&longs;i&longs;tit fractioni &longs;ecundùm Reciprocam Ratio­<lb/>nem longitudinum. </s> <s id="s.001444">Atqui Ratio motûs huju&longs;modi Potentiæ <lb/>conjunctæ e&longs;t &longs;ecundùm Rationem longitudinum, & ex <lb/>dictis Ratio Re&longs;i&longs;tentiæ in ordine ad huju&longs;modi motum e&longs;t <lb/>permutatim ac Reciprocè &longs;ecundùm eandem longitudinum <lb/>Rationem: igitur Ratio duplicatur, & re&longs;i&longs;tentia longioris <lb/>ad re&longs;i&longs;tentiam brevioris e&longs;t &longs;ecundùm &longs;ubduplicatam Ratio­<lb/>nem longitudinum reciprocè &longs;umptarum. </s> <s id="s.001445">Id quod etiam <lb/>hinc con&longs;tat, quia cùm &longs;ingula illius longitudinis puncta <lb/>&longs;uam habeant gravitatem, &longs;ua omnibus in&longs;unt momenta pro <lb/>Ratione di&longs;tantiæ à puncto quod e&longs;t veluti centrum motûs; <lb/>ergo aggregata momentorum &longs;unt ut &longs;ectores ab illis longi­<lb/>tudinibus tanquam à Radiis de&longs;cripti: &longs;unt autem &longs;imiles <lb/>&longs;ectores in duplicatâ Ratione Radiorum. </s> <s id="s.001446">Quare &longs;i longitudi­<lb/>nes &longs;int ut 3 ad 2, Re&longs;i&longs;tentia re&longs;pectiva longioris ad re&longs;i&longs;ten­<lb/>tiam brevioris e&longs;t ut 4 ad 9. Tota igitur &longs;olidorum re&longs;i&longs;ten­<lb/>tia, ne frangantur, componitur ex Rationibus Ba&longs;ium, & <lb/>Cra&longs;&longs;itierum, & ex &longs;ubduplicatâ Ratione longitudinum per­<lb/>mutatim ac reciprocè &longs;umptarum. </s> </p> <p type="main"> <s id="s.001447">Ex his itaque, quæ de &longs;olidorum re&longs;i&longs;tentiâ, ne frangan­<lb/>tur, hactenùs di&longs;putata &longs;unt, conjecturam facilè accipiet <pb pagenum="193" xlink:href="017/01/209.jpg"/>prudens machinator, quàm &longs;olida & cra&longs;&longs;a &longs;tatui debeant <lb/>quæque machinarum membra, quóve loco collocanda &longs;int, <lb/>ut & materia & forma re&longs;pondeant fini, in quem machinæ <lb/>de&longs;tinantur: neque enim &longs;atis e&longs;t concinno, & eleganti dia­<lb/>grammate machinam oculis repræ&longs;enta&longs;&longs;e, eju&longs;que vires ad <lb/>calculos revocâ&longs;&longs;e, quantum quidem ex machinæ figurâ col­<lb/>ligitur, &longs;i demùm, in&longs;tituto motu machina pondere prægra­<lb/>vata luxetur. </s> </p> <p type="main"> <s id="s.001448">Illud tamen præterea Machinator animadvertat, oportet, <lb/>quod &longs;pectat ad momenta virium, quas potentia movens <lb/>exercet; neque enim &longs;ola ponderis gravitas machinam, aut <lb/>corpus, cui machina alligatur, aut innititur, urget aut pre­<lb/>mit, &longs;ed & ip&longs;a potentia, dum adversùs ip&longs;um pondus co­<lb/>natur machinam movens, aliquando auget gravitatem ex <lb/>oppo&longs;itâ parte, adeò ut & huic & ponderi re&longs;i&longs;tere debeat <lb/>machina, aut id, quod machinam retinet. </s> <s id="s.001449">Si enim fuerit <lb/>vectis AB in­<lb/><figure id="id.017.01.209.1.jpg" xlink:href="017/01/209/1.jpg"/><lb/>nixus &longs;uper ba­<lb/>culum CD, ex <lb/>B pendeat glo­<lb/>bus plumbeus <lb/>E, & extremi­<lb/>tas A quie&longs;cat <lb/>aliquo corpore <lb/>retinente, ut &longs;i <lb/>fuerit parieti in­<lb/>fixa; &longs;olo globo E gravitante minus periculum &longs;ube&longs;t fractio­<lb/>nis tùm vectis, tùm baculi CD &longs;u&longs;tentantis, quàm &longs;i in A <lb/>&longs;it potentia F; cujus conatus deor&longs;um oppo&longs;itus conatui de­<lb/>or&longs;um ponderis E faciliùs curvitatem, aut etiam demùm <lb/>fractionem vectis efficere pote&longs;t in I, ut patet; immò & ba­<lb/>culus CD &longs;u&longs;tentans vectem, non &longs;olùm momenta ponderis E, <lb/>&longs;ed & momenta Potentiæ F, quæ in I uniuntur, in &longs;e recipit; <lb/>atque adeò utri&longs;que ferendis par e&longs;&longs;e debet. </s> </p> <p type="main"> <s id="s.001450">Simile quiddam ob&longs;ervare e&longs;t, &longs;i ex orbiculo O, in clavo <lb/>M &longs;u&longs;pen&longs;o, circà &longs;uum axem ver&longs;atili, dependeat pondus S, <lb/>& Potentia in R deor&longs;um conata cogat pondus S a&longs;cendere: <lb/>certum e&longs;t enim ab axe orbiculi, & à clavo M &longs;u&longs;tineri non <pb pagenum="194" xlink:href="017/01/210.jpg"/><figure id="id.017.01.210.1.jpg" xlink:href="017/01/210/1.jpg"/><lb/>&longs;olùm pondus S, &longs;ed & Poten­<lb/>tiam, quæ e&longs;t in R. <!-- KEEP S--></s> <s id="s.001451">Contrà ve­<lb/>rò &longs;i orbiculus V &longs;it adnexus pon­<lb/>deri T, funis autem orbiculo in­<lb/>&longs;ertus alligetur clavo in N, & po­<lb/>tentia P &longs;ur&longs;um trahat, con&longs;tat ab <lb/>axe quidem orbiculi &longs;u&longs;tineri &longs;o­<lb/>lum pondus T; à clavo verò N <lb/>non totum pondus T &longs;u&longs;tineri, <lb/>&longs;ed ejus &longs;emi&longs;&longs;em, nam etiam Po­<lb/>tentia P &longs;u&longs;tinet pondus. </s> <s id="s.001452">Validior <lb/>igitur e&longs;&longs;e debet clavus M quàm <lb/>clavus N, hic enim ponderis &longs;e­<lb/>mi&longs;&longs;em fert, ille verò plus quàm <lb/>duplum. </s> <s id="s.001453">Potentia enim R major <lb/>e&longs;t pondere S. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001454">Quòd &longs;i tàm pondera S & T, <lb/>quàm clavi M & N, atque Po­<lb/>tentiæ R & P non in plano Ver­<lb/>ticali, &longs;ed in Horizontali con&longs;tituantur, certum e&longs;t pondera <lb/>S & T non &longs;u&longs;pen&longs;a &longs;ed jacentia, nihil adversùs clavos M & <lb/>N; aut adversùs &longs;uorum orbiculorum O & V axes conari, im­<lb/>mò neque adversùs Potentias R & P; quandoquidem toto ni&longs;u <lb/>plano &longs;ubjecto incumbunt, nullámque exercent Activam Re­<lb/>&longs;i&longs;tentiam; &longs;ed Formalem tantummodo, quâ repugnent Po­<lb/>tentiis moventibus: quæ quidem re&longs;i&longs;tentia, tùm ex ip â pon­<lb/>derum gravitate, tùm ex attritu &longs;ubjecti plani componitur. </s> <lb/> <s id="s.001455">Clavorum igitur M & N ea &longs;it, oportet, &longs;oliditas atque firmi­<lb/>tas, quæ potentiarum R & P conatibus re&longs;pondeat; ne forte <lb/>clavi ip&longs;i frangantur faciliùs, aut revellantur, quàm pondera <lb/>&longs;uo loco dimoveantur. </s> <s id="s.001456">Sed hæc innui&longs;&longs;e &longs;at fuerit, ut &longs;ingula <lb/>diligenter à machinatore circum&longs;picienda e&longs;&longs;e intelligatur; ne­<lb/>que tamen in his ad nau&longs;eam diutiùs immorandum. <pb pagenum="195" xlink:href="017/01/211.jpg"/> </s> </p> <p type="main"> <s id="s.001457"><emph type="center"/>CAPUT VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001458"><emph type="center"/><emph type="italics"/>Præ&longs;tet-ne Machinam augere? </s> <s id="s.001459">an componere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001460">EX iis, quæ de Machinarum viribus di&longs;putata &longs;unt &longs;atis <lb/>liquet nullum dari finitum Pondus quod data Potentia mo­<lb/>vere non po&longs;&longs;it &longs;i congruens machina adhibeatur: cum etenim <lb/>data &longs;it Ratio Ponderis ad Potentiam, eo artificio Machina <lb/>di&longs;ponatur, ut Ratione illâ datâ fiat major Ratio motûs Potentiæ <lb/>ad motum Ponderis; & Pondus cedet Potentiæ moventi. </s> <s id="s.001461">Sic <lb/>vici&longs;&longs;im &longs;i oblata fuerit machina, examinandus primùm e&longs;t lo­<lb/>cus, ubi Potentia applicanda e&longs;t, ubi Pondus collocandum; <lb/>tùm utriu&longs;que motûs rationes ineundæ: & pronunciabis majo­<lb/>rem requiri rationem Potentiæ ad Pondus, quàm &longs;it Ratio mo­<lb/>tûs Ponderis ad motum Potentiæ. <!-- KEEP S--></s> <s id="s.001462">Sit enim ex. </s> <s id="s.001463">gr. <!-- REMOVE S-->motuum hu­<lb/>ju&longs;modi Ratio, quæ e&longs;t 3 ad 8; Potentia vim movendi habens <lb/>ut 3 non movebit Pondus, cujus vis re&longs;i&longs;tendi, & momentum, <lb/>&longs;it ut 8; &longs;ed opus e&longs;t, ut illa major &longs;it quàm 3. At neque Po­<lb/>tentiam augere potes, ut oportet, neque Ponderi quicquam de­<lb/>trahere: vide igitur utrum fieri po&longs;&longs;it, ut mutetur in machinâ <lb/>motuum Ratio, aut Potentiæ motum augendo, aut ponderis <lb/>motum minuendo. </s> </p> <p type="main"> <s id="s.001464">Hinc manife&longs;tum e&longs;t machinam majorem non plus afferre <lb/>facilitatis præ minore, &longs;i illæ quidem omninò &longs;imiles fuerint <lb/>(modò utraque &longs;atis &longs;olida &longs;it, ne fractioni &longs;it obnoxia) mo­<lb/>tuum enim Ratio eadem e&longs;t in utráque. </s> <s id="s.001465">Sic Vectis 100 pal­<lb/>morum &longs;i ita ab hypomochlio di&longs;tinguatur in partes ut hinc <lb/>palmos 20, hinc 80 relinquat, non majorem movendi faci­<lb/>litatem præbebit, quàm vectis palmorum quinque ita divi­<lb/>&longs;us ab hypomochlio, ut hinc palmus unus, hinc verò quatuor <lb/>relinquantur. </s> <s id="s.001466">Ut igitur longior ille Vectis utilior accidat, &longs;i <lb/>hypomochlium quidem transferri queat, remove illud à Po­<lb/>tentiâ, & admove Ponderi, motuumque Ratio augebitur; pa­<lb/>tet &longs;cilicet majorem e&longs;&longs;e Rationem 85 ad 15, quam 80 ad 20: <lb/>Quod &longs;i verò hypomochlium ita fixum &longs;it ac vecti adnexum, <pb pagenum="196" xlink:href="017/01/212.jpg"/>ut mutari loco nequeat, ab&longs;cinde palmos (5 15/17), adeò ut hinc &longs;int <lb/>palmi 80 ut priùs, hinc autem &longs;int palmi (14 2/17), & eadem erít <lb/>Ratio, quæ e&longs;t 85 ad 15. Quare breviore vecte plus ponderis <lb/>movebis, quàm longiore; vis enim, quæ longiore illo 100 pal­<lb/>morum movebat pondus librarum 100, breviore hoc palmo­<lb/>rum (94 2/17) movebit libras 141 2/3: Quia quamvis in utroque Vecte <lb/>hypomochlium habente po&longs;t palmum octuage&longs;imum, Potentia <lb/>eodem &longs;emper motu moveatur, non tamen idem e&longs;t ponderis <lb/>motus, qui in minore vecte minor e&longs;t, in majore major, ac <lb/>proinde motûs Potentiæ ad motum Ponderis Ratio major e&longs;t in <lb/>minore, minor in majore vecte. </s> <s id="s.001467">Quod &longs;i demùm nec hypo­<lb/>mochlium transferre, nec vecte mutilato uti liceat, licebit &longs;a­<lb/>nè fu&longs;tem, vel quid &longs;imile, firmiter ad alligatum Vecti adjun­<lb/>gere, potentiamque ab hypomochlio longiùs removere: opor­<lb/>teret autem additamentum huju&longs;modi e&longs;&longs;e palmorum 33 1/3; nam <lb/>ut 15 ad 85, ita 20 ad 113 1/3; adeóque totus vectis e&longs;&longs;et pal­<lb/>morum 133 1/3. </s> </p> <p type="main"> <s id="s.001468">Porrò hîc ob&longs;erva, quantò facilius &longs;it ponderis motum mi­<lb/>nuere, quàm potentiæ motum augere: in allato &longs;iquidem <lb/>exemplo, manente eodem potentiæ motu, minuitur ponderis <lb/>motus decurtato vecte ac diminuto palmis (5 15/17); manente au­<lb/>rem eodem ponderis motu augetur Potentiæ motus acuto vecte <lb/>palmis 33 1/3: Quia nimirum in Ratione majoris Inæqualitatis &longs;i <lb/>Con&longs;equens terminus minor minuatur, aut Antecedens termi­<lb/>nus major augeatur, fit adhuc major Inæqualitas; ut autem <lb/>eadem Ratio &longs;ervetur aucto Antecedente ac diminuto Con&longs;e­<lb/>quente, manife&longs;tum e&longs;t, quæ pars Con&longs;equentis integri e&longs;t <lb/>con&longs;equens diminutus, eam debere e&longs;&longs;e partem Antecedentis <lb/>aucti Antecedentem datum: atqui Antecedens datus e&longs;t major <lb/>dato Con&longs;equente; igitur plus addendum e&longs;t Antecedenti, <lb/>quàm dematur Con&longs;equenti. </s> <s id="s.001469">Sic data &longs;it Ratio 8 ad 6: Con­<lb/>&longs;equens bifariam &longs;ecetur, eju&longs;que &longs;emi&longs;&longs;is fiat novus Con&longs;e­<lb/>quens; erit Ratio 8 ad 3 majoris adhuc inæqualitatis; hæc enim <lb/>e&longs;t dupla &longs;uperbipartiens tertias, illa verò erat &longs;olùm &longs;e&longs;qui­<lb/>tertia. </s> <s id="s.001470">Ut igitur retento priori Con&longs;equente 6 fit eadem Ratio <lb/>dupla &longs;uperbipartiens tertias, &longs;icut Con&longs;equens fuit bifariam <lb/>divi&longs;us, ita datus Antecedens 8 e&longs;t duplicandus, ut &longs;it Ratio <pb pagenum="197" xlink:href="017/01/213.jpg"/>16 ad 6: plus autem e&longs;t totus antecedens major qui additur, <lb/>quàm &longs;it &longs;emi&longs;&longs;is Con&longs;equentis minoris qui demitur. </s> <s id="s.001471">In re au­<lb/>tem no&longs;trâ &longs;emper Ratio motûs Potentiæ per machinam vali­<lb/>dioris factæ ad motum dati ponderis e&longs;t Ratio Majoris inæqua­<lb/>litatis: Quapropter &longs;atius e&longs;t Ponderis motum minuere, quam <lb/>potentiæ motum auctâ machinâ augere. </s> </p> <p type="main"> <s id="s.001472">Hæc quidem, quæ in vecte propo&longs;ita facilè ac in promptu <lb/>e&longs;t per&longs;picere, in cæteris pariter mechanicis Facultatibus, ut <lb/>in Trochleis, Cochleâ, & reliquis intelligenda &longs;unt, ut ex iis, <lb/>quæ inferiùs dicentur, &longs;uo loco manife&longs;tum fiet. </s> <s id="s.001473">Sed quoniam <lb/>ad ponderis motum extenuandum certos quo&longs;dam fines ip&longs;a <lb/>machinarum materia præ&longs;cribit; neque enim quemadmodum <lb/>quantitatem omnem, & corporum molem in &longs;ubtiliores, ac <lb/>&longs;ubindè &longs;ubtiliores partes mente concidimus, ita etiam id re <lb/>ipsâ perficere atque in praxim deducere po&longs;&longs;umus: propterea <lb/>ut plurimum cogimur Potentiæ velociorem motum conciliare, <lb/>ut majorem obtineat Rationem ad motum Ponderis. <!-- KEEP S--></s> <s id="s.001474">Quis ete­<lb/>nim non inca&longs;&longs;um uti po&longs;&longs;it Vecte, cujus hypomochlium à <lb/>pondere &longs;atis gravi non ampliùs di&longs;tet, quàm per digiti &longs;emi&longs;­<lb/>&longs;em? </s> <s id="s.001475">aut Cochleam adhibere, cujus &longs;piras intervallum capilla­<lb/>ceum &longs;ecernat? </s> </p> <p type="main"> <s id="s.001476">Verùm cum id duplici methodo præ&longs;tare po&longs;&longs;imus, videlicet <lb/>aut Machinam ip&longs;am, &longs;pecie non mutatâ, augentes, aut illam <lb/>ex pluribus membris componentes, &longs;ive eju&longs;dem generis &longs;int, <lb/>&longs;ive diver&longs;i; operæ pretium fuerit perpendere, maju&longs;-ne in <lb/>augmento? </s> <s id="s.001477">an verò in compo&longs;itione? </s> <s id="s.001478">compendium inveniatur. <lb/><emph type="italics"/>Augmentum<emph.end type="italics"/> voco (ne ullus &longs;ub&longs;it æquivocandi locus) cum eju&longs;­<lb/>dem Facultatis &longs;pecies immutata permanet, factâ &longs;olum partis <lb/>alicujus acce&longs;&longs;ione; ut &longs;i, quia Vectis ju&longs;to brevior e&longs;t, Poten­<lb/>tiæ ab hypomochlio di&longs;tantiam longiorem facias; cum Tro­<lb/>chleæ adhibeantur oneri movendo impares, amplificatis locu­<lb/>lamentis orbiculorum numerum augeas; quia Cochlea ob &longs;pi­<lb/>rarum raritatem minùs valida e&longs;t quàm oporteat, lineam ip&longs;am <lb/>ita inclines, ut &longs;pi&longs;&longs;ioribus &longs;piris circumducatur. </s> <s id="s.001479">At verò <emph type="italics"/>Com­<lb/>po&longs;ita<emph.end type="italics"/> dicitur Machina, cum invalidæ Facultati membra alia <lb/>adjiciuntur, aut generis eju&longs;dem, ut cum Vectis Vecti, Co­<lb/>chleæ Cochlea, Trochleis Throchleæ adjunguntur; aut diver­<lb/>&longs;i generis, ut cum facultates ip&longs;æ permi&longs;centur, vecti trochleas, <pb pagenum="198" xlink:href="017/01/214.jpg"/>Cochleæ vectem, Trochleis Cochleam, & deinceps, adjun­<lb/>gendo. </s> <s id="s.001480">Prioris Compo&longs;itionis intrà idem genus &longs;pecimen ali­<lb/>quod exhibui in <emph type="italics"/>Terrâ Machinis motâ: Di&longs;&longs;ertat.<emph.end type="italics"/> 1. & inferius <lb/>&longs;uis locis de eâ redibit &longs;ermo: Po&longs;terioris autem Compo&longs;itionis <lb/>diver&longs;arum Facultatum, ubi de &longs;ingulis di&longs;purabimus, exem­<lb/>pla aliqua &longs;ubjiciemus, ut di&longs;cat Tyro Machinarum vires ritè <lb/>ad calculos revocare, &longs;olertiamque machinandi acquirat. </s> </p> <p type="main"> <s id="s.001481">Quamvis autem quæ&longs;tio hæc multò dilucidiùs explicaretur, <lb/>&longs;i unamquamque Facultatem &longs;ingillatim attingeremus, quàm <lb/>&longs;i unâ comprehen&longs;ione omnia complectamur; hîc tamen <lb/>doctrinæ ratio exigit, ut dimi&longs;&longs;is rivulis fontem ip&longs;um aperia­<lb/>mus, ex quo in Machinam Compo&longs;itam vis major, quàm in <lb/>Amplificatam, majore compendio derivatur. </s> <s id="s.001482">Et quidem cum <lb/>res tota ex potentiæ atque Ponderis motuum Ratione pendeat, <lb/>quamdiu in &longs;implici aliquâ facultate con&longs;i&longs;timus, motus Po­<lb/>tentiæ ad motum Ponderis &longs;implicem habet Rationem; &longs;i verò <lb/>Facultas una cum aliâ quâpiam facultate conjungitur, atque <lb/>connectitur, jam Potentiæ motus ad motum ponderis eam ha­<lb/>bet Rationem, quæ ex &longs;ingularum facultatum rationibus com­<lb/>ponitur. </s> <s id="s.001483">Voco autem <emph type="italics"/>&longs;ingularum Facultatum Rationem<emph.end type="italics"/> eam, quæ <lb/>inter ip&longs;os Potentiæ ac Ponderis motus intercederet, &longs;i facul­<lb/>tas illa &longs;olitaria adhiberetur; Atqui Ratio hæc motuum in &longs;in­<lb/>gulis Facultatibus modum recipit ex Facultatis ip&longs;ius partibus, <lb/>quarum altera ad Potentiam, àd Pondus altera &longs;pectare vide­<lb/>tur; ut per &longs;ingulas Facultates eunti con&longs;tabit. </s> <s id="s.001484">In Vecte enim <lb/>Ponderis ab hypomochlio di&longs;tantia pertinet ad Pondus, Poten­<lb/>tiæ autem di&longs;tantia ab eodem hypomochlio penes potentiam <lb/>e&longs;t: In Trochleis ip&longs;arum Trochlearum di&longs;tantia Pondus re&longs;pi­<lb/>cit; funis autem explicatio Potentiam: In Axe in Peritrochio <lb/>cra&longs;&longs;ities Axis Ponderi, Peritrochij amplitudo Potentiæ tribui­<lb/>tur: In Cuneo longitudo ad Potentiam &longs;pectat, cra&longs;&longs;ities ad <lb/>Pondus: In Cochleâ demùm &longs;piræ circumductæ perimeter ad <lb/>Potentiam attinet, extremitatum &longs;piralis lineæ intervallum, ad <lb/>Pondus. <!-- KEEP S--></s> <s id="s.001485">Manife&longs;tum e&longs;t igitur, ubi &longs;implex motuum Ratio in <lb/>&longs;ingulis Facultatibus augenda fuerit, manente eâ parte, quæ <lb/>ad Pondus &longs;pectat, nece&longs;&longs;ariò ita augendam e&longs;&longs;e partem reli­<lb/>quam, quæ Potentiæ tribuitur, ut majori illi motuum Rationi <lb/>re&longs;pondeat. </s> <s id="s.001486">Sic dato Vecte palmorum &longs;ex, quo potentia mo-<pb pagenum="199" xlink:href="017/01/215.jpg"/>veatur in quintuplâ Ratione ad Pondus, &longs;i maneat eadem pon­<lb/>deris ab hypomochlio di&longs;tantia, & motuum Ratio e&longs;&longs;e debeat <lb/>vigecupla, &longs;atis con&longs;tat totum vectem requiri palmorum 21, ut <lb/>unus Ponderi cedat, Potentiæ autem viginti. </s> </p> <p type="main"> <s id="s.001487">At verò &longs;i motuum Ratio ex Rationibus componenda &longs;it, &longs;a­<lb/>tisfuerit datæ Facultati minorem Rationem continenti, quàm <lb/>oporteat, Facultatem aliam adjicere, cujus Ratio cum priori <lb/>Ratione compo&longs;ita quæ&longs;itam Rationem con&longs;tituat. </s> <s id="s.001488">Sic dato <lb/>Vecti quintuplam rationem continenti adjunge aliam quamli­<lb/>bet facultatem quadruplæ Rationis; ex quadruplâ enim Ratio­<lb/>ne & quintuplâ componitur Ratio vigecupla quæ&longs;ita. </s> <s id="s.001489">Ita au­<lb/>tem &longs;ecunda hæc Facultas priori Facultati adnectenda e&longs;t, ut <lb/>quemadmodum duorum Magnetum oppo&longs;iti poli junguntur, <lb/>Au&longs;tralis videlicet unius Aquilonari alterius, &longs;ic duarum Fa­<lb/>cultatum oppo&longs;itæ partes connectantur, ut &longs;cilicet quo loco ad <lb/>priorem Facultatem applicanda e&longs;&longs;et Potentia, eidem admo­<lb/>veatur locus Ponderi in &longs;ecundâ Facultate de&longs;tinatus: proinde <lb/>&longs;iquidem &longs;e res habebit, atque &longs;i pondus diminutum pro Ra­<lb/>tione prioris facultatis, videlicet &longs;ub quintuplum, in &longs;ecun­<lb/>dam hanc Facultatem transferretur, in quâ ejus motus ad mo­<lb/>tum Potentiæ Rationem haberet &longs;ubquadruplam: re enim ve­<lb/>râ duabus hi&longs;ce Facultatibus junctis, Potentiæ motus vigecu­<lb/>plus e&longs;t ad motum Ponderis; nam Pondus in vectis extremita­<lb/>te alterâ con&longs;titutum quintuplo tardiùs movetur, quàm reli­<lb/>qua vectis extremitas; hæc autem po&longs;teriori Facultati loco <lb/>Ponderis adjuncta quadruplo tardiùs movetur quàm Poten­<lb/>tia; igitur Ponderis motus vigecuplo tardior e&longs;t motu Po­<lb/>tentiæ. </s> </p> <p type="main"> <s id="s.001490">Statuamus exempli gratiâ &longs;ecundam hanc Facultatem Vecti <lb/>adjunctam e&longs;&longs;e pariter Vectem eju&longs;dem generis quinque pal­<lb/>morum ita ab hypomochlio di&longs;tinctum in partes, ut hæ in qua­<lb/>druplâ &longs;int Ratione: Ecce quanto compendio rem a&longs;&longs;equamur; <lb/>id enim quod &longs;implici Vecte palmorum 21 præ&longs;tandum e&longs;&longs;et, <lb/>compo&longs;itis vectibus duobus altero palmorum &longs;ex, altero palm. </s> <lb/> <s id="s.001491">quinque perficimus, &longs;ervatâ &longs;emper eâdem Ponderis ab hypo­<lb/>mochlio di&longs;tantiâ, nimirum palmi unius. </s> <s id="s.001492">Hæc tamen de duo­<lb/>bus hi&longs;ce vectibus dicta ita intelliges velim, ut ad motum &longs;im­<lb/>pliciter pertineant; non verò ad motûs quantitatem; &longs;atis enim <pb pagenum="200" xlink:href="017/01/216.jpg"/>&longs;cio non ad eam di&longs;tantiam promoveri po&longs;&longs;e Pondus adhibito <lb/>&longs;ecundo hoc vecte, ad quam promoveretur Vecte palmorum 21: <lb/>Verùm hîc &longs;ola movendi facilitas con&longs;ideratur. </s> <s id="s.001493">Quòd &longs;i non <lb/>alterum Vectem adhibeas; &longs;ed aliud facultatis genus, ut Tro­<lb/>chleas binis orbiculis in&longs;tructas, & Vecti in loco Potentiæ ad­<lb/>nexas, multò adhuc faciliùs movebitur Pondus, cujus motus <lb/>erit &longs;ubvigecuplus motûs Potentiæ funem Trochlearum tra­<lb/>hentis, & tantus erit Ponderis motus, quantus e&longs;&longs;et, &longs;i extre­<lb/>mitati Vectis palmorum &longs;ex apponeretur Potentia quadrupla <lb/>datæ Potentiæ. <!-- KEEP S--></s> <s id="s.001494">Idem planè de cæteris dicendum Faculta­<lb/>tibus. </s> </p> <p type="main"> <s id="s.001495">Hinc manife&longs;tum e&longs;t compo&longs;itis tribus, quatuorve, aut plu­<lb/>ribus Facultatibus, Rationem Compo&longs;itam motus potentiæ ad <lb/>motum Ponderis fieri multò majorem; cui &longs;i æqualem Ratio­<lb/>nem habere velimus unicâ atque &longs;implici Facultate, hujus <lb/>magnitudinem aliquando enormem fieri nece&longs;&longs;e e&longs;&longs;et; ut &longs;uis <lb/>locis infrà declarabitur. </s> </p> <p type="main"> <s id="s.001496">In eo igitur elucebit Machinatoris indu&longs;tria, &longs;i Facultates <lb/>ip&longs;as aptè congruenterque di&longs;ponat, atque permi&longs;ceat, &longs;pecta­<lb/>tâ materiæ &longs;oliditate, &longs;patij amplitudine, Ponderis po&longs;itione, <lb/>Potentiæ virtute, temporis ad movendum conce&longs;&longs;i opportuni­<lb/>tate: hæc enim omnia attenti&longs;&longs;imè perpendenda &longs;unt; ne, dum <lb/>nimis &longs;ollicitè laborem imminuere &longs;tudet, motum plus æquo <lb/>imminuens, tardioremque efficiens temporis jacturam faciat, <lb/>aut totum &longs;patium machina implens in eas angu&longs;tias Potentiam <lb/>moventem conjiciat, ut motum expeditè perficere nequeat. <lb/></s> </p> <p type="main"> <s id="s.001497"><emph type="center"/>CAPUT VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001498"><emph type="center"/><emph type="italics"/>Cur majores Rotæ motum juvent præ minoribus.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001499">ONera &longs;i ex alio in alium locum deportanda fuerint, gemi­<lb/>no labore opus e&longs;t, conatu videlicet, quo &longs;u&longs;tineantur, <lb/>& impetu, quo transferantur: proptereà &longs;atius e&longs;t ita res di&longs;po­<lb/>nere, ut vires omnes ad transferendum exerceantur, citrà co­<lb/>natum &longs;u&longs;tinendi; ut eâ ratione vel gravius onus vel idem mul-<pb pagenum="201" xlink:href="017/01/217.jpg"/>tò faciliùs à potentia moveatur, quàm &longs;i ea illud &longs;u&longs;tinere <lb/>pariter atque transferre cogeretur. </s> <s id="s.001500">Quoniam verò (cum one­<lb/>ra &longs;ubjecto plano impo&longs;ita illud premant, atque tùm onerum <lb/>tùm &longs;ubjecti plani facies, quæ &longs;e invicem contingunt, non ita <lb/>læves &longs;int, ut partes omnes in rectum directæ nihil habeant <lb/>a&longs;peritatis; quin immò ut plurimum, & &longs;alebris impedita via <lb/>&longs;it, & movendi corporis partes aliæ præ aliis extent atque emi­<lb/>neant) ex mutuo prominentium particularum tritu atque con­<lb/>flictu difficultas ad movendum oriretur; idcircò optimo con&longs;i­<lb/>lio factum e&longs;t, ut oneribus ip&longs;is &longs;ubjiciantur Cylindri aut Rotæ, <lb/>quæ dum in gyrum aguntur, conflictum illum partium tollunt, <lb/>qui vitari non po&longs;&longs;et, &longs;i onera &longs;uper plano raptarentur. </s> <s id="s.001501">Hinc Ci­<lb/>&longs;ia, Sarraca, Vehes, Carri & genus omne plau&longs;trorum. </s> <s id="s.001502">Id quod <lb/>etiam homines ip&longs;i, ut terre&longs;tre iter commodiùs habeant, & <lb/>minori jumentorum labore illud perficiant, quàm &longs;i iis in&longs;i­<lb/>dentes veherentur, &longs;uos in u&longs;us retulerunt: Hinc Belgæ &longs;ua <lb/>e&longs;&longs;eda, Galli petorita & rhedas, Hi&longs;pani pilenta, Itali carpen­<lb/>ta; & pro &longs;uâ qui&longs;que voluntate diver&longs;a vehiculorum genera <lb/>excogitârunt, quæ &longs;ubjectis rotis aguntur: dum enim Rota <lb/>convertitur, eju&longs;que curvaturæ partes aliis atque &longs;ubinde aliis <lb/>&longs;ubjectæ planitiei partibus aptantur, adeóque currus promove­<lb/>tur, &longs;olus rotæ modiolus axis ambitum axungiâ lubricum terit; <lb/>ex quo tritu aut nulla aut levis mora motui infertur. </s> </p> <p type="main"> <s id="s.001503">Illud autem e&longs;t omnibus explorati&longs;&longs;imum, & quotidiano ex­<lb/>perimento confirmatum, quo majoribus rotis in&longs;tructi currus <lb/>(ni&longs;i di&longs;crimen aliquod in cæteris intercedat) multò faciliùs <lb/>trahuntur, pa&longs;&longs;imque ob&longs;ervatur Romæ in vulgaribus illis vehi­<lb/>culis (ab antiquis Ci&longs;iis aut parum aut nihil di&longs;tant) quæ cum <lb/>ex celeberrimi Architecti Bonarotæ præ&longs;cripto duas ingentes <lb/>rotas habeant, tantis ponderibus onu&longs;ta cernuntur, ut miracu­<lb/>lo proximum videatur ab unico equo tam ingentia onera trahi <lb/>po&longs;&longs;e: id quod alibi neutiquam fieri pote&longs;t, ubi minoribus Rotis <lb/>vehicula huju&longs;modi in&longs;tructa longè minoribus oneribus defe­<lb/>rendis paria &longs;unt, &longs;i unicus equus adhibeatur. </s> </p> <p type="main"> <s id="s.001504">Hujus rei cau&longs;am indaganti acquie&longs;cendum non e&longs;t iis, qui <lb/>illam ex rationibus Vectis petendam e&longs;&longs;e exi&longs;timant, perinde <lb/>atque &longs;i rotæ majoris &longs;emidiameter e&longs;&longs;et longior Vectis, mino­<lb/>ris verò brevior; ac proptereà majore rotâ faciliùs moveretur <pb pagenum="202" xlink:href="017/01/218.jpg"/>vehiculum onu&longs;tum, quàm minore, quia & longiore vecte fa­<lb/>ciliùs pondera moventur, quàm breviore. </s> <s id="s.001505">Hoc, inquam, 1/4 <lb/>veritate abe&longs;&longs;e palam fiet, &longs;i animadvertamus potentiam tra­<lb/>hentem medio temone applicatam e&longs;&longs;e axi, cui pariter axi in­<lb/>nititur onus; atque adeò tùm onus tùm Potentiam concipi <lb/>qua&longs;i in Rotæ centro, cujus &longs;emidiametri altera extremitas hy­<lb/>pomochlij punctum de&longs;ignaret. </s> <s id="s.001506">Atqui Vectis, in quo Potentia <lb/>& onus ab hypomochlio eandem aut æqualem di&longs;tantiam ha­<lb/>bent, parùm aut nihil habet utilitatis: immò in Vecte, quâ <lb/>vectis e&longs;t, tria puncta diver&longs;a tribuenda &longs;unt Potentiæ, oneri, <lb/>& Hypomochlio, ut infrà, ubi de Vecte di&longs;putabitur: in Rotâ <lb/>autem duo tantummodo puncta con&longs;iderantur, &longs;cilicet cen­<lb/>trum & &longs;emidiametri extremitas. </s> <s id="s.001507">Igitur in Rotâ ratio Vectis <lb/>non invenitur, ideóque neque major Rota accipienda e&longs;t qua­<lb/>&longs;i longior Vectis. <!-- KEEP S--></s> <s id="s.001508">Aliundè itaque petendam e&longs;&longs;e cau&longs;am, cur <lb/>majores rotæ præ minoribus motum juvent, manife&longs;tum e&longs;t. </s> </p> <p type="main"> <s id="s.001509">Et primùm quidem, quod ad moram illam attinet, quæ ex <lb/>modioli Rotæ atque axis tritu oritur, eam minorem e&longs;&longs;e in ma­<lb/>joribus Rotis, &longs;atis con&longs;tàt, &longs;i attendamus axis cra&longs;&longs;itiem, non <lb/>Rotæ magnitudini re&longs;pondere, &longs;ed oneris gravitati, quam opus <lb/>e&longs;t &longs;u&longs;tinere; quapropter axi &longs;atis valido pro ratione ponderis <lb/>&longs;u&longs;tinendi parùm refert, utrùm Rota, cujus radij bipalmares <lb/>&longs;int, an verò tripalmares, infigatur: manente igitur codem axe <lb/>aut major, aut minor Rota vehiculo &longs;ubjici pote&longs;t. </s> <s id="s.001510">Sed quo­<lb/>niam Rota major, cujus diameter &longs;e&longs;quialtera e&longs;t minoris, dum <lb/>conver&longs;ionem unam perficit, &longs;patium quoque &longs;e&longs;quialterum <lb/>decurrit, eumdem tamen axem, quem minor Rota, terit, hinc <lb/>fit, per 8. lib. 5. eumdem axis ambitum ad majoris Rotæ peri­<lb/>metrum (hoc e&longs;t ad ejus motum) minorem habere rationem <lb/>quàm ad perimetrum minoris Rotæ (hoc e&longs;t ad minorem mo­<lb/>tum) atque adeò tritus ille modioli, & axis minùs impedit ma­<lb/>jorem motum quàm minorem. </s> </p> <p type="main"> <s id="s.001511">Deinde, ut cap.16. lib.1. &longs;ubindicatum e&longs;t &longs;uperiùs, majo­<lb/>res rotæ efficiunt, ut axis magis à terrâ di&longs;tet; ac proinde te­<lb/>mo, cui alligatus e&longs;t equus, vel &longs;ubjecto plano parallelus e&longs;t, <lb/>vel minimùm à paralleli&longs;mo recedit: ex quo fit tractionem aut <lb/>parallelam e&longs;&longs;e, aut &longs;altem minùs obliquam, quam &longs;i Rota mi­<lb/>nor e&longs;&longs;et, & axis depre&longs;&longs;ior: quò autem minor e&longs;t tractionis <pb pagenum="203" xlink:href="017/01/219.jpg"/>obliquitas, minorem quoque e&longs;&longs;e trahendi difficultatem loco <lb/>citato explicatum e&longs;t. </s> </p> <p type="main"> <s id="s.001512">Ad hæc viarum a&longs;peritatem impedimento e&longs;&longs;e nemo ne&longs;cit; <lb/>offendicula autem, in quæ vehiculorum Rotæ incurrunt, ma­<lb/>gis ob&longs;i&longs;tere minori Rotæ, quàm majori, facilè o&longs;tenditur; hîc <lb/>enim pariter (id quod de magnitudinibus demon&longs;trat Eucli­<lb/>des lib. 5. prop. 8.) idem majorem habet Rationem ad minus, <lb/>quàm ad majus. </s> <s id="s.001513">Nam &longs;i <lb/><figure id="id.017.01.219.1.jpg" xlink:href="017/01/219/1.jpg"/><lb/>Rotæ minoris &longs;emidiame­<lb/>ter CB fuerit, majoris au­<lb/>tem CD, & in planis pa­<lb/>rallelis BA, DE volvantur, <lb/>ut impedimentum &longs;imile &longs;i­<lb/>militerque po&longs;itum inve­<lb/>nient, multò majus e&longs;&longs;e <lb/>oportet illud, quod majori <lb/>Rotæ objicitur, quàm quod <lb/>minori. </s> <s id="s.001514">Sit enim minoris <lb/>offendiculum GI; ducatur <lb/>ex centro per I recta, quæ <lb/>&longs;it CIE &longs;ecans majoris Rotæ peripheriam in H: erit igitur ar­<lb/>cus IB &longs;imilis arcui HD, & ille quidem minor, hic verò ma­<lb/>jor, ut manife&longs;tum e&longs;t. </s> <s id="s.001515">Ducatur in planum perpendicularis <lb/>HF, & hoc erit impedimentum majoris Rotæ &longs;imile impedi­<lb/>mento minoris IG, nam &longs;imilem arcum à conver&longs;ione circà <lb/>centrum cum plani contactu impedit; nece&longs;&longs;e quippe e&longs;t Ro­<lb/>tam majorem converti circà punctum H, &longs;icut & minorem cir­<lb/>cà punctum I, ut tran&longs;grediantur ob&longs;i&longs;tens offendiculum. </s> <lb/> <s id="s.001516">Porrò lineam HF majorem e&longs;&longs;e quàm IG &longs;ic o&longs;tenditur. </s> <s id="s.001517">Quo­<lb/>niam AB & ED parallelæ &longs;unt, triangula CBA, & CDE <lb/>&longs;imilia &longs;unt: ergo per 4. lib.6. ut CB ad CD, hoc e&longs;t ut CI <lb/>ad CH, ita CA ad CE; & permutando ut CI ad CA, ita <lb/>CH ad CE; & dividendo ut CI ad IA, ita CH ad HE: at <lb/>CI minor e&longs;t quàm CH; igitur per 14. lib.5. etiam IA minor <lb/>e&longs;t quàm HE. <!-- KEEP S--></s> <s id="s.001518">Item quia AB & ED ex hypothe&longs;i parallelæ <lb/>&longs;unt, recta IE in illas incidens facit angulos IAG & HEF <lb/>æquales per 29. lib. 1. &longs;unt autem triangula IGA & HFE <lb/>rectangula ad G & F ex con&longs;tructione; &longs;unt igitur &longs;imilia, & <pb pagenum="204" xlink:href="017/01/220.jpg"/>per 4. lib. 6. ut. </s> <s id="s.001519">IA ad IG, ita HE ad HF: quare cum ex <lb/>dictis IA minor &longs;it quàm HE, erit per 14.lib.5. etiam IG mi­<lb/>nor quàm HF. </s> </p> <p type="main"> <s id="s.001520">Cum itaque HF major &longs;it quàm IG (a&longs;&longs;umptâ DM æqua­<lb/>li ip&longs;i IG, & ductâ perpendiculari MS, donec occurrat peri­<lb/>phæriæ in S) inter Tangentem ED & arcum circuli &longs;tatuatur <lb/>perpendicularis SL æqualis ip&longs;i IG; & ex centro C ducatur <lb/>per S recta CO. <!-- KEEP S--></s> <s id="s.001521">In triangulo igitur CEO angulus internus <lb/>E, per 16. lib. 1; minor e&longs;t externo SOL; igitur etiam angu­<lb/>lus SOL major e&longs;t quàm IAG: adde utrique angulum <lb/>rectum, ergo duo SLO, SOL &longs;imul majores &longs;unt duobus <lb/>IGA, IAG &longs;imul; ac propterea etiam externus LSC major <lb/>e&longs;t externo GIC per 32.lib.1. Quapropter &longs;emidiameter CS <lb/>obliquior incidit in offendiculum SL, quàm &longs;emidiameter CI <lb/>incidat in æquale offendiculum IG: minùs igitur impeditur <lb/>Rotæ majoris conver&longs;io, quàm minoris, quippe cui minus di­<lb/>rectè opponatur æquale offendiculum. </s> </p> <p type="main"> <s id="s.001522">Præterea cum trahendi difficultas hinc oriatur, quòd Rota <lb/>incurrens in ob&longs;tantem lapidem, aut quid &longs;imile, jam non cir­<lb/>cà &longs;uum centrum convoluta aptatur &longs;ubjecto plano, &longs;ed, dum <lb/>Rota adhæret atque in&longs;i&longs;tit offendiculo; nece&longs;&longs;e e&longs;t plau&longs;trum <lb/>cum impo&longs;ito onere elevari pro objecti impedimenti altitudi­<lb/>ne; faciliùs ab eâdem Potentia elevatur plau&longs;trum onu&longs;tum, &longs;i <lb/>major fuerit Rota, quàm &longs;i minor, quia videlicet motus Poten­<lb/>tiæ ad eandem elevationem majorem habet Rationem in Ro­<lb/>tâ majore quàm in minore, cum illâ enim plus movetur, <lb/>quàm cum i&longs;tâ. </s> <s id="s.001523">Sit majoris Rotæ impedimentum LS pla­<lb/>nè æquale impedimento GI minoris; producatur perpendicu­<lb/>laris LS in T, & perpendicularis GI in V: tùm intervallo SC <lb/>de&longs;cribatur arcus CT, & intervallo IC de&longs;cribatur arcus CV. <!-- KEEP S--></s> <lb/> <s id="s.001524">Certum e&longs;t in motu Rotæ majoris propter obicem LS manente <lb/>puncto S transferri centrum C in T, ita ut ST &longs;it Rotæ &longs;emi­<lb/>diameter æqualis &longs;emidiametro CD, & &longs;imiliter in motu Rotæ <lb/>minoris propter offendiculum GI manente puncto I transferri <lb/>centrum C in V, ita ut IV æqualis &longs;it &longs;emidiametro CB. <!-- KEEP S--></s> <s id="s.001525">Quo­<lb/>niam verò CD, VG, TL ad angulos rectos &longs;ubjecto plano in­<lb/>&longs;i&longs;tunt, & parallelæ &longs;unt, anguli alterni VIC, ICB æquales <lb/>&longs;unt per 29. lib.1, eorumque men&longs;uræ, arcus videlicet VC & <pb pagenum="205" xlink:href="017/01/221.jpg"/>IB, æquales &longs;unt; & ob eandem Rationem anguli alterni <lb/>TSC, SCD, eorumque men&longs;uræ arcus TC & SD, &longs;unt <lb/>æquales. </s> <s id="s.001526">Atqui arcus SD major e&longs;t quàm IB; igitur & arcus <lb/>TC major e&longs;t quàm VC; hi autem arcus TC & VC re&longs;pon­<lb/>dent motui Potentiæ trahentis: longiore igitur ac majore mo­<lb/>tu Potentiæ fit eadem elevatio, ac proinde faciliùs in Rotâ ma­<lb/>jore quàm in minore. </s> <s id="s.001527">Porrò arcum SD majorem e&longs;&longs;e arcu IB, <lb/>magi&longs;que di&longs;tare punctum S à puncto D, quàm punctum I à <lb/>puncto B, illicò manife&longs;tum fiet, &longs;i duos circulos datis duobus, <lb/>æquales de&longs;crip&longs;eris &longs;e intùs contingentes, & ad contactüs <lb/>punctum lineam Tangentem duxeris, quocumque enim po&longs;ito <lb/>minoris circuli offendiculo inter Tangentem, & circulum mi­<lb/>norem interjecto, illud idem offendiculum longiùs à con­<lb/>tactûs puncto removendum videbis, ut inter Tangentem <lb/>eandem, & circulum majorem interjici po&longs;&longs;it: Id quod adeò <lb/>manife&longs;tum e&longs;t, ut non &longs;it in eo explicando diutiùs immo­<lb/>randum. </s> </p> <p type="main"> <s id="s.001528">Quòd &longs;i ad calculos rem hanc curiosiùs revocare libeat, &longs;ic <lb/>ex gr. <!-- REMOVE S-->Rotæ minoris &longs;emidiameter CA pedum duorum, &longs;cilicet <lb/>digitorum 32, offendiculi verò DE <lb/><figure id="id.017.01.221.1.jpg" xlink:href="017/01/221/1.jpg"/><lb/>altitudo digitorum 4. Cum igitur <lb/>FD & CA parallelæ &longs;int, &longs;icut & <lb/>FC ac DA per 34. lib. 1. FD & <lb/>CA æquales &longs;unt, remanetque EF <lb/>digit.28, & e&longs;t Sinus anguli FCE, <lb/>quo cognito innote&longs;cit complemen­<lb/>tum, arcus &longs;cilicet quæ&longs;itus EA. </s> <lb/> <s id="s.001529">Fiat itaque ut CE ad EF, hoc e&longs;t <lb/>ut 32 ad 28, &longs;eu ut 8 ad 7, ita <lb/>100000. Radius ad 87500 Sinum <lb/>arcûs gr.61. 2′ 42″; erit enim quæ&longs;i­<lb/>tus arcus EA gr. <!-- REMOVE S-->28. 57′ 18″. <!-- KEEP S--></s> <s id="s.001530">Jam verò po&longs;itâ &longs;emidiametro <lb/>CA digitorum 32, fiat ut 113 ad 355, ita data &longs;emidiameter <lb/>digit. </s> <s id="s.001531">32 ad &longs;emiperipheriam circuli digitorum ferè 100 1/2, &longs;ci­<lb/>licet 100. 53″: ergo arcus EA e&longs;t proximè digitorum 16. </s> </p> <p type="main"> <s id="s.001532">At Rotæ majoris &longs;emidiameter BA &longs;it &longs;e&longs;quialtera (quic­<lb/>quid &longs;it quòd figura &longs;olùm exprimat &longs;e&longs;quiquartam) pedum <lb/>&longs;cilicet trium, hoc e&longs;t digitorum 48, & offendiculum GH <pb pagenum="206" xlink:href="017/01/222.jpg"/>pariter digit. </s> <s id="s.001533">4. Quare HI e&longs;t digit. </s> <s id="s.001534">44 Sinus anguli IBH, <lb/>ex quo innote&longs;cet arcus complementi HA. <!-- KEEP S--></s> <s id="s.001535">Fiat ut BH 48 ad <lb/>HI 44, &longs;eu ut 12 ad 11, ita Radius 100000 ad 91666 Sinum <lb/>arcûs gr. <!-- REMOVE S-->66. 26′. </s> <s id="s.001536">33″; & e&longs;t quæ&longs;itus arcus HA gr.23.33′.27″. <!-- KEEP S--></s> <lb/> <s id="s.001537">Jam &longs;it ut 113 ad 355, ita &longs;emidiameter 48 ad &longs;emiperiphe­<lb/>riam digitorum 150 4/5 ferè: igitur arcus HA e&longs;t proximè di­<lb/>gitorum 20. Cum itaque dum onus elevatur ut 4, Potentia <lb/>in minore Rotâ moveatur ut 16, in majore autem ut 20 <lb/>(ut paulò &longs;uperiùs o&longs;ten&longs;um e&longs;t motum centri æqualem e&longs;&longs;e <lb/>arcubus EA, & HA) facilitas movendi, quæ hinc oritur, erit <lb/>ut 5 ad 4. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001538">Ex his manife&longs;tum e&longs;t, in vehiculis, quæ quatuor rotis <lb/>in&longs;truuntur, quarum binæ, priores minores &longs;unt, po&longs;teriores <lb/>verò majores, faciliùs &longs;uperari impedimenta à po&longs;terioribus <lb/>rotis quàm à prioribus, ac propterea minori labore currum ab <lb/>equis trahi, quàm &longs;i po&longs;teriores prioribus e&longs;&longs;ent æquales. </s> <s id="s.001539">Id <lb/>quod opportunè factum e&longs;t, quia ut plurimum (quemadmo­<lb/>dum in antiquioribus Rhedis viatoriis cernere e&longs;t) in po&longs;te, <lb/>riorem potiùs, quàm in anteriorem currus partem, onus reji­<lb/>citur, atque adeò po&longs;terior axis magis premitur: quæren­<lb/>dum igitur fuit aliquod laboris compendium. </s> <s id="s.001540">Quamquam <lb/>non negarim alio pror&longs;us con&longs;ilio primùm excogitatam hanc <lb/>Rotarum inæqualitatem; ut nimirum onus con&longs;titutum qua&longs;i <lb/>in plano trahentem versùs inclinato, faciliùs quoque illum <lb/>ex impre&longs;&longs;o anterioris tractionis impetu &longs;equeretur, &longs;i in pla­<lb/>nitie quidem tractio fieret; ubi verò &longs;uperandus e&longs;&longs;et clivus, <lb/>ut minùs adversùs trahentem repugnaret onus &longs;e ip&longs;um in <lb/>proclive urgendo; nam &longs;i Rotæ æquales e&longs;&longs;ent, longè faciliùs <lb/>vehiculum in po&longs;teriora relaberetur, pro ip&longs;ius clivi inclina­<lb/>tione, cui parallelum e&longs;&longs;et planum oneri &longs;ubjectum in&longs;i&longs;tens <lb/>axibus æqualium Rotarum: at Rotis inæqualibus po&longs;itis, & <lb/>po&longs;terioribus quidem majoribus, planum, cui onus incumbe­<lb/>re intelligitur à po&longs;teriori axe ad anteriorem deductum minùs <lb/>inclinatur, quàm collis proclivitas ferat; ac propterea trahen­<lb/>tibus equis minùs repugnat. </s> <s id="s.001541">Licèt autem non &longs;emper a&longs;cen­<lb/>dendum &longs;it in colles & clivos, quorum a&longs;cen&longs;us manife&longs;tè ar­<lb/>duus e&longs;t atque difficilis, rarò tamen, aut ferè nunquam, adeò <lb/>æquata e&longs;t viarum planities, quin leviter &longs;altem inflexæ modò <pb pagenum="207" xlink:href="017/01/223.jpg"/>a&longs;cendere cogant, modò de&longs;cendere: in quâ a&longs;cen&longs;uum atque <lb/>de&longs;cen&longs;uum vici&longs;&longs;itudine non modicè utilis e&longs;t illa Rotarum <lb/>inæqualitas. </s> </p> <p type="main"> <s id="s.001542">Hinc manualia illa curricula (&longs;eu ru&longs;ticæ vehes) quæ binis <lb/>brachiis in&longs;tructa unicam habent in anteriore parte rotam & <lb/>&longs;ublevatis brachiis conver&longs;a Rotâ promoventur, faciliùs <lb/>con&longs;trui po&longs;&longs;ent, &longs;i propè vectorem duæ e&longs;&longs;ent Rotæ majores <lb/>illâ anteriore Rotâ, ita ut harum diameter triplex e&longs;&longs;et diame­<lb/>tri illius: hunc enim unicus homo multò majus pondus trans­<lb/>ferre pote&longs;t vel impellendo, cùm in planitie e&longs;t, aut clivum <lb/>a&longs;cendit, vel trahendo, cùm ex declivi de&longs;cendit; levatur &longs;i­<lb/>quidem labore &longs;u&longs;tinendi, & omnes vires exercet impellendo <lb/>aut trahendo; & illa Rotarum inæqualitas in causâ e&longs;t, cur fa­<lb/>ciliùs impellatur pondus versùs illam partem, in quam incli­<lb/>natur. </s> </p> <p type="main"> <s id="s.001543">Et quoniam in Rotarum inæqualium mentionem incidi, il­<lb/>lud hîc pariter ob&longs;ervandum videtur, commodiùs currum mo­<lb/>veri, cùm anteriores Rotæ à po&longs;terioribus aliquantulùm di&longs;tant, <lb/>quàm cùm valdè vicinæ &longs;unt (ubi tamen reliqua omnia paria <lb/>fuerint, neque aliud præter Rotarum di&longs;tantiam, intercedat <lb/>di&longs;crimen) &longs;i in planitie quidem, & viâ minimum flexuosâ de­<lb/>ducendus &longs;it. </s> <s id="s.001544">Quia nimirum quo propiores fuerint axes, pla­<lb/>num, cui onus incumbit, magis inclinatur, ac propterea an­<lb/>teriores Rotas premens adversùs &longs;ubjectam tellurem minus <lb/>obliquè conatur, ideóque pondus illam validiùs urgens majo­<lb/>rem creat movendi difficultatem: contrà verò &longs;i axes invicem <lb/>paulò remotiores fuerint, minùs inclinato plano, minor e&longs;t <lb/>priorum rotarum pre&longs;&longs;us in &longs;ubjectam tellurem. </s> <s id="s.001545">Sic &longs;i Rotæ <lb/>fuerinc A & B, pla­<lb/><figure id="id.017.01.223.1.jpg" xlink:href="017/01/223/1.jpg"/><lb/>num, cui onus in&longs;i­<lb/>det, e&longs;t AB, at &longs;i Ro­<lb/>tæ fuerint A & C, <lb/>planum e&longs;t AC, quod <lb/>utique minùs incli­<lb/>natum e&longs;t, magi&longs;que <lb/>accedit ad paralleli&longs;mum cum Horizonte DE, atque adeò <lb/>Rota B magis terram premit, quàm Rota C. <!-- KEEP S--></s> <s id="s.001546">Si enim in utro­<lb/>que plano pondus fuerit &longs;imiliter po&longs;itum (puta circà me-<pb pagenum="208" xlink:href="017/01/224.jpg"/>dium) linea directionis à centro gravitatis ponderis ducta ca­<lb/>det ad angulos magis inæquales in planum AB magis inclina­<lb/>tum, quàm in AC minùs inclinatum, atque momentum gra­<lb/>vitatis ponderis magis accedet ad B quàm ad C, ut infrà &longs;uo <lb/>loco explicabitur, & &longs;ubindicatum e&longs;t &longs;uperiùs lib.1. cap. 14. <lb/>§. <emph type="italics"/>Ex his fieri pote&longs;t.<emph.end type="italics"/></s> <s id="s.001547"> Hinc Hamburgen&longs;ia plau&longs;tra, quibus <lb/>merces Hamburgo Norimbergam devehuntur, longiora &longs;unt, <lb/>quia nec altiores clivi in itinere frequentes occurrunt, nec <lb/>angu&longs;tæ &longs;unt viarum flexiones, ex quibus oriatur aut a&longs;cen­<lb/>dendi, aut plau&longs;trum inflectendi difficultas. </s> <s id="s.001548">Quare illis & <lb/>majora onera imponi po&longs;&longs;unt, & &longs;ex equi non bini & bini, &longs;ed <lb/>&longs;inguli recto ordine adjunguntur; quo fit ut non in diver&longs;a <lb/>trahentes, omninò &longs;imili impetu currum deducant. </s> <s id="s.001549">Quòd &longs;i <lb/>viæ plus haberent difficultatis tùm ex clivis, tùm ex flexioni­<lb/>bus, non expediret tàm longa plau&longs;tra con&longs;truere, nec equos <lb/>tam longâ &longs;erie di&longs;ponere, ut cuique rem vel leviter con&longs;ide­<lb/>ranti &longs;tatim patebit. <lb/> </s> </p> <p type="main"> <s id="s.001550"><emph type="center"/>CAPUT IX.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001551"><emph type="center"/><emph type="italics"/>Quid Cylindri & Scytalæ ad faciliorem ponderis <lb/>motum præ&longs;tent.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001552">ADeò ingentia aliquando pondera transferenda proponun­<lb/>tur, ut ea carris imponere tran&longs;vehenda aut nimis opero­<lb/>&longs;um &longs;it, aut periculo non vacet, ne rotarum axes pondere præ­<lb/>gravati diffringantur, aut propter &longs;oli mollitudinem rotæ de­<lb/>vorentur: propterea rationem aliquam inire oportet, quâ voti <lb/>compotes &longs;imus, citrà huju&longs;modi pericula. </s> <s id="s.001553">Et quidem &longs;i cor­<lb/>pus teres &longs;it, nec viarum &longs;alebræ, aut angu&longs;tiæ impedimento <lb/>&longs;int, ip&longs;um ver&longs;ari in gyrum poterit &longs;imili artificio, quo ad <lb/>deportandos Ephe&longs;um ex lapicidinis &longs;capos columnarum cen­<lb/>tum viginti &longs;eptem altitudine pedum &longs;exaginta u&longs;us e&longs;t Cte&longs;i­<lb/>phon Gno&longs;&longs;ius (&longs;ic eum vocat Plinius lib. 7. cap. 37. cum Vi­<lb/>truvio lib.10. cap. 6, quem tamen idem Plinius lib.36. cap.14. <pb pagenum="209" xlink:href="017/01/225.jpg"/>cum Strabone vocat Cher&longs;iphronem) celeberrimo Dianæ <lb/>templo con&longs;truendo præfectus, & quidem felici eventu: ca­<lb/>pitibus enim &longs;caporum, ubi axis extremitates de&longs;inebant, &longs;ub­<lb/>&longs;cudis in modum in&longs;eruit, atque implumbavit ferreos axes: <lb/>tùm de materiâ trientali &longs;capos (hoc e&longs;t ligneos tigillos cra&longs;&longs;i­<lb/>tudinis unciarum quatuor pedis, &longs;eu pollicum quatuor) duos <lb/>longiores juxtà columnæ longitudinem, duo&longs;que breviores <lb/>tran&longs;ver&longs;arios ita compegit, ut parallelogrammum con&longs;tituen­<lb/>tes columnam po&longs;&longs;ent complecti; medii&longs;que tran&longs;ver&longs;ariis <lb/>ferreas armillas in&longs;eruit, quibus axes ferrei infigebantur, <lb/>adeò ut liberè ver&longs;ari po&longs;&longs;ent, cum boves traherent; quem­<lb/>admodum & in gyrum volvuntur cylindri marmorei aut la­<lb/>pidei, quorum u&longs;us e&longs;t in exæquandis ambulationibus. </s> <s id="s.001554">E&longs;t <lb/>autem maximè veri&longs;imile, & probabile, ita firmiter <lb/>ligneum illud parallelogrammum fui&longs;&longs;e compactum, ut non <lb/>&longs;olùm extremis tran&longs;ver&longs;ariorum capitibus anterioribus alli­<lb/>gari po&longs;&longs;ent boves; &longs;ed etiam per totam anterioris &longs;capi lon­<lb/>gitudinem di&longs;tribui, ut faciliùs columna transferretur. </s> </p> <p type="main"> <s id="s.001555">Pro&longs;perum exitum con&longs;ecuta &longs;caporum vectura animum <lb/>adjecit Methageni Cte&longs;iphontis filio, ut paternam in­<lb/>du&longs;triam æmularetur in Epi&longs;tyliis vehendis: cum enim ho­<lb/>rum figura non ea e&longs;&longs;et, quæ perinde atque cylindrica vol­<lb/>vi po&longs;&longs;et, duabus rotis pedum circiter duodenûm &longs;ingula <lb/>epi&longs;tylia firmiter inclu&longs;it; rotarumque centris ferreos axes <lb/>infixit, qui in armillis &longs;imilem haberent ver&longs;ationem, ac <lb/>dictum e&longs;t in &longs;caporum vecturâ. </s> <s id="s.001556">Cum enim boves ligneo <lb/>parallelogrammo alligati traherent, Rotæ volvebantur, at­<lb/>que cum illis pariter epi&longs;tylia Rotis cohærentia in gyrum <lb/>ver&longs;abantur; quippe quæ in &longs;ubjectum &longs;olum non incurre­<lb/>bant, cum &longs;olæ Rotæ terram attingerent. </s> <s id="s.001557">Hâc methodo <lb/>corporibus, quæ non &longs;unt ad volubilitatem rotundata, faci­<lb/>lem conyer&longs;ionem conciliare po&longs;&longs;umus; ex Rotis nimirum & <lb/>pondere moles una compingitur, cujus extremitatibus cylin­<lb/>dricis tota innititur, nihilque refert, cujus demum figuræ &longs;it <lb/>pars media, &longs;cilicet pondus, modò hæc à &longs;olo aliquantulum <lb/>di&longs;tans motum non impediat. </s> <s id="s.001558">Quâ autem ratione aut Rotæ <lb/>con&longs;truantur, aut illis onus includatur, artificis &longs;eu architecti <lb/>&longs;olertiæ relinquitur. </s> </p> <pb pagenum="210" xlink:href="017/01/226.jpg"/> <p type="main"> <s id="s.001559">Methagenis artificium imitatus Paconius, te&longs;te Vitruvio <lb/>lib. 10. cap. 6. lapideam ba&longs;im longam pedes duodecim, la­<lb/>tam pedes octo, & altam pedes &longs;ex Apollinis colo&longs;&longs;o re&longs;ti­<lb/>tuendam, duabus Rotis pedum circiter quindecim, &longs;imili­<lb/>ter inclu&longs;it: &longs;ed aliâ ratione ac Methagenes deducere &longs;tatuit. </s> <lb/> <s id="s.001560">A Rotâ ad Rotam circâ lapidem fu&longs;os &longs;extantales, hoc e&longs;t <lb/>cra&longs;&longs;itudinis pollicum duorum, ad circinum compegit ita, ut <lb/>fu&longs;us à fu&longs;o non di&longs;taret pedem unum. </s> <s id="s.001561">Tùm circà fu&longs;os fu­<lb/>nem involvit, qui bobus trahentibus explicabatur, & con­<lb/>vertebantur Rotæ. </s> <s id="s.001562">Verùm quia funis circumvoluti &longs;piræ ad <lb/>unam, aut ad alteram partem &longs;pectabant, non poterat viâ <lb/>rectâ ad lineam deduci moles illa; &longs;ed modò in hanc, mo­<lb/>dò in illam partem deflectebat, ut opus e&longs;&longs;et retroducere, <lb/>adeò ut ducendo & reducendo pecuniam contriverit, & ope­<lb/>ram lu&longs;erit Paconius. </s> <s id="s.001563">Potui&longs;&longs;et tamen huic malo occurrere, <lb/>nec &longs;ui inventi laude fraudari, &longs;i circà fu&longs;os non unicum, <lb/>&longs;ed duplicem funem ita involvi&longs;&longs;et, ut funium &longs;piris vel ab <lb/>extremitatibus fu&longs;orum, vel à medio, incipientibus, funis <lb/>uterque paribus &longs;emper intervallis à &longs;ibi proximâ Rotâ di&longs;ta­<lb/>rent; &longs;ic enim factum fui&longs;&longs;et, ut boves æqualiter utrumque <lb/>funem trahentes, æqualiterque evolventes, molem illam rectâ <lb/>viâ deducerent. </s> </p> <p type="main"> <s id="s.001564">Quamquam autem &longs;uâ laude non careant huju&longs;modi arti­<lb/>ficum inventa, expediti&longs;&longs;imè tamen, & citrà impendium, one­<lb/>ra ingentia traducuntur &longs;ubjectis cylindris, qui pondere pre&longs;&longs;i, <lb/>cùm illud trahitur, convertuntur. </s> <s id="s.001565">Palangas peculiari voca­<lb/>bulo Veterès dixere fre&longs;tes teretes, qui navibus &longs;ubjiciuntur, <lb/>cùm attrahuntur ad pelagus, vel cùm ad littora &longs;ubducuntur; <lb/>ut apud Nonium Marcellum legi&longs;&longs;e me memini. </s> <s id="s.001566">Neque aliud <lb/>quidpiam cen&longs;endus e&longs;t Cæ&longs;ar intellexi&longs;&longs;e, ubi lib. 3. Belli <lb/>Civil. </s> <s id="s.001567">&longs;cribit <emph type="italics"/>Quatuor biremes &longs;ubjectis &longs;cutulis<emph.end type="italics"/> (forta&longs;&longs;e <emph type="italics"/>&longs;cuta­<lb/>lis<emph.end type="italics"/>; hoc e&longs;t <emph type="italics"/>&longs;cytalis,<emph.end type="italics"/> antiquis enim Romanis <emph type="italics"/>is<emph.end type="italics"/> literam u&longs;upari <lb/>&longs;olitam. </s> <s id="s.001568">loco <emph type="italics"/>y<emph.end type="italics"/> literæ Græcæ notum e&longs;t) <emph type="italics"/>impul&longs;as vectibus in <lb/>interiorem partem tran&longs;duxit.<emph.end type="italics"/></s> <s id="s.001569"> Sunt autem &longs;cytalæ ut apud Sui­<lb/>dam, rotunda & polita ligna: aliquid tamen peculiare. </s> <s id="s.001570">ad­<lb/>dit Ari&longs;toteles in Mechan. quæ&longs;t. </s> <s id="s.001571">11. quærens, <emph type="italics"/>cur &longs;uper &longs;cy­<lb/>talas faciliùs portantur onera quàm &longs;uper currus, cum tamen ij <lb/>magnas habeant rotas, illæ verò pu&longs;illas<emph.end type="italics"/>? </s> <s id="s.001572">Scytalis nimirum pu-<pb pagenum="211" xlink:href="017/01/227.jpg"/>&longs;illas rotas adjectas intelligit, <lb/><figure id="id.017.01.227.1.jpg" xlink:href="017/01/227/1.jpg"/><lb/>non eas quidem circà axem, <lb/>&longs;ed cum axe ip&longs;o, cui adnectun­<lb/>tur, ver&longs;atiles; cuju&longs;modi e&longs;­<lb/>&longs;ent in hoc &longs;chemate rotulæ A <lb/>& B cum &longs;uo axe connexæ. </s> </p> <p type="main"> <s id="s.001573">Porrò duplicem huju&longs;modi &longs;cytalarum u&longs;um con&longs;idero: &longs;i <lb/>enim onus impo&longs;itum incumbat Rotulis ip&longs;is, vel quia plana <lb/>&longs;it ejus &longs;uperficies, vel quia tabulato fuerit &longs;uperpo&longs;itum, <lb/>perinde res &longs;e habet, atque &longs;i cylindrus e&longs;&longs;et, cujus diameter <lb/>idem e&longs;&longs;et cum rotularum diametro: neque tunc admodum <lb/>refert, cuju&longs;nam figuræ &longs;it axis, quem onus non tangit, &longs;i­<lb/>ve rotundus ille &longs;it, &longs;ive angulatus. </s> <s id="s.001574">At &longs;i onus ip&longs;i axi in­<lb/>cumbat, promineantque hinc & hinc rotulæ, omninò ne­<lb/>ce&longs;&longs;e e&longs;t axem rotundum e&longs;&longs;e, ut fieri po&longs;&longs;it rotularum con­<lb/>ver&longs;io, atque ita longum, ut inter rotulas onus laxè interci­<lb/>piatur; maximè quippe cavendum e&longs;t, ne rotulæ onus con­<lb/>tingant, alioquin ex mutuo conflictu mora non mediocris <lb/>motui crearetur. </s> <s id="s.001575">Ideò autem excogitatæ videntur huju&longs;mo­<lb/>di &longs;cytalæ, ut minimâ &longs;ui parte &longs;ecundùm extremitates tan­<lb/>gerent &longs;ubjectum planum, atque adeò in pauciora incurre­<lb/>rent offendicula, quàm cylindri totâ &longs;ua longitudine incum­<lb/>bentes plano. </s> <s id="s.001576">Sed illæ ab u&longs;u artificum jam diù intermi&longs;&longs;æ <lb/>locum &longs;implicibus cylindris conce&longs;&longs;ere, quippe qui ob con­<lb/>tinentem &longs;ibique &longs;emper &longs;imilem figuram &longs;olidiores &longs;unt, & <lb/>periculo carent, cui obnoxiæ &longs;unt &longs;cytalæ, ne videlicet Ro­<lb/>tulæ illæ labem aliquam faciant cum rotunditatis, atque adeò <lb/>etiam motûs, detrimento. </s> <s id="s.001577">Illud verò commodum, quod ex <lb/>offendiculorum evitatione oriebatur, obtinemus pariter, &longs;i <lb/>duplicem planorum tigillorum &longs;eriem &longs;ub&longs;ternamus capitibus <lb/>cylindrorum; hinc enim fit, ut viarum &longs;alebræ evitentur, & <lb/>Cylindri modicâ &longs;ui parte contingant &longs;ubjectos tigillos, qui <lb/>viam planam & æquabilem con&longs;tituentes moram nullam mo­<lb/>tui injiciunt. </s> </p> <p type="main"> <s id="s.001578">Sed & in hoc cylindrorum u&longs;u communiter cen&longs;etur ali­<lb/>quid ine&longs;&longs;e facilitatis majoris ad onera deducenda, quàm &longs;i <lb/>illa currui imponerentur; tùm quia currui &longs;ua ine&longs;t gravitas, <lb/>quæ unâ cum impo&longs;itâ &longs;arcinâ majus onus con&longs;tituit, ac <pb pagenum="212" xlink:href="017/01/228.jpg"/>propterea in utroque transferendo is, qui trahit, majorem <lb/>impendit laborem; at &longs;ubjectis oneri cylindris, horum gra­<lb/>vitas nihil officit trahenti: Tùm quia currûs Rotæ, cum &longs;int <lb/>circà &longs;uum axem, cui infiguntur, mobiles, aut hûc & illuc <lb/>nutant, &longs;i laxa &longs;int capita, nec clavo exqui&longs;itè coërceantur, <lb/>aut &longs;i arctiùs axi cohæreant, axem quem complectuntur, & <lb/>clavum quo coërcentur, validiùs terunt; & ex utroque hoc <lb/>capite movendi difficultas oritur, cùm aliquid impre&longs;&longs;i im­<lb/>petûs aut in illâ incon&longs;tantiâ, aut in hoc conflictu contera­<lb/>tur: nihil autem huju&longs;modi cylindris contingit. </s> <s id="s.001579">Tùm etiam <lb/>quia Rotæ modiolus ab axe premitur, & deor&longs;um pondere <lb/>urgente, & antror&longs;um impetu ad anteriora trahente; ex quo <lb/>quantum difficultatis in movendo oriatur, hinc manife&longs;tum <lb/>e&longs;t, quod ni&longs;i axungiâ aut amurcâ illinantur curruum axes, <lb/>ægrè convertuntur rotæ, & den&longs;o &longs;tridore, quantus &longs;it par­<lb/>tium tritus atque conflictus, te&longs;tatum faciunt. </s> <s id="s.001580">At Cylindri <lb/>quantumvis ab onere premantur, nullo pingui liquore obli­<lb/>nendi &longs;unt, ut lubrici fiant; nulla enim impo&longs;iti oneris a&longs;pe­<lb/>ritas cylindrorum conver&longs;ionem impedire pote&longs;t. </s> <s id="s.001581">Nam &longs;i fue­<lb/>rit ingens lapis AB cylin­<lb/><figure id="id.017.01.228.1.jpg" xlink:href="017/01/228/1.jpg"/><lb/>dris &longs;ubjectis impo&longs;itus, & <lb/>cylindri punctum C cen­<lb/>gruat puncto A lapidis, dia­<lb/>metri CD altera extremitas <lb/>D tangit &longs;ubjectum planum; <lb/>cum verò &longs;axum ex B ver­<lb/>sùs A propellitur, &longs;eu tra­<lb/>hitur ex A, ita cylindrus <lb/>convertitur, ut DF ar­<lb/>cus &longs;en&longs;im ad &longs;ubjectum <lb/>planum, contrà verò arcus CE ad impo&longs;itum &longs;axum accom. </s> <lb/> <s id="s.001582">modetur, citrà omnem &longs;axi & cylindri affrictum. </s> </p> <p type="main"> <s id="s.001583">Hinc tamen aliquid etiam incommodi cylindris adhæret, &longs;i <lb/>cum plau&longs;trorum rotis conferantur; hæ &longs;cilicet motum con­<lb/>tinuant, cum &longs;ine fine volvantur, quippe quæ axi infixæ, im­<lb/>po&longs;ito oneri pariter, ut ita loquar, cohærent; illos verò, ni­<lb/>mirum cylindros, onus dum promovetur, po&longs;t &longs;e relinquit; ac <lb/>proinde aut cylindrorum copia non exigua &longs;uppetere debet, <pb pagenum="213" xlink:href="017/01/229.jpg"/>qui longâ &longs;erie di&longs;po&longs;iti onus alij ex aliis excipiant, aut qui <lb/>relinquuntur, &longs;ubinde transferendi &longs;unt, ut iterùm oneri <lb/>&longs;ubjiciantur. </s> <s id="s.001584">Verùm hæc alterna cylindrorum tran&longs;latio non <lb/>adeò gravis e&longs;t; quin plus habeat adjumenti, quàm incom­<lb/>modi; cum enim plurimùm referat, utrùm qui &longs;ubjicitur cy­<lb/>lindrus, reliquis po&longs;terioribus cylindris parallelus, an obli­<lb/>quus &longs;tatuatur, ut onus ad lineam viâ rectâ deducatur, aut <lb/>motus &longs;ui ve&longs;tigium inflectat; facillimum e&longs;t opportunâ cylin­<lb/>dri tran&longs;lati collocatione parallelâ, aut obliqua, de&longs;tinatum <lb/>oneris motum admini&longs;trare. </s> </p> <p type="main"> <s id="s.001585">Illud autem non immeritò hîc examinandum occurrit, utrùm <lb/>majores cylindri minoribus potiores cen&longs;endi &longs;int, & an præ&longs;tet <lb/>&longs;ubjicere oneri cylindrum GI majorem, an verò minorem <lb/>GH. <!-- KEEP S--></s> <s id="s.001586">Et quidem &longs;i figuræ dumtaxat magnitudo atque parvi­<lb/>tas &longs;pectetur, hoc unum di&longs;crimen invenio, quòd ad certam <lb/>motûs men&longs;uram perficiendam crebriùs volvi oportet cylin­<lb/>drum minorem, quàm majorem; onus verò à &longs;ubjecto plano <lb/>di&longs;tare majoris diametri GI intervallo potiùs, quàm minoris <lb/>GH, non video, quid conferat ad motûs facilitatem; tantum <lb/>enim promovetur onus, quantus e&longs;t peripheriæ arcus, cui illud <lb/>in motu aptatur, eíque æqualis e&longs;t arcus oppo&longs;itus, qui plano <lb/>pariter in motu congruit: ac propterea parum refert, utrùm <lb/>eadem arcus men&longs;ura &longs;it majoris circuli pars minor, an minoris <lb/>circuli pars major. </s> </p> <p type="main"> <s id="s.001587">Verùm &longs;i qua inter motum occurrant offendicula, hæc <lb/>minùs officere majori cylindro, quàm minori, dicendum e&longs;t, <lb/>quemadmodum & de rotis majoribus dictum e&longs;t &longs;uperiori ca­<lb/>pite; &longs;iquidem majoris cylindri diameter obliquior incidit in <lb/>idem offendiculum, quod minùs directè opponitur motui, & <lb/>longiore motu Potentiæ fit eadem ponderis elevatio, ut ibi ex­<lb/>plicatum e&longs;t. </s> </p> <p type="main"> <s id="s.001588">Aliud e&longs;t præterea, nec &longs;anè nullius momenti, quod majo­<lb/>ri cylindro incitatiorem dat volubilitatem; quòd videlicet <lb/>(quemadmodum & globo majori contingit) major cylindrus, <lb/>quamvis Geometricam Rotunditatem non a&longs;&longs;equatur, tamen <lb/>propiùs accedit ad figuram exqui&longs;itè Rotundam, quàm mi­<lb/>nor: &longs;i enim à circulo Geometricè perfecto æqualiter recedant <lb/>utriu&longs;que cylindri majoris ac minoris ba&longs;es, non tamen æqua-<pb pagenum="214" xlink:href="017/01/230.jpg"/>liter angulata e&longs;t utraque ba&longs;is, &longs;ed in majori major e&longs;t angu­<lb/>lus, in minori minor, atque adeò ille magis, quàm hic, ad <lb/>rotunditatem accedit. </s> <s id="s.001589">In majori autem circulo angulum, qui <lb/>peripheriam complectitur, majorem e&longs;&longs;e palam e&longs;t, quia idem <lb/>exce&longs;&longs;us majori Radio additus con&longs;tituit &longs;ecantem anguli mi­<lb/>noris, quàm &longs;i minori Radio addatur; ac propterea angulus <lb/>Complementi major e&longs;t in majori, quàm in minori. </s> <s id="s.001590">Id quod, <lb/>per &longs;e quidem &longs;atis clarum, dilucidiùs explicabitur, &longs;i ex mi­<lb/><figure id="id.017.01.230.1.jpg" xlink:href="017/01/230/1.jpg"/><lb/>nore circulo extet particula, cu­<lb/>jus altitudo &longs;it ON, ex majore <lb/>autem circulo æqualis altitudo <lb/>emineat IM. </s> <s id="s.001591">Ductis Tangen­<lb/>tibus & Radiis, certum e&longs;t Se­<lb/>cantis exce&longs;&longs;um ON &longs;upra Ra­<lb/>dium LO minorem, habere <lb/>majorem Rationem ad &longs;uum <lb/>Radium, quàm habeat æqualis <lb/>exce&longs;&longs;us IM ad &longs;uum Radium <lb/>LI majorem ex 8.lib.5. <!--neuer Satz--> E&longs;t igi­<lb/>tur MLP angulus minor angulo NLS, & Complementum <lb/>LMP majus e&longs;t Complemento LNS quare totus angulus <lb/>VMP major e&longs;t toto angulo TNS, ac proinde magis ad ro­<lb/>tunditatem accedit. <lb/></s> </p> <p type="main"> <s id="s.001592"><emph type="center"/>CAPUT X.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001593"><emph type="center"/><emph type="italics"/>Circulorum Concentricorum motus explicatur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001594">CIrculi motus, ob id ip&longs;um quia circulus e&longs;t, circa &longs;uum <lb/>centrum perficitut eâ ratione, ut &longs;uperiores partes pro­<lb/>grediantur, inferiores retrocedant, anteriores de&longs;cendant, <lb/>po&longs;teriores a&longs;cendant, &longs;ervatâ &longs;emper pari oppo&longs;itorum pro­<lb/>gre&longs;sûs atque regre&longs;sûs, de&longs;censûs atque a&longs;censûs men&longs;urâ; <lb/>pro ut unicuique rem vel leviter con&longs;ideranti patet. </s> <s id="s.001595">Quare <lb/>dum in gyrum circulus agitur, centrum quidem manet, reli­<lb/>quæ verò partes ita &longs;ingulæ ex alio in alium locum &longs;ibi invi-<pb pagenum="215" xlink:href="017/01/231.jpg"/>cem &longs;uccedentes commeant, ut circulus totus &longs;patium, in quo <lb/>volvitur, omninò non mutet. </s> <s id="s.001596">Quemadmodum ob&longs;ervare e&longs;t <lb/>in Solis orbitâ, quam Eclipticam vocant; hæc enim diurnâ <lb/>conver&longs;ione circa Mundi axem Solem &longs;ecum rapiens à &longs;uo lo­<lb/>co non recedit, Sole ab ortu in Occa&longs;um commigrante: id <lb/>multò magis in &longs;ingulorum circulorum circà &longs;ua centra revo­<lb/>lutione manife&longs;tum apparet. </s> <s id="s.001597">Quod &longs;i circulus aut horizonti <lb/>parallelus, aut illi ad perpendiculum in&longs;i&longs;tens, raptetur; mo­<lb/>tus ille nihil habet circulari affine, cum circà centrum non <lb/>perficiatur, &longs;ed &longs;ingula circuli puncta &longs;olo motu recto unâ cum <lb/>centro moveantur. </s> </p> <p type="main"> <s id="s.001598">Sin autem axis circulo ver&longs;atili infixus trahatur, jam circu­<lb/>lus & cum-axe pariter movetur, & circa axem volvitur: atque <lb/>adeò &longs;ingularum circuli partium motus is e&longs;t, qui ex recto cen­<lb/>tri, & circulari ip&longs;ius orbitæ componitur. </s> <s id="s.001599">Hinc &longs;emicirculi <lb/>&longs;uperioris partes cum progrediantur versùs cumdem locum, ad <lb/>quem centrum tendit, &longs;uum motum motui centri addunt: <lb/>Contrà verò inferioris &longs;emicirculi partes retrocedentes &longs;uum <lb/>motum à centri motu detrahunt. </s> <s id="s.001600">Rotæ igitur puncta omnia, <lb/>dum currus trahitur, &longs;i non &longs;ummatim tota revolutio, &longs;ed par­<lb/>ticulatim, accipiatur, non æquali velocitate moventur. </s> <s id="s.001601">Sit <lb/>explicandi gratiâ, <lb/><figure id="id.017.01.231.1.jpg" xlink:href="017/01/231/1.jpg"/><lb/>circulus BD AE, <lb/>cujus centrum C <lb/>moveatur ver&longs;us F, <lb/>& &longs;it tangens GA, <lb/>cui in motu appli­<lb/>catur ip&longs;ius circu­<lb/>li orbita; in quâ <lb/>accipiatur &longs;extans <lb/>hinc & hinc AD, <lb/>& AE. <!-- KEEP S--></s> <s id="s.001602">Igitur in <lb/>Conver&longs;ione, dum <lb/>Centrum C trahitur ad F, punctum D venit in G, & arcus <lb/>DA æqualis e&longs;t rectæ GA, cui in motu &longs;ubinde per partes <lb/>congruit: atque adeò, quarum partium &longs;emidiameter CA <lb/>e&longs;t 21, earum arcus AD, & recta AG e&longs;t 22, & motus cen­<lb/>tri illi æqualis CF e&longs;t pariter 22. Quoniam verò in motu or-<pb pagenum="216" xlink:href="017/01/232.jpg"/>bitæ circa &longs;uum centrum, punctum A a&longs;cendens in E retroce­<lb/>dit juxta men&longs;uram &longs;inûs SE (qui ad Radium CA 21 e&longs;t ut 18) <lb/>hinc e&longs;t po&longs;t conver&longs;ionem, in qua D e&longs;t in G, punctum A <lb/>ita a&longs;cendi&longs;&longs;e, ut &longs;it in lineâ HE parallelâ Tangenti GA, &longs;ed <lb/>motui centri tantum detraxerit, quantus e&longs;t &longs;inus SE. <!-- KEEP S--></s> <s id="s.001603">Quia <lb/>igitur Radius CD ubi congruit punctis FG, &longs;ecat in H <lb/>rectam HE, &longs;umatur HI æqualis &longs;inui SE, & puncti A totus <lb/>progre&longs;&longs;us remanet SI partium 4, quarum SH, &longs;eu CF e&longs;t 22. <lb/>Quare A e&longs;t in I, quando D e&longs;t in G. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001604">Contrà verò in &longs;uperiore &longs;emicirculo &longs;umatur item ex B <lb/>hinc, & hic &longs;extans BK & BL; atque in conver&longs;ione ubi cen­<lb/>trum C venerit in F, & punctum orbitæ D in G, erit K in O, <lb/>& diameter DK &longs;ecabit parallelam KN in M. </s> <s id="s.001605">Igitur punctum <lb/>B ita de&longs;cendit ad parallelam NK, ut motui centri CF, hoc <lb/>e&longs;t BO &longs;eu RM, addiderit &longs;uum progre&longs;&longs;um juxta men&longs;uram <lb/>RL Sinum Sextantis BL, hoc e&longs;t 18. Venit igitur B in N; <lb/>atque additis RM 22, & MN 18, totus progre&longs;&longs;us puncti B <lb/>e&longs;t RN 40. Comparatis itaque invicem curvis lineis AI & <lb/>BN, manife&longs;tum e&longs;t puncta B & A non æque velociter mo­<lb/>veri, cum eodem temporis &longs;patio inæqualia loci &longs;patia per­<lb/>currant. </s> </p> <p type="main"> <s id="s.001606">Eadem erit methodus, &longs;i reliquorum orbitæ punctorum ve­<lb/>locitates aut tarditates con&longs;iderandæ &longs;int: &longs;i tamen adverteris <lb/>non eandem e&longs;&longs;e omnium circuli Quadrantum rationem in de­<lb/>terminandâ men&longs;ura motûs addendi, aut demendi motui cen­<lb/>tri. </s> <s id="s.001607">Nam in anteriori Quadrante &longs;uperioris &longs;emicirculi, & in <lb/>po&longs;teriori Quadrante inferioris &longs;emicirculi, men&longs;ura progre&longs;­<lb/>sûs addendi in illo, & regre&longs;&longs;us demendi in i&longs;to, attendenda <lb/>e&longs;t ex Sinu Recto arcûs, qui de&longs;cribitur in motu circa cen­<lb/>trum à puncto, cujus velocitas inquiritur, aut tarditas: Et <lb/>quidem integer Sinus Rectus accipitur, &longs;i punctum à &longs;ummo <lb/>vertice de&longs;cendens, vel ab infimo contactûs puncto a&longs;cendens <lb/>movetur, ut ex B vel ex A: &longs;in autem punctum con&longs;ideretur, <lb/>quod intrà eo&longs;dem Quadrantes di&longs;tet ab extremitatibus diame­<lb/>tri &longs;ubjecto plano in&longs;i&longs;tentis, puta L aut E, quæ moventur in <lb/>V, aut in P, progre&longs;sûs aut regre&longs;sûs men&longs;ura de&longs;umitur ex dif­<lb/>ferentiâ Sinuum Rectorum, qui re&longs;pondent arcubus BL & BV, <lb/>aut arcubus AE & AP. <!-- KEEP S--></s> <s id="s.001608">In po&longs;teriori verò Quadrante &longs;upe-<pb pagenum="217" xlink:href="017/01/233.jpg"/>rioris &longs;emicirculi, & in anteriori Quadrante inferioris &longs;emicir­<lb/>culi, progre&longs;&longs;us addendus, aut regre&longs;&longs;us demendus, motui <lb/>centri, men&longs;uram de&longs;umit ex Sinubus Ver&longs;is, aut ex eorum <lb/>differentiâ, pro ut puncti motus a&longs;cendens aut de&longs;cendens in­<lb/>cipit ab extremitate Quadrantis, aut à loco medio, ut facilè <lb/>cuique con&longs;tat: neque enim &longs;chema multiplici linearum de&longs;­<lb/>criptione ad confu&longs;ionem implere operæ pretium e&longs;t. </s> </p> <p type="main"> <s id="s.001609">Cum itaque in oppo&longs;itis Quadrantibus &longs;imilem men&longs;uram <lb/>recipiant incrementa atque decrementa &longs;ive à &longs;inubus Rectis, <lb/>&longs;ive à Ver&longs;is, addenda aut demenda motui centri, mani­<lb/>fe&longs;tum e&longs;t punctum quodlibet in integrâ conver&longs;ione demùm <lb/>progre&longs;&longs;um fui&longs;&longs;e pari men&longs;urâ cum motu centri. </s> <s id="s.001610">Si enim Al­<lb/>gebricè &longs;tatuatur motus Centri Z, incrementum in &longs;uperiore <lb/>&longs;emicirculo addendum +A, decrementum in inferiore &longs;emicir­<lb/>culo tollendum — A; manife&longs;tum e&longs;t totum motum, qui com­<lb/>ponitur, Z +A — A non e&longs;&longs;e ni&longs;i Z. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001611">His ita con&longs;titutis, quæ ita clara &longs;unt, ut nihil habere vi­<lb/>deantur dubitationis, nec in controver&longs;iam vocari queant, jam <lb/>eximendus e&longs;t &longs;crupulus, quem philo&longs;ophantibus injecit Ari­<lb/>&longs;toteles Mechanic. <!-- REMOVE S-->quæ&longs;t. </s> <s id="s.001612">24. de circulorum concentricorum <lb/>motu, quando alter ad alterius motum promoto communi cen­<lb/>tro movetur. </s> <s id="s.001613">Sit <lb/><figure id="id.017.01.233.1.jpg" xlink:href="017/01/233/1.jpg"/><lb/>enim major circu­<lb/>lus, cujus Radius <lb/>CB, minor autem, <lb/>cujus Radius CS; <lb/>quos tangant pa­<lb/>rallelæ BF & ST, <lb/>quibus item recta <lb/>per centrum ducta <lb/>parallela &longs;it CO, <lb/>quam videlicet per­<lb/>currit centrum, <lb/>dum trahitur. </s> <s id="s.001614">Ne­<lb/>gari non pote&longs;t in <lb/>hâc circulorum tractione & conver&longs;ione peripherias tùm ma­<lb/>joris, tùm minoris Circuli &longs;uis Tangentibus ita coaptari, ut <lb/>factâ Quadrantis BD conver&longs;ione, fiat pariter Quadrantis SI <pb pagenum="218" xlink:href="017/01/234.jpg"/>conver&longs;io, & ubi punctum D venerit in F, punctum I &longs;it in T, <lb/>& centrum C in O, atque adeò Radius CD matato &longs;itu factus <lb/>&longs;it OF. </s> <s id="s.001615">Major igitur Quadrans percurrit &longs;patium BF, & mi­<lb/>nor &longs;patium ST. <!-- KEEP S--></s> <s id="s.001616">At quia æquales rectæ OF & CB perpen­<lb/>diculares &longs;unt ad eandem rectam BF, ctiam &longs;unt parallelæ, <lb/>jungúntque parallelas ST & BF, quæ propterea etiam &longs;unt <lb/>æquales, ex 34. lib.1. Igitur arcus SI minor arcu BD, coap­<lb/>tatur &longs;patio æquali ip&longs;i arcui Quadrantis BD, cui &longs;upponitur <lb/>æqualis recta BF. <!-- KEEP S--></s> <s id="s.001617">Quarum itaque partium 7 e&longs;t Radius CB, <lb/>earum e&longs;t Quadrans BD, hoc e&longs;t recta BF 11, e&longs;tque pariter <lb/>ST 11. At quarum partium 7 e&longs;t Radius CB, earum &longs;it Ra­<lb/>dius CS 4; igitur Quadrans SI e&longs;t 6 3/7 multo minor quàm <lb/>recta ST, cui ip&longs;e Quadrans SI in motu congruit. </s> </p> <p type="main"> <s id="s.001618">Id enim verò tantum præ &longs;e fert difficultatis, ut mirum &longs;it, <lb/>quot Ixiones rota hæc torqueat, & quàm varias in partes &longs;e alij <lb/>aliter ver&longs;ent; quorum &longs;ententias &longs;i examinare liberet, in lon­<lb/>gum nimis &longs;ermonem me vocaret i&longs;ta di&longs;putatio, nec &longs;atis &longs;ci­<lb/>rem, utrùm plus aliquid lucis propo&longs;itæ quæ&longs;tioni affunderetur. </s> <lb/> <s id="s.001619">Quid igitur probabilius dicendum videatur, paucis expono. </s> </p> <p type="main"> <s id="s.001620">Priùs tamen ob&longs;erva in dictâ Quadrantis revolutione, quan­<lb/>do Centrum C venerit in O, & D in F, & in I in T, tunc <lb/>punctum B e&longs;&longs;e in E (e&longs;t enim OE æqualis Radio CB) atque <lb/>punctum S in V (e&longs;t &longs;cilicet OV æqualis Radio CS) ita <lb/>ut B a&longs;cendat per curvam BE, punctum autem S a&longs;cendat <lb/>per curvam SV, & &longs;imiliter punctum D de&longs;cendat per cur­<lb/>vam DF, punctum verò I de&longs;cendat per curvam IT. </s> <s id="s.001621">Ex quo <lb/>patet punctum S minoris circuli plus promoveri, quàm <lb/>punctum B majoris circuli; hujus enim progre&longs;&longs;us e&longs;t CE, il­<lb/>lius autem e&longs;t CV: & pari ratione con&longs;tat magis ad anterio­<lb/>ra promoveri punctum I minoris circuli, cujus progre&longs;sûs men­<lb/>&longs;ura e&longs;t IO, quàm punctum D majoris circuli, cujus progre&longs;­<lb/>&longs;us e&longs;t DO. </s> </p> <p type="main"> <s id="s.001622">Et hæc quidem, quando centri motus legem accipit à pe­<lb/>ripheriâ majoris circuli; ad cujus motum minor circulus con­<lb/>centricus movetur; eo quod major circulus in&longs;i&longs;tit &longs;ubjecto pla­<lb/>no, cui orbita &longs;ubinde coaptatur rectam lineam &longs;ibi æqualem <lb/>de&longs;ignans ex hypothe&longs;i, dumque movetur, &longs;ecum rapit interio­<lb/>rem circulum. </s> </p> <pb pagenum="219" xlink:href="017/01/235.jpg"/> <p type="main"> <s id="s.001623">Quod &longs;i minor circulus in&longs;i&longs;tat &longs;ubjecto &longs;ibi plano, legem­<lb/>que det motui centri; quia minor peripheria de&longs;ignat rectam <lb/>&longs;ibi æqualem, res contrario modo procedit, quia dum ad mi­<lb/>noris circuli motum circulus major movetur, hujus orbita de­<lb/>&longs;ignat in plano &longs;ubjecto lineam minori peripheriæ æqualem. </s> <lb/> <s id="s.001624">Hinc &longs;i arcus SI de&longs;ignat rectam SG &longs;ibi æqualem, ubi I ve­<lb/>nerit in G, etiam D erit in H, atque totus Quadrans BD de­<lb/>&longs;ignabit &longs;olùm rectam BH æqualem rectæ SG. <!-- KEEP S--></s> <s id="s.001625">Erit igitur <lb/>recta SG æqualis Quadranti SI 6 2/7; cui pariter æqualis e&longs;t <lb/>BH: Ex quo fit punctum B, quia di&longs;tat à centro C partibus 7, <lb/>non &longs;olùm non procedere in revolutione Quadrantis; &longs;ed re­<lb/>trocedere per 5/7 interea, dum commune centrum C promove­<lb/>tur per 6 2/7. </s> </p> <p type="main"> <s id="s.001626">Non ab&longs;imili ratione punctorum B, & S jam in E & V <lb/>tran&longs;latorum motus per con&longs;equentes circuli Quadrantes, do­<lb/>nec integra revolutio perficiatur, con&longs;iderandus e&longs;t: & quæ <lb/>de uno puncto cuju&longs;que circuli deprehenduntur, de &longs;ingulis <lb/>eju&longs;dem orbitæ punctis dicta faciliùs intelliguntur, quàm ut <lb/>uberiori explicatione opus &longs;it. </s> </p> <p type="main"> <s id="s.001627">Ex his apertè liquet eam lineam rectam in &longs;ubjecto plano de­<lb/>&longs;ignari à peripheriâ tùm majoris, tùm minoris circuli, quæ <lb/>æqualis &longs;it motui centri, prout ille legem accipit à majore aut <lb/>à minore orbitâ, ad cujus motum altera movetur; ac proinde <lb/>modò longiori, modò breviori lineæ rectæ in motu coaptantur <lb/>ambæ peripheriæ; ut enim rectè loquitur Ari&longs;toteles loc. </s> <s id="s.001628">cit. <lb/><emph type="italics"/>Quando hic quidem movet, ille verò movetur ab i&longs;to, quantum uti­<lb/>que moverit alter, tantum alter movebitur.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.001629">Cur igitur parem lineam rectam de&longs;ignat in plano utraque <lb/>orbita major & minor? </s> <s id="s.001630">con&longs;tat ex dictis: quia nimirum cu­<lb/>ju&longs;libet circuli quodlibet punctum dum trahitur &longs;imul, & vol­<lb/>vitur, promovetur non ni&longs;i pro ratione motûs centri: &longs;ed con­<lb/>centricorum circulorum unum & idem e&longs;t centrum; ergo uni­<lb/>cus e&longs;t centri motus, & &longs;ecundùm unam eandemque men&longs;u­<lb/>ram motûs centri, omnia puncta tùm majoris, tùm minoris or­<lb/>bitæ, demum ab&longs;olutâ conver&longs;ione, promota &longs;unt; &longs;ingulorum <lb/>enim incrementa, dum &longs;uperiorem &longs;emiperipheriam motu <lb/>de&longs;cribunt, ab oppo&longs;itis decrementis eli&longs;a in inferioris &longs;emipe-<pb pagenum="220" xlink:href="017/01/236.jpg"/>ripheriæ de&longs;criptione, &longs;olum centri motum relinquunt. </s> <s id="s.001631">Nil <lb/>itaque mirum, &longs;i tres lineæ, quarum primam centrum percur­<lb/>rit, &longs;ecundam orbita minor de&longs;ignat, tertiam orbita major, pla­<lb/>nè æquales &longs;unt; pendent enim ab unico & communi motu <lb/>centri, cui nihil additur, aut demitur ex integrâ conver&longs;ione <lb/>circa centrum, &longs;ivè illa latiùs excurrat in majore circulo, &longs;ivè <lb/>arctiùs in minore coërceatur. </s> </p> <p type="main"> <s id="s.001632">At, inquis, difficile e&longs;t cogitatione a&longs;&longs;equi, & oratione ex­<lb/>plicare, quî fieri po&longs;&longs;it, ut peripheriâ utráque &longs;ubjectum &longs;ibi <lb/>planum &longs;emper tangente, nullóque puncto manente &longs;ine mo­<lb/>tu, ita ut plana &longs;ubjecta ab aliis &longs;ubinde atque aliis punctis tan­<lb/>gantur, pauciora puncta minoris peripheriæ totidem punctis <lb/>rectæ lineæ coaptentur, ac plura puncta majoris peripheriæ. </s> </p> <p type="main"> <s id="s.001633">Sunt qui difficultatem hanc declinant ad&longs;truentes infinita <lb/>puncta tùm in circulorum peripheriis, tùm in lineis rectis, ne­<lb/>ganté&longs;que inter infinitas multitudines, quæ invicem compa­<lb/>rentur, affirmari po&longs;&longs;e totidem in unâ infinitâ multitudine, ac <lb/>in aliâ pariter infinitâ unitates reperiri, nulla enim e&longs;t infiniti <lb/>ad infinitum Ratio, ac proinde nulla fieri pote&longs;t, perinde ac in <lb/>multitudinibus finitis, comparatio minoris, aut majoris, aut <lb/>propriè, &, ut aiunt, po&longs;itivè æqualis. </s> <s id="s.001634">Hæc tamen (quamvis <lb/>quod ad infinita Ratione carentia &longs;pectat, à me ultrò admit­<lb/>tantur, Rationem &longs;cilicet habere dicuntur inter &longs;e magnitudi­<lb/>nes, idem & de multitudinibus dicendum, quæ po&longs;&longs;unt mul­<lb/>tiplicatæ &longs;e mutuò &longs;uperare, ut definit Euclides lib.5. ubi au­<lb/>tem nullus e&longs;t terminus, ut in infinito, nullus pariter exce&longs;&longs;us <lb/>intercedere pote&longs;t quavis factâ multiplicatione) non facient <lb/>&longs;atis comparanti omnia puncta unius lineæ cum omnibus <lb/>punctis alterius lineæ, non quâ infinitæ punctorum multitudi­<lb/>nes &longs;unt, &longs;ed quâ finitæ magnitudines ex punctis illis quan­<lb/>tumvis infinitis con&longs;tituuntur: finitas autem magnitudines <lb/>comparari invicem po&longs;&longs;e, ac Rationem inter&longs;e habere nemo <lb/>negaverit. </s> <s id="s.001635">Supere&longs;t igitur explicandum, quomodo peripheria <lb/>minor coaptetur lineæ rectæ æquali illi eidem, cui commen&longs;u­<lb/>ratur peripheria major. </s> </p> <p type="main"> <s id="s.001636">Propterea, duce Galilæo Dialog.1. de motu, ob&longs;ervant &longs;imi­<lb/>lium polygonorum concentricorum motum ac conver&longs;ionem, <lb/>in quâ polygonum, ex quo centri motus legem accipit, &longs;ingu-<pb pagenum="221" xlink:href="017/01/237.jpg"/>la latera ita æqualibus lineæ rectæ partibus accommodat, ut in <lb/>integrâ conver&longs;ione linea recta &longs;ubjecti plani &longs;it æqualis peri­<lb/>metro polygoni: at non item partes omnes lineæ, cui alterum <lb/>polygonum in motu coaptatur, &longs;i unica comprehen&longs;ione &longs;u­<lb/>mantur, lineam æqualem polygoni majoris perimetro con&longs;ti­<lb/>tuunt. </s> <s id="s.001637">Res, clarita­<lb/><figure id="id.017.01.237.1.jpg" xlink:href="017/01/237/1.jpg"/><lb/>tis gratia, explicetur <lb/>in Hexagonis, quo­<lb/>rum commune cen­<lb/>trum &longs;it A, & latera <lb/>BC, DE incumbant <lb/>parallelis lineis BH, <lb/>DK. <!-- KEEP S--></s> <s id="s.001638">Det primùm le­<lb/>gem motui centri po­<lb/>lygonum exterius, & majus, fiatque conver&longs;io circa punctum <lb/>C, demùm latus CF congruet rectæ CH, & centrum A per <lb/>arcum AF erit tran&longs;latum in F; latus verò minoris polygoni <lb/>EG congruet parti IK, intactam relinquens partem EI, ita <lb/>tamen; ut tota EK æqualis &longs;it ip&longs;i CH. <!-- KEEP S--></s> <s id="s.001639">Id quod e&longs;t mani­<lb/>fe&longs;tum, quia factâ tran&longs;latione centri in F, &longs;emidiameter, quæ <lb/>ex F pertingit ad H, e&longs;t parallela ip&longs;i AC, cum ad &longs;imiles an­<lb/>gulos incidat in &longs;ubjectam lineam; &longs;unt autem parallelæ etiam <lb/>AF, DK, & BH; igitur tres lineæ AF, EK, CH &longs;unt æqua­<lb/>les, ex 34. lib.1. Atqui quod uni lateri contingit, etiam reli­<lb/>quis lateribus commune e&longs;t; igitur factá integrâ conver&longs;ione <lb/>Hexagonum majus de&longs;ignabit lineam &longs;extuplicem ip&longs;ius CH <lb/>æqualem toti perimetro, & Hexagonum minus percurret li­<lb/>neam &longs;imiliter ip&longs;ius EK &longs;extuplicem, quæ æqualis e&longs;t perime­<lb/>tro majoris Hexagoni, &longs;umendo tàm partes lineæ DK, quas <lb/>intactas relinquit, quàm quæ tangunrur. </s> <s id="s.001640">Cæterùm &longs;i eæ &longs;o­<lb/>lùm, quæ ab Hexagono minore tanguntur, accipiantur, patet <lb/>illas &longs;imul &longs;umptas non e&longs;&longs;e majores perimetro eju&longs;dem mino­<lb/>ris Hexagoni. </s> </p> <p type="main"> <s id="s.001641">Deinde polygonum interius & minus det legem motui cen­<lb/>tri, & conver&longs;io fiat circa punctum E, po&longs;tquam latus EG <lb/>congruit lineæ EI, & centrum e&longs;t in G (in hoc enim exem­<lb/>plo ad vitandam in Schemate confu&longs;ionem literarum a&longs;&longs;ump-<pb pagenum="222" xlink:href="017/01/238.jpg"/>tum e&longs;t Hexagonum minus &longs;ubquadruplum majoris, latera &longs;ci­<lb/>licet minotis &longs;ubdupla &longs;unt laterum majoris) cum interim <lb/>punctum C retroce&longs;&longs;erit in L, & demum latus CF congruat <lb/>lineæ LM. <!-- KEEP S--></s> <s id="s.001642">Igitur majus polygonum &longs;olùm de&longs;ignat in motu, <lb/>quo progreditur, lineam CM æqualem lateri minoris polygoni <lb/>EI; & factâ integrâ conver&longs;ione, de&longs;ignata erit linea &longs;extuplex <lb/>ip&longs;ius CM & ip&longs;ius EI; atque adeò utrumque polygonum <lb/>æqualem lineam progrediendo de&longs;ignat. </s> </p> <p type="main"> <s id="s.001643">Hæc quæ de Hexagonis concentricis exempli gratiâ dicta <lb/>&longs;unt, de omnibus &longs;imilibus atque concentricis polygonis dicta <lb/>intelliguntur, quotcumque &longs;int laterum. </s> <s id="s.001644">Jam verò Authores <lb/>illi concipiunt circulos tanquam polygona infinitorum late­<lb/>rum: & quemadmodum minus polygonum totidem &longs;patia &longs;ub­<lb/>jectæ lineæ intacta relinquit, totidemque tangit, quot habet <lb/>latera; ita pariter in circuli minoris conver&longs;ione, infinita &longs;pa­<lb/>tia vacua non-quanta (ne &longs;cilicet &longs;i quanta e&longs;&longs;ent, opus e&longs;&longs;et <lb/>lineâ infinitâ) intermi&longs;ta &longs;patiis, quæ tanguntur, ad&longs;truunt, <lb/>adeò ut demùm ex omnibus &longs;patiis tactis &longs;imul & intactis coa­<lb/>le&longs;cat linea æqualis ei, quæ tangitur à majore peripheriâ ma­<lb/>joris circuli. </s> </p> <p type="main"> <s id="s.001645">Mihi tamen arridere non pote&longs;t illa loquendi formula, quæ <lb/>circulum polygonum infinitorum (& quidem infinitorum &longs;im­<lb/>pliciter) laterum dicit. </s> <s id="s.001646">Polygonum enim utique regulare cir­<lb/>culus e&longs;&longs;et; polygonum autem e&longs;&longs;e non pote&longs;t illud, quod angu­<lb/>lis caret; neque anguli e&longs;&longs;e po&longs;&longs;unt, ubi non e&longs;t lineæ ad li­<lb/>neam inclinatio; in peripheriâ verò circuli linea nulla e&longs;&longs;e po­<lb/>te&longs;t, e&longs;&longs;ent &longs;iquidem infinitæ lineæ æquales invicem, quæ uti­<lb/>que con&longs;tituerent exten&longs;ionem &longs;impliciter infinitam. </s> <s id="s.001647">Quod &longs;i <lb/>infinita dixeris puncta; non e&longs;t puncti ad punctum inclinatio, <lb/>quæ po&longs;&longs;it angulum con&longs;tituere, ac proinde circulus non e&longs;t po­<lb/>lygonum infinitorum laterum, ni&longs;i vocabulis ad opinandi li­<lb/>centiam immoderatè abutamur. </s> <s id="s.001648">Adde quod omnia diametri <lb/>puncta ad omnia puncta peripheriæ e&longs;&longs;ent in Ratione, quam <lb/>Archimedes <emph type="italics"/>lib.de dimen&longs;ione circuli<emph.end type="italics"/> definivit contineri inter Ra­<lb/>tionem 7 ad 22, & Rationem 71 ad 223: non igitur infinita e&longs;&longs;e <lb/>po&longs;&longs;unt aut diametri, aut peripheriæ, aut utriu&longs;que puncta; ab <lb/>infinitis enim Rationem omnem ablegant iidem Authores. </s> <s id="s.001649">Si <pb pagenum="223" xlink:href="017/01/239.jpg"/>itaque circulus polygonus non e&longs;t, adhuc indiget explicatione, <lb/>quomodo ad circulos concentricos traducantur ea, quæ de po­<lb/>lygonorum concentricorum conver&longs;ione con&longs;iderata &longs;unt. </s> </p> <p type="main"> <s id="s.001650">Quòd &longs;i circulum ita in polygonum convertamus, ut nec <lb/>illi fixum definitumque laterum numerum tribuamus, nec &longs;im­<lb/>pliciter infinitum; &longs;ed liceat minora &longs;emper atque minora late­<lb/>ra concipere, ut laterum ip&longs;orum numerus &longs;emper augeatur, <lb/>ita ut non &longs;impliciter infinitus, &longs;ed indefinitus dicatur, non <lb/>abnuo: propo&longs;ita enim difficultas &longs;atis commodè hâc ratione <lb/>explicabitur. </s> <s id="s.001651">Verùm in hac laterum extenuatione, &longs;i ad mini­<lb/>mam exten&longs;ionem deveniamus, quæ à puncto phy&longs;icè non dif­<lb/>ferat; non infinitus e&longs;t huju&longs;modi punctorum numerus, &longs;ed <lb/>certus e&longs;t atque definitus: Necip&longs;is punctis, &longs;eu minimis Phy­<lb/>&longs;icis &longs;ua figura detrahenda e&longs;t, in majori enim peripheriâ mi­<lb/>nùs curvantur interiùs, minú&longs;que convexa &longs;unt exteriùs, pro­<lb/>piú&longs;que ad lineam rectam accedunt; in minori autem orbitâ <lb/>puncta hæc circularia curvantur magis, magi&longs;que convexa &longs;unt <lb/>exteriùs, & à rectitudine magis deflectentia ita ab&longs;unt à &longs;ub­<lb/>jectâ rectâ lineâ, ut, dum conver&longs;io fit circuli, & trahitur, de&longs;­<lb/>cribant in motu lineam curvam magis ob&longs;ecundantem motui <lb/>centri, quàm quæ de&longs;cribitur à punctis &longs;imiliter po&longs;itis in ma­<lb/>jore peripheriâ. </s> </p> <p type="main"> <s id="s.001652">Cærerùm cavendum e&longs;t maximè ab eo, quod quia &longs;ube&longs;t <lb/>æquivocationi, difficultatem in hâc quæ&longs;tione auget; illud au­<lb/>tem e&longs;t, quod punctum peripheriæ cum puncto lineæ Tangen­<lb/>tis perperam comparatur, qua&longs;i in contactu coæquarentur; id <lb/>quod à veritate longè abe&longs;t; &longs;e enim contingunt circulus & li­<lb/>nea incommen&longs;urabiliter, &longs;i contactus præcisè &longs;pectetur: at &longs;i <lb/>contactus & motus componantur, jam quædam exten&longs;io conci­<lb/>pitur, quæ aliquâ ratione comparari pote&longs;t cum &longs;patio lineæ, <lb/>quæ tangitur, quatenùs huic aut illi parti lineæ in motu coapta­<lb/>tur circulus, aut ejus pars. </s> <s id="s.001653">Quare circuli minoris, qui ad ma­<lb/>joris circuli motum movetur, &longs;ingula puncta non aptè compa­<lb/>rantur cum &longs;ingulis &longs;ubjectæ rectæ lineæ punctis, qua&longs;i circuli <lb/>punctum, quod e&longs;t tertium à contactu, antequam incipiat mo­<lb/>tus, in conver&longs;ione tangat tertium rectæ lineæ punctum; &longs;ed <lb/>tanget forta&longs;&longs;e quintum aut &longs;extum pro ratione magnitudinis <pb pagenum="224" xlink:href="017/01/240.jpg"/>aut parvitatis ip&longs;ius circuli; pro ut in polygonis concentricis <lb/>obiervare e&longs;t; quò enim majus e&longs;t interius polygonum, eò <lb/>etiam minora &longs;unt intervalla, quæ intacta relinquuntur. </s> <s id="s.001654">Ex <lb/>quamvis in circuli contactu intervalla huju&longs;modi intacta non <lb/>admittantur, non e&longs;t tamen abs re puncto circuli, quod volui­<lb/>tur &longs;imul & trahitur cum ip&longs;o circulo, vim tribuere tangendi <lb/>plus quàm unum &longs;ubjectæ rectæ lineæ punctum, quemadmo­<lb/>dum majoris peripheriæ punctum in motu contingit ex punctis <lb/>&longs;ubjectæ lineæ rectæ non communicantibus minus quàm unum, <lb/>&longs;i ad interioris circuli motum circulus exterior moveatur: nam <lb/>ad majoris, & exterioris motum minor, & interior promovetur; <lb/>ad minoris verò & interioris motum major & exterior circulus <lb/>retroagitur. </s> <s id="s.001655">Quapropter &longs;i interior circulus in primo ca&longs;u ve­<lb/>lociùs, & exterior in &longs;ecundo ca&longs;u tardiùs movetur comparatè <lb/>ad &longs;patium collocatum cum eorum peripheriis, nil mirum in <lb/>motu perfici ab illius puncto Phy&longs;ico plus &longs;patij, quàm ferat <lb/>ejus magnitudo, ab hujus autem puncto Phy&longs;ico minus &longs;patij: <lb/>in continuâ enim quantitate partes minores &longs;ubinde ac minores <lb/>vera, ut opinor, Philo&longs;ophia admittit. </s> <s id="s.001656">Sed quia hæc e&longs;&longs;et in­<lb/>finita, concertationumque plena di&longs;putatio, &longs;atis ea &longs;int, quæ <lb/>diximus, & ad utiliora gradum faciamus. <lb/><figure id="id.017.01.240.1.jpg" xlink:href="017/01/240/1.jpg"/></s> </p> <pb pagenum="225" xlink:href="017/01/241.jpg"/> <figure id="id.017.01.241.1.jpg" xlink:href="017/01/241/1.jpg"/> <p type="main"> <s id="s.001657"><emph type="center"/>MECHANICORUM<emph.end type="center"/><!-- REMOVE S--><emph type="center"/>LIBER TERTIUS.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.001658"><emph type="center"/><emph type="italics"/>De Libra.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001659">EXPLICATIS &longs;uperiore Libro Cau&longs;is motûs Ma­<lb/>chinalis, ordinis ratio po&longs;tularet, ut ad ip&longs;as Ma­<lb/>chinas, &longs;eu, ut ab Antiquioribus apud Pappum <lb/>lib.8. Collect. Mathem. prop.10. vocantur, Facul­<lb/>tates, ad quas Machinamenta ab artificibus exco­<lb/>gitata reducuntur, aut ex quibus hæc componuntur, exami­<lb/>nandas & explicandas progrederemur: Et fortè alicui videatur <lb/>ab in&longs;tituto no&longs;tro alienum libram hîc con&longs;iderare, quippe quæ <lb/>non ad motum oneribus conciliandum inventa e&longs;t, ideóque <lb/>nec inter Facultates enumeratur, &longs;ed u&longs;um omnem habet in <lb/>motu prohibendo, ubi factum fuerit ponderibus æquilibrium. </s> <lb/> <s id="s.001660">Nec eo quidem con&longs;ilio libræ momenta hic expendo, ut indè <lb/>Vectis rationes explicentur (quemadmodum non paucis placet) <lb/>non enim Vectis vires ad libræ Rationes revocandas exi&longs;timo, <lb/>cum &longs;ua cuique Facultati cau&longs;a in&longs;it, communis illa quidem, <lb/>&longs;ed quæ perinde in Vecte reperitur, atque &longs;i nulla pror&longs;us <lb/>exi&longs;teret libra. </s> <s id="s.001661">Verùm eatenus libram Mechanicæ contem­<lb/>plationi in&longs;erendam cen&longs;eo, quatenus non minoris artis e&longs;t ea, <lb/>quæ in motum prona &longs;unt, cohibere & &longs;i&longs;tere, quàm onera <lb/>quie&longs;centia per vim &longs;uo loco dimovere: Cum maximè ad libram <lb/>pertineat Statera, in qua modicum pondus multò majori pon­<lb/>deri æquipollet, æquatis in di&longs;pari gravitate gravitationum <pb pagenum="226" xlink:href="017/01/242.jpg"/>momentis, ut infra in loco o&longs;tendetur. </s> <s id="s.001662">Præterquam quod <lb/>explicato æquilibrio, faciliùs declaratur in motu Machinali, <lb/>quid præ&longs;tet major illa Ratio momentorum agendi ad momen­<lb/>ta re&longs;i&longs;tendi, quàm &longs;it reciproca Ratio gravitatum, &longs;eu vi­<lb/>rium oppo&longs;itarum, ab&longs;olutè &longs;umptarum extrà machinam; ex <lb/>qua majore Ratione momentorum, etiam Potentiæ moventis <lb/>virtus innote&longs;cit. </s> <s id="s.001663">Nihil autem officit libræ dignitati, quod <lb/>Cain authorem agno&longs;cere videatur, qui, ut Jo&longs;ephus lib. 1. <lb/><expan abbr="Antiq.">Antique</expan> Jud. <!-- REMOVE S-->cap.2. loquitur, <emph type="italics"/>Simplicem hactenus vivendi rationem <lb/>excogitatis men&longs;uris & ponderibus immutavit, pri&longs;linamque &longs;inceri­<lb/>tatem & genero&longs;itatem ignaram talium artium, in novam quan­<lb/>dam vir&longs;utiam depravavit.<emph.end type="italics"/></s> <s id="s.001664"> Quid enim &longs;i quis præclaro artifi­<lb/>cio ex naturæ the&longs;auris deprompto abutatur? </s> <s id="s.001665">Dolos & fallacias, <lb/>aut errores, quibus in&longs;ici pote&longs;t libræ u&longs;us, ideò retegemus: <lb/>ut nimirum quod Ju&longs;titiæ commutativæ &longs;ymbolum datur, om­<lb/>ni inju&longs;titiæ &longs;u&longs;picione vacet. </s> <s id="s.001666">Cæterùm quæ nobis ine&longs;t arbi­<lb/>trij libertas, poti&longs;&longs;ima naturæ rationis compotis prærogativa, <lb/>libræ, aut &longs;tateræ jure merito comparatur, quâ iniqui abuten­<lb/>tes dicuntur P&longs;alm. <!-- REMOVE S-->61. <emph type="italics"/>Mendaces filij hominum in &longs;tateris:<emph.end type="italics"/> ubi <lb/>S. <!-- REMOVE S-->Ba&longs;ilius hom. </s> <s id="s.001667">in P&longs;alm. <!-- REMOVE S-->61. ait <emph type="italics"/>Cuilibet no&longs;trûm intus &longs;tatera <lb/>quædam e&longs;t à Conditore omnium apparata, per quam rerum naturam <lb/>po&longs;&longs;is probè digno&longs;cere.<emph.end type="italics"/> & infra: <emph type="italics"/>Tibi namque propria datur libra, <lb/>quæ &longs;ufficiens di&longs;crimen boni, ac mali demon&longs;trat. </s> <s id="s.001668">Corporea enim <lb/>pondera in libræ lancibus probamus; quæ verò ad in&longs;tituendam vi­<lb/>tam eligenda veniunt, per liberum arbitrium di&longs;cernimus: quod & <lb/>&longs;tateram nominavit, quòd momentum æquale ad utrumlibet po&longs;&longs;it <lb/>capere.<emph.end type="italics"/><lb/></s> </p> <p type="main"> <s id="s.001669"><emph type="center"/>CAPUT I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001670"><emph type="center"/><emph type="italics"/>Libræ forma, & natura exponitur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001671">EO con&longs;ilio in&longs;tituta e&longs;t libra, ut certis, ac notis ponderi­<lb/>bus, ignotæ gravitatis quantitas indagetur, quæ demùm <lb/>innote&longs;cit, cum æquatis hinc & hinc ponderum libræ adnexo­<lb/>rum momentis, neutro prævalente, libra con&longs;i&longs;tit. </s> <s id="s.001672">In hoc <pb pagenum="227" xlink:href="017/01/243.jpg"/>in&longs;trumento con&longs;ideratur pri­<lb/><figure id="id.017.01.243.1.jpg" xlink:href="017/01/243/1.jpg"/><lb/>mùm <emph type="italics"/>Iugum<emph.end type="italics"/>, &longs;eu <emph type="italics"/>&longs;capus,<emph.end type="italics"/> &longs;eu <lb/><emph type="italics"/>librile<emph.end type="italics"/> AB: hoc bifariam divi­<lb/>ditur in C, quod, <emph type="italics"/>Centrum<emph.end type="italics"/> li­<lb/>bræ dicitur, non quia &longs;it ne­<lb/>ce&longs;&longs;ariò Centrum gravitatis li­<lb/>bræ, &longs;ed quia e&longs;t Centrum, <lb/>circa quod agitur, &longs;eu ver&longs;a­<lb/>tur jugum, infixo nimirum in C axiculo, qui & <emph type="italics"/>Agina<emph.end type="italics"/> Latinis, <lb/>Græcis apud Ari&longs;totelem in quæ&longs;t. </s> <s id="s.001673">Mechan. <emph type="italics"/>Spartum<emph.end type="italics"/> dicitur. </s> <lb/> <s id="s.001674">Partes autem jugi videlicet CA, & CB. <emph type="italics"/>Brachia, Radij,<emph.end type="italics"/> aut <lb/>etiam ab aliquibus <emph type="italics"/>Librilia<emph.end type="italics"/> vocantur. </s> <s id="s.001675">Ex medio jugi ad per­<lb/>pendiculum a&longs;&longs;urgit lingula CD, quæ in&longs;eritur an&longs;æ EF com­<lb/>plectenti capita axiculi, adeò ut &longs;u&longs;pensâ ex F an â, quæ ho­<lb/>rizonti ad perpendiculum immineat, tùm demùm intelligatur <lb/>factum æquilibrium, cum lingula an&longs;æ congruit, & jugum <lb/>con&longs;i&longs;tit horizonti parallelum. </s> <s id="s.001676">Utrùm autem <emph type="italics"/>Trutina<emph.end type="italics"/> dicenda <lb/>&longs;it ip&longs;a lingula, an verò an&longs;a, non conveniunt Authores: li­<lb/>tem Grammaticis dirimendam relinquo. </s> </p> <p type="main"> <s id="s.001677">Extremis brachiorum punctis A & B adnectitur utrumque <lb/>pondus, tam notum, quod e&longs;t alterius men&longs;ura, quàm igno­<lb/>tum; cujus gravitas examinatur. </s> <s id="s.001678">Nihil autem refert, an pon­<lb/>dera uncinis adnexa dependeant, an verò lancibus indè pen­<lb/>dentibus imponantur; id quod vulgare e&longs;t magi&longs;que u&longs;itatum, <lb/>& libræ fecit nomen <emph type="italics"/>Bilanci.<emph.end type="italics"/></s> <s id="s.001679"> Illud enim præcipuum e&longs;t, ac <lb/>maximè attendendum, quòd omnia hinc & hinc æqualia &longs;int, <lb/>nimirum pondus unius lancis cum funiculis &longs;eu catenulis æqua­<lb/>le &longs;it ponderi alterius lancis cum &longs;uis appendiculis (pondus, in­<lb/>quam, ponderi æquale &longs;it; nil enim intere&longs;t æquales ne? </s> <s id="s.001680">an <lb/>inæquales fuerint utriu&longs;que lancis funiculi &longs;ecundùm longitu­<lb/>dinem, modò in æquali di&longs;tantiâ à centro adnectantur) & bra­<lb/>chium alterum majus non &longs;it reliquo brachio non &longs;olùm quoad <lb/>gravitatem, quæ materiæ jugi ine&longs;t, &longs;ed poti&longs;&longs;imùm quoad <lb/>ip&longs;orum brachiorum longitudinem. </s> </p> <p type="main"> <s id="s.001681">Porrò hæc brachiorum longitudo non e&longs;t de&longs;umenda, ut ita <lb/>loquar, materialiter, à centro jugi ad extremitatem, ubi mate­<lb/>ria de&longs;init, ex quâ con&longs;tat, &longs;ivè ferrum &longs;it, &longs;ivè lignum, &longs;ivè <lb/>aliud quidpiam: &longs;ed brachiorum longitudinem definiunt <pb pagenum="228" xlink:href="017/01/244.jpg"/>puncta jugi; ex quibus pondera dependent: horum etenim <lb/>di&longs;tantiam à centro omnino æqualem e&longs;&longs;e oportet. </s> <s id="s.001682">Huju&longs;modi <lb/>autem puncta non alia &longs;unt, quàm puncta contactûs jugi & an­<lb/>nulorum &longs;eu uncinorum illi infixorum, quibus deinde lances <lb/>aut pondera adnectuntur. </s> <s id="s.001683">Hoc illud e&longs;t, in quo maxima arti­<lb/>ficis indu&longs;tria, atque diligentia collocanda e&longs;t, ut exacti&longs;&longs;imam <lb/>brachiorum æqualitatem a&longs;&longs;equatur. </s> </p> <p type="main"> <s id="s.001684">Data itaque hac, quam diximus, brachiorum æqualitate, &longs;i <lb/>æqualia pondera hinc & hinc addantur, manife&longs;tum e&longs;t jugum <lb/>libræ ex aginâ &longs;u&longs;pen&longs;um ad neutram partem inclinari, &longs;ed ma­<lb/>nere horizonti parallelum; fieri namque non pote&longs;t, ut extremi­<lb/>tas altera de&longs;cendat, quin oppo&longs;ita extremitas cum adnexo pon­<lb/>dere a&longs;cendat, & quidem æquali motu propter brachiorum <lb/>æqualitatem. </s> <s id="s.001685">Finge enim pondus B de&longs;cendere in F, utique <lb/><figure id="id.017.01.244.1.jpg" xlink:href="017/01/244/1.jpg"/><lb/>pondus A a&longs;cendet in E, at­<lb/>que de&longs;cribent arcus BF & <lb/>AE æquales, quippe qui <lb/>æqualibus angulis ad verti­<lb/>cem in C &longs;ubtenduntur, & <lb/>ab æqualibus radiis CB, CA <lb/>de&longs;cribuntur. </s> <s id="s.001686">At æqualis e&longs;t in B vis de&longs;cendendi atque in A <lb/>repugnantia ad a&longs;cendendum; illa igitur præpollere non pote&longs;t. </s> <lb/> <s id="s.001687">Siquidem vis de&longs;cendendi componitur ex ponderis gravitate, <lb/>& non impeditâ motûs naturalis velocitate; repugnantia verò <lb/>ad a&longs;cendendum componitur & ex ponderis contranitentis gra­<lb/>vitate, & ex velocitate motûs præter naturam: &longs;unt autem gra­<lb/>vitates ex hypothe&longs;i æquales, motus etiam per arcus BF & AE <lb/>e&longs;&longs;ent æquales; ac proinde vis tendendi deor&longs;um inveniens <lb/>æqualem oppo&longs;itam repugnantiam ad motum &longs;ur&longs;um nequit illi <lb/>imprimere impetum, quo per vim moveatur: ut enim &longs;equa­<lb/>tur motus, aut gravitates di&longs;pares e&longs;&longs;e oportet, aut motuum Po­<lb/>tentiæ moventis & Ponderis moti velocitates inæquales, ut ma­<lb/>jor &longs;it Ratio huju&longs;modi velocitatum, quàm &longs;it reciproca Ratio <lb/>gravitatum: alioquin nulla e&longs;&longs;et virium movendi & re&longs;i&longs;tentiæ <lb/>inæqualitas, ubi omnia e&longs;fent æqualia. </s> <s id="s.001688">Cum itaque in librâ &longs;ic <lb/>con&longs;titutâ intercedat omnimoda æqualitas & brachiorum, qui­<lb/>bus definitur motus, & gravitatum, quæ &longs;ibi invicem æquali­<lb/>ter ob&longs;i&longs;tunt, ac proinde eadem &longs;it reciproca Ratio gravitatum <pb pagenum="229" xlink:href="017/01/245.jpg"/>& motuum, jugum libræ horizonti parallelum con&longs;i&longs;tere ne­<lb/>ce&longs;&longs;e e&longs;t; & in alteram partem &longs;i inclinerur, manife&longs;tum e&longs;t in <lb/>illâ lance plus ponderis fui&longs;&longs;e impo&longs;itum, quàm in reliquâ. </s> </p> <p type="main"> <s id="s.001689">Ut autem quàm exacti&longs;&longs;imè ponderum ignota gravitas exa­<lb/>minari queat, opus e&longs;t ut axiculus jugo infixus (&longs;altem in &longs;upe­<lb/>riore parte, cui &longs;capus incumbit) exqui&longs;itè cylindricam figu­<lb/>ram obtineat; hinc enim fiet, ut cum rotundo foramine &longs;capi <lb/>contactus fiat in lineâ, quamcumque tandem po&longs;itionem ha­<lb/>beat ip&longs;e &longs;capus: nam quemadmodum ex prop. 13. lib. 3. duo <lb/>circuli &longs;e intùs contingentes tangunt in puncto, ita duæ &longs;uper­<lb/>ficies cylindricæ, cava altera, altera convexa, &longs;e tangunt in li­<lb/>neà. </s> <s id="s.001690">Id &longs;i fiat facilè ab æquilibrio deflectet &longs;capus, &longs;i vel modi­<lb/>ca intercedat ponderum inæqualitas. </s> <s id="s.001691">At &longs;i angulatus fuerit axi­<lb/>culus, vel &longs;uperior foraminis pars rotunditatem non fuerit a&longs;&longs;e­<lb/>cuta, jam non in unâ lineâ, &longs;ed in pluribus contactus fieret, at­<lb/>que adeò iners e&longs;&longs;et ad motum &longs;eapus, etiam&longs;i non omninò <lb/>æqualia e&longs;&longs;ent pondera lancibus impo&longs;ita. </s> </p> <p type="main"> <s id="s.001692">Quare artifices illos non probo, qui axem ita ef&longs;ormant, ut <lb/>&longs;uperior pars in aciem de&longs;inat, illud &longs;ibi per&longs;uadentes, quod <lb/>minore partium conflictu &longs;e tangentes axis & &longs;capus faciliorem <lb/>relinquant in alterutram partem motum libræ. </s> <s id="s.001693">Id quod ut ve­<lb/>rum &longs;it, non tamen vacat periculo, ne, dum axis capita in&longs;e­<lb/>runtur an&longs;æ, acies illa planè &longs;ursùm non dirigatur, &longs;ed modi­<lb/>cum in alterutram partem vergat: quæ declinatio &longs;i contingat, <lb/>foramen autem exactè rotundum fuerit, miraculo proximum <lb/>cen&longs;e, &longs;i libra vacua æquilibrium con&longs;tituat, ita ut lingula ritè <lb/>collocata congruat an&longs;æ; acies &longs;i quidem illa dividit inæquali­<lb/>ter &longs;capi longitudinem, & brachium alterum altero longius e&longs;t, <lb/>atque præponderat. </s> <s id="s.001694">Hoc vitium ubi libra contraxerit, inepti <lb/>artifices nihil &longs;u&longs;picati ab axe malè conformato, aut perperam <lb/>di&longs;po&longs;ito, ortum duxi&longs;&longs;e, vel brachium extenuant, vel lancem <lb/>immutant, donec æquilibrium inveniant. </s> <s id="s.001695">Verùm libram hu­<lb/>ju&longs;modi dolo&longs;am e&longs;&longs;e inferiùs con&longs;tabit propter brachiorum in­<lb/>æqualitatem: quæ quidem levem infert ponderum differen­<lb/>tiam in rebus exigui momenti contemnendam; &longs;ed in iis, quæ <lb/>exqui&longs;itam ponderis men&longs;uram exigunt, non leve damnum <lb/>hinc pote&longs;t emergere. </s> </p> <p type="main"> <s id="s.001696">Quod &longs;i axis non &longs;it an&longs;æ, &longs;ed &longs;capo, firmiter infixus, volua-<pb pagenum="230" xlink:href="017/01/246.jpg"/>turautem in an&longs;æ foraminibus (id quod artificibus non paucis <lb/>magis arridet) jam non &longs;uperior; &longs;ed inferior axiculi pars at­<lb/>tendenda e&longs;t; quippe quæ inferiorem foraminum an&longs;æ partem <lb/>contingit; & eadem, quæ de &longs;uperiore parte dicebantur, ob­<lb/>&longs;ervanda &longs;unt. </s> <s id="s.001697">Illud tamen præterea in an&longs;æ foraminibus ob­<lb/>&longs;ervandum venit, quod eorum infima pars ita &longs;it con&longs;tituta, ut <lb/>axis illis incumbens parallelus &longs;it horizonti, quando an&longs;a &longs;u&longs;­<lb/>penditur, ut liberè pendeat, vel ita collocatur, ut ad perpen­<lb/>diculum horizonti immineat: alioquin axe inclinato, jugum <lb/>urgeret aiteram an&longs;æ partem, ab alterâ recederet; ex quo jugi <lb/>cuman, conflictu aliqua motui difficultas crearetur. </s> </p> <p type="main"> <s id="s.001698">Jam verò quod ad pondera attinet, &longs;upervacaneum e&longs;t mo­<lb/>nere non omnia pondera omnibus libris convenire: quamvis <lb/>enim libra, quâ libra e&longs;t, nuliam pror&longs;us re&longs;puat ponderum gra­<lb/>vitatem, &longs;ed omnem quorumcumque ponderum æqualitatem <lb/>apta &longs;it indicare &longs;uo æquilibrio; quia tamen ex materiâ con&longs;tat, <lb/>quæ definitam habet &longs;oliditatem atque partium firmitatem (ut <lb/>nihil dicam de certis atque definitis viribus retinentis an&longs;am, <lb/>& cum ansâ libram, ac utrumque pondus) fieri pote&longs;t, ut adeò <lb/>gravia lancibus imponantur onera, quæ brachiorum rectitudi­<lb/>nem inflectant, & eorum æqualitatem corrumpant: Quare te­<lb/>nuioribus libris parva pondera examinantur, cra&longs;&longs;ioribus ma­<lb/>jora. </s> <s id="s.001699">Illud potiùs cavendum e&longs;t, ne pondera, quibus tanquam <lb/>men&longs;urâ utimur, fallacia &longs;int, quia fal&longs;a, aut excedendo legi­<lb/>timam gravitatis quantitatem, aut ab illâ deficiendo. </s> </p> <p type="main"> <s id="s.001700">Quamvis autem tot pondera minimæ men&longs;uræ adhibere po&longs;­<lb/>&longs;emus, quot numerare oporteret ad explorandam propo&longs;itæ <lb/>gravitatis ignotæ quantitatem, hoc tamen valde incommodum <lb/>e&longs;&longs;et: quid enim, &longs;i lanius carnem in macello vendens grana <lb/>numerare cogeretur, quæ æquilibrium cum carne con&longs;tituunt? <lb/></s> <s id="s.001701">&longs;ed & inutilis e&longs;&longs;et labor, nam multa &longs;unt, quorum quantitas <lb/>non e&longs;t ad vivum re&longs;ecanda, & minuti&longs;&longs;imæ particulæ fru&longs;tra <lb/>inve&longs;tigantur. </s> <s id="s.001702">Subtilitas hæc relinquatur gemmariis, aurifici­<lb/>bus, auríque monetalis cu&longs;oribus, quibus damnum e&longs;&longs;et minu­<lb/>tias contemnere. </s> <s id="s.001703">Quamquam nec i&longs;tis author fuerim, ut &longs;in­<lb/>gularibus granis uterentur, &longs;ed potiùs ponderibus, quæ pluri­<lb/>bus granis æquivalerent; &longs;i enim &longs;ingula grana à legitimo pon­<lb/>dere dificiunt per cente&longs;imam grani partem, quæ facilè &longs;ensûs <pb pagenum="231" xlink:href="017/01/247.jpg"/>aciem fugit, additis centum huju&longs;modi granis error e&longs;t inte­<lb/>gri grani deficientis; & in uncia libræ Romanæ ponderalis ad <lb/>monetam pertinentis cum grana 576 contineantur, in uncia <lb/>auri error e&longs;&longs;et granorum ferè &longs;ex deficientium, & in integrâ <lb/>librâ, quæ e&longs;t granorum 6912, e&longs;&longs;et error granorum 69; qui <lb/>tamen error vix contingat, &longs;i a&longs;&longs;umatur integra uncia, aut li­<lb/>bra: illud &longs;i quidem, quod &longs;olitarium præ &longs;ua tenuitate in con­<lb/>&longs;pectum non cadit, cum pluribus &longs;imilibus conjunctum evadit <lb/>demum notabile atque con&longs;picuum. </s> <s id="s.001704">Quare ad paranda pon­<lb/>dera huju&longs;modi &longs;ubtiliora, a&longs;&longs;ume laminam metallicam ponde­<lb/>re unius libræ, &longs;ed æquabiliter exten&longs;am, eju&longs;que duodecimam <lb/>partem accipe; hæc erit Uncia, quam &longs;epones. </s> <s id="s.001705">Alterius Unciæ <lb/>octavam partem a&longs;&longs;umens habebis Draclimam. </s> <s id="s.001706">Drachmæ pars <lb/>tertia dabit &longs;crupulum. </s> <s id="s.001707">Scrupuli &longs;emi&longs;&longs;is e&longs;t obolus. </s> <s id="s.001708">Oboli <lb/>triens e&longs;t &longs;iliqua. </s> <s id="s.001709">Demùm &longs;iliquæ quadrans e&longs;t Granum. <!-- KEEP S--></s> <s id="s.001710">Ex <lb/>hac minutâ divi&longs;ione &longs;atis con&longs;tat, quàm obnoxiæ errori &longs;int <lb/>minores particulæ præ majoribus; idemque error, qui in unciâ <lb/>fingularis e&longs;&longs;et, & ut nullus con&longs;ideraretur, toties repetitus, <lb/>quot grana in unciâ continentur, jam non e&longs;&longs;et contemnen­<lb/>dus. </s> <s id="s.001711">Id autem dictum intelligatur etiam in majoribus ponde­<lb/>ribus, ubi unciæ non reputantur, &longs;atius e&longs;&longs;e majora pondera <lb/>habere, quàm minimam men&longs;uram &longs;æpiùs multiplicatam a&longs;­<lb/>&longs;umere. </s> </p> <p type="main"> <s id="s.001712">Sed quoniam adhuc incommodum accideret tot habere <lb/>men&longs;uras, quæ juxta &longs;eriem naturalem numerorum cre&longs;cerent, <lb/>ut propo&longs;itæ paucitatis examinandæ quantitas indagetur, ob­<lb/>&longs;ervatum e&longs;t non leve compendium, quod offert progre&longs;&longs;io <lb/>Geometrica ab unitate incipiens, & in Ratione dupla aut tri­<lb/>plâ progrediens. </s> <s id="s.001713">Nam maximum terminum progre&longs;&longs;ionis du­<lb/>plæ &longs;ibimet ip&longs;i additum &longs;i mulctaveris unitate, & in progre&longs;­<lb/>&longs;ione triplâ maximo termino unitate mulctato &longs;i re&longs;idui &longs;emi&longs;­<lb/>&longs;em addideris, numerum habebis gravitatum omnium, quæ <lb/>paucis illis ponderibus examinari po&longs;&longs;unt. </s> <s id="s.001714">Sic dentur octo pon­<lb/>dera in Ratione duplâ incipiendo ab uncia 1; octavum e&longs;t <lb/>unc. </s> <s id="s.001715">128: hunc numerum duplica, & à 256 aufer unitatem, <lb/>reliquus numerus 255 indicat octo illis ponderibus po&longs;&longs;e in li­<lb/>brâ examinari omnes gravitates ab uncia 1 ad uncias 255. Si­<lb/>mili modo in Ratione triplâ dentur quatuor pondera 1. 3. 9. 27. <pb pagenum="232" xlink:href="017/01/248.jpg"/>aufer ab ultimo unitatem, remanet 26, cujus &longs;emi&longs;&longs;is 13 addi­<lb/>tus numero 27 dat 40: cujus igitur gravitatis e&longs;t primum pon­<lb/>dus ut 1, tot gravitates u&longs;que ad 40 examinari po&longs;&longs;unt illis &longs;olis <lb/>quatuor ponderibus. </s> <s id="s.001716">Præ&longs;tat autem uti ponderibus in Ratio­<lb/>ne duplâ, quia licèt plura pondera requirantur, omnia tamen <lb/>&longs;eor&longs;im in propriâ libræ lance collocantur: at &longs;i Ratio ponde­<lb/>rum &longs;it tripla, aliquâ commutatione uti nece&longs;&longs;e e&longs;t, ut in ad­<lb/>jecta Tabella ob&longs;ervabis, quæ u&longs;que ad numerum 40. exten­<lb/>ditur: Ubi etiam vides in Ratione triplâ &longs;ufficere quatuor pon­<lb/>dera 1. 3.9. 27, at in duplâ exigi &longs;ex videlicet 1. 2. 4. 8. 16. 32. </s> </p> <p type="table"> <s id="s.001717">TABELLE WAR HIER<!-- KEEP S--></s> </p> <pb pagenum="233" xlink:href="017/01/249.jpg"/> <p type="table"> <s id="s.001718">TABELLE WAR HIER<!-- KEEP S--></s> </p> <p type="main"> <s id="s.001719">At contingere pote&longs;t paratis hi&longs;ce ponderibus in Ratione <lb/>duplâ aut triplâ aliquid abundare, & maximum terminum cæ­<lb/>teris additum excedere quæ&longs;itum numerum, (ut hic, &longs;i opus <lb/>e&longs;&longs;et provenire &longs;olum ad 40, maximus terminus 32 e&longs;t abun­<lb/>dans) proptereà retentâ cæterorum &longs;ummâ adde aliud pondus, <lb/>ut quæ&longs;itum numerum compleat, & e&longs;t illud, quo opus e&longs;t; <lb/>&longs;ic 1. 2. 4. 8. 16. conficiunt &longs;ummam 31; aufer 31 ex 40, re&longs;i­<lb/>duum e&longs;t 9; &longs;it igitur &longs;extum pondus 9, & &longs;atis erit u&longs;que ad <lb/>40; quia cum habeantur reliquis ponderibus omnes numeri <lb/>infra 31, jam ex 23 & 9 fit 32, ex 24 & 9 fit 33, & &longs;ic de re­<lb/>liquis deinceps. </s> <s id="s.001720">Idem dic de aliâ qualibet &longs;ummâ majore <lb/>quàm ferant data pondera, minore tamen quàm opus &longs;it, &longs;i <lb/>adhuc unum pondus in eâdem progre&longs;&longs;ione adderetur; &longs;ufficit <lb/>enim re&longs;iduum. </s> <s id="s.001721">Exemplum habes in &longs;uperiore Tabella pon­<lb/>derum in Ratione triplâ, ubi quatuor conficiunt 40, &longs;ed &longs;i ad­<lb/>deretur quintum in eadem Ratione 81, e&longs;&longs;et nimis magnum, <pb pagenum="234" xlink:href="017/01/250.jpg"/>&longs;i &longs;olùm habere velimus pondera infra 121: quæratur u&longs;que ad <lb/>52, & quia inter 40 & 52 differentia e&longs;t 12, quintum pondus <lb/>ut 12 &longs;ufficiet. </s> <s id="s.001722">Hinc quia ad libram requiruntur &longs;olum 24 &longs;e­<lb/>munciæ, ad unciam 24 &longs;crupuli, ad &longs;crupulum 24 grana, &longs;i <lb/>pondera &longs;int in Ratione triplâ, &longs;ufficiunt tria ponderâ 1. 3.9. <lb/>quæ conficiunt 13, & quartum pondus &longs;it 11, ut compleatur <lb/>&longs;umma 24: & in Ratione duplâ &longs;ufficiunt quatuor pondera <lb/>1. 2. 4. 8. quæ conficiunt 15, & quintum pondus 9 complens <lb/>&longs;ummam 24. illud e&longs;t, quod requiritur, ut ex adjectis Tabel­<lb/>lis liquet. </s> </p> <p type="table"> <s id="s.001723">TABELLE WAR HIER<!-- KEEP S--></s> </p> <p type="table"> <s id="s.001724">TABELLE WAR HIER<!-- KEEP S--></s> </p> <p type="main"> <s id="s.001725">Unum hîc, ubi de Ponderibus &longs;ermo e&longs;t, obiter moneo, libræ <lb/>nomen apud Romanos æquivocum fui&longs;&longs;e, alia enim erat libra <lb/>Ponderalis aridorum, alia Men&longs;uralis liquidorum (& poti&longs;&longs;i­<lb/>mum olei, quod cornu librali metiebantur) quam inci&longs;is & in­<lb/>&longs;culptis lineis in uncias 12 partiebantur, quemadmodum & li­<lb/>bra pondo in uncias pariter 12 di&longs;tinguebatur: &longs;ed inter utram­<lb/>que libram, &longs;i materia ip&longs;a ad pondus revocabatur, non exi­<lb/>guum erat di&longs;crimen; ut enim ex proprio experimento te&longs;ta-<pb pagenum="235" xlink:href="017/01/251.jpg"/>tur Galenus lib. 6. cap. 8. <emph type="italics"/>de compo&longs;itione medicam. </s> <s id="s.001726">per genera. <emph.end type="italics"/><!--neuer Satz--> <lb/>Libra men&longs;ura &longs;olùm uncias decem continebat, quarum li­<lb/>bra pondo erat duodecim: quapropter uncia men&longs;uralis ad un­<lb/>ciam ponderalem erat ut 5 ad 6 &longs;pectatâ gravitate & quantita­<lb/>te materiæ. <lb/></s> </p> <p type="main"> <s id="s.001727"><emph type="center"/>CAPUT II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001728"><emph type="center"/><emph type="italics"/>Libra inæqualium brachiorum expenditur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001729">USus libræ brachiorum inæqualium minùs nece&longs;&longs;arius e&longs;t, <lb/>ac propterea neque communis aut vulgaris, ni&longs;i quatenus <lb/>ad &longs;tateram traductus e&longs;t: illam tamen hîc con&longs;iderare erit <lb/>operæ pretium, ut æquilibrij rationes magis innote&longs;cant. </s> <s id="s.001730">Sit <lb/>libra AB, cujus centro C <lb/><figure id="id.017.01.251.1.jpg" xlink:href="017/01/251/1.jpg"/><lb/>dividatur jugum in brachia <lb/>inæqualia CA & CB. <!-- KEEP S--></s> <lb/> <s id="s.001731">Certum e&longs;t, etiam &longs;i nul­<lb/>lum addatur pondus, ju­<lb/>gum ex centro C &longs;u&longs;pen­<lb/>fum retinere non po&longs;&longs;e po­<lb/>&longs;itionem AB horizonti pa­<lb/>rallelam; quia licet punctum C &longs;it centrum motûs libræ, non <lb/>e&longs;t tamen centrum gravitatis illius; hoc enim e&longs;t in puncto ju­<lb/>gum (quod hîc æquabiliter ductum ponitur) bifariam dividen­<lb/>te, videlicet in I, quod æquales gravitates IA & IB cir­<lb/>cum&longs;tant. </s> <s id="s.001732">Verùm interim ex hypothe&longs;i fingamus lineam AB <lb/>omni gravitate carentem; & in ip&longs;is libræ extremitatibus &longs;ta­<lb/>tuamus pondera eam inter &longs;e reciprocè Rationem habentia, <lb/>quæ e&longs;t Ratio brachiorum, & ut CA ad CB, ita &longs;it pondus B <lb/>ad pondus A. <!-- KEEP S--></s> <s id="s.001733">Pondera hæc, quæ in lancibus libræ vulgaris <lb/>æqualium brachiorum magnam momentorum inæqualitatem <lb/>haberent, quia inæqualiter gravia, hîc æquilibrium con&longs;ti­<lb/>tuunt, quamvis inæquales &longs;int eorum gravitates ab&longs;olutæ, quia <lb/>libræ brachia reciprocè: &longs;ecundùm eandem Rationem in­<lb/>æqualia: quatenus enim alligantur pondera hæc extremita-<pb pagenum="236" xlink:href="017/01/252.jpg"/>tibus libræ, æqualia obtinent momenta, nec jugum AB <lb/>pote&longs;t in alterutram partem inclinari, cum neutrum pon­<lb/>dus po&longs;&longs;it ab altero a&longs;&longs;umere vim, qua &longs;ursùm moveatur, <lb/>majorem oppo&longs;itâ virtute innatá de&longs;cendendi, qua repu­<lb/>gnat, ne elevetur. </s> <s id="s.001734">Sit CA ad CB ut 1 ad 4, & vici&longs;&longs;im pon­<lb/>dus B ut 1 ad pondus A ut 4. Si gravitates dumtaxat con­<lb/>&longs;iderentur, virtus ponderis A e&longs;t ut 4, virtus verò ponderis B <lb/>ut 1: &longs;ed quia à centro motûs C retinentur, nec liberè rectâ viâ <lb/>moveri po&longs;&longs;unt, impedimentum recipiunt pro brachiorum lon­<lb/>gitudine, minû&longs;que impeditur de&longs;cen&longs;us aut a&longs;cen&longs;us rectus <lb/>ponderis, quod longiori brachio adjacet, magis, quod brevio­<lb/>ri. </s> <s id="s.001735">Illud igitur pondus, quod majori brachio adnectitur, &longs;i <lb/>de&longs;cendat, magis de&longs;cendit, &longs;i a&longs;cendat, magis a&longs;cendit; quod <lb/>verò breviori, &longs;i a&longs;cendat, minùs a&longs;cendit, & &longs;i de&longs;cendat, <lb/>minùs de&longs;cendit: atque adeò &longs;i B de&longs;cenderet in E, men&longs;ura <lb/>de&longs;censùs e&longs;&longs;et perpendicularis EG, a&longs;&longs;en&longs;um autem ponderis <lb/>A in D metiretur perpendicularis DF: idem dic &longs;i A de&longs;cen­<lb/>deret, & B a&longs;cenderet. </s> <s id="s.001736">Porrò DF & EG &longs;unt in Ratione <lb/>brachiorum CA & CB ut patet, quia triangula rectangula <lb/>CFD, & CGE, præter rectos angulos ad F & G æquales, ha­<lb/>bent etiam æquales ad C angulos ad verticem, & per 32. lib. 1. <lb/>&longs;unt æquiangula; igitur per 4 lib. 6. ut CD ad CE, ita DF <lb/>ad EG; at CD æqualis e&longs;t ip&longs;i CA, & CE ip&longs;i CB (e&longs;t enim <lb/>eadem linea, quæ mutatâ po&longs;itione AB venit in DE) igitur <lb/>ut CA ad CB ita DF ad EG. <!-- KEEP S--></s> <s id="s.001737">Quare ratione po&longs;itionis pon­<lb/>dus B vim habet de&longs;cendendi, & re&longs;i&longs;tit a&longs;cen&longs;ui, ut 4, pon­<lb/>dus autem A vim habet de&longs;cendendi, ac proinde etiam re­<lb/>&longs;i&longs;tendi, ne a&longs;cendat, &longs;olùm ut 1. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001738">Cum itaque momentum de&longs;cendendi (idem e&longs;to judicium <lb/>de momento repugnantiæ, ne a&longs;cendat) componatur tùm ex <lb/>gravitate ponderis, tùm ex propen&longs;ione ad motum, hoc e&longs;t ex <lb/>motûs, qui con&longs;equi po&longs;&longs;et, velocitate, manife&longs;tum e&longs;t gravi­<lb/>tatem ut 4, cujus motus e&longs;&longs;et ut 1, nec po&longs;&longs;e vincere gravitatem <lb/>ut 1, cujus motus e&longs;&longs;et ut 4, nec vici&longs;&longs;im po&longs;&longs;e ab illâ vinci; <lb/>e&longs;t &longs;iquidem inter gravitatem quadruplum &longs;emel, & gravita­<lb/>tem &longs;ubquadruplam quater Ratio æqualitatis; victoria autem <lb/>obtineri non pote&longs;t, ni&longs;i intercedat virium inæqualitas. </s> <s id="s.001739">Si <lb/>enim pondera e&longs;&longs;ent æqualia, ponderis A re&longs;i&longs;tentia ratione <pb pagenum="237" xlink:href="017/01/253.jpg"/>motûs e&longs;&longs;et &longs;ubquadrupla, &longs;ed quadruplicatur ratione gravita­<lb/>tis, ergo re&longs;i&longs;tentia e&longs;t æqualis: item &longs;i longitudines e&longs;&longs;ent <lb/>æquales, re&longs;i&longs;tentia ponderis B e&longs;&longs;et &longs;ubquadrupla ratione <lb/>gravitatis, &longs;ed quadruplicatur ratione di&longs;tantiæ CB; ergo in B <lb/>e&longs;t æqualis. </s> </p> <p type="main"> <s id="s.001740">Neutrum igitur pondus pote&longs;t oppo&longs;ito ponderi impetum <lb/>imprimere, quo elevetur; quia nimirum unaquæque gravitas <lb/>majorem impetum alteri communicare non pote&longs;t, quàm po&longs;­<lb/>&longs;it ip&longs;a concipere, ac propterea impetus gravitatis B, quæ e&longs;t <lb/>ut CA, potens conari deor&longs;um ut GE, &longs;i imprimeretur gravi­<lb/>tati A, quæ e&longs;t ut CB, deberet illam elevare ut FD: Atqui <lb/>gravitas ip&longs;ius A, quæ e&longs;t ut CB, conatur deorsùm ut FD, & <lb/>ejus impetus &longs;i gravitati B, quæ e&longs;t ut CA, imprimeretur, il­<lb/>lam elevare deberet ut GE: igitur in unaquâque gravitate <lb/>æqualis e&longs;&longs;et eju&longs;dem conatus deorsùm & vis illata nitens &longs;ur­<lb/>sùm, nec plus præ&longs;tare po&longs;&longs;et impetus impre&longs;&longs;us, quàm innatus. </s> <lb/> <s id="s.001741">Utraque igitur con&longs;i&longs;tere debet, & neutra impetum acquirit, <lb/>aut ab alterâ impetum accipit, quia fru&longs;tra e&longs;&longs;et impetus acqui­<lb/>&longs;itus aut impre&longs;&longs;us, quem nullus con&longs;equi pote&longs;t motus. </s> <s id="s.001742">Quare <lb/>cum eadem &longs;it gravitatum Ratio ut CA ad CB, atque motuum <lb/>reciprocè ut FD ad GE, ex 16 lib. 6. rectangulum &longs;ub extre­<lb/>mis CA, hoc e&longs;t pondere B, ut 1, & motu GE, ut 4, æquale <lb/>e&longs;t rectangulo &longs;ub mediis CB, hoc e&longs;t pondere A ut 4, & mo­<lb/>tu FD ut 1: &longs;unt igitur æqualia momenta, quæ componuntur <lb/>ex gravitate ut 1 & motu ut 4, atque ex gravitate ut 4 & <lb/>motu ut 1. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001743">Ex his aperti&longs;&longs;imè liquet, cur &longs;uperiori capite tantopere in­<lb/>culcata &longs;it brachiorum æqualitas in libræ jugo, ut ex æquili­<lb/>brio innote&longs;cat propo&longs;iti ponderis ignota gravitas; hæc enim <lb/>æqualis cen&longs;etur notæ gravitati, ubi cùm oblato pondere illa <lb/>æquâ lance libratur: quia &longs;cilicet, &longs;i inæqualia e&longs;&longs;ent brachia, <lb/>inæquales e&longs;&longs;ent propen&longs;iones ad motum, &longs;eu motuum veloci­<lb/>tates, quæ ad componendam momentorum Rationem concur­<lb/>runt; adeóque fieri non po&longs;&longs;et, ut æquales e&longs;&longs;ent gravitates in <lb/>lancibus; nam minor gravitas ex brachio longiore plus habet <lb/>momenti, quàm ex breviore, pro ratione inæqualitatis brachio­<lb/>rum. </s> <s id="s.001744">Verum e&longs;t libram huju&longs;modi brachiorum inæqualium <lb/>vacuam po&longs;&longs;e priùs ad æquilibritatem reduci, deinde, illâ &longs;ic <pb pagenum="238" xlink:href="017/01/254.jpg"/>æquilibri con&longs;titutâ po&longs;&longs;e lancibus imponi Reciprocè pondera <lb/>pro Ratione inæqualium brachiorum, & ex æquilibrio argui <lb/>ponderum illorum Rationem, non tamen æqualitatem: &longs;edar­<lb/>tificium hoc, quod peritioribus nihil officeret, an&longs;am non mo­<lb/>dicam furacibus, & dolo&longs;is mercatoribus præberet decipiendi <lb/>imperitos; quamvis enim libræ huju&longs;modi æquilibri impo&longs;itis, <lb/>hinc & hinc ponderibus adhuc fieret æquilibrium, &longs;ignum <lb/>quidem e&longs;&longs;et æqualibus momentis addita e&longs;&longs;e æqualia momen­<lb/>ta gravitatis, non tamen verùm e&longs;&longs;et additas e&longs;&longs;e æquales gra­<lb/>vitates, ut rudioribus forta&longs;&longs;e videretur. </s> <s id="s.001745">Hinc e&longs;t libram bra­<lb/>chiorum inæqualium in u&longs;u non e&longs;&longs;e, ne locus pateat dolis. </s> </p> <p type="main"> <s id="s.001746">Dixi autem expre&longs;sè priùs &longs;tatuendam e&longs;&longs;e libræ vacuz <lb/>æquilibritatem, deinde &longs;umenda pondera reciprocè pro Ratio­<lb/>ne longitudinis brachiorum: ni&longs;i etenim priùs æquilibritas illa <lb/>&longs;tatueretur, &longs;i pondera impo&longs;ita e&longs;&longs;ent reciprocè in Ratione <lb/>longitudinis brachiorum, &longs;emper pondus minus additum bra­<lb/>chio longiori præponderaret, quia etiam ip&longs;a brachij longioris <lb/>gravitas &longs;ua habet momenta, & quidem non modica, majora <lb/>momentis brachij brevioris, quæ omninò computanda &longs;unt: <lb/>nam &longs;i ponderum in ea Ratione reciprocè po&longs;itorum momenta <lb/>&longs;int æqualia, illi&longs;que adjiciantur inæqualia gravitatis bra­<lb/>chiorum momenta, manife&longs;tum e&longs;t momentorum &longs;ummam, cui <lb/>plus additur, majorem e&longs;&longs;e reliquâ, cui additur minus. </s> </p> <p type="main"> <s id="s.001747">Sed quænam &longs;unt, & quanta utriu&longs;que brachij momenta? </s> <lb/> <s id="s.001748">Ut hæc inve&longs;tigemus, & certâ ratione definiamus, ponamus <lb/>jugum ip&longs;um &longs;ecundùm &longs;uas omnes partes uniu&longs;modi, & gravi­<lb/>tatem æquabiliter fu&longs;am per totam illius longitudinem. </s> <s id="s.001749">Sit igi­<lb/><figure id="id.017.01.254.1.jpg" xlink:href="017/01/254/1.jpg"/><lb/>tur datum pri&longs;ma AB, quod <lb/>in quinque partes æquales <lb/>dividatur, &longs;ingulas pondoli­<lb/>bram unam; & per &longs;ingula <lb/>gravitatis centra ducatur <lb/>recta <emph type="italics"/>a u<emph.end type="italics"/>: fiatque &longs;ecun­<lb/>dùm rectam HI, à qua pars <lb/>una C ab&longs;cinditur à reliquis, totius pri&longs;matis &longs;u&longs;pen&longs;io, ita ut <lb/>centrum motûs &longs;it in S. <!-- KEEP S--></s> <s id="s.001750">Proculdubio unaquæque pars à cæteris <lb/>&longs;ejuncta &longs;i appenderetur &longs;ecundùm longitudinem jugi <emph type="italics"/>a u,<emph.end type="italics"/><lb/>quod infigeretur per centra gravitatum <emph type="italics"/>a, e, i, o, u,<emph.end type="italics"/> obtineret <pb pagenum="239" xlink:href="017/01/255.jpg"/>fuum momentum juxtà di&longs;tantiam centri &longs;uæ gravitatis à <lb/>centro motûs. </s> <s id="s.001751">Quid autem refert (quod quidem attinet ad <lb/>hanc momentorum Rationem) &longs;i in unum continuum corpus <lb/>unitæ illæ partes coagmententur, an verò divi&longs;æ &longs;olo contactu <lb/>&longs;ibi invicem adhæreant? </s> <s id="s.001752">eadem quippe e&longs;t gravitas &longs;ingulis in­<lb/>&longs;ita, eadem &longs;ingularum à centro di&longs;tantia. </s> <s id="s.001753">Cum itaque centra <lb/>gravitatum <emph type="italics"/>a<emph.end type="italics"/> & <emph type="italics"/>e<emph.end type="italics"/> æqualiter di&longs;tent ab S centro motûs, partes <lb/>C & D æquiponderant: at di&longs;tantia <emph type="italics"/>S i<emph.end type="italics"/> tripla e&longs;t di&longs;tantiæ <emph type="italics"/>S a<emph.end type="italics"/>; <lb/>ergo momentum partis E triplum e&longs;t momenti partis C; &longs;imi­<lb/>lique ratione pars F habet momentum quintuplum, & pars G <lb/>&longs;eptuplum. </s> <s id="s.001754">Igitur componendo, momentum totius aggregati <lb/>quatuor partium D, E, F, G, e&longs;t &longs;edecuplum momenti partis <lb/>C; neque enim &longs;ingulæ partes ex hoc quod cum cæteris pen­<lb/>deant, illi&longs;que cohæreant, &longs;uum amittunt momentum. </s> <s id="s.001755">Hinc <lb/>fit momenta brachiorum e&longs;&longs;e inter &longs;e ut Quadrata longitudi­<lb/>num eorumdem brachiorum: &longs;iquidem o&longs;tenditur &longs;ingularum <lb/>partium momentum cre&longs;cere &longs;ecundùm Rationem numero­<lb/>rum imparium, prout &longs;ecundùm eandem Rationem cre&longs;cunt <lb/>di&longs;tantiæ centrorum gravitatis illarum. </s> <s id="s.001756">Sic brachiorum <lb/>longitudines &longs;i e&longs;&longs;ent in Ratione 2 ad 7, illorum momenta <lb/>ratione &longs;uæ gravitatis innatæ & ratione po&longs;itionis e&longs;&longs;ent ut 4 <lb/>ad 49. </s> </p> <p type="main"> <s id="s.001757">Hæc Ratio momentorum in Ratione Quadratorum longi­<lb/>tudinis, &longs;i res attentè perpendatur, omnibus e&longs;t manife&longs;ta: <lb/>Nam &longs;ingulorum brachiorum gravitates juxta hypothe&longs;im <lb/>æquabiliter fu&longs;æ per totum libræ jugum Rationem inter &longs;e <lb/>habent, quam illorum longitudinis propen&longs;iones ad motum, <lb/>&longs;eu, quod eòdem recidit, di&longs;tantiæ à centro motûs eandem <lb/>pariter Rationem habent, quam brachiorum longitudines: <lb/>Quoniam igitur (ut &longs;æpiùs dictum e&longs;t, &longs;æpiú&longs;que iterùm <lb/>inculcandum) momenta componuntur ex gravitatibus ratio­<lb/>ne materiæ, & ex propen&longs;ionibus ad motum ratione &longs;itûs &longs;eu <lb/>po&longs;itionis, componuntur duæ Rationes longitudinum; atque <lb/>adeó momentum unius brachij ad momentum alterius bra­<lb/>chij e&longs;t in duplicata Ratione &longs;uarum longitudinum, hoc <lb/>e&longs;t, ut ip&longs;arum longitudinum Quadrata. </s> <s id="s.001758">Id quod adhuc ul­<lb/>teriùs &longs;ic explicari po&longs;&longs;e videtur. </s> <s id="s.001759">Sit libræ jugum M. N, & <lb/>motûs centrum O: intelligatur moveri, ut obtineat po&longs;itio-<pb pagenum="240" xlink:href="017/01/256.jpg"/>nem PR. </s> <s id="s.001760">Momentum brachij minoris OM referre videtur <lb/>&longs;ector MOP, momentum verò brachij majoris ON referre <lb/><figure id="id.017.01.256.1.jpg" xlink:href="017/01/256/1.jpg"/><lb/>videtur &longs;ector NOR; &longs;ingularum <lb/>quippe partium motus ab arcu <lb/>de&longs;criptus illarum momentum ob <lb/>oculos ponit, & totius brachij mo­<lb/>mentum illius motus, &longs;cilicet &longs;ector <lb/>in motu de&longs;criptus. </s> <s id="s.001761">At ob æquali­<lb/>tatem angulorum ad verticem in <lb/>O, &longs;ectores MOP, NOR &longs;unt &longs;i­<lb/>miles, &, quia uterque &longs;ector e&longs;t <lb/>&longs;imilis pars &longs;ui circuli, eam inter &longs;e habent &longs;ectores Rationem, <lb/>quæ e&longs;t circulorum, per 15.lib.5. circuli autem &longs;unt in dupli­<lb/>catâ Ratione diametrorum, ex 2.lib.12. &longs;eu Radiorum OM <lb/>& ON; igitur & &longs;ectores &longs;unt in duplicatâ Ratione OM ad <lb/>ON, hoc e&longs;t quadrati OM ad quadratum ON. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001762">At quæris. </s> <s id="s.001763">In propo&longs;ito pri&longs;mate AB, momentum brachij <lb/>SA ad momentum brachij SB e&longs;t ut 1 ad 16: An, ut ha­<lb/>beatur æquilibrium in S, addendum erit in A pondus libra­<lb/>rum 15? quandoquidem pars C e&longs;t libræ unius, reliquum au­<lb/>tem brachium lib. 4, & longitudo SB e&longs;t quadrupla longitu­<lb/>dinis SA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001764">Hoc &longs;anè non e&longs;t iis, quæ dicta &longs;unt, con&longs;equens, necex illis <lb/>efficitur: aliud quippe e&longs;t momenta brachiorum e&longs;&longs;e ut 1 ad 16, <lb/>aliud verò perinde &longs;e habere, atque &longs;i ex brachiorum gravita­<lb/>te carentium extremitatibus penderent libræ 1 & 16, ut ad <lb/>æquilibrium con&longs;tituendum opus &longs;it breviori brachio addere <lb/>libras 15. Primum illud verum e&longs;t, etiam &longs;i extremitatibus ad­<lb/>necti intelligamus hinc quidem libræ &longs;emi&longs;&longs;em; hinc verò li­<lb/>bras octo, mane &longs;cilicet eadem Ratio 1 ad 16. Alterum à for­<lb/>mà veritatis prorsùs alienum videtur, nam licet libræ 4 in ex­<lb/>tremitate B po&longs;itæ æquivaleant libræ unciæ &longs;imul cum pondere <lb/>lib.15. in extremitate A; non e&longs;t tamen eadem ratio librarum 4 <lb/>&longs;ecundùm longitudinem brachij SB di&longs;tributarum; quo enim <lb/>propiores &longs;unt partes centro motûs, eò minus habent mo­<lb/>menti: non igitur libræ 4 &longs;ic di&longs;tributæ æquivalent libris <lb/>16, nec addendum erit pondus librarum 15 in oppo&longs;itâ extre­<lb/>mitate ad æquilibrium con&longs;tituendum, quandoquidem nec ip&longs;a <pb pagenum="241" xlink:href="017/01/257.jpg"/>unica libra partis C tantumdem habet momenti, quantum ha­<lb/>beret &longs;i totâ ex A penderet. </s> </p> <p type="main"> <s id="s.001765">Equidem ex his, quæ paulò ante dicebam de &longs;ectoribus re­<lb/>ferentibus momenta brachiorum, aliquando eò deveni, ut &longs;u&longs;­<lb/>picarer totam gravitatem brachij ON (idem dic de reliquo <lb/>OM) intelligendam e&longs;&longs;e ibi exercere totum momentum, ubi <lb/>e&longs;t qua&longs;i centrum omnium &longs;uorum momentorum, hoc e&longs;t, ubi <lb/>momenta bifariam dividuntur. </s> <s id="s.001766">Si autem &longs;ector NOR refert <lb/>totum momentum brachij ON; non e&longs;t intelligendum cen­<lb/>trum hoc momentorum e&longs;&longs;e punctum L, ubi e&longs;t &longs;emi&longs;&longs;is bra­<lb/>chij ON; quia Sector LOQ ad Sectorem NOR e&longs;t in Ra­<lb/>tione Quadrati OL ad Quadratum ON, quod e&longs;t illius qua­<lb/>druplum. </s> <s id="s.001767">Quod &longs;i inter OL & ON &longs;umatur media propor­<lb/>tionalis OV, jam &longs;ector VOT e&longs;t ad Sectorem NOR in du­<lb/>plicatâ Ratione Radiorum OV, & ON, hoc e&longs;t ut OL ad <lb/>ON, hoc e&longs;t ut 1 ad 2; ac propterea Sector VOT æqualis e&longs;t <lb/>Trapezio NVTR; proinde in V videbantur divi&longs;a æqualiter <lb/>momenta, Hinc arguebam vel totam brachij gravitatem cen­<lb/>&longs;endam e&longs;&longs;e &longs;ua exercere momenta in puncto di&longs;tantiæ à centro <lb/>motûs mediæ proportionalis inter &longs;emi&longs;&longs;em brachij & totam <lb/>brachij longitudinem, vel in extremitate brachij cen&longs;en­<lb/>dam e&longs;&longs;e pendere gravitatem, quæ medio loco proportiona­<lb/>lis &longs;it inter totam brachij eju&longs;dem gravitatem & ejus &longs;e­<lb/>mi&longs;&longs;em. </s> </p> <p type="main"> <s id="s.001768">Verùm, ut quod res e&longs;t &longs;incerè eloquar, quamvis in Secto­<lb/>ribus illis, quos paulò ante commemorabam, imaginem <lb/>quandam momentorum gravitatis &longs;ecundùm brachiorum <lb/>longitudinem di&longs;tributæ agno&longs;cerem, non tamen in re <lb/>Phy&longs;icâ &longs;atis fidebam Geometricæ illi commentationi: quip­<lb/>pe qui ob&longs;ervabam à Sectoribus quidem poni ob oculos Ra­<lb/>tionem momentorum &longs;ingulorum brachiorum ex motu, qui <lb/>idem e&longs;t, &longs;ivè multa, &longs;ivè modica &longs;it gravitas, &longs;ivè in uno, <lb/>&longs;ivè in alio puncto con&longs;tituta intelligatur, non tamen defi­<lb/>niri ip&longs;ius gravitatis momenta. </s> <s id="s.001769">Quare &longs;atius duxi ad experi­<lb/>menta potiùs confugere, ut hinc lux aliqua &longs;uboriretur, qua <lb/>gravitatis quæ&longs;ita momenta innote&longs;cerent. </s> </p> <p type="main"> <s id="s.001770">Primùm igitur a&longs;&longs;umptus e&longs;t ligneus cylindrus, cujus dia­<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"/>pondere D unciarum 40 1/2 collocatus e&longs;t in æquilibrio, quod <lb/>factum e&longs;t in B puncto. </s> <s id="s.001772">Fuit autem longitudo BA unciarum <lb/><figure id="id.017.01.258.1.jpg" xlink:href="017/01/258/1.jpg"/><lb/>pedis Romani 7 2/5 BC ve­<lb/>rò unc.(42 17/50). Re&longs;ecto de­<lb/>mùm &longs;ubtili&longs;&longs;imè cylindro, <lb/>repertum e&longs;t pondus AB <lb/>unciarum 2 1/8, pondus an­<lb/>tem BC unc. </s> <s id="s.001773">13 1/2. Hisob­<lb/>&longs;ervatis cum nullus dubitarem, quin momenta brachiorum <lb/>e&longs;&longs;ent ut quadrata longitudinum, ip&longs;as longitudines AB <lb/>unc. </s> <s id="s.001774">7 2/5, & BC unc.(42 17/50) ad unicam <expan abbr="denomination&etilde;">denominationem</expan> reduxi, vi­<lb/>delicet (370/50) & (2117/50): & a&longs;&longs;umptis numeratorum Quadratis 136900 <lb/>atque 4481689 hanc po&longs;ui Rationem momentorum. </s> <s id="s.001775">Tùm &longs;ic <lb/>ratiocinatus &longs;um Algebricè; ut 136900 ad 4481689, ita mo­<lb/>mentum BA 1 ℞ ad 32.73″ ℞ momentum BC. <!-- KEEP S--></s> <s id="s.001776">Cum igitur <lb/>æqualitas e&longs;&longs;et inter momentum brachij BC, & momentum <lb/>brachij BA plus ip&longs;o pondere D; hæc enim con&longs;tituebant <lb/>æquilibrium, æquatio Algebricè e&longs;t inter momentum BC <lb/>32. 73″ ℞ & BA + D, hoc e&longs;t 1 ℞ + unc. </s> <s id="s.001777">40 1/2: & per An­<lb/>tithe&longs;im demptâ utrinque 1 ℞, æquatio e&longs;t inter 37. 73″ ℞ & <lb/>unc. </s> <s id="s.001778">40 1/2. Factâ itaque numeri ab&longs;oluti 40 1/2 divi&longs;ione per nu­<lb/>merum Radicum prodit pretium 1 ℞ pondo unc.1.27″, quod e&longs;t <lb/>momentum brachij BA; ac proinde momentum brachij BC: <lb/>e&longs;t pondo unc.41. 57″. <!-- KEEP S--></s> <s id="s.001779">Quare perinde e&longs;t atque &longs;i gravitas <lb/>unc. </s> <s id="s.001780">1. 27″ poneretur in extremitate Alineæ Mathematicæ, ac <lb/>in extremitate C poneretur gravitas unc. </s> <s id="s.001781">41. 57″. <!-- KEEP S--></s> <s id="s.001782">At in A fuit <lb/>additum pondus unc. </s> <s id="s.001783">40 1/2: ergo momentum brachij BC æqui­<lb/>valet ponderi D, & præterea unc.1.07″, qui e&longs;t &longs;emi&longs;&longs;is gravitatis <lb/>brachij AB ob&longs;ervatæ unc. </s> <s id="s.001784">2 1/8, hoc e&longs;t in cente&longs;imis paulò ul­<lb/>tra 2. 12″. <!-- KEEP S--></s> <s id="s.001785">Si verò momentis brachij BA pondo unc. </s> <s id="s.001786">1.27″ ad­<lb/>datur gravitas D pondo unc. </s> <s id="s.001787">40. 50″, fit aggregatum 41.77″, <lb/>quod excedit inventum momentum brachij BC unc.41.57″. <!-- REMOVE S--><lb/>exce&longs;&longs;u (20/100) unciæ: quæ di&longs;crepantia facillimè potuit oriri ex <lb/>aliquâ exili, ac minime notabili differentiâ vel in dimetiendis <lb/>brachiorum longitudinibus, vel in ponderandis eorum gravi­<lb/>tatibus; cum maximè re&longs;egmina illa, & &longs;cobs, non computa­<lb/>rentur in gravitate. </s> <s id="s.001788">Quod &longs;i fiat ut longitudo BC 2117 ad <pb pagenum="243" xlink:href="017/01/259.jpg"/>longitudinem AB 370, ita pondus in A unc.41.77″ ad pon­<lb/>dus in B unc. </s> <s id="s.001789">7. 30″, con&longs;tat e&longs;&longs;e ferè &longs;emi&longs;&longs;em gravitatis <lb/>unc. </s> <s id="s.001790">13 1/2: &longs;ed e&longs;t exce&longs;&longs;us &longs;emunciæ ob minùs accuratam ob­<lb/>&longs;ervationem. </s> </p> <p type="main"> <s id="s.001791">Qua propter aliud experimentum quàm accurati&longs;&longs;imè in&longs;ti­<lb/>tui ligneo parallelepipedo, cujus longitudo palmorum Roma­<lb/>norum 7. unc.6. 566‴, ejus verò pondus lib. 1. unc.1 1/4. Alte­<lb/>ri extremitati additus e&longs;t <lb/><figure id="id.017.01.259.1.jpg" xlink:href="017/01/259/1.jpg"/><lb/>plumbeus cylindrus ad per­<lb/>pendiculum pendens, cujus <lb/>pondus unc. </s> <s id="s.001792">20. Impo&longs;itum <lb/>e&longs;t parallelepipedum rotun­<lb/>do claviculo ferreo, qui horizonti parallelus erat, & factum <lb/>e&longs;t æquilibrium in puncto, ubi tota longitudo in duas partes <lb/>dividebatur, quarum minor ponderi adhærens fuit men&longs;urâ <lb/>unc. </s> <s id="s.001793">18 1/6, partes verò major fuit men&longs;urâ palm. </s> <s id="s.001794">6. unc.2/5. Cum <lb/>itaque longitudo CB ob&longs;ervata fuerit unciarum men&longs;uralium <lb/>72. 40″, & AC unciarum men&longs;uralium 18. 16″, in eadem <lb/>pariter Ratione ponuntur brachiorum gravitates ab&longs;olutæ. </s> <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″. <!-- KEEP S--></s> <lb/> <s id="s.001798">Igitur ut longitudinis BC quadratum 52417600 ad longitudi­<lb/>nis AC quadratum 3297856, ita momentum BC 1 ℞ ad <lb/>(3297856/52417600) ℞ momentum brachij AC: cui additur cylindrus D <lb/>unc.20: E&longs;t ergo æquatio inter AC + D, hoc e&longs;t (3297856/52417600) ℞ + <lb/>unc. </s> <s id="s.001799">20.00″ & 1 ℞; & factâ Antithe&longs;i e&longs;t æquatio inter <lb/>unc. </s> <s id="s.001800">20.00″ & (49119744/52417600) ℞: demum in&longs;titutâ divi&longs;ione con&longs;urgit <lb/>pretium 1 ℞, hoc e&longs;t momentum BC, unc. </s> <s id="s.001801">21. 342‴ & paulo <lb/>amplius: atque momentum brachij AC e&longs;t pondo unc.1.343‴, <lb/>cui additâ gravitate cylindri fit &longs;umma unc. </s> <s id="s.001802">21. 343‴ planè <lb/>æqualis momento brachij BC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001803">Et ut hanc operandi methodum confirmarem, iterum in&longs;ti­<lb/>tui argumentationem a&longs;&longs;umendo quadrata gravitatum utriu&longs;­<lb/>que brachij, &longs;unt enim ex hypothe&longs;i gravitates in Ratione lon­<lb/>gitudinum. </s> <s id="s.001804">Cum igitur &longs;it CB pondo unc. </s> <s id="s.001805">10. 50″; & AC <lb/>pondo unc. </s> <s id="s.001806">2. 66.″ fiat ut quadratum CB 1121481 ad quadra­<lb/>tum AC 70756, ita ip&longs;ius CB momentum 1 ℞ ad (70756/1121481) ℞ <lb/><expan abbr="momentũ">momentum</expan> ip&longs;ius AC. <!-- KEEP S--></s> <s id="s.001807">Quoniam verò AC + D hoc e&longs;t (70756/1121481) ℞ <pb pagenum="244" xlink:href="017/01/260.jpg"/>+ unc. </s> <s id="s.001808">20.00″ æquatur momento BC hoc e&longs;t 1 ℞, factâ per <lb/>Antithe&longs;in communi &longs;ubtractione (70756/1121481) ℞, remanet æquatio <lb/>inter pondus unc. </s> <s id="s.001809">20.00″ & (1050725/1121481) ℞, & factâ divi&longs;ione emer­<lb/>git pretium 1 ℞, hoc e&longs;t momentum BC pondo unc. </s> <s id="s.001810">21. 347‴. </s> <lb/> <s id="s.001811">atque adeò momentum ip&longs;ius AC e&longs;t pondo unc. </s> <s id="s.001812">1. 347″; cui <lb/>&longs;i addatur cylindri D gravitas unc. </s> <s id="s.001813">20, totum momentum in A <lb/>e&longs;t unc. </s> <s id="s.001814">21. 347‴, omnino æquale momento ip&longs;ius B: id quod <lb/>ab initio vix &longs;perare audebam, cum hæc operatio à &longs;uperiore <lb/>differat &longs;olùm per (1/1000). <!--neuer Satz--> Hîc pariter brachij AC gravitas ab&longs;o­<lb/>luta pondo unc. </s> <s id="s.001815">2. 66″. <!-- REMOVE S-->habet momentum unc. </s> <s id="s.001816">1. 347‴, cum <lb/>ejus &longs;emi&longs;&longs;is &longs;it unc. </s> <s id="s.001817">1. 330‴, quæ e&longs;t minima atque prorsùs <lb/>contemnenda differentia: quî enim fieri potuit, ut, quantali­<lb/>bet adhiberetur diligentia in metiendo, & ponderando, ne <lb/>pilum quidem à verò aberrarem? </s> <s id="s.001818">aut quis omninò certus &longs;it <lb/>omnes parallelepipedi partes æquali prorsùs fui&longs;&longs;e præditas gra­<lb/>vitate, itaut quæ pars ad arboris radicem vergebat, non fuerit <lb/>paulò den&longs;ior, aut interiùs nodulum aliquem latentem habue­<lb/>rit, quo factum fuerit, ut vera gravitas in&longs;tituto calculo non <lb/>exacti&longs;&longs;imè re&longs;ponderet? </s> <s id="s.001819">&longs;imili ratione &longs;emi&longs;&longs;is gravitatis bra­<lb/>chij BC intelligitur in extremitate B: nam fiat ut longitudo <lb/>BC 72. 40″ ad longitudinem AC 18.16″, ita reciprocè pon­<lb/>dus in A unc. </s> <s id="s.001820">21. 347‴ ad pondus in B unc. </s> <s id="s.001821">5. 354‴: erat au­<lb/>tem brachij BC gravitas ab&longs;oluta unc. </s> <s id="s.001822">10. 59″ cujus, &longs;emi&longs;&longs;is <lb/>5. 295‴. </s> <s id="s.001823">differt ab invento pondere &longs;olùm per (50/1000) unciæ, hoc <lb/>e&longs;t ferè &longs;e&longs;qui&longs;crupulum, &longs;eu grana 34. </s> </p> <p type="main"> <s id="s.001824">Ex his quidem &longs;atis apparebat brachij gravitatem in libræ <lb/>jugo intelligendam e&longs;&longs;e, qua&longs;i ejus &longs;emi&longs;&longs;is in ipsâ extremitate <lb/>con&longs;titueretur, &longs;eu, quod idem e&longs;t, tota gravitas brachij ad <lb/>mediam longitudinem applicaretur (eadem &longs;iquidem e&longs;&longs;e mo­<lb/>menta totius gravitatis in dimidiatâ di&longs;tantiâ, ac dimidiæ gra­<lb/>vitatis in totâ di&longs;tantiâ, ex &longs;æpiùs dictis e&longs;t manife&longs;tum) mihi <lb/>tamen &longs;atisfactum non exi&longs;timabam, ni&longs;i ulteriore experimento <lb/>veritatis ve&longs;tigia per&longs;equerer. </s> <s id="s.001825">Quare eundem plumbeum cy­<lb/>lindrum, cujus longitudo erat palmi 1. unc. </s> <s id="s.001826">1. (9/10), ita in extre­<lb/>mitate A collocavi, ut &longs;uper AI jaceret, & factum e&longs;t æquili­<lb/>brium in E, eratque EA longitudo unc. (22 4/10). Tùm divi&longs;o bi­<lb/>fariam in O &longs;patio AI, quod cylindrus jacens occupabat, ex <pb pagenum="245" xlink:href="017/01/261.jpg"/>puncto O &longs;u&longs;pendi cylindrum, & factum e&longs;t pariter æquili­<lb/>brium exacti&longs;&longs;imè in E, &longs;icut priùs, cum jacebat &longs;uper AI. <!-- KEEP S--></s> <lb/> <s id="s.001827">Deinde cylindrum eumdem iterum parallelepipedo impo&longs;ui ja­<lb/>centem, &longs;ed ea ratione illum ultrò citróque promovebam, ut <lb/>omnino propè fulcrum con&longs;i&longs;teret, donec demùm factum e&longs;t <lb/>æquilibrium in H, & fuit HA palm.2. unc.(10 7/10): Factâ verò <lb/>&longs;u&longs;pen&longs;ione cylindri ex L, ita ut HL e&longs;&longs;et dimidiata cylin­<lb/>dri jacentis longitudo, æquilibrium pariter in H factum e&longs;t. </s> </p> <p type="main"> <s id="s.001828">Relictâ igitur illâ &longs;ectorum analogiâ, deprehendi per illas <lb/>quidem ob oculos poni motum, non verò momentum, &longs;eu pro­<lb/>pen&longs;ionem ad motum, quæ ex di&longs;tantiâ à centro motûs in ipsâ <lb/>longitudine definienda e&longs;t: & quod ad gravitatem attinet, nul­<lb/>lus mihi relictus e&longs;t dubitandi locus ita computandam e&longs;&longs;e to­<lb/>tius brachij gravitatem per ip&longs;um æquabiliter diffu&longs;am, qua&longs;i <lb/>tota in dimidiatâ di&longs;tantiâ à centro motûs collocaretur: quam­<lb/>vis enim particularum gravium, quæ ultrâ &longs;emi&longs;&longs;em longitudi­<lb/>nis magis à centro removentur, momentum cre&longs;cat pro Ratio­<lb/>ne di&longs;tantiæ, reliquarum tamen numero æqualium citrà longi­<lb/>tudinis &longs;emi&longs;&longs;em centro propiorum momentum &longs;imiliter pro <lb/>Ratione minoris di&longs;tantiæ minuitur; ac proptereà tantùm i&longs;ta <lb/>momenta &longs;imul &longs;umpta decre&longs;cunt, quantum illa &longs;imul &longs;umpta <lb/>augentur. </s> <s id="s.001829">Ex quo oritur quædam qua&longs;i æqualitas, perinde at­<lb/>que &longs;i momenta omnia majora & minora in illam particulam <lb/>confluerent, quæ media e&longs;t Arithmeticè inter extrema (mo­<lb/>menta &longs;i quidem ratione di&longs;tantiæ Arithmeticè cre&longs;cunt, prout <lb/>Arithmeticè ip&longs;a di&longs;tantia cre&longs;cit) hæc autem e&longs;t in &longs;emi&longs;&longs;e <lb/>longitudinis brachij. </s> <s id="s.001830">Ex quo iterum confirmatur momenta <lb/>brachiorum e&longs;&longs;e ut quadrata longitudinum; &longs;unt enim in du­<lb/>plicatâ Ratione illarum; &longs;emi&longs;&longs;es quippè &longs;unt in Ratione inte­<lb/>grarum longitudinum, gravitates &longs;unt in Ratione earumdem <lb/>longitudinum, ergo Ratio compo&longs;ita e&longs;t duplicata eju&longs;dem Ra­<lb/>tionis longitudinum. </s> </p> <p type="main"> <s id="s.001831">Hinc datâ jugi æquabilis, & uniformis gravitate ab&longs;olutâ, <lb/>& datâ Ratione longitudinum brachiorum inæqualium libræ, <lb/>dividatur data gravitas &longs;ecundùm datam Rationem brachio­<lb/>rum: tùm fiat ut longitudo minor ad longitudinem majorem, <lb/>ita dimidia gravitas majoris brachij ad aliud, ex quo quarto ter-<pb pagenum="246" xlink:href="017/01/262.jpg"/>mino invento &longs;i auferatur dimidia gravitas brachij minoris, re­<lb/>&longs;iduum indicabit pondus addendum extremitati brachij mino­<lb/>ris, ut fiat æquilibrium cum &longs;olâ gravitate brachij longioris. </s> <lb/> <s id="s.001832">Vel potiùs fiat ut quadratum longitudinis brachij minoris ad <lb/>differentiam inter quadrata brachiorum, ita &longs;emi&longs;&longs;is gravitatis <lb/>brachij minoris ad pondus ip&longs;i addendum. <lb/> </s> </p> <p type="main"> <s id="s.001833"><emph type="center"/>CAPUT III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001834"><emph type="center"/><emph type="italics"/>Quomodò corporum æquilibria explicentur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001835">QUamvis libro primo plura de Gravitatis centro, prout hu­<lb/>jus operis in&longs;tituto congruebat, di&longs;putata &longs;int, eorum ta­<lb/>men plenior explicatio ex his, quæ duobus præcedentibus ca­<lb/>pitibus dicta &longs;unt, petenda e&longs;t, &longs;i quidem Phy&longs;icam æquilibrij <lb/>cau&longs;am no&longs;&longs;e velimus. </s> <s id="s.001836">Neque enim Gravitatis centrum illud <lb/>e&longs;t, quod æquales gravitates, &longs;ed quod æquales gravitationes, <lb/>aut æqualia gravitatis momenta, hoc e&longs;t æquales ad de&longs;cen­<lb/>dendum propen&longs;iones ac vires circum&longs;tant. </s> <s id="s.001837">Nam gravitas câ <lb/>Ratione per univer&longs;um corpus grave di&longs;tribuitur, quâ Ratio­<lb/>ne materia ip&longs;a, cui illa ine&longs;t, diffu&longs;a intelligitur; quæ &longs;i uniu&longs;­<lb/>modi &longs;it & homogenea, ibi centrum habet, ubi e&longs;t molis ip&longs;ius <lb/>centrum; ubi &longs;iquidem bifariam moles & materia, ibi pariter <lb/>gravitas illi in&longs;ita bifariam dividitur. </s> <s id="s.001838">Quoniam verò fieri po­<lb/>te&longs;t, ac &longs;æpiùs contingit, materiam quidem corporis & molem <lb/>invariatam permanere, figuram autem mutari; ex quo nunc in <lb/>hanc, nunc in illam partem migrat gravitatis centrum, quia <lb/>alia atque alia fiunt gravitatis momenta pro variâ corporis &longs;e­<lb/>cundùm &longs;uas partes po&longs;itiones; proptereà huju&longs;modi momento; <lb/>rum æqualitas ex libræ Rationibus de&longs;umenda e&longs;t, &longs;ivè æqua­<lb/>lium, &longs;ivè inæqualium brachiorum libra intelligatur, prout va­<lb/>ria corporis gravis &longs;u&longs;pen&longs;io aut &longs;u&longs;tentatio contingit. </s> </p> <p type="main"> <s id="s.001839">Sed quia in communi u&longs;u non adeò frequens e&longs;t illa &longs;u&longs;pen­<lb/>&longs;io, qua corpus pendeat qua&longs;i ex puncto lineæ directionis tran­<lb/>&longs;euntis per centrum gravitatis, & ad univer&longs;i centrum de­<lb/>ductæ, aut illa &longs;u&longs;tentatio, qua corpus grave acuti&longs;&longs;imo apici <pb pagenum="247" xlink:href="017/01/263.jpg"/>incumbat, cui immineat idem gravitatis centrum; quinimmò <lb/>ita plerumque &longs;u&longs;penditur, aut &longs;u&longs;tinetur corpus, ut ductâ per <lb/>Gravitatis centrum lineâ, aut ex hujus extremitatibus tan­<lb/>quam polis illud &longs;u&longs;pendatur, aut &longs;ubjecto fulcro lineæ huic <lb/>parallelo illud &longs;u&longs;tineatur; ideò huju&longs;modi lineam per centrum <lb/>gravitatis ductam liceat appellare <emph type="italics"/>Diametrum Gravitatis<emph.end type="italics"/>; quæ <lb/>diameter qua&longs;i in librâ locum Axis &longs;eu Aginæ obtinet, corporis <lb/>verò partes hinc & hinc po&longs;itæ rationem habent brachiorum <lb/>libræ, atque pro di&longs;tantiarum &longs;eu longitudinum Ratione &longs;ua <lb/>habent momenta. </s> <s id="s.001840">Sit propo&longs;itum Trapezium, cujus gravita­<lb/>tis centrum C puncto re&longs;pondeat, & <lb/><figure id="id.017.01.263.1.jpg" xlink:href="017/01/263/1.jpg"/><lb/>&longs;u&longs;tineatur &longs;ecundùm rectam lineam <lb/>ACN (&longs;imilis e&longs;&longs;et philo&longs;ophandi <lb/>ratio, &longs;i a&longs;&longs;umeretur recta RCS) quæ <lb/>proptereà <emph type="italics"/>Diameter Gravitatis<emph.end type="italics"/> à me <lb/>dicitur, quia &longs;icut circuli diameter <lb/>per centrum ducta illum in &longs;emicircu­<lb/>los æquales di&longs;tinguit, ita hæc per <lb/>gravitatis centrum tran&longs;iens dividit <lb/>Trapezium in momenta æqualia, itaut in neutram partem in­<lb/>clinetur, juxta dicta de centro Gravitatis. <!-- KEEP S--></s> <s id="s.001841">Sed cur fiat æquili­<lb/>brium intelliges ex Rationibus libræ Brachiorum inæqualium: <lb/>ducatur enim ad rectam AN per C perpendicularis DCE, & <lb/>fiunt brachia CD, CE inæqualia; &longs;unt igitur momenta CE <lb/>longioris majora momentis CD brevioris. </s> <s id="s.001842">Ductis verò ip&longs;i <lb/>DE parallelis BF & ML, &longs;ecatur diameter gravitatis AN in <lb/>punctis H & I: quare inæqualia &longs;unt brachia HB longius, & <lb/>HF brevius, & vici&longs;&longs;im IM e&longs;t brevius, & IL longius: Ex quo <lb/>fit momenta in L & E majora e&longs;&longs;e momentis in M & D, at mo­<lb/>mentum in F minus e&longs;&longs;e momento in B; atque adeò compo­<lb/>nendo majora cum minoribus ex eâdem parte, fieri compo&longs;i­<lb/>tum momentum unius partis æquale toti momento oppo&longs;itæ <lb/>partis. </s> <s id="s.001843">Vel &longs;i non placeat particulatim Trapezium di&longs;tinguere <lb/>qua&longs;i in tot libras, quot ductæ intelliguntur parallelæ, dic to­<lb/>tius gravitatis ADN &longs;emi&longs;&longs;em intelligi in D, & totius gravi­<lb/>tatis AEN &longs;emi&longs;&longs;em intelligi in E; & quamvis pars ADN ab­<lb/>&longs;olutè & &longs;eor&longs;im accepta major &longs;it & gravior parte AEN ab&longs;o­<lb/>lutè &longs;umptâ, quia tamen &longs;unt reciprocè in Ratione di&longs;tantia-<pb pagenum="248" xlink:href="017/01/264.jpg"/>rum CE & CD, propterea æquilibrium con&longs;tituere; pars enim <lb/>minùs gravis ex po&longs;itione majorem habet propen&longs;ionem ad mo­<lb/>tum, qui e&longs;&longs;et velocior; partis verò gravioris minor e&longs;t propen­<lb/>&longs;io ad motum, qui e&longs;&longs;et tardior; atque adeò hæc minùs re&longs;i&longs;tit <lb/>ratione motûs, magis autem ratione gravitatis; at illa ex adver­<lb/>&longs;o magis re&longs;i&longs;tit ratione motûs, &longs;ed minùs ratione gravitatis, <lb/>&longs;ervatâ reciprocè eâdem Ratione inter gravitates & motus. </s> <s id="s.001844">Nil <lb/>igitur mirum &longs;i æquatis hinc & hinc viribus agendi, & re&longs;i&longs;ten­<lb/>di &longs;equatur con&longs;i&longs;tentia. </s> </p> <p type="main"> <s id="s.001845">Hinc manife&longs;tum e&longs;t, cur mutatâ figurâ centrum gravitatis <lb/>ad eam partem transferatur, quæ longiùs à &longs;u&longs;tentationis vel <lb/>&longs;u&longs;pen&longs;ionis loco recedit; quia nimirum cre&longs;cunt ex illâ parte <lb/>comparatè ad oppo&longs;itam momenta ratione di&longs;tantiæ majoris, ac <lb/>proinde, ut fiat momentorum æqualitas, centrum ad illam par­<lb/>tem &longs;ecedit. </s> <s id="s.001846">Sic ce&longs;pitantes à naturâ docentur in partem op­<lb/>po&longs;itam illi, in quam inclinantur, brachium illicò extendere, <lb/>ut brachij gravitas longiùs à corpore tran&longs;lata plus habeat mo­<lb/>menti, quàm cùm reliquo corpori adhæret, atque hinc &longs;equa­<lb/>tur centri gravitatis in illam partem tran&longs;latio. </s> <s id="s.001847">Veritas hæc &longs;a­<lb/>tis nota e&longs;t ip&longs;is funambulis, cùm corpus univer&longs;um &longs;uper ex­<lb/>tento fune librant; neque enim temerè crura & brachia exten <lb/>dunt aut contrahunt, &longs;ed certâ lege, ut centrum momento­<lb/>rum gravitatis totius corporis hac vel illâ ratione di&longs;po&longs;iti im­<lb/>mineat, & incumbat funi. </s> <s id="s.001848">Sic plumbeæ virgæ rectæ ex medio <lb/>&longs;u&longs;pen&longs;æ, & in æquilibrio manentis, &longs;i brachium alterum in­<lb/>flexeris, fieri non pote&longs;t, ut reliquum brachium rectum &longs;ervet <lb/>po&longs;itionem horizonti parallelam, &longs;ed deor&longs;um inclinabitur, qun <lb/>cum longius &longs;it brachio inflexo, majora habet momenta ac <lb/>prævalet. </s> <s id="s.001849">Quod &longs;i ob inæqualem virgæ cra&longs;&longs;itiem non planè <lb/>ad mediam illius longitudinem facta &longs;it &longs;u&longs;pen&longs;io, &longs;ed æquili­<lb/>britas contingat in puncto, quod propius e&longs;t cra&longs;&longs;iori extremi­<lb/>tati virgæ, factâ alterutrius brachij inflexione tollitur æquili­<lb/>brium, quia non jam ampliùs eadem e&longs;t reciprocè Ratio longi­<lb/>tudinum, quæ & gravitatum. </s> </p> <p type="main"> <s id="s.001850">Ex his pariter con&longs;equens e&longs;t aliquando minimam virtutem <lb/>&longs;atis e&longs;&longs;e ad dimovenda ab æquilibrio ingentia corpora, &longs;i ita <lb/>&longs;u&longs;tineantur, ut fulcrum vel in puncto, vel in lineâ contingant: <lb/>quoniam &longs;i corpus grave in&longs;i&longs;tat apici coni, aut pyramidis, aut <pb pagenum="249" xlink:href="017/01/265.jpg"/>angulo &longs;olido, aut portioni &longs;phæricæ, quam contingat idem <lb/>corpus &longs;ive planâ, &longs;ive &longs;phæricè cavâ, &longs;ive &longs;phæricam æmulan­<lb/>te &longs;uperficie, contactus in puncto efficitur, ac propterea qua­<lb/>cunque in extremitate corporis addatur vis movendi, æquili­<lb/>brium tollitur, & quidem eò faciliùs, quo magis à puncto con­<lb/>tactûs extremitas illa removetur; in illâ quippe di&longs;tantiâ vis mo­<lb/>vendi apta velociorem motum efficere, quàm &longs;i propior e&longs;&longs;et, <lb/>plus habet momenti: Id quod adhuc faciliùs accidit, &longs;i ab ex­<lb/>tremitate, ubi vis movendi applicatur, ductâ per contingentis <lb/>fulcri punctum rectâ lineâ ad oppo&longs;itam extremitatem, inæqua­<lb/>liter divi&longs;a &longs;it in puncto contactûs, & vis ip&longs;a movendi in magis <lb/>di&longs;tante extremitate con&longs;tituta fuerit; tunc enim non &longs;ua tan­<lb/>tùm momenta addit, &longs;ed illa multiplicat pro Ratione exce&longs;sûs <lb/>&longs;uæ di&longs;tantiæ; quemadmodum de inæqualibus libræ brachiis <lb/>dictum e&longs;t. </s> <s id="s.001851">Sin autem fulcrum &longs;u&longs;tinens, quod horizonti paral­<lb/>lelum ponitur, &longs;it acies pri&longs;matis, aut latus pyramidis jacentis, <lb/>aut portio cylindrica &longs;eu conica jacens; tunc in lineâ fit con­<lb/>tactus, &longs;i vel plana &longs;it, vel circulariter concava corporis in­<lb/>&longs;i&longs;tentis &longs;uperficies: &longs;ed &longs;i vis movendi, quantacumque &longs;it, ad­<lb/>datur &longs;ecundùm rectam lineam, quæ efficit Gravitatis diame­<lb/>trum, puta in A vel N, non mutat æquilibritatem, &longs;i fulcrum <lb/>congruit toti diametro AN: &longs;i verò fulcrum brevius e&longs;t quàm <lb/>AN, & ex. </s> <s id="s.001852">gr. <!-- REMOVE S-->congruit &longs;olùm ip&longs;i AI, jam centrum motûs e&longs;t <lb/>I, & oportet vim movendi tantam e&longs;&longs;e in N, ut aggregatum ex <lb/>parte MLN ac virtute additâ in N habeat ad partem MAL re­<lb/>liquam majorem Rationem, quàm &longs;it Ratio di&longs;tantiæ IA ad <lb/>di&longs;tantiam IN. </s> <s id="s.001853">Quare in huju&longs;modi contactu lineari vis mo­<lb/>vendi, æquilibrium facilè tollens, e&longs;&longs;e debet ad latus diametri <lb/>gravitatis, & pro ratione di&longs;tantiæ majus erit momentum; ma­<lb/>ximum autem erit momentum in E di&longs;tantiâ maximâ. </s> </p> <p type="main"> <s id="s.001854">Non igitur facilè inter fabulas rejicienda &longs;unt, quæ Atlas <lb/>Sinicus pag.32. de Montibus circa urbem Peking loquens ait, <lb/><emph type="italics"/>Púon mons alti&longs;&longs;imus ac præruptus varios attollens vertices, in cujus <lb/>&longs;ummitate ingens e&longs;t lapis, qui minimo contactu movetur ac titubat:<emph.end type="italics"/><lb/>fieri &longs;iquidem potuit, ut lapis ille in infimâ parte excavatus in­<lb/>nitatur &longs;ubjecto &longs;axo, à quo vel in puncto, vel in lineâ tanga­<lb/>tur, &longs;icuti dictum e&longs;t; & cum &longs;it perfectè libratus, modico im­<lb/>pul&longs;u tangentis, quâ &longs;altem parte ad illum patet acce&longs;&longs;us, po-<pb pagenum="250" xlink:href="017/01/266.jpg"/>te&longs;t ab æquilibrio dimoveri: quòd &longs;i u&longs;quequaque circum­<lb/>obeundo lapidem quâcumque in parte tangatur, &longs;equitur illius <lb/>trepidatio, &longs;ignum e&longs;t contactum &longs;ubjecti fulcri e&longs;&longs;e in puncto. </s> <lb/> <s id="s.001855">Simili ratione explicanda &longs;unt, quæ idem Atlas Sinicus in XI <lb/>Provincia Fokien habet pag. </s> <s id="s.001856">125, ubi ait, <emph type="italics"/>Versùs Vrbis <lb/>Changcheu Orientalem partem mons e&longs;t Cio dictus, in quo lapida, <lb/>e&longs;&longs;e &longs;cribunt altum perticas quinque, cra&longs;&longs;um decem & octo, qui quo­<lb/>ties tempe&longs;tas imminet, titubat omninò, ac movetur:<emph.end type="italics"/> hic enim la­<lb/>pis in perfecto æquilibrio con&longs;titutus &longs;uprà fulcrum, à quo in <lb/>puncto, vel in lineâ tangatur, & forta&longs;&longs;e etiam ab eodem fulcro <lb/>di&longs;tinctus in longitudines inæquales, violento impul&longs;u hali­<lb/>tuum aut infernè &longs;ubeuntium, aut ex &longs;uperiore nubium parte <lb/>obliquè reflexorum, facilè moveri pote&longs;t ac titubare, &longs;i extre­<lb/>mitas à fulcro remotior impellatur. </s> </p> <p type="main"> <s id="s.001857">Et quoniam de Sinen&longs;ibus mentio incidit, non injucundum <lb/>fuerit hîc aliud addere pertinens ad eorum indu&longs;triam in &longs;er­<lb/>vando æquilibrio. </s> <s id="s.001858">Idem Atlas Sinicus, cum &longs;ermo e&longs;t de Pro­<lb/>vincia Peking, ubi &longs;olum e&longs;&longs;e areno&longs;um atque plani&longs;&longs;imum <lb/>te&longs;tatur, hæc habet pag.28. <emph type="italics"/>Modus itineris faciendi hi&longs;ce locis <lb/>non infrequens, nec incommodus e&longs;t. </s> <s id="s.001859">Plau&longs;trum adhibent cum unâ <lb/>rotâ ita con&longs;titutum, ut uni illius medium occupandi, & qua&longs;i equo <lb/>in&longs;idendi &longs;it locus, aliis duobus ab utroque latere ad&longs;identibus; auri­<lb/>ga plau&longs;trum retro ligneis vectibus urget ac promovet non &longs;ecurè mi­<lb/>nùs, quàm velociter.<emph.end type="italics"/></s> <s id="s.001860"> Si rem conjecturis indagare liceat, ego ro­<lb/>tam concipio ita inclu&longs;am ligneo loculamento majoris &longs;egmen­<lb/>ti circuli figuram habente, ut huic in&longs;itus &longs;it rotæ axis, ad dex­<lb/>tram autem & ad lævam extantia tabulata tantæ latitudinis, <lb/>ut quis modò propè rotam, modò longiùs ad&longs;idere queat ad <lb/>æquilibrium con&longs;tituendum inter duos viatores inæqualiter <lb/>graves: Aurigæ locus e&longs;t in &longs;uprema parte loculamenti, cui <lb/>qua&longs;i equitans in&longs;idet, bino&longs;que contos, &longs;eu vectes concinnè <lb/>locatos, ut manubrium ante &longs;e habeat, extremitas altera (for­<lb/>ta&longs;sè in acumen de&longs;inens, ut leviter &longs;olo infigatur) po&longs;t &longs;e ter­<lb/>ram re&longs;piciat, utrâque manu apprehendens &longs;olum obliquè pre­<lb/>mit, & currum in anteriora velociter promovet. </s> <s id="s.001861">Id quod nemi­<lb/>ni difficile videatur, qui &longs;æpiùs ob&longs;ervaverit à puero fabri <lb/>lignarij aut ferrarij rotam curulem identidem impul&longs;am per <lb/>urbis vias velociter deduci; quæ dum impre&longs;&longs;o impetu veloci-<pb pagenum="251" xlink:href="017/01/267.jpg"/>ter conver&longs;a in anteriora promovetur, licet huc atque illuc <lb/>nutabunda inclinetur, ob velocem conver&longs;ionem immunis e&longs;t à <lb/>ca&longs;u: quemadmodum etiam &longs;tanneum aut argenteum orbem <lb/>apici cultri impo&longs;itum, &longs;i in gyrum velociter agatur, à ca&longs;u im­<lb/>munem videmus, etiam&longs;i punctum &longs;u&longs;tentationis non exacti&longs;&longs;i­<lb/>mè centro re&longs;pondeat. </s> <s id="s.001862">Sic aliquis &longs;uppo&longs;itam &longs;phærulam altero <lb/>pede, etiam &longs;ummis digitis premens, celeriter in gyrum totum <lb/>corpus contorquet, qui non ita facilè citrà cadendi periculum <lb/>eidem &longs;phærulæ in&longs;i&longs;tens quietus con&longs;i&longs;teret; ipsâ nimirum <lb/>conver&longs;ionis celeritate gravitatis propen&longs;ionem eludente. </s> <s id="s.001863">Non <lb/>ab&longs;imili igitur ratione in huju&longs;modi rotæ Sinici plau&longs;tri conver­<lb/>&longs;ione veloci deteritur, quicquid in alterutram partem inclinatio­<lb/>nis oriretur vel ex modicâ viæ inæqualitate, vel ex æquilibrio <lb/>non adeò exactè &longs;ervato, ut etiam con&longs;i&longs;tente plau&longs;tro in&longs;iden­<lb/>tes viatores con&longs;i&longs;terent æqualiter librati ab&longs;que alicujus artifi­<lb/>cij &longs;ub&longs;idio: Quod artificium in promptu e&longs;&longs;e non dubito; ne­<lb/>que enim Sinen&longs;es ita &longs;ibi præfidentes exi&longs;timo, ut aliquâ ratio­<lb/>ne &longs;ibi non præcaveant à periculo casûs, &longs;i fortè rotun in obicem <lb/>incurrente plau&longs;trum &longs;eu loculamentum in anteriorem, aut in <lb/>po&longs;teriorem partem improvisâ inclinatione convertatur. </s> <s id="s.001864">Sed <lb/>&longs;ingula per&longs;equi nec otium e&longs;t, nec operæ pretium: quapropter <lb/>generatim dicendum corporis æquilibrium ibi fieri, ubi in duas <lb/>partes ita di&longs;tinguitur, ut illarum gravitates &longs;int reciprocè in <lb/>Ratione longitudinum &longs;eu di&longs;tantiarum à puncto &longs;u&longs;pen&longs;ionis <lb/>&longs;eu &longs;u&longs;tentationis, quemadmodum in librâ dictum e&longs;t. </s> <s id="s.001865">Quare &longs;i <lb/>tota moles propo&longs;ita eâdem gravitatis &longs;pecie prædita fuerit, nec <lb/>facile &longs;it in illâ centrum gravitatis invenire, quia nimis irregu­<lb/>laris e&longs;t, di&longs;tingue illam in duas partes, & &longs;ingularum inventa <lb/>centra gravitatis junge rectâ lineâ, quæ qua&longs;i libræ jugum divi­<lb/>datur in reciprocâ Ratione illarum partium; e&longs;t enim punctum <lb/>illud, in quod cadit divi&longs;io, punctum æquilibrij, & centrum gra­<lb/>vitatis totius. </s> <s id="s.001866">Sic Trapezij, NPMQ in­<lb/><figure id="id.017.01.267.1.jpg" xlink:href="017/01/267/1.jpg"/><lb/>venies punctum æquilibrij, &longs;i duorum <lb/>triangulorum NQM, NPM, in quæ di­<lb/>viditur, &longs;ingularia centra gravitatis inve­<lb/>nias O & B: hæc jungantur rectâ OB; <lb/>tum fiat ut triangulum NQM ad trian­<lb/>gulum NPM, ita reciprocè BD ad DO, <pb pagenum="252" xlink:href="017/01/268.jpg"/>& e&longs;t D punctum æquilibrij, &longs;eu centrum gravitatis Trapezij <lb/>quæ&longs;itum. </s> <s id="s.001867">At &longs;i Trapezio addatur triangulum NLP eju&longs;dem <lb/>&longs;pecificæ gravitatis, emergit Pentagonum irregulare LPMQN: <lb/>inveniatur additi trianguli centrum &longs;ingulare gravitatis A, & <lb/>jungatur recta AD; tùm fiat ut Trapezium ad triangulum ad­<lb/>ditum, ita reciprocè AS ad SD, & e&longs;t punctum S centrum <lb/>commune gravitatis totius Pentagoni, in quo fit æquilibrium; <lb/>perinde enim e&longs;t ac &longs;i in jugo libræ AD inæqualiter di&longs;tributæ <lb/>appenderetur ex A quidem triangulum NLP; ex D verò Tra­<lb/>pezium NQMP, quæ in illis di&longs;tantiis à centro motûs æqualia <lb/>haberent momenta. </s> </p> <p type="main"> <s id="s.001868">Quòd &longs;i tota moles propo&longs;ita con&longs;tet partibus non eju&longs;dem <lb/>&longs;pecificæ gravitatis, non jam &longs;atis e&longs;t inveni&longs;&longs;e &longs;ingularia cen­<lb/>tra, ut ducatur jugum libræ illa connectens, & notam e&longs;&longs;e Ra­<lb/>tionem molis ad molem; &longs;ed prætereà opus e&longs;t notam habere <lb/>Rationem gravitatis &longs;pecificæ ad gravitatem &longs;pecificam; quiz <lb/>Ratio gravitatum ab&longs;olutarum componitur ex Rationibus <lb/>quantitatum, & gravitatum &longs;ecundùm &longs;peciem. </s> <s id="s.001869">Quamobrem <lb/>&longs;i additum triangulum habeat &longs;pecificam gravitatem majorem <lb/>gravitate &longs;pecificâ Trapezij, quia hoc ligneum e&longs;t, illud fer­<lb/>reum, non cadet in S punctum æquilibrij, &longs;ed accedet ad <lb/>punctum A, quia factâ huju&longs;modi Rationum compo&longs;itione, <lb/>minor e&longs;t inæqualitas gravitatum ab&longs;olutarum; &longs;i enim Trape­<lb/>zium excedit mole Triangulum, cedit illi &longs;pecificâ gravitate. </s> <lb/> <s id="s.001870">Ponamus namque Rationem molis Trapezij ad molem Trian­<lb/>guli e&longs;&longs;e ut & ad 2; &longs;pecificæ verò gravitatis Rationem ut 5 ad <lb/>42, gravitas ab&longs;oluta Trapezij lignei e&longs;t ut 35, gravitas Trian­<lb/>guli ferrei ut 84: &longs;unt igitur gravitates in Ratione 5 ad 12: di­<lb/>vidatur itaque jugum AD in I reciprocè, ut &longs;it AI 5, ID 12, <lb/>& erit I centrum gravitatis compo&longs;itæ, ac punctum æquilibrij, <lb/>quia ab illo inæquales gravitates habent &longs;uas di&longs;tantias in Ra­<lb/>tione reciprocâ ip&longs;arum gravitatum. </s> <s id="s.001871">Eadem e&longs;t in corporibus <lb/>omnibus Ratio, & methodus deprehendendi punctum æqui­<lb/>librij, &longs;eu centrum gravitatis, per quod deinde duci pote&longs;t dia­<lb/>meter gravitatis, ut fiat opportuna &longs;u&longs;pen&longs;io. </s> </p> <p type="main"> <s id="s.001872">Quia tamen aliquando evenit &longs;u&longs;pen&longs;um corpus aut &longs;u&longs;ten­<lb/>tatum, dum po&longs;itionem horizonti parallelam &longs;ervare contendit, <lb/>aliquod incommodum &longs;ubire in motu corporis, cui innititur; <pb pagenum="253" xlink:href="017/01/269.jpg"/>proptereà huic occurrendum e&longs;t artificio, quo &longs;itum eumdem <lb/>perpetuò &longs;ervet. </s> <s id="s.001873">Rem exemplo declaro. </s> <s id="s.001874">In pyxide nauticâ in­<lb/>&longs;i&longs;tit cu&longs;pidi acus magnetica æqualibus momentis librata, ut <lb/>horizonti parallela jaceat, quamcumque in partem dirigatur. </s> <lb/> <s id="s.001875">Si alicui navis plano pyxis ip&longs;a adhæreret ita, ut infimâ &longs;ui par­<lb/>te illi congrueret, quamcumque in partem navis inclinaretur, <lb/>ip&longs;um pariter pyxidis fundum inclinari manife&longs;tum e&longs;t, & alte­<lb/>ri acûs magneticæ po&longs;itionem horizonti parallelam &longs;ervantis <lb/>extremitati occurrens illius motum impediret, aut &longs;altem retar­<lb/>daret. </s> <s id="s.001876">Ut igitur &longs;emper pyxis tùm acui magneticæ, tùm hori­<lb/>zonti parallela con&longs;i&longs;tat, &longs;u&longs;pendenda fuit, non quidem funi­<lb/>culo, ne incertis motibus jactaretur, &longs;ed duobus polis, &longs;uper <lb/>quibus opportunè ver&longs;aretur æqualiter librata. </s> <s id="s.001877">Verùm duobus <lb/>hi&longs;ce polis non tollitur omne incommodum; &longs;i etenim poli <lb/>re&longs;piciant navis latera, elevatâ aut depre&longs;sâ prorâ juvant, &longs;ed <lb/>navi in dextrum aut in &longs;ini&longs;trum latus inclinatâ, alter deprime­<lb/>retur, alter elevaretur, ni&longs;i & ip&longs;i infigerentur circulo &longs;uper <lb/>alios polos proram & puppim re&longs;picientes ver&longs;atili. </s> <s id="s.001878">Sit pyxis <lb/>ip&longs;a ABCD, in qua venti de&longs;­<lb/><figure id="id.017.01.269.1.jpg" xlink:href="017/01/269/1.jpg"/><lb/>cripti &longs;int, & in centro O acus <lb/>magnetica volubilis in&longs;i&longs;tat: py­<lb/>xidem circulus EIFH com­<lb/>plectatur, cui poli D & B facilè <lb/>ver&longs;atiles infigantur, ut inclinatâ <lb/>navi in A vel in C pyxis horizon­<lb/>ti parallela maneat; & ut eumdem <lb/>paralle i&longs;mum &longs;ervet, etiam &longs;i na­<lb/>vis in B aut D inclinetur, circu­<lb/>lus ille EIFH duos pariter polos <lb/>facilè ver&longs;atiles habeat in E & F <lb/>externæ pyxidi immobili infixos: <lb/>hac enim ratione fiet, ut in quacumque navis inclinatione <lb/>pyxis nautica à &longs;uo paralleli&longs;mo & æquilibrio non recedat. </s> </p> <p type="main"> <s id="s.001879">Hoc eodem artificio con&longs;truitur luceina ferreo aut æneo <lb/>globo inclu&longs;a multipliciter perforato, ut fumo exitus pateat, <lb/>quæ citrà effu&longs;ionem olci in &longs;olo rotata non extinguitur; e&longs;t &longs;i­<lb/>quidem va&longs;culum plumbeum, ut &longs;ua gravitate &longs;ecuriùs deor­<lb/>&longs;um vergat, polis ver&longs;atilibus &longs;u&longs;pen&longs;um in circulo, qui pariter <pb pagenum="254" xlink:href="017/01/270.jpg"/>polos in&longs;erit &longs;ecundo circulo, &longs;ecundus &longs;imiliter tertio, tertius <lb/>demum &longs;caphio, &longs;eu inferiori hemi&longs;phærio globi, cui includi­<lb/>tur, eâ di&longs;po&longs;itione, ut quemadmodum pyxidis nauticæ hic <lb/>de&longs;criptæ ambitus in quatuor partes di&longs;tinguitur à polis, ita lu <lb/>cernæ hujus ambitus in octo partes à polis di&longs;tribuatur, atque <lb/>proinde facilior &longs;it globi in omnem partem volutatio citrà peri­<lb/>culum inclinationis va&longs;culi oleum cum ellychnio continentis. </s> </p> <p type="main"> <s id="s.001880">Nec pluribus opus e&longs;t hîc explicare, quàm proclive &longs;it arti­<lb/>ficium hoc ad plura traducere, quorum u&longs;us e&longs;t in plano hori­<lb/>zontali, ne libellâ &longs;emper & normâ indigeamus, ut illa ritè <lb/>collocentur: ut &longs;i horologium horizontale &longs;tatuendum &longs;it quo­<lb/>cumque in plano, &longs;it illud pyxidi inclu&longs;um cum circulo, quem­<lb/>admodum de pyxide nauticâ dictum e&longs;t: &longs;i lectulum viatorium <lb/>in rhedâ &longs;ternere oporteat, in quo citrà jactationem, etiam viâ <lb/>&longs;alebrosâ, quie&longs;cere liceat, ferreo parallelogrammo complecte­<lb/>re lectulum ex polis &longs;u&longs;pen&longs;um circâ medium eo loco, ut cor­<lb/>pus in lectulo jacens &longs;it horizonti parallelum, ip&longs;um verò paral­<lb/>lelogrammum polis rhedæ infixis & ver&longs;atilibus ad caput & ad <lb/>pedes &longs;u&longs;pendatur: & alia huju&longs;modi, quæ facilè pro rerum <lb/>opportunitate excogitari po&longs;&longs;unt. </s> </p> <p type="main"> <s id="s.001881">Verùm quàm facilè e&longs;t &longs;uper polos in æquilibrio con&longs;tituere <lb/>corpora gravitatis centrum habentia vel in ipsâ &longs;u&longs;tentationis <lb/>lineâ, vel infrà illam, tam multis difficultatibus implicitum <lb/>opus e&longs;t in æquilibrio &longs;tatuere corpus, cujus gravitatis cen­<lb/>trum in parte &longs;uperiori reperitur, & quidem maximè &longs;i mul­<lb/>tùm inde removeatur; tunc enim &longs;u&longs;&longs;icit vel minima inclinatio, <lb/>ut totum corpus revolvatur, cum ex alterâ parte &longs;int plura gra­<lb/>vitatis momenta, quàm in oppo&longs;itâ. </s> </p> <p type="main"> <s id="s.001882">Nam &longs;i corpus BC, cujus centrum gravitatis &longs;it A, &longs;u&longs;pen­<lb/>datur &longs;uper polis in I, quando axi &longs;u&longs;tentanti ad perpendiculum <lb/><figure id="id.017.01.270.1.jpg" xlink:href="017/01/270/1.jpg"/><lb/>re&longs;pondet centrum gravitatis A, ma­<lb/>net æquilibrium, &longs;ed factâ corporis <lb/>inclinatione, ut A recedat à perpen­<lb/>diculo, jam versùs C plures &longs;unt <lb/>partes gravitatis de&longs;cendentes, quàm <lb/>versùs B &longs;int partes a&longs;cendentes, & <lb/>illæ velociùs moventur deor&longs;um, <lb/>quàm hæ &longs;ur&longs;um; quapropter illæ <pb pagenum="255" xlink:href="017/01/271.jpg"/>majora habent momenta, quibus deorium urgentibus corpus <lb/>revolvitur. </s> <s id="s.001883">Id quod multò magis contingit in Acrobarycis, quæ <lb/>nimirum gravitatem in &longs;ummitate habent, ut &longs;i corpori BC in <lb/>&longs;uperiori parte adnexa e&longs;&longs;et pyramis D; cum enim totius com­<lb/>po&longs;itæ molis ex &longs;olido BC, & pyramide D, centrum commu­<lb/>ne gravitatis non e&longs;&longs;et in A, &longs;ed adhuc &longs;uperius procul à polo <lb/>I, qui e&longs;t centrum motûs, factâ levi inclinatione multo plus <lb/>gravitatis e&longs;&longs;et ex parte C, quàm ex oppo&longs;ità B, ut con&longs;tat: <lb/>nam quò altius & remotius e&longs;t centrum gravitatis, eò faciliùs <lb/>linea directionis cadit extra punctum vel lineam &longs;u&longs;tentationis, <lb/>facta pari inclinatione. </s> </p> <p type="main"> <s id="s.001884">Liceat autem hîc obiter, qua&longs;i cerollarij loco, attingere <lb/>æquilibria corporum humido in&longs;identium, & Acrobary corum <lb/>fluitantium, in quibus pariter Rationes libræ agno&longs;centur, &longs;i <lb/>rectè perpendatur, ubi fiat &longs;u&longs;tentatio. </s> <s id="s.001885">In omni igitur corpo­<lb/>re fluitante duplex pars con&longs;ideranda e&longs;t, & quæ intrá humi­<lb/>dum mergitur, & quæ in aëre extat: illa quidem utpote &longs;ecun­<lb/>dùm &longs;peciem minùs gravis, quàm humor, levitat, hæc verò <lb/>aëre gravior gravitat: Quare & illa &longs;uum habet centrum levi­<lb/>tatis, & hæc centrum gravitatis; nec po&longs;&longs;et corpus datam po&longs;i­<lb/>tionem &longs;ervare, ni&longs;i in eâdem lineâ perpendiculari ad univer&longs;i <lb/>centrum tendente e&longs;&longs;et utrumque centrum & levitatis & gra­<lb/>vitatis; cumque par &longs;it virtus a&longs;cendendi virtuti de&longs;cendendi, <lb/>neutrâ prævalente, & &longs;ibi vici&longs;&longs;im utrâque ob&longs;i&longs;tente, con&longs;i&longs;tit <lb/>corpus. </s> <s id="s.001886">Quòd &longs;i non in eodem perpendiculo &longs;it utrumque <lb/>centrum, utrumque &longs;uâ viâ pergere pote&longs;t, illud a&longs;cendendo, <lb/>hoc de&longs;cendendo. </s> <s id="s.001887">Sic baculum rectum in aquam immittens, <lb/>manúque retinens, ne in alterutram partem inclinetur, mergi <lb/>quidem illum videbis pro Ratione &longs;pecificæ &longs;uæ gravitatis, quæ <lb/>minor e&longs;t &longs;pecificâ gravitate aquæ, &longs;ed erectus non manebit, <lb/>ni&longs;i quandiù retinueris; nam ubi illum dimi&longs;eris, &longs;tatim cen­<lb/>trum gravitatis de&longs;cendet, & levitatis centrum a&longs;cendet, quia <lb/>vel exiguus aquæ motus partem immer&longs;am inclinans &longs;atis e&longs;t, <lb/>ut centra illa non eidem perpendiculo re&longs;pondeant; ac prop­<lb/>terea demùm baculus jacens innatabit. </s> </p> <p type="main"> <s id="s.001888">Quie&longs;cente igitur corpore in humoris &longs;uperficie, mani­<lb/>fe&longs;tum e&longs;t centrum gravitatis partis extantis in eodem perpen­<lb/>diculo e&longs;&longs;e cum centro levitatis partis demer&longs;æ. </s> <s id="s.001889">Quare &longs;i <pb pagenum="256" xlink:href="017/01/272.jpg"/>ligneum pri&longs;ina AC aquæ imponatur, & immergatur ita, ut <lb/>pars demer&longs;a & levitans &longs;it EC, pars verò extans in aëre & <lb/><figure id="id.017.01.272.1.jpg" xlink:href="017/01/272/1.jpg"/><lb/>gravitans &longs;it AF, centrum gravi­<lb/>tatis e&longs;t G, centrum levitatis e&longs;t <lb/>H, quæ &longs;ibi directè adver&longs;antia <lb/>in oppo&longs;itas partes conantur <lb/>æqualibus viribus, atque prop­<lb/>terea nullus &longs;equitur motus. </s> <lb/> <s id="s.001890">Quòd &longs;i aut H recederet versùs <lb/>D, aut G versùs B, & hoc po&longs;&longs;et <lb/>de&longs;cendere, & illud a&longs;cendere <lb/>neutro contranitente. </s> </p> <p type="main"> <s id="s.001891">Jam verò quie&longs;centi pri&longs;mati imponatur aliquod pon­<lb/>dus, certum e&longs;t partem in aëre extantem, conflatam ex <lb/>parte pri&longs;matis & ex addito pondere, graviorem e&longs;&longs;e, ac <lb/>proinde prævalere viribus partis in aquâ levitantis, illam­<lb/>que deprimere, quoadu&longs;que fiat æqualitas inter levitatem <lb/>& gravitatem. </s> <s id="s.001892">Sed multùm intere&longs;t, utrùm additi pon­<lb/>deris centrum gravitatis in eodem perpendiculo &longs;it cum cen­<lb/>tro gravitatis G, ut rectâ deprimatur pri&longs;ma infrà &longs;uperfi­<lb/>ciem aquæ; an verò &longs;it extrà illud perpendiculum; id <lb/>quod &longs;i accidat, commune centrum gravitatis transfertur ver­<lb/>&longs;us A, aut B. </s> <s id="s.001893">Sit ex. </s> <s id="s.001894">gr. <!-- REMOVE S-->ad partes A propè S; cumque non <lb/>immineat puncto H centro levitatis, de&longs;cendit pri&longs;ma ad partes <lb/>A, & oppo&longs;ita pars a&longs;cendit, ita ut E deprimatur infrà &longs;uperfi­<lb/>ciem aquæ, F veró emergat. </s> <s id="s.001895">Sed dum ad partes CF pri&longs;ma <lb/>emergit ex aquâ, ad partes autem DE deprimitur, centrum levi­<lb/>tatis non manet in H, &longs;ed ad majorem partem depre&longs;&longs;am &longs;ecedit, <lb/>donec fiat V, atque in eodem <expan abbr="perp&etilde;diculo">perpendiculo</expan> &longs;it cum centro gravi­<lb/>tatis S; & tunc quie&longs;cit pri&longs;ma, nec amplius demergitur in E, <lb/>aut emergit ex F. <!-- KEEP S--></s> <s id="s.001896">Su&longs;tinetur itaque centrum gravitatis S à cen­<lb/>tro levitatis V, & vici&longs;&longs;im centrum levitatis V retinetur à cen­<lb/>tro gravitatis S; & fit tùm inter gravitates, tùm inter levitates <lb/>æquilibrium, quia gravitas in A major minùs di&longs;tat à puncto, <lb/>vel potiusà lineâ &longs;u&longs;tentationis factâ à plano tran&longs;eunte per V, <lb/>& gravitas in B minor magis di&longs;tat; ideóque neutra prævalet: <lb/>& &longs;imiliter ievitas in DE major minùs di&longs;tat à lineâ detentio­<lb/>nis facta à plano tran&longs;eunte per S, ac levitas minor in C magis <pb pagenum="257" xlink:href="017/01/273.jpg"/>di&longs;tat; quare vis tardiùs a&longs;cendendi major prævalere non po­<lb/>re&longs;t minori virtuti repugnanti ad de&longs;cendendum velociùs. </s> </p> <p type="main"> <s id="s.001897">Quemadmodum verò &longs;i tantum ponderis adderetur in A, ut <lb/>centrum commune gravitatis non po&longs;&longs;et imminere centro levi­<lb/>tatis partis demer&longs;æ, nemo non intelligit futuram omnimodam <lb/>depre&longs;&longs;ionem partis A infrà &longs;uperficiem aquæ, & omnimodam <lb/>emer&longs;ionem oppo&longs;itæ partis C; ita in Acrobarycis fluitantibus <lb/>manife&longs;tum e&longs;t, quò altiùs attollitur gravitas, eò faciliùs factâ <lb/>inclinatione transferri commune centrum gravitatis ultrà per­<lb/>pendiculum, in quo e&longs;t centrum levitatis partis demer&longs;æ. </s> <s id="s.001898">Sic <lb/>&longs;i ju&longs;to longior &longs;it in navi malus, factâ ex fluctibus inclinatione <lb/>in latus, aut &longs;altem impul&longs;u venti &longs;uprema carba&longs;a implentis, <lb/>facilis erit navis &longs;ubmer&longs;io, quia plus momentorum gravitatis <lb/>e&longs;t ex alterâ parte, quàm ex oppo&longs;itâ, tran&longs;lato in navis latus, <lb/>aut ultra illud, centro gravitatis totius partis extantis in aëre. </s> <lb/> <s id="s.001899">Sed de his, Deo dante, pleniùs in Hydro&longs;taticis di&longs;&longs;erendum <lb/>erit, ubi o&longs;tendetur ad navium &longs;tabilitatem nece&longs;&longs;ariam e&longs;&longs;e <lb/>eam centrorum di&longs;po&longs;itionem, ut centrum gravitatis totius na­<lb/>vis cum omnibus impo&longs;itis &longs;it infrà centrum levitatis partis de­<lb/>mer&longs;æ in eodem perpendiculo, in quo pariter erit centrum gra­<lb/>vitatis partis extantis. <lb/> </s> </p> <p type="main"> <s id="s.001900"><emph type="center"/>CAPUT IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001901"><emph type="center"/><emph type="italics"/>An, & cur libra ab æquilibrio dimota ad illud <lb/>redeat.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001902">NEmini dubium e&longs;&longs;e pote&longs;t æquilibrium tolli ob momento­<lb/>rum gravitatis inæqualitatem, vel quia in una libræ æqui­<lb/>libris lance additum e&longs;t pondus, vel quia altera jugi extremi­<lb/>tas, alicujus elevantis aut deprimentis vi, recedit à po&longs;itione <lb/>horizonti parallelâ. </s> <s id="s.001903">Illud in quæ&longs;tionem revocati pote&longs;t, an <lb/>&longs;ublato ponderis exce&longs;&longs;u, aut ce&longs;&longs;ante impul&longs;u extrin&longs;eco, li­<lb/>bra redeat ad æquilibrium, & po&longs;itionem horizonti parallelam <lb/>&longs;ibi ip&longs;a re&longs;tituat. </s> <s id="s.001904">Certè Keplerus in A&longs;tronomiâ Opticâ cap.1. <pb pagenum="258" xlink:href="017/01/274.jpg"/>prop. 20. a&longs;&longs;erit eum, qui negat libram brachiorum æqualium <lb/>ad horizontis æquilibrium redituram, <emph type="italics"/>non antiquitati tantum, <lb/>&longs;ed rerum naturæ, &longs;ed utilitati generis humani bellum indicere.<emph.end type="italics"/></s> <s id="s.001905"> At <lb/>ex adver&longs;o Authores ferè omnes, qui de his accuratiùs &longs;crip&longs;e­<lb/>runt, triplicem libræ &longs;peciem di&longs;tinguentes unam tantummo­<lb/>do agno&longs;cunt, quæ &longs;e re&longs;tituat horizonti parallelam. </s> <s id="s.001906">Hoc &longs;i­<lb/>quidem tanquam certum a&longs;&longs;umunt, corpus quodcumque gra­<lb/>ve, quod &longs;u&longs;pen&longs;um, aut &longs;u&longs;tentatum liberè in aëre pendeat, <lb/>in cò tantum &longs;itu quie&longs;cere, in quo gravitatis centrum cum &longs;u&longs;­<lb/>pen&longs;ionis aut &longs;u&longs;tentationis puncto in eâdem directionis lineâ <lb/>reperiatur; de&longs;cendit enim quantum pote&longs;t, neque ei opponi­<lb/>tur punctum &longs;u&longs;pen&longs;ionis aut &longs;u&longs;tentationis, ni&longs;i in eodem per­<lb/>pendiculo ad univer&longs;i centrum ducto utrumque &longs;it. </s> <s id="s.001907">Cumitaque <lb/>libra &longs;it corpus grave &longs;u&longs;pen&longs;um, & &longs;uum habeat centrum gra­<lb/>vitatis, tunc demùm quie&longs;cet, ubi eam po&longs;itionem obtinuerit, <lb/>in quâ &longs;u&longs;pen&longs;ionis punctum, & gravitatis centrum in eâdem <lb/>&longs;int directionis lineâ. </s> <s id="s.001908">Punctum verò &longs;upen&longs;ionis libræ non il­<lb/>lud hîc intelligitur, ex quo pendet an&longs;a, cui libra in&longs;eritur, &longs;ed <lb/>ip&longs;a Agina, &longs;eu &longs;partum, ut Ari&longs;totelico vocabulo utar, e&longs;t &longs;u&longs;­<lb/>pen&longs;ionis punctum; ex illo enim proximè libra &longs;u&longs;penditur. </s> </p> <p type="main"> <s id="s.001909">Hinc oritur triplex libræ &longs;pecies, quia tripliciter componi <lb/>po&longs;&longs;unt centrum motûs, & centrum gravitatis; primò &longs;cilicet <lb/>po&longs;&longs;unt in uno eodemque puncto convenire, deinde centrum <lb/>motûs pote&longs;t e&longs;&longs;e &longs;uperius, demum inferius centro gravitatis. </s> </p> <p type="main"> <s id="s.001910">Et quidem &longs;i unum idemque punctum &longs;it motûs & gravita­<lb/>tis centrum A, & æqualibus brachiis AB, AC æqualia &longs;int <lb/><figure id="id.017.01.274.1.jpg" xlink:href="017/01/274/1.jpg"/><lb/>adnexa pondera B & C, uti­<lb/>que æquilibrium horizonta­<lb/>le manet, propter momento­<lb/>rum æqualitatem tùm ratio­<lb/>ne gravitatum æqualium, <lb/>tùm ratione æqualium pro­<lb/>pen&longs;ionum ad motum. </s> <s id="s.001911">Si <lb/>igitur applicatâ manu in B <lb/>deprimatur libra, ut &longs;it DE; <lb/>amotâ manu, cur redeat libra ad priorem po&longs;itionem BC? </s> <s id="s.001912"><lb/>adhuc enim momenta utrinque &longs;unt æqualia, & tantumdem <lb/>a&longs;cendere deberet D, quantum de&longs;cenderet E: par igitur e&longs;t <pb pagenum="259" xlink:href="017/01/275.jpg"/>re&longs;i&longs;tentia ip&longs;ius D propen&longs;ioni ad motum ip&longs;ius E: neutro ita­<lb/>que prævalente fiet in eo &longs;itu DE con&longs;i&longs;tentia. </s> </p> <p type="main"> <s id="s.001913">Attamen huic argumentationi, quamvis legitimæ, non ac­<lb/>quie&longs;cunt nonnulli, qui libram huju&longs;modi in quácumque po&longs;i­<lb/>tione quie&longs;centem &longs;e vi&longs;uros de&longs;perant, quia nunquam vide­<lb/>runt: quare potiùs cau&longs;am inquirunt, cur ad æquilibrium re­<lb/>deat libra æqualium brachiorum, quamvis ex medio jugo &longs;u&longs;­<lb/>pendatur. </s> <s id="s.001914">Exi&longs;timant aliqui po&longs;&longs;e vim argumenti eludi, &longs;i con­<lb/>cedant quidem in uno eodemque puncto convenire centrum <lb/>motûs & centrum gravitatis jugi, non tamen libræ: nam &longs;i <lb/>præter jugum a&longs;&longs;umantur etiam uncini aut lances, quibus ad­<lb/>nectuntur aut imponuntur pondera, multò magis &longs;i eadem pon­<lb/>dera a&longs;&longs;umantur, centrum gravitatis huju&longs;ce molis compo&longs;itæ <lb/>reperiri a&longs;&longs;erunt infrà ip&longs;um jugum, ac propterea nullam e&longs;&longs;e <lb/>huju&longs;modi primam &longs;peciem libræ. </s> </p> <p type="main"> <s id="s.001915">Sit libræ jugum AB; centrum motûs & gravitatis jugi &longs;it C: <lb/>pendeant lances D & E, &longs;ingularúmque cum &longs;uis appendiculis <lb/>gravitas &longs;it æqualis gra­<lb/><figure id="id.017.01.275.1.jpg" xlink:href="017/01/275/1.jpg"/><lb/>vitati jugi, ut facere con­<lb/>&longs;ueverunt accuratiores <lb/>monetarij. </s> <s id="s.001916">Lancium igi­<lb/>tur &longs;imul &longs;umptarum <lb/>commune gravitatis cen­<lb/>trum e&longs;t in F: jungantur <lb/>centra gravitatum C & <lb/>F; & erit demum totius <lb/>libræ vacuæ DABE <lb/>commune gravitatis cen­<lb/>trum in G. <!-- KEEP S--></s> <s id="s.001917">Quod &longs;i lan­<lb/>cibus D & E imponan­<lb/>tur æqualia pondera, <lb/>commune centrum gravitatis erit inter G & F, atque quò gra­<lb/>viora erunt pondera, eò propiùs accedet ad F. <!-- KEEP S--></s> <s id="s.001918">E&longs;t igitur ma­<lb/>nife&longs;tum centra motûs & gravitatis totius libræ non in eodem <lb/>puncto convenire, &longs;ed gravitatis centrum e&longs;&longs;e infrà centrum <lb/>motûs, &longs;eu &longs;partum C. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001919">Verum effugium hoc nullum e&longs;&longs;e cen&longs;eo: inclinetur enim <lb/>libra, & acquirat po&longs;itionem HI, jam HM & IN lineæ di-<pb pagenum="260" xlink:href="017/01/276.jpg"/>rectionis lancium &longs;unt æquales, quia cædem cum AD & BE, <lb/>& &longs;unt parallelæ, quia ambæ perpendiculares ad horizontem; <lb/>ac propterea ex 33. lib.1. æquales &longs;unt ac parallelæ HI & MN. </s> <lb/> <s id="s.001920">Cumque CF linea directionis centri gravitatis jugi &longs;it ii&longs;dem <lb/>HM & IN parallela, & exeat ex C medio rectæ HI, cadet <lb/>pariter in medium rectæ MN ex 34 lib.1. & idem punctum F <lb/>e&longs;t commune centrum gravitatum M & N; atque proinde li­<lb/>bræ MHIN commune centrum gravitatis erit in eadem rectâ <lb/>lineá CF. <!-- KEEP S--></s> <s id="s.001921">Si itaque quie&longs;cit corpus grave &longs;u&longs;pen&longs;um, quando <lb/>in eâdem directionis linea e&longs;t punctum &longs;u&longs;pen&longs;ionis, & gravi­<lb/>tati, centrum, etiam in po&longs;itione HI deberet libra quie&longs;cere, <lb/>e&longs;to in C non conveniant contra motûs & gravitatis totius <lb/>libræ. </s> </p> <p type="main"> <s id="s.001922">Nicolaus Tartalea lib. 8. quæ&longs;ito 32. ideo libram ad paralle­<lb/>li&longs;mum horizontis redire exi&longs;timat, quia in inclinatione jugi <lb/>putat majora e&longs;&longs;e momenta brachij elevati, quàm depre&longs;&longs;i. </s> <lb/> <s id="s.001923">Id quod hâc methodo conatur o&longs;tendere. </s> <s id="s.001924">Si ex C æqualiter <lb/><figure id="id.017.01.276.1.jpg" xlink:href="017/01/276/1.jpg"/><lb/>di&longs;tent pondera æqualia A & B, <lb/>fuerintque ab æquilibrio remota, <lb/>de&longs;cribunt circulum, in quo <lb/>&longs;umptis partibus æqualibus, dum <lb/>A de&longs;cendit ex F in A, vis de­<lb/>&longs;cendendi e&longs;t NO, at ex A in G <lb/>vis de&longs;cendendi e&longs;t OP major, <lb/>quàm NO, ut con&longs;tat ex doctri­<lb/>nâ Sinuum. <!-- KEEP S--></s> <s id="s.001925">Similiter vis de&longs;cen­<lb/>dendi ip&longs;ius B ex I in B e&longs;t KL <lb/>major, quàm LM vis de&longs;cenden­<lb/>di ex B in H. <!-- KEEP S--></s> <s id="s.001926">E&longs;t autem KL ip&longs;i OP, & LM ip&longs;i ON <lb/>æqualis; igitur OP e&longs;t etiam major, quàm LM. <!-- KEEP S--></s> <s id="s.001927">Cum itaque <lb/>in &longs;itu ACB pondus B gravitet &longs;olùm ut LM, & pondus A <lb/>gravitet ut OP, major e&longs;t potentia ip&longs;ius A, quàm ip&longs;ius B: <lb/>igitur ad æquilibrium de&longs;cendere oportet pondus A. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001928">Sed peccat hæc Tartaleæ argumentatio, quia in pondere B <lb/>non e&longs;t con&longs;ideranda vis de&longs;cendendi in H, &longs;ed repugnantia <lb/>ad a&longs;cendendum in I, &longs;ecundùm quam ob&longs;i&longs;tit oppo&longs;ito pon­<lb/>deri A; hujus autem re&longs;i&longs;tentiæ men&longs;ura e&longs;t LK æqualis ip&longs;i <lb/>OP potentiæ &longs;eu propen&longs;ioni ip&longs;ius A ad de&longs;cendendum: <pb pagenum="261" xlink:href="017/01/277.jpg"/>æquatur ergo potentia re&longs;i&longs;tentiæ, nec ullus fieri pote&longs;t motus, <lb/>quamdiu hæc æqualitas permanet. </s> </p> <p type="main"> <s id="s.001929">Joannes Keplerus A&longs;tronomiæ Opticæ loco citato, cur libræ <lb/>brachia revolvantur ad æquilibrium, infert ex eo, quòd altero <lb/>brachiorum prægravato additione ponderis, ita jugum libræ <lb/>con&longs;i&longs;tit, ut quod e&longs;t gravius non planè imum locum petat, <lb/>& quod e&longs;t levius, non planè in apicem attollatur. </s> <s id="s.001930">Cujus rei <lb/>cau&longs;am inquirens &longs;tatuit libræ jugum <lb/><figure id="id.017.01.277.1.jpg" xlink:href="017/01/277/1.jpg"/><lb/>CD bifariam in A divi&longs;um; & centro <lb/>A de&longs;cripto circulo ducit perpendicu­<lb/>lum BAF: ex quo manife&longs;tum e&longs;t <lb/>neutrum pondus po&longs;&longs;e deprimi infra F, <lb/>aut attolli &longs;upra B. </s> <s id="s.001931">Sed quia pondus D <lb/>ponitur gravius, quàm pondus C, & <lb/>utrumque naturâ &longs;uâ ad imum tendit, <lb/>contenduntque invicem, partiuntur <lb/>inter &longs;e de&longs;cen&longs;um BF in proportione, <lb/>quâ ip&longs;a &longs;unt: adeò ut BH de&longs;cen&longs;us <lb/>ponderis C &longs;it ad BG de&longs;cen&longs;um ponderis D, ut pondus C ad <lb/>pondus D. <!-- KEEP S--></s> <s id="s.001932">E&longs;t autem FG linea æqualis lineæ BH, quia ex <lb/>æqualibus AB & AF auferuntur æqualia latera AH & AG, <lb/>cum enim triangula CHA, DGA rectangula &longs;int, & angu­<lb/>los ad verticem A æquales habeant, & latera AC, AD æqua­<lb/>lia; etiam per 26. lib.1. latus AH e&longs;t æquale lateri AG. <!-- KEEP S--></s> <s id="s.001933">Igitur <lb/>ut pondus C ad pondus D, ita FG ad GB. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001934">Ducatur ex F ad AD perpendicularis FK: &longs;imiliter triangula <lb/>AGD, AKF rectangula, & <expan abbr="cõmunem">communem</expan> angulum in A habentia, <lb/>cum latere AF æquali lateri AD, per eandem 26.lib.1. <expan abbr="hab&etilde;tla-tera">habent la­<lb/>tera</expan> AG & AK æqualia: ergo & re&longs;idua FG, DR æqualia &longs;unt. </s> <lb/> <s id="s.001935">Igitur propter <expan abbr="æqualitat&etilde;">æqualitatem</expan> <expan abbr="diametrorũ">diametrorum</expan> FB & DC, erit etiam GB <lb/>linea æqualis lineæ KC. </s> <s id="s.001936">Quare ut <expan abbr="põdus">pondus</expan> D ad pondus C, ita GB <lb/>ad GF, hoc e&longs;t ita KC ad KD: ac propterea factâ jugi &longs;u&longs;pen­<lb/>&longs;ione in K pondera C & D inæqualia &longs;ecundùm Rationem bra­<lb/>chiorum reciprocè po&longs;ita æquiponderabunt & con&longs;i&longs;tent. </s> <s id="s.001937">Cum <lb/>igitur in hac eâdem Ratione &longs;it de&longs;cen&longs;us BH & BG, ut e&longs;t <lb/>pondus C ad pondus D, fiet con&longs;i&longs;tentia in &longs;itu CAD. <emph type="italics"/>Ergo <lb/>per &longs;ub&longs;umptionem patet,<emph.end type="italics"/> &longs;ubdit Keplerus, cujus &longs;uperiorem <lb/>doctrinam conatus &longs;um paulo clariùs exponere, <emph type="italics"/>cur libræ brachia <emph.end type="italics"/><pb pagenum="262" xlink:href="017/01/278.jpg"/><emph type="italics"/>revolvuntur ad æquilibrium; cum enim æque ponderent, æquales etiam <lb/>in circulo fieri de&longs;cen&longs;us par e&longs;t.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.001938">Meam hebetudinem di&longs;&longs;imulare non po&longs;&longs;um, qui huju&longs;ce <lb/>Keplerianæ argumentationis vim &longs;atis a&longs;&longs;equi non valeo: quid <lb/>enim, &longs;i fieret æquilibrium horizontale ponderum, facta in K <lb/>&longs;u&longs;pen&longs;ione? </s> <s id="s.001939">an propterea con&longs;equens e&longs;t fieri æquilibrium <lb/>etiam in &longs;itu CAD, ni&longs;i aliunde probetur? </s> <s id="s.001940">&longs;ed quod ad rem <lb/>no&longs;tram attinet, pondera alligata, & adnexa libræ non ita con­<lb/>&longs;ideranda &longs;unt, ut ambo de&longs;cendant, &longs;i comparatè &longs;umantur, <lb/>&longs;ed alterius propen&longs;io ad motum deor&longs;um comparanda e&longs;t cum <lb/>alterius repugnantiâ ad motum &longs;ur&longs;um, & vici&longs;&longs;im hujus pro­<lb/>pen&longs;io ad de&longs;cendendum cum illius re&longs;i&longs;tentiâ, ne a&longs;cendat. </s> <lb/> <s id="s.001941">Quapropter &longs;i ex D pondere majore auferatur exce&longs;&longs;us &longs;upra <lb/>pondus C, & fiant æqualia pondera, non po&longs;&longs;unt ad æquili­<lb/>brium horizontale redire, ni&longs;i C de&longs;cendat, D verò a&longs;cendat: <lb/>Cum autem hujus a&longs;cen&longs;us GA &longs;it æqualis de&longs;cen&longs;ui HA, nul­<lb/>la e&longs;t ratio, cur propen&longs;io ponderis C vincere debeat æqualem <lb/>ponderis D re&longs;i&longs;tentiam. </s> </p> <p type="main"> <s id="s.001942">Deinde quid intelligendum e&longs;t, cum dicitur ip&longs;ius C de&longs;cen­<lb/>&longs;us e&longs;&longs;e BH, ip&longs;ius verò D de&longs;cen&longs;us e&longs;&longs;e BG? </s> <s id="s.001943">ex B enim non <lb/>utrumque de&longs;cendit, &longs;ed alterutrum: & &longs;i pondus D de&longs;cendi&longs;­<lb/>&longs;et ex B, ex adver&longs;o pondus C a&longs;cendi&longs;&longs;et ex F; cúmque illius <lb/>de&longs;cen&longs;us e&longs;&longs;et BG, hujus a&longs;cen&longs;us e&longs;&longs;et FH; &longs;unt autem BG <lb/>& FH æquales. </s> <s id="s.001944">Quòd &longs;i non motus præcedens, &longs;ed &longs;ola pro­<lb/>pen&longs;io ad de&longs;cendendum & repugnantia ad a&longs;cendendum con­<lb/>&longs;ideretur pro ratione po&longs;itionis, pondus D habet men&longs;uram <lb/>propen&longs;ionis ad de&longs;cendendum, non motum (qui forta&longs;&longs;e tran­<lb/>&longs;iit) ex B in D, &longs;ed quem in eo &longs;itu po&longs;&longs;et perficere ex D in F: <lb/>atque adeò ip&longs;ius D de&longs;cen&longs;us e&longs;t GF, eju&longs;que re&longs;i&longs;tentia, ne <lb/>a&longs;cendat u&longs;que ad &longs;ummum e&longs;t GB, & vici&longs;&longs;im ponderis C pro­<lb/>pen&longs;io ad de&longs;cendendum non e&longs;t ex B in C, &longs;ed ex C in F, &longs;i <lb/>u&longs;que ad imum de&longs;cendat, habens men&longs;uram HF, ejus verò <lb/>repugnantiam ad a&longs;cendendum metitur HB. <!-- KEEP S--></s> <s id="s.001945">E&longs;t igitur mani­<lb/>fe&longs;tum uniu&longs;cuju&longs;que ponderis propen&longs;ionem habere oppo&longs;i­<lb/>tam re&longs;i&longs;tentiam æqualem (e&longs;t enim propen&longs;io GF æqualis re­<lb/>&longs;i&longs;tentiæ HB, & propen&longs;ioni HF æquali e&longs;t re&longs;i&longs;tentia GB) <lb/>ac proinde nullum &longs;equi po&longs;&longs;e motum ponderum æqualium à <lb/>centro A æqualiter di&longs;tantium. </s> <s id="s.001946">At, inquis, quid cau&longs;æ e&longs;t, <pb pagenum="263" xlink:href="017/01/279.jpg"/>cur &longs;imilem libram in quácumque po&longs;itione quie&longs;centem non <lb/>habemus? </s> <s id="s.001947">&longs;ed omnis libra ea e&longs;t, ut vel ad æquilibrium redeat, <lb/>vel omninò quantum pote&longs;t de&longs;cendat, qua parte habet bra­<lb/>chium inclinatum Re&longs;pon&longs;io in promptu e&longs;t; quia &longs;cilicet dif­<lb/>ficillimum e&longs;t duo illa puncta exqui&longs;itè convenire, hoc e&longs;t cen­<lb/>trum motus & centrum gravitatis, nimirùm punctum illud, <lb/>quod brachiorum longitudinem di&longs;criminat. </s> <s id="s.001948">Quod &longs;i vel mi­<lb/>nimum duo illa centra di&longs;crepent, natura omnes &longs;ui juris api­<lb/>ces exacti&longs;&longs;imè per&longs;equitur, & e&longs;t &longs;partum non in medio, &longs;ed <lb/>aut in &longs;uperiore, aut in inferiore parte jugi (&longs;i quidem brachia <lb/>&longs;int æqualia; nam &longs;i ad latus e&longs;&longs;et in eadem recta linea, libra e&longs;­<lb/>&longs;et inæqualium brachiorum, & tunc non adnexorum ponderum <lb/>æqualitas e&longs;&longs;et con&longs;ideranda, &longs;ed eorum Ratio, &longs;umpta recipro­<lb/>cè brachiorum Ratione) ex quo &longs;equitur aut reditus ad æquili­<lb/>brium, aut ulterior de&longs;cen&longs;us brachij inclinati. </s> </p> <p type="main"> <s id="s.001949">Hinc e&longs;t de illâ duplici tantummodo libræ &longs;pecie locutum <lb/>fui&longs;&longs;e Ari&longs;totelem in Mechan. <expan abbr="q.">que</expan> 2. omi&longs;sá priore hac, quæ vi­<lb/>detur &longs;peculantis intellectûs terminis coërceri, nunquam in <lb/>praxim ni&longs;i fortuito deducenda. </s> <s id="s.001950">Non enim &longs;atis e&longs;t accurati&longs;­<lb/>&longs;imè inquirere centrum gravitatis jugi, ut illud &longs;it pariter cen­<lb/>trum motûs, &longs;ed nece&longs;&longs;e e&longs;t punctum hoc in eádem rectá lineâ <lb/>e&longs;&longs;e, quæ jungit puncta contactuum jugi & annulorum, ex <lb/>quibus lances dependent: nam ni&longs;i hoc contingat, centrum il­<lb/>lud gravitatis a&longs;&longs;umptum non e&longs;t punctum, à quo brachiorum <lb/>longitudines di&longs;criminantur, ut inferiùs con&longs;tabit dilucidiùs <lb/>ex iis, quæ de librâ curvâ dicentur. </s> </p> <p type="main"> <s id="s.001951">Quærendum e&longs;t itaque, cur libra aginam habens in &longs;upe­<lb/>riore loco, &longs;i ab æquilibrio horizontali dimoveatur, ad illud re­<lb/>deat. </s> <s id="s.001952">Et ne locus æquivocationi pateat, dum ad hoc de­<lb/>mon&longs;trandum a&longs;&longs;umuntur puncta notabili intervallo inter &longs;e <lb/>di&longs;tantia (ne videlicet linearum brevitas confu&longs;ionem aut ob­<lb/>&longs;curitatem pariat) ob&longs;erva lingulæ nomine non eam &longs;olùm par­<lb/>tem intelligi, quæ &longs;upra libræ jugum intrà an&longs;am excurrens <lb/>extat; &longs;ed lingulæ, &longs;eu, ut aliis placet, trutinæ pars e&longs;t etiam <lb/>linea, quæ in ip&longs;a jugi cra&longs;&longs;itie de&longs;cripta intelligitur perpendi­<lb/>cularis ad lineam longitudinis brachiorum, & tran&longs;iens per <lb/>centrum motûs. </s> <s id="s.001953">Quare hujus lineæ pars intercepta inter cen­<lb/>trum motûs, & lineam longitudinis brachiorum, &longs;ivè exigua <pb pagenum="264" xlink:href="017/01/280.jpg"/>&longs;it, &longs;ivè valde notabilis (quod quidem ad præ&longs;entem con&longs;ide­<lb/>rationem attinet) nihil intere&longs;t, nam eadem planè &longs;emper e&longs;t <lb/>ratio, atque demon&longs;tratio. </s> <s id="s.001954">Sit libra æqualium brachiorum <lb/><figure id="id.017.01.280.1.jpg" xlink:href="017/01/280/1.jpg"/><lb/>AB, cujus puncto medio C in­<lb/>&longs;i&longs;tat perpendicularis CD, & &longs;it <lb/>in ipsâ jugi cra&longs;&longs;itie centrum mo­<lb/>tûs punctum D, impo&longs;iti&longs;que <lb/>æqualibus ponderibus in A & B, <lb/>maneat in æquilibrio horizonta­<lb/>li AB. <!-- KEEP S--></s> <s id="s.001955">Deprimatur extremitas A, <lb/>ut veniat in E, reliqua extremitas <lb/>B a&longs;cendit in F, & C venit in G. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001956">Non pote&longs;t igitur manere libra in po&longs;itione EF &longs;ublato de­<lb/>primente in E, &longs;ed manentibus æqualibus ponderibus redit ad <lb/>æquilibrium, séque re&longs;tituit in AB; tùm quia centrum gravi­<lb/>tatis non e&longs;t in lineâ directionis tran&longs;eunte per D punctum <lb/>&longs;u&longs;pen&longs;ionis, tùm poti&longs;&longs;imum quia momenta ip&longs;ius F majora <lb/>&longs;unt momentis ip&longs;ius E ratione po&longs;itionis & propen&longs;ionis ad <lb/>motum; pote&longs;t enim F de&longs;cendere juxta men&longs;uram FH, dum <lb/>E a&longs;cendit juxta men&longs;uram EI; e&longs;t autem major Ratio motûs <lb/>FH ad motum EI, quam &longs;it Ratio ponderum, quæ e&longs;t Ratio <lb/>æqualitatis, nimirum ut FG ad GE. <!-- KEEP S--></s> <s id="s.001957">Nam per 8 lib.5. FO ad <lb/>GE majorem habet Rationem quàm FG ad GE, & FO ad <lb/>OE majorem habet Rationem quàm FO ad GE; ergo multo <lb/>major e&longs;t Ratio FO ad OE, quàm FG ad GE. <!-- KEEP S--></s> <s id="s.001958">At &longs;imilia <lb/>&longs;unt triangula FHO, EIO, quia æquiangula (nam propter <lb/>paralleli&longs;mum linearum directionis FH & IE, alterni E & F, <lb/>& alterni I & H, qui etiam recti ponuntur, & qui ad verticem <lb/>O, æquales &longs;unt) igitur per 4.lib. 6. ut FO ad OE, ita FH <lb/>ad EI. <!-- KEEP S--></s> <s id="s.001959">E&longs;t igitur major Ratio de&longs;censûs FH ad a&longs;cen&longs;um EI, <lb/>quàm &longs;it Ratio ponderum, quæ e&longs;t ut FG ad GE. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001960">Hinc patet clara &longs;olutio quæ&longs;tionis à Keplero propo&longs;itæ: <lb/>quia &longs;i pondus E majus &longs;it pondere F, illud non ad imum lo­<lb/>cum de&longs;cendet, &longs;ed ibi libra obliquè &longs;ub&longs;i&longs;tet, ubi pondera <lb/>erunt in Ratione reciprocâ motuum; quando &longs;cilicet ratione <lb/>po&longs;itionis ita propen&longs;io ad de&longs;cendendum ponderis F erit ad <lb/>re&longs;i&longs;tentiam ponderis E, ne a&longs;cendat, ut e&longs;t vici&longs;&longs;im pondus E <lb/>ad pondus I: & tunc perpendicularis linea directionis ex D <pb pagenum="265" xlink:href="017/01/281.jpg"/>pancto &longs;u&longs;pen&longs;ionis demi&longs;&longs;a cadet in centrum gravitatis compo­<lb/>&longs;itæ libræ & ponderum. </s> <s id="s.001961">Cujus rei argumentum e&longs;t mani­<lb/>fe&longs;tum, quod libra quie&longs;cens in po&longs;itione EF &longs;i moveatur ab <lb/>aliquo deprimente ulteriùs aut elevante, &longs;ibi relicta non minùs <lb/>redit ad eumdem &longs;itum obliquum, quam redeat ad æquilibrium <lb/>horizontale, &longs;i pondera &longs;int æqualia. </s> <s id="s.001962">Quæ omnia ex dictis pla­<lb/>na &longs;unt & aperta; &longs;ed an hoc idem rite probaverit Keplerus, <lb/>viderint alij. </s> </p> <p type="main"> <s id="s.001963">Eadem philo&longs;ophandi ratio erit in librâ brachiorum inæqua­<lb/>lium LM, in qua &longs;int pondera L & M (computatis ip&longs;orum <lb/>brachiorum gravitatibus juxta <lb/><figure id="id.017.01.281.1.jpg" xlink:href="017/01/281/1.jpg"/><lb/>momenta, quæ habent in illâ eâ­<lb/>dem longitudine, ut dictum cap.2. <lb/>hujus libri) reciprocè in Ratione <lb/>brachiorum NM & NL. </s> <s id="s.001964">Depri­<lb/>matur L in P, & elevabitur M in <lb/>Q, & N in V. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001965">Dico libram &longs;ummoto deprimen­<lb/>te, ad æquilibrium LM redituram. </s> <lb/> <s id="s.001966">Ducantur perpendiculares PT & QR, productâ LM horizon­<lb/>tali, &longs;i opus fuerit. </s> <s id="s.001967">Triangula SQR, SPT &longs;unt &longs;imilia; igitur <lb/>per 4 lib.6. ut QS ad SP, ita ponderis Q propen&longs;io ad de&longs;cen­<lb/>dendum QR, ad ponderis P re&longs;i&longs;tentiam, ne a&longs;cendat, PT. </s> <lb/> <s id="s.001968">E&longs;t autem major Ratio QR ad PT, quàm &longs;it ponderis P ad <lb/>pondus <expan abbr="q;">que</expan> igitur pondus Q prævalebit. </s> <s id="s.001969">Majorem autem e&longs;&longs;e <lb/>Rationem &longs;ic o&longs;tenditur. </s> <s id="s.001970">Pondus P ad pondus Q e&longs;t ut NM <lb/>ad NL ex hypothe&longs;i, hoc e&longs;t ut QV ad VP: &longs;ed per 8. lib. 5. <lb/>major e&longs;t Ratio QS ad VP, quàm QV ad VP, & major Ra­<lb/>tio QS ad SP, quàm QS ad VP: igitur major e&longs;t Ratio QS <lb/>ad SP, quàm QV ad VP, hoc e&longs;t quàm pondus P ad pon­<lb/>dus <expan abbr="q.">que</expan> E&longs;t autem demon&longs;tratum ita e&longs;&longs;e QS ad SP, ut QR <lb/>ad PT; igitur major e&longs;t Ratio de&longs;censûs QR ad a&longs;cen&longs;um PT, <lb/>quàm &longs;it Ratio ponderis P ad pondus Q: Ergo vis de&longs;cendendi <lb/>major e&longs;t; quàm oppo&longs;ita re&longs;i&longs;tentia, ac proptereà re&longs;tituet &longs;e <lb/>libra in æquilibrio horizontali. </s> </p> <p type="main"> <s id="s.001971">Ex his manife&longs;tum e&longs;t rem contrario modo &longs;e habere, quan­<lb/>do &longs;partum e&longs;t in cra&longs;&longs;itie jugi ira collocatum, ut &longs;it infra li­<lb/>neam, quæ con&longs;tituit longitudinem brachiorum; tunc enim al-<pb pagenum="266" xlink:href="017/01/282.jpg"/>tero brachiorum inclinato, tantum abe&longs;t, ut libra revertatur ad <lb/>priorem paralleli&longs;mum cum horizonte, ut potiùs, nullo ulteriùs <lb/>deprimente, brachium inclinatum de&longs;cendat omninò, donec <lb/>impediatur ab ansá, in quam incurrit alterum brachium eleva­<lb/>tum: quod &longs;i &longs;uperiori aut inferiori brachio nullum occurreret <lb/>impedimentum, ita fieret totius libræ conver&longs;io & revolutio, <lb/>ut &longs;partum e&longs;&longs;et in loco &longs;uperiore, & tunc demùm in æquili­<lb/>brio horizontali jugum quie&longs;ceret. </s> <s id="s.001972">Quæ omnia licet per&longs;picua <lb/>&longs;int, &longs;i &longs;uperiores duæ figuræ invertantur, clarioris tamen ex­<lb/><figure id="id.017.01.282.1.jpg" xlink:href="017/01/282/1.jpg"/><lb/>plicationis gratiâ, &longs;it iterum jugum AB <lb/>æqualiter divi&longs;um in C, & in perpen­<lb/>diculari CD &longs;it axis, & centrum mo­<lb/>tûs inferiùs in D: po&longs;itis æqualibus <lb/>ponderibus A & B &longs;it æquilibrium ho­<lb/>rizontale: & quoniam æqualia &longs;unt <lb/>pondera, atque æquales ad motum pro­<lb/>pen&longs;iones, centrumque gravitatis e&longs;t <lb/>in eâdem perpendiculari lineâ di­<lb/>rectionis cum puncto &longs;u&longs;tentationis D, manent in æquilibrio. </s> <lb/> <s id="s.001973">Deprimatur A in E, elevatur pariter B in F, & C deprimitur <lb/>in G. <!-- KEEP S--></s> <s id="s.001974">Dico libram, &longs;i &longs;ibi ip&longs;a dimittatur, non redituram ad po­<lb/>&longs;itionem AB &longs;upra punctum D; &longs;ed pondus E ulteriùs de&longs;cen­<lb/>&longs;urum. </s> <s id="s.001975">Ductis enim perpendicularibus EI & FH, propen&longs;io <lb/>ponderis F ad motum deor&longs;um, ut &longs;e re&longs;tituat in priore æqui­<lb/>librio, e&longs;t FH, re&longs;i&longs;tentia ponderis E ad motum &longs;ur&longs;um e&longs;t <lb/>EI. <!-- KEEP S--></s> <s id="s.001976">E&longs;t autem major Ratio re&longs;i&longs;tentiæ EI ad propen&longs;ionem <lb/>deor&longs;um FH, quàm &longs;it Ratio ponderis F ad pondus E, aut vi­<lb/>ci&longs;&longs;im; hæc enim æqualia &longs;unt ex hypothe&longs;i, & e&longs;t eorum Ra­<lb/>tio ut AC ad CB, hoc e&longs;t ut EG ad GF: Non igitur pote&longs;t à <lb/>pondere F, cujus momenta minora &longs;unt elevari pondus E, cu­<lb/>jus momenta &longs;unt majora ex di&longs;po&longs;itione ad motum. </s> <s id="s.001977">Con&longs;tat <lb/>verò major Ratio re&longs;i&longs;tentiæ EI ad propen&longs;ionem FH, quàm <lb/>ponderis F ad pondus E, quia in triangulis OIE, & OHF &longs;i­<lb/>milibus eâdem e&longs;t Ratio EI ad FH, quæ e&longs;t EO ad OF; &longs;ed <lb/>ex 8 lib.5. EO ad OF majorem habet Rationem quam EG ad <lb/>GF: igitur major e&longs;t Ratio EI ad FH, quam EG ad GF, hoc <lb/>e&longs;t ponderis ad pondus. </s> <s id="s.001978">De&longs;cendet itaque E, & nullo occur­<lb/>rente obice ea fiet totius libræ revolutio circà centrum D, ut <pb pagenum="267" xlink:href="017/01/283.jpg"/>demum jugum EF &longs;it infrà punctum D, & quod inito fuit <lb/>punctum &longs;u&longs;tentationis, fiat punctum &longs;u&longs;pen&longs;ionis libræ. </s> <s id="s.001979">Ea­<lb/>dem dicta intelligantur de librâ brachiorum inæqualium, quæ <lb/>&longs;upervacaneum e&longs;t iterum inculcare. </s> </p> <p type="main"> <s id="s.001980">Oblata itaque librâ facilè digno&longs;ces, cujus &longs;peciei illa &longs;it, <lb/>quamvis ob punctorum propinquitatem, &longs;cilicet centri mo­<lb/>tûs, & puncti brachiorum longitudinem di&longs;criminantis, non <lb/>valeat oculus dijudicare: impo&longs;itis enim æqualibus ponderi­<lb/>bus, ut habeat æquilibrium horizontale, aliquantulum depri­<lb/>me alterutrum brachiorum, & &longs;ublato deprimente, &longs;i quidem <lb/>man&longs;erit obliqua (id quod rari&longs;&longs;imè continget) pronunciabis <lb/>centrum motûs convenire cum puncto brachiorum longitudi­<lb/>nem di&longs;criminante: &longs;in autem ad æquilibrium redierit, cen­<lb/>trum motûs erit in &longs;uperiore loco; &longs;i ulteriùs de&longs;cenderit, cen­<lb/>trum motûs erit infra lineam longitudinis brachiorum. </s> <s id="s.001981">Vel <lb/>etiam facto æquilibrio horizontali, adde pondus alteri lanci; <lb/>&longs;i de&longs;cendat ita, ut jugum oblique con&longs;i&longs;tat aut magis aut mi­<lb/>nùs, prout major aut minor factus e&longs;t exce&longs;&longs;us ponderis, pro­<lb/>nunciabis centrum motûs e&longs;&longs;e in &longs;uperiore loco: at &longs;i factâ <lb/>ponderum inæqualitate lanx gravior u&longs;que ad imum deprima­<lb/>tur, quantùm pote&longs;t, indicabit centrum motûs e&longs;&longs;e in inferio­<lb/>re loco, aut convenire cum puncto brachia di&longs;criminante: &longs;ed <lb/>hoc ultimum temerè non affirmabis, ni&longs;i re&longs;titutâ ponderum <lb/>æqualitate, &longs;equatur quies in quacumque po&longs;itione, aut con­<lb/>versâ deor&longs;um ansâ non contingat obliqua jugi con&longs;i&longs;tentia: <lb/>&longs;i enim factâ an&longs;æ &longs;u&longs;pen&longs;ione centrum illud fui&longs;&longs;et in inferio­<lb/>re loco, factâ conver&longs;ione e&longs;&longs;et in &longs;uperiore loco, & continge­<lb/>ret æquilibrium in po&longs;itione obliquâ. <lb/></s> </p> <p type="main"> <s id="s.001982"><emph type="center"/>CAPUT V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.001983"><emph type="center"/><emph type="italics"/>An fieri po&longs;sit libra Curva.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.001984">QUamvis ad ponderum examen in&longs;tituendum rarò contin­<lb/>gere po&longs;&longs;it, ut librâ Curvâ uti cogamur, quia tamen in <lb/>machinamentis aliquibus ita aut loci angu&longs;tiæ, aut opportuna <pb pagenum="268" xlink:href="017/01/284.jpg"/>corporum movendorum di&longs;po&longs;itio, exigunt collocari ponde­<lb/>ra, ut & libræ Rationes &longs;erventur, & tamen jugi rectitudo nul­<lb/>la appareat; non erit hî inutile libram curvam examinare, ut, <lb/>&longs;i quando eâ uti contigerit, innote&longs;cat, quænam &longs;int brachio­<lb/>rum, & motuum Rationes. </s> <s id="s.001985">Libram autem curvam voco, quæ <lb/>a communi formâ deflectens latera habet non in directum po&longs;i­<lb/>ta, &longs;ed in angulum concurrentia, aut in arcum &longs;inuata, quo­<lb/>rum extremitates &longs;ivè &longs;ur&longs;um, &longs;ivè deor&longs;um re&longs;piciunt: factâ <lb/>enim &longs;u&longs;pen&longs;ione &longs;ive ubi angulum latera con&longs;tituunt, &longs;ivè in <lb/>aliquo arcûs puncto, ea fieri pote&longs;t hinc & hinc ponderum ad­<lb/>ditio, quam horizontale æquilibrium con&longs;equatur. </s> <s id="s.001986">Sed quia <lb/>imperitis fucum facere po&longs;&longs;et apparens hæc laterum longitudo, <lb/>caveant, ne ex illis jugum libræ deductum intelligant: contin­<lb/>gere &longs;cilicet pote&longs;t, ut planè varia &longs;it huju&longs;modi libræ forma, <lb/>& magnitudo, idem tamen &longs;it &longs;emper libræ jugum, in quo <lb/>brachia de&longs;umenda &longs;unt. </s> </p> <p type="main"> <s id="s.001987">Sint enim in angulum compacta duo latera recta AB & <lb/>AC; non e&longs;t tota jugi magnitudo computanda ex horum late­<lb/><figure id="id.017.01.284.1.jpg" xlink:href="017/01/284/1.jpg"/><lb/>rum longitudinibus; &longs;ed ex ipsâ extre­<lb/>mitatum B & C di&longs;tantiâ BC; quæ &longs;em­<lb/>per eadem e&longs;t, &longs;ivè &longs;it arcus BEFC, <lb/>&longs;ivè alia &longs;int latera DB & DC, aut <lb/>GB & GC, atque &longs;u&longs;pen&longs;io fiat &longs;ivè <lb/>in A, &longs;ivè in D, &longs;ivè in G, &longs;ivè in quo­<lb/>cumque alio puncto, quod &longs;it intra &longs;pa­<lb/>tium à lineis AB, AC, BC comprehen&longs;um. </s> <s id="s.001988">E&longs;t igitur idem <lb/>jugum BC, quia in B & C adnexa intelliguntur pondera, eo­<lb/>rúmque di&longs;tantia, prout libræ adnectuntur, ca e&longs;t, quæ jugi <lb/>longitudinem determinat. </s> <s id="s.001989">Verùm an libra æqualium &longs;it po­<lb/>tiùs, quàm inæqualium brachiorum, definiendum e&longs;t ex <lb/>puncto &longs;u&longs;pen&longs;ionis, à quo ad extremitates B & C deducen­<lb/>dæ &longs;unt rectæ lineæ; quæ &longs;i æquales fuerint, libra e&longs;t æqualium <lb/>brachiorum; &longs;in autem inæquales, inæqualium. </s> <s id="s.001990">Hinc &longs;i late­<lb/>ra AB & AC jungantur tran&longs;ver&longs;ario HI, in eoque &longs;umatur <lb/>punctum &longs;u&longs;pen&longs;ionis D, nil refert æqualia-ne, an inæqualia <lb/>&longs;int latera AB & AC? </s> <s id="s.001991">&longs;ed attendenda e&longs;t æqualitas aut in­<lb/>æqualitas linearum ex D ductarum ad extremitates B & C. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.001992">Neque me arguas, quòd dixerim jugum e&longs;&longs;e BC, & attenden-<pb pagenum="269" xlink:href="017/01/285.jpg"/>dam æqualitatem aut <expan abbr="inæqualitat&etilde;">inæqualitatem</expan> <expan abbr="linearũ">linearum</expan> ex puncto &longs;u&longs;pen&longs;io­<lb/>nis ductarum, puta DB & DC; brachia &longs;iquidem in ip&longs;o jugo <lb/>con&longs;ideranda &longs;unt; illæ <expan abbr="aut&etilde;">autem</expan> lineæ nihil habent cum jugo com­<lb/>mune præter puncta extrema B & C. <!-- KEEP S--></s> <s id="s.001993">Quamvis enim lineæ hu­<lb/>ju&longs;modi brachia libræ non &longs;int, &longs;i res proprie con&longs;ideretur, in&longs;e­<lb/>runt tamen æqualitatem aut inæqualitatem brachiorum, qua­<lb/>tenus ex puncto &longs;u&longs;pen&longs;ionis D ducta intelligitur ad BC jugum <lb/>perpendicularis DM, quæ jugum dividit in partes BM & CM <lb/>æquales aut inæquales. </s> <s id="s.001994">Nam quia triangula BMD & CMD <lb/>&longs;unt rectangula, quadrato BD, ex 47. lib.1. æqualia &longs;unt duo <lb/>quadrata DM & MB, & quadrato DC æqualia &longs;unt duo qua­<lb/>drata DM & MC. <!-- KEEP S--></s> <s id="s.001995">Si igitur lineæ DB & DC æquales &longs;unt, <lb/>carum pariter quadrata &longs;unt æqualia; ex quibus dempto com­<lb/>muni quadrato DM, remanent quadrata BM & CM æqualia, <lb/>ac proinde lineæ MB & MC æquales. </s> <s id="s.001996">Si verò lineæ BD & <lb/>CD &longs;unt inæquales, quadrata carum &longs;unt inæqualia; ex qui­<lb/>bus dempto communi quadrato DM, re&longs;idua &longs;unt quadrata <lb/>BM & CM inæqualia, eorumque latera (&longs;cilicet lineæ MB & <lb/>MC) inæqualia erunt pronuncianda. </s> </p> <p type="main"> <s id="s.001997">Brachia itaque hujus libræ curvæ propriè &longs;umpta non illa <lb/>&longs;unt, quæ apparent, & quia ex illis libræ curvæ moles con&longs;tat, <lb/>vulgariter hoc vocabulo donantur; &longs;ed &longs;unt &longs;egmenta lineæ <lb/>jungentis extremitates, quibus pondera adnectuntur; in quæ <lb/>&longs;egmenta dividitur à perpendiculo, quod ad illam ducitur ex <lb/>puncto, quod e&longs;t motûs centrum. </s> <s id="s.001998">Cum igitur punctum hoc, <lb/>quod tanquam centrum legem dat motui, &longs;it extrà lineam ex­<lb/>tremitates illas jungentem, aut in &longs;uperiore, aut in inferiore <lb/>loco crit; ac proptereà altera erit ex duabus illis &longs;peciebus li­<lb/>bræ, de quibus capite &longs;uperiore &longs;ermo fuit, habentibus &longs;par­<lb/>tum aut &longs;uprà, aut infrà; & huic curvæ ea omnia convenient, <lb/>quæ ibi dicta &longs;unt, ut fiat æquilibrium horizontale, aut obli­<lb/>quum. </s> <s id="s.001999">Si enim &longs;it libræ &longs;ca­<lb/><figure id="id.017.01.285.1.jpg" xlink:href="017/01/285/1.jpg"/><lb/>pus rectus AB bifariam divi­<lb/>&longs;us, centrum motûs habens <lb/>in C & pondera adnexa in D <lb/>& E æqualia, habet æquilibrium horizontale, ad quod redit, &longs;i <lb/>ab illo dimoveatur; & &longs;i pondera D & E &longs;int inæqualia, ha­<lb/>bet æquilibrium obliquum pro Ratione di&longs;criminis ponderum, <pb pagenum="270" xlink:href="017/01/286.jpg"/>quia &longs;cilicet centrum motûs C e&longs;t &longs;upra lineam DE jungentem <lb/>puncta contactuum, quibus pondera adnectuntur. </s> <s id="s.002000">Facta au­<lb/>tem figuræ conver&longs;ione, ut C &longs;it in inferiore loco, & linea DE <lb/>in &longs;uperiore, in &longs;olo æquilibrio horizontali manet, à quo &longs;i re­<lb/>moveatur, ad illud non redit, neque ullum habet æquilibrium <lb/>in po&longs;itione obliquâ, ut dictum e&longs;t. </s> <s id="s.002001">Jam ex jugo AB omnia <lb/>&longs;uperflua re&longs;ecentur, & remaneant virgulæ CD & CE con­<lb/>nexæ in C centro motûs: manife&longs;tum e&longs;t non e&longs;&longs;e immutata <lb/>ponderum momenta, & eundem e&longs;&longs;e motum libræ curvæ DCE <lb/>ac rectæ AB; &longs;ivè C intelligatur in parte &longs;uperiori, &longs;ivè in in­<lb/>feriori. </s> <s id="s.002002">Quare & de hac curvâ, quod ad æquilibrium &longs;pectat, <lb/>eadem dicenda &longs;unt, quæ de librâ &longs;partum &longs;uperiùs aut inferiùs <lb/>habente &longs;unt dicta. </s> </p> <p type="main"> <s id="s.002003">Et quidem &longs;i latera illa, quibus libra curva con&longs;tat, &longs;ecun­<lb/>dùm longitudinem æqualia &longs;int, & paris gravitatis, additis <lb/>hinc & hinc æqualibus ponderibus fiet æquilibrium horizonta­<lb/>le; quia vera linea jugi in &longs;egmenta æqualia dividitur, &longs;unt au­<lb/>tem omnes Rationes Æqualitatis, omninò &longs;imiles. </s> <s id="s.002004">At &longs;i late­<lb/>ra illa &longs;int inæqualia, non erunt addenda reciprocè pondera <lb/>(etiam computatâ ip&longs;orum laterum gravitate) in Ratione illa­<lb/>rum longitudinum; &longs;ed in Ratione &longs;egmentorum jugi, ut fiat <lb/>æquilibrium: quia ex laterum illorum inæqualitate &longs;tatim qui­<lb/>dem infertur etiam veram lineam jugi dividi in &longs;egmenta in­<lb/>æqualia; &longs;ed non illico con&longs;equens e&longs;t &longs;imilem e&longs;&longs;e Rationem <lb/>Inæqualitatis: Immò &longs;i inæqualia &longs;int illa latera, fieri omnino <lb/>non pote&longs;t, ut &longs;egmenta, quæ fiunt à perpendiculari cadente <lb/>in ba&longs;im, videlicet in lineam jugi, &longs;int in eâdem Ratione; alio­<lb/>quin &longs;i ba&longs;is &longs;egmenta e&longs;&longs;ent in Ratione laterum adjacentium, <lb/>angulus, ex quo perpendicularis demittitur, e&longs;&longs;et bifariam <lb/>&longs;ectus, per 3 lib.6. atque adeò duo triangula haberent duos an­<lb/>gulos duobus angulis æquales, nimirum rectum & acutum, at­<lb/>que latus haberent commune; ergo per 26.lib.1. & reliqua late­<lb/><figure id="id.017.01.286.1.jpg" xlink:href="017/01/286/1.jpg"/><lb/>ra e&longs;&longs;ent æqualia, contra hy­<lb/>pothe&longs;im. </s> <s id="s.002005">Sit enim libra cur­<lb/>va laterum inæqualium BAC, <lb/>linea recta BC e&longs;t vera linea <lb/>jugi, in quam cadens perpen­<lb/>diculum AD definit brachio-<pb pagenum="271" xlink:href="017/01/287.jpg"/>rum DB & DC longitudinem. </s> <s id="s.002006">Non e&longs;t autem DB ad DC <lb/>ut BA ad AC, alioquin angulus BAC e&longs;&longs;et bifariam &longs;ectus, <lb/>& duo triangula DAB, DAC haberent præter rectos ad D, <lb/>ctiam acutos ad A æquales, atque latus AD commune, ac <lb/>proinde e&longs;&longs;ent etiam latera BA & AC æqualia contra hypo­<lb/>the&longs;im. </s> </p> <p type="main"> <s id="s.002007">Sunt igitur anguli ad A inæquales, & minor e&longs;t, qui adja­<lb/>cet minori lateri AC, quàm qui adjacet majori lateri AB: quia <lb/>in triangulo BAC major e&longs;t angulus C oppo&longs;itus majori lateri <lb/>BA, quàm angulus B oppo&longs;itus minori lateri AC, ex 18.lib.1. <lb/>igitur in triangulis BDA, CDA rectangulis ad D, comple­<lb/>mentum CAD minus e&longs;t complemento BAD. </s> <s id="s.002008">Qua propter <lb/>&longs;i angulus BAC &longs;it bifariam dividendus, recta AE auferet ali­<lb/>quid ex majore angulo BAD, & con&longs;tituens angulum BAE <lb/>cadet in ba&longs;im inter B & D. <!-- KEEP S--></s> <s id="s.002009">E&longs;t itaque, per 3.lib.6. ut BA ad <lb/>AC, ita BE ad EC: &longs;ed minor e&longs;t Ratio BE ad EC quàm BD <lb/>ad EC, & multo minor quàm BD ad DC. per 8.lib.5. igitur <lb/>minor e&longs;t Ratio BA ad AC, quàm &longs;it Ratio brachij BD ad <lb/>brachium DC. <!-- KEEP S--></s> <s id="s.002010">Si igitur pondera in C & B e&longs;&longs;ent reciprocè ut <lb/>BA ad AC, haberent minorem Rationem, quàm BD ad DC, <lb/>ac propterea non e&longs;&longs;ent apta ad con&longs;tituendum æquilibrium <lb/>horizontale. </s> <s id="s.002011">Retento igitur pondere B, augendum e&longs;&longs;et pon­<lb/>dus C, vel retento pondere C, minuendum e&longs;&longs;et pondus B, ut <lb/>e&longs;&longs;ent in reciprocâ Ratione brachiorum BD & DC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002012">Hinc etiam con&longs;tat retentis eodem latere AB eadémque li­<lb/>neâ horizontali BC cum eodem angulo B, &longs;i velis uti minori <lb/>pondere, quod cum pondere B faciat æquilibrium, addendum <lb/>e&longs;&longs;e in A latus majus latere AC, puta latus AF, itaut tota BF <lb/>&longs;it jugi longitudo, & brachia &longs;int BD & DF. <!-- KEEP S--></s> <s id="s.002013">Manife&longs;tum e&longs;t <lb/>autem ex 8.lib.5. majorem Rationem e&longs;&longs;e eju&longs;dem BD ad DC <lb/>minorem, quàm ad DF majorem; ad pondera debent e&longs;&longs;e in F <lb/>& B ut BD ad DF; igitur minus pondus in F æquivalet eidem <lb/>ponderi B, cui in C æquivalet pondus majus. </s> <s id="s.002014">Porrò nemini <lb/>dubium e&longs;&longs;e pote&longs;t, an latus AF majus &longs;it latere AC, quippe <lb/>quod in triangulo CAF opponitur angulo obtu&longs;o ACF, per <lb/>19.lib.1. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002015">Sed &longs;i res fuerit in praxim deducenda, indicare oportet, quâ <lb/>methodo utendum &longs;it, ut quæ&longs;itam ponderum Rationem, hoc <pb pagenum="272" xlink:href="017/01/288.jpg"/>e&longs;t ip&longs;a jugi &longs;egmenta inveniamus, quippe quod &longs;olá mente <lb/>concipitur ad laterum extremitates jungedas deductum. </s> <s id="s.002016">Hæc <lb/>autem e&longs;&longs;e poterit praxis. </s> <s id="s.002017">Laterum AB & AC longitudine <lb/>metire, tùm ex B ad C extentum funiculum ad &longs;imilem men­<lb/>&longs;uram revoca. </s> <s id="s.002018">His paratis certum e&longs;t hanc jugi longitudinem <lb/>communiter majorem e&longs;&longs;e longitudine &longs;ingulorum laterum, <lb/>&longs;emper tamen &longs;altem alterius, tanto exce&longs;&longs;u, ut po&longs;&longs;it ab ea au­<lb/>ferri pars, de quâ mox dicetur; debet &longs;cilicet excedere me­<lb/>diam proportionalem inter aggregatum laterum, & eorum dif­<lb/>ferentiam. </s> <s id="s.002019">Cum enim linea jugi à perpendiculo cadente ex <lb/>angulo verticali dividenda &longs;it, utrumque latus cum jugo facit <lb/>angulos acutos; alioquin &longs;i alteruter angulorum rectus e&longs;&longs;et, <lb/>aut linea jugi non e&longs;&longs;et parallela horizonti, aut latus e&longs;&longs;et idem <lb/>perpendiculum; & &longs;i obtu&longs;us e&longs;&longs;et, perpendiculum caderet ex­<lb/>tra lineam extremitates jungentem. </s> <s id="s.002020">Debet igitur tanta e&longs;&longs;e <lb/>jugi longitudo, ut differentia partium, in quas dividitur ad <lb/>differentiam laterum &longs;it ut &longs;umma laterum ad totum jugum. </s> </p> <p type="main"> <s id="s.002021">Quare fiat ut jugi longitudo funiculo deprehen&longs;a ad laterum <lb/>&longs;ummam, ita laterum differentia ad partem auferendam ex <lb/>longitudine jugi; cujus re&longs;iduum bifariam divi&longs;um dabit mi­<lb/>noris brachij longitudinem. </s> <s id="s.002022">Hujus operationis ratio manife&longs;ta <lb/>e&longs;t ex corollario primo prop. 36.lib.3, & ex 3. eju&longs;dem lib.3. Sit <lb/>exempli gratia latus AB partium 20, latus AC partium 9, <lb/>di&longs;tantia BC partium 23. Fiat ut 23 ad 29 &longs;ummam laterum, <lb/>ita laterum differentia 11 ad (13 20/23) partem auferendam ex jugi <lb/>longitudine 23: Re&longs;iduum partium (9 3/23) bifariam dividatur, & <lb/>ejus &longs;emi&longs;&longs;is (4 13/23) e&longs;t longitudo brachij minoris DC; quod reli­<lb/>quum e&longs;t jugi partium (18 10/23) dat longitudinem alterius brachij <lb/>majoris BD. <!-- KEEP S--></s> <s id="s.002023">E&longs;t igitur brachiorum (atque adeò etiam ponde­<lb/>rum reciprocè) Ratio ut 424 ad 105. </s> </p> <p type="main"> <s id="s.002024">Quod &longs;i his cognitis inve&longs;tigare oporteat, quanta &longs;it hujus <lb/>lineæ horizontalis BC di&longs;tantia à puncto &longs;u&longs;pen&longs;ionis A, ni­<lb/>mirum quanta &longs;it perpendicularis AD, &longs;tatim ex 47. lib.1. in­<lb/>note&longs;cet, &longs;i ex quadrato lateris AC 81 auferas brachij DC <lb/>quadratum (20 445/529); nam re&longs;iduum (60 84/529) e&longs;t quadratum perpen­<lb/>diculi AD, quod proinde e&longs;t partium (7 17/23) proximè. </s> </p> <p type="main"> <s id="s.002025">At &longs;i pro ratione tui in&longs;tituti nimia &longs;it hujus perpendiculi <pb pagenum="273" xlink:href="017/01/289.jpg"/>longitudo, & opportuniùs accidat jugum BC horizontale mi­<lb/>nus di&longs;tare à puncto &longs;u&longs;pen&longs;ionis A, jam con&longs;tat latera AB <lb/>& AC explicanda in majorem angulum; quapropter etiam <lb/>major erit jugi longitudo, ex 24.lib.1. Sit ergo definita per­<lb/>pendiculi AD altitudo partium 4: hujus quadratum 16 aufer <lb/>ex 81 quadrato lateris AC, & re&longs;iduum 65 e&longs;t quadratum bra­<lb/>chij minoris DC, quod idcircò e&longs;t partium (8 1/16) &longs;erè. </s> <s id="s.002026">Simili­<lb/>ter ip&longs;ius AD quadratum 16 aufer ex 400 quadrato lateris AB, <lb/>& re&longs;iduum 384 e&longs;t quadratum brachij majoris BD, quod e&longs;t <lb/>partium (19 23/39) proximè; & totum jugum BC e&longs;t partium (27 25/39). <lb/>Quare brachi BD ad brachium DC Ratio e&longs;&longs;et ut 764 ad 314, <lb/>quæ reciprocè e&longs;&longs;et & ponderum. </s> </p> <p type="main"> <s id="s.002027">Ex quibus per&longs;picuum e&longs;t, po&longs;itis ii&longs;dem libræ curvæ late­<lb/>ribus, di&longs;parem e&longs;&longs;e ponderum Rationem: in priore enim po&longs;i­<lb/>tione Ratio e&longs;t 424 ad 105, hoc e&longs;t proxime ut 4 ad 1. in po&longs;te­<lb/>riore po&longs;itione, ubi in majorem angulum latera explicantur, <lb/>Ratio e&longs;t 764 ad 314, hoc e&longs;t ut 2. 43 ad 1; quæ minor e&longs;t <lb/>Ratio, quàm prior ut 4 ad 1. Si autem latera eadem e&longs;&longs;ent in <lb/>directum con&longs;tituta, e&longs;&longs;et ponderum Ratio ut 20 ad 9, hoc e&longs;t <lb/>ut 2. 22′ ad 1; quæ e&longs;t minima Ratio omnium, quæ intercede­<lb/>re po&longs;&longs;unt inter pondera æquilibrium horizontale con&longs;tituen­<lb/>tia ex illorum laterum extremitatibus: quæ extremitates quo­<lb/>minus di&longs;tabunt, inflexis &longs;ubinde lateribus, eo majus pondus <lb/>requiretur in extremitate lateris brevioris, ut æquè ponderet <lb/>cum uno eodemque pondere collocato in extremitate lateris <lb/>longioris. </s> </p> <p type="main"> <s id="s.002028">Porrò ubi de ponderum Ratione &longs;ermo e&longs;t, cave ne ip&longs;orum <lb/>laterum inæqualium libræ curvæ gravitatem contemnas; &longs;i <lb/>enim æqualia illa e&longs;&longs;ent, æqualia quoque e&longs;&longs;ent eorum mo­<lb/>menta tùm ratione gravitatis, tum ratione po&longs;itionis, nam per­<lb/>pendiculum caderet in medium jugum, & latera e&longs;&longs;ent &longs;imi­<lb/>liter inclinata, ac proinde &longs;ola ponderum æqualitas &longs;pectaretur: <lb/>at laterum huju&longs;modi inæqualium momenta &longs;unt ex utroque <lb/>capite inæqualia, videlicet & ratione gravitatis in&longs;itæ, quæ ex <lb/>hypothe&longs;i &longs;ingulis lateribus ine&longs;t pro Ratione molis inæqualis, <lb/>& ratione po&longs;itionis, quæ valde diver&longs;a e&longs;t, cùm non &longs;int late­<lb/>ra illa &longs;imili angulo ad perpendiculum inclinata; &longs;ed magis in-<pb pagenum="274" xlink:href="017/01/290.jpg"/>clinatur latus longius faciens cum perpendiculo majorem an­<lb/>gulum: pro variâ autem inclinatione ip&longs;am eju&longs;dem lateris gra­<lb/>vitatem varia obtinere momenta manife&longs;tum videtur. </s> <s id="s.002029">Pona­<lb/><figure id="id.017.01.290.1.jpg" xlink:href="017/01/290/1.jpg"/><lb/>mus laminam metallicam AB clavo <lb/>infixam in A, circa quem qua&longs;i cen­<lb/>trum de&longs;cribat &longs;emicirculum BDC. </s> <lb/> <s id="s.002030">Si obtineat perpendicularem po&longs;itio­<lb/>nem AB, tota gravitas innititur clavo <lb/>A &longs;u&longs;tinenti, & nullam vim habet de­<lb/>&longs;cendendi; &longs;imiliter in perpendiculari <lb/>po&longs;itione AC tota gravitas retinetur à <lb/>clavo A, nec pote&longs;t de&longs;cendere. </s> <s id="s.002031">At &longs;i <lb/>po&longs;itionem habeat AD horizonti pa­<lb/>rallelam, omnino nec &longs;u&longs;tinetur, nec <lb/>retinetur à clavo, &longs;ed toto conatu &longs;uas <lb/>de&longs;cendendi vires exerit. </s> <s id="s.002032">In locis igi­<lb/>tur intermediis partim &longs;u&longs;tinetur aut <lb/>retinetur à clavo A, partim conatum <lb/>deor&longs;um exercet: &longs;ic ex B veniens in E &longs;u&longs;tinetur juxta men­<lb/>&longs;uram FE, & deor&longs;um tendit juxta men&longs;uram GE; at ex B ve­<lb/>niens in H &longs;u&longs;tinetur juxta men&longs;uram IH, & deor&longs;um tendit <lb/>juxta men&longs;uram KH. <!-- KEEP S--></s> <s id="s.002033">Simili modo contingit in quadrante in­<lb/>feriore; nam in po&longs;itione AL retinetur juxta men&longs;uram IL, <lb/>nec de&longs;cen&longs;um pote&longs;t habere ni&longs;i ut LM; atque in O impedi­<lb/>mentum à retinente e&longs;t ut FO, conatum deor&longs;um metitur ON. <!-- KEEP S--></s> <lb/> <s id="s.002034">Quia &longs;cilicet &longs;i ab aliquo &longs;u&longs;tineatur in L, perinde &longs;e habet ac <lb/>&longs;i e&longs;&longs;et in plano habente inclinationis angulum CAL; in quo <lb/>plano gravitatio e&longs;t ad gravitationem in perpendiculo ut Ra­<lb/>dius ad &longs;ecantem, &longs;eu ut Sinus Complementi ad Radium, hoc <lb/>e&longs;t ut IL ad AL: ac propterea vires clavi retinentis in eâ in­<lb/>clinatione ad vires retinentis in perpendiculo debent e&longs;&longs;e ut IL <lb/>ad AC, hoc e&longs;t ad AL: At gravitatio, quâ urgetur planum <lb/>inclinatum, e&longs;t ut PC Sinus Ver&longs;us anguli inclinationis, qui <lb/>planè æqualis e&longs;t ip&longs;i LM. <!-- KEEP S--></s> <s id="s.002035">Cùm autem hîc nullum habeatur <lb/>&longs;ubjectum planum, quod prematur à gravitante laminâ metal­<lb/>licâ, exerit hunc conatum deor&longs;um adversùs aliud oppo&longs;itum <lb/>pondus, quod elevare conatur, vel cui conanti re&longs;i&longs;tit, ne ab <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"/>contra clavum A exercet momenta totius gravitatis deor&longs;um <lb/>nitentis, & in AL impeditur, ac retinetur &longs;ecundum men&longs;u­<lb/>ram IL, fiat ut AC ad IL, ita tota gravitas laminæ ad aliud, <lb/>& prodibit quantitas gravitationis contra retinentem, re&longs;i­<lb/>duumque LM erit illa gravitatio, quæ con&longs;ideranda e&longs;t in eâ <lb/>po&longs;itione inclinata AL. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002037">Sed quoniam AL à centro motûs A di&longs;tantiam habet AI, <lb/>comparanda erit hæc di&longs;tantia cum di&longs;tantia oppo&longs;iti lateris li­<lb/>bræ, ut habeantur momenta invicem comparata. </s> <s id="s.002038">Ob&longs;ervan­<lb/>dum tamen e&longs;t non rem perinde &longs;e habere, ac &longs;i tota gravita­<lb/>tio laminæ inclinatæ AL po&longs;ita e&longs;&longs;et in L, atque adeò in di&longs;tan­<lb/>tià AI; &longs;ed quia di&longs;tribuitur &longs;ecundùm totam ip&longs;am longitudi­<lb/>nem AL, & partes remotiores plus habent momenti, quàm <lb/>propiores centro, juxtà Rationem di&longs;tantiarum, proptereà vel <lb/>tota gravitas lateris AL, quæ e&longs;t LM, intelligenda e&longs;t in me­<lb/>dia di&longs;tantiâ inter A & I, vel &longs;emi&longs;&longs;is gravitationis AL, hoc e&longs;t <lb/>&longs;emi&longs;&longs;is ip&longs;ius LM, intelligendus e&longs;t in I, quemadmodum hu­<lb/>jus libri 3. cap. 2. dictum e&longs;t totam gravitatem AD intelligen­<lb/>dam in mediâ di&longs;tantiâ inter A & D, aut ejus &longs;emi&longs;&longs;em in ex­<lb/>tremitate D. <!-- KEEP S--></s> <s id="s.002039">Quamvis autem ex inclinatione CAL oriatur <lb/>di&longs;tantia AI, hæc tamen venire pariter in computationem <lb/>debet, quia comparari debent hæc momenta cum momentis <lb/>di&longs;tantiæ oppo&longs;itæ, quæ momenta orta ex Ratione di&longs;tantiarum <lb/>eadem &longs;unt, &longs;ive AL &longs;it lamina, &longs;ive trabs; quamquam valde <lb/>di&longs;pares &longs;int gravitates, quæ a&longs;&longs;umendæ &longs;unt ex eâdem inclina­<lb/>tione; ac propterea & LM indicans gravitationem comparatè <lb/>ad totam gravitatem ab&longs;olutam, & AI definiens momentum <lb/>ex di&longs;tantiâ, con&longs;iderari debent. </s> <s id="s.002040">Hoc pacto habetur totum <lb/>momentum lateris AL; &longs;imiliterque habebitur momentum la­<lb/>teris oppo&longs;iti. </s> <s id="s.002041">Ex quo patet laterum inclinatorum in librâ cur­<lb/>vâ momenta componi & ex Ratione di&longs;tantiarum, & ex Ratio­<lb/>ne momenti, quod habent &longs;ingula latera ex inclinatione ad <lb/>perpendiculum. </s> </p> <p type="main"> <s id="s.002042">At &longs;ubdubitas, utrùm i&longs;ta, quæ hîc dicuntur, cum iis aptè <lb/>cohæreant, quæ lib.1. cap.15. dicta &longs;unt, ubi ponderis in L <lb/>con&longs;tituti vires ad de&longs;cendendum definiri diximus à Sinu an­<lb/>guli declinationis à perpendiculo CAL, qui æqualis e&longs;t ip&longs;i <lb/>AI: hîc verò laminæ AL gravitationem con&longs;tituimus ex <pb pagenum="276" xlink:href="017/01/292.jpg"/>Sinu complementi eju&longs;dem anguli CAL, nimirum ex li­<lb/>neâ IL. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002043">Quapropter ob&longs;erva non eandém e&longs;&longs;e rationem gravitationis <lb/>lateris AL libræ, atque ponderis adnexi in extremitate L; hu­<lb/>jus enim momenta perinde computantur, ac &longs;i e&longs;&longs;et in I; quia <lb/>&longs;cilicet AI æqualis e&longs;t brachio libræ PL, & planum inclina­<lb/>tum, in quo pondus L con&longs;titutum intelligitur, non e&longs;t AL, <lb/>&longs;ed Tangens in L ad angulos rectos, ut loco citato explicatum <lb/>e&longs;t. </s> <s id="s.002044">At libræ latus AL &longs;uam habens gravitatem aliter &longs;e habet: <lb/>nam quemadmodum &longs;i inniteretur clavo in A, non tamen illi <lb/>infigeretur, atque ab aliquo &longs;u&longs;tineretur in puncto L, certum <lb/>e&longs;t planum inclinatum, in quo moveretur, e&longs;&longs;e AL, contra <lb/>quæ momenta de&longs;cendendi in plano inclinato reluctatur clavus <lb/>in A po&longs;itus, & retinens; ita &longs;ublato &longs;u&longs;tinente in L, & po&longs;ito <lb/>contranitente reliquo latere libræ, non tollitur munus clavi A <lb/>retinentis, &longs;ed &longs;ub&longs;tituitur latus illud oppo&longs;itum loco &longs;u&longs;tinen­<lb/>tis in L: igitur contra illud latus hoc latus AL exercet eadem <lb/>momenta gravitationis, quæ exerceret adversùs &longs;u&longs;tinentem <lb/>in L, hoc e&longs;t in planum inclinatum; quæ momenta ea &longs;unt, <lb/>quæ remanent demptis IL momentis gravitationis in plano in­<lb/>clinato, nimirum re&longs;iduum LM. <!-- KEEP S--></s> <s id="s.002045">Quia verò qui &longs;u&longs;tineret la­<lb/>tus AL in L, non e&longs;&longs;et unicum &longs;u&longs;tinens, &longs;ed planum inclina­<lb/>tum e&longs;t AL, & ita latus retinetur in clavo A, ut etiam ab eo <lb/>aliquatenus &longs;u&longs;tineatur, atque adeò lamina inclinata &longs;u&longs;tinea­<lb/>tur à duobus in A & L, retineaturque &longs;olùm ab A; propterea <lb/>non totum momentum LM, &longs;ed ejus &longs;emi&longs;&longs;em accipiendum <lb/>diximus, ut habeantur momenta, quibus contranititur oppo­<lb/>&longs;itum latus, &longs;i addantur momenta, quæ oriuntur ex di&longs;tantiâ à <lb/>centro motûs, ut dictum e&longs;t. </s> </p> <p type="main"> <s id="s.002046">Hæc autem ut exemplo clariora fiant, &longs;int eadem, quæ priùs <lb/>in præcedente figurâ po&longs;ita &longs;unt, latera libræ curvæ BAC, lon­<lb/>gius BA partium 20, brevius CA partium 9, & quidem in eâ <lb/>po&longs;itione, ut perpendiculum AD cadens in jugum &longs;it partium <lb/>(7 17/23), & brachium jugi DC adjacens minori lateri &longs;it partium <lb/>(4 13/23), reliquum verò jugi brachium DB partium (18 10/23). Primùm <lb/>quære momenta laterum ex eorum inclinatione: Cumque per­<lb/>pendiculum AD &longs;it æquale Sinui Complementi anguli incli-<pb pagenum="277" xlink:href="017/01/293.jpg"/>nationis DAC, po&longs;ito Radio AC, notus e&longs;t Sinus Ver&longs;us eju&longs;­<lb/>dem anguli inclinationis, &longs;cilicet differentia inter AD & AC, <lb/>quæ e&longs;t partium (1 6/23): & &longs;imili methodo Sinus Ver&longs;us anguli in­<lb/>clinationis DAB e&longs;t partium (12 6/23). Ratio igitur gravitationis <lb/>lateris AB ad gravitationem lateris AC ex inclinatione e&longs;t ut <lb/>282 ad 29; Ratio momentorum ex di&longs;tantiâ à centro, ut &longs;upra <lb/>diximus, e&longs;t ut 424 ad 105. Compo&longs;itis igitur duabus hi&longs;ce <lb/>Rationibus, e&longs;t totius momenti lateris AB ad totum momen­<lb/>tum lateris AC Ratio ut 119568 ad 3045, hoc e&longs;t in minimis <lb/>terminis ut 39. 267″ ad 1. Sit igitur gravitas ab&longs;oluta lateris <lb/>AB unciarum 20; gravitatio re&longs;pondens &longs;emi&longs;&longs;i Sinus Ver&longs;i an­<lb/>guli inclinationis e&longs;t unciarum (6 3/23). Item gravitas ab&longs;oluta la­<lb/>teris AC &longs;it unc. </s> <s id="s.002047">9: gravitatio re&longs;pondens &longs;emi&longs;&longs;i Sinus Ver&longs;i <lb/>anguli inclinationis e&longs;t unc. (29/46). Hæc gravitatio (29/46) ducatur in <lb/>di&longs;tantiam à perpendiculo partium (4 13/23), & e&longs;t momentum <lb/>2.878‴. </s> <s id="s.002048">Similiter gravitatio unc. (6 3/23) ducatur in di&longs;tantiam à <lb/>perpendiculo partium (18 10/23), & e&longs;t momentum 113.013‴. </s> <s id="s.002049">Di­<lb/>vi&longs;o itaque majore numero 113013 per minorem 2878, in mi­<lb/>nimis terminis Ratio e&longs;t ut 39.268″ ad 1: quæ minimùm differt <lb/>à priore illa Ratione propter neglectas fractiunculas in divi­<lb/>&longs;ionibus. </s> </p> <p type="main"> <s id="s.002050">Nunc inquiramus, quantum ponderis addendum &longs;it lateri <lb/>minori, ut fiat æquilibrium cum &longs;olâ majoris lateris gravitate. </s> <lb/> <s id="s.002051">Statuatur pondus addendum Algebricè 1 ℞, cujus di&longs;tantia à <lb/>perpendiculo cum &longs;it partium (4 13/23), ponderis additi momentum <lb/>e&longs;t (105/23) ℞ addendum momento lateris minoris invento. </s> <s id="s.002052">Quare <lb/>2.878‴ + (105/23) ℞ æquantur momento 113.013‴ lateris majoris: <lb/>& utrinque demptis 2.878‴, remanet æquatio inter (105/23) ℞ & <lb/>110.135‴. </s> <s id="s.002053">Demum in&longs;titutâ divi&longs;ione prodit <expan abbr="pretiũ">pretium</expan> 1 ℞, hoc e&longs;t <lb/>ponderis addendi, unciarum 24 1/8. Huic itaque ponderi additâ <lb/>gravitatione lateris minoris AC unc. (29/46) hoc e&longs;t in mille&longs;imis <lb/>630‴, erit in C totum pondus unc. </s> <s id="s.002054">24.755‴; & in B intelli­<lb/>gitur gravitas unc. (6 3/23), hoc e&longs;t in mille&longs;imis unc. </s> <s id="s.002055">6.130‴ ferè. </s> <lb/> <s id="s.002056">Vides igitur hæc pondera e&longs;&longs;e reciprocè po&longs;ita in Ratione <lb/>di&longs;tantiarum DB & DC: & quamvis demum in his Ratio­<lb/>nibus non &longs;ibi exacti&longs;&longs;imè re&longs;pondeant numeri, &longs;atis pa-<pb pagenum="278" xlink:href="017/01/294.jpg"/>tet exiguum hoc di&longs;crimen oriri ex neglectis fractiun­<lb/>culis. </s> </p> <p type="main"> <s id="s.002057">Cæterùm hæc tam minutè per&longs;equi in librâ curvâ, cujus <lb/>latera non adeò notabili gravitate &longs;unt prædita, labor quidem <lb/>videtur inutilis: &longs;ed quoniam huju&longs;modi libræ præcipuus u&longs;us <lb/>e&longs;&longs;e pote&longs;t in machinationibus, ubi latera libræ &longs;unt tigilli cra&longs;­<lb/>&longs;iores non mediocris gravitatis, operæ pretium fuit indicare, <lb/>quâ methodo ip&longs;orum laterum gravitates & momenta compu­<lb/>tari oporteat, ut non ca&longs;u, &longs;ed ex certâ ratione pondera collo­<lb/>centur, & æquipondia &longs;tatuantur. <lb/> </s> </p> <p type="main"> <s id="s.002058"><emph type="center"/>CAPUT VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002059"><emph type="center"/><emph type="italics"/>Quænam libræ &longs;int omnium exacti&longs;simæ.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002060">IN&longs;trumenti cuju&longs;que bonitas æ&longs;timatur ex fine, ad quem fuit <lb/>in&longs;titutum, prout ad illum a&longs;&longs;equendum aptum fuerit, aut <lb/>ineptum, eóque melius cen&longs;etur in&longs;trumentum, quò certiùs <lb/>per illud propo&longs;itus finis obtinetur; quemadmodum per &longs;ingu­<lb/>la eunti facilè con&longs;tabit. </s> <s id="s.002061">Ut igitur exacti&longs;&longs;imum libræ genus <lb/>innote&longs;cat, &longs;atis patet inquirendum e&longs;&longs;e, quænam libra facilli­<lb/>mè ab æquilibrio recedat; quo rece&longs;&longs;u indicans vel minimam <lb/>ponderum inæqualitatem, etiam &longs;uo æquilibrio exqui&longs;itam <lb/>ponderum æqualitatem o&longs;tendit; id quod per libram ve&longs;tiga­<lb/>mus. </s> <s id="s.002062">Hîc autem de librâ æqualium brachiorum &longs;ermo e&longs;t, quâ <lb/>communiter uti &longs;olemus: quamquam aliqua etiam ad libram <lb/>inæqualium brachiorum proportione traduci queant. </s> <s id="s.002063">Ex du­<lb/>plici capite libram, quà libra e&longs;t, ponderum gravitates præ aliis <lb/>libris exqui&longs;itè examinare contingit, videlicet aut ex brachio­<lb/>rum longitudine, aut ex &longs;parti, &longs;eu centri motûs, po&longs;itione; <lb/>reliqua enim impedimenta, aut adjumenta materiam potiùs &longs;e­<lb/>quuntur, quàm libræ formam. </s> </p> <p type="main"> <s id="s.002064">Et quidem quod ad brachiorum longitudinem &longs;pectat, adeò <lb/>certum Ari&longs;toteli videtur majoribus libris, majori &longs;cilicet bra­<lb/>chiorum longitudine præditis, accuratiùs examinari ponde­<lb/>rum æqualitatem, ut in Mechanicis quæ&longs;tionibus hoc primum <pb pagenum="279" xlink:href="017/01/295.jpg"/>ab eo quæratur, <emph type="italics"/>Cur majores libræ exactiores &longs;unt minoribus?<emph.end type="italics"/> Cau­<lb/>&longs;am autem ex eo de&longs;umendam putat, quòd &longs;partum &longs;it cen­<lb/>trum, brachia verò qua&longs;i lineæ à centro exeuntes; & quia Ra­<lb/>dij longiores ab eodem centro cum brevioribus exeuntes &longs;i pa­<lb/>riter moveantur, majorem arcum de&longs;cribunt, propterea etiam <lb/>citius moveri nece&longs;&longs;e e&longs;t extremitatem libræ, quò plus à &longs;parto <lb/>di&longs;ce&longs;&longs;erit. </s> <s id="s.002065">Hinc e&longs;t in minore librâ po&longs;&longs;e aliquando ex tenui <lb/>inæqualitate ponderum fieri motum non con&longs;picuum, atque <lb/>adeò illam occultè di&longs;cedere ab æquilibrio; id quod in majore <lb/>librâ contingere non pote&longs;t, quia longioris brachij extremitas <lb/>notabili motu inclinatur. </s> <s id="s.002066">Sit enim li­<lb/><figure id="id.017.01.295.1.jpg" xlink:href="017/01/295/1.jpg"/><lb/>bra longior AB, cujus &longs;partum &longs;it C; <lb/>moveatur, & de&longs;cribat arcus BG, & <lb/>AF, qui &longs;unt multò magis con&longs;picui <lb/>& majores, quàm qui à librâ minore <lb/>DE habente idem motûs centrum C, <lb/>de&longs;cribantur arcus EI & DH. <!-- KEEP S--></s> <s id="s.002067">Con­<lb/>&longs;tat igitur motum puncti E pror&longs;us fugere omnem oculorum <lb/>aciem, &longs;i motus extremitatis B vix &longs;it con&longs;picuus. </s> <s id="s.002068">Ex quo il­<lb/>lud etiam con&longs;equens e&longs;t, quod major libra clariùs indicat <lb/>æquilibrium. </s> </p> <p type="main"> <s id="s.002069">Verùm &longs;i hæc ita accipiantur, prout communi huic inter­<lb/>pretationi &longs;ube&longs;t Ari&longs;toteles, vix aliquid habent momenti: <lb/>quis enim pondera vix inæqualia bilance &longs;ubtiliter examinans <lb/>jugi extremitates re&longs;picit, ut videat, an lineæ horizonti paral­<lb/>lelæ congruat jugum? </s> <s id="s.002070">& non potiùs lingulam CO con&longs;iderat, <lb/>an cum ansâ perpendiculari illa conveniat? </s> <s id="s.002071">Quod &longs;i lingula at­<lb/>tendatur, idem e&longs;t ejus motus &longs;ive longior &longs;it libra AB, &longs;ive <lb/>brevior DE; factâ enim inclinatione aut majore motu BG, aut <lb/>minore motu EI, eadem e&longs;t lingulæ po&longs;itio CS. <!-- KEEP S--></s> <s id="s.002072">Hoc tantùm <lb/>habent emolumenti brachia longiora, quod faciliùs dividuntur <lb/>bifariam æqualiter quàm breviora: & &longs;i minimum aliquod di&longs;­<lb/>crimen intercedat, hoc minorem habet Rationem ad bra­<lb/>chium longiùs, quàm ad brevius. </s> <s id="s.002073">Quare aliâ ratione acci­<lb/>pienda e&longs;t libra: nam &longs;i in uno eodemque puncto C conveniant <lb/>&longs;partum & jugi divi&longs;io, aut &longs;partum &longs;it inferius, &longs;ive longiora, <lb/>&longs;ive breviora &longs;int brachia, ponderum inæqualitas illicò inno­<lb/>te&longs;cit, quia extremitas præponderans, ad imum locum, quan-<pb pagenum="280" xlink:href="017/01/296.jpg"/>tum pote&longs;t, de&longs;cendit. </s> <s id="s.002074">Locutus igitur videtur Ari&longs;toteles de <lb/>librâ &longs;partum habente in &longs;uperiore jugi loco extrà lineam, quæ <lb/>jugi longitudinem definit. </s> </p> <p type="main"> <s id="s.002075">Sit iterum libra longior AB, & brevior DE, utraque bifa­<lb/>riam divi&longs;a in C; & &longs;it linea lingulæ perpendicularis CK, in <lb/><figure id="id.017.01.296.1.jpg" xlink:href="017/01/296/1.jpg"/><lb/>quâ &longs;umatur &longs;partum, &longs;eu motús <lb/>centrum O, & re&longs;iduum OK &longs;it <lb/>lingula, ex cujus declinatione à <lb/>perpendiculo an&longs;æ, digno&longs;citur <lb/>&longs;ublatum æquilibrium. </s> <s id="s.002076">Sit pondus <lb/>A ad pondus B ut 5 ad 3: centrum <lb/>gravitatis jugi & ponderum commune non pote&longs;t e&longs;&longs;e C, quod <lb/>brachia CA & CB æqualia con&longs;tituit; &longs;ed erit ut pondus A ad <lb/>pondus B, ita reciprocè longitudo BG ad longitudinem GA, <lb/>eritque punctum G centrum gravitatis, nec libra con&longs;i&longs;tet, ni­<lb/>&longs;i recta GOH fiat perpendicularis horizonti: lingula igitur <lb/>OK declinabit à perpendiculo an&longs;æ juxta angulum HOK. </s> <lb/> <s id="s.002077">Eadem pondera transferantur in minorem libram DE; & &longs;i <lb/>fiat ut pondus D 5 ad pondus E 3, ita EF ad FD, erit F cen­<lb/>trum gravitatis libræ DE & ponderum: quare libra non con­<lb/>&longs;i&longs;tet, ni&longs;i recta FOI &longs;it horizonti perpendicularis, & tunc à <lb/>perpendiculo declinabit lingula OK juxta angulum IOK. </s> <lb/> <s id="s.002078">Quoniam verò e&longs;t ut 4 ad 1, ita AC ad CG, ita DC ad CF, <lb/>& AC major e&longs;t quàm DC, erit etiam ex 14 lib.5. GC major <lb/>quàm FC; igitur angulus COF minor e&longs;t angulo COG, pars <lb/>minor toto; ac proinde ad verticem angulus KOI minor e&longs;t <lb/>angulo KOH. </s> <s id="s.002079">Po&longs;itis igitur ponderibus ii&longs;dem in libræ lon­<lb/>gioris AB extremitatibus, declinabit lingula à perpendiculo, <lb/>cum eo con&longs;tituens angulum majorem, quàm &longs;it angulus ab <lb/>eadem lingulâ con&longs;titutus cum perpendiculo, quando ponde­<lb/>ra illa inæqualia adnectuntur libræ breviori DE. <!-- KEEP S--></s> <s id="s.002080">Hinc e&longs;t <lb/>quòd &longs;i inæqualitas ponderum exigua &longs;it, centrum gravitatis <lb/>in utrâque librâ non multùm recedat à puncto C, parùm in ma­<lb/>jore, minimùm in minore, ac proinde lingulæ deflexio forta&longs;&longs;e <lb/>inob&longs;ervabilis erit in minore librâ, quæ in majore evadet nota­<lb/>bilis atque con&longs;picua. </s> <s id="s.002081">Hinc etiam patet, cur extremitas A <lb/>de&longs;cendens magis moveatur, quàm extremitas D minoris li­<lb/>bræ; quia &longs;cilicet angulus OGA, per 16. lib.1. major e&longs;t quàm <pb pagenum="281" xlink:href="017/01/297.jpg"/>angulus OFD, ac propterea ubi OG facta &longs;it perpendicularis, <lb/>linea AG cum illà faciens obtu&longs;iorem angulum, magis depri­<lb/>metur infrà lineam AB horizontalem. </s> </p> <p type="main"> <s id="s.002082">Sed jam inquirendum e&longs;t, utrùm expediat centrum motûs <lb/>magis di&longs;tare à lineâ jugi, an verò illi propiùs admoveri, ut <lb/>clariùs innote&longs;cat rece&longs;&longs;us jugi ab æquilibrio horizontali: illa <lb/>quippe &longs;parti po&longs;itio eligenda e&longs;t, quæ etiam minimum mo­<lb/>tum indicet notabili lingulæ declinatione. </s> <s id="s.002083">Dico itaque &longs;par­<lb/>tum lineæ jugi proximum utilius e&longs;&longs;e, quàm remotum. </s> <s id="s.002084">Sit <lb/>enim libra AB bifariam in C di­<lb/><figure id="id.017.01.297.1.jpg" xlink:href="017/01/297/1.jpg"/><lb/>vi&longs;a, & ex hoc puncto exeat per­<lb/>pendicularis CI; in quâ pro cen­<lb/>tro motûs eligatur punctum S; <lb/>ponantur verò pondera A & B ita <lb/>e&longs;&longs;e inæqualia, ut centrum gravi­<lb/>tatis commune &longs;it D. <!-- KEEP S--></s> <s id="s.002085">Igitur DSR <lb/>e&longs;t linea, quæ facta perpendicularis con&longs;tituit cum lingulâ SI <lb/>angulum ISR. </s> <s id="s.002086">Deinde reliquis omnibus manentibus, &longs;it cen­<lb/>trum motûs O remotius à lineâ jugi, & linea DOV facta per­<lb/>pendicularis declinabit à lingulâ OI juxta angulum IOV, <lb/>quem con&longs;tat e&longs;&longs;e minorem angulo ISR; nam angulus DSC <lb/>externus major e&longs;t interno DOS, per 16. lib. 1. e&longs;t autem huic <lb/>ad verticem IOV, & illi ad verticem ISR; igitur ISR angu­<lb/>lus e&longs;t major angulo IOV. </s> </p> <p type="main"> <s id="s.002087">Quòd &longs;i centrum motûs adhuc propiùs admoveatur medio <lb/>jugi puncto C, adhuc majorem angulum con&longs;tituet cum lin­<lb/>gulâ, ac proptereà adhuc multò notabilior erit deflexio lingu­<lb/>læ à perpendiculo, etiam &longs;i exiguus &longs;it motus ex eo, quod cen­<lb/>trum gravitatis D proximè accedat ad punctum C: e&longs;t &longs;iqui­<lb/>dem extrà controver&longs;iam, quò minor e&longs;t ponderum inæquali­<lb/>tas, eò etiam minorem e&longs;&longs;e puncti D à puncto C di&longs;tantiam. </s> <lb/> <s id="s.002088">Ex quo manife&longs;tum evadit exiguam ponderum differentiam <lb/>non digno&longs;ci, &longs;i &longs;partum notabili intervallo rece&longs;&longs;erit à lineâ ju­<lb/>gi; hæc enim &longs;parti di&longs;tantia habet rationem Radij, di&longs;tantia <lb/>centri gravitatis à medio jugi locum obtinet Tangentis; igitur <lb/>&longs;i fiat major &longs;parti di&longs;tantia, eadem Tangens ad majorem Ra­<lb/>dium minorem Rationem habebit, atque adeò &longs;ubtendet mul­<lb/>tò acutiorem angulum, qui proptereà minùs ob&longs;ervari poterit. <pb pagenum="282" xlink:href="017/01/298.jpg"/>Quare pro eâdem ponderum inæqualitate digno&longs;cendâ, &longs;i con­<lb/>currant minima &longs;parti à jugo di&longs;tantia, & ob longitudinem ma­<lb/>jorem brachiorum libræ major centri gravitatis di&longs;tantia à me­<lb/>dio jugi puncto, patet multò faciliùs digno&longs;ci inæqualia e&longs;&longs;e <lb/>pondera, quia majore angulo linea deflectit à perpendiculo; & <lb/>po&longs;ito minimo Radio Tangens major angulo majori opponitur. </s> </p> <p type="main"> <s id="s.002089">Hæc quidem de libra &longs;partum habente &longs;uprà lineam jugi <lb/>dicta accommodari po&longs;&longs;unt libræ &longs;partum habenti infrà jugi li­<lb/>neam, &longs;i eadem &longs;chemata inver&longs;o &longs;itu po&longs;ita intelligantur: quò <lb/>enim maiore angulo deflectit à perpendiculo linea jungens gra­<lb/>vitatis centrum, & centrum motus, eò faciliùs brachium, in <lb/>quo e&longs;t gravitatis centrum, inclinatur. </s> <s id="s.002090">Verùm &longs;i duplex hæc <lb/>libræ &longs;pecies, quæ &longs;uprà, & quæ infra jugi lineam &longs;partum ha­<lb/>bet, invicem comparetur, &longs;atis apertum e&longs;t multò faciliùs à <lb/>po&longs;teriore h`c &longs;pecie indicari ponderum inæqualitatem; quia <lb/>videlicet &longs;i centrum gravitatis in alterutram partem vel mini­<lb/>mùm recedat à medio jugi, non ampliùs imminet &longs;parto in eo­<lb/>dem perpendiculo, neque pote&longs;t &longs;u&longs;tineri, &longs;ed illicò, quantùm <lb/>pote&longs;t ad imum locum de&longs;cendit. </s> <s id="s.002091">At in priore illa &longs;pecie libræ <lb/>&longs;partum in &longs;uperiore loco habentis, recedente in alterutram <lb/>partem centro gravitatis, de&longs;cendit illud quidem; &longs;ed non ni&longs;i <lb/>pro ratione exce&longs;sûs ponderis; qui de&longs;cen&longs;us inob&longs;ervabilis erit, <lb/>&longs;i exigua &longs;it ponderum differentia. </s> <s id="s.002092">Hinc non &longs;emel animadver­<lb/>ti accurati&longs;&longs;imas bilances, quibus aurearum monetarum ponde­<lb/>ra examinantur, eas e&longs;&longs;e, quæ &longs;partum in inferiore loco habent; <lb/>lanx enim, quæ pondere prægravatur, ad imum, quantùm po­<lb/>te&longs;t de&longs;cendit: factâ autem libræ conver&longs;ione ita, ut an&longs;a infe­<lb/>riùs &longs;u&longs;tentata libram &longs;u&longs;tineat, ii&longs;demque ponderibus impo&longs;i­<lb/>tis, lanx prægravata non de&longs;cendit ad imum locum; &longs;ed manet <lb/>libra in obliquâ po&longs;itione, quæ ponderum inæqualitati congruè <lb/>re&longs;pondet; &, &longs;i ea &longs;it ponderum inæqualitas, quæ omnem ob­<lb/>&longs;ervantis &longs;ubtilitatem effugiat, videtur libra in æquilibrio hori­<lb/>zontali po&longs;ita, cum tamen in priore &longs;itu, antequam libra inver­<lb/>teretur, non po&longs;&longs;et in ullo æquilibrio con&longs;i&longs;tere. </s> </p> <p type="main"> <s id="s.002093">Non ita tamen hæc dicta intelligi velim, ut nulla &longs;it habenda <lb/>ratio materiæ, ex qua libra con&longs;tat; hæc &longs;iquidem tantæ gravi­<lb/>tatis e&longs;&longs;e pote&longs;t, ut axem vehementiùs premens motum aliqua­<lb/>tenus impediat, ac propterea levis illa virtus effectiva motus, <pb pagenum="283" xlink:href="017/01/299.jpg"/>qui ponderum adnexorum inæqualitatem cæteroqui con&longs;eque­<lb/>retur, ex hâc pre&longs;&longs;ione, & prominularum particularum &longs;e vi­<lb/>ci&longs;&longs;im contingentium conflictu elidatur, atque jugi æquili­<lb/>brium horizontale permaneat. </s> <s id="s.002094">Gravitatem autem motui im­<lb/>pedimento e&longs;&longs;e ex eo con&longs;tat, <emph type="italics"/>quòd faciliùs quando &longs;ine pondere <lb/>e&longs;t, movetur libra, quàm cum pondus habet,<emph.end type="italics"/> ut ob&longs;ervavit <lb/>Ari&longs;toteles 9. 10. Mechan. <!--neuer Satz-->Cui tamen in a&longs;&longs;ignandâ hujus <lb/>difficultatis causa non aquie&longs;co, licet ultrò concedam <emph type="italics"/>in con­<lb/>trarium ei, ad quod vergit onus, movere difficile e&longs;&longs;e<emph.end type="italics"/>; &longs;i enim libræ <lb/>vacuæ lances minùs graves &longs;unt, impo&longs;ito autem pondere fiunt <lb/>graviores, & proptereà lanx elevanda facta gravior difficiliùs <lb/>movetur contra in&longs;itam gravitati propen&longs;ionem, etiam vici&longs;&longs;im <lb/>lanx deprimenda facta gravior ex adnexo pondere faciliùs ob­<lb/>&longs;ecundat naturali gravium propen&longs;ioni, atque adeò augere de­<lb/>beret movendi facilitatem, vel &longs;altem hanc imminui non per­<lb/>mitteret. </s> <s id="s.002095">Non aliunde igitur ortum ducere videtur huju&longs;mo­<lb/>di difficultas movendi libram onu&longs;tam, quàm ex majore pre­<lb/>mentis gravitatis conatu: pre&longs;&longs;ione autem motum impediri quis <lb/>neget, &longs;i &longs;uper planam &longs;uperficiem continuo lævore lubricam <lb/>ducat regulam metallicam exqui&longs;ite politam, quam nunc te­<lb/>nui, nunc validiori conatu premat? </s> <s id="s.002096">utique percipiet pro vario <lb/>prementis conatu aliam atque aliam e&longs;&longs;e trahendæ regulæ me­<lb/>tallicæ difficultatem. </s> </p> <p type="main"> <s id="s.002097">Adde graviori libræ cra&longs;&longs;iorem axem, ut ei proportione <lb/>re&longs;pondeat, nece&longs;&longs;ariò adjungi; hic autem &longs;i non &longs;it exqui&longs;itè <lb/>cylindricus, quâ parte fit contactus, &longs;ed aliquatenùs angulatus <lb/>duobus in locis contingat, &longs;atis manife&longs;tè apparet magis impe­<lb/>diri motum libræ, quàm &longs;i axis tenuior e&longs;&longs;et, atque &longs;ubtilior; <lb/>licet enim hic pariter &longs;imilique ratione angulatus e&longs;&longs;et, quia <lb/>tamen anguli minùs di&longs;tarent invicem, quàm in axe cra&longs;&longs;iore, <lb/>minùs etiam libræ conver&longs;ionem impedirent. </s> <s id="s.002098">Idem accidit, &longs;i <lb/>axis quidem cylindricus, foramen autem, cui axis in&longs;eritur, <lb/>non exqui&longs;itè rotundum &longs;ed angulatum fuerit. </s> <s id="s.002099">Cur autem libræ <lb/>conver&longs;io impediatur, &longs;i fiat contactus in duobus punctis, pa­<lb/>làm e&longs;t; quia nimirum quamdiu centrum gravitatis compo&longs;itæ <lb/>interjicitur inter duos illos contactus (vel &longs;altem linea directio­<lb/>nis per illud centrum ducta tran&longs;it per intervallum illud duo­<lb/>rum contactuum) non pote&longs;t fieri libræ in alterutram partem <pb pagenum="284" xlink:href="017/01/300.jpg"/>conver&longs;io; quæ proinde ut convertatur, tantum ponderis alte­<lb/>ri lanci addi nece&longs;&longs;e e&longs;t, ut centrum gravitatis omninò cadat <lb/>extrà illud &longs;patium, quod à contactibus comprehenditur. </s> </p> <p type="main"> <s id="s.002100">Hinc patet, cur libræ cra&longs;&longs;iores, & majores ingentibus &longs;ar­<lb/>cinis onu&longs;tæ inertes fiant ad motum, etiam &longs;i adnexis ponderi­<lb/>bus in&longs;it aliquot unciarum, aliquando forta&longs;&longs;e etiam librarum, <lb/>di&longs;paritas. </s> <s id="s.002101">Contrà verò aurificibus, & gemmariis, quibus mi­<lb/>nutias contemnere damno e&longs;&longs;et, valdè exiguæ libræ in u&longs;u <lb/>&longs;unt; quippè quæ &longs;ubtili&longs;&longs;imo axe contentæ &longs;unt, & levi jugo <lb/>con&longs;tant, cujus gravitati æqualis e&longs;t &longs;ingularum lancium gra­<lb/>vitas: quare cum nec vehemens pre&longs;&longs;io contingat, nec axis <lb/>adeò tenuis facilè angulos admittat, exilioribus huju&longs;modi li­<lb/>bris etiam minima ponderum inæqualitas exploratur, &longs;i cæte­<lb/>róqui fuerint ritè con&longs;tructæ. </s> </p> <p type="main"> <s id="s.002102">At quærat hîc qui&longs;piam. </s> <s id="s.002103">Proponitur libra, quæ vacua æqui­<lb/>librium o&longs;tendit, nec ita gravis e&longs;t, ut de validiore axis pre&longs;­<lb/>&longs;ione dubitetur: ut inquiratur, quàm facilè mobilis illa &longs;it, alte­<lb/>ri lanci &longs;ingula &longs;ubinde grana delicatè imponuntur, quot &longs;atis <lb/>&longs;int ad primò tollendum æquilibrium, tùm aliâ librâ tenuiori <lb/>examinatum granorum omnium pondus (rejecto ultimo grano, <lb/>cujus additione primò facta e&longs;t libræ inclinatio) deprehendi­<lb/>tur unciæ unius, exempli gratiâ. </s> <s id="s.002104">Quæritur, an, &longs;i eidem lanci <lb/>imponantur merces, & oppo&longs;itæ lanci legitima pondera, &longs;it <lb/>&longs;emper numeranda uncia una amplius, ut verum mercis pon­<lb/>dus habeatur; quandoquidem deprehen&longs;um e&longs;t non mutari <lb/>æquilibrium, ni&longs;i uncia addatur. </s> </p> <p type="main"> <s id="s.002105">Ut quæ&longs;tioni &longs;atisfaciam, tanquam certum &longs;tatuamus hanc <lb/>libræ inertiam non oriri ex multâ jugi & lancium gravitate <lb/>axem premente; &longs;i enim ex huju&longs;modi pre&longs;&longs;ione oriretur, ad­<lb/>ditis hinc & hinc ponderibus multò major fieret pre&longs;&longs;io, ex <lb/>quâ movendi difficultas major crearetur; & &longs;i minorem pre&longs;&longs;io­<lb/>nem vix unius unciæ exce&longs;&longs;us vincit, utique majorem pre&longs;&longs;io­<lb/>nem non ni&longs;i plurium unciarum exce&longs;&longs;us vincere poterit. </s> <s id="s.002106">De­<lb/>finire autem huju&longs;modi pre&longs;&longs;ionum vires motum libræ retar­<lb/>dantes, meæ tenuitatis non e&longs;t; quippè qui nec divinare au­<lb/>deo, nec certam rationem pre&longs;&longs;iones illas dimetiendi invenio. </s> <lb/> <s id="s.002107">Illud igitur reliquum e&longs;t, &longs;eclusâ pre&longs;&longs;ione, quòd axis con­<lb/>tactus non omninò in unico puncto, &longs;ed in pluribus fiat, ac <pb pagenum="285" xlink:href="017/01/301.jpg"/>propterea alterutri vacuæ libræ lanci imponendam unciam, ut <lb/>primò di&longs;po&longs;ita &longs;it libra ad recedendum ab æquilibrio. </s> <s id="s.002108">Hoc au­<lb/>tem indicat, libræ pror&longs;us vacuæ centrum gravitatis e&longs;&longs;e inter <lb/>extrema puncta contactûs axis; &longs;ed additâ unciâ compo&longs;itæ gra­<lb/>vitatis centrum convenire cum extremo puncto contactûs <lb/>axis. </s> </p> <p type="main"> <s id="s.002109">Quærendum e&longs;t igitur, quo intervallo extremum hoc <lb/>punctum, quod etiam e&longs;t gravitatis centrum, di&longs;tet à medio <lb/>jugi puncto. </s> <s id="s.002110">Id quod ut innote&longs;cat, ob&longs;ervetur jugi & lan­<lb/>cium gravitas; tùm in extremitatibus jugi intelligatur &longs;emi&longs;&longs;is <lb/>&longs;ingulorum brachiorum, & addatur &longs;ingularum lancium gra­<lb/>vitas: &longs;int autem hinc & hinc ex. </s> <s id="s.002111">gr. <!-- REMOVE S-->unciæ duodecim tota gra­<lb/>vitas: alteri addatur uncia, & erunt hinc quidem unciæ 12; <lb/>hinc verò unciæ 13. Quare jugum reciprocè di&longs;tinguatur in <lb/>duas partes, quarum altera &longs;it 13, altera 12: igitur punctum <lb/>hoc divi&longs;ionis jugi di&longs;tat à medio jugi puncto parte unâ quin­<lb/>quage&longs;imá totius longitudinis eju&longs;dem jugi: hæc &longs;iquidem lon­<lb/>gitudo di&longs;tincta intelligitur in partes 25 æquales; punctum <lb/>medium ab extremitate di&longs;tat partibus 12 1/2, centrum gravita­<lb/>tis compo&longs;itæ di&longs;tat partibus 12; igitur punctorum i&longs;torum in­<lb/>tervallum e&longs;t (1/50). </s> </p> <p type="main"> <s id="s.002112">Jam imponatur alteri lanci merx, quæ cum pondere le­<lb/>gitimo lib. 2. faciat æquilibrium: aio non po&longs;&longs;e pronuncia­<lb/>ri mercem e&longs;&longs;e unc. </s> <s id="s.002113">25: nam &longs;i ponatur merx unc. </s> <s id="s.002114">25: ad­<lb/>ditâ gravitate lancis & brachij unc. </s> <s id="s.002115">12 ex hypothe&longs;i, hinc <lb/>quidem e&longs;&longs;ent unciæ 37, hinc verò unciæ 36; igitur divi­<lb/>&longs;o jugo in partes 73, centrum gravitatis di&longs;taret à medio jugi <lb/>puncto parte (1/146). At punctum extremum contactûs axis & jugi <lb/>di&longs;tat parte (1/50), igitur multo majus pondus &longs;upra unciam adden­<lb/>dum e&longs;t merci, ut æquilibrium exqui&longs;itè faciat cum pondere <lb/>legitimo lib. 2. Nimirum in&longs;tituenda e&longs;t analogia ut 12 ad 13, <lb/>ita unciæ 36 ad uncias 39; dempto igitur pondere lancis & bra­<lb/>chij libræ, quantitas mercis e&longs;t unc. </s> <s id="s.002116">27. Ex quo liquet, quò <lb/>majora pondera lancibus imponuntur, eò majorem e&longs;&longs;e diffe­<lb/>rentiam à pondere legitimo. </s> <s id="s.002117">Hinc ulteriùs patet huju&longs;modi <lb/>librâ &longs;atius e&longs;&longs;e multam mercem &longs;imul ponderare, quàm per <lb/>partes: pone enim e&longs;&longs;e uncias 12 legitimi ponderis, cum quo <pb pagenum="286" xlink:href="017/01/302.jpg"/>æquilibrium con&longs;tituitur, merx erit unicarum 14, quia ut 12 <lb/>ad 13, ita unc. </s> <s id="s.002118">24 ad 26, & demptis unciis 12 ad brachium & <lb/>lancem &longs;pectantibus, remanent mercis unciæ 14: quare bis <lb/>facta ponderatione erit differentia unc. </s> <s id="s.002119">4; unica autem ponde­<lb/>ratio dabat tantum uncias 3: quia videlicet &longs;ingulis vicibus ad­<lb/>ditui id, quod re&longs;pondet gravitati lancis oppo&longs;itæ; atque adeò <lb/>differentia &longs;æpiùs repetita major e&longs;t, quàm &longs;implex: &longs;ic qua­<lb/>tuor libris ponderis legitimi re&longs;ponderent in altera lance mer­<lb/>cis lib.4.unc. </s> <s id="s.002120">5; quòd &longs;i quatuor vicibus operando &longs;ingulas libras <lb/>expendi&longs;&longs;es, differentia demùm e&longs;&longs;et unciarum 8. </s> </p> <p type="main"> <s id="s.002121">Unum ad huc &longs;upere&longs;&longs;e videtur hîc ob&longs;ervandum, quoniam <lb/>longioribus brachiis exqui&longs;itiùs indicari æquilibrium diximus: <lb/>cavendum &longs;cilicet, ne in aliud incommodum incidamus, quo <lb/>illud idem pereat, quod per&longs;equimur. </s> <s id="s.002122">Si enim longiora fiant <lb/>brachia, additur gravitas, quæ magis axem premens motui ali­<lb/>quam difficultatem creat: quod &longs;i retentâ eâdem brachiorum <lb/>gravitate illorum cra&longs;&longs;ities extenuetur, & in longitudinem ex­<lb/>tendantur, vide ne nimis exilia evadant ita, ut flexioni obnoxia <lb/>&longs;int, vel &longs;uâ ip&longs;orum, vel expendendorum ponderum gravita­<lb/>te. </s> <s id="s.002123">Præterquam quod longiora brachia plus habere videntur <lb/>momenti ad premendum axem, etiam &longs;i par &longs;it longiorum at­<lb/>que breviorum libræ brachiorum gravitas ab&longs;oluta; cujus &longs;e­<lb/>mi&longs;&longs;is in extremitate brachij longioris plùs habet momenti ad <lb/>de&longs;cendendum, quàm in extremitate brevioris. </s> <s id="s.002124">Et &longs;i longior <lb/>ha&longs;ta ex medio &longs;u&longs;pen&longs;a faciliùs &longs;ponte &longs;uâ flectitur circa me­<lb/>dium (id quod breviori non accidit) indicio e&longs;t obicem reti­<lb/>nentem magis premi; idem igitur & axi libræ contingere po­<lb/>te&longs;t, cujus pre&longs;&longs;io major e&longs;&longs;e videtur ex longioribus brachiis, <lb/>etiam&longs;i in cæteris nullum intercedat di&longs;crimen. </s> <s id="s.002125">Sic Ari&longs;tote­<lb/>les quærit quæ&longs;t. </s> <s id="s.002126">27. Mechan. <emph type="italics"/>Cur &longs;i valde procerum fuerit idem <lb/>pondus, difficiliùs &longs;uper humeros ge&longs;&longs;atur, etiam &longs;i medium qui&longs;piam <lb/>illud ferat, quam &longs;i brevius &longs;it?<emph.end type="italics"/> cujus difficultatis cau&longs;am ille tri­<lb/>buit validiori vibrationi extremitatum magis di&longs;tantium ab hu­<lb/>mero &longs;u&longs;tinente: &longs;ed hoc non ni&longs;i in motu contingit, & cùm <lb/>flexile e&longs;t pondus, cuju&longs;modi e&longs;&longs;et longior ha&longs;ta aut bractea <lb/>ferrea mediocris cra&longs;&longs;itiei. </s> <s id="s.002127">Certè longiori columnæ marmoreæ <lb/>jacenti, cujus medio recens fulcrum &longs;ubjectum fuit, jam pu<lb/>tre&longs;centibus extremis fulcris, &longs;ua longitudo obfuit, ut frange-<pb pagenum="287" xlink:href="017/01/303.jpg"/>retur: id quod æqualis ponderis columnæ breviori ex graviore <lb/>&longs;ecundum &longs;peciem marmore non ita facilè accidi&longs;&longs;et: non ni&longs;i <lb/>quia gravitas magis à fulcro di&longs;tans plùs habet momenti, etiam­<lb/>&longs;i non contingat vibratio corporis, quemadmodum in motu. </s> </p> <p type="main"> <s id="s.002128">Illud po&longs;tremò non omittendum, quod ad lingulam perti­<lb/>net, hanc enim longiu&longs;culam e&longs;&longs;e præ&longs;tat, quàm brevem, ut <lb/>vel levi inclinatione libræ, apex lingulæ magis con&longs;picuo mo­<lb/>tu extra an&longs;am ad latus &longs;ecedat, & &longs;ublatum æquilibrium indi­<lb/>cet. </s> <s id="s.002129">Dum tamen lingulæ longitudinem affectas, cavendum, <lb/>ne illa momentum addat &longs;uâ gravitate brachio, quod inclina­<lb/>tur; quamvis enim hoc nihil referat, ubi &longs;ublatum horizontale <lb/>æquilibrium indicatur; in librâ tamen, quæ in æquilibrio obli­<lb/>quo pote&longs;t con&longs;i&longs;tere, videretur indicare majorem ponderum <lb/>inæqualitatem, quàm revera &longs;it. </s> <s id="s.002130">Cæterùm communiter libræ <lb/>hoc periculo vacant; &longs;ola enim ponderum æqualitas horizonta­<lb/>li æquilibrio inquiritur, non ponderum Ratio obliquo æquili­<lb/>brio inve&longs;tiganda proponitur: quare communiter nil de lingu­<lb/>læ gravitate timendum e&longs;t, quod nos &longs;olicitos habeat. </s> </p> <p type="main"> <s id="s.002131">Quare præter exqui&longs;itam brachiorum æqualitatem, & accu­<lb/>ratam lingulæ cum ip&longs;o jugo po&longs;itionem ad angulos rectos, ad <lb/>libram exacti&longs;&longs;imam con&longs;tituendam concurrunt brachiorum & <lb/>lingulæ longitudo, jugi & lancium modica gravitas, axis &longs;ub­<lb/>tilitas, &longs;parti & jugi quàm maxima propinquitas, & ip&longs;ius <lb/>&longs;parti infrà jugi lineam po&longs;itio. </s> <s id="s.002132">Quæ tamen omnia cum rectâ <lb/>ratione &longs;unt admini&longs;tranda, ut ponderibus examinandis pro­<lb/>portione re&longs;pondeant libræ partes; majoribus enim &longs;arcinis va­<lb/>lidior axis, & cra&longs;&longs;iora libræ brachia conveniunt; & &longs;ic de <lb/>reliquis. <lb/></s> </p> <p type="main"> <s id="s.002133"><emph type="center"/>CAPUT VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002134"><emph type="center"/><emph type="italics"/>Libræ dolo&longs;æ vitia reteguntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002135">LIbram dolo&longs;am voco, quæ &longs;olitariè accepta &longs;inè ponderi­<lb/>bus ju&longs;ta apparet, & æquilibrium o&longs;tentat, re tamen verâ <lb/>inju&longs;ta e&longs;t, quia adnexis ponderibus &longs;uo æquilibrio non tribuit <pb pagenum="288" xlink:href="017/01/304.jpg"/>æqualitatem, vel quia ponderum æqualitatem non indicat ve­<lb/>to æquilibrio. </s> <s id="s.002136">Quare nullus mihi &longs;ermo de iniquorum vendi­<lb/>torum &longs;ycophantiis, quibus, ju&longs;tam licèt libram adhibentes, <lb/>rudem ac &longs;implicem emptorem circumveniunt, aut imprimen­<lb/>do impetum &longs;ur&longs;um brachio, cui legitimum pondus adnectitur, <lb/>ut merx præponderare videatur, aut ponderibus iniquis & ju&longs;to <lb/>minoribus utendo, aut &longs;ubjectam men&longs;am, cui lanx mercis in­<lb/>cumbit, materiâ aliquatenus tenaci illinendo, ut &longs;ublatâ in <lb/>aërem librâ priùs attollatur lanx ponderis quàm mercis, quæ <lb/>omninò præponderans apparet, &longs;i libra &longs;partum habeat infra <lb/>jugum, aut &longs;imiles impo&longs;turas excogitando: &longs;ed de illis tantum <lb/>deceptionibus agendum, quæ ex ip&longs;ius libræ con&longs;tructione, <lb/>aut po&longs;itione ortum habere po&longs;&longs;unt. </s> </p> <p type="main"> <s id="s.002137">Et primò quidem &longs;e offert dolus, cujus meminit Ari&longs;toteles <lb/>quæ&longs;t.1.Mechan. familiaris eo tempore vendentibus purpuram, <lb/>& ea, quorum modica quantitas pretium exigebat non contem­<lb/>nendum: hi enim librâ utebantur, quæ brachiis non omninò <lb/>paribus con&longs;tabat, ita tamen, ut hæc inæqualitas non &longs;e oculis <lb/>&longs;tatim proderet. </s> <s id="s.002138">Ut autem lateret dolus, &longs;capum &longs;eu jugum <lb/>libræ ex ligno con&longs;truebant, cujus partes omnes non eandem <lb/>&longs;pecificam gravitatem obtinerent, quamvis nulla &longs;ecundùm <lb/>molem diver&longs;itas intuenti occurreret: quia enim nodi, & partes <lb/>radici propiores, ut potè magis den&longs;æ, graviores &longs;unt, quàm <lb/>reliquæ partes à radice remotiores & nodis carentes, partem il­<lb/>lam graviorem breviori brachio tribuebant, vel &longs;i materia pla­<lb/>nè uniu&longs;modi e&longs;&longs;et, & æquabili gravitate prædita, breviori <lb/>brachio aliquid plumbi infundebant, ut materiæ gravitate mo­<lb/>mentum, quod ratione po&longs;itionis deerat, &longs;upplente, appareret <lb/>æquilibrium lancium in vacuâ librâ. </s> <s id="s.002139">Sed ubi demum merx <lb/>lanci longioris brachij imponebatur, hæc erat ju&longs;to minor, <lb/>quamvis cum oppo&longs;ito pondere e&longs;&longs;et æquilibris; non enim erat <lb/>illi æqualis, &longs;ed in Ratione reciprocâ longitudinis brachij mi­<lb/>noris ad longitudinem majoris. </s> <s id="s.002140">Hûc &longs;pectat inæqualitas bra­<lb/>chiorum orta ex eo, quòd jugi ferrei pars altera ex validiore, & <lb/>diuturniore percu&longs;&longs;ione mallei facta den&longs;ior, etiam gravior e&longs;t; <lb/>nam puncto longitudinem jugi bifariam dividenti non re&longs;pon­<lb/>det centrum gravitatis; &longs;ed recedit à medio versùs extremita­<lb/>tem den&longs;iorem, atque graviorem; ac proptereà, ut æquili-<pb pagenum="289" xlink:href="017/01/305.jpg"/>brium appareat, centrum motûs inæqualiter dividit longitudi­<lb/>nem jugi. </s> <s id="s.002141">Similiter &longs;i jugi quidem materia æquabiliter &longs;it gra­<lb/>vis, &longs;ed brachiorum inæqualitatem &longs;uppleat lancium gravitas <lb/>reciprocè inæqualis; æquilibris erit libra vacua; &longs;ed damno <lb/>emptoris merx longiori brachij adnectitur. </s> <s id="s.002142">Quare ut pateat <lb/>dolus, facto æquilibrio inter mercem ac pondus, &longs;tatim com­<lb/>muta lances, & pondus majus ex longiore brachio multò plus <lb/>habebit momenti, quàm merx ex brachio breviore: idcircò, <lb/>&longs;i ex pondere dematur, quantùm &longs;atis &longs;it ad æquilibrium cum <lb/>merce iterum &longs;tatuendum, plus mercis habebit emptor, quàm <lb/>pro oppo&longs;iti ponderis men&longs;urâ. </s> </p> <p type="main"> <s id="s.002143">Secundò &longs;it jugi materia planè æquabilis, & ab axe jugum <lb/>dividatur omnino bifariam: &longs;ed puncta contactuum annulo­<lb/>rum, ex quibus pendent lances, non æqualiter di&longs;tent à me­<lb/>dio: etiam&longs;i lancis propioris gravitas &longs;uppleat momentum, quod <lb/>dee&longs;t ratione &longs;itûs, & æquilibris appareat libra vacua, non ta­<lb/>men æqualia pondera lancibus impo&longs;ita con&longs;tituent æquili­<lb/>brium, &longs;ed illud gravius apparebit, quod ex di&longs;tantia majore <lb/>appendetur: & &longs;i pondera æquilibrium faciant, inæqualia <lb/>erunt reciprocè juxtà Rationem inæqualitatis di&longs;tantiarum à <lb/>medio. </s> <s id="s.002144">Similiter igitur facto ponderum æquilibrio, lances <lb/>commuta, & quidem &longs;i po&longs;t commutationem iterum æquili­<lb/>brium fiat, ju&longs;ta e&longs;t libra, &longs;ecùs verò &longs;i alterum gravius appa­<lb/>reat, quod priùs æquale videbatur. </s> </p> <p type="main"> <s id="s.002145">At quæris, quá methodo po&longs;&longs;is deprehendere, quanta &longs;it bra­<lb/>chiorum inæqualitas, quando quidem non habetur æquili­<lb/>brium po&longs;t factam lancium commutationem, & planè ignora­<lb/>tur, quanta &longs;it mercis gravitas. </s> <s id="s.002146">Ut quæ&longs;tioni &longs;atisfaciam, acci­<lb/>pio legitima pondera, & primùm facto æquilibrio ob&longs;ervo legi­<lb/>timi ponderis quantitatem: Commuto deinde lances, & cum <lb/>non fiat æquilibrium cum eâdem merce, tantum accipio legi­<lb/>timi ponderis, quantum requiritur ad æquilibrium. </s> <s id="s.002147">Demum <lb/>inter hæc duo pondera legitima invenio terminum medio loco <lb/>proportionalem, & hoc e&longs;t mercis pondus, quod collatum cum <lb/>alterutro ex legitimis ponderibus dat reciprocè longitudinis <lb/>brachiorum Rationem. <!-- KEEP S--></s> <s id="s.002148">Hanc methodum e&longs;&longs;e certam patet, <lb/>quia cum bis fiat æquilibrium, bis inter pondera e&longs;t eadem Ra­<lb/>tio reciproca brachiorum. </s> <s id="s.002149">Sint brachia, quæ brevitatis gratia <pb pagenum="290" xlink:href="017/01/306.jpg"/>vocemus R & S; igitur ut R ad S ita primum pondus legiti­<lb/>mum in S ad mercem in R: & factâ commutatione ponitur <lb/>merx in S, & iterum fit ut R ad S, ita reciprocè merx eadem <lb/>in S ad &longs;ecundum pondus legitimum in R: igitur, per 11.lib.5. <lb/>ut primum pondus ad mercem, ita merx ad &longs;ecundum pondus: <lb/>&longs;unt autem nota duo pondera legitima; igitur & innote&longs;cit mer­<lb/>cis gravitas: quæ &longs;i comparetur ut con&longs;equens terminus cum <lb/>primo pondere, aut ut Antecedens cum &longs;ecundo pondere, ha­<lb/>bebitur Ratio R ad S. <!-- KEEP S--></s> <s id="s.002150">Sit itaque ex. </s> <s id="s.002151">gr. <!-- REMOVE S-->in primo æquilibrio <lb/>primum pondus legitimum unc. </s> <s id="s.002152">72, in &longs;ecundo æquilibrio &longs;e­<lb/>cundum pondus legitimum &longs;it unc. (69 18/100). E&longs;t ergo merx me­<lb/>dio loco proportionalis unc. (70 576/1000); ac propterea R ad S e&longs;t <lb/>ut 72 ad (70 1576/1000), aut ut (70 576/1000) ad (69 18/100), hoc e&longs;t ut 4500 ad <lb/>4411. Sit demum totius jugi longitudo di&longs;tincta in partes 200: <lb/>addantur termini Rationis inventæ, & fiat ut 8911 ad 4411 <lb/>ita 200 ad 99, & hæc e&longs;t longitudo brachij brevioris, erit au­<lb/>tem longioris brachij longitudo partium 101: di&longs;tat ergo &longs;par­<lb/>tum à puncto medio per unam ducente&longs;imam partem totius ju­<lb/>gi. </s> <s id="s.002153">Quòd &longs;i res &longs;ubtili&longs;&longs;imè ad calculos revocanda e&longs;&longs;et, hujus <lb/>ducente&longs;imæ partis gravitas, quæ e&longs;t &longs;emi&longs;&longs;is gravitatis diffe­<lb/>rentiæ brachiorum e&longs;&longs;et computanda, atque &longs;ubducenda, vel <lb/>addenda, ut mercis pondus exqui&longs;itè innote&longs;cat. </s> </p> <p type="main"> <s id="s.002154">Tertiò. <!-- KEEP S--></s> <s id="s.002155">Accidere pote&longs;t lingulam ex medio libræ &longs;capo a&longs;­<lb/>&longs;urgere ad angulos rectos, lineamque lingulæ tran&longs;euntem per <lb/>centrum motûs ita occurrere lineæ jungenti puncta, ex quibus <lb/>lances pendent, ut eam bifariam æqualiter dividat, in eam ta­<lb/>men ad angulos inæquales cadat. </s> <s id="s.002156">Aio nec brachia e&longs;&longs;e verè <lb/>æqualia, nec lingulam, quamvis an&longs;æ congruens videatur, in­<lb/>dicare æquilibrium horizontale, e&longs;&longs;e veram lingulam, etiam&longs;i <lb/>pondera in eo æquilibrio con&longs;i&longs;tentia &longs;int æqualia, & non in <lb/>Ratione brachiorum. </s> </p> <p type="main"> <s id="s.002157">Sit &longs;capus libræ AB, ex quo perpendicularis a&longs;&longs;urgat lingula <lb/><figure id="id.017.01.306.1.jpg" xlink:href="017/01/306/1.jpg"/><lb/>CD, & ex D per O centrum mo­<lb/>tûs ducta recta linea occurrat li­<lb/>neæ SV tangenti extrema puncta, <lb/>ex quibus lances pendent, eam­<lb/>que bifariam dividat in I: &longs;ed quo­<lb/>niam punctum S e&longs;t paulò altiùs <pb pagenum="291" xlink:href="017/01/307.jpg"/>quàm punctum V, fiat angulus SIO minor, & VIO major. </s> <lb/> <s id="s.002158">Dico lineam SV e&longs;&longs;e quidem jugum, &longs;ed brachia non e&longs;&longs;e æqua­<lb/>lia, non enim &longs;unt IS & IV: quandoquidem ductis rectis OS <lb/>& OV, e&longs;t libra curva SOV latera habens inæqualia, SO <lb/>minus, & VO majus. </s> <s id="s.002159">Nam in triangulis SIO, VIO latus <lb/>IS ex hypothe&longs;i e&longs;t æquale lateri IV, latus IO commune e&longs;t, <lb/>angulus SIO e&longs;t ex hypothe&longs;i minor, quàm angulus VIO; <lb/>ergo per 24.lib.1. ba&longs;is SO minor e&longs;t ba&longs;i VO. </s> <s id="s.002160">Igitur ex O <lb/>perpendicularis linea cadens in jugum SV dividit illud in bra­<lb/>chia inæqualia, & perpendiculum ex O cadit inter S & I, pu­<lb/>ta in H, quia ex hypothe&longs;i angulus SIO e&longs;t acutus. </s> <s id="s.002161">Vera <lb/>igitur lingula non e&longs;t ID, &longs;ed linea, quæ ad angulos rectos <lb/>in&longs;i&longs;tens jugo SV ex H per O ducitur. </s> <s id="s.002162">Quare &longs;i CD con­<lb/>gruit an&longs;æ perpendicularis horizonti, jugum SV non e&longs;t ho­<lb/>rizonti parallelum, non e&longs;t igitur æquilibrium horizontale, &longs;ed <lb/>obliquum: quia tamen e&longs;t I centrum commune gravitatis pon­<lb/>derum æqualium in S & V, ac per illud tran&longs;it perpendicu­<lb/>lum ex O cadens in horizontem, proptereà po&longs;&longs;unt e&longs;&longs;e ponde­<lb/>ra æqualia, & æquilibrium o&longs;tendere, quod modicá obliquita­<lb/>te inclinatum mentiatur æquilibrium horizontale. </s> <s id="s.002163">At &longs;i alia <lb/>fieret hypothe&longs;is, &longs;cilicet lineam jugi SV non dividi æqualiter, <lb/>pondera non e&longs;&longs;ent æqualia, &longs;ed e&longs;&longs;ent reciprocè in Ratione <lb/>motuum, quos perficere po&longs;&longs;ent extremitates S & V, juxta &longs;u­<lb/>periùs dicta cap. 4. hujus lib. 3. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002164">Vitium igitur hujus libræ non in eo con&longs;i&longs;tit, quòd ponde­<lb/>ra non &longs;int æqualia, &longs;ed quòd indicet æquilibrium horizontale, <lb/>cum &longs;it obliquum, & pondera æqualia nunquam po&longs;&longs;int ad <lb/>æquilibrium horizontale devenire; ut enim hoc fieret, ponde­<lb/>ra e&longs;&longs;e oporteret inæqualia reciprocè in Ratione brachiorum <lb/>SH & HV. </s> <s id="s.002165">Quòd &longs;i contingat punctum O centrum motûs, <lb/>e&longs;&longs;e idem cum puncto I, pondera æqualia verè habebunt æqui­<lb/>librium horizontale; &longs;ed lingula CD declinabit ab ansâ, qua&longs;i <lb/>æquilibrium non e&longs;&longs;et. </s> <s id="s.002166">Libræ huju&longs;modi vitium deprehendi <lb/>non pote&longs;t ponderum commutatione in lancibus; quia cùm <lb/>æqualia ex hypothe&longs;i &longs;int pondera, eadem utrobique habent <lb/>momenta, &longs;ervant quippè eamdem di&longs;tantiam, & æqualiter <lb/>&longs;unt ad motum di&longs;po&longs;ita. </s> <s id="s.002167">Rarò tamen continget jugum SV <lb/>planè æqualiter dividi à lineâ lingulæ ad angulos obliquos in-<pb pagenum="292" xlink:href="017/01/308.jpg"/>cidente, quæ tamen ad &longs;capum perpendicularis appareat: <lb/>proptereà facta ponderum in lancibus commutatione prodet &longs;e <lb/>momentorum inæqualitas. </s> </p> <p type="main"> <s id="s.002168">Quartò. <!-- KEEP S--></s> <s id="s.002169">Libra, quam diuti&longs;&longs;imè ju&longs;tam expertus es, pote&longs;t <lb/>momento à &longs;ua ju&longs;titiâ deficere, &longs;i vel modicum inflectatur al­<lb/>terutrum brachiorum, vel &longs;i utrumque non æqualiter flectatur; <lb/>hinc enim oritur brachiorum inæqualitas; quam deprehendes <lb/>commutatis ponderibus in utrâque lance; quæ &longs;cilicet æquili­<lb/>brium con&longs;tituebant propter reciprocam Rationem brachio­<lb/>rum, quibus adnectebantur, non ampliùs eandem &longs;ervant in <lb/>aliâ po&longs;itione Rationem. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002170">Quintò. <!-- KEEP S--></s> <s id="s.002171">Axis, qui duobus in punctis contingat (&longs;cio con­<lb/>tactum fieri in linea; &longs;ed puncta a&longs;&longs;umo in ip&longs;is lineis, per quæ <lb/>tran&longs;it planum perpendiculare ad horizontem, in quo e&longs;t linea <lb/>jugi) vel quia ip&longs;e e&longs;t angulatus, vel quia foramen, cui in&longs;eri­<lb/>tur, non exqui&longs;itè rotundum, quâ &longs;altem parte fit contactus, <lb/>libram con&longs;tituit dolo&longs;am: quia videlicet duo illa puncta axis <lb/>perinde &longs;e habent, ac &longs;i duo e&longs;&longs;ent centra motûs. </s> <s id="s.002172">Manife&longs;tum <lb/>e&longs;t autem eandem jugi lineam non po&longs;&longs;e in duobus punctis <lb/>æqualiter dividi. </s> <s id="s.002173">Tripliciter pote&longs;t hoc fieri. </s> <s id="s.002174">Primò unum ex <lb/>his punctis pote&longs;t exactè re&longs;pondere medio jugi; &longs;ecundò po­<lb/>te&longs;t utrumque hoc punctum æqualiter à medio jugi di&longs;tare; <lb/>Tertiò po&longs;&longs;unt ab eodem medio hinc & hinc inæqualiter <lb/>di&longs;tare. </s> </p> <p type="main"> <s id="s.002175">Sit linea jugi AB, cujus medium C: puncta contactuum <lb/>axis, ex quibus ad jugum ducitur perpendicularis, ea &longs;int pri­<lb/>mò, ut re&longs;pondeant in jugo punctis <lb/><figure id="id.017.01.308.1.jpg" xlink:href="017/01/308/1.jpg"/><lb/>C & D. <!-- KEEP S--></s> <s id="s.002176">Si lanci in B imponatur le­<lb/>gitimum pondus, tùm in A ponatur <lb/>merx u&longs;que ad æquilibrium, à quo <lb/>proximè recederet, &longs;i aliquid am­<lb/>plius mercis adderetur, fiet æqualitas, quia ex C puncto æqua­<lb/>liter ab extremitatibus di&longs;tante fit &longs;u&longs;pen&longs;io libræ. </s> <s id="s.002177">At &longs;i po&longs;itâ <lb/>primùm merce in A, deinde legitima pondera addantur in B, <lb/>utique plura pondera, quàm par &longs;it, addentur: quia videlicet <lb/>non inclinabitur libra infrà B, ni&longs;i ponderum ad mercem Ra­<lb/>tio excedat Rationem reciprocam brachiorum AD ad DB; e&longs;t <lb/>enim D qua&longs;i centrum motûs. </s> </p> <pb pagenum="293" xlink:href="017/01/309.jpg"/> <p type="main"> <s id="s.002178">Deinde puncta illa contactuum axis po&longs;&longs;unt re&longs;pondere jugi <lb/>punctis E & D æqualiter à medio C di&longs;tantibus: & tunc, ut <lb/>tollatur æquilibrium, nece&longs;&longs;e e&longs;t tantum ponderis uni lanci ad­<lb/>dere, ut pondera &longs;int in majori Ratione, quàm &longs;it Ratio reci­<lb/>proca brachiorum; erit &longs;i quidem extremitas A proxime di&longs;po­<lb/>&longs;ita, ut facto additamento gravitatis inclinetur, &longs;i fuerit ut BE <lb/>ad EA, ita pondus in A ad pondus in B; & vici&longs;&longs;im extremitas <lb/>B erit proximè di&longs;po&longs;ita, ut auctà gravitate inclinetur, &longs;i ut AD <lb/>ad DB ita pondus in B ad pondus in A. <!-- KEEP S--></s> <s id="s.002179">Quia autem ex hypo­<lb/>the&longs;i DC & EC æquales &longs;unt, etiam re&longs;idua EA & DB æqua­<lb/>lia &longs;unt, item AD & BE: quapropter ut AD ad DB, ita BE <lb/>ad EA; ex quo con&longs;equens e&longs;t ex &longs;olâ lancium commutatione <lb/>(&longs;i centrum motûs modò &longs;it D, modò &longs;it E) non po&longs;&longs;e digno&longs;ci <lb/>hoc libræ vitium, &longs;icut digno&longs;ceretur in primo ca&longs;u, &longs;i ut AD <lb/>ad DB, ita pondus in B ad pondus in A; factâ enim lancium <lb/>commutatione, pondus ex B in A tran&longs;latum præponderaret <lb/>ex centro motûs C, cum tamen in priori po&longs;itione circa cen­<lb/>trum motûs D non tolleret æquilibrium. </s> </p> <p type="main"> <s id="s.002180">Similiter in tertio ca&longs;u, quando puncta contactuum axis e&longs;­<lb/>&longs;ent F & D à medio C inæqualiter di&longs;tantia, & ut AF ad FB, <lb/>ita pondus in B ad pondus in A daret æquilibrium; factá pon­<lb/>derum in lancibus commutatione non maneret æquilibrium, <lb/>quia pondus tran&longs;latum in B ad pondus tran&longs;latum in A po&longs;t <lb/>hanc commutationem adhuc e&longs;&longs;et ut BF ad FA; &longs;ed ad æqui­<lb/>librium circa D centrum motûs deberet e&longs;&longs;e ut AD ad DB, <lb/>e&longs;t autem BF prima major, quàm AD tertia, & FA &longs;ecunda <lb/>minor e&longs;t, quàm DB quarta; igitur e&longs;t major Ratio BF ad FA, <lb/>quàm AD ad DB: igitur pondus, quod priùs erat in B, tran&longs;la­<lb/>tum in A impar e&longs;t ad æquilibrium con&longs;tituendum. </s> </p> <p type="main"> <s id="s.002181">Ad digno&longs;cendum, an libra hoc vitio laboret, uti poteris hac <lb/>methodo. </s> <s id="s.002182">Lancibus impone pondera, ut fiat æquilibrium: tùm <lb/>lances commuta; & &longs;iquidem iterum fiat æquilibrium, adde <lb/>alteri lanci aliquid ponderis, à quo &longs;i libra inclinetur, aufer ad­<lb/>ditum pondus, & oppo&longs;itæ lanci impone; quæ &longs;i per&longs;i&longs;tat non <lb/>inclinata, adde adhuc pondus, quantum ferre pote&longs;t citrà in­<lb/>clinationem: iterum commutatis lancibus, nullo pacto manere <lb/>æquilibrium videbis, & indicio erit contactum axis fieri in <lb/>duobus punctis, quorum alterum re&longs;pondet medio jugi &longs;iqui-<pb pagenum="294" xlink:href="017/01/310.jpg"/>dem in primâ lancium commutatione man&longs;it æquilibrium; & <lb/>e&longs;t primus ca&longs;us. </s> <s id="s.002183">Quòd &longs;i facto æquilibrio, alterutri lancium <lb/>addas pondus, & æquilibrium maneat, adde quantum &longs;atis e&longs;t, <lb/>ut libra &longs;it proximè inclinanda in eam partem, &longs;i adhuc pondus <lb/>adderetur, tùm oppo&longs;itæ lanci &longs;imiliter additum pondus &longs;i non <lb/>tollat æquilibrium, indicat inter puncta contactuum axis e&longs;&longs;e <lb/>medium punctum C, quod bifariam dividit jugum: & videbis <lb/>po&longs;&longs;e &longs;ine &longs;ine alternis additamentis augeri pondera &longs;ingularum <lb/>lancium, quia commune centrum gravitatis modò migrat ad <lb/>unum punctum contactûs, modò ad aliud extremum. </s> <s id="s.002184">Sed ad <lb/>interno&longs;cendum, utrùm puncta hæc æqualiter, an inæqualiter <lb/>à puncto C medio di&longs;tent, ob&longs;erva additamenta illa, æqualia ne <lb/>&longs;int? </s> <s id="s.002185">an inæqualia? </s> <s id="s.002186">Nam ut centrum gravitatis migret ex D in <lb/>E, & iterum ex E in D, æqualia addenda &longs;unt primùm in B, <lb/>deinde in A, pondera. </s> <s id="s.002187">At ut migret gravitatis centrum ex D <lb/>in F, plus addendum e&longs;t ponderis in A, quàm addatur in B, ut <lb/>migret ex F in D; quia &longs;cilicet B magis di&longs;tat à D centro mo­<lb/>tús, quàm A di&longs;tet ab F centro motûs: igitur plus ponderis ad­<lb/>dendum e&longs;t in A, ut habeat momentum æquale momento pon­<lb/>deris additi in B. </s> <s id="s.002188">Hoc vitium minoribus libris, quarum exilis <lb/>e&longs;t axis, non facilè inerit; majores libræ, quæ cra&longs;&longs;iori axe in­<lb/>digent, illi obnoxiæ e&longs;&longs;e po&longs;&longs;unt, ni&longs;i artificis indu&longs;tria in eo ex <lb/>poliendo &longs;olicita fuerit. </s> <s id="s.002189">Sed quid &longs;i axis, quâ parte contingit, <lb/>in angulum &longs;implicem de&longs;inat, non tamen in eum cadat per­<lb/>pendicularis linea lingulæ, quæ jugum bifariam dividit? </s> <s id="s.002190">Jam <lb/>con&longs;tat à centro motûs dividi jugum in brachia inæqualia, ac <lb/>proptereà æquilibrium horizontale e&longs;&longs;e non po&longs;&longs;e, inter pon­<lb/>dera verè æqualia. </s> </p> <p type="main"> <s id="s.002191">Sextò. </s> <s id="s.002192">Si libra exacti&longs;&longs;imè habens brachia æqualia, & lin­<lb/>gulam perpendicularem, & lances æquales, & funiculorum <lb/>pondera æqualia, habeat tamen funiculum alterum altero lon­<lb/>giorem, incumbátque plano horizontali, impo&longs;itis æqualibus <lb/>ponderibus non apparebit æquilibrium, &longs;i centrum motûs fue­<lb/>rit in medio jugi puncto, vel infrà illud; &longs;ed ad illam partem <lb/>inclinabitur, quæ breviorem funiculum habuerit. </s> <s id="s.002193">Hoc ideò <lb/>accidit, quia libram attollens extendit breviorem funiculum <lb/>longiori adhuc langue&longs;cente, ac proinde pondus huic lanci im­<lb/>po&longs;itum non re&longs;i&longs;tit &longs;ur&longs;um trahenti, ni&longs;i cum funiculus i&longs;te <pb pagenum="295" xlink:href="017/01/311.jpg"/>fuerit extentus: quare libræ jugum ex hâc parte a&longs;cendit &longs;ine <lb/>re&longs;i&longs;tentiâ, dum ex alterâ, quæ funiculum habet breviorem, <lb/>invenit re&longs;i&longs;tentiam; atque alterâ extremitate manente, alterâ <lb/>a&longs;cendente, jugum inclinatur, extento demùm utroque funi­<lb/>culo lanx utraque attollitur. </s> <s id="s.002194">Sed quia ex hypothe&longs;i omnia &longs;unt <lb/>æqualia, vel remanet jugum in eâdem po&longs;itione inclinatum, <lb/>&longs;i punctum libræ brachia di&longs;terminans congruat centro motûs, <lb/>vel pars inclinata ulteriùs de&longs;cendit, &longs;i &longs;partum &longs;it inferiùs po­<lb/>&longs;itum. </s> </p> <p type="main"> <s id="s.002195">Hinc pondera apparent inæqualia, quamvis verè æqualia <lb/>&longs;int; & non rarò accidit monetas aliquas aureas tanquam le­<lb/>ves rejici, quamvis reverâ &longs;int ju&longs;ti & legitimi ponderis; quia <lb/>lancis, cui imponuntur, funiculus longior e&longs;t, & libra ad hanc <lb/>partem, in quâ e&longs;t pondus, inclinatur; ideóque tribuitur mo­<lb/>netæ levitas, quia libra vacua in aëre &longs;u&longs;pen&longs;a ju&longs;ti&longs;&longs;ima appa­<lb/>ret. </s> <s id="s.002196">Vici&longs;&longs;im igitur pote&longs;t fieri, ut moneta levis appareat præ­<lb/>ponderans, in librâ &longs;partum inferiùs habentè, &longs;i moneta levis <lb/>fuerit impo&longs;ita lanci, cujus funiculus brevior e&longs;t; factâ &longs;cilicet <lb/>jam jugi ad hanc partem inclinatione, cum po&longs;tea lanx utra­<lb/>que à plano &longs;eparatur, legitimum pondus, quod gravius qui­<lb/>dem e&longs;t, non pote&longs;t de&longs;cendere, ni&longs;i attollat oppo&longs;itam lan­<lb/>cem, cujus a&longs;cendentis motus major e&longs;&longs;e deberet motu legitimi <lb/>ponderis de&longs;cendentis; ac proptereà ni&longs;i &longs;it major Ratio pon­<lb/>deris ad monetam, quàm motûs monetæ a&longs;cendentis ad motum <lb/>ponderis de&longs;cendentis, moneta videbitur præponderans: & <lb/>tanti&longs;per latebit dolus, dum facta fuerit in lancibus ponderis, <lb/>& monetæ commutatio: apparebit &longs;iquidem levius id, quod <lb/>in lance pendet ex funiculo longiore. </s> <s id="s.002197">Quòd &longs;i libra huju&longs;modi <lb/>funiculis inæqualibus in&longs;tructa &longs;partum haberet in loco &longs;upe<lb/>riore, initio quidem impo&longs;ita æqualia pondera apparerent in­<lb/>æqualia, quia non viderentur æquilibria, &longs;ed demùm &longs;e libra in <lb/>æquilibrio con&longs;titueret, &longs;i verè omnia æqualia &longs;int, ut fert hy­<lb/>pothe&longs;is. </s> <s id="s.002198">At &longs;i, ut non paucis venditoribus vulgare e&longs;t, ita li­<lb/>bra &longs;it con&longs;tituta, ut lanx altera, cui legitimum pondus impo­<lb/>nitur juxtà quæ&longs;itam mercis quantitatem, &longs;ubjecto piano in­<lb/>&longs;i&longs;tat, altera merci de&longs;tinata in aëre pendeat, lingulâ an&longs;æ <lb/>congruente, quæ æquilibrium o&longs;tendit; &longs;it verò funiculus lan­<lb/>cis plano incumbentis forta&longs;sè non &longs;atis extentus (quia ita con-<pb pagenum="296" xlink:href="017/01/312.jpg"/>textus, ut majore vi extendatur, quâ ce&longs;&longs;ante &longs;e iterum con­<lb/>trahat) merx videbitur præponderans, etiam&longs;i non &longs;it major <lb/>legitimo pondere; quia deor&longs;um &longs;uá gravitate connitens, dum <lb/>pondus ex alterâ parte re&longs;i&longs;tit, inclinat lingulam, & oppo&longs;itæ <lb/>lancis funiculum extendit. </s> </p> <p type="main"> <s id="s.002199">Septimò. </s> <s id="s.002200">Ex ip&longs;o plano, cui libra incumbit, antequam at­<lb/>tollatur, oriri pote&longs;t fallacia æqualibus ponderibus inæqualita­<lb/>tem tribuens, etiam&longs;i nullum libræ in&longs;it vitium aut ratione in­<lb/>æqualitatis brachiorum, aut ratione lingulæ perperam inclina­<lb/>tæ ad jugum, aut ratione axis angulati, aut ratione funiculo­<lb/>rum inæqualium. </s> <s id="s.002201">Nam &longs;i planum ab horizonte deflectat, & ad <lb/>illum inclinetur; cùm ad perpendiculum an&longs;a attollitur, funi­<lb/>culi pariter horizonti perpendiculares intelliguntur, & quia <lb/>æquales &longs;unt, jugum libræ e&longs;t parallelum plano, ac proptereà <lb/>perpendiculum an&longs;æ ad angulos inæquales incidit tùm in ju­<lb/>gum libræ, tùm in planum inclinatum; lingula igitur, quæ ju­<lb/>go in&longs;i&longs;tit ad angulos rectos, declinat ab ansâ, & &longs;ublatâ in <lb/>aërem librâ, inclinatur lingula ad depre&longs;&longs;iorem plani partem, <lb/>manetque inclinata, quamvis pondera æqualia &longs;int, &longs;i centrum <lb/>motûs & punctum brachia di&longs;terminans in codem puncto con­<lb/>veniant; &longs;i verò &longs;partum inferius &longs;it, adhuc magis inclinatur, <lb/>videturque lanx illa omninò præponderans: at &longs;i &longs;partum in &longs;u­<lb/>periore loco fuerit, libra primùm inclinata, demùm in aëre &longs;u&longs;­<lb/>pen&longs;a 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 ip&longs;ius <lb/>ponderis &longs;ingulare centrum gravitatis non omninò in eodem <lb/>perpendiculo &longs;it cum puncto jugi, ex quo lanx illa dependet, <lb/>æquilibrium non indicabit æqualitatem ponderum in utráque <lb/>lance po&longs;itorum: Nam &longs;i linea directionis per huju&longs;modi cen­<lb/>trum gravitatis tran&longs;iens incurrat in jugi punctum, quod &longs;it <lb/>centro motûs vicinius, quàm punctum extremum brachij, op­<lb/>po&longs;itæ lancis pondus erit minus; &longs;in autem occurrat lineæ jugi <lb/>(quæ producta intelligitur) remotiùs à centro motûs, oppo&longs;itæ <lb/>lancis pondus erit majus; quia &longs;cilicet hæc centri gravitatis <lb/>ponderis collocatio perinde &longs;e habet, atque &longs;i brachium illud <lb/>aut imminutum &longs;it, aut auctum: quapropter etiam pondera <lb/>æquilibria &longs;unt in Ratione reciprocâ brachiorum, ut ex &longs;æpius <lb/>dictis liquet. </s> <s id="s.002204">Hinc &longs;i pondus præter opinionem gravius aut le-<pb pagenum="297" xlink:href="017/01/313.jpg"/>vius appareat, eju&longs;que pars maxima extrà lancem extet, illud <lb/>aliter in lance di&longs;pone, ut centro gravitatis ponderis facilè im­<lb/>mineat punctum jugi, ex quo lanx illa &longs;u&longs;penditur; & tunc <lb/>certior fies, an verè gravitas illa ponderi in&longs;it, an verò irrep­<lb/>&longs;erit fallacia ex ineptâ ip&longs;ius ponderis po&longs;itione priori. </s> <s id="s.002205">Hoc <lb/>tamen intellige, quando ex huju&longs;modi po&longs;itione &longs;equeretur in­<lb/>æqualis velocitas motuum oppo&longs;itorum ponderum. <lb/></s> </p> <p type="main"> <s id="s.002206"><emph type="center"/>CAPUT VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002207"><emph type="center"/><emph type="italics"/>Stateræ natura & forma explicatur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002208">HActenùs de librâ &longs;ermo fuit, in quâ, cum brachia æqua­<lb/>lia &longs;int, legitimum pondus e&longs;t æquale gravitati rei, cujus <lb/>quantitatem ex gravitate inve&longs;tigamus: & quidem quando exi­<lb/>gua, vel etiam mediocria &longs;unt pondera, res commodè huju&longs;­<lb/>modi bilance perficitur; at ubi ingentium &longs;arcinarum quanti­<lb/>tas examinanda e&longs;t, prorsùs incommodum e&longs;&longs;et opportunas bi­<lb/>lances aut habere, aut adhibere: quot enim & quanta pondera <lb/>parare oporteret, ut centenas aliquot fæni libras, &longs;eu mercato­<lb/>rios fa&longs;ces, &longs;eu &longs;accos farinæ plenos expenderemus? </s> <s id="s.002209">& ex alio <lb/>in alium locum &longs;i transferenda e&longs;&longs;et libra cum legitimis ponde­<lb/>ribus tantæ gravitatis, nonne opus e&longs;&longs;et plau&longs;tro, ut tàm in­<lb/>gens onus in de&longs;tinatum locum tran&longs;veheretur? </s> <s id="s.002210">Quare Statera <lb/>excogitata e&longs;t tanquam libra brachiorum inæqualium, in quâ <lb/>pondus minus longiori brachio adnexum æqualia habet mo­<lb/>menta cum majori pondere, quod ex breviore brachio &longs;u&longs;pen­<lb/>ditur. </s> <s id="s.002211">Sed ne varia pondera in promptu habere cogeremur, <lb/>quæ longioris brachij extremitati adnecterentur, pro variâ <lb/>oneris gravitate explorandâ, &longs;apienti&longs;&longs;imè à majoribus &longs;ta­<lb/>tera con&longs;tructa e&longs;t quæ eodem æquipondio modò in majo­<lb/>re, modò in minore di&longs;tantiâ à centro motûs, æquilibrium <lb/>con&longs;titueret. </s> <s id="s.002212">Ex quo fit &longs;tateram eandem vires &longs;ubire plu­<lb/>rium librarum, prout plura longioris brachij puncta percur­<lb/>rit æquipondium; mutantur &longs;iquidem Rationes di&longs;tantiarum <lb/>ponderum, manente eâdem mercium à &longs;parto di&longs;tantiâ, ac <pb pagenum="298" xlink:href="017/01/314.jpg"/>proinde etiam idem æquipondium variam habet Rationem ad <lb/>merces inæquales. </s> </p> <p type="main"> <s id="s.002213">Sunt autem &longs;tateræ partes Jugum, An&longs;a, Uncus aut lanx, <lb/>Æquipondium, quod aliis Sacoma, aliis Cur&longs;orium dicitur. </s> <lb/> <s id="s.002214">Jugum e&longs;t, quod in partes inæquales divi&longs;um ab axe, qui An­<lb/>&longs;æ in&longs;eritur, definit Rationem ponderum, quæ momentis <lb/>æqualibus librantur. </s> <s id="s.002215">An&longs;a e&longs;t, ex quâ &longs;u&longs;penditur &longs;tatera, ut <lb/>liberè utramque in partem ver&longs;etur. </s> <s id="s.002216">Uncus, aut lanx, oneri <lb/>&longs;u&longs;tinendo de&longs;tinatur; quæ enim facilè molem unam efficiunt, <lb/>po&longs;&longs;unt ex Unco &longs;u&longs;pendi; &longs;ed quæ ex pluribus non facilè in <lb/>unam molem coëuntibus con&longs;tant, lance &longs;ubjectá recipi oportet. <lb/></s> <s id="s.002217">Æquipondium e&longs;t certæ gravitatis pondus, ex quo oppo&longs;itæ <lb/>gravitatis Ratio innote&longs;cit. </s> </p> <p type="main"> <s id="s.002218">Sit AB jugum ab axe inæqualiter in C divi&longs;um, &longs;itque CA <lb/>brachium minùs, cujus extremitati A catena aut funis adnecti­<lb/><figure id="id.017.01.314.1.jpg" xlink:href="017/01/314/1.jpg"/><lb/>tur cum unco aut lance E, & CB <lb/>brachium majus, cujus longitu­<lb/>dinem pro opportunitate percurrit <lb/>æquipondium F. <!-- KEEP S--></s> <s id="s.002219">An&longs;a re&longs;pondens <lb/>lingulæ CD, ip&longs;ius axis extremi­<lb/>tates recipit, ut facilè convolvi <lb/>po&longs;&longs;it. </s> <s id="s.002220">In minoribus & mediocri­<lb/>bus &longs;tateris lingula cra&longs;&longs;iu&longs;cula ad­<lb/>ditur, quæ an&longs;æ intercapedinem ita impleat, eíque congruat, <lb/>ut tamen nullo partium conflictu impediatur motus; in majori­<lb/>bus & longioribus &longs;tateris aliquando lingula omittitur, vel quia <lb/>&longs;partum e&longs;t infrà rectam lineam jugi, quod non ni&longs;i horizonta­<lb/>liter con&longs;i&longs;tit, vel quia &longs;i &longs;partum e&longs;t in &longs;uperiore loco, non <lb/>multùm à vero pondere aberrare permittit ip&longs;a brachij longitu­<lb/>do, quæ facilè prodit paralleli&longs;mum aut inclinationem ad ho­<lb/>rizontem; mediocris autem error in mercibus, quæ huju&longs;modi <lb/>magnis &longs;tateris expenduntur, neque emptori, neque venditori <lb/>incommodo e&longs;t; quapropter in iis &longs;ubtilitatem &longs;crupulosè per­<lb/>&longs;equi inutile e&longs;t, & ineptum. </s> <s id="s.002221">Quæ in librâ circà Axem, lin­<lb/>gulam, An&longs;am ob&longs;ervanda monuimus, &longs;tateræ pariter commu­<lb/>nia &longs;unt, neque hîc iterum inculcanda. </s> </p> <p type="main"> <s id="s.002222">Poti&longs;&longs;imum, quod in &longs;taterâ ob&longs;ervandum e&longs;t, pertinet ad <lb/>divi&longs;ionem longioris brachij in minutiores particulas, ut exqui-<pb pagenum="299" xlink:href="017/01/315.jpg"/>&longs;itiùs innote&longs;cat Ratio mercis ad æquipondium, quæ denota­<lb/>tur ab inci&longs;is in brachio notis indicantibus Rationem brachij <lb/>longioris ad brevius; e&longs;t &longs;cilicet minoris brachij longitudo <lb/>transferenda in alterum brachium, quoties fieri pote&longs;t; & quia <lb/>hoc longius produci pote&longs;t infinitè, proptereà &longs;tatera vocari <lb/>pote&longs;t libra qua&longs;i infinita brachiorum inæqualium. </s> <s id="s.002223">Sic di&longs;tan­<lb/>tia AC tran&longs;lata in brachium CB ex. </s> <s id="s.002224">gr. <!-- REMOVE S-->quater, facit ut pon­<lb/>dus in E po&longs;&longs;it e&longs;&longs;e quadruplum æquipondij F, &longs;i æquipondium <lb/>&longs;it in extremitate B: quia, ut dictum e&longs;t de librâ brachiorum <lb/>inæqualium, ut AC ad CB, ita pondus in B ad pondus in A: <lb/>& &longs; æquilibrium contingat &longs;acomate exi&longs;tente in G, erit ut <lb/>AC ad CG ita Sacoma in G ad pondus in E. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002225">Hîc animadvertendum e&longs;t di&longs;tantiam AC, &longs;i &longs;it valdè nota­<lb/>bilis, capacem e&longs;&longs;e multiplicis divi&longs;ionis, ac proptereà æqua­<lb/>lem partem HG po&longs;&longs;e &longs;ubtiliùs dividi, ut non &longs;olùm uncias, <lb/>&longs;ed & unciæ quadrantes, aut etiam drachmas o&longs;tendat, &longs;i tran­<lb/>&longs;itus ex H in G &longs;it nota unius libræ. </s> <s id="s.002226">Verum e&longs;t in brachio CB <lb/>huju&longs;modi majores partes minori brachio æquales non multas <lb/>e&longs;&longs;e po&longs;&longs;e: &longs;ed huic malo occurritur in adversâ parte jugi; con­<lb/>ver&longs;a enim &longs;tatera aliam habet an&longs;am, puta SV, quæ minùs <lb/>di&longs;tat ab extremitate A; hæc autem di&longs;tantia &longs;æpiùs iterata plu­<lb/>res exhibet partes, & factâ &longs;u&longs;pen&longs;ione VS, æquipondium in <lb/>extremitate B po&longs;itum æquilibratur cum majori pondere, quàm <lb/>cùm ex DC &longs;tatera &longs;u&longs;penditur; e&longs;t &longs;cilicet major Ratio BS ad <lb/>SA, quàm BC ad CA; nam ad eandem CA, majorem Ratio­<lb/>nem habet BS major, quàm BC minor, & eadem BS majo­<lb/>rem Rationem habet ad SA minorem, quàm ad CA majorem <lb/>ex 8 lib. 5. manife&longs;tum e&longs;t igitur majorem e&longs;&longs;e Rationem BS <lb/>ad SA, quàm BC ad CA. <!-- KEEP S--></s> <s id="s.002227">Si igitur pondera &longs;unt reciprocè ut <lb/>brachiorum longitudines, idem æquipondium in extremitate B <lb/>po&longs;itum minorem habet Rationem ad pondus in A, quando <lb/>brachia &longs;unt BS & SA, quàm cùm brachia &longs;unt BC & CA: <lb/>ac propterea tunc pondus in A e&longs;t majus. </s> </p> <p type="main"> <s id="s.002228">Verùm hactenùs de &longs;taterâ perinde locutus &longs;um, ac &longs;i nulla <lb/>illi ine&longs;&longs;et gravitas; quæ tamen omninò contemnenda non e&longs;t, <lb/>quantumvis minuta &longs;it ip&longs;a &longs;tatera atque exilis, hac enim mi­<lb/>norum ponderum gravitatem &longs;crupulo&longs;iùs exploramus: ideò <lb/>autem gravitatem à materiâ mente præcidere &longs;atius duxi, ut <pb pagenum="300" xlink:href="017/01/316.jpg"/>&longs;tatim appareat vis momentorum, quæ pro variâ di&longs;tantiâ obti­<lb/>net æquipondium; prout ad majorem, aut ad minorem motum <lb/>comparatè cum motu ponderis in A, e&longs;t di&longs;po&longs;itum. </s> <s id="s.002229">Cæterùm <lb/>pondus in A, quod æquilibrium facit cum &longs;acomate F, majus <lb/>e&longs;t quàm pro Ratione di&longs;tantiarum reciprocè &longs;umptâ; quia vi­<lb/>delicet ip&longs;ius brachij longioris gravitas &longs;ua habet momenta ma­<lb/>jora momentis brachij brevioris, ac propterea præter pondus, <lb/>quod Sacomati re&longs;pondet, addendum e&longs;t etiam pondus, quod <lb/>re&longs;pondeat exce&longs;&longs;ui momentorum brachij majoris &longs;uprà mo­<lb/>menta brachij minoris. </s> <s id="s.002230">Cùm itaque ex dictis cap.2. hujus lib. <!-- REMOVE S--><lb/>momenta brachiorum &longs;ingulorum perinde &longs;e habeant, atque <lb/>&longs;i &longs;emi&longs;&longs;is gravitatis &longs;ingulorum e&longs;&longs;et in extremitatibus, po&longs;ito <lb/>jugo æquabilis cra&longs;&longs;itiei, &longs;i nota &longs;it totius jugi gravitas, & bra­<lb/>chiorum Ratio, &longs;ingulorum quoque gravitas innote&longs;cit; cujus <lb/>&longs;emi&longs;&longs;is per &longs;ibi congruum terminum Rationis ductus exhibet <lb/>&longs;ingulorum momenta. </s> <s id="s.002231">Sit AB jugum lib.5. unc.10, hoc e&longs;t <lb/>omninò unc.70: Ratio AC ad CB &longs;it ut 2 ad 5; igitur gravi­<lb/>tas AC e&longs;t unc. </s> <s id="s.002232">20, & CB unc.50: &longs;emi&longs;&longs;is AC unc.10 ductus <lb/>per 2 (qui e&longs;t terminus Rationis illi congruens) dat momen­<lb/>tum 20: &longs;emi&longs;&longs;is CB unc. </s> <s id="s.002233">25 ductus per 5, dat momentum 125: <lb/>differentia momentorum e&longs;t 105 dividenda per terminum Ra­<lb/>tionis congruum di&longs;tantiæ AC, videlicet per 2: Quare ut fiat <lb/>æquilibrium cum &longs;olâ gravitate brachij longioris, addendæ <lb/>&longs;unt extremitati A unciæ 52 1/2: igitur addito &longs;emi&longs;&longs;e gravita­<lb/>tis AC, intelliguntur in A unciæ 62 1/2; & in B unciæ 25: &longs;unt <lb/>autem 62 1/2 ad 25, ut 5 ad 2, quæ e&longs;t Ratio reciproca brachio­<lb/>rum. </s> <s id="s.002234">Quare &longs;i jugum AB æquabile &longs;it, ut fert hypothe&longs;is, & <lb/>in extremitate B &longs;it Sacoma lib.2, pondus in A (computatâ <lb/>etiam gravitate catenæ & unci AE) non erit &longs;olùm lib.5. ut <lb/>exigit Ratio longitudinis brachiorum, &longs;ed prætereà unc.52 1/2, <lb/>hoc e&longs;t omnino lib.9. unc.4 1/2. </s> </p> <p type="main"> <s id="s.002235">Quia verò aliquando accidit properatâ ad &longs;ubitum u&longs;um &longs;ta­<lb/>terâ uti, videlicet cra&longs;&longs;iore tigillo, cujus gravitas non e&longs;t planè <lb/>contemnenda, &longs;ed valdè notabilis; proptereà hîc brevem <lb/>praxim adjicere placet, quæ etiam minùs peritis u&longs;ui e&longs;&longs;e po&longs;&longs;it, <lb/>ut &longs;tatim inveniant gravitatis quantitatem, quæ &longs;oli gravitati <lb/>brachij longioris re&longs;pondet. </s> <s id="s.002236">Sit tigillus AB, in quo intelliga-<pb pagenum="301" xlink:href="017/01/317.jpg"/>tur ip&longs;i AC brachio minori æqualis pars CH; e&longs;t igitur bra­<lb/>chiorum differentia HB. <!-- KEEP S--></s> <s id="s.002237">Ponamus totam jugi longitudinem <lb/>e&longs;&longs;e di&longs;tinctam in partes 22, quarum AC &longs;it 4, CB 18, ac dif­<lb/>ferentia HB 14. Sit verò tigilli pondus lib.84, cujus &longs;emi&longs;&longs;em <lb/>lib.42 accipio. </s> <s id="s.002238">Tum fiat ut longitudo brachij minoris 4 ad dif­<lb/>ferentiam brachiorum 14, ita &longs;emi&longs;&longs;is gravitatis jugi lib.42 ad <lb/>aliud, & provenient lib.147 addendæ brachio minori, ut fiat <lb/>æquilibrium cum &longs;olâ gravitate longioris. </s> <s id="s.002239">Sic in &longs;uperiore <lb/>exemplo, ubi brachia erant ut 2 ad 5, differentia 3, pondus ju­<lb/>gi unc.70, cujus &longs;emi&longs;&longs;is unc.35; fiat ut 2 ad 3, ita unc.35 ad <lb/>uncias 52 1/2, quod e&longs;t pondus ibi inventum pluribus calculis. </s> <lb/> <s id="s.002240">Ex his infertur jugum æquabilis cra&longs;&longs;itiei &longs;i &longs;u&longs;pendatur ex <lb/>quartâ parte &longs;uæ longitudinis, &longs;u&longs;tinere &longs;inè æquipondio pon­<lb/>dus additum minori brachio, cujus gravitas æqualis &longs;it gravita­<lb/>ti totius jugi. </s> <s id="s.002241">Si ex &longs;extâ parte &longs;u&longs;pendatur, &longs;u&longs;tinet pondus <lb/>duplex gravitatis ip&longs;ius jugi: &longs;i ex octavâ parte, &longs;u&longs;tinet pon­<lb/>dus triplex gravitatis jugi; &longs;i ex decima parte, &longs;u&longs;tinet pondus <lb/>quadruplex; &longs;i ex duodecimâ, &longs;u&longs;tinet pondus quintuplex, & <lb/>&longs;ic deinceps. </s> </p> <p type="main"> <s id="s.002242">Ut igitur ex ratione & certâ methodo con&longs;trueretur &longs;tatera <lb/>exqui&longs;itè di&longs;tincta in &longs;uas particulas, oporteret brachium mi­<lb/>nus cum adnexis appendiculis, catenâ, unco, &longs;eu lance, tantæ <lb/>gravitatis e&longs;&longs;e, ut cum &longs;olâ longioris brachij gravitate æquili­<lb/>brium con&longs;titueretur: tùm di&longs;tantia inter punctum, ex quo <lb/>onus &longs;u&longs;penditur, & centrum motûs transferenda e&longs;&longs;et ex eo­<lb/>dem centro motûs in brachium longius, quoties fieri po&longs;&longs;et, & <lb/>&longs;ingula intervalla in certas partes minores dividenda, vel pro <lb/>libito vel (quod magis rationi congruum e&longs;t) in partes pro­<lb/>prias men&longs;uræ, quæ adhibetur, ut &longs;i libra &longs;it in uncias, &longs;i un­<lb/>cia, in drachmas. </s> <s id="s.002243">Hoc autem pendet ex gravitate &longs;acomatis, <lb/>quod eligitur: nam &longs;i libram unam pendat unà cum &longs;uo annu­<lb/>lo æquipondium, tot erunt ponderis libræ, quot partes minori <lb/>brachio æquales intercipiuntur inter &longs;partum & ip&longs;um æqui­<lb/>pondium: at &longs;i bilibre &longs;it &longs;acoma, jam partes illæ a&longs;&longs;umptæ <lb/>æquales minori brachio &longs;unt bifariam dividendæ, ut &longs;ingula­<lb/>rum librarum notæ in jugo habeantur. </s> <s id="s.002244">Quod &longs;i con&longs;tructá jam <lb/>hoc modo &longs;taterâ, & majoribus partibus di&longs;tinctis in particulas <lb/>ex libito a&longs;&longs;umptas, velis apponere æquipondium majus, quàm <pb pagenum="302" xlink:href="017/01/318.jpg"/>fortè ab artifice de&longs;tinaretur, licebit; modò memineris reci­<lb/>procam e&longs;&longs;e di&longs;tantiarum Rationem & ponderum, quæ in æqui­<lb/>librio &longs;unt. </s> </p> <p type="main"> <s id="s.002245">At &longs;i contigerit ea omnia, quæ breviori brachio adhærent, <lb/>non con&longs;tituere æquilibrium cum brachio longiore &longs;eor&longs;im <lb/>&longs;umpto ab&longs;que &longs;acomate, vel quia graviora &longs;unt, vel quia mi­<lb/>nùs gravia; &longs;atis apparet æquipondium in di&longs;tantia à &longs;parto du­<lb/>plà brachij minoris non habere duplum momentum, &longs;ed inve­<lb/>niendum e&longs;&longs;e aliud punctum, à quo di&longs;tantiæ men&longs;ura de&longs;u­<lb/>matur. </s> </p> <p type="main"> <s id="s.002246">Sit &longs;tatera ACB, quæ in C &longs;u&longs;pendatur: gravitas brachio­<lb/>rum ita &longs;e habet, ac &longs;i illius &longs;emi&longs;&longs;is in &longs;ua cuju&longs;que brachij <lb/><figure id="id.017.01.318.1.jpg" xlink:href="017/01/318/1.jpg"/><lb/>extremitate poneretur. </s> <s id="s.002247">Huju&longs;modi &longs;e­<lb/>mi&longs;&longs;es gravitatum repræ&longs;ententur à li­<lb/>neis BD & AE, quæ &longs;unt utique invi­<lb/>cem in Ratione brachiorum (quoniam ju­<lb/>gum æquabile & uniforme ponitur) & ut <lb/>AC ad CB, ita AE ad BD. <!-- KEEP S--></s> <s id="s.002248">Sed ut fiat <lb/>æquilibrium debet e&longs;&longs;e vici&longs;&longs;im ut AC <lb/>ad CB, ita BD gravitas in B ad AF gra­<lb/>vitatem in A: E&longs;t igitur AE ad AF in <lb/>duplicatâ Ratione brachiorum AC ad <lb/>CB, hoc e&longs;t ut Quadratum AC ad Qua­<lb/>dratum CB: Ergo etiam dividendo, per 17. lib.5. ut Quadra­<lb/>tum CB minus Quadrato AC ad Quadratum AC, ita AF <lb/>minùs AE ad AE; hoc e&longs;t ut, differentia Quadratorum utriu&longs;­<lb/>que brachij ad Quadratum brachij minoris, ita FE pondus ad­<lb/>dendum, ad AE &longs;emi&longs;&longs;em gravitatis brachij minoris, ut fiat <lb/>æquilibrium cum &longs;emi&longs;&longs;e gravitatis, & momento brachij CB <lb/>longioris. </s> <s id="s.002249">Id &longs;i factum fuerit, a&longs;&longs;umantur in CB, incipiendo à <lb/>puncto C, partes æquales ip&longs;i CA, & tunc ad mercem addi­<lb/>tam in F habebit gravitas &longs;acomatis H eam Rationem, quam <lb/>habuerit AC ad di&longs;tantiam eju&longs;dem &longs;acomatis à puncto C, ut <lb/>&longs;uperiùs dicebatur. </s> </p> <p type="main"> <s id="s.002250">Verùm &longs;i præter AE gravitatem re&longs;pondentem minori bra­<lb/>chio AC, pendere intelligatur ex A non &longs;olùm gravitas EF, <lb/>quæ &longs;ufficiat ad æquilibrium cum longiore brachio CB, &longs;ed <lb/>præterea &longs;it etiam gravitas FG, ita ut tota gravitas addita &longs;it <pb pagenum="303" xlink:href="017/01/319.jpg"/>EG; tunc a&longs;&longs;umpto æquipondio H notæ gravitatis, debet fieri <lb/>ut pondus H ad pondus FG exce&longs;&longs;um &longs;uprà id, quod requiri­<lb/>tur ad æquilibrium, ita di&longs;tantia AC ad aliud ex. </s> <s id="s.002251">gr. <!-- REMOVE S-->CI: & <lb/>ex I initium &longs;umere debet divi&longs;io transferendo in longius bra­<lb/>chium, & iterando di&longs;tantiam CA ita, ut AC æqualis &longs;it ip&longs;i <lb/>IN: &longs;i enim in G addatur tantum mercis, cujus gravitas GM <lb/>&longs;it ad æquipondium H, ut IN ad AC, fiet in N æquilibrium. </s> <lb/> <s id="s.002252">Quia &longs;cilicet ut FG gravitas ad gravitatem H, ita IC di&longs;tan­<lb/>tia ad di&longs;tantiam CA ex con&longs;tructione; & ut gravitas H ad <lb/>gravitatem GM, ita CA di&longs;tantia ad di&longs;tantiam IN; erit ex <lb/>æqualitate per 22. lib.5. ut gravitas FG ad gravitatem GM, <lb/>ita di&longs;tantia CI ad di&longs;tantiam IN; Ergo componendo, per 18. <lb/>lib.5. ut FM ad GM, ita CN ad IN; &longs;ed ut GM ad H, ita <lb/>IN ad CA ex hypothe&longs;i; igitur ex æqualitate ut FM gravitas <lb/>ad gravitatem H, ita CN di&longs;tantia ad di&longs;tantiam CA. <!-- KEEP S--></s> <s id="s.002253">Cùm <lb/>itaque pondera addita ultrà æquilibrium, quod addità gravita­<lb/>te EF fit in C puncto &longs;u&longs;pen&longs;ionis, &longs;int in Ratione reciprocâ <lb/>di&longs;tantiarum à &longs;parto C, nece&longs;&longs;ariò &longs;equitur æquilibrium in N. <!-- KEEP S--></s> <lb/> <s id="s.002254">Idem dicendum de cæteris deinceps punctis iterando di&longs;tan­<lb/>tiam IN, prout brachij longitudo ferre pote&longs;t, nam duplicatâ <lb/>di&longs;tantiâ IN, poterit in G addi gravitas dupla gravitatis æqui­<lb/>pondij H. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002255">Quod &longs;i demùm partes minori brachio CA adjacentes non <lb/>e&longs;&longs;ent tantæ gravitatis, ut fieret cum longiore brachio CB <lb/>æquilibrium, quemadmodum &longs;i e&longs;&longs;ent ut OE ad EA &longs;emi&longs;&longs;em <lb/>gravitatis brachij minoris; primò ob&longs;erva, quantum de&longs;it gra­<lb/>vitatis, ut fiat æquilibrium, &longs;cilicet &longs;it quantitas OF, quæ po­<lb/>natur minor gravitate æquipondij H: intelligatur itaque gravi­<lb/>tas æqualis gravitati æquipondij H, & &longs;it exce&longs;&longs;us FG. </s> <s id="s.002256">Quare <lb/>&longs;icuti paulò antè dicebatur, fiat ut pondus H ad gravitatem <lb/>FG, ita AC ad CI, & erit I punctum à quo incipienda e&longs;t di­<lb/>vi&longs;io jugi, ita tamen ut facto æquilibrio in I intelligatur addita <lb/>merx æqualis gravitatis cum æquipondio H, & erit ex. </s> <s id="s.002257">gr. <!-- REMOVE S-->pri­<lb/>ma libra. </s> <s id="s.002258">At verò &longs;i OE tam modica gravitas e&longs;&longs;et, ut etiam <lb/>addita gravitas æqualis gravitati &longs;acomatis H, nondum adæ­<lb/>quaret gravitatem EF, addatur duplex, triplex, quadruplex <lb/>gravitas &longs;acomatis H ita, ut demum excedat gravitatem EF <lb/>nece&longs;&longs;ariam ad æquilibrium cum &longs;olo brachio longiore; tum fiat <pb pagenum="304" xlink:href="017/01/320.jpg"/>&longs;icuti priùs, ut pondus H ad exce&longs;&longs;um illum, &longs;cilicet ad FG, <lb/>ita AC ad CI, & e&longs;t I punctum quæ&longs;itum, ex quo incipit divi­<lb/>&longs;io, & in quo &longs;i fiat æquilibrium mercis cum &longs;acomate, indicat <lb/>mercis gravitatem e&longs;&longs;e duplam, triplam, quadruplam gravita­<lb/>tis &longs;acomatis H, prout hanc duplicare oportuit, aut triplicare. </s> </p> <p type="main"> <s id="s.002259">Sed quas habemus communes &longs;tateras ab hác &longs;edulitate pro­<lb/>cul remotas e&longs;&longs;e omnibus con&longs;tabit, &longs;i ob&longs;ervaverint amplitu­<lb/>dines priorum divi&longs;ionum non omninò re&longs;pondere brachij mi­<lb/>noris longitudini, hoc e&longs;t, intervallo, quo pondus di&longs;tat à &longs;par­<lb/>to; neque id &longs;olùm, quia artifices tantam adhibere diligentiam <lb/>recu&longs;ant pro tenui mercede; verùm etiam ne adeò graves <lb/>exi&longs;tant majores &longs;tateræ, &longs;i minori brachio tanta e&longs;&longs;et addita <lb/>gravitas, quæ longioris brachij momenta æquaret. </s> <s id="s.002260">Propterea <lb/>jugum con&longs;truunt, uncum &longs;eu lancem cum &longs;uis catenulis ad­<lb/>nectunt, ex ansâ &longs;u&longs;pendunt, &longs;acoma non certi ponderis &longs;ed ex <lb/>arbitrio eligunt, quod tamen additæ lanci, aut unco aliquate­<lb/>nus re&longs;pondeat juxta minoris brachij longitudinem; nam &longs;i hoc <lb/>valde breve &longs;it, augent lancis pondus, & minuunt æquipon­<lb/>dium; & ex adver&longs;o, &longs;i illud longiu&longs;culum &longs;it, minuunt lancem, <lb/>augent &longs;acoma; quia nimirum in illâ brevitate brachij minoris <lb/>majora &longs;unt momenta brachij longioris, & minus æquipon­<lb/>dium plus habet momenti; contrà verò auctâ minoris brachij <lb/>longitudine decre&longs;cunt momenta tùm longioris brachij tùm <lb/>æquipondij. </s> </p> <p type="main"> <s id="s.002261">His paratis &longs;tatuunt in lance legitimum aliquod pondus jux­<lb/>tà denominationem men&longs;uræ, quam a&longs;&longs;umunt tribuendam &longs;ta­<lb/>teræ, puta libram (idem dic de majoribus ponderibus in aversâ <lb/>&longs;tateræ parte in&longs;cribendis, ut lib.25 aut 100 juxtà regionis mo­<lb/>rem) deinde tanti&longs;per &longs;acoma adducunt vel reducunt, dum fiat <lb/>exqui&longs;itè æquilibrium; & punctum adnotant, in quo &longs;acoma <lb/>quie&longs;cit. </s> <s id="s.002262">Tùm aliam adhuc libram, aut, primâ &longs;ublatâ, bilibre <lb/>pondus, lanci imponunt, & &longs;acoma retrahunt, ut magis à mo­<lb/>tûs centro di&longs;tet; iterumque facto æquilibrio punctum notant. </s> <lb/> <s id="s.002263">Demum intervallum inter hæc duo notata puncta in jugo ite­<lb/>rant, quoties po&longs;&longs;unt; & ut uncias habeant, &longs;ingula intervalla <lb/>in duodecim æquales particulas di&longs;tinguunt, quæ in minu&longs;cu­<lb/>lis &longs;tateris ad huc minores divi&longs;iones recipiunt. </s> </p> <p type="main"> <s id="s.002264">Quod &longs;i adhuc pondera infrà libram unam, hoc e&longs;t infra un-<pb pagenum="305" xlink:href="017/01/321.jpg"/>cias 12, hac &longs;taterâ examinare libeat, inter punctum primò no­<lb/>tatum atque &longs;partum minu&longs;culas illas divi&longs;iones transferunt, <lb/>incipiendo ab illo puncto. </s> </p> <p type="main"> <s id="s.002265">Quid autem hîc meminerim puncta huju&longs;modi omnia in ju­<lb/>gi acie, &longs;eu angulo &longs;olido &longs;uperiore notari, majores autem di­<lb/>vi&longs;iones certis lineis ad latus ductis &longs;ignificari? </s> <s id="s.002266">hæc enim vul­<lb/>garia &longs;unt. </s> <s id="s.002267">Illud potius notandum e&longs;t, quod in unâ eâdemque <lb/>&longs;taterâ trium regionum &longs;tateras habere po&longs;&longs;umus: quia enim <lb/>&longs;tateræ &longs;capus communiter quadrangularis e&longs;t, & in &longs;uperiore <lb/>angulo libras hujus regionis in&longs;culp&longs;it artifex, in duobus angu­<lb/>lis hinc, & hinc libras duabus regionibus, cum quibus com­<lb/>mercia mi&longs;centur, peculiares in&longs;cribere licebit (nam pondera <lb/>&longs;imili nomine in pluribus regionibus donata, non e&longs;&longs;e inter &longs;e <lb/>æqualia docemur experientiâ, quæ libras Pari&longs;ien&longs;em, Ro­<lb/>manam, Venetam inæquales e&longs;&longs;e o&longs;tendit) & æquipondij an­<lb/>nulus unâ eâdemque operâ in tribus angulis diver&longs;arum regio­<lb/>num pondus eju&longs;dem mercis indicabit. </s> </p> <p type="main"> <s id="s.002268">Hîc verò curiosiùs inquirenti, præ&longs;tantiorne dicenda &longs;it &longs;ta­<lb/>tera? </s> <s id="s.002269">an libra? </s> <s id="s.002270">vix poterit qui&longs;quam ab&longs;olutè re&longs;pondere: nam <lb/>minoribus ponderibus, ut gemmis, aureis monetis, & &longs;imili­<lb/>bus examinandis parùm opportuna e&longs;t &longs;tatera; at ingentibus <lb/>oneribus hæc apti&longs;&longs;ima e&longs;t, libra autem incommoda. </s> <s id="s.002271">Compen­<lb/>dium habet &longs;tatera unico &longs;acomate contenta; pluribus ponderi­<lb/>bus eget libra. </s> <s id="s.002272">Vici&longs;&longs;im in librâ &longs;ecuriùs artifices laborem im­<lb/>pendunt, quia faciliùs æqualitatem a&longs;&longs;equuntur brachiorum, <lb/>quàm proportionem ju&longs;to æquilibrio nece&longs;&longs;ariam; & in librâ <lb/>quidem &longs;i æqualitatem perfectam &longs;emel &longs;tatuant, nil e&longs;t quæ­<lb/>rendum ampliùs; &longs;ed in &longs;taterâ &longs;ingula divi&longs;ionum puncta &longs;uam <lb/>habent Rationem, &longs;uamque expo&longs;cunt diligentiam; in pluribus <lb/>verò aliquando peccare proclivius e&longs;t, quàm in uno. </s> <s id="s.002273">Quòd &longs;i <lb/>libræ perfecta æqualitas de&longs;it, &longs;altem lancium & ponderum <lb/>commutatione, ut &longs;uperiùs monuimus, deprehenditur error; <lb/>at &longs;i fal&longs;a &longs;it &longs;tatera, non aliter innote&longs;cet, quàm &longs;i pondus idem <lb/>iterùm librâ examinemus, ut appareat, an &longs;ibi con&longs;tet eadem <lb/>gravitas: quis enim aliter iniqui venditoris impo&longs;turam rete­<lb/>gat, qui, ut major appareat mercis gravitas, ex æquipondio, <lb/>aut ex capite longioris brachij, qua&longs;i nitidiùs illa expoliens, <lb/>notabilem aliquam gravitatis particulam limâ abra&longs;it? </s> <s id="s.002274">cum ta-<pb pagenum="306" xlink:href="017/01/322.jpg"/>men à minore brachio expoliendo manum ab&longs;tinuerit; quippe <lb/>qui &longs;atis notat id fieri non po&longs;&longs;e citrà ip&longs;ius venditoris damnum: <lb/>con&longs;titutâ &longs;iquidem &longs;taterâ, nihil ex hac aut ex illâ parte de­<lb/>mendum, nihil addendum, ne mutetur Ratio, quæ intercedit <lb/>inter ip&longs;orum brachiorum momenta, aut ne æquipondium di­<lb/>minutis momentis magis removendum &longs;it à &longs;parto, quàm pro <lb/>gravitate mercis. </s> <s id="s.002275">Siverò hoc acciderit, occultum manet &longs;tate­<lb/>ræ vitium, nec ip&longs;a &longs;e prodit. </s> </p> <p type="main"> <s id="s.002276">Et quoniam de &longs;tateræ vitio &longs;ermo incidit, cavendum vendi­<lb/>tori e&longs;t, ne illâ utatur, &longs;i facta fuerit curva; cùm enim recta <lb/>fuerit ab artifice &longs;uas in partes ritè di&longs;tincta, & quidem juxta <lb/>Rationem brachiorum, curva non eandem &longs;ervat Rationem, <lb/>ut o&longs;ten&longs;um e&longs;t hîc cap.5. & venditoris damno plus mercis ad­<lb/>dendum e&longs;&longs;et lanci, ut haberetur æquilibrium; ut ex ibi dictis <lb/>con&longs;tat. <lb/></s> </p> <p type="main"> <s id="s.002277"><emph type="center"/>CAPUT IX.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002278"><emph type="center"/><emph type="italics"/>Antiquorum Statera examinatur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002279">DUbitatur à non paucis, utrùm no&longs;træ, quâ nunc utimur, <lb/>&longs;tateræ &longs;imilis e&longs;&longs;et Antiquorum, &longs;altem Græcorum, &longs;ta­<lb/>tera. </s> <s id="s.002280">Dubitationi locum fecit Ari&longs;toteles in quæ&longs;t. </s> <s id="s.002281">20. Mechan. <lb/>quærens, <emph type="italics"/>Cur &longs;tatera, quâ carnes ponderantur, pauco appendiculo <lb/>magna ponderat onera?<emph.end type="italics"/> quæ&longs;tioni autem &longs;atisfaciens plurium <lb/>&longs;partorum mentionem fecit. <emph type="italics"/>Quemadmodum autem &longs;i una li­<lb/>bra multa &longs;int libræ; &longs;ic talia in&longs;unt &longs;parta multa in eju&longs;modi li­<lb/>brâ; quorum uniu&longs;cuju&longs;que quod intrin&longs;ecùs e&longs;t ad appendicu­<lb/>lum, &longs;tateræ e&longs;t dimidium.<emph.end type="italics"/> & po&longs;t pauca. <emph type="italics"/>Huju&longs;modi autem <lb/>exi&longs;tens multæ &longs;unt libræ, totque, quot fuerint &longs;parta. </s> <s id="s.002282">Semper au­<lb/>tem quod lanci propinquius e&longs;t &longs;partum appen&longs;oque oneri, majus <lb/>trahit pondus.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.002283">Plura hæc &longs;parta, quorum Ari&longs;toteles meminit, Blancano in <lb/>locis Mathem. Ari&longs;t. occa&longs;ionem præbuerunt &longs;tateram quan­<lb/>dam commini&longs;cendi, qua&longs;i illa fuerit Antiquorum &longs;tatera: cu­<lb/>jus &longs;ententiam probare non potui, cum Mechanicam doctri-<pb pagenum="307" xlink:href="017/01/323.jpg"/>nam anno labentis &longs;æculi 54 in Collegio Romano explicans, <lb/>publici juris facerem hæc eadem, quæ nunc po&longs;t annos vigin­<lb/>ti &longs;cribo. </s> <s id="s.002284">Quoniam verò quæ tunc Blancano oppo&longs;ui, video <lb/>placui&longs;&longs;e Authori Magiæ Naturalis P. <!-- REMOVE S-->Ga&longs;pari Schoto tunc ibi <lb/>degenti (eaque cum aliis quibu&longs;dam in &longs;uam Magiam &longs;taticam <lb/>tran&longs;tulit, me identidem &longs;uprà meritum, pro &longs;uâ humanitate, <lb/>laudato) hîc iterum proferre non gravabor, ut meliùs &longs;tateræ <lb/>natura innote&longs;cat. </s> </p> <p type="main"> <s id="s.002285">Statuit itaque Blancanus &longs;tateram illam &longs;ui&longs;&longs;e ha&longs;tam oblon­<lb/>gam AB in certas partes di&longs;tributam inter &longs;e æquales, puta 12, <lb/>ex quibus exirent trutinæ diver&longs;æ, ut modò ex hâc, modò ex <lb/>illâ &longs;u&longs;penderetur &longs;tatera, prout carnis vendendæ quantitas <lb/>po&longs;tulabat, &longs;inguli&longs;que trutinis in&longs;culptam fui&longs;&longs;e <expan abbr="notã">notam</expan> ponderis <lb/>mercis. </s> <s id="s.002286">In extremitate A <lb/><figure id="id.017.01.323.1.jpg" xlink:href="017/01/323/1.jpg"/><lb/><expan abbr="p&etilde;debat">pendebat</expan> lanx capax mer­<lb/>cis, in oppo&longs;itâ extremita­<lb/>te B æquipondium, <emph type="italics"/>quod <emph.end type="italics"/><lb/>ut ille ait, <emph type="italics"/>debet habere <lb/>tantum pondus, quantum <lb/>e&longs;t in lance nudâ, ut &longs;ic tota <lb/>&longs;tatera &longs;it per &longs;e &longs;olam <lb/>æquilibralis; & præterea debet habere pondus &longs;tatum ac legitimum, <lb/>ex. </s> <s id="s.002287">gr. <!-- REMOVE S-->unius libræ, aut duarum, aut trium, prout magis trutinandæ <lb/>merci idoneum erit, & hoc erit proprium æquipondij pondus. </s> <s id="s.002288">Pona­<lb/>mus æquipondium e&longs;&longs;e librarum<emph.end type="italics"/> 12. <emph type="italics"/>Dico quod trutina C dabit in <lb/>lance pondus mercis<emph.end type="italics"/> 12 <emph type="italics"/>lib. &longs;i ex eâ fiat æquilibrium; e&longs;t enim ut AC <lb/>ad CB, it a permutatim æquipondium<emph.end type="italics"/> 12 <emph type="italics"/>ad mercem; &longs;ed AC ip&longs;i <lb/>CB e&longs;t æqualis; ergo etiam æquipondium<emph.end type="italics"/> 12 <emph type="italics"/>erit merci æquale, hoc <lb/>e&longs;t utrinque erit<emph.end type="italics"/> 12 <emph type="italics"/>lib. <!-- KEEP S--></s> <s id="s.002289">Similiter &longs;i fieret æquilibrium ex trutinâ D, <lb/>e&longs;&longs;et ut AD<emph.end type="italics"/> 3 <emph type="italics"/>ad DB<emph.end type="italics"/> 9, <emph type="italics"/>ita<emph.end type="italics"/> 12 <emph type="italics"/>ad<emph.end type="italics"/> 36. <emph type="italics"/>Tandem trutinâ E æquilibrante, <lb/>e&longs;&longs;et ut AE<emph.end type="italics"/> 9 <emph type="italics"/>ad EB<emph.end type="italics"/> 3, <emph type="italics"/>ita<emph.end type="italics"/> 12 <emph type="italics"/>ad<emph.end type="italics"/> 4. <emph type="italics"/>Si igitur trutina C notetur<emph.end type="italics"/> 12 <lb/><emph type="italics"/>numero, trutina D numero,<emph.end type="italics"/> 36, <emph type="italics"/>trutina E numero<emph.end type="italics"/> 4, <emph type="italics"/>& idem de cæteris, <lb/>&longs;tatim facile erit quodlibet pondus per huju&longs;modi &longs;tateram exhibere. </s> <lb/> <s id="s.002290">Vnde videas contrario ab illis modo in no&longs;tris &longs;tateris æquipondium <lb/>totam ha&longs;tam percurrere, in illis verò manente æquipondio trutinam <lb/>quodammodo per ha&longs;tam moveri.<emph.end type="italics"/> Hæc ille. </s> </p> <p type="main"> <s id="s.002291">Plures ha&longs;ce trutinas &longs;ic expo&longs;itas, qua&longs;i &longs;olidas an&longs;as ha&longs;tæ <lb/>infixas, quæ pro opportunitate apprehenderentur, nunquam <pb pagenum="308" xlink:href="017/01/324.jpg"/>potui in animum inducere, ut mihi per&longs;uaderem fui&longs;&longs;e anti­<lb/>quis in u&longs;u; cùm enim non po&longs;&longs;ent &longs;ummis digitis &longs;u&longs;pendi ob <lb/>nimiam mercis gravitatem, puta lib.36 (& multò plurium, &longs;i <lb/>ex F &longs;tatera penderet) manu fui&longs;&longs;ent validè apprehendendæ; <lb/>quis autem non videt, quibus dolis obnoxia fui&longs;&longs;et &longs;tatera ex <lb/>levi&longs;&longs;imâ manûs inclinatione æquilibrium mentiente? </s> <s id="s.002292">Neque <lb/>plicatiles fui&longs;&longs;e huju&longs;modi trutinas, videlicet funiculos forami­<lb/>nibus in&longs;itos in divi&longs;ionum locis, exi&longs;timo, quia vel nimis fre­<lb/>quentes e&longs;&longs;e debui&longs;&longs;ent, vel, ni&longs;i æquipondium fui&longs;&longs;et levi&longs;&longs;i­<lb/>mum, non potui&longs;&longs;ent, citrà venditoris, aut emptorum incom­<lb/>modum non leve, exhibere quæ&longs;itum pondus. </s> <s id="s.002293">Si enim (ut in­<lb/>&longs;i&longs;tam ratiocinantis Blancani ve&longs;tigiis) in D exhibentur libræ <lb/>36 mercis, in G exhiberentur libræ 60, quia ut AG 2 ad <lb/>GB 10, ita æquipondium 12 ad mercem 60: quâ igitur ratio­<lb/>ne innote&longs;cere poterat pondus mercis, &longs;i deprehendebatur e&longs;&longs;e <lb/>majus quidem libris 36, &longs;ed minus libris 60? </s> <s id="s.002294">Et &longs;i æquilibrium <lb/>fui&longs;&longs;et inter F & G, pondus fui&longs;&longs;et majus libris 60, minus li­<lb/>bris 132: quàm latè igitur patui&longs;&longs;et campus erroribus in tantâ <lb/>ponderum differentiâ? </s> </p> <p type="main"> <s id="s.002295">Quare &longs;i hoc &longs;tateræ genere utendum e&longs;&longs;et, in quâ manen­<lb/>te æquipondio &longs;partum percurreret jugi longitudinem, in&longs;e­<lb/>renda potius e&longs;&longs;et ha&longs;ta annulo &longs;olidè firmato, intrà quem ha&longs;ta <lb/>ip&longs;a ultrò citróque promoveretur, donec haberetur æquili­<lb/>brium; eâ enim ratione in minutiores particulas po&longs;&longs;et ha&longs;ta <lb/>di&longs;tingui; & plurima e&longs;&longs;ent &longs;parta, &longs;eu centra motûs. </s> <s id="s.002296">Aut <lb/><figure id="id.017.01.324.1.jpg" xlink:href="017/01/324/1.jpg"/><lb/>etiam jugum parari <lb/>po&longs;&longs;et cra&longs;&longs;ioris lami­<lb/>næ in &longs;peciem, cuju&longs;­<lb/>modi e&longs;&longs;et MO, per <lb/>cujus longitudinem <lb/>ductâ inci&longs;urâ &longs;eu cre­<lb/>nâ SI excurrere po&longs;&longs;et <lb/>axis exqui&longs;itè cylin­<lb/>dricus infixus an&longs;æ <lb/>DE cujus an&longs;æ extremitas in apicem E de&longs;inens indicaret par­<lb/>ticulas in lineâ MO notatas. </s> <s id="s.002297">Verùm quia adversùs ha&longs;ce &longs;tate­<lb/>ras faciunt pleræque rationes mox contrà Blancani &longs;tateram <lb/>afferendæ, proptereà illas ut parùm aptas rejicio. </s> </p> <pb pagenum="309" xlink:href="017/01/325.jpg"/> <p type="main"> <s id="s.002298">Et primùm quidem difficile videatur, quâ ratione fieri po&longs;­<lb/>&longs;et, ut in C puncto medio indicetur mercis pondus lib.12, &longs;i ex <lb/>illo &longs;tatera ip&longs;a e&longs;t per &longs;e &longs;olam æquilibralis, ut Blancanus loqui­<lb/>tur, po&longs;itâ lance æqualis gravitatis cum æquipondio: A&longs;&longs;umen­<lb/>da fui&longs;&longs;et trutina quarta H, quia ut AH 4 ad HB 8, ita 12 ad <lb/>24, & &longs;ubductâ gravitate lancis 12, reliquæ fui&longs;&longs;ent lib.12 <lb/>mercis. </s> <s id="s.002299">Hinc patet neque in D indicari pondus mercis lib.36; <lb/>hoc enim e&longs;t pondus mercis & lancis &longs;imul &longs;umptarum; quare <lb/>merx &longs;olum e&longs;&longs;et lib.24; & ut haberentur mercis lib.36, opor­<lb/>teret &longs;partum accipere, quod ha&longs;tam divideret in partes, qua­<lb/>rum proxima lanci e&longs;&longs;et 1, reliqua 4, quia ut 1 ad 4, ita 12 ad <lb/>48, & demptâ lancis gravitate lib.12 remanerent mercis lib.36. <lb/>Sed illud à veritate longi&longs;&longs;imè abe&longs;t, quod à Blancano additur, <lb/>ex trutinâ E indicari mercem lib.4. Immò addo nullum po­<lb/>tui&longs;&longs;e ibi fieri æquilibrium, & maximam partem illarum truti­<lb/>narum futuram fui&longs;&longs;e pror&longs;us inutilem; nam &longs;i lanx A æquè <lb/>gravis e&longs;t ac æquipondium B, lanx cum merce gravior e&longs;t æqui­<lb/>pondio; igitur lanx cum merce in di&longs;tantiâ majore, quàm &longs;it <lb/>æquipondij di&longs;tantia majora habet momenta quàm æquipon­<lb/>dium, cum quo nunquam poterit æquilibrium con&longs;tituere. </s> <lb/> <s id="s.002300">Quare omnes trutinæ inter B & C, & ip&longs;a trutina C inutiles <lb/>&longs;unt, &longs;i lanx æqualis gravitatis &longs;it cum æquipondio B: proptereà <lb/>lancem multò leviorem e&longs;&longs;e oporteret, ut cum impo&longs;itâ merce <lb/>po&longs;&longs;et habere ad æquipondium Rationem reciprocam di&longs;tantia­<lb/>rum à &longs;parto. </s> <s id="s.002301">Sed &longs;i lanx levior &longs;it æquipondio, ut inter C & B <lb/>haberi po&longs;&longs;it æquilibrium; jam non omnes quidem; &longs;ed aliquæ <lb/>tantum trutinæ inter B & C inutiles evadent; ubi enim ha&longs;ta <lb/>dividitur reciprocè in Ratione <expan abbr="gravitatũ">gravitatum</expan> lancis, & æquipondij, <lb/>ibi e&longs;&longs;et &longs;tatera per &longs;e &longs;olam æquilibralis, juxtà Blancani ratio­<lb/>cinium: igitur nulla trutina inter illud punctum, & B e&longs;&longs;et uti­<lb/>lis; quia diminutâ æquipondij à &longs;parto di&longs;tantiâ, ejus momenta <lb/>decre&longs;cunt, & auctâ lancis ab eodem &longs;parto di&longs;tantiâ, ip&longs;ius lan­<lb/>cis momenta augentur; igitur multò magis augentur facto pon­<lb/>deris in lance additamento; ac proinde fieri non poterit æqui­<lb/>librium. </s> </p> <p type="main"> <s id="s.002302">Verùm forta&longs;&longs;e Author ille, cùm &longs;tateram dixit per &longs;e &longs;olam <lb/>æquilibralem ex lancis, & æquipondij gravitatibus æqualibus, <lb/>hoc tantùmmodo voluit (& ex eju&longs;dem verbis inferendum vi-<pb pagenum="310" xlink:href="017/01/326.jpg"/>detur) ut æquipondium ultrà libras 12 &longs;ibi peculiares, tantam <lb/>prætereà haberet gravitatem, quæ &longs;i &longs;olitariè a&longs;&longs;umeretur, po&longs;­<lb/>&longs;et cum lance vacuâ æquilibrium facere in C: quo pacto lanx <lb/>non e&longs;&longs;et lib.12; &longs;ed levior. </s> <s id="s.002303">Per hæc tamen non omne incom­<lb/>modum &longs;ublatum e&longs;&longs;et, neque Blancani dicta con&longs;i&longs;terent; quia <lb/>&longs;it lanx unius libræ, & item æquipondium ultrà libras 12 habeat <lb/>libram unam; in C quidem e&longs;&longs;et æquilibrium cum merce <lb/>lib.12; quia merx cum lance, item æquipondium totum &longs;unt <lb/>lib.13. At facto æquilibrio in D, di&longs;tantiæ e&longs;&longs;ent ut 3 ad 9, igi­<lb/>tur æquipondium ad mercem cum lance ut 13 ad 39; & &longs;ub­<lb/>ductâ lancis gravitate lib.1, e&longs;&longs;et merx lib.38, non verò 36. Sic <lb/>in E facto æquilibrio, di&longs;tantiæ e&longs;&longs;ent ut 9 ad 3, igitur æquipon­<lb/>dium ad mercem cum lance ut 13 ad 4 1/3, & lancis gravitate <lb/>lib.1. demptâ, e&longs;&longs;et merx lib.3 1/3 non autem lib.4. Et in ultima <lb/>trutinâ prope B e&longs;&longs;et ut 11 ad 1, ita 13 ad (1 2/11), & lance &longs;ublatâ <lb/>lib.1, e&longs;&longs;et merx lib. (2/11), cum juxta Blancani ratiocinium debe­<lb/>ret e&longs;&longs;e &longs;olum lib. (1/11). </s> </p> <p type="main"> <s id="s.002304">Deinde jugi brachia &longs;ua habent gravitatis momenta, quæ pro <lb/>variâ longitudine inæqualitatem &longs;ubirent; & hæc in huju&longs;mo­<lb/>di &longs;taterâ modò majora, modò minora e&longs;&longs;ent, aliquando adden­<lb/>da lanci, aliquando æquipondio. </s> <s id="s.002305">Nam &longs;i &longs;partum &longs;it in D, ab­<lb/>&longs;cindens quartam jugi partem, &longs;ola brachij DB gravitas &longs;u&longs;ti­<lb/>net in A pondus æquale gravitati totius jugi; ac proinde facto <lb/>in D æquilibrio, pondus totum additum in A e&longs;t non &longs;olùm tri­<lb/>plum æquipondij, ut fert reciproca di&longs;tantiarum Ratio; &longs;ed e&longs;t <lb/>præterea æquale gravitati jugi. </s> <s id="s.002306">At &longs;i &longs;partum in F ab&longs;cindat ju­<lb/>gi partem duodecimam, non &longs;olùm pondus unâ cum lance e&longs;t <lb/>æquipondij undecuplum, &longs;ed etiam quintuplum gravitatis jugi: <lb/>& &longs;ic de cæteris. </s> <s id="s.002307">Contra verò &longs;i quando æquilibrium fieret in­<lb/>ter C & B, ex æquipondio demenda e&longs;&longs;et gravitas re&longs;pondens <lb/>momento brachij oppo&longs;iti; tum ex re&longs;iduo colligeretur gravitas <lb/>lancis cum merce, & &longs;ubductâ demùm lance, gravitas mercis <lb/>innote&longs;ceret. </s> <s id="s.002308">Sic in E facto æquilibrio, quia EB e&longs;t quarta pars <lb/>jugi, ex æquipondio B lib.12 auferenda e&longs;t gravitas jugi ex.gr. <!-- REMOVE S--><lb/>lib.4, remanent lib. 8: igitur ut AE 3 ad EB 1, ita lib. 8 ad <lb/>lib. 2 2/3: &longs;i demas pondus lancis, quæ utique valde levis e&longs;&longs;e de­<lb/>bet, vide quanta gravitas &longs;it demùm tribuenda merci. </s> <s id="s.002309">At &longs;i lanx <pb pagenum="311" xlink:href="017/01/327.jpg"/>adeò levis &longs;it, manife&longs;tum e&longs;t, quantò plus mercis apponen­<lb/>dum &longs;it, quando &longs;partum à medio &longs;ecedit ver&longs;us lancem A. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002310">Quare patet genus hoc &longs;tateræ, ut pote parùm utile, reji­<lb/>ciendum, nec potui&longs;&longs;e Antiquis u&longs;itatum e&longs;&longs;e, quin facilè de­<lb/>prehenderetur erroribus non levibus obnoxium; cum præ&longs;er­<lb/>tim oblongam fui&longs;&longs;e ha&longs;tam (non utique levi&longs;&longs;imam) commi<lb/>ni&longs;catur Blancanus, & qui eum ducem &longs;equuti &longs;unt. </s> <s id="s.002311">Non ne­<lb/>gârim quidem po&longs;&longs;e à perito mathematico ita iniri rationes, ut <lb/>certis mercium ponderibus &longs;ua puncta in jugo in&longs;criberentur, in <lb/>quibus æquilibrium fieret cum æquipondio manente in extre­<lb/>mitate jugi: &longs;ed hunc laborem &longs;ubii&longs;&longs;e antiquos Mathematicos, <lb/>ut &longs;tateras carnem in macello vendentibus pararent, &longs;uaderi <lb/>non pote&longs;t; artificibus autem tantum fui&longs;&longs;e indu&longs;triæ, omnem <lb/>fidem &longs;uperat. </s> <s id="s.002312">Ex his mihi certi&longs;&longs;imum videtur aliam prorsùs <lb/>adhibendam e&longs;&longs;e Ari&longs;totelicis verbis interpretationem: Nam <lb/>ponamus &longs;tateram illam, de quâ Ari&longs;toteles loquitur, planè &longs;i­<lb/>milem fui&longs;&longs;e no&longs;træ &longs;tateræ, quis neget unam libram brachio­<lb/>rum inæqualium e&longs;&longs;e multas libras, hoc ip&longs;o quod æquipon­<lb/>dium in multis di&longs;tantiis ab eodem puncto varias brachiorum <lb/>Rationes con&longs;tituit? </s> <s id="s.002313">&longs;unt autem plura &longs;parta, quia punctum <lb/>idem di&longs;terminans brachia varias Rationes habentia æquivalet <lb/>multis, & quàm multas Rationes brachiorum definire pote&longs;t, <lb/>tàm multas con&longs;tituit libras. </s> <s id="s.002314">Demùm quamvis lancis à &longs;parto <lb/>eadem materialiter &longs;it di&longs;tantia, non e&longs;t tamen eadem formali­<lb/>ter, neque enim &longs;olitariè accipienda e&longs;t, &longs;ed comparatè cum <lb/>di&longs;tantiâ æquipondij à &longs;parto; ac propterea cum major æqui­<lb/>pondij di&longs;tantia ad eandem lancis & oneris di&longs;tantiam majo­<lb/>rem habeat Rationem, pote&longs;t etiam dici tunc &longs;partum e&longs;&longs;e lan­<lb/>ci & oneri propinquius; nam &longs;i in unâ æquipondij di&longs;tantiâ bra­<lb/>chia &longs;int, ut 2 ad 5, & remoto æquipondio Ratio di&longs;tantiarum <lb/>&longs;it ut 2 ad 6, patet comparatè ad æquipondij di&longs;tantiam, e&longs;&longs;e <lb/>minorem priore po&longs;teriorem hanc lancis à &longs;parto di&longs;tantiam. </s> <lb/> <s id="s.002315">Cùm itaque nulla hîc intercedat violenta interpretatio, nil pro­<lb/>hibet exi&longs;timare Ari&longs;totelem de &longs;taterâ no&longs;tris non di&longs;&longs;imili lo­<lb/>cutum fui&longs;&longs;e. <pb pagenum="312" xlink:href="017/01/328.jpg"/> </s> </p> <p type="main"> <s id="s.002316"><emph type="center"/>CAPUT X.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002317"><emph type="center"/><emph type="italics"/>Libræ & &longs;tateræ u&longs;us extenditur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002318">QUæ &longs;emel aliquem in finem excogitata &longs;unt, non ea &longs;unt, <lb/>ut illis tantùm terminis coërceantur, &longs;ed ad plura extendi <lb/>po&longs;&longs;unt; & fundamentis po&longs;itis alia &longs;uper&longs;trui licet, modò non <lb/>de&longs;it artificis indu&longs;tria atque &longs;olertia. </s> <s id="s.002319">Quos in u&longs;us libra & &longs;ta­<lb/>tera à vulgo de&longs;tinentur, omnes nôrunt; &longs;ed ad quos alios tra­<lb/>duci po&longs;&longs;int, iis manife&longs;tum e&longs;t, qui illarum naturam diligen­<lb/>tiùs &longs;crutati &longs;unt. </s> <s id="s.002320">Qua propter ut aliquâ ratione indu&longs;triis arti­<lb/>ficibus præeam, qui &longs;imilia, & multò meliora commini&longs;ci po­<lb/>terunt, pauca quædam hoc capite innuam, quibus libræ & &longs;ta­<lb/>teræ u&longs;us extenditur. </s> </p> <p type="main"> <s id="s.002321">Di&longs;tinctionis autem atque claritatis gratiâ, in plures propo­<lb/>&longs;itiones caput hoc tribuere commodum accidet. </s> </p> <p type="main"> <s id="s.002322"><emph type="center"/>PROPOSITIO I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002323"><emph type="italics"/>Libram con&longs;truere, quâ innatantium &longs;olidorum in humido &longs;peci­<lb/>ficam levitatem, & ip&longs;orum humidorum &longs;pecificam gravita­<lb/>tem inve&longs;tigare po&longs;&longs;umus.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.002324">ERigatur tigillus AB fulcro ritè in&longs;tructus in B, ut firmiter <lb/>con&longs;titui po&longs;&longs;it horizonti perpendicularis: tran&longs;ver&longs;a juga <lb/><figure id="id.017.01.328.1.jpg" xlink:href="017/01/328/1.jpg"/><lb/>duo CD, & EF bifariam æqua­<lb/>liter divi&longs;a, & circà &longs;uos axes <lb/>ver&longs;atilia in&longs;erantur tigillo, pro­<lb/>ut opportunius fuerit, ita tamen, <lb/>ut in eâdem perpendiculari li­<lb/>neâ VS &longs;int axes, & inferiori <lb/>jugo addatur exteriùs axis capi­<lb/>ti in&longs;ertus index GI, qui ubi <lb/>convenerit cum perpendiculari <lb/>lineâ VS in facie tigilli de&longs;crip-<pb pagenum="313" xlink:href="017/01/329.jpg"/>tâ, æquilibrium horizontale jugorum CD & EF indicet. </s> <lb/> <s id="s.002325">Tum extremitates C & E vel &longs;olido, vel plicatili vinculo CE <lb/>connectantur, & in D quidem addatur lanx; in C verò mo­<lb/>mentum plumbi, ut æquilibrium &longs;uâ gravitate con&longs;tituant. </s> <lb/> <s id="s.002326">Po&longs;tquam in F adnexus fuerit &longs;tylus in triplicem cu&longs;pidem de­<lb/>&longs;inens, ut faciliùs deprimatur corpus &longs;olidum H infrà humo­<lb/>rem, in quo levitat, addatur pariter in E aliquid plumbi, ut <lb/>jugum EF in æquilibrio maneat; ni&longs;i fortè tanta &longs;it ip&longs;ius vin­<lb/>culi CE gravitas, ut plumbum addere non &longs;it opus. </s> <s id="s.002327">Demum <lb/>habeatur vas humore implendum, quod &longs;ubjici po&longs;&longs;it extremi­<lb/>tati F, unà cum &longs;olido H innatante. </s> </p> <p type="main"> <s id="s.002328">Primò quæritur levitas &longs;olidi H in aquâ. </s> <s id="s.002329">Expendatur &longs;oli­<lb/>dum H exactè in aëre librâ communi & con&longs;ucta; eju&longs;que pon­<lb/>dus adnotetur: deinde imponatur va&longs;i aquæ pleno, ita ut &longs;oli­<lb/>dum totum immergatur; id quod tunc &longs;olùm fiet, cùm lanci in <lb/>D fuerit impo&longs;itum pondus congruum, nam de&longs;cenden­<lb/>te D, a&longs;cendit C, & &longs;ecum trahit E &longs;ur&longs;um, ac proin­<lb/>de F deprimit &longs;olidum H infrà aquam. </s> <s id="s.002330">Ubi lingula GI in­<lb/>dicaverit æquilibrium &longs;olido H aquæ pror&longs;us immer&longs;o, ob&longs;er­<lb/>va pondus lanci D impo&longs;itum: hoc adde ponderi priùs in­<lb/>vento eju&longs;dem &longs;olidi H in aëre; & pronunciabis, ut hæc &longs;um­<lb/>ma ponderum ad pondus &longs;olidi in aëre, ita e&longs;&longs;e gravitatem <lb/>&longs;pecificam aquæ ad gravitatem &longs;pecificam propo&longs;iti &longs;olidi. </s> <lb/> <s id="s.002331">Fuerit pondus in aëre unc. </s> <s id="s.002332">20; additæ &longs;int in lance D unciæ 5; <lb/>igitur ut 25 ad 20, hoc e&longs;t ut 5 ad 4, ita gravitas &longs;pecifica <lb/>aquæ ad gravitatem &longs;pecificam &longs;olidi. </s> </p> <p type="main"> <s id="s.002333">Veritas o&longs;tenditur ex iis, quæ in Hydro&longs;taticis certa &longs;unt. </s> <lb/> <s id="s.002334">Si enim ponamus aquæ gravitatem ad &longs;olidi H gravitatem &longs;e­<lb/>cundùm &longs;peciem e&longs;&longs;e ut 5 ad 4, emergit ex aquâ pars quinta <lb/>&longs;olidi gravitans ut 4; reliquæ quatuor infrà aquam levitant &longs;in­<lb/>gulæ ut 1, quæ e&longs;t differentia &longs;pecificarum gravitatum: igitur <lb/>pars quinta &longs;olidi extans e&longs;t unc. </s> <s id="s.002335">4, quia totum in aëre e&longs;t <lb/>unc. </s> <s id="s.002336">20; & pars immer&longs;a levitat tanto ni&longs;u, ut æqualis &longs;it <lb/>contrario conatui unc. </s> <s id="s.002337">4. Igitur &longs;i quinque partes demergan­<lb/>tur, re&longs;i&longs;tent unciis quinque, quæ &longs;olido &longs;uperimponeren­<lb/>tur; idem autem e&longs;t, &longs;i unciæ quinque imponantur lanci D; <lb/>eandem enim deprimendi vim habent. </s> <s id="s.002338">Si igitur &longs;olidum gra­<lb/>ve in aëre ut 20, levitat in aquâ ut 5, aquæ moles æqualis <pb pagenum="314" xlink:href="017/01/330.jpg"/>e&longs;t 25, atque adeò aqua ad &longs;olidum e&longs;t ut 25 ad 20 &longs;ecundùm <lb/>gravitatis &longs;peciem. </s> </p> <p type="main"> <s id="s.002339">Secundò comparandi &longs;int humores, uter gravior &longs;it. </s> <lb/> <s id="s.002340">Idem &longs;olidum H notæ gravitatis in aëre unc. </s> <s id="s.002341">20, quod <lb/>priori aquæ immer&longs;um requirebat in lance D uncias 5, im­<lb/>mergatur eodem modo alteri aquæ, ita, ut in lance &longs;int <lb/>unc. </s> <s id="s.002342">4. drachmæ 5: igitur &longs;olidi gravitati in aere unc. </s> <s id="s.002343">20. <lb/>addantur unc. </s> <s id="s.002344">4. drach. </s> <s id="s.002345">5. & erit aquæ &longs;ecundùm molem <lb/>æqualis &longs;pecifica gravitas unc. </s> <s id="s.002346">24 5/8; hæc ergo po&longs;terior aqua <lb/>ad priorem aquam e&longs;t ut 197 ad 200. </s> </p> <p type="main"> <s id="s.002347">Tertiò. <!-- KEEP S--></s> <s id="s.002348">Notâ &longs;olidi &longs;ecundùm &longs;peciem gravitate com­<lb/>paratâ cum gravitate &longs;pecificâ humoris, cogno&longs;cere po&longs;&longs;umus <lb/>alterius molis eju&longs;dem &longs;peciei gravitatem in aëre. </s> <s id="s.002349">Sit cogni­<lb/>ta Ratio gravitatum &longs;ecundùm &longs;peciem ut 4 ad 5. Requi­<lb/>ratur in lance D pondus unc. </s> <s id="s.002350">8, ut infrà aquam deprima­<lb/>tur &longs;olidum. </s> <s id="s.002351">Fiat ut differentia &longs;pecificarum gravitatum 1, <lb/>ad &longs;pecificam gravitatem &longs;olidi 4, ita unciæ 8, ad unc. </s> <s id="s.002352">32: <lb/>E&longs;t ergo &longs;olidum in aëre unciarum 32, & aquæ moles æqualis <lb/>unc. </s> <s id="s.002353">40. </s> </p> <p type="main"> <s id="s.002354">Placeat forta&longs;&longs;e alicui rem hanc aliter perficere. </s> <s id="s.002355">Libræ ju­<lb/>gum EF ita firmetur in G, ut alteri extremitati E ad­<lb/><figure id="id.017.01.330.1.jpg" xlink:href="017/01/330/1.jpg"/><lb/>nexus funiculus a&longs;cendat <lb/>orbiculo X circumvolu­<lb/>tus, & appo&longs;itâ lance D, <lb/>atque in F &longs;tylo tricu&longs;pide, <lb/>omnia &longs;int æquilibrata, ad­<lb/>dito, &longs;i opus fuerit, in F <lb/>plumbi momento: Pondus <lb/>etiam lanci impo&longs;itum &longs;ur­<lb/>&longs;um trahens E deprimit F, <lb/>& pariter &longs;olidum &longs;ubjecto <lb/>humori innatans à &longs;tylo deprimitur, & immergitur. </s> </p> <pb pagenum="315" xlink:href="017/01/331.jpg"/> <p type="main"> <s id="s.002356"><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002357"><emph type="center"/><emph type="italics"/>Horologium arenarium ex librâ con&longs;truere, quod horæ minu­<lb/>ta indicet.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002358">JUgum libræ æqualium brachiorum AB paretur, &longs;partum O <lb/>in &longs;uperiore loco habens: huic enim tantummodo libræ &longs;pe­<lb/><figure id="id.017.01.331.1.jpg" xlink:href="017/01/331/1.jpg"/><lb/>ciei convenire pote&longs;t æqui­<lb/>librium obliquum. </s> <s id="s.002359">Lingu­<lb/>lam OI habeat longiu&longs;cu­<lb/>lam, quæ indicis munere <lb/>fungi po&longs;&longs;it, & quam levi&longs;­<lb/>&longs;ima &longs;it. </s> <s id="s.002360">Tum a&longs;&longs;umpta <lb/>lanx, quæ figuram conicam <lb/>æmuletur, in imâ parte, quâ <lb/>apex de&longs;init, foramen ha­<lb/>beat exiguum, ex quo po&longs;­<lb/>&longs;it &longs;en&longs;im arena fluere; cu­<lb/>ju&longs;modi ea e&longs;t, quâ in vul­<lb/>garibus horologiis arenariis <lb/>utimur. </s> <s id="s.002361">Su&longs;pendatur lanx <lb/>&longs;eor&longs;im à jugo, & impleatur <lb/>arenâ, quæ in &longs;ubjectum vas defluat &longs;patio horæ unius: horâ <lb/>elapsâ &longs;ervetur arena, quam vas excepit, reliqua, quæ in lan­<lb/>ce, rejiciatur. </s> </p> <p type="main"> <s id="s.002362">Sed quoniam ubi multum erat arenæ in lance, plus defluxit, <lb/>quàm par e&longs;t, iterum arena hæc va&longs;is &longs;ubjecti in lancem infun­<lb/>datur, & toties experimentum repetatur rejiciendo reliquam, <lb/>quoties opus fuerit, ut certi &longs;imus arenæ defluxum exqui&longs;itè <lb/>metiri unius horæ longitudinem. </s> </p> <p type="main"> <s id="s.002363">Habitâ jam congruâ arenæ quantitas diligenter &longs;ervetur, ne <lb/>pereat aliquid illius, & novum laborem &longs;ubire cogamur. </s> <s id="s.002364">Hujus <lb/>arenæ gravitas examinetur librâ exacti&longs;&longs;imâ: item lancis cum <lb/>&longs;uis appendiculis pondus inquiratur: quibus cognitis inter gra­<lb/>vitatem &longs;olius lancis C vacuæ, & gravitatem lancis congruâ <lb/>arenâ plenæ inveniatur terminus medio loco proportionalis, qui <lb/>dabit gravitatem ponderis D ex oppo&longs;ito libræ brachio appen­<lb/>dendi. </s> </p> <pb pagenum="316" xlink:href="017/01/332.jpg"/> <p type="main"> <s id="s.002365">Demùm intervallo OI longitudinis lingulæ, quæ &longs;cilicet à <lb/>&longs;parto incipit, de&longs;cribatur vel in lamellâ, vel in cra&longs;&longs;iore papy­<lb/>ro &longs;extans circularis limbi EIF, qui divi&longs;us in partes 60 ita ap­<lb/>tandus e&longs;t, ut lingula &longs;uo apice notata puncta percurrens me­<lb/>dio puncto I congruat, ubi libræ jugum AB horizontale fuerit. </s> <lb/> <s id="s.002366">Quare cum lingulæ apex I erit in E, declinabit lingula à per­<lb/>pendiculo angulo gr. <!-- REMOVE S-->30: id quod pariter in oppo&longs;itâ parte con­<lb/>tinget, quando lingulæ apex venerit in F. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002367">Cum igitur jugum &longs;imiliter inclinari debeat, ut æquilibrium <lb/>&longs;imiliter obliquum fiat hinc lancis C arenâ plenæ depre&longs;&longs;æ cum <lb/>pondere D elevato, hinc ponderis D depre&longs;&longs;i cum lance vacuâ <lb/>elevatâ; con&longs;tat eandem e&longs;&longs;e oportere Rationem gravitatis lan­<lb/>cis C arenâ plenæ ad pondus D, quæ e&longs;t ponderis D ad gravi­<lb/>tatem lancis vacuæ: E&longs;t igitur ponderis D gravitas medio loco <lb/>proportionalis inter gravitates lancis vacuæ, & lancis plenæ. </s> <lb/> <s id="s.002368">Sit deprehen&longs;a gravitas lancis vacuæ pondo unc. </s> <s id="s.002369">5 5/9, lancis au­<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 e&longs;t, quantum di&longs;tare oporteat &longs;partum à li­<lb/>neâ jugi, ut fiat huju&longs;modi æquilibrium obliquum gr. <!-- REMOVE S-->30. Sit <lb/><figure id="id.017.01.332.1.jpg" xlink:href="017/01/332/1.jpg"/><lb/>CD libra, & in C pondus <lb/>unc. </s> <s id="s.002373">18. in D unc. </s> <s id="s.002374">10; & <lb/>fiat æquilibrium ita, ut OI <lb/>lingula faciat cum perpen­<lb/>diculo HS angulum HOI <lb/>gr. <!-- REMOVE S-->30. Ergo in S e&longs;t cen­<lb/>trum gravitatis, & e&longs;t reci­<lb/>procè ut pondus C ad pon­<lb/>dus D, ita longitudo DS. <lb/>ad longitudinem SC: igi­<lb/>tur quarum partium tota <lb/>CD e&longs;t 28, & CG 14, <lb/>earum partium e&longs;t GS 4. In triangulo igitur OGS rectan­<lb/>gulo, GS e&longs;t Sinus gr. <!-- REMOVE S-->30, & GO e&longs;t Sinus gr. <!-- REMOVE S-->60; ac propterea. <lb/></s> <s id="s.002375">&longs;i GS e&longs;t 4, GO e&longs;t 6. 928‴: tanta itaque debet e&longs;&longs;e di&longs;tantia <lb/>&longs;parti O à lineâ jugi. </s> </p> <p type="main"> <s id="s.002376">Hîc autem ob&longs;ervabis lineam jugi inclinatam, cum lineâ ho­<lb/>rizontali, quam &longs;ecat, con&longs;tituere angulum æqualem angulo <lb/>declinationis lingulæ à perpendiculo; nam angulo lingulæ cum <pb pagenum="317" xlink:href="017/01/333.jpg"/>perpendiculari HOI æqualis e&longs;t ad verticem angulus SOG: <lb/>& quia horizontalis VR &longs;ecat perpendiculum HS ad angulos <lb/>rectos in B, duo triangula OGS, & EBS rectangula, & com­<lb/>munem angulum ad S habentia, &longs;unt æquiangula, atque adeò <lb/>angulò SOG, æqualis e&longs;t angulus SEB, cui ad verticem <lb/>æqualis e&longs;t angulus DER, qui proptereà æqualis e&longs;t ip&longs;i <lb/>HOI. </s> </p> <p type="main"> <s id="s.002377">Sed quoniam GS e&longs;t 4, & GO e&longs;t 6. 928‴, per 47. lib.1. <lb/>innote&longs;cit OS partium 7. 999‴ ex quâ aufertur OB æqualis <lb/>ip&longs;i GO (e&longs;t enim di&longs;tantia &longs;parti ab horizontali æqualis <lb/>di&longs;tantiæ eju&longs;dem &longs;parti à jugo) remanet BS partium 1. 071‴. </s> <lb/> <s id="s.002378">In triangulis igitur SGO, SBE &longs;imilibus ut GS 4 ad SO <lb/>7.999‴, ita BS 1. 071‴ ad SE partium 2. 142″: remanet <lb/>igitur EG partium 1. 858‴. </s> <s id="s.002379">Quare tota DE e&longs;t partium 15. <lb/>858″, angulus E in triangulo EMD rectangulo e&longs;t gr.30, ut <lb/>o&longs;ten&longs;um e&longs;t; igitur DM altitudo, ad quam elevatur pondus <lb/>e&longs;t partium 7. 929‴. </s> <s id="s.002380">Et &longs;imiliter quia EC e&longs;t partium 12.142‴, <lb/>depre&longs;&longs;io NC e&longs;t partium 6. 071‴. </s> <s id="s.002381">Ex quo habetur &longs;ub­<lb/>jectum vas, quod cadentem arenam excipit, hoc &longs;altem inter­<lb/>vallo depre&longs;&longs;um e&longs;&longs;e infrà lancem pendentem ex jugo horizon­<lb/>tali po&longs;ito. </s> </p> <p type="main"> <s id="s.002382">Et ut &longs;ubjecti va&longs;is longitudinem invenias, quâ po&longs;&longs;it caden­<lb/>tem arenam excipere, invenienda e&longs;t di&longs;tantia lancis à per­<lb/>pendiculo HS, & cùm in &longs;ummâ depre&longs;&longs;ione e&longs;t, & cùm e&longs;t <lb/>maximè elevata: Cùm depre&longs;&longs;a e&longs;t, di&longs;tat intervallo BN, cùm <lb/>horizontalis e&longs;t, di&longs;tat intervallo BV, cùm demùm e&longs;t elevata, <lb/>di&longs;tat intervallo æquali ip&longs;i BM. </s> <s id="s.002383">Sunt inve&longs;tigandæ di&longs;tantiæ <lb/>BN & BM: Et quia in triangulo EMD rectangulo angulus <lb/>e&longs;t gr. <!-- REMOVE S-->30, & Radius ED e&longs;t partium 15. 858‴; Sinus Com­<lb/>plementi EM e&longs;t partium 13. 733‴. </s> <s id="s.002384">Et in &longs;imili triangulo <lb/>ENC, quia EC Radius e&longs;t partium 12. 142‴, Sinus Comple­<lb/>menti EN e&longs;t partium 10. 515‴. </s> <s id="s.002385">Et iterum in &longs;imili triangu­<lb/>lo EBS, quia ES Radius inventus e&longs;t partium 2. 142‴, Sinus <lb/>Complementi EB e&longs;t partium 1. 855″. <!-- KEEP S--></s> <s id="s.002386">Itaque ex EN aufer <lb/>EB, remanet BN 8. 660‴, ip&longs;i verò EM adde EB, e&longs;t BM <lb/>partium 15. 588‴. </s> <s id="s.002387">Demum ex BM aufer BN, & re&longs;iduum <lb/>partium 6. 928‴ e&longs;t longitudo, quam percurrit lanx a&longs;cenden­<lb/>do, & e&longs;t æqualis di&longs;tantiæ &longs;parti à lineâ jugi; ac propterea vas <pb pagenum="318" xlink:href="017/01/334.jpg"/>excipiendæ arenæ de&longs;tinatum longitudinem habeat nece&longs;&longs;e e&longs;t, <lb/>quæ &longs;altem &longs;it quarta pars longitudinis totius jugi, quæ ex da­<lb/>tis e&longs;t partium 28. </s> </p> <p type="main"> <s id="s.002388">Hæc quæ hactenus dicta &longs;unt, eo con&longs;ilio attuli, ut &longs;i quis <lb/>velit rem ex certâ ratione peragere, intelligat, quâ &longs;it illi uten­<lb/>dum methodo: Cæterùm nemini author fuerim, ut hæc omnia <lb/>calculis indagare eligat, cùm po&longs;&longs;it citrà laborem citi&longs;&longs;imè a&longs;­<lb/>&longs;equi propo&longs;itum finem Statutis enim ponderibus, &longs;cilicet <lb/>lance, arenâ, & æquipondio (quod, ut dixi, medio loco pro­<lb/>portionale e&longs;&longs;e oportet inter vacuam lancem, & lancem ean­<lb/>dem cum arenâ) a&longs;&longs;umatur libræ jugum quodcumque, modò <lb/>&longs;it æqualium brachiorum, & &longs;partum in &longs;uperiore loco habeat, <lb/>tùm adnexis hinc lance cum arenâ, hinc æquipondio, libra <lb/>con&longs;i&longs;tat obliqua; & in plano Verticali libræ proximo notetur <lb/>punctum, cui lingulæ apex congruit: deinde extractâ arenâ <lb/>vacuam lancem relinquat, & librâ con&longs;i&longs;tente notetur pariter <lb/>punctum in plano, quod apici lingulæ re&longs;pondet; & hæc &longs;unt <lb/>extrema puncta arcûs, qui à circumductâ lingulâ de&longs;cribi po­<lb/>te&longs;t in eodem plano verticali, & dividi in quæ&longs;itas partes 60, ut <lb/>horæ minuta indicentur. </s> <s id="s.002389">Quò autem propius ad jugi lineam <lb/>accedet &longs;partum, & longior fuerit lingula, major quoque erit <lb/>huju&longs;modi arcus, & faciliùs in partes 60 dividetur. </s> <s id="s.002390">Va&longs;is de­<lb/>mum longitudinem ip&longs;a libræ po&longs;itio duplex & cum arenâ, & <lb/>&longs;ine arenâ &longs;tatim o&longs;tendet. </s> <s id="s.002391">Hîc verò ubi de arcûs divi&longs;ione in <lb/>partes 60 &longs;ermo e&longs;t, liceat mihi di&longs;&longs;imulare partes illas, &longs;i res <lb/>&longs;ubtili&longs;&longs;imè examinetur, non e&longs;&longs;e omninò inter &longs;e æquales; &longs;ed <lb/>in re Phy&longs;icâ &longs;ubtilitatem hanc per&longs;equi inutile e&longs;t. </s> </p> <p type="main"> <s id="s.002392"><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002393"><emph type="center"/><emph type="italics"/>Ex Libræ Rationibus aliquod Motûs perpetui <lb/>rudimentum proponere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002394">HOc &longs;axum jamdiu multi ver&longs;ant; &longs;ua cuique cogitata pla­<lb/>cent; quem corporibus tribuere nondum potuerunt arti­<lb/>fices perpetuum motum, hunc &longs;ibi vendicant Philo&longs;ophorum <lb/>mentes inquietâ vertigine illius ve&longs;tigiis in&longs;i&longs;tentes; &longs;ed nimis <pb pagenum="319" xlink:href="017/01/335.jpg"/>fugacem nunquam a&longs;&longs;equentes. </s> <s id="s.002395">Liceat & mihi hîc aliquid <lb/>proponere qua&longs;i rudimentum naturæ motum perpetuum effice­<lb/>re condi&longs;centis. </s> <s id="s.002396">Videtur autem omninò certum, ut motus &longs;e­<lb/>mel in&longs;titutus &longs;ine fine per&longs;everet (&longs;eclusâ materiæ corruptio­<lb/>ne, quæ ævo confecta tabe&longs;cit) opus e&longs;&longs;e alterno quodam vi­<lb/>rium incremento atque decremento, ut idem viribus auctis <lb/>prævaleat, viribus diminutis minùs re&longs;i&longs;tat: propterea &longs;impli­<lb/>ci&longs;&longs;imam machinulam, qua&longs;i duplicem libram æqualium bra­<lb/>chiorum ad angulos rectos compactam aliquando excogitavi, <lb/>in quâ alterna hæc vici&longs;&longs;itudo contingere po&longs;&longs;e videtur. </s> </p> <p type="main"> <s id="s.002397">Scapi duo AB & DE ad angulos rectos in C compingantur, <lb/>& &longs;it in C axis, circa quem facilè ver&longs;ari po&longs;&longs;int: quia verò <lb/>oppo&longs;ita brachia. </s> <s id="s.002398">ex hypothe&longs;i <lb/><figure id="id.017.01.335.1.jpg" xlink:href="017/01/335/1.jpg"/><lb/>æqualia &longs;unt, & centrum motûs <lb/>planè in medio congruens cen­<lb/>tro gravitatis ponitur, in quâ­<lb/>cumque po&longs;itione æqualibus mo­<lb/>mentis librata quie&longs;cunt. </s> <s id="s.002399">Sint <lb/>autem &longs;ingula brachia tubi in <lb/>morem excavata ab extremitate <lb/>u&longs;que ad decu&longs;&longs;ationis locum <lb/>æqualiter, ita tamen, ut ex uno <lb/>brachio in aliud brachium &longs;ivè <lb/>oppo&longs;itum, &longs;ivè proximum nul­<lb/>lus pateat exitus: extremum autem tubi o&longs;culum congruâ co­<lb/>chleâ po&longs;&longs;it exqui&longs;itè claudi. </s> <s id="s.002400">Hæc, inquam, omnia ea &longs;int, quæ <lb/>æquilibrium in quâcumque po&longs;itione con&longs;tituant: id quod im­<lb/>probus labor accurati artificis a&longs;&longs;equi &longs;e po&longs;&longs;e non de&longs;perat. </s> </p> <p type="main"> <s id="s.002401">Duplici hac librâ &longs;ic paratâ, &longs;ingulis brachiis certa & om­<lb/>ninò æqualis quantitas Argenti Vivi infundatur, aut major aut <lb/>minor pro ratione magnitudinis & gravitatis tuborum, ita ta­<lb/>men ut non &longs;it immodica quantitas. </s> <s id="s.002402">Occlu&longs;is diligenti&longs;&longs;imè tu­<lb/>borum o&longs;culis, erigatur DE ad perpendiculum. </s> </p> <p type="main"> <s id="s.002403">Utique hydrargyrus in &longs;uperiore brachio DC totus quie&longs;cit <lb/>propè C, in inferiore brachio CE totus e&longs;t in extremitate E: <lb/>in brachiis autem CA & CB horizonti parallelis &longs;e æqualiter <lb/>librat juxtà brachiorum longitudinem; quare libra tota manet <lb/>immota, cum &longs;int hinc & hinc æqualia momenta, tùm ratione <pb pagenum="320" xlink:href="017/01/336.jpg"/>brachiorum æqualium, tùm ratione argenti vivi æqualis, & <lb/>æqualiter ad motum di&longs;po&longs;iti: illud verò quod e&longs;t propè C, & <lb/>propè E, non pote&longs;t mutare æquilibrium, ut patet. </s> <s id="s.002404">Incline­<lb/>tur extremitas B aliquantulum deor&longs;um; illicò totus hydrargy­<lb/>rus brachij CB confluit ad extremitatem B, contrà verò qui <lb/>e&longs;t in brachio CA, totus confluit propè centrum C: Facta e&longs;t <lb/>igitur libra inæqualium brachiorum, & æqualia argenti vivi <lb/>pondera inæqualiter di&longs;tant à centro motûs; ac proinde juxtà <lb/>naturam libræ &longs;partum in ipsâ jugi lineâ habentis extremitas B <lb/>de&longs;cendit quantum pote&longs;t. </s> <s id="s.002405">Cum autem grave quodcumque <lb/>&longs;ponte &longs;ua de&longs;cendens acquirat impetum non &longs;tatim pereun­<lb/>tem, &longs;ed qui adhuc juxta priorem directionem ad ea&longs;dem par­<lb/>tes ferat corpus grave etiam contra naturæ propen&longs;ionem, ut in <lb/>perpendiculo a&longs;cendente e&longs;t manife&longs;tum, quid prohibeat ex­<lb/>tremitatem B hydrargyro prægravatam, ex concepto impetu <lb/>dum de&longs;cendit, vel, modicum quid tran&longs;ilire perpendicularem <lb/>po&longs;itionem ultrà punctum E? <!-- KEEP S--></s> <s id="s.002406">Id quod &longs;i accidat, extremitas D, <lb/>dum tota libra convertitur, inclinata infrà horizontalem AB <lb/>totum hydrargyrum habet non jam in C; &longs;ed in D, quare & <lb/>illa &longs;imili modo de&longs;cendit, nam hydrargyrus, qui erat in E, <lb/>elevato brachio CE &longs;upra horizontalem AB, totus confluit <lb/>prope C: neque difficilis e&longs;t de&longs;cen&longs;us; quia B, ubi tran&longs;ilierit <lb/>perpendiculum DE, ulteriùs ex concepto impetu &longs;ponte a&longs;cen­<lb/>deret; &longs;ed multò magis a&longs;cendit ex impetu impre&longs;&longs;o brachij <lb/>de&longs;cendentis, à quo urgetur. </s> </p> <p type="main"> <s id="s.002407">Fateor equidem in primâ conver&longs;ione po&longs;t quietem, hydrar­<lb/>gyrum E reluctari, nec juvare quicquam ad motum; quia &longs;ci­<lb/>licet, cùm debeat a&longs;cendere ex &longs;olo impetu impre&longs;&longs;o brachij <lb/>CB de&longs;cendentis, nihil confert ad motum, ni&longs;i quatenus E <lb/>initio &longs;ui a&longs;censûs modicum a&longs;cendit, B verò initio &longs;ui de&longs;cen­<lb/>sûs multum de&longs;cendit, ac propterea plus imprimi pote&longs;t impe­<lb/>tûs, ratione cujus, cre&longs;cente quamvis a&longs;cen&longs;uum men&longs;urâ, ha­<lb/>betur aliquid facilitatis ex prævio impul&longs;u. </s> <s id="s.002408">Hinc e&longs;t in primis <lb/>conver&longs;ionibus opus e&longs;&longs;e manûs adjumento, quæ &longs;ur&longs;um pellat <lb/>infimum brachium CE: concepto autem jam impetu, nondum <lb/>video, cur motus ce&longs;&longs;aturus &longs;it. </s> <s id="s.002409">Nam &longs;i nullâ factâ ponderum <lb/>alternâ tran&longs;latione (quæ &longs;emper novum motûs principium af­<lb/>fert) &longs;ed ponderibus &longs;emper in extremitate brachiorum manen-<pb pagenum="321" xlink:href="017/01/337.jpg"/>tibus, po&longs;t aliquot conver&longs;iones externo impul&longs;u factas &longs;ponte <lb/>&longs;ua diu convertitur rota, aut etiam &longs;implex &longs;capus, non ni&longs;i ex <lb/>impre&longs;&longs;o impetu tamdiu permanente, quidni per&longs;everet in mo­<lb/>tu, &longs;i in &longs;ingulis conver&longs;ionibus novum impetum concipiat? </s> <s id="s.002410">Sed <lb/>hæc indica&longs;&longs;e &longs;ufficiat, ut &longs;altem longiorem motum, &longs;i non per­<lb/>petuum, quis a&longs;&longs;equi po&longs;&longs;it &longs;uo in&longs;tituto atque propo&longs;ito op­<lb/>portunum: mihi enim &longs;atis e&longs;t rationes libræ huju&longs;modi com­<lb/>mentatione aliquantò uberiùs explicare. </s> <s id="s.002411">Unum tamen hîc ad­<lb/>dere fuerit operæ pretium, videlicet, &longs;i non placuerit &longs;capos <lb/>AB & DE invicem ad angulum rectum compactos excavare, <lb/>&longs;ed &longs;olidos retinere volueris, po&longs;&longs;e &longs;ingulis brachiis æquales <lb/>tubulos hydrargyri quantitate æquali impletos adalligari, ita ta­<lb/>men, ut &longs;imilem brachij faciem contingant, ex quo fiet, ut &longs;int <lb/>ip&longs;i tubuli alternatim di&longs;po&longs;iti, qui &longs;ibi ex adver&longs;o re&longs;pondent, <lb/>nimirum alter &longs;uperior, alter inferior, alter ad dexteram, alter <lb/>ad &longs;ini&longs;tram. </s> </p> <p type="main"> <s id="s.002412"><emph type="center"/>PROPOSITIO IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002413"><emph type="center"/><emph type="italics"/>Dato unico pondere legitimo examinare bilance <lb/>gravitatem multiplicem materiæ dividuæ.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002414">REs e&longs;t facilis, non tamen omittenda, ne fortè quis &longs;ibi per­<lb/>&longs;uadeat non ni&longs;i longi&longs;&longs;imâ operâ id perfici po&longs;&longs;e. </s> <s id="s.002415">Datum <lb/>&longs;it unicum pondus legitimum, ex. </s> <s id="s.002416">gr. <!-- REMOVE S-->uncia, & oblata &longs;it ma­<lb/>teria dividua, quæ particulatim examinari po&longs;&longs;it, ut &longs;al, & cæ­<lb/>tera minuta. </s> <s id="s.002417">Non &longs;unt &longs;ingulæ unciæ ponderandæ; &longs;ed pri­<lb/>mò quidem fiat cum unciâ æquilibrium &longs;alis; deinde in lancem <lb/>eandem cum pondere legitimo transferatur &longs;al; iterum cum <lb/>alio &longs;ale fiat æquilibrium, & hic in lancem ponderis refundatur, <lb/>totié&longs;que &longs;imili methodo repetatur ponderatio, donec oblatæ <lb/>materiæ plus quàm &longs;emi&longs;&longs;em exhau&longs;eris; & adnota, quoties <lb/>operam illam repetieris; tot enim termini in Ratione duplâ in­<lb/>cipiendo ab unitate a&longs;&longs;umpti, & in &longs;ummam redacti, dabunt <lb/>gravitatem &longs;alis jam examinati. </s> <s id="s.002418">Sint ex. </s> <s id="s.002419">gr. <!-- REMOVE S-->decem termini; <lb/>po&longs;tremus e&longs;t 512, cujus duplum demptâ unitate e&longs;t &longs;umma <lb/>omnium; &longs;unt igitur unciæ 1023, hoc e&longs;t libræ 85 1/4. Quod re-<pb pagenum="322" xlink:href="017/01/338.jpg"/>&longs;iduum e&longs;t &longs;alis, iterum &longs;imili ratione examinetur, donec <lb/>habeas plus quàm &longs;emi&longs;&longs;em illius re&longs;idui, acceptí&longs;que ite­<lb/>rum tot terminis progre&longs;&longs;ionis duplæ habebis ejus quantita­<lb/>tem: & &longs;ic deinceps, donec totius propo&longs;itæ molis pondus <lb/>innote&longs;cat. </s> </p> <p type="main"> <s id="s.002420">Quòd &longs;i certam &longs;alis men&longs;uram extrahere ex totâ illa mole <lb/>de&longs;ideras, ex. </s> <s id="s.002421">gr. <!-- REMOVE S-->libras tres, hoc e&longs;t uncias 36, ob&longs;erva quot <lb/>terminis progre&longs;&longs;ionis duplæ proximè accedas ad propo&longs;itam <lb/>quantitatem, & erunt quinque termini, quorum po&longs;tremus <lb/>e&longs;t 16, & tota &longs;umma 31. Quare operatio, ut &longs;uprà, quinquies <lb/>repetenda e&longs;t, & habentur unciæ 31: quibus &longs;epo&longs;itis inqui­<lb/>rantur unciæ 5 addendæ, nam duplici operatione &longs;ingulas un­<lb/>cias accipiens in eandem lancem cum unciâ legitimâ repones, <lb/>& facto demum æquilibrio reliquas tres uncias habebis, ut <lb/>&longs;umma conficiatur 36. unc. </s> </p> <p type="main"> <s id="s.002422"><emph type="center"/>PROPOSITIO V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002423"><emph type="center"/><emph type="italics"/>Libram æqualium brachiorum con&longs;truere ad plura <lb/>pondera tùm multiplicia tùm &longs;ubmultiplicia <lb/>eju&longs;dem æquipondij examinanda.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002424">ILlud, in quo præ&longs;tat libræ &longs;tatera, e&longs;t, quòd uno eo­<lb/>demque &longs;tateræ æquipondio plura pondera examinamus. </s> <lb/> <s id="s.002425">Non di&longs;&longs;imile compendium invenire po&longs;&longs;umus in libra æqua­<lb/>lium brachiorum, quæ tamen &longs;partum in &longs;uperiore loco <lb/>habeat; hæc enim pro diversâ ponderum inæqualitate va­<lb/>riam habet inclinationem, in quâ quie&longs;cat obliquè po&longs;ita. </s> <lb/> <s id="s.002426">Expedit autem &longs;partum à lineâ jugi aliquanto intervallo <lb/>di&longs;tare. </s> <s id="s.002427">Sit &longs;capus planus, in quo linea jugi recta AB <lb/>bifariam dividatur in C; ex quo ad angulos rectos a&longs;&longs;ur­<lb/>gat firmiter adnexa qua&longs;i lingula CD, ita tamen ut in <lb/>D &longs;tatuatur Axis an&longs;æ in&longs;ertus, circà quem ver&longs;anda e&longs;t <lb/>libra; & ex axe pendeat perpendiculum DE, cujus lon­<lb/>gitudo tanta e&longs;&longs;e debet, ut non &longs;it minor intervallo DA aut <lb/>DB. <!-- KEEP S--></s> <s id="s.002428">Tum ex A &longs;umatur totius lineæ jugi AB tertia pars <pb pagenum="323" xlink:href="017/01/339.jpg"/>A 2, quarta A 3, quinta A 4, &longs;exta A 5, & &longs;ic dein­<lb/>ceps, quate­<lb/><figure id="id.017.01.339.1.jpg" xlink:href="017/01/339/1.jpg"/><lb/>nus commo­<lb/>dè fieri po­<lb/>terit: quæ <lb/>eædem par­<lb/>tes ex B in <lb/>alterum bra­<lb/>chium <expan abbr="tran&longs;-ferãtur">tran&longs;­<lb/>ferantur</expan> quàm <lb/>accurati&longs;&longs;i­<lb/>mè. </s> <s id="s.002429">Demum <lb/>ex A & B <lb/>æquales lan­<lb/>ces pendeant, quæ æquilibrium con&longs;tituant. </s> </p> <p type="main"> <s id="s.002430">Hujus libræ u&longs;us e&longs;t ad multiplicia vel &longs;ubmultiplicia pon­<lb/>dera cum uno eodemque æquipondio comparata invenienda: <lb/>Nam ubi æquipondium legitimum &longs;tatueris in lance B, mer­<lb/>cem verò in lance A, &longs;i æqualitas intercedat, ita jugum ma­<lb/>net, ut perpendiculum DE congruat puncto C: &longs;i merx ma­<lb/>jor &longs;it æquipondio, inclinatur deor&longs;um lanx A, & perpendicu­<lb/>lum DE ad angulos inæquales &longs;ecans lineam jugi congruit ali­<lb/>cui ex punctis notatis inter C & A, &longs;cilicet in 2. &longs;i fuerit dupla, <lb/>in 3 &longs;i tripla, & &longs;ic de reliquis: &longs;i demum merx fuerit minor <lb/>æquipondio, lanx B inclinabitur, & perpendiculum DE con­<lb/>gruet alicui ex punctis inter C & B notatis, indicabitque mer­<lb/>cem e&longs;&longs;e æquipondij aut &longs;emi&longs;&longs;em, aut trientem, aut quadran­<lb/>tem, &c. </s> <s id="s.002431">Hinc &longs;i volueris plures uncias, aut unciæ partem ali­<lb/>quotam habere, &longs;tatue in B legitimum unciæ pondus; &longs;i verò <lb/>plures libras, aut libræ partem aliquotam quæ&longs;ieris, &longs;tatue in B <lb/>libram legitimam. </s> <s id="s.002432">Verùm poti&longs;&longs;ima hujus libræ utilitas &longs;e pro­<lb/>det, ubi dati ponderis, cujus gravitas &longs;ecundùm legitimas men­<lb/>&longs;uras ignota e&longs;t, quæritur pars aliquota, aut illius multiplex <lb/>pondus. </s> <s id="s.002433">Hujus autem libræ con&longs;tructio innititur &longs;uperiùs dictis, <lb/>& manife&longs;ta e&longs;t ratio, quia ex ponderum inæqualitate centrum <lb/>commune gravitatis re&longs;pondet jugi puncto, quod congruit <lb/>perpendiculo pendenti ex eodem puncto &longs;u&longs;pen&longs;ionis libræ. </s> </p> <pb pagenum="324" xlink:href="017/01/340.jpg"/> <p type="main"> <s id="s.002434"><emph type="center"/>PROPOSITIO VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002435"><emph type="center"/><emph type="italics"/>Staterâ examinare pondus majus, quàm ip&longs;a communiter ferat.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002436">CErtum e&longs;&longs;e pondus, quod unaquæque &longs;tatera ferat pro ra­<lb/>tione &longs;uæ magnitudinis, & gravitatis æquipondij, omni­<lb/>bus mani&longs;e&longs;tum e&longs;t: & quidem &longs;i oblatum pondus dividuum <lb/>&longs;it, explorari pote&longs;t per partes ejus gravitas, ut tota demùm in­<lb/>note&longs;cat; &longs;ed &longs;i moles quædam &longs;olida &longs;it, quæ &longs;e dividi non pa­<lb/>tiatur, &longs;tatera autem &longs;it impar tanto oneri, artificium aliquod <lb/>adhiberi pote&longs;t, quo gravitatem illam majorem hâc eâdem &longs;ta­<lb/>terâ inve&longs;tigemus. </s> <s id="s.002437">Et primò quidem ponamus &longs;tateram ita <lb/>fui&longs;&longs;e con&longs;tructam, ut lancis gravitas &longs;uis momentis æquet mo­<lb/>menta brachij longioris, adeò ut, dempto æquipondio &longs;tateræ <lb/>jugum con&longs;i&longs;tat in æquilibrio horizontaliter. </s> <s id="s.002438">Tunc certum <lb/>e&longs;t æquipondium ad onus e&longs;&longs;e reciprocè in Ratione di&longs;tantia­<lb/>rum oneris, & æquipondij à centro motûs. </s> <s id="s.002439">Quare eadem &longs;ta­<lb/>tera poterit quodammodò multiplex fieri, &longs;i nimirum æquipon­<lb/>dium duplicetur, aut triplicetur; poterit enim duplex aut tri­<lb/>plex pondus &longs;taterâ examinari; ut, &longs;i proprium &longs;tateræ æqui­<lb/>pondium &longs;it unius libræ, & brachium longius &longs;it brevioris bra­<lb/>chij quindecuplex, examinari poterit pondus ut &longs;ummum li­<lb/>brarum quindecim; a&longs;&longs;umptum verò æquipondium novum bi­<lb/>libre habebit momentum æquale libris 30; &longs;i trilibre &longs;it no­<lb/>vum æquipondium, momentum erit æquale libris 45; & &longs;ic de <lb/>reliquis, etiam &longs;i æquipondium hoc novum non e&longs;&longs;et ad anti­<lb/>quum omninò in Ratione multiplici; &longs;ed in quâcumque alia <lb/>Ratione etiam &longs;uper particulari, aut &longs;uperpartiente; ducto <lb/>enim pondere novi æquipondij per numerum notatum in &longs;ta­<lb/>teræ brachio, habebitur quantitas ponderis, quod pote&longs;t exa­<lb/>minari; &longs;ic &longs;i æquipondium novum &longs;it ad antiquum ut unc. </s> <s id="s.002440">20. <lb/>ad unc. </s> <s id="s.002441">12. ducto 20 per 15, fit pondus unc. </s> <s id="s.002442">300, hoc e&longs;t lib.25, <lb/>quibus novum æquipondium in extremitate &longs;tateræ po&longs;itum <lb/>æquivalet. </s> </p> <p type="main"> <s id="s.002443">Verùm illud e&longs;t incommodum, quòd huju&longs;modi æquipon­<lb/>dio majori non po&longs;&longs;umus exploratam habere gravitatem pon­<lb/>deris, &longs;i fortè gravitas æquipondij non &longs;it illius pars aliquotae <pb pagenum="325" xlink:href="017/01/341.jpg"/>nam &longs;i novum æquipondium &longs;it bilibre, non indicabit nume­<lb/>rum di&longs;parem librarum ponderis in punctis libras denotantibus <lb/>(&longs;ed &longs;olummodo in punctis &longs;elibrarum) vel &longs;altem &longs;ingulas un­<lb/>cias non indicabit, quia omnes numeri in &longs;taterâ notati dupli­<lb/>candi e&longs;&longs;ent: &longs;imiliter dicendum de æquipondio triplici, quo <lb/>adhibito omnes numeri triplicandi e&longs;&longs;ent. </s> </p> <p type="main"> <s id="s.002444">Propterea, ut huic incommodo occurratur, retineatur anti­<lb/>quum æquipondium in jugo &longs;tateræ, &longs;ed &longs;imul novum æqui­<lb/>pondium in jugi extremitate apponatur duplum, vel triplum, <lb/>vel quadruplum antiqui æquipondij, prout proximè requiri­<lb/>tur ad explorandam dati oneris gravitatem; tùm antiquum <lb/>æquipondium in jugo &longs;tateræ admoveatur vel removeatur, <lb/>quatenus opus fuerit ad æquilibrium con&longs;tituendum. </s> <s id="s.002445">Nam &longs;i <lb/>numerus librarum novi æquipondij ducatur per numerum om­<lb/>nium librarum, quas ferre pote&longs;t &longs;tatera, huicque addatur nu­<lb/>merus ab antiquo æquipondio indicatus, habebitur ip&longs;a pon­<lb/>deris gravitas, quæ inquiritur. </s> <s id="s.002446">Proponatur pondus aliquod <lb/>gravitatis ignotæ, quod &longs;tateræ lanci imponatur, & æquipon­<lb/>dium antiquum ac proprium &longs;tateræ in extremitate po&longs;itum <lb/>non valeat pondus elevare ad æquilibrium; addatur æquipon­<lb/>dium duplum, hoc adhuc impar e&longs;t; addatur triplum, neque <lb/>hoc &longs;atis e&longs;t; addatur quadruplum, & hoc unà cum antiquo <lb/>æquipondio in extremitate brachij po&longs;ito præponderans illud <lb/>e&longs;t, quod requiritur; manente enim novo hoc æquipondio qua­<lb/>druplo in extremitate, antiquum æquipondium admoveatur <lb/>versùs &longs;partum, & fiat æquilibrium in puncto lib. 7. unc. </s> <s id="s.002447">9: <lb/>quia &longs;tateræ numerus extremus e&longs;t ex hypothe&longs;i lib.15, & æqui­<lb/>pondium novum e&longs;t lib.4, jam &longs;unt lib.60; adde lib.7. unc. </s> <s id="s.002448">9. <lb/>tota gravitas ponderis quæ&longs;ita e&longs;t lib.67. unc.9. </s> </p> <p type="main"> <s id="s.002449">At quæris, an eodem hoc artificio uti liceat in communibus <lb/>&longs;tateris, quas no&longs;tratibus artificibus con&longs;truere &longs;olemne e&longs;t; in <lb/>quibus nec &longs;tatera e&longs;t per &longs;e &longs;olam æquilibris, nec æquipondij <lb/>&longs;tateræ jugo ita innexi, ut inde pro libito auferri nequeat, gra­<lb/>vitatem indagare po&longs;&longs;umus, ut æquipondium illius multiplex <lb/>eligamus. </s> <s id="s.002450">Opportunè utique dubitas; nam pondus & æquipon­<lb/>dium in vulgaribus &longs;tateris non &longs;unt omninò in reciprocâ Ra­<lb/>tione di&longs;tantiarum à &longs;parto, ut &longs;uperiùs &longs;uo loco dictum e&longs;t. </s> <lb/> <s id="s.002451">Propterea uti quidem po&longs;&longs;umus eodem artificio, &longs;ed certâ ratio-<pb pagenum="326" xlink:href="017/01/342.jpg"/>ne: quia enim antiquum æquipondium cum &longs;tateræ notis lon­<lb/>gè aliter &longs;e habet, ac in &longs;taterá &longs;uperiùs a&longs;&longs;umpta, hoc retinea­<lb/>tur, quod antiquum æquipondium indicabit gravitatem ponde­<lb/>ris juxta notas, &longs;tateræ impre&longs;&longs;as; &longs;ed æquipondium novum a&longs;­<lb/>&longs;umatur certæ ac notæ gravitatis proxime tàm &longs;ubmultiplicis <lb/>ponderis examinandi, quàm &longs;ubmultiplex brachij longioris e&longs;t <lb/>brachium minus &longs;tateræ; & hoc æquipondium adnectatur non <lb/>planè in &longs;tateræ extremitate, &longs;ed in puncto, in quod cadit lon­<lb/>gitudo multiplex brachij minoris. </s> <s id="s.002452">Sit ex. gr. <!-- REMOVE S-->&longs;tatera communis, <lb/>quæ elevet pondus lib.15; &longs;ed comparato breviore brachio cum <lb/>longiore, hoc non e&longs;t illius omnino quindecuplum; a&longs;&longs;umo, <lb/>quoties a&longs;&longs;umi pote&longs;t brachium minus, ex. </s> <s id="s.002454">gr. <!-- REMOVE S-->quaterdecies; & <lb/>in illo puncto &longs;tatuendum erit novum æquipondium notæ gra­<lb/>vitatis; & quoniam &longs;u&longs;picor propo&longs;itam gravitatem non mul­<lb/>tum abe&longs;&longs;e à lib.50, a&longs;&longs;umo æquipondium lib.3. quæ in notato <lb/>puncto æquivaleat libris 42 (nam ter 14 dant 42) & promoto <lb/>versùs &longs;partum antiquo æquipondio, fit æquilibrium in puncto <lb/>lib.5. unc.3: erit igitur propo&longs;ita gravitas lib.47. unc.3. Id quod <lb/>e&longs;t manife&longs;tum, quia antiquum æquipondium cum notis &longs;tate­<lb/>ræ impre&longs;&longs;is indicat gravitatem ponderis habitâ ratione mo­<lb/>mentorum brachij &longs;tateræ & cæterarum illius partium, quas <lb/>&longs;emel attendere opus e&longs;t; reliquæ gravitatis momenta non ni&longs;i <lb/>ratione di&longs;tantiarum con&longs;ideranda &longs;unt. </s> </p> <p type="main"> <s id="s.002455">Quòd &longs;i plurium æquipondiorum &longs;upellectile careas, & ur­<lb/>geat nece&longs;&longs;itas &longs;tatim explorandi gravitatem illam majorem, ob­<lb/>vium aliquod pondus, puta lapidem, vel quid eju&longs;modi, &longs;tate­<lb/>râ tuâ expende, ut ejus gravitas innote&longs;cat: hoc &longs;u&longs;pende ex <lb/>opportuno &longs;tateræ puncto, de quo dictum e&longs;t, & ejus gravita­<lb/>tem duc per 14 (vel alium quemlibet numerum minorem aut <lb/>majorem, prout opportuna ejus &longs;u&longs;pen&longs;io, aut &longs;tateræ longitu­<lb/>do feret) ut habeas gravitatem huic novo æquipondio re&longs;pon­<lb/>dentem: Cætera ut priùs ab&longs;olve. </s> <s id="s.002456">Non videtur autem nece&longs;&longs;a­<lb/>riò monendus hîc lector po&longs;&longs;e plura nova æquipondia vel di­<lb/>ver&longs;æ, vel paris gravitatis, addi in diver&longs;is di&longs;tantiis à &longs;parto; <lb/>ut &longs;i æquipondium lib.3. in di&longs;tantia 14, & aliud lib.2 in di&longs;tan­<lb/>tia 11 &longs;imul apponantur, æquivalebunt lib.42 & 22, hoc e&longs;t li­<lb/>bris 64; hæc enim clariora &longs;unt, quàm indigeant uberiori ex­<lb/>plicatione. </s> </p> <pb pagenum="327" xlink:href="017/01/343.jpg"/> <p type="main"> <s id="s.002457"><emph type="center"/>PROPOSITIO VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002458"><emph type="center"/><emph type="italics"/>Stateram parare ad minu&longs;culas gravitates expendendas.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002459">STateræ hujus jugum non differt à vulgaribus; &longs;ed æquipon­<lb/>dij & ponderis e&longs;t contraria po&longs;itio; pondus enim longiori <lb/>brachio, breviori æquipondium adnectitur, & quò levius fue­<lb/>rit pondus, eò magis à &longs;parto removetur. </s> <s id="s.002460">Paretur jugum cum <lb/>lance adnexâ, quæ &longs;uâ gravitate æquet momenta brachij lon­<lb/>gioris, & in perfecto æquilibrio con&longs;i&longs;tat. </s> <s id="s.002461">Tum brevioris bra­<lb/>chij longitudo accuratè transferatur in brachium majus, quod <lb/>minoris &longs;altem decuplum vellem, & &longs;ingulas partes iterum in <lb/>decem minores particulas tribuerem, ut totum longius bra­<lb/>chium in centum particulas di&longs;tingueretur. </s> <s id="s.002462">Sit &longs;tateræ jugum <lb/>AB ita in C à <lb/><figure id="id.017.01.343.1.jpg" xlink:href="017/01/343/1.jpg"/><lb/>&longs;parto divi&longs;um, <lb/>ut CB &longs;it de­<lb/>cuplex ip&longs;ius <lb/>CA: ex A au­<lb/>tem pendeat lanx D &longs;uâ gravitate æquè librans momenta bra­<lb/>chij CB longioris; quod di&longs;tinctum in longitudines decem <lb/>æquales brachio minori CA, in &longs;ingulis divi&longs;ionibus indicabit <lb/>Rationem ponderis ad æquipondium. </s> <s id="s.002463">Collocetur enim æqui­<lb/>pondium in lance D, pondus examinandum &longs;i leviu&longs;culum &longs;it <lb/>ita, ut &longs;erico crudo &longs;u&longs;pendi po&longs;&longs;it, jugo CB imponatur, & à <lb/>&longs;parto removeatur, donec fiat æquilibrium: nam &longs;i in primo <lb/>puncto divi&longs;ionis con&longs;i&longs;tat, erit æqualis gravitatis cum æqui­<lb/>pondio; &longs;i in &longs;ecundo puncto, erit &longs;emi&longs;&longs;is gravitatis æquipon­<lb/>dij; &longs;i in tertio, erit triens, &longs;i in quarto, quadrans, & &longs;ic de cæ­<lb/>teris. </s> <s id="s.002464">At &longs;ingulis divi&longs;ionibus minori brachio æqualibus ite­<lb/>rum in decem particulas di&longs;tinctis, indicabitur gravitas à <lb/>fractione, cujus numerator e&longs;t 10, denominator e&longs;t numerus <lb/>particularum omnium, quæ inter &longs;partum C & locum ponde­<lb/>ris æquèlibrati intercipiuntur: ut &longs;i ex. </s> <s id="s.002465">gr. <!-- REMOVE S-->æquilibrium fiat in F, <lb/>hoc e&longs;t in tertiâ particulâ po&longs;t duas integras divi&longs;iones priores, <lb/>jam &longs;unt particulæ 23; igitur pondus e&longs;t (10/23); ip&longs;ius æquipondij <lb/>in D po&longs;iti; ut con&longs;tat ex co, quòd ut di&longs;tantia CF 23 ad <pb pagenum="328" xlink:href="017/01/344.jpg"/>di&longs;tantiam CA 10, ita æquipondium in D ad pondus in <lb/>F (10/23). </s> </p> <p type="main"> <s id="s.002466">Hujus &longs;tateræ utilitas &longs;atis latè patet, quia non alligatur cer­<lb/>to æquipondio, &longs;ed in lance D &longs;tatui pote&longs;t &longs;ivè drachma, &longs;i­<lb/>vè uncia, &longs;ivè libra, & ponderis minoris gravitas examinabitur; <lb/>quæ quidem habebitur &longs;ecundùm Rationem partis ad a&longs;&longs;em, <lb/>&longs;ed deinde ad certam ponderis men&longs;uram, &longs;ivè &longs;crupula, &longs;ivè <lb/>grana revocabitur. </s> </p> <p type="main"> <s id="s.002467">Quòd &longs;i pondus examinandum non facilè &longs;u&longs;pendi po&longs;&longs;it &longs;e­<lb/>rico crudo, ut dictum e&longs;t, paratam habeto lancem minu&longs;culam, <lb/>cui imponi po&longs;&longs;it pondus; & demùm facto æquilibrio, gravita­<lb/>te ponderis inventâ, atque ad homogeneam cum æquipondio <lb/>men&longs;uram redactâ, &longs;ubducenda e&longs;t hujus lancis cum &longs;uo funi­<lb/>culo gravitas, ut &longs;ola ponderis impo&longs;iti gravitas habeatur. </s> <s id="s.002468">Ex <lb/>quo patet adhibitâ huju&longs;modi lance, quæ percurrat &longs;tateræ ju­<lb/>gum, po&longs;&longs;e expendi gravitatem multò minorem: propterea lan­<lb/>cis hujus gravitas minor e&longs;&longs;e deberet, quàm &longs;ubdecupla gravi­<lb/>tatis æquipondij impo&longs;iti lanci D, ut in extremo &longs;tateræ puncto <lb/>B fieri po&longs;&longs;et æquilibrium: verùm &longs;i æquipondium in D &longs;it un­<lb/>cia, aut aliquid unciâ minus, majus tamen decimâ ejus parte, <lb/>&longs;atius fuerit lancem illam excipiendo ponderi de&longs;tinatam e&longs;&longs;e <lb/>decimam unciæ partem. </s> </p> <p type="main"> <s id="s.002469">Ponatur enim æquipondium uncia, lanx ponderis cur&longs;o­<lb/>ria (1/10) unciæ: impo&longs;itum pondus faciat æquilibrium in F puncto <lb/>particulæ 23: e&longs;t igitur pondus cum &longs;uâ lance (10/23) unciæ, aufer <lb/>ratione lancis (1/10) unciæ, re&longs;iduum (77/230) unciæ e&longs;t gravitas ponde­<lb/>ris; hoc e&longs;t &longs;crupulorum 8. Similiter fiat æquilibrium in <lb/>puncto 99; ergo pondus cum lance e&longs;t (10/99) unciæ; aufer (1/10), re&longs;i­<lb/>duum e&longs;t (1/990) unciæ, quod e&longs;t levi&longs;&longs;imum pondus paulò majus <lb/>&longs;emi&longs;&longs;e grani. </s> <s id="s.002470">Si in parte 98, pondus erit (1/490) unciæ, hoc e&longs;t <lb/>grani 1 1/6, &longs;i in puncto 97, pondus erit (3/970) hoc e&longs;t ferè grani <lb/>1 4/5: & &longs;ic de cæteris. </s> </p> <pb pagenum="329" xlink:href="017/01/345.jpg"/> <p type="main"> <s id="s.002471"><emph type="center"/>PROPOSITIO VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002472"><emph type="center"/><emph type="italics"/>Ad ingentia onera examinanda &longs;tateras communes <lb/>componere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002473">SI opportunas &longs;tateras parare oporteret ingentibus oneribus <lb/>examinandis pares, cuju&longs;modi e&longs;&longs;et æs campanum, aut <lb/>bellicum tormentum majus, eas e&longs;&longs;e debere aut longi&longs;&longs;imas, <lb/>aut immani æquipondio in&longs;tructas, manife&longs;tum e&longs;t. </s> <s id="s.002474">Fac <lb/>enim tormentum e&longs;&longs;e lib. 17000 circiter, & &longs;tateram habe­<lb/>re uncialem di&longs;tantiam &longs;parti ab extremitate, cui pondus <lb/>adnectitur, æquipondium verò e&longs;&longs;e lib. 25; utique ut 25 <lb/>ad 17000, ita uncia pedis ad uncias 680, hoc e&longs;t pedes 56. <lb/>tinc. </s> <s id="s.002475">8: atque adeò tota &longs;tatera e&longs;&longs;et ped. <!-- REMOVE S-->56 3/4 ut minimum: <lb/>cui longitudini &longs;i congrua cra&longs;&longs;ities re&longs;pondeat, an non ma­<lb/>chinâ opus e&longs;t, ut &longs;ola &longs;tatera transferatur? </s> <s id="s.002476">præterquam <lb/>quod ip&longs;a longioris brachij gravitas momenta non exigua <lb/>haberet. </s> <s id="s.002477">Quòd &longs;i, ut non paucis &longs;olemne e&longs;t, ita trabem <lb/>ex mediâ longitudine &longs;u&longs;pendas, ut æquilibris maneat, cùm <lb/>alteri extremitati propo&longs;itum onus adnectas, oppo&longs;itæ au­<lb/>tem extremitati plura minora pondera adjicias, donec æqui­<lb/>librium fiat, quorum &longs;ingulæ gravitates in &longs;ummam redactæ <lb/>propo&longs;iti oneris gravitatem manife&longs;tam reddant, non &longs;olùm <lb/>methodus hæc artificio caret, &longs;ed & fal&longs;itatis periculo non <lb/>vacat, incertum quippe e&longs;t an trabis centrum gravitatis planè <lb/>in mediâ longitudine &longs;it, cùm pars radici proxima gravior &longs;it <lb/>reliquâ, ac proinde libra &longs;it inæqualium brachiorum, quæ <lb/>cen&longs;etur æqualium. </s> </p> <p type="main"> <s id="s.002478">Satius igitur fuerit &longs;tateras plures minores componere, <lb/>ut indicatum e&longs;t lib. 2. cap. 7, quàm ingentem &longs;tateram <lb/>con&longs;truere. </s> <s id="s.002479">A&longs;&longs;umantur tres &longs;tateræ AB, DE, GH, qua­<lb/>rum brachium minus &longs;it majoris &longs;ubdecuplum, & ita om­<lb/>nes ex &longs;uperiore loco &longs;u&longs;pendantur, ut orbiculi M & N <lb/>facilè ver&longs;atiles inferiùs firmati excipere po&longs;&longs;int funiculos <lb/>BMD, & ENG, quibus extremitates junguntur: ex quo <lb/>fiet, ut dum H vi æquipondij deprimitur, extremitas E, at­<lb/>que extremitas B pariter deprimantur, pondus verò in A ad-<pb pagenum="330" xlink:href="017/01/346.jpg"/>nexum elevetur. </s> <s id="s.002480">Motus autem &longs;taterarum non &longs;unt æquales: <lb/>nam &longs;icut depre&longs;&longs;io ip&longs;ius H e&longs;t decupla elevationis ip&longs;ius G, <lb/><figure id="id.017.01.346.1.jpg" xlink:href="017/01/346/1.jpg"/><lb/>cui elevationi æqualis e&longs;t depre&longs;&longs;io extremitatis E, ita hæc eju&longs;­<lb/>dem E depre&longs;&longs;io decupla e&longs;t elevationis ip&longs;ius D: quare depre&longs;­<lb/>&longs;io H e&longs;t centupla elevationis D; ac propterea quia depre&longs;&longs;io <lb/>B æqualis elevationi D e&longs;t decupla elevationis A, depre&longs;&longs;io <lb/>æquipondij in H e&longs;t millecupla elevationis ponderis in A <lb/>con&longs;tituti. </s> <s id="s.002481">Ex quo &longs;equitur æquipondium in H æquivalere <lb/>ponderi millecuplo, quod in A appendatur. </s> <s id="s.002482">Igitur æquipon­<lb/>dium lib.17 æquivalebit ponderi lib.17000. </s> </p> <p type="main"> <s id="s.002483">Quod autem hactenus de &longs;tateris æqualibus dictum e&longs;t, etiam <lb/>de inæqualibus dictum intelligatur, componendo Rationes, <lb/>quas &longs;ingularum &longs;taterarum brachia habent. </s> <s id="s.002484">Hinc &longs;i Ratio <lb/>AC ad CB &longs;it 1 ad 10, Ratio DF ad FE &longs;it 1 ad 8, Ratio GI <lb/>ad IH &longs;it 1 ad 12, Ratio compo&longs;ita e&longs;t 1 ad 960, quæ pote&longs;t <lb/>intercedere inter æquipondium & onus. </s> <s id="s.002485">Hinc manife&longs;tum e&longs;t <lb/>plures addi po&longs;&longs;e &longs;tateras, quot opus fuerit, quocumque tan­<lb/>dem ordine collocentur, &longs;ive &longs;ecundùm rectam lineam, &longs;ive <lb/>invicem parallelæ, prout commodius accidet, & loci oppor­<lb/>tunitas feret. </s> </p> <p type="main"> <s id="s.002486">Si &longs;tateræ i&longs;tæ fuerint ita con&longs;tructæ, ut jugum dempto <lb/>æquipondio æquilibre &longs;it, quia extremitas brachij minoris gra­<lb/>vitate tantâ prædita e&longs;t, ut gravitati longioris brachij æquipol­<lb/>leat, res plani&longs;&longs;ima e&longs;t, quia &longs;ola brachiorum longitudinis Ra­<lb/>tio attendenda e&longs;t; & præterea æquipondium in H augeri po&longs;­<lb/>&longs;et, aut minui. </s> <s id="s.002487">Immò hîc etiam adhiberi po&longs;&longs;et artificium, <lb/>de quo prop. 6. dicebatur, addendo novum æquipondium cer­<lb/>tæ gravitatis, ut &longs;i præter æquipondium H etiam e&longs;&longs;et L lib. 2; <lb/>quod in puncto jugi &longs;eptimo æquivaleret ponderi &longs;eptingenties <lb/>majori, hoc e&longs;t lib. 1400. Quâ methodo addi po&longs;&longs;unt etiam <lb/>plura æquipondia in punctis jugi diver&longs;is: quod &longs;anè e&longs;&longs;et egre-<pb pagenum="331" xlink:href="017/01/347.jpg"/>gium compendium, & ut plurimum duabus &longs;tateris pondera­<lb/>tio ip&longs;a perficeretur. </s> </p> <p type="main"> <s id="s.002488">At &longs;i &longs;tateræ cuju&longs;que jugum non fuerit æquilibre, contem­<lb/>nenda non e&longs;t brachij longioris gravitas, ut dati ponderis gra­<lb/>vitas ritè examinetur: Nam &longs;emi&longs;&longs;is gravitatis brachij IH in <lb/>extremitate H con&longs;titutus æquivalet ponderi decuplo in G mi­<lb/>nùs gravitate &longs;emi&longs;&longs;is brachij IG. </s> <s id="s.002489">Igitur perinde e&longs;t, atque &longs;i <lb/>huju&longs;modi pondus additum fui&longs;&longs;et in E, ubi habet momentum <lb/>decuplum æqualis ponderis in D, & centuplum æqualis pon­<lb/>deris in A. <!-- KEEP S--></s> <s id="s.002490">Quare momentum brachij IH e&longs;t ut 50, & mo­<lb/>mentum IG ut 1/2, atque adeò momentum ut 49 1/2 intelligitur <lb/>additum in E, quod propterea comparatum cum extremitate <lb/>A habet momentum ut 4950. Sic momentum gravitatis FE <lb/>comparatum cum extremitate A e&longs;t ut 495; & momentum bra­<lb/>chij CB e&longs;t ut 49 1/2. Tota igitur momentorum, quæ ex bra­<lb/>chiorum gravitate oriuntur (&longs;i illa æqualiter ducta intelligan­<lb/>tur) &longs;umma e&longs;t 5494 1/2, &longs;ive &longs;int unciæ, &longs;ive libræ, prout &longs;ta­<lb/>terarum moles requirit. </s> <s id="s.002491">Id quod quia ægrè innote&longs;cit, &longs;i ju­<lb/>gum non fuerit æquabiliter ductum, idcircò expeditius fuerit <lb/>&longs;tateris ritè di&longs;po&longs;itis, ac dempto æquipondio, addere in A tan­<lb/>tum gravitatis, ut juga &longs;int horizonti parallela (cujus paralle­<lb/>li&longs;mi indicium poti&longs;&longs;imum dabit extremæ &longs;tateræ lingula, quæ <lb/>plus cæteris movetur) quæ gravitas ubi innotuerit, addenda <lb/>erit gravitati, quam deinde æquipondium indicabit, cùm onus <lb/>ip&longs;um in A additum fuerit. </s> <s id="s.002492">Sic pone momenta illa 5494 1/2 e&longs;&longs;e <lb/>uncias, hoc e&longs;t lib. 457. unc. </s> <s id="s.002493">10 1/2, & expendendo onus addi­<lb/>tum in A, æquipondium librale H indicet æquilibrium in <lb/>puncto &longs;eptimo, hoc e&longs;t lib.700, addantur lib.457. unc. </s> <s id="s.002494">10 1/2, <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 &longs;tateræ, quibus communiter utimur <lb/>ad majorum ponderum gravitatem examinandam, non ita &longs;unt <lb/>fabrefactæ, ut brachium longius in partes aliquotas minori bra­<lb/>chio æquales di&longs;tinguatur, propterea minoris brachij longitu­<lb/>do, quoties fieri id poterit, transferatur in brachium longius, <lb/>ut inveniatur punctum, cui adnectendus e&longs;t funiculus, quo <lb/>cum alterius proximæ &longs;tateræ extremitate connectitur. </s> <s id="s.002497">Sed an­<lb/>tequam opus aggrediaris, amoto æquipondio &longs;ecundæ & ter-<pb pagenum="332" xlink:href="017/01/348.jpg"/>tiæ &longs;tateræ, vide quantum ponderis &longs;ingulæ, quantum con­<lb/>nexæ requirant ad æquilibrium cum longiore brachio, ut inno­<lb/>te&longs;cat, quantum adhuc gravitati, oneri tribuendum &longs;it, præter <lb/>illam, quæ ab æquipondio indicatur. </s> </p> <figure id="id.017.01.348.1.jpg" xlink:href="017/01/348/1.jpg"/> <p type="main"> <s id="s.002498">Sit ex. </s> <s id="s.002499">gr. <!-- REMOVE S-->&longs;ecunda &longs;tatera CD, cujus brachium longius <lb/>SD æquivaleat lib. 42, & tertiæ &longs;tateræ FG longius brachium <lb/>VG æquivaleat libris 37. Ponamus tertiæ &longs;tateræ (cui onus <lb/>erit adnectendum) brachium minus FV duodecies contineri <lb/>in longiore brachio u&longs;que ad I, ubi funiculus connectit illud <lb/>cum extremitate C &longs;ecundæ &longs;tateræ. </s> <s id="s.002500">Item &longs;ecundæ &longs;tateræ <lb/>CD brachium minus CS tredecies &longs;umi po&longs;&longs;it in brachio lon­<lb/>giore u&longs;que ad punctum E, ubi illam funiculus connectit cum <lb/>primæ &longs;tateræ extremitate A. <!-- KEEP S--></s> <s id="s.002501">Igitur quia momentum brachij <lb/>SD æquivalet libris 42 ex hypothe&longs;i, & intelligitur tran&longs;latum <lb/>in I, ubi duodecuplo velociùs movetur quàm punctum F, du­<lb/>cantur lib. 42 per 12, & æquivalet libris 504, quibus addenda <lb/>&longs;unt momenta brachij VG lib.37, & ponderi invento ex æqui­<lb/>pondio demum addendæ erunt lib. 541. Jam &longs;tatuamus æqui­<lb/>pondium H primæ &longs;tateræ AB con&longs;tituere æquilibrium in <lb/>puncto indicante libras 14: perinde igitur e&longs;t, atque &longs;i libræ 14 <lb/>ponerentur in E; & quia ES ad SC e&longs;t ut 13 ad 1, libræ 14 in <lb/>E æquivalent ponderi in C librarum 182, quæ in I po&longs;itæ (quia <lb/>IV ad VF e&longs;t ut 12 ad 1) æquivalent ponderi in F librarum <lb/>2184. Quod &longs;i punctum illud, in quo æquipondium H con­<lb/>&longs;i&longs;tit, non e&longs;&longs;et nota librarum &longs;implicium 14, &longs;ed ponderum, <lb/>quæ &longs;ingula libras 25 continent (ut nobis Italis præ&longs;ertim in <lb/>Galliâ Ci&longs;alpinâ &longs;olemne e&longs;t) utique onus in F adnexum e&longs;&longs;et <lb/>lib. 54600, quibus adhuc addendæ e&longs;&longs;ent libræ 541, propter <lb/>momenta brachiorum &longs;ecundæ & tertiæ &longs;tateræ, & e&longs;&longs;et tota <lb/>gravitas lib. 55141. </s> </p> <p type="main"> <s id="s.002502">At &longs;i &longs;tateras communes habeas, nec po&longs;&longs;is æquipondia jugo <lb/>in&longs;erta amovere, ut inquirere po&longs;&longs;is momenta gravitatis brachij <pb pagenum="333" xlink:href="017/01/349.jpg"/>longioris, hoc unum in &longs;ecundâ, & in tertiá &longs;taterâ, aut etiam <lb/>pluribus, &longs;i opus fuerit, ob&longs;erva, quoties nimirum brachium <lb/>minus in longiore contineatur, ut punctum I & E innote&longs;cat, <lb/>quod cum proximæ &longs;tateræ extremitate C & A connectendum <lb/>e&longs;t: in &longs;ingulis autem &longs;tateris &longs;ua æquipondia admoveantur, <lb/>vel removeantur, donec fiat æquilibrium. </s> <s id="s.002503">Non e&longs;t autem ne­<lb/>ce&longs;&longs;e &longs;ingulas &longs;tateras in&longs;culptas e&longs;&longs;e notis homogeneis gravi­<lb/>tatum; prima enim AB pote&longs;t habere notas indicantes quar­<lb/>tam partem Centenarij, hoc e&longs;t lib. 25, &longs;ecunda verò & ter­<lb/>tia po&longs;&longs;unt indicare tantum &longs;ingulas libras cum &longs;uis unciis. </s> <s id="s.002504">Fac <lb/>enim con&longs;tituto æquilibrio, æquipondium H e&longs;&longs;e in puncto <lb/>pond. </s> <s id="s.002505">9. lib.7. duc 9 per 25, & &longs;unt lib. 225, & additis lib.7, <lb/>&longs;unt lib. 232; quæ ducuntur primò per Rationem &longs;ecundæ &longs;ta­<lb/>teræ 13 ad 1, & fiunt 3016, quæ ductæ per Rationem tertiæ <lb/>&longs;tateræ 12 ad 1, dant demum lib. 36192. Deinde æquipon­<lb/>dium &longs;ecundæ &longs;tateræ CD &longs;it in puncto lib. 7. unc. </s> <s id="s.002506">8: hæ du­<lb/>cendæ &longs;unt per Rationem tertiæ &longs;tateræ 12 ad 1, & fiunt <lb/>lib. 92. Demum æquipondium tertiæ &longs;tateræ indicet lib. 5. <lb/>unc. </s> <s id="s.002507">6, addantur hi tres numeri 36192, 92, & 5. unc. </s> <s id="s.002508">6; tota <lb/>gravitas oneris in F adnexi erit lib.36289. unc. </s> <s id="s.002509">6. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002510">Ideò autem inquirenda dixi puncta I & E, ut longitudines <lb/>VI & SE &longs;int multiplices longitudinum brachiorum minorum <lb/>FV & CE, atque fractionum mole&longs;tia evitetur. </s> <s id="s.002511">Cæterùm &longs;i <lb/>volueris extremitates ip&longs;as G & D cum extremitatibus C & A <lb/>connectere, omnino licebit, ubi innotuerit, quota pars brachij <lb/>minoris &longs;it IG & ED. <!-- KEEP S--></s> <s id="s.002512">Nam &longs;i Ratio DS ad SC deprehen­<lb/>datur ut 13 2/5 ad 1, Ratio autem GV ad VF ut 12 1/4 ad 1, gra­<lb/>vitas indicata ab æquipondio H ducenda primùm erit per 13 2/5, <lb/>deinde numerus productus per 12 3/4 ductus dabit quæ&longs;itam <lb/>oneris gravitatem re&longs;pondentem æquipondio H, quod, ex hy­<lb/>pothe&longs;i &longs;uperiùs con&longs;titutá, indicans pond. </s> <s id="s.002513">9. lib.7, hoc e&longs;t <lb/>lib.232, monet ducendas libras 232 per 13 2/5, & fit 3108 4/5, qui <lb/>numerus ducatur per 12 3/4, & fiunt lib.39637 1/5. Deinde æqui­<lb/>pondium &longs;ecundæ &longs;tateræ po&longs;itum in puncto lib.7. unc. </s> <s id="s.002514">8 indi­<lb/>cat has ducendas per 12 3/4, & erunt lib. 97 3/4: quibus &longs;i adda­<lb/>tur numerus primæ &longs;tateræ, & numerus quem dat tertia &longs;tatera <lb/>lib.5. unc. </s> <s id="s.002515">6, &longs;umma erit omnino lib.39740. unc. </s> <s id="s.002516">5. <!-- KEEP S--></s> </p> <pb pagenum="334" xlink:href="017/01/350.jpg"/> <p type="main"> <s id="s.002517"><emph type="center"/>PROPOSITIO IX.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002518"><emph type="center"/><emph type="italics"/>In librâ brachiorum æqualium po&longs;&longs;e non æqualia e&longs;&longs;e ponderum <lb/>æqualium momenta.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002519">SIt libra AB, cujus centrum C, pror&longs;us in medio, jugum in <lb/>brachia dividat æqualia: &longs;int autem in brachiorum extre­<lb/><figure id="id.017.01.350.1.jpg" xlink:href="017/01/350/1.jpg"/><lb/>mitatibus annuli <lb/>vel unci, quibus <lb/>adnectenda &longs;unt <lb/>pondera, quæ a&longs;­<lb/>&longs;umantur gravi­<lb/>tatis exqui&longs;itè æ­<lb/>qualis, computatâ <lb/>etiam funiculo­<lb/>rum gravitate. </s> <s id="s.002520">Sed <lb/>alterum quidem <lb/>pondus D unco <lb/>adnectatur unà <lb/>cum &longs;uo funicu­<lb/>lo; alterius verò ponderis E funiculus &longs;uâ extremitate inferiùs <lb/>in F paxillo alligetur, & tran&longs;iens per annulum, vel uncum <lb/>&longs;u&longs;pendat connexum pondus E. <!-- KEEP S--></s> <s id="s.002521">Experimento di&longs;ces pondus E <lb/>&longs;emper prævalere æquali ponderi D, &longs;i per annulum vel uncum <lb/>funiculus liberè valeat excurrere, de&longs;cendente ip&longs;o pon­<lb/>dere E. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002522">Sed rei primâ facie admiratione dignæ cau&longs;am inquirenti illa <lb/>&longs;e &longs;tatim offert, quæ Machinalium motionum cau&longs;a à nobis af­<lb/>fertur; quia videlicet pondus E de&longs;cendens duplo velociùs de­<lb/>&longs;cendit, quàm pondus D a&longs;cendat; ubi enim pondus E vene­<lb/>rit in F, extremitas libræ B ibi con&longs;i&longs;tet, ubi duplicatus e&longs;t fu­<lb/>niculus, mediâ nimirum viâ; atqui extremitas A non ni&longs;i tan­<lb/>tumdem a&longs;cendit, & cum eâ pondus D; igitur pondus E velo­<lb/>ciùs de&longs;cendens potiora habet momenta, nec erit æquilibrium, <lb/>ni&longs;i pondus E &longs;it ponderis D &longs;ubduplum. </s> </p> <p type="main"> <s id="s.002523">Cave tamen exi&longs;times &longs;emper e&longs;&longs;e motuum Rationem du­<lb/>plam; id enim tunc &longs;olùm accidit, cum funiculus extentus e&longs;t <pb pagenum="335" xlink:href="017/01/351.jpg"/>horizonti perpendicularis, cuju&longs;modi e&longs;t FE: at &longs;i fuerit in­<lb/>clinatus, non e&longs;t eadem motuum Ratio, &longs;ed ut duplex funicu­<lb/>li GE longitudo ad altitudinem perpendicularem EF, ita &longs;e <lb/>habet motus ponderis E ad differentiam, quâ excedit motum <lb/>ponderis D, &longs;eu depre&longs;&longs;ionis libræ B. </s> <s id="s.002524">Sit funiculus GE, alti­<lb/>tudo perpendicularis, per quam de&longs;cendit pondus E, &longs;it EF; <lb/>di&longs;tantia GF: de&longs;cendente pondere E, ubi hoc attigerit pla­<lb/>num horizontale in F, funiculus, qui erat GE, factus e&longs;t GIF; <lb/>igitur libra deprimitur u&longs;que in I, & e&longs;t IF differentia motuum <lb/>EF & EI. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002525">Quare cùm GI &longs;it GE minùs IF, quadratum GI æquale <lb/>e&longs;t quadrato GE plus quadrato IF, minùs rectangulo &longs;ub GE <lb/>& IF bis comprehen&longs;o. </s> <s id="s.002526">At eidem quadrato GI æqualia &longs;unt <lb/>quadrata IF & GF &longs;imul &longs;umpta ex 47. lib. 1: propterea aufe­<lb/>ratur utrinque quadratum IF, & remanet quadratum GE, mi­<lb/>nùs rectangulo bis &longs;ub GE & IF comprehen&longs;o æquale quadra­<lb/>to GF: Addatur utrinque rectangulum &longs;ub GE & IF bis, & <lb/>utrinque dematur quadratum GF, & e&longs;t quadratum GE mi­<lb/>nus quadrato GF (hoc e&longs;t quadratum EF ex 47. lib.1.) æqua­<lb/>le rectangulo bis &longs;ub GE & IF. <!-- KEEP S--></s> <s id="s.002527">Igitur ex 17 lib 6. ut bis GE <lb/>ad EF, ita EF ad IF. <!-- KEEP S--></s> <s id="s.002528">Ponderis itaque motus deor&longs;um EF <lb/>comparatus cum a&longs;cen&longs;u ponderis D, e&longs;t ad differentiam mo­<lb/>tuum IF, ut duplex longitudo funiculi GE ad altitudinem <lb/>perpendicularem EF, per quam de&longs;cendit pondus E. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002529">Ex quo ulteriùs colligitur, quò obliquior e&longs;t funiculus, eò <lb/>minorem e&longs;&longs;e differentiam IF, ac propterea minorem e&longs;&longs;e Ra­<lb/>tionem de&longs;censûs EF ad a&longs;cen&longs;um ponderis oppo&longs;iti, ideóque <lb/>etiam minus habere virium ad prævalendum. </s> <s id="s.002530">Hinc ex diver­<lb/>sâ funiculi longitudine & obliquitate, &longs;i æquilibrium fiat, lice­<lb/>bit arguere ip&longs;am ponderum inæqualitatem, ratione habitâ mo­<lb/>tuum reciprocè &longs;umptorum; qui motus cum habere non po&longs;&longs;int <lb/>Rationem multiplicem majorem duplâ, ut con&longs;tat funiculi ip&longs;ius <lb/>flexionem con&longs;ideranti, neque pondus D pote&longs;t e&longs;&longs;e minus pon­<lb/>dere E, neque eodem majus quàm duplum, &longs;i fiat æquilibrium; <lb/>minus autem erit quàm duplum, &longs;i funiculus &longs;it obliquus, & ex <lb/>motuum differentiâ, quæ &longs;ingulas funiculi obliquitates con&longs;e­<lb/>queretur, etiam ip&longs;a ponderum inæqualium differentia in­<lb/>fertur, </s> </p> <pb pagenum="336" xlink:href="017/01/352.jpg"/> <p type="main"> <s id="s.002531"><emph type="center"/>PROPOSITIO X.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002532"><emph type="center"/><emph type="italics"/>Æqualia pondera &longs;imilis figuræ, &longs;ed diver&longs;æ &longs;ub&longs;tantiæ, &longs;imili­<lb/>bus & æqualibus pyxidibus inclu&longs;a di&longs;cernere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002533">SInt duo globi, alter ferreus H, alter argenteus S, inclu&longs;i <lb/>æqualibus & &longs;imilibus pyxidibus AB & CD ita æqualis <lb/><figure id="id.017.01.352.1.jpg" xlink:href="017/01/352/1.jpg"/><lb/>ponderis, ut pyxides vacuæ librà <lb/>examinatæ æquiponderent, & ad­<lb/>jecti globi pariter &longs;int æquales ra­<lb/>tione ponderis, quamvis moles <lb/>inæquales &longs;int, major enim e&longs;t <lb/>ferreus, minor argenteus. </s> <s id="s.002534">Opor­<lb/>teat igitur di&longs;cernere, utra pyxis <lb/>argenteum globum contineat. </s> <lb/> <s id="s.002535">Singularum pyxidum longitudo <lb/>bifariam dividatur in F & E, ex <lb/>quibus punctis fiat &longs;u&longs;pen&longs;io; quâ factâ utique de&longs;cendent ex­<lb/>tremitates B & D. <!-- KEEP S--></s> <s id="s.002536">Addantur tum in A, tum in C pondera, <lb/>ut fiat æquilibrium. </s> <s id="s.002537">Pondus majus indicabit ibi e&longs;&longs;e globum <lb/>argenteum. </s> <s id="s.002538">Vel &longs;i unico æquipondio uti placeat, invento <lb/>æquilibrio unius pyxidis, idem æquipondium ad alteram pyxi­<lb/>dem transferatur: &longs;i enim appo&longs;ita extremitas præponderet, ibi <lb/>e&longs;t argentum, &longs;i &longs;ur&longs;um attollatur, ibi e&longs;t ferrum. </s> <s id="s.002539">Manife&longs;ta au­<lb/>tem e&longs;t ratio, quia majoris globi centrum gravitatis propius e&longs;t <lb/>medio pyxidis, ex quo &longs;it &longs;u&longs;pen&longs;io, ac propterea minus habet <lb/>momenti, quàm minor globus, cujus centrum magis di&longs;tat. </s> </p> <p type="main"> <s id="s.002540">Quamvis verò &longs;u&longs;pen&longs;io facta fuerit ex medio, nihil refert, <lb/>etiam&longs;i ad alterutram extremitatem accedat ut in K, dummodo <lb/>æqualis a&longs;&longs;umatur di&longs;tantia in L; eadem enim &longs;emper ratio pro <lb/>inæqualitate momentorum militat, inæqualis &longs;cilicet di&longs;tantia <lb/>centrorum gravitatis. </s> </p> <p type="main"> <s id="s.002541">At &longs;i non ea e&longs;&longs;et pyxidum longitudo, ut extremitatibus A <lb/>& C facilè adnectatur æquipondium, a&longs;&longs;ume regulam BZ lon­<lb/>giorem ipsâ pyxide, eamque alliga funiculo per K tran&longs;eunte, <lb/>& in Z æquipondium &longs;tatuatur: deinde regulam eandem &longs;imi­<lb/>liter alliga alteri pyxidi, ut &longs;it DX, & funiculus per L tran&longs;eat: <pb pagenum="337" xlink:href="017/01/353.jpg"/>nam idem æquipondium in X &longs;i nimis leve &longs;it, indicat ibi ar­<lb/>gentum e&longs;&longs;e; id quod pariter indicabit æquipondium majus <lb/>faciens æquilibrium. </s> </p> <p type="main"> <s id="s.002542">Quòd &longs;i pondus idem utrobique faceret æquilibrium, indicio <lb/>e&longs;&longs;et aut inclu&longs;a corpora non e&longs;&longs;e &longs;ecundùm molem &longs;imilia, aut <lb/>&longs;i &longs;imilia fuerint non e&longs;&longs;e in pyxidibus &longs;imiliter po&longs;ita in extre­<lb/>mitate, contrà hypothe&longs;im. </s> <s id="s.002543">Id quod ut deprehendas, ita pyxi­<lb/>des converte, ut ad latus con&longs;tituatur pars, quæ priùs erat infi­<lb/>ma; tunc enim ponderis aliqua diver&longs;itas apparebit. </s> <s id="s.002544">Si autem <lb/>adhuc æquilibrium con&longs;tituatur, minorem molem ita ex arte <lb/>collocatam fui&longs;&longs;e, ut centrum gravitatis æqualem di&longs;tantiam <lb/>habeat à puncto &longs;u&longs;pen&longs;ionis, ac moles major in alterâ pyxide, <lb/>manife&longs;tum e&longs;t. </s> <s id="s.002545">Tunc igitur utraque pyxis intrà aquam pon­<lb/>deranda e&longs;t; quæ enim minùs gravis apparebit, continet ar­<lb/>gentum; hoc quippe minus &longs;patij occupans quàm ferrum, ma­<lb/>jori aëris moli in pyxide locum relinquit: major autem aëris <lb/>moles plus deterit ponderis pyxidi intra aquam: pyxidum &longs;ci­<lb/>licet moles ponuntur æquales. <lb/></s> </p> <p type="main"> <s id="s.002546"><emph type="center"/>CAPUT XI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002547"><emph type="center"/><emph type="italics"/>Fundamenta præmittuntur ad explicandum, cur <lb/>gravia &longs;u&longs;pen&longs;a modò præponderent, modò <lb/>æquilibria &longs;int.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002548">LOcus hic e&longs;t ob&longs;trictam non &longs;emel in &longs;uperioribus fidem <lb/>liberandi, cùm me o&longs;ten&longs;urum &longs;u&longs;cepi in corporibus &longs;u&longs;­<lb/>pen&longs;is aliquando minùs gravia gravioribus prævalere, nec ta­<lb/>men ullum libræ aut Vectis ve&longs;tigium deprehendi, neque mo­<lb/>tum propriè circularem tribui po&longs;&longs;e potentiæ moventi, quæ vi <lb/>&longs;uæ gravitatis juxtà directionis lineam deor&longs;um conatur, atque <lb/>movetur motu recto, &longs;ur&longs;um a&longs;cendente rectà corpore gravio­<lb/>re, quod per vim elevatur. </s> <s id="s.002549">Sed ut res tota capite &longs;equenti cla­<lb/>riùs & breviùs explicari valeat, propo&longs;itiones aliquot hîc lem­<lb/>matum loco præmittendæ videntur, & problemata, quibus cer-<pb pagenum="338" xlink:href="017/01/354.jpg"/>ta methodus præ&longs;cribatur, ut pro in&longs;tituto corpora ip&longs;a gravia <lb/>eligantur, atque &longs;uis quæque locis di&longs;ponantur. </s> </p> <p type="main"> <s id="s.002550"><emph type="center"/>PROPOSITIO I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002551"><emph type="center"/><emph type="italics"/>Exce&longs;&longs;us &longs;ecantis cuju&longs;cumque anguli &longs;upra Radium, minor e&longs;t <lb/>Tangente eju&longs;dem anguli.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002552">SIt datus angulus quilibet DBC, ejus Tangens DC, &longs;ecans <lb/>BD, & exce&longs;&longs;us &longs;ecantis &longs;upra Radium DE. <!-- KEEP S--></s> <s id="s.002553">Dico DE <lb/><figure id="id.017.01.354.1.jpg" xlink:href="017/01/354/1.jpg"/><lb/>minorem e&longs;&longs;e Tangente DC. <!-- KEEP S--></s> <s id="s.002554">Ducatur <lb/>recta CE dato angulo &longs;ubten&longs;a faciens <lb/>angulos ad ba&longs;im æquales ex 5. lib. 1. ac <lb/>proinde acutos: igitur angulus DEC <lb/>complementum ad duos rectos e&longs;t ob­<lb/>tu&longs;us, & maximus in triangulo DEC, <lb/>ac propterea ex 19. lib. 1. maximum la­<lb/>tus e&longs;t, quod illi opponitur, nimirum <lb/>Tangens DC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002555"><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002556"><emph type="center"/><emph type="italics"/>Cuju&longs;libet anguli Tangens e&longs;t media proportionalis inter exce&longs;&longs;um <lb/>&longs;ecantis &longs;upra Radium, & aggregatum ex Radio & <lb/>&longs;ecante eju&longs;dem anguli.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002557">DAtus &longs;it idem angulus DBC, Tangens DC, exce&longs;&longs;us <lb/>&longs;ecantis DE: producatur recta DB u&longs;que in A, & e&longs;t <lb/>recta DA aggregatum ex Radio BA & &longs;ecante BD. <!-- KEEP S--></s> <s id="s.002558">Dico <lb/>Tangentem DC e&longs;&longs;e mediam proportionalem inter ED & <lb/>DA. <!-- KEEP S--></s> <s id="s.002559">Cùm enim ex 36. lib. 3. rectangulum &longs;ub ED & DA <lb/>æquale &longs;it quadrato, quod à Tangente CD de&longs;cribitur, per 17. <lb/>lib. 6. &longs;unt tres continuè proportionales ED, DC, DA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002560">Hinc &longs;equitur exce&longs;&longs;um &longs;ecantis &longs;upra Radium ad aggrega­<lb/>tum ex Radio & &longs;ecante habere Rationem duplicatam Ratio-<pb pagenum="339" xlink:href="017/01/355.jpg"/>nis, quam idem exce&longs;&longs;us habet ad Tangentem, hoc e&longs;t, &longs;e ha­<lb/>bere ut quadratum ED ad quadratum DC, ita ED ad DA, <lb/>igitar & dividendo ut quadratum ED ad differentiam quadra­<lb/>torum ED & DC, ita exce&longs;&longs;us ED ad Radij duplum EA, dif­<lb/>ferentiam inter ED & DA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002561"><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002562"><emph type="italics"/>Dato angulo, ad cujus &longs;ecantis exce&longs;&longs;um &longs;upra Radium &longs;ua <lb/>Tangens habet Rationem datam, cuju&longs;cumque anguli mino­<lb/>ris Tangens ad exce&longs;&longs;um &longs;uæ &longs;ecantis habet Rationem majo­<lb/>rem datâ; cuju&longs;cumque autem anguli majoris Tangens ad <lb/>exce&longs;&longs;um &longs;uæ &longs;ecantis habet Rationem minorem datâ Ratione.<emph.end type="italics"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002563">ANgulus DBC &longs;it datus, & illius Tangens DC ad DE ex­<lb/>ce&longs;&longs;um &longs;uæ &longs;ecantis habeat datam aliquam Rationem. <!-- KEEP S--></s> <s id="s.002564">Pri­<lb/>mò &longs;it minor angulus FBC. </s> <s id="s.002565">Dico ejus Tangentem FC ad &longs;uæ <lb/>&longs;ecantis exce&longs;&longs;um FZ habere majorem Rationem quàm DC ad <lb/>DE. <!-- KEEP S--></s> <s id="s.002566">Quia angulus CFB exterior major e&longs;t interno CDB ex <lb/>16. lib.1. fiat huic æqualis angulus CFG, eruntque ex 28.lib.1. <lb/>parallelæ lineæ DB & FG, & ex 29.lib.1. DBF & GFH al­<lb/>terni æquales: &longs;unt autem BHE & FHG æquales per 15. <lb/>lib.1. ut pote ad verticem; ergo & reliquus angulus BEH e&longs;t <lb/>reliquo angulo FGH æqualis. </s> <s id="s.002567">Similia itaque &longs;unt triangula, <lb/>& per 4. lib. 6. ut EB ad BH, ita GF ad FH: e&longs;t autem EB <lb/>major quàm BH (nam BH minor e&longs;t Radio BZ, cui æqualis <lb/>e&longs;t Radius BE) igitur & GF major e&longs;t quàm FH; ergo & mul­<lb/>tò major quàm FZ. </s> <s id="s.002568">Sed quoniam GF & ED &longs;unt parallelæ, & <lb/>triangula CFG, CDE &longs;unt æquiangula, ex 4. lib.6. eadem e&longs;t <lb/>Ratio CF ad FG, quæ e&longs;t CD ad DE: CF autem ad FG <lb/>majorem ex 8. lib. 5. habet minorem Rationem quàm ad FZ <lb/>minorem; ergo CF Tangens anguli minoris habet ad FZ <lb/>exce&longs;&longs;um &longs;uæ &longs;ecantis &longs;upra Radium, Rationem majorem quàm <lb/>CD ad DE. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002569">Secundò &longs;it angulus IBC major dato angulo DBC: Dico <lb/>illius Tangentem CI ad &longs;uæ &longs;ecantis exce&longs;&longs;um KI habere mi-<pb pagenum="340" xlink:href="017/01/356.jpg"/>norem Rationem, quàm CD ad DE. <!-- KEEP S--></s> <s id="s.002570">Quoniam externus an­<lb/>gulus CDB major e&longs;t interno CIB, fiat illi æqualis angulus <lb/>CIO, & lineæ CE productæ occurrat in O, linea IO, quæ <lb/>parallela e&longs;t lineæ BD; & &longs;unt anguli OIL & EBL alterni <lb/>æquales, quemadmodum & anguli ad verticem in L æquales <lb/>&longs;unt. </s> <s id="s.002571">Quapropter in triangulis IOL & EBL æquiangulis per <lb/>4. lib.6. ut LB ad BE, ita LI ad IO: e&longs;t autem LB major <lb/>quàm BE (nam LB major e&longs;t Radio BK) ergo etiam LI, & <lb/>multo magis KI major e&longs;t quàm IO. <!-- KEEP S--></s> <s id="s.002572">Sed ut CD ad DE ita <lb/>CI ad IO; ergo minor e&longs;t Ratio CI ad IK majorem, quàm <lb/>&longs;it CI ad IO minorem; ergo e&longs;t minor Ratio Tangentis CI ad <lb/>exce&longs;&longs;um &longs;uæ &longs;ecantis KI, quàm &longs;it Ratio CD ad DE. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002573"><emph type="center"/>PROPOSITIO IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002574"><emph type="center"/><emph type="italics"/>Differentia inter Tangentes duorum quorumlibet angulorum <lb/>major e&longs;t, quàm differentia inter eorum &longs;ecantes.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002575">SInt anguli BAC, BAD, eorum Tangentes BC & BD, <lb/>quarum differentia CD: angulorum &longs;ecantes AC & AD, <lb/><figure id="id.017.01.356.1.jpg" xlink:href="017/01/356/1.jpg"/><lb/>&longs;ecantium differentia (a&longs;&longs;umptâ AG <lb/>æquali ip&longs;i AC) e&longs;t DG. <!-- KEEP S--></s> <s id="s.002576">Dico CD <lb/>e&longs;&longs;e majorem quàm DG. <!-- KEEP S--></s> <s id="s.002577">Ducatur <lb/>recta CG, & e&longs;t triangulum CAG <lb/>i&longs;o&longs;celes, ideóque angulus CGA <lb/>acutus, & qui e&longs;t illi deinceps, CGD <lb/>obtu&longs;us, & maximus in triangulo <lb/>CGD: quare per 18. lib. 1. major e&longs;t <lb/>CD Tangentium differentia quàm <lb/>DG &longs;ecantium differentia. </s> </p> <p type="main"> <s id="s.002578"><emph type="center"/>PROPOSITIO V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002579"><emph type="center"/><emph type="italics"/>Ratio differentiæ Tangentium ad differentiam &longs;ecantium fit <lb/>&longs;emper minor.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002580">ESto anguli BAC Tangens BC, anguli BAK Tangens <lb/>BK; de&longs;cripto arcu COG, differentia &longs;ecantium e&longs;t KO, <pb pagenum="341" xlink:href="017/01/357.jpg"/>& Tangentium differentia e&longs;t CK. <!-- KEEP S--></s> <s id="s.002581">Item anguli BAD Tan­<lb/>gens BD, & de&longs;cripto arcu KH, differentia Tangentium <lb/>BK & BD e&longs;t KD, atque &longs;ecantium AK & AD differentia <lb/>e&longs;t HD. </s> <s id="s.002582">Dico majorem Rationem e&longs;&longs;e CK ad KO, quàm <lb/>KD ad DH. <!-- KEEP S--></s> <s id="s.002583">Ducantur rectæ CG & KH. <!-- KEEP S--></s> <s id="s.002584">In triangulis i&longs;o­<lb/>&longs;celibus CAG & KAH, anguli ad ba&longs;im CG minores &longs;unt <lb/>angulis ad ba&longs;im KH, quia angulus CAG major e&longs;t angulo <lb/>KAH: quapropter angulo CGA fiat æqualis angulus IHA. <lb/><!--neuer Satz-->Cum itaque ex 28.lib.I. IH & CG &longs;int parallelæ, per 2.lib.6. <lb/>ut CI ad HG, hoc e&longs;t ad KO, ita ID ad DH: atqui CK ma­<lb/>jor e&longs;t quàm CI; ergo major e&longs;t Ratio CK ad KO quàm CI <lb/>ad KO ex 8.lib.5. hoc e&longs;t quàm ID ad DH. <!-- KEEP S--></s> <s id="s.002585">Sed ID e&longs;t ma­<lb/>jor quàm KD; ergo per 8.lib. 5. major e&longs;t Ratio ID ad DH, <lb/>quàm KD ad DH; ergo multò major e&longs;t Ratio CK ad KO, <lb/>quàm KD ad DH. <!-- KEEP S--></s> <s id="s.002586">Idem de cæteris con&longs;equentibus angulis <lb/>nec di&longs;&longs;imili methodo demon&longs;trari poterit, minorem &longs;cilicet <lb/>fieri Rationem differentiæ Tangentium ad differentiam &longs;e­<lb/>cantium. </s> </p> <p type="main"> <s id="s.002587"><emph type="center"/>PROPOSITIO VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002588"><emph type="center"/><emph type="italics"/>Dato Radio, & datâ Ratione Tangentis ad exce&longs;&longs;um &longs;ecantis, <lb/>invenire Tangentem & &longs;ecantem, earúmque angulum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002589">DAtus Radius &longs;it B, data Ratio Tangentis ad exce&longs;&longs;um &longs;e­<lb/>cantis &longs;uprà Radium &longs;it R ad S. <!-- KEEP S--></s> <s id="s.002590">Oportet Tangentem <lb/>ip&longs;am atque &longs;ecantem inve­<lb/><figure id="id.017.01.357.1.jpg" xlink:href="017/01/357/1.jpg"/><lb/>nire. </s> <s id="s.002591">Tangens e&longs;to A: ut R <lb/>ad S ita A ad (A in S/R) exce&longs;&longs;um <lb/>&longs;ecantis &longs;upra Radium; igitur <lb/>&longs;ecans integra e&longs;t B + (A in S/R); <lb/>hujus quadratum e&longs;t B quad. <lb/>+ (2 B in A in S/R) + (A quad. </s> <s id="s.002592">in S quad./R quadr.)quod <lb/>ex 47. lib. 1. æquale e&longs;t qua­<lb/>dratis Radij & Tangentis &longs;i­<lb/>mul, hoc e&longs;t B quad. + A quad. </s> <s id="s.002593">Utrinque dempto B quad. </s> <lb/> <s id="s.002594">tum omnibus per A divi&longs;is, deinde omnibus ductis per R quad. <pb pagenum="342" xlink:href="017/01/358.jpg"/>demum factâ Antithe&longs;i A in R quad. </s> <s id="s.002595">—— A in S quad. </s> <s id="s.002596">æqua­<lb/>tur 2 S in B in R. <!-- KEEP S--></s> <s id="s.002597">Quare revocatâ ad Analogiam æquatione, <lb/>e&longs;t 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/>Tangentem quæ&longs;itam. </s> <s id="s.002600">Tum fiat ut R ad S ita A inventa ad <lb/>aliud, & erit exce&longs;&longs;us &longs;ecantis, qui additus Radio B dabit quæ­<lb/>&longs;itam &longs;ecantem. </s> </p> <p type="main"> <s id="s.002601">Sit R 3, S 2: horum quadratorum 9 & 4 differentia e&longs;t 5; <lb/>duplum rectangulum &longs;ub R & S e&longs;t 12. Igitur ut 5 ad 12, ita B <lb/>Radius 100000 ad 240000 Tangentem gr. <!-- REMOVE S-->67. 22′.48′. </s> <s id="s.002602">Iterum <lb/>ut 3 ad 2 ita 240000 ad 160000 exce&longs;&longs;um &longs;ecantis; Igitur ad­<lb/>dito Radio, Secans quæ&longs;ita e&longs;t 260000; quæ etiam in Canone <lb/>re&longs;pondet eidem angulo. </s> </p> <p type="main"> <s id="s.002603">Itaque generatim loquendo, fiat ut differentia inter quadra­<lb/>ta terminorum datæ Rationis ad rectangulum bis &longs;ub ii&longs;dem <lb/>terminis comprehen&longs;um, ita datus Radius ad aliud, & prove­<lb/>niet Tangens quæ&longs;ita; quæ habita facile dabit &longs;ecantis exce&longs;­<lb/>&longs;um in Ratione datâ. </s> </p> <p type="main"> <s id="s.002604">Quòd &longs;i rem Geometricè perficere velis, circà majorem Ra­<lb/>tionis datæ terminum R de&longs;cribe &longs;emicirculum, & in eo ac­<lb/>commoda minorem Rationis terminum S; nam linea T dabit <lb/>quadratum, quod e&longs;t differentia quadratorum ex R & ex S, ut <lb/>e&longs;t manife&longs;tum ex eo, quod angulus in &longs;emicirculo e&longs;t rectus <lb/>per 31.lib.3.& ex 47 lib.1. quadratum unius lateris circa rectum <lb/>e&longs;t differentia quadratorum hypothenu&longs;æ & reliqui lateris. </s> <lb/> <s id="s.002605">Deinde inter alterutrum terminorum duplicatum, & reliquum <lb/>terminum quære mediam proportionalem, & &longs;it V potens qua­<lb/>dratum æquale duplo rectangulo &longs;ub terminis datis. </s> <s id="s.002606">Quoniam <lb/>verò ex 20.lib. 6. quadrata &longs;unt in duplicatâ Ratione laterum, <lb/>& T quadratum ad V quadratum e&longs;t in duplicatâ Ratione T <lb/>ad V; inveniatur tertia proportionalis X. <!-- KEEP S--></s> <s id="s.002607">Demum ut T ad X <lb/>ita fiat B ad Z, quæ e&longs;t quæ&longs;ita Tangens, & ad angulum rectum <lb/>con&longs;tituta cum Radio B dabit hypothenu&longs;am &longs;ecantem quæ&longs;i­<lb/>tam, quæ cum Radio con&longs;tituet quæ&longs;itum angulum. </s> </p> <p type="main"> <s id="s.002608">Vel etiam ex corollario prop.2. fiat ut differentia quadrato­<lb/>rum ex R & ex S ad S quadratum, ita duplum Radij B ad exce&longs;­<lb/>&longs;um &longs;ecantis: deinde hic exce&longs;&longs;us inventus ad Tangentem <lb/>quæ&longs;itam fiat ut S ad R; & &longs;umma ex dato Radio atque exce&longs;­<lb/>&longs;u invento dabit quæ&longs;itam &longs;ecantem. </s> </p> <pb pagenum="343" xlink:href="017/01/359.jpg"/> <p type="main"> <s id="s.002609"><emph type="center"/>PROPOSITIO VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002610"><emph type="center"/><emph type="italics"/>Datá Tangente communi duorum circulorum inæqualium, & datis <lb/>Rationibus exce&longs;&longs;um Secantium ad eandem Tangentem, <lb/>invenire Circulorum Radios.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002611">SIt &longs;uper lineam CD indefinitam erecta ad perpendiculum <lb/>recta AB, quam in B oporteat tangere duos circulos inæ­<lb/>quales, ita ut &longs;it Tangens <lb/><figure id="id.017.01.359.1.jpg" xlink:href="017/01/359/1.jpg"/><lb/>duorum angulorum inæ­<lb/>qualium, exce&longs;&longs;us autem <lb/>&longs;ecantis unius &longs;it ad da­<lb/>tam Tangentem ut E ad <lb/>G, alterius verò &longs;ecantis <lb/>exce&longs;&longs;us &longs;it ad eandem ut <lb/>F ad G: & huju&longs;modi <lb/>circulorum &longs;emidiame­<lb/>tros invenire oporteat. </s> </p> <p type="main"> <s id="s.002612">Fiat ut G ad E ita AB <lb/>data ad H; & ut H ad <lb/>AB ita AB ad MS, ex <lb/>quâ dematur MO ip&longs;i H <lb/>æqualis, reliquæ OS &longs;e­<lb/>mi&longs;&longs;i RS æqualis &longs;uma­<lb/>tur BD pro Radio circuli BL. </s> <s id="s.002613">Item fiat ut G ad F ita AB <lb/>data ad I; & ut I ad AB ita AB ad NT, ex quâ dematur <lb/>NP æqualis ip&longs;i I, & reliquæ PT &longs;emi&longs;&longs;i VT æqualis &longs;tatua­<lb/>tur BC &longs;emidiameter circuli BK. </s> <s id="s.002614">Junctis CA, & DA erunt <lb/>exce&longs;&longs;us &longs;ecantium &longs;uprà &longs;uos Radios ad Tangentem, videlicet <lb/>KA & LA ad AB in datis Rationibus. </s> </p> <p type="main"> <s id="s.002615">Quia enim recta TP &longs;ecta e&longs;t bifariam in V, & adjecta e&longs;t <lb/>illi PN, per 6. lib.2. quadratum NV e&longs;t æquale quadrato VT <lb/>(hoc e&longs;t quadrato CB) unà cum rectangulo TNP: huic au­<lb/>tem rectangulo, ex 17. lib. 6. æquale e&longs;t quadratum AB, quæ <lb/>ex con&longs;tructione e&longs;t media proportionalis inter PN, hoc e&longs;t I, <lb/>& NT. </s> <s id="s.002616">At ii&longs;dem quadratis CB & BA &longs;imul &longs;umptis æquale <pb pagenum="344" xlink:href="017/01/360.jpg"/>e&longs;t quadratum CA ex 47. lib.1. igitur quadratum CA æquatur <lb/>quadrato NV, & linea CA æqualis e&longs;t lineæ NV. </s> <s id="s.002617">Sunt au­<lb/>tem VP & CK æquales (nam & æquales &longs;unt lineis VT & <lb/>CB) ergo etiam KA reliqua æqualis e&longs;t reliquæ PN, hoc <lb/>e&longs;t I. <!-- KEEP S--></s> <s id="s.002618">Cum itaque I ad AB &longs;it ut F ad G ex con&longs;tructione, <lb/>etiam KA ad AB e&longs;t in eâdem datâ Ratione F ad G. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002619">Nec di&longs;&longs;imili methodo utendum erit ad o&longs;tendendum LA <lb/>ad AB e&longs;&longs;e in datâ Ratione E ad G: id quod indica&longs;&longs;e &longs;uffi­<lb/>ciat, nec pluribus e&longs;t opus. </s> <s id="s.002620">Quare CB & DB &longs;unt quæ&longs;ito­<lb/>rum circulorum &longs;emidiametri. </s> </p> <p type="main"> <s id="s.002621"><emph type="center"/>PROPOSITIO VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002622"><emph type="italics"/>Datis duobus inæqualibus circulis &longs;e contingentibus in B, da­<lb/>ti&longs;que eorum Radiis CB & DB, invenire Tangentem com­<lb/>munem BA, ad quam &longs;ecantium exce&longs;&longs;us habeant datas Ra­<lb/>tiones E ad G, & F ad G.<emph.end type="italics"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002623">OPortet &longs;ecantis exce&longs;&longs;um, qui ad Tangentem habet majo­<lb/>rem Rationem, quàm alter exce&longs;&longs;us; pertinere ad mino­<lb/>rem circulum; qui verò minorem Rationem habet, pertinere <lb/>ad majorem circulum. </s> <s id="s.002624">Cum enim rectangula &longs;ub exce&longs;&longs;ibus <lb/>& aggregatis &longs;uarum &longs;ecantium &longs;uorúmque Radiorum &longs;int in­<lb/>ter &longs;e æqualia, ut pote ex 36. lib.3. eidem Tangentis quadrato <lb/>æqualia, erit per 16.lib. 6. ut exce&longs;&longs;us &longs;ecantis majoris circuli <lb/>ad exce&longs;&longs;um minoris, ita aggregatum ex &longs;ecante & Radio mi­<lb/>noris ad aggregatum ex &longs;ecante & Radio majoris. </s> <s id="s.002625">Sicut ergo <lb/>eadem Tangens habet majorem Rationem ad Radium minoris <lb/>circuli quàm ad Radium majoris, &longs;ubtendítque majorem angu­<lb/>lum in circulo minori quàm in majori; ita &longs;uæ &longs;ecantis exce&longs;&longs;us <lb/>habet majorem Rationem ad eandem Tangentem, quàm ex­<lb/>ce&longs;&longs;us &longs;ecantis 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/>ad circulum minorem. </s> <s id="s.002627">Fiat ut F ad G ita G ad QX, ex quâ <lb/>dematur QZ æqualis ip&longs;i F. <!-- KEEP S--></s> <s id="s.002628">Tum fiat ut XZ ad ZQ, ita mi­<lb/>noris Radij duplum TP ad PN: & inter PN & NT invenia­<lb/>tur media proportionalis BA, quam ex B ad perpendiculum <pb pagenum="345" xlink:href="017/01/361.jpg"/>erectam jungat cum centro C rectâ CA: nam KA ad Tan­<lb/>gentem AB habet datam Rationem F ad G. <!-- KEEP S--></s> <s id="s.002629">Cùm enim ea­<lb/>dem AB, quæ ex con&longs;tructione e&longs;t media inter PN & NT, &longs;it <lb/>etiam ex 36. lib.3. & 17. lib.6. Media inter KA & ACB, & <lb/>extremarum NT & ACB exce&longs;&longs;us &longs;upra &longs;ibi re&longs;pondentes <lb/>extremas PN & KA &longs;int ex con&longs;tructione æquales (&longs;unt &longs;cili­<lb/>cet PT & KCB duplum Radij CB) etiam ip&longs;æ extremæ &longs;unt <lb/>æquales, nimirum NT æqualis ip&longs;i ACB, & PN, æqualis <lb/>KA. <!-- KEEP S--></s> <s id="s.002630">Atqui ut XZ ad ZQ, ita ex con&longs;tructione TP ad PN, <lb/>& componendo atque convertendo ut ZQ ad QX ita PN ad <lb/>NT; ergo etiam ut ZQ ad QX ita KA ad ACB. </s> <s id="s.002631">Quare &longs;i­<lb/>cuti ZQ ad QX e&longs;t duplicata Rationis F ad G ex con&longs;tructio­<lb/>ne, etiam KA ad ACB e&longs;t eju&longs;dem Rationis F ad G duplica­<lb/>ta; ergo KA ad mediam AB, hoc e&longs;t Exce&longs;&longs;us &longs;ecantis ad Tan­<lb/>gentem, e&longs;t ut F ad G. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002632">Eádem methodo fiat ut E ad G ita G ad Y <emph type="italics"/>a,<emph.end type="italics"/> ex quâ dema­<lb/>tur Y <emph type="italics"/>b<emph.end type="italics"/> æqualis ip&longs;i E: & fiat ut <emph type="italics"/>ab<emph.end type="italics"/> ad <emph type="italics"/>b<emph.end type="italics"/> Y, ita Radij majoris <lb/>BD duplum SO ad OM; atque inter OM & MS erit media <lb/>proportionalis eadem AB: &longs;imilique ratiocinatione o&longs;tendetur <lb/>exce&longs;&longs;um LA ad Tangentem AB e&longs;&longs;e in datâ Ratione E ad G. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002633">Ut in praxim res faciliùs deduci queat, exemplo illu&longs;tretur. </s> <lb/> <s id="s.002634">Sit Radius minor CD 12, F ad G ut 16 ad 35: inveniatur his <lb/>tertia proportionalis QX (76 9/16). Dematur F 16, remanet XZ <lb/>(60 9/16). Fiat ut (60 9/16) ad 16, ita Radij duplum TP 24 ad <lb/>PN 6 2/5 proximè. </s> <s id="s.002635">E&longs;t ergo NT 30 2/5. Inter 6 2/5 & 30 2/5 me­<lb/>dia e&longs;t 14. </s> </p> <p type="main"> <s id="s.002636">Item &longs;it Radius major BD 18, E ad G ut 12 ad 35: inve­<lb/>niatur his tertia proportionalis Y <emph type="italics"/>a<emph.end type="italics"/> 102 1/2, & auferatur E 12, <lb/>remanet <emph type="italics"/>a b<emph.end type="italics"/> 90 1/2. Fiat ut 90 1/2 ad 12, ita Radij duplum SO 36 <lb/>ad OM 4 7/9. E&longs;t ergo MS 40 7/9. Inter 4 7/9 & 40 7/9 e&longs;t media <lb/>proportionalis 14: in his autem exemplis neglectæ &longs;unt <lb/>fractiunculæ. </s> </p> <pb pagenum="346" xlink:href="017/01/362.jpg"/> <p type="main"> <s id="s.002637"><emph type="center"/>PROPOSITIO IX.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002638"><emph type="italics"/>Si duorum circulorum &longs;e exteriùs contingentium centra jungat <lb/>recta linea, & ab unius centro ad alterius convexam peri­<lb/>pheriam rectæ ducantur, &longs;ubten&longs;a arcus ab&longs;ci&longs;&longs;i major e&longs;t <lb/>quàm differentia linearum angulum in illo centro con&longs;ti­<lb/>tuentium.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.002639">DUorum circulorum centra &longs;int A & B, qui &longs;e tangant in C, <lb/>& jungat centra recta AB. <!-- KEEP S--></s> <s id="s.002640">Ex centro A in alterius con­<lb/><figure id="id.017.01.362.1.jpg" xlink:href="017/01/362/1.jpg"/><lb/>vexam peripheriam ducatur <lb/>recta AD ab&longs;cindens arcum <lb/>CD. <!-- KEEP S--></s> <s id="s.002641">Dico linearum AD & <lb/>AC angulum in centro A <lb/>con&longs;tituentium differentiam <lb/>ED minorem e&longs;&longs;e &longs;ubtensâ <lb/>CD. <!-- KEEP S--></s> <s id="s.002642">Quia ex 20. lib. 1. duæ <lb/>lineæ AC & CD &longs;imul majores &longs;unt rectâ AD; auferantur <lb/>AC & AE æquales, remanet CD major quàm ED. <!-- KEEP S--></s> <s id="s.002643">Simili <lb/>ratione CI major e&longs;t quam IF. & &longs;i &longs;umatur angulus IAD, <lb/>etiam ID major e&longs;t quàm DH differentia inter AI & AD, <lb/>quia in triangulo AID duo latera AI & ID majora &longs;unt reli­<lb/>quo DA, demptí&longs;que æqualibus AI & AH remanet ID ma­<lb/>jor quàm DH. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002644"><emph type="center"/>PROPOSITIO X.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002645"><emph type="italics"/>Si duo circuli &longs;e exteriùs contingant, & in uno æquales arcus <lb/>&longs;umantur, ad quorum extremitates ducantur rectæ à centro <lb/>alterius circuli; differentia &longs;inuum arcûs &longs;impli & dupli ad <lb/>differentiam Exce&longs;&longs;uum harum rectarum &longs;upra &longs;uum Radium <lb/>habet minorem Rationem, quàm &longs;inus arcûs &longs;impli ad Exce&longs;­<lb/>&longs;um lineæ ad ip&longs;um ductæ.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.002646">SInt duo circuli, quorum centra A & B, &longs;e contingentes in C, <lb/>&longs;umantur æquales arcus CI & ID, ad quos ex centro B <pb pagenum="347" xlink:href="017/01/363.jpg"/>ducantur rectæ BI & BD &longs;ecantes circulum CE in F & E: <lb/>Radium B excedunt exce&longs;&longs;ibus FI & ED, qui ex 8. lib. 3. in­<lb/>æquales &longs;unt, & ma­<lb/><figure id="id.017.01.363.1.jpg" xlink:href="017/01/363/1.jpg"/><lb/>jor e&longs;t ED quàm FI <lb/>differentiâ KD. </s> <s id="s.002647">Ar­<lb/>cuum &longs;ubten&longs;æ CI <lb/>& ID æquales &longs;unt, <lb/>&longs;inuum IH & DG <lb/>differentia e&longs;t LD. <!-- KEEP S--></s> <lb/> <s id="s.002648">Dico majorem Ra­<lb/>tionem e&longs;&longs;e HI ad IF, quàm LD ad DK. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002649">Primò ducantur rectæ EF, KI: EF autem producatur ita, <lb/>ut occurrat rectæ DI productæ in O. <!-- KEEP S--></s> <s id="s.002650">Quia triangula BFE & <lb/>BIK &longs;unt i&longs;o&longs;celia, & angulus BEF æqualis e&longs;t angulo BKI, <lb/>rectæ EO & KI ex 28.lib.1. &longs;unt parallelæ: igitur ex 2 lib. 6. <lb/>in triangulo DOE ut DI ad IO, ita DK ad KE: Atqui DI <lb/>major e&longs;t quàm IO, ergo etiam DK major quàm KE. </s> <s id="s.002651">Proba­<lb/>tur autem DI majorem e&longs;&longs;e quàm IO; quia DI æqualis e&longs;t <lb/>ip&longs;i CI ex hypothe&longs;i; punctum verò O e&longs;t extra circulum CE, <lb/>quem linea EFO &longs;ecat: ergo linea EF producta occurrit li­<lb/>neæ IC citrà punctum C in S. <!-- KEEP S--></s> <s id="s.002652">Sed quoniam angulus BEF e&longs;t <lb/>acutus, qui e&longs;t illi deinceps DEO e&longs;t obtu&longs;us; ergo per 16. <lb/>lib.1. externus DOS multo magis e&longs;t obtu&longs;us: ergo per 19 <lb/>lib.1. major e&longs;t IS quàm IO, ergo multò major e&longs;t IC quàm <lb/>IO, hoc e&longs;t ID major e&longs;t quàm IO. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002653">Deinde angulus MCI major e&longs;t angulo NID, majori enim <lb/>arcui MDI ille in&longs;i&longs;tit, hic autem minori ND ex 33. lib. 6: <lb/>triangula verò HIC & LDI rectangula æquales habent hy­<lb/>pothenu&longs;as, hoc e&longs;t Radios CI & ID, ergo majoris anguli <lb/>HCI major e&longs;t &longs;inus HI; minoris verò anguli LID minor e&longs;t <lb/>&longs;inus LD. <!-- KEEP S--></s> <s id="s.002654">Igitur ex 8. lib.5. HI major ad KE, hoc e&longs;t ad IF, <lb/>habet majorem Rationem quàm ad eandem KE habeat LD <lb/>minor: & eadem LD habet minorem Rationem ad DK ma­<lb/>jorem quàm ad KE minorem: Ergo HI ad IF majorem habet <lb/>Rationem, quàm LD ad DK. <pb pagenum="348" xlink:href="017/01/364.jpg"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002655"><emph type="center"/>CAPUT XII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002656"><emph type="center"/><emph type="italics"/>Præponderatio & Æquilibritas gravium fune <lb/>&longs;u&longs;pen&longs;orum con&longs;ideratur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002657">PRopo&longs;itum e&longs;t lib.2. capit. </s> <s id="s.002658">5. Experimentum, cujus hîc <lb/>&longs;ymptomata explicanda, cau&longs;am afferendo omninò con&longs;o­<lb/>nam iis, quæ &longs;æpiùs inculcata &longs;unt. </s> <s id="s.002659">Funiculi extremitatibus al­<lb/>ligantur pondera prorsùs æqualia; tùm claviculis duobus à &longs;e <lb/>invicem aliquo intervallo disjunctis, &longs;ed in eádem horizontali <lb/>lineâ con&longs;titutis (exqui&longs;itè tamen, quoad ejus fieri poterit ro­<lb/>tundis atque politis, ne &longs;uâ a&longs;peritate motui impedimento &longs;int) <lb/>funiculus imponitur. </s> <s id="s.002660">Deinde tertium pondus a&longs;&longs;umitur duo­<lb/>bus illis &longs;imul acceptis levius, aut &longs;ingulis illis æquale, aut etiam <lb/>illis minus, & funiculo inter utrumque claviculum adnectitur: <lb/>hoc &longs;ibi dimi&longs;&longs;um ita duobus illis ponderibus, quæ ob gravita­<lb/>tis æqualitatem &longs;ibi mutuo ni&longs;u ob&longs;i&longs;tebant, ne moverentur, <lb/>prævalet, ut ip&longs;um de&longs;cendens vi &longs;uæ gravitatis cogat utrum­<lb/>que illud a&longs;cendere. </s> <s id="s.002661">Id quod admiratione carere non pote&longs;t, <lb/>cum duo majora pondera, &longs;uum æqualem conatum &longs;ingula vi­<lb/>ci&longs;&longs;im elidentia, conjunctis viribus minori gravitati præ&longs;tare <lb/>non valeant. </s> </p> <p type="main"> <s id="s.002662">Funiculo CABD jungantur æquales gravitates C & D ex <lb/>claviculis A & B pendentes, quæ æqualiter deor&longs;um conniten­<lb/><figure id="id.017.01.364.1.jpg" xlink:href="017/01/364/1.jpg"/><lb/>tes, &longs;ibique æqualiter repugnantes <lb/>ne a&longs;cendant, quie&longs;cunt. </s> <s id="s.002663">Adnecta­<lb/>tur in E pondus: huic etiam&longs;i mi­<lb/>nori illæ gravitates C & D omnino <lb/>ob&longs;i&longs;tere non po&longs;&longs;unt, quin ex E <lb/>de&longs;cendat in F ex.gr. <!-- REMOVE S-->& funicu­<lb/>lum trahens cogat illas a&longs;cendere <lb/>C quidem in I, D verò in K. <!-- KEEP S--></s> <s id="s.002664">Qua­<lb/>propter funiculo EBD æqualis e&longs;t funiculus FBK, & funicu­<lb/>lo EAC æqualis e&longs;t funiculus FAI: cum autem rectæ BE & <lb/>BG æquales &longs;int (nam centro B, intervallo BE de&longs;criptus e&longs;t <pb pagenum="349" xlink:href="017/01/365.jpg"/>arcus) his ablatis, BD æquatur ip&longs;i FG plus BK; & demptâ <lb/>communi BK, remanet GF æqualis ip&longs;i DK. <!-- KEEP S--></s> <s id="s.002665">Eadem ratione <lb/>HF o&longs;tenditur æqualis ip&longs;i CI. <!-- KEEP S--></s> <s id="s.002666">E&longs;t igitur men&longs;ura motús pon­<lb/>derum C & D a&longs;cendentium HFG, ponderis verò intermedij <lb/>de&longs;cendentis EF. <!-- KEEP S--></s> <s id="s.002667">At ex prop.1. capitis &longs;uperioris Tangens EF <lb/>major e&longs;t &longs;ecantis BF exce&longs;&longs;u GF, item &longs;ecantis AF exce&longs;&longs;u <lb/>HF: contingit autem aliquam Tangentem majorem e&longs;&longs;e utro­<lb/>que exce&longs;&longs;u &longs;imul &longs;umpto: pote&longs;t igitur gravitas minor velociùs <lb/>de&longs;cendens præ&longs;tare utrique ponderi tardiùs a&longs;cendenti. </s> </p> <p type="main"> <s id="s.002668">Quamdiu itaque &longs;patium de&longs;cendentis per Tangentem ma­<lb/>jus e&longs;t &longs;patio a&longs;cendentium, quod metitur exce&longs;&longs;us &longs;ecantium, <lb/>ita ut Ratio motûs de&longs;cendentis ad motum a&longs;cendentium major <lb/>e&longs;&longs;e po&longs;&longs;it Ratione, quam habent pondera extrema ad pondus <lb/>intermedium; hoc minore illa majora præponderantur. </s> <s id="s.002669">Ubi ve­<lb/>rò eò ventum &longs;it, ut jam neutra Ratio alteri præ&longs;tet, tunc pon­<lb/>dera &longs;ub&longs;i&longs;tunt, & quies e&longs;t. </s> <s id="s.002670">Si demùm ponderi intermedio <lb/>pondus addatur, vel vis aliqua inferatur ponderis vicem &longs;ubiens, <lb/>utique adhuc de&longs;cendit, quia Ratio ponderum extremorum ad <lb/>pondus intermedium auctum facta e&longs;t minor; &longs;ed &longs;ublato hoc <lb/>ponderis additamento, illa extrema majorem habent Rationem <lb/>ad pondus intermedium, quàm po&longs;&longs;it e&longs;&longs;e motuum reciprocè <lb/>&longs;umptorum Ratio; ac proinde illa de&longs;cendentia hoc tanti&longs;per <lb/>elevant, dum fiat Rationum æqualitas. </s> </p> <p type="main"> <s id="s.002671">Non e&longs;t autem hîc opus ea, quæ uberiùs &longs;uperiore libro ex­<lb/>plicata &longs;unt, replicare, videlicet, gravium re&longs;i&longs;tentiam, ne mo­<lb/>veantur, non e&longs;&longs;e attendendam penès ip&longs;am gravitatem dum­<lb/>taxat, verùm etiam motús, qui &longs;itum ip&longs;um atque po&longs;itionem <lb/>con&longs;equeretur, velocitate aut tarditate dimetiendam; hanc ve­<lb/>rò unius tarditatem cum alterius velocitate comparari non po&longs;­<lb/>&longs;e ni&longs;i ex longitudine &longs;patiorum, quæ utrumque codem tempo­<lb/>ris intervallo percurreret. </s> <s id="s.002672">Ex quo manife&longs;tâ con&longs;equutione con­<lb/>ficitur &longs;atis e&longs;&longs;e, &longs;i &longs;patiorum inæqualitas aut æqualitas o&longs;tenda­<lb/>tur; ut præponderatio aut æquilibritas innote&longs;cat: ac propterea <lb/>&longs;atis e&longs;t hîc &longs;ecantium exce&longs;&longs;us cum Tangente comparare; hæc <lb/>enim ponderis intermedij, illi ponderum extremorum motum <lb/>definiunt. </s> </p> <p type="main"> <s id="s.002673">Quapropter animum in rem ip&longs;am attentiùs intendentes ob­<lb/>&longs;ervamus de&longs;cendentis ponderis intermedij funiculum BFA <pb pagenum="350" xlink:href="017/01/366.jpg"/>cum horizontali lineá BA angulos con&longs;tituere ad B & A pri<lb/>mum quidem acuti&longs;&longs;imos, deinde majores & majores; ac <lb/>propterea Tangentis ad Exce&longs;&longs;um &longs;ecantis Rationem &longs;emper mi­<lb/>nui ex propo&longs;. </s> <s id="s.002674">3. ideóque tandem ad eam deveniri Rationem, <lb/>quæ non &longs;it major Ratione ponderum reciprocè &longs;umptorum. </s> <lb/> <s id="s.002675">Quid igitur mirum, &longs;i tandem fiat quies, ubi non e&longs;t Ratio­<lb/>num inæqualitas? </s> <s id="s.002676">Vici&longs;&longs;im autem quia ponderum certa e&longs;t Ra­<lb/>tio; certa e&longs;t etiam Ratio Tangentis ad Exce&longs;&longs;um &longs;ecantis certi <lb/>cuju&longs;dam anguli; igitur ex eádem prop.3. minoris anguli Tan­<lb/>gens ad Exce&longs;&longs;um &longs;uæ &longs;ecantis majorem habet Rationem, quam <lb/>&longs;it Ratio ponderum reciprocè: ideóque pondus in E con&longs;titu­<lb/>tum po&longs;itionem habens, ex quâ aliquis major motus deor&longs;um <lb/>con&longs;equi pote&longs;t, quàm a&longs;cendant extrema pondera, de&longs;cendit, <lb/>& &longs;uperat eorum re&longs;i&longs;tentiam. </s> <s id="s.002677">Sed quoniam &longs;uppo&longs;ita extre­<lb/>mis ponderibus manu ita elevare ea po&longs;&longs;umus, ut pondus inter­<lb/>medium de&longs;cendens funiculumque intendens con&longs;tituat ad B <lb/>& A angulos, quorum communis Tangens EF habeat ad Ex­<lb/>ce&longs;&longs;um &longs;ecantium HFG Rationem minorem, quàm &longs;it reci­<lb/>procè Ratio ponderum extremorum ad pondus intermedium, <lb/>&longs;atis con&longs;tat, cur illa extrema præponderent, cùm & plus gra­<lb/>vitatis & majora momenta, hoc e&longs;t propen&longs;ionem ad majorem <lb/>motum, obtineant. </s> <s id="s.002678">Quamvis enim ex prop.4. differentia inter <lb/>Tangentes duorum in eodem circulo arcuum inæqualium ma­<lb/>jor &longs;emper &longs;it differentia, quæ inter eorumdem &longs;ecantes inter­<lb/>cedit; quia tamen ex prop.5. Ratio hæc &longs;emper fit minor, quò <lb/>anguli augentur, idcircò &longs;i Tangens &longs;it duobus circulis com­<lb/>munis, fieri pote&longs;t, ut utriu&longs;que circuli &longs;ecantium differentiæ <lb/>&longs;imul &longs;umptæ majores &longs;int ipsá Tangente, vel &longs;altem Tangens <lb/>ad illas &longs;imul &longs;umptas eam habeat Rationem, quæ minor &longs;it Ra­<lb/>tione ponderum reciprocè. </s> </p> <p type="main"> <s id="s.002679">Et ut veritas exemplis ante omnium oculos po&longs;ita nullum du­<lb/>bitationi locum relinquat, data &longs;it Ratio extremorum ponde­<lb/>rum ad pondus intermedium, & inquiratur Tangens &longs;imilem <lb/>Rationem habens ad utriu&longs;que &longs;ecantis Exce&longs;&longs;um: intelligatur <lb/>autem hîc facilitatis gratià punctum E omninò æqualiter <lb/>di&longs;tans ab A & B ita, ut æquales etiam &longs;int &longs;ecantium exce&longs;&longs;us <lb/>HF & GF. </s> <s id="s.002680">Et primò quidem ponatur pondus medium æqua­<lb/>le &longs;ingulis extremis. </s> <s id="s.002681">E&longs;t igitur quæ&longs;ita Ratio dupla Tangentis <pb pagenum="351" xlink:href="017/01/367.jpg"/>EF ad Exce&longs;&longs;uum &longs;ummam HFG, cujus &longs;ummæ &longs;emi&longs;&longs;is e&longs;t <lb/>GF, atque adeò Ratio Tangentis EF ad GF e&longs;t quadrupla, <lb/>hoc e&longs;t ut 4 ad 1. </s> <s id="s.002682">Ergo ex corollar. prop. 2. ut quadratum Ex­<lb/>ce&longs;sus ad differentiam inter quadrata Exce&longs;sûs & Tangentis <lb/>(&longs;unt autem quadrata 1 & 16) hoc e&longs;t ut 1 ad 15, ita exce&longs;&longs;us <lb/>&longs;ecantis ad duplum Radij BE. </s> <s id="s.002683">Quare Exce&longs;&longs;us &longs;ecantis ad Ra­<lb/>dium BE e&longs;t ut 1 ad 7 1/2. Po&longs;ito igitur Radio BE 100000, Ex­<lb/>ce&longs;&longs;us &longs;ecantis GF e&longs;t 13333 1/3, & ejus quadrupla Tangens EF <lb/>53333 1/3 dat angulum EBF gr. <!-- REMOVE S-->28. 4′. </s> <s id="s.002684">21″, cujus &longs;ecans BF e&longs;t <lb/>113333 1/3. Di&longs;tantia AB &longs;tatuatur pedum quatuor, hoc e&longs;t di­<lb/>gitorum 64: e&longs;t BE dig. </s> <s id="s.002685">32. Igitur ut BE 100000 ad GF <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/>ce&longs;sus quadrupla erit de&longs;cen&longs;us EF dig. </s> <s id="s.002688">17, a&longs;cen&longs;us verò DK <lb/>aut CI dig. </s> <s id="s.002689">4 1/4 &longs;inguli, & ambo &longs;imul 8 1/2. In omnibus igitur <lb/>angulis minoribus angulo gr. <!-- REMOVE S-->28. 4′. </s> <s id="s.002690">21″. <!-- KEEP S--></s> <s id="s.002691">Ratio Tangentis ad <lb/>Exce&longs;&longs;uum &longs;ecantium &longs;ummam major e&longs;t Ratione duplâ, quæ e&longs;t <lb/>ponderum Ratio, in angulis verò majoribus minor e&longs;t Ratione <lb/>duplâ: ac propterea ibi pondus intermedium &longs;uperat extrema, <lb/>hic &longs;uperatur ab illis, & quie&longs;cunt in invento angulo gr. <!-- REMOVE S-->28. <lb/>4′. </s> <s id="s.002692">21″. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002693">Generaliter autem ut invenias, quantum a&longs;cendere po&longs;&longs;int <lb/>extrema pondera vi ponderis medij de&longs;cendentis, &longs;it nota Ra­<lb/>tio ponderum: tùm minoris termini Rationis datæ &longs;emi&longs;&longs;em ac­<lb/>cipe (quia unicus Exce&longs;&longs;us hîc &longs;umitur, & pondus medium <lb/>æquali intervallo di&longs;tat ab A & B) & hujus &longs;emi&longs;&longs;is quadratum <lb/>deme ex quadrato termini majoris: Deinde fiat ut hæc quadra­<lb/>torum differentia ad quadratum illius &longs;emi&longs;&longs;is, ita duplum Ra­<lb/>dij, hoc e&longs;t tota claviculorum di&longs;tantia AB ad aliud, & erit Ex­<lb/>ce&longs;&longs;us unius &longs;ecantis, quæ e&longs;t men&longs;ura a&longs;censûs æqualis pon­<lb/>derum DK aut CI. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002694">Ponderum extremorum Ratio &longs;imul &longs;umptorum ad interme­<lb/>dium &longs;it ex. </s> <s id="s.002695">gr. <!-- REMOVE S-->ut 7 ad 6: termini minoris 6 &longs;emi&longs;&longs;is e&longs;t 3, cu­<lb/>jus quadratum 9 ex 49 quadrato termini majoris 7 deme, & e&longs;t <lb/>differentia 40. Di&longs;tantia claviculorum A & B &longs;it digitorum 80; <lb/>fiat igitur ut 40 ad 9 ita 80 ad 18, & vi ponderis illius interme­<lb/>dij poterunt extrema pondera a&longs;cendere dig. </s> <s id="s.002696">18. Ut verò inno­<lb/>te&longs;cat, quantum de&longs;cendat pondus medium, inter Exce&longs;&longs;um <pb pagenum="352" xlink:href="017/01/368.jpg"/>&longs;ecantis 18, & 98 &longs;ummam &longs;ecantis & Radij, quære mediam <lb/>proportionalem, & ex prop.2. hæc e&longs;t Tangens dig.42: dupli­<lb/>catus autem 18 pro utroque exce&longs;&longs;u &longs;ecantis dat 36, atque mo­<lb/>tuum Ratio 42 ad 36 eadem e&longs;t cum reciprocá Ratione ponde­<lb/>rum 7 ad 6. Quòd &longs;i angulum EBF tantummodo quæris, quem <lb/>funiculus FB con&longs;tituit cum horizontali AB, fiat &longs;imiliter ut <lb/>40 ad 9 ita Radij duplum 200000 ad 45000 Exce&longs;&longs;um Radio <lb/>addendum, ut habeatur &longs;ecans 145000 gr.46. 24′. </s> </p> <p type="main"> <s id="s.002697">Ex his facilè intelligitur cur pro majore claviculorum A & B <lb/>intervallo pondus medium magis de&longs;cendat, quia &longs;cilicet atten­<lb/>denda e&longs;t anguli magnitudo, ex quâ pendet Tangentis & &longs;e­<lb/>cantis Ratio; ubi verò major e&longs;t Radius, majorem quoque e&longs;&longs;e <lb/>&longs;imilis anguli Tangentem atque &longs;ecantem manife&longs;tum e&longs;t. </s> <lb/> <s id="s.002698">Quare &longs;i exiguum &longs;it pondus medium, & vix appareat, an ab <lb/>illo extrema pondera eleventur, atque dubitetur, an ideò &longs;o­<lb/>lùm illud de&longs;cendat, quia funiculum magis intendit; adhibe <lb/>longiorem funiculum, cui eadem pondera adnectas, & augea­<lb/>tur, quantum opus fuerit, claviculorum A & B intervallum; <lb/>demum enim apparebit extremorum ponderum a&longs;cendentium <lb/>motus: acuti&longs;&longs;imus &longs;cilicet angulus in majore circulo habet &longs;e­<lb/>cantis Exce&longs;&longs;um &longs;uprà Radium faciliùs notabilem quàm in mi­<lb/>nore. </s> <s id="s.002699">Sic vides po&longs;ito Radio habente unitatem cum &longs;eptem <lb/>cyphris, non inveniri Exce&longs;&longs;um &longs;ecantis ni&longs;i gr.0. 1′. </s> <s id="s.002700">10. uni­<lb/>tatem: at po&longs;ito Radio cum quindecim cyphris, habetur eju&longs;­<lb/>dem anguli &longs;ecantis Exce&longs;&longs;us &longs;upra Radium partium 57585857: <lb/>immò habetur etiam unius &longs;ecundi &longs;ecans, cujus Exce&longs;&longs;us &longs;u­<lb/>pra Radium e&longs;t 11752. </s> </p> <p type="main"> <s id="s.002701">Hinc etiam de&longs;ines mirari, cur longiores funes aut catenæ <lb/>nullâ vi ita intendi po&longs;&longs;int, ut in lineâ horizonti parallelâ <lb/>rectam po&longs;itionem habentes con&longs;i&longs;tant, &longs;ed aliquantulum &longs;al­<lb/>tem inflectantur; quia nimirum in&longs;itum funi aut catenæ pon­<lb/>dus idem præ&longs;tat, quod in hoc experimento pondus in medio <lb/>appen&longs;um. </s> <s id="s.002702">Id quod nautæ non ignorantes &longs;æpius malunt uni <lb/>anchoræ funem duplo longiorem adnectere, quàm duabus an­<lb/>choris &longs;implici & &longs;ubduplo fune in&longs;tructis navem firmare: nô­<lb/>runt &longs;iquidem longè majore vi opus e&longs;&longs;e ut funis longitudinem <lb/>habens ducentorum cubitorum intendatur, quàm &longs;i centum <lb/>tantummodo cubitorum longitudo e&longs;&longs;et; ac proinde undarum <pb pagenum="353" xlink:href="017/01/369.jpg"/>impetum longior funis faciliùs eludit, eóque minùs timendum <lb/>e&longs;t, ne dirumpatur, quò difficiliùs intendi pote&longs;t. </s> </p> <p type="main"> <s id="s.002703">Simile quiddam dicendum videtur, cùm longiorum pri&longs;ma­<lb/>tum aut cylindrorum extremitates &longs;ubjectis fulcris totam longi­<lb/>tudinem horizonti parallelam in aëre qua&longs;i &longs;u&longs;pen&longs;am &longs;u&longs;tinent; <lb/>&longs;uo enim pondere &longs;i non franguntur, &longs;altem curvantur; id quod <lb/>brevioribus cylindris aut pri&longs;matis non contingit. </s> <s id="s.002704">Quia vide­<lb/>licet ex ipsá po&longs;itione partes, quæ in mediá longitudine locum <lb/>obtinent, & quæ his proximæ &longs;unt, aptæ &longs;unt velociùs moveri <lb/>quàm remotiores: & quemadmodum pondus in medio po&longs;itum <lb/>de&longs;cendens vincit re&longs;i&longs;tentiam extremorum ponderum a&longs;cen­<lb/>dentium, ita vis harum partium mediarum &longs;uperat vim, quâ <lb/>partes invicem nectuntur, ac proinde di&longs;tractæ flectuntur &longs;al­<lb/>tem, & demum &longs;eparantur. </s> </p> <p type="main"> <s id="s.002705">Sed antequam planè ex animo effluat, unum hîc ob&longs;ervan­<lb/>dum (de quo forta&longs;&longs;e malui&longs;&longs;es initio præmoneri) aliud e&longs;&longs;e <lb/>quod ex naturæ in&longs;tituto, aliud quod ex iis, quæ accidunt, con­<lb/>tingit. </s> <s id="s.002706">Quæ hactenus diximus de Ratione motuum &longs;pectatis <lb/>ponderum gravitatibus, intelligenda &longs;unt, ni&longs;i quid interveniat, <lb/>quod legem hanc infringat; cuju&longs;modi e&longs;t aliqua funiculi re­<lb/>mi&longs;&longs;io, vel minor inten&longs;io, ita ut hic faciliùs à medio pondere <lb/>de&longs;cendente adhuc intendatur, quàm extrema pondera eleven­<lb/>tur; ubi enim eò devenerit pondus medium, ut intentus funi­<lb/>culus cum lineá horizonti parallelá angulum faciat, cujus Tan­<lb/>gens ad &longs;ecantium Exce&longs;&longs;us Rationem habet reciprocam pon­<lb/>derum, ibi &longs;ub&longs;i&longs;tit, etiam&longs;i extrema pondera elevata non &longs;ue­<lb/>rint ni&longs;i juxtâ men&longs;uram differentiæ &longs;ecantium duorum angu­<lb/>lorum, ejus videlicet quem demum funiculus con&longs;tituit, & ejus <lb/>qui funiculi remi&longs;&longs;ionem ip&longs;o motûs initio con&longs;equitur: quia <lb/>ulterior de&longs;cen&longs;us ad ulteriorem a&longs;cen&longs;um non haberet majo­<lb/>rem Rationem, &longs;ed minorem Ratione ponderum reciprocè <lb/>&longs;umptorum. </s> <s id="s.002707">Quòd &longs;i valde inæqualia fuerint pondera, eveni­<lb/>re pote&longs;t totam vim de&longs;cendendi, quam pondus medium habet, <lb/>ab&longs;umi in funiculo intendendo, nec quicquam virium &longs;upere&longs;­<lb/>&longs;e ad extrema pondera attollenda. </s> </p> <p type="main"> <s id="s.002708">Húc etiam &longs;pectat impedimentum, quod ex funiculi clavi­<lb/>culos terentis conflictu oritur; cùm enim de&longs;cendentis ponde­<lb/>ris medij momentum &longs;emper decre&longs;cat, ut ex prop.5. con&longs;tat, <pb pagenum="354" xlink:href="017/01/370.jpg"/>adeò extenuari pote&longs;t, ut jam &longs;uperare non valeat extremorum <lb/>ponderum a&longs;cendentium momenta aucta momento, quod ex <lb/>partium conflictu oritur; qui conflictus &longs;i non ade&longs;&longs;et, pergeret <lb/>illud adhuc de&longs;cendendo. </s> <s id="s.002709">Propterea &longs;i claviculos ip&longs;os con­<lb/>gruentibus rotulis in&longs;eras, adeò ut funiculus excavatæ ab&longs;idi <lb/>in&longs;ideat, longè majorem motum faciliú&longs;que perfici videbis; <lb/>minùs enim rotula cum &longs;uo axe confligit, quàm funiculus <lb/>cum claviculo, &longs;i illum terat; & quidem quò major fuerit ro­<lb/>tula, circa eundem axem faciliùs volvitur, minor &longs;iquidem <lb/>partium tritus &longs;it, &longs;i cætera omnia &longs;int paria. </s> <s id="s.002710">Simili modo &longs;i <lb/>pondus medium plus æquo per vim deprimas, faciliùs &longs;uum <lb/>in locum redibit adhibitis rotulis, quàm &longs;i funiculus clavicu­<lb/>lis in&longs;i&longs;teret: quia pondera extrema &longs;uperare non valent & gra­<lb/>vitatem ponderis medij & impedimentum, quod oritur ex ma­<lb/>jori tritu funiculi & claviculorum, quàm rotularum & axium. </s> <lb/> <s id="s.002711">Ob&longs;ervabis etiam adhibitis rotulis pondus medium &longs;ibi re­<lb/>lictum tanto impetu à lineâ horizonti parallelâ de&longs;cendere, ut <lb/>ex concepto impetu fines &longs;uos tran&longs;iliat, ac idcirco de&longs;inente <lb/>impetu, quem in motu acqui&longs;ivit, iterum &longs;ur&longs;um trahi ab ex­<lb/>tremis ponderibus, quæ &longs;icut minorem Rationem habebant ad <lb/>gravitatem ponderis medij auctam impetu acqui&longs;ito, ita ma­<lb/>jorem Rationem habent ad eandem &longs;poliatam illo impetu. </s> </p> <p type="main"> <s id="s.002712">Porrò hæc quæ hactenus de pondere in mediâ planè di&longs;tan­<lb/>tiâ inter claviculos aut rotulas con&longs;tituto dicta &longs;unt, intelli­<lb/>genda &longs;unt pariter de pondere claviculorum intervallum inæ­<lb/>qualiter dividente, quod quidem &longs;pectat ad æquilibrium aut <lb/>præponderationem propter Rationum æqualitatem aut inæqua­<lb/>litatem. </s> <s id="s.002713">Peculiare tamen aliquid ob&longs;ervandum e&longs;t, videlicet <lb/>aliquando contingere, ut hoc pondere medio de&longs;cendente <lb/>pondus proximum a&longs;cendat, remotum verò de&longs;cendat, utró­<lb/>que autem pondere extremo a&longs;cendente magis a&longs;cendere quod <lb/>proximum e&longs;t, minùs quod remotum. </s> <s id="s.002714">Hujus inæqualis a&longs;cen­<lb/>sûs (&longs;i pondus medium rectâ ad perpendiculum de&longs;cendat) <lb/>cau&longs;a in promptu e&longs;t ex iis, quæ prop. 8. indicata &longs;unt, nam <lb/>eju&longs;dem Tangentis quadrato æqualia &longs;unt, atque adeò & in­<lb/>ter &longs;e æqualia, rectangula, quæ fiunt &longs;ub Exce&longs;&longs;u &longs;ecantis & <lb/>aggregato &longs;ecantis & Radij: &longs;unt igitur ex 14. lib.6. Exce&longs;&longs;us <lb/>&longs;ecantium reciprocè in Ratione aggregatorum &longs;ecantis & Ra-<pb pagenum="355" xlink:href="017/01/371.jpg"/>dij: quapropter ubi major e&longs;t Radius & &longs;ecans, ibi minor e&longs;t <lb/>&longs;ecantis Exce&longs;&longs;us, hoc e&longs;t remoti ponderis a&longs;cen&longs;us, & <lb/>contra ubi minor e&longs;t Radius & &longs;ecans, ibi major e&longs;t &longs;ecan­<lb/>tis Exce&longs;&longs;us, hoc e&longs;t ponderis proximi a&longs;cen&longs;us. </s> </p> <p type="main"> <s id="s.002715">Cur autem aliquando proximum pondus a&longs;cendat, atque <lb/>remotum de&longs;cendat, quando nimirum valde inæquales &longs;unt <lb/>ponderis medij à claviculis di&longs;tantiæ, hinc fit, quod idem pon­<lb/>dus ex longiore funiculo majorem habet vim de&longs;cendendi, quàm <lb/>ex breviore; cui majori momento cum re&longs;i&longs;tere debeat pondus <lb/>proximum, faciliùs cedit de&longs;cendenti, atque adeò non rectâ de­<lb/>or&longs;um tendit pondus medium, &longs;ed obliquè, accedendo ad pon­<lb/>dus remotum, quod propterea de&longs;cendit. </s> <s id="s.002716">Sic po&longs;itum pondus in <lb/>E valde inæqualia habet mo­<lb/><figure id="id.017.01.371.1.jpg" xlink:href="017/01/371/1.jpg"/><lb/>menta comparatum cum ex­<lb/>tremis ponderibus D & C, <lb/>quæ in punctis B & A exer­<lb/>cent &longs;uas vires adversùs pon­<lb/>dus medium; quod ubi infrà <lb/>horizontalem AB <expan abbr="de&longs;c&etilde;derit">de&longs;cenderit</expan>, <lb/>illico inæquales angulos cum <lb/>horizontali linea AB con&longs;ti­<lb/>tuit inflexus funiculus; ut &longs;i intelligatur pondus ex E veni&longs;&longs;e in <lb/>F, angulus FBA major e&longs;t angulo FAB ex 18. lib.1. quia latus <lb/>AF e&longs;t majus latere FB. </s> <s id="s.002717">Igitur angulus FBD, quem funiculus <lb/>inflexus FB facit cum perpendiculari BD minor e&longs;t angulo <lb/>FAC; ergo ex dictis lib.1. cap. 15. pondus in F minora habet <lb/>momenta ad de&longs;cendendum versùs perpendiculum BD, quàm <lb/>ad de&longs;cendendum versùs AC, & quidem duplici titulo, &longs;cilicet <lb/>anguli FBD minoris, & funiculi FB brevioris. </s> <s id="s.002718">Cum itaque pon­<lb/>dus illicò ac ex E de&longs;cendit magis pronum &longs;it ad de&longs;cendendum <lb/>versùs perpendiculum AC, non per rectam EF <expan abbr="perpendicular&etilde;">perpendicularem</expan> <lb/>de&longs;cendit; &longs;ed obliquè per lineam EG, ita ut funiculus GA bre­<lb/>vior &longs;it funiculo EA, ac propterea cedit ponderi C deor&longs;um tra­<lb/>henti. </s> <s id="s.002719">Et quia funiculus GB longior e&longs;t funiculo FB, & multo <lb/>magis funiculo EB, propterea aliquando contingere pote&longs;t pon­<lb/>dus D magis a&longs;cendere, quàm a&longs;cenderet, &longs;i E fui&longs;&longs;et planè in <lb/>mediâ di&longs;tantiâ inter A & B. </s> <s id="s.002720">Ex quo etiam fit de&longs;cen&longs;um per­<lb/>pendicularem ponderis medij minorem e&longs;&longs;e; nam punctum G <pb pagenum="356" xlink:href="017/01/372.jpg"/>minùs di&longs;tat ab horizontali AB, quàm punctum F, & tamen <lb/>major e&longs;t differentia inter EB & GB; ideò minor e&longs;t Ratio IG <lb/>ad Exce&longs;&longs;um GL, quàm EF ad Exce&longs;&longs;um FO. </s> </p> <p type="main"> <s id="s.002721">Hanc momentorum inæqualitatem per&longs;picies, &longs;i pondus me­<lb/>dium &longs;ingulis extremis æquale inter claviculos æqualiter con&longs;ti­<lb/>tutum de&longs;cendere permittas, &longs;uóque in loco <expan abbr="cõ&longs;i&longs;tere">con&longs;i&longs;tere</expan>; cùm enim <lb/>æqualis &longs;it funiculorum illud &longs;u&longs;tinentium longitudo, & æqua­<lb/>les faciat angulos tùm cum horizontali, tùm cum perpendicula­<lb/>ribus, contra utrumque extremum æqualibus momentis pugnat, <lb/>ac rectâ ad perpendiculum de&longs;cendit. </s> <s id="s.002722">Tum alteri extremorum <lb/>aliquid adde ponderis; hoc utique de&longs;cendens &longs;ecum rapit & <lb/>ponderis medij & reliqui extremi gravitates, quas cogit a&longs;cen­<lb/>dere, donec ea fiat funiculorum inæqualitas, ut momenta, quæ <lb/>pondus medium habet ad de&longs;cendendum ratione di&longs;tantiæ á cla­<lb/>viculo remotiori, jam &longs;uperari non valeant à pondere illo ex­<lb/>tremo cum &longs;uo additamento. </s> </p> <p type="main"> <s id="s.002723">Nec di&longs;par e&longs;t philo&longs;ophandi methodus, cum funiculi extre­<lb/>mitas alterutri claviculo alligatur, unico pondere in alterâ extre­<lb/>mitate pendente ex altero claviculo: pondus enim inter clavi­<lb/>culos funiculo adnexum, quia velociùs movetur de&longs;cendendo, <lb/>quàm reliquum pondus a&longs;cendendo, &longs;uperare pote&longs;t illius gra­<lb/><figure id="id.017.01.372.1.jpg" xlink:href="017/01/372/1.jpg"/><lb/>vitatem. </s> <s id="s.002724">Sit enim funiculus alligatus <lb/>in A, & pendeat pondus D ex clavicu­<lb/>lo B: pondus (utrùm æquale &longs;it, an ma­<lb/>jus, an minus, parum refert) adnectatur <lb/>in C: utique de&longs;cendens de&longs;cribit ar­<lb/>cum CI circa centrum A; e&longs;t autem <lb/>funiculus IB longior quàm CB ex 8. <lb/>lib.3. Sed quoniam duo latera BC & <lb/>CI &longs;imul majora &longs;unt reliquo latere IB ex 20.lib. 1. major e&longs;t <lb/>recta CI, & multo magis arcus CI &longs;patium quod percurrit pon­<lb/>dus medium de&longs;cendens) quàm IE Exce&longs;&longs;us lateris IB &longs;upra <lb/>CB, hoc e&longs;t men&longs;ura motûs ponderis D a&longs;cendentis. </s> <s id="s.002725">Quia verò <lb/>ponderis medij de&longs;cendentis circa centrum A momenta decre&longs;­<lb/>cunt ex dictis lib.1. cap.15. circa centrum autem B decre&longs;cunt <lb/>quidem, quia minor fit angulus declinationis à perpendiculo <lb/>GBD, &longs;ed decrementum hoc temperatur, quia momenta cre&longs;­<lb/>cunt ratione longitudinis funiculi, quæ &longs;emper augetur ex 8. <pb pagenum="357" xlink:href="017/01/373.jpg"/>lib.3.propterea ad momentorum æqualitatem venit, ubi demùm <lb/>quie&longs;cit. </s> <s id="s.002726">Quantum autem de&longs;cendat, pendet ex ip&longs;ius ponderis <lb/>gravitate ab&longs;olutâ &longs;ive majori, &longs;ive minori, &longs;ive æquali compara­<lb/>tâ cum pondere D, & ex di&longs;tantiâ à centro A: &longs;i enim valde pro­<lb/>pinquum &longs;it centro, parùm de&longs;cendit, etiam&longs;i cæteroqui gravius <lb/>&longs;it; & &longs;i per vim adhuc deprimatur, ut veniat in G, ce&longs;&longs;ante vi <lb/>extrin&longs;ecùs illatâ pondus D de&longs;cendens illud iterum attollit. </s> </p> <p type="main"> <s id="s.002727">Cave tamen ponderis medij de&longs;cendentis momenta metiaris <lb/>ex arcu, quem de&longs;cribit, &longs;ed potiùs illa definienda &longs;unt ex ip&longs;o <lb/>de&longs;cen&longs;u perpendiculari, cum moveatur vi &longs;uæ gravitatis. </s> <s id="s.002728">Quo­<lb/>niam verò æqualibus arcubus de&longs;criptis non re&longs;pondent paria <lb/>perpendicularium linearum incrementa ex prop.10.&longs;ed &longs;emper <lb/>minora fiunt; contra verò incrementa &longs;ecantium augentur, hinc <lb/>e&longs;t deveniri ad momentorum æqualitatem, ita ut pondus me­<lb/>dium gravius pondere extremo aptum &longs;it minùs de&longs;cendere <lb/>quàm illud a&longs;cenderet &longs;ecundùm <expan abbr="reciprocã">reciprocam</expan> <expan abbr="Ration&etilde;">Rationem</expan> gravitatum. </s> </p> <p type="main"> <s id="s.002729">Hinc elici pote&longs;t compendium aliquod in attollendo ponde­<lb/>re cæteroqui valde gravi; &longs;it enim pondus P attollendum fune <lb/>circumducto rotulæ A: quò longior <lb/><figure id="id.017.01.373.1.jpg" xlink:href="017/01/373/1.jpg"/><lb/>funis pote&longs;t alligari in B, eò faciliùs <lb/>&longs;equetur motus, &longs;i ad &longs;ervandam in <lb/>mediâ di&longs;tantiâ po&longs;itionem poten­<lb/>tiæ moventis &longs;implicem trochleam <lb/>aut annulum in C addideris, cui in­<lb/>&longs;eratur funis BA: nam applicata po­<lb/>tentia in D deor&longs;um trahens multo <lb/>faciliùs attollet pondus P, quàm &longs;i <lb/>arreptâ funis extremitate B idem <lb/>onus elevare conaretur ad eam al­<lb/>titudinem, ad quam attolleretur à pondere in C adnexo, quod <lb/>æqualibus viribus præditum e&longs;&longs;et cum potentiâ in D trahente. </s> <lb/> <s id="s.002730">Ubi jam &longs;it attollendi difficultas, &longs;uppone aliquid ponderi P, cui <lb/>illud incumbat, nec contra funem conetur: tùm iterum funem <lb/>intende, & alliga in B, ut &longs;it AB horizonti parallelus, & ite­<lb/>rum in D deor&longs;um trahens priorem facilitatem experieris: id <lb/>quod toties iterari poterit, quoties opus fuerit. </s> </p> <p type="main"> <s id="s.002731">Ex his omnibus, quæ toto hoc capite di&longs;putata &longs;unt, mani­<lb/>fe&longs;tum e&longs;t non referendas e&longs;&longs;e machinarum vires ad Rationes <pb pagenum="358" xlink:href="017/01/374.jpg"/>circuli aut Vectis, quandoquidem hic videmus minori pondere <lb/>majus pondus moveri ab&longs;que ullo motu circulari. <lb/></s> </p> <p type="main"> <s id="s.002732"><emph type="center"/>CAPUT XIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002733"><emph type="center"/><emph type="italics"/>An aliqua &longs;it Libræ obliquæ utilitas.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002734">LIbram obliquam vocat Simon Stevinus Static. <!-- REMOVE S-->lib.3.prop.6. <lb/>rotulam L funiculi in excavatâ ap&longs;ide capacem pondus <lb/><figure id="id.017.01.374.1.jpg" xlink:href="017/01/374/1.jpg"/><lb/>cum æquipondio jungentis, & in &longs;uo lo­<lb/>culamento facillimè ver&longs;atilem, cujus par­<lb/>ticula extans E po&longs;&longs;it pro re natâ eximi, at­<lb/>que iterum in&longs;eri foraminibus, quibus <lb/>exactè congruat, tigilli P firmè infixi pedi <lb/>&longs;atis gravi, ne valeat à ponderis examinan­<lb/>di gravitate rapi & inclinari. </s> <s id="s.002735">Hanc ille ad <lb/>ponderum obliquorum momenta inve&longs;ti­<lb/>ganda utilem exi&longs;timavit, eamque &longs;æpiùs <lb/>ingerit Static.lib.1.prop.19.& &longs;eqq quam­<lb/>vis &longs;emper illam cum elevante directo conjunctam adhibeat. </s> <lb/> <s id="s.002736">Propterea, an aliquid ex illâ emolumenti, &longs;i &longs;olitaria adhibeatur, <lb/>capere po&longs;&longs;imus in ponderum momentis inve&longs;tigandis &longs;ivè &longs;u&longs;­<lb/>pen&longs;orum, &longs;ivè in plano inclinato jacentium, hîc examinare ope­<lb/>ræ pretium fuerit; nam & à &longs;uperioris capitis argumento non <lb/>aliena videtur e&longs;&longs;e præ&longs;ens di&longs;putatio. </s> </p> <p type="main"> <s id="s.002737">Antequam verò rem aggrediar, monendum te cen&longs;eo, Ami­<lb/>ce Lector, opportunius accidere, &longs;i tigilli perforati loco cylin­<lb/>drum in cochleam efformatum &longs;tatueris, cui congruat in &longs;imi­<lb/>lem helicem excavatum foramen S in rotulæ L loculamento: <lb/>&longs;ic enim faciliùs elevabitur aut deprimetur rotula, prout exiget <lb/>ip&longs;ius ponderis po&longs;itio. </s> </p> <p type="main"> <s id="s.002738">Dupliciter itaque contingere pote&longs;t ponderis obliquitas, &longs;eu <lb/>quia &longs;u&longs;pen&longs;um non in codem perpendiculo, in quo e&longs;t punctum <lb/>&longs;u&longs;pen&longs;ionis, habet centrum &longs;uæ gravitatis, &longs;eu quia plano incli­<lb/>nato incumbit; utroque enim in ca&longs;u momenta habet ad de&longs;cen­<lb/>dendum, quæ communi librâ aut &longs;taterâ ve&longs;tigare utique non <lb/>po&longs;&longs;umus: an libræ obliquæ ope id a&longs;&longs;equemur? </s> <s id="s.002739">Et primò qui­<lb/>dem &longs;i pondus examinandum è funiculo &longs;u&longs;pen&longs;um fuerit, eju&longs;­<lb/>que momenta pro variá declinatione à &longs;uo perpendiculo inqui-<pb pagenum="359" xlink:href="017/01/375.jpg"/>rantur, res manet incerta, &longs;i in praxim deducatur, quia plurimum <lb/>intere&longs;t, quâ obliquitate inclinetur, atque à &longs;uo perpendiculo de­<lb/>flectat funiculus libræ obliquæ, &longs;i maxime cum diversa obliqui­<lb/>tate jungatur di&longs;par funiculi illius longitudo. </s> <s id="s.002740">Nam ex A &longs;u&longs;­<lb/>pendatur pondus B habens BAC <expan abbr="angulũ">angulum</expan> <lb/><figure id="id.017.01.375.1.jpg" xlink:href="017/01/375/1.jpg"/><lb/>declinationis à &longs;uo perpendiculo AC; & <lb/>primùm &longs;it libra obliqua D, itaut <expan abbr="æquip&etilde;-dium">æquipen­<lb/>dium</expan> E retineat pondus B in eodem &longs;itu: <lb/>deinde transferatur libra obliqua ex D in <lb/>F, & æquipondium G retineat pariter in <lb/>codem &longs;itu pondus B cum declinationis <lb/>angulo BAC. <!-- KEEP S--></s> <s id="s.002741">Si in eâdem rectâ lineâ &longs;int <lb/>BDF, nulla e&longs;t momentorum inæqualitas, <lb/>quamvis di&longs;paritas intercedat inter funi­<lb/>culi longitudines BD, & BF. <!-- KEEP S--></s> <s id="s.002742">Sin autem F <lb/>paulo &longs;uperior fuerit aut paulo inferior, jam BD & BF angulum <lb/>in B con&longs;tituunt, & momenta mutantur. </s> <s id="s.002743">Quoniam enim IE & <lb/>HG <expan abbr="perp&etilde;diculares">perpendiculares</expan> &longs;unt parallelæ, in ea&longs;que incidit recta BDF <lb/>producta, anguli BIE, & BHG &longs;unt æquales per 29.lib.1. at verò <lb/>&longs;i libra obliqua F non planè in eâdem rectâ lineâ, &longs;ed &longs;uperiore <lb/>loco collocaretur, angulum con&longs;titueret cum perpendiculo HG <lb/>acutiorem, & inferiùs po&longs;ita angulum efficeret minùs acutum. </s> <lb/> <s id="s.002744">Quare pondus B, quò acutior e&longs;t angulus, & magis accedit ad <lb/><expan abbr="perpendiculũ">perpendiculum</expan> FG, eò etiam magis conatur contra F, & ad æqui­<lb/>librium exigit majorem gravitatem in G, quàm cum angulus e&longs;t <lb/>minù, acutus. </s> <s id="s.002745">Id quod experimento allato &longs;uperiori capite ma­<lb/>nife&longs;tum &longs;it; &longs;i enim funiculi extremitates jungant pondera inæ­<lb/>qualia, pondus intermedium magis accedit ad perpendiculum, in <lb/>quo e&longs;t major gravitas. </s> <s id="s.002746">Hinc quia valde incertum e&longs;t in praxi <lb/>utrùm B, D, & F in eâdem &longs;int rectá lineâ, propterea <expan abbr="etiã">etiam</expan> <expan abbr="incertũ">incertum</expan> <lb/>erit ex gravitate ponderis G inferre, quanta &longs;int ponderis B <expan abbr="mo-m&etilde;ta">mo­<lb/>menta</expan> cum declinatione BAC: Ni&longs;i fortè <expan abbr="duplic&etilde;">duplicem</expan> in&longs;tituas libræ <lb/>obliquæ po&longs;itionem in D, & in F atque <expan abbr="eod&etilde;">eodem</expan> &longs;emper <expan abbr="põdere">pondere</expan> tam <lb/>in E quàm in G retineatur pondus B in po&longs;itione <expan abbr="câd&etilde;">eâdem</expan>. </s> <s id="s.002747">Ita <expan abbr="tam&etilde;">tamen</expan> <lb/>collocanda e&longs;t libra obliqua, ut angulus ABD &longs;it rectus; ex illo <lb/>quippe æ&longs;timatur <expan abbr="planũ">planum</expan> <expan abbr="inclinatũ">inclinatum</expan>, in quo pondus B conatur de&longs;­<lb/>cendere, ut <expan abbr="dictũ">dictum</expan> e&longs;t lib.1.cap 15.alioquin &longs;i acutus fuerit aut ob­<lb/>tu&longs;us ille angulus, quamvis in eâdem declinatione BAC reti­<lb/>neatur, valde inæqualia apparebunt momenta. </s> <s id="s.002748">Quis autem de <pb pagenum="360" xlink:href="017/01/376.jpg"/>anguli illius rectitudine certus fuerit? </s> <s id="s.002749">cùm maximè rectam DB <lb/>oporteat ad perpendiculum in&longs;i&longs;tere lineæ jungenti punctum A <lb/>&longs;u&longs;pen&longs;ionis cum centro gravitatis ponderis B. </s> <s id="s.002750">Ex pondere ita­<lb/>que, quod e&longs;t in E, aut in G, nemo pote&longs;t certò definire mo­<lb/>menta ponderis B &longs;u&longs;pen&longs;i. </s> </p> <p type="main"> <s id="s.002751">At &longs;i dato quopiam plano inclinato jaceat pondus, veli&longs;que li­<lb/>brâ huju&longs;modi obliquâ explorare, quanta habeat pro eâ plani in­<lb/>clinatione ad de&longs;cendendum momenta, ego &longs;anè nihil certi affir­<lb/>mare auderem; quippè qui &longs;emper incertus <expan abbr="hærer&etilde;">hærerem</expan>, an æquipon­<lb/>dium libræ obliquæ indicaret ip&longs;a <expan abbr="mom&etilde;ta">momenta</expan> ponderis in plano in­<lb/>clinato pro ratione inclinationis; nam plani &longs;ubjecti non omnino <lb/>lubrica &longs;uperficies, & ponderis illi incumbentis a&longs;peritas impe­<lb/>dientes motum, non nihil detrahunt momenti ad <expan abbr="de&longs;cendendũ">de&longs;cendendum</expan>. </s> <lb/> <s id="s.002752">Cum verò pro diversa inclinatione <expan abbr="planũ">planum</expan> inæqualiter prematur <lb/>ab in&longs;i&longs;tente <expan abbr="põdere">pondere</expan>, adhuc eadem <expan abbr="&longs;uperficierũ">&longs;uperficierum</expan> &longs;e <expan abbr="contingentiũ">contingentium</expan> <lb/>a&longs;peritas magis ob&longs;i&longs;tit motui, quò major e&longs;t plani inclinatio de­<lb/>clinans à perpendiculo. </s> <s id="s.002753">Quare adhuc magis incerta e&longs;&longs;ent mo­<lb/>menta, quæ ab æquipondio libræ obliquæ indicarentur. </s> </p> <p type="main"> <s id="s.002754">Nihil aliud itaque commodi hinc &longs;perari pote&longs;t præter <expan abbr="notitiã">notitiam</expan> <lb/>momenti, quod planorum a&longs;peritas detrahit <expan abbr="mom&etilde;to">momento</expan> de&longs;cenden­<lb/>di. </s> <s id="s.002755">Si enim nota &longs;it ponderis dati gravitas ab&longs;oluta, & plani incli­<lb/>natio innotuerit, videlicet angulus, quem planum <expan abbr="inclinatũ">inclinatum</expan> cum <lb/>plano horizontali con&longs;tituit, fiat ut Radius ad Sinum noti anguli <lb/>inclinationis, ita gravitas ab&longs;oluta dati ponderis ad <expan abbr="mom&etilde;ta">momenta</expan>, quæ <lb/>habet in plano inclinato: Tum librâ obliquâ exploretur, quanto <lb/>æquipondio opus &longs;it ad <expan abbr="retin&etilde;dum">retinendum</expan> pondus in plano inclinato, ne <lb/>deor&longs;um labatur: nam differentia inter gravitatem æquipondij, & <lb/>momenta inventa pro tali inclinatione indicabit, quantum impe­<lb/>dimenti oriatur ex <expan abbr="planorũ">planorum</expan> &longs;e contingentium a&longs;peritate, &longs;i æqui­<lb/>pondij gravitas minor &longs;it momentis, quæ ab huju&longs;modi inclina­<lb/>tione exiguntur. </s> <s id="s.002756">Sic ex.gr. &longs;it ponderis dati ab&longs;oluta gravitas un­<lb/>ciarum 30, inclinationis angulus dati plani cum plano <expan abbr="horizõtali">horizontali</expan> <lb/>&longs;it gr.60.fiat ut 10000 Radius ad 86603 Sinum gr.60.ita 30 ad <lb/>25.98. <!--neuer Satz-->Si applicata libra obliqua æquipondium habeat &longs;olùm <lb/>unc. 24, manife&longs;tum e&longs;t à planorum a&longs;peritate detrahi momenti <lb/>partem ferè decimam tertiam, cùm de&longs;int ju&longs;to æquipondio ferè <lb/>unciæ 2. <!--neuer Satz-->Verùm & hîc ob&longs;ervandum, opus e&longs;&longs;e funiculi, à quo <lb/>pondus retinetur, paralleli&longs;mum cum plano inclinato, prout ex <lb/>iis, quæ de obliquis tractionibus lib. 1. cap.16. dicta &longs;unt, &longs;atis <lb/>con&longs;tat. </s> </p> <pb pagenum="361" xlink:href="017/01/377.jpg"/> <figure id="id.017.01.377.1.jpg" xlink:href="017/01/377/1.jpg"/> <p type="main"> <s id="s.002757"><emph type="center"/>MECHANICORUM<emph.end type="center"/><!-- REMOVE S--><emph type="center"/>LIBER QUARTUS.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.002758"><emph type="center"/><emph type="italics"/>De Vecte.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002759">HACTENUS de in&longs;trumentis ad movenda pondera <lb/>idoneis nihil, ni&longs;i forta&longs;sè obiter, dictum e&longs;t: Jam <lb/>ad illa explicanda accedimus, quibus veteres facul­<lb/>tatibus nomen indiderunt. </s> <s id="s.002760">Quamvis autem in <lb/>quinque facultatibus enumerandis primum locum <lb/>Vecti Pappus lib. 8. Collect. Math. non tribuat, placuit tamen <lb/>de Vecte ante cæteras facultates di&longs;&longs;erere, e&longs;t &longs;iquidem paratu <lb/>facillimus, & ad &longs;ubitum u&longs;um prompti&longs;&longs;imus, atque cen&longs;eri <lb/>pote&longs;t, ut idem Pappus loquitur, <emph type="italics"/>forta&longs;&longs;e præmeditatio motús cir­<lb/>ca excedentia pondera: &longs;tatuentes enim quidam magna pondera mo­<lb/>vere [quoniam primùm à terrâ attollere oportet, an&longs;as autem non <lb/>habebant) quòd omnes partes ba&longs;is ip&longs;ius ponderis &longs;olo incumberent, <lb/>paulum &longs;uffodientes, & ligni longi extremitatem &longs;ubjicientes &longs;ub <lb/>onus, adducebant ex alterâ extremitate, &longs;upponentes ligno propè <lb/>ip&longs;um onus lapidem, qui Hypomochlium appellatur. </s> <s id="s.002761">Cúmque illis vi­<lb/>&longs;us e&longs;&longs;et hic motus valde facilis, exi&longs;imaverunt fieri pe&longs;&longs;e, ut hoc <lb/>pacio magna pondera moveretur. </s> <s id="s.002762">Vocatur autem tale lignum Vectis, <lb/>&longs;ive quadratum &longs;it, &longs;ive rotundum, & quanto propinquius oneri poni­<lb/>tur hypomochlium, tanto faciliùs pondus movetur.<emph.end type="italics"/> Hæc illo vectis <lb/>ortum & procreationem quodammodo indigitans. </s> </p> <p type="main"> <s id="s.002763">Contingere quidem pote&longs;t, ut Vecte aliquando utamur ad <lb/>&longs;u&longs;tinendum ingens pondus, non autem ad movendum, adeò <lb/>ut potentia exigua &longs;u&longs;tinens, in alterâ vectis extremitate po&longs;ita. <pb pagenum="362" xlink:href="017/01/378.jpg"/>habeat rationem æquipondij retinentis pondus in oppo&longs;itá ex­<lb/>tremitate collocatum: & tunc locum habet Ari&longs;totelis &longs;enten­<lb/>tia Mechan. quæ &longs;t.3.dicentis, <emph type="italics"/>Ip&longs;e vectis e&longs;t in causá libra exi&longs;tens, <lb/>&longs;partum inferne habens, in inæqualia divi&longs;a; hypomoclion enim e&longs;t <lb/>&longs;partum, ambo namque &longs;unt ut centrum.<emph.end type="italics"/></s> <s id="s.002764"> Verùm cùm propriè, & <lb/>pre&longs;sè tunc facultas e&longs;&longs;e non videatur, neque exerceat munus <lb/>vectis, quia non movet, &longs;ed &longs;it qua&longs;i jugum &longs;tateræ; fru&longs;tra <lb/>Vectis quâ vectis e&longs;t, ad libram revocatur: præ&longs;ertim cùm ali­<lb/>quod vectis genus &longs;it, in quo nullum libræ ve&longs;tigium depre­<lb/>hendi pote&longs;t, etiam&longs;i pondus cæteroqui ruiturum &longs;u&longs;tineat; &longs;i <lb/>nimirum pondus ip&longs;um inter vectis extremitates con&longs;titutum <lb/>&longs;u&longs;tineatur, aut potentia ip&longs;a &longs;u&longs;tentans medium locum occu­<lb/>pet inter pondus & hypomochlium, ut infra dicetur. </s> <s id="s.002765">Quid <lb/>enim pariter non revocetur libra aut &longs;tatera ad Vectem, &longs;i ex <lb/>altera jugi extremitate pondus addatur, quod ad oppo&longs;itum <lb/>pondus majorem habeat Rationem, quàm libræ, aut &longs;tateræ <lb/>brachia reciprocè &longs;umpta? </s> <s id="s.002766">tunc enim (qua&longs;i &longs;tateræ aut libræ <lb/>centrum motûs e&longs;&longs;et hypomochlium) &longs;equitur motus prout ex <lb/>vecte. </s> <s id="s.002767">Quemadmodum igitur libra aut &longs;tatera ad ponderum <lb/>æquilibrium in&longs;titute, non verò ad eorum motum, libræ aut <lb/>&longs;tateræ munus non exercent in motu, quâ motus e&longs;t; ita pari­<lb/>ter vectis hypomochlium inter extremitates habens non exer­<lb/>cet munus vectis in quiete: alioquin & vectis ad libram, & vi­<lb/>ci&longs;&longs;im libra ad vectem ab&longs;urdo circulo revocaretur. </s> <s id="s.002768">Adde verò <lb/>genus hoc vectis hypomochlium inter extremitates habentis, &longs;i <lb/>adhibeatur ad onus in plano horizontali movendum, non verò <lb/>ad illud &longs;u&longs;tentandum, nihil habere commercij cum librâ, onus <lb/>&longs;i quidem nullam exercet vim &longs;uæ gravitatis adversùs ip&longs;um <lb/>vectem, nam ce&longs;&longs;ante potentia onus illicò quie&longs;cit; at in libra <lb/>&longs;ublato æquipondio pondus de&longs;cendit. </s> <s id="s.002769">Quid &longs;i vecte utamur <lb/>ad corpus leve infra aquam deprimendum? </s> <s id="s.002770">an erit illa libra in­<lb/>ver&longs;a? </s> <s id="s.002771">Non igitur me fru&longs;tra conficiam labore enitens rationes <lb/>libræ in vecte recogno&longs;cere, &longs;ed ip&longs;um per &longs;e con&longs;iderans, quæ <lb/>opportuniora cen&longs;uero, di&longs;putabo. <pb pagenum="363" xlink:href="017/01/379.jpg"/> </s> </p> <p type="main"> <s id="s.002772"><emph type="center"/>CAPUT I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002773"><emph type="center"/><emph type="italics"/>Vectis forma, & vires explicantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002774">VEctis ob id ip&longs;um quia Vectis e&longs;t & Facultas mechanica, <lb/>longitudo quædam e&longs;t, in qua tria puncta a&longs;&longs;ignantur, pri­<lb/>mum Potentiæ moventi, alterum Ponderi movendo, tertium <lb/>Fulcro, &longs;eu Hypomochlio, cui innixus vectis tanquam ex <lb/>centro duos arcus de&longs;cribens duplicem motum definit, Poten­<lb/>tiæ videlicet & Ponderis, pro variâ illorum ab codem fulcro <lb/>di&longs;tantiâ. </s> <s id="s.002775">Hinc quia tripliciter in hac longitudine tria hæc <lb/>puncta di&longs;poni po&longs;&longs;unt, tria oriuntur vectis genera. </s> <s id="s.002776">Primum <lb/>e&longs;t vectis genus, cùm extremi­<lb/>tates occupantur à Potentia A <lb/><figure id="id.017.01.379.1.jpg" xlink:href="017/01/379/1.jpg"/><lb/>& Pondere B, medius locus <lb/>Hypomochlio C cedit. </s> <s id="s.002777">Secun­<lb/>dum genus e&longs;t, cum extremi­<lb/>tati alteri F innititur vectis, al­<lb/>teri Potentia D adjungitur, & <lb/>inter utramque extremitatem <lb/>collocatur Pondus E. <!-- KEEP S--></s> <s id="s.002778">Tertium <lb/>genus e&longs;t, cum Potentia & Pondus loca &longs;ecundi generis invi­<lb/>cem permutant, Potentia G videlicet in medio, Pondus H in <lb/>extremitate con&longs;tituitur, manente alterâ extremitate I tan­<lb/>quam motuum centro. </s> <s id="s.002779">Cum itaque nulla alia fieri po&longs;&longs;it trium <lb/>huju&longs;modi punctorum diver&longs;a di&longs;po&longs;itio, patet tria &longs;olùm <lb/>Vectis genera excogitari potui&longs;&longs;e. </s> <s id="s.002780">quod enim quartum Vectis <lb/>genus, &longs;cilicet inflexum RSV commini&longs;ci quibu&longs;dam placuit, <lb/>omnino ineptum e&longs;t, quippe quod à primo genere nihil differt, <lb/>ni&longs;i quia, loco &longs;ubjecti fulcri, adnexum habet hypomochlium <lb/>inter extremitates con&longs;titutum in S, ubi &longs;inuatur in angulum, <lb/>cui in motu innititur. </s> </p> <p type="main"> <s id="s.002781">Quemadmodum autem inter hæc tria Vectis genera di&longs;&longs;imi­<lb/>litudo, ita non modica inter eorum vires di&longs;crepantia interce-<pb pagenum="364" xlink:href="017/01/380.jpg"/>dit. </s> <s id="s.002782">Primum enim genus, &longs;i ab hypomochlio inæqualiter di­<lb/>vidatur longitudo vectis, ut ab eo plus di&longs;tet Potentia, quàm <lb/>Pondus, juvat Potentiam; &longs;ecus verò, &longs;i Potentia & Pondus <lb/>æqualibus intervallis ab hypomochlio ab&longs;int, aut propior &longs;it <lb/>Potentia quàm Pondus; Potentiæ etenim tunc vectis vel nihil <lb/>affert adjumenti, vel plurimum detrimenti. </s> <s id="s.002783">Secundum genus <lb/>Potentiæ laborem &longs;emper minuit, Tertium &longs;emper auget. </s> <s id="s.002784">Quo­<lb/>nam id pacto contingat, manife&longs;tum fiet, &longs;i vectis vires unde <lb/>ortum habeant, aperiamus. </s> </p> <p type="main"> <s id="s.002785">Certum e&longs;t fieri non po&longs;&longs;e, ut pondus aliquod per vim mo­<lb/>veatur, ni&longs;i potentiæ moventis virtus &longs;uperet ponderis re&longs;i&longs;ten­<lb/>tiam; &longs;i enim pari conatu confligerent, anceps e&longs;&longs;et victoria, <lb/>& nullus e&longs;&longs;et motus; multo minùs à potentiâ infirmiore, quàm <lb/>par &longs;it, vinci poterit innata ponderis propen&longs;io. </s> <s id="s.002786">Hoc igitur <lb/>ip&longs;o quod motus efficitur, argumento e&longs;t potentiæ virtutem re­<lb/>&longs;i&longs;tentiâ ponderis e&longs;&longs;e majorem: Quod verò pondus eodem <lb/>temporis intervallo plus &longs;patij aut minus decurrat, pro ratione <lb/>exce&longs;sûs virium potentiæ &longs;upra ponderis re&longs;i&longs;tentiam definitur; <lb/>nam &longs;i perexiguus fuerit exce&longs;&longs;us, movebitur quidem pondus, <lb/>&longs;ed tardè; &longs;in autem potentiæ virtus longè excedat ponderis <lb/>vires, eam celerior motus con&longs;equetur. </s> <s id="s.002787">Et hæc quidem intel­<lb/>ligi hactenus velim, quando potentia & pondus juxta æqualem <lb/>&longs;patij longitudinem pari velocitate promoventur, ut ip&longs;a expe­<lb/>rientia omnibus manife&longs;tum facit; nemo &longs;iquidem dubitat, an <lb/>currus à validioribus equis celerius quàm à debilibus canthe­<lb/>riis trahatur; & à robu&longs;tiore bajulo citiùs quàm ab imbecillio­<lb/>re onus in de&longs;tinatum locum transferri quotidie videmus. </s> </p> <p type="main"> <s id="s.002788">Ut igitur vecte pondus moveri valeat, lex hæc eadem &longs;tabi­<lb/>lis & firma permaneat, nece&longs;&longs;e e&longs;t, ut ponderis re&longs;i&longs;tentia mi­<lb/>nor &longs;it virtute potentiæ moventis. </s> <s id="s.002789">Quia verò re&longs;i&longs;tentia com­<lb/>ponitur ex innatâ ponderis gravitate, & ex motûs violenti tar­<lb/>ditate aut velocitate, hoc e&longs;t ex motûs huju&longs;modi quantitate <lb/>intra datam temporis men&longs;uram; propterea ita duo hæc tempe­<lb/>rari oportet, ut quod alteri additur, alteri dematur; ne adeò <lb/>re&longs;i&longs;tentia augeatur, ut jam minor non &longs;it virtute potentiæ. </s> <lb/> <s id="s.002790">Quare in vecte, cujus extremitati A potentia applicatur certæ <lb/>virtutis, ita &longs;tatuendus e&longs;t hypomochlio C locus, ut compara­<lb/>to motu potentiæ in A cum motu ponderis in B, ea &longs;it motûs B <pb pagenum="365" xlink:href="017/01/381.jpg"/>tarditas; quæ addita gravitati ponderis B re&longs;i&longs;tentiam compo­<lb/>nat minorem virtute movendi potentiæ A. <!-- KEEP S--></s> <s id="s.002791">Quoniam enim, <lb/>manente puncto C tanquam centro motús potentiæ de&longs;cenden­<lb/>tis & ponderis a&longs;cendentis, manife&longs;tum e&longs;t eam e&longs;&longs;e motuum <lb/>Rationem, quæ e&longs;t Radiorum CA & CB idcirco quò major <lb/>erit huju&longs;modi Radiorum inæqualitas, eò etiam major erit Ra­<lb/>tio motûs potentiæ ad motum ponderis, cujus tarditas gravi­<lb/>tatem compen&longs;ans minuet re&longs;i&longs;tentiam, ut virtuti potentiæ, pro­<lb/>portione re&longs;pondeat. </s> </p> <p type="main"> <s id="s.002792">Hic verò, &longs;i rem paulò attentiùs intro&longs;picias, deprehendes <lb/>tamdiu &longs;olùm admirationi e&longs;&longs;e machinarum vires, quamdiu <lb/>cau&longs;a occulta manet; quæ &longs;i in medium proferatur, admiratio­<lb/>ni nobis e&longs;t ip&longs;a no&longs;tra admiratio. </s> <s id="s.002793">Aio igitur potentiam tan­<lb/>tumdem plane motûs in pondere efficere cum vecte conjunctam <lb/>(idem de cæteri pariter Facultatibus intelligatur, ne idem &longs;æ­<lb/>piùs ad nau&longs;eam inculcare oporteat) ac &longs;i &longs;olitaria eodem cona­<lb/>tu pondus aliquod &longs;ecum pari velocitate adduceret, aut eleva­<lb/>ret. </s> <s id="s.002794">Sit potentia A æqualiter, ac pondus B, di&longs;tans à fulcro C; <lb/>& quo conatu movetur potentia de&longs;cendens &longs;patio digitorum <lb/>decem, dum arteria bis pul&longs;at; cogat oppo&longs;itum pondus libræ <lb/>unius a&longs;cendere pariter eodem tempore per digitos decem; e&longs;&longs;e <lb/>enim æquales oppo&longs;itos huju&longs;modi motus, qui ex æqualibus <lb/>Radiis arcus æquales de&longs;cribunt, certum e&longs;t. </s> <s id="s.002795">Jam manente Ra­<lb/>dio CA, finge Radium CB mutilum atque decurtatum adeò, <lb/>ut &longs;ola ejus pars decima reliqua &longs;it, & CB ponderis di&longs;tantia <lb/>ab hypomochlio &longs;it &longs;ubdecupla di&longs;tantiæ CA potentiæ ab eo­<lb/>dem hypomochlio: erit igitur motus in B &longs;ubdecuplus motûs <lb/>in A. <!-- KEEP S--></s> <s id="s.002796">Quare pondus unius libræ in hac &longs;ubdecuplâ di&longs;tantiâ <lb/>cùm &longs;ubdecuplo tardius moveatur (percurrit enim tempore eo­<lb/>dem &longs;patium &longs;ubdecuplum) indiget &longs;olùm &longs;ubdecuplo impetu <lb/>ejus, quem prius exigebat, ut æqualiter cum potentiâ move­<lb/>retur. </s> <s id="s.002797">Totus igitur impetus ille, quem potentia ponderi unius <lb/>libræ imprimebat, ut æquali velocitate pariter moverentur, illa <lb/>de&longs;cendendo, hoc a&longs;cendendo, &longs;i decem ponderibus &longs;imilibus <lb/>di&longs;tribuatur, &longs;atis e&longs;t, ut omnia illa moveantur &longs;ubdecuplâ ve­<lb/>locitate. </s> <s id="s.002798">Quia autem duorum arteriæ pul&longs;uum &longs;patio &longs;ingula <lb/>a&longs;cendunt digitum unum, & &longs;unt decem a&longs;cen&longs;us digitales, <lb/>dum potentia de&longs;cendit digitos decem, & dum potentia primo <pb pagenum="366" xlink:href="017/01/382.jpg"/>arteriæ pul&longs;u decurrit digitos quinque, decem illa pondera <lb/>motum quinque digitorum perficiunt, &longs;ingula videlicet per &longs;e­<lb/>midigitum (id quod pariter ob&longs;ervari facilè poterit in &longs;ingulis <lb/>minutioribus temporis particulis) tantumdem motus perficit <lb/>potentia ac pondus, &longs;ive toto impetu uni libræ impre&longs;&longs;o libra <lb/>una habeat motum decem digitorum, &longs;ive decimâ impetûs par­<lb/>te &longs;ingulis libris impre&longs;sâ, &longs;ingulæ habeant motum digitalem: <lb/>utrobique &longs;cilicet &longs;unt decem motus digitales, &longs;ive unius pon­<lb/>deris, &longs;ive decem ponderum eodem tempore. </s> <s id="s.002799">Quis verò mi­<lb/>retur, &longs;i ille idem, qui decem aureis nobili ho&longs;piti &longs;plendidio­<lb/>res epulas parare po&longs;&longs;et, decem hominibus frugalem men&longs;am <lb/>in&longs;trueret &longs;ingulis aureis in &longs;ingulos homines tributis? </s> <s id="s.002800">De&longs;inat <lb/>igitur pariter mirari, &longs;i potentia eadem, quæ decem impetûs <lb/>particulis libram unam &longs;ecum pari velocitate movet, &longs;ingulis <lb/>particulis in &longs;ingulas libras tributis moveat decem libras, &longs;in­<lb/>gulas &longs;ubdecupla velocitate; neque enim hic plus conatûs, <lb/>quàm ibi, requiritur. </s> </p> <p type="main"> <s id="s.002801">In hoc itaque Vectis vires &longs;itæ &longs;unt, quod ex Potentiæ & <lb/>Ponderis po&longs;itione ita temperantur motus, ut impetûs quem <lb/>potentia ponderi imprimere valet, aut re ip&longs;a imprimit, inten­<lb/>&longs;io re&longs;pondeat tarditati aut velocitati motûs ip&longs;ius ponderis. </s> <lb/> <s id="s.002802">Hinc &longs;i Potentia, & Pondus æqualibus intervallis ab hypomo­<lb/>chlio di&longs;tent; motus æquales &longs;unt; & perinde ac &longs;i potentia &longs;o­<lb/>litaria &longs;ine vecte (&longs;i illa quidem vivens &longs;it) attolleret pondus, <lb/>vectis nihil juvat potentiam, quia pondus hoc recipit totam <lb/>impetûs inten&longs;ionem, quam illa efficere pote&longs;t. </s> <s id="s.002803">Sin autem <lb/>Potentia quidem magis, Pondus verò minùs à fulcro ab&longs;it, tar­<lb/>dior ponderis motus minorem exigit impetûs inten&longs;ionem; ac <lb/>proinde entitas eadem impetûs, quæ e&longs;t inten&longs;ivè minor, po­<lb/>te&longs;t fieri exten&longs;ivè major, & communicari ponderi majori, ac <lb/>priùs. </s> <s id="s.002804">Quare pro Ratione tarditatis motûs extenuatur impetûs <lb/>inten&longs;io, atque ideò pro eadem Ratione augeri pote&longs;t ponderis <lb/>exten&longs;io, hoc e&longs;t gravitas; ut quæ Ratio e&longs;t velocitatis motûs <lb/>in pondere æqualis velocitati motûs in potentiâ, ad tarditatem <lb/>motûs in pondere minoris motu in potentiâ, eadem &longs;it directè <lb/>Ratio inten&longs;ionis impetûs in pondere æquè veloci ad inten&longs;io­<lb/>nem impetûs in pondere tardiori, & reciprocè eadem &longs;it Ratio <lb/>ponderis tardioris majoris ad pondus illud minus, quod æquè <pb pagenum="367" xlink:href="017/01/383.jpg"/>velociter cum potentia moveri pote&longs;t. </s> <s id="s.002805">Quòd &longs;i potentia pro­<lb/>pior fuerit hypomochlio, quàm pondus, potentia tardiùs, pon­<lb/>dus movetur velocius: plus igitur inten&longs;ionis impetûs requiri­<lb/>tur in pondere quàm in potentiâ, adeò ut impetus, qui in po­<lb/>tentiâ non vivente e&longs;t exten&longs;ivè major, inten&longs;ivè minor, con­<lb/>tra in pondere &longs;it exten&longs;ivè minor, inten&longs;ivè major: ac <lb/>propterea pondus tantò levius e&longs;&longs;e oportet pondere, quod æquè <lb/>velociter cum potentiâ moveretur, quantò velociùs movetur <lb/>præ illo æquè veloci. </s> <s id="s.002806">Non igitur vectis juvat potentiam, ut <lb/>faciliùs moveat, &longs;ed movendi difficultatem auget. </s> <s id="s.002807">Id quod in <lb/>Tertio vectis genere &longs;emper contingit, in quo potentia G mi­<lb/>nus ab hypomochlio I di&longs;tat, quam pondus H, & tardiùs mo­<lb/>vetur. </s> <s id="s.002808">Accidit autem hoc idem etiam in Primo genere, cum <lb/>vectis inæqualiter ab hypomochlio di&longs;tinguitur in partes, &longs;i lo­<lb/>ca permutentur, ut potentia propior &longs;it, quàm pondus. </s> <s id="s.002809">His <lb/>tamen uti po&longs;&longs;umus, quoties quidem viribus abundamus, &longs;ed <lb/>&longs;patium, in quo potentia moveatur, angu&longs;tum e&longs;t, oportet au­<lb/>tem ponderi velocem motum conciliare. </s> <s id="s.002810">Contra verò in vecte <lb/>Secundi generis potentia à fulcro &longs;emper remotior e&longs;t, quàm <lb/>pondus; idcirco &longs;emper juvat potentiam; quia quo tardior e&longs;t <lb/>ponderis motus, eò minorem ponderis pars, quæ æqualis &longs;it <lb/>ponderi æquè veloci, exigit impetûs inten&longs;ionem; ac propterea <lb/>quod reliquum e&longs;t impetûs à potentia producendi, pluribus <lb/>aliis &longs;imilibus ponderis partibus impertiri pote&longs;t; atque adeò <lb/>ab&longs;olutè majus e&longs;t pondus, quàm quod æquè velociter mo­<lb/>veretur. </s> </p> <p type="main"> <s id="s.002811">Hæc eadem, quæ de ponderibus vecte movendis dicta &longs;unt, <lb/>intelligi pariter oportet de ponderibus vecte &longs;u&longs;tentandis citra <lb/>motum; eo tantùm ob&longs;ervato di&longs;crimine, quod ad motum ma­<lb/>jor requiritur potentiæ virtus, quàm &longs;it ponderis re&longs;i&longs;tentia, in <lb/>&longs;u&longs;tentatione verò par re&longs;i&longs;tentiæ ponderis e&longs;t virtus potentiæ. </s> <lb/> <s id="s.002812">Re&longs;i&longs;tentia autem in &longs;u&longs;tentatione non ex motûs tarditate aut <lb/>velocitate, quæ re ip&longs;a &longs;it, &longs;ed ex eâ, quæ e&longs;&longs;et, &longs;i motus fieret, <lb/>quatenus pondus e&longs;t vecti connexum, definienda e&longs;t; & pro <lb/>huju&longs;modi momentorum Ratione, quibus pondus deor&longs;um co­<lb/>natur, etiam impetûs contranitentis inten&longs;ionem dimetiri ne­<lb/>ce&longs;&longs;e e&longs;t. </s> <s id="s.002813">Quia igitur pondus cum vecte connexum quo propiùs <lb/>ad hypomochlium accedit, eo tardiùs &longs;ibi relictum de&longs;cenderet; <pb pagenum="368" xlink:href="017/01/384.jpg"/>propterea etiam minorem contranitentis impetûs inten&longs;ionem <lb/>requirit: Ex quo fit eodem potentiæ conatu, quo illa pondus &longs;i­<lb/>ne vecte &longs;u&longs;tineret, po&longs;&longs;e majorem ponderis gravitatem &longs;u&longs;tineri <lb/>adhibito vecte, eóque majorem, quo major e&longs;t Ratio di&longs;tantiæ <lb/>potentiæ ad <expan abbr="di&longs;tantiã">di&longs;tantiam</expan> ponderis à fulcro; & vici&longs;&longs;im potentia mi­<lb/>nore conatu idem pondus &longs;u&longs;tinebit, &longs;i hoc propius admoveatur <lb/>ad hypomochlium, quàm priùs, cum opus erat majore conatu. </s> </p> <p type="main"> <s id="s.002814">Porrò conatum potentiæ de indu&longs;tria dixi, ut vocabulo ute­<lb/>rer, quo tum potentia vivens, tùm inanimata æquè comprehen­<lb/>deretur; quia aliquando quidem potentia conatum adhibet in­<lb/>natâ &longs;uâ gravitate, aliquando autem præter, aut contra gravitatis <lb/>propen&longs;ionem. </s> <s id="s.002815">Gravitate utitur, quæ inanima e&longs;t, & vires &longs;uas <lb/>exerit totas, quodcunque demum pondus vecte movendum aut <lb/>&longs;u&longs;tentandum proponatur. </s> <s id="s.002816">Potentia verò vivens &longs;uo con&longs;ulens <lb/>commodo, ne &longs;e inani conficiat labore, non plus operæ confert, <lb/>quàm opus fuerit, &longs;ed vires ex opportunitate admini&longs;trat, modò <lb/>majores, modò minores impendens, quippe quæ mu&longs;culorum <lb/>contentione voluntarios motus perficit, & non &longs;olùm deor&longs;um <lb/>premendo, &longs;ed etiam &longs;ur&longs;um connitendo, aut in tran&longs;ver&longs;um ur­<lb/>gendo, vecte uti pote&longs;t: At inanimata potentia non ni&longs;i de&longs;cen­<lb/>dendo vi &longs;uæ gravitatis cogere pote&longs;t adver&longs;um pondus ad a&longs;cen­<lb/>dendum; atque &longs;i primum vectis genus demas, cui pote&longs;t illa <lb/>proximè admoveri, in cæteris generibus, &longs;i attollendum &longs;it pon­<lb/>dus, artificium aliquod excogitandum e&longs;t, quo interjecto, aut <lb/>potentiæ virtus, aut ip&longs;um pondus ad vectem applicetur, ut <lb/>propo&longs;itum finem a&longs;&longs;equamur; conatus enim potentiæ & pon­<lb/>deris, licet inæquales, non tamen oppo&longs;iti &longs;unt, &longs;ed ad eandem <lb/>partem &longs;ua gravitate contendunt. </s> </p> <p type="main"> <s id="s.002817">Sic vecte RS, cujus fulcrum &longs;it in extremitate R, non pote&longs;t <lb/><figure id="id.017.01.384.1.jpg" xlink:href="017/01/384/1.jpg"/><lb/>pondus V attolli à po­<lb/>tentia inanimata P, &longs;i <lb/>proximè illi adjungatur <lb/>in S; ac propterea rotu­<lb/>la in T figenda e&longs;t ver­<lb/>&longs;atilis, & funiculo STP <lb/>jungenda potentia P, <lb/>quæ deor&longs;um connitens <lb/>elevat vectem in S, at-<pb pagenum="369" xlink:href="017/01/385.jpg"/>que adeò etiam pondus V. <!-- KEEP S--></s> <s id="s.002818">Simili ratione &longs;it vectis Secundi <lb/>generis MN, & hypomochlium in M, locus autem ponderis <lb/>in H: &longs;i potentia N inanimata vecti proximè adnectatur, uti­<lb/>que elevare non poterit pondus in H collocatum: quare &longs;ta­<lb/>tuatur in loco &longs;uperiore rotula K, & funiculo HKL jungatur <lb/>pondus L cum puncto H; nam potentia N &longs;ua gravitate de&longs;cen­<lb/>dens deprimendo punctum H vectis elevabit pondus L. <!-- KEEP S--></s> <s id="s.002819">Idem <lb/>continget, &longs;i vectis MN &longs;it tertij generis, & N &longs;it pondus at­<lb/>tollendum, potentia verò inanimata collocanda &longs;it in H. <!-- KEEP S--></s> <s id="s.002820">Nihil <lb/>utique præ&longs;tabit de&longs;cendendo in H; ut igitur punctum H <lb/>a&longs;cendat, rotula K adhibeatur, & à potentia L de&longs;cendente <lb/>elevabitur idem punctum H, ac proinde etiam pondus N. <!-- KEEP S--></s> <s id="s.002821">Vel <lb/>&longs;i in vecte RS tertij generis &longs;tatuatur potentia V, illa de&longs;cen­<lb/>dens deprimet velociter extremitatem S, & pari velocitate <lb/>a&longs;cendet pondus P. <!-- KEEP S--></s> <s id="s.002822">Quid hoc &longs;implex artificium aliquando in <lb/>&longs;cenicis motionibus præ&longs;tare po&longs;&longs;it emolumenti, facilè prudens <lb/>machinator intelligit. </s> </p> <p type="main"> <s id="s.002823">Ex his, quæ de Vectis viribus explicata &longs;unt, apertè liquet <lb/>omnino veritati con&longs;entanea e&longs;&longs;e ea, quæ lib. 2. cap. 8. diximus, <lb/>in rotis curruum inveniri non po&longs;&longs;e rationem vectis, quia duo <lb/>tantummodo &longs;unt puncta, &longs;cilicet extremitas Radij &longs;ubjectam <lb/>tellurem tangentis, & rotæ centrum, cui & innititur pondus, <lb/>& medio temone applicatur potentia. </s> <s id="s.002824">Cum igitur potentia & <lb/>pondus eandem habeant po&longs;itionem, & æquali velocitate mo­<lb/>veantur, nullum habetur ex Vectis rationibus compendium. </s> <lb/> <s id="s.002825">Eatenus enim Vectis in Mechanicarum Facultatum cen&longs;u nu­<lb/>meratur, quoad potentia & pondus di&longs;pari celeritate moventur, <lb/>vel quia potentia &longs;e velociter movens exiguo conatu tardè mo­<lb/>vet pondus, ut in primo & &longs;ecundo genere vectis, vel quia po­<lb/>tentia &longs;e tardè movens multo conatu celeriter movet pondus, <lb/>ut in tertio genere. </s> <s id="s.002826">Quare &longs;emper in motu ponderis per vectem <lb/>aliquid lucri habetur, nimirum aut major ponderis gravitas, <lb/>quæ movetur, aut &longs;altem major velocitas, qua movetur. <pb pagenum="370" xlink:href="017/01/386.jpg"/> </s> </p> <p type="main"> <s id="s.002827"><emph type="center"/>CAPUT II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002828"><emph type="center"/><emph type="italics"/>Quid in hypomochlij collocatione &longs;it oh&longs;ervandum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002829">TRia in Vecte, ut dictum e&longs;t, puncta con&longs;tituuntur & de­<lb/>&longs;ignantur duo quæ moventur, tertium illorum motuum <lb/>centrum, quod alicui corpori innititur, ut vectis con&longs;i&longs;tat, nec <lb/>à ponderis gravitate, aut à potentiæ vi abripiatur: huic cor­<lb/>pori <emph type="italics"/>Hypomochlio<emph.end type="italics"/> nomen inditum e&longs;t à Græcis, qua&longs;i (&longs;i verbum <lb/>è verbo volumus) <emph type="italics"/>&longs;ubvectis,<emph.end type="italics"/> nam ut plurimum vecti &longs;ubjicitur, <lb/>nos <emph type="italics"/>Fulcrum<emph.end type="italics"/> dicimus, quia vectem &longs;ibi incumbentem fulcit. </s> <lb/> <s id="s.002830">Cæterùm non e&longs;t hæc con&longs;tans, & perpetua hujus corporis <lb/>po&longs;itio, ut &longs;ub vecte &longs;it, quamvis &longs;emper Hypomochlij aut Ful­<lb/>cri nomine donetur; quandoquidem in vecte tertij generis, ubi <lb/>pondus in extremitate e&longs;t, potentia medium locum obtinet, &longs;i <lb/>infra alteram vectis extremitatem e&longs;&longs;et corpus huju&longs;modi, uti­<lb/>que à potentia nequiret attolli pondus, ut patet: in &longs;uperiore <lb/>igitur parte &longs;it oportet, ut potentiâ &longs;ur&longs;um conante, pondere <lb/>deor&longs;um contranitente, impediatur altera vectis extremitas, ne <lb/>fiat totius vectis conver&longs;io ob&longs;ecundans aut potentiæ conatui, <lb/>aut gravitati ponderis, quod e&longs;&longs;et attollendum. </s> <s id="s.002831">Quod &longs;i hoc <lb/>vecte tertij generis deprimendum e&longs;&longs;et infra aquam per vim <lb/>corpus aliquod leve, tunc &longs;ub vecte con&longs;titueretur hypomo­<lb/>chlium: contrà vectis primi & &longs;ecundi generis &longs;i ad premen­<lb/>dum aut deprimendum adhibeatur, exigit hypomochlium in <lb/>&longs;uperiori parte. </s> <s id="s.002832">Similiter non e&longs;t &longs;ub vecte, &longs;ed ad latus adja­<lb/>cet, quoties pondus e&longs;t movendum in plano horizontali, &longs;ive in <lb/>eodem plano &longs;it vectis, &longs;ive in plano verticali, ut cùm duo mar­<lb/>mora non elevanda &longs;unt, &longs;ed immi&longs;&longs;o inter illa vecte invicem <lb/>disjungenda. </s> <s id="s.002833">Quemadmodum igitur lapis à lædendo pedem <lb/>vocabulum habet, etiam&longs;i non lapides omnes pedem lædant; <lb/>ita corpus illud, cui punctum vectis quie&longs;cens innititur, hypo­<lb/>mochlij & fulcri nomen retinet, quamvis non &longs;emper &longs;ub vecte <lb/>&longs;it, illúmque &longs;uffulciat. </s> <s id="s.002834">Quid autem profuerit immutare vo-<pb pagenum="371" xlink:href="017/01/387.jpg"/>cabula, ubi rem ip&longs;am tenemus? </s> <s id="s.002835">Immò punctum ip&longs;um vectis <lb/>quie&longs;cens, quod hypomochlio re&longs;pondet, non raro ab iis hy­<lb/>pomochlium dicitur, aut fulcrum, qui verborum compendio <lb/>claritati con&longs;ultum volunt; mihíque hanc loquendi facultatem, <lb/>ubi res tulerit, re&longs;ervo. </s> </p> <p type="main"> <s id="s.002836">Quie&longs;cens autem voco punctum vectis, quod e&longs;t centrum <lb/>motuum potentiæ, & ponderis; non quia &longs;emper omnino <lb/>quie&longs;cat, &longs;ed quia &longs;i aliquo motu moveatur, tardi&longs;&longs;imum certè <lb/>e&longs;t omnium punctorum; cætera quippe vectis puncta circa hoc <lb/>tanquam circa centrum de&longs;cribunt lineam inflexam ac recur­<lb/>vam: alioquin &longs;i punctum hoc plus moveretur quàm pondus, <lb/>mutatæ fui&longs;&longs;ent vices, & quod pondus dicitur, e&longs;&longs;et reip&longs;a hy­<lb/>pomochlium, corpus verò, quod hypomochlium dicitur, e&longs;&longs;et <lb/>pondus, quod à potentiâ poti&longs;&longs;imum moveretur. </s> <s id="s.002837">Ob&longs;ervandum <lb/>enim e&longs;t non pondus &longs;olum, verùm etiam hypomochlium acci­<lb/>pere vim externam potentiæ vectem agitantis, re&longs;i&longs;tente vide­<lb/>licet pondere, ex quo fit illud premi; quod &longs;i inæqualiter re­<lb/>&longs;i&longs;tant, licet utrumque moveatur, in illud potiùs exercet vir­<lb/>tutem &longs;uam potentia, quod languidiùs re&longs;i&longs;tit, altero validiore <lb/>hypomochlij rationem habente. </s> <s id="s.002838">Sic vecti ad attollendum mar­<lb/>mor applicato &longs;i glebam, hypomochlij loco, &longs;uppo&longs;ueris, non <lb/>marmor attolles, &longs;ed glebam vecte conteres: marmor igitur e&longs;t <lb/>hypomochlium vecti &longs;uperpo&longs;itum, & glebæ e&longs;t pondus contri­<lb/>tum vecte &longs;ecundi generis: At &longs;i pro gleba lignum &longs;ubjicias, <lb/>quod non frangatur, &longs;ed aliquantulum cedens comprimatur, & <lb/>vectis ve&longs;tigium recipiat, ita tamen, ut marmor moveatur, du­<lb/>plex vectis genus hic intercedit, prout duplex effectus poten­<lb/>tiæ conatum con&longs;equitur; ad comprimendum &longs;cilicet lignum <lb/>vectis e&longs;t &longs;ecundi generis hypomochlium habens impo&longs;itum <lb/>marmor, ad elevandum autem marmor vectis e&longs;t primi generis, <lb/>cujus hypomochlium e&longs;t &longs;ubjectum lignum. </s> <s id="s.002839">Cuju&longs;modi &longs;it hy­<lb/>pomochlium, &longs;ive &longs;it funis vectem retinens, &longs;ive axis infixus, <lb/>circa quem volvatur vectis, &longs;ive quodcumque aliud corpus, cui <lb/>ille incumbat, aut innitatur, modò ab&longs;it incommodi periculum <lb/>ex ejus fragilitate, parum refert: &longs;atis e&longs;t, &longs;i par fuerit ferendo <lb/>oneri, quod vecte elevatur. </s> <s id="s.002840">Ex ponderis autem gravitate hy­<lb/>pomochlij &longs;oliditas atque materies definienda e&longs;t; ex motùs <pb pagenum="372" xlink:href="017/01/388.jpg"/>qualitate (&longs;pectatâ loci, in quo perficiendus e&longs;t, po&longs;itione) <lb/>forma hypomochlij &longs;tatuatur. </s> </p> <p type="main"> <s id="s.002841">Illud examinandum videtur, quandó nam præ&longs;ter uti vecte <lb/>primi generis, quando vecte Secundi generis, hoc e&longs;t an plus <lb/>commodi a&longs;&longs;erat fulcrum in vectis extremitate collocatum, ut <lb/>in &longs;ecundo genere, an verò inter pondus atque potentiam in­<lb/>terjectum, ut in primo genere. </s> <s id="s.002842">Propo&longs;ita &longs;it vectis longitudo <lb/>decem palmorum, quo oporteat pondus ita attollere, ut ejus <lb/>motus &longs;it re&longs;pondens arcui de&longs;cripto ex Radio duorum palmo­<lb/>rum. </s> <s id="s.002843">Si vectis &longs;it primi generis, pondus & potentia &longs;unt in <lb/>vectis extremitatibus, hypomochlium dividit totam longitudi­<lb/>nem in partes duas, quarum major ad potentiam &longs;pectans e&longs;t <lb/>quadrupla minoris &longs;pectantis ad pondus; e&longs;t &longs;cilicet illa octo, <lb/>hæc duorum palmorum. </s> <s id="s.002844">At &longs;i vectis fuerit &longs;ecundi generis, <lb/>hypomochlium & potentia illius extremitates occupant, pon­<lb/>dus ab hypomochlio di&longs;tat palmos duos: quare potentiæ di&longs;tan­<lb/>tia ab hypomochlio cum &longs;it tota vectis longitudo, e&longs;t quintu­<lb/>pla di&longs;tantiæ ponderis. </s> <s id="s.002845">Cum igitur ponderis motus cum po­<lb/>tentiæ motu comparatus hic quintuplo tardior &longs;it, ibi verò &longs;o­<lb/>lum quadruplo tardior, minore impetu indiget, ut moveatur <lb/>vecte &longs;ecundi generis. </s> <s id="s.002846">Cæterùm con&longs;iderato hoc duplici vectis <lb/>genere, ob&longs;ervandum e&longs;t in &longs;ecundo genere à potentia elevan­<lb/>dum non &longs;olum pondus &longs;ed etiam vectem ip&longs;um, qui &longs;i valde <lb/>gravis &longs;it (ut aliquando contingere pote&longs;t trabem fungi vectis <lb/>munere) auget potentiæ movendi difficultatem: Contra verò <lb/>in vecte primi generis ip&longs;a vectis gravitas juvat potentiam; & <lb/>quidem &longs;i homo &longs;it, qui vectem premat, ip&longs;a corporis gravitas <lb/>acce&longs;&longs;ionem facit, ad impetum, qui à vitali conatu oritur: præ­<lb/>terquam quod hic liberè & facillimè potentiam inanimatam <lb/>adhibere po&longs;&longs;umus, & aliam atque aliam adjicere prout opus <lb/>fuerit; at non item in vecte &longs;ecundi generis, ni&longs;i adhibito arti­<lb/>ficio, de quo &longs;uperiori capite dictum e&longs;t. </s> </p> <p type="main"> <s id="s.002847">Datâ igitur ponderis movendi gravitate, & datâ potentiæ <lb/>virtute (quæ videlicet tanto conatu adhibito pote&longs;t certam gra­<lb/>vitatem &longs;ola &longs;ine vecte movere in &longs;imili plano &longs;ive horizontali, <lb/>&longs;ive inclinato, &longs;ive verticali) di&longs;tinguatur vectis in duas partes <lb/>ita, ut vel pars ad partem, &longs;i &longs;it primi generis, vel totus ad par-<pb pagenum="373" xlink:href="017/01/389.jpg"/>tem, &longs;i &longs;it &longs;ecundi generis, eandem Rationem habeat, quæ <lb/>e&longs;t dati ponderis ad pondus, quod à potentia &longs;olâ &longs;ine vecte po­<lb/>te&longs;t moveri. </s> <s id="s.002848">Sic data Potentia virtutem habeat movendi pon­<lb/>dus lib. 6. certo conatu, oporteat autem hoc codem conatu <lb/>movere lib. 30: quia virtus potentiæ e&longs;t &longs;ubquintupla pon­<lb/>deris dati, propo&longs;itus vectis intelligatur primùm di&longs;tinctus <lb/>in partes &longs;ex, quarum una tribuatur di&longs;tantiæ ponderi, ab <lb/>hypomochlio, reliquæ quinque tribuantur di&longs;tantiæ poten­<lb/>tiæ, ita ut reciprocè &longs;it di&longs;tantia potentiæ ad di&longs;tantiam <lb/>ponderis, ut pondus datum ad virtutem potentiæ: & hic <lb/>e&longs;t vectis primi generis. </s> <s id="s.002849">Deinde ut habeatur vectis &longs;ecun­<lb/>di generis, di&longs;tinguatur totus vectis in partes quinque, & <lb/>una ex illis &longs;it di&longs;tantia ponderis ab hypomochlio in vectis <lb/>extremitate con&longs;tituto. </s> <s id="s.002850">In utroque enim ca&longs;u motus poten­<lb/>tiæ e&longs;t quintuplus motûs ponderis, atque adeò potentia <lb/>poterit vecte movere pondus quintuplum ponderis, quod &longs;o­<lb/>la pote&longs;t movere. </s> </p> <p type="main"> <s id="s.002851">Potentiæ virtutem dixi, non potentiæ gravitatem, tùm <lb/>quia non omnis potentia vim movendi habet ex gravitate, <lb/>tum quia potentiæ gravitas movere non pote&longs;t gravitatem <lb/>æqualem, &longs;ed minorem, nam cum æquali facit æquili­<lb/>brium, & &longs;olùm pote&longs;t illam &longs;u&longs;pendere. </s> <s id="s.002852">Quare &longs;i poten­<lb/>tia vi &longs;uæ gravitatis moveat, non &longs;atis erit, &longs;i fiat ut po­<lb/>tentiæ gravitas ad ponderis gravitatem, ita reciprocè pon­<lb/>deris di&longs;tantia à centro motús ad di&longs;tantiam potentiæ ab eo­<lb/>dem centro; &longs;ed di&longs;tantia ponderis ad di&longs;tantiam potentiæ <lb/>exigit habere minorem Rationem. <!-- KEEP S--></s> <s id="s.002853">Hinc &longs;i potentia &longs;it pon­<lb/>deris &longs;ubquintupla ratione &longs;uarum gravitatum, pondus ab <lb/>hypomochlio di&longs;tare debet minus quàm parte quinta di&longs;tan­<lb/>tiæ potentiæ ab eodem hypomochlio. </s> <s id="s.002854">Quod &longs;i vectis is e&longs;&longs;et, <lb/>cujus gravitas notabile momentum adderet potentiæ, tunc <lb/>di&longs;tantia ponderis, quæ e&longs;&longs;et &longs;ubquintupla di&longs;tantiæ potentiæ, <lb/>&longs;ufficeret, minor enim e&longs;&longs;et Ratione potentiæ adæquatè ac­<lb/>ceptæ ad Pondus. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002855">Ubi verò ponderis gravitatem con&longs;iderare oportet, non <lb/>&longs;atis e&longs;t illam notam habere, ac &longs;i &longs;taterâ expenderetur, <lb/>&longs;ed con&longs;iderandum e&longs;t planum, in quo illud movendum e&longs;t; <lb/>neque enim eadem habet momenta, &longs;i &longs;ur&longs;um elevandum &longs;it <pb pagenum="374" xlink:href="017/01/390.jpg"/>in plano Verticali, ac &longs;i urgendum &longs;it in plano inclinato, <lb/>aut propellendum in horizontali: propterea in Ratione a&longs;­<lb/>&longs;ignandà partibus vectis non e&longs;t attendenda gravitas ab&longs;oluta <lb/>ponderis, &longs;ed quatenus in propo&longs;ito plano. </s> <s id="s.002856">Idem e&longs;t de gravi­<lb/>tate potentiæ dicendum. </s> </p> <p type="main"> <s id="s.002857">Ex dictis patet non quamcumque vectis longitudinem &longs;em­<lb/>per opportunam e&longs;&longs;e, quamvis verum &longs;it quemlibet vectem <lb/>po&longs;&longs;e &longs;ecundùm quamcumque Rationem in partes di&longs;tingui, <lb/>atque proinde quodcumque pondus à quacumque datâ po­<lb/>tentiâ po&longs;&longs;e moveri, &longs;i ritè applicari po&longs;&longs;et. </s> <s id="s.002858">Unum enim <lb/>e&longs;t incommodum, quod, quo propiùs ad centrum motuum <lb/>admovetur pondus, eo minor e&longs;t illius motus: & continge­<lb/>re pote&longs;t adeò exiguam e&longs;&longs;e ponderis ab hypomochlio di&longs;tan­<lb/>tiam, ut motus adeò tenuis nulli futurus &longs;it u&longs;ui. </s> <s id="s.002859">Qua­<lb/>propter longiori vecte utendum erit, ut, &longs;ervatâ eâdem <lb/>di&longs;tantiarum Ratione, intervallum inter pondus & centrum <lb/>motuum &longs;it notabile & con&longs;picuum, ex quo motus &longs;uffi­<lb/>ciens obtineri po&longs;&longs;it. </s> <s id="s.002860">Quid enim juvaret, &longs;i vecte palmo­<lb/>rum 25 tentares attollere pondus centuplum virtutis poten­<lb/>tiæ? </s> <s id="s.002861">an ut pondus ab hypomochlio di&longs;tans per digitum (&longs;u­<lb/>mo digitos quatuor pro &longs;ingulis palmis) elevaretur ad altitu­<lb/>dinem unius aut alterius grani hordei? </s> <s id="s.002862">Præterquam quod <lb/>tam ingens pondus ægrè po&longs;&longs;et in tantillo &longs;patio ad vectem <lb/>opportunè applicari. </s> </p> <p type="main"> <s id="s.002863">Quod autem ad hypomochlium attinet, curandum maxi­<lb/>mè e&longs;t, ut qua parte vectem contingit, minimum &longs;it, &, &longs;i <lb/>fieri pote&longs;t, proximè in aciem de&longs;inat; ut &longs;cilicet eandem <lb/>&longs;emper in motu vectis partem contingat; &longs;i enim alia atque <lb/>alia vectis pars hypomochlio in&longs;i&longs;tat, mutantur ponderis at­<lb/>que potentiæ momenta, ideoque augeri pote&longs;t movendi diffi­<lb/><figure id="id.017.01.390.1.jpg" xlink:href="017/01/390/1.jpg"/><lb/>cultas. </s> <s id="s.002864">Sit vectis &longs;ecundi generis AB <lb/>innixus &longs;axo, quod contingit in C, <lb/>& centri gravitatis ponderis locus &longs;it <lb/>D: utique quia DC minore&longs;t quàm <lb/>DB, major e&longs;t Ratio AB ad DC mi­<lb/>norem, quàm eju&longs;dem AB ad DB <lb/>majorem, per 8. lib. 5. At elevato <lb/>vecte, ut habeat po&longs;itionem FE, &longs;i-<pb pagenum="375" xlink:href="017/01/391.jpg"/>cut A venit in F, ita B venit in E, ubi &longs;axo innititur, & pon­<lb/>dus D venit in G. <!-- KEEP S--></s> <s id="s.002865">E&longs;t igitur FE ad GE, ut AB ad DB; ergo <lb/>etiam FE ad GE habet minorem Rationem quàm AB ad DC. <!-- KEEP S--></s> <lb/> <s id="s.002866">Quo autem minor e&longs;t motuum Ratio, eò etiam minus e&longs;t po­<lb/>tentiæ momentum ad momentum ponderis; igitur &longs;i Ratio AB <lb/>ad DB minor &longs;it, quàm AC ad DC, etiam Ratio FE ad GE <lb/>minor erit quàm Ratio AC ad DC. <!-- KEEP S--></s> <s id="s.002867">Quare tunc &longs;olùm ea­<lb/>dem movendi facilitas manebit (quod quidem &longs;pectat ad ra­<lb/>tionem hypomochlij quicquid &longs;it an ex alio capite mutetur, ut <lb/>infra) quando CB pars extrema vectis, quæ innititur hypo­<lb/>mochlio, ea e&longs;t, ut eadem &longs;it Ratio AB ad DB, quæ e&longs;t AC <lb/>ad DC: Hoc autem fieri omnino non pote&longs;t, quia AB & DB <lb/>&longs;unt idem ac AC, atque DC, &longs;i his utri&longs;que addatur eadem <lb/>pars CB. <!-- KEEP S--></s> <s id="s.002868">Si ergo ut AC plus CB ad DC plus CB e&longs;&longs;et ut <lb/>AC ad DC, etiam permutando, & dividendo, & iterum per­<lb/>mutando, per 16. & 17. lib. 5. e&longs;&longs;et ut AC ad DC ita CB ad <lb/>CB, ac propterea AC totum æquale e&longs;&longs;et parti DC. <!-- KEEP S--></s> <s id="s.002869">Non <lb/>igitur fieri pote&longs;t, ut maneat in motu eadem facilitas ratione <lb/>hypomochlij, &longs;i accidat, ut vectis po&longs;itiones in motu &longs;e de­<lb/>cu&longs;&longs;ent; id quod evenit, &longs;i alia atque alia pars vectis hypomo­<lb/>chlium tangat. </s> <s id="s.002870">Et quia major e&longs;t Ratio totius AB ad totam <lb/>DB, quàm &longs;it ablatæ CB ad ablatam CB, erit etiam, per 33. <lb/>lib. 5. reliquæ AC ad reliquam DC major Ratio quàm totius <lb/>AB ad totam DB, hoc e&longs;t major Ratio quàm FE ad GE. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002871">Similiter in vecte primi generis, &longs;i fulcrum &longs;it cylindricum, <lb/>tangit quidem in puncto, &longs;ed dum vectis deor&longs;um urgetur, <lb/>aliud atque aliud ejus punctum aliis cylindri punctis congruit: <lb/>nam &longs;i fuerit potentia in C, & pondus <lb/><figure id="id.017.01.391.1.jpg" xlink:href="017/01/391/1.jpg"/><lb/>in E, vectis autem tangat in F, in con­<lb/>ver&longs;ione cum E venerit in I, & C in L, <lb/>jam contactus fit in H ita, ut HL minor <lb/>&longs;it quàm FC, contrà verò HI major &longs;it <lb/>quàm FE. <!-- KEEP S--></s> <s id="s.002872">Decre&longs;cunt ergo potentiæ <lb/>momenta, cujus di&longs;tantia à motûs centro <lb/>minuitur, augentur autem ponderis momenta, cujus di&longs;tan­<lb/>tiæ à motûs centro aliquid &longs;emper accedit. </s> <s id="s.002873">Et quidem quò <lb/>cra&longs;&longs;ior fuerit cylindrus, factâ pari vectis inclinatione, major <lb/>etiam oritur di&longs;tantiarum differentia; ut facilè demon&longs;tratur, <pb pagenum="376" xlink:href="017/01/392.jpg"/>&longs;i duo circuli &longs;e intus contingant in O, ubi vectem &longs;u&longs;tinent, <lb/>& deinde vectis inclinetur, ut faciat angulum OIG tangens <lb/><figure id="id.017.01.392.1.jpg" xlink:href="017/01/392/1.jpg"/><lb/>cylindrum minorem <lb/>in G, aut faciat angu­<lb/>lum OHS illi æqua­<lb/>lem tangens cylin­<lb/>drum majorem in S: <lb/>duo &longs;i quidem trian­<lb/>gula IRG & HMS <lb/>&longs;unt æquiangula, quia <lb/>vectes CK & BD &longs;unt <lb/>paralleli ex hypothe­<lb/>&longs;i, lineæ verò à centris R & M ad puncta contactuum G & S <lb/>ductæ cadunt ad angulos rectos, ex 18. lib. 3. quapropter & an­<lb/>guli ad centra R & M &longs;unt æquales: igitur etiam arcus OG <lb/>& OS &longs;unt &longs;imiles in Ratione &longs;uarum &longs;emidiametrorum OR <lb/>& OM: major ergo e&longs;t arcus OS quàm arcus OG, ac <lb/>propterea illi major quàm huic vectis pars in conver&longs;ione apta­<lb/>tur, adeóque di&longs;tantia ponderis ab hypomochlio minùs auge­<lb/>tur ab O in G, quàm ab O in S, factâ æquali vectis inclina­<lb/>tione. </s> <s id="s.002874">Illud tamen habetur compendij, &longs;i cra&longs;&longs;ior cylindrus <lb/>vecti &longs;upponatur, quod non adeò inclinandus &longs;it vectis, ut ad <lb/>certam altitudinem attollatur pondus, ac illum inclinare opor­<lb/>teret, &longs;i exilior cylindrus fulcri munere fungeretur. </s> </p> <p type="main"> <s id="s.002875">Quæ de cylindro dicta &longs;unt, manife&longs;ta quoque apparent, &longs;i <lb/>hypomochlium planum &longs;it, ut OS: e&longs;t nimirum longè alia <lb/><figure id="id.017.01.392.2.jpg" xlink:href="017/01/392/2.jpg"/><lb/>Ratio VO ad OR atque XS ad ST; <lb/>nam additur ip&longs;i OR longitudo OS, ut <lb/>habeatur ST. <!-- KEEP S--></s> <s id="s.002876">Cum ergo minor &longs;it poten­<lb/>tiæ di&longs;tantia XS, quàm VO, minora &longs;unt <lb/>potentiæ momenta: contra verò cum ma­<lb/>jor &longs;it ponderis di&longs;tantia TS, quàm RO, <lb/>majora pariter &longs;unt ponderis momenta. </s> <s id="s.002877">Ut itaque in vectis mo­<lb/>tu momentorum Ratio &longs;tabilis ac firma per&longs;everet, &longs;atius e&longs;t <lb/>hypomochlium vecti objicere aciem anguli, in quem duæ &longs;ub­<lb/>jecti corporis facies concurrunt, aut vecti axem infigi, circa <lb/>quem ille convolvatur. <pb pagenum="377" xlink:href="017/01/393.jpg"/> </s> </p> <p type="main"> <s id="s.002878"><emph type="center"/>CAPUT III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002879"><emph type="center"/><emph type="italics"/>Qua Ratione &longs;tatuendus &longs;it ponderi locus in Vecte <lb/>primi generis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002880">QUoniam pondus vecte movendum non e&longs;t corpus aliquod <lb/>planè individuum, &longs;ed partes habet, quarum aliæ &longs;unt <lb/>puncto fulcri, hoc e&longs;t, centro motûs, propiores, aliæ remotio­<lb/>res; animum diligenter advertere opus e&longs;t, cuinam vectis <lb/>puncto intelligendum &longs;it adjunctum onus, ut ex eo ad fulcrum <lb/>di&longs;tantia determinetur. </s> <s id="s.002881">Et quidem vix cuiquam dubium e&longs;&longs;e <lb/>pote&longs;t, an inter omnia ponderis puncta illud unum eligendum <lb/>&longs;it, in quo gravitas vires &longs;uas omnes exercere intelligitur, vi­<lb/>delicet circa quod paribus momentis deor&longs;um nititur, &longs;i ip&longs;a &longs;ibi <lb/>relinquatur: hoc autem e&longs;t Gravitatis centrum ip&longs;i ponderi in­<lb/>&longs;itum, in quod &longs;ingularum partium conatus confluere, & &longs;e­<lb/>cundùm quod per directionis lineam deor&longs;um vectem urgeri <lb/>concipimus. </s> </p> <p type="main"> <s id="s.002882">Sit enim pondus P, quod vecti AB infixum, & longitudini <lb/>AC congruens, &longs;uo gravitatis centro I deor&longs;um nititur per li­<lb/>neam directionis IH. <!-- KEEP S--></s> <s id="s.002883">Dico vectem <lb/>perinde à toto pondere urgeri, atque <lb/><figure id="id.017.01.393.1.jpg" xlink:href="017/01/393/1.jpg"/><lb/>&longs;i tota ejus gravitas e&longs;&longs;et in puncto I, <lb/>atque ideò di&longs;tantiam ponderis ab <lb/>hypomochlio D e&longs;&longs;e, neque AD ma­<lb/>ximam, neque CD minimam, &longs;ed <lb/>ID mediam: quia, et&longs;i partibus &longs;in­<lb/>gulis &longs;ua in&longs;it gravitas, & &longs;ingula pro &longs;uâ à puncto D di&longs;tantia <lb/>&longs;ua habeant momenta, ita majora momenta remotiorum parti­<lb/>cularum à minoribus vicinarum compen&longs;antur, ut intelligenda <lb/>&longs;it vel tota gravitas in media di&longs;tantia ID vel &longs;emi&longs;&longs;is gravitatis <lb/>in extrema di&longs;tantia AD, prout lib. 3. cap. 2. de momentis bra­<lb/>chiorum inæqualium libræ o&longs;ten&longs;um e&longs;t. </s> <s id="s.002884">Hoc autem, quod <lb/>de pondere &longs;ecundùm molem & gravitatem æquabili dicitur, <lb/>etiam de ponderibus, quorum anomala e&longs;t figura, vel ex diver-<pb pagenum="378" xlink:href="017/01/394.jpg"/>&longs;is &longs;ecundùm &longs;peciem gravitatibus compo&longs;ita, intelligendum <lb/>e&longs;t, &longs;i eorum centro gravitatis congruat vectis longitudo; nam <lb/>ponderis di&longs;tantia non e&longs;t Arithmeticè media inter maximam & <lb/>minimam, &longs;ed e&longs;t intervallum, quod inter fulcrum & centrum <lb/>gravitatis interjicitur. </s> </p> <p type="main"> <s id="s.002885">Sed quia non rarò pondus aut vecti totum incumbit, aut plu­<lb/>ribus funiculis firmiter alligatum ex illo &longs;u&longs;penditur, propterea <lb/>ob&longs;ervandum e&longs;t, in quod vectis punctum incidat Directionis <lb/>linea ex centro gravitatis ponderis ducta; hæc enim definiet <lb/>di&longs;tantiam ponderis ab hypomochlio, & innote&longs;cent momenta, <lb/>quibus illud re&longs;i&longs;tit potentiæ elevanti. </s> <s id="s.002886">Id quod per libram <lb/>æqualium brachiorum (ne illorum inæqualitas aliquam pariat <lb/>difficultatem) in&longs;tituto æquilibrio facillimè experiri poteris, &longs;i <lb/>laminas ligneas, aut metallicas, in varias figuras conformave­<lb/>ris, in quibus centrum gravitatis inventum fuerit, & ita &longs;ingu­<lb/>las &longs;ecundùm unum latus immobiliter uni brachio aptaveris, ut <lb/>illi congruant, atque in oppo&longs;itâ jugi extremitate æquipon­<lb/>dium addideris; facto enim æquilibrio, & demi&longs;&longs;o perpendicu­<lb/>lo per centrum gravitatis notatum tran&longs;eunte, apparebit, cui­<lb/>nam libræ puncto re&longs;pondeat; atque inter hoc punctum, & cen­<lb/>trum motús libræ, di&longs;tantia erit ad reliqui brachij totam lon­<lb/>gitudinem, ut æquipondij gravitas ad ponderis examinati gra­<lb/>vitatem. </s> </p> <p type="main"> <s id="s.002887">Quod &longs;i pondus ex unico fune pendulum adnectatur vecti, <lb/>&longs;atis con&longs;tat, ex quo vectis puncto de&longs;umatur ejus di&longs;tantia, ni­<lb/>mirum ex puncto &longs;u&longs;pen&longs;ionis; intentus enim funis à pendente <lb/>gravitate lineam Directionis o&longs;tendit. </s> <s id="s.002888">Quamvis autem &longs;i hujus <lb/>puncti tantummodo ratio habeatur, eadem videantur futura <lb/>ponderis momenta, quæcumque tandem fuerit vectis po&longs;itio <lb/>&longs;ive horizonti parallela, &longs;ive obliqua, examinandum tamen <lb/>erit inferius cap.8. utrum ratione anguli, &longs;ecundùm quem pon­<lb/>dus deor&longs;um 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/>unico fune pendulum, con&longs;ideremus, &longs;ive vecti incumbat, &longs;ive <lb/>infra vectem collocetur; hoc nimirum e&longs;t illud, in quo, propo­<lb/>&longs;itis majoribus ponderibus, non videtur connivendum; neque <lb/>enim nihil refert, utrùm infra, an &longs;upra vectem &longs;it movendæ <lb/>gravitatis centrum, quantóque intervallo hoc ab illo ab&longs;it, ibi <pb pagenum="379" xlink:href="017/01/395.jpg"/>&longs;i quidem gravitas collocata intelligitur, ubi &longs;uas omnes vires <lb/>omnium partium con&longs;piratione exercet. </s> <s id="s.002890">Quapropter, ut pon­<lb/>deris momenta innote&longs;cant, centri gravitatis motum perpen­<lb/>dere, ac dimetiri oportet. </s> <s id="s.002891">Hinc e&longs;t pondus firmiter adnexum <lb/>vecti perinde &longs;e habere, atque &longs;i vectis quidam curvus in an­<lb/>gulum inflexus ad punctum hypomochlij, &longs;i &longs;it vectis primi ge­<lb/>neris, extremitatem alteram in centro gravitatis ponderis, al­<lb/>teram in potentiâ haberet. </s> </p> <p type="main"> <s id="s.002892">Sit Vectis rectus AB horizonti parallelus, hypomochlium <lb/>habens in C, & in parte inferiore &longs;tabili nexu adjungatur pon­<lb/>dus, cujus gravitatis centrum I. <lb/><figure id="id.017.01.395.1.jpg" xlink:href="017/01/395/1.jpg"/><lb/>Ex I in vectem horizontalem cadat <lb/>perpendicularis linea directionis <lb/>IE; hoc enim perpendiculum de­<lb/>finit di&longs;tantiam gravitatis à vecte. </s> <lb/> <s id="s.002893">E&longs;t igitur potentia in A, & pondus <lb/>in I perinde, atque &longs;i e&longs;&longs;et vectis <lb/>ACI; & ut pondus atque potentia <lb/>in eâdem linea horizontali con&longs;i&longs;tant, non e&longs;t attendenda <lb/>vectis po&longs;itio AB, &longs;ed rectæ lineæ AI jungentis centrum po­<lb/>tentiæ A cum centro gravitatis ponderis I; quæ linea AI &longs;imul <lb/>ut æquè ab horizonte di&longs;tabit, & linea CH ad angulos rectos <lb/>cadens in eandem lineam AI congruens erit rectæ lineæ jun­<lb/>genti punctum hypomochlij C cum centro terræ, æquilibrium <lb/>indicabit; eademque definiet Rationem ponderis ad potentiam <lb/>&longs;u&longs;tinentem horizontaliter, juxta reciprocam eorumdem <lb/>di&longs;tantiam à puncto H; pro ut lib.3. cap.5. de librâ curvâ ex­<lb/>plicatum e&longs;t. </s> <s id="s.002894">In po&longs;itione autem obliqua AI, quando recta ex <lb/>C ad centrum terræ ducta e&longs;t CG cadens &longs;uper AI ad angu­<lb/>los inæquales, potentia &longs;u&longs;tinens e&longs;t ad pondus, ut IG ad GA. <!-- KEEP S--></s> <lb/> <s id="s.002895">Cum igitur &longs;it IG minor quàm IH, contrà verò GA &longs;it ma­<lb/>jor quàm HA, erit minor Ratio IG ad GA, quàm IH <lb/>ad HA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002896">Quoniam verò linea directionis ponderis IE perpendicula­<lb/>ris e&longs;t ad vectem AB horizontalem ex hypothe&longs;i, & parallela <lb/>lineæ CG, e&longs;t ut AG ad GI, ita AC ad CE, per 2. lib.6. ac <lb/>propterea, in &longs;itu vectis parallelo horizonti, locus ponderis e&longs;t <lb/>in vecte determinatus à lineâ directionis ponderis occurrente <pb pagenum="380" xlink:href="017/01/396.jpg"/>ip&longs;i vecti. </s> <s id="s.002897">Et quia major e&longs;t Ratio AG ad GI, quàm &longs;it AH <lb/>ad HI, etiam major e&longs;t Ratio AC ad CE, quàm &longs;it AH ad <lb/>HI: Ergo convertendo EC ad CA minorem habet Ratio­<lb/>nem, quàm IH ad HA, per 26. lib. 5. Atqui potentia &longs;u&longs;ti­<lb/>nens pondus datum, quando recta AI æquè di&longs;tat ab horizon­<lb/>te, e&longs;t ad pondus ut IH ad HA; quando autem pondus e&longs;t in­<lb/>fra lineam BA illud cum potentiâ jungentem horizonti paral­<lb/>lelam e&longs;t ut EC ad CA. <!-- KEEP S--></s> <s id="s.002898">Igitur potentia &longs;u&longs;tinens in horizon­<lb/>tali pondus habet majorem Rationem ad illud, quàm ad idem <lb/>pondus habeat potentia &longs;u&longs;tinens illud infra horizontalem. </s> <lb/> <s id="s.002899">Ergo, ex 8. lib. 5. potentiâ &longs;u&longs;tinens pondus infra horizonta­<lb/>lem minor e&longs;t potentiâ illud &longs;u&longs;tinente in horizontali. </s> <s id="s.002900">Finge <lb/>enim e&longs;&longs;e libram curvam ACI habentem &longs;partum in C: uti­<lb/>que &longs;i in A e&longs;&longs;et æquipondium, quod ad pondus I e&longs;&longs;et ut IH <lb/>ad HA, non maneret in eadem po&longs;itione obliqua, &longs;ed A de&longs;cen­<lb/>deret ad po&longs;itionem horizontalem, ut dictum e&longs;t lib. 3. cap.4. <lb/>ut igitur obliqua maneat, æquipondium A debet e&longs;&longs;e minus. </s> <lb/> <s id="s.002901">Ad &longs;u&longs;tinendum autem pondus, hîc in vecte idem à Potentiâ <lb/>præ&longs;tatur, ac ab æquipondio in librâ brachiorum inæqualium. </s> </p> <p type="main"> <s id="s.002902">Simili omnino methodo o&longs;tendetur pondus idem vecti AB <lb/>horizontali impo&longs;itum, cujus centrum gravitatis &longs;it D, linea <lb/>directionis DI occurrens vecti in E, e&longs;&longs;e ad potentiam A, ut <lb/>e&longs;t AC ad CE; at &longs;i recta AD jungens potentiam cum cen­<lb/>tro gravitatis D e&longs;&longs;et horizonti parallela, pondus ad potentiam <lb/>e&longs;&longs;et ut AL ad LD, quam Rationem determinat CL cadens <lb/>ad angulos rectos in rectam AD. <!-- KEEP S--></s> <s id="s.002903">Quia enim DE & IE &longs;unt <lb/>æquales ex hypothe&longs;i, cum &longs;it idem pondus, & latus EA e&longs;t <lb/>commune, anguli verò ad E &longs;unt recti, etiam, per 4. lib. 1. <lb/>lineæ AD & AI, item anguli EAD & EAI &longs;unt æquales. </s> <lb/> <s id="s.002904">Præterea in triangulis CHA, CLA rectangulis ad H & L, <lb/>latus CA e&longs;t commune, & anguli ad A &longs;unt æquales; igitur, <lb/>per 26. lib. 1. lineæ AL & AH &longs;unt æquales, igitur & re&longs;i­<lb/>duæ LD & HI &longs;unt æquales. </s> <s id="s.002905">Quapropter ut AH ad HI, ita <lb/>AL ad LD: quia igitur Ratio AH ad HI o&longs;ten&longs;a e&longs;t &longs;uperiùs <lb/>minor Ratione AC ad CE, etiam minor e&longs;t Ratio AL ad LD, <lb/>quàm AC ad CE. <!-- KEEP S--></s> <s id="s.002906">Sed ut AC ad CE, ita AO ad OD, per <lb/>2. lib. 6. propter paralleli&longs;mum linearum CO & ED; ergo mi­<lb/>nor e&longs;t 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"/>parallela e&longs;t horizonti, pondus D impo&longs;itum vecti ad poten­<lb/>tiam A &longs;u&longs;tinentem e&longs;t ut AL ad LD, in po&longs;itione verò obli­<lb/>quâ AD e&longs;t idem pondus ad potentiam &longs;u&longs;tinentem ut AO <lb/>ad OD; ergo in priori po&longs;itione horizontali pondus ad poten­<lb/>tiam habet minorem Rationem, quàm in po&longs;teriori po&longs;itione <lb/>obliqua: ergo per 8. lib. 5. in priori e&longs;t major potentia, quàm <lb/>in po&longs;teriori. </s> </p> <p type="main"> <s id="s.002908">Quamvis autem, cùm vectis e&longs;t horizonti parallelus, pon­<lb/>dus &longs;ive illi impo&longs;itum, &longs;ive &longs;uppo&longs;itum fuerit, ii&longs;dem momen­<lb/>tis reluctetur potentiæ &longs;u&longs;tinenti, non ita tamen &longs;e res habet, <lb/>&longs;i idem vectis <lb/><figure id="id.017.01.397.1.jpg" xlink:href="017/01/397/1.jpg"/><lb/>AB, fulcrum <lb/>habens in C, <lb/>elevetur &longs;upra <lb/>lineam hori­<lb/>zontalem RT: <lb/><expan abbr="plurimũ">plurimum</expan> enim <lb/>intere&longs;t, <expan abbr="utrũ">utrum</expan> <lb/>ponderi &longs;ub­<lb/>jectus &longs;it ve­<lb/>ctis, an vecti pondus. </s> <s id="s.002909">Sint, ut prius, gravitatis ponderis cen­<lb/>tra D &longs;uperius, & I inferius, ex quibus in vectem perpendi­<lb/>culares cadunt DE & IE, quæ, ex 14. lib. 1. &longs;unt una recta li­<lb/>nea DI. <!-- KEEP S--></s> <s id="s.002910">Jungantur centra potentiæ & ponderis rectâ AD, <lb/>quæ &longs;ecat rectam tran&longs;euntem per fulcrum C & terræ centrum <lb/>in puncto M. </s> <s id="s.002911">Quare ex dictis de librâ curva, &longs;i &longs;int æqualia <lb/>momenta ponderis atque potentiæ, erit ut AM ad MD, ita <lb/>pondus D ad potentiam A. <!-- KEEP S--></s> <s id="s.002912">Ducatur ex D linea directionis <lb/>DN parallela perpendiculari MC; & per 2. lib. 6; e&longs;t ut AM <lb/>ad MD, ita AC ad CN: e&longs;t autem CN minor quàm CE, <lb/>ergo, ex 8. lib. 5. major e&longs;t Ratio AC ad CN, quàm AC ad <lb/>CE. <!-- KEEP S--></s> <s id="s.002913">Atqui in vecte horizontali potentia ad pondus e&longs;t ut EC <lb/>ad CA; hic autem ut NC ad CA; igitur minor e&longs;t potentia <lb/>&longs;u&longs;tinens pondus impo&longs;itum vecti obliquo &longs;upra horizontem, <lb/>quàm potentia &longs;u&longs;tinens pondus idem vecte parallelo hori­<lb/>zonti. </s> </p> <p type="main"> <s id="s.002914">At &longs;i pondus vecti &longs;ubjiciatur, & &longs;it ejus gravitatis centrum <lb/>I, ducatur recta AI &longs;ecans perpendiculum ex C ductum ad <pb pagenum="382" xlink:href="017/01/398.jpg"/>centrum terræ in V. <!-- KEEP S--></s> <s id="s.002915">Igitur &longs;i æqualia &longs;unt momenta ponde­<lb/>ris I & potentiæ A, e&longs;t pondus ad potentiam ut AV ad VI. <!-- KEEP S--></s> <lb/> <s id="s.002916">Ex I centro gravitatis linea directionis IB parallela lineæ CV <lb/>occurrat vecti in B; igitur, ex 2. lib.6. ut AV ad VI, ita AC <lb/>ad CB: e&longs;t autem CB major quàm CE; ergo AC ad CB ha­<lb/>bet, ex 8. lib. 5. minorem Rationem, quàm AC ad CE. <!-- KEEP S--></s> <s id="s.002917">Cum <lb/>itaque in vecte horizontali potentia ad pondus e&longs;&longs;et ut EC ad <lb/>CA, hic autem in vecte obliquo &longs;it ut BC ad eandem CA, <lb/>major potentia &longs;u&longs;tinens hîc requiritur. </s> <s id="s.002918">Quare tantumdem <lb/>cre&longs;cit &longs;u&longs;tinendi difficultas in pondere infra vectem adjuncto, <lb/>quantum decre&longs;cit in &longs;u&longs;tinendo pondere &longs;upra vectem po&longs;ito. </s> <lb/> <s id="s.002919">Cum enim triangula BEI, DEN &longs;int æquiangula (quia BI <lb/>& DN, per 30. lib. 1. &longs;unt parallelæ, adeóque per 29. lib. 1. <lb/>alterni anguli ad B & N, & alterni ad I & D &longs;unt æquales, & <lb/>reliquus reliquo, per 32. lib.1.) e&longs;t, per 4. lib. 6. ut IE ad ED, <lb/>ita BE ad EN: &longs;unt autem ex hypothe&longs;i DE & IE æquales, <lb/>igitur & BE æqualis e&longs;t ip&longs;i EN, illa refert incrementum po­<lb/>tentiæ, hæc decrementum; ergo æqualiter ibi cre&longs;cit, hic de­<lb/>cre&longs;cit difficultas &longs;u&longs;tinendi pondus. </s> </p> <p type="main"> <s id="s.002920">Contraria &longs;unt momenta, quæ ponderibus accidunt, vecte <lb/>cum pondere infra horizontalem lineam inclinato: concipe <lb/>enim hoc idem &longs;chema ita conver&longs;um, ut potentia A &longs;it in &longs;u­<lb/>periore loco, pondera autem I & D &longs;int infra horizontalem <lb/>RT. </s> <s id="s.002921">Jam pondus I incumbit vecti, pondus verò D illi &longs;ub­<lb/>jectum adnectitur. </s> <s id="s.002922">Igitur pondus I vecti impo&longs;itum majora mo­<lb/>menta habet vecte cum pondere infra horizontem inclinato, <lb/>quàm vecte horizonti parallelo: in hoc autem eodem &longs;itu in­<lb/>clinato pondus &longs;ubjectum D minora habet momenta, nam pon­<lb/>dus I ad potentiam A &longs;u&longs;tinentem e&longs;t ut AC ad CB majorem, <lb/>quæ e&longs;t minor Ratio quàm AC ad CE minorem, ex 8. lib. 5: <lb/>è contrario D pondus ad potentiam A &longs;u&longs;tinentem e&longs;t ut AC <lb/>ad CN minorem, quæ e&longs;t major Ratio, quàm AC ad CE ma­<lb/>jorem. </s> <s id="s.002923">Hinc e&longs;t momenta ponderis vecti ex primo genere im­<lb/>po&longs;iti infra horizontem majora e&longs;&longs;e, &longs;upra horizontem minora; <lb/>contrà autem ponderis vecti &longs;ubjecti infra horizontem minora <lb/>e&longs;&longs;e, &longs;upra horizontem majora. </s> </p> <p type="main"> <s id="s.002924">Et hæc <expan abbr="quid&etilde;">quidem</expan> eatenus dicta intelligantur, quatenus concipitur <lb/>Potentia vi &longs;uæ gravitatis rectâ deor&longs;um connitens, adeò ut Di-<pb pagenum="383" xlink:href="017/01/399.jpg"/>rectione, <expan abbr="Pot&etilde;tiæ">Potentiæ</expan> atque Ponderis &longs;int parallelæ, propterea enim <lb/>con&longs;iderata e&longs;t linea per <expan abbr="centrũ">centrum</expan> motûs, hoc e&longs;t punctum fulcri, <lb/>ducta ad <expan abbr="centrũ">centrum</expan> terræ utrique Directioni parallela. </s> <s id="s.002925">At &longs;i linea Di­<lb/>rectionis Potentiæ non e&longs;&longs;et parallela Directioni gravitatis Pon­<lb/>deris &longs;i res &longs;crupulo&longs;ius agatur, paulo aliter con&longs;ideranda vide­<lb/>tur linea per punctum fulcri tran&longs;iens, quæ determinet partes li­<lb/>neæ jungentis Potentiam & Centrum gravitatis ponderis, linea <lb/>videlicet per fulcrum ducta ex puncto, in quo concurrunt di­<lb/>rectiones Potentiæ atque <lb/><figure id="id.017.01.399.1.jpg" xlink:href="017/01/399/1.jpg"/><lb/>Ponderis. <!-- KEEP S--></s> <s id="s.002926">Sit Vectis AB <lb/>in&longs;i&longs;tens fulcro C depre&longs;­<lb/>&longs;us in A infra horizontem, <lb/>ut &longs;u&longs;tineat pondus D in­<lb/>cumbens vecti, à quo di&longs;tat <lb/>per lineam DE. <!-- KEEP S--></s> <s id="s.002927">Directio <lb/>gravitatis ponderis e&longs;t per­<lb/>pendicularis DR, at di­<lb/>rectio Potentiæ non &longs;it per­<lb/>pendicularis AT, verùm <lb/>obliqua AR faciens cum <lb/>vecte angulum BAR. </s> <lb/> <s id="s.002928">Concurrunt itaque di­<lb/>rectiones Ponderis, & Potentiæ in R. <!-- KEEP S--></s> <s id="s.002929">Quare &longs;icuti quando <lb/>&longs;unt directiones DR & AT parallelæ, premunt fulcrum C <lb/>juxta perpendicularem CV, quæ rectam AD &longs;ecat in M, ita <lb/>directiones DR & AR videntur premere fulcrum C juxta <lb/>rectam CR, quæ producta &longs;ecat rectam AD in S: ac propterea <lb/>Ratio Potentiæ &longs;u&longs;tinentis ad Pondus non e&longs;t ut DM ad MA, <lb/>&longs;ed ut DS ad SA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002930">Hinc e&longs;t lineam directionis Potentiæ, quò majorem angu­<lb/>lum con&longs;tituit cum vecte in A, eò minorem angulum efficere <lb/>cum perpendiculari lineâ directionis ponderis DR productâ, <lb/>atque proinde cum illa concurrere multo remotiùs quàm in R, <lb/>& lineam ex puncto concursûs directionum ductam ad C, & <lb/>ulterius productam &longs;ecare lineam AD inter M & S, adeò ut <lb/>aliquando facilè citra notabilem errorem a&longs;&longs;umi po&longs;&longs;it punctum <lb/>M: Cum enim DR & MV &longs;int parallelæ, angulus DRC in­<lb/>ternus æqualis e&longs;t externo MCS, ex 29. lib. 1. idémque di-<pb pagenum="384" xlink:href="017/01/400.jpg"/>cendum de quolibet angulo con&longs;tituto cum perpendiculari <lb/>DR à lineâ ex puncto concur&longs;us directionum ducta per C <lb/>punctum fulcri: ideò quo minor fit angulus ad B, minor quo­<lb/>que e&longs;t ad C, & punctum in lineâ AD notatum magis acce­<lb/>dit ad M. </s> </p> <p type="main"> <s id="s.002931">Hinc pro determinanda Ratione momentorum potentiæ ad <lb/>momenta ponderis pro diversâ vectis inclinatione duplici me­<lb/>thodo uti poteris. </s> <s id="s.002932">Prima e&longs;t, fi ex centro gravitatis ponderis <lb/>lineam directionis ducas, punctum enim, in quo hæc occurrit <lb/>vecti, illud e&longs;t, quod definit locum ponderis, in quo &longs;ua exer­<lb/>cet momenta. </s> <s id="s.002933">Secunda e&longs;t, &longs;i tam ex Potentiæ quàm ex Pon­<lb/>deris centro gravitatis lineam ducas ad perpendiculum in li­<lb/>neam horizontalem, quæ tran&longs;it per C punctum fulcri; nam <lb/>partes hujus lineæ horizontalis interceptæ inter puncta, in quæ <lb/>cadunt perpendiculares, & punctum C, illæ &longs;unt, quæ reci­<lb/>procè &longs;umptæ o&longs;tendunt Rationem ponderis ad potentiam. </s> <s id="s.002934">In <lb/>&longs;itu namque horizontali vectis punctum E congruit puncto S, <lb/>& potentia A congruit puncto X: e&longs;t igitur ut AC ad CE ita <lb/>XC ad CS: in po&longs;itione autem obliquá ex A in horizontalem <lb/>perpendicularis cadit in Z, ex D cadit in K, ex I verò in O. <!-- KEEP S--></s> <lb/> <s id="s.002935">Quia igitur triangula AZC & NKC &longs;unt æquiangula, vide­<lb/>licet rectangula ad Z & K, angulos ad verticem C, ex 15.lib.1; <lb/>æquales habent, &, ex 32 lib. 1. reliquum reliquo, e&longs;t per 4. <lb/>lib. 6. ut AC ad CN ita ZC ad CK. <!-- KEEP S--></s> <s id="s.002936">Similiter triangula <lb/>BOC & AZC rectangula ad O & Z angulos ad verticem C <lb/>æquales habent, & reliquum reliquo, adeóque &longs;unt &longs;imilia, & <lb/>ut AC ad CB, ita ZC ad CO, Quare in hac obliquâ vectis <lb/>po&longs;itione momentum ponderis D ad momentum potentiæ &longs;u&longs;ti­<lb/>nentis e&longs;t ut ZC ad CK, & momentum ponderis I ad momen­<lb/>tum potentiæ &longs;u&longs;tinentis e&longs;t ut ZC ad CO. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002937">Ex his, quæ de potentia &longs;u&longs;tentante dicta &longs;unt, &longs;atis apparet <lb/>potentiam paulo validiorem &longs;atis e&longs;&longs;e ad pondus movendum. </s> <lb/> <s id="s.002938">Verùm licèt in vecte primi generis ad pondus &longs;u&longs;tentandum <lb/>opportunè animum adverterimus ad libram curvam, hæc ta­<lb/>men in vecte &longs;ecundi generis locum habere non po&longs;&longs;unt; <lb/>propterea ad aliam explicandi rationem confugiendum e&longs;t, <lb/>quæ utrique generi communis &longs;it; nec difficile erit ea, quæ &longs;ta­<lb/>tim capite &longs;equenti &longs;ubjiciam pro &longs;ecundo vectis genere ad pri-<pb pagenum="385" xlink:href="017/01/401.jpg"/>mum traducere. </s> <s id="s.002939">Con&longs;ideratur nimirum motus ponderis com­<lb/>paratus cum eodem motu potentiæ: &longs;i enim potentia &longs;it &longs;uâ <lb/>gravitate de&longs;cendens, ejus de&longs;cen&longs;um metitur ZA: pondus <lb/>vecti impo&longs;itum a&longs;cendit, ut &longs;it &longs;upra horizontalem altitudine <lb/>KD; &longs;ed ex hac demenda e&longs;t centri gravitatis di&longs;tantia DE, <lb/>qua eminebat &longs;upra horizontalem, ut habeatur ejus motus <lb/>DK minùs DE, hoc e&longs;t GK. </s> <s id="s.002940">Contra verò pondus vecti &longs;ub­<lb/>jectum erat infra horizontalem di&longs;tantiâ IE, quæ &longs;i addatur al­<lb/>titudini OI, dabit OH motum ip&longs;ius ponderis. </s> <s id="s.002941">Major e&longs;t au­<lb/>tem motus OI plus IE, hoc e&longs;t plus DE, quàm &longs;it motus KD <lb/>minùs DE; nam po&longs;ita obliquitate lineæ DI, facto centro D, <lb/>intervallo DE circulus de&longs;criptus tran&longs;it per G punctum de­<lb/>pre&longs;&longs;ius quàm E, & ex I intervallo IE de&longs;criptus tran&longs;it per H <lb/>punctum altius quàm E: ergo motus ZA ad minorem motum <lb/>habet majorem Rationem, quàm ad majorem motum, atque <lb/>adeò major e&longs;t movendi facilitas. <lb/></s> </p> <p type="main"> <s id="s.002942"><emph type="center"/>CAPUT IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.002943"><emph type="center"/><emph type="italics"/>Momenta ponderis in Vecte &longs;ecundi generis <lb/>con&longs;iderantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.002944">IN Vecte &longs;ecundi generis circa extremitatem, ubi e&longs;t ful­<lb/>crum, de&longs;cribuntur à pondere proximo & à potentiâ remotâ <lb/>duo circulorum arcus tanquam circa commune centrum. </s> <s id="s.002945">Et <lb/>quidem &longs;i in eadem rectâ lineâ &longs;int punctum fulcri, centrum <lb/>gravitatis ponderis, & ip&longs;a virtus potentiæ &longs;ur&longs;um a&longs;cendentis, <lb/>motus potentiæ & ponderis &longs;unt in eadem Ratione, in qua &longs;unt <lb/>di&longs;tantiæ ab hypomochlio, &longs;ive pondus &longs;upra horizontalem <lb/>tran&longs;euntem per fulcrum, &longs;ive à loco inferiore ad horizontalem <lb/>elevetur; quia videlicet tam pondus quàm potentia per &longs;imi­<lb/>les arcus ab horizontali æqualiter remotos moventur; ac pro­<lb/>inde eorum arcuum Sinus, qui metiuntur elevationem, ha­<lb/>bent inter &longs;e Rationem eandem, quæ e&longs;t radiorum, &longs;ive di­<lb/>&longs;tantiarum. </s> </p> <pb pagenum="386" xlink:href="017/01/402.jpg"/> <p type="main"> <s id="s.002946">At verò &longs;i centrum gravitatis ponderis &longs;it extra lineam <lb/>rectam jungentem punctum fulcri cum puncto virtutis poten­<lb/>tiæ exi&longs;tentis in alterâ vectis extremitate, &longs;ive &longs;upra vectem, <lb/>&longs;ive infra illum &longs;it, non manet eadem Ratio motuum, quæ e&longs;t <lb/>di&longs;tantiarum potentiæ & ponderis (quatenus ponderis di&longs;tan­<lb/>tia &longs;umitur à puncto, in quod à centro gravitatis cadit in <lb/>vectem perpendicularis) quia a&longs;cen&longs;us & elevationes non &longs;er­<lb/>vant eandem Rationem; ex eo quod, licèt in vectis conver&longs;io­<lb/>ne tam centrum gravitatis ponderis quàm centrum potentiæ <lb/>de&longs;cribant in motu arcus &longs;imiles, hi tamen arcus non &longs;unt &longs;i­<lb/>militer po&longs;iti, hoc e&longs;t &longs;imili modo ab horizontali di&longs;tantes: ac <lb/>propterea (ut patet ex doctrina Sinuum) differentiæ Sinuum, <lb/>qui conveniunt arcubus &longs;upra vel infra horizontem, ubi incipit <lb/>quadrans circuli, æqualiter cre&longs;centibus, non &longs;unt æquales: hæ <lb/>autem differentiæ metiuntur motum elevationis, qui maximè <lb/>attenditur, quatenus opponitur innatæ propen&longs;ioni gravitatis. </s> </p> <p type="main"> <s id="s.002947">Sit in C fulcrum vectis CA, & in A &longs;it potentia movens. </s> <lb/> <s id="s.002948">Si centrum gravitatis ponderis &longs;it in eadem rectâ CBA, &longs;em­<lb/><figure id="id.017.01.402.1.jpg" xlink:href="017/01/402/1.jpg"/><lb/>per motus ponderis & <lb/>potentiæ &longs;unt omnino <lb/>&longs;imiles, & ut CB ad <lb/>ad CA; illud enim de&longs;­<lb/>cribit arcum BG, hæc <lb/>verò arcum AS, & ele­<lb/>vatio ponderis ex B in <lb/>G e&longs;t BR, a&longs;cen&longs;us po­<lb/>tentiæ e&longs;t AP; & prop­<lb/>ter triangulorum rectan­<lb/>gulorum CRB & CPA &longs;imilitudinem e&longs;t ut CB ad CA, <lb/>ita BR ad AP. <!-- KEEP S--></s> <s id="s.002949">Et quamvis, divi&longs;o arcu BG in partes ali­<lb/>quot æquales, & in totidem æquales partes divi&longs;o arcu <lb/>&longs;imili AS, non &longs;int in &longs;ingulis eju&longs;dem arcûs partibus æquales <lb/>a&longs;cen&longs;us) nam BH minor e&longs;t quàm HI, hic minor quàm IK, <lb/>& hic minor quàm KR, &longs;imiliterque AL minor quàm LM, <lb/>hic minor quàm MN, & hic minor quàm NP) comparatis ta­<lb/>men &longs;ingulis a&longs;cen&longs;ibus in minore arcu BG, cum &longs;ingulis <lb/>a&longs;cen&longs;ibus in arcu majore AS &longs;ibi invicem re&longs;pondentibus, ma­<lb/>net eadem Ratio, & ut BH ad AL, ita HI ad LM, & &longs;ic de <pb pagenum="387" xlink:href="017/01/403.jpg"/>reliquis (ut ex Sinuum doctrinâ manife&longs;tum e&longs;t, nec opus e&longs;t <lb/>hic o&longs;tendere) &longs;unt enim omnes in Ratione Radij CB ad Ra­<lb/>dium CA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002950">Longè aliter &longs;e res habet, quando extra rectam lineam jun­<lb/>gentem punctum fulcri cum potentiâ e&longs;t centrum gravitatis <lb/>ponderis. </s> <s id="s.002951">Nam &longs;i Vecti CBA impo&longs;itum &longs;it pondus, cujus <lb/>centrum gravitatis &longs;it D, potentiâ A de&longs;cribente arcum AQ <lb/>centrum gravitatis ponderis de&longs;cribit arcum DE, qui licèt <lb/>æqualis &longs;it arcui BD; habet tamen a&longs;cen&longs;um HI majorem <lb/>quàm BH: igitur a&longs;cen&longs;us AL ad HI majorem, habet mino­<lb/>rem Rationem quàm ad BH minorem, ex 8.lib.5. igitur in hoc <lb/>motu Potentia ad Ponderis motum habet minorem Rationem, <lb/>quàm &longs;i centrum gravitatis ponderis e&longs;&longs;et in B; ergo majorem <lb/>experitur in movendo difficultatem. </s> </p> <p type="main"> <s id="s.002952">Contrà verò &longs;i pondus &longs;it vecti CBA &longs;ubjectum, ejú&longs;que <lb/>centrum gravitatis &longs;it O; dum potentia A de&longs;cribit arcum AQ, <lb/>centrum gravitatis O de&longs;cribit arcum OB, eju&longs;que a&longs;cen&longs;us <lb/>e&longs;t OV; atqui OV minor e&longs;t quàm BH; ergo AL a&longs;cen&longs;us <lb/>potentiæ ad OV minorem e&longs;t in majori Ratione quàm ad BH <lb/>majorem; e&longs;t autem HI major quàm BH; ergo AL ad OV <lb/>multo majorem Rationem habet quàm ad HI. <!-- KEEP S--></s> <s id="s.002953">Ergo datâ eâ­<lb/>dem vectis po&longs;itione, eodemque motu, major facilitas erit in <lb/>elevando pondere habente centrum gravitatis infra vectem in <lb/>O, quàm &longs;i illud habeat &longs;upra vectem in D. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002954">Eadem erit demon&longs;trandi methodus in cæteris a&longs;cen&longs;ibus: <lb/>nam potentia percurrens arcum AT habet a&longs;cen&longs;um AM, <lb/>centrum D percurrit arcum DF, cujus a&longs;censûs men&longs;ura e&longs;t <lb/>HK; centrum autem O percurrens arcum OD habet a&longs;cen­<lb/>&longs;um OX: cùm igitur OX minor &longs;it quàm BI, & hic minor <lb/>quàm HK, etiam AM ad OX minorem e&longs;t in majore Ratione <lb/>quàm ad HK majorem. </s> </p> <p type="main"> <s id="s.002955">Et hæc quidem hactenus dicta intelliguntur de vecte infra <lb/>lineam horizonti parallelam depre&longs;&longs;o; nam vecte &longs;upra hori­<lb/>zontalem lineam elevato, contraria pror&longs;us accidere ex dictis <lb/>demon&longs;tratur. </s> <s id="s.002956">Concipe vectem AC elevatum &longs;upra horizon­<lb/>tem, pondus OB e&longs;t illi impo&longs;itum, pondus DB e&longs;t &longs;ubjectum: <lb/>quando potentia a&longs;cendens per arcum QA habet a&longs;cen&longs;um <lb/>LA, centrum gravitatis O de&longs;cribit arcum BO, & a&longs;censûs <pb pagenum="388" xlink:href="017/01/404.jpg"/>men&longs;ura e&longs;t VO; at centrum gravitatis D de&longs;cribens arcum <lb/>ED habet a&longs;cen&longs;um IH. <!-- KEEP S--></s> <s id="s.002957">Cum igitur o&longs;ten&longs;um &longs;it majorem <lb/>Rationem e&longs;&longs;e LA ad VO, quàm ad IH, ctiam &longs;upra horizon­<lb/>tem elevato vecte major erit facilitas in movendo pondere vecti <lb/>impo&longs;ito, quàm in elevando pondus habens centrum gravita­<lb/>tis infra vectem. </s> </p> <p type="main"> <s id="s.002958">Ut autem innote&longs;cat, qua Ratione in progre&longs;&longs;u motûs cre&longs;­<lb/>cat difficultas, aut minuatur, ob&longs;erva ex Canone in arcubus <lb/>æqualiter cre&longs;centibus Sinuum differentias ab initio quadran­<lb/>tis progrediendo u&longs;que ad finem Quadrantis &longs;emper decre&longs;ce­<lb/>re, harum verò differentiarum differentias, hoc e&longs;t differen­<lb/>tias &longs;ecundas, &longs;emper augeri. </s> <s id="s.002959">Hinc e&longs;t ita RK Sinum arcûs <lb/>GF majorem e&longs;&longs;e quàm differentiam KI, & KI majorem quàm <lb/>IH, & IH majorem quàm HB, ut differentia inter Sinum <lb/>RK & differentiam KI minor &longs;it quàm differentia inter KI <lb/>& IH, hæc verò differentia minor &longs;it quàm differentia inter <lb/>IH & HB. <!-- KEEP S--></s> <s id="s.002960">Idem dicendum de &longs;imilibus differentiis inter Si­<lb/>num PN, & differentias NM, & ML, & LA. <!-- KEEP S--></s> <s id="s.002961">In ii&longs;dem li­<lb/>neis PA & RB particulas a&longs;&longs;umptas donavi vocabulo Sinuum <lb/>aut differentiarum, non qua&longs;i ignorans illas particulas non e&longs;&longs;e <lb/>Sinus aut differentias Sinuum arcubus æqualiter cre&longs;centibus <lb/>re&longs;pondentium, &longs;ed claritatis gratia abutens vocabulo; quan­<lb/>doquidem illis æquales &longs;unt, cum a&longs;&longs;umantur per lineas Radio <lb/>CS parallelas. </s> </p> <p type="main"> <s id="s.002962">His po&longs;itis intelligatur vectis totus CA cum pondere B intrà <lb/>aquam, potentia verò &longs;it cortex &longs;uberis, aut uter inflatus, &longs;eu <lb/>ve&longs;ica, aut quid huju&longs;modi levitans. </s> <s id="s.002963">Potentiæ motum metiri <lb/>oportet ex naturalibus a&longs;cen&longs;ibus AL, LM, & reliquis. </s> <s id="s.002964">Quia <lb/>autem e&longs;t ut AL ad LM, ita BH ad HI; etiam vici&longs;&longs;im, per <lb/>16. lib. 5. ut AL ad BH, ita LM ad HI, & &longs;ic de cæteris, &longs;ive <lb/>infra, &longs;ive &longs;upra horizontalem: propterea eadem &longs;emper manet <lb/>facilitas aut difficultas elevandi pondus in aquâ gravitans, cu­<lb/>jus gravitatis centrum congruat vecti CA. <!-- KEEP S--></s> <s id="s.002965">Idem dic &longs;i Poten­<lb/>tia S in aqua gravitans deprimeret per vim pondus G, quod in <lb/>aquâ levitaret: nam PN de&longs;cen&longs;us naturalis potentiæ ad RK <lb/>depre&longs;&longs;ionem ponderis, eandem Rationem haberet, quam <lb/>de&longs;cen&longs;us NM ad depre&longs;&longs;ionem KI. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002966">Si vectis &longs;it CA, cui pondus incumbat habens centrum gra-<pb pagenum="389" xlink:href="017/01/405.jpg"/>vitatis D, atque tam pondus quàm potentia &longs;int in medio, in <lb/>quo alterum levitet, alterum gravitet, utriu&longs;que motum qua­<lb/>tenus naturalis e&longs;t auz violentus, metitur linea perpendicularis <lb/>in horizontalem cadens: & ut particulæ ip&longs;æ invicem compa­<lb/>rentur, Sinuum differentiæ AL, LM &c. </s> <s id="s.002967">BH, HI &c. </s> <s id="s.002968">con­<lb/>&longs;iderandæ &longs;unt. </s> <s id="s.002969">Cum itaque differentia inter BH, & HI ma­<lb/>jor &longs;it quàm differentia inter HI, & IK, utique BH magis de­<lb/>ficit ab æqualitate cum HI, quàm HI cum IK; ideóque mi­<lb/>nor e&longs;t Ratio BH ad HI, quàm HI ad IK: Atqui eadem e&longs;t <lb/>Ratio BH ad HI, quæ e&longs;t AL ad LM; igitur minor e&longs;t etiam <lb/>Ratio AL ad LM, quàm HI ad IK, & vici&longs;&longs;im, per 27. lib. 5. <lb/>minor e&longs;t Ratio AL ad HI, quàm &longs;it LM ad IK. <!-- KEEP S--></s> <s id="s.002970">Igitur &longs;i po­<lb/>tentia A levitet, & pondus, cujus centrum gravitatis D, gravi­<lb/>tet, a&longs;cendendo ad horizontalem, quæ per fulcrum C tran&longs;it, <lb/>acquirit movendi facilitatem. </s> </p> <p type="main"> <s id="s.002971">Jam figuram inverte, ut vectis moveatur &longs;upra horizontalem: <lb/>vecte congruente lineæ horizontali CS, ponderis impo&longs;iti cen­<lb/>trum gravitatis erit in F, & a&longs;cendet juxta men&longs;uram KI & IH, <lb/>cum potentiæ a&longs;cen&longs;us erit PN & NM. </s> <s id="s.002972">Quia igitur differentia <lb/>inter Sinum RK & differentiam KI minor e&longs;t, quàm differentia <lb/>inter KI & IH, utique RK minùs excedit æqualitatem cum <lb/>KI, quàm KI cum IH: ideóque minor e&longs;t Ratio RK ad KI, <lb/>quàm KI ad IH. <!-- KEEP S--></s> <s id="s.002973">E&longs;t autem eadem Ratio RK ad KI, quæ e&longs;t <lb/>PN ad NM; igitur minor e&longs;t Ratio PN ad NM, quàm KI <lb/>ad IH, & vici&longs;&longs;im minor e&longs;t Ratio PN ad KI, quàm NM ad <lb/>IH: Igitur a&longs;cendendo magis & recedendo ab horizontali <lb/>cre&longs;cit movendi facilitas. </s> </p> <p type="main"> <s id="s.002974">Demum &longs;i vecti CA &longs;ubjectum &longs;it pondus, cujus centrum <lb/>gravitatis O, & potentiæ motum metiatur perpendicularis AP <lb/>a&longs;cendendo versùs horizontalem; quia differentia inter OV, & <lb/>VX major e&longs;t quàm differentia inter VX & HI, adeóque OV <lb/>magis deficit ab æqualitate cum VX, quàm VX cum HI, prop­<lb/>terea OV ad VX habet minorem Rationem quàm VX ad HI: <lb/>&longs;ed ut VX, hoc e&longs;t BH, ad HI, ita AL ad LM; ergo minor e&longs;t <lb/>Ratio OV ad VX quàm AL ad LM; & vici&longs;&longs;im minor e&longs;t Ra­<lb/>tio OV ad AL quàm VX ad LM; ideóque faciliùs elevatur ex <lb/>O in B, quàm ex B in D. <!-- KEEP S--></s> <s id="s.002975">Factâ autem figuræ conver&longs;ione, ut <lb/>a&longs;cen&longs;us Potentiæ &longs;it PA, & a&longs;cen&longs;us Ponderis &longs;it RB, &longs;i poten-<pb pagenum="390" xlink:href="017/01/406.jpg"/>tia &longs;it in Z, centrum gravitatis ponderis &longs;ubjecti e&longs;t in G, & <lb/>dum potentia a&longs;cendit per NM & ML de&longs;cribens arcum ZQ, <lb/>pondus a&longs;cendit per RK & KI. <!-- KEEP S--></s> <s id="s.002976">Atqui RK ad KI habet mi­<lb/>norem Rationem quàm KI ad IH, ut &longs;uperiùs o&longs;ten&longs;um e&longs;t, & <lb/>ut KI ad IH, ita NM ad ML; ergo minor e&longs;t Ratio RK ad <lb/>KI, quàm NM ad ML, & vici&longs;&longs;im minor e&longs;t Ratio RK ad <lb/>NM quàm KI ad ML; ergo faciliùs movetur per RK a&longs;cen­<lb/>dendo, quàm per KI, adeóque cre&longs;cit difficultas elevandi <lb/>pondus &longs;ubjectum vecti &longs;uprà horizontalem, &longs;i comparentur <lb/>inter &longs;e partes elevationis. </s> </p> <p type="main"> <s id="s.002977">Quare, ut in &longs;ummam ea, quæ dicta &longs;unt, referantur, &longs;i pon­<lb/>dus &longs;it infra vectem &longs;ecundi generis, faciliùs elevatur eodem <lb/>vectis motu versùs horizontalem, quàm &longs;i fuerit &longs;upra vectem: <lb/>Contrà verò &longs;upra horizontalem faciliùs eodem vectis motu <lb/>elevatur pondus vecti impo&longs;itum, quàm vecti &longs;ubjectum. </s> <s id="s.002978">Con­<lb/>&longs;ideratis autem particulatim &longs;ingulis elevationibus, divi&longs;o &longs;cili­<lb/>cet in æquales particulas univer&longs;o motu eju&longs;dem ponderis, &longs;i <lb/>pondus &longs;it in eâdem rectâ lineâ cum fulcro & potentia, eadem <lb/>&longs;emper e&longs;t movendi facilitas aut difficultas: Si pondus &longs;it &longs;upra <lb/>vectem, & motus infra horizontalem incipiat, &longs;emper cre&longs;cit <lb/>movendi facilitas non &longs;olùm u&longs;que ad horizontalem, verùm <lb/>etiam &longs;upra illam: At &longs;i pondus &longs;it infra vectem, motú&longs;que in­<lb/>fra horizontalem incipiat, augetur &longs;emper difficultas movendi <lb/>tùm u&longs;que ad horizontalem, tùm &longs;upra illam. </s> </p> <p type="main"> <s id="s.002979">Hæc omnia confirmari po&longs;&longs;unt, &longs;i lineam directionis per cen­<lb/>trum gravitatis ponderis ductam produci intelligamus u&longs;que ad <lb/>horizontalem lineam, quæ per fulcrum tran&longs;it; Secabit enim <lb/>vectem, & in &longs;ectionis puncto quodammodo con&longs;titutum pon­<lb/><figure id="id.017.01.406.1.jpg" xlink:href="017/01/406/1.jpg"/><lb/>dus concipere po&longs;&longs;umus. </s> <s id="s.002980">Sit enim <lb/>infra horizontalem CR, vectis <lb/>CA, & ad punctum B illi in&longs;i&longs;tat <lb/>perpendiculariter linea à centro <lb/>gravitatis ducta, &longs;cilicet DB &longs;u­<lb/>pra, & OB infra. </s> <s id="s.002981">Quando vectis <lb/>CA congruet lineæ CR, & erit <lb/>horizonti parallelus, pondus con­<lb/>cipietur niti in B contra vectem: <lb/>at infra horizontalem centrum D nititur in S, & centrum O <pb pagenum="391" xlink:href="017/01/407.jpg"/>in T, juxta lineas directionis DS & OT. </s> <s id="s.002982">Quia igitur punctum <lb/>S magis di&longs;tat à fulcro C quàm punctum T, pondus infra <lb/>vectem faciliùs &longs;u&longs;tinetur &longs;ub horizontali, quàm pondus &longs;upra <lb/>vectem. </s> <s id="s.002983">Contra autem &longs;upra horizontalem centrum O nititur <lb/>in I remotiùs à fulcro C, & centrum D in H propiùs; ergo <lb/>&longs;upra horizontalem faciliùs &longs;u&longs;tinetur pondus vecti impo&longs;itum, <lb/>quàm illi &longs;ubjectum. </s> </p> <p type="main"> <s id="s.002984">Quoniam verò triangula rectangula CNT, & OBT, an­<lb/>gulos ad verticem T æquales habent, & reliquum reliquo <lb/>æqualem, erit, ex 4. lib. 6. ut CT ad TN, ita OT ad TB. <!-- KEEP S--></s> <lb/> <s id="s.002985">Igitur prout ex elevatione vectis minuitur angulus ACN, <lb/>etiam minuitur angulus TOB, ac propterea T recedit à ful­<lb/>cro C ver&longs;us B, & augetur &longs;u&longs;tinendi atque movendi difficul­<lb/>tas. </s> <s id="s.002986">I&longs;ti autem acce&longs;&longs;us versùs B &longs;unt inæquales, etiam &longs;i æqua­<lb/>lia &longs;int anguli TOB decrementa, prout decre&longs;cunt angulo­<lb/>rum ad O factorum Tangentes, po&longs;ito Radio OB. </s> <s id="s.002987">Porrò ex <lb/>Canone Tangentium con&longs;tat illarum differentias &longs;emper ma­<lb/>jores fieri, &longs;i augeatur angulus, minores fieri, &longs;i minuatur an­<lb/>gulus. </s> <s id="s.002988">Igitur recedente lineâ directionis Centri gravitatis O à <lb/>fulcro C, augetur difficultas &longs;u&longs;tinendi & elevandi pondus <lb/>vecti &longs;ubjectum: & quia &longs;upra horizontalem &longs;emper magis re­<lb/>cedit ab eodem fulcro C ultrà punctum B versùs A potentiam, <lb/>puta, ut &longs;it OI, multo adhuc major e&longs;t &longs;u&longs;tinendi atque mo­<lb/>vendi difficultas. </s> <s id="s.002989">Con&longs;ideratis autem particulatim motibus, <lb/>quia infra horizontalem differentiæ rece&longs;&longs;uum à puncto C fiunt <lb/>&longs;emper minores; propterea cre&longs;cit quidem difficultas, &longs;ed inæ­<lb/>qualibus & minoribus incrementis; quia verò &longs;upra horizon­<lb/>talem differentiæ rece&longs;&longs;uum à fulcro C fiunt &longs;emper majores, <lb/>cre&longs;cit adhuc difficultas, & quidem &longs;emper majoribus incre­<lb/>mentis. </s> <s id="s.002990">At &longs;i pondus &longs;it D vecti impo&longs;itum, linea directionis <lb/>DS accedit versùs B u&longs;que ad horizontalem, &longs;upra quam re­<lb/>cedit à B versùs C, ut &longs;it ex. </s> <s id="s.002991">gr. <!-- REMOVE S-->DH: &longs;emper igitur faciliùs <lb/>movetur, quamquam non æqualibus facilitatis incrementis; <lb/>fiunt enim incrementa infra horizontalem &longs;en&longs;im minora, &longs;u­<lb/>pra autem fiunt &longs;emper majora. </s> <s id="s.002992">Sed hic unum explicandum <lb/>e&longs;t, quod forta&longs;&longs;e alicui animum minùs attentè advertenti dif­<lb/>ficultatem pariat adversùs ea, quæ &longs;uperiùs dicta &longs;unt: videlicet <lb/>o&longs;ten&longs;um e&longs;t pondus vecti impo&longs;itum, &longs;i motus incipiat infra <pb pagenum="392" xlink:href="017/01/408.jpg"/>horizontalem, majori difficultate moveri, quàm pondus vecti <lb/>&longs;ubjectum. </s> <s id="s.002993">Si enim, inquis, linea Directionis DS magis ac <lb/>magis accedit ad B, utique cre&longs;cit movendi facilitas; contra <lb/>verò lineâ directionis OT accedente ad B cre&longs;cit movendi <lb/>difficultas. </s> </p> <p type="main"> <s id="s.002994">Ut nodum hunc &longs;olvas, ob&longs;erva triangula SBD, & TBO <lb/>rectangula ad B, quia DS & TO &longs;unt parallelæ, e&longs;&longs;e æquian­<lb/>gula & &longs;imilia, immò æqualia, quia ut DB ad OB &longs;ibi ex hy­<lb/>pothe&longs;i æqualem, ita SB ad TB. <!-- KEEP S--></s> <s id="s.002995">Igitur qua Ratione minuitur <lb/>angulus ACR, etiam minuitur anguius SDB, & angulus <lb/>TOB: igitur Tangentium differentiæ fiunt &longs;emper minores. </s> <lb/> <s id="s.002996">Quare in primo motu tam linea directionis DS, quàm linea <lb/>directionis OT, magis accedit ad B quàm in &longs;ecundo motu, <lb/>& magis in &longs;ecundo, quàm in tertio; acce&longs;&longs;us tamen utriu&longs;­<lb/>que lineæ directionis ex eodem vectis motu &longs;unt æquales; & <lb/>qua men&longs;urâ augetur rece&longs;&longs;us ponderis D vecti impo&longs;iti, à Po­<lb/>tentia A, eâdem pariter men&longs;ura augetur rece&longs;&longs;us ponderis O <lb/>vecti &longs;ubjecti, à fulcro C. <!-- KEEP S--></s> <s id="s.002997">Itaque cre&longs;cit quidem illius facili­<lb/>tas, hujus difficultas, &longs;i ponderum &longs;ingulorum motus particu­<lb/>latim accipiantur, eju&longs;démque ponderi, motûs pars cum par­<lb/>te conferatur: at verò &longs;i utriu&longs;que ponderis motus invicem <lb/>comparentur, utique pondus D difficiliùs movetur, cùm ejus <lb/>linea directionis e&longs;t citra punctum B versùs potentiam, quàm <lb/>moveatur pondus O, quamdiu ejus linea directionis e&longs;t ultra <lb/>idem punctum B. </s> </p> <p type="main"> <s id="s.002998">Ex his, quæ de vecte &longs;ecundi generis dicta &longs;unt, quid de <lb/>vecte tertij generis dicendum &longs;it, faciliùs innote&longs;cit, quàm ut <lb/>illud pluribus explicari oporteat; potentia &longs;i quidem & pondus <lb/>invicem loca commutant, &longs;ed motuum Ratio eadem e&longs;t, & quæ <lb/>in vecte &longs;ecundi generis e&longs;t Ratio motûs Potentiæ ad motum <lb/>Ponderis, vice versâ in vecte tertij generis e&longs;t Ratio motûs <lb/>Ponderis ad motum Potentiæ. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.002999">Hoc te monitum velim, Amice Lector, con&longs;ideratum hacte­<lb/>nus vectem ad movenda &longs;ur&longs;um pondera gravia, aut deprimen­<lb/>da deor&longs;um levia, & quidem à Potentia, quæ vi &longs;uæ gravitatis <lb/>aut levitatis moveatur, cujus propterea a&longs;cen&longs;um aut de&longs;cen­<lb/>&longs;um con&longs;ideravimus. </s> <s id="s.003000">Nam &longs;i in plano horizontali à Potentia <lb/>vivente movendum &longs;it pondus, utique Potentiæ motus circu-<pb pagenum="393" xlink:href="017/01/409.jpg"/>laris ob&longs;ervatur, & attendendum e&longs;t vectis punctum, in quod <lb/>cadit linea, quæ à centro gravitatis ponderis in vectem per­<lb/>pendicularis ducitur, ut ponderis locus &longs;tatuatur, & momen­<lb/>ta definiantur. </s> <s id="s.003001">Naturâ quippe comparatum e&longs;t, ut &longs;i vectis non <lb/>occurrat huic perpendiculari, non moveatur totum pondus, <lb/>&longs;ed fiat ponderis conver&longs;io vel circa gravitatis centrum, vel <lb/>circa aliud punctum quod maneat immotum, aut &longs;altem mino­<lb/>re motu moveatur. <lb/> </s> </p> <p type="main"> <s id="s.003002"><emph type="center"/>CAPUT V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003003"><emph type="center"/><emph type="italics"/>Quæ &longs;it Ratio Vectis Hypomochlium mobile <lb/>habentis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003004">NOn hîc hypomochlium mobile illud intelligo, quod &longs;imul <lb/>cum pondere à potentiâ &longs;u&longs;tentato ad ea&longs;dem partes pro­<lb/>movetur; cuju&longs;modi &longs;unt manualia bajulorum vehicula, quæ <lb/>unicâ rotâ in&longs;truuntur, & habentia rationem vectis &longs;ecundi ge­<lb/>neris; nam fulcrum habent in axe rotæ, & potentiam in extre­<lb/>mitate manubriorum, quibus illa &longs;u&longs;tinet pondus transferen­<lb/>dum: cui propterea addita e&longs;t rota illa ver&longs;atilis, ut etiam hy­<lb/>pomochlium citra difficultatem, quin atterat &longs;ubjectam plani­<lb/>tiem, &longs;imul cum pondere jam elevato, atque &longs;u&longs;tentato pro­<lb/>moveatur. </s> </p> <p type="main"> <s id="s.003005">Huju&longs;modi pariter e&longs;t novitium vehiculi genus, cui Sellæ <lb/>Rotatæ nomen fecerunt, hoc uno à lecticâ viatoriâ di&longs;crepans, <lb/>quòd loco po&longs;terioris jumenti &longs;u&longs;tinentis additus e&longs;t axis dua­<lb/>bus rotis infixus, cui innituntur vectes ab anteriore equo <lb/>&longs;u&longs;tentanti unâ cum pondere intermedio. </s> <s id="s.003006">Hic e&longs;t vectis &longs;ecun­<lb/>di generis, cujus hypomochlium &longs;equitur potentiam trahen­<lb/>tem pariter ac &longs;u&longs;tentantem impo&longs;itum pondus, non mutatâ <lb/>Ratione momenti potentiæ &longs;u&longs;tinentis, &longs;ive hypomochlium <lb/>moveatur, &longs;ive &longs;tabile &longs;it ac fixum. </s> <s id="s.003007">Cæterùm quò pondus ma­<lb/>gis à rotis di&longs;tat, magis equum gravat, minùs autem &longs;ub&longs;ilit, <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"/>movente potentiâ atque conante adversùs pondus, re&longs;i&longs;tit qui­<lb/>dem vecti, &longs;ed & &longs;imul loco cedit ita, ut pondus & hypomo­<lb/>chlium in oppo&longs;itas partes immi&longs;&longs;o inter illa vecte moveantur. </s> <lb/> <s id="s.003009">Sic contingere pote&longs;t fulcrum deprimi, dum pondus elevatur, <lb/>aut fulcrum elevari, dum pondus deprimitur, aut &longs;i utrumque <lb/>in plano horizontali moveatur, in oppo&longs;itas plagas recedere. </s> <lb/> <s id="s.003010">Loquor autem de vecte primi & &longs;ecundi generis, quibus com­<lb/>muniter utimur; nam in vecte tertij generis, &longs;i hypomochlium <lb/>cedat, movetur ad ea&longs;dem partes cum pondere & potentia, &longs;ed <lb/>tardiùs. </s> <s id="s.003011">Hinc &longs;i vecte inter duo pondera non immodicè inæ­<lb/>qualia interjecto alterutrum movere coneris, reliquum etiam <lb/>movetur; ita tamen ut neutrum tantum motûs perficiat, quan­<lb/>tum haberent &longs;ingula, &longs;i &longs;olitariè moverentur, reliquo manen­<lb/>te immoto. </s> </p> <p type="main"> <s id="s.003012">Sit vectis AB inter duos lapides C & D interjectus, qui la­<lb/>pidem C non dimovebit, ni&longs;i eum tangat in puncto cui occur­<lb/><figure id="id.017.01.410.1.jpg" xlink:href="017/01/410/1.jpg"/><lb/>rit linea ex C gravitatis centro <lb/>ducta (aut potiùs planum per <lb/>idem gravitatis centrum C <lb/>tran&longs;iens) ad perpendiculum <lb/>in vectem, & &longs;it linea CE; ni&longs;i <lb/>enim in E lapis à vecte tanga­<lb/>tur, movebitur quidem lapis circa centrum C, donec congruat <lb/>vecti, &longs;ed non propelletur totus lapis. </s> <s id="s.003013">Idem dic de lapide D, <lb/>ni&longs;i tangatur in F occurrente lineæ perpendiculari DF. <!-- KEEP S--></s> <s id="s.003014">Qua­<lb/>re pondera intelligantur in E & F: & quoniam F re&longs;i&longs;tit vecti, <lb/>ut E propellatur versùs C, & vici&longs;&longs;im E re&longs;i&longs;tit vecti, ut F <lb/>propellatur versùs D, propterea ad movendum pondus C, <lb/>vectis AE e&longs;t primi generis, & ad movendum pondus D, <lb/>vectis AE e&longs;t &longs;ecundi generis; atque pondera illa vici&longs;&longs;im ha­<lb/>bent rationem hypomochlij, quia vectis alteri innititur, ut al­<lb/>terum moveat. </s> </p> <p type="main"> <s id="s.003015">Cæterùm &longs;ingulorum lapidum ab&longs;oluta & &longs;impliciter &longs;umpta <lb/>re&longs;i&longs;tentia tum ex eorum ingenitâ gravitate, tum ex &longs;uperfi­<lb/>cierum &longs;e tangentium a&longs;peritate atque conflictu definitur: <lb/>Comparatè verò ad vectem non &longs;ic accipienda e&longs;t &longs;ingulorum <lb/>re&longs;i&longs;tentia, qua&longs;i motûs centra e&longs;&longs;ent E aut F: experimento <lb/>enim manife&longs;to deprehenditur motum potentiæ A ad motum <pb pagenum="395" xlink:href="017/01/411.jpg"/>ponderis C non e&longs;&longs;e ut AF ad FE, neque eju&longs;dem potentiæ A <lb/>æqualem motum e&longs;&longs;e ad motum ponderis D ut AE ad FE. <!-- KEEP S--></s> <lb/> <s id="s.003016">Nam &longs;i punctum E vectis fixum e&longs;&longs;et, & potentiæ motus e&longs;&longs;et <lb/>AL, motus ponderis F e&longs;&longs;et FH: Si verò punctum F mane<lb/>ret immotum, & potentiæ <lb/><figure id="id.017.01.411.1.jpg" xlink:href="017/01/411/1.jpg"/><lb/>motus e&longs;&longs;et AI æqualis ip&longs;i <lb/>AL, motus ponderis e&longs;&longs;et <lb/>EG. <!-- KEEP S--></s> <s id="s.003017">Tunc autem motus AI <lb/>æqualis e&longs;t motui AL, quan­<lb/>do ut AF ad AE, ita vici&longs;&longs;im <lb/>angulus AEL ad angulum <lb/>AFI: æqualium &longs;i quidem <lb/>angulorum in circulis inæ­<lb/>qualibus arcus &longs;unt ut Radij; <lb/>ergo &longs;i fuerint anguli reciprocè ut Radij, &longs;cilicet minor in ma­<lb/>jore circulo, & major angulus in minore, erunt æquales arcus <lb/>illis oppo&longs;iti: Sic anguli AFR æqualis angulo AEL arcus AR <lb/>e&longs;t ad AL, ut Radius FA ad Radium EA; &longs;ed ut FA ad EA, <lb/>ita arcus AR ad arcum AI ex con&longs;tructione; ergo ut AR ad <lb/>AL ita AR ad AI: ergo per 9.lib.5. AI & AL &longs;unt æquales. </s> </p> <p type="main"> <s id="s.003018">Quoniam igitur tam E quàm F ex hypothe&longs;i in oppo&longs;itas <lb/>partes moventur circumacto vecte, punctum aliquod e&longs;t inter E <lb/>& F, quod e&longs;t veluti centrum motuum tam potentiæ quàm pon­<lb/>derum, in quo centro quodammodo divi&longs;a intelligitur re­<lb/>&longs;i&longs;tentia, quæ componitur tùm ex eorum innatâ gravitate, <lb/>tùm ex eorum motu, &longs;pectatâ po&longs;itione ad vectem. </s> <s id="s.003019">Hinc ma­<lb/>nife&longs;tum e&longs;t &longs;ingula pondera minùs moveri, quàm &longs;i &longs;ingula <lb/>moverentur reliquo manente immoto; quia videlicet &longs;ingula <lb/>minùs di&longs;tant à centro, circa quod moventur. </s> <s id="s.003020">Sic ponderum <lb/>E & F gravitas ponatur æqualis: &longs;i intelligatur centrum mo­<lb/>tûs ab utroque æqualiter di&longs;tare, ut &longs;it KE æqualis ip&longs;i KF, <lb/>motus potentiæ factus intervallo AK æqualem habet Ratio­<lb/>nem ad motum, qui fit à &longs;ingulis ponderibus. </s> </p> <p type="main"> <s id="s.003021">Quare potentiæ momentum perinde &longs;e habet, atque &longs;i utrum­<lb/>que pondus e&longs;&longs;et in E, aut utrumque in F, hypomochlium verò <lb/>in K. <!-- KEEP S--></s> <s id="s.003022">Ponamus enim EF e&longs;&longs;e partium 6, quarum partium 7 e&longs;t <lb/>FA: igitur EK e&longs;t 3, & KA 10; & potentia &longs;ine vecte movens <lb/>lib.3, vecte AKE movebit lib.10 in E. <!-- KEEP S--></s> <s id="s.003023">Similiter KF e&longs;t 3, & <pb pagenum="396" xlink:href="017/01/412.jpg"/>KA e&longs;t 10; igitur potentia ut 3 in A, movebit in F pondus ut <lb/>10: igitur etiam in A potentia ut 6, facto motûs centro K, mo­<lb/>vebit vel utrumque pondus ut 10 in E & F, vel unicum pondus <lb/>ut 20 &longs;ive in E, &longs;ive in F. <!-- KEEP S--></s> <s id="s.003024">Con&longs;tituatur itaque potentiæ virtus <lb/>ut 6, &longs;i hypomochlium e&longs;&longs;et F immotum, non moveret ni&longs;i pon­<lb/>dus grave ut 7 po&longs;itum in E; & facto hypomochlio &longs;tabili E <lb/>moveret pondus grave 13 po&longs;itum in F; adeóque univer&longs;um <lb/>pondus e&longs;&longs;et librarum 20. Quare idem pondus lib. 20 movetur <lb/>ab eâdem potentia, &longs;ed non eodem motu: Nam hîc amborum <lb/>&longs;imul ponderum motus circa centrum K e&longs;t ut 6; at &longs;i potentiæ <lb/>motus AI &longs;it 10 (quemadmodum motus potentiæ circa cen­<lb/>trum K e&longs;t 10) circa F centrum, motus EG e&longs;t 8 4/7; & &longs;i po­<lb/>tentiæ motus AL &longs;it pariter 10 circa centrum E motus FH <lb/>e&longs;t (4 8/13). </s> </p> <p type="main"> <s id="s.003025">Hinc patet &longs;ingulorum ponderum motum, quando utrum­<lb/>que &longs;imul movetur, minorem e&longs;&longs;e, quàm &longs;i &longs;ingula &longs;olitariè <lb/>moverentur, adeóque totum motum, qui ex duobus motibus <lb/>coale&longs;cit, minorem e&longs;&longs;e &longs;ummâ, quæ conflatur ex motu EG <lb/>& motu FH. <!-- KEEP S--></s> <s id="s.003026">Præterea manife&longs;tum e&longs;t cæteris paribus move­<lb/>ri faciliùs pondus F, quod e&longs;t Potentiæ A proximum, quàm <lb/>pondus E ab eádem remotum; minor enim differentia e&longs;t in­<lb/>ter (4 8/13) & 3, quàm inter 8 4/7 & 3. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003027">Quod &longs;i duorum ponderum E & F ab&longs;oluta re&longs;i&longs;tentia, quæ <lb/>ex gravitate oritur, inæqualis fuerit, inæqualem pariter e&longs;&longs;e <lb/>oportet re&longs;i&longs;tentiam ex motûs velocitate, quæ unicuique pon­<lb/>deri conveniat, &longs;ed reciprocè, ut fiat totius re&longs;i&longs;tentiæ æquali­<lb/>tas. </s> <s id="s.003028">Cum enim utrumque pondus movendum &longs;it, par e&longs;t ita re­<lb/>&longs;i&longs;tentiam dividi, ut æqualibus momentis adver&longs;entur poten­<lb/>tiæ contranitenti; quod &longs;cilicet gravius e&longs;t, difficiliùs movetur, <lb/>quod minus grave, faciliùs: igitur illius motus minor e&longs;t, hu­<lb/>jus major. </s> <s id="s.003029">Propterea centrum motuum iis intervallis ab utro­<lb/>que pondere aberit, ut quæ Ratio e&longs;t gravioris ponderis ad mi­<lb/>nus grave, ea &longs;it Ratio di&longs;tantiæ centri motûs à minùs gravi ad <lb/>di&longs;tantiam eju&longs;dem centri à graviore. </s> <s id="s.003030">Sit ex. </s> <s id="s.003031">gr. <!-- REMOVE S-->pondus E lib.8. <lb/>& pondus F lib.12; di&longs;tantia EF eadem quæ priùs, hoc e&longs;t, 6; & <lb/>FA 7. Cum igitur pondera &longs;int ut 2 ad 3, dividatur EF in quin­<lb/>que partes, & propè gravius F a&longs;&longs;umantur duæ FM, reliquæ <pb pagenum="397" xlink:href="017/01/413.jpg"/>tres ME &longs;pectent ad minus grave E. <!-- KEEP S--></s> <s id="s.003032">Si itaque circa centrum <lb/>M moveantur pondera E & F, habent æqualia re&longs;i&longs;tentiæ mo­<lb/>menta; nam lib. 12 moventur ut 2, & lib. 8 moventur ut 3. <lb/>Quare AM e&longs;t ad ME ut 9 2/5 ad 3 3/5, & AM ad MF e&longs;t ut 9 2/5 <lb/>ad 2 2/5. Fiat igitur ut AM ad ME, ita reciprocè pondus E <lb/>lib. 8 ad virtutem potentiæ A movendi &longs;ine vecte libras (3 3/47): & <lb/>ut AM ad MF, ita reciprocè pondus F lib.12 ad eju&longs;dem po­<lb/>tentiæ A virtutem movendi &longs;inè vecte libras (3 3/47). In hac ita­<lb/>que ponderum inæqualium di&longs;po&longs;itione paulo plus virium re­<lb/>quiritur in potentia (hoc e&longs;t vis movendi lib. (6 6/47)) quàm &longs;i e&longs;­<lb/>&longs;ent æqualia, eandemque gravitatis &longs;ummam lib. 20 con&longs;ti­<lb/>tuerent. </s> </p> <p type="main"> <s id="s.003033">At &longs;i vice versâ pondus E e&longs;&longs;et lib. 12, & F lib. 8, centrum <lb/>motuum e&longs;&longs;et N, atque AN e&longs;&longs;et 10 3/5: ac propterea ut AN <lb/>10 3/5 ad NE 2 2/5, ita pondus E lib. 12 ad virtutem potentiæ &longs;i­<lb/>ne vecte moventis libras (2 38/53); & ut AN 10 3/5 ad NF 3 3/5, ita <lb/>pondus F lib. 8 ad virtutem potentiæ A moventis &longs;ine vecte li­<lb/>bras (2 38/53). Tota igitur virtus potentiæ in hac eorumdem pon­<lb/>derum inæqualium collocatione &longs;ufficiet, &longs;i fuerit vis movendi <lb/>lib. (5 23/53), quæ minor e&longs;t eâ, quæ requiritur; quando pondera <lb/>&longs;unt æqualia, & differt à virtute, quæ requiritur, quando F <lb/>gravius e&longs;t 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/>nus fuerit intervallum inter E & F, etiam faciliùs duo illa pon­<lb/>dera eodem vecte moveri. </s> <s id="s.003035">Nam &longs;i idem vectis AE 13 adhi­<lb/>beatur, atque pondera E & F æqualia fuerint, intervallum ve­<lb/>rò EF &longs;it 4, centrum motuum di&longs;tabit ab A intervallo 11, & <lb/>à &longs;ingulis ponderibus intervallo 2: Quare potentia ut 4 move­<lb/>bit pondera &longs;ingula ut 11: vel &longs;i ponantur ut prius &longs;ingula <lb/>lib. 10, fiat ut 11 ad 2, ita lib. 10 ad potentiam &longs;ine vecte mo­<lb/>ventem lib. (1 9/11); atque ideò tota potentia &longs;ufficiens ad movenda <lb/>duo pondera æqualia &longs;imul &longs;umpta lib. 20, erit vis movendi &longs;ine <lb/>vecte lib. (3 7/11). Quod &longs;i E fuerit lib. 8, & F lib.12, E di&longs;tabit à <lb/>à centro motuum partibus 2 2/5, F verò part. </s> <s id="s.003036">1 3/5, & potentia A <lb/>di&longs;tabit part. </s> <s id="s.003037">10 3/5: Ex quo fit &longs;ingula moveri po&longs;&longs;e à potentia <pb pagenum="398" xlink:href="017/01/414.jpg"/>habente virtutem movendi &longs;ine vecte lib. (1 43/53), & ambo &longs;imul à <lb/>potentia habente vim movendi lib. (3 33/53). At verò &longs;i vici&longs;&longs;im E <lb/>fuerit lib. 12, & F lib. 8, di&longs;tantia potentiæ à centro motuum <lb/>erit part. </s> <s id="s.003038">11 2/5, ac propterea &longs;ingula pondera exigent virtutem <lb/>movendi lib. (1 39/57), & tota potentia ad utrumque &longs;imul moven­<lb/>dum &longs;ufficiens erit vis movendi &longs;ine vecte lib. (3 21/57), quæ deficit <lb/>à vi movendi lib. (3 33/53), ea virtute, quæ requireretur ad moven­<lb/>dum uncias (3 1/20), atque à vi movendi lib. (3 7/11) deficit per uncias <lb/>3 1/5 ferè. </s> </p> <p type="main"> <s id="s.003039">Que de corpore gravi dimovendo dicta &longs;unt, intelligantur <lb/>pariter, &longs;i vectis inter duo corpora flectenda, aut divellenda, <lb/>interjiceretur; quemadmodum objectos caveæ &longs;i quæras fran­<lb/>gere clathros: quod enim hîc gravitas, ibi ferreæ virgæ aut <lb/>lignei tigilli &longs;oliditas impedimentum affert motui. </s> </p> <p type="main"> <s id="s.003040">Porrò in vecte tertij generis, quando potentia inter utrum­<lb/>que pondus mobile con&longs;tituitur, aliter res &longs;e habet: adhoc &longs;ci­<lb/>licet, ut aliquam vectis Rationem habeat, requiritur aut inæ­<lb/>qualitas ponderum, aut &longs;altem inæqualitas di&longs;tantiarum po­<lb/>tentiæ à ponderibus in utrâque extremitate con&longs;titutis, ita ta­<lb/>men ut hæ di&longs;tantiæ non &longs;int in reciprocâ Ratione ponderum: <lb/>nam &longs;i planè æqualiter di&longs;taret potentia ab æqualibus ponderi­<lb/>bus, aut inæquales di&longs;tantiæ e&longs;&longs;ent in reciprocâ Ratione inæ­<lb/>qualium ponderum, ita utrumque traheretur, aut impellere­<lb/>tur, ut pondera &longs;ingula æquè moverentur ac potentia: ad Ra­<lb/>tiones autem vectis &longs;pectat inæqualiter moveri potentiam ac <lb/>pondus, &longs;i vectis quidem obtineat vim Facultatis Mechanicæ. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003041">Quoniam igitur in huju&longs;modi vecte tertii generis oportet <lb/>utrumque pondus opponi motui potentiæ; vel quia utrumque <lb/>impellitur, vel quia alterum trahitur, alterum impellitur, &longs;it <lb/><figure id="id.017.01.414.1.jpg" xlink:href="017/01/414/1.jpg"/><lb/>vectis AB, in cujus extremitati­<lb/>bus pondera re&longs;pondeant punctis <lb/>A & B: Si potentia fuerit in C <lb/>æquè di&longs;tans ab A & B, pondera <lb/>autem fuerint æqualia; potentia <lb/>ex C versùs D mota nullum ha­<lb/>beret &longs;ui motûs centrum, &longs;ed pariter traheret aut impelleret <lb/>ad partes D utrumque pondus; nam æquè utrumque re&longs;i&longs;teret <pb pagenum="399" xlink:href="017/01/415.jpg"/>tùm ratione gravitatis, tùm ratione po&longs;itionis & di&longs;tantiæ, quæ <lb/>legem daret motui, ac proinde utrumque æqualiter cederet <lb/>virtuti potentiæ. </s> <s id="s.003042">At &longs;i pondus A minus fuerit, quàm pondus B, <lb/>&longs;ed reciprocam Rationem habeant di&longs;tantiæ potentiæ exi&longs;tentis <lb/>in E, ut &longs;it EB ad EA, in Ratione ponderis A ad pondus B; <lb/>adhuc æquales &longs;unt re&longs;i&longs;tentiæ; &longs;icut enim in plano Verticali <lb/>potentiæ in E &longs;u&longs;tineret utrumque pondus in æquilibrio, ita in <lb/>plano horizontali trahens aut impellens utrumque æqualiter <lb/>moveret. </s> </p> <p type="main"> <s id="s.003043">Sint igitur pondera A & B &longs;ive æqualis gravitatis, &longs;ive inæ­<lb/>qualis, & ita potentia &longs;it in E, ut EB ad EA non &longs;it in Ratio­<lb/>ne ponderis A ad pondus B: utique &longs;i B moveri non po&longs;&longs;et, <lb/>potentia E circa B, tanquam circa centrum, de&longs;criberet ar­<lb/>cum EI, & pondus A arcum AF: &longs;imiliter &longs;i pondus A immo­<lb/>tum maneret, potentia circa A, tanquam circa centrum, de&longs;­<lb/>criberet arcum EH, & pondus B arcum BG, ex hypothe&longs;i <lb/>æqualem arcui AF. <!-- KEEP S--></s> <s id="s.003044">Potentia igitur in E faciliùs cæteris pari­<lb/>bus moveret pondus B &longs;ibi proximum, quàm pondus A remo­<lb/>tum, &longs;i &longs;ingula &longs;ingillatim movenda e&longs;&longs;ent; quia, cum arcus <lb/>EH major &longs;it arcu EI, arcus autem BG, & AF &longs;int æquales, <lb/>major e&longs;t Ratio EH ad EI quàm BG ad AF; & per 27. lib.5. <lb/>vici&longs;&longs;im EH ad BG habet majorem Rationem quàm EI ad <lb/>AF. <!-- KEEP S--></s> <s id="s.003045">Cum itaque neutra extremitas immota maneat, &longs;ed ambo <lb/>pondera moveantur, minùs movetur A, quod difficiliùs, ma­<lb/>gis B, quod faciliùs: ac propterea A &longs;impliciter fungitur mu­<lb/>nere hypomochlij ad motum ponderis B: hoc verò vici&longs;&longs;im ad <lb/>ponderis A motum, quamvis minorem, &longs;ubit vicem fulcri: <lb/>Neque enim hic unum tribus motibus, potentiæ videlicet & <lb/>duorum ponderum, commune centrum reperire e&longs;t, quia ad <lb/>eandem partem omnium motus dirigitur. </s> <s id="s.003046">Hinc &longs;i fune alligato <lb/>in E trahas vectem cum ponderibus, punctum E neque omni­<lb/>no versùs I, neque omnino versùs H dirigetur, quamquam ad <lb/>H potiùs, quàm ad I inclinabitur; quia faciliùs A vectis <lb/>punctum re&longs;pondens ponderi convertitur circa centrum gravi­<lb/>tatis ponderis, quàm propellat aut trahat totum pondus, pro <lb/>ut ferunt, & ip&longs;ius gravitas, & eju&longs;dem di&longs;tantia ab E, quæ il­<lb/>lius re&longs;i&longs;tentiam componunt. <pb pagenum="400" xlink:href="017/01/416.jpg"/></s> </p> <p type="main"> <s id="s.003047"><emph type="center"/>CAPUT VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003048"><emph type="center"/><emph type="italics"/>Quænam &longs;int momenta Vectis pondus fune <lb/>connexum trahentis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003049">COntingere pote&longs;t oblato ponderi &longs;uper planum horizonta­<lb/>le, aut inclinatum, trahendo non e&longs;&longs;e parem Potentiam: <lb/>hujus imbecillitati opem ferre licebit Vecte poti&longs;&longs;imùm &longs;ecun­<lb/>di generis, cujus extremitas altera fixa & &longs;tabilis maneat in pla­<lb/>no, in quo pondus jacet, alteram extremitatem Potentia mo­<lb/>veat, & loco intermedio alligetur funis cum pondere connexus, <lb/>qui dum vecte movetur, &longs;ecum rapiat & pondus. </s> <s id="s.003050">Verùm non <lb/>leviter hallucinaretur, qui&longs;quis momenta vectis ex alligati fu­<lb/>nis loco &longs;impliciter & ab&longs;olutè definiret; cum potiùs ponderis <lb/>re&longs;i&longs;tentiam ex ip&longs;ius motu computare oporteat. </s> <s id="s.003051">Quoniam verò <lb/>duplex e&longs;&longs;e pote&longs;t vectis motus, nimirum aut in plano Verticali, <lb/>aut in Horizontali, propterea uterque &longs;eor&longs;im con&longs;iderandus <lb/>e&longs;t; diver&longs;as enim lineas in plano, in quo jacet, pondus percur­<lb/>rit; rectam &longs;cilicet, &longs;i vectis in plano Verticali agitatur; curvam <lb/>verò, &longs;i in plano horizontali aut inclinato eodem, cui pondus <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/>gruere intelligatur, & concipiatur pondus in puncto C; vecti <lb/><figure id="id.017.01.416.1.jpg" xlink:href="017/01/416/1.jpg"/><lb/>autem in D alligetur funis ita <lb/>connectens pondus cum vecte, <lb/>ut parti vectis DA æqualis &longs;it <lb/>funis DC. <!-- KEEP S--></s> <s id="s.003053">Attollatur in plano <lb/>Verticali vectis, ut &longs;it AE <lb/>de&longs;cribens arcum BE; etiam <lb/>punctum D a&longs;cendit in F, ac <lb/>propterea funis e&longs;t FG, & <lb/>ponderis motus e&longs;t CG. <!-- KEEP S--></s> <s id="s.003054">Ite­<lb/>rum attollatur æqualiter vectis <lb/>ex E in H; funis caput venit in I, & pondus in K. <!-- KEEP S--></s> <s id="s.003055">Similiter <pb pagenum="401" xlink:href="017/01/417.jpg"/>vecte in L &longs;ublato, funis venit in M, & pondus in N. <!-- KEEP S--></s> <s id="s.003056">Sunt <lb/>igitur tre, ponderis motus, CG, GK, KN, inter &longs;e inæqua­<lb/>les, qui &longs;emper majores fiunt; motus autem potentiæ BE, EH, <lb/>HL ex hypothe&longs;i &longs;unt æquales; igitur major e&longs;t Ratio motûs <lb/>BE ad motum CG, quàm motûs EH ad motum GK, & hæc <lb/>Ratio major e&longs;t Ratione motûs HL ad motum KN. </s> <s id="s.003057">Cum ita­<lb/>que motibus BE, EH, HL &longs;imiles &longs;int motus DF, FI, IM, <lb/>manife&longs;tum e&longs;t motum ponderis non &longs;ervare Rationem &longs;ecun­<lb/>dùm quam dividitur vectis ab alligati funis capite, eadem quip­<lb/>pe &longs;emper e&longs;t Ratio EA ad AF, & HA ad AI, & LA <lb/>ad AM. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003058">Motus autem illos ponderis CG, GK & KN &longs;emper e&longs;&longs;e <lb/>majores hinc con&longs;tat, quia pars vectis inter funem alligatum <lb/>atque hypomochlium A ex hypothe&longs;i e&longs;t æqualis ip&longs;i funi con­<lb/>nectenti pondus: &longs;unt igitur triangula I&longs;o&longs;celia æqualium &longs;em­<lb/>per laterum, &longs;ed quæ majores & majores angulos ad ba&longs;im ha­<lb/>bent, ideóque minorem & minorem angulum verticalem con­<lb/>tinent. </s> <s id="s.003059">Atqui angulorum ad centrum in circulis æqualibus, <lb/>vel in eodem circulo, &longs;emper æqualiter decre&longs;centium &longs;ubten­<lb/>&longs;æ minores fiunt eâ lege, ut decrementa illa, hoc e&longs;t, &longs;ubten&longs;a­<lb/>rum diminutarum differentiæ augeantur, ut ex Canone Si­<lb/>nuum con&longs;tat. </s> <s id="s.003060">Cum itaque AG &longs;it &longs;ubten&longs;a anguli AFG, & <lb/>AK &longs;it &longs;ubten&longs;a anguli AIK minoris, & AN &longs;it &longs;ubton&longs;a an­<lb/>guli AMN adhuc minoris; harum &longs;ubten&longs;arum differentiæ, <lb/>videlicet CG (differentia inter diametrum circuli AC & &longs;ub­<lb/>ten&longs;am AG) GK & KN motus ponderis &longs;emper augentur. </s> </p> <p type="main"> <s id="s.003061">Id quod ut manife&longs;tum fiat, triangula ip&longs;a ad calculos revo­<lb/>cemus &longs;ingulorum ba&longs;im inquirentes: ponamus verò ex. </s> <s id="s.003062">gr. <!-- REMOVE S-->ar­<lb/>cus BE, EH, HL &longs;ingulos grad. <!-- REMOVE S-->15, & latera &longs;ingula AF & <lb/>GF e&longs;&longs;e partium 100. Igitur angulus AFG e&longs;t grad. <!-- REMOVE S-->150, & <lb/>ba&longs;is AG deprehenditur partium (193 18/100). E&longs;t autem AC ex <lb/>hypothe&longs;i 200, adeóque CG part. (6 82/100). In triangulo AIK la­<lb/>tera &longs;unt eadem, anguli ad ba&longs;im &longs;inguli grad. <!-- REMOVE S-->30, angulus ver­<lb/>ticalis grad. <!-- REMOVE S-->120; ergo ba&longs;is AK part. (173 20/100): & inter AK at­<lb/>que AG differentia GK e&longs;t (19 98/100). Deinde in triangulo AMN <lb/>anguli ad ba&longs;im &longs;inguli &longs;unt grad. <!-- REMOVE S-->45; igitur angulus vertica­<lb/>lis grad. <!-- REMOVE S-->90, & ba&longs;is AN part. (141 42/100), & inter AN & AK dif-<pb pagenum="402" xlink:href="017/01/418.jpg"/>ferentia KN e&longs;t (31 78/100). Et &longs;i vectem elevando pergas, idem in <lb/>con&longs;equentibus triangulis deprehendes, augeri &longs;cilicet diffe­<lb/>rentia, u&longs;que ad A. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003063">Hinc patet eò faciliorem e&longs;&longs;e, cæteris paribus, motum, quò <lb/>majorem angulum funis cum vecte con&longs;tituit, nam ab æquali <lb/>potentiæ motu minor ponderis motus efficitur, quam &longs;i major <lb/>e&longs;&longs;et angulus elevationis vectis. </s> <s id="s.003064">Quare faciliùs promovebitur <lb/>ad de&longs;tinatum locum pondus quod trahitur, &longs;i po&longs;t aliqualem <lb/>vectis elevationem, iterum inclinato, quàm maximè fieri po­<lb/>terit, vecte, extremitatem A, hoc e&longs;t hypomochlium &longs;ubinde <lb/>promoveas, quantum feret funis longitudo: tunc enim fune <lb/>maximè inclinato tractio ponderis minùs obliqua juvabit mo­<lb/>tum, qui etiam minor e&longs;t, quàm &longs;i pergeres vectem elevando. </s> </p> <p type="main"> <s id="s.003065">Non e&longs;t tamen nece&longs;&longs;e &longs;ervari hanc, quam claritatis gratiâ <lb/>propo&longs;ui, æqualitatem funis FG, & partis vectis FA; &longs;ed a&longs;&longs;u­<lb/>mi pote&longs;t vel longior, vel brevior funis, adeò ut ex vectis par­<lb/>te, ex fune, & ex di&longs;tantiâ ponderis ab hypomochlio fiat trian­<lb/>gulum &longs;calenum: in quo &longs;i funis fuerit longior parte vectis, eo­<lb/>dem potentiæ motu minùs accedet pondus ad hypomochlium, <lb/>quàm &longs;i funis fuerit brevior eâdem vectis parte; atque quo lon­<lb/>gior fuerit funis, etiam minor erit, adeóque facilior, ponderis <lb/>motu, cæteris paribus, nam & tractio minùs obliqua erit. </s> <lb/> <s id="s.003066">Statue igitur ex. </s> <s id="s.003067">gr. <!-- REMOVE S-->AD e&longs;&longs;e partium 73, & DC part. </s> <s id="s.003068">100: <lb/>quare vecte jacente, di&longs;tantia AC e&longs;t 173. </s> </p> <p type="main"> <s id="s.003069">Sit angulus FAG iterum gr. <!-- REMOVE S-->15; invenitur angulus AFG <lb/>gr. <!-- REMOVE S-->154. m. </s> <s id="s.003070">7, & ba&longs;is AG part. (168 66/100); igitur CG (4 34/100). In <lb/>triangulo AIK latera AI 73, IK 100 ut priùs, angulus IAK <lb/>gr. <!-- REMOVE S-->30: invenitur angulus verticalis AIK gr. <!-- REMOVE S-->128. m. </s> <s id="s.003071">36, & <lb/>ba&longs;is AK part. (156 30/100): igitur GK e&longs;t (12 36/100). Demum in trian­<lb/>gulo AMN latera &longs;unt eadem ut priùs, angulus MAN gr.45: <lb/>ex quibus datis invenitur angulus AMN gr.103. m. </s> <s id="s.003072">55, & ba­<lb/>&longs;is AN part. (137 27/100): igitur KN (19 3/100), & totus motus CN e&longs;t <lb/>part. (35 73/100). </s> </p> <p type="main"> <s id="s.003073">Sed vici&longs;&longs;im &longs;tatue AD partium 100, DC verò funem <lb/>part. </s> <s id="s.003074">73; quibus æqualia &longs;unt trianguli AFG latera AF 100 <lb/>& FG 73; angulus autem FAG e&longs;t gr. <!-- REMOVE S-->15: invenitur angulus <lb/>FGA gr. <!-- REMOVE S-->20. m. </s> <s id="s.003075">46, & angulus AFG gr. <!-- REMOVE S-->144. m. </s> <s id="s.003076">46: quare <pb pagenum="403" xlink:href="017/01/419.jpg"/>AG e&longs;t part. (164 85/100), & motus CG part. (8 15/100), qui tamen &longs;u­<lb/>periùs, quando DC major erat, quàm AD, deprehen&longs;us e&longs;t <lb/>&longs;olum (4 34/100). In Triangulo AIK &longs;imiliter datur AI 100, IK 73, <lb/>angulus IAK gr. <!-- REMOVE S-->30: invenitur angulus IKA gr. <!-- REMOVE S-->43. m. </s> <s id="s.003077">14, <lb/>ac proinde angulus verticalis AIK gr. <!-- REMOVE S-->106. m. </s> <s id="s.003078">46, & ba&longs;is AK <lb/>part. (139 79/100): igitur GK part. (25 6/100), quæ tamen priùs erat <lb/>(12 36/100). <!--neuer Satz-->Deinde in triangulo AMN latera &longs;int eadem, & an­<lb/>gulus MAN gr. <!-- REMOVE S-->45: invenitur angulus MNA gr. <!-- REMOVE S-->75. m. </s> <s id="s.003079">37, <lb/>& verticalis AMN gr.59.m.23: quare ba&longs;is AN e&longs;t par. (88 85/100), <lb/>& motus KN part. (50 94/100), qui priùs erat (19 3/100). Verùm ele­<lb/>vari vectis poterit &longs;olùm, ut funis fiat perpendicularis horizon­<lb/>ti, &longs;cilicet facto angulo ad A gr. <!-- REMOVE S-->46. m. </s> <s id="s.003080">53; & ba&longs;is erit di&longs;tan­<lb/>tia ab A part. (68 35/100). </s> </p> <p type="main"> <s id="s.003081">Ut autem innote&longs;cat, quid contingat fune adhuc longiore <lb/>quàm part. </s> <s id="s.003082">100, po&longs;itâ eádem vectis parte AD part. </s> <s id="s.003083">73. non <lb/>pigeat iterum examinare triangula. </s> <s id="s.003084">Sit ergo funis DC <lb/>part. </s> <s id="s.003085">200, quarum AD e&longs;t 73; anguli elevationis vectis &longs;int <lb/>iidem, qui &longs;uperiùs. </s> <s id="s.003086">In Triangulo AFG, angulus FAG e&longs;t <lb/>gr. <!-- REMOVE S-->15, latus AF 73, latus FG 200: invenitur angulus FGA <lb/>gr. <!-- REMOVE S-->5. m. </s> <s id="s.003087">25, & angulus AFG gr. <!-- REMOVE S-->159. m. </s> <s id="s.003088">35: ac proinde ba&longs;is <lb/>AG part. (269 56/100), & motus CG part. (3 44/100), qui, po&longs;ito fune <lb/>FG 100, erat (4 34/100). In triangulo AIK latera &longs;unt eadem, <lb/>angulus IAK e&longs;t gr. <!-- REMOVE S-->30; ergo angulus IKA gr. <!-- REMOVE S-->10. m. </s> <s id="s.003089">31, & <lb/>verticalis AIK gr. <!-- REMOVE S-->139. m. </s> <s id="s.003090">29; atque ba&longs;is AK part. (259 87/100); ac <lb/>propterea GK part. (9 69/100), quæ priùs fuit (12 36/100). Denique in <lb/>Triangulo AMN eadem latera 73 & 200 cum angulo MAN <lb/>gr. <!-- REMOVE S-->45, dant angulum MNA gr. <!-- REMOVE S-->14. m. </s> <s id="s.003091">57, & verticalem <lb/>AMN gr. <!-- REMOVE S-->120. m. </s> <s id="s.003092">3; atque ba&longs;im AN part. (244 82/100): quare <lb/>motus KN e&longs;t (15 5/100), qui in priore hypothe&longs;i erat (19 3/100). Lon­<lb/>gior itaque funis dat minorem & faciliorem motum pon­<lb/>deris. </s> </p> <p type="main"> <s id="s.003093">Quemadmodum verò elevando vectem à po&longs;itione hori­<lb/>zontali u&longs;que ad perpendiculum difficultas trahendi augetur, <lb/>quia pondus velociùs movetur, ita ex adver&longs;o, &longs;i vectis hori­<lb/>zonti perpendicularis inclinetur ad partem oppo&longs;itam ponderi <pb pagenum="404" xlink:href="017/01/420.jpg"/>(adeò ut vectis &longs;it inter potentiam & pondus) cre&longs;cit trahendi <lb/>facilitas, quia pondus &longs;emper tardiùs movetur, quo magis <lb/><figure id="id.017.01.420.1.jpg" xlink:href="017/01/420/1.jpg"/><lb/>vectis ad horizontem de­<lb/>primitur. </s> <s id="s.003094">Sit enim pon­<lb/>dus in P, vectis perpendi­<lb/>cularis CB, funis OP <lb/>utique longior parte vectis <lb/>OC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003095">Inclinetur vectis per <lb/>Quadrantis trientem, ut O <lb/>veniat in S; funis erit ST, <lb/>& pondus veniet ex P in T. Æquali inclinatione deprimatur <lb/>vectis, ut S veniat in M; funis erit MV, & ponderis motus <lb/>TV. </s> <s id="s.003096">Demum vectis horizonti congruat, ut M veniat in N; <lb/>funis erit NI, motú&longs;que ponderis VI. <!-- KEEP S--></s> <s id="s.003097">Cum igitur &longs;emper tar­<lb/>diùs moveatur pondus, quia &longs;patia PT, TV, VI &longs;emper de­<lb/>cre&longs;cunt, æquales autem potentiæ motus ill's re&longs;pondeant, <lb/>etiam cre&longs;cit trahendi facilitas. </s> </p> <p type="main"> <s id="s.003098">Illa autem ba&longs;ium CP, CT, CV decrementa in triangulis <lb/>COP, CST, CMV &longs;emper minui con&longs;tabit ex Trigonome­<lb/>tria; dantur enim in omnibus eadem duo latera, &longs;cilicet funis <lb/>longitudo, & pars vectis, datúrque in &longs;ingulis æqualiter <lb/>cre&longs;cens angulus funi oppo&longs;itus; quare inveniuntur & ba&longs;es, <lb/>quarum differentiæ &longs;emper minores fiunt. </s> <s id="s.003099">Sic in triangulo <lb/>OCP rectangulo &longs;it perpendiculum OC partium 73, & hypo­<lb/>thenu&longs;a OP part. </s> <s id="s.003100">100; igitur CP ba&longs;is e&longs;t part. (68 34/100). Deinde <lb/>quia CS e&longs;t part. </s> <s id="s.003101">73, ST part. </s> <s id="s.003102">100, & angulus SCT gr. <!-- REMOVE S-->120, <lb/>invenitur CT part. (40 97/100): ergo PT e&longs;t part. (27 37/100). Similiter <lb/>MC e&longs;t 73, MV 100, angulus MCV gr. <!-- REMOVE S-->150; igitur CV in­<lb/>venitur part. (29 91/100); ac proinde TV e&longs;t part. (11 6/100). Demum <lb/>quia NC e&longs;t 73, & NI e&longs;t 100, remanet CI part. </s> <s id="s.003103">27: & <lb/>ablatâ CI ex CV, relinquitur IV part. (2 91/100). Totus itaque <lb/>motus PI e&longs;t part. (41 34/100). Fac autem OC 73 e&longs;&longs;e quartam to­<lb/>tius vectis partem, qui proinde erit part. </s> <s id="s.003104">292. Et quia Radius <lb/>ad Quadrantem peripheriæ circuli e&longs;t ut 7 ad 11, &longs;i fiat ut 7 <lb/>ad 11, ita 292 ad 458 6/7, potentia in vectis extremitate po&longs;ita <pb pagenum="405" xlink:href="017/01/421.jpg"/>motum habet part. </s> <s id="s.003105">458, dum pondus movetur &longs;olum per &longs;pa­<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/>quam &longs;tatuamus OP ex. </s> <s id="s.003108">gr. <!-- REMOVE S-->partium 200, quarum OC e&longs;t 73. <lb/>igitur CP e&longs;t 186 1/5; & quia NC e&longs;t 73, atque NI ex hypo­<lb/>the&longs;i e&longs;t 200, remanet CI part. </s> <s id="s.003109">127; atque adeò totus motus <lb/>PI e&longs;t part. </s> <s id="s.003110">59 1/5: ad quem motum idem potentiæ motus 458 <lb/>habet minorem Rationem quàm ad motum part.41, quem dat <lb/>minor funis longitudo. </s> </p> <p type="main"> <s id="s.003111">Supere&longs;t adhuc tertia quædam ponderis po&longs;itio, vecte agita­<lb/>to in plano verticali; quando nimirum initio motûs &longs;tatuitur <lb/>pondus proximum hypomochlio, à quo in motu recedat: hu­<lb/>ju&longs;modi &longs;cilicet vecte uti po&longs;&longs;umus, cùm aliquid modicè qui­<lb/>dem movendum in plano horizontali proponitur, &longs;ed multa e&longs;t <lb/>difficultas. </s> <s id="s.003112">Similiter &longs;i longiu&longs;cula ferrea bractea e&longs;&longs;et &longs;uis ex­<lb/>tremitatibus validè connexa cum aliquo corpore, & circa me­<lb/>dium eam flecti oporteret, ut cuneus vel aliquid &longs;imile inter <lb/>corpus & bracteam in&longs;eri po&longs;&longs;et; funi adnecteretur uncus <lb/>bracteam apprehendens, qui elevato vecte, &longs;ive depre&longs;&longs;o il­<lb/>lam aliquantulum flecteret. </s> </p> <p type="main"> <s id="s.003113">Sit vectis hypomochlium in R, funis in K alligatus, & funis <lb/>longitudo KS 73; pars verò vectis RK 100: quare RS di&longs;tan­<lb/>tia ponderis S ab hypomo­<lb/><figure id="id.017.01.421.1.jpg" xlink:href="017/01/421/1.jpg"/><lb/>chlio R e&longs;t 27. Moveatur <lb/>vectis &longs;ur&longs;um, & faciat angu­<lb/>lum LRK gr. <!-- REMOVE S-->15 funis e&longs;t LI <lb/>part. </s> <s id="s.003114">73, & LR part. </s> <s id="s.003115">100; igi­<lb/>tur ex his datis invenitur an­<lb/>gulus RIL gr. <!-- REMOVE S-->159. m. </s> <s id="s.003116">14, & <lb/>RLI gr. <!-- REMOVE S-->5. m. </s> <s id="s.003117">46; adeóque <lb/>ba&longs;is RI part. (28 34/100); igitur mo­<lb/>tus ex S in I e&longs;t (1 34/100). Quod <lb/>&longs;i ponatur RL e&longs;&longs;e quarta pars vectis, totus Radius e&longs;t part.400, <lb/>& arcus ab extremitate vectis de&longs;criptus gr. <!-- REMOVE S-->15, e&longs;t part. <lb/>(104 32/100): Ex quo vides motum potentiæ ad motum ponderis e&longs;&longs;e <lb/>proximè ut 78 ad 1. At verò &longs;i funis LI longior ponatur, ut &longs;it <lb/>part. </s> <s id="s.003118">90, & reliqua &longs;int ut priùs, invenitur angulus RIL gr. <!-- REMOVE S-->163. <pb pagenum="406" xlink:href="017/01/422.jpg"/>m. </s> <s id="s.003119">17, & angulus RLI gr. <!-- REMOVE S-->1. m. </s> <s id="s.003120">43′: quare ba&longs;is RI e&longs;t part. <lb/>(10 4s/100): & quia RS ex hypothe&longs;i e&longs;t &longs;olum part. </s> <s id="s.003121">10, motus SI <lb/>e&longs;t (42/100) multo minor quàm cum funis brevior ponitur, ac prop­<lb/>terea etiam facilior e&longs;t motus, quippe qui minorem Rationem <lb/>habet ad motum potentiæ. </s> </p> <p type="main"> <s id="s.003122">Pergendo autem in elevatione vectis adhuc per gr. <!-- REMOVE S-->15, ita ut <lb/>angulus PRO &longs;it gr. <!-- REMOVE S-->30, PR e&longs;t 100, PO e&longs;t 73: invenitur <lb/>angulus ROP gr. <!-- REMOVE S-->136. m. </s> <s id="s.003123">46, & angulus RPO gr. <!-- REMOVE S-->13. m. </s> <s id="s.003124">14, <lb/>atque ba&longs;is RO part. (33 42/100): quare motus IO e&longs;t part. (5 8/100). Et <lb/>iterum elevando vectem per gr. <!-- REMOVE S-->15, ita ut angulus QRT &longs;it <lb/>gr. <!-- REMOVE S-->45, invenitur angulus QTR gr. <!-- REMOVE S-->104. m. </s> <s id="s.003125">23, & angulus <lb/>RQT gr. <!-- REMOVE S-->30. m. </s> <s id="s.003126">37, ba&longs;is autem RT part. (52 57/100): ex quo fit <lb/>motum OT e&longs;&longs;e partium (19 15/100). Hinc patet æqualibus poten­<lb/>tiæ motibus inæquales, &longs;empérque majores ponderis motus <lb/>re&longs;pondere, ac proinde cre&longs;cere movendi difficultatem; cum <lb/>enim pondus &longs;uâ gravitate in&longs;i&longs;tat &longs;ubjecto plano, in quo trahi­<lb/>tur, quò magis elevatur vectis, etiam funis magis obliquus e&longs;t, <lb/>minu&longs;que valida fit tractio, quæ magis obliqua e&longs;t. </s> </p> <p type="main"> <s id="s.003127">Quas hactenus recen&longs;uimus tractiones, fieri per lineam <lb/>rectam vel accedendo ad punctum hypomochlij, vel ab eo re­<lb/>cedendo, &longs;atis con&longs;tat; quia, dum vectis in plano Verticali <lb/>movetur, pondus non recedit ab illo eodem plano Verticali <lb/>&longs;emper &longs;uâ gravitate in&longs;i&longs;tens plano horizontali, atque idcirco <lb/>motus illius e&longs;t in communi horum planorum &longs;ectione, hoc <lb/>e&longs;t, in lineâ rectâ. </s> <s id="s.003128">Sin autem motus vectis fuerit in eodem <lb/>plano horizontali, in quo e&longs;t pondus fune trahendum, quia <lb/>vectis circulariter movetur, illum &longs;equitur pondus per lineam <lb/>curvam, &longs;ed quo ad ejus fieri po&longs;&longs;it, brevi&longs;&longs;imam, ut quàm mi­<lb/>nimam patiatur violentiam. </s> <s id="s.003129">Certum quippe e&longs;t oportere fu­<lb/>nem vecti congruentem citra quamlibet anguli inclinationem, <lb/>e&longs;&longs;e breviorem parte illâ vectis, quæ inter hypomochlium, & <lb/>locum alligati funis, intercipitur; &longs;i enim æqualis e&longs;&longs;et, cir­<lb/>cumducto vecte pondus fulcro proximum non moveretur; <lb/>multo minùs, &longs;i longior e&longs;&longs;et funis. </s> <s id="s.003130">Cum itaque brevior &longs;it, <lb/>nece&longs;&longs;e e&longs;t pondus quoque circumduci, &longs;ed non eâ ratione, <lb/>qua moveretur, &longs;i funis eundem &longs;emper angulum cum vecte <lb/>con&longs;titueret; quemadmodum contingeret, &longs;i vecti loco funis <pb pagenum="407" xlink:href="017/01/423.jpg"/>flexilis rigidum brachium adjaceret, cur pondus adnecteretur. </s> <lb/> <s id="s.003131">Verùm quia pondus &longs;uâ gravitate re&longs;i&longs;tit, dum vectis movetur, <lb/>recinetur aliquantulum funis à pondere, & angulum &longs;ubinde <lb/>majorem cum vecte efficit; trahitur tamen pondus, &longs;ed ita, ut <lb/>violentiam &longs;ubeat quàm minimam pro ratione po&longs;itionis; ac <lb/>propterea lineam curvam helici &longs;imilem de&longs;cribit, quo ad fu­<lb/>nis certum angulum acutum (pro ut funis, aut pars vectis lon­<lb/>giores fuerint, &longs;ive breviores) cum vecte con&longs;tituat; quo de­<lb/>inde angulo manente pondus in gyrum ducitur per circuli am­<lb/>bitum. </s> <s id="s.003132">Ob&longs;ervabis enim po&longs;itâ eadem vectis parte, quò bre­<lb/>vior fuerit funis, eò majorem e&longs;&longs;e angulum illum, ad quem de­<lb/>venitur, & in quo con&longs;i&longs;titur nec illum augendo, nec minuendo. </s> <lb/> <s id="s.003133">Quod &longs;i in funem eâdem vectis parte longiorem ita di&longs;ponas, ut <lb/>non vecti congruat, &longs;ed cum illo angulum efficiat, circumducto <lb/>vecte ita pondus per helicem moveri videbis, ut diminuto &longs;ub­<lb/>inde angulo, demum funis vecti in eadem rectâ lineâ congruat, <lb/>& ponderis ultra hypomochlium manentis tractio de&longs;inat: quia <lb/>videlicet ponderis gravitas re&longs;i&longs;tens licèt trahatur, retinet ta­<lb/>men funem, & minuitur angulus, u&longs;que dum omnis angulus <lb/>pereat. </s> </p> <p type="main"> <s id="s.003134">Huju&longs;modi motûs cau&longs;am deprehendes, &longs;i attentè in&longs;picias <lb/>pondus, cum vecti, in gyrum moveri incipit, ita trahi, ut etiam <lb/>aliquantulum circa &longs;uum centrum gravitatis, aut circa aliud <lb/>punctum (neque enim hic locus e&longs;t punctum illud definiendi) <lb/>volvatur; ex qua conver&longs;ione fit minore motu opus e&longs;&longs;e, ut <lb/>pondus con&longs;equatur trahentem: ubi verò tanta facta fuerit <lb/>ponderis circa &longs;uum centrum conver&longs;io, ut &longs;i in hanc, vel in <lb/>illam partem adhuc converteretur, majorem &longs;ubiret in tractio­<lb/>ne violentiam, hoc e&longs;t, cogeretur majorem motum perficere, <lb/>quàm &longs;it motus eju&longs;dem nulla factâ circa &longs;uum illud centrum <lb/>conver&longs;ione, tunc manet angulus funis cum vecte, nec jam au­<lb/>getur aut minuitur. </s> <s id="s.003135">Quia autem, cæteris paribus, quò bre­<lb/>vior e&longs;t funis, cò major fit ponderis circa &longs;uum centrum con­<lb/>ver&longs;io; propterea ad majorem angulum demum inclinatur fu­<lb/>nis in vectem. </s> <s id="s.003136">Sed in hoc non e&longs;t diutiùs immorandum; rarus <lb/>quippe e&longs;t huju&longs;modi tractionis u&longs;us. <pb pagenum="408" xlink:href="017/01/424.jpg"/></s> </p> <p type="main"> <s id="s.003137"><emph type="center"/>CAPUT VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003138"><emph type="center"/><emph type="italics"/>Quid conferat Potentiæ moventis applicatio <lb/>ad vectem.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003139">QUoniam duplex e&longs;t Potentiæ genus, alia &longs;iquidem inani­<lb/>ma e&longs;t, cujus conatus &longs;ecundùm rectam lineam in cen­<lb/>trum, vel à centro, dirigitur, prout gravis e&longs;t aut levis, alia <lb/>e&longs;t vivens, quæ pro variâ mu&longs;culorum intentione, ac mem­<lb/>brorum inflexione in aliam atque aliam partem dirigi pote&longs;t; <lb/>idcirco, cuju&longs;modi potentiâ uti liceat, con&longs;iderandum e&longs;t, ut <lb/>opportunum vectis genus eligatur. </s> <s id="s.003140">Nam &longs;i vectis primi gene<lb/>ris depre&longs;&longs;ione attollendum &longs;it pondus, & potentia &longs;it vivens, <lb/>ut homo, major e&longs;t movendi facilitas, tum quia ip&longs;um vectis <lb/>pondus potentiam juvat &longs;uâ gravitate, tum quia in huju&longs;modi <lb/>depre&longs;&longs;ione vectis non &longs;olùm brachiorum, &longs;ed etiam quando­<lb/>que totius humani corporis vecti incumbentis, vel ex vecte <lb/>pendentis gravitas momentum non leve addit contentioni, qua <lb/>virtus movendi impetum vecti imprimen connititur. </s> <s id="s.003141">Sin au­<lb/>tem vecte &longs;ecundi, aut tertij generis elevandum &longs;it pondus <lb/>idem, ip&longs;a vectis gravitas officit, quam pariter cum ip&longs;o pon­<lb/>dere attollere oportet, & majore virium contentione opus e&longs;t, <lb/>ut experientiâ docemur. </s> </p> <p type="main"> <s id="s.003142">Verùm illud, quod hîc poti&longs;&longs;imum examinandum proponi­<lb/>tur, e&longs;t ip&longs;a potentiæ, quæcumque illa &longs;it, applicatio ad <lb/>vectem: neque enim &longs;atis e&longs;t, &longs;i illa extremitati vectis adjun­<lb/>gatur, aut certo quodam loco in vecte tertij generis collocata <lb/>intelligatur; &longs;ed maximè attendendum e&longs;t, &longs;ecundùm quam <lb/>lineam potentiæ motus dirigatur; diver&longs;a quippe &longs;unt poten­<lb/>tiæ momenta pro alia atque aliâ huju&longs;modi motûs directione, <lb/>quatenus cum vecte comparatur. </s> <s id="s.003143">Quemadmodum enim &longs;i po­<lb/>tentia vectem urgeat, aut trahat, juxta eju&longs;dem vectis in hy­<lb/>pomochlij puncto firmati longitudinem, nihil prorsùs in pon­<lb/>dere efficit; ita quoquè &longs;i in vectis longitudinem obliquè inci-<pb pagenum="409" xlink:href="017/01/425.jpg"/>dat impetûs a potentiâ concepti directio, pro ratione obliqui­<lb/>tatis minuitur potentiæ momentum; quod integrum manet, &longs;i <lb/>ad angulos rectos vecti occurrat linea motus, quam init po­<lb/>tentia. </s> </p> <p type="main"> <s id="s.003144">Sit vectis primi generis AB habens hypomochlium in C, &longs;i­<lb/>ve &longs;ecundi aut tertij generis DE habens hypomochlium in D. <!-- KEEP S--></s> <lb/> <s id="s.003145">Si potentia con&longs;tituta in B, aut <lb/>E aut F, motum &longs;uum dirigat <lb/><figure id="id.017.01.425.1.jpg" xlink:href="017/01/425/1.jpg"/><lb/>&longs;ecundùm eandem rectam li­<lb/>neam BA aut ED, &longs;ive urgen­<lb/>do vectem versùs C aut D, &longs;ive <lb/>illum inde retrahendo, mani­<lb/>fe&longs;tum e&longs;t, puncto hypomo­<lb/>chlij C aut D manente, pondus <lb/>in A, aut in F, aut in E con&longs;ti­<lb/>tutum nihil pror&longs;us moveri, nam totus potentiæ conatus irri­<lb/>tus e&longs;t, nec vectem movet. </s> <s id="s.003146">Oportet igitur lineam, &longs;ecundùm <lb/>quam dirigitur motus potentiæ, con&longs;tituere cum vecti, longi­<lb/>tudine angulum aut rectum, aut recto minorem, aut majorem. </s> <lb/> <s id="s.003147">Si angulum acutum HBA efficiat, movetur quidem potentia <lb/>& vectis, &longs;ed cùm urgeatur vectis versùs hypomochlium C, <lb/>impeditur potentia, nec movet vectem pro ratione impetûs, <lb/>quem illa concipit. </s> <s id="s.003148">Similiter &longs;i directio motûs potentiæ &longs;it &longs;e­<lb/>cundùm lineam BG, & fiat angulus obtu&longs;us GBA, quamvis <lb/>vectem moveat, minùs tamen illum flectit aut deprimit, quàm <lb/>requirat impetûs concepti inten&longs;io, quia conatur vectem re­<lb/>trahere ab hypomochlio C, & ab illo retinetur. </s> <s id="s.003149">Cùm autem, <lb/>quò acutior aut obtu&longs;ior e&longs;t angulus, eò etiam majus &longs;it impe­<lb/>dimentum, hinc e&longs;t pariter plus laboris à potentiâ impendi. </s> <lb/> <s id="s.003150">Quare, cùm nullum &longs;it huju&longs;modi impedimentum, quando ad <lb/>rectos cum vecte angulos potentiæ motus dirigitur, ut IBA, <lb/>propterea tunc &longs;olùm potentia obtinet omnia momenta, quæ <lb/>concepto impetui re&longs;pondent: nihil enim impetûs deteritur ab <lb/>impedimento, quod vectis inferat, quippe qui nec versùs hy­<lb/>pomochlium urgetur, nec ab illo retrahitur. </s> </p> <p type="main"> <s id="s.003151">Porrò ob&longs;erva longè aliam e&longs;&longs;e lineam motûs potentiæ, à li­<lb/>neâ &longs;ecundùm quam eju&longs;dem potentiæ motus dirigitur; nam <lb/>potentia in B applicata movetur de&longs;cribendo arcum circa C <pb pagenum="410" xlink:href="017/01/426.jpg"/>punctum hypomochlij, &longs;ed pro varià directione modò majo­<lb/>rem, modò minorem arcum de&longs;cribit eodem tempore ex vi <lb/>eju&longs;dem impetûs concepti. </s> <s id="s.003152">Hinc e&longs;t, ni&longs;i potentia &longs;uum mo­<lb/>tum in gyrum dirigat, fieri non po&longs;&longs;e, ut in motu eadem &longs;er­<lb/>vet virium momenta: Nam licèt eandem directionem &longs;ervaret, <lb/>quatenus horizontem re&longs;picit, aut certum aliquod punctum, <lb/>non e&longs;&longs;et tamen eadem directio comparata cum vecte; alium <lb/>quippe atque alium cum vecte angulum con&longs;titueret illa ea­<lb/>dem directionis linea: id quod manife&longs;tò con&longs;tat, cùm vectis <lb/>à potentiâ gravi deprimitur; linea enim directionis in centrum <lb/>gravium directa &longs;emper obliquior incidit in vectem, qui depri­<lb/>mitur. </s> </p> <p type="main"> <s id="s.003153">Voco autem <emph type="italics"/>Directionem motûs<emph.end type="italics"/> lineam illam, quam potentia <lb/>ex vi concepti impetûs &longs;ponte percurreret, ni&longs;i ab illâ deflecte­<lb/>re cogeretur, quia cum vecte connectitur. </s> <s id="s.003154">Sic potentia B li­<lb/>neam BH ex. </s> <s id="s.003155">gr. <!-- REMOVE S-->percurreret, ni&longs;i vectis in C firmati &longs;oliditas <lb/>ob&longs;taret, cogerétque arcum BR de&longs;cribere: idem de cæteris <lb/>lineis dicendum. </s> <s id="s.003156">Hinc &longs;i longitudo BH concipiatur &longs;patium, <lb/>quod à potentiâ libera vi &longs;ui impetûs certo tempore perficere­<lb/>tur, illa utique non recederet à lineâ AB ni&longs;i pro ratione Si­<lb/>nûs Recti angulo HBA convenientis po&longs;ito Radio BH, &longs;cili­<lb/>cet per HO. <!-- KEEP S--></s> <s id="s.003157">Similiter &longs;i directio motûs &longs;it BG angulum obtu­<lb/>&longs;um GBA con&longs;tituens, potentia non recederet ab eâdem lineâ <lb/>AB ni&longs;i pro ratione GK &longs;inus Recti eju&longs;dem anguli GBA ob­<lb/>tu&longs;i po&longs;ito Radio BG, qui ex hypothe&longs;i æqualis e&longs;t Radio BH, <lb/>ponitur enim utrobique æqualis impetus potentiæ. </s> <s id="s.003158">Quare cùm <lb/>idem &longs;it Sinus Rectus anguli acuti, atque obtu&longs;i, quorum &longs;um­<lb/>ma æquatur duobus rectis, eadem pariter momenta virium <lb/>exercet potentia, &longs;ive ad acutum, &longs;ive ad obtu&longs;um cum vecte <lb/>angulum dirigatur. </s> <s id="s.003159">Hoc tamen intercedit di&longs;crimen, quando <lb/>potentia eandem &longs;ervat ad horizontem directionem, quod acu­<lb/>tus angulus procedente motu fit major accedens ad Rectum, <lb/>augetúrquo ejus Sinus; obtu&longs;us verò angulus fit obtu&longs;ior magis <lb/>recedens à Recto, minuitúrque ejus Sinus; ac proinde ibi au­<lb/>getur, hîc minuitur movendi facilitas. </s> </p> <p type="main"> <s id="s.003160">Potentia itaque motum &longs;uum dirigens ad acutum angulum <lb/>per lineam BH, vi &longs;ui impetûs de&longs;cribit circa centrum C ar­<lb/>cum BR; ad angulum rectum per lineam BI de&longs;cribit arcum <pb pagenum="411" xlink:href="017/01/427.jpg"/>BN; ad anguium demum obtu&longs;um per lineam BG de&longs;cribit <lb/>arcum BL, qui e&longs;t æqualis arcui BR, &longs;i angulu; obtu&longs;us GBA <lb/>&longs;it &longs;upplementum ad duo, recto, anguli acuti HBA, major <lb/>autem, aut mino, eodem arcu BR, &longs;i angulus obtu&longs;us &longs;it mi­<lb/>nor aut major codem Supplemento ad duos recto. </s> <s id="s.003161">Hæc tamen <lb/>ita dicta intelligas velim, ut huju&longs;modi arcus toti atque integri <lb/>non motum ip&longs;um exprimant, qui revera fiat, &longs;ed virium Ra­<lb/>tionem pro diversa potentiæ applicatione in minimâ arcû de&longs;­<lb/>cripti particula; neque enim &longs;ingulis temporis momentis æqua­<lb/>lis pars arcus eidem impetui re&longs;pondet, &longs;ingulis nimirum mo­<lb/>mentis mutatur vectis inclinatio, & manet eadem motús di­<lb/>rectio, atque varia e&longs;t potentiæ ad vectem applicatio, ni&longs;i il­<lb/>la impetum concipiat, quo &longs;ua &longs;ponte in gyrum ageretur, <lb/>etiam&longs;i ad motum circularem non determinaretur à vecte. </s> <lb/> <s id="s.003162">Sed quoniam arcus eodem Radio CB à potentiâ de&longs;criptus <lb/>e&longs;t &longs;imilis arcui eodem Radio CA de&longs;cripto à pondere, <lb/>proinde non mutatur Ratio motuum, &longs;ive potentia de&longs;cri­<lb/>bat codem impetu minorem, &longs;ive majorem arcum eju&longs;dem <lb/>circuli: propterea non mutatur quidem momentum poten­<lb/>tiæ cum pondere ab&longs;olutè comparatæ, mutatur tamen &longs;ubinde <lb/>momentum potentiæ, quatenus &longs;ecum ip&longs;a comparatur, faci­<lb/>liú&longs;que movere pondus tunc dicitur, quando eodem conatu <lb/>majorem motum ponderi æquali tempore conciliat; id quod <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 &longs;unt, <lb/>in reliquis pariter duobus generibus locum habent, nec opus <lb/>e&longs;t illa iterum inculcare. </s> <s id="s.003164">Unum in his ob&longs;ervandum vide­<lb/>tur, quando potentia movens e&longs;t à vecte &longs;ejuncta, illúm­<lb/>que trahendo movet certo in loco firmiter con&longs;tituta, vectem <lb/>in motu propiùs accedere ad potentiam trahentem, ac proin­<lb/>de diligenter attendendam e&longs;&longs;e ip&longs;ius potentiæ po&longs;itionem, ut <lb/>innote&longs;cat, utrùm angulus, quem &longs;ubinde cum vecte funicu­<lb/>lus efficit, accedat magis ad rectum, an verò recedat à recto, <lb/>quia in motu acutior aut obtu&longs;ior evadat. </s> <s id="s.003165">Id quod &longs;atis fuerit <lb/>&longs;ubindicá&longs;&longs;e; præ&longs;tat &longs;iquidem laborem in motu minui, quàm <lb/>augeri. </s> </p> <p type="main"> <s id="s.003166">Sic dato vecte CB &longs;ecundi generis habente hypomo­<lb/>chlium in C, &longs;tatue quantum moveri debeat, ex. </s> <s id="s.003167">gr. <!-- REMOVE S-->per <pb pagenum="412" xlink:href="017/01/428.jpg"/>arcum BD. <!-- KEEP S--></s> <s id="s.003168">Erit igitur vectis po&longs;itio CD. <!-- KEEP S--></s> <s id="s.003169">Excitetur ex D <lb/><figure id="id.017.01.428.1.jpg" xlink:href="017/01/428/1.jpg"/><lb/>perpendicularis DE, & in ali­<lb/>quo rectæ lineæ DE puncto, pu­<lb/>ta in E, &longs;tatuatur potentia, quæ <lb/>funiculo EB trahens vectem ita <lb/>vecti intelligatur applicata, ut <lb/>majorem &longs;ubinde angulum effi­<lb/>ciat, donec ad rectum CDE de­<lb/>veniat. </s> <s id="s.003170">Sic facilior erit motus, & <lb/>labor minuetur. </s> <s id="s.003171">Quod &longs;i poten­<lb/>tia movendo pergeret adhuc trahens vectem, jam augeretur <lb/>labor, quia applicaretur ad angulum obtu&longs;um. </s> <s id="s.003172">Porrò in recta <lb/>DE eligendum e&longs;&longs;e punctum, quoad fieri poterit, proximum <lb/>puncto D, ut funiculus EB minùs acutum angulum cum vecte <lb/>CB con&longs;tituat, apertius e&longs;t, quàm ut oporteat id pluribus <lb/>hic o&longs;tendere. </s> <s id="s.003173">Eximendus tamen e&longs;t omnis &longs;crupulus, o&longs;ten­<lb/>dendo angulum &longs;emper majorem fieri, quando di&longs;tantia po­<lb/>tentiæ E ab hypomochlio C major e&longs;t longitudine dati <lb/>vectis CB. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003174">Intelligatur de&longs;criptus integer circulus BGIL à vecte CB <lb/>circumducto, & producatur BC in I, atque funiculus EB &longs;e­<lb/><figure id="id.017.01.428.2.jpg" xlink:href="017/01/428/2.jpg"/><lb/>cet peripheriam in G. <!-- KEEP S--></s> <s id="s.003175">Tum vectis <lb/>po&longs;itio fiat CO, & funiculus EO <lb/>&longs;ecet peripheriam in H. <!-- KEEP S--></s> <s id="s.003176">Mani­<lb/>fe&longs;tum ex 33. lib. 6. angulum <lb/>COE majorem e&longs;&longs;e angulo CBE: <lb/>nam, productâ OC in L, angulus <lb/>CBE in&longs;i&longs;tit arcui GI, angulus <lb/>autem COE in&longs;i&longs;tit arcui HL, qui <lb/>major e&longs;t arcu GI: omnes autem <lb/>acuti &longs;unt, quia in&longs;i&longs;tunt periphe­<lb/>riæ minori, quàm &longs;it &longs;emicirculus <lb/>(&longs;unt enim, ex 20.lib.3, &longs;ubdupli <lb/>&longs;uorum angulorum ad centrum <lb/>ii&longs;dem peripheriis in&longs;i&longs;tentium) <lb/>donec funiculus ED &longs;it Tangens circuli, & ex 18.lib.3. angu­<lb/>lum rectum con&longs;tituat in D. <!-- KEEP S--></s> <s id="s.003177">Quòd &longs;i vectis adhuc trahatur à <lb/>potentia E, & veniat in H & in G, con&longs;tat ex 21.lib.1. angu-<pb pagenum="413" xlink:href="017/01/429.jpg"/>lum CDE minorem e&longs;&longs;e angulo CHE, hunc verò minorem <lb/>angulo CGE, atque ita deinceps. </s> </p> <p type="main"> <s id="s.003178">Idem contingit, &longs;i di&longs;tantia potentiæ R ab hypomochlio C <lb/>omnino æquali &longs;it longitudini vectis CB; nimirum trahendo <lb/>vectem ex B in S, angulus RSC major e&longs;t angulo RBC, & &longs;ic <lb/>deinceps trahendo ex S versùs R: quamvis enim &longs;emper &longs;it an­<lb/>gulus acutus, major tamen &longs;ubinde fit & major; quia manente <lb/>eadem di&longs;tantia RC æquali longitudini vectis, tam triangulum <lb/>CBR quàm CSR, & reliqua omnia &longs;unt I&longs;o&longs;celia; quò ergo <lb/>minor fit angulus ad C, eò major fit angulus ad ba&longs;iin in R, <lb/>cui, per 5. lib. 1. æqualis e&longs;t reliquus angulus ad eandem ba­<lb/>&longs;im. </s> <s id="s.003179">At &longs;i di&longs;tantia potentiæ ab hypomochlio minor fuerit lon­<lb/>gitudine vectis, utique locus potentiæ e&longs;t intra circulum à <lb/>vecte circumducto de&longs;criptum. </s> <s id="s.003180">Con&longs;ideranda e&longs;t igitur varia <lb/>funiculi ad vectem inclinatio: pro qua explicanda hæc præ­<lb/>mitto lemmata. </s> </p> <p type="main"> <s id="s.003181">LEMMA I. </s> <s id="s.003182">Si intra circulum a&longs;&longs;umptum fuerit punctum <lb/>E, in quo duæ rectæ lineæ BF & GH æqualiter à cen­<lb/>tro di&longs;tantes, ideóque ex 14.lib.3.æquales, &longs;e invicem &longs;e­<lb/>cent; & à centro C ducantur Radij CB & CG; anguli <lb/>CBF & CGH &longs;unt æquales. </s> </p> <p type="main"> <s id="s.003183">Producatur BC in M, & <lb/><figure id="id.017.01.429.1.jpg" xlink:href="017/01/429/1.jpg"/><lb/>GC in N, ducantúrque rectæ <lb/>FM & HN. </s> <s id="s.003184">Quia MB & NG <lb/>&longs;unt diametri, anguli in &longs;emicir­<lb/>culo BFM & GHN, ex 31. <lb/>lib.3. &longs;unt recti: igitur quadra­<lb/>ta BF & FM &longs;imul &longs;umpta <lb/>&longs;unt æqualia quadratis GH & <lb/>HN &longs;imul &longs;umptis, cum, ex <lb/>47. lib. 1. æqualia &longs;int qua­<lb/>drato diametri. </s> <s id="s.003185">E&longs;t autem qua­<lb/>dratum BF æquale quadrato <lb/>GH, nam rectæ BF & GH <lb/>ex hypothe&longs;i &longs;unt æquales; <lb/>igitur quadrata FM & HN &longs;unt æqualia, ideoque rectæ <pb pagenum="414" xlink:href="017/01/430.jpg"/>FM & HN &longs;unt æquales; ergo ex 28. lib. 3. &longs;ubtendunt <lb/>æquales peripherias, FHM & HMN; ergo ex 27.lib.3. <lb/>anguli FBM & HGN æqualibus peripheriis in&longs;i&longs;tentes <lb/>æquales &longs;unt. </s> </p> <p type="main"> <s id="s.003186">Invenitur autem recta linea tran&longs;iens per E, quæ æqua­<lb/>lis &longs;it rectæ BF, &longs;i facto centro E, intervallo EF, de&longs;cri­<lb/>batur circulus FRG &longs;ecans datum circulum in G; nam ex <lb/>G per E ducitur recta GH quæ&longs;ita: e&longs;t enim, per 35.lib.3, <lb/>rectangulum GEH æquale rectangulo FEB; &longs;unt autem GE <lb/>& FE æquales Radij eju&longs;dem circuli ex con&longs;tructione; igitur <lb/>per 1. lib. 6, etiam EH & EB &longs;unt æquales; ergo tota GH <lb/>toti FB e&longs;t æqualis. </s> </p> <p type="main"> <s id="s.003187">LEMMA II. <!-- KEEP S--></s> <s id="s.003188">Si in puncto E intra circulum a&longs;&longs;ump­<lb/>to &longs;ecent &longs;e invicem duæ rectæ BF & IO inæquales, <lb/>ac proinde ut colligitur ex 15.lib. 3. inæqualiter à cir­<lb/>culi centro di&longs;tantes, ducantúrque ex centro Radij <lb/>CB, & CI; angulus factus à Radio cum lineâ remo­<lb/>tiore major e&longs;t angulo facto à Radio cum lineâ pro­<lb/>pinquiore. </s> </p> <p type="main"> <s id="s.003189">Perficiantur triangula BFM & IOS rectangula ad F & O <lb/>ex 31.lib.3, quia MB & SI &longs;unt diametri. </s> <s id="s.003190">Quadrata BF & FM <lb/>&longs;imul &longs;umpta, ex 47. lib. 1, &longs;unt æqualia quadratis IO & OS <lb/>&longs;imul &longs;umptis: Quia autem ex hypothe&longs;i recta IO remotior e&longs;t <lb/>à centro quàm BF, e&longs;t etiam minor, ut con&longs;tat ex 15. lib. 3: <lb/>igitur quadratum IO minus e&longs;t quadrato BF, adeóque qua­<lb/>dratum reliquum OS majus e&longs;t reliquo quadrato FM, & <lb/>linea OS major e&longs;t linea FM. </s> <s id="s.003191">Quapropter etiam OS &longs;ub­<lb/>tendit majorem arcum OMS, & FM &longs;ubtendit minorem <lb/>arcum FOM, & angulus SIO factus à Radio cum lineâ re­<lb/>motiore major e&longs;t angulo MBF facto à Radio cum lineâ pro­<lb/>pinquiore. </s> </p> <p type="main"> <s id="s.003192">LEMMA III. <!-- KEEP S--></s> <s id="s.003193">Si in circulo ab extremitate diametri B <lb/>exeat recta linea BC circulum &longs;ecans, in qua a&longs;&longs;umatur <lb/>punctum D eam bifariam æqualiter dividens, & per <pb pagenum="415" xlink:href="017/01/431.jpg"/>punctum D alia recta circulum &longs;ecans ducatur, hæc <lb/>vicinior e&longs;t centro, & major. </s> </p> <p type="main"> <s id="s.003194">Ducatur ex centro S <lb/><figure id="id.017.01.431.1.jpg" xlink:href="017/01/431/1.jpg"/><lb/>recta SD, quæ per 3.lib.3. <lb/>facit angulum SDC rec­<lb/>tum: Tum per D alia <lb/>quædam linea EF tran­<lb/>&longs;eat, quæ utique cum rec­<lb/>ta SD facit angulum <lb/>SDF minorem recto, & <lb/>SDE majorem recto: <lb/>nam &longs;i angulos faceret <lb/>rectos, e&longs;&longs;et SD utrique <lb/>lineæ BC, & EF perpendicularis, &longs;ecaret EF bifariam in D <lb/>per 3.lib. 3. adeóque duæ rectæ BC & EF &longs;e mutuo bifariam <lb/>&longs;ecarent, contra 4. lib. 3. Igitur in rectam EF perpendicu­<lb/>laris ducta ex centro S erit SG cadens ad partes anguli acu­<lb/>ti. </s> <s id="s.003195">Quapropter in triangulo SGD rectangulo ad G ma­<lb/>jor e&longs;t hypothenu&longs;a SD, quàm perpendiculum SG. <!-- KEEP S--></s> <s id="s.003196">Ma­<lb/>gis ergo di&longs;tat linea BC quàm linea EF à centro, ac proinde <lb/>per 15.lib.3. illa e&longs;t minor, hæc major. </s> </p> <p type="main"> <s id="s.003197">LEMMA IV. <!-- KEEP S--></s> <s id="s.003198">Si in eâdem rectâ BC a&longs;&longs;umatur punctum <lb/>I inter extremitatem B & punctum medium D, atque, <lb/>ex centro directâ rectâ SIV, inter V & B alia quæ­<lb/>piam per I tran&longs;eat recta HL circulum &longs;ecans, quæ <lb/>& &longs;ecet perpendicularem SD, ex. </s> <s id="s.003199">gr. <!-- REMOVE S-->in puncto K; <lb/>hæc pariter HL centro propinquior e&longs;t quàm BC, ac <lb/>proinde major. </s> </p> <p type="main"> <s id="s.003200">Angulus KDI e&longs;t rectus, angulus DKI, & qui e&longs;t illi ad <lb/>verticem, SKL e&longs;t acutus, igitur perpendicularis ex centro S <lb/>in rectam HL ducta cadit inter K & L, puta in M. </s> <s id="s.003201">In trian­<lb/>gulo igitur rectangulo SMK major e&longs;t SK quàm SM, ex <lb/>18. lib. 1: ergo multò major e&longs;t SD quàm SM, ac prop­<lb/>terea ex 15. lib. 3. HL vicinior e&longs;t centro, & major <lb/>quàm BC. <!-- KEEP S--></s> </p> <pb pagenum="416" xlink:href="017/01/432.jpg"/> <p type="main"> <s id="s.003202">LEMMA V. <!-- KEEP S--></s> <s id="s.003203">Si in rectà BC ducta ab extremitate dia­<lb/><figure id="id.017.01.432.1.jpg" xlink:href="017/01/432/1.jpg"/><lb/>metri a&longs;&longs;umatur punctum N <lb/>ultra punctum medium D, at­<lb/>que ex S centro ductâ per N <lb/>lineâ rectâ SO, productâque <lb/>perpendiculari SD in P, tran­<lb/>&longs;eat per N alia quæpiam recta <lb/>QR inter P & B circulum &longs;e­<lb/>cans in <expan abbr="q;">que</expan> hæc pariter &longs;ecat in <lb/>T perpendicularem productam, <lb/>& e&longs;t à centro S remotior quàm recta BC atque pro­<lb/>inde minor. </s> </p> <p type="main"> <s id="s.003204">Quia in Triangulo NDT rectangulo ad D, angulus DTN <lb/>e&longs;t acutus, utique in lineâ TS a&longs;&longs;umpto puncto S, ex hoc ca­<lb/>det in lineam QR perpendicularis inter puncta T, & R; quam <lb/>dico majorem e&longs;&longs;e perpendiculari SD. <!-- KEEP S--></s> <s id="s.003205">Nam &longs;i ip&longs;a recta SN <lb/>perpendicularis fuerit ad RQ, e&longs;t triangulum SDN rectan­<lb/>gulum, adeóque hypothenu&longs;a SN major e&longs;t quàm latus SD: <lb/>Sin autem perpendicularis ad RQ cadat in V &longs;ecans rectam <lb/>BC in Z, utique SZ &longs;ubtendens angulum rectum SDZ ma­<lb/>jor e&longs;t quàm SD; e&longs;t autem SV major quàm SZ, ergo & <lb/>multo maior, quàm SD: ergo linea QR remotior e&longs;t quàm <lb/>BC, & minor. </s> </p> <p type="main"> <s id="s.003206">Deinde &longs;i linea per N tran&longs;iens, & circulum &longs;ecans, extre­<lb/>mitatem alteram habeat non inter punctum P terminum per­<lb/>pendicularis SD productæ, atque B terminum rectæ BC; Vel <lb/>dividitur in N bifariam, & linea SN per 3.lib.3. e&longs;t perpendi­<lb/>cularis ad illam, quæ major e&longs;t quàm SD, ut pote oppo&longs;ita an­<lb/>gulo recto SDN: Vel dividitur inæqualiter. </s> <s id="s.003207">Si &longs;egmentum <lb/>majus &longs;it in parte &longs;uperiori, hoc inter N & arcum OP, utique <lb/>perpendicularis ex S centro ducta in illam lineam cadens &longs;eca­<lb/>bit lineam BC inter puncta N & D, ac propterea o&longs;tendetur <lb/>major quàm SD, ut &longs;upra o&longs;ten&longs;um e&longs;t de linea RQ At &longs;i in <lb/>parte &longs;uperiori, hoc e&longs;t inter N & arcum OP &longs;it &longs;egmentam <lb/>minus, perpendicularis ex S in lineam ducta cadet in&longs;ra <pb pagenum="417" xlink:href="017/01/433.jpg"/>punctum N, & à &longs;egmento majore ab&longs;cindet particulam in­<lb/>ter N & punctum perpendiculi interceptam. </s> <s id="s.003208">Hæc particula &longs;i <lb/>fuerit æqualis particulæ ND, linea BC & linea ducta &longs;unt <lb/>æqualiter à centro remotæ; &longs;in illa particula minor fuerit quàm <lb/>ND, linea ducta remotior erit quàm BC; &longs;i demùm major fue­<lb/>rit quàm ND, linea ducta propinquior centro erit quam BC. <!-- KEEP S--></s> <lb/> <s id="s.003209">Finge &longs;cilicet ductam e&longs;&longs;e rectam PNX, & &longs;egmentum majus <lb/>e&longs;&longs;e NX; utique perpendicularis ex S bifariam &longs;ecans totam <lb/>PX cadit inter N & X, puta in Y. </s> <s id="s.003210">E&longs;t igitur SYN triangu­<lb/>lum rectangulum in Y, & per 47. lib.1. quadratum SN æqua­<lb/>le e&longs;t quadratis NY & YS; atqui etiam triangulum SDN e&longs;t <lb/>rectangulum ex hypothe&longs;i, eandemque habet hypothenu&longs;am <lb/>SN; igitur quadrata ND & DS æqualia &longs;unt quadratis NY <lb/>& YS. </s> <s id="s.003211">Quare &longs;i particulæ NY & ND æquales &longs;unt, æqualia <lb/>&longs;unt & earum quadrata, ac idcircò etiam æqualia &longs;unt quadra­<lb/>ta YS & DS, atque eorum latera æqualia &longs;unt, & lineæ BC <lb/>atque PX &longs;unt æqualiter remotæ. </s> <s id="s.003212">Quod &longs;i particula NY mi­<lb/>nor e&longs;t quàm ND, etiam illius quadratum minus e&longs;t quadrato <lb/>hujus; ergo reliquum quadratum YS majus e&longs;t reliquo qua­<lb/>drato DS, atque adeò linea SY major e&longs;t quàm linea SD, & <lb/>linea ducta PX remotior e&longs;t atque minor quàm BC. <!-- KEEP S--></s> <s id="s.003213">Si demum <lb/>NY major e&longs;t quàm ND, etiam illius quadratum majus e&longs;t hu­<lb/>jus quadrato, & reliquum quadratum YS minus e&longs;t reliquo <lb/>quadrato DS: igitur linea YS minor e&longs;t quàm linea DS, ac <lb/>propterea linea ducta PX propinquior e&longs;t centro, & major <lb/>quàm BC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003214">His præmi&longs;&longs;is facilis e&longs;t &longs;olutio propo&longs;itæ difficultatis, ut in­<lb/>note&longs;cat, utrùm in tractione minuatur labor, an augeatur, <lb/>quando potentiæ trahentis di&longs;tantia ab hypomochlio e&longs;t minor <lb/>longitudine vectis. </s> <s id="s.003215">Dato &longs;i quidem loco potentiæ datur eju&longs;­<lb/>dem di&longs;tantia tùm ab hypomochlio, tum ab extremitate vectis, <lb/>cum qua funiculus connectitur; &longs;ed & datur ip&longs;ius vectis longi­<lb/>tudo: quare per Trigonometriam innote&longs;cit quantitas anguli, <lb/>cui opponitur vectis. </s> <s id="s.003216">Nam &longs;i ille rectus e&longs;t, ut SDB, per lem­<lb/>ma 3. in tractione funiculus fit pars lineæ centro propinquio­<lb/>ris, quàm primò a&longs;&longs;umpta DB: igitur in tractione angulus fu­<lb/>niculi cum vecte fit &longs;en&longs;im acutior ex lemm. </s> <s id="s.003217">2. augeturque <lb/>difficultas trahendi. </s> <s id="s.003218">Si angulus vecti oppo&longs;itus &longs;it obtu&longs;us, ut <pb pagenum="418" xlink:href="017/01/434.jpg"/>SIB, in tractione funiculus evadit pars lineæ propinquioris <lb/>centro, quam prima IB ex lemm. </s> <s id="s.003219">4. & &longs;imiliter ex lemm. </s> <s id="s.003220">2. <lb/>fit angulus magis acutus, atque trahentis labor augetur. </s> <s id="s.003221">Si de­<lb/>mum angulus vecti oppo&longs;itus &longs;it acutus, ut SNB, ex lemm.3. <lb/>minuitur labor trahentis u&longs;que ad certum terminum, quandiu <lb/>&longs;cilicet vectis non &longs;ecat perpendiculariter primam funiculi po­<lb/>&longs;itionem NB, hoc e&longs;t vectis circumductus nondum e&longs;t SP; <lb/>tandiu enim funiculus e&longs;t pars lineæ à centro remotioris, & fa­<lb/>cit per lemm. </s> <s id="s.003222">2. cum vecte angulum majorem. </s> <s id="s.003223">Ubi autem <lb/>vectis fuerit SP, tunc ob&longs;ervandum e&longs;t, utrùm angulus SNP <lb/>rectus &longs;it, an obtu&longs;us, an acutus; & eadem methodo proce­<lb/>dendum e&longs;t, qua&longs;i prima funiculi po&longs;itio e&longs;&longs;et NP, ut innote&longs;­<lb/>cat, utrùm funiculus in ulteriori tractione fiat pars lineæ re­<lb/>motioris, an verò propinquioris, ac proinde fiat angulus &longs;ub­<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/>râ potentiam habente hactenus exempli gratia explicata &longs;unt, <lb/>facilè referuntur ad vectem, quando hypomochlium, aut po­<lb/>tentia inter extremitates collocantur; &longs;emper enim attendenda <lb/>e&longs;t hypomochlij di&longs;tantia à potentia trahente, ut potentiæ obli­<lb/>què trahentis momenta innote&longs;cant; angulus &longs;cilicet funiculi <lb/>cum vecte pendet ab hypomochlij puncto, circa quod fit vectis <lb/>conver&longs;io. </s> </p> <p type="main"> <s id="s.003225">Quoniam autem hujus capitis initio momentorum Rationem <lb/>juxta diver&longs;am potentiæ applicationem ex arcubus vi eju&longs;dem <lb/>impetùs de&longs;criptis æ&longs;timandam e&longs;&longs;e dictum e&longs;t, & quis forta&longs;&longs;e <lb/>&longs;u&longs;picetur arduum e&longs;&longs;e huju&longs;modi arcus inter &longs;e comparare; <lb/>animadvertat ex Tabulis Trigonometricis eju&longs;dem arcûs Si­<lb/>num & Tangentem ii&longs;dem planè numeris definiri, quando ar­<lb/>cus valde exiguus e&longs;t. </s> <s id="s.003226">Quapropter cum quilibet arcus minor <lb/>&longs;it &longs;uâ Tangente, & major Sinu, arcuum minorum Rationem <lb/>citra ullum erroris periculum explicare po&longs;&longs;umus per eorum <lb/>Sinus. </s> <s id="s.003227">Cum verò hîc, ubi de potentiæ ad vectem &longs;ecundùm <lb/>diver&longs;os angulos applicatæ momentis &longs;ermo e&longs;t, non ni&longs;i mini­<lb/>mi arcus a&longs;&longs;umendi &longs;int, eorum Ratio eadem a&longs;&longs;umitur, quæ <lb/>Sinuum. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003228">Quare &longs;i vectis &longs;it AB, hypomochlium C, vis potentiæ & <lb/>directio motûs potentiæ BH: loco arcûs BK, qui in motu vi <pb pagenum="419" xlink:href="017/01/435.jpg"/>talis impetús cum hac directione de&longs;cribitur; a&longs;&longs;umi pote&longs;t an­<lb/>guli HBO, Radio BH, Sinus HO, <lb/><figure id="id.017.01.435.1.jpg" xlink:href="017/01/435/1.jpg"/><lb/>qui e&longs;t æqualis Sinui arcus BK Ra­<lb/>dio CB. <!-- KEEP S--></s> <s id="s.003229">E&longs;t autem minimus ar­<lb/>cus longe minor quàm arcus BK, <lb/>&longs;ed claritatis gratia arcum notabi­<lb/>lem & con&longs;picuum a&longs;&longs;umere opor­<lb/>tuit. </s> <s id="s.003230">E&longs;t igitur potentiæ ad an­<lb/>gulum rectum in B applicatæ mo­<lb/>mentum, ad eju&longs;dem potentiæ ad <lb/>angulum HBO acutum applicatæ momentum, ut Radius BH <lb/>ad acuti anguli Sinum HO. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003231">Jam intellige vectem AB converti, & lineam BH produci, <lb/>donec in D ad angulos rectos occurrat vecti habenti po&longs;itio­<lb/>nem EF. <!-- KEEP S--></s> <s id="s.003232">Dico potentiæ ad perpendiculum & obliquè appli­<lb/>catæ momenta invicem comparata ita e&longs;&longs;e, ac &longs;i eadem poten­<lb/>tia tam in F quàm in D ad angulum rectum applicaretur, quia <lb/>ut BH ad HO, ita e&longs;t FC ad CD. <!-- KEEP S--></s> <s id="s.003233">Ducatur enim ex D ad CB <lb/>perpendicularis DG, quæ e&longs;t parallela ip&longs;i HO. quare per 4. <lb/>lib. 6. ut BH ad HO, ita BD ad DG, & per 8. lib. 6. ut BD <lb/>ad DG, ita BC, hoc e&longs;t FC, ad CD: igitur per 11. lib. 5. ut <lb/>BH ad HO, ita FC ad CD; hoc e&longs;t ut Radius ad Sinum angu­<lb/>li, &longs;ecundùm quem potentia dirigitur, ita momentum poten­<lb/>tiæ perpendiculariter applicatæ ad momentum eju&longs;dem obli­<lb/>què ad angulum acutum, vel obtu&longs;um applicatæ. </s> <s id="s.003234">Nam &longs;i po­<lb/>tentia in B dirigat &longs;uum motum &longs;ecundum lineam BI, utique <lb/>po&longs;ito Radio BI, Sinus anguli ABI obtu&longs;i e&longs;t IL, & Ratio mo­<lb/>menti potentiæ in B applicatæ &longs;ecundùm angulum rectum, ad <lb/>momentum eju&longs;dem potentiæ in B applicatæ &longs;ecundùm angu­<lb/>lum obtu&longs;um ABI, e&longs;t ut BI ad IL. <!-- KEEP S--></s> <s id="s.003235">Producatur IB, donec in <lb/>D perpendicularis cadat &longs;upra CF rectam æqualem ip&longs;i CB. <!-- KEEP S--></s> <lb/> <s id="s.003236">Quia <expan abbr="triāgula">triangula</expan> BIL & BCD rectangula ad L & D, & æquales <lb/>angulos ad verticem B habentia, &longs;imilia &longs;unt, e&longs;t ut BI ad IL, <lb/>ita BC ad CD per 4. lib. 6. Perinde igitur in extremitate B ad <lb/>angulum obtu&longs;um ABI applicata potentia operatur, atque &longs;i ad <lb/>angulum rectum applicaretur in D puncto vectis EF, qui idem <lb/>ponitur e&longs;&longs;e ac vectis AB: & momenta potentiæ &longs;unt ut FC <lb/>ad CD. <!-- KEEP S--></s> </p> <pb pagenum="420" xlink:href="017/01/436.jpg"/> <p type="main"> <s id="s.003237">Dato itaque angulo, &longs;ecundum quem potentia applicatur <lb/>ad vectem, &longs;i angulus &longs;it Rectus, momentum e&longs;t ut Radius; &longs;in <lb/>autem angulus acutus &longs;it vel obtu&longs;us, momentum e&longs;t ut Sinus <lb/>eju&longs;dem anguli; atque adeò comparando inter &longs;e huju&longs;modi <lb/>angulos, Ratio illorum erit eadem, quæ e&longs;t Sinuum. <!-- KEEP S--></s> <s id="s.003238">Hinc <lb/><figure id="id.017.01.436.1.jpg" xlink:href="017/01/436/1.jpg"/><lb/>datus vectis AB hypomochlium ha­<lb/>bens in C, &longs;i fuerit ita inclinatus, ut <lb/>po&longs;itionem habeat EF, potentia in E <lb/>deor&longs;um premens per rectam EG per­<lb/>pendicularem ad CB, momentum ha­<lb/>bet ut CG; & &longs;i <expan abbr="po&longs;ition&etilde;">po&longs;itionem</expan> habeat IH, <lb/>potentia in I deor&longs;um premens aut tra­<lb/>hens juxta rectam IK, quæ producta <lb/>incidat perpendicularis ad rectam AB <lb/>in L, momentum habet ut CL. </s> <s id="s.003239">Quare <lb/>ex E in B augentur prementis aut trahentis momenta, quæ ex <lb/>B in I minuuntur. </s> </p> <p type="main"> <s id="s.003240">Id quod iis etiam, qui campanas pul&longs;ant, manife&longs;tum e&longs;t: &longs;i <lb/>enim intelligatur vecti CB adhærere campanam, cujus cen­<lb/>trum gravitatis &longs;it O, utique dum B deprimitur, O elevatur, <lb/>&longs;ed elevandi difficultas cre&longs;cit, tum quia centrum gravitatis O <lb/>arcum de&longs;cribens circa punctum C, æqualibus temporibus in­ <lb/>quales, atque &longs;emper majores habet a&longs;cen&longs;us juxta incremen­<lb/>ta Sinuum Ver&longs;orum, tum quia ex depre&longs;&longs;ione vectis ex B in I <lb/>facto angulo funis & vectis &longs;emper obtu&longs;iore, momenta poten­<lb/>tiæ minuuntur: & licet in reditu ex I in B cre&longs;cerent, &longs;i quis <lb/>vectem &longs;ur&longs;um traheret, hoc nihil juvat potentiam deor&longs;um <lb/>trahentem ad elevandam campanam, quæ &longs;ponte &longs;ua de&longs;cen­<lb/>dens elevat vectis caput, cui funis adnectitur. </s> <s id="s.003241">Propterea majo­<lb/>ribus gravioribú&longs;que campanis non &longs;implicem vectem CB &longs;ed <lb/>rotam, aut rotæ &longs;egmentum adjungunt, cujus excavatæ peri­<lb/>metro funis in&longs;eritur; qui dum trahitur, &longs;emper e&longs;t Tangens <lb/>circuli; atque ideo ad Radium circuli, qua&longs;i e&longs;&longs;et novus atque <lb/>novus vectis, applicatur potentia trahens ad angulum rectum. <pb pagenum="421" xlink:href="017/01/437.jpg"/></s> </p> <p type="main"> <s id="s.003242"><emph type="center"/>CAPUT VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003243"><emph type="center"/><emph type="italics"/>Oneris ex Vecte pendentis momentum inquiritur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003244">COntingit aliquando pondus vecte elevandum fune con­<lb/>necti, & pendulum ex vecte &longs;u&longs;pendi. </s> <s id="s.003245">Nemo dubitat, an <lb/>gravitas ponderis ibi &longs;ua exerceat momenta, ubi cum vecte <lb/>connectitur; funis &longs;i quidem intentus congruit lineæ directio­<lb/>nis, qua pondus ip&longs;um nititur in centrum gravium: verùm non <lb/>eandem percipi in elevando difficultatem experientia te&longs;tatur <lb/>pro varia vectis inclinatione. </s> <s id="s.003246">Si enim ex vecte AB horizontali <lb/>hypomochlium in extremitate B ha­<lb/><figure id="id.017.01.437.1.jpg" xlink:href="017/01/437/1.jpg"/><lb/>bente, pondus D &longs;u&longs;pen&longs;um ex C <lb/>pendeat ad angulos rectos, omnia &longs;ua <lb/>momenta exercet pro ratione di&longs;tan­<lb/>tiæ CB ab hypomochlio. </s> <s id="s.003247">At &longs;i ele­<lb/>vatus vectis po&longs;itionem habeat EB, <lb/>& C venerit in F, pondus verò pen­<lb/>dulum D venerit in G, ita ut linea <lb/>Directionis in centrum gravium congruat funiculo &longs;u&longs;penden­<lb/>ti FG; etiam&longs;i FB æqualis &longs;it ip&longs;i CB, non eadem tamen mo­<lb/>menta habet pondus adversùs eandem <expan abbr="pot&etilde;tiam">potentiam</expan> ex A <expan abbr="tran&longs;latã">tran&longs;latam</expan> <lb/>in E; quia &longs;cilicet angulus GFB e&longs;t acutus, DCB autem rectus. </s> </p> <p type="main"> <s id="s.003248">Id explicare ex iis, quæ &longs;uperiori capite di&longs;putata &longs;unt, non <lb/>erit difficile, &longs;i animadvertamus in vectibus &longs;ecundi & tertij <lb/>generis utrumque genus conjungi: quemadmodum enim po­<lb/>tentia conatur adversùs gravitatem ponderis, ita pondus cona­<lb/>tur adversùs vim potentiæ: & in hoc conatu vici&longs;&longs;im exercent <lb/>munus potentiæ & ponderis. </s> <s id="s.003249">Finge &longs;iquidem duos homines ap­<lb/>plicari vecti AB, alterum quidem in A, alterum verò in C, &longs;ed <lb/>in adver&longs;a conantes; uterque e&longs;t potentia, uterque e&longs;t pondus, <lb/>dum &longs;ibi reluctantur. </s> <s id="s.003250">Anne ita hæc vocabula intra certos fines <lb/>coërceri exi&longs;timas, ut potentiæ nomine illum &longs;olum donandum <lb/>putes, qui reliquum vincit? </s> <s id="s.003251">&longs;ed quid, &longs;i horum hominum co-<pb pagenum="422" xlink:href="017/01/438.jpg"/>natus &longs;int reciprocè ut eorum di&longs;tantiæ ab hypomochlio B, & <lb/>A quidem conetur ut CB, C autem conetur ut AB; utique <lb/>neuter &longs;uperat; nec tamen negari pote&longs;t ideò contingere mo­<lb/>mentorum æqualitatem inter inæquales conatus, quia vecti ap­<lb/>plicantur: &longs;unt igitur &longs;ibi vici&longs;&longs;im potentia & pondus. </s> <s id="s.003252">Si ita­<lb/>que A e&longs;t potentia, & C pondus; vectis e&longs;t &longs;ecundi generis: <lb/>Si verò C e&longs;t potentia, & A pondus, vectis e&longs;t tertij generis. </s> <lb/> <s id="s.003253">Illud igitur quod de hominibus dicitur, de reliquis omnibus <lb/>vim movendi habentibus dictum intelligitur: nihil &longs;i quidem <lb/>intere&longs;t, utrùm animata &longs;int, an inanima, quæ vecti applican­<lb/>tur, & in oppo&longs;ita partes conantur. </s> <s id="s.003254">Et quamvis non interce­<lb/>dat inter ip&longs;os conatus momentorum æqualitas, quam con&longs;e­<lb/>quatur quies, &longs;ed efficiatur motus; ita tamen id quod prævalet, <lb/>e&longs;t potentia ad motum efficiendum, ut id quod vincitur, & re­<lb/>&longs;i&longs;tit, &longs;it potentia ad motum retardandum. </s> <s id="s.003255">In omni itaque <lb/>vecte &longs;ive &longs;ecundi, &longs;ive tertij generi &longs;it, utrumque genus ami­<lb/>co, nec &longs;olubili fœdere copulantur. </s> <s id="s.003256">In vecte autem primi ge­<lb/>neris idem genus manet, licèt vici&longs;&longs;im habeant rationem po­<lb/>tentiæ & ponderis ad movendum & retardandum, &longs;imiliter <lb/>enim potentiæ & ponderi, licet inæqualibus intervallis, inter­<lb/>jacet hypomochlium. </s> </p> <p type="main"> <s id="s.003257">Hîc itaque, ubi oneris ex vecte pendentis momentum inqui­<lb/>ritur, con&longs;iderandus e&longs;t vectis tertij generis, in quo gravitas in <lb/>C, aut in F po&longs;ita exercet munus potentiæ conantis deprimere <lb/>vim &longs;ur&longs;um connitentem in A, aut in E. <!-- KEEP S--></s> <s id="s.003258">Quare in po&longs;itione vectis <lb/>horizontali, cum &longs;it angulus rectus DCB, neque gravitas illa <lb/>vectem versùs hypomochlium B urgeat, aut cum ab illo re­<lb/>trahat, omnia &longs;ua momenta obtinet, quæ in hac à fulcro di&longs;tantiâ <lb/>gravitati huic convenire po&longs;&longs;unt. </s> <s id="s.003259">At elevato vecte ita, ut fiat an­<lb/>gulus acutus GFB, licet eadem maneat gravitas, eadémque ab <lb/>hypomochlio di&longs;tantia, non tamen eadem manent momenta, &longs;ed <lb/>decre&longs;cunt pro ratione Sinûs anguli, ut &longs;uperiori capite dictum <lb/>e&longs;t. </s> <s id="s.003260">Producta igitur intelligatur linea directionis FG u&longs;que ad <lb/>horizontalem in H: po&longs;ito Radio BF, hoc e&longs;t BC, e&longs;t BH Si­<lb/>nus anguli GFB; ac proinde ut BC ad BH, ita momentum <lb/>oneris pendentis ex vecte horizontali, ad momentum eju&longs;dem <lb/>oneris pendentis ex codem vecte inclinato. </s> <s id="s.003261">Hinc e&longs;t, inclinato <lb/>vecte EB, tantumdem conatûs adhibendum e&longs;&longs;e in E ad &longs;u&longs;ti-<pb pagenum="423" xlink:href="017/01/439.jpg"/>nendum onus G, quanto conatu opus e&longs;&longs;et in vecte horizontali <lb/>AB ad &longs;u&longs;tinendum idem onus, &longs;i penderet ex H. <!-- KEEP S--></s> <s id="s.003262">Quoniam igi­<lb/>tur di&longs;tantia BH minor e&longs;t quàm BC, major e&longs;t Ratio AB ad <lb/>BH, quàm eju&longs;dem AB ad BC, ex 8.lib.5. ideóque faciliùs &longs;u&longs;ti­<lb/>netur idem onus vecte inclinato, quàm vecte horizontali. </s> </p> <p type="main"> <s id="s.003263">Quod &longs;i ex G centro gravitatis oneris ductam intelligas ad <lb/>vectem EB rectam perpendicularem GI, habes &longs;imiliter mo<lb/>mentorum differentiam, quæ &longs;cilicet intercedit inter FG, & GI, <lb/>&longs;i FG repræ&longs;entet omnia momenta in vecte horizontali: &longs;unt <lb/>enim triangula FIG & FHB rectangula, communem angulum <lb/>ad F habentia, adeóque &longs;imilia, & ut FB ad BH, ita FG ad GI. </s> <lb/> <s id="s.003264">Cave autem ne putes (ut non pauci hallucinantur) ita ex I ter­<lb/>mino rectæ CI perpendicularis de&longs;umendam e&longs;&longs;e men&longs;uram <lb/>decrementi momentorum, ut perinde &longs;e habeat, qua&longs;i pondus <lb/>e&longs;&longs;et in I: hoc enim à veritate longi&longs;&longs;imè abe&longs;&longs;e deprehendes, &longs;i <lb/>manente eadem vectis inclinatione, & eadem oneris gravitate, <lb/>funiculo longiore onus &longs;u&longs;penderis; quandoquidem ctiam <lb/>punctum I magis accedet ad hypomochlium B, nec tamen adhi­<lb/>bito longiore funiculo adeò minuuntur momenta; alioquin <lb/>tam longo funiculo &longs;u&longs;pendere po&longs;&longs;es onus, ut recta ex one­<lb/>ris centro ducta ad vectem EB perpendicularis caderet in B, <lb/>atque ideo nullum e&longs;&longs;et gravitatis momentum, qua&longs;i onus e&longs;&longs;et <lb/>in B: id autem omnino fal&longs;um e&longs;t. </s> </p> <p type="main"> <s id="s.003265">Quando autem dicitur faciliùs à potentiâ &longs;u&longs;tineri idem onus <lb/>&longs;u&longs;pen&longs;um vecte inclinato, quàm vecte horizontali, ita intelli­<lb/>gendum e&longs;t, ut linea directionis motus potentiæ &longs;u&longs;tinentis <lb/>eundem &longs;emper faciat cum vecte angulum: nam &longs;i hæc linea <lb/>alium atque alium efficiat angulum, etiam potentiæ momenta <lb/>variantur, quæ cum oneris momentis comparanda &longs;unt. </s> <s id="s.003266">Hinc <lb/><figure id="id.017.01.439.1.jpg" xlink:href="017/01/439/1.jpg"/><lb/>e&longs;t in vecte primi generis CD, <lb/>cujus hypomochlium O, &longs;i po­<lb/>tentia & pondus &longs;int gravia M <lb/>& N, licèt inclinato vecte, ut <lb/>habeat po&longs;itionem RS, receden­<lb/>tibus angulis à rectitudine, &longs;in­<lb/>gulorum momenta minora fiant, <lb/>non tamen mutari momentorum <lb/>potentiæ & ponderis invicem <pb pagenum="424" xlink:href="017/01/440.jpg"/>comparatorum Rationem; quia &longs;cilicet &longs;ingulorum momenta <lb/>proportionaliter minuuntur. </s> <s id="s.003267">Cum enim gravia &longs;emper nitan­<lb/>tur juxta &longs;uas lineas directionis in centrum gravium, huju&longs;mo­<lb/>di lineæ parallelæ cen&longs;entur, & cum vecte duos angulos effi­<lb/>ciunt duobus rectis æquales, ac proinde &longs;i alter acutus fuerit, <lb/>alter e&longs;t obtu&longs;us &longs;upplementum acuti ad duos rectos. </s> <s id="s.003268">Sicut au­<lb/>tem in eodem circulo idem e&longs;t Sinus anguli acuti, atque obtu&longs;i, <lb/>qui compleat duos rectos; ita in diver&longs;is circulis huju&longs;modi an­<lb/>gulorum Sinus proportionales &longs;unt &longs;uis Radiis. </s> <s id="s.003269">Quapropter in­<lb/>clinato vecte, ut &longs;it RS, gravia nituntur deor&longs;um juxta lineas <lb/>directionis ST & RV parallelas, quæ occurrunt perpendicu­<lb/>lares horizontali in Z & V. <!-- KEEP S--></s> <s id="s.003270">Momentum igitur gravis T ad <lb/>momentum æqualis, &longs;eu eju&longs;dem gravis N e&longs;t ut OZ ad OD, <lb/>& momentum gravis V ad momentum æqualis, &longs;eu eju&longs;dem <lb/>gravis M e&longs;t ut OV ad OC. <!-- KEEP S--></s> <s id="s.003271">Quare in vecte RS inclinato <lb/>momenta gravium pendentium &longs;unt ut OZ ad OV. <!-- KEEP S--></s> <s id="s.003272">Quia ve­<lb/>rò triangula RVO, SZO rectangula, & angulos ad verticem <lb/>O æquales habentia, &longs;unt &longs;imilia, per 4. lib. 6. ut OS ad OR, <lb/>hoc e&longs;t ut OD ad OC, ita OZ ad OV. <!-- KEEP S--></s> <s id="s.003273">Manet itaque ea­<lb/>dem momentorum Ratio invicem comparatorum, &longs;ive integra <lb/>in vecte horizontali, &longs;ive diminuta in vecte inclinato &longs;int &longs;ingu­<lb/>lorum momenta. </s> </p> <p type="main"> <s id="s.003274">At in vecte &longs;ecundi aut tertij generis, &longs;i potentia non fuerit <lb/>vivens, fieri non pote&longs;t ut eadem &longs;ervetur momentorum Ratio <lb/>inter potentiam & pondus, ni&longs;i fortè in eodem medio ho­<lb/>rum alterutrum grave e&longs;&longs;et, alterum leve; ut &longs;i vectis AK <lb/><figure id="id.017.01.440.1.jpg" xlink:href="017/01/440/1.jpg"/><lb/>intra aquam con&longs;titutus adnexum <lb/>haberet in K inflatum ut rem V, <lb/>in L verò pendulus e&longs;&longs;et lapis: <lb/>tunc enim, &longs;i uter a&longs;cendens trahat <lb/>vectem in B, elevabit lapidem <lb/>pendulum, ut &longs;it angulus ICA <lb/>acutus, & angulus ABF obtu­<lb/>&longs;us; qui cum æqualis &longs;it alterno <lb/>BCI (&longs;unt enim FBE & CI paral­<lb/>lelæ, quia utraque ad horizontem <lb/>perpendicularis e&longs;t) &longs;imilem habet Sinum Sinui acuti ICA &longs;e­<lb/>cundum Rationem Radiorum BA & CA, hoc e&longs;t KA & LA; <pb pagenum="425" xlink:href="017/01/441.jpg"/>atque ut BA ad CA, ita e&longs;t EA ad IA; &longs;imilia quippe &longs;unt <lb/>triangula ABE & ACI. </s> <s id="s.003275">Cæterùm &longs;i rotulæ X in&longs;i&longs;tens funi­<lb/>culus jungeret vectis extremitatem K & pondus aliquod ad­<lb/>nexum in fungens munere potentiæ elevantis; hoc de&longs;cen­<lb/>dens ex S in M elevaret vectem in B, & lapidem, qui penderet <lb/>ex C: &longs;ed angulus ABX e&longs;&longs;et multò obtu&longs;ior quàm Supple­<lb/>mentum acuti ICA ad duos rectos, ac proinde Sinus anguli <lb/>ABX e&longs;&longs;et multo minor quàm EA Sinus anguli obtu&longs;i ABF: <lb/>igitur multo minor e&longs;&longs;et Ratio momenti potentiæ applicatæ in <lb/>B ad momentum ponderis in C, quàm &longs;it Ratio BA ad CA, <lb/>hoc e&longs;t KA ad LA. <!-- KEEP S--></s> <s id="s.003276">Sola igitur potentia vivens pote&longs;t ita &longs;ui <lb/>motûs directionem inflectere, ut eundem faciat cum vecte an­<lb/>gulum, ideóque elevans vectem acquirat majorem &longs;u&longs;tinendi <lb/>facilitatem. </s> </p> <p type="main"> <s id="s.003277">His con&longs;equens e&longs;t, quantò altiùs &longs;upra horizontem eleva­<lb/>tur vectis cum pondere pendulo, tantò validiùs a pondere pre­<lb/>mi aut urgeri hypomochlium A. <!-- KEEP S--></s> <s id="s.003278">Nam quemadmodum in vecte <lb/>horizontali AK pondus in L &longs;u&longs;pen&longs;um magis premit hypomo­<lb/>chlium A vicinum quàm potentiam K remotam ex hypothe&longs;i, <lb/>ita elevato vecte multo magis premitur hypomochlium, quia <lb/>quodammodo propiùs illi admovetur pondus in I quàm in L, <lb/>&longs;uáque innatá gravitate in vecte elevato conatur versù hypo­<lb/>mochlium qua&longs;i &longs;ecedens à potentia; ut nihil dicam de vecte <lb/>ip&longs;o, cujus gravitas, maximam partem, innititur fulcro. <lb/></s> </p> <p type="main"> <s id="s.003279"><emph type="center"/>CAPUT IX.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003280"><emph type="center"/><emph type="italics"/>An duo pondus ge&longs;tantes æqualiter premantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003281">HActenus di&longs;putatis proximè affinis e&longs;t præ&longs;ens quæ&longs;tio, qua <lb/>inquirimus, utrùm æqualis &longs;it labor duorum in codem pon­<lb/>dere ge&longs;tando con&longs;entientium. </s> <s id="s.003282">Et quidem &longs;i movendum &longs;it <lb/>pondus atque trahendum, cur duo &longs;imul faciliùs illud mo­<lb/>veant, quàm &longs;inguli, omnes intelligunt; quia plus impetûs à <lb/>duobus producitur, quàm à &longs;ingulis; & quem impetum mo-<pb pagenum="426" xlink:href="017/01/442.jpg"/>vendo oneri parem &longs;inguli multo conatu producerent, &longs;ingu­<lb/>lis in parte impetûs efficiendâ minùs conantibus, totus produ­<lb/>citur, totique oneri imprimitur. </s> <s id="s.003283">At in pondere &longs;u&longs;tentando, <lb/>cujus gravitas in partes non dividitur, quomodo hæc &longs;ingulis <lb/>levior accidat, &longs;i plures in &longs;u&longs;tentando con&longs;pirent, quàm &longs;i &longs;in­<lb/>gulis imponeretur, tunc maximè cùm nullum impetum &longs;ur&longs;um <lb/>gravitatis conatui adver&longs;antem producant, non ita explicatu <lb/>facile exi&longs;timant aliqui. </s> <s id="s.003284">Verum ex rationibus vectis &longs;atis ma­<lb/>nife&longs;ta &longs;olutio eruitur. </s> <s id="s.003285">Claritatis autem gratiâ, ob&longs;ervandum <lb/>e&longs;t, an onus palangâ (ut cum bajuli dolium ex funibus &longs;u&longs;pen­<lb/>&longs;um transferunt) an verò &longs;ubjectis humeris &longs;u&longs;tineatur. </s> </p> <p type="main"> <s id="s.003286">Et primo &longs;it palanga AB, cujus extremitates à ge&longs;tatoribus <lb/>&longs;u&longs;tineantur; &longs;it autem onus in C. <!-- KEEP S--></s> <s id="s.003287">Duplex effectus hîc con&longs;i­<lb/><figure id="id.017.01.442.1.jpg" xlink:href="017/01/442/1.jpg"/><lb/>derandus e&longs;t, videlicet oneris &longs;u&longs;ten­<lb/>tatio, & ge&longs;tatorum pre&longs;&longs;io; Si pri­<lb/>mum re&longs;picias, ge&longs;tatores A & B ratio­<lb/>nem habent potentiæ efficientis &longs;u­<lb/>&longs;tentationem, atque impedientis motum oneris &longs;uá gravitate <lb/>deor&longs;um conantis: Si &longs;ecundum, idem onus C munus potentiæ <lb/>pre&longs;&longs;ionem efficientis exercet, dum &longs;ecum palangam deor&longs;um <lb/>trahens, oppo&longs;itos ge&longs;tatorum humeros comprimit, aut &longs;i ma­<lb/>nibus palanga ge&longs;tatur contentos contracto&longs;que brachiorum <lb/>mu&longs;culos, quantum pote&longs;t, di&longs;trahit, atque relaxat. </s> <s id="s.003288">Sunt <lb/>enim duo conatus, ge&longs;tatorum &longs;cilicet & oneris, motum in op­<lb/>po&longs;itas partes efficere valentes, ni&longs;i &longs;ibi mutuo impedimento <lb/>e&longs;&longs;ent: hinc &longs;i ge&longs;tatores conari ce&longs;&longs;ent, onus de&longs;cendit; Si ex <lb/>improvi&longs;o abruptis funibus onus à palangâ &longs;ejungatur, ge&longs;tato­<lb/>res palangam &longs;ur&longs;um attollunt, &longs;ive æqualiter, &longs;ive inæqualiter, <lb/>pro ut æquales aut inæquales &longs;unt eorum conatus. </s> <s id="s.003289">Quare æ&longs;ti­<lb/>manda res e&longs;t ex motu, quem &longs;inguli conantes efficerent tum <lb/>in &longs;e, tum in oppo&longs;ito conante, ni&longs;i prohiberentur momento­<lb/>rum æqualitate. </s> <s id="s.003290">Sic potentia in A &longs;uo conatu elevaret pondus <lb/>in C po&longs;itum, & circa centrum B arcum de&longs;criberent; &longs;imili­<lb/>ter potentia in B &longs;uo conatu elevaret pondus idem in C po&longs;i­<lb/>tum, & circa centrum A &longs;uos motus perficerent. </s> <s id="s.003291">Quod ita­<lb/>que ad &longs;u&longs;tentationem &longs;pectat, ge&longs;tatores A & B vici&longs;&longs;im ha­<lb/>bent rationem potentiæ & fulcri; nam &longs;i A e&longs;t potentia, ful­<lb/>crum e&longs;t B; atque vici&longs;&longs;im &longs;i B &longs;it potentia, fulcrum e&longs;t A; & <pb pagenum="427" xlink:href="017/01/443.jpg"/>e&longs;t duplex vectis &longs;ecundi generis, &longs;cilicet AB & BA. Q: od <lb/>verò ad pre&longs;&longs;ionem attinet, in qua onus C e&longs;t potentia premens, <lb/>ge&longs;tatores vici&longs;&longs;im habent rationem fulcri atque ponderi pre&longs;­<lb/>&longs;i; & e&longs;t duplex vectis tertij generis, quo utitur unica poten­<lb/>tia, &longs;icut in duplici vecte &longs;ecundi generis unicum e&longs;t pondus, <lb/>quod &longs;u&longs;tinent duæ potentiæ. </s> </p> <p type="main"> <s id="s.003292">In vecte igitur &longs;ecundi generis po&longs;ito fulcro B, momentum <lb/>potentiæ A &longs;ur&longs;um nitentis, ad momentum ponderis C deor­<lb/>&longs;um conantis, e&longs;t ut AB ad CB; ac propterea potentia &longs;u&longs;tinere <lb/>valens &longs;ine vecte pondus C, ad potentiam vecte AB &longs;u&longs;tinentem <lb/>idem pondus C, e&longs;t ut AB ad CB; quanto igitur CB minor e&longs;t <lb/>quàm AB, tanto minor potentia requiritur in A, quàm require­<lb/>retur in C, &longs;i in C pondus &longs;ine vecte &longs;u&longs;tineretur. </s> <s id="s.003293">Idem quod <lb/>de potentiâ A, po&longs;ito fulcro B, dictum e&longs;t, dic vici&longs;&longs;im de poten­<lb/>tiâ B; po&longs;ito fulcro A; Requiritur enim in B potentia ut CA, ad <lb/>potentiam, quæ e&longs;&longs;et ut BA, &longs;i &longs;ine vecte pondus in C &longs;u&longs;tinere­<lb/>tur. </s> <s id="s.003294">Hinc e&longs;t vires &longs;u&longs;tinendi requiri reciprocè tantas, quanta <lb/>e&longs;t Ratio di&longs;tantiarum à pondere ip&longs;orum &longs;u&longs;tinentium: vires &longs;i <lb/>quidem in A requiruntur ut CB, & vires in B ut CA. <!-- KEEP S--></s> <s id="s.003295">Si itaque <lb/>æquali intervallo pondus medium di&longs;tet à ge&longs;tatoribus, æqua­<lb/>liter eos conari oportet, ut illud &longs;u&longs;tineant in C: at &longs;i inæquali­<lb/>ter ab iis remotum &longs;it, ut in D, requiruntur in A vires tam eò ma­<lb/>jores quàm in B, quantò major e&longs;t di&longs;tantia DB quàm DA. <!-- KEEP S--></s> <lb/> <s id="s.003296">Quapropter datá virium inæqualitate; &longs;tatim innote&longs;cet, in quo <lb/>palangæ puncto adnectendum &longs;it onus; &longs;i nimirum palangæ lon­<lb/>gitudo dividatur &longs;ecundùm Rationem virium, & ge&longs;tatores re­<lb/>ciprocè collocentur. </s> <s id="s.003297">Sint enim ex. </s> <s id="s.003298">gr. <!-- REMOVE S-->duo, quorum alter vires <lb/>habeat ut 3, alter ut 2: concipe totam longitudinem AB in <lb/>quinque partes di&longs;tinctam, & hinc accipe duas AD, hinc verò <lb/>tres BD: locus ponderi debitus e&longs;t punctum D, in quod cadit <lb/>divi&longs;io in duas partes juxta datam Rationem: locus debilioris <lb/>ge&longs;tatoris e&longs;t in palangæ extremitate B, ad quam &longs;pectat major <lb/>di&longs;tantia ab onere in C po&longs;ito. </s> <s id="s.003299">Similiter virium inæqualitatem <lb/>deprehendes, &longs;i pondere in medio puncto C po&longs;ito, alter &longs;e præ­<lb/>gravari &longs;entiat: palanga enim ita promota, ut pondere in D <lb/>con&longs;tituto neuter &longs;e ultra vires prægravatum experiatur, indi­<lb/>cabit vires ge&longs;tatoris A e&longs;&longs;e ut DB, ad vires ge&longs;tatoris B, quæ <lb/>&longs;unt ut DA. <!-- KEEP S--></s> </p> <pb pagenum="428" xlink:href="017/01/444.jpg"/> <p type="main"> <s id="s.003300">At &longs;i vectem tertij generis, quatenus ge&longs;tatorum pre&longs;&longs;io ab <lb/>onere efficitur, con&longs;ideremus; po&longs;ito fulcro in A, potentia in C <lb/>aut in D exi&longs;tens non premit ge&longs;tatorem B perinde, atque &longs;i <lb/>nullo intercedente vecte ge&longs;tator e&longs;&longs;et pariter in C aut in D, &longs;ed <lb/>tanto minus, quanto minor e&longs;t CA aut DA, quam BA: idem­<lb/>que de ge&longs;tatore A, po&longs;ito fulcro in B, dicendum e&longs;t. </s> <s id="s.003301">Quare re­<lb/>ciprocæ &longs;unt pre&longs;&longs;iones di&longs;tantiis ge&longs;tatorum ab onere, & A <lb/>premitur ut CB aut DB, B autem premitur ut CA aut DA. <!-- KEEP S--></s> <lb/> <s id="s.003302">Cum enim vis ip&longs;a gravitatis oneris deor&longs;um conantis apta &longs;it <lb/>circa centrum A moveri pro ratione di&longs;tantiæ CA aut DA, uti­<lb/>que in B motum efficere debet pro ratione di&longs;tantiæ BA: e&longs;t <lb/>autem CA major quam DA ex hypothe&longs;i, igitur, ex 8.lib.5.ma­<lb/>jor e&longs;t Ratio momenti CA quàm momenti DA ad idem mo­<lb/>mentum BA, ac proinde major pre&longs;&longs;io ge&longs;tatoris B efficitur ab <lb/>onere in C, quam in D, collocato. </s> <s id="s.003303">Contra verò ge&longs;tatorem A <lb/>magis premit onus in D quam in C po&longs;itum, quia motus re&longs;pi­<lb/>cit centrum B, atque adeo ad eandem di&longs;tantiam AB major <lb/>e&longs;t Ratio di&longs;tantiæ DB majoris, quam CB minoris di&longs;tantiæ: <lb/>eadem autem e&longs;t di&longs;tantiarum, & motuum Ratio, ac proinde <lb/>momentorum. </s> </p> <p type="main"> <s id="s.003304">Ob&longs;erva autem mihi ideò de ge&longs;tatoribus oneris &longs;ermonem <lb/>fui&longs;&longs;e, ut duplicem effectum &longs;u&longs;tentationis atque pre&longs;&longs;ionis ex­<lb/>pre&longs;&longs;ius recogno&longs;cerem; qui enim onus ge&longs;tando &longs;u&longs;tentant, <lb/>mu&longs;culorum contentione conantes elidunt impetum oneris de­<lb/>or&longs;um nitentis, & aliquid efficientes, dum Activè re&longs;i&longs;tunt, no­<lb/>men Potentiæ merentur. </s> <s id="s.003305">At &longs;i onus palangæ connexum &longs;u&longs;ti­<lb/>neretur à duobus fulcris in extremitate po&longs;itis, hæc utique cùm <lb/>oneris gravitati nullo conatu adver&longs;arentur, &longs;olam re&longs;i&longs;tentiam <lb/>Formalem &longs;uâ &longs;oliditate exercerent, impediendo ne onus cum <lb/>palangâ de&longs;cenderet, &longs;ed nullam haberent <expan abbr="Re&longs;i&longs;tentiã">Re&longs;i&longs;tentiam</expan> Activam, <lb/>quæ illis Potentiæ vocabulum tribueret. </s> <s id="s.003306">In his unicus pre&longs;&longs;ionis <lb/>effectus attendendus e&longs;t, & validiori fulcro propiùs admoven­<lb/>dum e&longs;t onus, ne fortè fulcrum infirmius nimiâ pre&longs;&longs;ione coga­<lb/>tur &longs;uccumbere. </s> </p> <p type="main"> <s id="s.003307">Unum adhuc in palangæ ge&longs;tatoribus attendendum e&longs;t, &longs;i vi­<lb/>rium inæqualium fuerint, & onus non ita &longs;it palangæ applica­<lb/>tum, ut ejus di&longs;tantiæ à ge&longs;tatoribus &longs;int permutatim ut eorum­<lb/>dem vires; nimirum contingere po&longs;&longs;e, ut validior ge&longs;tator dum <pb pagenum="429" xlink:href="017/01/445.jpg"/>juxta &longs;uas vires conatur adversùs onus, magis premat infirmio­<lb/>rem ge&longs;tatorem, quam premeretur fulcrum infirmius, &longs;i unâ <lb/>cum validiore fulcro eandem gravitatem &longs;u&longs;tineret. </s> <s id="s.003308">Quia vide­<lb/>licet in eâdem palanga AB vectem primi generis con&longs;iderare <lb/>po&longs;&longs;umus, in quo onus C deor&longs;um nitens contra vim ge&longs;tatoris <lb/>habeat rationem hypomochlij, & validior ge&longs;tator A &longs;it poten­<lb/>tia repellens ge&longs;tatorem infirmiorem B conantem adver&longs;us <lb/>onus, ac proinde illum premat: quemadmodum &longs;i funi deor&longs;um <lb/>firmiter alligato in&longs;ereretur palanga, cui humeros &longs;ubjicerent <lb/>duo inæqualibus viribus &longs;ur&longs;um conantes; con&longs;tat enim infir­<lb/>miorem à validiore premi, & e&longs;&longs;e vectem primi generis. </s> <s id="s.003309">Ex quo <lb/>vides, cur bajuli dato invicem &longs;igno curent, ne alter alterum <lb/>præveniat in elevanda palanga; ne &longs;cilicet qui &longs;egnior fuerit, <lb/>pre&longs;&longs;ionem, non ab onere adhuc jacente & nondum elevato, &longs;ed <lb/>à &longs;ocio diligentiùs &longs;uam palangæ extremitatem elevante, reci­<lb/>piat. </s> </p> <p type="main"> <s id="s.003310">Hæc eadem, quæ de onere &longs;ublevando &longs;unt dicta, de eodem <lb/>trahendo pariter intelligantur, &longs;i vecti illigatum &longs;it onus, & <lb/>vectis extremitatibus jungantur trahentes: horum enim cona­<lb/>tus e&longs;&longs;e oportet permutatim in Ratione di&longs;tantiarum ab onere, <lb/>hoc e&longs;t à vectis puncto, cui onus adnectitur. </s> <s id="s.003311">Id non &longs;ine jucun­<lb/>dâ quadam animi titillatione vidi aliquando ob&longs;ervatum à ru&longs;ti­<lb/>co, qui alterius equorum currum trahentium defatigati laborem <lb/>mi&longs;eratus, tran&longs;ver&longs;arium, cui ambo adjungebantur, ita tran&longs;tu­<lb/>lit, ut in partes inæquales à temone di&longs;tingueretur, & longior <lb/>tran&longs;ver&longs;arij pars ad debiliorem equum &longs;pectaret. </s> <s id="s.003312">Inerant &longs;iqui­<lb/>dem tran&longs;ver&longs;ario tria foramina, per quæ temoni nectebatur fer­<lb/>reo clavo; unum quidem plane æqualiter ab extremitatibus abe­<lb/>rat, reliqua duo hinc & hinc à medio di&longs;tabant modico quidem <lb/>&longs;ed congruo intervallo, ut &longs;i equus dexter defatigaretur, clavus <lb/>immitteretur &longs;ini&longs;tro foramini, aut contra dextro, &longs;i &longs;ini&longs;ter <lb/>equus languidiùs traheret. </s> <s id="s.003313">Verùm cautè modica intervalla de­<lb/>finierat, ne nimia fieret momentorum inæqualitas; quod enim <lb/>alteri equorum laboris demebatur, addebatur reliquo. </s> </p> <p type="main"> <s id="s.003314">Quando autem non palangâ defertur onus, &longs;ed ip&longs;um imme­<lb/>diate à duobus &longs;u&longs;tinetur, eadem pror&longs;us e&longs;t philo&longs;ophandi ra­<lb/>tio, quandoquidem e&longs;t quodammodo onus vecti conjunctum, <lb/>atque juxta vectis longitudinem di&longs;tributum. </s> <s id="s.003315">Quamvis verò <pb pagenum="430" xlink:href="017/01/446.jpg"/>&longs;ingulis partibus &longs;ua gravitas in&longs;it, quia tamen in unam coale&longs;­<lb/>cunt gravitatem, ideò totius molis gravitas ibi intelligenda e&longs;t, <lb/>ubi e&longs;t centrum gravitatis; vectis autem longitudo æ&longs;timanda <lb/>e&longs;t in lineá jungente puncta, quibus ge&longs;tatores aut &longs;u&longs;tinentes <lb/>applicantur: Ex quibus punctis &longs;i ponamus exire lineas paral­<lb/>lelas lineæ directionis exeunti ex centro gravitatis, cadent om­<lb/>nes ad perpendiculum in lineam horizontalem tran&longs;euntem per <lb/>centrum gravitatis, aut illi parallelam. </s> <s id="s.003316">Harum igitur parallela­<lb/>rum, quæ directionem conatûs oppo&longs;iti gravitationi ponderis <lb/>referunt, di&longs;tantia à lineâ directionis centri gravitatis, ip&longs;orum <lb/>deferentium conatum in &longs;u&longs;tinendo, permutatim &longs;umpta de­<lb/>finiet; quæcumque demùm &longs;it oneris figura. </s> </p> <p type="main"> <s id="s.003317">Sit onus deferendum, cujus centrum gravitatis E; linea per <lb/><figure id="id.017.01.446.1.jpg" xlink:href="017/01/446/1.jpg"/><lb/>ge&longs;tatores tran&longs;iens, haben&longs;que rationem vectis, &longs;it BC, quæ <lb/>intelligatur horizonti parallela. </s> <s id="s.003318">In hanc igitur ad angulos <lb/>rectos cadit linea directionis EF; & ge&longs;tatores in quocumque <lb/>puncto lineæ BC fuerint, permutatim habent momenta &longs;u&longs;ti­<lb/>nendi, aut recipiunt momenta pre&longs;&longs;onis pro Ratione di&longs;tan­<lb/>tiarum à puncto F, in quod cadit linea directionis: cùm enim <lb/>linea BC ex hypothe&longs;i &longs;it horizonti parallela, omnium ip&longs;i FE <pb pagenum="431" xlink:href="017/01/447.jpg"/>parallelarum di&longs;tantia ab eádem FE, de&longs;umenda e&longs;t ex inter­<lb/>valli, ge&longs;tatorum & puncti F. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003319">At verò &longs;i recta BC non fuerit horizonti parallela, vel quia <lb/>deferentes onus non &longs;unt æquè alti, vel quia in clivo con&longs;i&longs;tunt, <lb/>utique linea directionis centri gravitatis E non cadit amplius in <lb/>rectam BC ad angulos rectos in F, &longs;ed obliquè incidit in H. <!-- KEEP S--></s> <lb/> <s id="s.003320">Concipiatur itaque per H tran&longs;iens linea GI horizonti paral­<lb/>lela, & ip&longs;i EH &longs;int parallelæ CD & BK: &longs;unt igitur di&longs;tan­<lb/>tiæ HD & HK, quæ eandem inter &longs;e habent Rationem, quæ <lb/>reperitur inter HC & HB; &longs;unt enim triangula HDC & <lb/>HKB rectangula ad D & K, æquales angulos ad verticem H <lb/>habentia, ac proinde &longs;imilia, & per 4. lib.6 ut HD ad HK, <lb/>ita HC ad HB. <!-- KEEP S--></s> <s id="s.003321">Quo igitur magis ab horizonte removetur <lb/>punctum B præ puncto C, etiam linea directionis ex E propior <lb/>cadit puncto C; atque adeò qui inferior e&longs;t, magis gravatur <lb/>ab onere. </s> </p> <p type="main"> <s id="s.003322">Id quod ex iis, quæ hujus libri cap.4. dicta &longs;unt, confirma­<lb/>tur: Nam vectis CB habens hypomochlium B, & pondus E <lb/>vecti impo&longs;itum, e&longs;t infra horizontem inclinatus; igitur plus la­<lb/>boris potentia impendit, quàm in horizontali po&longs;itione vectis. </s> <lb/> <s id="s.003323">Similiter vectis BC habens hypomochlium C & pondus E <lb/>vecti impo&longs;itum, e&longs;t elevatus &longs;upra horizontalem; igitur mi­<lb/>nus laborat potentia quàm in po&longs;itione horizontali. </s> <s id="s.003324">Itaque &longs;i <lb/>po&longs;ita lineá BC horizonti parallelâ æqualiter premebantur <lb/>ge&longs;tatores in B & in C, facta inclinatione ad horizontem, mi­<lb/>nùs premitur B &longs;uperior quàm C loco inferior. </s> </p> <p type="main"> <s id="s.003325">Quòd &longs;i ge&longs;tatores non &longs;u&longs;tineant onus &longs;ubjectis humeris, &longs;ed <lb/>illud manibus arreptum qua&longs;i &longs;u&longs;pen&longs;um retineant in M & N; <lb/>&longs;imili ratione attendenda e&longs;t di&longs;tantia illorum à puncto, in quod <lb/>cadit linea directionis centri gravitatis E; quæ utique ad angu­<lb/>los recto incidit in rectam MN, &longs;i hæc &longs;uerit horizonti pa­<lb/>rallela, & labor ge&longs;tatorum e&longs;t permutatim ut eorum di&longs;tantia <lb/>à pancto S. <!-- KEEP S--></s> <s id="s.003326">At verò &longs;i linea MN fuerit ad horizontem incli­<lb/>nata, & linea directionis &longs;it EO; utique minor e&longs;t di&longs;tantia à <lb/>&longs;uperiore M, quam ab inferiore N, ideoque plus laborabit &longs;u­<lb/>perior retinendo, quam inferior. </s> <s id="s.003327">Id quod pariter ex dictis <lb/>cap.4. confirmatur; nam pondus e&longs;t vecti &longs;ubjectum, & vectis <lb/>MN habens hypomochlium N e&longs;t &longs;upra horizontalem lineam, <pb pagenum="432" xlink:href="017/01/448.jpg"/>ac propterea potentia plus laborat quam in horizontali: contra <lb/>autem vectis NM habens hypomochlium M e&longs;t in&longs;ta horizon­<lb/>talem lineam depre&longs;&longs;us, ideoque minus potentia laborat quam <lb/>in horizontali. </s> <s id="s.003328">Quæ omnia tam aperte re&longs;pondent quotidia­<lb/>no experimento, ut mirum videatur potui&longs;&longs;e aliquos authores <lb/>idem planè opinari, &longs;ive ge&longs;tatores &longs;u&longs;tineant impo&longs;itum onus, <lb/>&longs;ive illud &longs;u&longs;pen&longs;um retineant in po&longs;itione vectis declivi; Si <lb/>enim ducti per O lineâ horizonti parallelâ, ducantur ex M & <lb/>N rectæ MT, & NV parallelæ lineæ directionis centri gravi­<lb/>tatis EO, utique di&longs;tantiæ &longs;unt TO & VO: atqui TO ad <lb/>VO e&longs;t ut MO ad NO propter triangulorum OTM & <lb/>OVN &longs;imilitudinem; & MO ad ON habet minorem Ratio­<lb/>nem quàm MS ad SN ex 8.lib.5. igitur etiam TO ad OV <lb/>habet minorem Rationem quàm MS ad SN: igitur in po&longs;itio­<lb/>ne vectis declivi, M &longs;uperior laborabit ut ON, atque N infe­<lb/>rior laborabit ut OM. </s> </p> <p type="main"> <s id="s.003329">Ex his unu&longs;qui&longs;que intelligit non ad duos tantùm ge&longs;tato­<lb/>res, &longs;ed etiam ad plures referenda e&longs;&longs;e, quæ hactenus diximus, <lb/>habitâ &longs;cilicet di&longs;tantiarum ratione, quibus &longs;inguli ab&longs;unt a <lb/>pondere, adeò ut qui æqualibus intervallis à pondere di&longs;tant, <lb/>æqualem conatum impendant in eo &longs;u&longs;tinendo. </s> <s id="s.003330">Sic &longs;i à pon­<lb/><figure id="id.017.01.448.1.jpg" xlink:href="017/01/448/1.jpg"/><lb/>dere P æqualiter di&longs;tent A & B, æqua­<lb/>liter premuntur: item C & D æquali­<lb/>ter di&longs;tantes ab eodem pondere P æqua­<lb/>lem pre&longs;&longs;ionem recipiunt: Et &longs;i compa­<lb/>rentur invicem D & B, aut C & A, <lb/>manife&longs;tum e&longs;t propinquiores premi præ remotioribus; ac <lb/>propterea, &longs;i &longs;olùm po&longs;itionis ratio haberetur, qui robu&longs;tiores <lb/>&longs;unt, collocandi e&longs;&longs;ent in C & D, infirmiores verò in A & B: <lb/>&longs;ed quoniàm contingit inter plures &longs;odales aliquem aliquando <lb/>connivere, ideò ut plurimum extremi A & B validiores &longs;unt, <lb/>ut &longs;i fortè mediorum aliquis languidiùs conetur &longs;u&longs;tinendo, <lb/>illi faciliùs muneri &longs;uo &longs;atisfaciant. <pb pagenum="433" xlink:href="017/01/449.jpg"/></s> </p> <p type="main"> <s id="s.003331"><emph type="center"/>CAPUT X.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003332"><emph type="center"/><emph type="italics"/>An vis Ela&longs;tica ad aliquod Vectis genus <lb/>pertineat.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003333">QUoniam Græcis <foreign lang="greek">e)lasma\</foreign> tùm laminam, tùm plicam &longs;eu <lb/>flexum &longs;ignificat, atque <foreign lang="greek">e)/lastrw=n</foreign> e&longs;t id, quod impellit; &longs;æ­<lb/>pius autem chalybeas laminas in machinulis ita di&longs;ponimus, ut <lb/>primùm flexæ, deinde &longs;ibi dimi&longs;&longs;æ, dum &longs;e&longs;e re&longs;tituunt, aliud <lb/>corpus impellant, cui motum concilient; propterea Ela&longs;mata, <lb/>&longs;eu Ela&longs;mos, huju&longs;modi laminas dicimus, quas Itali <emph type="italics"/>Su&longs;te <emph.end type="italics"/> aut <lb/><emph type="italics"/>molle<emph.end type="italics"/> vocamus; & facultatem illam, qua &longs;ibi congruentem fi­<lb/>guram atque po&longs;itionem hæ laminæ reparant, Vim Ela&longs;ticam <lb/>appellamus. </s> <s id="s.003334">Quamquam non &longs;olis laminis, &longs;ed cæteris quoque <lb/>corporibus per vim inflexis, & ad &longs;ibi debitam rectitudinem <lb/>redeuntibus, facultas hæc Ela&longs;tica tribuenda e&longs;t, quemadmo­<lb/>dum flexilibus virgultorum ramis, à quibus &longs;ecundus in &longs;ylvâ <lb/>&longs;ibi cavere debet, & perticæ, quâ toreutæ utuntur in toreu­<lb/>mate elaborando, dum tornum circumagunt circumducto fu­<lb/>niculo, qui depre&longs;&longs;o &longs;uppedanco perticam flectit, hæc enim, <lb/>ce&longs;&longs;ante pedis pre&longs;&longs;ione, funiculum retrahens &longs;uam &longs;ibi repa­<lb/>rat rectitudinem. </s> <s id="s.003335">An verò ignis atque aër &longs;ive externá com­<lb/>pre&longs;&longs;ione, &longs;ive alieno frigore concretus, & in exigua &longs;patia con­<lb/>tractus, ubi ce&longs;&longs;ante vi, aut abeunte frigore, extenuatus am­<lb/>pliorem locum occupat, proximumque corpus pellens à &longs;uá &longs;e­<lb/>de removet, facultate Ela&longs;ticâ præditus dicendus &longs;it, quæ&longs;tio <lb/>Grammaticis dirimenda relinquatur: hæc enim fluida corpora, <lb/>nullam partium texturam habentia, nec certis figuræ, quam <lb/>expetant, terminis &longs;uapte naturà circum&longs;cripta, vix quicquam <lb/>cum Ela&longs;mate commune habere videntur. </s> </p> <p type="main"> <s id="s.003336">Cum itaque inæquales deprehendantur ela&longs;matis eju&longs;dem <lb/>vires pro diversâ &longs;uarum partium po&longs;itione juxta longitudinem, <lb/>animum &longs;ubiit cupido examinandi, an forte in eo aliqua vectis <lb/>&longs;pecies reperiatur, ut propterea & vectis rationibus illa virium <pb pagenum="434" xlink:href="017/01/450.jpg"/>inæqualitas definienda &longs;it. </s> <s id="s.003337">Et quidem manife&longs;tum e&longs;t aliquam <lb/>ela&longs;matis partem fixam e&longs;&longs;e atque manentem, &longs;ive illa extrema <lb/>&longs;it, ut in perticâ toreutæ, &longs;ive media, ut in arcu bali&longs;tæ, &longs;ive <lb/>utraque extremitate manente pars media flectatur in &longs;inum, ut <lb/>citharæ nervis contingit. </s> <s id="s.003338">Quî enim fieri po&longs;&longs;et, ut per vim la­<lb/>mina flecteretur, &longs;i partes omnes æqualiter moverentur? </s> <s id="s.003339">Ut igi­<lb/>tur externam vim recipiat, & flectatur, aliquam ejus partem <lb/>oportet aut omnino immotam manere, aut &longs;altem languidiùs <lb/>moveri. </s> </p> <p type="main"> <s id="s.003340">Hinc e&longs;t ela&longs;matis motum, dum inflectitur, circa partem <lb/>manentem perfici, ac proinde particulas, quæ ad cavam qui­<lb/>dem &longs;uperficiem &longs;pectant, per vim comprimi, quæ verò ad con­<lb/>vexam, intendi. </s> <s id="s.003341">Quòd &longs;i particulæ illæ non ita tenaci nexu in­<lb/>ter &longs;e invicem cohærerent, ut facilè di&longs;traherentur contentæ, <lb/>& exprimerentur compre&longs;&longs;æ, quemadmodum plumbeæ laminæ, <lb/>quæ in figuram quamlibet conformatur, accidit, ami&longs;&longs;am recti­<lb/>tudinem non recuperarent. </s> <s id="s.003342">Sed quoniam arcti&longs;&longs;imo vinculo <lb/>conjunguntur, quod ni&longs;i validioribus viribus revelli non pote&longs;t, <lb/>ut in chalybeâ laminâ ob&longs;ervamus; ce&longs;&longs;ante externâ vi, quæ <lb/>contentæ fuerant, &longs;e contrahunt, quæ compre&longs;&longs;æ, &longs;elatiùs ex­<lb/>plicant; atque adeò his debitam po&longs;itionem &longs;ibi reparantibus, <lb/>lamina ad pri&longs;tinam formam eò vehementiùs redit, quò majo­<lb/>rem violentiam patiebatur. </s> <s id="s.003343">Quare Potentia movens &longs;unt ip&longs;æ <lb/>particulæ illatam vim excutientes, & ad &longs;ibi debitam po&longs;itio­<lb/>nem redeuntes. </s> </p> <p type="main"> <s id="s.003344">Licet igitur in arcu bali&longs;tæ intento duplex ela&longs;ma hinc atque <lb/>hinc con&longs;iderare; media &longs;i quidem pars arcûs bali&longs;tæ manubrio <lb/>infixa manet, & &longs;ingula cornua &longs;inuantur, &longs;ed eò difficiliùs, <lb/>quò breviora &longs;unt, cæteris paribus, attentâ eorum cra&longs;&longs;itie, & <lb/>ferri temperatione; pari enim flexione paucioribus minoris ar­<lb/>cûs particulis major violentia inferenda e&longs;t, quippe quas ma­<lb/>gis comprimi, magí&longs;que intendi oportet, quàm in longiore ar­<lb/>cu, ubi minore plurium partium compre&longs;&longs;ione & intentione <lb/>flexio eadem habetur. </s> <s id="s.003345">Præterquam quod in ip&longs;a flexione adhi­<lb/>betur quodammodo vectis &longs;ecundi generis, quem ip&longs;a longitu­<lb/>do repræ&longs;entat, pars manens vicem hypomochlij &longs;ubit, & par­<lb/>tes intermediæ, quas per vim coarctari aut dilatari oportet, lo­<lb/>cum obtinent ponderis: nihil igitur mirum, &longs;i Potentia extre-<pb pagenum="435" xlink:href="017/01/451.jpg"/>mitatem arcûs ad &longs;e nervo adnexo trahens faciliùs moveat par­<lb/>ticulas ea&longs;dem, quò, longiùs ab&longs;ens ab hypomochlio, faciliùs <lb/>movetur. </s> </p> <p type="main"> <s id="s.003346">Hîc autem ubi arcûs mentio incidit, in ip&longs;o nervo illud ela&longs;­<lb/>matis genus occurrit, quod utramque extremitatem habet ma­<lb/>nentem; curvato enim arcu nervus inflexus intenditur; po&longs;tea <lb/>cùm dimittitur, pars media, cui &longs;agitta aut globus excutiendus <lb/>aptatur, plus movetur quàm ejus extremitates arcûs cornibus <lb/>cohærentes. </s> <s id="s.003347">Univer&longs;a autem violentia, quam nervus conten­<lb/>tus &longs;ubit, con&longs;i&longs;tit in &longs;uarum particularum intentione, quæ, <lb/>dum &longs;e contrahentes aliquid juvant ad nervum ip&longs;um juxta <lb/>rectam lineam extendendum, aliquid etiam impetûs &longs;agittæ ex­<lb/>cu&longs;&longs;æ imprimunt. </s> </p> <p type="main"> <s id="s.003348">Quod verò ad ip&longs;a arcûs cornua attinet, &longs;atis liquet illa &longs;i­<lb/>milis cra&longs;&longs;itiei, paris longitudinis, æquali&longs;que temperationis <lb/>e&longs;&longs;e debere, ut æqualis fiat hinc & hinc compre&longs;&longs;io atque in­<lb/>tentio partium, ex qua æquales oriantur vires &longs;e&longs;e in pri&longs;tinam <lb/>formam re&longs;tituendi. </s> <s id="s.003349">Si enim alterutra pars arcûs majorem <lb/>violentiam pa&longs;&longs;a velociùs atque validiùs præ reliquâ &longs;e move­<lb/>ret, à de&longs;tinato &longs;copo &longs;agitta aberraret in dexteram aut in &longs;i­<lb/>ni&longs;tram declinans. </s> </p> <p type="main"> <s id="s.003350">Ut igitur hi&longs;ce prænotatis ad propo&longs;itam quæ&longs;tionem acce­<lb/>damus, non e&longs;t hîc &longs;ermo de laminâ in &longs;piram multiplicem in­<lb/>flexâ, atque &longs;pi&longs;sè per vim contorta, quæ amoto repagulo &longs;e&longs;e <lb/>in ampliores gyros explicans &longs;ecum rapit aliud corpus extremi­<lb/>tati mobili adnexum; cuju&longs;modi e&longs;t Ela&longs;ma in Automatis ho­<lb/>ras indicantibus, cujus extremitati adnectitur tympanum &longs;pi­<lb/>ram illam includens; dum enim ex dilatatione Ela&longs;matis in am­<lb/>pliorem &longs;piram, circumagitur tympanum, adnexam catenu­<lb/>lam conum circumplexam trahit, totique machinulæ motum <lb/>conciliat. </s> <s id="s.003351">Hîc &longs;iquidem, uti nulla longitudo in con&longs;ideratio­<lb/>nem cadere pote&longs;t, nullam vectis &longs;peciem habere po&longs;&longs;u­<lb/>mus; nam facultas movendi non ratione po&longs;itionis exte­<lb/>nuatur, ut in vecte, &longs;ed vires initio validæ &longs;en&longs;im lan­<lb/>gue&longs;cunt, quia ela&longs;matis partes compre&longs;&longs;æ atque conten­<lb/>tæ, pro ratione violentiæ, quam &longs;ubeunt, excutiendæ, ve­<lb/>hementiùs primùm, deinde remi&longs;&longs;iùs conantur. </s> <s id="s.003352">Quare con­<lb/>trover&longs;ia in illo e&longs;t, utrùm in ela&longs;mate, cujus aliqua <pb pagenum="436" xlink:href="017/01/452.jpg"/>longitudo de&longs;ignari pote&longs;t, aliqua vectis &longs;pecies repe­<lb/>riatur. </s> </p> <p type="main"> <s id="s.003353">Et ut majori in luce quæ&longs;tio ver&longs;etur, perticam toreutæ, <lb/><figure id="id.017.01.452.1.jpg" xlink:href="017/01/452/1.jpg"/><lb/>oculis &longs;ubjiciamus, quæ &longs;it AB, & <lb/>in A fixa atque immota per&longs;everet, <lb/>quamvis extremitas B deprimatur, <lb/>ut veniat in C. <!-- KEEP S--></s> <s id="s.003354">In hac perticæ <lb/>flexione partes, quæ circa D ex. </s> <lb/> <s id="s.003355">gr. <!-- REMOVE S-->intelliguntur, maximam violentiam patiuntur, nam inter <lb/>eas, quæ ad cavitatem &longs;pectantes compre&longs;&longs;ione coarctantur, <lb/>illæ præ cæteris hinc atque hinc cohærentibus urgentur ma­<lb/>gis; inter eas verò, quæ convexitatem re&longs;picientes di&longs;tentu <lb/>explicantur, quæ ibi &longs;unt, præ reliquis à &longs;ummo flexu paulò <lb/>remotioribus vehementiùs tenduntur. </s> <s id="s.003356">Hinc licèt particulæ <lb/>omnes in hac flexione vim pa&longs;&longs;æ, dum nituntur &longs;ingulæ pri&longs;ti­<lb/>num &longs;tatum &longs;ibi reparare, conatus &longs;uos exerant, majores aut <lb/>minores pro ratione majoris aut minoris violentiæ; poti&longs;&longs;ima ta­<lb/>men vis ela&longs;tica ibi con&longs;ideranda e&longs;t, ubi &longs;umma inflexio &longs;um­<lb/>mam vim particulis infert; ibi enim majore conatu quàm alibi <lb/>violentiam excutit natura. </s> <s id="s.003357">Quamvis igitur vis ela&longs;tica per uni­<lb/>ver&longs;am ela&longs;matis longitudinem, quàm particulæ compre&longs;&longs;æ at­<lb/>que contentæ obtinent, extendatur, ibi tamen poti&longs;&longs;imùm col­<lb/>locata intelligitur, ubi in &longs;ummo flexu puta in D, validiùs co­<lb/>natur. </s> </p> <p type="main"> <s id="s.003358">Jam verò quis ignorat in tornando plurimum intere&longs;&longs;e, utrùm <lb/>funiculus in ipsâ extremitate B, an verò in E adnectatur? </s> <s id="s.003359">Si­<lb/>quidem, &longs;icut ex E difficiliùs flectitur pertica, quàm ex B, æqua­<lb/>li flexione, ita cæteris paribus in E validiùs retrahitur funicu­<lb/>lus, & minor motus perficitur quàm in B. </s> <s id="s.003360">E&longs;t igitur hîc ratio <lb/>Vectis tertij generis, in quo hypomochlium e&longs;t A pars fixa & <lb/>immota; Potentia movens (&longs;cilicet particulæ vim illatam ex­<lb/>cutientes) e&longs;t poti&longs;&longs;imùm in D; pondus, quod movetur, e&longs;t <lb/>ultra D, &longs;ive in extremitate B, &longs;ive in aliqua ex partibus inter­<lb/>mediis, ut in E. <!-- KEEP S--></s> <s id="s.003361">Quare in collocatione corporis, quod ope <lb/>ela&longs;matis movendum e&longs;t, attendere oportet, quanto motu opus <lb/>&longs;it, ut in majore &longs;eu minore di&longs;tantiâ à puncto ela&longs;matis ma­<lb/>nente, & immoto applicetur: quo enim minor e&longs;t di&longs;tantia, <lb/>minus &longs;patium percurrit. </s> </p> <pb pagenum="437" xlink:href="017/01/453.jpg"/> <p type="main"> <s id="s.003362">Quamvis autem ela&longs;matis vires ad impellendum vel trahen­<lb/>dum corpus ex huju&longs;modi di&longs;tantiâ pendeant, & comparatis <lb/>inter &longs;e duabus po&longs;itionibus E atque B, validiùs operetur in E <lb/>quàm B, non tamen eâdem vi motus (quicumque demum ille <lb/>&longs;it &longs;ive major, &longs;ive minor) inchoatur, atque procedit, ut &longs;upra <lb/>innuimus; natura quippe remi&longs;&longs;iore ni&longs;u reluctatur, ubi mino­<lb/>rem patitur violentiam, ac proinde &longs;en&longs;im attenuatur conatus, <lb/>quatenus particularum violenta compre&longs;&longs;io atque contentio di­<lb/>minuitur. </s> </p> <p type="main"> <s id="s.003363">Hæc quæ de ela&longs;mate pror&longs;us recto explicata &longs;unt, etiam de <lb/>incurvo intelliguntur; cuju&longs;modi e&longs;&longs;et lamina chalybea RT <lb/>inflexa in S, cujus manens & immota ex­<lb/><figure id="id.017.01.453.1.jpg" xlink:href="017/01/453/1.jpg"/><lb/>tremitas e&longs;&longs;et R: dum enim pars ST pro­<lb/>pellitur versùs R, particulæ, poti&longs;&longs;imùm <lb/>quæ in S, comprimuntur atque intendun­<lb/>tur. </s> <s id="s.003364">Quod &longs;i pars RS paulo longior fuerit, <lb/>contingere pote&longs;t, ut facilius &longs;it illam inflecti &longs;altem leviter, <lb/>quàm particulas in S ulteriùs comprimi, aut intendi. </s> <s id="s.003365">Quare <lb/>particulæ ip&longs;ius RS &longs;e&longs;e re&longs;tituentes impellunt S, particulæ au­<lb/>tem ip&longs;ius ST impellunt T. <!-- KEEP S--></s> <s id="s.003366">Semper autem Potentiam minùs <lb/>moveri, quàm corpus, quod impellitur, con&longs;tat, quemadmo­<lb/>dum ratio vectis tertij generis exigit. </s> </p> <p type="main"> <s id="s.003367">Neque his, quæ dicta &longs;unt, adver&longs;antur percu&longs;&longs;iones, quæ <lb/>in extremitate longioris ela&longs;matis per vim inflexi, &longs;tatimque di­<lb/>mi&longs;&longs;i, validiores fiunt, quam in partibus mediis; &longs;icut ip&longs;e te <lb/>docere potes, &longs;i longiu&longs;culi virgulti inflexi atque dimi&longs;&longs;i pri­<lb/>mùm parti mediæ deinde extremitati manum in eodem plano <lb/>verticali con&longs;titutam opponas, quam percutiat, magis enim ex <lb/>&longs;ecundâ quàm ex primâ percu&longs;&longs;ione dolebis. </s> <s id="s.003368">Quia &longs;cilicet non <lb/>impetus &longs;olùm primo productus, &longs;ed & velocitas percutientis <lb/>cum impetu acqui&longs;ito ex motu ante percu&longs;&longs;ionem (ut &longs;uo loco <lb/>dicetur) attenditur, ut validior &longs;it ictus: majorem autem e&longs;&longs;e <lb/>partis extremæ quàm mediarum velocitatem con&longs;tat, quamvis <lb/>initio illæ impetu eodem, aut æquali moveantur. </s> <s id="s.003369">Quando ve­<lb/>rò ela&longs;matis vires prope partem manentem majores e&longs;&longs;e, quam <lb/>procul ab illâ, dictum e&longs;t, non e&longs;t habita ratio percu&longs;&longs;ionis, <lb/>quæ prævium percutientis motum requirit, &longs;ed tractionis aut <lb/>impul&longs;ionis, quæ nullum trahentis aut impellentis prævium <pb pagenum="438" xlink:href="017/01/454.jpg"/>motum exigunt, eóque faciliores accidunt, quò tardiores &longs;unt; <lb/>minùs enim re&longs;i&longs;tit corpus, quod tardè movetur, ac proinde <lb/>validiùs trahitur aut impellitur, quo minorem potentia invenit <lb/>re&longs;i&longs;tentiam: Contrà quàm accidat in percu&longs;&longs;ione, quæ vali­<lb/>diorem facit ictum, quo majorem invenit re&longs;i&longs;tentiam; hæc <lb/>autem major e&longs;t, quò velociùs moveri deberet corpus percu&longs;­<lb/>&longs;um, ut percutientis motui ob&longs;ecundaret, cui magis re&longs;i&longs;tens <lb/>majorem ictum recipit; cum tamen hic languidior e&longs;&longs;et atque <lb/>infirmior, &longs;i manum &longs;en&longs;im &longs;ubduceres virgulto percutienti. </s> <lb/> <s id="s.003370">Quare pars ela&longs;matis extrema validiùs percutit, quia majorem <lb/>invenit re&longs;i&longs;tentiam, pars media validiùs trahit aut impellit, <lb/>quia minùs illi re&longs;i&longs;titur. <lb/></s> </p> <p type="main"> <s id="s.003371"><emph type="center"/>CAPUT XI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003372"><emph type="center"/><emph type="italics"/>Cur longiora corpora faciliùs flectantur, difficiliùs <lb/>&longs;u&longs;tineantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003373">PRæ&longs;ens di&longs;putatio non di&longs;tat ab iis, quæ ab Ari&longs;totele in­<lb/>quiruntur in Mechanicis quæ&longs;t.14. <emph type="italics"/>Cur eju&longs;dem magnitu­<lb/>dinis lignum faciliùs genu frangitur, &longs;i qui&longs;piam æquè diductis ma­<lb/>nibus extrema comprehendens fregerit, quàm &longs;i juxta genu: & &longs;i <lb/>terræ illud applicans pede &longs;uperimpo&longs;ito manu longè diductâ confre­<lb/>gerit, quàm propè.<emph.end type="italics"/> & quæ&longs;t. </s> <s id="s.003374">16. <emph type="italics"/>Cur quanto longiora &longs;unt ligna, <lb/>tanto imbecilliora fiunt: & &longs;i tollantur, inflectuntur magis; tamet&longs;i <lb/>quod breve quidem e&longs;t, ceu bicubitum, fuerit tenue; quod verò cu­<lb/>bitorum centum, cra&longs;&longs;um?<emph.end type="italics"/> & quæ&longs;t. </s> <s id="s.003375">26. <emph type="italics"/>Cur difficilius e&longs;t longa <lb/>ligna ab extremo &longs;uper humeros ferre, quàm &longs;ecundùm medium, <lb/>æquali exi&longs;tente pondere?<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.003376">Et quidem quod ad primum, &longs;cilicet ad flexionem &longs;pectat, <lb/>quam demum con&longs;equitur fractio, quemadmodum flectendi <lb/>corpus aliquod longum aut frangendi difficultas oritur ex com­<lb/>plexione atque copulatione particularum, quibus con&longs;tat, ægrè <lb/>di&longs;&longs;olubili, ita illud inflectitur, atque frangitur, cum earum­<lb/>dem particularum coagmentatio vehementi impul&longs;ione labe-<pb pagenum="439" xlink:href="017/01/455.jpg"/>factatur, his quidem per vim compre&longs;&longs;is, his verò validè in di­<lb/>ver&longs;a di&longs;tractis. </s> <s id="s.003377">Quo igitur faciliùs compre&longs;&longs;io hæc atque di­<lb/>&longs;tractio perficitur, eò etiam faciliùs flectitur corpus, aut fran­<lb/>gitur. </s> <s id="s.003378">Hanc autem particularum compre&longs;&longs;ionem atque di­<lb/>&longs;tractionem faciliùs contingere longiori corpori quàm breviori <lb/>manife&longs;tum e&longs;t; quia videlicet motus ille particularum ad fle­<lb/>xionem aut fractionem nece&longs;&longs;arius minorem Rationem habet <lb/>ad motum Potentiæ longiùs applicatæ, quàm ad motum Po­<lb/>tentiæ propioris. </s> </p> <p type="main"> <s id="s.003379">Potentia movens bifariam con&longs;iderari pote&longs;t, &longs;ivè in ip&longs;o cor­<lb/>pore inclu&longs;a, cuju&longs;modi e&longs;t illi in&longs;ita atque ingenita gravitas, <lb/>vi cujus &longs;ponte &longs;uâ flectitur; &longs;ivè extrin&longs;ecùs adhibita, ut &longs;i <lb/>onus aliquod grave deor&longs;um premens adjiciatur, aut potentia <lb/>vivens ad motum in quamcumque po&longs;itionis differentiam apta: <lb/>utrobique tamen e&longs;t eadem ratio; ubi &longs;cilicet a&longs;&longs;umpta atque <lb/>adventitia potentia applicatur, ibi operatur; atque ibi innata <lb/>gravitas intelligitur &longs;ua exercere momenta, ubi partis ultra &longs;ub­<lb/>jectum fulcrum extantis centrum gravitatis reperitur; illáque <lb/>e&longs;t à fulcro di&longs;tantia potentiæ flectentis, aut etiam frangentis. </s> <lb/> <s id="s.003380">Sit enim pri&longs;ma AB, cujus pars AC <lb/>infixa &longs;it parieti, extra quem emineat <lb/><figure id="id.017.01.455.1.jpg" xlink:href="017/01/455/1.jpg"/><lb/>horizonti parallela pars CB &longs;uâ gravi­<lb/>tate deor&longs;um connitens; quæ &longs;ane non <lb/>e&longs;t intelligenda in B, &longs;ed qua&longs;i tota <lb/>con&longs;tituta e&longs;&longs;et in D, ubi e&longs;t centrum gravitatis non totius cor­<lb/>poris AB, &longs;ed partis extantis CB. <!-- KEEP S--></s> <s id="s.003381">Quod &longs;i brevius e&longs;&longs;et pri&longs;­<lb/>ma AE, partis CE minor e&longs;&longs;et gravitas, quàm partis CB, & <lb/>prætereà minus abe&longs;&longs;e à fulcro C intelligeretur, quippe cujus <lb/>centrum gravitatis e&longs;&longs;et F multo propius quàm D. <!-- KEEP S--></s> <s id="s.003382">Plura igi­<lb/>tur momenta habet CB quàm CE ad flectendum pri&longs;ma parie­<lb/>ti infixum, juxta ea, quæ uberiùs dicta &longs;unt lib.2. cap.6. ubi &longs;o­<lb/>lidorum Re&longs;i&longs;tentiam re&longs;pectivam con&longs;ideravimus, nec vacat <lb/>hic iterum inculcare. </s> </p> <p type="main"> <s id="s.003383">Unum hîc con&longs;iderandum e&longs;t, quod ad rationes vectis atti­<lb/>net, videlicet, &longs;i &longs;uperiori pri&longs;matis parti, quæ re&longs;pondet ip&longs;i <lb/>AC, incumberet onus, quod faciliùs loco moveri po&longs;&longs;it, quàm <lb/>particularum complexio labefactari, aut omnino di&longs;&longs;olvi, pri&longs;<lb/>ma neque frangi, aut forta&longs;&longs;e ne flecti quidem contingeret; <pb pagenum="440" xlink:href="017/01/456.jpg"/>&longs;ed rationem vectis primi generis haberet, cujus fulcrum e&longs;&longs;et <lb/>in C, pondus in A, potentia in D. <!-- KEEP S--></s> <s id="s.003384">At &longs;i onus impo&longs;itum nul­<lb/>latenus dimoveri queat, quemadmodum cùm pri&longs;ma parieti in­<lb/>figitur, &longs;i CB ejus &longs;it longitudinis, ut vis gravitatis ad de&longs;cen­<lb/>dendum tali intervallo CD &longs;ejuncta à fulcro C plus habeat <lb/>momenti, quàm particularum coagmentatio, ne compriman­<lb/>tur, aut di&longs;trahantur; tunc vectis e&longs;t &longs;ecundi generis, fulcrum <lb/>quidem habens in C, quatenus totum &longs;egmentum AC retine­<lb/>tur pror&longs;us immotum, & potentia in D, pondus verò, cujus <lb/>vires vincuntur, eo loco, ubi maxima fit particularum com­<lb/>pre&longs;&longs;io atque di&longs;tractio. </s> </p> <p type="main"> <s id="s.003385">Hinc factum videtur &longs;atis Ari&longs;toteli quærenti, cur faciliùs <lb/>flectatur lignum cra&longs;&longs;um cubitorum centum, quàm tenue bi­<lb/>cubitum; quia nimirum in cra&longs;&longs;o ligno cubitorum centum è <lb/>pariete extantium, &longs;i ponatur &longs;imilem atque æquabilem cra&longs;&longs;i­<lb/>tiem juxta totam longitudinem habere, gravitatis centrum <lb/>di&longs;tat à fulcro cubitis quinquaginta, tenue verò atque exile <lb/>lignum &longs;imilis figuræ atque materiæ centrum habet uno tantùm <lb/>cubito di&longs;tans à fulcro, & gravitas illius ad hujus gravitatem in <lb/>eâ e&longs;t Ratione, quam habent inter &longs;e ip&longs;orum lignorum moles, <lb/>quæ &longs;cilicet ex Rationibus ba&longs;ium atque longitudinum compo­<lb/>nitur. </s> <s id="s.003386">Cum itaque longitudo ad longitudinem &longs;it ut 50 ad 1, <lb/>&longs;i ba&longs;ium &longs;imilium latera homologia &longs;int ut 10 ad 1, ba&longs;ium <lb/>Ratio e&longs;t ut 100 ad 1: quare cùm longioris ligni gravita­<lb/>tis Ratio ad gravitatem brevioris componatur ex Ratione ba­<lb/>&longs;ium ut 100 ad 1, & ex ratione longitudinum ut 50 ad 1, gravi­<lb/>tas longioris ad gravitatem brevioris e&longs;t ut 5000 ad 1. Atqui <lb/>momenta ad de&longs;cendendum componuntur ex gravitate & di­<lb/>&longs;tantia à fulcro; igitur momenta longioris ad momenta brevio­<lb/>ris &longs;unt ut 250000 ad 1. At verò re&longs;i&longs;tentia ab&longs;oluta, ne flectan­<lb/>tur, aut frangantur huju&longs;modi ligna, e&longs;t in Ratione compo&longs;itâ <lb/>ex Rationibus ba&longs;ium atque cra&longs;&longs;itierum; ac proinde &longs;i ba&longs;es <lb/>&longs;int &longs;imiles, & &longs;imiliter po&longs;itæ, Ratio e&longs;t triplicata Rationis la­<lb/>terum homologorum, hoc e&longs;t Rationis 10 ad 1; atque adeò re­<lb/>&longs;i&longs;tentia longioris ad re&longs;i&longs;tentiam brevioris e&longs;t ut 1000 ad 1. <lb/>Patet igitur momenta ut 250.000 ad re&longs;i&longs;tentiam ut 1000 ma­<lb/>jorem habere Rationem, quàm momenta ut 1 ad re&longs;i&longs;tentiam <lb/>ut 1: faciliùs ergo illa quàm hæc momenta re&longs;i&longs;tentiam &longs;ibi con-<pb pagenum="441" xlink:href="017/01/457.jpg"/>gruentem &longs;uperant, atque faciliùs lignum cra&longs;&longs;um longius <lb/>flectitur, aut frangitur, quàm brevius. </s> </p> <p type="main"> <s id="s.003387">Quæ autem de ligno parieti &longs;ecundum alteram extremitatem <lb/>infixo dicta &longs;unt, &longs;ervatâ analogiâ de eodem dicantur, &longs;i circa <lb/>medium fulcro alicui in&longs;i&longs;tat ita, ut hinc atque hinc habeat <lb/>gravitatis momenta compo&longs;ita ex ip&longs;arum partium gravitate & <lb/>ex di&longs;tantia centrorum gravitatis à fulcro, cui innititur: eâdem <lb/>enim ratiocinatione colligitur in longiore ligno majorem e&longs;&longs;e <lb/>Rationem momentorum gravitatis ad re&longs;i&longs;tentiam ortam ex <lb/>partium complexione, ne flectatur, quàm in breviore. </s> <s id="s.003388">Quod <lb/>verò &longs;pectat ad longioris ligni faciliorem flexionem, quando <lb/>utraque extremitas innixa e&longs;t &longs;ubjecto fulcro, non videtur pro­<lb/>priè hujus loci, &longs;ed de eâ dictum e&longs;t &longs;uperiùs lib.3. cap.12. </s> </p> <p type="main"> <s id="s.003389">Ex his, quæ de pri&longs;mate extra parietem extante, quod faci­<lb/>liùs flectitur, hactenus diximus, ulteriùs patet, cur ex contra­<lb/>rio longius lignum ut AB, etiam&longs;i parem cum breviore AE <lb/>cra&longs;&longs;itiem habeat, alterâ extremitate æqualiter in AC ap­<lb/>prehen&longs;um difficiliùs &longs;u&longs;tineatur. </s> <s id="s.003390">Nam quod longius e&longs;t ad il­<lb/>lud, quod brevius e&longs;t, &longs;ecundùm gravitatem, quæ deor&longs;um ni­<lb/>titur, eam habèt Rationem, quæ e&longs;t longitudinis majoris CB <lb/>ad longitudinem minorem CE: & præterea momenta, quæ ex <lb/>di&longs;tantia oriuntur, &longs;unt ut CD ad CF, hoc e&longs;t ut CB ad CE, <lb/>&longs;i quidem ex hypothe&longs;i centrum gravitatis intelligatur in me­<lb/>diâ longitudine; &longs;ecus autem, univer&longs;aliter juxta di&longs;tantias cen­<lb/>tri gravitatis à fulcro. </s> <s id="s.003391">Quare tota momentorum Ratio ea e&longs;t <lb/>quæ componitur ex Rationibus gravitatum re&longs;pondentium <lb/>moli ultrà fulcrum proten&longs;æ, & di&longs;tantiarum centri gravitatis. </s> <lb/> <s id="s.003392">Cum itaque in longiore ligno plus inveniatur gravitatis, & ma­<lb/>gis à fulcro di&longs;tet centrum gravitatis, quàm in breviore ligno, <lb/>nil mirum, &longs;i vis in A po&longs;ita, ut contranitatur momentis lon­<lb/>gioris ligni innixi fulcro C, major e&longs;&longs;e debeat, quàm ut re&longs;i&longs;te­<lb/>ret momentis ligni brevioris. </s> </p> <p type="main"> <s id="s.003393">De&longs;inant igitur mirari, qui &longs;ari&longs;&longs;am decem cubitorum per­<lb/>pendicularem extremo digiti apice &longs;u&longs;tineri, eandem verò ho­<lb/>rizontaliter jacentem non ni&longs;i valido conatu elevari vident. </s> <lb/> <s id="s.003394">Res enim ex dictis per&longs;picua e&longs;t; quia dum ha&longs;ta perpendicu­<lb/>laris digito incumbit, centrum gravitatis rectá deor&longs;um urgens <lb/>digito motum &longs;ibi æqualem præ&longs;cribit, ac proinde vici&longs;&longs;im <pb pagenum="442" xlink:href="017/01/458.jpg"/>æqualis e&longs;t digiti & ha&longs;tæ motus &longs;ur&longs;um, &longs;i digitus &longs;ur&longs;um co­<lb/>netur: hinc e&longs;t &longs;olam Rationem gravitatis comparatæ ad vires <lb/>&longs;u&longs;tinendi attendendam e&longs;&longs;e, ideóque &longs;i &longs;ari&longs;&longs;æ pondus &longs;it ex. </s> <lb/> <s id="s.003395">gr. <!-- REMOVE S-->lib.10, &longs;olo ni&longs;u opus e&longs;t, quo libræ 10 &longs;u&longs;tineantur. </s> <s id="s.003396">Cum <lb/>verò ha&longs;ta obliqua e&longs;t, & horizonti parallela, &longs;ive ad illum in­<lb/>clinata, jam non idem &longs;eu æqualis convenit motus manni <lb/>ha&longs;tam elevanti, atque centro gravitatis, &longs;ed hoc ad motum <lb/>multo majorem incitatur; ac propterea momentorum Ratio <lb/>non ex &longs;olâ gravitate pendet, verùm etiam ex motuum Ratio­<lb/>ne componitur. </s> </p> <p type="main"> <s id="s.003397">Sit ha&longs;ta horizontaliter jacens AI cubitorum 10; pars manu <lb/><figure id="id.017.01.458.1.jpg" xlink:href="017/01/458/1.jpg"/><lb/>apprehen&longs;a &longs;it IC quinta <lb/>fermè pars cubiti adeò ut <lb/>IC ad CA &longs;it ut 1 ad 49: <lb/>punctum I re&longs;pondet ex­<lb/>tremæ parti metacarpij, <lb/>quâ carpo adhæret arti­<lb/>culatio minimi digiti: <lb/>punctum autem C re&longs;pondet &longs;ecundo indicis articulo; motú&longs;­<lb/>que elevationis ha&longs;tæ perficitur deprimendo I & elevando C, <lb/>ac motûs centrum e&longs;t in juncturâ manûs cum o&longs;&longs;e cubiti; quod <lb/>centrum propterea intelligitur re&longs;pondere &longs;ari&longs;&longs;æ ex. </s> <s id="s.003398">gr. <!-- REMOVE S-->in O <lb/>inter C & I. <!-- KEEP S--></s> <s id="s.003399">Quapropter &longs;i facultas in I deprimens con&longs;idere­<lb/>tur, vectis e&longs;t primi generis, &longs;in autem vis in C elevans atten­<lb/>datur, vectis e&longs;t tertij generis; pondus verò movendum e&longs;t &longs;i­<lb/>ve tota gravitas longitudinis OA in centro gravitatis E, &longs;ive <lb/>&longs;emi&longs;&longs;is gravitatis in extremitate A, ut con&longs;tat ex iis, quæ di&longs;­<lb/>putata &longs;unt lib.3. cap.2. de brachiis libræ. </s> </p> <p type="main"> <s id="s.003400">Intelligatur itaque, facilioris explicationis gratiâ, centrum <lb/>motû, in O planè medium inter C & I; eritque tam AO ad <lb/>OC, quàm AO ad OI, ut 99 ad 1. Gravitas igitur partis OA <lb/>e&longs;t lib. (9 9/10) ex hypothe&longs;i; illius &longs;emi&longs;&longs;is e&longs;t lib.(4 19/20); cujus mo­<lb/>mentum in A ad momentum, quod haberet illa eadem in Caut <lb/>in I, e&longs;t ut 99 ad 1. Cùm autem potentia in I deprimens æqui­<lb/>valeat potentiæ elevanti in C, quippe illarum di&longs;tantia ab O <lb/>centro motûs ex hypothe&longs;i e&longs;t æqualis, perinde e&longs;t atque &longs;i in C <lb/>unica potentia totum pondus elevans po&longs;ita e&longs;&longs;et æquivalens <lb/>duplici illi potentiæ in I & in C. <!-- KEEP S--></s> <s id="s.003401">Quare potentia in C elevans <pb pagenum="443" xlink:href="017/01/459.jpg"/>pondus perpendiculare lib.(9 9/10) ad potentiam in C pariter con&longs;ti­<lb/>tutam elevantem lib.(4 10/20) in di&longs;tantia, quæ exigat motum unde<lb/>centuplum erit ut (9 9/10) ad 490, hoc e&longs;t, tàm valida e&longs;&longs;e debet, ut <lb/>po&longs;&longs;et perpendiculariter elevare libras 490. </s> </p> <p type="main"> <s id="s.003402">Porrò elevatâ ha&longs;tâ ita ut A veniat in F, jam non intelligi­<lb/>tur &longs;emi&longs;&longs;is gravitatis in A, &longs;ed in G puncto, quod definitur à <lb/>perpendiculari cadente ex F in horizontalem: & idcirco gra­<lb/>vitas (4 19/20) ducenda e&longs;t in di&longs;tantiam GO minorem quàm AO; <lb/>atque ita deinceps minuitur, u&longs;que dum ha&longs;ta fiat in O hori­<lb/>zonti perpendicularis, & facillimè &longs;u&longs;tineatur, aut attollatur. </s> <s id="s.003403">Si <lb/>autem in hac ratiocinatione tibi, Lector, placuerit non negli­<lb/>gere momentum illud exiguum, quod potentiæ elevanti addi­<lb/>tur à gravitate particulæ OI, non abnuo, &longs;i operæ pretium te <lb/>facturum exi&longs;times. </s> </p> <p type="main"> <s id="s.003404">Quòd &longs;i punctum I concipiatur omnino immotum, illud e&longs;t <lb/>centrum motûs, & vis elevans in C aliam habet Rationem; <lb/>nam potentiæ motus ad motum &longs;emi&longs;&longs;is ponderis ha&longs;tæ in A e&longs;t <lb/>ut 1 ad 50; &longs;unt igitur lib.5 ex hypothe&longs;i, quæ moventur motu <lb/>quinquagecuplo; ac propterea vis elevandi datam ha&longs;tam po&longs;i­<lb/>ta in C, quando ha&longs;ta e&longs;t horizonti parallela, ea e&longs;&longs;e debet, quæ <lb/>po&longs;&longs;it elevare libras 250 perpendiculares. </s> <s id="s.003405">Hinc e&longs;t quod, &longs;i <lb/>ha&longs;tam eandem lib.10. humero ita imponas in C, ut apprehen­<lb/>&longs;um calcem in I manus retineat, & CI &longs;it pars decima totius <lb/>longitudinis ha&longs;tæ parallelæ horizonti, &longs;emi&longs;&longs;is (&longs;cilicet lib.4 1/2) <lb/>reliquæ ha&longs;tæ ultra humerum intelligitur in A, & ut IC ad <lb/>CA, hoc e&longs;t ut 1 ad 9, ita lib.4 1/2 ad lib.40 1/2, quibus æquiva­<lb/>lere debet partis CI momentum & vis manûs deor&longs;um urgen­<lb/>tis, atque in I retinentis ha&longs;tam horizonti parallelam. </s> <s id="s.003406">Perinde <lb/>itaque humerus in C premitur ab ha&longs;tâ &longs;ic po&longs;itâ, & à manu <lb/>deor&longs;um urgente, atque &longs;i ponderis librarum 81 centrum gra­<lb/>vitatis immineret humero; nam &longs;i loco manûs deor&longs;um trahen­<lb/>tis adderes in I pondus faciens æquilibrium, e&longs;&longs;e oporteret <lb/>lib.40; &longs;iquidem partis CI momentum e&longs;t lib. 1/2 in I. <!-- KEEP S--></s> <s id="s.003407">At &longs;i ex­<lb/>tremitas I retineatur quidem, &longs;ed nemine deor&longs;um urgente <lb/>(quemadmodum &longs;i in parietis foramen inferatur, & à &longs;uperio­<lb/>re foraminis &longs;axo impediatur, ne po&longs;&longs;it elevari) in C verò &longs;u&longs;ti­<lb/>neatur ab humero; tunc humeri pre&longs;&longs;io &longs;oli gravitati ha&longs;tæ tri-<pb pagenum="444" xlink:href="017/01/460.jpg"/>buenda e&longs;t; ha&longs;ta quippe e&longs;t vectis &longs;ecundi generis hypomo­<lb/>chlium in I habens, pondus movendum, hoc e&longs;t, humerum <lb/>premendum in C, potentiam verò, hoc e&longs;t lib.5. &longs;emi&longs;&longs;em gra­<lb/>vitatis ha&longs;tæ, in A, ita ut AI di&longs;tantia &longs;it decupla di&longs;tantiæ CI: <lb/>premitur ergo humerus, qua&longs;i &longs;u&longs;tineat libras 50. </s> </p> <p type="main"> <s id="s.003408">Demum, ne intacta relinquatur Ari&longs;totelis quæ&longs;tio 14. de <lb/>ligno, quod terræ applicatum pede impo&longs;ito faciliùs frangitur <lb/>manu longè diducta quàm prope, dic ligni partem, quæ inter <lb/>pedem impo&longs;itum, & terram &longs;ubjectam interjicitur, e&longs;&longs;e pror­<lb/>&longs;us &longs;imilem parti pri&longs;matis infixi parieti, ne moveatur, manum <lb/>verò e&longs;&longs;e potentiam, quæ longiùs applicata majora habet mo­<lb/>menta ad vincendum nexum particularum ligni; e&longs;t enim lon­<lb/>gior vectis. </s> <s id="s.003409">Similiter applicato ad genu ligno, & æquè di­<lb/>ductis manibus; duo &longs;unt vectes hinc atque hinc, fulcrum ad <lb/>genu, &longs;cilicet ad duo puncta contactuum, habentes, eóque <lb/>longiores, quò magis diductæ fuerint manus, ac proinde faci­<lb/>liùs di&longs;trahentes particulas extimas ligni, quod circa genu <lb/>curvatur, faciliú&longs;que comprimentes particulas eju&longs;dem ligni <lb/>ad cavam faciem pertinentes; quæ dum &longs;ibi vici&longs;&longs;im ob&longs;i&longs;tunt, <lb/>uberiorem reliquarum di&longs;tractionem juvant: longiorem autem <lb/>vectem præ breviori eligendum e&longs;&longs;e quis ne&longs;ciat? </s> <s id="s.003410">ac propterea <lb/>&longs;i ad genu propiùs admoverentur manus ligno, cùm minor e&longs;­<lb/>&longs;et illarum motûs Ratio ad motum particularum ligni di&longs;trahen­<lb/>darum, quàm &longs;it Ratio motûs illarum longiùs diductarum, uti­<lb/>que difficiliùs frangeretur lignum; ideóque longiùs diducun­<lb/>tur manus, ut longiores &longs;int vectes. <lb/></s> </p> <p type="main"> <s id="s.003411"><emph type="center"/>CAPUT XII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003412"><emph type="center"/><emph type="italics"/>Vnde oriantur forcipum & forficum vires.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003413">FOrcipum duplex e&longs;t u&longs;us; primus quidem ad corpus ali­<lb/>quod firmiter apprehendendum, &longs;ecundus verò ad evel­<lb/>lendum illud faciliùs, vel &longs;uâ è &longs;ede dimovendum; id quod <lb/>adhibito huju&longs;modi in&longs;trumento faciliùs perficitur, quàm nudâ <pb pagenum="445" xlink:href="017/01/461.jpg"/>manu. </s> <s id="s.003414">Hinc Ari&longs;toteles mechan. </s> <s id="s.003415">quæ&longs;t. </s> <s id="s.003416">21. quærit, <emph type="italics"/>Cur medi­<lb/>ci faciliùs dentes extrahunt denti forcipis onere adjecto, quàm &longs;i &longs;o­<lb/>lâ utantur manu?<emph.end type="italics"/> Quia nimirum infixum mandibulæ dentem <lb/>extrahendum vix &longs;ummis duobus digitis, quibus non multa vis <lb/>ine&longs;t, arripere valent, & ob carnis mollitudinem facilè è tra­<lb/>hentibus digitis elabitur lubricus dens: at forcipulam in os im­<lb/>mittere potiùs, quàm digitos, &longs;æpe facilius e&longs;t, validiû&longs;que <lb/>trahit manus in pugnum con&longs;tricta forcipi dentem per vim <lb/>educenti applicata, quàm digitorum extremitates dentem <lb/>adhuc in gingivâ hærentem evellere valeant. </s> <s id="s.003417">Præterquam <lb/>quod in dentiforcipe, cuju&longs;cumque tandem figuræ &longs;it, ratio <lb/>vectis intercedit ad dentem firmiùs apprehendendum, dum <lb/>pre&longs;&longs;o manubrio arctiùs con&longs;tringitur: nec facilè Chirurgus <lb/>operam ludit, ubi dens forcipem &longs;ubterfugere nullatenus po­<lb/>te&longs;t. </s> <s id="s.003418">Totam igitur vectis vim in dentiforcipe agno&longs;co ad &longs;trin­<lb/>gendum dentem, ut medica manus illum faciliùs evellat: ne­<lb/>que enim eâdem ratione à medicis (ni&longs;i fortè veterinariis) ex­<lb/>trahuntur dentes, quâ fabri lignarij revellunt infixos tabulæ <lb/>clavos, de quibus mox erit &longs;ermo. </s> </p> <p type="main"> <s id="s.003419">Similiter quia ad &longs;tringendam exilem aliquam materiam <lb/>inepta e&longs;&longs;et digitorum cra&longs;&longs;itudo, minutorum opu&longs;culorum fa­<lb/>bricatores forcipulis utuntur, quibus illam apprehendentes fir­<lb/>miter, aut limæ &longs;ubjiciunt, aut opportunè collocant. </s> <s id="s.003420">Et quia <lb/>candens ferrum manu tractari nequit, ut in quamcumque par­<lb/>tem ver&longs;etur, incudique impo&longs;itum ni&longs;i retineretur, &longs;æpè &longs;e­<lb/>cundis aut tertiis malleorum ictibus &longs;e &longs;ubduceret, propterea <lb/>fabri ferrarij forcipes adhibent, quarum author & inventor <lb/>Cinyra Cyprius Agriopæ filius &longs;cribitur à Plinio lib. 7 cap. 56; <lb/>ideóque forcipes, qua&longs;i forvicapes, dictæ &longs;unt, quòd iis for­<lb/>va, ide&longs;t calida, capiantur. </s> </p> <p type="main"> <s id="s.003421">Vis autem forcipum in eo &longs;ita e&longs;t, quòd duo vectes primi <lb/>generis AB, & CD in E connexi <lb/><figure id="id.017.01.461.1.jpg" xlink:href="017/01/461/1.jpg"/><lb/>commune hypomochlium E habent; <lb/>potentia verò in B & D longiorum <lb/>brachiorum extremitates adducens eò <lb/>validius &longs;tringit ferrum brevioribus brachiis EA & EC ap­<lb/>prehen&longs;um, quo major fuerit Ratio BE ad EA: támque firma <lb/>retentio e&longs;&longs;e pote&longs;t, ut modico pueri conatu extremitates B & D <pb pagenum="446" xlink:href="017/01/462.jpg"/>vici&longs;&longs;im comprimentis, robu&longs;ti&longs;&longs;imi cuju&longs;que vires elidantur, <lb/>ne arreptum ferrum ex AC po&longs;&longs;it eximere. </s> <s id="s.003422">Quòd &longs;i forcipibus <lb/>BA & DC utatur aliquis veterinarius vice Po&longs;tomidis (&longs;eu, ut <lb/>aliquibus Grammaticis placet Pa&longs;tomidis) equi nares, ad frænan­<lb/>dam ejus tenaciam, ut loquitur Fe&longs;tus, inter longiora brachia <lb/>BE & DE contingens; jam BE & DE vectes &longs;unt &longs;ecundi ge­<lb/>neris, cum illud, quod ponderis vicem &longs;ubit, inter hypomo­<lb/>chlium & potentiam interjiciatur. </s> </p> <p type="main"> <s id="s.003423">Huju&longs;modi forcipibus vectis in EC & EA non ab&longs;imile fui&longs;&longs;e <lb/>exi&longs;timo in&longs;trumentum antiquioribus Græcis ad frangendas <lb/>ab&longs;que ictu percutientis mallei nuces familiare, ut ex Ari&longs;tote­<lb/>le Mechan. quæ&longs;t. </s> <s id="s.003424">22. colligitur: quod forta&longs;&longs;e vel in alterutrâ, <lb/>vel in utraque interiori facie breviorum brachiorum modicè <lb/>excavatá frangendæ nuci locum de&longs;ignabat; adducto enim in <lb/>oppo&longs;itas partes utroque vecte BA & DC, quo propior erat <lb/>nux communi hypomochlio, puncto &longs;cilicet connexionis E, eò <lb/>faciliùs frangebatur, quia eò major erat Ratio motûs poten­<lb/>tiæ ad motum particularum nucis ex compre&longs;&longs;ione dividenda­<lb/>rum, quàm e&longs;&longs;et Ratio re&longs;i&longs;tentiæ ex earumdem particularum <lb/>complexione ortæ, ad vim motivam potentiæ. </s> <s id="s.003425">Et quoniam in <lb/>nucum mentionem incidi, ne levitati mihi tribuas, quòd hîc <lb/>puerile inventum à me puero, & tunc quidem admiratione <lb/>ob&longs;tupefacto, ob&longs;ervatum commemorare non erube&longs;cam. </s> <s id="s.003426">Vide­<lb/>bam pueros clande&longs;tinis jentaculis indulgentes, ut citra multi­<lb/>plicis percu&longs;&longs;ionis &longs;trepitum nuces confringerent, eas inter <lb/>po&longs;tium angulos & fores collocare; tum adductis foribus levi&longs;­<lb/>&longs;imo negotio unâ operâ confringere. </s> <s id="s.003427">Erat &longs;cilicet vectis primi <lb/>generis, cuius majorem longitudinem definiebat foris latitudo, <lb/>minorem ip&longs;ius foris cra&longs;&longs;itudo, ita ut vectis e&longs;&longs;et in angulum <lb/>inflexus, cujus hypomochlium cardinibus re&longs;pondebant. </s> <lb/> <s id="s.003428">U&longs;que adeò natura ip&longs;a Mechanicen, u&longs;umque vectis, vel pue­<lb/>ros docet. </s> </p> <p type="main"> <s id="s.003429">His adde acutas forcipulas, quibus catenularum fabricatores <lb/>extremitatem fili ferrei inflectunt: ratio enim vectis poti&longs;&longs;i­<lb/>mùm con&longs;i&longs;tit in validâ & firmâ ip&longs;ius fili ferrei apprehen&longs;io­<lb/>ne; nam quo ad eju&longs;dem inflexionem &longs;pectat, non e&longs;t, cur nos <lb/>torqueamus, ut aliquam demum vectis umbram venemur: &longs;a­<lb/>tis e&longs;t, &longs;i manubrij amplitudinem con&longs;iderantes, eámque cum <pb pagenum="447" xlink:href="017/01/463.jpg"/>tenui apice forcipulæ, circa quem filum ferreum contorquetur, <lb/>comparantes motum potentiæ manubrio applicatæ longè ma­<lb/>jorem motu particularum fili ferrei, quod flectitur, deprehen­<lb/>damus; hinc quippe aucta potentiæ momenta cogno&longs;cimus. </s> </p> <p type="main"> <s id="s.003430">Aliud forcipum genus frequentiùs u&longs;urpatur, quarum poti&longs;­<lb/>&longs;imus u&longs;us e&longs;t in eximendis clavis, <lb/><figure id="id.017.01.463.1.jpg" xlink:href="017/01/463/1.jpg"/><lb/>& minora brachia AE & CE non <lb/>recta &longs;unt, &longs;ed curva; non &longs;olùm ut <lb/>clavus tenaciùs apprehendatur ex­<lb/>cepto ejus capite intra forcipum &longs;i­<lb/>num, verùm etiam ut forcipes aliam <lb/>exerceant vectis curvi rationem: cùm enim arrepto inter A & <lb/>C clauo inclinantur forcipes, ut punctum H tangat &longs;ubjectum <lb/>planum, &longs;ive paries &longs;it, &longs;ive tabula, jam hypomochlium e&longs;t <lb/>in H, & momenta potentiæ in B ad re&longs;i&longs;tentiam clavi evellen­<lb/>di, &longs;unt ut BH ad HA, cùm circa punctum H perficiatur <lb/>motus. </s> <s id="s.003431">Quare ad con&longs;tringendum clavum momentorum Ra­<lb/>tio e&longs;t ut BE ad EA (perinde atque &longs;i ab E ad A ducta e&longs;&longs;et <lb/>recta linea) ad revellendum verò momentorum Ratio e&longs;t ut <lb/>recta ex B ad H ducta ad rectam, quæ ex H ad A ducitur; ne­<lb/>que enim curva linea ex H ad B, aut ex H ad A, &longs;ed recta le­<lb/>gem con&longs;tituit motibus potentiæ in B, & ponderis in A. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003432">Id quod pariter contingit cùm aver&longs;am mallei partem &longs;ubti­<lb/>liorem clavo &longs;ubmittimus, & in oppo&longs;itam partem manu­<lb/>brium retrahimus, ut clavus extrahatur: e&longs;t &longs;iquidem curvus <lb/>quidam vectis fulcrum habens in E, circa <lb/><figure id="id.017.01.463.2.jpg" xlink:href="017/01/463/2.jpg"/><lb/>quod punctum manens uterque motus perfi­<lb/>citur; & motus potentiæ in H ad motum <lb/>clavi in I habet Rationem rectæ HE ad <lb/>rectam EI. <!-- KEEP S--></s> <s id="s.003433">Ex quo patet pro majori manu­<lb/>brij longitudine augeri etiam potentiæ mo­<lb/>menta. </s> </p> <p type="main"> <s id="s.003434">Quoniam verò aliquando forcipes huju&longs;­<lb/>modi curvæ aciem habent in A & C, ut id, <lb/>quod con&longs;tringitur vehementiùs, etiam &longs;cin­<lb/>datur, non e&longs;t alia philo&longs;ophandi ratio, quod quidem &longs;pectat <lb/>ad momenta potentiæ duplici illi vecti applicatæ, hoc uno dif­<lb/>ferunt, quod vis &longs;cindendi orta ex acie ferri pertinet ad ratio-<pb pagenum="448" xlink:href="017/01/464.jpg"/>nes Cunei, de quo inferiùs &longs;uo loco. </s> <s id="s.003435">Idem dicendum de for­<lb/>ficibus, quarum acies pariter ex rationibus Cunei vim &longs;cin­<lb/>dendi habent; majora autem momenta potentiæ, quæ faci­<lb/>liùs &longs;cindat, petenda &longs;unt ex rationibus vectis; &longs;unt enim <lb/>hîc pariter duo vectes in oppo&longs;itas partes commoti, commu­<lb/>ne hypomochlium in puncto connexionis habentes; & quo <lb/>majorem Rationem manubriorum longitudo habet ad di&longs;tan­<lb/>tiam rei &longs;cindendæ à puncto connexionis, eò etiam facilior <lb/>contingit &longs;ci&longs;&longs;io. </s> <s id="s.003436">Idcirco quæ duriora &longs;unt, prope connexio­<lb/>nis punctum applicantur, quia eadem manubriorum longi­<lb/>tudo ad minorem di&longs;tantiam habet majorem Rationem quàm <lb/>ad di&longs;tantiam majorem; & quæ ad hæc duriora &longs;cindenda <lb/>in&longs;titutæ &longs;unt forfices, breviora habent brachia, quæ ad <lb/>&longs;cindendum exacuuntur, longiora verò ea, quibus poten­<lb/>tia movens applicatur; cuju&longs;modi &longs;unt forfices, quibus fa­<lb/>bri ferrarij ad æreas aut ferreas laminas &longs;cindendas utuntur. </s> <lb/> <s id="s.003437">In harum u&longs;u illud etiam ob&longs;ervare poteris, &longs;atis e&longs;&longs;e, &longs;i <lb/>duorum vectium communi fulcro connexorum, ita ut de­<lb/>cu&longs;&longs;ati exi&longs;tant, alterum moveatur manente altero: hoc <lb/>enim poti&longs;&longs;imum attenditur, quo pacto potentia validiùs <lb/>applicetur, ubi multâ opus e&longs;t virtute; cum autem uni­<lb/>cum huju&longs;modi forficum brachium movetur, tota illi ma­<lb/>nus applicatur, & reliquo deor&longs;um connitente corpore va­<lb/>lidè premit. <lb/></s> </p> <p type="main"> <s id="s.003438"><emph type="center"/>CAPUT XIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003439"><emph type="center"/><emph type="italics"/>Cur Tollenones juxta puteos con&longs;tituantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003440">QUi Tollenones Latinis (Ciconias aliqui vocant) Græcis <lb/><foreign lang="greek">*kelo/nia</foreign> dicuntur, familiaria ru&longs;ticis & olitoribus in&longs;tru­<lb/>menta ad hauriendas ex puteis non admodum altis aquas, ali­<lb/>qua habent explicatu digna, quæ ex Vectis doctrinâ petenda <lb/>&longs;unt, nec vi&longs;um e&longs;t Ari&longs;toteli quæ&longs;tione 28. hanc eandem di&longs;­<lb/>putationem in&longs;tituere indecorum, aut homini Philo&longs;opho mi-<pb pagenum="449" xlink:href="017/01/465.jpg"/>nùs conveniens. </s> <s id="s.003441">Et primùm quidem ip&longs;a Tollenonis con­<lb/>&longs;tructio pendet ex rationibus vectis primi generis, habet &longs;i qui­<lb/>dem fulcrum medium inter potentiam moventem & pondus <lb/><figure id="id.017.01.465.1.jpg" xlink:href="017/01/465/1.jpg"/><lb/>elevatum. </s> <s id="s.003442">Erecto enim tigno DA imponitur tran&longs;ver&longs;a ha&longs;ta <lb/>CE, infigitúrque axi in A, circa quem liberè converti po&longs;&longs;it. </s> <lb/> <s id="s.003443">Tum extremitati E puteo appo&longs;itæ alligatur plumbum, aut &longs;a­<lb/>xum, &longs;ive grave aliud quodpiam; ab extremitate autem C, <lb/>quæ puteo re&longs;pondet, funis CF pendet (&longs;eu ha&longs;ta fune con­<lb/>vexa in C, &longs;ed tamen facilè mobilis) cui in F &longs;itula adnectitur. </s> </p> <p type="main"> <s id="s.003444">Jam, verò duplex motus in hauriendâ aquá con&longs;iderandus <lb/>e&longs;t, alter, quo hydria vacua in puteum demittitur, alter, quo <lb/>eadem hydria aquæ plena è puteo extrahitur. </s> <s id="s.003445">Priori motui <lb/>utique non favet tolleno, faciliùs quippe hydria de&longs;cende­<lb/>ret, &longs;i nullum e&longs;&longs;et onus in E, quod depre&longs;sâ hydria e&longs;&longs;et <lb/>elevandum; hujus enim gravitas major e&longs;t hydriæ gravita­<lb/>te; ac propterea præter eju&longs;dem hydriæ gravitatem alia po­<lb/>tentia deprimens requiritur in F, ut major &longs;it Ratio gra­<lb/>vitatis & potentiæ in F ad gravitatem ponderis in E, quàm <lb/>&longs;it reciprocè Ratio di&longs;tantiæ AE ad di&longs;tantiam AC. <!-- KEEP S--></s> <s id="s.003446">Quare <lb/>&longs;i AC longitudo multo major &longs;it longitudine AE, facilior <pb pagenum="450" xlink:href="017/01/466.jpg"/>erit hydriæ vacuæ depre&longs;&longs;io; contra verò deprimendi difficul­<lb/>tas augebitur, quo magis pondus E di&longs;tabit à fulcro A. <!-- KEEP S--></s> <lb/> <s id="s.003447">Sed hæc eadem, quæ deprimendi difficultatem augent, ju­<lb/>vant ad extrahendam faciliùs hydriam: pondus enim E quò <lb/>longiùs aberit à fulcro A, eò plura habebit momenta adversùs <lb/>gravitatem aquæ & hydriam pendentes ex C. <!-- KEEP S--></s> <s id="s.003448">Hinc e&longs;t poten­<lb/>tiæ atque ponderis vices permutari; in depre&longs;&longs;ione nimirum <lb/>pondus in E exi&longs;tens attollitur, & potentia in C de&longs;cendit; at <lb/>in elevatione vici&longs;&longs;im pondus elevatur in C, & potentia in E <lb/>de&longs;cendit. </s> <s id="s.003449">Prudenter itaque providere oporteÏ„, ut & ha&longs;tæ <lb/>CE longitudo opportunè di&longs;tinguatur in partes CA, AE, & <lb/>pondus in E neque ita leve &longs;it, ut parum adjumenti afferat in <lb/>extrahendâ aquâ, neque ita grave, ut detrimento &longs;it in depri­<lb/>mendâ hydriâ. </s> <s id="s.003450">Præ&longs;tat tamen plus aliquid laboris &longs;u&longs;cipere in <lb/>deprimenda hydriâ, ut ea deinde elevetur majore compen­<lb/>dio: nemo quippe dubitat, quin longè faciliùs &longs;it homini <lb/>funem FC deor&longs;um trahenti attollere pondus E, quàm pa­<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 &longs;e ru&longs;ticis <lb/>ridendum præbeat, ubi pro altitudine putei a&longs;&longs;umpto fune CF, <lb/>ha&longs;tam CE æquo longiorem con&longs;tituerit præter rationem inter­<lb/>valli inter tigillum DA & puteum; contingeret enim, ut <lb/>ha&longs;ta in putei labra incurrens nece&longs;&longs;ariam funis longitudinem <lb/>minueret. </s> <s id="s.003452">Quapropter tria hæc nece&longs;&longs;e e&longs;t &longs;ibi invicem pro­<lb/>portione re&longs;pondere, videlicet ha&longs;tæ CE longitudinem, tigilli <lb/>AD altitudinem, ejú&longs;que à puteo di&longs;tantiam; ut erecta ferè ad <lb/>perpendiculum ha&longs;ta eam admittat funis longitudinem, quæ <lb/>& facile hydriæ jungi po&longs;&longs;it, & putei altitudinem exæquet. </s> </p> <p type="main"> <s id="s.003453">Cur autem tùm in deprimendo Tollenone, ut hydria im­<lb/>mergatur, tùm in attollendo, ut aqua è puteo eximatur, non <lb/>parem &longs;emper & æquabilem experiamur facilitatem, ratio in <lb/>promptu e&longs;t; quia &longs;cilicet varia e&longs;t potentiæ medio fune FC <lb/>tollenonem agitantis applicatio; quo enim acutior fuerit angu­<lb/>lus FCA, eò minora &longs;unt potentiæ trahentis momenta, quæ <lb/>cre&longs;cente angulo pariter augentur, ut tunc maxima &longs;int, cùm <lb/>funis FC, & ha&longs;ta CA angulum rectum con&longs;tituerint. </s> <s id="s.003454">Et qui­<lb/>dem licèt, ubi funis ab angulo recto ad obtu&longs;um de&longs;ci erit, ite­<lb/>rum momenta potentiæ decre&longs;cant, &longs;i applicationis potentiæ <pb pagenum="451" xlink:href="017/01/467.jpg"/>eju&longs;dem tantummodo habeatur ratio; fieri tamen pote&longs;t, ut <lb/>ponderis in E momenta minuantur, quo altiùs attollitur, &longs;i il­<lb/>lud fuerit ha&longs;tæ impo&longs;itum, cum eju&longs;dem linea directionis ca­<lb/>dat in ha&longs;tæ punctum, quod magis ad fulcrum A accedat, jux­<lb/>ta ea, quæ hujus libri cap. 3. dicta &longs;unt; atque adeò deprimendi <lb/>facilitas, quæ hinc &longs;umit incrementum, diminutâ ponderis E re­<lb/>&longs;i&longs;tentiâ &longs;uppleat decrementum, quod obliquam potentiæ ap­<lb/>plicationem con&longs;equitur. </s> </p> <p type="main"> <s id="s.003455">Nec ab&longs;imilis momentorum varietas contingit ex di&longs;parili <lb/>angulorum amplitudine, quos lineæ directionis gravitatum <lb/>tum aquæ attollendæ, tum ponderis E, cum ha&longs;tâ CE con&longs;ti­<lb/>tuunt. </s> <s id="s.003456">Nam depre&longs;sâ ha&longs;tâ, & pondere maximè elevato, hu­<lb/>jus momenta initio minora &longs;unt, & &longs;ubinde augentur receden­<lb/>te à fulcro A lineâ directionis centri gravitatis, &longs;i illud quidem <lb/>ha&longs;tæ incumbat: Pondere igitur E minùs conante adversùs <lb/>aquam cum hydriâ attollendam, plus laborandum e&longs;t homini <lb/>funem &longs;ur&longs;um trahenti; cujus deinde labor minuitur auctis gra­<lb/>vitatis E momentis; & tunc poti&longs;&longs;imùm præ&longs;tare videntur, cum <lb/>angulus FCA ex recto in acutum tran&longs;it; tunc enim aquæ de­<lb/>or&longs;um connitentis ac oppo&longs;ito ponderi re&longs;i&longs;tentis momenta de­<lb/>cre&longs;cere incipiunt, ac infirmiora fieri. </s> </p> <p type="main"> <s id="s.003457">Ex his non parum lucis affulget &longs;cenicis machinationibus, <lb/>in quibus non planè ad perpendiculum, &longs;ed obliquè a&longs;cenden­<lb/>dum e&longs;t aut de&longs;cendendum, &longs;i enim &longs;tatuatur vectis ZX ha­<lb/>bens in P fulcrum, & fune <lb/><figure id="id.017.01.467.1.jpg" xlink:href="017/01/467/1.jpg"/><lb/>ZN pendeat corpus demit­<lb/>tendum, utique obliquus erit <lb/>de&longs;cen&longs;us ex N in M, & vici&longs;­<lb/>&longs;im obliquus a&longs;cen&longs;us ex M in <lb/>N: momenta autem ponderis <lb/>X, aut S, pro variâ po&longs;itione, <lb/>ut dictum e&longs;t, di&longs;&longs;imilia atque <lb/>di&longs;paria &longs;unt: Quapropter <lb/>temperanda &longs;unt pro motûs <lb/>in&longs;tituendi opportunitate; atque &longs;i pondus X levius &longs;it corpore <lb/>demittendo ex N, hoc &longs;ponte de&longs;cendet; &longs;i verò in S augea­<lb/>tur pondus, ut corporis in M gravitatem &longs;uperet, hoc ex M <lb/>in N elevabitur. </s> <s id="s.003458">Quod autem de &longs;cenicâ machinatione hîc <pb pagenum="452" xlink:href="017/01/468.jpg"/>innui, ad alias motiones corporum elevandorum (ut &longs;i ex navi <lb/>in altiorem fluminis ripam onus transferendum e&longs;&longs;et) facilè <lb/>traduci po&longs;&longs;e ita manife&longs;tum e&longs;t, ut pluribus non &longs;it opus, &longs;i <lb/>accuratè examinetur altitudo, ad quam deducendum e&longs;t, & <lb/>amplitudo &longs;eu di&longs;tantia parallelarum, intra quas obliquus mo­<lb/>tus perficiendus e&longs;t, ut vecti congrua longitudo &longs;tatuatur, & <lb/>opportuno loco collocetur, ubi eam anguli RPZ inclinatio­<lb/>nem habeat, cui Sinus Ver&longs;us RN re&longs;pondeat. <lb/></s> </p> <p type="main"> <s id="s.003459"><emph type="center"/>CAPUT XIV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003460"><emph type="center"/><emph type="italics"/>Remorum vires in agendâ navi expenduntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003461">REmum, quo naves aguntur, Copen&longs;ibus, & Platæen&longs;ibus <lb/>debemus, ut Plinius lib. 7 cap. 56. &longs;cribens ait, <emph type="italics"/>Remum <lb/>Copæ, latitudinem ejus Platææ<emph.end type="italics"/> (utraque e&longs;t Bœotiæ urbs) inve­<lb/>nerunt; & nomen ip&longs;um ab inventoribus inditum videtur, nam <lb/>Græcis <foreign lang="greek">xw/ph</foreign> Remus, <foreign lang="greek">pla/th</foreign> Palmula, latior &longs;cilicet remi pars <lb/>dicitur. </s> <s id="s.003462">Ratem &longs;iquidem conto propellere rudis adhuc ars nau­<lb/>tica noverat, ubi fluminis non admodum alti fundum perticâ <lb/>pertentare licebat; at ubi uberior unda prohibet, ne fundum <lb/><figure id="id.017.01.468.1.jpg" xlink:href="017/01/468/1.jpg"/><lb/>attingatur, operam luderet, qui navim <lb/>conto AB impellere &longs;e po&longs;&longs;e &longs;ibi per­<lb/>&longs;uaderet, ni&longs;i fortè extremitati B <expan abbr="ligneã">ligneam</expan> <lb/>tabellam adjungeret, ita levem, ut <lb/>&longs;ponte &longs;uâ innataret; illam enim per <lb/>vim velociter immergenti obliquam, <lb/>aqua re&longs;i&longs;teret, & navis aliquantulum <lb/>promoveretur: cæterum ingens e&longs;&longs;et <lb/>labor in conto retrahendo, & tabellâ ex <lb/>aquis extrahendâ, etiam&longs;i &longs;calmo tigil­<lb/>lus longiu&longs;culus EF ad perpendiculum <lb/>infigeretur, ex cujus &longs;ummo vertice <lb/>funis EI penderet, fune autem contus <lb/>medius in I &longs;u&longs;penderetur. </s> <s id="s.003463">Quare opus fuit in&longs;trumentum <pb pagenum="453" xlink:href="017/01/469.jpg"/>moliri, quo & facile uteremur, & aliquod laboris compen­<lb/>dium inveniremus. </s> </p> <p type="main"> <s id="s.003464">Ratione &longs;uæ longitudinis ad unum aliquod vectis genus re­<lb/>ferendus e&longs;t remus, ad cujus caput applicatur potentia, videli­<lb/>cet remex; extrema palmula immergitur aquæ, & circa me­<lb/>dium innititur &longs;calmo: &longs;ed aquæ ne? </s> <s id="s.003465">an &longs;calmo? </s> <s id="s.003466">ratio fulcri <lb/>conveniat, di&longs;putatur. </s> <s id="s.003467">Si Ari&longs;toteles audiendus e&longs;&longs;et mechan. </s> <lb/> <s id="s.003468">quæ&longs;t. </s> <s id="s.003469">4. <emph type="italics"/>hypomochlion fit &longs;calmus, &longs;tat enim ille, pondus verò mare <lb/>e&longs;t, quod propellit remus, vectem autem movens ip&longs;e e&longs;t remex.<emph.end type="italics"/></s> <s id="s.003470"> Id <lb/>quidem verum e&longs;&longs;et, &longs;i quis anchoris nondum &longs;olutis, & &longs;tante <lb/>navi, adumbratâ ad &longs;peciem remigatione &longs;e exerceret; nil enim <lb/>præ&longs;taret præter aquarum impul&longs;ionem. </s> <s id="s.003471">Cæterùm nautæ re­<lb/>morum pul&longs;u non aquam verberare, &longs;ed navim impellere con­<lb/>tendunt. </s> <s id="s.003472">Igitur aqua, cui remi palmula immergitur, divi&longs;io­<lb/>ni re&longs;i&longs;tens, atque impediens motum palmulæ, hypomochlij, <lb/>cui vectis, hoc e&longs;t remus, innititur, rationem habet; navis ve­<lb/>rò ip&longs;a, quæ promovetur, quatenus e&longs;t &longs;calmo conjuncta, uti­<lb/>que e&longs;t pondus, ex cujus movendi, non ex aquæ repellendæ <lb/>difficultate æ&longs;timandus e&longs;t nautarum labor: alioquin eodem <lb/>remo, qui &longs;calmo &longs;imiliter in&longs;i&longs;teret, æqualis labor e&longs;&longs;et, &longs;ive <lb/>actuarium, &longs;ive corbitam impellere oporteat; pari &longs;iquidem <lb/>aquæ occurrit utrobique palmula. </s> <s id="s.003473">Manife&longs;tum e&longs;t igitur pon­<lb/>dus vecte promovendum navim e&longs;&longs;e, non aquam, ac propterea <lb/>hypomochlij vices aquam &longs;ubire, adeóque remum cen&longs;endum <lb/>e&longs;&longs;e vectem &longs;ecundi generis, cujus extremitates potentia & <lb/>fulcrum occupant. </s> </p> <p type="main"> <s id="s.003474">Hinc e&longs;t aliquod &longs;emper haberi laboris compendium, pon­<lb/>deris enim motus, qui vecte perficitur, minor e&longs;t motu poten­<lb/>tiæ remi capiti applicatæ, illud enim minùs, hæc magis ab hy­<lb/>pomochlio di&longs;tat. </s> <s id="s.003475">Motus, inquam, qui vecte perficitur, mi­<lb/>nor e&longs;t motu potentiæ; fieri enim contingit, ut vi impre&longs;&longs;i im­<lb/>petûs, etiam ce&longs;&longs;ante remigis impul&longs;ione, navis promoveatur, <lb/>adeò ut pro ratione impetûs multo major &longs;it navis motus, quàm <lb/>potentiæ impellentis. </s> <s id="s.003476">Verum hoc non ex vecte ob idip&longs;um, <lb/>quia vectis e&longs;t, oritur, &longs;ed quia navis innatans aquæ non eam <lb/>invenit à corpore fluido re&longs;i&longs;tentiam, quam cæteroqui ex mu­<lb/>tuo tritu inveniunt pondera corpori &longs;olido in&longs;i&longs;tentia, etiam&longs;i <lb/>vecte horizontaliter moveantur; ac proinde impre&longs;&longs;us impetus, <pb pagenum="454" xlink:href="017/01/470.jpg"/>ce&longs;&longs;ante vi externâ, non &longs;tatim perit. </s> <s id="s.003477">Id autem intelligendum <lb/>e&longs;t, cum in lacu vel tranquillo mari navigatur, cum &longs;cilicet <lb/>aqua &longs;uo cur&longs;u non adver&longs;atur motui navis: nam &longs;i adver&longs;o flu­<lb/>mine promovendum &longs;it navigium, contrarius aquæ impul&longs;us <lb/>impetum à remige impre&longs;&longs;um elidit, fieríque pote&longs;t, ut utilius <lb/>accidat navim trahere, quàm remigando impellere, ne &longs;ubla­<lb/>tis ex aquâ remis navigium vi aquæ fluentis retro-actum eò re­<lb/>deat, unde di&longs;ce&longs;&longs;it, & alternâ remorum immer&longs;ione atque ex­<lb/>tractione opera ludatur: præterquam quod quò magis immer&longs;a <lb/>remi palmula ab adver&longs;o flumine repellitur, eò amplius detrahi­<lb/>tur motui navis. </s> <s id="s.003478">Ideò quamvis navim trahens plus laboris &longs;im­<lb/>pliciter impendat, quàm remigans, facit tamen operæ pretium, <lb/>qui enim navim adversùs profluentem trahit, etiam retinet, ne <lb/>retror&longs;um agatur; at qui remo impellit, &longs;ublatâ ex undis pal­<lb/>mulâ, rece&longs;&longs;um impedire non valet. </s> </p> <p type="main"> <s id="s.003479">Cum itaque remus vectis &longs;it &longs;ecundi generis, remigis vires <lb/>æ&longs;timandæ &longs;unt ex Ratione, quam longitudo remi habet ad il­<lb/>lam eju&longs;dem remi partem, quæ aquæ & &longs;calmo interjecta e&longs;t; <lb/>hæc &longs;iquidem Ratio e&longs;t motuum, ac proinde & momentorum, <lb/>ut &longs;æpiùs dictum e&longs;t. </s> <s id="s.003480">Remi autem longitudinem non ab&longs;olutam <lb/>intelligas; &longs;ed primùm ea demenda e&longs;t palmulæ particula, quæ <lb/>aquæ immergitur; quippe quæ aquam repellens qua&longs;i hypomo­<lb/>chlio incumbit. </s> <s id="s.003481">Deinde attendendum e&longs;t, quam remi partem <lb/>remex apprehendat; &longs;i enim plures eundem remum agitent, ut <lb/>in triremibus, non &longs;unt æqualia momenta &longs;ingulorum, &longs;ed ejus, <lb/>qui &longs;calmo propior e&longs;t, minora &longs;unt (perinde atque &longs;i remo <lb/>adeò brevi uteretur) ejus, qui remi caput apprehendit, maxi­<lb/>ma &longs;unt momenta; medij autem medio modo &longs;e habent. </s> </p> <p type="main"> <s id="s.003482">Quare longitudo vectis in remo definitur intervallo, quod in­<lb/>ter aquam, & remigis manum interjectum e&longs;t; ponderis di&longs;tan­<lb/>tiam ab hypomochlio metitur intervallum, quo &longs;calmus ab aquâ <lb/>palmulam excipiente disjungitur. </s> <s id="s.003483">Si igitur intervallum illud <lb/>e&longs;t hujus intervalli duplum aut &longs;e&longs;quialterum, momenta Po­<lb/>tentiæ ad momenta ponderis Rationem habent duplam aut &longs;e&longs;­<lb/>quialteram, & quatuor remiges ad promovendam navim tan­<lb/>tumdem ferè valent ac &longs;ex aut octo, qui pari conatu navim <lb/>eandem &longs;ine remis propellerent, aut traherent. </s> <s id="s.003484">Dixi, <emph type="italics"/>ferè,<emph.end type="italics"/> quia <lb/>cum motus cuju&longs;libet vectis &longs;it circularis circa punctum hypo-<pb pagenum="455" xlink:href="017/01/471.jpg"/>mochlij, remex, qui in dextero navis latere remigat, &longs;ecun­<lb/>dùm vectis naturam arcum de&longs;cribit ad dexteram inclinatum, <lb/>id quod pariter contingit &longs;ini&longs;tro remigi arcum &longs;ini&longs;tror&longs;um <lb/>de&longs;cribenti: Cum autem navis non ni&longs;i unico motu moveri po&longs;­<lb/>&longs;it, ex his duobus circularibus &longs;ibi adver&longs;antibus re&longs;ultat tertius <lb/>medius, &longs;cilicet rectus, qui proinde tantus e&longs;&longs;e non pote&longs;t, <lb/>quantus e&longs;&longs;et, &longs;i &longs;ex aut octo homine, æquali ni&longs;u navim &longs;ine <lb/>remis impellerent aut traherent; quia contrariæ illæ directio­<lb/>nes ad dexteram & ad &longs;ini&longs;tram nequeant in tertiam mixtam <lb/>directionem coale&longs;cere, &longs;inè aliquo impetûs detrimento. </s> </p> <p type="main"> <s id="s.003485">Quod &longs;i remiges omnes non con&longs;entirent in deprimendo, im­<lb/>pellendo, atque extrahendo remo, &longs;ed alij alios præverterent, <lb/>non &longs;olùm id incommodi accideret, quod ab in&longs;tituto itinere de­<lb/>flecteret navis in alterutram partem, ni&longs;i æqualis utrinque e&longs;&longs;et <lb/>impul&longs;us, verùm etiam retardaretur motus, tùm quia minor im­<lb/>pul&longs;us à paucioribus navi imprimitur, tum quia remi tardiores, <lb/>reliquis elevatis, adhuc immer&longs;i dum communi navis motu <lb/>moventur, minùs impellunt aquam po&longs;t &longs;e fugientem, & pal­<lb/>mulæ latitudo occurrenti aquæ obver&longs;a moram infert, ut eam <lb/>dividat; ex quo fit, ut aliquid impetûs ab aliis remigibus im­<lb/>pre&longs;&longs;i deteratur, qui citra hoc impedimentum adhuc per&longs;evera­<lb/>ret. </s> <s id="s.003486">Sunt &longs;cilicet plures remi plures vectes, quibus idem pon­<lb/>dus movetur; & ni&longs;i remiges omnes con&longs;piraverint, aut navis <lb/>tardiùs movetur, aut aliquorum labor augetur: haud &longs;ecus ac <lb/>&longs;i plures homines uni vecti ad pondus aliquod elevandum appli­<lb/>carentur, uno aut altero ce&longs;&longs;ante reliquorum ni&longs;us augendus <lb/>e&longs;&longs;et, &longs;upplementum de&longs;idio&longs;orum. </s> </p> <p type="main"> <s id="s.003487">Ex rationibus igitur vectis &longs;atisfit quæ&longs;tioni ab Ari&longs;totele <lb/>propo&longs;itæ, <emph type="italics"/>Cur <02>, qui in navis medio &longs;unt remiges, maxime navim <lb/>movent?<emph.end type="italics"/> Allatam à Philo&longs;opho re&longs;pon&longs;ionem intactam relinquo; <lb/>an &longs;atis commoda &longs;it, alij examinent. </s> <s id="s.003488">Remigum alij in puppi <lb/>con&longs;tituuntur, qui Thranitæ dicebantur, ut e&longs;t apud Suidam, <lb/>alij in prorâ, qui Thalamij &longs;eu Thalamitæ, alij in navis medio, <lb/>qui Zygitæ: & quamvis omnes ad promovendam navim &longs;uum <lb/>conatum conferant, non tamen omnium æqualis e&longs;t labor, aut <lb/>par in movendo efficacitas; quia non &longs;ecundùm eandem Ratio­<lb/>nem &longs;ingulorum remorum longitudo in partes à &longs;calmo di&longs;tin­<lb/>guitur; &longs;ed quia puppis altior e&longs;t, & &longs;patium angu&longs;tum, major <pb pagenum="456" xlink:href="017/01/472.jpg"/>remi pars extra navim e&longs;t, parúmque à &longs;calmo di&longs;tat remex; <lb/>ideò motus potentiæ ad motum ponderis minorem habet ratio­<lb/>nem, quàm &longs;i brevior e&longs;&longs;et inter palmulam, & &longs;calmum, lon­<lb/>giórque inter &longs;calmum & remigem di&longs;tantia, ut contingit in <lb/>medio, ubi navis depre&longs;&longs;ior e&longs;t, & maximam habet latitudi­<lb/>nem; pondere enim minùs di&longs;tante ab hypomochlio, majora <lb/>&longs;unt potentiæ momenta, cum eadem ponatur utrobique vectis <lb/>longitudo. </s> <s id="s.003489">Quæ autem de puppi dicta &longs;unt, &longs;altem quo ad &longs;pa­<lb/>tij angu&longs;tias, etiam de prorâ intelligenda &longs;unt, quæ quia de­<lb/>pre&longs;&longs;ior e&longs;t puppi, & aliquanto altior quàm circa medium, <lb/>propterea Thalamiorum labor medius e&longs;t inter Thranitarum & <lb/>Zygitarum laborem. </s> <s id="s.003490">Dicuntur autem remiges, qui in navis <lb/>medio &longs;unt, maximè movere vim, non quia navis motus, qui <lb/>circa hypomochlium tanquam circa centrum fit, ibi &longs;it major <lb/>motu, qui fit in puppi, &longs;i remiges parem arcum de&longs;cribant, <lb/>nam potiùs oppo&longs;itum contingit; &longs;ed quia remex in medio mi­<lb/>norem inveniens ponderis movendi re&longs;i&longs;tentiam plus navim <lb/>impellit, quàm &longs;i in puppi pariter conaretur, ubi eodem ni&longs;u <lb/>non pote&longs;t eodem temporis &longs;patio tam amplum arcum de&longs;cribe­<lb/>re. </s> <s id="s.003491">Propterea forti&longs;&longs;imi remiges ad puppim &longs;tatuuntur, ut ma­<lb/>jore impetu producto vincant majorem re&longs;i&longs;tentiam; ideóque <lb/>Thranitis præter publicum &longs;tipendium etiam extraordinarium <lb/>datum commemorat Thucydides lib. 6. Hæc verò, quæ de An­<lb/>tiquorum navibus magis propriè dicuntur, quarum forma à <lb/>no&longs;tris di&longs;&longs;idebat, no&longs;tris tamen celocibus aut triremibus &longs;erva­<lb/>tâ analogiâ accommodari po&longs;&longs;unt; nam etiam apud nos &longs;cal­<lb/>mus ad proram & ad puppim a&longs;cendit, & in medio major e&longs;t <lb/>navigij amplitudo, ita ut, licèt remorum capita in eâdem rectâ <lb/>lineâ juxta navigij longitudinem con&longs;tituantur, di&longs;pari tamen <lb/>Ratione à &longs;calmo di&longs;tinguantur in partes. </s> </p> <p type="main"> <s id="s.003492">Sed præ&longs;tat ip&longs;um navis motum paulo attentiùs con&longs;iderares <lb/>quandoquidem &longs;i hypomochlium e&longs;&longs;et pror&longs;us immobile, & <lb/>aqua locum non daret palmulæ urgenti, utique motus navis <lb/>ad motum capitis remi in eâ e&longs;&longs;et Ratione, quæ intercedit <lb/>inter di&longs;tantias &longs;calmi, & capitis remi ab aquâ. </s> <s id="s.003493">Nam &longs;i pal­<lb/>mula B immota maneret, & &longs;calmus e&longs;&longs;et in C, motus remi­<lb/>gis AD ad motum navis CE e&longs;&longs;et ut AB ad CB. <!-- KEEP S--></s> <s id="s.003494">Contra <lb/>verò &longs;i aqua nihil pror&longs;us ob&longs;i&longs;teret remo (&longs;icuti continge-<pb pagenum="457" xlink:href="017/01/473.jpg"/>ret, &longs;i ille admodum lentè moveretur, aut palmula nimis <lb/>obliqua aquam finderet) tantúmque palmula retrogrederetur <lb/>per BH, quantum remex per AD <lb/><figure id="id.017.01.473.1.jpg" xlink:href="017/01/473/1.jpg"/><lb/>progreditur, immota maneret na­<lb/>vis in C: id quod etiam continge­<lb/>ret, &longs;i regre&longs;&longs;us BH ad progre&longs;­<lb/>&longs;um AD e&longs;&longs;et in Ratione CB ad <lb/>CA. <!-- KEEP S--></s> <s id="s.003495">Quod &longs;i palmula à profluen­<lb/>te rapta ex B in L majus &longs;patium <lb/>conficeret, quàm remex ex A in D <lb/>(aut &longs;altem BL ad AD e&longs;&longs;et in <lb/>majore Ratione quàm CB ad CA) <lb/>utique navis ip&longs;a retrocederet, & <lb/>&longs;calmus ex C veniret in F. <!-- KEEP S--></s> <s id="s.003496">Cum <lb/>igitur promoveatur navis, & aqua <lb/>palmulæ ob&longs;i&longs;tens &longs;it hypomochlium mobile, nece&longs;&longs;e e&longs;t pro­<lb/>gre&longs;&longs;u remigis AD minorem e&longs;&longs;e palmulæ regre&longs;&longs;um BI, ut <lb/>&longs;calmus ex C propellatur in E. <!-- KEEP S--></s> <s id="s.003497">Quare quo magis aqua re&longs;i&longs;tit, <lb/>minú&longs;que palmula movetur in oppo&longs;itam remigis motui par­<lb/>tem, magis promovetur navis, quia majorem impul&longs;um recipit. </s> <lb/> <s id="s.003498">Majorem autem aquæ re&longs;i&longs;tentiam efficere pote&longs;t aut velocior <lb/>remi motio, aut major palmulæ immer&longs;io: con&longs;tat &longs;i quidem, &longs;i <lb/>baculo aquam lentè dividas, vix percipi in illa &longs;cindendâ labo­<lb/>rem; at &longs;i velociter baculum immer&longs;um agitare libeat, multò <lb/>validiùs illam re&longs;i&longs;tere: &longs;imiliter quò major palmulæ immer&longs;æ <lb/>pars plus aquæ propul&longs;at, eò majorem invenit re&longs;i&longs;tentiam, dif­<lb/>ficiliùs enim multa, quàm modica aqua dividitur. </s> <s id="s.003499">Verùm cum <lb/>fe&longs;tinato opus e&longs;t, &longs;atius e&longs;t velociter remum movere, & parum <lb/>immergere palmulam, ut frequentiori remorum percu&longs;&longs;ione <lb/>plus impetûs navi imprimatur. </s> </p> <p type="main"> <s id="s.003500">Quod demum &longs;pectat ad remi motum, unum &longs;upere&longs;t ob&longs;er­<lb/>vandum, videlicet, non eum tantummodo motum capiti remi <lb/>tribuendum, qui re&longs;pondet partibus navis, quatenus ex remigis <lb/>mu&longs;culorum contentione atque membrorum inclinatione pen­<lb/>det, cujus men&longs;uram definiret perpendiculum à capite remi in <lb/>&longs;ubjectum navis planum de&longs;cendens, & in eo remi iter de&longs;cri­<lb/>bens; &longs;ed præterea addendus e&longs;t motus navis, qui omnibus in <lb/>navi exi&longs;tentibus communis e&longs;t, adeò ut navis vi remorum acta <pb pagenum="458" xlink:href="017/01/474.jpg"/>moveatur à motore tran&longs;lato. </s> <s id="s.003501">Quapropter &longs;i AD e&longs;t univer&longs;us <lb/>capitis remi, &longs;eu manús remigis motus, demendus ex illo e&longs;t <lb/>navis progre&longs;&longs;us CE, & re&longs;iduus motus à remigis conatu, qua­<lb/>tenus remum impellit, pendet. </s> </p> <p type="main"> <s id="s.003502">Sed antequàm ab hac remorum contemplatione animum <lb/>avertamus, placet innuere, quæ de Sinen&longs;ium remis attigit <lb/>Atlas Sinicus in Præfatione pag. </s> <s id="s.003503">10, ubi de Præfectorum navi­<lb/>bus, quæ no&longs;tris triremibus æquales &longs;unt, hæc habet. <emph type="italics"/>Dum <lb/>ce&longs;&longs;ant ventorum flatus, ad&longs;unt de&longs;tinati, qui remulco trahant, aut <lb/>remis moles tota impellitur motis ad modum caudæ pi&longs;cium, methodo <lb/>facili, & compendiosâ; quippe &longs;ine ulla aquæ percu&longs;&longs;ione, extractio­<lb/>neve remi, vel remo unico propellitur & dirigitur navis; adeòque <lb/>unus hic &longs;ex aut octo no&longs;tratibus nautis æquivalet.<emph.end type="italics"/></s> <s id="s.003504"> Po&longs;tremum hoc <lb/>de uno remige &longs;ex aut octo no&longs;tratibus nautis æquivalente, <lb/>adeò magnificè dictum videtur, &longs;ed & adeò jejunè expo&longs;itum, <lb/>ut verba mihi dari non facilè patiar, nec me libenter præbeam <lb/>credulum: fundamentum con&longs;tituendæ fidei fui&longs;&longs;et remigan­<lb/>di ordo de&longs;criptus, remorum forma atque po&longs;itio verbis aut <lb/>iconi&longs;mo propo&longs;ita, ut, quanta &longs;int remigis Sinici momenta, <lb/>innote&longs;cerent; aliam enim utique à no&longs;tratibus remorum for­<lb/>mam e&longs;&longs;e nece&longs;&longs;e e&longs;t, quippe quos flexiles e&longs;&longs;e oporteat, ut ad­<lb/>modum caudæ pi&longs;cium moveantur; hi &longs;cilicet po&longs;tremam cor­<lb/>poris &longs;ui partem flectunt priùs atque contorquent, ut caudam <lb/>po&longs;tmodum velociter porrigentes aquam verberent, qua re­<lb/>&longs;i&longs;tente conceptus impetus totum corpus promoveat, quandiu <lb/>ille per&longs;everat. </s> <s id="s.003505">Ubi animadvertendum e&longs;t, quàm &longs;apienti na­<lb/>turæ in&longs;tituto factum &longs;it, ut pi&longs;ces caudam lentiùs inflectant, <lb/>&longs;ed velociùs explicent, inflectant obliquam, explicent erectam; <lb/>&longs;i enim erectam caudam velociter flecterent, ita aqua re&longs;i&longs;te­<lb/>ret, ut potiùs retrocederent, quemadmodum A&longs;taci fluviati­<lb/>les (cammaros, alij cancros fluviatiles vocant, rectè ne? </s> <s id="s.003506">an <lb/>perperam? </s> <s id="s.003507">non e&longs;t hujus loci examinare) quando timent, cau­<lb/>dâ aquam validè percutientes, ac qua&longs;i ad &longs;e velociter trahen­<lb/>tes non procedunt, &longs;ed retror&longs;um curvatâ caudâ &longs;ecedunt. </s> <s id="s.003508">Sic <lb/>etiam contingeret cymbæ, &longs;i quis in puppi &longs;tans ligneam tabel­<lb/>lam extremæ perticæ infixam aquæ à tergo po&longs;itæ immitteret, <lb/>perticámque ad &longs;e velociter traheret, nam cymba retror&longs;um <lb/>agi videretur. </s> <s id="s.003509">Cùm autem pi&longs;ces caudam & obliquè & lentiùs <pb pagenum="459" xlink:href="017/01/475.jpg"/>inflectant, minorem aquæ re&longs;i&longs;tentiam percipiunt. </s> <s id="s.003510">Quare, ut <lb/>remus &longs;uo motu imitetur motum caudæ pi&longs;cium, opus e&longs;t <lb/>erectam palmulam (hoc e&longs;t, in plano verticali longitudinem <lb/>navis obliquè, aut ad rectos angulos, &longs;ecante exi&longs;tentem) aquæ <lb/>occurrere, ut aquâ re&longs;i&longs;tente propellatur navis, eandem verò <lb/>palmulam po&longs;teà obliquam fieri, ne dum, intra aquam retrahi­<lb/>tur ad iterandum impul&longs;um, tantam inveniat re&longs;i&longs;tentiam, &longs;ed <lb/>aquam faciliùs findat. </s> <s id="s.003511">Hinc conjecturâ aliquâ ducebar ali­<lb/>quando ad &longs;u&longs;picandum, an ita remi palmula reliquæ remi lon­<lb/>gitudini adnexa e&longs;&longs;et fibulâ plicatili, ut, cùm remi caput pup­<lb/>pim versùs, palmula verò in oppo&longs;itam partem impelleretur, <lb/>hæc occurrenti aquæ cederet, eámque obliquè finderet modi­<lb/>câ manûs remigis deflexione remum interim contorquentis. </s> <lb/> <s id="s.003512">Sed, ut quod res e&longs;t eloquar, vereor, ne argutum nimis, víx­<lb/>que aliquid habens compendij, artificium hoc videatur: nam <lb/>& no&longs;trates cymbularij communi remo cymbam ex puppi <lb/>agentes eam propellunt, & dirigunt aquam non percutientes, <lb/>nec remum extrahentes, cujus varia inclinatione, loco guber­<lb/>naculi, cymbæ motum temperant remigando. </s> <s id="s.003513">Ut quid ergo <lb/>remum in duas partes, quæ fibulâ jungantur, divi&longs;um adhibe­<lb/>re: quippe qui noceat potiùs; nam remigis motus in proram <lb/>directus nullum impul&longs;um imprimit navi, ni&longs;i quando iterum <lb/>rectus factus fuerit remus. </s> <s id="s.003514">Sed flectatur & dirigatur remus in <lb/>morem caudæ pi&longs;cium; quid hoc, ut unus remex &longs;ex aut octo <lb/>no&longs;tratibus nautis æquivaleat? </s> <s id="s.003515">Hac autem oblatâ occa&longs;ione <lb/>cùm varias excogitaverim rationes utendi remis aquæ &longs;emper <lb/>immer&longs;is, liceat mihi per lectoris patientiam unum proponere, <lb/>quod forta&longs;&longs;e nec incommodum, nec inutile accideret, &longs;i in <lb/>u&longs;um deduceretur, tunc maximè, cum plures hinc & hinc re­<lb/>miges adhibentur, qui navis æquilibrio non officerent: neque <lb/>enim facilè author e&longs;&longs;em, ut levioribus cymbis methodus hæc <lb/>communis e&longs;&longs;et: quippe quæ deficiente ponderis hinc & hinc <lb/>æqualitate in alterutram partem nimis inclinarentur, nec citra <lb/>casûs nautæ, aut ever&longs;ionis naviculæ periculum. </s> <s id="s.003516">Remiges <lb/>&longs;tatuo hinc & hinc &longs;calmo in&longs;i&longs;tentes; id quod incommo­<lb/>dum non erit; quandoquidem additis extrin&longs;ecùs opportu­<lb/>nis fulcris cra&longs;&longs;iorem &longs;atí&longs;que firmam tabulam impono me­<lb/>diocris latitudinis à &longs;calmo di&longs;tantem tanto intervallo, quan-<pb pagenum="460" xlink:href="017/01/476.jpg"/>to opus e&longs;t, ut interjici valeat remus, commodéque agitari: <lb/>externam autem additæ tabulæ oram ambiat limbus, ne facilè <lb/>pes labatur; alterum enim pedem tabulæ, alterum &longs;calmo com­<lb/>mode imponere poterit remex. </s> <s id="s.003517">Remi longitudinem definit al­<lb/><figure id="id.017.01.476.1.jpg" xlink:href="017/01/476/1.jpg"/><lb/>titudo &longs;calmi ferè &longs;upra navis fundum, <lb/>addita mediocri hominis altitudine; <lb/>ipsíque remi capiti cylindrulus tran&longs;­<lb/>ver&longs;arius injicitur, ita tamen, ut in eo­<lb/>dem plano inveniatur cum remi pal­<lb/>mulâ: apprehen&longs;o &longs;i quidem utrâque <lb/>remigis manu hinc & hinc cylindrulo, <lb/>palmulæ planities aquæ obvertitur, <lb/>eámque impellit; apprehensâ autem <lb/>alterâ tantum extremitate, &longs;ive A, &longs;i­<lb/>ve B, quando retrahitur remus, pal­<lb/>mula DE aquam findit, & e&longs;t quo­<lb/>dammodo parallela carinæ. </s> <s id="s.003518">Duplicem <lb/>igitur motum remo conciliare oportet, <lb/>alterum quidem à puppi ad proram, & vici&longs;&longs;im, cùm &longs;cilicet im­<lb/>pellitur, & retrahitur, alterum verò circa &longs;uum axem longitu­<lb/>dinis, ut convertatur nunc ad impellendam, nunc ad findendam <lb/>aquam. </s> <s id="s.003519">Primus motus perficitur, &longs;i ferreo circulo HI remus in­<lb/>&longs;eratur duos polos habenti; quorum alter &longs;calmum, alter additam <lb/>tabulam ingrediatur (&longs;ivè potiùs excavatæ congruæ crenæ in­<lb/>cumbant, ut extrahi pro libito po&longs;&longs;int) síntque facilè ver&longs;atiles: <lb/>remus enim in eodem plano verticali &longs;emper exi&longs;tens deprimi <lb/>pote&longs;t, ut horizontem versùs inclinetur, atque iterum elevari ac <lb/>etiam in oppo&longs;itam partem inclinari. </s> <s id="s.003520">Secundus autem motus fa­<lb/>cillimè habetur, &longs;i remo ferreus rotundus claviculus F adjicia­<lb/>tur circuli HI &longs;uperiorem partem contingens; impedit enim, <lb/>ne remus deor&longs;um prolabatur, adeóque nullo labore converti­<lb/>tur circa axem &longs;uæ longitudinis remus, dimi&longs;sâ alterutrâ cylin­<lb/>druli AB extremitate, quando retrahitur. </s> <s id="s.003521">Inventum hoc rudi­<lb/>ter propo&longs;itum expolire, atque accuratiùs excolere poteris, in­<lb/>genio&longs;e Lector, qui forta&longs;&longs;e tuâ indu&longs;triä con&longs;equeris artem <lb/>mihi ignotam remos ita di&longs;ponendi, ut remex unus pluribus <lb/>nautis æquivaleat, quemadmodum de Sinen&longs;ibus narratur. <pb pagenum="461" xlink:href="017/01/477.jpg"/></s> </p> <p type="main"> <s id="s.003522"><emph type="center"/>CAPUT XV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003523"><emph type="center"/><emph type="italics"/>Quomodo Naves à gubernaculo moveantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003524">REs e&longs;t, cui a&longs;&longs;iduus u&longs;us admirationem detraxi&longs;&longs;e videtur, <lb/>non tamen propterea minus habet admirabilitatis, motus <lb/>&longs;cilicet navium, quæ à gubernaculo reguntur, cùm magnum <lb/>pondus temporis momento moveatur. </s> <s id="s.003525">Nam & Apo&longs;tolus S. <!-- REMOVE S-->Ja­<lb/>cobus in Canonicâ Epi&longs;tola cap.3.ait, <emph type="italics"/>Ecce & naves cum magnæ <lb/>&longs;int, & à ventis validis minentur, circumferuntur à modico guber­<lb/>naculo, ubi impetus dirigentis voluerit.<emph.end type="italics"/></s> <s id="s.003526"> Et Ari&longs;toteles Mechan. <lb/>quæ&longs;t. </s> <s id="s.003527">5. inquirit. <emph type="italics"/>Cur parvum exi&longs;tens gubernaculum, & in ex­<lb/>tremo navigio, tantas habet vtres, ut ab exiguo temone, & ab homi­<lb/>nis unius viribus alioquin modicè utentis, magnæ navigiorum movean­<lb/>tur moles?<emph.end type="italics"/> Partes duas in gubernaculo invenimus; alteram ex­<lb/>trin&longs;ecùs navi adjectam, ligneam videlicet alam, &longs;ive cardini­<lb/>bus, circa quos converti pote&longs;t, po&longs;tremæ puppis parti affixam, <lb/>&longs;ive ad latus puppi adjacentem, tignóque, quod ex &longs;calmo a&longs;­<lb/>&longs;urgit, adalligatam, prout maritimo vel fluviatili itineri de&longs;ti­<lb/>nata &longs;unt navigia; alteram intra navim, temonis in morem; ex <lb/>cujus conver&longs;ione aut pars illa externa in oppo&longs;itam plagam <lb/>convertitur, ita ut deductâ temonis extremitate ad dexteram, <lb/>gubernaculi ala extremæ puppi adjacens in &longs;ini&longs;tram circa &longs;uos <lb/>cardines deflectat, & vici&longs;&longs;im hæc ad dexteram, temone in &longs;i­<lb/>ni&longs;tram conver&longs;o: aut in navigiis, quorum in fluminibus u&longs;us <lb/>e&longs;t, gubernaculum ad puppis latus habentibus, depre&longs;&longs;o temo­<lb/>ne &longs;uperior extremitas alæ in triangulum &longs;ubjecto cylindro <lb/>infixum conformatæ propiùs accedit ad navim, temone autem <lb/>elevato ab illa recedit: contra verò &longs;i ala infra cylindrum con&longs;ti­<lb/>tuatur, ejus extremitas inferior ad navim accedit temone ele­<lb/>vato, à navi recedit temone depre&longs;&longs;o. </s> </p> <p type="main"> <s id="s.003528">Quemadmodum autem ad propellendam navim in&longs;tituti &longs;unt <lb/>remi, ita ad eju&longs;dem cur&longs;um dirigendum, atque pro opportu­<lb/>nitate in dexteram aut in &longs;ini&longs;tram inclinandum, clavus ad-<pb pagenum="462" xlink:href="017/01/478.jpg"/>jectus e&longs;t. </s> <s id="s.003529">Quamquam enim remorum pul&longs;u navis acta proram <lb/>in dexteram obvertere po&longs;&longs;it, &longs;i remiges &longs;ini&longs;tri ce&longs;&longs;ent, atque <lb/>è contrario in &longs;ini&longs;tram ce&longs;&longs;antibus dexteris; aut etiam vento <lb/>navim impellente fieri po&longs;&longs;it hæc in alterutram partem decli­<lb/>natio modò pa&longs;&longs;o, modò contracto velo, ut me ob&longs;erva&longs;&longs;e me­<lb/>mini, cum ex in&longs;ula Seelandia per fretum Ore&longs;unticum in Sca­<lb/>niam (El&longs;ingorâ &longs;cilicet El&longs;emburgum) navigiolo transfreta­<lb/>rem: id tamen longè faciliùs, atque ad unius gubernatoris ar­<lb/>bitrium perficitur conver&longs;o opportunè clavo, ut quotidiano <lb/>experimento docemur. </s> </p> <p type="main"> <s id="s.003530">Porrò dupliciter gubernaculi motum con&longs;iderare po&longs;&longs;umus; <lb/>neque enim eadem e&longs;t ratio, cùm navis quie&longs;cit, nullu&longs;que e&longs;t <lb/>aquæ motus, atque cùm navis vento &longs;eu remis agitur, aut aqua <lb/>ip&longs;a movetur. </s> <s id="s.003531">Et quidem &longs;i navigium in aquà immotâ quie&longs;­<lb/>cat, qui gubernaculi temonem movet, e&longs;t potentia applicata <lb/>vecti, cujus hypomochlium e&longs;t aqua, &longs;i navis non &longs;it tantæ gra­<lb/>vitatis, ut faciliùs ip&longs;a moveatur, quam tota aqua propellatur <lb/>ab alâ gubernaculi; & tunc e&longs;t vectis &longs;ecundi generis, nam <lb/>puppis, aquâ re&longs;i&longs;tente, &longs;ecedit ad dexteram aut ad &longs;ini&longs;tram <lb/>&longs;equens temonis conver&longs;ionem. </s> <s id="s.003532">At &longs;i tanta &longs;it navis gravitas, <lb/>ut multo faciliùs tota aqua propellatur, quàm navis loco mo­<lb/>veatur, vectis e&longs;t primi generis habens hypomochlium in car­<lb/>dinibus, circa quos gubernaculi ala convertitur, pondus au­<lb/>tem, quod movetur, e&longs;t aqua, quæ eò faciliùs, minori &longs;cilicet <lb/>labore, propellitur, quò longior e&longs;t temo; tunc enim potentia <lb/>plus habet momenti. </s> <s id="s.003533">Hinc duplex vectis ratio invenitur, cùm <lb/>aliquâ ex parte aqua, aliquâ ex parte puppis movetur; quo in <lb/>motu &longs;atis con&longs;tat neque motum puppis fieri circa aquam extre­<lb/>mæ gubernaculi alæ re&longs;pondentem, neque motum aquæ re&longs;pon­<lb/>dentis extremo gubernaculo fieri circa cardines puppi inhæren­<lb/>tes; &longs;ed conver&longs;ionem fieri circa punctum aliquod intermedium <lb/>reciprocè acceptum pro Ratione re&longs;i&longs;tentiarum aquæ & navis, <lb/>juxta dicta cap.5. hujus libri. </s> <s id="s.003534">Cùm autem re&longs;i&longs;tentia aquæ æ&longs;ti­<lb/>manda &longs;it ex magnitudine & figura alæ gubernaculi aquam <lb/>ip&longs;am impellentis, & re&longs;i&longs;tentia navis pariter definienda &longs;it tùm <lb/>ex ejus gravitate, tum ex aquæ propellendæ quantitate, dum <lb/>navis in dexteram aut in &longs;ini&longs;tram convertitur, patet nul­<lb/>lum certum punctum navibus omnibus commune &longs;tatui po&longs;&longs;e; <pb pagenum="463" xlink:href="017/01/479.jpg"/>&longs;unt &longs;iquidem huju&longs;modi re&longs;i&longs;tentiæ multiplici varietati ob­<lb/>noxiæ. </s> </p> <p type="main"> <s id="s.003535">At verò cùm navis in motu e&longs;t, & vento impellente &longs;eu re­<lb/>mis agitur, potentia quidem pro temonis longitudine &longs;ua ha­<lb/>bet momenta, & ad navis conver&longs;ionem juvat, magis tamen <lb/>accipiendo vim externam & ferendo, quàm agendo & facien­<lb/>do, hoc e&longs;t retinendo gubernaculum in illa obliquâ po&longs;itione <lb/>adversùs vim aquæ in contrarium nitentis, aut re&longs;i&longs;tentis. </s> <s id="s.003536">Quò <lb/>enim velociùs fertur navis, obviam aquam prorâ &longs;cindens illam <lb/>ità dividit, ut ad navis latera hinc atque hinc velociùs refluat <lb/>in puppim; ubi &longs;i gubernaculi alam inveniat rectam, pergit na­<lb/>vis recto itinere; Sed &longs;i aqua refluens obli­<lb/>quum gubernaculum offendat, ut &longs;i exi&longs;ten­<lb/><figure id="id.017.01.479.1.jpg" xlink:href="017/01/479/1.jpg"/><lb/>te carina AB fuerit gubernaculum obli­<lb/>quum CD, aqua in alam AD incurrens <lb/>dum illam urget, puppim cogit declinare <lb/>ex A in E, & prora obvertitur versùs F. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003537">Hæc tamen, quæ de aquá ad navis late­<lb/>ra refluente dicta &longs;unt, non ita accipi velim, <lb/>ut non ni&longs;i ab ejus impetu flecti navis cur­<lb/>&longs;um exi&longs;times; &longs;ed hæc deflexio præcipuè <lb/>tribuenda e&longs;t re&longs;i&longs;tentiæ ip&longs;ius aquæ, in <lb/>quam incurrit gubernaculum obliquum, <lb/>dum navis tota impellitur; eò autem major <lb/>e&longs;t aquæ re&longs;i&longs;tentia, quò velociùs illam &longs;cindi oportet, ut &longs;æ­<lb/>piùs dictum e&longs;t. </s> <s id="s.003538">Ideò quo validiore venti aut remorum impul­<lb/>&longs;u agitur navis, faciliùs flectitur ope gubernaculi, majorem <lb/>quippe invenit re&longs;i&longs;tentiam. </s> <s id="s.003539">Cum verò re&longs;i&longs;tentia hæc ex al­<lb/>terutra tantùm parte inveniatur, nece&longs;&longs;e e&longs;t proram in eandem <lb/>obverti plagam. </s> <s id="s.003540">Cujus rei obvium experimentum &longs;umere qui&longs;­<lb/>que pote&longs;t, &longs;i corpus aliquod angulatum (cuju&longs;modi e&longs;&longs;et nor­<lb/>ma, qua ad angulum rectum de&longs;cribendum utimur) in plano <lb/>inclinato æquabili ac polito de&longs;cendere permittat: nam &longs;i, quod <lb/>ponè &longs;equitur, brachium in offendiculum aliquod incurrat, il­<lb/>lico reliquum brachium ad eam partem inclinari videbit, im­<lb/>petu &longs;cilicet promovente corpus, atque objectum impedimen­<lb/>tum declinante. </s> </p> <p type="main"> <s id="s.003541">Quare id, quod navim maximè movet in dexteram aut &longs;i-<pb pagenum="464" xlink:href="017/01/480.jpg"/>ni&longs;tram, e&longs;t impetus ab ip&longs;o vento aut à remigibus navi impre&longs;­<lb/>&longs;us; gubernaculum autem infert moram & impedimentum, ne <lb/>motus omnino fiat juxta directionem impetûs ab impellente im­<lb/>pre&longs;&longs;i: quamdiu verò impedimentum per&longs;everat, navis magis <lb/>aut minùs obliquè fertur, pro ut modificata impetûs directio <lb/>exigit. </s> <s id="s.003542">Juvat autem, ut dictum e&longs;t, aqua, quæ à prorâ dividi­<lb/>tur, & ad latera refluit, maximè &longs;i adver&longs;o flumine, aut contra <lb/>marini æ&longs;tus cur&longs;um navigatio in&longs;tituatur; aucto enim impedi­<lb/>mento faciliùs flectitur in&longs;tituta progre&longs;&longs;io; &longs;ed idcircò etiam <lb/>navis motus retardatur magis. </s> </p> <p type="main"> <s id="s.003543">Quod quidem &longs;pectat ad gubernaculum extremæ puppis pla­<lb/>næ faciei adhærens, ut in majoribus navigiis maritimo itineri <lb/>de&longs;tinatis, &longs;atis jam explicatum e&longs;t: unum addendum videtur, <lb/>quod in navigiis ad devehendas merces fabricatis in fo&longs;sâ qua­<lb/>dam manufactâ aliquando ob&longs;erva&longs;&longs;e me memini; ex puppi vi­<lb/>delicet extremâ in acutum a&longs;&longs;urgente, qua&longs;i caudæ in morem, <lb/>clavus longiùs protendebatur apici puppis in&longs;i&longs;tens remo ab&longs;i­<lb/>milis tantùm, quatenus palmula paulò latior, nec juxta &longs;capi <lb/>longitudinem directa, &longs;ed inflexa intra aquam immergebatur; <lb/>caput autem temonis fune jungebatur navigij plano ita, ut gu­<lb/>bernaculi pars externa &longs;uo pondere recidere nequiret, ac fun­<lb/>dum alvei non peteret, &longs;ed palmula paulò infra aquæ &longs;uperfi­<lb/>ciem con&longs;i&longs;teret. </s> <s id="s.003544">Hinc enim fiebat, ut temonis capite in alteru­<lb/>tram partem adducto in eandem puppis recederet aquâ re&longs;i&longs;ten­<lb/>te palmulæ, ac proinde prora in oppo&longs;itam partem obvertere­<lb/>tur: perinde atque in cymbis gubernaculo de&longs;titutis, cùm remi <lb/>ad latus extremæ puppis directè immer&longs;i caput ad &longs;e retrahit <lb/>nauta, puppim ip&longs;am impellit, ac proram in oppo&longs;itam partem <lb/>convertit. </s> <s id="s.003545">Huc &longs;pectare po&longs;&longs;unt, quæ habet Atlas Sinicus <lb/>pag.123 in XI Provincia Fokien loquens de flumine Min, quod <lb/>ex Puching ad oppidum u&longs;que Xuiken per valles & &longs;axa ingen­<lb/>ti impetu ac violentiâ volvitur, inde placidi&longs;&longs;imum flumen e&longs;t; <lb/>& quantumcumque violentum enavigatur tamen à Sinis con­<lb/>&longs;uetâ illorum indu&longs;triâ, ac parvarum navicularum artificio: hæ <lb/>enim naves clavum, ut aliæ non habent, &longs;ed duos longi&longs;&longs;imè <lb/>porrectos, ad puppim unum, ad proram alterum: his per &longs;axa <lb/>ac &longs;copulos prominentes facillime ac veloci&longs;&longs;imè naves, ac &longs;i <lb/>fræno equos continerent, dirigunt. </s> <s id="s.003546">Hæc ibi. </s> </p> <pb pagenum="465" xlink:href="017/01/481.jpg"/> <p type="main"> <s id="s.003547">Sed ut aliquid etiam de guberna culo ad puppis latus con&longs;ti­<lb/>tuto in navigiis, quorum poti&longs;&longs;imus u&longs;us e&longs;t in fluviis, dicatur, <lb/>animadvertendum e&longs;t huju&longs;modi navigia non &longs;olùm proram, <lb/>&longs;ed & puppim habere, quæ obliquè a&longs;&longs;urgentes in acutum de­<lb/>&longs;inunt, gubernaculum autem con&longs;tare ex cylindro obliquè <lb/>de&longs;cendénte juxta puppis longitudinem, atque ex alâ triangu­<lb/>lari ut plurimùm &longs;ur&longs;um re&longs;piciente, cujus latus unum cylin­<lb/>dro congruit, cui ïnfixum e&longs;t. </s> <s id="s.003548">Quandiu ala &longs;ur&longs;um re&longs;picit, <lb/>nihil impedit navis motum, æqualiter enim aqua hinc & hinc <lb/>fluit, ac proinde navis fertur juxta impetûs à vento aut à remi­<lb/>gibus impre&longs;&longs;i directionem (idem dic, cùm navis trahitur) <lb/>quam &longs;equitur, ni&longs;i aliquid fortuito interveniat, à quo turbe­<lb/>tur motus, & præter nautarum voluntatem aliò flectatur. </s> <s id="s.003549">Quod <lb/>&longs;i convoluto circà &longs;uum axem cylindro, ala in hanc aut illam <lb/>partem vertatur, jam occurrit aquæ, ex cujus e&longs;i&longs;tentiâ impe­<lb/>dimentum objicitur navi, ne recta feratur, &longs;ed in alteram par­<lb/>tem detorquetur: nam &longs;i depre&longs;&longs;o temone, qui priùs erat hori­<lb/>zonti parallelus, ala versùs navim inclinetur, aqua inter guber­<lb/>naculum & navim intercepta re&longs;i&longs;tit, atque interfluens conatur <lb/>alam gubernaculi in directum re&longs;tituere: quapropter puppim in <lb/>dexteram trahens, illíque ad dexteram re&longs;i&longs;tens (clavus &longs;cilicet <lb/>dextero puppis lateri adjacet) proram obvertit ad &longs;ini&longs;tram. </s> <s id="s.003550">At <lb/>&longs;i gubernaculi ala in oppo&longs;itam navi partem extror&longs;um verta­<lb/>tur, obviam habet aquam externam, qua re&longs;i&longs;tente repellitur <lb/>puppis in &longs;ini&longs;tram, & prora in dexteram convertitur. </s> <s id="s.003551">Quod &longs;i <lb/>alam triangularem placeat potiùs cylindro &longs;ubjicere, elevato te­<lb/>mone ala accedit ad navim, & depre&longs;&longs;o temone ala recedit à na­<lb/>vi: quapropter ibi puppis repellitur in &longs;ini&longs;tram, hîc ab aquâ in­<lb/>tercurrente trahitur in dexteram, motù&longs;que oppo&longs;iti proræ <lb/>conveniunt. </s> </p> <p type="main"> <s id="s.003552">Ex his facilè innote&longs;cit, quid præ&longs;tet gubernaculum inter <lb/>puppes duorum pontonum, quos impo&longs;itus pons jungit, vali­<lb/>dú&longs;que rudens congruæ longitudinis retinet, ne &longs;ecundo flu­<lb/>mine rapiantur; prout enim in hanc vel illam partem guberna­<lb/>culi ala vertitur, obvium habet interjectarum aquarum impe­<lb/>tum, quo propellitur in adver&longs;am partem, eáque ratione traji­<lb/>citur flumen, ut in Pado & aliis Galliæ Ci&longs;alpinæ fluviis pa&longs;&longs;im <lb/>videre e&longs;t. </s> </p> <pb pagenum="466" xlink:href="017/01/482.jpg"/> <p type="main"> <s id="s.003553">Illud po&longs;tremò con&longs;ideratione dignum e&longs;t, quod ad ip&longs;ius <lb/>navis conver&longs;ionem attinet nimirùm quodnam &longs;it punctum <lb/>circa quod convertitur: Manife&longs;tum e&longs;t enim neque circa pup­<lb/>pim tanquam circa centrum de&longs;cribi arcum à prorâ, neque vi­<lb/>ci&longs;&longs;im circa proram qua&longs;i centrum arcum à puppi de&longs;cribi, quia <lb/>aquæ quantitas re&longs;pondens longitudini carinæ plurimum re­<lb/>&longs;i&longs;tit, ne circulariter moveatur tota ad eandem partem, coge­<lb/>retur &longs;cilicet nimis amplum arcum de&longs;cribere, nimí&longs;que veloci­<lb/>ter moveri in latus, ut per de&longs;tinatum navigationis Rumbum <lb/>nova loxodromia in&longs;titueretur: faciliùs igitur convertitur na­<lb/>vis, &longs;i dum pars anterior proræ aquam in dexteram propellit, <lb/>reliqua pars po&longs;terior puppi proxima aquam repellat in &longs;i­<lb/>ni&longs;tram, utraque enim extremitas minore arcu de&longs;cripto ad ma­<lb/>jorem angulum carinam inclinat atque deflectit à lineâ prioris <lb/>cursûs, & minore velocitate aquam urgens minorem invenit <lb/>re&longs;i&longs;tentiam. </s> <s id="s.003554">Fit igitur conver&longs;io circa punctum aliquod me­<lb/>dium inter proram & puppim; illud autem e&longs;t, circa quod na­<lb/>tura faciliùs a&longs;&longs;equitur propo&longs;itum, & minore motu removetur <lb/>impedimentum, quod ab aquâ occurrente infertur, quæ cùm <lb/>dividatur à prorâ, refluátque juxtà navis latera, æqualiter qui­<lb/>dem à prorâ di&longs;pertitur, &longs;ed ubi navis ventrem, hoc e&longs;t ampli&longs;­<lb/>&longs;imam navigij partem prætergre&longs;&longs;a e&longs;t, offendens ex alterâ <lb/>parte gubernaculi alam fluere non pote&longs;t, qua velocitate flue­<lb/>ret nullo objecto offendiculo; propterea aquæ refluenti ex ad­<lb/>ver&longs;o navis latere objiciendum e&longs;t obliquè puppis latus, ut illa <lb/>pariter lentiùs fluat, divisóque impedimento æquales aquæ por­<lb/>tiones ex utroque puppis latere fluant. </s> <s id="s.003555">Quare probabili con­<lb/>jecturâ exi&longs;timo conver&longs;ionem fieri circa illud carinæ punctum, <lb/>quod re&longs;pondet maximæ navigij amplitudini; pars quippe na­<lb/>vigij anterior juxta &longs;uam latitudinem occurrens aquæ invenit <lb/>re&longs;i&longs;tentiam; aqua igitur incurrens in gubernaculum movet <lb/>partem po&longs;teriorem in latus, ubi non e&longs;t tam valida aquæ re­<lb/>&longs;i&longs;tentia. </s> <s id="s.003556">Cum autem in majoribus navigiis præcipuus malus <lb/>&longs;tatuatur in maxima navigij amplitudine, hoc e&longs;t, ubi carinæ <lb/>longitudo be&longs;&longs;em relinquit puppim ver&longs;us, & trientem versùs <lb/>proram, carina ad proram &longs;pectans minùs movetur quàm quæ <lb/>ad puppim; &longs;ed propter notabilem proræ projecturam &longs;i pars na­<lb/>vis &longs;uprema in&longs;piciatur, malus ille e&longs;t circa mediam totius na-<pb pagenum="467" xlink:href="017/01/483.jpg"/>vis longitudinem, ibíque fit conver&longs;io. </s> <s id="s.003557">Cæterùm quicumque <lb/>navis formam, tormentorumque bellicorum di&longs;po&longs;itionem ac <lb/>numerum ob&longs;ervet, utique centrum gravitatis inter puppim & <lb/>malum præcipuum interjectum e&longs;&longs;e affirmabit; præ&longs;ertim cum <lb/>id nece&longs;&longs;e &longs;it, ne ventorum vi prora nimis deprimatur; id quod <lb/>multo manife&longs;tiùs innote&longs;cit in minoribus navigiis, &longs;i fortè ve­<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/>marius in G, centrum gravitatis navigij <lb/><figure id="id.017.01.483.1.jpg" xlink:href="017/01/483/1.jpg"/><lb/>ex. </s> <s id="s.003559">gr. <!-- REMOVE S-->in K. <!-- KEEP S--></s> <s id="s.003560">Duo &longs;unt principia moven­<lb/>tia; unum e&longs;t ventus in G, alterum e&longs;t <lb/>aqua refluens in AD: duo pariter &longs;unt hy­<lb/>pomochlia, &longs;eu impedimenta; vento re&longs;i&longs;tit <lb/>gubernaculum AD, propterea navim mo­<lb/>tione tran&longs;versâ promovens transfert cen­<lb/>trum gravitatis K ver&longs;us H: aquæ refluenti <lb/>re&longs;i&longs;tit vis venti in G, ita ut non valeat na­<lb/>vim retror&longs;um agere, propterea puppim ex <lb/>A transfert in E, & centro gravitatis K im­<lb/>petum imprimit dirigentem versùs I, cui <lb/>tamen prævalente impetu venti dirigente <lb/>versùs H, obliquus navis motus efficitur. </s> <s id="s.003561">Quare duplex e&longs;t <lb/>vectis &longs;ecundi generis; aqua in AD ad re&longs;i&longs;tentiam centri gra­<lb/>vitatis K habet momentum ut AG, &longs;eu DG, ad KG: Ventus <lb/>in G ad eju&longs;dem centri gravitatis K re&longs;i&longs;tentiam habet momen­<lb/>tum ut GA ad KA. <!-- KEEP S--></s> <s id="s.003562">Ex quo con&longs;tat majorem quidem e&longs;&longs;e Ra­<lb/>tionem AG ad KG minorem, quàm ad KA majorem; &longs;ed <lb/>multo validiorem potentiam e&longs;&longs;e ventum, quàm aquam re­<lb/>fluentem, ni&longs;i fortè addatur naturalis fluxus aquæ, qui aliquan­<lb/>do prævalere digno&longs;citur ex occultis Maris Currentibus, quæ <lb/>navim aliquando retror&longs;um agunt contrà vim venti. </s> <s id="s.003563">Sed quo­<lb/>niam tam varia & multiplex e&longs;t navigiorum forma, nec in iis <lb/>con&longs;truendis omnes artifices eandem &longs;ervant partium membro­<lb/>rúmque Rationem, nulla a&longs;&longs;ignari pote&longs;t certa Ratio, quæ in­<lb/>tercedat inter di&longs;tantiam centri gravitatis ab extremitate pup­<lb/>pis, atque di&longs;tantiam puncti, circa quod fit conver&longs;io, ab ea­<lb/>dem extremitate. </s> </p> <p type="main"> <s id="s.003564">Hìc autem (ne quis facilè &longs;imiliter labatur) fateor me ali-<pb pagenum="468" xlink:href="017/01/484.jpg"/>quando veri quadam &longs;pecie deceptum exi&longs;tima&longs;&longs;e intervallum <lb/>inter extremam proram & punctum conver&longs;ionis ad quartam <lb/>totius longitudinis partem proximè &longs;tatuendum e&longs;&longs;e; ducebar <lb/>&longs;cilicet quadam analogiâ de&longs;umpta ex cylindro ligneo innatan­<lb/>te, cujus quie&longs;centis extremitatem &longs;i tanto impetu percu&longs;&longs;eris, <lb/>quo certum &longs;patium percurrat, videbar mihi ritè inferre <lb/>punctum, circa quod convertitur, di&longs;tare ab extremitate per­<lb/>cu&longs;sâ ad totius longitudinis dodrantem: &longs;atis enim ip&longs;o u&longs;u in­<lb/>note&longs;cebat, nec punctum medium, &longs;cilicet centrum gravitatis, <lb/>nec oppo&longs;itam extremitatem e&longs;&longs;e centrum conver&longs;ionis. </s> <s id="s.003565">Vide­<lb/>batur autem naturæ &longs;ua jura tueri conanti valde con&longs;enta­<lb/>neum, &longs;i corpus amans quietis externo impul&longs;ui ita ob&longs;ecun­<lb/>det, ut quam minimo totius corporis motu impre&longs;&longs;us impetus <lb/>partem percu&longs;&longs;am pro &longs;uæ inten&longs;ionis modo transferat. </s> </p> <p type="main"> <s id="s.003566">Sit enim Cylindrus AB, cujus medium atque centrum gra­<lb/><figure id="id.017.01.484.1.jpg" xlink:href="017/01/484/1.jpg"/><lb/>vitatis C: AE verò &longs;it <lb/>totius longitudinis do­<lb/>drans: percutiatur extre­<lb/>mitas A tanto impetu, <lb/>quanto illa ferri po&longs;&longs;et <lb/>per &longs;patium AD, &longs;i mo­<lb/>veretur circa centrum C. <!-- KEEP S--></s> <lb/> <s id="s.003567">Ducatur igitur per C <lb/>recta DO æqualis toti <lb/>cylindro; qui &longs;i movere­<lb/>tur circa punctum C, utique &longs;uo motu de&longs;criberet duos Secto­<lb/>res, ACD, & BCO. </s> <s id="s.003568">Item per E ducatur ip&longs;i DO parallela <lb/>FI ita, ut ip&longs;i EA æqualis &longs;it EF, & ip&longs;i EB æqualis &longs;it EI. <!-- KEEP S--></s> <s id="s.003569">Cy­<lb/>lindrus igitur AB conver&longs;ione factâ circa punctum E de&longs;cribe­<lb/>ret Sectores AEF & BEI duobus prioribus &longs;imiles. </s> <s id="s.003570">Sunt au­<lb/>tem Sectores &longs;imiles, ut quadrata Radiorum; quemadmodum <lb/>facilè colligitur ex 2 lib. 12: atque ideò, cum Radius AC ad <lb/>Radium AE &longs;it ut 2 ad 3, Sector ACD ad Sectorem AEF e&longs;t <lb/>ut 4 ad 9: & quia Radius BC ad Radium BE e&longs;t ut 2 ad 1, <lb/>Sector BCO ad Sectorem BEI e&longs;t ut 4 ad 1. Motus igitur cy­<lb/>lindri circa centrum C ad motum circa centrum E e&longs;t ut 8 ad <lb/>10, &longs;i Sectores &longs;imiles de&longs;cribantur. </s> <s id="s.003571">Atqui impetus impre&longs;&longs;us <lb/>&longs;olùm pote&longs;t extremitatem A transferre per &longs;patium æquale <pb pagenum="469" xlink:href="017/01/485.jpg"/>ip&longs;i AD (e&longs;t autem arcus AD ad arcum AF &longs;imilem, ut Ra­<lb/>dius AC ad Radium AE, hoc e&longs;t ut 2 ad 3) igitur eandem <lb/>transfert &longs;olùm per AG be&longs;&longs;em arcûs AF, ac proinde motus e&longs;t <lb/>per Sectores AEG & BEH, qui ex ult. </s> <s id="s.003572">lib. 6. &longs;unt bes duo­<lb/>rum Sectorum AEF & BEI, quorum &longs;umma e&longs;t 10; ip&longs;ius au­<lb/>tem 10 bes e&longs;t 6 2/3. Motus igitur circa centrum E minor e&longs;t <lb/>motu circa centrum C, & impetus impre&longs;&longs;us æqualiter transfert <lb/>extremitatem A. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003573">Fateor potui&longs;&longs;e &longs;tatui AE mediam proportionalem inter to­<lb/>tam longitudinem AB & ejus &longs;emi&longs;&longs;em AC, & motus fui&longs;&longs;et <lb/>paulo minor. </s> <s id="s.003574">Ponatur enim tota AB 200, AC 100, e&longs;t AE <lb/>141 2/5 proximè: igitur ut quadratum AC ad quadratum AE <lb/>mediæ proportionalis, hoc e&longs;t ut 10000 ad 19994, ita Sector <lb/>ACD ad &longs;ectorem AEF &longs;imilem; & ut ip&longs;ius CB 100 qua­<lb/>dratum 10000, ad ip&longs;ius EB 59 proximè quadratum 3481, ita <lb/>Sector BCO ad Sectorem &longs;imilem BEI. </s> <s id="s.003575">Quare &longs;umma Secto­<lb/>rum ACD, BCO e&longs;t 20000, Sectorum verò AEF, BEI e&longs;t <lb/>23475. Sed quia ut AC ad AE, ita arcus AD ad arcum AF, <lb/>quarum partium AD e&longs;t 100, AF e&longs;t 141 proximè: & a&longs;&longs;ump­<lb/>to arcu AG 100, &longs;umma Sectorum AEG & BEH, ad &longs;um­<lb/>mam Sectorum AEF & BEI erit ut 100 ad 141, hoc e&longs;t, ut <lb/>16649 ad 23475: minor igitur e&longs;t quàm &longs;umma Sectorum <lb/>ACD & BCO, quæ e&longs;t 20000. At &longs;i AE &longs;it 150, & EB 50, <lb/>&longs;umma Sectorum AEF, BEI ut 25000, bes autem 16666 2/3, <lb/>qui excedit numerum &longs;uperiùs inventum 16649 adeò modico <lb/>intervallo, ut contemnendum &longs;it; cùm maximè impetus per ar­<lb/>cum AF aliquantulo majorem motum efficiat quàm per cir­<lb/>cumferentiam circuli minoris, ac propterea cen&longs;endus &longs;it arcus <lb/>AG aliquantulum major quàm arcus AD; idcircò vero pro­<lb/>pior e&longs;t AE dodrans totius longitudinis AB, quàm AE media <lb/>proportionalis inter AC & AB. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003576">Verùm quæ de cylindro in aquâ quie&longs;cente dicuntur &longs;atis <lb/>probabiliter, non omninò congruere po&longs;&longs;unt motui navis, quæ <lb/>præter motum aquæ percutientis gubernaculum promovetur à <lb/>vento aut à remigibus, & præterea non habet æquabili ductu <lb/>con&longs;titutam figuram, quemadmodum cylindrus: propterea <lb/>huic analogiæ non acquie&longs;cendum duxi. </s> <s id="s.003577">Sed & illud adden-<pb pagenum="470" xlink:href="017/01/486.jpg"/>dum, quod neque de cylindro &longs;atis certus e&longs;&longs;e po&longs;&longs;um; nam &longs;i <lb/>alia fiat hypothe&longs;is, & ad totius longitudinis be&longs;&longs;em &longs;tatuatur <lb/>punctum conver&longs;ionis ita ut &longs;emi&longs;&longs;is AC &longs;it 3, AE verò &longs;it 4, <lb/>& EB &longs;it 2; Sector ACD ad Sectorem AEF e&longs;t ut 9 ad 16, <lb/>& Sector BCO ad Sectorem BEI e&longs;t ut 9 ad 4; igitur &longs;umma <lb/>priorum ad &longs;ummam po&longs;teriorum e&longs;t ut 18 ad 20. Atqui Sector <lb/>AEG ad Sectorem AEF e&longs;t ut 3 ad 4 (id quod de &longs;imili <lb/>Sectore BEH ad Sectorem BEI intelligendum e&longs;t) igitur, cum <lb/>AEF &longs;it 16, AEG e&longs;t 12, & cum BEI &longs;it 4, BEH e&longs;t 3, ac <lb/>propterea &longs;umma Sectorum AEG & BEH e&longs;t ut 15 ad &longs;um­<lb/>mam Sectorum ACD & BCO ut 18. <!--neuer Satz-->In prima autem hypo­<lb/>the&longs;i quando erat AC ut 2 & AE ut 3, erat motuum Ratio ut <lb/>8 ad 6 2/3, quæ e&longs;t planè eadem cum Ratione 18 ad 15. <!--neuer Satz-->Cum <lb/>itaque eadem motuum Ratio &longs;equatur, &longs;ive AE &longs;it bes, &longs;ive <lb/>dodrans totius longitudinis AB, cur dodrantem potiùs quàm <lb/>be&longs;&longs;em pronunciemus, ni&longs;i aliunde doceamur? <lb/></s> </p> <p type="main"> <s id="s.003578"><emph type="center"/>CAPUT XVI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003579"><emph type="center"/><emph type="italics"/>An malus in motu navis habeat Rationem vectis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003580">NAvim impelli ventorum vi certum e&longs;t, qui velum implent <lb/>ex antennâ &longs;u&longs;pen&longs;um atque expa&longs;&longs;um, funibù&longs;que, quos <lb/>Propedes vocant, po&longs;teriori navis parti alligatum. </s> <s id="s.003581">Quoniam <lb/>verò, ut quotidiano u&longs;u didicimus, quò altiùs evecta fuerit an­<lb/>tenna, eò validiùs, cæteris paribus, navis à vento impellitur, <lb/>quæritur ab Ari&longs;totele quæ&longs;t.6. <emph type="italics"/>Cur quando antenna &longs;ublimior <lb/>fuerit, ii&longs;dem velis, & codem vento, celeriùs feruntur navigia?<emph.end type="italics"/><lb/>Cau&longs;am ille ex vectis rationibus petendam opinatur, qua&longs;i ma­<lb/>lus &longs;it vectis habens hypomochlium in ea carinæ parte, cui in­<lb/>figitur; potentia movens &longs;it ventus velum implens &longs;upremæ <lb/>mali parti applicatus, ubi antenna cum malo connectitur; pon­<lb/>dus verò &longs;it navigium: quò igitur potentia magis ab hypomo­<lb/>chlio abe&longs;t, plus habere momenti manife&longs;tum e&longs;t. </s> <s id="s.003582">Verùm non <lb/>ego hîc inani labore &longs;u&longs;cepto, ut Philo&longs;ophi dicto aliquam veri <pb pagenum="471" xlink:href="017/01/487.jpg"/>&longs;imilitudinem adjiciam, tempus conteram: Cui otium e&longs;t, <lb/>Authores legat. </s> </p> <p type="main"> <s id="s.003583">Si quæ&longs;tio e&longs;&longs;et, Cur longiores mali &longs;int magis obnoxij peri­<lb/>culo fractionis, facilè invenirem rationem vectis, quia ponde­<lb/>ris vicem &longs;ubeunt particulæ ip&longs;æ, quarum nexus per vim &longs;ol­<lb/>vendus e&longs;t in fractione; quò autem longior e&longs;t malus, ad mo­<lb/>tum partium, quæ dividuntur, majorem Rationem habet mo­<lb/>tus venti applicati longiori malo, quàm motus eju&longs;dem brevio­<lb/>ri malo applicati. </s> <s id="s.003584">Sed hîc, ubi de navis motu quæ&longs;tio e&longs;t, &longs;ive <lb/>altè a&longs;&longs;urgat malus, &longs;ive brevis &longs;it, &longs;emper eadem e&longs;t Ratio mo­<lb/>tûs venti velo excepti, atque navis. </s> <s id="s.003585">Quomodo enim pterna, <lb/>ide&longs;t ima mali calx, e&longs;&longs;e pote&longs;t extremus vectis in carinâ, cui <lb/>in&longs;eritur, velut in hypomochlio quie&longs;cens, navis verò tota &longs;i­<lb/>mul mota æquali motu, rationem habet ponderis à vecte impul­<lb/>&longs;i? </s> <s id="s.003586">nonne hypomochlium, pondus, & potentia æquali planè <lb/>motu moventur? </s> <s id="s.003587">neque enim velociùs movetur ventus velo ex­<lb/>ceptus, quàm ip&longs;um velum, nec velum velociùs quàm navis; <lb/>& cum ip&longs;a navi planè æqualiter movetur carinæ punctum, cui <lb/>malus infigitur. </s> <s id="s.003588">Quis autem motus per Vectem, qua Vectis e&longs;t <lb/>Facultas mechanica, huju&longs;modi æqualitatem admittit? </s> <s id="s.003589">Non <lb/>igitur malus in motu, quo navis progreditur, Rationem vectis <lb/>habere dicendus e&longs;t. </s> </p> <p type="main"> <s id="s.003590">Sit malus CD cujus pterna C in&longs;eratur carinæ AB, car­<lb/>che&longs;io autem D applicetur an­<lb/><figure id="id.017.01.487.1.jpg" xlink:href="017/01/487/1.jpg"/><lb/>tenna cum velo pendente, cujus <lb/>imæ extremitates navis lateribus <lb/>opportunè ad ventum excipien­<lb/>dum jungantur. </s> <s id="s.003591">Certum e&longs;t ma­<lb/>lum CD moveri &longs;emper &longs;ibi pa­<lb/>rallelum (ni&longs;i fortè aliquanto <lb/>plus extremitas D moveatur, &longs;i­<lb/>cut in homine plus caput move­<lb/>tur quàm pedes &longs;upra &longs;phæricam <lb/>terræ vel aquæ &longs;uperficiem, &longs;ed <lb/>hoc nihil refert) neque po&longs;&longs;e obtinere rationem vectis ni&longs;i <lb/>comparatè ad eum motum, quo circa C tanquam circa centrum <lb/>fieret conver&longs;io; quemadmodum &longs;i deprimenda e&longs;&longs;et prora & <lb/>elevanda puppis, ut carina AB non e&longs;&longs;et horizonti parallela, <pb pagenum="472" xlink:href="017/01/488.jpg"/>&longs;ed B deprimeretur infra planum horizontale, & A &longs;upra illud <lb/>elevaretur. </s> <s id="s.003592">Verùm (præterquam quod non hic e&longs;t navis mo­<lb/>tus, de quo di&longs;putatur) ob&longs;ervandum e&longs;t in navigiis minoribus, <lb/>quibus movendis unicus malus adhibetur, hunc &longs;tatui non in <lb/>medio navigio, &longs;ed magis accedere ad proram, in majoribus <lb/>autem navibus, quæ plures malos habent, maximum quidem <lb/>malum, cujus validi&longs;&longs;imæ &longs;unt vires, aliquanto quidem magis <lb/>ad puppim quàm ad proram accedere (&longs;i longitudo in &longs;uperiori <lb/>parte attendatur) ut in Oceano videre e&longs;t Anglicas, Gallicas, <lb/>Hollandicas naves, comparatè tamen ad carinam, majorem <lb/>carinæ partem puppim, minorem proram re&longs;picere. </s> <s id="s.003593">Id autem <lb/>eo con&longs;ilio factum e&longs;t, ne malus centro gravitatis navis re&longs;pon­<lb/>deat, neque exercere po&longs;&longs;it munus vectis deprimendo proram, <lb/>puppímque elevando. </s> <s id="s.003594">Quando enim magis ad proram accedit <lb/>malus, major pars navigij inter malum & puppim interjecta re­<lb/>nititur &longs;ua gravitate, ne elevetur, quando verò à medio pup­<lb/>pim versùs recedit, major pars navis, quæ deprimenda e&longs;&longs;et, <lb/>majorem aquæ re&longs;i&longs;tentiam invenit; ac proinde &longs;ervatà carinæ <lb/>po&longs;itione horizontali faciliùs navis movetur. </s> <s id="s.003595">Hinc tardiorem <lb/>fieri navigij cur&longs;um contingit, vel quia perperam collocatus e&longs;t <lb/>malus, vel quia pondera in navi non &longs;unt ritè di&longs;tributa, adeò <lb/>ut à malo vix ab&longs;it centrum gravitatis navigij onu&longs;ti; tunc enim <lb/>depre&longs;sâ prorâ & carinâ ad horizontem inclinatâ major vis ob­<lb/>viæ aquæ re&longs;i&longs;tit. </s> <s id="s.003596">Quare tantum abe&longs;t malus à ratione vectis, <lb/>vi cujus progrediatur navigium, ut potius caveatur, ne vectis <lb/>munus ille exerceat, motum aliquem efficiendo, qui celeritati <lb/>non parum officeret. </s> </p> <p type="main"> <s id="s.003597">In motu autem majoris navigij pluribus malis in&longs;tructi non <lb/>&longs;olus malus, qui præcipuus e&longs;t & maximus, attenditur, &longs;ed etiam <lb/>reliqui: potior tamen ad provehendam navim e&longs;t malus, qui à <lb/>medio ad proram accedit, quippe qui navim trahit; nam qui â <lb/>centro gravitatis puppim versùs recedit, navim impellit potiùs, <lb/>quàm trahat: quamquam ille, qui ad puppim proximè &longs;pectat, <lb/>& velum habet triangulare, maximè juvat, ut gubernatoris pro­<lb/>po&longs;ito, qui clavum regit, ob&longs;ecundet ad navis cur&longs;um in alteru­<lb/>tram partem dirigendum. </s> <s id="s.003598">Verum quicumque malus con&longs;ide­<lb/>retur, in nullo rationem vectis reperies, &longs;ive ad impellendam, <lb/>&longs;ive ad trahendam navim. </s> </p> <pb pagenum="473" xlink:href="017/01/489.jpg"/> <p type="main"> <s id="s.003599">At, inquis, &longs;i adver&longs;o flumine deducendum &longs;it navigium &longs;i­<lb/>ve à nautica turbâ &longs;ive ab equis trahentibus, cur funis ma­<lb/>lo, non autem proræ, alligatur, &longs;i nihil confert facilitatis appli­<lb/>catio potentiæ trahentis medio fune ad majorem altitudinem à <lb/>carinâ? </s> <s id="s.003600">Ego verò ex te, qui&longs;quis hæc objicis, quæro, cur jidem <lb/>nautæ &longs;i remulco navim trahere aggrediantur, funem navi non <lb/>tam altè alligant; &longs;i ex vectis rationibus illa altitudo aliquod af­<lb/>fert compendium laboris in trahendo. </s> <s id="s.003601">Sed &longs;atis utrique quæ&longs;tio­<lb/>ni factum videbis, &longs;i ob&longs;erves non planè æqualem e&longs;&longs;e in uni­<lb/>ver&longs;o alveo aquæ altitudinem, ac proinde neque po&longs;&longs;e navim <lb/>æquè &longs;emper abe&longs;&longs;e à fluminis ripâ, in qua trahentes progre­<lb/>diuntur; idcircò longiore fune opus e&longs;t, qui &longs;uo pondere &longs;pon­<lb/>te curvatus aquam &longs;ecaret, & trahentium laborem augeret, aut <lb/>in occultum aliquem &longs;ub aquis latentem obicem incurreret non <lb/>&longs;ine gravi incommodo, &longs;i funis extremitas depre&longs;&longs;iori loco navi­<lb/>gij alligaretur; propterea malo altiùs adnectitur, eo quoque <lb/>con&longs;ilio, ut &longs;i quæ virgulta aut arbu&longs;culæ &longs;ecundùm fluminis <lb/>ripam occurrant, minori impedimento &longs;int funi obliquè incli­<lb/>nato, quàm &longs;i horizonti e&longs;&longs;et ferè parallelus. </s> <s id="s.003602">Qui verò navim <lb/>remulco trahunt, non adeò longè ab illa abe&longs;&longs;e coguntur, nec <lb/>huju&longs;modi impedimentis obnoxij &longs;unt; ideò breviore fune <lb/>utuntur, quem proræ alligant. </s> <s id="s.003603">Cæterùm nullæ vectis vires <lb/>exercentur; non enim prora infra aquam deprimi, & puppis <lb/>elevari pote&longs;t: id quod &longs;i contingeret, prora magis demer&longs;a <lb/>plus inveniret re&longs;i&longs;tentiæ ab aquâ dividendâ. </s> </p> <p type="main"> <s id="s.003604">Quid igitur, ais, cau&longs;æ e&longs;t, quòd antennâ u&longs;que ad carche­<lb/>&longs;ium D elevatâ, magis promovetur navis, quàm &longs;i tantummodo <lb/>u&longs;que ad E attolleretur? </s> <s id="s.003605">quandoquidem nulla vectis ratio hîc <lb/>habetur. </s> <s id="s.003606">Eos, qui cum Ari&longs;totele &longs;entiunt, æquivocatione la­<lb/>borare facilè o&longs;tenditur: quid enim refert, utrùm antenna ma­<lb/>gis an minùs elevetur, &longs;i potentia, videlicet ventus velum im­<lb/>plens, illi mali parti applicata intelligeretur, cui antenna ad­<lb/>nectitur? </s> <s id="s.003607">hæc autem funibus, quos <foreign lang="greek">mesou<gap/>i/as</foreign> vocant, &longs;ur&longs;um <lb/>trahitur, &longs;emperque, &longs;ive altior, &longs;ive depre&longs;&longs;ior &longs;it, adnectitur <lb/>carche&longs;io in D: quemadmodum nauta fune in D alligato na­<lb/>vim trahens, &longs;emper in D applicatus intelligitur, quamvis hu­<lb/>miliore in loco, quàm D, con&longs;tituatur. </s> <s id="s.003608">Verùm non ibi vis <lb/>venti præcisè intelligenda e&longs;t, ubi antenna e&longs;t, &longs;ed toti malo <pb pagenum="474" xlink:href="017/01/490.jpg"/>aut ejus parti applicatur, quæ re&longs;pondet velo non &longs;olùm anten­<lb/>næ cornibus, &longs;ed etiam navis lateribus alligato: velum autem <lb/>in humiliore loco minus recipit venti, quia alta majorum na­<lb/>vium puppis (ni&longs;i ventus ex latere &longs;piret) vento oppo&longs;ita illum <lb/>&longs;ubtrahit velo, & præterea ventus, qui in navis puppim & la­<lb/>tera illiditur, reflectitur, & proximas venti partes turbat, atque <lb/>alior&longs;um dirigit, vel &longs;altem illarum impetum imminuit; ex quo <lb/>oritur minori vi impelli velum. </s> <s id="s.003609">At partes venti &longs;ublimiores ab <lb/>his inferioribus reflexis, vel nihil, vel mitiùs turbantur, atque <lb/>adeò plures ad implendum velum majore vi accurrunt. </s> </p> <p type="main"> <s id="s.003610">Adde (his etiam mente &longs;eclu&longs;is) ventum in &longs;ublimiore loco <lb/>multo validiorem e&longs;&longs;e, quàm in inferiore, ac propterea quò al­<lb/>tius attollitur velum, non &longs;olùm majorem, &longs;ed etiam validio­<lb/>rem ventum excipit, quo fit, ut incitatior &longs;it navigij motus. </s> <lb/> <s id="s.003611">Neque de hoc venti di&longs;crimine dubitare poterit cui contin­<lb/>gat iter habere in ampla planitie arboribus & ædificiis va­<lb/>cuâ vento flante; &longs;i enim ex equo de&longs;iliat, & humi &longs;edeat, <lb/>manife&longs;tè percipiet, quanto minore vi impetatur à vento. </s> <s id="s.003612">Id <lb/>quod pariter ex ipsâ veli figurâ arguitur; &longs;ive enim velum trian­<lb/>gulare fuerit, & obliquâ antennâ erigatur ita ut qua&longs;i aurem <lb/>leporis imitetur, altiori vento, utpote vehementiori, pars veli <lb/>&longs;trictior objicitur; &longs;ive pluribus quadrangularibus velis in&longs;trua­<lb/>tur navigium ita, ut alia &longs;uperiora, &longs;cilicet dolones, alia infe­<lb/>riora &longs;int, videlicet Acatia; quæ &longs;upra Corbem &longs;tatuuntur, <lb/>non &longs;olum minora &longs;unt inferiore velo, &longs;ed etiam eorum &longs;upre­<lb/>ma pars longè &longs;trictior e&longs;t ba&longs;i, ut nimirum minus recipiat <lb/>venti validioris: propterea ingruente tempe&longs;tate primùm &longs;u­<lb/>periora vela deprimuntur, at majori ventorum vi &longs;ubducan­<lb/>tur; eriguntur autem celeritatis cau&longs;a, ut &longs;i quando effusè fu­<lb/>gere opus &longs;it. </s> </p> <p type="main"> <s id="s.003613">Ecce igitur citra omnem vectis rationem, <emph type="italics"/>Cur quando anten­<lb/>na &longs;ublimior fuerit, ii&longs;dem velis, & vento codem, celeriùs feruntur <lb/>navigia:<emph.end type="italics"/> quia &longs;cilicet velum altiùs &longs;ublatum & plus venti, & <lb/>validiorem ventum recipit. </s> <s id="s.003614">Quod &longs;i ad vectis rationes confu­<lb/>giendum e&longs;&longs;et, non quæreretur, cur celeriùs ferantur navigia, <lb/>&longs;ed, cur faciliùs? </s> <s id="s.003615">Nam vectis longitudo (ni&longs;i fortè in vecte ter­<lb/>tij generis, cujus nullum ve&longs;tigium deprehenditur in malo na­<lb/>vis) non celeritatem motûs ponderi conciliat, &longs;ed facilitatem, <pb pagenum="475" xlink:href="017/01/491.jpg"/>ita ut po&longs;ito longiore vecte potentia &longs;ervans eandem &longs;ui motûs <lb/>velocitatem faciliûs quidem moveat propo&longs;itum pondus &longs;ed <lb/>tardiùs quàm breviore vecte, po&longs;ità eádem ponderis ab hypo­<lb/>mochlio di&longs;tantiâ: Ac propterea, &longs;i in hoc navis motu, de quo <lb/>quæ&longs;tio e&longs;t, intercederet ratio vectis, idem ventus eadem vela <lb/>altiùs &longs;ublata implens eâdem quidem velocitate moveretur, &longs;ed <lb/>tardiùs navim moveret, quamquam faciliùs, hoc e&longs;t magis <lb/>onu&longs;tam. </s> <s id="s.003616">Id autem à vero longi&longs;&longs;imè abe&longs;&longs;e te&longs;tatur experien­<lb/>tia; quæ idcirco confirmat navigij malo nihil e&longs;&longs;e cum vecte <lb/>commercij ad navim promovendam. <lb/></s> </p> <p type="main"> <s id="s.003617"><emph type="center"/>CAPUT XVII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003618"><emph type="center"/><emph type="italics"/>An ex vectis rationibus pendeat u&longs;us anchoræ.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003619">QUandoquidem nauticas aliquot quæ&longs;tiones cum Ari&longs;tote­<lb/>le &longs;uperioribus capitibus examinare placuit, liceat & hanc <lb/>addere, quæ ad u&longs;um anchoræ &longs;pectat in firmanda navi, ne à <lb/>fluctibus, aut à vento abripiatur: tranquillo enim mari, aut in <lb/>lacu quie&longs;cente, &longs;ua &longs;ponte &longs;ub&longs;i&longs;tit navis, nec anchoræ ope in­<lb/>diget, ut &longs;ua in &longs;tatione permaneat. </s> <s id="s.003620">Et quidem ip&longs;a navis gra­<lb/>vitas cum &longs;uis in&longs;trumentis, & onus quod illa ferre pote&longs;t (cu­<lb/>jus gravitas æquat navigij gravitatem) &longs;atis per &longs;e re&longs;i&longs;tunt, <lb/>nec facilè cuilibet auræ aut fluctui cedunt. </s> <s id="s.003621">Quare major e&longs;&longs;e <lb/>debet vis venti, aut fluctuum, aut profluentis, quàm ut illi ob­<lb/>&longs;i&longs;tere valeat univer&longs;um navigij pondus, ad hoc ut &longs;it opus an­<lb/>chorâ, qua navigium firmetur. </s> </p> <p type="main"> <s id="s.003622">In anchorâ autem &longs;pectanda e&longs;t & gravitas, & figura; utra­<lb/>que enim juvat: aliquando &longs;i quidem &longs;olum anchoræ pondus <lb/>&longs;ufficit, ne placidiores fluctus, aut fluminis impetus, aut lenis <lb/>flatus navim &longs;ecum rapiant. </s> <s id="s.003623">Sic legi&longs;&longs;e me memini navim à <lb/>naufragio anchoris omnibus de&longs;titutam in &longs;tatione totam <lb/>noctem quievi&longs;&longs;e &longs;ecuram firmatam &longs;acculo, quo mille trecen­<lb/>ti Hi&longs;pani Crucigeri (octo Reales &longs;ingulis Crucigeris tribuun­<lb/>tur) continebantur, rudentis autem munus &longs;upplebat evolutus <pb pagenum="476" xlink:href="017/01/492.jpg"/>telæ &longs;capus: qui enim fluctus navim aliò propellere potui&longs;&longs;ent, <lb/>non &longs;atis habebant virium, ut etiam illud argenti pondus ma­<lb/>ris fundo incumbens & navi connexum pariter trahere po&longs;&longs;ent. </s> <lb/> <s id="s.003624">Simili igitur ratione anchora, licèt duriori &longs;olo dentem infige­<lb/>re non valeat, aliquando &longs;uo pondere navim firmabit. </s> <s id="s.003625">Re&longs;pon­<lb/>det autem anchoræ gravitas oneri, quod ferre pote&longs;t navis, ea <lb/>Ratione, ut pro oneris libris 40000 (hoc e&longs;t 20 Amphoris aut <lb/>doliis, &longs;ingulorum quippe doliorum gravitas &longs;tatuitur librarum <lb/>bis mille, & &longs;ingulis libris unciæ &longs;exdecim tribuuntur) ferri li­<lb/>bras centum & decem habeat primaria & maxima anchora, <lb/>&longs;ecunda habeat primæ dodrantem, tertia be&longs;&longs;em, quarta &longs;emi&longs;­<lb/>&longs;em. </s> <s id="s.003626">Rudentis vero, cui anchora adnectitur, pondus ferè <lb/>ponitur duplum &longs;e&longs;quiquartum gravitatis &longs;uæ anchoræ. </s> <s id="s.003627">Quam­<lb/>quam non omnino &longs;ervetur hæc Ratio ponderis anchoræ in in­<lb/>gentibus navigiis, quæ nimirum &longs;uâ gravitate maximè re­<lb/>&longs;i&longs;tunt fluctuum impul&longs;ioni, ac proinde minore anchorâ opus <lb/>habent. </s> </p> <p type="main"> <s id="s.003628">Primariæ anchoræ poti&longs;&longs;imus u&longs;us communiter e&longs;t, cùm va­<lb/>lidior tempe&longs;tas navim aggreditur; &longs;ecundæ, ut navis in &longs;ta­<lb/>tione quie&longs;cat; tertiam adhibent nautæ, ut duabus anchoris ad <lb/>diver&longs;a plagas con&longs;titutis (puta, alterâ ad Sub&longs;olanum, alterâ <lb/>ad Boream aut ad Borrhapeliotem,) vento & fluctibus navis re­<lb/>&longs;i&longs;tat validiùs, nec abrepta à fluctibus anchoram pariter &longs;ecum <lb/>rapiat, &longs;ed tantum alternis motibus qua&longs;i circa centrum agite­<lb/>tur: Quartam demum lintre transferunt procul à navi juxta lon­<lb/>gitudinem funis adnexi ducentorum circiter cubitorum, quem <lb/>machinâ ad id de&longs;tinatâ colligentes accedunt ad anchoram, & <lb/>&longs;tationem commutant, aut portum intrant, &longs;eu ab illo exeunt, <lb/>ubi ce&longs;&longs;at ventus, aut adver&longs;us &longs;pirat. </s> </p> <p type="main"> <s id="s.003629">Ad firmandam verò navim plurimum habet momenti longi­<lb/>tudo ip&longs;a rudentis; &longs;atis enim manife&longs;tum e&longs;t, quantâ vi opus <lb/>&longs;it, ut longior funis intendatur, qui ce&longs;&longs;ante externâ vi illico <lb/>&longs;inuatur; ac propterea vehementi conatu ventorum ac fluctuum <lb/>navis impellenda e&longs;t, ut rudens intentus anchoram trahat. </s> <lb/> <s id="s.003630">Varia e&longs;t autem Rudentis longitudo pro anchorarum Ratione; <lb/>longitudo &longs;i quidem rudentis anchoræ primariæ cubitos habet <lb/>centum viginti, &longs;ecundæ cubitos centum, tertiæ cubitos octo­<lb/>ginta: quò enim adversùs validiorem impetum repugnandum <pb pagenum="477" xlink:href="017/01/493.jpg"/>e&longs;t, eò longior adhibetur rudens, ut difficiliùs intendatur, ac <lb/>idcirco fractionis periculo minùs obnoxius &longs;it, & venti <lb/>fluctuumve impetus in rudente intendendo eli&longs;us minus vi­<lb/>rium habeat ad rapiendam &longs;imul cum navi anchoram. </s> <s id="s.003631">Hine <lb/>ingentes bellicæ naves in Oceano ferè &longs;emper primariam an­<lb/>choram demittunt, & tres aut quatuor rudentes capitibus in­<lb/>vicem firmiter colligatis in unam longitudinem productos adji­<lb/>ciunt; vix enim tanta e&longs;&longs;e pote&longs;t fluctuum aut venti vis, quæ <lb/>valeat tam longum rudentem intendere atque dirumpere, ni&longs;i <lb/>fortè ad navis latus aut ad &longs;copulum colli&longs;us atteratur. </s> <s id="s.003632">Id quod <lb/>aliis quoque nautis placet tùm propter eandem cau&longs;am, tùm ut <lb/>longiùs à littore con&longs;i&longs;tere po&longs;&longs;it navis, & anchora arenæ infi­<lb/>gi, etiam&longs;i altior &longs;it aqua. </s> <s id="s.003633">Mihi &longs;anè contingit nautarum incu­<lb/>riam experiri in Albi fluvio; cùm enim anchoram breviore ru­<lb/>dente demi&longs;i&longs;&longs;ent, nocturno maris æ&longs;tu intume&longs;centibus undis <lb/>ita elevatum e&longs;t navigium, ut ex prorâ penderet &longs;u&longs;pen&longs;a an­<lb/>chora, nó&longs;que dormientes æ&longs;tus abriperet; quos demum exci­<lb/>tavit fragor ex colli&longs;ione cum altero navigio, in quod tanto im­<lb/>petu impacti fuimus, ut abrupto fune &longs;capham ami&longs;erimus. </s> </p> <p type="main"> <s id="s.003634">Sed quod ad anchoræ formam attinet, non eadem omnibus <lb/>e&longs;t figura; navigia enim, quorum in majoribus fluminibus u&longs;us <lb/>e&longs;t, ut noctu in medio alveo &longs;ub&longs;i&longs;tant, anchoram habent qua­<lb/>tuor aduncis brachiis in&longs;tructam; cuju&longs;modi pariter &longs;unt trire­<lb/>mium anchoræ. </s> <s id="s.003635">At in Oceano navium anchoræ non ni&longs;i duo <lb/>habent brachia ad angulum acutum inflexa cum &longs;capo; ne ve­<lb/>tò demi&longs;&longs;a anchora pror&longs;us jaceat in maris fundo, &longs;capo prope <lb/>annulum adnectitur ligneum tran&longs;ver&longs;arium (cujus gravitas <lb/>e&longs;t ferè &longs;ubquintupla gravitatis anchoræ, &longs;i tamen etiam fer­<lb/>reos clavos, quibus firmatur, in computationem admittas) eju&longs;­<lb/>dem cum Scapo longitudinis, adeò ut jacente utroque brachio <lb/>Scapus tran&longs;ver&longs;ario &longs;ecundum extremitatem innixus obliquè <lb/>inclinetur. </s> <s id="s.003636">Ex quo etiam fit, ut extremæ brachiorum palmulæ <lb/>obliquè occurrentes maris fundo faciliùs in &longs;ubjectum &longs;olum <lb/>penetrent. </s> <s id="s.003637">Quando igitur vehementior e&longs;t fluctuum impetus, <lb/>aut venti impul&longs;us validior, quàm ut illi re&longs;i&longs;tere po&longs;&longs;it ip&longs;a an­<lb/>choræ gravitas, intento rudente tanti&longs;per abripit cum navi an­<lb/>choram, quæ maris fundum &longs;ulcans, ubi brachiorum palmu­<lb/>læ arenis aliquantulum immer&longs;æ inæqualem invenerint &longs;ubjecti <pb pagenum="478" xlink:href="017/01/494.jpg"/>&longs;oli re&longs;i&longs;tentiam (quocumque tandem ex capite oriatur hæc re­<lb/>&longs;i&longs;tentiæ inæqualitas) po&longs;itionem mutat, nec ampliùs jacet <lb/>utrumque brachium, &longs;ed illud, cui minùs ob&longs;i&longs;titur, elevatur, <lb/>adnitente etiam ligneo tran&longs;ver&longs;ario, cui naturalis e&longs;t in aquâ <lb/>po&longs;itio horizonti parallela, quam acquirens ita anchoram con­<lb/>vertit, ut Dens maris fundo inhærens magis in illud infigatur <lb/>tùm urgente deor&longs;um ip&longs;ius anchoræ gravitate, tùm trahente <lb/>ip&longs;a navi, quam fluctus aut ventus impellit; cùm etenim bra­<lb/>chium cum &longs;capo acutum angulum con&longs;tituat, non ad perpen­<lb/>diculum, &longs;ed obliquè fundum ingreditur, & idcirco in illud <lb/>profundiùs penetrat. </s> </p> <p type="main"> <s id="s.003638">Cum itaque anchoræ &longs;capo alij duplicem, alij triplicem tri­<lb/>buant alterius brachij longitudinem, hæc utique major e&longs;t, <lb/>quàm di&longs;tantia inter extremos anchoræ dentes, non enim bra­<lb/>chia cum &longs;capo rectum &longs;ed acutum angulum, ut dictum e&longs;t, <lb/>con&longs;tituunt. </s> <s id="s.003639">Ad hanc igitur extremorum dentium di&longs;tantiam <lb/>major tran&longs;ver&longs;arij longitudo majorem habet Rationem, quàm <lb/>minor; e&longs;t autem longitudini &longs;capi par tran&longs;ver&longs;arij longitudo; <lb/>quare longioris anchoræ tran&longs;ver&longs;arium longius e&longs;t, ejú&longs;que <lb/>conver&longs;io, ut &longs;e horizonti parallelum &longs;tatuat, magis juvat an­<lb/>choræ conver&longs;ionem, ut dens inferior profundiùs in arenam <lb/>infigatur. </s> </p> <p type="main"> <s id="s.003640">Sit primùm anchoræ &longs;capus AB duplex longitudinis bra­<lb/><figure id="id.017.01.494.1.jpg" xlink:href="017/01/494/1.jpg"/><lb/>chij AC, & prope an­<lb/>nulum in B æquale tran&longs;­<lb/>ver&longs;arium EF adjiciatur <lb/>ad angulos rectos, ea ta­<lb/>men conditione, ut ja­<lb/>centibus brachiis AC <lb/>& AD in plano hori­<lb/>zontali, tran&longs;ver&longs;arium <lb/>&longs;it in plano verticali, ejú&longs;­<lb/>que altera extremitas, ex. </s> <lb/> <s id="s.003641">gr.F.maris fundum con­<lb/>tingat, altera E &longs;ublimis <lb/>&longs;it, ac proinde &longs;capus AB <lb/>inclinetur ad horizon­<lb/>tem grad. <!-- REMOVE S-->30. Vento, aut fluctu, navim impellente intenditur <pb pagenum="479" xlink:href="017/01/495.jpg"/>rudens, & &longs;capi extremitas B annulo proxima elevatur, nec <lb/>ampliùs tran&longs;ver&longs;arium in F incumbit arenæ; propterea bra­<lb/>chiorum palmulæ C & D in triangulum conformatæ, dum &longs;i­<lb/>mul cum navi trahuntur, &longs;e &longs;e profundiùs in arenam in&longs;inuant: <lb/>&longs;ed &longs;i inæqualem offendant re&longs;i&longs;tentiam, aut altera, ex.gr. <!-- REMOVE S-->C, <lb/>profundiùs infigatur præ reliquâ (&longs;ive ex &longs;ubjecti &longs;oli diver&longs;i­<lb/>tate, &longs;ive quia navis in tran&longs;ver&longs;um acta trahit anchoræ caput B <lb/>in latus, & brachij alterius extremitas de&longs;cribens circa A in <lb/>&longs;olo arcum versùs navim profundiùs infigitur, atque adeò re­<lb/>liqua extremitas oppo&longs;iti brachij in contrarium mota circa A, <lb/>vix terram mordet) vis in B trahens, neque valens pariter <lb/>utrumque brachium trahere, cogitur circa C, tanquam circa <lb/>centrum, &longs;eu potiùs tanquam circa polum, moveri. </s> <s id="s.003642">Et quia <lb/>punctum B &longs;ublimius e&longs;t puncto C, nece&longs;&longs;e e&longs;t ita huju&longs;modi <lb/>conver&longs;ionem fieri, ut oppo&longs;ita extremitas D elevetur, atque <lb/>ex fundo extrahatur. </s> <s id="s.003643">Cúmque jam tran&longs;ver&longs;arium non æqua­<lb/>liter hinc & hinc retineatur per vim in plano verticali, &longs;ed ejus <lb/>&longs;uperior pars BE versùs C inclinetur, conatur po&longs;itionem ho­<lb/>rizontalem acquirere, ejú&longs;que inferior pars BF ad latus decli­<lb/>nans a&longs;cendit, juvátque ip&longs;ius brachij AD a&longs;cen&longs;um; ex quo <lb/>fit demum centrum gravitatis totius anchoræ imminere palmu­<lb/>læ C, quæ propterea etiam urgente gravitate profundiùs in­<lb/>figitur. </s> </p> <p type="main"> <s id="s.003644">In hac lignei tran&longs;ver&longs;arij conver&longs;ione ob&longs;ervandum e&longs;t par­<lb/>tem alteram &longs;ublimiorem BE per vim in aquâ deprimi, partem <lb/>autem inferiorem BF in aquâ &longs;ponte a&longs;cendere, ac proinde, <lb/>propter intermediam gravitatem in B, illam re&longs;i&longs;tere huic &longs;ur­<lb/>&longs;um conánti, atque ideò illam habere rationem hypomochlij, <lb/>hanc potentiæ, pondus verò e&longs;&longs;e in B, quod & convertitur: <lb/>non quidem quia totum pondus &longs;it in B, &longs;ed quia totius ancho­<lb/>ræ centrum gravitatis e&longs;t in &longs;capo AB, adeóque intelligitur <lb/>applicatum puncto B, quamvis ip&longs;ius centri gravitatis conver­<lb/>&longs;io fiat circa extremitatem C manentem. </s> </p> <p type="main"> <s id="s.003645">At verò &longs;i &longs;capus AK fuerit triplex brachij AC, etiam tran&longs;­<lb/>ver&longs;arium HI &longs;capo æquale e&longs;t eju&longs;dem brachij triplex: hinc <lb/>fit ip&longs;ius longioris tran&longs;ver&longs;arij HI vim, qua &longs;e horizontale <lb/>&longs;tatuat in aquâ, majorem e&longs;&longs;e, quàm brevioris EF; lignum <lb/>enim longiùs difficiliùs in aquâ erectum retinetur. </s> <s id="s.003646">Quamvis <pb pagenum="480" xlink:href="017/01/496.jpg"/>autem eadem &longs;it Ratio FE ad BE, quæ e&longs;t IH ad KH, ta­<lb/>men major e&longs;t Ratio motûs ip&longs;ius K ad motum centri gravitatis <lb/>circa extremitatem C manentem, quàm &longs;it Ratio motûs ip&longs;ius B <lb/>ad motum centri gravitatis circa idem punctum C: in illa enim <lb/>conver&longs;ione centrum gravitatis exi&longs;tens in aliquo puncto lon­<lb/>gitudinis AK elevari vix pote&longs;t ad majorem altitudinem, quàm <lb/>&longs;it CL; quia in longiore anchorâ AK centrum gravitatis ma­<lb/>gis recedens ab extremitate A, quàm in anchorâ breviore AB, <lb/>magis imminet palmulæ C, eámque profundiùs in arenam in­<lb/>figit; ideóque &longs;i fortè &longs;it inter L & K, atque ex inclinatione <lb/>&longs;capi ad horizontem paulò altius exi&longs;teret quàm CL, &longs;i C ma­<lb/>neret in &longs;uperficie fundi maris, ip&longs;a depre&longs;&longs;io puncti C infra il­<lb/>lam &longs;uperficiem demit aliquid ex altitudine. </s> </p> <p type="main"> <s id="s.003647">Nam quod &longs;pectat ad centrum gravitatis anchoræ longioris, <lb/>certum e&longs;t illud non removeri ab extremitate Scapi A &longs;ecun­<lb/>dùm eandem Rationem, &longs;ecundùm quam ejus longitudo pro­<lb/>ducitur: &longs;i enim &longs;capus e&longs;&longs;et longitudo pari & æquabili cra&longs;&longs;itie <lb/>ducta, utique &longs;icut AK e&longs;t ip&longs;ius AB &longs;e&longs;quialtera, etiam cen­<lb/>tri gravitatis di&longs;tantia ab A in &longs;capo longiore e&longs;&longs;et &longs;e&longs;quialtera <lb/>di&longs;tantiæ centri gravitatis ab A in Scapo breviore. </s> <s id="s.003648">Quoniam <lb/>verò & pars BK aliquanto decremento deficit à cra&longs;&longs;itie reli­<lb/>quæ partis BA, & pro centro gravitatis totius anchoræ atten­<lb/>denda e&longs;t non &longs;olius &longs;capi gravitas, &longs;ed & brachiorum, mani­<lb/>fe&longs;tum e&longs;t centrum gravitatis anchoræ longioris removeri ab A <lb/>minùs, quàm in Ratione &longs;e&longs;quialtera. </s> <s id="s.003649">Atqui circa punctum A <lb/>(quando jacent brachia, & elevari incipit extremitas altera &longs;ca­<lb/>pi) moventur B & K pro Ratione di&longs;tantiarum, hoc e&longs;t in Ra­<lb/>tione &longs;e&longs;quialtera; igitur motus ip&longs;ius K ad motum &longs;ui centri <lb/>gravitatis e&longs;t in majore Ratione, quàm motus puncti B ad mo­<lb/>tum &longs;ui centri gravitatis. </s> </p> <p type="main"> <s id="s.003650">Hinc e&longs;t intento rudente faciliùs pro rata portione elevari <lb/>extremitatem K longioris &longs;capi, quam B brevioris, & centrum <lb/>gravitatis inter A & K, hoc e&longs;t inter hypomochlium & poten­<lb/>tiam, habere rationem ponderis, quod elevatur vecte &longs;ecundi <lb/>generis AK. <!-- KEEP S--></s> <s id="s.003651">Quia autem facta elevatione puncti K jacentibus <lb/>adhuc brachiis, po&longs;tea fieri debet conver&longs;io circa palmulam C <lb/>manentem, tunc punctum C habet rationem hypomochlij, & <lb/>pondus intelligitur e&longs;&longs;e centrum gravitatis interjectum inter K <pb pagenum="481" xlink:href="017/01/497.jpg"/>& C, &longs;i minus &longs;it intervallum inter K & centrum gravitatis, <lb/>quàm inter K & hypomochlium C, cuju&longs;modi e&longs;&longs;et, &longs;i cen­<lb/>trum gravitatis e&longs;&longs;et citra L versùs K, & e&longs;&longs;et vectis curvus &longs;e­<lb/>cundi generis. </s> <s id="s.003652">Quòd &longs;i magis di&longs;tat centrum gravitatis à <lb/>puncto K, quàm ab eodem puncto K di&longs;tet punctum C, vectis <lb/>e&longs;t curvus primi generis. </s> <s id="s.003653">Quid autem, inquis, &longs;i pari interval­<lb/>lo di&longs;ter punctum K à puncto C, atque à centro gravitatis? </s> <s id="s.003654">cu­<lb/>ju&longs;modi generis vectis erit? </s> <s id="s.003655">primi-ne? </s> <s id="s.003656">an &longs;ecundi? </s> </p> <p type="main"> <s id="s.003657">Re&longs;pondeo in vecte hoc curvo, cujus altera extremitas ma­<lb/>net, & pondus non ad perpendiculum, neque motu recto in <lb/>plano verticali, &longs;ed conver&longs;ione elevatur, attendenda e&longs;&longs;e pla­<lb/>na, in quibus tùm potentia, tùm pondus propriam conver&longs;io­<lb/>nem perficiunt; his autem planis parallelum concipe aliud pla­<lb/>num, quod per extremitatem C manentem tran&longs;eat, quod pla­<lb/>num &longs;i interjectum fuerit inter illa plana conver&longs;ionum, vectis <lb/>erit primi generis, quia hypomochlium e&longs;t inter potentiam & <lb/>pondus; &longs;in autem hoc extremum fuerit, & medium locum ob­<lb/>tineat planum, in quo convertitur centrum gravitatis, vectis <lb/>erit &longs;ecundi generis. </s> </p> <p type="main"> <s id="s.003658">Facta demùm conver&longs;ione ita, ut tran&longs;ver&longs;arium ligneum <lb/>po&longs;itionem habeat horizontalem, & utrumque brachium in <lb/>eodem &longs;it plano verticali; quia faciliùs elevatur K quàm B, & <lb/>tran&longs;ver&longs;arium HI longius majorem habet vim &longs;u&longs;tinendi, <lb/>quàm tran&longs;ver&longs;arium EF brevius, hinc e&longs;t brachium AC ma­<lb/>gis inclinari ad &longs;ubjectum maris planum horizontale, ac prop­<lb/>terea etiam validiùs in arenam infigi, quando à navi trahitur <lb/>anchora. <lb/></s> </p> <p type="main"> <s id="s.003659"><emph type="center"/>CAPUT XVIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003660"><emph type="center"/><emph type="italics"/>Plures Vectis u&longs;us exponuntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003661">QUod &longs;uperiore libro præ&longs;titimus libræ atque &longs;tateræ u&longs;um <lb/>extendentes, & hîc præ&longs;tare operæ pretium fuerit, tum <lb/>ut vectis natura ex uberiori utilitate innote&longs;cat, tum ut fax ali-<pb pagenum="482" xlink:href="017/01/498.jpg"/>qua tyronibus præferatur viam common&longs;trando, qua &longs;imiles <lb/>u&longs;us po&longs;&longs;int pro opportunitate excogitare. </s> </p> <p type="main"> <s id="s.003662"><emph type="center"/>PROPOSITIO I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003663"><emph type="center"/><emph type="italics"/>Duplex Vectis genus in uno vecte conjungere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003664">SÆpi&longs;&longs;ime contingit unico quidem vecte nos uti, re tamen <lb/>vera duplicem e&longs;&longs;e vectem; quemadmodum cùm ingentis <lb/>alicujus &longs;axi extremitati vectem &longs;ubjicimus, quo extremitatem <lb/>illam attollimus. </s> <s id="s.003665">Sit enim &longs;axum, cujus centrum gravitatis S, <lb/><figure id="id.017.01.498.1.jpg" xlink:href="017/01/498/1.jpg"/><lb/>& &longs;ubjecto vecte AB ha­<lb/>bente hypomochlium in <lb/>C attollatur extremitas F, <lb/>manente extremitate E: <lb/>utique vectis primi gene­<lb/>ris e&longs;t AB; &longs;ed &longs;i rem at­<lb/>tentiùs perpendamus, <expan abbr="etiã">etiam</expan> <lb/>longitudo FE, aut potiùs <lb/>BE vectis e&longs;t &longs;ecundi ge­<lb/>neris habens impo&longs;itum pondus S, & fulcrum in E; at­<lb/>que quò magis &longs;upra horizontem elevatur, linea Directio­<lb/>nis SD magis accedit versùs E, ex quo oritur movendi fa­<lb/>cilitas; quam juvat Potentiæ A depre&longs;&longs;io, ex qua fit ut B ma­<lb/>gis accedens ad F, magis etiam recedat ab hypomochlio E. <!-- KEEP S--></s> <lb/> <s id="s.003666">Manife&longs;tum e&longs;t autem pondere accedente ad hypomochlium, <lb/>& potentiâ ab eodem recedente, majorem fieri Rationem mo­<lb/>tûs Potentiæ ad motum Ponderis, atque adeò augeri movendi <lb/>facilitatem. </s> <s id="s.003667">Quare momenta potentiæ in A &longs;u&longs;tinentis &longs;axum <lb/>ea &longs;unt, quæ componuntur ex Ratione AC ad CB, & Ratio­<lb/>ne BE ad EI. <!-- KEEP S--></s> <s id="s.003668">Sed de hoc nullus mihi hîc &longs;ermo; quia vel duo <lb/>vectes &longs;unt, ut explicatum e&longs;t, alter quidem ab ip&longs;o pon­<lb/>dere non &longs;ejunctus FE, alter verò ab eo di&longs;tinctus AB; vel <lb/>&longs;i unicus intelligatur vectis, qui ponderi applicatur, hic <lb/>&longs;anè ad unum pertinet genus non ad duo, ut hæc propo­<lb/>&longs;itio exigit. </s> </p> <p type="main"> <s id="s.003669">Sit igitur dati vectis longitudo CD, in cujus medio hypo­<lb/>mochlium O bifariam æqualiter dividat totam longitudinem, <pb pagenum="483" xlink:href="017/01/499.jpg"/>& &longs;it pondus in P. Erit, peri 7. lib. 5. eadem Ratio CO ad <lb/>OP, atque DO ad OP; & <lb/><figure id="id.017.01.499.1.jpg" xlink:href="017/01/499/1.jpg"/><lb/>&longs;i in C &longs;it potentia depri­<lb/>mens, in D autem potentia <lb/>elevans, æqualia habent mo­<lb/>menta ad elevandum pondus <lb/>in P. <!-- KEEP S--></s> <s id="s.003670">E&longs;t ergo CP vectis primi generis, & DO vectis &longs;e­<lb/>cundi generis, cui cum primo commune e&longs;t hypomochlium <lb/>O, & communis pars OP. </s> <s id="s.003671">Quòd &longs;i Potentiæ inæquales fue­<lb/>rint, utraque autem valeat &longs;ive deprimere, &longs;ive elevare, di­<lb/>vidatur longitudo CD in duas partes, quarum Ratio eadem <lb/>&longs;it ac Potentiarum, & in puncto divi&longs;ionis &longs;tatuatur fulcrum: <lb/>tum in extremitatibus reciprocè collocentur Potentiæ, vali­<lb/>dior &longs;cilicet propior &longs;it fulcro, debilior verò remotior, ut æqua­<lb/>lium &longs;int momentorum. </s> </p> <p type="main"> <s id="s.003672">Datæ Potentiæ &longs;int ut 5 ad 3. Dividatur CD partium <lb/>octo ita in P (ubi &longs;tatuendum e&longs;t fulcrum) ut CP &longs;it 5, <lb/>PD &longs;it 3; & Potentia robu&longs;tior, quæ e&longs;t ut 5 &longs;it in D; <lb/>infirmior verò, quæ e&longs;t ut 3, &longs;it in C; & pondus &longs;it in R, <lb/>quoniam CP ad PR e&longs;t ut 5 ad 1, & DP ad PR e&longs;t ut <lb/>3 ad 1. Igitur &longs;i pondus R &longs;it lib. 30, attolletur à Poten­<lb/>tia C potente &longs;ine vecte attollere lib. 3, & à Potentia D <lb/>potente &longs;ine vecte elevare lib. 5: utriu&longs;que enim momenta <lb/>&longs;ingillatim accepta &longs;unt 15 compo&longs;ita ex virtute movendi & <lb/>motûs velocitate. </s> <s id="s.003673">At &longs;i pondus P &longs;it lib. 30, & fulcrum in <lb/>O, &longs;it autem CO ad OP, atque DO ad OP ut 4 ad 1, <lb/>&longs;atis e&longs;t &longs;i &longs;ingulæ Potentiæ æquales C & D po&longs;&longs;int &longs;ine vecte <lb/>attollere lib. 3. unc. </s> <s id="s.003674">9. </s> </p> <p type="main"> <s id="s.003675">Porrò &longs;i inæqualium potentiarum altera po&longs;&longs;it &longs;olùm depri­<lb/>mendo vectem elevare pondus, manife&longs;tum e&longs;t ad illam per­<lb/>tinere vectem primi. </s> <s id="s.003676">generis: ac propterea &longs;i illa &longs;it potentia <lb/>validior, eidem tribuetur minor di&longs;tantia ab hypomochlio; &longs;in <lb/>autem illa &longs;it imbecillior, ip&longs;i tribuetur di&longs;tantia major, atque <lb/>illam inter ac pondus &longs;tatuetur fulcrum. </s> <s id="s.003677">Hinc facilè pote­<lb/>rit potentia vivens uti ope potentiæ inanimatæ, quæ vi &longs;uæ <lb/>gravitatis deor&longs;um premat oppo&longs;itam extremitatem propo&longs;iti <lb/>vectis. </s> </p> <p type="main"> <s id="s.003678">Huc &longs;pectare videtur facillimum genus antliæ &longs;implicis, <pb pagenum="484" xlink:href="017/01/500.jpg"/>qua ex depre&longs;&longs;iore loco in altiorem aquas attollimus. </s> <s id="s.003679">Sit enim <lb/>modiolus B, cui aptè in&longs;eratur congruens embolus medio <lb/><figure id="id.017.01.500.1.jpg" xlink:href="017/01/500/1.jpg"/><lb/>ha&longs;tili CD connexus cum tran&longs;ver&longs;ario EF ver&longs;atili circa <lb/>axem infixum in I: cujus tran&longs;ver&longs;arij extremitatem E occu­<lb/>pet ma&longs;&longs;a plumbea opportunæ gravitatis ad deprimendum em­<lb/>bolum intra modiolum, po&longs;tquam elevatus fuerit à potentia fu­<lb/>nem FG trahente adnexum in altera extremitate F. <!-- KEEP S--></s> <s id="s.003680">Vectis FE <lb/>e&longs;t primi generis duplex habens pondus, alterum in E, alterum <lb/>in D, utrumque enim per vim elevatur. </s> <s id="s.003681">At vectis EI e&longs;t &longs;e­<lb/>cundi generis, in quo E e&longs;t potentia deprimens embolum, & <lb/>quo magis di&longs;tabit ab hypomochlio I, minor ma&longs;&longs;a plumbea <lb/>eadem obtinebit momenta. </s> <s id="s.003682">Quòd &longs;i IF con&longs;tet materiâ &longs;atis <lb/>gravi, jam habet rationem ponderis, ac propterea di&longs;tantia <lb/>centri gravitatis illius H à puncto I determinabit ejus momen­<lb/>ta. </s> <s id="s.003683">Quare potentia E vecte EI &longs;ecundi generis deprimet em­<lb/>bolum, & vecte EH primi generis attollet pondus brachij IF. <!-- KEEP S--></s> <lb/> <s id="s.003684">Hinc e&longs;t commodius accidere, &longs;i longitudo EK ferrea &longs;it, in K <pb pagenum="485" xlink:href="017/01/501.jpg"/>verò in&longs;eratur, ut firmiter cohæreat, &longs;atis validus baculus <lb/>ligneus KF; poterit enim longior e&longs;&longs;e, & faciliorem efficere <lb/>antliæ agitationem, quin gravitas nimia indigeat multo plum­<lb/>bo in E, ut præponderetur. </s> </p> <p type="main"> <s id="s.003685">Quòd &longs;i non placeret addere plumbum in E, & &longs;olo vecte <lb/>primi generis FD uti velles, recurrendum e&longs;&longs;et ad vim ela&longs;ti­<lb/>cam, qua vel arcûs X in &longs;uperiore loco firmati nervo, vel <lb/>extremæ perticæ AM longiu&longs;culæ (ut Toreuticen exercen­<lb/>tibus &longs;olemne e&longs;t) adnecteretur funis pertingens ad F, ut <lb/>ex tractione Potentiæ GF curvatus arcus, vel inflexa per­<lb/>tica, ce&longs;&longs;ante potentiâ, iterum &longs;e &longs;uum in &longs;tatum re&longs;titueret, <lb/>&longs;ursúmque traheret extremitatem F, ac proinde embolum <lb/>intra modiolum B deprimeret. </s> <s id="s.003686">Tunc enim e&longs;&longs;et FD vectis <lb/>primi generis, cujus extremitati F applicarentur duæ Poten­<lb/>tiæ, altera deor&longs;um, altera vici&longs;&longs;im alterno conatu &longs;ur&longs;um <lb/>trahens. </s> </p> <p type="main"> <s id="s.003687">At &longs;i fortè duplicem antliam velis &longs;imul componere, dupli­<lb/>cémque potentiam viventem alternis operis conantem adhi­<lb/>bere, jugo RS ver&longs;atili cir­<lb/><figure id="id.017.01.501.1.jpg" xlink:href="017/01/501/1.jpg"/><lb/>ca axem X adde duo, le­<lb/>viora quidem, &longs;ed &longs;atis fir­<lb/>ma, manubria RO & SM, <lb/>quorum extremitates aut <lb/>premi, aut adjecto fune <lb/>trahi deor&longs;um valeant: <lb/>nam depre&longs;sâ extremitate <lb/>O deprimitur pariter ha&longs;ti­<lb/>le infixum in R, & e&longs;t OX <lb/>vectis &longs;ecundi generis, at­<lb/>que attollitur ha&longs;tile ad­<lb/>nexum in S, & e&longs;t OS <lb/>vectis primi generis. </s> <s id="s.003688">Simi­<lb/>liter MX vectis e&longs;t &longs;ecundi generis, movens pondus po&longs;itum <lb/>in S, atque MR e&longs;t vectis primi generis attollens pondus po­<lb/>&longs;itum in R. <!-- KEEP S--></s> <s id="s.003689">Propterea autem leviora dixi adjecta manubria <lb/>RO & SM, ne &longs;uâ gravitate movendi difficultatem augeant. </s> <lb/> <s id="s.003690">Verùm &longs;i &longs;olus volueris antliam utramque agitare, unus &longs;it <lb/>continuus funis ex O per rotulas P & Q tran&longs;iens, atque in M <pb pagenum="486" xlink:href="017/01/502.jpg"/>connexus: quacumque enim in parte con&longs;titutus fueris, tan­<lb/>tumdem funis &longs;equitur a&longs;cendentem extremitatem, quantum <lb/>trahitur deprimendo alteram extremitatem: &longs;ic trahendo fu­<lb/>nem PO deprimitur extremitas O, deinde trahendo funem <lb/>PQ deprimitur extremitas M. </s> <s id="s.003691">Aut etiam &longs;it unicum ma­<lb/>nubrium SM, & erit RM: atque premens in M attollet <lb/>ha&longs;tulam R, elevans aut in M attollet ha&longs;tulam S. <!-- KEEP S--></s> <s id="s.003692">Ut au­<lb/>tem faciliùs attollatur M, &longs;it in &longs;uperiore loco orbiculus, <lb/>per quem tran&longs;eat funis connexus in M, alteram ením ex­<lb/>tremitatem deor&longs;um trahens attollet manubrium M. </s> <s id="s.003693">Quod <lb/>&longs;i ab oculis remotam volveris antliam, fac per parietis fora­<lb/>men in proximum conclave exire funem MI orbiculi H <lb/>excavatæ ab&longs;idi in&longs;ertum, & per orbiculos P, Q, tran&longs;ire <lb/>funem OP QL; connexis enim funium extremitatibus I <lb/>& L modò hunc modò illum funem trahendo utramque <lb/>antliam agitabis. </s> </p> <p type="main"> <s id="s.003694"><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003695"><emph type="center"/><emph type="italics"/>Antliam opportuno vecte in&longs;truere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003696">UT aliquam &longs;peciem vectis curvi oneri movendo de&longs;tina­<lb/>ti exhibeam, placet in antlia, qua ad hauriendas aquas <lb/>utimur, exemplum ponere, quod facilè in reliquis pro re <lb/>nata imitari po&longs;&longs;imus. </s> <s id="s.003697">E&longs;t in antliâ loco ponderis aqua, quæ <lb/>adducto embolo attrahitur in modiolum, eóque reducto ex­<lb/>primitur, & prætereà conflictus ip&longs;e emboli cum modiolo; <lb/>&longs;uperanda quippe e&longs;t difficultas, quæ ex mutuo horum con­<lb/>tactu oritur, & aqua per vim elevanda e&longs;t, &longs;ive &longs;olùm at­<lb/>trahatur, ut ex modiolo per emboli reducti foramen &longs;ubin­<lb/>de erumpens effluat, &longs;ive in modiolo compre&longs;&longs;a ab embolo, <lb/>cùm reducitur, exprimatur in tubum, ut adhuc altiùs a&longs;cen­<lb/>dat, juxta ea, quæ in Hydrotechnicis fu&longs;iùs dicuntur. </s> <s id="s.003698">Id <lb/>quidem fieret &longs;i ha&longs;tili, quod embolo infigitur, ip&longs;a poten­<lb/>tia proximè applicaretur; &longs;ed ut minus laboris illa &longs;ubeat, <lb/>additur vectis, ut multo major &longs;it potentiæ motus, quàm <lb/>emboli. </s> </p> <pb pagenum="487" xlink:href="017/01/503.jpg"/> <p type="main"> <s id="s.003699">Sit enim embolus A congruens modiolo B, illíque in­<lb/>fixum ha&longs;tile CA; quo elevato aqua <lb/><figure id="id.017.01.503.1.jpg" xlink:href="017/01/503/1.jpg"/><lb/>per &longs;ubjectum modiolo tubum F at­<lb/>trahitur in modiolum ip&longs;um B, quo <lb/>depre&longs;&longs;o aqua cogitur ex eodem mo­<lb/>diolo exire. </s> <s id="s.003700">Sed ut minore operâ id <lb/>totum perficiatur, additur in C vectis <lb/>curvus CDE ver&longs;atilis circa axem D <lb/>infixum parieti interjecto inter an­<lb/>tliam & potentiam moventem. </s> <s id="s.003701">Nam <lb/>extremitatem E arripiens potentia &longs;i <lb/>vectem urgeat versùs parietem inter­<lb/>medium, elevatur embolus, & aqua <lb/>modiolum implet, &longs;i verò vectis ex­<lb/>tremitatem à pariete removeat, deprimitur embolus, & <lb/>compre&longs;&longs;a aqua exprimitur. </s> <s id="s.003702">Hìc vectem primi generis agno­<lb/>&longs;cis habentem hypomochlium in D, &longs;cilicet in axe, circa <lb/>quem ver&longs;atur vectis; & pro Ratione longitudinis DE ad <lb/>longitudinem DC e&longs;t Ratio momentorum potentiæ ad re­<lb/>&longs;i&longs;tentiam ponderis, hoc e&longs;t tantò magis augentur potentiæ <lb/>vires, quò major e&longs;t Ratio DE ad DC: &longs;umitur autem DE <lb/>recta linea non computato flexu DGE, qui eatenus ad&longs;trui­<lb/>tur, quatenus parietis cra&longs;&longs;ities ob&longs;taret, ne commodè utere­<lb/>mur vecte EDC inflexo in D. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003703">Quia verò faciliùs ab homine urgetur vectis in E, quàm <lb/>ip&longs;a extremitas E retrahatur, ideò in antliâ &longs;olùm attrahen­<lb/>te utendo hoc vecte primi generis curvo minus e&longs;t laboris, <lb/>nam in deprimendo embolo minus e&longs;t difficultatis quàm in <lb/>elevando. </s> <s id="s.003704">At &longs;i aqua altiùs elevanda e&longs;&longs;et &longs;upra antliam non <lb/>attrahentem &longs;olum, &longs;ed etiam expellentem, faciliùs attol­<lb/>leretur embolus, quàm deprimeretur, propter majorem aquæ <lb/>re&longs;i&longs;tentiam, cùm exprimitur, juxtà altitudinem perpendi­<lb/>cularem, ad quam expellitur: propterea tunc mutanda e&longs;­<lb/>&longs;et po&longs;itio, ut e&longs;&longs;et vectis &longs;ecundi generis; hypomochlium <lb/>enim &longs;tatuendum e&longs;&longs;et in C, & ha&longs;tile emboli adnecten­<lb/>dum in D. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003705">Quod &longs;i potentia viribus abundet, poterit duplicem an­<lb/>tliam agitare, cuju&longs;modi e&longs;&longs;et &longs;i jugum RS bifariam divi&longs;um <pb pagenum="488" xlink:href="017/01/504.jpg"/>in X jungeretur in R & S duplici ha&longs;tili, centrum autem <lb/><figure id="id.017.01.504.1.jpg" xlink:href="017/01/504/1.jpg"/><lb/>motûs re&longs;ponderet puncto X, cui <lb/>firmiter adnecteretur manu­<lb/>brium XZ, quod agitaretur pa-<lb/>rallelum plano, per quod tran&longs;it <lb/>axis jungens jugum RS cum <lb/>manubrio ip&longs;o: dum enim Z <lb/>versùs P movetur, deprimitur <lb/>R & attollitur S, atque vici&longs;&longs;im <lb/>remeans in Q deprimit embo­<lb/>lum re&longs;pondentem jugi extremi­<lb/>tati S, & oppo&longs;itum attollit. </s> <s id="s.003707">Sunt <lb/>autem duo vectes curvi ZXR <lb/>& ZXS primi generis partem <lb/>unam, videlicet manubrium XZ, <lb/>habentes communem. </s> </p> <p type="main"> <s id="s.003708">Sed quoniam po&longs;ito longiore manubrio ZX, vel DE, faci­<lb/>liùs quidem attollitur aqua, quàm &longs;i illud brevius e&longs;&longs;et, major <lb/>tamen corporis agitatio requiritur, & multâ membrorum incli­<lb/>natione laborio&longs;a exercitatio &longs;u&longs;cipienda e&longs;t, propterea &longs;atius <lb/>e&longs;t uti vecte recto, ut prop. 1. dictum e&longs;t, quem etiam &longs;edens <lb/>modico labore commovere poteris adnexum extremitati fu­<lb/>nem deor&longs;um trahendo. </s> </p> <p type="main"> <s id="s.003709"><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003710"><emph type="center"/><emph type="italics"/>Rotam in profluente po&longs;itam, quæ aquam faciliùs elevet ex <lb/>vectis Rationibus, con&longs;tituere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003711">AQuam ex depre&longs;&longs;iore loco in altiorem provehi va&longs;culis ab­<lb/>&longs;idi rotæ circum circa alligatis, quæ in infimâ rotæ parte <lb/>&longs;ubjectam aquam immer&longs;a hauriunt, & circumactâ rotâ, ubi <lb/>circuli &longs;emi&longs;&longs;em a&longs;cendendo perfecerint, de&longs;cendendo effun­<lb/>dunt, quibus per Helvetios iter facere contigit, per&longs;pectum <lb/>e&longs;t; &longs;i in Tigurinâ Urbe, quam lacus Limagum fluvium exci­<lb/>piens interluit, ob&longs;ervârunt ab utrâque ripâ ductum ex palis <lb/>confertim den&longs;atis obicem obliquum u&longs;que ad medium alveum, <pb pagenum="489" xlink:href="017/01/505.jpg"/>ut ex angu&longs;tiis erumpens aqua cæteroqui leniter defluens, ve­<lb/>lociùs fluere cogatur, & validiùs in prominentes rotæ palmulas <lb/>incurrens ingentem illam rotam cum adjunctis va&longs;culis aquâ <lb/>plenis faciliùs circumagat, atque adeò in &longs;ubjectum vas ponti <lb/>impo&longs;itum effu&longs;a aqua per urbem univer&longs;am dividatur. </s> <s id="s.003712">Verùm <lb/>quia pondus, &longs;cilicet aqua va&longs;culis contenta, &longs;emper à centro <lb/>rotæ intervallo eodem abe&longs;t, aliud rotæ genus excogitari po­<lb/>te&longs;t, quod aquam facilius elevet, nec adnexa, &longs;ed congenita <lb/>habeat va&longs;cula. </s> </p> <p type="main"> <s id="s.003713">In plano ex a&longs;&longs;eribus rite conjunctis compacto, centro A, in­<lb/>tervallo AB, intelli­<lb/><figure id="id.017.01.505.1.jpg" xlink:href="017/01/505/1.jpg"/><lb/>gatur de&longs;criptus cir­<lb/>culus, cujus &longs;emidia­<lb/>meter aliquanto ma­<lb/>jor &longs;it altitudine, ad <lb/>quam aqua evehen­<lb/>da e&longs;t, dividatúrque <lb/>de&longs;cripti circuli peri­<lb/>pheria in quotlibet <lb/>æquales partes, ex. </s> <lb/> <s id="s.003714">gr. <!-- REMOVE S-->duodecim, aut <lb/>plures. </s> <s id="s.003715">Tum a&longs;&longs;ump­<lb/>tâ palmulæ congruâ <lb/>altitudine BD, alius <lb/>interior circulus eo­<lb/>dem centro A, in­<lb/>tervallo AD de&longs;cribatur, qui à ductis per centrum A dia­<lb/>metris &longs;imiliter in totidem æquales partes dividitur. </s> <s id="s.003716">A&longs;&longs;ump­<lb/>tâ itaque CF æquali ip&longs;i DB, &longs;tatuatur CE intervallum op­<lb/>portunæ amplitudinis, ut aqua facilè ingredi po&longs;&longs;it. </s> <s id="s.003717">Et ductâ <lb/>rectâ lineâ BE, re&longs;ecetur particula exterior, ut &longs;it BE CF: <lb/>idémque de cæteris partibus intelligatur, prout adjectum <lb/>&longs;chema refert. </s> </p> <p type="main"> <s id="s.003718">Duo huju&longs;modi plana parentur omnino æqualia, &longs;imilitérque <lb/>denticulata, quæ cylindro (&longs;ive pri&longs;mati &longs;imilem ba&longs;im haben­<lb/>ti cum polygono ab initio de&longs;cripto) hoc e&longs;t axi in&longs;erantur in <lb/>A, & parallela &longs;int. </s> <s id="s.003719">Planorum autem intervallum definiant a&longs;­<lb/>&longs;eres æquè lati, qui perpendiculares in&longs;i&longs;tant lineis GBE, & <pb pagenum="490" xlink:href="017/01/506.jpg"/>&longs;imilibus; quorum a&longs;&longs;erum latitudo palmulis quoque DB, CF, <lb/>& reliquis latitudinem &longs;tatuet. </s> <s id="s.003720">Omnibus ritè firmatis, ac ob­<lb/>&longs;tructis accuratè rimulis, rota &longs;uper polos axi infixos collocetur <lb/>in profluente, ità ut palmula tota in aquam immergatur, quæ per <lb/>apertum o&longs;culum CE ingrediens impleat &longs;patium EBD. </s> </p> <p type="main"> <s id="s.003721">Impetu igitur profluentis dum rota convertitur, aqua inclu&longs;a <lb/>paulatim versùs rotæ centrum &longs;ecedit, donec quadrantem cir­<lb/>culi a&longs;cendendo tran&longs;gre&longs;&longs;a proxima fiat axi: cùm enim B vene­<lb/>rit in H, aqua erit in I, cùm verò ex H in S venerit, jam aqua <lb/>in &longs;ubjectum vas effluet. </s> <s id="s.003722">Quare, licèt æqualium conver&longs;ionum <lb/>non &longs;int æquales a&longs;cen&longs;us in eâdem circuli peripheriâ, &longs;ed ab <lb/>imo puncto u&longs;que ad finem Quadrantis cre&longs;cant, quia tamen <lb/>centrum gravitatis aquæ &longs;e in æquilibrio &longs;tatuentis &longs;en&longs;im cen­<lb/>trum versùs recedit, ejus a&longs;cen&longs;us minor e&longs;t, quàm &longs;i eodem <lb/>&longs;emper intervallo abe&longs;&longs;et à centro rotæ. </s> </p> <p type="main"> <s id="s.003723">E&longs;t itaque vectis curvus primi generis, cujus hypomochlium <lb/>re&longs;pondet centro A, Potentia movens duplex e&longs;t, &longs;cilicet vis <lb/>profluentis applicata in B, atque vis aquæ de&longs;cendentis exi&longs;tens <lb/>in S: pro variâ autem centri gravitatis aquæ elevatæ di&longs;tantiâ <lb/>ab hypomochlio A, diver&longs;a etiam e&longs;t motuum Ratio & momen­<lb/>torum. </s> <s id="s.003724">Aqua enim in &longs;uperiore &longs;emicirculo &longs;upra RS in &longs;ingu­<lb/>lis loculamentis &longs;ibi invicem hinc atque hinc re&longs;pondentibus <lb/>æqualiter di&longs;po&longs;ita obtinet æqualia gravitatis momenta. </s> <s id="s.003725">Qua­<lb/>propter totus profluentis conatus impenditur in elevandâ aquâ, <lb/>quæ loculamentis inter B & R interceptis continetur. </s> <s id="s.003726">Quare <lb/>&longs;i multæ &longs;int profluentis vires, cra&longs;&longs;ior rota &longs;tatui pote&longs;t, ut, <lb/>planis magis di&longs;tantibus, major aquæ copia &longs;ingulis loculamen­<lb/>tis hauriatur: quo fiet, ut palmula latior majorem incurrentis <lb/>aquæ impetum recipiat. </s> <s id="s.003727">Quòd &longs;i placuerit palmulas addere la­<lb/>tiores, quàm &longs;it rotæ cra&longs;&longs;itudo, non abnuo: hæc enim, & cæ­<lb/>tera, quæ con&longs;tructionis facilitatem juvent, prudentis machi­<lb/>natoris arbitrio relinquuntur: mihi &longs;atis e&longs;t innui&longs;&longs;e, quid com­<lb/>pendij ex vectis rationibus peti po&longs;&longs;it. </s> </p> <pb pagenum="491" xlink:href="017/01/507.jpg"/> <p type="main"> <s id="s.003728"><emph type="center"/>PROPOSITIO IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003729"><emph type="center"/><emph type="italics"/>A pluribus hominibus ingens pondus transferri po&longs;&longs;e ita, ut <lb/>omnes æqualiter ferant.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003730">ONus ingens palangâ transferri pluribus hinc atque hinc <lb/>longo ordine &longs;uccollantibus, notum e&longs;t: &longs;ed quoniam non <lb/>omnium æqualis e&longs;t di&longs;tantia à pondere (ni&longs;i fortè bini & bini <lb/>æquè di&longs;tarent à centro gravitatis) non &longs;unt æqualia momenta; <lb/>&longs;ed qui propiores &longs;unt magis premuntur, cæteris paribus, quàm <lb/>remotiore, maximè &longs;i quis &longs;en&longs;im &longs;e &longs;ubducat oneri adeò, ut <lb/>inæqualis fiat oneris di&longs;tantia ab iis, qui illud &longs;u&longs;tentant. </s> <s id="s.003731">Prop­<lb/>terea methodus aliqua excogitanda e&longs;t, qua fiat ut &longs;inguli pa­<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/>re&longs;pondeat punctum C, <lb/><figure id="id.017.01.507.1.jpg" xlink:href="017/01/507/1.jpg"/><lb/>atque æqualia &longs;int in­<lb/>tervalla AC & BC. <!-- KEEP S--></s> <s id="s.003733">Si <lb/>centrum gravitatis pon­<lb/>deris re&longs;pondens puncto <lb/>C vectis non fuerit pla­<lb/>nè in mediâ eju&longs;dem <lb/>ponderis longitudine, <lb/>neque fuerit vectis val­<lb/>dè longior ip&longs;o ponde­<lb/>re, non poterunt plu­<lb/>res ita æqualiter di&longs;po­<lb/>ni, ut ad ferendum <lb/>æqualiter pondus &longs;in­<lb/>gulis anterioribus &longs;in­<lb/>guli po&longs;teriores re&longs;pon­<lb/>deant æquè à puncto C <lb/>di&longs;tantes, impediente <lb/>videlicet ipsâ ponderis <lb/>longitudine. </s> </p> <pb pagenum="492" xlink:href="017/01/508.jpg"/> <p type="main"> <s id="s.003734">Quare tam in A quàm in B duo alij vectes DE, & FG <lb/>bifariam æqualiter divi&longs;i &longs;u&longs;tineant vectem AB: atque adeò <lb/>quemadmodum in A (idem dic de B) &longs;u&longs;tinetur &longs;emi&longs;&longs;is <lb/>totius gravitatis, in D &longs;u&longs;tinetur tantùm quadrans, &longs;icut <lb/>& in E. <!-- KEEP S--></s> <s id="s.003735">Su&longs;tineatur &longs;imiliter extremitas D alio vecte HI <lb/>(id quod præ&longs;titum intellige pariter in extremitatibus E, F, G) <lb/>& in H &longs;u&longs;tinetur octava pars: item extremitas H &longs;u&longs;ti­<lb/>neatur vecte KL, & extremitas K vecte MN; percipitur <lb/>in K gravitatis pars decima &longs;exta, & in M pars trige&longs;ima <lb/>&longs;ecunda. </s> <s id="s.003736">Poterunt igitur hac ratione di&longs;poni homines 32, <lb/>qui &longs;i po&longs;&longs;int &longs;inguli deferre lib. 100, transferent pondus <lb/>lib. 3200, & erit æqualiter inter illos di&longs;tributa gravitas. </s> <lb/> <s id="s.003737">Quòd &longs;i &longs;patium non prohibeat adhuc vectem &longs;ingulis ex­<lb/>tremitatibus adjungere, numerus hominum deferentium du­<lb/>plicabitur, & vel &longs;ingulorum labor dimidiatus erit, vel du­<lb/>plicatum pondus transferre poterunt. </s> <s id="s.003738">Porrò vectem vecti <lb/>e&longs;&longs;e firmo vinculo connectendum, ne fortè in motu, vecte <lb/>aliquo &longs;e &longs;ubducente, luxetur machina, non opus e&longs;t mo­<lb/>nere, cum per &longs;e res ip&longs;a loquatur. </s> <s id="s.003739">Illud ob&longs;erva, quod <lb/>vectium inter &longs;e æqualitatem, &longs;ive longitudo, &longs;ive cra&longs;&longs;i­<lb/>ties &longs;pectetur, non opus e&longs;t &longs;tudiosè accurare, dum­<lb/>modo &longs;inguli vectes æqualiter bifariam dividantur: im­<lb/>mò po&longs;tremi, & breviores e&longs;&longs;e po&longs;&longs;unt, ut minus &longs;pa­<lb/>tij requiratur, & graciliores, minùs quippe urgentur à <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"/>In ferendis oneri­<lb/>bus &longs;citißimi &longs;unt Sinæ, ac rustici illic non parvum &longs;anè &longs;ta­<lb/>ticis no&longs;tris &longs;peculatoribus face&longs;&longs;erent negotium ad cau&longs;as inve­<lb/>niendas ac rationes, &longs;i viderent illos tormenta etiam majora, <lb/>ac &longs;imilia pondera, ita vectibus utrinque &longs;u&longs;pendentes, ut per <lb/>arcti&longs;&longs;imas etiam montium fauces facillimè transferant; ac li­<lb/>cèt præcedant alij, alij &longs;ub&longs;equantur, multí&longs;que pa&longs;&longs;ibus à pon­<lb/>dere &longs;u&longs;pen&longs;o di&longs;tent, ita tamen illud vectibus ac funibus ex æquo <lb/>nôrunt dividere, ut quilibet æquale ferè &longs;entiat onus, &longs;eu paulò re­<lb/>motior &longs;it, &longs;ive vicinior. </s> <s id="s.003742">Hoc pacto ingentia marmora, atque inte­<lb/>gras etiam arbores facilè videas humeris ge&longs;tare Sinas.<emph.end type="italics"/> Hæc ibi. </s> <lb/> <s id="s.003743">Sed quonam id artificio in praxim deducatur, nullum planè <lb/>apparet ve&longs;tigium. </s> </p> <pb pagenum="493" xlink:href="017/01/509.jpg"/> <p type="main"> <s id="s.003744">Si igitur funibus &longs;u&longs;penditur pondus, & deferentes alij <lb/>propiores &longs;unt, alij remotiores, duo ob&longs;ervanda &longs;unt. </s> <s id="s.003745">Pri­<lb/>mum e&longs;t, quòd &longs;u&longs;pen&longs;io non e&longs;t perpendicularis &longs;ed obli­<lb/>qua, ac proinde plus virium requiritur, ut con&longs;tat ex iis, <lb/>quæ dicta &longs;unt tum lib. 1. cap. 16. de elevationibus obli­<lb/>quis, tum lib. 3. cap. 12, de præponderatione, & æquili­<lb/>britate gravium fune &longs;u&longs;pen&longs;orum. </s> <s id="s.003746">Verùm hoc momento­<lb/>rum augmentum in elevatione & &longs;u&longs;pen&longs;ione obliquâ, ubi <lb/>operis abundamus, non con&longs;ideratur; videtur quippe &longs;atis <lb/>leve incommodum, quod facilitate transferendi onus com­<lb/>pen&longs;atur. </s> <s id="s.003747">Secundum e&longs;t, quòd &longs;i omnes ferè æqualiter la­<lb/>borant, non di&longs;&longs;imiles e&longs;&longs;e oportet, &longs;ed proximè ea&longs;dem <lb/>obliquitates funium, ex quibus onus &longs;u&longs;pen&longs;um defer­<lb/>tur: manife&longs;tum enim e&longs;t in minori obliquitate &longs;u&longs;pen­<lb/>&longs;ionis minus virium requiri, quàm in majori obliqui­<lb/>tate. </s> </p> <p type="main"> <s id="s.003748">Quare &longs;i hanc Sinarum indu&longs;triam æmulari conarer, pri­<lb/>mùm oneris transferendi extremitatibus (vel &longs;altem in pa­<lb/>ri di&longs;tantiâ à centro gravitatis, quantùm conjecturâ a&longs;&longs;e­<lb/>qui po&longs;&longs;em) vectes tran&longs;ver&longs;os firmin mè alligarem, ut <lb/>vectium horum capitibus jungerem funes, quibus &longs;u&longs;pen­<lb/>&longs;um onus deferatur. </s> <s id="s.003749">Horum autem tran&longs;ver&longs;orum vectium <lb/>longitudinem ita definirem, ut in lineâ vectibus parallelâ, <lb/>& æquali quatuor &longs;altem homines commodè collocari <lb/>queant, quin &longs;ibi ullum impedimentum progredientes in­<lb/>ferant. </s> <s id="s.003750">Deinde &longs;atis validos funes utrique vectium extre­<lb/>mitati adnexos tantæ longitudinis &longs;tatuerem, quantâ opus <lb/>&longs;it, ut (tribus hominibus ante onus &longs;ibi ordine recto &longs;uc­<lb/>cedentibus ac mediocriter di&longs;tantibus, quin po&longs;terior prio­<lb/>ris calcem progrediendo feriat) ad tertij humerum pertin­<lb/>gere po&longs;&longs;it; hæc enim videtur minima obliquitas &longs;u&longs;pen&longs;io­<lb/>nis, & quæ proximè accedat ad &longs;u&longs;pen&longs;ionem perpendicu­<lb/>larem: Si verò major fuerit funium longitudo, majori labo­<lb/>re deferetur onus, &longs;i maximè ita elevetur, ut multum di&longs;tet <lb/>à &longs;ubjecto &longs;olo, major enim erit obliquitas &longs;u&longs;pen&longs;ionis. </s> <lb/> <s id="s.003751">Tum extremitati funis alius vectis alligetur, qui vecti­<lb/>bus aliis &longs;u&longs;tentetur eâ methodo, quam paulo &longs;uperiùs in­<lb/>dicavi. </s> </p> <pb pagenum="494" xlink:href="017/01/510.jpg"/> <p type="main"> <s id="s.003752">Sit onus transferendum P; extremitati anteriori (omnia <lb/><figure id="id.017.01.510.1.jpg" xlink:href="017/01/510/1.jpg"/><lb/>eadem in alterâ extre­<lb/>mitate po&longs;ita intelli­<lb/>gantur) adnectatur vec. </s> <lb/> <s id="s.003753">tis AB, cui in A & B <lb/>jungantur funes AD <lb/>& BC &longs;ufficientis lon­<lb/>gitudinis, quibus in D <lb/>& C alligetur vectis <lb/>ab aliis vectibus, ut <lb/>paulo &longs;uperiùs indica­<lb/>tum e&longs;t, &longs;u&longs;tentatus, <lb/>adeò ut quarto vecti <lb/>duo homines facilè <lb/>humeros &longs;upponere va­<lb/>leant, & &longs;ingulorum <lb/>funium extremitates D <lb/>& C à &longs;exdecim ho­<lb/>minibus &longs;u&longs;tineantur. </s> <lb/> <s id="s.003754">Quare &longs;i totidem fu­<lb/>nes atque homines po­<lb/>&longs;teriori ponderis parti <lb/>&longs;imili ratione applicen­<lb/>tur, totum pondus ab hominibus 64 æqualiter laborantibus <lb/>&longs;u&longs;tinetur. </s> </p> <p type="main"> <s id="s.003755">Ex quo fit non adeò difficile e&longs;&longs;e in exercitu, ubi non e&longs;t <lb/>hominum &longs;uccollantium inopia, bombardas ex loco in locum <lb/>transferre, &longs;i nimis arduum &longs;it iter, nec equis trahi po&longs;&longs;int: <lb/>Nam majoribus bombardis pro &longs;ingulis globi ferrei libris me­<lb/>talli libræ 150 aut 160 dimidiatis Cartois, ut vocant, in &longs;in­<lb/>gulas globi libras, metalli libræ 180 aut 190, campe&longs;tribus & <lb/>minoribus bombardis metalli libræ 238 u&longs;que ad 266 in &longs;ingu­<lb/>las globi libras communiter tribuuntur. </s> </p> <p type="main"> <s id="s.003756">Quòd &longs;i eæ &longs;int viarum angu&longs;tiæ, quæ octo homines pariter <lb/>incedentes non capiant, adhibeatur longior funis, duos, aut <lb/>etiam tres, aut plures vectes connectens ita invicem di&longs;tan­<lb/>tes, ut intentus funis rectus &longs;it, & propiores quidem &longs;uum <lb/>vectem aut manu apprehen&longs;um &longs;u&longs;tentent, aut fune &longs;u&longs;pen&longs;um <pb pagenum="495" xlink:href="017/01/511.jpg"/>alio vecte parallelo humeris ge&longs;tent, remoti&longs;&longs;imi verò humeros <lb/>&longs;uo vecti &longs;ubjiciant. </s> <s id="s.003757">Sic di&longs;ponatur funis AH, ut intentus <lb/>pertingat ad humeros <lb/><figure id="id.017.01.511.1.jpg" xlink:href="017/01/511/1.jpg"/><lb/>eorum, qui in E & F <lb/>&longs;u&longs;tentant vectes DE <lb/>& FI. <!-- KEEP S--></s> <s id="s.003758">Quoniam ve­<lb/>rò vectis MN longè <lb/>depre&longs;&longs;ior e&longs;t, quàm <lb/>humeri eorum, qui tam <lb/>propè ab&longs;unt à ponde­<lb/>re; propterea vel &longs;olis <lb/>manibus apprehen&longs;um <lb/>vectem &longs;u&longs;tentent, vel, <lb/>quod &longs;atius e&longs;t, alium <lb/>præterea vectem hu­<lb/>meris ge&longs;tent paralle­<lb/>lum vecti MN, ita ut <lb/>ex illo funibus ad per­<lb/>pendiculum intentis <lb/>&longs;u&longs;pendantur extremi­<lb/>tates M & N. <!-- KEEP S--></s> <s id="s.003759">Id quod <lb/>etiam de reliquis, at­<lb/>que de con&longs;equentibus <lb/>vectibus dictum intel­<lb/>ligatur. </s> <s id="s.003760">Omnes autem <lb/>æqualiter conari palàm <lb/>e&longs;t, quia intento fune <lb/>AH eadem e&longs;t obliqua <lb/>&longs;u&longs;pen&longs;io ponderis, & <lb/>paria &longs;unt momenta ad­<lb/>versùs &longs;ingulos vectes, <lb/>quos funis connectit. </s> <lb/> <s id="s.003761">Illud tamen negari non <lb/>pote&longs;t, quod pro majore <lb/>funis AH longitudine <lb/>major e&longs;t &longs;u&longs;pen&longs;ionis <lb/>obliquitas, ac proinde, <lb/>& major &longs;u&longs;tentandi labor. </s> </p> <pb pagenum="496" xlink:href="017/01/512.jpg"/> <p type="main"> <s id="s.003762">Unum adhuc hìc addere (ne quid intactum relinquatur) fuerit <lb/>operæ pretium, videlicet, &longs;i ponderis transferendi cra&longs;&longs;ities <lb/>&longs;eu altitudo mediocris &longs;altem fuerit, ita ut non &longs;olùm infi­<lb/>mo plano &longs;ubjici vectes po&longs;&longs;int, &longs;ed etiam &longs;upremæ aut me­<lb/>diæ parti adnecti, po&longs;&longs;e eidem lateri duos aut etiam tres fu­<lb/>nes, non quidem omninò, &longs;ed proximè parallelos alligari, <lb/>quibus duæ, aut tres, ferè &longs;imiles obliquæ &longs;u&longs;pen&longs;iones fiant, <lb/>& deferentes pondus alij aliis remotiores &longs;int, ferè tamen <lb/>æqualiter conantes. </s> <s id="s.003763">Sic ingentis &longs;axi altitudo &longs;it FG, & al­<lb/><figure id="id.017.01.512.1.jpg" xlink:href="017/01/512/1.jpg"/><lb/>ligatus in F funis connectantur cum vecte in S aliis vectibus <lb/>&longs;u&longs;tentato, ut &longs;upra. </s> <s id="s.003764">Item in E & in G alij funes paralleli &longs;imili­<lb/>ter jungantur cum vectibus in T & V, ut homines ibi &longs;uccollan­<lb/>tes vectibú&longs;que &longs;ubjecti &longs;ibi invicem impedimento non &longs;int. </s> <s id="s.003765">Si <lb/>igitur &longs;ingulis lateribus ad B, C, D tres funes hac ratione addan­<lb/>tur, erunt 12 funes, & &longs;i homines 16 &longs;ingulis funibus applicen­<lb/>tur methodo &longs;uperiùs indicatâ, pondus ge&longs;tabitur à viris 192: <lb/>con&longs;tat igitur quàm ingens onus facilè transferri vectibus <lb/>queat. </s> </p> <p type="main"> <s id="s.003766"><emph type="center"/>PROPOSITIO V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003767"><emph type="center"/><emph type="italics"/>Multiplici vecte moventis vires augere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003768">PRo vectis longitudine majori, in eâdem ab hypomochlio <lb/>di&longs;tantiâ ponderis, potentiæ momenta augeri, quia Ratio <pb pagenum="497" xlink:href="017/01/513.jpg"/>motûs potentiæ ad motum ponderis augetur, &longs;atis manife&longs;tum <lb/>e&longs;t ex dictis. </s> <s id="s.003769">Verùm quia non rarò tam longus vectis, quanto <lb/>opus e&longs;&longs;et, in promptu non e&longs;t, aut ip&longs;a longitudo illum redde­<lb/>ret fractioni, aut &longs;altem flexioni, magis obnoxium, aut, &longs;i peri­<lb/>culo huic occurratur, tam immanis e&longs;t vectis moles, ut non levi <lb/>incommodo &longs;it eo utentibus: propterea ars aliqua excogitanda <lb/>e&longs;t, qua oblati vectis brevitatem compen&longs;atione aliquâ &longs;up­<lb/>pleamus. </s> </p> <p type="main"> <s id="s.003770">Et primò quidem &longs;i oblatus &longs;it vectis AB, habens hypomo­<lb/>chlium in C, & pondus in B tam <lb/><figure id="id.017.01.513.1.jpg" xlink:href="017/01/513/1.jpg"/><lb/>grave, ut unica potentia in A non <lb/>&longs;atis &longs;it ad vincendam oneris re&longs;i­<lb/>&longs;tentiam, utique &longs;i altero, aut ter­<lb/>tio movente opus &longs;it, non omnes in <lb/>extremitate A vectem apprehen­<lb/>dere valent, &longs;ed alter in D, tertius <lb/>in E; qui propterea, licèt &longs;inguli <lb/>æquali robore polleant, non tamen <lb/>æqualia habent momenta, &longs;ed pri­<lb/>mus ut AC, &longs;ecundus ut DC, ter­<lb/>tius ut EC. <!-- KEEP S--></s> <s id="s.003771">Quapropter alter vectis GH adnectatur extremi­<lb/>tati A ad angulos rectos, ut huic applicati motores plus ha­<lb/>beant momenti. </s> <s id="s.003772">Si enim AC ad CB fuerit ut 10 ad 1, perinde <lb/>e&longs;t, atque &longs;i decima ponderis pars à duabus in G & H æquali­<lb/>ter ab A di&longs;tantibus movenda e&longs;&longs;et, ac propterea &longs;inguli &longs;emi&longs;­<lb/>&longs;em decimæ partis re&longs;i&longs;tentiæ percipiunt, hoc e&longs;t, habent &longs;imul <lb/>&longs;umpti momentum ut 20 ad 1: qui autem in A e&longs;&longs;et &longs;olus, habe­<lb/>ret momentum ut 10, & qui in D haberet momentum ex.gr. <!-- REMOVE S-->ut <lb/>9; qui idcircò &longs;imul &longs;umpti minùs po&longs;&longs;unt quàm G & H. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003773">At &longs;i volueris tres homines in extremitatibus vectis GH di­<lb/>&longs;tribuere in potentias æqualiter conantes, di&longs;tingue GH in tres <lb/>partes, & &longs;it AH triens totius longitudinis GH: tum duo ap­<lb/>plicentur extremitati H, tertius verò extremitati G: ut enim <lb/>potentia duplex in H ad potentiam in G, ita reciprocè duplex <lb/>di&longs;tantia GA ad di&longs;tantiam AH. </s> <s id="s.003774">Nam quemadmodum de <lb/>&longs;u&longs;tinentibus pondus vecte &longs;ive æqualiter, &longs;ive inæqualiter di­<lb/>vi&longs;o dictum e&longs;t, ita hìc pariter de Prementibus dicendum, qui <lb/>in H & in G &longs;unt vici&longs;&longs;im Potentia & Hypomochlium: ex eo <pb pagenum="498" xlink:href="017/01/514.jpg"/>&longs;cilicet quòd potentia in H premit, habet rationem hypomo­<lb/>chlij, dum re&longs;i&longs;tit, ne potentia in G premens elevet ip&longs;am ex­<lb/>tremitatem H, ac propterea deprimat pondus in A exi&longs;tens; & <lb/>vici&longs;&longs;im potentia in G premens habet rationem hypomochlij <lb/>re&longs;i&longs;tendo, ne elevetur à potentiâ premente in H; quæ prop­<lb/>terea deprimit pondus in A. <!-- KEEP S--></s> <s id="s.003775">E&longs;t itaque veluti duplex vectis &longs;e­<lb/>cundi generis; & AG ad AH &longs;i fuerit ut 2 ad 1, momentum <lb/>potentiæ in G ad re&longs;i&longs;tentiam ponderis in A e&longs;t ut 3 ad 1, hoc <lb/>e&longs;t ut GH ad AH: & momentum unius potentiæ in H ad re­<lb/>&longs;i&longs;tentiam ponderis in A e&longs;t ut 3 ad 2, hoc e&longs;t, ut HG ad AG. <!-- KEEP S--></s> <lb/> <s id="s.003776">Sed quia in H ex hypothe&longs;i &longs;unt duæ potentiæ, duplicatæ po­<lb/>tentiæ in H momentum erit ut 6 ad 2, hoc e&longs;t, &longs;ingularum <lb/>momentum ut 3 ad 1. Igitur qui e&longs;t in G habet momentum at­<lb/>que conatum, qua&longs;i &longs;ine vecte moveret trige&longs;imam partem pon­<lb/>deris in B exi&longs;tentis; & duo, qui in H, &longs;inguli habent momen­<lb/>tum æquale atque conatum &longs;imilem. </s> <s id="s.003777">Ponatur pondus B lib.60; <lb/>in A percipitur ponderis (1/10), hoc e&longs;t, lib. 6. Igitur potentia in <lb/>G percipit re&longs;i&longs;tentiam lib.2, & duæ potentiæ in H &longs;imul lib.4, <lb/>hoc e&longs;t &longs;ingulæ lib.2. </s> </p> <p type="main"> <s id="s.003778">Quod &longs;i non placuerit longitudinem GH habere tanquam <lb/>vectem, qui non alternâ quadam motione & quiete extremita­<lb/>tum perficitur motus, &longs;ed G, & A, & H omnino &longs;imul & <lb/>æquali motu moventur, non admodum contendo: perinde erit, <lb/>atque &longs;i tres potentiæ in A e&longs;&longs;ent con&longs;titutæ, quarum &longs;ingulæ <lb/>tertiam partem ponderis moveant conatu &longs;ubdecuplo illius co­<lb/>natus, quo tertia illa pars &longs;ine vecte movenda e&longs;&longs;et. </s> </p> <p type="main"> <s id="s.003779">Hinc &longs;altem con&longs;tat, quo virium atque conatûs compendio <lb/>valeat unicus homo oblati vectis momenta augere: Nam &longs;i idem <lb/>&longs;it primi generis vectis AB, & AC ad CB &longs;it ut 10 ad 1, adhi­<lb/>be vectem &longs;ecundi generis GH, & alterâ extremitate fixâ, ut <lb/>ibi &longs;it hypomochlium, idem augebit momenta juxta Rationem <lb/>totius longitudinis GH ad di&longs;tantiam ip&longs;ius A ab hypomo­<lb/>chlio: Quare &longs;i Ratio &longs;it dupla, aut tripla, æquivalebit duobus <lb/>aut tribus, qui in A moventes haberent momentum decuplum; <lb/>nam A movetur decuplo velociùs quàm B, & po&longs;ito hypomo­<lb/>chlio H, movetur potentia G duplo aut triplo velociùs quàm A, <lb/>hoc e&longs;t vigecuplo aut trigecuplo velociùs quàm B. </s> <s id="s.003780">Id quod <lb/>u&longs;um habet non &longs;olùm, quando vectis AB movendus e&longs;t in pla-<pb pagenum="499" xlink:href="017/01/515.jpg"/>no Verticali, &longs;ed etiam in plano horizontali, ut &longs;i duo marmora <lb/>disjungenda e&longs;&longs;ent, aut clathri di&longs;&longs;ipandi. </s> <s id="s.003781">Juxta autem loci op­<lb/>portunitatem adjungendus e&longs;t &longs;ecundus vectis GH aut proxi­<lb/>mè ip&longs;i primo vecti, aut remotè medio fune extremitatem A <lb/>connectente cum &longs;ecundo <lb/>vecte. </s> <s id="s.003782">Sic inter duo mar­<lb/><figure id="id.017.01.515.1.jpg" xlink:href="017/01/515/1.jpg"/><lb/>mora immi&longs;&longs;us ferreus cla­<lb/>vus SR jungitur vecti TV <lb/>fune SO, & potentia in <lb/>T habet momentum com­<lb/>po&longs;itum ex Rationibus <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/>bere po&longs;&longs;umus, tunc maximè, cùm ingenti oneri exiguus <lb/>motus tribuendus e&longs;t. </s> <s id="s.003784">Sit enim marmor P attollendum &longs;ub­<lb/>jecto vecte AB &longs;ecundi <lb/><figure id="id.017.01.515.2.jpg" xlink:href="017/01/515/2.jpg"/><lb/>generis habente hypo­<lb/>mochlium in B, ac pon­<lb/>dere incumbente illi in <lb/>C: & AB ad CB &longs;it <lb/>ut 7 ad 1. Quia vectis <lb/>attollendus e&longs;t, &longs;ubji­<lb/>ce illi in A vectem al­<lb/>terum DE, ut. </s> <s id="s.003785">ED ad <lb/>AD &longs;it in Ratione 3 ad 1. Item extremitati E &longs;ubjice tertium <lb/>vectem FG, & &longs;it GF ad EF ut 8 ad 1. Igitur A movetur &longs;ep­<lb/>tuplò velociùs quàm C, & E triplo velociùs quàm A, atque <lb/>G octuplo velociùs quàm E. <!-- KEEP S--></s> <s id="s.003786">Quare motus potentiæ in G ad <lb/>motum ponderis in C e&longs;t ut 168 ad 1. Quàm difficile autem <lb/>accideret, &longs;i tam longum vectem parare oporteret, cujus lon­<lb/>gitudo e&longs;&longs;et ad CB ut 168 ad 1! </s> </p> <figure id="id.017.01.515.3.jpg" xlink:href="017/01/515/3.jpg"/> <p type="main"> <s id="s.003787">Adde non &longs;olùm vectibus rectis hoc <lb/>momentorum incrementum acquiri <lb/>po&longs;&longs;e, &longs;ed etiam pro loci opportunita­<lb/>te vectibus curvis aut angulatis. </s> <s id="s.003788">Si <lb/>enim in &longs;uperiore loco fuerit vectis <lb/>&longs;ecundi generis MN oneri &longs;ubjectus, <lb/>aut oneri inferiùs po&longs;ito junctus fune <pb pagenum="500" xlink:href="017/01/516.jpg"/>in O, non &longs;olùm po&longs;&longs;umus extremitatem M fune connectere <lb/>cum vecte recto &longs;uperiùs po&longs;ito, &longs;ed etiam &longs;ubjicere illi po&longs;&longs;u­<lb/>mus vectem curvum KL fixum in I, & extremitas L fune LR <lb/>trahi pote&longs;t deor&longs;um, ut IK elevetur, atque illo motu attol­<lb/>lat extremitatem M, quantum ferre pote&longs;t flexus IK. <!-- KEEP S--></s> <s id="s.003789">Non <lb/>e&longs;t autem opus monere inæqualia &longs;en&longs;im fieri momenta, prout <lb/>&longs;ubjectus vectis curvus KIL in alio atque alio puncto contingit <lb/>vectem MN, pro variâ &longs;cilicet di&longs;tantiâ ab hypomochlio. </s> </p> <p type="main"> <s id="s.003790">In vecte tertij generis majorem e&longs;&longs;e ponderis motum mo­<lb/>tu potentiæ, ac proinde majores requiri potentiæ vires ad at­<lb/>tollendum onus, &longs;i illa conjuncta ac &longs;ociata &longs;it cum huju&longs;mo­<lb/>di vecte, quàm &longs;i ip&longs;a &longs;olitaria manum admoveret ponderi <lb/>&longs;ublevando, manife&longs;tum e&longs;t; propterea infirmiori potentiæ &longs;ub­<lb/>&longs;idium aliquod indu&longs;triâ comparare po&longs;&longs;umus, & propo&longs;itum <lb/>vectem in aliam vectis &longs;peciem qua&longs;i convertere, <expan abbr="etiã">etiam</expan> &longs;i &longs;patij an­<lb/>gu&longs;tiis 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/><figure id="id.017.01.516.1.jpg" xlink:href="017/01/516/1.jpg"/><lb/>adnectendum &longs;it <lb/>pondus ex. </s> <s id="s.003792">gr. <!-- REMOVE S--><lb/>lib. 200: poten­<lb/>tia autem appli­<lb/>cari nequeat ni­<lb/>&longs;i in E, ita ut <lb/>EC &longs;it quarta <lb/>pars totius vectis <lb/>CD. Igitur, cum <lb/>motus in D &longs;it quadruplus motûs in E, ut potentia &longs;ublevet onus <lb/>D, tanta &longs;it, oportet, ut ip&longs;a &longs;e &longs;ola valeat quadruplum onus, &longs;ci­<lb/>licet lib.800 attollere: id quod valdè incommodum accideret, <lb/>&longs;i adeò validam potentiam invenire opus e&longs;&longs;et. </s> <s id="s.003793">Perfode igitur in <lb/>&longs;uperiore parte B parietem, illíque immitte vectem FG facile in <lb/>B hypomochlio ver&longs;atilem, ita ut BF pars imminens &longs;ubjecto <lb/>vecti &longs;it æqualis parti EC, hoc e&longs;t, di&longs;tantiæ potentiæ E ab hy­<lb/>pomochlio C, & fune FE connectantur: pars verò ultra parie­<lb/>tem in proximum conclave extans BG ad partem BF &longs;it in qua­<lb/>cumque Ratione. <!-- KEEP S--></s> <s id="s.003794">Tum in inferiore loco, prout opportunius ac­<lb/>ciderit, vectem alium &longs;tatue HI, cui junge &longs;uperioris Vectis ex­<lb/>tremitatem G fune GM: nam Ratio compo&longs;ita ex Rationibus <pb pagenum="501" xlink:href="017/01/517.jpg"/>IH ad MH, & GB ad BF dabit momentum potentiæ in I po­<lb/>&longs;itæ ad attollendum pondus in D con&longs;titutum per vectem da­<lb/>tum CD habentem potentiam in E. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003795">Hîc habes tria vectis genera; nam IH e&longs;t &longs;ecundi generis, quia <lb/>pondus intelligitur in M inter potentiam I & hypomochlium H; <lb/>GF e&longs;t primi generis, quia hypomochlium B e&longs;t inter potentiam <lb/>G & pondus in F; CD e&longs;t tertij generis, quemadmodum ab ini­<lb/>tio con&longs;titutum e&longs;t. </s> <s id="s.003796">Si itaque in E requireretur vis attollendi <lb/>lib.800, & &longs;it GB dupla ip&longs;ius BF, requiritur in G vis attollendi <lb/>lib.400. Si verò I H ad MH &longs;it quadrupla, requiritur in I vis <lb/>elevandi lib.100. Quare & uteris vecte tertij generis CD, quo <lb/>&longs;atis notabiliter movetur pondus D; & potentiæ momenta <lb/>auxi&longs;ti adeò, ut non &longs;olùm non requiratur potentia major <lb/>pondere attollendo, &longs;ed &longs;ufficiat potentia minor, habet quippe <lb/>motum duplo majorem, quàm &longs;it motus ponderis D; nam mo­<lb/>tus extremitatis F & puncti E &longs;unt æquales; motus potentiæ I <lb/>e&longs;t quadruplus motus ip&longs;ius M; hoc e&longs;t extremitatis G; hæc ve­<lb/>rò motum habet duplum motus ip&longs;ius F: igitur motus poten­<lb/>tiæ I e&longs;t octuplus motûs puncti E, quod movetur motu &longs;ubqua­<lb/>druplo extremitatis D: Motus igitur potentiæ I ad motum <lb/>ponderis D e&longs;t ut 8 ad 4, hoc e&longs;t ut 2 ad 1. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003797"><emph type="center"/>PROPOSITIO VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003798"><emph type="center"/><emph type="italics"/>Stateræ vires addito Vecte augere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003799">PAretur ha&longs;ta AB, atque extremitati A addatur annulus, cui <lb/>in&longs;eri valeat &longs;tateræ CD uncus, & extremitas B ita confor­<lb/><figure id="id.017.01.517.1.jpg" xlink:href="017/01/517/1.jpg"/><lb/>metur, ut notabile &longs;it & con&longs;picuum punctum, quod hypomo­<lb/>chlio re&longs;pondeat; &longs;itque certa nota, qua digno&longs;catur vec is pa­<lb/>rallelú&longs;ne &longs;it horizonti, an inclinatus. </s> <s id="s.003800">Tum di&longs;tantia BA di-<pb pagenum="502" xlink:href="017/01/518.jpg"/>vidatur primùm in duas partes, deinde in tres, & &longs;ic deinceps, <lb/>quatenus commodè fieri id poterit citra confu&longs;ionem, quando <lb/>opus fuerit huic aut illi puncto adnectere onus expendendum, <lb/>adeò ut certi &longs;cimus, quotuplex &longs;it totius vectis AB longitudo <lb/>comparata cum di&longs;tantiâ ponderis ab hypomochlio B. </s> </p> <p type="main"> <s id="s.003801">Hoc vecte ad u&longs;um parato, examinetur &longs;taterâ communi, <lb/>quantum ille gravitet parallelus horizonti: & &longs;it æquipondium <lb/>&longs;tateræ ex. </s> <s id="s.003802">gr. <!-- REMOVE S-->in H indicans lib. 2: id quod memoriâ retinen­<lb/>dum e&longs;t, ut, cùm ponderis gravitas explorabitur, ex numero, <lb/>qui in &longs;tateræ jugo indicabitur ab æquipondio, dematur ip&longs;a <lb/>vectis gravitas deprehen&longs;a, &longs;cilicet lib. 2. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003803">Propo&longs;ita igitur gravitate majori, quàm ut expendi valeat <lb/>communi &longs;taterâ CD, adnecte onus vecti in aliquo ex adno­<lb/>tatis punctis, ex. </s> <s id="s.003804">gr. <!-- REMOVE S-->in puncto 6, prout commodius acciderit: <lb/>tum reduc tanti&longs;per æquipondium &longs;tateræ, dum eju&longs;dem &longs;ta­<lb/>teræ jugum & vectis æquè ab horizonte di&longs;tent, & con&longs;i&longs;tat <lb/>æquipondium, puta, in puncto I indicante lib.12. unc. </s> <s id="s.003805">8. de­<lb/>me lib. 2. gravitatem vectis, remanent lib. 10. unc. </s> <s id="s.003806">8. Quia <lb/>autem onus ex hypothe&longs;i adnexum e&longs;t in puncto 6, multiplica <lb/>per 6 lib. 10. unc. </s> <s id="s.003807">8, & habebis lib. 64 gravitatem oneris quæ­<lb/>&longs;itam. </s> <s id="s.003808">Quod &longs;i plane in medio puncto 2 con&longs;titutum fui&longs;&longs;et <lb/>pondus, duplicanda e&longs;&longs;et gravitas indicata à &longs;taterâ. </s> <s id="s.003809">Mani­<lb/>fe&longs;ta e&longs;t hujus operationis ratio; &longs;iquidem æquipondium &longs;ta­<lb/>teræ in puncto I &longs;u&longs;tinet lib. 12. unc. </s> <s id="s.003810">8 adnexas extremitati D. <!-- KEEP S--></s> <lb/> <s id="s.003811">At vis &longs;u&longs;tinendi in vecte &longs;ecundi generis po&longs;ita in A &longs;u&longs;tinet <lb/>vectem, cujus momentum e&longs;t lib. 2, & præterea &longs;u&longs;tinet pon­<lb/>dus in puncto 6 po&longs;itum, quod ad reliquam potentiæ virtutem <lb/>in A, hoc e&longs;t lib.10. unc. </s> <s id="s.003812">8, habet Rationem, quæ &longs;it AB ad <lb/>B 6. igitur convertendo ut 1 ad 6, ita lib.10. unc.8. ad lib.64. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.003813">Quoniam verò accidere pote&longs;t, ut oblatum pondus exce­<lb/>dat quidem datæ &longs;tateræ vires, &longs;ed ejus gravitas minor &longs;it quàm <lb/>dupla ejus, cui æquipondium in extremo &longs;tateræ jugo re&longs;pon­<lb/>det; propterea divi&longs;iones eædem, quæ ex 2 ad B adnotatæ &longs;unt, <lb/>transferantur ex 2 versùs A, ut habeamus diver&longs;a puncta in <lb/>vecte, quibus applicari po&longs;&longs;it onus ponderandum. </s> <s id="s.003814">Ex numero <lb/>igitur adnotato, cui adnectitur pondus, fiat numerator fractionis, <lb/>cujus Denominator &longs;it unitate minor ip&longs;o numeratore; & per <lb/>hanc fractionem multiplicetur numerus à &longs;taterâ indicatus <pb pagenum="503" xlink:href="017/01/519.jpg"/>(demptâ priùs vectis gravitate, ut &longs;uperiùs dictum e&longs;t) & habe­<lb/>bitur oneris gravitas. </s> <s id="s.003815">Sit igitur inter A & 2 adnexum pondus in <lb/>puncto 7, & &longs;tatera indicet lib.13. unc.7: demo vectis gravita­<lb/>tem, quæ e&longs;t lib. 2 ex hypothe&longs;i, remanent lib.11. unc.7. mul­<lb/>tiplicandæ per 7/6, & dabitur oneris gravitas lib. 13. unc. </s> <s id="s.003816">6 1/6. <lb/>Quare in hoc ca&longs;u forta&longs;&longs;e nullum habetur ex vecte adjecto <lb/>compendium, potui&longs;&longs;et enim ex ipsâ &longs;tatera immediatè cogno­<lb/>&longs;ci eadem gravitas. </s> <s id="s.003817">Quod &longs;i eundem numerum indica&longs;&longs;et &longs;ta­<lb/>tera, &longs;ed onus adjunctum fui&longs;&longs;et in puncto 3, per 3/2 multiplica­<lb/>tis lib. 11. unc. </s> <s id="s.003818">7, proveni&longs;&longs;et gravitas oneris quæ&longs;ita lib. 17 <lb/>unc. </s> <s id="s.003819">4 1/2, quæ ex hypothe&longs;i major e&longs;t, quàm ut &longs;olâ &longs;taterâ <lb/>oblatâ expendi po&longs;&longs;it. </s> <s id="s.003820">Vel &longs;i rem breviùs expedire placuerit, <lb/>numeri &longs;taterâ inventi accipe partem denominatam à numero <lb/>vectis unitate minore, eámque illi numero invento adde, & <lb/>idem obtinebis. </s> <s id="s.003821">Sic quia in puncto 3 appen&longs;um fuit onus, ac­<lb/>cipe librarum 11. unc. </s> <s id="s.003822">7. partem denominatam à 2, &longs;cilicet lib.5. <lb/>unc. </s> <s id="s.003823">9 1/2, eámque adde libris 11. unc. </s> <s id="s.003824">7 inventis, & habebis, <lb/>ut priùs, lib. 17 unc. </s> <s id="s.003825">4 1/2. Cur hac methodo operandum &longs;it, <lb/>manife&longs;tò con&longs;tat ex ip&longs;a vectis divi&longs;ione; nam AB ad A 3 e&longs;t <lb/>ut 3 ad 1 ex con&longs;tructione, atque ideò AB ad 3 B e&longs;t ut 3 ad 2: <lb/>igitur ut 2 ad 3, ita numerus à &longs;tatera indicatus (demptâ <lb/>vectis gravitate) ad numerum quæ&longs;itum, quo ponderis gravitas <lb/>innote&longs;cit. </s> </p> <p type="main"> <s id="s.003826">Generatim itaque atque universè oblato quocumque vecte <lb/>ad &longs;ubitum u&longs;um properato utere, etiam&longs;i nullæ in eo divi&longs;io­<lb/>nes adnotatæ fuerint, examinato tamen priùs ip&longs;ius vectis ho­<lb/>rizonti paralleli gravitatis momento, quatenus ad &longs;tateram <lb/>comparatur: Tum datum pondus ibi alliga, ubi commodè à <lb/>&longs;taterâ extremo vecti applicatâ elevari po&longs;&longs;it. </s> <s id="s.003827">Facto demum <lb/>æquilibrio, &longs;tateræ numerum (dempto priùs vectis momen­<lb/>to) multiplica per Rationem, quam habet vectis longitudo ad <lb/>di&longs;tantiam ponderis ab hypomochlio; & propo&longs;itum obtine­<lb/>bis. </s> <s id="s.003828">Hìc habes maximum compendium ad ingentium ponde­<lb/>rum gravitatem explorandam: etiam&longs;i enim vectis non &longs;it <lb/>adeò cra&longs;&longs;us, quia tamen non procul ab extremitate illius, ubi <lb/>e&longs;t hypomochlium, alligatur onus, validè re&longs;i&longs;tit fractioni; <lb/>& quo major e&longs;t Ratio longitudinis vectis ad di&longs;tantiam pon-<pb pagenum="504" xlink:href="017/01/520.jpg"/>eris ab hypomochlio, tanto majore incremento augentur &longs;ta­<lb/>teræ vires. </s> </p> <p type="main"> <s id="s.003829">Quod &longs;i fortè unicus vectis &longs;atis non fuerit, nihil prohibet <lb/>plures adhiberi vectes multo majore compendio, quàm &longs;i uni­<lb/>cum longiorem adhiberes. </s> <s id="s.003830">Nam &longs;i vectis AB non ita &longs;tateræ <lb/><figure id="id.017.01.520.1.jpg" xlink:href="017/01/520/1.jpg"/><lb/>vires multiplicet, <lb/>ut tormentum æ­<lb/>neum in C <expan abbr="alli-gatũ">alli­<lb/>gatum</expan> elevari po&longs;­<lb/>&longs;it ab æquipondio <lb/>&longs;tateræ, alium <lb/>vectem EF &longs;tatue ip&longs;i AB parallelum, habeátque in E hypo­<lb/>mochlium, & &longs;tatera in F adnectatur, qua primùm ip&longs;orum <lb/>vectium fune HI conjunctorum & po&longs;itionem horizonti paral­<lb/>lelam habentium gravitatis momentum expendatur. </s> <s id="s.003831">Deinde <lb/>facto æquilibrio dematur vectium momentum, & reliquus li­<lb/>brarum numerus à &longs;taterâ indicatus multiplicetur primò per <lb/>Rationem FE ad HE, & quod ex hac multiplicatione con&longs;ur­<lb/>git, &longs;ecundò multiplicetur per Rationem IB ad CB; habebitur <lb/>enim demum tormenti ænei gravitas quæ&longs;ita. </s> <s id="s.003832">Sit ex. </s> <s id="s.003833">gr. <!-- REMOVE S-->IB ad <lb/>CB ut 10 ad 1, & FE ad HE ut 12 ad 1, atque &longs;tatera, dempto <lb/>vectium momento, indicet libras 100: igitur 100 per 12 dat 1200, <lb/>& 1200 per 10 dat lib.12000 gravitatem ænei tormenti. </s> </p> <p type="main"> <s id="s.003834">His autem indicatis &longs;tatim occurit animo non duos tantum­<lb/>modo &longs;ed plures vectes po&longs;&longs;e ita di&longs;poni, ut &longs;emper fiat major <lb/>Ratio, quæ ex illorum Rationibus componitur: &longs;i nimirum in­<lb/>ter duas trabes in &longs;olo ad perpendiculum firmatas, & æquali in­<lb/>tervallo à &longs;e invicem di&longs;&longs;itas interjiciantur vectes alterna hypo­<lb/>mochlia habentes in axibus, circa quos facilè converti po&longs;&longs;int, <lb/>& &longs;imili ratione jungantur, ac de duobus vectibus AB & EF <lb/>dictum e&longs;t: Ex &longs;ingulorum enim vectium Rationibus una Ra­<lb/>tio componitur, per quam multiplicandus e&longs;t numerus à &longs;tate­<lb/>râ indicatus, dempto priùs vectium momento. </s> <s id="s.003835">Id quod paulo <lb/>latiùs explicatum e&longs;t in <emph type="italics"/>Terra machinis mota.<emph.end type="italics"/> di&longs;&longs;ert.-1-n. </s> <s id="s.003836">16. nec <lb/>opus e&longs;t hìc tran&longs;cribere. </s> </p> <pb pagenum="505" xlink:href="017/01/521.jpg"/> <figure id="id.017.01.521.1.jpg" xlink:href="017/01/521/1.jpg"/> <p type="main"> <s id="s.003837"><emph type="center"/>MECHANICORUM <emph.end type="center"/><!-- REMOVE S--><emph type="center"/>LIBER QUINTUS.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.003838"><emph type="center"/><emph type="italics"/>De Axe in Peritrochio.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003839">QUÆ de Vecte ejú&longs;que viribus &longs;uperiore libro <lb/>di&longs;putata &longs;unt, illa quidem vera &longs;unt, & ad­<lb/>mirabilia, &longs;ed, ni&longs;i vectis admodum longus <lb/>&longs;it, exiguus motus conciliatur ponderi, adeò <lb/>ut, &longs;i ad notabilem aliquam altitudinem attol­<lb/>lendum illud &longs;it, oporteat &longs;ubinde & ip&longs;i ponderi fulcrum <lb/>&longs;upponere, ne recidat, & ip&longs;i vecti hypomochlium altiùs <lb/>&longs;ubjicere, ut congruo loco &longs;tatuatur. </s> <s id="s.003840">Præterquam quod pro <lb/>variâ ip&longs;ius vectis inclinatione, onerí&longs;que illi impo&longs;iti, aut <lb/>&longs;ubjecti po&longs;itione, varia quoquè &longs;unt momenta potentiæ <lb/>vectem urgentis. </s> <s id="s.003841">Hinc alia Facultas excogitata e&longs;t, quæ, <lb/>ut pluribus placet, vectis quidam &longs;it perpetuus, citrà in­<lb/>commoda, quæ in &longs;implici Vecte, ut innuebam, occur­<lb/>runt. <emph type="italics"/>Vectem<emph.end type="italics"/> autem appellant, quia ad vectis Rationes il­<lb/>lius vim revocant; <emph type="italics"/>perpetuum<emph.end type="italics"/> verò, quia nullâ opus e&longs;t <lb/>hypomochlij mutatione: proprio tamen, tritóque jam ve­<lb/>tu&longs;tate vocabulo, communiter dicitur <emph type="italics"/>Axis in Peritrochio,<emph.end type="italics"/><lb/>qua&longs;i <emph type="italics"/>Axis in Rota,<emph.end type="italics"/> ut quidam interpretantur; &longs;ed forta&longs;sè <lb/>clariùs, pleniú&longs;que vocabuli vim a&longs;&longs;equeremur, &longs;i <emph type="italics"/>Axem <lb/>Convolutum<emph.end type="italics"/> vocaremus; neque enim &longs;emper ade&longs;t Rota, <lb/>cum tamen &longs;emper inter&longs;it Convolutio, &longs;imul quippe vol­<lb/>vitur, & Axis ip&longs;e, & id, cum quo Axis conjungitur. <!--neuer Satz--><pb pagenum="506" xlink:href="017/01/522.jpg"/>Neque hic &longs;umitur Axis quemadmodum in Cono, Cylin­<lb/>dro, atque Sphærâ, pro linea rectâ, circa quam immo­<lb/>tam corpora illa in gyrum aguntur; &longs;ed e&longs;t corpus &longs;uâ <lb/>præditum cra&longs;&longs;itie, cui Axis nomen inditum e&longs;t, quia ro­<lb/>tarum axem imitatur, non tamen circà illum fit convolu­<lb/>tio, &longs;ed ip&longs;e circa idem centrum volvitur minore motu, <lb/>circa quod potentia motu majore rotatur, quatenus illi ap­<lb/>plicatur, ut ex his, quæ dicentur, manife&longs;tum fiet. <lb/></s> </p> <p type="main"> <s id="s.003842"><emph type="center"/>CAPUT I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003843"><emph type="center"/><emph type="italics"/>Axis in Peritrochio forma, & vires <lb/>de&longs;cribuntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003844">AXis in Peritrochio forma à Pappo Alexandrino circa <lb/>finem lib. 8. Collect. Mathem. de&longs;cribitur, quadratum <lb/><figure id="id.017.01.522.1.jpg" xlink:href="017/01/522/1.jpg"/><lb/>&longs;cilicet lignum tympano quadra­<lb/>to foramen AB eidem ligno con­<lb/>gruens circa &longs;uum centrum ha­<lb/>benti in&longs;eritur, ut &longs;imul verti <lb/>po&longs;&longs;int: ligni autem partes è <lb/>tympano prominentes in cylin­<lb/>dricam rotunditatem conforman­<lb/>tur; & lignum horizonti parallelum &longs;uper polos æreos, aut <lb/>ferreos (choinicidas Pappus vocat) congruis fulcris in&longs;i&longs;ten­<lb/>tes &longs;tatuitur. </s> <s id="s.003845">Extremæ verò tympani orbitæ infiguntur Ra­<lb/>dij CD, EF, &c, quos Pappus <emph type="italics"/>Scytalas,<emph.end type="italics"/> Ari&longs;toteles <emph type="italics"/>Col­<lb/>lopes<emph.end type="italics"/> nominat, longiores &longs;cilicet paxilli, quibus arreptis <lb/>ver&longs;atur tympanum, & cum eo Axis, quem ductarius fu­<lb/>nis HI in convolutione circumplectens attollit adnexam in <lb/>I &longs;arcinam; atque hæc tantumdem attollitur, quantus funis <lb/>Axem circumplicat ex convolutione. </s> </p> <p type="main"> <s id="s.003846">Ut Axis huju&longs;modi vires explicentur, communiter in eo <lb/>agno&longs;cunt Vectis Rationes: cum enim CB &longs;it &longs;emidiame­<lb/>ter cylindri, quem funis complectitur, & CE &longs;emidiame-<pb pagenum="507" xlink:href="017/01/523.jpg"/>ter tympani circumpo&longs;iti, EA verò longitudo Radij, conci­<lb/>piunt AB qua&longs;i Vectem <lb/>primi generis habentem hy­<lb/><figure id="id.017.01.523.1.jpg" xlink:href="017/01/523/1.jpg"/><lb/>pomochlium in C, adeò ut <lb/>ex Ratione AC ad CB mo­<lb/>mentum potentiæ in A ap­<lb/>plicatæ computetur. </s> <s id="s.003847">Quæ <lb/>quidem vera e&longs;&longs;e non nega­<lb/>verim, &longs;i hoc unum intelli­<lb/>gatur, quòd Ratio AC ad <lb/>CB &longs;imilis &longs;it Rationi, quam <lb/>haberet æqualis Vectis &longs;imi­<lb/>lem habens po&longs;itionem Po­<lb/>tentiæ, Hypomochlij, & <lb/>Ponderis. <!-- KEEP S--></s> <s id="s.003848">Verùm cur primi <lb/>potiùs quàm &longs;ecundi gene­<lb/>ris vectis dicatur Axis in Pe­<lb/>ritrochio, cùm æquè attolla­<lb/>tur pondus P, &longs;i Radij extre­<lb/>mitas D elevetur &longs;ur&longs;um, ac &longs;i extremitas Radij A deprimatur <lb/>deor&longs;um? </s> <s id="s.003849">E&longs;to facilior &longs;it depre&longs;&longs;io, quàm elevatio. </s> <s id="s.003850">Quid, <lb/>&longs;i Axis &longs;tatueretur horizonti perpendicularis, tympanum au­<lb/>tem horizonti parallelum, non ad attollendum, &longs;ed ad trahen­<lb/>dum pondus? </s> <s id="s.003851">Utique par e&longs;&longs;et trahendi facilitas, &longs;ive impel­<lb/>latur D versùs H, &longs;ive A versùs I: adeóque nulla e&longs;&longs;et ratio, <lb/>cur primi potiùs quàm &longs;ecundi generis Vectis diceretur: an <lb/>utrique generi a&longs;cribendus e&longs;t? </s> </p> <p type="main"> <s id="s.003852">Sed quid Axem ad Vectem revocare opus e&longs;t? </s> <s id="s.003853">cùm eodem <lb/>ex fonte ita utriu&longs;que vires emanent, ut etiam&longs;i Vectem extra <lb/>omnem Naturæ facultatem po&longs;itum, atque inter <foreign lang="greek">a)/dunata</foreign> re­<lb/>cen&longs;endum e&longs;&longs;e fingeremus, adhuc Axi &longs;ua permanerent mo­<lb/>menta: E&longs;t nimirum, &longs;i &longs;ecundum velocitatem comparentur, <lb/>motûs potentiæ ad motum Ponderis Ratio major, quàm gravi­<lb/>tatis ponderis ad virtutem potentiæ: dum enim funis ductarius <lb/>&longs;emel cylindrum circumplectitur, potentia &longs;emel percurrit &longs;pa­<lb/>tium æquale peripheriæ circuli ab extremo Radio de&longs;cripti; <lb/>cùm autem &longs;int peripheriæ circulorum in Ratione &longs;emidiame­<lb/>trorum, motus potentiæ A ad motum ponderis P e&longs;t ut AC ad <pb pagenum="508" xlink:href="017/01/524.jpg"/>CB. <!-- KEEP S--></s> <s id="s.003854">Quare potentiæ Peritrochium ver&longs;antis conatus, ad cona­<lb/>tum potentiæ &longs;ine machinâ attollentis pondus P, erit in Ratio­<lb/>ne CB ad AC; quò enim minor &longs;ecundùm velocitatem e&longs;t <lb/>motus ponderis comparatus cum motu potentiæ, eo minor e&longs;t <lb/>eju&longs;dem re&longs;i&longs;tentia; minorem autem re&longs;i&longs;tentiam minor cona­<lb/>tus &longs;uperat. </s> <s id="s.003855">Quæ ita ex dictis tùm lib.2.cap.5. tum lib.4.cap.1. <lb/>clara &longs;unt, ut uberior explicatio &longs;upervacanea cen&longs;enda &longs;it. </s> </p> <p type="main"> <s id="s.003856">Hinc apparet, quid juvet ip&longs;ius rotæ adjunctæ magnitudo, <lb/>aut infixarum &longs;cytalarum longitudo; quò enim fuerit major <lb/>Potentiæ di&longs;tantia à centro motûs, eò pariter major erit mo­<lb/>vendi facilitas. </s> <s id="s.003857">Quo circa &longs;i eidem Peritrochio placuerit dupli­<lb/>cem applicare potentiam, atque ideò &longs;cytalas non exteriori ab­<lb/>&longs;idi tympani infigas, &longs;ed potiùs extremam tympani oram &longs;cy­<lb/>talis ad ejus planum perpendicularibus transfigas; tunc ad au­<lb/>genda Potentiæ momenta nequicquam prode&longs;t &longs;cytalæ longitu­<lb/>do, &longs;ed à foramine, cui illa infigitur, u&longs;que ad centrum de&longs;u­<lb/>menda e&longs;t potentiæ di&longs;tantia, quæ ut major fiat, tympani dia­<lb/>meter augenda e&longs;t. </s> <s id="s.003858">Id quod pariter dicendum e&longs;t, quando ma­<lb/>nubrium (unicus &longs;cilicet paxillus tympani plano infixus) appo­<lb/>nitur, quod moventis manu perpetuò in conver&longs;ione retine­<lb/>tur; ejus enim di&longs;tantia à centro perinde con&longs;ideratur, atque &longs;i <lb/>potentia illi tympani parti fui&longs;&longs;et proximè applicata, cui manu­<lb/>brium infigitur. </s> </p> <p type="main"> <s id="s.003859">Cavendum tamen hìc videtur, ne quis majorem aliquam ro­<lb/>tam ultrà manubrium excurrentem cylindro circumpo&longs;itam <lb/>con&longs;iderans, quæ aliquando plus habere videtur momenti, <lb/>quàm &longs;i rota non major e&longs;&longs;et, quàm ferat manubrij à centro <lb/>di&longs;tantia, exi&longs;timet non ex hac di&longs;tantiâ computandum e&longs;&longs;e po­<lb/>tentiæ manubrio applicatæ momentum. </s> <s id="s.003860">Ob&longs;ervet, oportet, hoc <lb/>non contingere in immanibus & colo&longs;&longs;icoteris ponderibus, im­<lb/>mò neque in mediocribus movendis, &longs;ed in iis tantummodo, <lb/>quæ leviore negotio & velociter moveri po&longs;&longs;unt: Rota enim, <lb/>cujus &longs;emidiameter major e&longs;t, quàm manubrij à centro di&longs;tan­<lb/>tia, impre&longs;&longs;um à movente potentiâ impetum concipit, qui le­<lb/>vem nactus re&longs;i&longs;tentiam non &longs;tatim perit, &longs;ed aliquanti&longs;per per­<lb/>&longs;everans motum rotæ unà cum novo potentiæ conatu efficit <lb/>majorem, quàm pro &longs;olitariis potentiæ viribus: immò tanta fie­<lb/>ri pote&longs;t impetûs impre&longs;&longs;i acce&longs;&longs;io, ut po&longs;t aliquod tempus, <pb pagenum="509" xlink:href="017/01/525.jpg"/>etiam dimi&longs;&longs;o à potentiâ manubrio, vi eju&longs;dem impre&longs;&longs;i impe­<lb/>tûs adhuc &longs;e rota in gyrum contorqueat. </s> <s id="s.003861">Hinc e&longs;t aliquando <lb/>eju&longs;dem rotæ diametrorum extremitatibus addi plumbeas ma&longs;­<lb/>&longs;as, quæ plus impetûs concipientes, atque diutiùs retinentes, <lb/>rotæ conver&longs;ionem validiùs promoveant, etiam ce&longs;&longs;ante poten­<lb/>tiâ. </s> <s id="s.003862">Sed hìc non unica e&longs;t potentia, quæ manubrio applicatur, <lb/>cujus momenta ex di&longs;tantiâ manubrij à centro definimus; &longs;ed <lb/>præterea impetus ille per&longs;everans rationem habet alterius po­<lb/>tentiæ applicatæ illis rotæ partibus, quibus ine&longs;t; & pro variâ <lb/>à centro di&longs;tantiâ, alia pariter atque alia &longs;unt particularum ip­<lb/>&longs;ius impetûs impre&longs;&longs;i momenta ad rotam convertendam. </s> <s id="s.003863">Quo­<lb/>niam verò rotæ &longs;emidiameter ex hypothe&longs;i major e&longs;t, quàm <lb/>manubrij à centro di&longs;tantia, nil mirum, &longs;i particulæ impetûs ex­<lb/>tremæ rotæ impre&longs;&longs;i multum habeant momenti, quippe quæ <lb/>magis di&longs;tant, & velociorem motum efficiunt. </s> </p> <p type="main"> <s id="s.003864">Quod verò ad cylindrum &longs;pectat, quem funis ductarius cir­<lb/>cumplicat, non e&longs;t nece&longs;&longs;e illum e&longs;&longs;e exactè & Geometricè ro­<lb/>tundum, &longs;ed &longs;atis e&longs;t &longs;i cylindricam figuram æmuletur: catenus <lb/>&longs;iquidem rotundum axem con&longs;truimus, quatenus eadem volu­<lb/>mus in convolutione &longs;ervari momenta: &longs;i verò angulatus e&longs;&longs;et <lb/>axis, perpendiculum, in quo e&longs;&longs;et pondus, modò vicinum cen­<lb/>tro e&longs;&longs;et, modò ab eo remotum, ac propterea eju&longs;dem remoti <lb/>majora e&longs;&longs;ent momenta, quàm vicini. </s> <s id="s.003865">Sit enim ex. </s> <s id="s.003866">gr. <!-- REMOVE S-->qua­<lb/>dratus Axis BDHG: utique per­<lb/>pendiculum, in quo e&longs;t funis reti­<lb/><figure id="id.017.01.525.1.jpg" xlink:href="017/01/525/1.jpg"/><lb/>nens pondus quod attollitur, va­<lb/>riam habet à centro C di&longs;tantiam; <lb/>nam quando latus BD congruit fu­<lb/>ni perpendiculari, di&longs;tantia à cen­<lb/>tro C æqualis e&longs;t &longs;emi&longs;&longs;i lateris GB, <lb/>& e&longs;t CI; cum verò latus BD in <lb/>conver&longs;ione fit obliquum, di&longs;tan­<lb/>tia perpendiculi fit major, & e&longs;t <lb/>CE, ita ut demùm di&longs;tantia maxi­<lb/>ma &longs;it æqualis ip&longs;i CB; quæ iterum decre&longs;cit, donec funis <lb/>congruat lateri BG. <!-- KEEP S--></s> <s id="s.003867">Potentiæ autem à centro di&longs;tantia eadem <lb/>&longs;emper manet AC, ideòque momentorum potentiæ ad mo­<lb/>menta ponderis Ratio &longs;ubinde mutatur. </s> <s id="s.003868">Quòd &longs;i non quadra-<pb pagenum="510" xlink:href="017/01/526.jpg"/>tus &longs;it Axis, &longs;ed plurium angulorum, ita ut latera brevi&longs;&longs;ima <lb/>&longs;int, &longs;icuti vix di&longs;tat à rotunditate cylindri, ita vix momento­<lb/>rum di&longs;paritatem infert. </s> </p> <p type="main"> <s id="s.003869">Illud quidem animadver&longs;ione dignum e&longs;t, quòd non te­<lb/>merè &longs;tatuenda &longs;it Axi cra&longs;&longs;itudo, &longs;ed adeò validus e&longs;&longs;e de­<lb/>bet ac firmus, ut ponderis gravitati ob&longs;i&longs;tere po&longs;&longs;it, quin <lb/>flectatur, aut di&longs;&longs;iliat; &longs;i enim incurve&longs;ceret, augeretur mo­<lb/>vendi difficultas, quia nimirum in conver&longs;ione majorem <lb/>ambitum de&longs;criberet, quàm pro ejus &longs;oliditate. </s> <s id="s.003870">Sed neque <lb/>idcircò præter modum cra&longs;&longs;us Axis eligi debet, quia quò <lb/>major ille e&longs;t atque cra&longs;&longs;ior, eò major etiam e&longs;t potentiæ <lb/>moventis labor, ni&longs;i pariter majus illi addatur Peritro­<lb/>chium. </s> <s id="s.003871">Hinc fit contingere po&longs;&longs;e, ut in attollendo ponde­<lb/>re augeatur labor potentiæ circa finem motûs; quia vide­<lb/>licet, &longs;i ductarij funis &longs;piræ jam univer&longs;am cylindri faciem <lb/>circumplectantur, & &longs;equentes &longs;piræ non cylindro cohæ­<lb/>reant, &longs;ed &longs;ubjecto funi, jam intelligitur &longs;emidiameter axis <lb/>aucta cra&longs;&longs;itudine funis &longs;ubjecti, ac proinde &longs;ecundus hic <lb/>&longs;pirarum ordo majorem funis longitudinem exigit, adeóque <lb/>etiam infert majorem ponderis motum, quo tempore poten­<lb/>tia motum non majorem perficit: quare diminutâ Ratione mo­<lb/>tûs potentiæ ad motum ponderis, minora fiunt illius momenta <lb/>ad attollendum pondus. </s> </p> <p type="main"> <s id="s.003872">Porrò non e&longs;t omnino nece&longs;&longs;e, ut ad pondus attollendum <lb/>Axis &longs;tatuatur in &longs;uperiore loco, &longs;ed fieri pote&longs;t, ut longè al­<lb/>tiùs elevetur pondus &longs;upra locum Axis; &longs;i nimirum funis <lb/>ductarius tran&longs;eat per orbiculum &longs;uperiùs firmatum: Ve­<lb/>rùm ita firmiter &longs;tabilienda e&longs;t machina, ut hæc à nimiâ <lb/>ponderis gravitate non rapiatur &longs;ur&longs;um. </s> <s id="s.003873">Cæterùm cùm fu­<lb/>nis immediatè nectitur ponderi inferiùs po&longs;ito, ip&longs;a ponde­<lb/>ris gravitas &longs;tabilit machinam &longs;uis fulcris in&longs;i&longs;tentem &longs;olo. </s> <lb/> <s id="s.003874">Hactenus quidem Axem rectum, prout magis communiter <lb/>u&longs;urpatur, &longs;tatuimus; pro opportunitate tamen adhiberi etiam <lb/>pote&longs;t curvatus. </s> <s id="s.003875">Quemadmodum &longs;i ex profluente aquam &longs;ur­<lb/>&longs;um antliâ propellere velimus, rotæ BC congruis prunis <lb/>in&longs;tructæ, in quas aqua incurrens vim &longs;uam exerceat, addi­<lb/>tur cra&longs;&longs;ior ferreus &longs;tylus centro A infixus, curvatú&longs;que <lb/>ADEFGHIK (&longs;i IK &longs;it alter polus, cui machina incum-<pb pagenum="511" xlink:href="017/01/527.jpg"/>bit, nam &longs;i fulcrum &longs;it propè A inter A & D, &longs;ufficit &longs;i in H <lb/>terminetur) ita ut ip&longs;i DE <lb/>æqualis &longs;it particula HI, utri­<lb/><figure id="id.017.01.527.1.jpg" xlink:href="017/01/527/1.jpg"/><lb/>u&longs;que autem dupla FG, atque <lb/>inter EF & GH annulo in&longs;e­<lb/>ritur ha&longs;ta adnexa embolo, ita <lb/>ut dum alter embolus attolli­<lb/>tur, alter deprimatur. </s> <s id="s.003876">Hìc <lb/>attendenda e&longs;t Ratio &longs;emidia­<lb/>metri rotæ, &longs;eu di&longs;tantiæ poten­<lb/>tiæ à centro, ad DE, quæ e&longs;t <lb/>&longs;emidiameter cylindri, qui ex <lb/>ejus convolutione gignitur; <lb/>perinde atque &longs;i e&longs;&longs;et cylin­<lb/>drus, cujus tota diameter e&longs;&longs;et <lb/>FG: atque ideo non ex ip&longs;ius <lb/>ferrei &longs;tyli cra&longs;&longs;itudine, &longs;ed ex flexu æ&longs;timanda e&longs;t Axis &longs;e­<lb/>midiameter; eatenus quippe cra&longs;&longs;ior, aut exilis ferreus &longs;tylus <lb/>eligitur, quatenus majore aut minore vi opus e&longs;t in attollendo <lb/>atque deprimendo embolo. <lb/></s> </p> <p type="main"> <s id="s.003877"><emph type="center"/>CAPUT II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003878"><emph type="center"/><emph type="italics"/>Succulæ & Ergatæ u&longs;us con&longs;ideratur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003879">PEritrochij u&longs;us quidem frequens e&longs;t, &longs;ed & &longs;æpiùs &longs;ine rotâ <lb/>idem præ&longs;tatur, vel addito Axi manubrio, vel Radiis Axi in­<lb/>fixis; & machina Latinis <emph type="italics"/>Succula<emph.end type="italics"/> dicitur; parvulus autem paxil­<lb/>lus, cui funis ductarij caput adnectitur, <emph type="italics"/>Porculus<emph.end type="italics"/> nominatur: Si <lb/>tamen paxilli loco annulum cylindro adnectas, cui funis caput <lb/>in&longs;ertum firmetur, perinde e&longs;t. </s> <s id="s.003880">Hu­<lb/><figure id="id.017.01.527.2.jpg" xlink:href="017/01/527/2.jpg"/><lb/>ju&longs;modi e&longs;t cylindrus AB &longs;uis polis <lb/>in&longs;i&longs;tens congruo pegmati, infixo&longs;­<lb/>que habens Radios CD, EF; quibus <lb/>manu arreptis cylindrus volvitur, & <lb/>funis adnexus paxillo I circumduci­<lb/>tur cylindro, atque connexum in H <lb/>onus attollitur. </s> <s id="s.003881">Dupliciter autem <pb pagenum="512" xlink:href="017/01/528.jpg"/>&longs;ucculâ uti licet, aut Radiis perpetuò infixis, aut qui cylindri <lb/>foraminibus dum ver&longs;atur, &longs;ubinde in&longs;erantur: Si perpetuò in­<lb/>fixi maneant, unus pote&longs;t Axem convertere alio po&longs;t alium Ra­<lb/>dio arrepto: at verò &longs;i idem Radius in aliud atque aliud fora­<lb/>men immittendus &longs;it, duo &longs;int, oportet, qui alternâ operâ &longs;uum <lb/>Radium deprimentes Axem convolvant; alioquin, ni&longs;i artificio <lb/>aliquo retineatur, dum ex uno foramine extrahitur Radius, ut <lb/>in aliud immittatur, pondus &longs;uâ gravitate deor&longs;um relaberetur. </s> <lb/> <s id="s.003882">Licet tamen hanc duplicis potentiæ nece&longs;&longs;itatem utilitate aliâ <lb/>compen&longs;are; ubi enim duo &longs;int, quorum vici&longs;&longs;itudine circum­<lb/>agatur Axis immi&longs;&longs;itio huju&longs;modi Radio, hic pote&longs;t e&longs;&longs;e multo <lb/>longior, quàm &longs;i eidem Axi infixus maneret; oporteret &longs;iqui­<lb/>dem plures Radios perpetuò manentes infigere; id quod, &longs;i lon­<lb/>giores e&longs;&longs;ent, non careret incommodo. </s> <s id="s.003883">Quo autem longior <lb/>Radius fuerit, eò pariter faciliùs potentia movebit, quippe quæ <lb/>motum multò velociorem motu ponderis habebit, pro Ratione <lb/>longitudinis Radij plus &longs;emidiametro cylindri, ad eandem cy­<lb/>lindri &longs;emidiametrum. </s> </p> <p type="main"> <s id="s.003884">Ad hoc forta&longs;&longs;e genus revocari po&longs;&longs;unt Scytalæ oneribus <lb/>promovendis &longs;ubjectæ, de quibus dictum e&longs;t lib. 2. cap.9; cùm <lb/>harum capitibus aptè perforatis immittuntur ferrei aut lignei <lb/>vectes, quorum ope &longs;cytalæ ip&longs;æ convertuntur, atque incum­<lb/>bens onus dum ad aliam atque aliam orbitæ partem accommo­<lb/>datur, promovetur. </s> <s id="s.003885">Quo enim longioribus vectibus utimur, <lb/>potentia circa cylindri centrum multò velociùs movetur <lb/>quàm impo&longs;itum &longs;axum, cujus motus æqualis e&longs;t conver&longs;ioni <lb/>peripheriæ. </s> <s id="s.003886">Nam quod motus ab&longs;olutè &longs;umptus &longs;it aliquantulo <lb/>major, quia centrum ip&longs;um promovetur, nihil refert, quia <lb/>motus hic & cylindro &longs;ubjecto, & oneri, & Potentiæ commu­<lb/>nis e&longs;t. </s> </p> <p type="main"> <s id="s.003887">Præter Succulam Radiis infixis in&longs;tructam, cuju&longs;modi ea <lb/>e&longs;t, quæ ad hauriendas è puteis aquas vulgò u&longs;urpatur (quam­<lb/>quam ob radiorum brevitatem & ip&longs;ius Axis cra&longs;&longs;itudinem non <lb/>admodum potentiæ momenta augeantur) forma alia cæmenta­<lb/>riis maximè familiaris e&longs;t ad attollenda &longs;axa, lateres, & calcem, <lb/>duplici manubrio in oppo&longs;itas partes di&longs;po&longs;ito, ut quædam co­<lb/>natuum con&longs;tans &longs;imilitudo &longs;ervetur, dum altero &longs;uum manu­<lb/>brium deprimente, &longs;uum alter elevat: cùm enim vi brachio <pb pagenum="513" xlink:href="017/01/529.jpg"/>deor&longs;um connitentium facilior contingat depre&longs;&longs;io, quàm ele­<lb/>vatio, &longs;i manubriorum inflexio ad eandem partem collocaretur, <lb/>uterque &longs;imul deprimendo faciliùs axem converteret, at <lb/>uterque &longs;imul elevans aliquid amplius laboris &longs;ubiret; alternis <lb/>autem elevationibus atque depre&longs;&longs;ionibus labor temperatur. </s> <lb/> <s id="s.003888">Cæterum quod ad potentiæ momenta attinet, parum intere&longs;t, <lb/>quam po&longs;itionem manubria habeant vici&longs;&longs;im comparata; <lb/>&longs;pectatur videlicet &longs;ingulorum longitudo & cuju&longs;modi motum <lb/>potentia manubrio applicata de&longs;cribat: Sic manubrij longitu­<lb/>do GH, hoc e&longs;t potentiæ apprehen­<lb/>dentis HO di&longs;tantia perpendicula­<lb/>ris ab axe cylindri, qui convolvitur, <lb/><figure id="id.017.01.529.1.jpg" xlink:href="017/01/529/1.jpg"/><lb/>attendenda e&longs;t, & cùm ip&longs;ius cylin­<lb/>dri, &longs;emidiametro comparanda, ut <lb/>Ratio motûs Potentiæ ad ponderis <lb/>motum innote&longs;cat, ac proinde Po­<lb/>tentiæ momentum de&longs;iniatur. </s> </p> <p type="main"> <s id="s.003889">Hinc apparet pro ip&longs;ius GH longitudine ad cylindri axem <lb/>productum perpendiculari augeri momenta potentiæ; perinde <lb/>namque &longs;e habet, ac &longs;i infixus e&longs;&longs;et cylindro Radius KI ip&longs;i <lb/>GH æqualis; quia ut KI ad KE &longs;emidiametrum, ita GH ad <lb/>KE, & ambitus à potentia in H de&longs;criptus ad eju&longs;dem cylin­<lb/>dri ambitum. </s> <s id="s.003890">Quare non leviter allucinantur, qui manubrij <lb/>longitudinem GH non rectam, &longs;ed in hemicyclum curvatam <lb/>volunt qua&longs;i hinc plus aliquid momenti potentiæ conferretur; <lb/>quamvis enim circuli &longs;emiperipheria &longs;it <expan abbr="&longs;alt&etilde;">&longs;altem</expan> diametri &longs;e&longs;quial­<lb/>tera, potentiæ applicatæ motui non &longs;emiperipheria, &longs;ed diame­<lb/>ter legem &longs;tatuit: alioquin &longs;i ex ip&longs;a manubrij inflexione mo­<lb/>menta augerentur, &longs;atius e&longs;&longs;et non tantùm &longs;emiperipheriæ, &longs;ed <lb/>majori circuli &longs;egmento &longs;imile e&longs;&longs;e <expan abbr="manubriũ">manubrium</expan>; id quod &longs;i expe­<lb/>riri voluerint, tantum abe&longs;t, ut movendi facilitatem acquirant, ut <lb/>potiùs momenta minui &longs;entiant; nam in circulo maximam <expan abbr="lineã">lineam</expan> <lb/>e&longs;&longs;e diametrum, & quò <expan abbr="majorũ">majorum</expan> <expan abbr="egmentorũ">&longs;egmentorum</expan> arcus majores fiunt, <lb/>minui &longs;ubten&longs;as chordas, ex 15. lib.3. nôrunt ip&longs;i Elementarij. </s> </p> <p type="main"> <s id="s.003891">Hac igitur manubrij longitudine perpensâ, non &longs;olùm non <lb/>e&longs;t eorum æqualitas religiosè &longs;ervanda, verùm author e&longs;&longs;em <lb/>cætementariis, ut manubriorum alterum paulò longius con&longs;ti­<lb/>tuerent; cùm enim ut plurimum inæquales &longs;int operarum vi-<pb pagenum="514" xlink:href="017/01/530.jpg"/>res, &longs;i æqualia &longs;int manubria, qui infirmior e&longs;t, plus &longs;ubit <lb/>laboris, quàm ferre po&longs;&longs;it: at &longs;i alterum paulo longius &longs;it, de­<lb/>biliorem illi applicari oportebit, ut minore incommodo præ­<lb/>&longs;criptum opus perficiat. </s> <s id="s.003892">Quòd &longs;i contingat ab unico homine <lb/>convertendam e&longs;&longs;e &longs;ucculam, non erit contemnendum laboris <lb/>compendium, &longs;i po&longs;&longs;it longiore manubrio uti. </s> </p> <p type="main"> <s id="s.003893">Quamvis autem nullus &longs;tatuatur finis conver&longs;ioni, quia funis <lb/>ductarius &longs;ucculam non circumplectatur, eadem manet Ratio. <!-- KEEP S--></s> <lb/> <s id="s.003894">Si enim axi polygono in&longs;i&longs;tens catena &longs;ingulis palmaribus, aut <lb/>majoribus intervallis adnexos globulos aut di&longs;cos habeat tubo, <lb/>per quem tran&longs;eunt, congruentes, qui intra tubum aquam <expan abbr="in-tercipi&etilde;tes">in­<lb/>tercipientes</expan> dum ex &longs;ucculæ conver&longs;ione attolluntur, aquam pa­<lb/>riter elevant, &longs;ecúmque rapiunt, perpetua fieri pote&longs;t conver­<lb/>&longs;io; pondus autem, quod movetur, e&longs;t aqua tubum implens. </s> <lb/> <s id="s.003895">Ubi aliquorum imperitiam ca&longs;tigare oporteret, qui manubrij <lb/>longitudinem (quæ ip&longs;e non &longs;ine in&longs;citiæ admiratione vidi, <lb/>narro) minorem &longs;emidiametro axis, cui catena in&longs;i&longs;tit, con&longs;ti­<lb/>tuunt, & operarum laborem fru&longs;tra augent, dum minor e&longs;t <lb/>potentiæ motus, quàm ponderis. </s> <s id="s.003896">Quid enim paulo majorem <lb/>longitudinem manubrio non tribuunt? </s> <s id="s.003897">minùs &longs;cilicet laboran­<lb/>tes operæ concitatiùs axem volverent, & globuli celeriùs ele­<lb/>vati minus aquæ elabi &longs;inerent. </s> </p> <p type="main"> <s id="s.003898">Jam verò ad Ergatam, quæ modicum à &longs;ucculâ differt, tran­<lb/>&longs;eamus, cujus u&longs;us poti&longs;&longs;imùm e&longs;t in trahendis oneribus, quan­<lb/>quam illâ etiam, adhibitâ videlicet trochleâ, ad onera attollen­<lb/>da uti po&longs;&longs;imus, & frequenter utamur. </s> <s id="s.003899">Quemadmodum autem <lb/>in &longs;ucculæ po&longs;itione e&longs;t cylindrus ut plurimùm horizonti pa­<lb/>rallelus, ita in Ergatâ &longs;tatuitur horizonti perpendicularis. </s> <s id="s.003900">Cy­<lb/><figure id="id.017.01.530.1.jpg" xlink:href="017/01/530/1.jpg"/><lb/>lindro enim DC ita fir­<lb/>mato, ut vel circa extre­<lb/>mos polos, vel in locula­<lb/>mento congruo converti <lb/>po&longs;&longs;it, additur vectis GEF <lb/>(aut etiam plures vectes <lb/>eidem cylindro infigun­<lb/>tur) cui applicata Poten­<lb/>tia dum cylindrum circa <lb/>&longs;uum axem ver&longs;at, fu-<pb pagenum="515" xlink:href="017/01/531.jpg"/>nemque convolvit, adnexam &longs;arcinam adducit. </s> <s id="s.003901">Æ&longs;timatur au­<lb/>tem potentiæ momentum ex eju&longs;dem Potentiæ di&longs;tantiâ ab axe <lb/>cylindri, comparata cum ip&longs;ius cylindri &longs;emidiametro: quan­<lb/>tus nimirum funis cylindrum circumplectitur, tantus e&longs;t one­<lb/>ris adducti motus, qui ad potentiæ motum eam habet Ratio­<lb/>nem, quæ inter cylindri ambitum circularem, & peripheriam <lb/>Radio EF, aut EG, de&longs;criptam intercedit. </s> <s id="s.003902">Cum verò huju&longs;­<lb/>modi vecti EF tanta tribui po&longs;&longs;it longitudo, quantam ferre po&longs;­<lb/>&longs;it &longs;patium; in quo Potentia movetur, patet longiore vecte mo­<lb/>mentum potentiæ pro arbitratu augeri po&longs;&longs;e. </s> <s id="s.003903">Verum quidem <lb/>e&longs;t plures potentias eidem vecti EE applicatas inæqualia ha­<lb/>bere momenta pro Ratione inæqualium di&longs;tantiarum ab axe <lb/>cylindri. </s> </p> <p type="main"> <s id="s.003904">Sed illud maximè commodum accidit in Ergatâ, quod hìc <lb/>jumentorum ope hominum laborem minuere licet, dum illa <lb/>extremo vecti alligata, & in gyrum acta cylindrum convolvunt; <lb/>à quibus tamen &longs;ub&longs;idium petere in &longs;ucculæ convolutione non <lb/>po&longs;&longs;umus; ni&longs;i fortè cylindrum horizontalem Verticali peritro­<lb/>chio in&longs;eramus, & extremam cra&longs;&longs;ioris peritrochij orbitam fu­<lb/>nis circumplectatur; qui dum jumento trahente evolvitur, co­<lb/>gat cylindrum converti, funémque, cui &longs;arcina adnectitur, cir­<lb/>ca cylindri orbitam convolutum attollere pondus. </s> <s id="s.003905">Id quod <lb/>etiam præ&longs;tare valemus, &longs;i <lb/><figure id="id.017.01.531.1.jpg" xlink:href="017/01/531/1.jpg"/><lb/>trahendum &longs;it onus, ne­<lb/>que in locum inducere li­<lb/>ceat jumentum: nam per­<lb/>pendiculari cylindro HI <lb/>peritrochium, &longs;eu tympa­<lb/>num LM horizonti paral­<lb/>lelum circumponitur, & <lb/>pluribus &longs;piris tympano cir­<lb/>cumducitur funis, quem <lb/>in O jumentum trahens <lb/>quamvis procul po&longs;itum <lb/>explicat, atque cylindrum <lb/>convertit, ac propterea <lb/>onus in N adnexum ad­<lb/>ducit. </s> </p> <pb pagenum="516" xlink:href="017/01/532.jpg"/> <p type="main"> <s id="s.003906">Porrò in funis ductarij circumvolutione circa &longs;ucculæ aut <lb/>Ergatæ cylindrum ob&longs;ervandum e&longs;t, non e&longs;&longs;e nece&longs;&longs;e totum <lb/>funem circumvolvi, illique adnecti; nimis enim multus ali­<lb/>quando e&longs;&longs;et, & non leve afferret incommodum; ut &longs;atis <lb/>con&longs;tat, cùm &longs;olvendæ &longs;unt anchoræ, &longs;i cra&longs;&longs;um illum ruden­<lb/>tem totum cylindro circumduci opus e&longs;&longs;et, ut anchora è maris <lb/>fundo extrahatur. </s> <s id="s.003907">Satis igitur e&longs;t, &longs;i funis duplici aut triplici <lb/>&longs;pirâ cylindrum circumplectatur, quando ingentia pondera <lb/>movenda &longs;unt; hæc &longs;iquidem valdè re&longs;i&longs;tunt, & ita funis circa <lb/>ip&longs;um cylindrum con&longs;tringitur, ut illum validè premat, nec fa­<lb/>cilè po&longs;&longs;it excurrere, maximè &longs;i cylindrus non fuerit exqui&longs;itè <lb/>tornatus; nimius &longs;cilicet partium &longs;e &longs;e mutuo contingentium <lb/>affrictus, qui cum cylindri &longs;uperficie fieri deberet, perinde re­<lb/>&longs;i&longs;tit, atque &longs;i funis paxillo aut annulo e&longs;&longs;et idem cylindro ad­<lb/>nexus. </s> <s id="s.003908">Quare &longs;atis fuerit, &longs;i puer funem in conver&longs;ione expli­<lb/>catum colligat. </s> </p> <p type="main"> <s id="s.003909">Ex dictis tùm hoc, tùm &longs;uperiori capite, &longs;atis con&longs;tat, quæ­<lb/>nam longitudo &longs;tatuenda &longs;it Radio, cui potentia data applican­<lb/>da e&longs;t, &longs;i pariter cylindri &longs;emidiameter, & oneris gravitas detur. </s> <lb/> <s id="s.003910">Nam &longs;i fiat ut data Potentia ad datam ponderis gravitatem, ita <lb/>data cylindri &longs;emidiameter ad quæ&longs;itam Radij longitudinem, <lb/>habetur longitudo &longs;ufficiens ad &longs;u&longs;tinendum pondus in aëre <lb/>&longs;u&longs;pen&longs;um. </s> <s id="s.003911">Quare pro arbitratu augeatur longitudo Radij, &, <lb/>cùm facta jam &longs;it major Ratio motûs potentiæ ad motum pon­<lb/>deris, quàm &longs;it Ratio gravitatis ponderis ad virtutem potentiæ <lb/>&longs;u&longs;tinentis, illa poterit propo&longs;itum pondus movere. </s> <s id="s.003912">Sic quo­<lb/>niam in navibus ad proram jacet horizonti parallelus ver&longs;atilis <lb/>cylindrus (aut potiùs hexagonum &longs;eu octogonum pri&longs;ma) cu­<lb/>jus extremitas decre&longs;centibus crenis denticulata incumbentem <lb/>ligneam regulam &longs;ingulis &longs;ubinde crenis excipit, ne ponderis <lb/>vi in contrariam partem retroagi valeat, & cylindro circumdu­<lb/>citur rudens (<emph type="italics"/>Pi&longs;ma<emph.end type="italics"/> ab aliquibus dicitur) ex quo anchora pen­<lb/>det; nec habere pote&longs;t plures Radios perpetuò adnexos, quos <lb/>videlicet &longs;patij angu&longs;tiæ ferre non po&longs;&longs;ent, ideò foramina quæ­<lb/>dam habet, quibus, ubi opus fuerit, in&longs;eruntur vectes. </s> <s id="s.003913">Ut <lb/>vectium longitudo &longs;tatuatur, anchoræ gravitas cum adjecto <lb/>ligneo tran&longs;ver&longs;ario con&longs;ideranda e&longs;t, quæ e&longs;t ferè &longs;ub trecen­<lb/>tupla gravitatis navis vacuæ, ut con&longs;tat ex iis, quæ lib.4.cap.17. <pb pagenum="517" xlink:href="017/01/533.jpg"/>innuimus. </s> <s id="s.003914">Navis autem capacitas (hoc e&longs;t pondus, quod <lb/>navis ge&longs;tare valet, & æquale e&longs;t gravitati navis in aëre) <lb/>vel per dolia, &longs;eu amphoras aquæ, quam &longs;ine incommodo <lb/>ferre pote&longs;t, numeratur, ut &longs;olent Galli & Angli &longs;ingulis do­<lb/>liis navalibus libras bis mille tribuentes, vel per pondera, <lb/>quæ Hollandis atque Germanis <emph type="italics"/>La&longs;t<emph.end type="italics"/> dicuntur, &longs;ingula li­<lb/>brarum &longs;altem quatuor millibus definita (nam <emph type="italics"/>La&longs;t<emph.end type="italics"/> Ham­<lb/>burgi continet libras 4554, Am&longs;telodami, &longs;i &longs;it triticum <lb/>habet lib. 4800, &longs;in autem &longs;iligo lib. 4200, Stevinus verò <lb/>lib. 3. &longs;taticæ pop. </s> <s id="s.003915">10 &longs;ingulos modios definit lib. 360) & <lb/>&longs;ingulis libris unciæ &longs;exdecim, &longs;eu <emph type="italics"/>Lotones<emph.end type="italics"/> 32, hoc e&longs;t &longs;e­<lb/>munciæ tribuendæ &longs;unt. </s> <s id="s.003916">Quare data navis capacitas ex. </s> <s id="s.003917">gr. <!-- REMOVE S--><lb/>doliorum 400, multiplicetur per lib. 2000; & &longs;unt. </s> <s id="s.003918">lib. <!-- REMOVE S--><lb/>800000, quarum pars trecente&longs;ima lib. 2666 e&longs;t ferè pon­<lb/>dus anchoræ cum ligneo tran&longs;ver&longs;ario. </s> <s id="s.003919">Po&longs;&longs;unt autem non <lb/>plures applicari vectes quàm quatuor, ideóque &longs;inguli quar­<lb/>tam ponderis partem elevare debent, hoc e&longs;t lib. 666. Si <lb/>fuerit igitur cylindri &longs;emidiameter 3/4 pedis, & vis potentiæ <lb/>(quia ip&longs;a corporis gravitas vectem premit) &longs;it elevandi <lb/>lib. 100, fiat ut 100 ad 666, ita 3/4 pedis ad pedes ferè <lb/>quinque; & hæc erit quæ&longs;ita longitudo Radij, cui poten­<lb/>tia applicanda e&longs;t. <lb/></s> </p> <p type="main"> <s id="s.003920"><emph type="center"/>CAPUT III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003921"><emph type="center"/><emph type="italics"/>Tympani à calcante circumacti vires <lb/>expenduntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003922">TYmpana, quæ Græcis <foreign lang="greek">geranoi</foreign>, Latinis retentâ vocabuli in­<lb/>terpretatione <emph type="italics"/>Grues<emph.end type="italics"/> dicuntur, hoc differunt à Succulâ, <lb/>quòd ab hominibus non brachiorum contentione, &longs;ed cor­<lb/>poris calcantis gravitate moventur. </s> <s id="s.003923">Horum autem frequen­<lb/>ti&longs;&longs;imus e&longs;t u&longs;us tum in Hollandiâ tùm in Germaniâ juxta <lb/>fluvios navigabiles, ut ingentia pondera è navibus extra­<lb/>hant, & in ripa deponant: quamquam & ad alios u&longs;us fa-<pb pagenum="518" xlink:href="017/01/534.jpg"/>cilè traduci po&longs;&longs;int, &longs;i apto loco collocentur. </s> <s id="s.003924">Cylindro AB <lb/><figure id="id.017.01.534.1.jpg" xlink:href="017/01/534/1.jpg"/><lb/>ver&longs;atili, & horizon­<lb/>ti parallelo; ac ritè <lb/>firmato, ampliorem <lb/>rotæ peripheriam <lb/>CDE circumponi­<lb/>mus ex latioribus a&longs;­<lb/>&longs;eribus compactam, <lb/>ut unus &longs;altem ho­<lb/>mo ingredi valeat; <lb/>qui dum ex C in D <lb/>a&longs;cendere conatur, &longs;ua gravitate deprimens tympanum, cy­<lb/>lindrum pariter convertit, ductariúmque funem convolvit, <lb/>qui per orbiculos F & G &longs;uperiori trabi exporrectæ connexos <lb/>tran&longs;iens, cum onere in P connectitur, atque adeò ex Cylin­<lb/>dri conver&longs;ione attollitur pondus, etiam&longs;i à machinâ ip&longs;a ab­<lb/>&longs;it, quantum trabs exporrigitur. </s> <s id="s.003925">Si placuerit eidem cylindro <lb/>duplex tympanum, aut unicum valdè amplum apponere, lice­<lb/>bit, ut plurium hominum operâ in elevando onere uti po&longs;­<lb/>&longs;imus. </s> </p> <p type="main"> <s id="s.003926">Machinæ hujus vires eò majores e&longs;&longs;e, quò major e&longs;t &longs;emi­<lb/>diametri rotæ ad cylindri &longs;emidiametrum Ratio, &longs;atis mani­<lb/>fe&longs;tum e&longs;t ex iis, quæ &longs;æpi&longs;&longs;imè dicta &longs;unt; major e&longs;t enim <lb/>potentiæ motus, quò amplior e&longs;t rota. </s> <s id="s.003927">Cavendum tamen ne, <lb/>quemadmodum in &longs;ucculâ atque Ergatâ, ita etiam hìc omnino <lb/>ex ip&longs;a &longs;emidiametrorum rotæ & Cylindri Ratione definiantur <lb/>potentiæ momenta: hìc &longs;cilicet potentia tympanum movens e&longs;t <lb/>in&longs;ita homini gravitas deor&longs;um connitens; in &longs;ucculâ autem <lb/>atque in Ergatâ potentia movens e&longs;t impul&longs;us ab animali facul­<lb/>tate impre&longs;&longs;us, ac in gyrum directis. </s> <s id="s.003928">Quapropter in &longs;ucculâ, at­<lb/>que in Ergatâ cùm eadem &longs;it potentiæ directio &longs;imiliter appli­<lb/>catæ in quocumque &longs;itu, eadem manent in conver&longs;ione poten­<lb/>tiæ momenta: at hominis tympanum calcantis non eædem &longs;em­<lb/>per &longs;unt vires, &longs;ed quo magis a&longs;cendit versùs D, augentur ejus <lb/>momenta; quia videlicet perinde e&longs;t, atque &longs;i à centro ad <lb/>punctum orbitæ, in quo e&longs;t gravitas calcans, ducta e&longs;&longs;et linea; <lb/>ibi enim momentum de&longs;cendendi e&longs;t ut Sinus declinationis à <lb/>perpendiculo, juxta dicta lib.1.cap.15. </s> </p> <pb pagenum="519" xlink:href="017/01/535.jpg"/> <p type="main"> <s id="s.003929">Sit enim rotæ CHG &longs;emidiameter EB, cylindri verò &longs;e­<lb/>midiameter EO. </s> <s id="s.003930">Si homo <lb/><figure id="id.017.01.535.1.jpg" xlink:href="017/01/535/1.jpg"/><lb/>tympanum ingre&longs;&longs;us con­<lb/>&longs;i&longs;tat in infimo loco H, in <lb/>quem &longs;cilicet cadit per­<lb/>pendiculum EH, utique <lb/>machinam non movet, à <lb/>qua ip&longs;e &longs;u&longs;tinetur: Simi­<lb/>liter &longs;i funi, OC per &longs;upe­<lb/>rioris trabis orbiculos tran­<lb/>&longs;iens fuerit intentus, etiam­<lb/>&longs;i ex H a&longs;cendat in D, in <lb/>quod punctum cadit recta <lb/>CO cylindrum tangens, <lb/>non movetur pondus, quod ex hypothe&longs;i excedit gravitatem <lb/>hominis tympanum calcantis: quia nimirum in D homo ra­<lb/>tione po&longs;itionis non habet de&longs;cendendi momenta majora, <lb/>quàm &longs;it &longs;emidiameter OE, quæ pariter &longs;unt momenta one­<lb/>ris funi OC adnexi: Cùm autem ratione po&longs;itionis momen­<lb/>ta ponderis atque potentiæ æqualia &longs;int, &longs;ed ratione gravita­<lb/>tis potentia infirmior &longs;it ex hypothe&longs;i quàm pondus, illa uti­<lb/>que elevare pondus non valebit. </s> <s id="s.003931">Procedet igitur a&longs;cenden­<lb/>do ex. </s> <s id="s.003932">gr. <!-- REMOVE S-->u&longs;que ad I, ubi obtinebit momenta ut VE (hoc <lb/>e&longs;t IK &longs;inus anguli declinationis IEH) ad momenta one­<lb/>ris ut OE, adeò ut quæ Ratio e&longs;t VE ad OE, eadem &longs;it <lb/>Ratio gravitatis oneris ad gravitatem hominis; ac proinde <lb/>a&longs;cendentis ex I in G momenta augebuntur, & in G erunt <lb/>ut SE. <!-- KEEP S--></s> <s id="s.003933">Cùm ergo SE ad OE Ratio major &longs;it quàm VE <lb/>ad OE, hoc e&longs;t gravitatis oneris ad gravitatem hominis, <lb/>jam prævalet potentia, & tympanum convertitur, de&longs;cen­<lb/>dítque illius punctum G in locum, ubi erat punctum I, in <lb/>quo fit con&longs;i&longs;tentia & quoddam æquilibrium &longs;u&longs;tentando <lb/>pondus, quod ut porrò moveatur, pergendum e&longs;t in per­<lb/>currendâ tympani orbitâ. </s> <s id="s.003934">Numquam igitur ratione po&longs;itio­<lb/>nis potentiæ momentum e&longs;t ut &longs;emidiameter rotæ, ni&longs;i ho­<lb/>mo ita a&longs;cenderet, ut ejus centrum gravitatis re&longs;ponderet <lb/>puncto B; ex momento enim, quod potentia obtineret in <lb/>B, demendum e&longs;t, quantum ab ip&longs;o centro retrahitur: in G <pb pagenum="520" xlink:href="017/01/536.jpg"/>autem retrahitur juxta men&longs;uram BS, & in I juxta men&longs;uram <lb/>BV, ac propterea ibi momentum remanet ut SE, hìc ut VE. </s> <lb/> <s id="s.003935">Perinde autem &longs;e habere momentum in G ad pondus, atque &longs;i <lb/>e&longs;&longs;et libra curva GEF, & ab alterâ extremitate F diametri cy­<lb/>lindri, duceretur recta FG &longs;ecans in R perpendicularem EH, <lb/>manife&longs;tum e&longs;t, quia ex 2. lib. 6. ut SE ad EF, hoc e&longs;t OE, <lb/>ita GR ad RF. </s> <s id="s.003936">Quòd &longs;i tympani orbitam limbus hinc & hinc <lb/>ambiat, cui teretes paxilli in&longs;erti veluti gradus &longs;calas con&longs;ti­<lb/>tuant, quibus homo non &longs;olùm in&longs;i&longs;tat pedibus, &longs;ed quos etiam <lb/>manibus apprehendat; ob&longs;ervare oportet, pedibu&longs;ne tantum <lb/>premat &longs;ubjectum tympanum, an ex manibus qua&longs;i &longs;u&longs;pen&longs;us <lb/>pendeat. </s> <s id="s.003937">Nam &longs;i in eodem perpendiculo non &longs;int paxillus, cui <lb/>in&longs;i&longs;tit, & is ex quo dependet, valde di&longs;paria &longs;unt momenta. </s> <s id="s.003938">Si <lb/>verò non planè rectum &longs;it corpus, &longs;ed qua&longs;i procumbens incli­<lb/>netur, tunc potentiæ locus definitur à perpendiculo tran&longs;eunte <lb/>per centrum gravitatis ip&longs;ius hominis. </s> <s id="s.003939">Id quod dicendum pari­<lb/>ter, quando tympano includitur canis (nam & à cane ingens <lb/>tympanum ver&longs;ari vidi, quo in Solarium attollebatur non me­<lb/>diocris ci&longs;ta linteis recens ablutis plena) cujus gravitatis cen­<lb/>trum &longs;pectandum e&longs;t, ejú&longs;que di&longs;tantia à perpendiculo tran­<lb/>&longs;eunte per tympani centrum. </s> </p> <p type="main"> <s id="s.003940">Hinc &longs;i ex navi aliæ atque aliæ &longs;arcinæ hac machinâ extrahan­<lb/>tur, is qui tympanum ver&longs;at, etiam&longs;i omnino non videat onus ex­<lb/>tra machinæ domunculam po&longs;itum, facilè pronuntiabit major­<lb/>ne? </s> <s id="s.003941">an minor &longs;it &longs;ecundæ &longs;arcinæ gravitas comparata cum prio­<lb/>re: quò enim magis a&longs;cendere cogitur in tympano, eò major e&longs;t <lb/>oneris gravitas; quærenda nimirum &longs;unt momenta majora ex <lb/>po&longs;itione, ut majore intervallo ab&longs;it à perpendiculo EH tran­<lb/>&longs;eunte per E centrum. </s> <s id="s.003942">Simili ratione, &longs;i inter duos homines <lb/>quæ&longs;tio oriatur uter illorum gravior &longs;it, facilè litem dirimes, &longs;i <lb/>alter po&longs;t alterum ingrediatur tympanum, ut idem onus attollat; <lb/>qui enim magis a&longs;cendere cogitur, minus habet gravitatis, ideò­<lb/>que majora momenta quærit ex po&longs;itione. </s> <s id="s.003943">Quanta autem &longs;it one­<lb/>ris gravitas, digno&longs;cetur ex artificio &longs;tatim indicando. </s> <s id="s.003944">Unum hìc <lb/>qua&longs;i per anticipationem addendum, quod ad funem ductarium <lb/>&longs;pectat; præ&longs;tat &longs;cilicet ejus extremitatem unco extremæ trabi <lb/>infixo adnecti, & per orbiculum cum onere conjunctum tran&longs;i­<lb/>re, atque hinc per orbiculos G & F ad cylindrum deduci: hac <pb pagenum="521" xlink:href="017/01/537.jpg"/>enim ratione attollendi facilitas geminatur, ut clariùs patebit ex <lb/>iis, quæ &longs;equenti libro de Trochleâ dicentur. </s> </p> <p type="main"> <s id="s.003945">Ut igitur innote&longs;cat, quanta &longs;it proximè oneris gravitas ob­<lb/>&longs;ervandus e&longs;t in tympano locus, ubi homo illud calcans facit <lb/>cum pondere æquilibrium: quando &longs;cilicet eò venerit, ut paulo <lb/>altiùs a&longs;cendens incipiat attollere pondus. </s> <s id="s.003946">Quoniam verò hu­<lb/>ju&longs;modi pondera ea &longs;unt, ut in iis exiguæ differentiæ contem <lb/>nantur, exqui&longs;ita quædam accuratio omnino &longs;upervacanea e&longs;­<lb/>&longs;et, &longs;i &longs;ingulas, aut pauculas libras ad calculos revocandas e&longs;&longs;e <lb/>cen&longs;eremus, cum &longs;æpè non ni&longs;i per centenas libras eorum gra­<lb/>vitas definiatur. </s> <s id="s.003947">Primùm ex centro E in ipsâ limbi cra&longs;&longs;itudine <lb/>de&longs;cribatur circuli peripheria BCH: id quod facilè fiet funi­<lb/>culo extento, & axem FRO complectente; quo funiculo cir­<lb/>cumducto &longs;tylus in extremitate colligatus de&longs;cribet <expan abbr="peripheriã">peripheriam</expan>. </s> </p> <p type="main"> <s id="s.003948">Deinde &longs;i nota non &longs;it accurata &longs;emidiametri men&longs;ura, quæ <lb/>peripheriæ &longs;extantem accipiat, punctum unum, quod placuerit, <lb/>&longs;tatue, ex quo peripheriam in partes aliquotas (qua&longs;cumque <lb/>tandem opportunitas dederit) dividere incipias: nam per nu­<lb/>merum partium divi&longs;is gradibus 360, &longs;tatim patebit, quot gra­<lb/>dus &longs;ingulæ partes contineant, quas aliquotas a&longs;&longs;ump&longs;i&longs;ti. </s> <s id="s.003949">Par­<lb/>tem igitur unam in gradus &longs;ibi congruentes tribue; eorum enim <lb/>men&longs;ura in con&longs;equentem arcum tran&longs;lata, quoties oportuerit, <lb/>demùm integrum Quadrantem in gradus 90 divi&longs;um dabit. </s> <s id="s.003950">Po­<lb/>namus commodam accidi&longs;&longs;e divi&longs;ionem peripheriæ in partes 15: <lb/>divi&longs;is gr. <!-- REMOVE S-->360 per 15, quotiens 24 indicat numerum graduum <lb/>parti decimæ quintæ tribuendorum. </s> <s id="s.003951">Quare partem unam bifa­<lb/>riam divide, & intervallum gr. <!-- REMOVE S-->12 inter punctum divi&longs;ionis & <lb/>a&longs;&longs;umptum punctum, ex quo divi&longs;io incipit, iterum divide bifa­<lb/>riam, ut parti uni cedant gradus 6: his iterum bifariam divi&longs;is, <lb/>habetur partis aliquotæ primò a&longs;&longs;umptæ pars octava gr.3. hanc <lb/>in tres æquales partes di&longs;tribue, & &longs;ingulorum graduum men­<lb/>&longs;ura manife&longs;ta e&longs;t. </s> <s id="s.003952">Acceptis itaque tribus partibus decimis <lb/>quintis addantur gradus 18, & habebitur integer peripheriæ <lb/>Quadrans in gradus 90 di&longs;tributus, qui adeò notabiles erunt, <lb/>ut etiam gradûs partes, cuju&longs;modi e&longs;t &longs;emi&longs;&longs;is, triens, & qua­<lb/>drans &longs;atis clarè digno&longs;ci queant. </s> </p> <p type="main"> <s id="s.003953">Tertiò. <!-- KEEP S--></s> <s id="s.003954">Quia non arcus HG, &longs;ed &longs;emidiametri pars ES con­<lb/>&longs;ideratur, ut dictum e&longs;t, concipe &longs;emidiametrum EB in partes <pb pagenum="522" xlink:href="017/01/538.jpg"/>aliquotas di&longs;tributam, primùm in duas, deinde in tres, in qua­<lb/>tuor, & deinceps, prout opportunum accidet, ita tamen, ut non <lb/>venias ad partem aliquotam minorem &longs;emidiametro Axis: Id <lb/>quod deprehendes, &longs;i a&longs;&longs;umptam chordam &longs;ubten&longs;am gradibus <lb/>60, in illud genus partium aliquotarum, de quo dubitas, divi&longs;e­<lb/>ris, & in &longs;emidiametro BE incipiendo ab extremitate B illas ac­<lb/>ceperis; &longs;i enim po&longs;trema pars aliquota re&longs;idua major &longs;it &longs;emi­<lb/>diametro Axis, aut illi æqualis, non e&longs;t ju&longs;to minor. </s> <s id="s.003955">Igitur ex <lb/>Canone Sinuum exquire arcum &longs;ingulis partibus, incipiendo à <lb/>centro tympani, congruentem, & in peripheriâ de&longs;criptâ atque <lb/>in gradus di&longs;tributa arcum inventum ex Canone de&longs;igna notâ <lb/>partis aliquotæ 1/2, 1/3, 1/4 &c. </s> <s id="s.003956">ut &longs;tatim appareat, quo loco intelli­<lb/>gatur po&longs;ita potentia &longs;ive in &longs;emi&longs;&longs;e, &longs;ive in triente, &longs;ive in qua­<lb/>drante &longs;emidiametri, &longs;ive in eju&longs;dem be&longs;&longs;e aut dodrante &c. </s> <lb/> <s id="s.003957">Factâ &longs;iquidem comparatione inter di&longs;tantiam potentiæ ` cen­<lb/>tro, & Axis &longs;emidiametrum, innote&longs;cet Ratio ponderis ad po­<lb/>tentiam in tympano calcantem. </s> <s id="s.003958">Ponamus itaque tympani &longs;emi­<lb/>diametrum di&longs;tinctam in partes 10, ita ut Axis &longs;emidiameter <lb/>EO ad EB &longs;it ut 1 ad 10: po&longs;&longs;unt commodè omnes partes intra <lb/>decimas reperiri, pro ut in adjectâ tabellâ oculis &longs;ubjicio, in <lb/>qualibèt minuta &longs;ecunda exprimantur, ut innote&longs;cat etiam alios <lb/>in u&longs;us, quibus Sinubus quinam arcus re&longs;pondeant: in præ&longs;enti </s> </p> <p type="table"> <s id="s.003959">TABELLE WAR HIER <pb pagenum="523" xlink:href="017/01/539.jpg"/>tamen opere pror&longs;us inutilis accideret tam exqui&longs;ita accuratio: <lb/>&longs;atis quippe e&longs;t circiter illum gradum ejú&longs;que minuta prima <lb/>rotam appingere, indicem partis, vel partium &longs;emidiametri <lb/>tympani. </s> </p> <p type="main"> <s id="s.003960">Quartò. <!-- KEEP S--></s> <s id="s.003961">Funiculum Axi in&longs;i&longs;tentem, & facilè excurrentem <lb/>ita di&longs;pone, ut plumbeus globus in ejus extremitate pendulus <lb/>intendat funiculum ip&longs;um, qui in tympani limbo de&longs;ignet <lb/>punctum, per quod tran&longs;it linea perpendicularis ab Axis cen­<lb/>tro in horizontem de&longs;cendens. </s> <s id="s.003962">Tum ab hoc puncto u&longs;que ad <lb/>punctum, ubi fit æquilibrium, &longs;umatur intervallum, atque <lb/>transferatur in Quadrantem gradibus di&longs;tinctum: Nam <lb/>punctum, in quod ab initio Quadrantis cadit altera ob&longs;ervati <lb/>intervalli extremitas, indicabit notâ in limbo prænotatâ, quotâ <lb/>&longs;emidiametri parte di&longs;tet à tympani centro gravitas calcans <lb/>ip&longs;um tympanum, ex. </s> <s id="s.003963">gr. <!-- REMOVE S-->3/5 aut 4/7. Cum igitur jam innotuerit <lb/>Ratio &longs;emidiametri Axis ad &longs;emidiametrum tympani, &longs;cilicet <lb/>ex hypothe&longs;i (1/10), fiat ut fractio index Rationis &longs;emidiametro­<lb/>rum, ad fractionem in limbo notatam, ita gravitas calcantis <lb/>tympanum ad gravitatem ponderis, cum quo facit æquilibrium, <lb/>videlicet ut (1/10) ad 3/5, ita gravitas hominis, puta lib. 250, ad gra­<lb/>vitatem oneris lib. 1500. Hinc patet in puncto D, per quod <lb/>tran&longs;it linea OD tangens Axem & parallela perpendiculari <lb/>EH ex centro demi&longs;&longs;æ, æquilibrium e&longs;&longs;e inter gravitates om­<lb/>nino æquales, ac proinde minimum pondus e&longs;&longs;e æquale gravi­<lb/>tati hominis calcantis: Nam &longs;i inter H & D fieret æquilibrium, <lb/>pondus levius e&longs;&longs;et quàm homo, & communi &longs;taterá facile po­<lb/>tes a&longs;&longs;equi illius gravitatem. </s> <s id="s.003964">Maximum autem pondus e&longs;t il­<lb/>lud, quod indicat &longs;emidiameter tympani ad &longs;emidiametrum <lb/>Axis, homine nimirum &longs;uæ gravitatis vires exercente in B, ac <lb/>propterea gravitas ponderis ad gravitatem hominis in B e&longs;&longs;et <lb/>ex. </s> <s id="s.003965">gr. <!-- REMOVE S-->in Ratione decuplâ. </s> </p> <p type="main"> <s id="s.003966">Illud tamen hìc perpende, quòd, &longs;i homo calcans in B, aut <lb/>indè pendens, non volvit Axem, atque adeò non attollit pon­<lb/>dus adnexum, non con&longs;tat, an &longs;it æquilibrium, idem enim ac­<lb/>cideret etiam, &longs;i pondus e&longs;&longs;et multò majus; ac proinde neque <lb/>con&longs;tat de eju&longs;dem ponderis gravitate ni&longs;i hoc, quod &longs;it ut mi­<lb/>nimum decupla gravitatis hominis; quia nimirum nunquam il-<pb pagenum="524" xlink:href="017/01/540.jpg"/>lud movebit; nam a&longs;cendens homo ex B versùs C minora &longs;em­<lb/>per obtinet momenta, quàm in B. </s> <s id="s.003967">Hoc autem ubi contigerit, <lb/>& velis exploratam habere oneris gravitatem, a&longs;&longs;ume pondus <lb/>aliquod notæ gravitatis, quod adnectere valeas oræ tympani <lb/>aut in B, aut eo loco, ut deinde homo calcans infra B, attollat <lb/>pondus: ubi enim demum fiat æquilibrium, duplex in&longs;titue ra­<lb/>tiocinium, alterum quidem ratione hominis, alterum verò ra­<lb/>tione gravitatis additæ: inventi &longs;iquidem termini &longs;imul additi <lb/>indicabunt quæ&longs;itam oneris gravitatem. </s> <s id="s.003968">Sic &longs;i a&longs;&longs;umptum pon­<lb/>dus &longs;it lib.36, & homo calcans &longs;it lib. 250, fiat autem æquili­<lb/>brium homine exi&longs;tente in X, pondere verò in G; &longs;umptis in­<lb/>tervallis XH & GH, atque tran&longs;latis in Quadrantem invenia­<lb/>tur pro homine (9/10), & pro pondere 4/5. Fiat primò ut (1/10) ad (9/10), ita <lb/>lib. 250 ad lib. 2250: deinde ut (1/10) ad 4/5, ita lib.36 ad lib. 288: <lb/>igitur &longs;umma lib. 2538 indicat ponderis gravitatem. </s> <s id="s.003969">Quòd &longs;i <lb/>funis ductarij extremitas &longs;it adnexa extremæ trabi, ut indica­<lb/>tum e&longs;t, atque tran&longs;eat per orbiculum ponderi conjunctum, <lb/>inventus numerus lib.2538 duplicandus e&longs;t & &longs;unt lib.5076. </s> </p> <p type="main"> <s id="s.003970">Ex his forta&longs;&longs;e alicui placeat &longs;tateræ L M vires augere addi­<lb/>to tympano huju&longs;modi EB ut &longs;upra prænotato. </s> <s id="s.003971">Nam firmatâ <lb/><figure id="id.017.01.540.1.jpg" xlink:href="017/01/540/1.jpg"/><lb/>in &longs;uperiore loco &longs;taterâ <lb/>LM, cujus an&longs;a &longs;it in N, <lb/>primùm ob&longs;ervetur, quan­<lb/>tum ponderis requiratur <lb/>in M, ut fiat æquilibrium <lb/>cum &longs;olo brachio NL; <lb/>hæc enim gravitas &longs;emper <lb/>addenda erit gravitati, quæ <lb/>invenietur ex ratiocina­<lb/>tione, qua componuntur <lb/>Rationes &longs;tateræ & tym­<lb/>pani. </s> <s id="s.003972">Deinde in tympani <lb/>limbo notetur punctum C, <lb/>cui congruit funis OL, quando &longs;tatera e&longs;t horizonti parallela; <lb/>ut hinc digno&longs;catur, quo in loco tympani, dum convertitur, <lb/>contingat æquilibrium. </s> <s id="s.003973">Demum cùm nota &longs;it Ratio MN ad <lb/>NL, componatur cum Ratione &longs;emidiametri Axis ad partem <pb pagenum="525" xlink:href="017/01/541.jpg"/>&longs;emidiametri tympani, ex. </s> <s id="s.003974">gr. <!-- REMOVE S-->EO ad ES; & habetur Ratio <lb/>gravitatis hominis tympanum calcantis ad gravitatem oneris, <lb/>quod in M expenditur. </s> <s id="s.003975">Sit EO ad ES ut (1/10) ad 4/5, & MN ad <lb/>NL ut 2 ad 25; Ratio compo&longs;ita e&longs;t ut 1/5 ad 20, hoc e&longs;t ut 1 ad <lb/>100. Igitur pondus in M expen&longs;um e&longs;t, ut minimum, centu­<lb/>plum gravitatis hominis; nam addenda præterea e&longs;t gravitas <lb/>re&longs;pondens gravitati brachij LN longioris ip&longs;ius &longs;tateræ. </s> </p> <p type="main"> <s id="s.003976">Si verò neque tympano, quod ab homine intùs calcante <lb/>premitur, neque adeò incerto &longs;acomate, cuju&longs;modi e&longs;t varia <lb/>hominum gravitas, uti volueris, aut potius non ingentes &longs;arci­<lb/>nas, &longs;ed onera mediocria expendere placuerit, paretur CBH <lb/>di&longs;cus ligneus parvulum axem habens ad centrum E, & in eo <lb/>de&longs;cripta &longs;it peripheria circuli CBH, atque adnotatum <lb/>punctum C, per quod funiculus OL tran&longs;it, quando &longs;tateræ <lb/>jugum LM e&longs;t horizonti parallelum. </s> <s id="s.003977">Tum conver&longs;o di&longs;co ita, <lb/>ut LO tran&longs;eat per C, dimi&longs;&longs;o perpendiculo in&longs;i&longs;tente Axi, <lb/>notetur in peripheria punctum H, per quod tran&longs;it perpendi­<lb/>cularis à centro E. <!-- KEEP S--></s> <s id="s.003978">Deinde ex H versùs B a&longs;cendendo acci­<lb/>piantur gradus juxta &longs;uperiorem tabellam, affixis notis indici<lb/>bus partium, &longs;imiliter ac de tympano dictum e&longs;t. </s> <s id="s.003979">Demum &longs;a­<lb/>coma certæ gravitatis, puta unius aut alterius libræ, ita di&longs;po­<lb/>natur, ut per di&longs;ci ambitum ex H versùs B excurrere po&longs;&longs;it, & <lb/>cochleâ firmari, ubi æquilibrium contigerit: Aut potiùs &longs;ingu­<lb/>lis Quadrantis gradibus, aut &longs;altem punctis partium notatis, <lb/>claviculos infige, qui in&longs;eri po&longs;&longs;int annulo &longs;acomatis. </s> <s id="s.003980">Nota <lb/>enim Ratio &longs;emidiametri axis EO ad di&longs;ci &longs;emidiametrum EB <lb/>indicabit, quid faciendum &longs;it juxta dicta, ut gravitas ponderis <lb/>in M &longs;u&longs;pen&longs;i innote&longs;cat. </s> <s id="s.003981">At &longs;i volueris Quadrantem HB in <lb/>&longs;uos 90 gradus di&longs;tribuere, & uti Canone Sinuum, priùs inno­<lb/>te&longs;cat Ratio &longs;emidiametri axis ad Radium in partibus Radij: <lb/>deinde fiat ut partes Radij axi congruentes, ad Sinum Rectum <lb/>graduum, ubi fit æquilibrium, ita Sacoma appen&longs;um ad gra­<lb/>vitatem ponderis, quod expenditur. <pb pagenum="526" xlink:href="017/01/542.jpg"/></s> </p> <p type="main"> <s id="s.003982"><emph type="center"/>CAPUT IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.003983"><emph type="center"/><emph type="italics"/>An Axis in Peritrochio inveniatur etiam &longs;ine <lb/>tractione.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.003984">HActenus ductarij funis conver&longs;ionem circa Axem convolu­<lb/>tum con&longs;ideravimus, ex quo oritur ponderis fune connexi <lb/>tractio: &longs;ed numquid non etiam ad hoc genus machinæ aliquæ <lb/>revocari po&longs;&longs;unt, quibus non quidem trahitur pondus, &longs;ed ali­<lb/>qua re&longs;i&longs;tentia &longs;uperatur? </s> </p> <p type="main"> <s id="s.003985">Occurrit autem primo loco antiquus &longs;ervorum metus Pi&longs;tri­<lb/>num, in quod detru&longs;i frumentum tundere cogebantur molâ <lb/>ver&longs;atili, &longs;ive in no&longs;trarum Moletrinarum &longs;peciem ac &longs;imili­<lb/>tudinem metam congruo Catillo impo&longs;itam manu truderent, ac <lb/>circumagerent, &longs;ive ingentem lapideum di&longs;cum perpendicula­<lb/>ri cylindro coagmentatum ver&longs;arent a&longs;ellorum vicarij laborio­<lb/>&longs;o muneri &longs;uccedentes, cum Vectis cylindro ad angulos rectos <lb/>infixi extremitatem aut traherent, aut propellerent: Cuju&longs;mo­<lb/>di machinæ genere nos quoque utimur in frendendis legumi­<lb/>nibus, & in contundendis &longs;eminibus, ex quibus demùm oleum <lb/>prælo exprimitur. </s> <s id="s.003986">Et hìc quidem non ip&longs;ius molaris lapidis <lb/>gravitatem movendam attendimus, quippe qui ip&longs;ius machinæ <lb/>pars e&longs;t; &longs;ed poti&longs;&longs;imum corporis à molâ compre&longs;&longs;i re&longs;i&longs;tentia <lb/>con&longs;ideranda e&longs;t, quæ nimirum vincenda proponitur. </s> <s id="s.003987">Oritur <lb/>autem hæc re&longs;i&longs;tentia ex corporis obterendi aut contundendi <lb/>duritiæ, in quod incurrit &longs;cabra molæ circumactæ &longs;uperficies; <lb/>cum verò illud incumbenti lapidi &longs;e &longs;ubducere non po&longs;&longs;it, à la­<lb/>pidis gravitate & potentiæ impetu cogitur di&longs;&longs;ilire in partes. </s> <lb/> <s id="s.003988">Quia igitur potentiæ cum machinâ connexæ motum impedit <lb/>illa granorum aut nucleorum frangendorum durities, compa­<lb/>randa e&longs;t di&longs;tantia potentiæ moventis à centro motûs, cum <lb/>di&longs;tantiâ corporis comminuendi; & quò major e&longs;t huju&longs;modi <lb/>intervallorum Ratio, majora pariter &longs;unt potentiæ momenta. </s> </p> <p type="main"> <s id="s.003989">Hinc vides, cur in tru&longs;atili mola (quam mediocrem e&longs;&longs;e <pb pagenum="527" xlink:href="017/01/543.jpg"/>oportet, ne nimio labore frangatur molitor in immani &longs;axo ver­<lb/>&longs;ando, catillus quidem planus e&longs;t, meta verò, quâ catillum <lb/>re&longs;picit, non omnino &longs;ubjecto plano congruit, &longs;ed cavam ob­<lb/>tu&longs;i&longs;&longs;imi coni &longs;uperficiem æmulatur: ut &longs;cilicet integra grana <lb/>per medium foramen immi&longs;&longs;a inter utrumque lapidem interci­<lb/>piantur non procul à centro, à quo potentia abe&longs;t, comminu­<lb/>ta autem peripheriam versùs accedant ad angu&longs;tiora &longs;patia, <lb/>quò magis obterantur: cùm enim integra grana magis fractio­<lb/>ni ob&longs;i&longs;tant, quàm comminuta, integris frangendis majora de­<lb/>bentur potentiæ momenta, comminutis in minu&longs;culas particu­<lb/>las redigendi, minores vires &longs;ufficiunt. </s> <s id="s.003990">In molâ autem A&longs;ina­<lb/>riâ ubi lapideus di&longs;cus in plano Verticali con&longs;titutus &longs;ub­<lb/>jectum catillum modicè excavatum vix extremo ambitu con­<lb/>tingit, eadem ferè e&longs;t &longs;emper di&longs;tantia à centro motûs, ni&longs;i <lb/>quatenus ip&longs;ius molæ cra&longs;&longs;itudo partem aliam centro motûs pro­<lb/>piorem, aliam remotiorem habet: porrò grana illa, quæ lapi­<lb/>dum contactui, vel qua&longs;i contactui, propiora &longs;unt, validiùs te­<lb/>runtur, quàm quæ magis ab eodem contactu recedunt: &longs;ed <lb/>hoc nihil ad præ&longs;entem di&longs;putationem attinet, ni&longs;i quatenus la­<lb/>pidis partes remotiores &longs;ubjecta grana agitantes, atque tunden­<lb/>tes cra&longs;&longs;iùs, majorem Rationem ad potentiæ motum habent in <lb/>&longs;uâ convolutione, quàm partes eju&longs;dem minùs à centro <lb/>remotæ. </s> </p> <p type="main"> <s id="s.003991">Haud di&longs;&longs;imili ratione, &longs;i ex chalybe ellipticum &longs;phæroides <lb/>obliquis &longs;triis modicè a&longs;perum, qua&longs;i in limæ &longs;peciem, congruo <lb/>loculamento interiùs pariter a&longs;perato includatur, ita tamen, ut <lb/>&longs;patium, quo &longs;phæroides à loculamento di&longs;tat, paulatim à lati­<lb/>tudine in angu&longs;tias &longs;e &longs;e contrahat; axi verò &longs;phæroidis &longs;upe­<lb/>riùs producto addatur manubrium, quo arrepto converti po&longs;­<lb/>&longs;it in gyrum; grana piperis, aut &longs;imilia &longs;uperiùs immi&longs;&longs;a levi&longs;­<lb/>&longs;imo negotio comminuentur: quorum &longs;cilicet durities &longs;i cum <lb/>potentiæ viribus conferatur, re&longs;i&longs;tentiam habet maximam pro <lb/>Ratione &longs;emidiametri tran&longs;ver&longs;æ Ellip&longs;is ad manubrij longitu­<lb/>dinem: initio autem, quia grana minùs ab axe di&longs;tant, minùs <lb/>re&longs;i&longs;tunt, &longs;i cætera fuerint paria, hoc e&longs;t, &longs;i modicè comminu­<lb/>torum durities, integrorum duritiei omnino re&longs;pondeat; nam <lb/>minor di&longs;tantia à centro motûs minorem habet Rationem, <lb/>quàm di&longs;tantia major ad eandem manubrij longitudinem. </s> </p> <pb pagenum="528" xlink:href="017/01/544.jpg"/> <p type="main"> <s id="s.003992">Par erit philo&longs;ophandi ratio, &longs;i tympanis non idem centrum <lb/>habentibus inclu&longs;a aqua ex interioris tympani conver&longs;ione ad <lb/>angu&longs;tias redigatur, atque compre&longs;&longs;a exprimatur ex tubo; cu­<lb/>ju&longs;modi forta&longs;&longs;e fuit veterum Hydraconti&longs;terium; de cujus for­<lb/>mâ non e&longs;t hìc di&longs;putandi locus; nam manubrij à potentiâ <lb/>commoti longitudo comparanda e&longs;t cum di&longs;tantia peripheriæ <lb/>tympani aquam comprimentis à centro, circa quod perficitur <lb/>motus, ut momentorum Ratio per&longs;pecta &longs;it; aqua &longs;cilicet dum <lb/>impellitur, atque exprimitur, re&longs;i&longs;tit. </s> </p> <p type="main"> <s id="s.003993">Ad hæc porrò inver&longs;us quidam Axis in peritrochio u&longs;us con­<lb/>&longs;iderandus e&longs;t, quando videlicet potentia non peritrochio, &longs;ed <lb/>ip&longs;i Axi, applicatur; id quod tunc poti&longs;&longs;imum contingit, cùm <lb/>potentia vitibus abundat, motui autem, qui efficiendus e&longs;t, <lb/>non admodum re&longs;i&longs;tit corpus, quod vel modicè impellendum <lb/>e&longs;t, vel in orbem circumducendum. </s> <s id="s.003994">Certum quippe e&longs;t poten­<lb/>tiam Axi applicatam tardiùs multò moveri, quàm peritrochij <lb/>peripheriam, pro Ratione &longs;emidiametrorum Axis & Peritro­<lb/>chij, ac proinde licèt impetus amplioris peripheriæ partibus <lb/>impre&longs;&longs;us imbecillior quodammodo &longs;it, ut pote di&longs;tractus, &longs;atis <lb/>tamen e&longs;&longs;e ad vincendam levem re&longs;i&longs;tentiam. </s> <s id="s.003995">Hinc quoniam <lb/>ferrum cotis tritu extenuatur, eóque faciliùs, quò celeriùs eos <lb/>movetur, qui re&longs;tituunt obtu&longs;as cultorum aut novacularum <lb/>acies, lapidem ex cotariâ eductum in di&longs;cum rotundant, ut cir­<lb/>ca axem centro infixum ver&longs;atilis circumagi po&longs;&longs;it. </s> <s id="s.003996">Quamvis <lb/>autem non rarò eidem axi cohæreat manubrium, quo circum­<lb/>ducto rotatur lapis, ut tamen minori labore adhuc etiam velo­<lb/>ciùs rotetur, &longs;apienter in&longs;tituerunt amplioris rotæ ab&longs;idi exca­<lb/>vatæ funem in&longs;i&longs;tere, qui rotulam eumdem cum lapide axem <lb/>habentem circumplectatur, ut minor hæc rotula amplioris ro­<lb/>tæ ductum &longs;equens &longs;ecum pariter rapiat cotem; cujus periphe­<lb/>ria, cùm adnexam rotulam valdè excedat, velociùs quoquè <lb/>movetur. </s> <s id="s.003997">Quantùm verò motus hic, celeritate &longs;uâ, potentiæ <lb/>motum &longs;uperet, facilè con&longs;tabit, &longs;i motuum &longs;ingulis membris <lb/>convenientium ratio ineatur. </s> <s id="s.003998">Sit ex. </s> <s id="s.003999">gr. <!-- REMOVE S-->manubrij longitudo <lb/>ad amplioris rotæ &longs;emidiametrum &longs;ubquadrupla, hæc autem &longs;e­<lb/>midiameter ad rotulæ &longs;emidiametrum &longs;it octupla: demum ro­<lb/>tulæ eundem cum cote axem habentis &longs;emidiameter &longs;it &longs;ubtri­<lb/>pla &longs;emidiametri ip&longs;ius cotis. </s> <s id="s.004000">Igitur puncti in cotis peripheriâ <pb pagenum="529" xlink:href="017/01/545.jpg"/>notati motus triplo velocior e&longs;t motu &longs;imilis puncti in rotæ <lb/>peripheriâ: rotulæ motum definit funis, qui in convolutione ex­<lb/>plicatur, hic enim pariter majoris rotæ motum metitur: octies <lb/>ergo rotula, & cum eâ lapis rotatur, dum amplior rota &longs;emel in <lb/>gyrum agitur. </s> <s id="s.004001">Quoniam verò rotæ &longs;emidiameter e&longs;t ad cotis <lb/>&longs;emidiametrum ut 8 ad 3 ex hypothe&longs;i, dum punctum in rotæ, <lb/>peripheriá notatum movetur velocitate ut 8, &longs;imile punctum <lb/>cotis movetur velocitate ut 24. Atqui motus rotæ cum motu <lb/>potentiæ manubrio applicatæ comparatus e&longs;t ut 4 ad 1, ex hy­<lb/>pothe&longs;i; igitur &longs;i duæ Rationes 1 ad 4, & 8 ad 24 componan­<lb/>tur, erit Ratio motus potentiæ ad motum cotis ut 1 ad 12. Sunt <lb/>hìc itaque duo Axes in Peritrochiis &longs;uis compo&longs;iti, & Potentia <lb/>Axibus applicata intelligitur; cùm enim in Verticali plano lapis <lb/>ip&longs;e ver&longs;etur &longs;uper polos læves atque politos, non admodum re­<lb/>pugnat motui; impre&longs;&longs;us autem impetus aliquandiu manens <lb/>potentiam ip&longs;am juvat. </s> </p> <p type="main"> <s id="s.004002">Simile quiddam ob&longs;ervandum occurrit in <expan abbr="horologiorũ">horologiorum</expan> motu, <lb/>quæ in turribus &longs;tatuuntur: nam cylindrum horizonti paralle­<lb/>lum circumplicat funis, quo vi ponderis de&longs;cendentis explica­<lb/>to, circumagitur rota eidem axi infixa: ex hac in con&longs;equentes <lb/>rotas derivatur motus &longs;emper velocior, qui demum temperatur <lb/>ex quadam motûs retardati & brevi&longs;&longs;imæ morulæ vici&longs;&longs;itudine, <lb/>cum po&longs;tremæ rotæ dentes in &longs;erræ modum conformati fu&longs;um, <lb/>cui Tempus adnectitur alternis motibus agunt. </s> <s id="s.004003">Primùm enim <lb/>dens rotæ &longs;uperior in pin­<lb/><figure id="id.017.01.545.1.jpg" xlink:href="017/01/545/1.jpg"/><lb/>nulam A incurrens eam <lb/>impellit, axémque HL <lb/>convertit unà cum tran&longs;­<lb/>ver&longs;ario CD & adjunctis <lb/>globulis plumbeis E & F, <lb/>qui &longs;imilem arcum de&longs;cri­<lb/>bunt, ac pinnula A, &longs;ed <lb/>longè majorem; propterea <lb/>pro ratione gravitatis glo­<lb/>bulorum, eorúmque di­<lb/>&longs;tantiæ ab axe, HL, etiam <lb/>major vis requiritur, adeóque impeditur, ac retardatur motus <lb/>rotæ dentatæ, & cum eâ reliquarum rotarum, atque ip&longs;ius pon-<pb pagenum="530" xlink:href="017/01/546.jpg"/>deris, à quo totius machinæ motus initium &longs;umit: qui &longs;i fuerit <lb/>ju&longs;to velocior, globuli E & F removentur ab L, &longs;in autem <lb/>ju&longs;to tardior, admoventur, ut modò major, modò minor &longs;it re­<lb/>&longs;i&longs;tentia. </s> <s id="s.004004">Deinde quia globuli E & F ex impul&longs;u pinnulæ A <lb/>impetum conceperunt ad certam plagam directum, pergerent <lb/>illor&longs;um moveri, quamdiu impre&longs;&longs;us impetus per&longs;everaret, ni&longs;i <lb/>in eâ conver&longs;ione inferior pinnula B occurreret inferiori denti <lb/>rotæ &longs;erratæ: hinc fit vi hujus impetûs brevi&longs;&longs;imam morulam in­<lb/>ferri conver&longs;ioni rotæ, quæ eandem pinnulam urgens ip&longs;os quo­<lb/>què globulos in contrariam plagam reflectit. </s> <s id="s.004005">Po&longs;&longs;e autem adeò <lb/>exilibus viribus morulam inferri tanto ponderi de&longs;cendenti, <lb/>paulò inferiùs manife&longs;tum fiet, ubi de Rotis dentatis in unam <lb/>machinam compactis di&longs;&longs;eretur; illud &longs;altem palam e&longs;t, &longs;i mo­<lb/>rulam nullam admittas, re&longs;i&longs;tentiam e&longs;&longs;e non &longs;olùm pro gravi­<lb/>tate globulorum, eorúmque di&longs;tantiâ, verùm etiam pro ratione <lb/>impetûs impre&longs;&longs;i in antecedenti impul&longs;ione. </s> <s id="s.004006">Quare globuli <lb/>iidem quando moventur impulsâ pinnulâ, rationem habent <lb/>ponderis peritrochio adnexi, & potentia impellens pinnulæ, <lb/>hoc e&longs;t axi, applicatur: Contrà verò ad retardandum, aut tan­<lb/>ti&longs;per coërcendum motum ponderis, quod pinnulæ applicatum <lb/>intelligitur, iidem globuli vi impetûs &longs;ibi impre&longs;&longs;i rationem ha­<lb/>bent potentiæ peritrochio applicatæ. </s> </p> <p type="main"> <s id="s.004007">Quoniam verò hìc horologiorum mentio incidit, cur in illis, <lb/>quæ &longs;ecum qui&longs;que ad perpetuum u&longs;um ferre pote&longs;t, catenula <lb/>&longs;eu nervus cono, non cylindro, circumducatur, manife&longs;tum <lb/>e&longs;t: quia &longs;cilicet chalybea lamella, à qua motûs origo e&longs;t, initio <lb/>in &longs;pi&longs;&longs;iorem &longs;piram contracta &longs;uam vim ela&longs;ticam exerens va­<lb/>lidiùs conatur &longs;e re&longs;tituere, trahén&longs;que catenulam, &longs;eu ner­<lb/>vum FE, totum conum DEC eíque con­<lb/><figure id="id.017.01.546.1.jpg" xlink:href="017/01/546/1.jpg"/><lb/>nexam rotulam dentatam AB in gyrum agit; <lb/>cum verò illa fuerit in paulò laxiores &longs;piras <lb/>explicata, languidiùs conatur, atque catenu­<lb/>lam, &longs;eu nervum, trahens jam non propè verti­<lb/>cem coni, &longs;ed magis ad ba&longs;im, eandem rotam <lb/>AB circumagit. </s> <s id="s.004008">Cùm igitur motus potentiæ <lb/>propè verticem coni, ad motum rotæ AB mi­<lb/>norem habeat Rationem, quàm ad eundem rotæ motum mo­<lb/>tus potentiæ in latiore coni parte (ibi enim breviorem, hìc ma-<pb pagenum="531" xlink:href="017/01/547.jpg"/>jorem gyrum perficit) ut quædam motûs æquabilitas in horo­<lb/>logio &longs;ervetur, opportunum fuit potentiæ validiùs conanti ma­<lb/>jorem opponi re&longs;i&longs;tentiam, minorem verò languidiùs conanti: <lb/>nam &longs;i catenula non conum, &longs;ed cylindrum circumplecte­<lb/>retur, eadem &longs;emper e&longs;&longs;et motuum men&longs;ura atque Ratio, <lb/>&longs;ed inæquales vires ela&longs;ticæ motum inæqualiter velocem ef­<lb/>ficerent. </s> </p> <p type="main"> <s id="s.004009">Huc pariter revocandas e&longs;&longs;e Terebrarum vires vix cui­<lb/>quam dubium e&longs;&longs;e pote&longs;t, quarum quò ampliora &longs;unt ma­<lb/>nubria, majores pariter e&longs;&longs;e vires con&longs;tat; quandoquidem <lb/>potentia ampliorem circulum de&longs;cribit, dum terebræ acies <lb/>minimo motu ligni aut metalli particulas, in quas incurrit, <lb/>ab&longs;cindit. </s> <s id="s.004010">Quæcumque demum &longs;it terebræ forma, five ejus <lb/>apex in cochleam &longs;triatam exacutam <lb/>de&longs;inat, ut AB manubrium habens <lb/><figure id="id.017.01.547.1.jpg" xlink:href="017/01/547/1.jpg"/><lb/>CD rectum, &longs;ive in aciem obli­<lb/>quam aut planam, aut modicè ex­<lb/>cavatam exeat ut EF, manubrium <lb/>autem LHI inflexum habeat circa <lb/>GI ver&longs;atile (quam Zerebram Gal­<lb/>licam aliqui Itali appellant) &longs;ive fer­<lb/>rea lamina in orbem convoluta, & <lb/>inferiùs denticulata, ut MN, ma­<lb/>nubrio tran&longs;ver&longs;o OP coaptetur: Si­<lb/>militer &longs;emper e&longs;t momentorum Ra­<lb/>tio de&longs;umenda aut ex tran&longs;ver&longs;arij <lb/>CD longitudine ad cra&longs;&longs;itiem co­<lb/>chleæ &longs;triatæ B, aut ex di&longs;tantiâ <lb/>puncti H à lineâ tran&longs;eunte per GIEF ad integram &longs;eu di­<lb/>midiatam aciei F latitudinem (prout extremum punctum F in <lb/>mediâ aut in extremâ latitudine po&longs;itionem habet) aut ex ma­<lb/>nubrij OP longitudine ad diametrum circularis &longs;erræ NM: <lb/>partes autem &longs;ubjecti corporis ab&longs;cindendæ, ut illud perfore­<lb/>tur, habent rationem ponderis movendi eò difficiliùs, quò va­<lb/>lidiore nexu illæ invicem conjunguntur. </s> </p> <p type="main"> <s id="s.004011">Eadem erit philo&longs;ophandi methodus in eo terebræ gene­<lb/>re, cui nos Itali proximè ad Græcum vocabulum <foreign lang="greek">tru/panon</foreign><lb/>accedentes nomen fecimus. </s> <s id="s.004012">Teretis baculi BC extremitati C <pb pagenum="532" xlink:href="017/01/548.jpg"/>additur chalybea cu&longs;pis CD ita in punctum de&longs;inens, ut ad <lb/><figure id="id.017.01.548.1.jpg" xlink:href="017/01/548/1.jpg"/><lb/>aliquam latitudinem obliquè a&longs;cendat pro <lb/>ratione &longs;emidiametri foraminis, quod ma­<lb/>ximum artificis animus de&longs;tinavit. </s> <s id="s.004013">Inferior <lb/>baculi pars infigitur &longs;phæroidi H, & tran&longs;­<lb/>ver&longs;arium GI in medio E ita perforatum <lb/>e&longs;t, ut facillimè per in&longs;ertum baculum ex­<lb/>currens &longs;ur&longs;um deor&longs;um agitari po&longs;&longs;it: <lb/>cum enim extremitates G & I adnexum <lb/>funiculum habeant pertingentem u&longs;que <lb/>in B, hoc circa baculum contorto tran&longs;ver&longs;arium non procul <lb/>abe&longs;t à B, quod &longs;i deprimatur, explicatur funiculus, & bacu­<lb/>lus in gyrum agitur pariter cum infixa cu&longs;pide; qua &longs;en&longs;im ac <lb/>leniter minimas &longs;ubjectæ laminæ metallicæ particulas abraden­<lb/>te, demùm &longs;æpiùs repetitâ tran&longs;ver&longs;arij &longs;ur&longs;um deor&longs;um agita­<lb/>tione, atque adeò celeri terebellæ conver&longs;ione, foramen patet. </s> <lb/> <s id="s.004014">Quamvis autem artificis manus applicetur medio tran&longs;ver&longs;ario <lb/>in E, quod deprimit; potentia tamen intelligitur applicata &longs;u­<lb/>perficiei baculi medio funiculo illum circumplexo, perinde at­<lb/>que &longs;i inter utramque palmam alternis motibus adductam atque <lb/>reductam idem baculus convolveretur: tantóque major e&longs;t po­<lb/>tentiæ &longs;ic applicatæ motus, quanto exce&longs;&longs;u baculi ambitus &longs;u­<lb/>perat terebellæ &longs;ubjectam laminam abradentis gyrum. </s> <s id="s.004015">Quo­<lb/>niam verò potentia, hoc e&longs;t manus, movetur de&longs;cendendo, <lb/>ejus motus comparandus e&longs;t cum multiplici convolutione ba­<lb/>culi, quæ fit, dum explicatur funis. </s> </p> <p type="main"> <s id="s.004016">Sed & alia potentia hìc con&longs;ideranda occurrit: adjunctum <lb/>enim &longs;phæroides H, quod mediocriter grave &longs;tatuitur, non &longs;o­<lb/>lùm &longs;uo pondere juvat, ut cu&longs;pis paulò pre&longs;&longs;iùs adhæreat &longs;ub­<lb/>jectæ laminæ, verùm etiam concepto in convolutione impetu <lb/>dum explicatur funis, pergit ad ea&longs;dem partes moveri, expli­<lb/>catúmque funem iterum circa baculum contorquens cogit <lb/>tran&longs;ver&longs;arium GI a&longs;cendere versùs B, quo vici&longs;&longs;im ab artificis <lb/>manu depre&longs;&longs;o in contrarias partes volvitur. </s> <s id="s.004017">Impetus igitur <lb/>&longs;phæroidi H impre&longs;&longs;us, dum illud movet, rationem habet po­<lb/>tentiæ cu&longs;pidem in gyrum contorquentis, cujus momenta ex <lb/>di&longs;tantiâ ab axe, circa quem efficitur motus, definienda &longs;unt. <pb pagenum="533" xlink:href="017/01/549.jpg"/></s> </p> <p type="main"> <s id="s.004018"><emph type="center"/>CAPUT V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004019"><emph type="center"/><emph type="italics"/>Axium in &longs;uis peritrochiis compo&longs;itione vires <lb/>augentur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004020">COntingere pote&longs;t, & quidem non rarò, ad &longs;ervandam pe­<lb/>ritrochij & axis cum pondere & potentiá analogiam, tam <lb/>ingens tympanum aut manubrium Axi coaptandum e&longs;&longs;e, ut <lb/>aut loci angu&longs;tiæ commodè illud non patiantur, aut non ni&longs;i <lb/>majore di&longs;pendio, quàm &longs;it operæ pretium, tam ampla ma­<lb/>china parari, aut congruè di&longs;poni queat. </s> <s id="s.004021">Quid enim, &longs;i &longs;pecta­<lb/>tâ potentiæ validioris virtute centuplum onus &longs;u&longs;tollendum <lb/>proponatur? </s> <s id="s.004022">an cra&longs;&longs;iori Axi, qui &longs;atis firmus &longs;it, rotam aut <lb/>tympanum cujus diameter centupla &longs;it diametri Axis, adjun­<lb/>gemus? </s> <s id="s.004023">quanto id incommodo futurum e&longs;&longs;et, quantáque &longs;ub­<lb/>&longs;idia comparare oporteret, ut tam immanis machina citrà luxa­<lb/>tionem &longs;ub&longs;i&longs;teret, & congruo pegmati inniteretur, nemo non <lb/>videt. </s> <s id="s.004024">Satius itaque fuerit, in&longs;i&longs;tendo iis, quæ lib.2 cap.7. dicta <lb/>&longs;unt, machinam, quam ad centuplam altitudinem augere in­<lb/>commodum accideret, componere, pluribus Axibus cum &longs;uis <lb/>peritrochiis invicem ritè coaptatis. </s> </p> <p type="main"> <s id="s.004025">Statuendus primùm e&longs;t Axis, cujus &longs;oliditas oneris gravi­<lb/>tati &longs;u&longs;tinendæ re&longs;pondeat, longitudo circumflexas ducta­<lb/>rij funis &longs;piras commodè capiat. </s> <s id="s.004026">Deinde tympanum eligatur, <lb/>cujus diameter ad con&longs;tituti cylindri diametrum eam habeat, <lb/>quæ placuerit, Rationem, modò illa &longs;it majoris inæqualitatis, <lb/>ut manife&longs;tum e&longs;t: &longs;it ex. </s> <s id="s.004027">gr. <!-- REMOVE S-->quintupla, & cylindri cra&longs;&longs;ities <lb/>palmaris ponatur. </s> <s id="s.004028">Sed quoniam propo&longs;ito ponderi attollendo <lb/>impar e&longs;t potentia applicata machinæ re&longs;i&longs;tentiam gravitatis in <lb/>Ratione &longs;olùm quintuplâ extenuanti, alium adhibe Axem &longs;uo <lb/>Peritrochio infixum (vel quem fors tulerit antè in alios u&longs;us <lb/>paratum, vel &longs;ecundùm de&longs;tinatam Rationem elaboratum) cu­<lb/>jus ductarius funis ip&longs;i quidem Axi congruo loco conjungatur, <lb/>&longs;ed tympanum prioris Axis circumplectatur, ut in convolutio-<pb pagenum="534" xlink:href="017/01/550.jpg"/>ne &longs;ecundi Axis evolutus tympano illi motum conciliet, adeò­<lb/>que etiam ponderi. </s> <s id="s.004029">Duas igitur Rationes, quas &longs;ingula Peritro­<lb/>chia ad &longs;uos Axes habent, compone, ut potentiæ &longs;ecundo Peri­<lb/>trochio applicatæ momenta innote&longs;cant. </s> <s id="s.004030">Sit Ratio hæc po&longs;te­<lb/>rior ex. </s> <s id="s.004031">gr. <!-- REMOVE S-->quadrupla; & Ratio, quæ ex quadruplâ & quintuplâ <lb/>componitur, e&longs;t vigecupla, quæ adhuc minor e&longs;t, quàm opor­<lb/>teat. </s> <s id="s.004032">Quare, cùm ex Ratione centuplâ Ratio vigecupla &longs;ubducta <lb/>relinquat Rationem quintuplam, tertium Axem cum manubrio <lb/>quintuplæ longitudinis ad eju&longs;dem Axis &longs;emidiametrum &longs;imili­<lb/>ter appone, & erit ex his tribus Rationibus compo&longs;ita Ratio <lb/>centupla quæ&longs;ita. </s> <s id="s.004033">Quia enim potentia manubrio huic applicata <lb/>movetur quintuplo velociùs, quàm punctum, cui illa in &longs;ecundi <lb/>Axis tympano applicaretur, hoc verò quadruplo velociùs, quàm <lb/>&longs;imile punctum in tympano prioris Axis, potentia movetur vi­<lb/>gecuplo velociùs, quàm &longs;i applicaretur tympano prioris Axis: <lb/>huic autem tympano applicata moveretur quintuplo velociùs <lb/>quàm pondus: igitur manubrium illud ver&longs;ans potentia move­<lb/>tur centuplo velociùs quàm pondus: id quod fieri oportebat, ut <lb/>propo&longs;ita gravitas in altum attolleretur. </s> </p> <p type="main"> <s id="s.004034">Placeat jam triplicem hunc Axem cum unico illo comparate, <lb/>qui &longs;olus adhiberetur, &longs;i machina &longs;implex e&longs;&longs;et, & non compo­<lb/>&longs;ita: ille &longs;iquidem &longs;i palmaris diametri e&longs;&longs;et, adjunctum tympa­<lb/>num haberet altitudinis palmorum centum; in quo con&longs;truen­<lb/>do quàm multâ materiâ opus e&longs;&longs;et, quantoque artificio compin­<lb/>genda, ne &longs;uâ mole labefactata di&longs;&longs;olveretur? </s> <s id="s.004035">Triplex autem hic <lb/>Axis cum &longs;uis duobus tympanis, & manubrio (præterquam quod <lb/>multipliciter di&longs;poni pote&longs;t pro loci opportunitate, & potentiæ <lb/>moventis commodo) non &longs;olùm ad altitudinem <expan abbr="palmorũ">palmorum</expan> viginti <lb/><expan abbr="nõ">non</expan> a&longs;&longs;urgeret, &longs;ed longè infra <expan abbr="illã">illam</expan> &longs;ub&longs;i&longs;teret, à <expan abbr="quocũque">quocunque</expan> artifice <lb/>nullo negotio con&longs;trueretur, ab alio in <expan abbr="aliũ">alium</expan> <expan abbr="locũ">locum</expan> facillimè <expan abbr="trãsfer-retur">transfer­<lb/>retur</expan>, levíque <expan abbr="di&longs;p&etilde;dio">di&longs;pendio</expan> pararetur, ut cuique <expan abbr="cõ&longs;iderãti">con&longs;ideranti</expan> <expan abbr="obviũ">obvium</expan> e&longs;t. </s> </p> <p type="main"> <s id="s.004036">Hocautem, quod in tribus Axibus explicatum e&longs;t, de pluribus <lb/>etiam <expan abbr="dictũ">dictum</expan> intelligatur: nam &longs;i Rationes &longs;ingulæ peritrochij ad <lb/>&longs;uum axem con&longs;iderentur, & &longs;imul componantur multiplicando <lb/>invicem omnes Rationum terminos Antecedentes, item omnes <lb/>Con&longs;equentes, ut habeatur novus Antecedens & novus Con&longs;e­<lb/>quens, apparebit Ratio motûs potentiæ ad <expan abbr="motũ">motum</expan> ponderis, adeó­<lb/>que gravitatis ponderis ad virtutem potentiæ. </s> <s id="s.004037">Ex quo patet quo&longs;-<pb pagenum="535" xlink:href="017/01/551.jpg"/>cumque Axes oblatos utiles e&longs;&longs;e po&longs;&longs;e, modò Peritrochiorum ad <lb/>&longs;uos Axes Ratio innote&longs;cat, &longs;ive &longs;imiles &longs;int, &longs;ive di&longs;&longs;imiles Ratio­<lb/>nes, &longs;ive multiplices, &longs;ive &longs;uperparticulares, &longs;ive &longs;uperpartientes: <lb/>demum enim, &longs;i quid de&longs;it ad quæ&longs;itam Rationem, addi pote&longs;t <lb/>certus Axis cum manubrio ita, ut Ratio quæ&longs;ita expleatur. </s> <s id="s.004038">Sint <lb/>quinque Axes in &longs;uis Peritrochiis omnino &longs;imiles, & &longs;inguli con­<lb/>tineant <expan abbr="Ration&etilde;">Rationem</expan> <expan abbr="decuplã">decuplam</expan>: quinque Rationes 10 ad 1 invicem du­<lb/>cantur, & erit motus <expan abbr="Pot&etilde;tiæ">Potentiæ</expan> ad <expan abbr="motũ">motum</expan> ponderis, ut 100000 ad 1; <lb/>ac propterea quo conatu Potentia &longs;olitaria moveret talentum, <lb/>ac machinâ compo&longs;itâ movebit centum millia talentorum. </s> <s id="s.004039">Sint <lb/>item quinque Rationes, 10 ad 1, 20 ad 7, 8 ad 3, 9 ad 2, 4 ad 1 <lb/>(quocumque ordine inter &longs;e di&longs;ponantur) omnes Antecedentes <lb/>invicem ducti faciunt novum Antecedentem 57600, omnes au­<lb/>tem Con&longs;equentes invicem ducti dant novum <expan abbr="Con&longs;equent&etilde;">Con&longs;equentem</expan> 42; <lb/>quare Ratio Compo&longs;ita e&longs;t 57600 ad 42, hoc e&longs;t 9600 ad 7: & <lb/>potentia valens attollere libras 7, hac machinâ compo&longs;itâ attol­<lb/>let libras 9600. Quod &longs;i oporteret moveri libras decies mille ab <lb/>hac <expan abbr="ead&etilde;">eadem</expan> Potentiâ, auferatur Ratio 9600 ad 7 ex Ratione 10000 <lb/>ad 7, & relinquitur Ratio 700 ad 672, hoc e&longs;t 25 ad 24: quare <lb/>addendus e&longs;&longs;et &longs;extus Axis cum manubrio, cujus longitudo ad <lb/>Axis &longs;emidiametrum e&longs;&longs;et ut 25 ad 24, & potentia eadem hu­<lb/>ju&longs;modi manubrio applicata attollere po&longs;&longs;et libras 10000. </s> </p> <p type="main"> <s id="s.004040">Illud habere videtur incom­<lb/><figure id="id.017.01.551.1.jpg" xlink:href="017/01/551/1.jpg"/><lb/>modi hæc Axium compo&longs;itio, <lb/>quod magnam vim funium tym­<lb/>pana circumplectentium exi­<lb/>git, qui &longs;cilicet &longs;ingulorum tym­<lb/>panorum motui re&longs;pondeant. </s> <lb/> <s id="s.004041">Cum enim tympani A diameter <lb/>&longs;it quintupla Axis BC ex hypo­<lb/>the&longs;i, ejus motus e&longs;t quintuplo <lb/>major motu ponderis P, ac pro­<lb/>inde funis, qui tympani limbum <lb/>complectitur, quintuplo longior <lb/>e&longs;&longs;e debet fune PD, hoc e&longs;t mo­<lb/>tu ponderis; cujus funis tympa­<lb/>no A circumducti caput cum <lb/>Axe EF connectitur, circa <pb pagenum="536" xlink:href="017/01/552.jpg"/>quem in motu convolvitur. </s> <s id="s.004042">Quoniam verò tympanum O ex <lb/>hypothe&longs;i diametrum habet quadruplam diametri &longs;ui Axis EF <lb/>eju&longs;que motus e&longs;t ad motum &longs;ui Axis quadruplus, funis circum­<lb/>plicatus tympano O quadruplus e&longs;t funis, qui circa Axem EF <lb/>convolvitur, ac propterea etiam vigecuplus funis DP; adeo ut, <lb/>ubi totus funis evolutus fuerit, atque circa axem HG convolu­<lb/>tus, pondus &longs;ublatum u&longs;que in D intelligatur. </s> <s id="s.004043">Ex quo fit poten­<lb/>tiam manubrio IL applicatam, quia IG longitudo e&longs;t quintu­<lb/>pla &longs;emidiametri Axis GH, adhuc quintuplo velociùs moveri <lb/>quàm tympanum O, cujus motum metitur evolutio funis illud <lb/>circumplectentis; atque idcirco Potentia in I movetur centu­<lb/>plo velociùs quàm pondus P. <!-- KEEP S--></s> <s id="s.004044">Quare &longs;i adhuc quartum Axem ad­<lb/>dere oporteret, & loco manubrij GI tympanum &longs;uo fune in­<lb/>&longs;tructum apponeretur, funis ille e&longs;&longs;et ip&longs;ius PD centuplus; at­<lb/>que ita deinceps pro tympanorum & Axium multiplicatione <lb/>juxta &longs;ingulorum Rationem augeretur funium longitudo. </s> </p> <p type="main"> <s id="s.004045">Verùm pro tantâ funium longitudine non e&longs;t tympanorum <lb/>limbo enormis amplitudo tribuenda, ut eos capiat; quia &longs;cilicet <lb/>quò longiores exiguntur huju&longs;modi funes, eò etiam tenuiores <lb/>atque exiliores e&longs;&longs;e po&longs;&longs;unt: Si enim funis DP oneri attollendo <lb/>par con&longs;tat funiculis contortis invicem ex. </s> <s id="s.004046">gr. <!-- REMOVE S-->centum, funis <lb/>qui tympanum A complectitur, non ni&longs;i quintam ponderis par­<lb/>tem re&longs;i&longs;tentem habet, hoc e&longs;t ip&longs;um pondus P &longs;ubquintuplo <lb/>minore re&longs;i&longs;tentiâ repugnans potentiæ per tympanum A reti­<lb/>nenti: ac proinde &longs;i funium firmitatem funiculorum numerus <lb/>metitur, &longs;atis validus erit funis con&longs;tans ex funiculis viginti. </s> <s id="s.004047">Si­<lb/>militer funis complectens tympanum O, quia pondus re&longs;i&longs;ten­<lb/>tiam &longs;ubvigecuplo minorem habet, &longs;atis firmus cen&longs;ebitur, &longs;iex <lb/>quinque funiculis invicem contortis confletur. </s> <s id="s.004048">Quod &longs;i ju&longs;to <lb/>tenuiores timeas huju&longs;modi funes, licebit adhuc paulò cra&longs;&longs;io­<lb/>res adhibere. </s> <s id="s.004049">Illud certè manife&longs;tum e&longs;t multo minores &longs;uffi­<lb/>cere, quàm &longs;it funis DP. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004050">Ne autem tympanis limbi amplitudinem funis circumducti <lb/>capacem temerè con&longs;tituas, &longs;ingulorum funium cra&longs;&longs;itudo con­<lb/>&longs;ideranda e&longs;t, ut eorum diameter innote&longs;cat, & longitudinis ra­<lb/>tione habitâ &longs;pirarum numerus inveniatur, per quem ducta fu­<lb/>nis diameter dabit nece&longs;&longs;ariam limbi amplitudinem; quæ &longs;i <lb/>ju&longs;to minor e&longs;&longs;et primum &longs;pirarum ordinem &longs;ecundus ordo cir-<pb pagenum="537" xlink:href="017/01/553.jpg"/>cumplecteretur: & quamvis initio hinc major aliqua movendi <lb/>facilitas oriretur (auctâ &longs;cilicet peritrochij diametro) tamen <lb/>evoluto &longs;ecundo hoc &longs;pirarum ordine diameter peritrochij di­<lb/>minuta majorem crearet movendi difficultatem; maxime &longs;i cir­<lb/>ca Axem convolutus funis &longs;pirarum ordinem pariter geminaret, <lb/>atque adeò Axis diametrum augeret. </s> <s id="s.004051">Funes itaque qua&longs;i cylin­<lb/>dri con&longs;iderandi &longs;unt, quorum cra&longs;&longs;itudinis diametri &longs;unt in <lb/>&longs;ubduplicatâ ba&longs;ium Ratione; ac propterea inter ip&longs;as cra&longs;&longs;itu­<lb/>dines numeris definitas inventendus e&longs;t numerus medio loco <lb/>proportionalis, & hic indicabit tenuioris funis diametrum, <lb/>quemadmodum primus numerus major ponitur pro diametro <lb/>cra&longs;&longs;ioris funis. </s> <s id="s.004052">Sic quoniam funis DP e&longs;t ex hypothe&longs;i ut 100, <lb/>& funis circa tympanum A &longs;ubquintuplæ cra&longs;&longs;itudinis e&longs;t ut <lb/>20, inveniatur inter 100 & 20 medius (44 72/100) proximè, & dia­<lb/>metri funium erunt proximè in Ratione 100 ad 45. Quare <lb/>quam amplitudinem requirunt 45 &longs;piræ maximi funis DP, <lb/>eandem exigunt 100 &longs;piræ minoris funis attributi tympano A, <lb/>&longs;i uteroue funis circa eumdem cylindrum convolvatur. </s> <s id="s.004053">Sed <lb/>quia perimeter tympani A quinquies continet perimetrum <lb/>Axis BC, unica tympani &longs;pira quinque Axis &longs;piris æquatur &longs;e­<lb/>cundùm longitudinem, & centum &longs;piræ tympani quingentis <lb/>Axis &longs;piris re&longs;pondent, &longs;i lineæ longitudo &longs;pectetur: &longs;atis au­<lb/>tem e&longs;t, &longs;i longitudinem &longs;pirarum 225 circa Axem, ille funis <lb/>tympani obtineat, quia longitudo illa e&longs;t ad funis DP longitu­<lb/>dinem 45 quintupla. </s> <s id="s.004054">Propterea tympani limbus minorem exi­<lb/>git amplitudinem, quàm &longs;it &longs;patium, quod in Axe BC occupa­<lb/>tur à convoluto fune DP: nimirum à limbo contineri oportet <lb/>&longs;ui funis circumplicati &longs;piras 45; &longs;atis igitur fuerit dimidiata <lb/>amplitudo. </s> </p> <p type="main"> <s id="s.004055">Simili methodo tympano O limbi amplitudinem definies: <lb/>quoniam enim funis cra&longs;&longs;itudo ad cra&longs;&longs;itudinem funis DP e&longs;t <lb/>&longs;ubvigecupla, inter 100 & 5 medio loco proportionalis (22 36/100) <lb/>proximè inveniatur; & e&longs;t diameter funis tympani O ad dia­<lb/>metrum funis DP ut (22 36/100) ad 100. Quare &longs;i circa eundem cy­<lb/>lindrum uterque funis convolveretur, quod &longs;patium &longs;piras fu­<lb/>nis DP 45 contineret, tenuioris hujus funis &longs;piras &longs;altem 201 <lb/>comprehenderet. </s> <s id="s.004056">Ponamus tympani O perimetrum e&longs;&longs;e ad pe-<pb pagenum="538" xlink:href="017/01/554.jpg"/>rimetrum Axis BC ut 4 ad 1: igitur limbus tympani O &longs;i ean­<lb/>dem habeat amplitudinem, quam funis DP occupat in &longs;uo Axe <lb/>BC, capiet &longs;ui funis &longs;piras 201, quæ in unam longitudinem <lb/>exten&longs;æ con&longs;tituunt longitudinem, quæ ad longitudinem &longs;pi­<lb/>rarum 45 funis DP e&longs;t ut 804 ad 45. Sed quia longitudo illius <lb/>funis e&longs;t vigecupla longitudinis funis DP, debet e&longs;&longs;e ut 900 <lb/>ad 45; ideò adhuc majorem exigit amplitudinem, ut adhuc &longs;pi­<lb/>ras 24 aut 25 &longs;upra ducentas obtineat. </s> <s id="s.004057">Quod &longs;i Axis EF &longs;ubti­<lb/>lior &longs;it quàm Axis BC, & tympani O diameter ad &longs;ui axis EF <lb/>diametrum quadrupla &longs;it, jam tympani perimeter ad perime­<lb/>trum Axis BC habebit minorem Rationem quàm 4 ad 1, ac <lb/>proinde ejus limbum adhuc ampliorem con&longs;titui nece&longs;&longs;e e&longs;t. </s> </p> <p type="main"> <s id="s.004058">Quare &longs;i Axis BC diameter &longs;it palmaris, &longs;piræ 45 funis DP <lb/>convoluti elevabunt pondus P ad altitudinem palmorum circi­<lb/>ter 141, quanta nimirum e&longs;&longs;et ip&longs;ius funis convoluti longitudo: <lb/>funis circa tympanum A longitudo e&longs;&longs;et palmorum &longs;altem 705, <lb/>& funis circa tympanum O longitudo palmorum 2820. Hinc <lb/>quamvis præter primum Axem BC oneri &longs;u&longs;tinendo parem, <lb/>reliqui con&longs;equentes Axes EF, & GH, & &longs;i qui alij adhuc &longs;int, <lb/>po&longs;&longs;int in minorem &longs;oliditatem extenuari; &longs;i ponderis re&longs;i&longs;ten­<lb/>tia attendatur, quia tamen, quò &longs;ubtiliores &longs;unt, frequentiori­<lb/>bus etiam &longs;piris circumplicantur, ex quo fit ut plures &longs;pirarum <lb/>ordines fiant, adeoque Axis diameter aucta minuat momento­<lb/>rum Rationem; præ&longs;tat exiles Axes non &longs;tudiosè quærere, ni&longs;i <lb/>fortè nece&longs;&longs;itas aliqua iis uti &longs;uadeat. </s> <s id="s.004059">Quamquam & huic in­<lb/>commodo occurri pote&longs;t, &longs;i, quemadmodum in Ergatæ u&longs;u <lb/>funem paucis aliquot &longs;piris circumductum, dum in conver&longs;io­<lb/>ne evolvitur, puer agglomerat, ita etiam hîc funem tympano­<lb/>rum A & O quatuor aut quinque &longs;piris circa Axes E & H con­<lb/>volutum puer colligeret: hoc enim pacto non contingeret, ut <lb/>primum &longs;pirarum ordinem alter &longs;pirarum ordo &longs;uperinductus <lb/>circumplecteretur. </s> </p> <p type="main"> <s id="s.004060">Porrò ne tantam funium vim comparare cogamur, & am­<lb/>pliore limbo tympana circum&longs;cribere, haud &longs;anè ineptum cen­<lb/>&longs;erem, &longs;i pro eorum more, qui novaculas obtu&longs;as acuunt, ut aliàs <lb/>innui, funis in &longs;e&longs;e rediens unâ aut altera (aut etiam triplici, &longs;i <lb/>opus fuerit) &longs;pirâ tùm Axem, tum &longs;ubjectum tympanum arctè <lb/>complecteretur: &longs;ic enim fieret, ut Axe convoluto etiam &longs;ub-<pb pagenum="539" xlink:href="017/01/555.jpg"/>jectum tympanum volveretur; idémque funis perpetuo ordine <lb/>à tympano in proximum Axem & ab Axe in tympanum &longs;uc­<lb/>cedens quantolibet motui perficiendo &longs;ufficeret. </s> <s id="s.004061">Et ut omne <lb/>periculum &longs;ubmoveatur, ne funis excurrat, &longs;atius e&longs;t tùm Axis, <lb/>tùm tympani ambitum non in cylindricam &longs;uperficiem expoli­<lb/>re, &longs;ed angulis a&longs;perum e&longs;&longs;e. </s> <s id="s.004062">Quod &longs;i aliquando languidior fu­<lb/>nis non adeò pre&longs;sè complecteretur Axem & tympanum, <lb/>&longs;pongiam aquâ imbutam ip&longs;i funi admove, & intentus fiet. </s> </p> <p type="main"> <s id="s.004063">Demùm in huju&longs;modi Axium compo&longs;itione non &longs;ine <lb/>animadver&longs;ione prætereundæ videntur mutua Axium po&longs;itio, <lb/>atque di&longs;tantia, qua &longs;ecundus Axis à primo tympano abe&longs;t. </s> <s id="s.004064">Sit <lb/>Axis A in Peritrochio CDE, at­<lb/><figure id="id.017.01.555.1.jpg" xlink:href="017/01/555/1.jpg"/><lb/>que ex fune perpendiculari BG <lb/>dependeat onus, & GB Tangens <lb/>cum Radio AB con&longs;tituat angu­<lb/>lum rectum ABG. </s> <s id="s.004065">Producatur <lb/>recta AB u&longs;que ad tympani peri­<lb/>pheriam in C, & &longs;it ad angulum <lb/>rectum Tangens CH, cui ad­<lb/>nexa intelligatur potentia per <lb/>axem S trahens, atque tympa­<lb/>num CDE convertens; ex cu­<lb/>jus conver&longs;ione convolvitur Axis <lb/>AB versùs I, & pondus a&longs;cendit. </s> <lb/> <s id="s.004066">Verùm &longs;ecundus Axis S cum pri­<lb/>mo tympano comparatus non <lb/>hanc &longs;olùm po&longs;itionem obtinere <lb/>pote&longs;t, ut &longs;uperior &longs;it, &longs;ed etiam <lb/>con&longs;titui pote&longs;t ad latus ita, ut <lb/>funis ductarius cadens in hori­<lb/>zontem ad perpendiculum &longs;it KL, Tangens verò HC &longs;it ho­<lb/>rizonti parallela; aut ita di&longs;poni po&longs;&longs;unt, ut Axis A &longs;uperiore <lb/>loco, Axis S inferiore loco &longs;tatuatur, & funis pondus retinens <lb/>&longs;it MN, cui parallelus &longs;it funis CH. <!-- KEEP S--></s> <s id="s.004067">Quamcumque ex his <lb/>tribus po&longs;itionem habeat Axis &longs;ecundus S, &longs;ivè &longs;uperior, &longs;ivè <lb/>ad latus, &longs;ive inferior &longs;it (modò linea CH vel parailela &longs;it li­<lb/>neis ductarij funis BG aut MN, vel parallela lineæ AK jun­<lb/>genti centrum Axis cum puncto contactûs perpendiculi KL) <pb pagenum="540" xlink:href="017/01/556.jpg"/>eadem habere videtur momenta; quia punctum C, cui appli­<lb/>cata intelligitur potentia, juxta potentiæ directionem &longs;imili <lb/>Ratione accedit versùs potentiam comparatè ad a&longs;cen&longs;um <lb/>puncti Axis, cui applicatur pondus, ac e&longs;t Ratio &longs;emidiametri <lb/>tympani ad &longs;emidiametrum Axis. <!-- KEEP S--></s> <s id="s.004068">Concipiamus enim in con­<lb/>volutione punctum C venire in F, punctum verò B in I: <lb/>punctum igitur C &longs;equens potentiæ directionem accedit versùs <lb/>potentiam juxta men&longs;uram Sinûs arcûs CF, hoc e&longs;t OF, <lb/>quemadmodum punctum B contrà directionem gravitatis pon­<lb/>deris a&longs;cendit juxta men&longs;uram Sinûs arcûs BI: &longs;unt autem hi <lb/>&longs;inus arcuum &longs;imilium &longs;imiliter po&longs;itorum in Ratione Radio­<lb/>rum AC ad AB. <!-- KEEP S--></s> <s id="s.004069">Atqui &longs;ive in K, &longs;ive in M intelligatur pon­<lb/>dus, a&longs;cen&longs;us illius e&longs;t æqualis a&longs;cen&longs;ui BI; ergo ad illos, ut po­<lb/>te huic æquales, acce&longs;&longs;us puncti C ad potentiam, qui e&longs;t OF, <lb/>eandem habet Rationem, quæ e&longs;t Radij AC ad Radium AB. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004070">At verò &longs;i Axis &longs;ecundus &longs;it T, potentia non intelligitur ap­<lb/>plicata tympano in C, &longs;ed in F, ubi circulum tangit recta TF; <lb/>nec ejus directio FT e&longs;t parallela directioni ponderis BG, &longs;ed <lb/>obliqua, adeò ut quamvis F veniat in Q per arcum æqualem <lb/>arcui CF, quia tamen non e&longs;t &longs;imiliter po&longs;itus, punctum F &longs;e­<lb/>quens directionem potentiæ accedit versùs potentiam acce&longs;&longs;u, <lb/>quem metitur RP; e&longs;t autem RP minor quàm AR, hoc e&longs;t <lb/>OF, ut ex doctrinâ Sinuum con&longs;tat; igitur acce&longs;&longs;us RP ad <lb/>a&longs;cen&longs;um BI habet minorem Rationem, quàm acce&longs;&longs;us OF ad <lb/>eundem a&longs;cen&longs;um BI. </s> <s id="s.004071">Potentia igitur volvens Axem T in li­<lb/>neâ TF obliquâ minora habet momenta, quàm in parallelâ <lb/>HC. <!-- KEEP S--></s> <s id="s.004072">Similiter &longs;i Axis fuerit V propior quàm Axis T; linea <lb/>VD tangit circulum in D puncto remotiore quàm F, à puncto <lb/>C; ac propterea datâ arcûs æqualitate adhuc minor e&longs;t acce&longs;&longs;us <lb/>in D quàm in F, multóque minor quàm in C, & idcircò mi­<lb/>norem habet trahendi facilitatem. </s> <s id="s.004073">Quare quò propior e&longs;t Axis <lb/>&longs;ecundus, &longs;i tractio &longs;it obliqua, ut TF & VD, plus laboris re­<lb/>quiritur in movendo. </s> </p> <p type="main"> <s id="s.004074">Neque hoc mihi incon&longs;iderantiæ tribuas, quod a&longs;&longs;ump&longs;erim <lb/>arcus CF & BI perinde atque &longs;i idem e&longs;&longs;et motus, ac quando <lb/>funis HC e&longs;&longs;et firmiter alligatus in C, & ejus caput veniret ex <lb/>C in F; cum tamen alia &longs;emper atque alia pars funis aliis &longs;ub­<lb/>inde peripheriæ tympani partibus re&longs;pondeat, in quibus fit ad <pb pagenum="541" xlink:href="017/01/557.jpg"/>angulum rectum cum diametro contactus, dum ille evolvitur. </s> <lb/> <s id="s.004075">Eatenus enim notabilem arcum a&longs;&longs;ump&longs;i, quatenus ob oculos <lb/>ponenda erat momentorum Ratio: Cæterùm &longs;atis &longs;cio non adeò <lb/>notabiles arcus, ut CF & BI, con&longs;iderandos e&longs;&longs;e, &longs;ed eorum <lb/>particulam minimam, &longs;ive cente&longs;imam dicas, &longs;ive mille&longs;imam <lb/>aut decies mille&longs;imam: eadem &longs;cilicet erit Ratio Sinuum, qui <lb/>re&longs;pondent minimis arcubus &longs;imilibus ac &longs;imiliter po&longs;itis, qui <lb/>nimirum incipiunt à C & B, atque Sinuum re&longs;pondentium <lb/>majoribus arcubus &longs;imilibus ab ii&longs;dem punctis C & B incipien­<lb/>tibus. </s> <s id="s.004076">Id quod pariter de punctis F & D comparatis cum <lb/>puncto B, aut K, aut M dicendum: Nam quæ inito &longs;emel mo­<lb/>tu intercedit momentorum Ratio inter potentiam & pondus <lb/>ratione po&longs;itionis, eadem in toto motu per&longs;everat. </s> <s id="s.004077">At &longs;i funis <lb/>non evolveretur, &longs;ed puncto C e&longs;&longs;et firmiter colligatus, in <lb/>tractione ex C in F &longs;ubinde mutarentur Potentiæ momenta, <lb/>fieréntque &longs;emper minora, adeò ut demum perirent, & nulla <lb/>e&longs;&longs;ent, ubi in rectam lineam coale&longs;cerent Radius AC & fu­<lb/>nis HC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004078">Hæc quæ de &longs;ecundo Axe funem primo tympano circum­<lb/>ductum evolvente dicta &longs;unt, facilè traduci po&longs;&longs;unt etiam ad <lb/>funem in &longs;e &longs;e redeuntem, cuju&longs;modi e&longs;&longs;et funis FTEF, aut <lb/>DVXD: ni&longs;i enim funis contingat tympanum in puncto &longs;e­<lb/>midiametri tran&longs;euntis per contactum Axis & funis ductarij <lb/>(hoc e&longs;t in C extremitate &longs;emidiametri AC tran&longs;euntis per B, <lb/>aut M) ita ut &longs;it funi ductario parallelus, aut in puncto &longs;emi­<lb/>diametri parallelæ funi ductario KL, con&longs;ultius erit, cæteris <lb/>paribus, axem &longs;ecundum e&longs;&longs;e remotum ut T, quàm proximum <lb/>ut V: in proximo enim lineæ DV & XV productæ coirent <lb/>in angulum majorem, quàm lineæ FT & ET, ac propterea <lb/>comprehen&longs;us arcus DX minor e&longs;t arcu FE. Cæteris, in­<lb/>quam, paribus; &longs;i videlicet in eadem rectâ lineâ intelligan­<lb/>tur trium Axium centra A, V, T; nam &longs;i in lineâ eadem jun­<lb/>gente centra AT non e&longs;&longs;et V, &longs;ed recederet ita, ut funis tym­<lb/>panum contingens minùs obliquus e&longs;&longs;et, &longs;ed magis accederet <lb/>ad paralleli&longs;mum cum lineâ BG, aut cum Radio Axis AK, <lb/>quamvis Axis V propior e&longs;&longs;et, quàm Axis T, plus tamen <lb/>haberet momenti ratione directionis potentiæ minùs obliquè <lb/>trahentis. </s> </p> <pb pagenum="542" xlink:href="017/01/558.jpg"/> <p type="main"> <s id="s.004079">Cave autem, ne hîc in latentem quendam æquivocationis <lb/>&longs;copulum incurras, &longs;i fortè permixtim accipias ponderis eleva­<lb/>tionem atque eju&longs;dem &longs;u&longs;pen&longs;i retentionem, ne recidat; hæc <lb/>enim duo oppo&longs;ito modo contingunt, & quæ minor funis obli­<lb/>quitas cau&longs;a e&longs;t facilioris elevationis, eadem difficiliorem ef­<lb/>ficit retentionem: nam pondus B retinetur à potentiâ C, <lb/>ab&longs;que eo quod potentia ullo conatu urgeat polos, quibus <lb/>axis & peritrochium incumbit, ideò pondus totas &longs;uas vi­<lb/>res exerit adversùs potentiam &longs;ur&longs;um directè trahentem: at <lb/>verò in F aut in D potentia &longs;ur&longs;um obliquè trahens tym­<lb/>panum ver&longs;us centrum quodammodo urget, & quidem eò <lb/>magis, quò magis obliqua e&longs;t tractio; ac proinde pondus <lb/>non &longs;olùm vincere debet potentiæ vires, &longs;ed etiam re&longs;i&longs;ten­<lb/>tiam ex illâ pre&longs;&longs;ione ortam; quæ quò major e&longs;t pro majo­<lb/>re declinatione à paralleli&longs;mo cum lineâ BG, aut Radio AK, <lb/>majorem quoque potentiæ tribuit retinendi facilitatem. </s> <s id="s.004080">Hinc <lb/>quando &longs;ecundus Axis e&longs;t in inferiore loco, & potentiæ tra­<lb/>hentis directio deor&longs;um tendit, magis premuntur poli, quin <lb/>& à potentia deor&longs;um conante, & à ponderis gravitate ur­<lb/>gentur, & quidem eò magis, quò magis potentiæ deor&longs;um <lb/>trahentis directio accedit ad lineam directioni ponderis MN <lb/>parallelam, aut ultra illam excurrit &longs;e quodammodo invicem <lb/>decu&longs;&longs;ando. </s> <s id="s.004081">An non exe&longs;a publicorum puteorum marmorea <lb/>labra aliquando ob&longs;erva&longs;ti, quæ diuturno atque frequenti&longs;­<lb/>&longs;imo u&longs;u à funibus, quibus aqua hauritur, detrita &longs;unt: <lb/>Utique aquam in &longs;itulâ &longs;ur&longs;um trahentis labor minor e&longs;&longs;et, <lb/>cæteris paribus, &longs;i &longs;olùm &longs;itulæ & aquæ gravitatem vince­<lb/>re oporteret, quàm &longs;i præter hanc etiam &longs;uperanda &longs;it re­<lb/>&longs;i&longs;tentia, quæ ex funis conflictu cum marmore oritur. </s> <s id="s.004082">Sed <lb/>quia deinde hoc eodem conflictu efficitur, ut quando tractio <lb/>alternis morulis interciditur, retentio minorem potentiæ co­<lb/>natum exigat; propterea etiam trahentes facilè patiuntur <lb/>re&longs;i&longs;tentiam augeri, ut aliquantulo laboris compendio gau­<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/>in opere, naturâ duce, quærere etiam in trahendâ &longs;ur&longs;um &longs;i­<lb/>tulâ non leve laboris compendium: &longs;i enim rectâ, intacto pu­<lb/>tei labro, funis &longs;ur&longs;um trahendus e&longs;&longs;et, id utique &longs;olâ brachio-<pb pagenum="543" xlink:href="017/01/559.jpg"/>rum contentione perfici po&longs;&longs;et; &longs;ed ubi funis labro innititur, <lb/>non &longs;olùm contentis brachiorum mu&longs;culis trahunt, &longs;ed etiam <lb/>inclinato retror&longs;um corpore hoc efficiunt, ut ip&longs;a corporis <lb/>gravitas nonnihil conferat, quo potentiæ animalis viribus <lb/>fiat additamentum. </s> <s id="s.004084">Ex quo manife&longs;tum e&longs;t re&longs;i&longs;tentiam il­<lb/>lam ex pre&longs;&longs;ione ortam & difficiliorem efficere tractionem, & <lb/>faciliorem retentionem: ac proinde lap&longs;um putarem, qui tra­<lb/>hentis potentiæ momenta æ&longs;timaret ex majore retinendi fa­<lb/>cultate. </s> </p> <p type="main"> <s id="s.004085">In his, quæ in po&longs;teriore hujus capitis parte di&longs;putata &longs;unt <lb/>de hac inæqualitate momentorum pro diver&longs;a po&longs;itione axis <lb/>&longs;ecundi, mihi videor &longs;atis probabiliter philo&longs;ophatus: verùm <lb/>&longs;i ad Rationes Vectis (ut pluribus placet) revocanda e&longs;&longs;et vis <lb/>Axis in Peritrochio, quamvis aliqua &longs;atis commodè explica­<lb/>ri po&longs;&longs;ent, ubi Vectis e&longs;t rectus, non omnia tamen, ubi Vectis <lb/>curvus intelligendus e&longs;t, congruam patiuntur explicationem, <lb/>ut cuilibet rem attentè con&longs;ideranti manife&longs;tum fiet; mihi <lb/>enim hìc non videtur operæ pretium in re parùm utili tempus <lb/>conterere; placuit tamen id obiter innuere, ut ip&longs;e tibi per&longs;ua­<lb/>deas inanem e&longs;&longs;e laborem, quo quis &longs;ingularum Facultatum <lb/>vires ad Vectem revocare conatur. <lb/></s> </p> <p type="main"> <s id="s.004086"><emph type="center"/>CAPUT VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004087"><emph type="center"/><emph type="italics"/>Tympanorum dentatorum u&longs;us & vires <lb/>exponuntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004088">QUæ hactenus tympana con&longs;ideravimus, fune circum­<lb/>ducto atque evoluto ver&longs;antur; nunc genus aliud, cujus <lb/>ampli&longs;&longs;imus u&longs;us e&longs;t, contemplari oportet, tympana videlicet <lb/>dentata, &longs;eu Rotas dentatas, in quibus &longs;ivè fuerint &longs;implices, <lb/>&longs;ivè compo&longs;itæ, aut nullo pror&longs;us fune indigemus aut illo tan­<lb/>tummodo, quo pondus proximè trahitur, aut attollitur: den­<lb/>tes enim majoris atque minoris tympani, ubi plura componun­<lb/>tur, &longs;e mutuâ collabellatione mordentes &longs;e vici&longs;&longs;im urgent, <pb pagenum="544" xlink:href="017/01/560.jpg"/>pro ut hoc aut illud tympanum habet originem motûs. </s> <s id="s.004089">Sit <lb/><figure id="id.017.01.560.1.jpg" xlink:href="017/01/560/1.jpg"/><lb/>chalybea lamina AB &longs;atis &longs;olida, in alte­<lb/>râ extremitate, quæ pondus re&longs;picit, mo­<lb/>dicè &longs;inuata, ut in A, & in validum un­<lb/>cum recurva, ut in C; latus autem DE <lb/>qua&longs;i &longs;erræ in morem &longs;it dentibus a&longs;pe­<lb/>rum. </s> <s id="s.004090">Tum rotula I paris &longs;altem cum la­<lb/>minâ cra&longs;&longs;itudinis paretur dentes habens <lb/>ita in orbem di&longs;po&longs;itos, ut hi in rotulæ <lb/>circa &longs;uum centrum conver&longs;ione denti­<lb/>bus laminæ &longs;ubinde congruant: colloca­<lb/>tis enim in apto loculamento rotulâ, at­<lb/>que laminâ (cujus tamen pars DC extet) <lb/>adeò, ut hæc ex illius conver&longs;ione liberè <lb/>promoveri, illa circa &longs;uum axem, cui fir­<lb/>miter infixa &longs;it, facilè ver&longs;ari valeat, circumducto axis manu­<lb/>brio ad latus extra loculamentum extante, urgeri poterit pon­<lb/>dus, aut trahi: Nimirum &longs;i rotulæ conver&longs;io fiat ex H in I, <lb/>propellitur extra loculamentum lamina, ejú&longs;que extremitas A <lb/>recedens à rotulâ urget pondus obvium: contrà verò &longs;i rotula <lb/>convertatur ex I in H, laminam ad &longs;e intra loculamentum re­<lb/>trahit, & pondus unco C connexum ad &longs;e rapit. </s> <s id="s.004091">Hinc clariùs <lb/>vides, quàm ut monendus &longs;is, oportere in attollendo, aut pro­<lb/>pellendo pondere loculamentum aut ponderi &longs;uppo&longs;itum firmo <lb/>&longs;olo in&longs;i&longs;tere, aut ponderi objectum &longs;olido repagulo inniti; in <lb/>trahendo autem pondere, quod uncus C apprehendit, oppo&longs;i­<lb/>tam loculamenti extremitatem valido fune retineri. </s> </p> <p type="main"> <s id="s.004092">Illud potiùs attentè perpendendum, quod in &longs;tatuendis tùm <lb/>laminæ, tùm rotulæ dentibus plurimum refert, utrùm rari, an <lb/>&longs;pi&longs;&longs;iores &longs;int laminæ dentes, ac proinde utrùm pauci, an plures <lb/>in&longs;int ip&longs;i rotulæ, cujus peripheria in conver&longs;ione aptatur lami­<lb/>næ; hæc enim juxta numerum dentium rotulæ, quibus &longs;ubinde <lb/>coaptatur, promovetur, & cum ipsâ pondus pari velocitate aut <lb/>tarditate movetur. </s> <s id="s.004093">Præ&longs;tare autem pondus tardè; potentiam <lb/>velociter moveri, quid opus e&longs;t iterùm inculcare? </s> <s id="s.004094">Igitur quò <lb/>minor erit rotula & paucioribus dentibus in&longs;tructa, eodem ma­<lb/>nente manubrio, faciliùs movebitur pondus; quia ut &longs;emidia­<lb/>meter rotulæ ad manubrij longitudinem, ita motus ponderis, <pb pagenum="545" xlink:href="017/01/561.jpg"/>ad motum potentiæ, & reciprocè ita potentiæ vis movendi, ad <lb/>pondus. </s> <s id="s.004095">Quare ulteriùs manife&longs;tum e&longs;t, &longs;i majore potentiæ vir­<lb/>tute opus fuerit, &longs;pectatâ ponderis movendi difficultate, po&longs;&longs;e <lb/>augeri manubrium, ut majora &longs;int potentiæ momenta. </s> <s id="s.004096">Quo­<lb/>niam verò non &longs;emper in promptu e&longs;t opportunum manubrium, <lb/>&longs;uaderem extremum axis caput, quod manubrio in&longs;eritur, qua­<lb/>dratum fieri, & longiu&longs;culum e&longs;&longs;e: loco autem vulgaris manu­<lb/>brij habeatur cra&longs;&longs;ioris cylindri fru&longs;tulum MN, in cujus imâ <lb/>ba&longs;i circa centrum excavatum &longs;it quadratum foramen S ad exci­<lb/>piendum caput axis, & ip&longs;ius cylindri &longs;capum penetrent fora­<lb/>mina rotunda R, T, quibus pro opportunitate in&longs;eri po&longs;&longs;int ba­<lb/>culi &longs;ive longiores, &longs;ive breviores. </s> <s id="s.004097">Porrò cylindri cra&longs;&longs;itiem <lb/>nihil obe&longs;&longs;e apertè con&longs;tat, &longs;i quidem &longs;ola rotulæ &longs;emidiameter <lb/>attenditur ad definiendum ponderis motum comparata cum ba­<lb/>culi longitudine, quatenus potentia ab axe rotulæ di&longs;tat. </s> </p> <p type="main"> <s id="s.004098">Quod &longs;i uno eodémque tempore duo pondera in oppo&longs;itas <lb/>partes di&longs;pellere, aut &longs;ibi invicem propiora fieri oporteat, &longs;imi­<lb/>lem alteram laminam priori parallelam in eodem plano &longs;ed con­<lb/>trario modo po&longs;itam (ut &longs;cilicet extremitas &longs;imilis ip&longs;i ACD <lb/>re&longs;piciat prioris laminæ extremitatem B) in oppo&longs;itâ rotulæ <lb/>parte colloca ad I, ut pariter laminæ dentes rotulæ dentibus im­<lb/>plicentur: Quia enim circuli circa &longs;uum centrum circumacti <lb/>partes adver&longs;æ oppo&longs;itis motibus cientur, etiam laminarum ex­<lb/>tremitates, quæ pondus propellunt aut trahunt, in contrarias <lb/>partes à rotulâ circumactâ moventur, ita ut vel à &longs;e invicem re­<lb/>cedant, vel ad &longs;e mutuò accedant. </s> </p> <p type="main"> <s id="s.004099">Hoc idem quod laminæ rectæ dentatæ tympano &longs;imiliter den­<lb/>tato implicitæ contingit, accideret pariter, &longs;i illius in circulum <lb/>inflexæ extremam oram dentes ambirent: quemadmodum enim <lb/>recta lamina AB, tympani HI conver&longs;i ductum &longs;equitur, ita il­<lb/>la in circulum conformata circa &longs;uum centrum moveretur à <lb/>tympani dentibus impul&longs;a; eâ tamen ratione, ut duarum huju&longs;­<lb/>modi rotarum &longs;e invicem mordentium conver&longs;iones in oppo&longs;i­<lb/>tas plagas tenderent; &longs;i enim prioris rotæ pars &longs;uperior Occa­<lb/>&longs;um versùs converteretur, po&longs;terioris rotæ pars item &longs;uperior <lb/>Ortum versùs convolveretur; & &longs;i adhuc tertia rota dentata ad­<lb/>deretur, hæc iterum proximæ adver&longs;ata ad occa&longs;um pergeret; <lb/>atque ita deinceps alternis conver&longs;ionibus &longs;ibi vici&longs;&longs;im re&longs;pon­<lb/>dentibus. </s> </p> <pb pagenum="546" xlink:href="017/01/562.jpg"/> <p type="main"> <s id="s.004100">Ob&longs;ervandum e&longs;t autem huju&longs;modi tympanorum, quæ den­<lb/>tata vocamus, multiplicem e&longs;&longs;e po&longs;&longs;e formam, eámque eligen­<lb/>dam, quæ præ&longs;tituto motui magis congruere videbitur: non &longs;o­<lb/>lùm enim pro majoribus tympanis a&longs;&longs;umi pote&longs;t di&longs;cus extre­<lb/>mum limbum habens &longs;erræ in morem denticulatim inci&longs;um, ve­<lb/>rùm etiam in orbem infigi po&longs;&longs;unt paxilli dentium loco promi­<lb/>nentes, &longs;ive peripheriam ip&longs;am qua&longs;i radij exeuntes ambiant, &longs;i­<lb/>ve &longs;uprà di&longs;ci planum erigantur ad perpendiculum certis inter­<lb/>vallis di&longs;tributi. </s> <s id="s.004101">Hoc autem dentium in&longs;itorum genus non pa­<lb/>rum habet utilitatis præ dentibus illis qua&longs;i connatis: nam &longs;i lon­<lb/>go u&longs;u dens aliquis atteratur, aut excutiatur, facilè re&longs;titui po­<lb/>te&longs;t novo paxillo in prioris locum immi&longs;&longs;o; at non ita facilè re­<lb/>paratur pars illa limbi denticulata, quæ facta e&longs;t inutilis. </s> <s id="s.004102">Simi­<lb/>liter pro minoribus tympanis non &longs;olùm dentatas rotulas adhi­<lb/>bere po&longs;&longs;umus &longs;uis axibus infixas, &longs;ed etiam uti licet aut paulò <lb/>cra&longs;&longs;ioribus axibus &longs;triatis, quorum excavatæ &longs;triæ majoris tym­<lb/>pani dentibus congruant, aut vertebris pariter &longs;triatis &longs;ubtiliori <lb/>axi infixis. </s> <s id="s.004103">Quando verò majus tympanum paxillos habet pro <lb/>dentibus, tympanum minus illi re&longs;pondens e&longs;t Curriculus (<expan abbr="qu&etilde;">quem</expan> <lb/>alij ex Italico idiomate <emph type="italics"/>Rocchetum<emph.end type="italics"/> dicunt) aliquot virgulis, ut <lb/>plurimum ferreis ad firmitatem, capita duobus parallelis planis <lb/>infixa habentibus con&longs;tans, ita ut in majoris tympani conver&longs;io­<lb/>ne &longs;ingulos paxillos excipiant &longs;ingula virgularum intervalla, <lb/>quibus propul&longs;is curriculus convolvitur, & cum eo aut pondus <lb/>ip&longs;um, aut aliud tympanum movetur. </s> </p> <p type="main"> <s id="s.004104">Sic contingere pote&longs;t ut Axe AB attollendum &longs;it pondus, & <lb/><figure id="id.017.01.562.1.jpg" xlink:href="017/01/562/1.jpg"/><lb/>expediat uti jumento, quod tamen <lb/>non ni&longs;i in plano horizontali moveri <lb/>pote&longs;t; tympanum autem CD, in <lb/>quo e&longs;t Axis AB horizontalis, e&longs;t in <lb/>plano Verticali. </s> <s id="s.004105">Ad tympani CD <lb/>planam faciem aver&longs;am perpendicu­<lb/>lares paxillos in ambitu &longs;tatue, & <lb/>Curriculum EF circa &longs;uum axem <lb/>&longs;uperiùs atque inferiùs firmatum <lb/>ver&longs;atilem adjice, cujus virgulæ con­<lb/>gruis intervallis di&longs;tinctæ tympani <lb/>dentibus re&longs;pondeant. </s> <s id="s.004106">Aliud item tympanum HG horizonti <pb pagenum="547" xlink:href="017/01/563.jpg"/>parallelum dentes habens ex peripheriâ extantes, & Curriculi <lb/>EF virgulis aptè congruentes, infigatur axi perpendiculari IK, <lb/>cui opportuno loco addatur vectis LM, ita ut in M commodè <lb/>jungi po&longs;&longs;it jumentum: hoc enim progrediente, & tympanum <lb/>ex G versùs O convolvente, dens H incurrens in virgulam cur­<lb/>riculi EF illum convertit versùs tympanum CD, cujus pariter <lb/>denti occurrens alia virgula, atque impellens cogit infimam <lb/>tympani partem D a&longs;cendere, &longs;imúlque Axem AB converti, & <lb/>convoluto fune ductario RS attolli pondus. </s> </p> <p type="main"> <s id="s.004107">Hæc tympanorum duorum & curriculi intermedij complexio <lb/>&longs;i attentè perpendatur, non auget potentiæ momenta præter ea, <lb/>quæ obtineret proximè applicata tympano CD ad convolven­<lb/>dum Axem AB: Nam &longs;i ponatur vectis LM non longior &longs;emi­<lb/>diametro tympani dentati HG, perinde e&longs;t, ac &longs;i potentia in M <lb/>po&longs;ita exi&longs;teret in G, æquali &longs;cilicet motu cum tympani HG <lb/>peripheriâ movetur. </s> <s id="s.004108">Curriculi autem EF motus æquè velox e&longs;t <lb/>atque motus tympani HG; licèt enim hoc &longs;it majus, ille mi­<lb/>nor, tamen dum illud &longs;emel, hic &longs;æpiùs convolvitur pro ratione <lb/>diametrorum; adeò ut &longs;i tympanum HG habeat dentes viginti, <lb/>curriculus &longs;trias quinque, hic quater volvatur ex unicâ tympani <lb/>conver&longs;ione: quapropter quatuor curriculi &longs;ubquadrupli con­<lb/>volutiones uni conver&longs;ioni tympani HG æquantur. </s> <s id="s.004109">Similiter <lb/>& de tympano CD dicendum, cujus tantummodo dentes quin­<lb/>que re&longs;pondentes quinque &longs;triis aut virgulis curriculi EF ur­<lb/>gentur unicâ conver&longs;ione curriculi eju&longs;dem, & idcircò æqualis <lb/>e&longs;t utriu&longs;que motus, ac proinde etiam duo tympana CD & HG <lb/>æqualiter moventur, & potentiæ in M applicatæ momenta ea­<lb/>dem &longs;unt, quæ forent, &longs;i tympano CD proximè applicaretur. </s> <lb/> <s id="s.004110">Quamobrem, ut aliqua fiat momentorum acce&longs;&longs;io in potentiâ, <lb/>oportet vectem LM &longs;tatuere longiorem &longs;emidiametro tympani <lb/>HG: tunc enim ex Ratione longitudinis LM ad &longs;emidiame­<lb/>trum tympani, & Ratione diametri tympani CD ad diametrum <lb/>Axis AB, componitur Ratio, quæ definit momenta potentiæ; <lb/>e&longs;t &longs;cilicet Ratio motûs potentiæ ad motum ponderis. </s> </p> <p type="main"> <s id="s.004111">Ex quo &longs;atis vides eatenus addi tympanum HG, quatenus <lb/>quærendus e&longs;t jumento locus, ut in gyrum circumagi valeat: <lb/>cæterum &longs;i tympanum CD cum &longs;uo Axe AB ita in &longs;uperiore <lb/>aut inferiore loco collocari atque firmari po&longs;&longs;it, ut nulli impedi-<pb pagenum="548" xlink:href="017/01/564.jpg"/>mento &longs;it jumento in inferiore aut &longs;uperiore plano exi&longs;tenti & <lb/>circumacto, &longs;atius e&longs;t labori & &longs;umptibus parcere omi&longs;&longs;o tym­<lb/>pano HG, & circa cra&longs;&longs;iorem axem con&longs;truere curriculum EF, <lb/>cui axi opportunè adjungatur vectis LM, ut potentia ip&longs;um <lb/>curriculum immediatè convertat; erit &longs;iquidem major Ratio <lb/>ip&longs;ius vectis LM ad curriculi &longs;emidiametrum, quàm ad &longs;emi­<lb/>diametrum tympani HG; ac proinde minore conatu indige­<lb/>bit potentia, ut Curriculum cum adjacente tympano CD con­<lb/>vertat. </s> </p> <p type="main"> <s id="s.004112">Non alio quàm huju&longs;modi artificio videtur u&longs;us Anonymus, <lb/>qui de Rebus Bellicis &longs;crip&longs;it ad Theodo&longs;ium Augu&longs;tum ejú&longs;­<lb/>que filios Honorium, & Arcadium Cæ&longs;ares, ubi liburnam pro­<lb/>ponit <emph type="italics"/>navalibus idoneam bellis, quam pro magnitudine &longs;ui virorum <lb/>exerceri manibus quodammodo imbecillitas humana prohibeat, & <lb/>quocumque utilitas vocet, ad facilitatem cursús ingenij ope &longs;ubnixa <lb/>animalium virtus impellit. </s> <s id="s.004113">In cujus alveo, vel capacuate bini boves <lb/>machinis adjuncti, adhærentes rotas navis lateribus volvunt; qua­<lb/>rum &longs;upra ambitum vel rotunditatem extantes radij currentibus ii&longs;­<lb/>dem rotis in modum remorum aquam conatibus elidentes miro quodam <lb/>artis effectu operantur, impetu parturiente di&longs;cur&longs;um. </s> <s id="s.004114">Hæc cadens <lb/>tamen liburna pro mole &longs;ui, próque machinis in &longs;emet operantibus <lb/>tanto virium fremitu pugnam capeßit, ut omnes adver&longs;arias libur­<lb/>nas cominùs venientes facili attritu comminuat.<emph.end type="italics"/></s> <s id="s.004115"> Quamvis, quæ de­<lb/>mum machinæ e&longs;&longs;ent, quibus boves adjungebantur, Author <lb/>non exponat, facile tamen e&longs;t opinari boves in &longs;uperiore tabu­<lb/>lato circumacto versâ&longs;&longs;e axem carinæ perpendiculariter in­<lb/>&longs;i&longs;tentem, cui infra tabulatum rota dentata horizonti parallela <lb/>infixa e&longs;&longs;et dentes habens in alterutra tympani facie, quibus <lb/>&longs;ubinde apprehenderet virgulas curriculi in plano Verticali <lb/>convoluti, & infixi axi horizontali, qui utrumque navis latus <lb/>permearet, & in extantibus extremitatibus rotas haberet cum <lb/>palmulis prominentibus, quæ aquam in conver&longs;ione verbera­<lb/>rent. </s> <s id="s.004116">Potuerunt autem huju&longs;modi machinæ juxta liburnæ lon­<lb/>gitudinem multiplicari, prout ip&longs;um &longs;chema ab Authore pro­<lb/>po&longs;itum exhibet. </s> <s id="s.004117">An verò hujus liburnæ, quam in Præfatione <lb/>dicit <emph type="italics"/>veloci&longs;&longs;imum liburnæ genus, decem navibus ingenij magi&longs;terio <lb/>prævalere,<emph.end type="italics"/> tantus impetus, tantáque velocitas e&longs;&longs;et, ut adver&longs;a­<lb/>riæ liburnæ venientes facilè comminuerentur, di&longs;piciat lector, <pb pagenum="549" xlink:href="017/01/565.jpg"/>cui otium fuerit, quemadmodum an no&longs;tris u&longs;ibus navalibus <lb/>artificium hoc aliquid utilitatis afferre po&longs;&longs;it. </s> </p> <p type="main"> <s id="s.004118">Hinc manife&longs;ta fit illarum machinarum vis, quæ Pancratia <lb/>Glo&longs;&longs;ocoma, Chari&longs;tia, & &longs;i quod e&longs;t aliud vocabuli genus, di­<lb/>cuntur, ex plurium tympanorum complexione minorum & ma­<lb/>jorum, ita ut à minore tympano, cui manubrium additur, inci­<lb/>piat motus, & deinceps minora majoribus con&longs;equentibus mo­<lb/>tum communicent. </s> <s id="s.004119">Quandoquidem rotula minor dentata &longs;i <lb/>comparetur cum rotâ majore, cum qua communem habet <lb/>axem, utique tardius movetur, quàm peripheria rotæ majoris, <lb/>cum qua connectitur: At verò &longs;i cum rotâ majore con&longs;equente <lb/>comparetur, cujus dentes apprehendit, utique æqualis e&longs;t ip&longs;a­<lb/>rum motûs velocitas, nam plures minoris conver&longs;iones æquales <lb/>&longs;unt uni conver&longs;ioni majoris, quam efficiunt. </s> <s id="s.004120">Sit rota dentata <lb/><figure id="id.017.01.565.1.jpg" xlink:href="017/01/565/1.jpg"/><lb/>AB, cujus axi firmiter infixo additum &longs;it manubrium eju&longs;dem <lb/>rotæ &longs;emidiametri ex. </s> <s id="s.004121">gr. <!-- REMOVE S-->quintuplum, ac propterea potentia <lb/>manubrio applicata quintuplo velociùs movetur, quàm <lb/>punctum in rotæ AB peripheriâ notatum. </s> <s id="s.004122">Addatur rota ma­<lb/>jor BC, cujus dentes implicentur dentibus rotulæ BA: ex hu­<lb/>jus conver&longs;ione illa pariter convolvitur; &longs;ed &longs;i diameter BA &longs;it <lb/>diametri BC &longs;ubtripla, ter rotula BA volvitur, ut rota BC <lb/>compleat integram conver&longs;ionem, ac proinde potentia quin­<lb/>tuplo velociùs movetur, quàm rota BC. <!-- KEEP S--></s> <s id="s.004123">Rota hæc major &longs;ibi <lb/>connexam habeat in eodem axe minorem SD, quæ apprehen­<lb/>dat dentes &longs;ecundæ majoris rotæ DE; quæ &longs;imiliter in eodem <lb/>axe conjunctam habeat minorem FR: hujus dentes mor-<pb pagenum="550" xlink:href="017/01/566.jpg"/>deant peripheriam tertiæ majoris rotæ FG, in qua e&longs;t Axis <lb/>IH, ex cujus convolutione ductarius funis LO trahit <lb/>pondus. </s> </p> <p type="main"> <s id="s.004124">Ut potentiæ momenta habeantur, ejus motum cum ponderis <lb/>motu collatum ad calculos revoca componendo Rationes, quas <lb/>&longs;ingulæ majores rotæ ad &longs;uas minores habent quo ad diame­<lb/>trum. </s> <s id="s.004125">Quare &longs;i manubrium ad &longs;emidiametrum rotulæ AB ha­<lb/>beat Rationem quintuplam, diameter BC ad diametrum SD <lb/>triplam, DE ad RF item triplam, & FG ad IH &longs;imiliter tri­<lb/>plam, compo&longs;itis tribus Rationibus triplis cum Ratione quin­<lb/>tuplâ, oritur Ratio 135 ad 1: atque adeò ut po&longs;trema rota FG <lb/>& cum eâ Axis IH &longs;emel volvatur, prima rotula AB facit 27 <lb/>conver&longs;iones; potentia autem manubrio applicata movetur <lb/>quintuplo velociùs quàm &longs;uæ rotæ AB peripheria; igitur pon­<lb/>deris motus ad motum potentiæ e&longs;t ut 1 ad 135. Porrò fieri 27 <lb/>conver&longs;iones manubrij apertè con&longs;tat, quia hoc ter volvi po­<lb/>nitur, ut rota BC, atque adeò etiam SD illi connexa, &longs;emel <lb/>convolvatur: & quia ex hypothe&longs;i rotula SD tres conver&longs;io­<lb/>nes habet, ut toti peripheriæ rotæ DE congruat, manubrium <lb/>novies in gyrum agitur, ut rota major DE & minor RF &longs;emel <lb/>convertatur: demum quia pariter ex hypothe&longs;i rota minor RF <lb/>triplici convolutione indiget, ut toti peripheriæ rotæ majoris <lb/>FG re&longs;pondeat, ut hæc unicam circuitionem perficiat, viginti <lb/>&longs;eptem manubrij conver&longs;ionibus opus e&longs;t. </s> <s id="s.004126">Quare momentum <lb/>Potentiæ manubrio applicatæ comparatæ cum rotulâ AB e&longs;t <lb/>ut 5, cum &longs;equenti rotulâ SD ut 15, cum rotulâ RF ut 45, <lb/>cum Axe IH ut 135. </s> </p> <p type="main"> <s id="s.004127">Ne verò artifex huju&longs;modi rotas dentatas majores atque mi­<lb/>nores parare ju&longs;&longs;us inutili demùm labore &longs;e torqueat, monendus <lb/>e&longs;t, ut animum diligenter advertat, utrùm omnes majores ro­<lb/>tas, item omnes minores, inter &longs;e æquales &longs;tatuere velit, an in­<lb/>æquales; ex hoc enim ip&longs;arum rotarum collocationem definiet, <lb/>ne &longs;ibi vici&longs;&longs;im impedimento &longs;int. </s> <s id="s.004128">Finge enim rotulas DS & <lb/>FR æquales e&longs;&longs;e, item majores BC & DE, atque alterno or­<lb/>dine po&longs;itas, ita ut &longs;i rotula DS fuerit in parte anteriore &longs;uæ <lb/>rotæ BC, vici&longs;&longs;im rotula FR &longs;it in parte aversâ rotæ DE: <lb/>quemadmodum dentes minoris DS implicantur dentibus <lb/>majoris DE, ita pariter dentes majoris BC implicantur <pb pagenum="551" xlink:href="017/01/567.jpg"/>dentibus minores FR: igitur unica conver&longs;io majoris rotæ <lb/>BC ter convolveret minorem rotam FR: atqui cum minore <lb/>rotâ FR &longs;imul converteretur major DE in eodem axe; igitur <lb/>dum &longs;emel converteretur major rota BC, ter convolveretur <lb/>rota major DE: hoc autem omnino fieri nequit, quia unica ro­<lb/>tæ BC conver&longs;io e&longs;t etiam unica conver&longs;io rotulæ minoris DS, <lb/>hujus autem unica conver&longs;io re&longs;pondet &longs;olùm tertiæ parti con­<lb/>ver&longs;ionis majoris rotæ DE: plurimum igitur abe&longs;t à trinâ con­<lb/>ver&longs;ione. </s> <s id="s.004129">Quare &longs;i huju&longs;modi æqualitas intercederet tùm inter <lb/>majores, tùm inter minores rotas dentatas, oporteret minores <lb/>rotas ad eandem partem re&longs;picere, ne rota major con&longs;equenti <lb/>minori rotæ motum ullum communicare po&longs;&longs;it. </s> <s id="s.004130">Verùm hoc <lb/>forta&longs;&longs;e alicui videatur incommodum, quod non ita aptè in &longs;uo <lb/>loculamento huju&longs;modi rotæ collocari valeant, &longs;i rotarum ma­<lb/>jorum con&longs;equentium plana &longs;uperimponantur planis antece­<lb/>dentium; id quod exigit ip&longs;a minorum rotularum po&longs;itio, &longs;i <lb/>omnes partem eandem re&longs;piciant. </s> <s id="s.004131">Propterea inæquales fiant <lb/>rotæ ita, ut alternatim po&longs;itæ minores rotulæ occurrant qui­<lb/>dem &longs;ingulæ peripheriæ con&longs;equentis majoris, non attingantur <lb/>autem à dentibus majoris rotæ antecedentis: hoc enim pacto <lb/>in &longs;uo loculamento pre&longs;&longs;iùs firmantur, & &longs;unt qua&longs;i duo plana <lb/>parallela, in quibus hinc rota minor inter duas majores, hinc <lb/>verò rota major inter duas minores con&longs;picitur. </s> </p> <p type="main"> <s id="s.004132">Porrò minore, rotæ in eodem axe cum majoribus dupliciter <lb/>di&longs;poni po&longs;&longs;unt: primùm ut major rota minori proxima &longs;it, & <lb/>earum plana &longs;e contingant; deinde ut aliquo inter &longs;e ab&longs;int in­<lb/>tervallo. </s> <s id="s.004133">Si minor majori cohæreat, &longs;uaderem minores rotas ex <lb/>lamina paulò cra&longs;&longs;iore fieri quàm majores; &longs;ic enim in locula­<lb/>mento ita di&longs;ponuntur, ut plana majorum &longs;e omninò non con­<lb/>tingant, ac proinde nullus &longs;it partium &longs;e vici&longs;&longs;im terentium con­<lb/>flictus, qui moram inferat motui. </s> <s id="s.004134">Sin autem quæ in eodem axe <lb/>&longs;unt rotæ, major & minor invicem di&longs;tent, nullum quidem &longs;ub­<lb/>e&longs;t periculum ex mutuo affrictu majorum, verùm cavendum <lb/>e&longs;t, ne axis longior, quàm par fuerit, etiam &longs;it infirmior; &longs;i ni­<lb/>mirum axis longioris extremitates loculamento infixæ volvan­<lb/>tur. </s> <s id="s.004135">Propterea aliter etiam di&longs;poni po&longs;&longs;unt, ita ut loculamentum <lb/>validi&longs;&longs;imum &longs;it, nec fractioni obnoxium, & facilè ex alio in <lb/>alium locum transferri valeat. </s> <s id="s.004136">Parentur axes rotundi, &longs;ed utra-<pb pagenum="552" xlink:href="017/01/568.jpg"/>que extremitas quadrata &longs;it, ut in&longs;eratur quadrato foramini, <lb/>quod rotarum centro ine&longs;t: tum tigni pars accipiatur cra&longs;&longs;itu­<lb/>dinis tantæ, ut congruis foraminibus rotundi, excipiat axium <lb/>rotunditatem, extantibus hinc atque hinc extremitatibus qua­<lb/>dratis. </s> <s id="s.004137">Deinde quadratas axis extremitates excipiant rotæ den­<lb/>tatæ alterno ordine, ut qua parte prior axis habet rotam majo­<lb/>rem, &longs;ecundus axis habet rotam minorem, & vici&longs;&longs;im ille in op­<lb/>po&longs;itâ tigni parte habeat rotam minorem, hic majorem; illud <lb/>&longs;emper præcavendo, ne rota minor &longs;ecundi axis contingat pe­<lb/>ripheriam rotæ majoris primi axis; id quod fiet, &longs;i po&longs;teriores <lb/>rotæ majores etiam paulò majorem &longs;emidiametrum habeant. </s> <lb/> <s id="s.004138">Relinquitur autem artificis indu&longs;triæ ita foraminum extremi­<lb/>tates munire, ut nec axes ultrò citròque commeare valeant, <lb/>rotis ip&longs;is illos coërcentibus, nec nimio affrictu tigni faciem <lb/>rotæ circumactæ terant, interjecto inter tignum & rotam exiguo <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/>primam rotulam manubrio affixam, brevitatis gratiâ, exami­<lb/>nanda propo&longs;uerim, plura, & plura &longs;imilia addi po&longs;&longs;e e&longs;t mani­<lb/>fe&longs;tum, adeò ut omni arrogantiæ notâ vacent magnificæ illæ <lb/>Mechanicorum propo&longs;itiones, quibus &longs;e quodcumque etiam <lb/>immane pondus moturos &longs;pondent, immò tellurem ip&longs;am, &longs;i lo­<lb/>cus daretur &longs;tatuendæ machinæ idoneus. </s> <s id="s.004140">Illud tamen incom­<lb/>modum vitari nullatenus pote&longs;t, quod ex ponderis tarditate ori­<lb/>tur: quî enim fieri po&longs;&longs;it, ut gravitatis re&longs;i&longs;tentia ex motûs tar­<lb/>ditate minuatur, quin multo tempore opus &longs;it ad pondus mo­<lb/>vendum? </s> <s id="s.004141">Idcirco unâ eadémque operâ, qua potentiæ momen­<lb/>ta inquiris componendo Rationes, quas majora tympana ha­<lb/>bent ad minora &longs;ibi adjuncta, etiam motûs tarditatem notam fa­<lb/>cis; ac proinde con&longs;tituto intra certam temporis men&longs;uram po­<lb/>tentiæ motu, innote&longs;cit ponderis motus, quem eodem tempo­<lb/>re perficit. </s> <s id="s.004142">Fac e&longs;&longs;e decem Rationes quintuplas, quæ compo­<lb/>nendæ &longs;unt, &longs;i manubrium ad &longs;uæ rotulæ &longs;emidiametrum ha­<lb/>beat Rationem quintuplam, & &longs;imilis &longs;it Ratio majorum ad &longs;ua <lb/>minora tympana. </s> <s id="s.004143">Motus potentiæ manubrio applicatæ e&longs;t ad <lb/>motum ponderis ut 9.765625 ad 1: tot igitur &longs;patij pedes itera­<lb/>tis revolutionibus confici à potentiâ nece&longs;&longs;e e&longs;t, ut pondus pe­<lb/>dem unum percurrat. </s> <s id="s.004144">Quod &longs;i potentiam tantâ velocitate mo-<pb pagenum="553" xlink:href="017/01/569.jpg"/>veri ponamus, ut horis &longs;ingulis pedum quindecim millia per­<lb/>currat, indigebit horis (651 1/24), hoc e&longs;t diebus 27, horis 3. min. <!-- REMOVE S--><lb/>2 1/2, ut pondus à loco in locum pedis unius intervallo di&longs;tan­<lb/>tem transferat: atque ideò quis tantâ oculorum acie polleat, ut <lb/>ponderis motum digno&longs;cat, ni&longs;i po&longs;t aliquot horas? </s> <s id="s.004145">quando­<lb/>quidem unius horæ &longs;patio vix unius unciæ partem quinquage­<lb/>&longs;imam quartam perficit, &longs;cilicet (1/651) pedis. </s> <s id="s.004146">Verùm tam immane <lb/>pondus, quod ad gravitatem re&longs;pondentem potentiæ machinâ <lb/>de&longs;titutæ &longs;it ut 9. 765625 ad 1, movere, licet tardi&longs;&longs;imè, &longs;atius <lb/>e&longs;t, quàm nullo pacto movere. </s> </p> <p type="main"> <s id="s.004147">Ex his liquet, quid contingat, &longs;i potentiæ & ponderis loca <lb/>ita commutentur, ut potentia extremo tympano applicetur, <lb/>pondus verò movendum primæ rotulæ axi aut manubrio re&longs;­<lb/>pondeat; exiguus enim validioris potentiæ motus veloci&longs;&longs;imè <lb/>movet pondus, motúmque diu continuat; ut palam e&longs;t in au­<lb/>tomatis horas indicantibus, &longs;ive potentia movens &longs;it vis ela&longs;ti­<lb/>ca laminæ chalybeæ inflexæ, &longs;ivè gravitas ponderis axem ma­<lb/>ximæ rotæ volvens; ni&longs;i enim Tempus alternis motibus objice­<lb/>ret rotæ &longs;erratæ dentibus &longs;ui fu&longs;i pinnulas, quæ moram infer­<lb/>rent, rota ip&longs;a &longs;errata veloci&longs;&longs;imè volveretur. </s> <s id="s.004148">Sed quoniam ra­<lb/>rò contingit validi&longs;&longs;imam potentiam adhibere, ut leve pondus <lb/>moveatur, propterea non e&longs;t frequens huju&longs;modi locorum <lb/>commutatio inter pondus & potentiam: u&longs;um tamen aliquan­<lb/>do habere po&longs;&longs;et in rebus &longs;cenicis, maximè &longs;i æquabilis e&longs;&longs;e de­<lb/>beat motus; gravitas enim, quæ per unius aut alterius palmi <lb/>&longs;patium de&longs;cendat, non acquirit in motu notabile aliquod velo­<lb/>citatis incrementum, atque idcirco æquabilis apparet motus <lb/>tam ip&longs;ius gravitatis de&longs;cendentis, quàm ponderis illius virtu­<lb/>te a&longs;cendentis: hoc &longs;i non rectâ &longs;ur&longs;um trahatur, &longs;ed circum­<lb/>agatur, forta&longs;&longs;e impre&longs;&longs;us impetus velociorem circumvolutio­<lb/>nem efficere po&longs;&longs;it. </s> </p> <p type="main"> <s id="s.004149">Quod demum ad ip&longs;os rotarum dentes attinet, &longs;ingulæ qui­<lb/>dem rotæ à &longs;uis dentibus in partes æquales tribuuntur; &longs;ed hal­<lb/>lucinati videntur non pauci fru&longs;tra requirentes in omnibus ro­<lb/>tis invicem comparatis dentium æqualitatem, & ex dentium <lb/>numero potentiæ momenta metientes; qua&longs;i &longs;ervari nequiret <lb/>eadem momentorum Ratio, etiam&longs;i minoris tympani dentes <pb pagenum="554" xlink:href="017/01/570.jpg"/>non omninò &longs;imiles e&longs;&longs;ent, aut ut veriùs loquar, &longs;inguli non <lb/>e&longs;&longs;ent æquales &longs;ingulis dentibus majoris tympani in eodem axe <lb/>exi&longs;tentis. </s> <s id="s.004150">Dentium æqualitas in iis tantummodo rotis requiri­<lb/>tur, quæ &longs;ibi mutuâ collabellatione cohærentes in convolutione <lb/>dentem dentibus implicant; ni&longs;i enim ab unius rotæ dentium <lb/>intervallis alterius dentes &longs;ubinde reciperentur, fieri non po&longs;­<lb/>&longs;et utriu&longs;que rotæ conver&longs;io. </s> <s id="s.004151">Cæterùm nil prohibet, quomi­<lb/>nùs in plurium majorum tympanorum complexione alia rario­<lb/>res, alia &longs;pi&longs;&longs;iores dentes habeant, dummodo &longs;ingulis majori­<lb/>bus &longs;ingula minora, à quibus illa motum recipiunt, re&longs;pondeant <lb/>&longs;imilibus dentibus in&longs;tructa, etiam&longs;i hi di&longs;&longs;imiles &longs;int dentibus <lb/>tympani in eodem axe connexi. </s> <s id="s.004152">Si enim prioris majoris tym­<lb/>pani peripheria &longs;it in dentes 24 di&longs;tributa, minoris autem tym­<lb/>pani eidem axi infixi peripheria &longs;ex dentes habeat, &longs;ed eorum <lb/>diametri &longs;int ut 3 ad 2, utique eorum motus non aliam habent <lb/>Rationem quàm &longs;e&longs;quialteram, licèt dentium numeri &longs;int in <lb/>Ratione quadruplâ. </s> <s id="s.004153">Idem planè dicendum de &longs;ecundo tympa­<lb/>no majore, quod ad prioris motum convolvitur; hujus enim <lb/>motus pariter comparandus e&longs;t cum motu tympani minoris &longs;ibi <lb/>coniuncti &longs;pectatâ diametrorum Ratione, non dentium multi­<lb/>tudine, ut momenta innote&longs;cant. </s> <s id="s.004154">Quare illæ dentium multi­<lb/>tudines invicem comparatæ &longs;atis quidem faciunt quærenti, <lb/>quoties potentia manubrium circumagat, ut &longs;emel convertatur <lb/>Axis, quem ductarius funis complectitur; &longs;ed quibus momen­<lb/>tis id perficiat ip&longs;a potentia, &longs;ola diametrorum Ratio &longs;pectata <lb/>indicabit. </s> </p> <p type="main"> <s id="s.004155">Sed hìc ubi diametrorum incidit mentio (quamquam res <lb/>Mechanicæ in praxim deductæ tantâ &longs;ubtilitate non indigeant) <lb/>non e&longs;t di&longs;&longs;imulandum aliquam nece&longs;&longs;ariò intercedere momen­<lb/>torum inæqualitatem in ip&longs;o motu, quando tympanorum am­<lb/>bitus e&longs;t dentium inci&longs;uris a&longs;peratus: cum enim extremi den­<lb/>tium apices à centro magis ab&longs;int, quàm anguli, in quibus &longs;ibi <lb/>dentes occurrunt, non e&longs;t utrobique eadem movendi fa­<lb/>cultas, quippe quæ in majore à centro di&longs;tantiâ validior <lb/>e&longs;t, cæteris paribus. </s> <s id="s.004156">Eatenus &longs;cilicet rota rotam urget, qua­<lb/>tenus rota movens &longs;ui dentis apice contingit faciem den­<lb/>tis rotæ, quæ movetur: hic autem contactus primùm fit <lb/>propè angulum, hoc e&longs;t minùs procul à centro, & &longs;en-<pb pagenum="555" xlink:href="017/01/571.jpg"/>&longs;im dens rotæ moventis &longs;uo apice excurrens versùs extre­<lb/>mitatem dentis rotæ, quæ movetur, magis recedit à cen­<lb/>tro. </s> <s id="s.004157">Cum igitur rota movens &longs;uam vim exerceat apice den­<lb/>tis, integra &longs;emper illius diameter aut &longs;emidiameter con&longs;i­<lb/>deranda e&longs;t; at rota, quæ urgetur, cum non in eodem <lb/>puncto recipiat moventis impul&longs;ionem, non e&longs;t ab&longs;olu­<lb/>tè attendenda integra illius diameter aut &longs;emidiameter, <lb/>&longs;ed potiùs mediocris quædam inter maximam & mini­<lb/>mam à centro di&longs;tantiam eligenda e&longs;t, ut alter Rationis <lb/>terminus habeatur. </s> <s id="s.004158">Ex quo vides (&longs;i res &longs;ubtiliter elime­<lb/>tur) non parum intere&longs;&longs;e, utrùm minor rota majorem ur­<lb/>geat, an è contrario major minorem propellat. </s> <s id="s.004159">Concipe <lb/>enim majoris rotæ integram &longs;emidiametrum e&longs;&longs;e particula­<lb/>rum 100, & talem e&longs;&longs;e dentium inci&longs;uram, ut angulus, <lb/>in quo ip&longs;i dentes coëunt, di&longs;tet à centro particulis 94: <lb/>rotæ autem minoris, quæ &longs;uos dentes illius dentibus impli­<lb/>cat, &longs;emidiameter integra &longs;it &longs;imilium particularum 20, & <lb/>angulus concursûs dentium di&longs;tet à centro particulis 14. <!--neuer Satz-->Uti­<lb/>que &longs;i minor majorem urgeat illius Radius e&longs;t ut 20, hu­<lb/>jus verò e&longs;t ut 97: contra autem &longs;i major pellat minorem <lb/>illius Radius e&longs;t ut 100, hujus ut 17. <!--neuer Satz-->Quare &longs;ingulæ com­<lb/>paratæ cum iis, quæ &longs;ecum communem habent axem, di­<lb/>ver&longs;am con&longs;tituunt Rationem: &longs;i enim major rota urgea­<lb/>tur à minore &longs;ibi proximâ, adeò ut &longs;ecunda minor movea­<lb/>tur ad motum majoris in eodem axe, & Ratio &longs;it ut 20 ad <lb/>97, &longs;i majore proximâ urgente minorem moveretur major <lb/>ad motum minoris in eodem axe, hujus minoris motus ad <lb/>motum &longs;uæ majoris non e&longs;&longs;et pariter ut 20 ad 97, &longs;ed ut 17 <lb/>ad 100, quæ e&longs;t minor Ratio. <pb pagenum="556" xlink:href="017/01/572.jpg"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004160"><emph type="center"/>CAPUT VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004161"><emph type="center"/><emph type="italics"/>Mole&longs;trinarum artificium ex Axe in Peritrochio <lb/>pendet.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004162">ARtificia omnia, quæ ex Axe in Peritrochio pendent, re­<lb/>cen&longs;ere res e&longs;&longs;et non quidem injucunda, &longs;ed penè infi­<lb/>niti laboris, hi&longs;toriam potiùs redolens, quàm theoriam, cui <lb/>poti&longs;&longs;imùm in&longs;ervio Machinarum fontes indicans, ex quibus <lb/>ingenio&longs;us qui&longs;que Machinas &longs;uo in&longs;tituto opportunas moliri <lb/>queat. </s> <s id="s.004163">Placuit tamen in Molendinorum artificio pauli&longs;per im­<lb/>morari, ut quam uberem ab Axe in Peritrochio utilitatem ad <lb/>vitæ commoda percipiamus, innote&longs;cat. </s> <s id="s.004164">Quamvis autem po­<lb/>ti&longs;&longs;imùm in&longs;tituta &longs;int molendina ad comminuendum triticum <lb/>& alia &longs;emina, ut ex farinâ panis conficiatur, ad alios tamen <lb/>u&longs;us pars eorum aliqua de&longs;tinatur: omnibus quippe communis <lb/>e&longs;t rota exterior, quam aqua incurrens ver&longs;at, & Axis, qui <lb/>convolvitur. </s> <s id="s.004165">Si enim tundenda &longs;it lana, aut Cannabis; &longs;i in <lb/>pollinem redigenda elementa pulveris pyrij carbo, &longs;ulphur, <lb/>nitrum; &longs;i antiqua linteorum re&longs;egmina conterenda, & in mi­<lb/>nimas particulas di&longs;&longs;ipanda ad conficiendam chartam, Axi in­<lb/>fixæ &longs;unt pinnulæ, quæ in conver&longs;ione occurrentes alis pi&longs;til­<lb/>lorum illos elevant, atque dimittunt, & eorum gravitate reci­<lb/>dente &longs;ubjecta materia aut contunditur, aut conteritur. </s> </p> <p type="main"> <s id="s.004166">Nec di&longs;&longs;imili methodo di&longs;poni po&longs;&longs;ent pi&longs;tilli &longs;uis embolis <lb/>congruentes, qui à pinnulis Axis elevati aquam in embolum <lb/>attraherent, aut &longs;ponte irruentem admitterent per a&longs;&longs;arium, <lb/>tùm dimi&longs;&longs;i vi &longs;uæ gravitatis aquam exprimerent per tubum, <lb/>& in altiorem locum a&longs;cendere cogerent. </s> <s id="s.004167">Vel &longs;i non adeò gra­<lb/>ves pi&longs;tillos parare placuerit, velí&longs;que certiùs aquam in altio­<lb/>rem locum pellere, di&longs;pone binos pi&longs;tillos fune, aut catenâ, <lb/>per excavatum rotæ &longs;uperiùs po&longs;itæ ambitum tran&longs;eunte con­<lb/>nexos, aut potiùs tran&longs;ver&longs;ario, qua&longs;i libræ jugo conjunctos, <lb/>ita ut altero depre&longs;&longs;o alter elevetur, pinnula autem Axis de-<pb pagenum="557" xlink:href="017/01/573.jpg"/>primat pi&longs;tillum, vi cujus aqua in tubum a&longs;cendentem expri­<lb/>matur, & alter pi&longs;tillus attollatur aquam inferiùs po&longs;itam at­<lb/>trahens, qui pariter ab Axis pinnulâ ejus alæ re&longs;pondente &longs;ub­<lb/>inde deprimatur. </s> <s id="s.004168">Hinc fit po&longs;&longs;e longiorem Axem addi rotæ, <lb/>& plura huju&longs;modi pi&longs;tillorum paria di&longs;poni pinnulis in ambi­<lb/>tu Axis ita di&longs;tributis, ut non plures &longs;imul pi&longs;tillos, &longs;ed &longs;ingu­<lb/>los unum po&longs;t alium premant, &longs;i non adeò valida fuerit poten­<lb/>tia rotam ver&longs;ans; Sin autem validior illa fuerit, plures &longs;imul <lb/>deprimant, ií&longs;que conjugatos attollant. </s> <s id="s.004169">Ni&longs;i fortè magis arri­<lb/>&longs;erit duobus tantummodo pi&longs;tillis conjugatis uti, tot pinnulis <lb/>in Axe di&longs;po&longs;itis, ut in unâ eju&longs;dem Axis conver&longs;ione bis aut <lb/>ter pi&longs;tillus idem deprimatur. </s> </p> <p type="main"> <s id="s.004170">Huc pariter &longs;pectant, quæ pa&longs;&longs;im videre e&longs;t in officinis mal­<lb/>leatorum cupri aut ferri, ubi & rota exterior vi aquæ labentis <lb/>circumacta interiùs in conclavi qua&longs;i manubrium convolvit, <lb/>quod &longs;uperiori Axi horizonti parallelo infixum Radium, &longs;ibí­<lb/>que regulâ in juncturis plicatili connexum, dum attollit, at­<lb/>que deprimit, in alterâ eju&longs;dem Axis extremitate tran&longs;ver&longs;a­<lb/>rium hinc pariter attollens atque hinc deprimens follibus alter­<lb/>num motum conciliat: Et rota alia validiorem aquæ deciden­<lb/>tis impetum recipiens, &longs;uúmque Axem convolvens, pinnulis <lb/>axi infixis extremitatem alteram deprimit tigilli, cujus oppo&longs;i­<lb/>tæ extremitati elevatæ cohæret ingens ferreus malleus, qui præ­<lb/>terlapsä Axis pinnulâ &longs;ponte recidens tundit &longs;ubjectum cuprum <lb/>aut ferrum ignitum. </s> </p> <p type="main"> <s id="s.004171">In his omnibus rotæ quidem &longs;emidiameter attendenda e&longs;t, <lb/>in cujus extantes palmulas aqua incurrens vim potentiæ mo­<lb/>ventis obtinet; &longs;ed Axis &longs;emidiameter non &longs;olitariè accipienda <lb/>e&longs;t, verùm & addenda prominentis pinnulæ longitudo, ita ut <lb/>ex utrâque conficiatur unica &longs;emidiameter motûs, qui commu­<lb/>nicatur pi&longs;tillo, aut depre&longs;&longs;æ extremitati mallei. </s> <s id="s.004172">Depre&longs;&longs;æ, in­<lb/>quam, extremitati mallei, nam mallei elevatio aliquanto major <lb/>e&longs;t, quam illa depre&longs;&longs;io, ut validior ictus &longs;equatur; neque enim <lb/>tigillus à &longs;uo axe, cui innititur, omnino æqualiter dividitur, <lb/>&longs;ed ab eo aliquantulo remotior e&longs;t malleus, quàm oppo&longs;ita ex­<lb/>tremitas, quæ deprimitur: ac proinde vis illam deprimens ma­<lb/>jor e&longs;t, quàm &longs;i tigillus in partes æquales di&longs;tingueretur. </s> <s id="s.004173">Simi­<lb/>liter in follium motu primùm comparanda e&longs;t rotæ &longs;emidiame-<pb pagenum="558" xlink:href="017/01/574.jpg"/>ter cum adhærente manubrio, deinde Radius Axi &longs;uperiori in­<lb/>fixus comparandus e&longs;t cum &longs;emi&longs;&longs;e tran&longs;ver&longs;arij, cui folles jun­<lb/>guntur; & ex his duabus Rationibus componitur Ratio mo­<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/>rizontali parallela circumagenda e&longs;t, & quidem velociter, ut <lb/>granum in farinam di&longs;&longs;olvatur, non &longs;atis e&longs;t exterior rota aquæ <lb/>impetum recipiens & Axem &longs;ibi infixum volvens, &longs;ed etiam in­<lb/>terior rota denticulata in eodem Axe requiritur; & ne Machi­<lb/>næ membra fru&longs;trà multiplicentur, ita molares lapides com­<lb/>muniter di&longs;ponuntur, ut ferreus axis metam &longs;u&longs;tinens, & cur­<lb/>riculo in&longs;tructus, inferiorem locum obtineat, ac proinde cur­<lb/>riculus ip&longs;e proximè attingat &longs;uperiorem partem interioris rotæ <lb/>in &longs;uo plano denticulatæ eundem cum exteriore rotâ axem ha­<lb/>bentis. </s> <s id="s.004175">Quod &longs;i molares lapides collocari non po&longs;&longs;int in plano, <lb/>infra vel &longs;upra quod volvatur rota interior denticulata, &longs;ed &longs;o­<lb/>lùm paulò infra, aut &longs;upra axem eju&longs;dem rotæ; quia Vertebra <lb/>&longs;triata proximè molari lapidi cohærens; adeóque lapidem ip&longs;um <lb/>volvens, di&longs;tat, à rotâ denticulatâ, hæc autem commodè non <lb/>admittit tam longos dentes, qui eju&longs;dem Vertebræ aut curri­<lb/>culi virgulis aptè commi&longs;ceri valeant, propterea exigitur alius <lb/>Axis horizonti perpendicularis curriculo & rotæ infixus, quem <lb/>convertat rota interior curriculi hujus virgulas &longs;uis dentibus <lb/>impellens; &longs;imul enim rota dentata horizonti parallela, eidem <lb/>Axi perpendicularis infixa volvitur, & curriculum molæ con­<lb/>junctum circumagit. </s> </p> <p type="main"> <s id="s.004176">Hìc quoque plures Rationes componendæ &longs;unt; prima e&longs;t <lb/>Ratio diametri rotæ exterioris ad diametrum rotæ interioris in <lb/>eodem axe; deinde Ratio diametri curriculi molæ adhærentis <lb/>ad ip&longs;ius molæ circumactæ diametrum (&longs;ive integra diameter <lb/>accipienda &longs;it, &longs;ive illa tantum pars, quæ e&longs;t diameter circuli <lb/>in rotatione molæ de&longs;cripti à puncto inter centrum & periphe­<lb/>riam intermedio) & &longs;i, ut in &longs;ecundo ca&longs;u, interjectus fuerit <lb/>Axis perpendicularis, prætereà in compo&longs;itionem venit Ratio <lb/>diametri curriculi ad diametrum rotæ denticulatæ in eodem <lb/>Axe perpendiculari. </s> <s id="s.004177">Ex quibus apparet præ&longs;tare rotæ interio­<lb/>ris diametrum minorem e&longs;&longs;e diametro rotæ exterioris, ut aquæ <lb/>hanc impellentis momenta validiora &longs;int: &longs;ed & cavendum, ne <pb pagenum="559" xlink:href="017/01/575.jpg"/>illa ita minor &longs;tatuatur, ut ejus dentium numerus vix excedat <lb/>numerum virgularum curriculi molæ adhærentis, hæc enim <lb/>nimis tardè moveretur; & &longs;i intermedius fuerit Axis perpen­<lb/>dicularis, po&longs;itâ hac dentium æqualitate & virgularum cur­<lb/>riculi, unica rotæ exterioris conver&longs;io &longs;emel tantùm con­<lb/>volveret rotam denticulatam horizonti parallelam, atque <lb/>idcirco eodem tempore mola toties &longs;olùm converteretur, <lb/>quoties numerus virgularum ejus curriculi contineretur in <lb/>numero dentium rotæ denticulatæ infixæ Axi perpendi­<lb/>culari. </s> </p> <p type="main"> <s id="s.004178">Ut autem convolutionem molæ numerum augeas, cave ne <lb/>movendi difficultas pariter plus ju&longs;to augeatur, &longs;i nimirum <lb/>in axe perpendiculari diameter curriculi &longs;it immodicè mi­<lb/>nor diametro rotæ denticulatæ in eodem axe: potentia &longs;i <lb/>quidem curriculo applicata multo tardiùs moveretur, quàm <lb/>pondus extremis rotæ dentibus applicatum, ac proinde mo­<lb/>vendi difficultas augeretur. </s> <s id="s.004179">Quare omnia prudenter admi­<lb/>ni&longs;tranda, ut neque potentiæ moventis vires fru&longs;tra con­<lb/>terantur, neque mola tardiùs aut velociùs, quàm par &longs;it, <lb/>moveatur. </s> </p> <p type="main"> <s id="s.004180">Quod &longs;i non placuerit, aut loci di&longs;po&longs;itio non tulerit, <lb/>axem illum intermedium &longs;tatui perpendicularem, &longs;ed hori­<lb/>zonti parallelus commodior accidat, tunc rotæ interioris <lb/>eundem cum exteriore rotâ axem habentis dentes non pla­<lb/>no infixi, &longs;ed in extremo ambitu defixi requiruntur, ut &longs;u­<lb/>perioris axis curriculum (&longs;ive majorem, &longs;ive minorem, prout <lb/>opus fuerit) convertant, & cum eo rotam non in ambitu, &longs;ed <lb/>in plano, denticulatam, à qua molæ curriculus convolvatur. </s> <lb/> <s id="s.004181">Neque aliter, ac priùs, momentorum Ratio componitur, ex <lb/>Rationibus videlicet tympanorum, quæ communem Axem <lb/>habent, ut &longs;atis con&longs;tat ex dictis. </s> </p> <p type="main"> <s id="s.004182">Hinc quoniam potentia movens e&longs;t aqua, ob&longs;ervamus non <lb/>omnino eandem e&longs;&longs;e forman rotæ aquam excipientis; quæ <lb/>enim in profluente collocantur rotæ, nimis incommodæ e&longs;­<lb/>&longs;ent, &longs;i valdè amplam diametrum haberent; aut modico aquæ <lb/>labentis impetu pellerentur, &longs;i palmulis exiguis in&longs;truerentur: <lb/>propterea rotæ huju&longs;modi mediocrem quidem habent diame-<pb pagenum="560" xlink:href="017/01/576.jpg"/>trum, &longs;ed valdè notabilem axis partem occupant palmulis <lb/>adeò juxtà axis longitudinem expan&longs;is, ut à multâ aquâ in <lb/>illas incurrente validiore impul&longs;u circumagantur. </s> <s id="s.004183">Sic in Pa­<lb/>do communiter Rotæ hujus longitudo e&longs;t cubitorum 10, <lb/>diameter tota cubitorum 6; interior rota diametrum habet <lb/>cubit. </s> <s id="s.004184">5 1/2, dentes 108 plano infixos, & molæ curriculus in <lb/>fu&longs;os 9 di&longs;tinguitur; lapis autem molaris in cra&longs;&longs;itudine nu­<lb/>merat uncias 6 aut 7, in diametro cubitos 2 1/2. Quia ve­<lb/>rò aquæ ex alto cadentis motus major e&longs;t quàm profluen­<lb/>tis, propterea rotarum diameter amplior &longs;tatui pote&longs;t, &longs;i <lb/>opus fuerit, & palmularum latitudo valde mediocris &longs;uffi­<lb/>cit, quippe inclu&longs;a canali, per quem aqua decidens labi­<lb/>tur: modica &longs;cilicet aqua per planum magis elevatum pro­<lb/>lap&longs;a majora habet momenta, quàm per planum ferè ho­<lb/>rizontale: & præterea rota amplioris diametri faciliùs vol­<lb/>vitur etiam à minore aquâ, nam ad interiorem rotam, <lb/>cæteris paribus, habet majorem Rationem. <!-- KEEP S--></s> <s id="s.004185">Porrò palmu­<lb/>læ communiter quidem planæ &longs;unt, aut non ni&longs;i mo­<lb/>dicè &longs;inuatæ, ita ut aqua hinc atque hinc diffluat; ali­<lb/>quando tamen limbo ex utraque parte concluduntur, & <lb/>qua&longs;i va&longs;cula aquam aliquandiu continent, ut ip&longs;ius aquæ <lb/>inclu&longs;æ gravitas conver&longs;ionem juvet deor&longs;um urgendo. </s> <s id="s.004186">Ad­<lb/>de in ip&longs;o canali inclinato majores e&longs;&longs;e vires aquæ in parte <lb/>inferiore, quàm in &longs;uperiore propè initium casûs; quia vide­<lb/>licet aqua naturaliter de&longs;cendens motum habet acceleratum, <lb/>& ex antecedente de&longs;cen&longs;u acqui&longs;ivit impetum. </s> </p> <p type="main"> <s id="s.004187">Hactenus Molendina, quæ aquarum vi aguntur con&longs;ide­<lb/>ravimus, nihil addentes de iis, quæ ab hominibus, aut ab <lb/>animalibus volvuntur, nihil enim hæc habent peculiare præ­<lb/>terquàm quod axis primæ rotæ, quæ cæteris con&longs;equentibus <lb/>membris motum conciliat, e&longs;t horizonti perpendicularis, quia <lb/>potentia faciliùs in plano horizontali movetur, quàm in tym­<lb/>pano Verticali, quod calcaretur, & loco exterioris rotæ ab <lb/>aquâ propul&longs;æ vectis axi infigitur, quem aut jumenta trahunt, <lb/>aut homines urgent. </s> </p> <p type="main"> <s id="s.004188">Aliquid tamen innuendum de Molendinis, quæ vento <lb/>aguntur, &longs;ive ad comminuendas fruges, &longs;ive etiam ad agi-<pb pagenum="561" xlink:href="017/01/577.jpg"/>tandas antlias, quibus aquæ depre&longs;&longs;ioribus campis, in&longs;iden­<lb/>tes exhauriuntur. </s> <s id="s.004189">Quod enim attinet ad interius artificium <lb/>rotarum & curriculorum, &longs;imillimum e&longs;t iis, quæ in no&longs;tra­<lb/>tibus molendinis aquâ urgente commotis reperiuntur, ni&longs;i <lb/>quod in illis, ut pote à &longs;ubjectâ planitie remotis (locus &longs;i­<lb/>quidem amplo ventilabro opportunus tribuendus e&longs;t, & cap­<lb/>tandus ventus) per &longs;calas a&longs;cenditur, & in &longs;uperiorem lo­<lb/>cum comportandæ &longs;unt fruges, quas commolere oportet, <lb/>atque farina inde transferenda: quo labore levari pote&longs;t <lb/>molitor, &longs;i operâ eâdem, qua ventus axem primarium cum <lb/>rotis ver&longs;at, &longs;accos tritico aut farinâ plenos attollat, aut de­<lb/>ponat, fune ductario circa ip&longs;um Axem convoluto, aut evo­<lb/>luto. </s> <s id="s.004190">Illud poti&longs;&longs;imum in hoc molendinorum genere atten­<lb/>dendum e&longs;t, quod ad ip&longs;a flabella, quibus ventus excipi­<lb/>tur, &longs;pectat; neque enim quemadmodum juxta aquæ cur­<lb/>&longs;um rotæ planum dirigitur, etiam ventilabrum flabella habet <lb/>ita di&longs;po&longs;ita, ut venti ductum &longs;equantur: &longs;ed &longs;uperior do­<lb/>munculæ pars, qua Axis cum rotâ denticulatâ continetur, <lb/>u&longs;que adeò convertitur, ut ventilabrum flanti vento adver­<lb/>&longs;um &longs;tatuatur. </s> </p> <p type="main"> <s id="s.004191">Sunt autem flabella qua&longs;i quatuor &longs;calæ in primarij Axis <lb/>extremitate conjunctæ, quibus obducitur &longs;ingulis linteum, <lb/>ut vento re&longs;i&longs;tat; qui &longs;i ju&longs;to validior fuerit, lintei pars <lb/>complicata aliquem vento exitum præbet. </s> <s id="s.004192">Non tamen fla­<lb/>bella hæc ita ex æquo collocantur, ut in uno eodémque pla­<lb/>no Verticali con&longs;tituantur, &longs;ed &longs;ingulorum flabellorum pla­<lb/>num modicè obliquum &longs;tatuitur latere altero &longs;e paulatim &longs;ub­<lb/>ducente à vento. </s> <s id="s.004193">Ex quo fit ventum inter quatuor flabel­<lb/>lorum intervalla intercurrentem repellere in latus, & qua&longs;i <lb/>cubito percutere ip&longs;a flabella, atque adeò Axem converti <lb/>juxta flabellorum inclinationem. </s> <s id="s.004194">Nam &longs;i nulla e&longs;&longs;et flabel­<lb/>lorum obliquitas, & omnia qua&longs;i unicum planum efficerent, <lb/>in quod Axis e&longs;&longs;et perpendicularis, incertum e&longs;&longs;et, quam <lb/>in partem fieret conver&longs;io. </s> <s id="s.004195">Quod ad latitudinem aut longi­<lb/>tudinem huju&longs;modi flabellorum obliquè po&longs;itorum attinet, <lb/>non dubitatur, quin eorum latitudo maximè juvet motum; <lb/>quia eâdem obliquitate po&longs;itâ, major aëris pars incurrit in <pb pagenum="562" xlink:href="017/01/578.jpg"/>amplius quàm in &longs;trictius linteum; & in vehementiori ven­<lb/>to, ne nimia &longs;it machinæ velocitas, experimur aliquando <lb/>non ni&longs;i dimidium velum expandi. </s> <s id="s.004196">An verò fuerit operæ <lb/>pretium horum longitudinem augere, incertum e&longs;t: quam­<lb/>vis enim potentia magis à centro motûs di&longs;tans plus habeat <lb/>momenti, tamen quia longiorum flabellorum extremitates <lb/>valde inter &longs;e di&longs;tarent, ventus ampliora &longs;patia nactus mi­<lb/>nus haberet virium; &longs;icut & aqua fluens, velociùs atque <lb/>majore conatu per angu&longs;tias, quàm per patentem alveum <lb/>currit. </s> <s id="s.004197">Propterea in huju&longs;modi flabellis non auderem omni­<lb/>no definire, quo loco potentiæ moventis vires &longs;tatuendæ <lb/>&longs;int qua&longs;i in centro virtutis; nam prope Axem, cui infixa <lb/>&longs;unt, modica e&longs;t di&longs;tantia, & ventus qua&longs;i eorum objectu <lb/>compre&longs;&longs;us velociùs &longs;pirat, procul autem ab Axe in majo­<lb/>re intervallo faciliùs elabens minùs incitat cur&longs;um. </s> <s id="s.004198">Cum <lb/>verò non &longs;it temerè &longs;tatuendum venti compre&longs;&longs;ionem om­<lb/>nino re&longs;pondere mutuis flabellorum di&longs;tantiis, quæ in eâ­<lb/>dem Ratione &longs;unt ac di&longs;tantiæ ab Axe; neque facilè a&longs;&longs;eri <lb/>pote&longs;t eâdem Ratione decre&longs;cere vim venti ex compre&longs;&longs;io­<lb/>ne, qua eju&longs;dem momenta cre&longs;cunt ex di&longs;tantiâ ab Axe: <lb/>Ex quo fieret momenta compo&longs;ita ex di&longs;tantia ab Axe, <lb/>& ex vi compre&longs;&longs;ionis, e&longs;&longs;e per totam flabelli longitudi­<lb/>nem æqualiter diffu&longs;a, ac proinde in mediâ longitudine e&longs;­<lb/>&longs;e Centrum virtutis moventis. </s> <s id="s.004199">Omnibus tamen ritè per­<lb/>pen&longs;is, exi&longs;timarem centrum hoc virtutis, cui applicata <lb/>potentia intelligitur, haud procul abe&longs;&longs;e à mediâ fibelli lon­<lb/>gitudine: Ni&longs;i fortè flabella ip&longs;a talia e&longs;&longs;ent, ut eorum la­<lb/>titudo ab Axe recedens augeretur; &longs;ic enim diminutâ in <lb/>extremitatibus flabellorum di&longs;tantiâ, etiam venti compre&longs;&longs;io <lb/>augeretur. </s> </p> <p type="main"> <s id="s.004200">Quod &longs;i occurrendum putares incommodo, quod &longs;ubire <lb/>nece&longs;&longs;e e&longs;t ædiculam polo innixam ita convertendo, ut fla­<lb/>bella adver&longs;um ventum excipiant, haud abs re e&longs;&longs;e duce­<lb/>rem, &longs;i quis in &longs;upremo domûs fa&longs;tigio, loco patente & <lb/>ventis omnibus expo&longs;ito, cra&longs;&longs;um &longs;atí&longs;que validum axem ho­<lb/>rizonti perpendicularem &longs;tatueret, quem rota denticulata <lb/>horizonti parallela complecteretur, ex cujus conver&longs;ione de-<pb pagenum="563" xlink:href="017/01/579.jpg"/>mum mola circumageretur. </s> <s id="s.004201">At flabellorum latitudo juxta <lb/>Axis longitudinem in eju&longs;dem &longs;upremo capite extra tectum <lb/>collocanda e&longs;&longs;et, ut incurrentis venti impul&longs;­um exciperent, <lb/>perindè atque fluentis aquæ impetum recipiunt palmulæ ro­<lb/>tarum. </s> <s id="s.004202">Sed quoniam plana flabella parùm apta videntur ad <lb/>conver&longs;ionem continuandam, quia, quæ &longs;unt à diametro <lb/>oppo&longs;ita, demùm venti viribus exponerentur æqualiter, nec <lb/>dexterum potiùs quàm &longs;ini&longs;trum impellendum e&longs;&longs;et, adeó­<lb/>que ce&longs;&longs;aret conver&longs;io; propterea flabella con&longs;truenda e&longs;­<lb/>&longs;ent modicè incurva; hac enim ratione fieret, ut oppo&longs;ita <lb/>inæqualiter urgerentur, & dextri quidem convexam, &longs;i­<lb/>ni&longs;tri verò cavam faciem ventus impeteret inæqualibus vi­<lb/>ribus, illud &longs;cilicet qua&longs;i &longs;e &longs;ubducit vento, nec admo<lb/>dum ejus impul&longs;ui opponitur extremitas juxtà venti directio­<lb/>nem inflexa; hoc autem cavo &longs;inu ventum excipiens to­<lb/>tum ejus impul&longs;um recipit. </s> <s id="s.004203">Adde quod venti particula in <lb/>duo proxima flabella incurrens à convexâ unius facie in ca­<lb/>vam proximi faciem reflectitur, & auget impul&longs;ionem. </s> <lb/> <s id="s.004204">Quod &longs;i placuerit non quatuor, &longs;ed quinque flabella &longs;ta­<lb/>tuere, ne unquam duo ex diametro opponantur, non ab­<lb/>nuo. </s> <s id="s.004205">Illud certum e&longs;t huju&longs;modi flabellorum tùm longi­<lb/>tudinem, tùm latitudinem plurimùm juvare, quo enim <lb/>ampliora &longs;unt, plus venti excipiunt, & quò longiora, ut <lb/>pote à motûs centro magis &longs;ejuncta, plus habent momen­<lb/>ti. </s> <s id="s.004206">Quomodo autem &longs;i&longs;tenda &longs;it machina, explicanda aut <lb/>complicanda vela, ne præter molitoris voluntatem agitentur <lb/>flabella, nil refert hìc pluribus di&longs;putare, ubi tantummodo <lb/>vis movendi con&longs;ideratur. </s> <s id="s.004207">Neque &longs;olùm hujus molendini <lb/>u&longs;us e&longs;&longs;et in comminuendis tritici aut leguminum granis, <lb/>&longs;ed etiam in attollendis atque alio derivandis aquis, ut palus <lb/>ex&longs;iccetur, & cæteris huju&longs;modi, quæ præ&longs;ente &longs;emper cor­<lb/>pore movendo, non certo tempori alligantur, quemadmodum, <lb/>opus molendi, quod non perpetuò exercetur. <pb pagenum="564" xlink:href="017/01/580.jpg"/></s> </p> <p type="main"> <s id="s.004208"><emph type="center"/>CAPUT VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004209"><emph type="center"/><emph type="italics"/>Axis cum Vecte compo&longs;itus auget Potentiæ <lb/>momenta.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004210">TAnta e&longs;t aliquando ponderis gravitas, ut datæ poten­<lb/>tiæ vires illi movendo impares &longs;int, aut de oblatæ ma­<lb/>chinæ &longs;oliditate ac firmitate dubitetur: propterea oppor­<lb/>tunum accidet Vectem cum Axe in Peritrochio componere. <lb/><figure id="id.017.01.580.1.jpg" xlink:href="017/01/580/1.jpg"/><lb/>Primùm dato Vecte <lb/>AB &longs;ecundi gene­<lb/>ris, cujus hypomo­<lb/>chlium &longs;it B, & <lb/>pondus con&longs;titutum <lb/>in C, Potentia, quæ <lb/>in extremitate A ap­<lb/>plicanda e&longs;t, minor <lb/>&longs;it, quàm pro gra­<lb/>vitate ponderis, da­<lb/>tâ vectis Ratione <lb/>CB ad AB. <!-- KEEP S--></s> <s id="s.004211">Adhi­<lb/>beatur &longs;uccula EF <lb/>opportunè collata, ut funis ductarius in A alligatus Vectem <lb/>attollat: momenta enim potentiæ componuntur ex Ratio­<lb/>nibus radiorum &longs;ucculæ ad &longs;emidiametrum Axis, & di&longs;tan­<lb/>tiæ AB ad di&longs;tantiam CB in vecte. </s> <s id="s.004212">Hinc &longs;i Ratio AB <lb/>ad CB &longs;it ut 3 ad 1, Ratio autem radiorum ad Axis &longs;e­<lb/>midiametrum &longs;it ut 4 ad 1, unicus homo Succulam ver­<lb/>tens momentum habet æquale momentis quatuor hominum <lb/>in A vecti applicatorum, quorum &longs;inguli æquiparantur tri­<lb/>bus, qui pondus idem &longs;inè vecte attollere conarentur: atque <lb/>adeò unicus homo &longs;ucculam convertens æquat vires duode­<lb/>cim hominum ponderi ip&longs;i proximè applicatorum citra quod­<lb/>libet machinæ &longs;ub&longs;idium. </s> </p> <pb pagenum="565" xlink:href="017/01/581.jpg"/> <p type="main"> <s id="s.004213">Deinde in vecte primi generis, quando movendo pon­<lb/>deri velocitas aliqua concilianda e&longs;t, validiore potentiâ opus <lb/>e&longs;t, & tamen adjecto Axe infirmæ potentiæ adjumentum <lb/>comparare in promptu e&longs;t. </s> <s id="s.004214">Sit enim vectis IG, & hypo­<lb/>mochlium in H. Uti­<lb/><figure id="id.017.01.581.1.jpg" xlink:href="017/01/581/1.jpg"/><lb/>que potentia in I tan­<lb/>to major requiritur, <lb/>quanto major e&longs;&longs;e de­<lb/>bet ponderis motus &longs;u­<lb/>pra motum potentiæ, <lb/>hoc e&longs;t in Ratione <lb/>HG ad HI. <!-- KEEP S--></s> <s id="s.004215">Statua­<lb/>tur Axis RS, & fu­<lb/>nis ductarius Vectem <lb/>apprehendat in I. <!-- KEEP S--></s> <s id="s.004216">Tum axi infigatur Radius VT; nam <lb/>pro Ratione longitudinis VT ad Axis &longs;emidiametrum ita <lb/>augeri po&longs;&longs;unt potentiæ momenta, ut non &longs;olùm ponderis <lb/>gravitati paria &longs;int, &longs;ed & illam excedant. </s> <s id="s.004217">Fac enim IH <lb/>ad HG e&longs;&longs;e ut 1 ad 4, pondus verò in G e&longs;&longs;e lib. 200, <lb/>certè requireretur in I potentia major libris 800, ut &longs;uâ <lb/>virtute gravitati ponderis præ&longs;taret: At &longs;i Radius VT ad <lb/>Axis RS &longs;emidiametrum &longs;it ut 10 ad 1, jam potentia in T <lb/>motum habet ad motum ponderis in G ut 10 ad 4: igitur <lb/>reciprocè potentia in T ad pondus in G e&longs;&longs;et ut 4 ad 10, <lb/>ac proinde potentia habens vires attollendi ab&longs;que machinâ <lb/>libras 80, applicata in T attollet libras 200. Hæc quæ de <lb/>attollendo pondere dicta &longs;unt, intellige pariter &longs;i in plano <lb/>horizontali aut inclinato movendum e&longs;&longs;et; collocato &longs;cilicet <lb/>Axe non parallelo horizonti, &longs;ed vel perpendiculari, vel in­<lb/>clinato, pro ut loci opportunitas feret: hìc &longs;iquidem &longs;ola mo­<lb/>mentorum incrementa con&longs;iderantur ex harum duarum Fa­<lb/>cultatum compo&longs;itione. </s> </p> <p type="main"> <s id="s.004218">Quid autem opus e&longs;t monere idem virium compendium <lb/>haberi po&longs;&longs;e in Vecte pariter primi generis, quando pon­<lb/>dus tardè movendum e&longs;t? </s> <s id="s.004219">res enim per &longs;e clara e&longs;t, hypo­<lb/>mochlio &longs;cilicet magis ad extremitatem G accedente, quàm <lb/>ad extremitatem I, quæ potentiæ locus e&longs;t, ut &longs;i e&longs;&longs;et in L: <pb pagenum="566" xlink:href="017/01/582.jpg"/>id quod tunc poti&longs;&longs;imùm u&longs;urpari pote&longs;t, cùm elevatio pon­<lb/>deris ad aliquam non minimam altitudinem requiritur; opor­<lb/>tet enim hypomochlium à pondere intervallo notabili abe&longs;&longs;e, <lb/>unde & major movendi difficultas oritur, atque idcircò addi­<lb/>tâ &longs;ucculâ potentiam juvari nece&longs;&longs;e e&longs;t. </s> <s id="s.004220">Succulam verò po­<lb/>tiùs adhibendam proponere cen&longs;ui, quippe quæ & parabilior <lb/>e&longs;t, & commodior, nec multis impen&longs;is con&longs;truitur: Cæte­<lb/>rum nec Ergatam, nec tympana &longs;eu Grues, nec rotas denta­<lb/>tas, &longs;i placuerint, excludo. </s> </p> <p type="main"> <s id="s.004221">Ex his &longs;atis liquet, quid de Vecte tertij generis dicendum <lb/>&longs;it, in quo Potentia media inter pondus & hypomochlium <lb/>collocatur: Succula &longs;cilicet in &longs;uperiore loco &longs;tatuenda e&longs;t, <lb/>ita ut funis ductarius vectem apprehendat, ubi potentiæ lo­<lb/>cus a&longs;&longs;ignatur: &longs;ed quoniam minor e&longs;t potentiæ, quàm pon­<lb/>deris motus, & augenda &longs;unt potentiæ momenta, ut ponde­<lb/>ris gravitati elevandæ par &longs;it, Axi addendus e&longs;t Radius tantæ <lb/>longitudinis, ut potentia non jam Vecti, &longs;ed Radio applicata <lb/>velociùs moveatur, quàm pondus. </s> </p> <p type="main"> <s id="s.004222">Hactenus Axem in Peritrochio additum Vecti con&longs;ide­<lb/>ravimus, quatenus Vectem &longs;olitarium infirmior potentia <lb/>movere nequit: Nunc Vectem addere opportet Axi in <lb/>Peritrochio, ut hujus u&longs;us illo addito facilior accidat. </s> <s id="s.004223">Ma­<lb/>chinulam &longs;ecum deferunt communiter aurigæ in Germania, <lb/>qua rotam currûs, &longs;i fortè limo profundiùs infixa inhæ&longs;e­<lb/>rit, &longs;ublevant, ac proinde recte <emph type="italics"/>pancratium aurigarum<emph.end type="italics"/> dici <lb/>pote&longs;t. </s> <s id="s.004224">Lamina e&longs;t chalybea denticulata, cui rotula pariter <lb/>dentata congruit, cuju&longs;modi initio capitis 6. de&longs;crip&longs;imus: <lb/>parvula tamen e&longs;t rotula illa, &longs;ed centrum habens commu­<lb/>ne cum rotâ majore &longs;imiliter dentatâ, ex cujus conver&longs;ione <lb/>minor convolvitur, & laminam &longs;ur&longs;um propellit. </s> <s id="s.004225">Majoris <lb/>rotæ dentes apprehendit Axis &longs;triatus, cujus motûs princi­<lb/>pium ducitur à manubrio extra loculamentum ad latus ex­<lb/>tante. </s> <s id="s.004226">Quare duplex e&longs;t Ratio, videlicet manubrij ad &longs;emi­<lb/>diametrum axis &longs;triati, atque diametri rotæ majoris ad dia­<lb/>metrum rotulæ minoris concentricæ; ex quibus componi­<lb/>tur Ratio motûs Potentiæ manubrium ver&longs;antis, ad motum <lb/>ponderis &longs;ublevati. </s> <s id="s.004227">Quia autem fieri pote&longs;t, ut aut de lami-<pb pagenum="567" xlink:href="017/01/583.jpg"/>næ &longs;oliditate dubitetur, aut &longs;ubjectum rotæ &longs;olum non ad­<lb/>mittat congruam machinulæ po&longs;itionem; tunc rotæ elevan­<lb/>dæ capiti &longs;ubjiciatur validus fu&longs;tis alterâ extremitate incum­<lb/>bens telluri, alterâ innixus dentatæ laminæ; quæ eò minùs <lb/>à plau&longs;tri onere gravabitur, quò major erit Ratio totius lon­<lb/>gitudinis fu&longs;tis ad ejus partem inter rotæ caput, & &longs;olum, <lb/>cui innititur, interjectam. </s> </p> <p type="main"> <s id="s.004228">Hinc &longs;i Ratio vectis &longs;it ut 2 ad 1, machinæ lamina non <lb/>ni&longs;i à ponderis &longs;emi&longs;&longs;e gravatur; & Potentiæ manubrio Pan­<lb/>cratij applicatæ momenta geminantur. </s> <s id="s.004229">Nam &longs;i manubrij <lb/>longitudo ad Axis &longs;triati &longs;emidiametrum &longs;it ut 8 ad 1, ro­<lb/>tæ autem majoris diameter ad rotulæ concentricæ diame­<lb/>trum &longs;it ut 4 ad 1, potentiæ motus ad motum laminæ den­<lb/>tatæ e&longs;t ut 32 ad 1: &longs;ed appo&longs;ito vecte, cujus Ratio datur ut <lb/>2 ad 1, jam motus potentiæ ad motum ponderis elevati e&longs;t ut <lb/>64 ad 1, & potentiæ conatus, qui &longs;atis e&longs;&longs;et ad attollendas <lb/>&longs;ine machinâ libras 20, hoc Pancratio unâ cum Vecte attol­<lb/>leret libras 1280. </s> </p> <p type="main"> <s id="s.004230">Similiter &longs;i Ergatâ AB raptandum e&longs;&longs;et onus, & poten­<lb/>tia infirmior e&longs;­<lb/><figure id="id.017.01.583.1.jpg" xlink:href="017/01/583/1.jpg"/><lb/>&longs;et, quam ut in <lb/>extremitate Ra­<lb/>dij CD valeret <lb/>&longs;uperare oneris <lb/>re&longs;i&longs;tentiam, ad­<lb/>hibe Vectem <lb/>EF, & extre­<lb/>mitate E inni­<lb/>tente &longs;ubjecto <lb/>&longs;olo, potentia <lb/>applicetur ex­<lb/>tremitati F; nam <lb/>ejus momenta <lb/><expan abbr="componũtur">componuntur</expan> ex <lb/>Rationibus CD <lb/>Radij ad &longs;emidiametrum Axis AB, & vectis FE ad DE. <!-- KEEP S--></s> <lb/> <s id="s.004231">Pote&longs;t autem po&longs;t aliquantulum motum &longs;ubinde promoveri <pb pagenum="568" xlink:href="017/01/584.jpg"/>extremitas E vectis, ut manife&longs;tum e&longs;t. </s> <s id="s.004232">Quod &longs;i Ergatâ ipsâ <lb/>uteremur ad &longs;en&longs;im demittendum in plano inclinato onus <lb/>quoddam ingens, & timeretur, ne vis gravitatis vinceret co­<lb/>natum hominum in D reluctantium, ne præceps delabatur <lb/>onus; adhibeatur vectis EF, quo &longs;en&longs;im dimi&longs;&longs;o certiùs reti­<lb/>netur onus, & lentiùs de&longs;cendit. <lb/></s> </p> <p type="main"> <s id="s.004233"><emph type="center"/>CAPUT IX.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004234"><emph type="center"/><emph type="italics"/>Multiplex rotarum dentatarum u&longs;us innuitur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004235">QUanquam ea, quæ ad Mechanicam &longs;cientiam &longs;pectant <lb/>circa tympana dentata, &longs;atis in &longs;uperioribus explicata <lb/>&longs;int, quatenus ex iis &longs;ub&longs;idium petitur ad virium &longs;upplemen­<lb/>tum, & fontes indicati &longs;int, ex quibus unu&longs;qui&longs;que variam <lb/>huju&longs;modi tympanorum complexionem pro opportunitate ex­<lb/>cogitare po&longs;&longs;it; placuit tamen auctarium adjicere multiplicis <lb/>usûs, etiam aliquando citra momentorum potentiæ moven­<lb/>tis incrementum. </s> <s id="s.004236">Illud autem generatim ob&longs;ervandum e&longs;t, <lb/>ne pluribus membris di&longs;tinguatur machina, &longs;i pauciora &longs;uf­<lb/>ficiant: fieri &longs;iquidem non pote&longs;t, quin motui mora aliqua <lb/>inferatur, ubi plurium membrorum multiplex conflictus at­<lb/>que tritus contingit, etiam&longs;i omnia ritè di&longs;ponantur, & &longs;ibi <lb/>invicem proportione re&longs;pondeant. </s> </p> <p type="main"> <s id="s.004237"><emph type="center"/>PROPOSITIO I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004238"><emph type="center"/><emph type="italics"/>Anemo&longs;copium, Ventorum flantium indicem di&longs;cribere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004239">SI quis in conclavi manens cogno&longs;cere cupiat, quo vento <lb/>impellatur aër externus, & Anemo&longs;copium con&longs;truen­<lb/>dum curet, &longs;i hoc quidem in fornice, aut in laqueari de&longs;­<lb/>cribendum &longs;it, nullo opus e&longs;t artificio; &longs;ed &longs;atis e&longs;t intra <lb/>laminæ VK foramen erecto axi perpendiculari AB, qui <pb pagenum="569" xlink:href="017/01/585.jpg"/>nodo C laminæ in&longs;i&longs;tat facilè ver&longs;atilis, adjicere flabellum AD <lb/>&longs;upra tecti fa&longs;tigium loco apto ita eminens, ut directè, citra <lb/>reflexionum &longs;u&longs;picionem, cuju&longs;li­<lb/><figure id="id.017.01.585.1.jpg" xlink:href="017/01/585/1.jpg"/><lb/>bet auræ flantis impul&longs;um exci­<lb/>piens, & venti ductum &longs;equens <lb/>convertatur, atque ejus extremi­<lb/>tas D cœli plagam vento oppo&longs;i­<lb/>tam re&longs;piciat. </s> <s id="s.004240">In alterâ verò axis <lb/>extremitate B infra laquearis aut <lb/>fornicis faciem, in qua ritè juxta <lb/>horizontis po&longs;itionem de&longs;cripti <lb/>&longs;int ventorum cardines, adnecta­<lb/>tur index BF, ea lege, ut ex dia­<lb/>metro contrariam flabello AD <lb/>po&longs;itionem BF obtineat: hinc <lb/>enim fiet, ut quoniam ventus ex <lb/>A in D directus &longs;pirat, index F <lb/>eam horizontis partem, unde flat, <lb/>re&longs;piciat. </s> </p> <p type="main"> <s id="s.004241">Sin autem in plano Verticali <lb/>(aut etiam inclinato) de&longs;criben­<lb/>dum &longs;it Anemo&longs;copium, &longs;it axis <lb/>AH cum flabello AD tran&longs;iens <lb/>per C foramen, & acutâ cu&longs;pide <lb/>in&longs;i&longs;tens plano H, ut facillimè <lb/>converti queat, vertebram &longs;tria­<lb/>tam EG habens in octo æquales <lb/>&longs;trias di&longs;tinctam, quibus &longs;ubinde <lb/>exactè congruere po&longs;&longs;int rotæ MN dentes octo, in quos ferrea <lb/>lamina di&longs;tributa e&longs;t æqualiter, antequàm in circulum inflecte­<lb/>retur. </s> <s id="s.004242">Ex hujus rotæ centro infixus exeat axis R parietem per­<lb/>vadens, & in extremitate adnexum <expan abbr="indic&etilde;">indicem</expan> convolvens ad indi­<lb/>candos ventos in interiori, aut exteriori parietis facie de&longs;criptos. </s> </p> <p type="main"> <s id="s.004243">Verùm in ventorum de&longs;criptione cavendum, ne, quemad­<lb/>modum in Mappis Geographicis &longs;upremus locus Septentrioni, <lb/>infimus Au&longs;tro, dexter (qui &longs;cilicet e&longs;t ad dexteram a&longs;picien­<lb/>tis) Sub&longs;olano, &longs;ini&longs;ter Favonio tribuitur, ita hìc ordinem eun­<lb/>dem &longs;erves: quia enim vento flabellum impellente &longs;i vertebra <pb pagenum="570" xlink:href="017/01/586.jpg"/>&longs;triata convertatur ex G in I, rota dentata a&longs;cendit ex N in M, <lb/>& &longs;imiliter index convolvitur ex T in L; propterea &longs;i ventus ab <lb/>Arcto &longs;pirans in &longs;upremâ parte de&longs;criptus &longs;it in T, & in infimâ <lb/>qui à meridie in O, is, qui ab ortu flat, de&longs;cribendus e&longs;t ad &longs;i­<lb/>ni&longs;tram in L, & qui ab Occa&longs;u, ad dexteram in P. <!-- KEEP S--></s> <s id="s.004244">Quare mani­<lb/>fe&longs;tum e&longs;t, quo ordine reliquos intermedios de&longs;cribere oporteat. </s> </p> <p type="main"> <s id="s.004245"><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004246"><emph type="center"/><emph type="italics"/>Currûs motum metiri.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004247">QUæ Vitruvius lib. 10. cap. 14. &longs;crip&longs;it methodum innuens, <lb/>qua Veteres navi aut rhedâ vecti peractum iter dimetie­<lb/>bantur, plurium ingenia excitarunt (quandoquidem non paucis <lb/>Vitruvij verba ob&longs;curitate admodum laborare videbantur, quam <lb/>tamen notam illi inurere <expan abbr="nõ">non</expan> au&longs;im) ad varias rationes excogitan­<lb/>das, quibus hoc idem a&longs;&longs;equi &longs;e po&longs;&longs;e confidant. </s> <s id="s.004248">Maneat &longs;ua cui­<lb/>que Machinatori laus; neminis inventa improbo, aut a&longs;pernor: <lb/>Mihi <expan abbr="plani&longs;&longs;imã">plani&longs;&longs;imam</expan> inire viam &longs;emper placuit, qua putaverim ad id, <lb/>quod volumus, perveniri po&longs;&longs;e: quapropter nec certam rhedæ <lb/>formam, nec ver&longs;atilem cum affixis rotis axem præ&longs;cribo, &longs;ed ali­<lb/>quid vulgaribus rhedis aut curribus commune commini&longs;ci pla­<lb/>cuit, modò liceat alterius po&longs;teriorum rotarum (quippe anterio­<lb/>bus altiores &longs;unt) modiolo ad partem interiorem infigere bre­<lb/>viorem paxillum, quo rota ip&longs;a, dum convertitur, motum ma­<lb/>chinulæ currui alligatæ conciliet. </s> </p> <p type="main"> <s id="s.004249">Unum moneo, quod ad Vitruvium &longs;pectat (in quo nullam re­<lb/>perio ob&longs;curitatem) non arguendum e&longs;&longs;e o&longs;citantiæ, quòd rotæ <lb/>diametrum &longs;tatuerit pedum quaternum & &longs;extantis, deinde verò <lb/>totam rotæ ver&longs;ationem definiat pedibus duodecim; cùm tamen <lb/>ex Rationibus Cyclicis &longs;int ut minimum tredecim; atque adeò <lb/>quadringentæ ver&longs;ationes perficiant pedes 5200, hoc e&longs;t pa&longs;&longs;us <lb/>geometricos 40 &longs;upra milliare; ex quo integrâ die, qua milliaria <lb/>30 computarentur, error e&longs;&longs;et pa&longs;&longs;um 1200, qui milliaribus 30 ad­<lb/>dendi e&longs;&longs;ent. </s> <s id="s.004250">Contra verò &longs;i rotæ ambitus &longs;olùm peragat pedes <lb/>12; quadringentæ ver&longs;ationes dant pedes 4800, & pedes 200 de­<lb/>&longs;unt ad milliaris complementum: quare & hìc in milliarium 30 <lb/>computatione dee&longs;&longs;ent pa&longs;&longs;us 1200, qui milliaribus 30 demen­<lb/>di e&longs;&longs;ent. </s> <s id="s.004251">Ip&longs;e tamen Vitruvius quadringentis ver&longs;ationibus tri­<lb/>buit &longs;patia pedum 5000, hoc e&longs;t integri milliaris. </s> </p> <pb pagenum="571" xlink:href="017/01/587.jpg"/> <p type="main"> <s id="s.004252">Non e&longs;t, inquam, o&longs;citantiæ arguendus Vitruvius, quem i&longs;ta <lb/>latere non potuerunt, cùm &longs;int admodum obvia cuique vel levi­<lb/>ter Geometricis a&longs;per&longs;o; &longs;ed eo con&longs;ilio rotæ diametrum &longs;upra <lb/>quatuor pedes &longs;extante auxit, ut quod deteritur ex aliquâ rotæ <lb/>depre&longs;&longs;iore in &longs;olo, cui impre&longs;&longs;a ve&longs;tigia relinquit, hoc augmento <lb/>aliquâ ex parte re&longs;tituatur, adeóque tota ver&longs;atio con&longs;i&longs;tat intra <lb/>pedes 12 & 13; addere autem certam fractiunculam pedibus 12, <lb/>temerarium fui&longs;&longs;et, illa quippe valdè incon&longs;tans & incerta e&longs;t: <lb/>&longs;atis fuit demum in &longs;ummâ 400 ver&longs;ationum medium eligere <lb/>inter 5200, & 4800: neque enim error, qui notabilis e&longs;&longs;et, obre­<lb/>pere poterat. </s> </p> <p type="main"> <s id="s.004253">Non tamen placet Vitruvianum tympanum, cujus orbita in <lb/>quadringentos æquales denticulos e&longs;&longs;et di&longs;tributa, nimia quippe <lb/>& incommoda mihi videtur huju&longs;modi tympani magnitudo: &longs;i <lb/>enim ligneum fuerit tympanum, &longs;ingulorum <expan abbr="d&etilde;tium">dentium</expan> pedem, quo <lb/>orbitæ cohærent, vix puto minorem e&longs;&longs;e po&longs;&longs;e latitudine digitali, <lb/>hoc e&longs;t quatuor granorum hordei, &longs;i <expan abbr="quid&etilde;">quidem</expan> &longs;atis validi, & ad pe­<lb/>rennitatem con&longs;tructi intelligantur: &longs;in autem ferreum fuerit <lb/>tympanum, latitudo &longs;ingulorum &longs;altem æquabit duo grana hor­<lb/>dei. </s> <s id="s.004254">Quare orbita tympani in 400 huju&longs;modi dentes di&longs;tributa, <lb/>erit digitorum 400, aut 200, hoc e&longs;t, palmorum 100, aut 50; ac <lb/>proinde diameter erit palmorum ferè 32, aut 16. Commodius <lb/>igitur acciderit minora tympana componere, quàm adeò in­<lb/>gens tympanum con&longs;truere in tot dentes divi&longs;um. </s> </p> <p type="main"> <s id="s.004255">Sit it a que primùm rota denticulata A, cujus denti in&longs;i&longs;tens <lb/>ha&longs;tula CI axiculo in I jun­<lb/><figure id="id.017.01.587.1.jpg" xlink:href="017/01/587/1.jpg"/><lb/>gatur laminæ HI ita fixæ <lb/>in H, ut elateris, non tamen <lb/>admodùm validi, vice fun­<lb/>gatur. </s> <s id="s.004256">Tum ha&longs;tulæ CI <lb/>&longs;ubjiciatur ela&longs;ma D ali­<lb/>quanto validius, quantum <lb/>&longs;atis fuerit ad efficiendum <lb/>motum, quem &longs;tatim indi­<lb/>cabo. </s> <s id="s.004257">Alia pariter ha&longs;tula <lb/>FG cum &longs;uo ela&longs;mate E ita <lb/>di&longs;ponatur, ut denti G occurrens non permittat rotam retroagi <lb/>ex G versùs B, &longs;ed &longs;olùm converti po&longs;&longs;e ex G in C, atque à &longs;in-<pb pagenum="572" xlink:href="017/01/588.jpg"/>gulis dentibus elevata &longs;tatim vi ela&longs;matis E recidat, séque illis <lb/>objiciat, ne retrocedant. </s> <s id="s.004258">Additus igitur &longs;uniculus CS &longs;i trahatur, <lb/>dentem rotæ convertit ha&longs;tula CI impellens &longs;ubjectum ela&longs;ma <lb/>D, eademque operâ dens unus tran&longs;greditur ha&longs;tulam FG, quæ <lb/>vi ela&longs;matis E recidens prohibet, ne in contrarium fieri po&longs;&longs;it ro­<lb/>tæ conver&longs;io. </s> <s id="s.004259">Quia verò ha&longs;tula CI dum trahitur, dentem quo­<lb/>què &longs;ecum rapit, & ab eo inclinato demum liberatur, dimi&longs;&longs;o fu­<lb/>niculo, vi ela&longs;matis D &longs;ur&longs;um validè propellitur, & per obliquum <lb/>dentis latus excurrens extremitas C, obliquè pariter de&longs;inens, <lb/>repellit in I elaterem HI, donec ha&longs;tula ip&longs;a dentis apicem <lb/>tran&longs;gre&longs;&longs;a ab elatere HI &longs;e&longs;e re&longs;tituente coaptetur lateri &longs;upe­<lb/>riori dentis. </s> <s id="s.004260">Quo pacto &longs;inguli rotæ dentes &longs;ubinde convertun­<lb/>tur; atquè tandiu huju&longs;modi convolutio per&longs;everat, quandiu <lb/>trahitur, & dimittitur funiculus. </s> </p> <p type="main"> <s id="s.004261">Deinde rota altera pariter denticulata paretur, &longs;uóque axi in­<lb/>fixa ita di&longs;ponatur priori rotæ parallela (&longs;ed citra planorum con­<lb/>tactum) ut in ejus dentes incurrat paxillus L in rotæ A plano ad <lb/>perpendiculum erectus, quo po&longs;t integram prioris rotæ conver­<lb/>&longs;ionem dens unus &longs;ecundæ rotæ promoveatur. </s> <s id="s.004262">Ex quo fiet tot <lb/>prioris rotæ conver&longs;iones requiri ad po&longs;teriorem &longs;emel convol­<lb/>vendam, quot in po&longs;teriore rotâ dentes numerantur. </s> <s id="s.004263">Simili ra­<lb/>tione tertia, aut etiam, &longs;i opus fuerit, quarta rota denticulata pa­<lb/>retur, & ita pariter parallelæ di&longs;ponantur, ut paxillus &longs;ecundæ <lb/>rotæ tertiam, & tertiæ quartam convertat, paxillo videlicet den­<lb/>tium intervalla &longs;ubeunte po&longs;t integram &longs;uæ rotæ conver&longs;ionem. </s> </p> <p type="main"> <s id="s.004264">Hinc ut innote&longs;cat, quoties trahendus, atque dimittendus &longs;it <lb/>funiculus, ut rotæ convertantur, attendendus e&longs;t in &longs;ingulis rotis <lb/>dentium numerus: tùm numerus primæ per numerum &longs;ecundæ <lb/>ducendus; & qui producitur indicans numerum tractionum fu­<lb/>niculi, ut &longs;ecunda rota &longs;emel convertatur, per numerum dentium <lb/>tertiæ rotæ e&longs;t multiplicandus, ut &longs;ciamus, quot funiculi tractio­<lb/>nibus tertia rota gyrum integrum perficiat. </s> <s id="s.004265">Quod &longs;i hæc po&longs;tre­<lb/>ma non fuerit, &longs;ed & quarta rota adjiciatur, productus ex &longs;ecun­<lb/>dâ illâ multiplicatione numerus per numerum dentium quartæ <lb/>hujus rotæ multiplicabitur: ac demum innote&longs;cet, quoties funi­<lb/>culum trahere oporteat, ut quarta hæc rota totam circuli peri­<lb/>pheriam percurrat. </s> </p> <p type="main"> <s id="s.004266">Quod &longs;i paxillis, de quibus dictum e&longs;t, uti non placuerit, &longs;ed po-<pb pagenum="573" xlink:href="017/01/589.jpg"/>tiùs libeat &longs;ingulis rotis cra&longs;&longs;iu&longs;culos axes in&longs;erere, ex quibus <expan abbr="d&etilde;s">dens</expan> <lb/>unus promineat, qui po&longs;t integram &longs;uæ rotæ conver&longs;ionem den­<lb/>tibus &longs;equentis rotæ implicetur; omnino licebit, & forta&longs;&longs;e &longs;uo <lb/>commodo non carebit. </s> <s id="s.004267">Illud in rotarum collocatione intra &longs;uum <lb/>loculamentum e&longs;t diligenter animadvertendum, quod prioris ro­<lb/>tæ paxillus (aut axis dens) non ni&longs;i po&longs;t integram &longs;uæ rotæ con­<lb/>ver&longs;ionem incurrat in dentes po&longs;terioris; alioquin in errorem <lb/>non &longs;anè levem inducere nos po&longs;&longs;et index, qui extremitati axis <lb/>adnexus in exteriore loculamenti facie indicat &longs;ingularum ro­<lb/>tarum convolutiones. </s> </p> <p type="main"> <s id="s.004268">Affigatur itaque po&longs;teriori rhedæ parti opportuno loco regula <lb/>circa axem ver&longs;atilis, cujus &longs;uperior extremitas conjunctum ha­<lb/>beat funiculi CS trahendi caput S, inferior <expan abbr="aut&etilde;">autem</expan> extremitas oc­<lb/>currat paxillo, quem ab initio rotæ modiolo ad partem <expan abbr="interior&etilde;">interiorem</expan> <lb/>infixi&longs;ti: &longs;ic enim fiet, ut paxillo regulam impellente funiculus <lb/>trahatur, atque ad &longs;ingulas rotæ currûs conver&longs;iones, &longs;inguli den­<lb/>tes rotulæ A funiculum trahentem &longs;equantur: ac propterea in <lb/><expan abbr="loculam&etilde;ti">loculamenti</expan> facie index cum axe A convolutus indicabit, quoties <lb/>rota currûs <expan abbr="cõver&longs;a">conver&longs;a</expan> fuerit; & ab&longs;olutâ integrâ rotæ A conver&longs;io­<lb/>ne index &longs;equentis &longs;ecundæ rotulæ o&longs;tendet integras convolu­<lb/>tiones primæ; atque ita deinceps index tertiæ numerabit convo­<lb/>lutiones &longs;ecundæ, & index quartæ convolutiones tertiæ. </s> <s id="s.004269">Hinc &longs;i <lb/>rotulæ &longs;ingulæ &longs;int in dentes <expan abbr="dec&etilde;">decem</expan> di&longs;tributæ, numero, quem in­<lb/>dicat &longs;ecunda rotula, adde unicam cyphram 0, numero tertiæ ro­<lb/>tulæ adde duas cyphras 00, & numero à quarta rotula indicato <lb/>adde tres cyphras 000; &longs;tatímque manife&longs;tus fiet numerus con­<lb/>ver&longs;ionum rotæ currûs. </s> <s id="s.004270">Quare &longs;i po&longs;teriores currûs rotæ <expan abbr="habeãt">habeant</expan> <lb/>diametrum quinque pedum, rotæ ambitus e&longs;t trium <expan abbr="pa&longs;&longs;uũ">pa&longs;&longs;uum</expan> Geo­<lb/>metricorum (quod e&longs;t &longs;uper, negligitur, nam &longs;æpè rota <expan abbr="&longs;olũ">&longs;olum</expan> mol­<lb/>liu&longs;culum penetrans extenuat diametrum) atque adeò, ut &longs;emel <lb/>prima rotula <expan abbr="cõvertatur">convertatur</expan>, currûs rota decies <expan abbr="cõver&longs;a">conver&longs;a</expan> percurrit &longs;pa­<lb/>tium pa&longs;&longs;uum 30; ut &longs;ecunda unicam <expan abbr="conver&longs;ion&etilde;">conver&longs;ionem</expan> perficiat, rota <lb/>currûs centies volvitur, & conficit pa&longs;&longs;us 300; ut tertia gyrum ab­<lb/>&longs;olvat, rota currûs millies vertitur, & tria Italica milliaria percur­<lb/>rit. </s> <s id="s.004271">Ideò numerus ab indice quartæ rotulæ &longs;ignificatus, indicans <lb/>tertiæ rotulæ integras convolutiones, triplicandus e&longs;t, ut peracti <lb/>itineris men&longs;ura Italicis milliaribus definiatur. </s> <s id="s.004272">Ex quo fit quar­<lb/>tam rotulam in dentes decem di&longs;tributam &longs;ufficere ad numeran-<pb pagenum="574" xlink:href="017/01/590.jpg"/>da milliaria Italica 30: quod &longs;i plura velis numerare unicâ hu­<lb/>jus rotæ conver&longs;ione, in plures dentes, quàm decem, quartam <lb/>rotulam di&longs;tingue: &longs;ed non e&longs;t opus, quia unâ convolutione <lb/>ab&longs;olutâ, milliaribus indicatis addi po&longs;&longs;unt milliaria 30. </s> </p> <p type="main"> <s id="s.004273">Si ad navis cur&longs;um dimetiendum machinulam hanc eandem <lb/>traducere placeat, <expan abbr="adjici&etilde;da">adjicienda</expan> e&longs;t ad navis latus rota, ex cujus con­<lb/>ver&longs;ione integrâ ob&longs;ervatum fuerit, <expan abbr="quãtùm">quantùm</expan> navis promoveatur: <lb/>nam &longs;imiliter impellendo regulam, qua funiculus trahitur, rotæ <lb/><expan abbr="conver&longs;ionũ">conver&longs;ionum</expan> numerus innote&longs;cet, atque adeò <expan abbr="etiã">etiam</expan> itineris <expan abbr="&longs;patiũ">&longs;patium</expan>. </s> <lb/> <s id="s.004274">Cave tamen, ne in <expan abbr="error&etilde;">errorem</expan> incidas, qui facilè obrepere po&longs;&longs;et; cum <lb/>enim navis non &longs;emper æquè mergatur in aquâ (&longs;eu quia illa non <lb/>e&longs;t &longs;emper æquè onu&longs;ta, &longs;eu quia hæc <expan abbr="nõ">non</expan> e&longs;t &longs;emper æquè cra&longs;&longs;a, <lb/>aut tenuis) etiam rota inæqualiter mergitur, ac proinde una rotæ <lb/>hujus conver&longs;io non &longs;emper æquali itineris &longs;patio re&longs;pondet. </s> </p> <p type="main"> <s id="s.004275">Nec di&longs;&longs;imili ratione pede&longs;tria itinera metiri licebit, &longs;i parvu­<lb/>lam huju&longs;modi machinulam ita corpori alligaveris, ut funiculi <lb/>extremitas &longs;ub poplite adnectatur: nam ad &longs;ingulos pa&longs;&longs;us denti­<lb/>culus unus convertetur, & demum pa&longs;&longs;uum numerus innote&longs;cet. </s> </p> <p type="main"> <s id="s.004276"><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004277"><emph type="center"/><emph type="italics"/>Objecti procul vi&longs;i p&longs;eudographam &longs;peciem deformare.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004278">COntingit aliquando minùs attentos recti &longs;pecie decipi: pro­<lb/>pterea hac propo&longs;itione non inutile fuerit abu&longs;um <expan abbr="quendã">quendam</expan> <lb/>rotæ dentatæ, quæ facilè fucum faciat imperitis, indicare: ne for­<lb/>tè &longs;ibi qua&longs;i de præclaro invento inaniter gratulentur. </s> <s id="s.004279">Quadri­<lb/><figure id="id.017.01.590.1.jpg" xlink:href="017/01/590/1.jpg"/><lb/>laterum Pri&longs;ma AB eliga­<lb/>tur, cujus extremitas in te­<lb/>nuiorem cylindrum CD <lb/>de&longs;inat in&longs;erendum forami­<lb/>ni &longs;ubjecti plani cra&longs;&longs;ioris, <lb/>ita ut plano ad perpendicu­<lb/>lum in&longs;i&longs;tat pri&longs;ma, & &longs;er­<lb/>vatâ po&longs;itione perpendicu­<lb/>lari, facilè converti po&longs;&longs;it in <lb/>dexteram, & in &longs;ini&longs;tram. </s> <lb/> <s id="s.004280">Tum rotæ dentatæ &longs;emi&longs;&longs;is <lb/>MFN pri&longs;mati &longs;ecundùm <lb/>longitudinem excavato in&longs;eratur, atque circa axem A ductum per <pb pagenum="575" xlink:href="017/01/591.jpg"/>pri&longs;ma & rotæ dentatæ centrum circumagi po&longs;&longs;it. </s> <s id="s.004281">In eandem au­<lb/>tem pri&longs;matis fi&longs;&longs;uram infra rotæ dentatæ &longs;egmentum immitia­<lb/>tur regula HI &longs;uperiùs exa&longs;perata in crenas dentibus rotæ tan­<lb/>gentis congruentes, adeò ut ex rotæ conver&longs;ione regula HI ad­<lb/>ducatur, & reducatur: quæ in I calamum &longs;criptorium, aut lapi­<lb/>dem plumbarium habens (aut &longs;altem acutum &longs;tylum, quo certa <lb/>puncta lineis deinde jungenda notari valeant) in &longs;ubjecti char­<lb/>tâ lineas de&longs;cribit &longs;equens ductum radij optici per dioptram <lb/>MN excepti. </s> <s id="s.004282">Quare quot lineas in objecto procul vi&longs;o percurrit <lb/>radius opticus, totidem lineæ à &longs;tylo I de&longs;cribuntur in chartâ. </s> </p> <p type="main"> <s id="s.004283">Quando igitur magis altum, aut longiùs po&longs;itum objecti <lb/>punctum per dioptram a&longs;picitur, dioptræ extremitas oculo pro­<lb/>xima deprimitur, atque adeò rotæ dentatæ portio ita conver­<lb/>titur, ut versùs objectum promoveat regulam: contrà verò de­<lb/>pre&longs;&longs;ius, aut propius objecti punctum a&longs;piciens, proximam ocu­<lb/>lo extremitatem dioptræ elevat, & regulam ab objecto removet; <lb/>cuicumque tandem extremitati M, aut N oculum admoveas: Si <lb/>enim ex M a&longs;picias, deprimendo M propellis &longs;tylum I versùs pri&longs;­<lb/>ma, hoc e&longs;t versùs objectum; atque &longs;imiliter ex N <expan abbr="a&longs;pici&etilde;s">a&longs;piciens</expan>, depri­<lb/>mendo N removes &longs;tylum I à pri&longs;mate, & versùs objectum im­<lb/>pellis. </s> <s id="s.004284">At verò ubi tran&longs;ver&longs;um objecti latus a&longs;piciendum e&longs;t, <lb/>factâ circa cylindrulum CD conver&longs;ione, plurimum intere&longs;t, <lb/>utrùm ex M, an ex N a&longs;picias: Nam &longs;i oculus &longs;it in N, & radio <lb/>optico percurrat objecti latus à &longs;ini&longs;trâ in dexteram, <expan abbr="etiã">etiam</expan> &longs;tylus I <lb/>à &longs;ini&longs;trâ in dextram movetur unâ cum extremitate M objectum <lb/>re&longs;piciente. </s> <s id="s.004285">Sin autem oculus &longs;it in M, atque &longs;tylus I inter <expan abbr="oculũ">oculum</expan> <lb/>& pri&longs;ma, aut oculus inter &longs;tylum & pri&longs;ma interjectus &longs;it, con­<lb/>trariam po&longs;itionem habent puncta à &longs;tylo de&longs;cripta, & &longs;ini&longs;tra mi­<lb/>grant in dexteram, atque dextera in &longs;ini&longs;tram; &longs;tylus quippe ocu­<lb/>lum &longs;equitur, qui motum habet oppo&longs;itum motui alterius extre­<lb/>mitatis N objectum re&longs;picientis. </s> <s id="s.004286">Quamobrem expedit oculum <lb/>dioptræ in N admovere, & in <expan abbr="objectũ">objectum</expan> <expan abbr="&longs;tylũ">&longs;tylum</expan> I obvertere, ut dextra <lb/>dextris, & &longs;ini&longs;tra &longs;ini&longs;tris <expan abbr="re&longs;pondeãt">re&longs;pondeant</expan>, prout &longs;ub <expan abbr="a&longs;pectũ">a&longs;pectum</expan> cadunt. </s> </p> <p type="main"> <s id="s.004287">Verùm, licèt objecti vi&longs;i &longs;peciem aliquam hoc artificio adum­<lb/>brare liceat, <expan abbr="cavendũ">cavendum</expan> tamen, ne ip&longs;i nobis <expan abbr="a&longs;&longs;entãtes">a&longs;&longs;entantes</expan> qua&longs;i <expan abbr="exactã">exactam</expan> <lb/>Ichnographiam, & &longs;ubtilem, &longs;ervatis corporis <expan abbr="partiũ">partium</expan> Rationibus, <lb/>de&longs;criptionem nos compara&longs;&longs;e exi&longs;timemus: cuique &longs;cilicet rem <lb/>accuratè perpendenti manife&longs;tum e&longs;t, quandiu &longs;emicirculus in <pb pagenum="576" xlink:href="017/01/592.jpg"/>eodem plano Verticali <expan abbr="cõ&longs;i&longs;tit">con&longs;i&longs;tit</expan>, & dioptra elevatur, &longs;ive deprimi­<lb/>tur, lineam objecti, quam radius opticus percurrit in plano hori­<lb/>zontali, re&longs;pondere differentiæ Tangentium angulorum, quos <expan abbr="cũ">cum</expan> <lb/>perpendiculo AB con&longs;tituit radius opticus: At linea, quam &longs;ty­<lb/>lus I de&longs;cribit, re&longs;pondet quidem (&longs;altem proximè, & quatenus <lb/>&longs;en&longs;u in tantâ parvitate percipi pote&longs;t) differentiæ Tangentium <lb/>angulorum æquè differentium, quos cum perpendiculo eodem <lb/>AB con&longs;tituere intelligitur linea à centro A ad &longs;tylum I ducta. </s> <lb/> <s id="s.004288">Non tamen fieri pote&longs;t, ut deinde in omnibus po&longs;itionibus muta­<lb/>to Verticali eadem Ratio &longs;ervetur; quia linea à centro A ad &longs;ty­<lb/>lum I ducta, non e&longs;t parallela radio optico, &longs;ed angulum multo <lb/>minorem con&longs;tituit cum perpendiculo; ac proinde angulorum <lb/>minorum differentia, etiam&longs;i æqualis differentiæ angulorum ma­<lb/>jorum, non infert proportionalem <expan abbr="differentiã">differentiam</expan> Tangentium. </s> <s id="s.004289">Sta­<lb/>tuatur ex. </s> <s id="s.004290">gr. <!-- REMOVE S-->differentia angulorum duobus gradibus definita, & <lb/>in uno Verticali majores anguli à dioptrâ con&longs;tituti &longs;int gr.88. & <lb/>86, minores autem gr.58. & 56: in altero Verticali majores anguli <lb/>à dioptrâ con&longs;tituti &longs;int gr. <!-- REMOVE S-->73. & 71, minores verò gr.43, & 41. <lb/>Quia idem e&longs;t Radius AB, quarum partium 1000 e&longs;t Radius, in <lb/>primo Verticali differentia majorum Tangentium e&longs;t 14336, & <lb/>differentia Tangentium minorum e&longs;t 118: in &longs;ecuado Vertica­<lb/>li differentiæ Tangentium &longs;unt 367 majorum, & 63 minorum <lb/>angulorum: inter hos autem terminos non intercedere propor­<lb/>tionem manife&longs;tum e&longs;t. </s> </p> <p type="main"> <s id="s.004291">Quando verò, factâ circa cylindrum CD conver&longs;ione, fit tran­<lb/>&longs;itus ab uno plano Verticali ad aliud planum Verticale, linea, <lb/>quam radius opticus percurrit, & linea, quam &longs;tylus I de&longs;cribit, <lb/>&longs;ubtendunt quidem &longs;imiles arcus, opponuntur enim eidem an­<lb/>gulo Verticalium, &longs;ed &longs;unt in Ratione di&longs;tantiarum objecti vi&longs;i, <lb/>atque &longs;tyli à cylindrulo tanquam centro motûs. </s> <s id="s.004292">Porrò ha&longs;ce li­<lb/>neas differentiis illis Tangentium non e&longs;&longs;e analogas per&longs;picuum <lb/>e&longs;t. </s> <s id="s.004293">Quapropter de&longs;criptum &longs;chema non &longs;ervans objecti Ratio­<lb/>nes, cen&longs;endum e&longs;t p&longs;eudographum. </s> </p> <p type="main"> <s id="s.004294">Oporteret plano immobili, cui infigitur pri&longs;ma, adnectere con­<lb/>gruis cardinibus aut fibulis, tabellam, quæ &longs;emper parallela diop­<lb/>træ cum hac pariter elevaretur & deprimeretur (non tamen cum <lb/>eâ convolveretur) ut in chartâ tabellæ affixâ &longs;pecies magis cum <lb/>objecto conveniens de&longs;criberetur: Qua autem methodo? </s> <s id="s.004295">inge­<lb/>nio&longs;us lector di&longs;piciat. </s> </p> <pb pagenum="577" xlink:href="017/01/593.jpg"/> <figure id="id.017.01.593.1.jpg" xlink:href="017/01/593/1.jpg"/> <p type="main"> <s id="s.004296"><emph type="center"/>MECHANICORUM <emph.end type="center"/><!-- REMOVE S--><emph type="center"/>LIBER SEXTUS.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.004297"><emph type="center"/><emph type="italics"/>De Trochlea.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004298">NON &longs;emper commodum accidit Ergatâ, aut &longs;uc­<lb/>culâ, aut Tympano uti ad pondus aliquod moven­<lb/>dum: ut enim ex iis, quæ &longs;uperiore libro di&longs;puta­<lb/>ta &longs;unt, manife&longs;tum e&longs;t, &longs;i in altiorem locum eve­<lb/>hendum &longs;it pondus, ibi con&longs;truere oporteret peg­<lb/>ma, cui machina in&longs;i&longs;teret: &longs;æpè autem id fieri non po&longs;&longs;et &longs;ine <lb/>magna impen&longs;a, aut citrà incommodum &longs;ive propter loci an­<lb/>gu&longs;tias, &longs;ive propter temporis brevitatem pegmati con&longs;truendo <lb/>imparem. </s> <s id="s.004299">Hinc alia Facultas excogitata e&longs;t, cui <emph type="italics"/>Throchleæ<emph.end type="italics"/> no­<lb/>men inditum e&longs;t; quippè quæ communiter ex rotulis circà <lb/>axem in &longs;uo loculamento ver&longs;atilibus coagmentatur, ií&longs;que cir­<lb/>cumducitur funis ductarius, quo trahitur pondus trochleæ ad­<lb/>nexum. </s> <s id="s.004300">Trochleam autem, ut Vitruvius lib. 10 cap.2. te&longs;tatur <lb/>nonnulli <emph type="italics"/>Rechamum<emph.end type="italics"/> dicunt. </s> </p> <p type="main"> <s id="s.004301">Ex orbiculorum numero nomen ducit machina; nam &longs;i uni­<lb/>cus &longs;it orbiculus, Trochlea &longs;implex, aut Mono&longs;patos vocatur; <lb/>&longs;i duo fuerint orbiculi, Di&longs;pa&longs;tos; &longs;i tres Tri&longs;pa&longs;tos; atque ita <lb/>deinceps. </s> <s id="s.004302">In hac tamen nomenclaturâ ob&longs;ervandum e&longs;t, non <lb/>codem omnes vocabulo uti: aliqui enim cunctos orbiculos <lb/>utriu&longs;que loculamenti in unam &longs;ummam referunt, & ex eorum <lb/>numero vocabulum &longs;tatuunt; ut &longs;i alterius loculamenti duo &longs;int <lb/>orbiculi, alterius verò unicus, Tri&longs;pa&longs;ton appellant: Alij ta­<lb/>men nomen indunt ex orbiculis &longs;ingulorum loculamentorum; <lb/>nam &longs;i binos orbiculos &longs;ingula contineant, non Tetra&longs;pa&longs;ton, <lb/>&longs;ed Di&longs;pa&longs;ton vocant, quia communiter ambo loculamenta <lb/>æquali orbiculorum numero in&longs;truuntur, & ex alterius numero <lb/>reliqui, pariter numerus innote&longs;cit. </s> <s id="s.004303">Neque omnino abs re alte-<pb pagenum="578" xlink:href="017/01/594.jpg"/>rius tantummodo loculamenti orbiculos numerant, quia hujus <lb/>facultatis vires poti&longs;&longs;imùm habentur ex &longs;olis orbiculis locula­<lb/>menti, cui pondus trahendum adnectitur; reliquum &longs;cilicet lo­<lb/>culamentum cum &longs;uis rotulis proptereà adjicitur, ut funis ducta­<lb/>rius &longs;ingulos illius orbiculos complecti po&longs;&longs;it. </s> <s id="s.004304">Ex quo fit, po&longs;ito <lb/>inæquali orbiculorum numero, modò Mono&longs;pa&longs;ton, modò Di&longs;­<lb/>pa&longs;ton dici, prout pondus adnectitur loculamento unum, aut <lb/>duos orbiculos habenti. </s> <s id="s.004305">Cæterùm in vocabulis non e&longs;t hæren­<lb/>dum: Ego Trochleam voco loculamentum unum cum &longs;uis or­<lb/>biculis; & quando opus e&longs;t duplici loculamento uti, duplicem <lb/>Throchleam dico, atque orbiculos numero, ne ullus &longs;ube&longs;&longs;e <lb/>po&longs;&longs;it æquivocationi locus. </s> </p> <p type="main"> <s id="s.004306">Quantum autem Facultas hæc &longs;it Axe, aut Vecte utilior, hinc <lb/>&longs;altem con&longs;tat, quod etiam&longs;i plures potentiæ diver&longs;is funis <lb/>ductarij partibus applicentur, æqualia tamen obtinent momen­<lb/>ta; id quod non contingit pluribus eundem Succulæ Radium, <lb/>aut eumdem Vectem urgentibus; neque enim æqualibus à mo­<lb/>tûs centro intervallis ab&longs;unt. <lb/></s> </p> <p type="main"> <s id="s.004307"><emph type="center"/>CAPUT I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004308"><emph type="center"/><emph type="italics"/>Trochlearum forma, & vires exponuntur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004309">ALiquando &longs;implicem orbiculum, cujus excavatæ orbitæ <lb/>funis ductarius in&longs;i&longs;tit, adhibemus, ut onera &longs;ur&longs;um attol­<lb/>lamus: & quidem communiter in &longs;uperiore loco firmatur locu­<lb/>lamentum cum orbiculo ver&longs;atili, & alteram funis extremita­<lb/>tem apprehendit Potentia, alteri adnectitur pondus &longs;ublevan­<lb/>dum, quod a&longs;cendendo &longs;patium percurrit æquale &longs;patio, per <lb/>quod Potentia de&longs;cendendo movetur. </s> <s id="s.004310">Id quod eatenus excogi­<lb/>tatum e&longs;t, quatenus brachia deprimentibus in ponderis eleva­<lb/>tione in&longs;ita brachiorum gravitas vires addit, & minore lacerto­<lb/>rum contentione opus e&longs;t, quàm &longs;i pondus ip&longs;um &longs;ur&longs;um trahe­<lb/>remus brachia elevantes. </s> <s id="s.004311">Factus e&longs;t autem orbiculus circa &longs;uum <lb/>axem ver&longs;atilis, ut vitetur difficultas, quæ cæteroqui con&longs;eque­<lb/>retur mutuum tritum funis cum &longs;ubjecto corpore, cui in&longs;i&longs;teret, <lb/>&longs;i illud non variaretur. </s> <s id="s.004312">Quantus enim &longs;it huju&longs;modi funis cum <lb/>&longs;ubjecto corpore (&longs;i illud non convolvatur) conflictus, <expan abbr="manife&longs;tũ">manife&longs;tum</expan> <pb pagenum="579" xlink:href="017/01/595.jpg"/>e&longs;t in puteis, quibus ad hauriendam aquam non e&longs;t girgillus, <lb/>hoc e&longs;t, orbiculus ver&longs;atilis, adjectus, &longs;ed funis tran&longs;ver&longs;o fu&longs;ti <lb/>cylindrico, verùm immobili, in&longs;i&longs;tit; excavatur &longs;iquidem cylin­<lb/>der ille diuturno, & frequenti tritu funium. </s> <s id="s.004313">Cæterum &longs;i non ad <lb/>perpendiculum attollendum &longs;it pondus, &longs;ed in plano horizonta­<lb/>li, aut inclinato (non tamen lubrico) raptandum, vix, aut ne <lb/>vix quidem, ullum compendium con&longs;equeris, &longs;i funem per or­<lb/>biculum tran&longs;euntem trahas in plagam oppo&longs;itam plagæ, ver­<lb/>sùs quam pondus dirigitur, ac &longs;i pondus idem arrepto fune ad te <lb/>directè rapias: eadem quippe e&longs;t brachiorum contentio, quo­<lb/>rum in&longs;ita gravitas non juvat potentiam, ni&longs;i quando hæc deor­<lb/>&longs;um tendit. </s> <s id="s.004314">Adhiberi tamen huju&longs;modi orbiculus in planitie <lb/>poterit, &longs;i commodiùs Potentia con&longs;i&longs;tat in loco, ubi jacet pon­<lb/>dus, quàm ibi, quò illud adducendum e&longs;t. </s> </p> <p type="main"> <s id="s.004315">Quamquam verò orbiculus &longs;tabili loculamento infixus non <lb/>&longs;it aptus ad augendas Potentiæ vires, prout ad Machinæ ratio­<lb/>nem pertinet; &longs;i tamen loculamentum ip­<lb/><figure id="id.017.01.595.1.jpg" xlink:href="017/01/595/1.jpg"/><lb/>&longs;um adnectatur ponderi, quod cum illo mo­<lb/>veatur, geminantur Potentiæ momenta, non <lb/>enim æqualis e&longs;t Potentiæ & Ponderis mo­<lb/>tus, &longs;ed illa duplo velociùs movetur. </s> <s id="s.004316">Sit <lb/>pondus attollendum &longs;ivè raptandum A, cui <lb/>adnectatur loculamentum orbiculi B; funis <lb/>autem ductarius firmetur in C, & funis <lb/>extremitatem reliquam apprehendat Po­<lb/>tentia in D: utique Potentia ut adducat <lb/>orbiculum u&longs;que in C, tantumdem pro­<lb/>gredi debet ultra C, quantum orbiculus <lb/>B di&longs;tat à puncto C; oportet &longs;iquidem <lb/>totum funem DBC explicari. </s> <s id="s.004317">Igitur po­<lb/>tentia ex D venit primùm in E, deinde <lb/>in F: e&longs;t autem di&longs;tantia DE æqualis in­<lb/>tervallo BC; &longs;ed tunc, cùm illa e&longs;t in <lb/>E, orbiculus &longs;olùm e&longs;t in I, & demum <lb/>hic e&longs;t in C, quando potentia e&longs;t in F. <!-- KEEP S--></s> <lb/> <s id="s.004318">Motus itaque potentiæ DF e&longs;t duplus mo­<lb/>tus orbiculi BC. <!-- KEEP S--></s> <s id="s.004319">Porrò cum orbiculo pa­<lb/>riter trahitur pondus A adnexum; igitur <pb pagenum="580" xlink:href="017/01/596.jpg"/>duplo velocior e&longs;t potentiæ motus præ motu ponderis. </s> <s id="s.004320">Qua­<lb/>re potentia valens trahere motu &longs;ibi æquali pondus aliquod <lb/>&longs;ine orbiculo, hoc addito valebit trahere pondus duplo ma­<lb/>jore gravitate præditum. </s> </p> <p type="main"> <s id="s.004321">Ex quibus manife&longs;tum e&longs;t, quantum inter&longs;it, utrùm ex­<lb/>tremitati funis adnectatur pondus, & orbiculi loculamen­<lb/>tum &longs;tabile &longs;it, an verò, funis extremitate manente atque <lb/>immotâ, ponderi adnectatur loculamentum, quod cum ip&longs;o <lb/>pondere moveatur, immò veriùs, cujus motum con&longs;equatur <lb/>motus ponderis: nam in &longs;ecundo hoc ca&longs;u potentiæ motus <lb/>duplus e&longs;t ad motum ponderis; in primâ autem po&longs;itione mo­<lb/>tus utriu&longs;que &longs;unt planè æquales. </s> </p> <p type="main"> <s id="s.004322">Hinc ulteriùs con&longs;tat, quando duæ Trochleæ &longs;implici <lb/>orbiculo in&longs;tructæ adhibentur, ita ut altera fixa maneat, al­<lb/>tera cum pondere moveatur, nihil addi momenti Potentiæ <lb/>&longs;i funis extremitas alligetur trochleæ &longs;tabili, aut loco alicui <lb/>extra trochleas. </s> <s id="s.004323">Nam &longs;i in G po&longs;ita &longs;it Trochlea manens <lb/>immota H, & altera funis extremitas illi jungatur in O, <lb/>&longs;eu extra illam clavo, aut paxillo in C, Potentia in L ap­<lb/>plicata æqualiter movetur cum puncto D: at punctum D <lb/>movetur duplo velocius, quàm Trochlea B; igitur Poten­<lb/>tia L movetur &longs;olum duplo velociùs quàm pondus, perinde <lb/>atque &longs;i non fui&longs;&longs;et addita trochlea H. <!-- KEEP S--></s> <s id="s.004324">Eatenus igitur additur <lb/>Trochlea H, quatenus Potentiam & Pondus in oppo&longs;itas pla­<lb/>gas moveri oportet, aut potentia deor&longs;um conari debet, ut <lb/>pondus a&longs;cendat. </s> </p> <p type="main"> <s id="s.004325">Sin autem extremitas funis alligetur Trochleæ mobili, <lb/>cui pariter adnectitur pondus, & primùm funis ab unco <lb/>trochleæ mobilis deducatus ad orbiculum trochleæ immotæ, <lb/>deinde ad orbiculum eju&longs;dem Trochleæ mobilis, jam Po­<lb/>tentia triplo velociùs movetur quàm Pondus; quia videli­<lb/>cet etiam ip&longs;a funis extremitas movetur trahentem &longs;equens <lb/>unâ cum pondere. </s> <s id="s.004326">Concipe enim pondus A &longs;ejunctum à <lb/>Trochleâ B, quæ ita firmetur, ut immota maneat, pondus <lb/>verò intelligatur tran&longs;latum in G, atque Trochlea H jam <lb/>&longs;it mobilis: utique Potentia funem in L arreptum trahens <lb/>in motu progreditur ultra B, quanta e&longs;t longitudo funis ex­<lb/>plicati OBDH, quæ longitudo dupla e&longs;t intervalli OB: <pb pagenum="581" xlink:href="017/01/597.jpg"/>igitur potentia L accedens ad B &longs;emel percurrit interval­<lb/>lum OB, & præterea adhuc duplum &longs;patium ultrà B, <lb/>dum punctum O venit ad B &longs;imul cum pondere ad­<lb/>nexo in G: triplo igitur velociùs movetur Potentia quàm <lb/>Pondus. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004327">Simili omnino ratione ac de Trochleis &longs;implicibus phi­<lb/>lo&longs;ophamur, etiam ratiocinari oportet in Trochleis plures <lb/>orbiculos habentibus; &longs;i enim &longs;ingulæ duos habeant orbi­<lb/>culos, attendendum e&longs;t, an funis extremitas adnectatur <lb/>Trochleæ immotæ, an verò mobili: &longs;i immotæ, potentia <lb/>movetur quadruplo velociùs quàm pondus; &longs;in autem mo­<lb/>bili, movetur quintuplo velociùs. </s> <s id="s.004328">Generatim igitur nume­<lb/>ra orbiculos trochleæ mobilis, cui &longs;cilicet jungitur pondus, <lb/>& pro &longs;ingulis orbiculis duplica potentiæ momenta. </s> <s id="s.004329">Hinc &longs;i <lb/>tres fuerint orbiculi, momentum Potentiæ e&longs;t &longs;extuplum; &longs;i <lb/>quatuor, octuplum; & &longs;ic deinceps. </s> <s id="s.004330">At &longs;i eidem Trochleæ <lb/>mobili adnectatur extremitas funis, adhuc adde unitatem, <lb/>& momentum erit &longs;eptuplum, aut noncuplum. </s> <s id="s.004331">Funis &longs;iqui­<lb/>dem uni trochleæ alligatus primùm in&longs;i&longs;tit orbiculo primo <lb/>reliquæ trochleæ; inde flectitur ad orbiculum primum tro­<lb/>chleæ, cui adnectitur: po&longs;tmodum ad &longs;ecundum orbicu­<lb/>lum alterius trochleæ tran&longs;it, & rediens ad priorem tro­<lb/>chleam in&longs;i&longs;tit orbiculo ejus &longs;ecundo; atque ita deinceps, al­<lb/>terno ex trochleâ in trochleam excur&longs;u, donec orbiculis om­<lb/>nibus in&longs;i&longs;tat. </s> <s id="s.004332">Quod &longs;i duabus Trochleis non in&longs;it æqualis <lb/>orbiculorum numerus, &longs;ed altera alteram unitate &longs;uperet, <lb/>nece&longs;&longs;e e&longs;t funem alligari trochleæ pauciorum orbiculorum <lb/>Quare attendendus pariter e&longs;t numerus orbiculorum trochleæ <lb/>mobilis, quæ &longs;i pauciores habeat orbiculos, utique illi ad­<lb/>nectitur extremitas funis; atque adeò duplicato ejus orbi­<lb/>culorum numero addenda e&longs;t unitas: ut, &longs;i duos habeat or­<lb/>biculos, motus Potentiæ e&longs;t quintuplus motus Ponderis. <!-- KEEP S--></s> <s id="s.004333">At <lb/>&longs;i trochlea mobilis plures habeat orbiculos quàm trochlea im­<lb/>mota, duplicandus &longs;olùm e&longs;t illorum numerus, ut habeatur <lb/>denominatio momenti; ut, &longs;i tres fuerint orbiculi, motus po­<lb/>tentiæ ad ponderis motum e&longs;t &longs;extuplus. </s> </p> <p type="main"> <s id="s.004334">In huju&longs;modi Trochleis plures rotulas habentibus ob&longs;er­<lb/>vandum e&longs;t interiores rotulas minores &longs;tatui, exteriores verò <pb pagenum="582" xlink:href="017/01/598.jpg"/>majores: nam A & C minores &longs;unt, B & D majores, ne fu­<lb/>nium ductus &longs;e invicem intercipiant, ac mo­<lb/><figure id="id.017.01.598.1.jpg" xlink:href="017/01/598/1.jpg"/><lb/>tum mutuo tritu retardent, ni&longs;i etiam &longs;e&longs;e vici&longs;­<lb/>&longs;im atterentes funes di&longs;rumpantur. </s> <s id="s.004335">Quare pro­<lb/>bare non po&longs;&longs;um Trochleas, quæ plures orbi­<lb/>culos parallelos uni & eidem axi infixos intra <lb/>congruum loculamentum habent; quamvis <lb/>enim Trochleis huju&longs;modi valde inter &longs;e di&longs;tan­<lb/>tibus non adeò appareat incommodum funium <lb/>&longs;e&longs;e perfricantium, ubi tamen illæ propiores <lb/>factæ fuerint, hoc manife&longs;tò apparet: præ­<lb/>terquam quod funis obliquè in&longs;i&longs;tens extremæ <lb/>ip&longs;arum rotularum orbitæ, quam contingit, <lb/>non adeò facilè movetur, ac &longs;i illis exactè con­<lb/>grueret, ut fit, quando &longs;ingulæ rotulæ &longs;uos ha­<lb/>bent axes. </s> </p> <p type="main"> <s id="s.004336">Et quidem quod ad axes rotularum &longs;pectat, <lb/>quamvis nec admodum longi &longs;int, & rotula <lb/>&longs;uo loculamento proximè adhæreat, atque adeò <lb/>non &longs;int facilè obnoxij fractionis periculo, ca­<lb/>vendum tamen e&longs;t, ne nimis exiles &longs;int, aut <lb/>ex materia non &longs;atis &longs;olidâ; ne fortè ponderis <lb/>attollendi gravitas illos labefactet. </s> <s id="s.004337">Verum qui­<lb/>dem e&longs;t non e&longs;&longs;e nece&longs;&longs;e &longs;ingulos axes &longs;tatuere <lb/>&longs;u&longs;tinendo oneri pares; cum enim plures &longs;int, adver&longs;us &longs;ingu­<lb/>los minor conatus ponderis exercetur. </s> <s id="s.004338">Si verò illi exqui&longs;itè <lb/>læves atque politi fuerint, faciliorem fore rotularum iis infixa­<lb/>rum revolutionem apertiùs con&longs;tat, quàm ut moneri artificem <lb/>oporteat. </s> </p> <p type="main"> <s id="s.004339">Prætereà rotularum facies optimè lævigatas velim, & lo­<lb/>culamentum ip&longs;um non placet ita amplum, ut maximam ro­<lb/>tularum partem includat: &longs;atis e&longs;t, &longs;i ità firmum ac &longs;olidum <lb/>&longs;it, ut axes contineat, & in extremitatibus validos uncos ha­<lb/>beat, quibus & funis, & onus alligari queant: Quò &longs;cilicet <lb/>minorem rotularum partem tangit, minus cum illis confligit, <lb/>adeóque facilior e&longs;t motus: neque enim leviora hæc compen­<lb/>dia omninò contemnenda &longs;unt. </s> </p> <p type="main"> <s id="s.004340">Demum funis ductarij cra&longs;&longs;itudo &longs;tatuenda e&longs;t, quæ reti-<pb pagenum="583" xlink:href="017/01/599.jpg"/>nendo ponderi re&longs;pondeat: &longs;ed quia plures &longs;unt funis à trochleâ <lb/>in trochleam ductus, ideò qua&longs;i plures funes reputantur, inter <lb/>quos quodammodo di&longs;tribuitur &longs;u&longs;tentatio ponderis, perinde <lb/>ferè, atque &longs;i ex pluribus illis ductibus funis unicus compone­<lb/>retur. </s> <s id="s.004341">Hinc &longs;i pondus fuerit adnexum trochleæ I, &longs;u&longs;tinetur à <lb/>quatuor funibus; &longs;in autem trochlea I in &longs;uperiore loco firmata <lb/>fuerit, & pondus trochleæ H alligatum dependeat, &longs;u&longs;tinetur <lb/>à quinque funibus, nam etiam Potentia in O &longs;u&longs;tinet fune RO. <!-- KEEP S--></s> <lb/> <s id="s.004342">Ex funis autem cra&longs;&longs;itudine definitur rotularum altitudo, ut ni­<lb/>mirum orbitæ excavatæ in&longs;i&longs;tere po&longs;&longs;it funis, quin interiorem <lb/>loculamenti faciem contingat, ne perpetuo affrictu atteratur <lb/>cum di&longs;ruptionis periculo, & non levi celeritatis detrimento, <lb/>auctâ trahendi difficultate. </s> <s id="s.004343">Porrò cùm excavatam dico rotu­<lb/>larum orbitam, nolim intelligas qua&longs;i crenam perimetro pro­<lb/>fundiùs inci&longs;am; &longs;ed &longs;atius fuerit orbitam ip&longs;am e&longs;&longs;e modicè <lb/>&longs;inuatam; hoc enim pacto facilius excurrit funis, etiam&longs;i paulò <lb/>cra&longs;&longs;ior aliquando adhibendus &longs;it, qui cæteroqui inter crenæ <lb/>inci&longs;æ labra depre&longs;&longs;us non &longs;ine labore ex illis angu&longs;tiis eximere­<lb/>tur in rotulæ conver&longs;ione. </s> </p> <p type="main"> <s id="s.004344">Cum itaque ea &longs;it Trochlearum di&longs;po&longs;itio, ut pondus tardiùs <lb/>moveatur, potentia velocius (&longs;i videlicet alteri Trochlearum <lb/>non Potentia, &longs;ed Pondus adnectatur, alioquin &longs;i loca permu­<lb/>tarent, res contrario prorsùs modo &longs;e haberet) manife&longs;tum e&longs;t <lb/>re&longs;i&longs;tentiam ponderis minui ex tarditate; poterit igitur augeri <lb/>ex gravitate: &longs;æpiùs quippe dictum e&longs;t adæquatum re&longs;i&longs;tentiæ <lb/>momentum componi ex in&longs;ita gravitate, & ex di&longs;po&longs;itione ad <lb/>motûs velocitatem, aut tarditatem. </s> <s id="s.004345">Potentia igitur valens &longs;u­<lb/>perare re&longs;i&longs;tentiam ponderis alicujus certæ gravitatis, &longs;i cum <lb/>illa æqualiter movendum &longs;it, poterit eodem impetu, atque co­<lb/>natu &longs;uperare re&longs;i&longs;tentiam majoris ponderis, &longs;i ex collocatione, <lb/>quatenus cum Potentiâ connectitur, ita minus velociter movea­<lb/>tur, ut quæ Ratio e&longs;t æqualis illius velocitatis ad minorem ve­<lb/>locitatem, eadem &longs;it Ratio majoris ponderis ad pondus illud <lb/>æquè velox cum potentiâ; e&longs;t enim omnino par re&longs;i&longs;tentia; <lb/>quia quantum addit major velocitas minori ponderi, tantum­<lb/>dem addit majus pondus minori velocitati. </s> </p> <p type="main"> <s id="s.004346">Quamvis autem ponderis motus non &longs;it æquè velox ac motus <lb/>potentiæ, tamen ponderis motus entitativè acceptus æqualis e&longs;t <pb pagenum="584" xlink:href="017/01/600.jpg"/>motui potentiæ, ac proindè mirum non e&longs;t, &longs;i potentia eadem <lb/>impetu eodem æqualem motum producat, atque efficiat. </s> <s id="s.004347">Pone <lb/>enim in O gravitatem paulò majorem libris 100; utique &longs;i in S <lb/>&longs;tatuerentur libræ 100 gravitas O prævaleret, & gravitatem S <lb/>elevaret: igitur illa eadem gravitas O elevabit libras 400 in I <lb/>adnexas Trochleæ, nam I movetur quadruplo tardiùs quam <lb/>O, ex dictis, S autem movetur æqualiter ac O; ergo ratione <lb/>motûs tardioris quadruplo minùs re&longs;i&longs;tit pondus lib.400 in I, <lb/>licet ratione gravitatis quadruplo magis re&longs;i&longs;tat. </s> <s id="s.004348">Si itaque li­<lb/>bræ 100 in S intra certum tempus percurrant unà cum Poten­<lb/>tia O &longs;patij pedes 40, eodem tempore &longs;ingulæ libræ 100 gra­<lb/>vitatis in I adnexæ percurrunt pedes 10: at &longs;unt libræ 400; <lb/>igitur &longs;unt quatuor motus pedum 10, & illarum omnium mo­<lb/>tus e&longs;t pedum 40. Quare potentia O idem planè efficit, ac &longs;i <lb/>moveret in S libras 100; id quod præ&longs;tare pote&longs;t ab&longs;que ulla <lb/>machina. </s> <s id="s.004349">Et quidem &longs;i res attentè perpendatur, nec vulga­<lb/>ribus vocabulis notionem minus propriam &longs;ubjiciamus, non <lb/>e&longs;t dicendum manente eodem conatu, & eadem velocitate Po­<lb/>tentiæ augeri per Machinam potentiæ momenta, aut vires, <lb/>&longs;emper enim Potentia vincit æqualem re&longs;i&longs;tentiam &longs;ive adhibi­<lb/>tâ machinâ, &longs;ive ab&longs;que illâ, quamvis non &longs;emper vincat ean­<lb/>dem gravitatem. </s> <s id="s.004350">Quemadmodum in libra nil refert, utrum <lb/>corpus expendendum habeat majorem gravitatem &longs;ecundum <lb/>&longs;peciem, &longs;ed molem minorem, an verò minorem gravitatem <lb/>&longs;pecificam &longs;ub mole majori, modò reciprocè &longs;it ut gravitas <lb/>&longs;pecifica ad &longs;pecificam gravitatem, ita moles ad molem; e&longs;t &longs;i­<lb/>quidem par gravitas ab&longs;oluta, quæ componitur ex gravitate &longs;pe­<lb/>cificâ & mole. </s> <s id="s.004351">Ita pariter æqualis e&longs;t ab&longs;oluta ponderis re­<lb/>&longs;i&longs;tentia, quæ ex gravitate, & velocitate componitur, &longs;i fuerit <lb/>inter eas reciproca Ratio. <pb pagenum="585" xlink:href="017/01/601.jpg"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004352"><emph type="center"/>CAPUT II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004353"><emph type="center"/><emph type="italics"/>An Trochlea ad Vectem revocanda &longs;it.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004354">UT Machinalis motûs cau&longs;a meliùs innote&longs;cat, neque opus <lb/>e&longs;&longs;e Facultates omnes ad Vectem revocare, ut non pauci <lb/>hactenus conati &longs;unt, & adhuc conantur, hìc poti&longs;&longs;imùm <lb/>quæ&longs;tionem huju&longs;modi examinare placuit in Trochleâ. </s> <s id="s.004355">Aiunt <lb/>&longs;iquidem in &longs;implici orbiculo, quando ejus centrum immotum <lb/>manet, & alteram funis extremitatem potentia apprehendit, ex <lb/>alterâ dependet pondus, Vectem e&longs;&longs;e primi generis, cujus hy­<lb/>pomochlium e&longs;t in centro orbiculi, potentia & pondus in ex­<lb/>tremitatibus diametri; quæ cùm à centro æqualibus intervallis <lb/>ab&longs;int, vectis ille nil juvat potentiam. </s> <s id="s.004356">Quando verò ponderi <lb/>adnectitur theca, cui orbiculus includitur, adeóque ejus cen­<lb/>trum unà cum pondere movetur, jam pondus re&longs;pondet orbicu­<lb/>li centro, & extremitatem alteram diametri obtinet potentia <lb/>trahens funem; quapropter hypomochlium cen&longs;endum e&longs;t in <lb/>oppo&longs;ita diametri extremitate. </s> <s id="s.004357">Quapropter cùm pondus &longs;it in­<lb/>ter potentiam, & hypomochlium, vectis e&longs;t &longs;ecundi generis: & <lb/>quia pondus e&longs;t in vectis medio, potentiæ momentum duplum <lb/>e&longs;t momenti ponderis, &longs;i po&longs;itio ip&longs;a &longs;pectetur. </s> <s id="s.004358">Sit orbiculus, <lb/>cujus centrum C, eju&longs;que loculamento <lb/>adnexum pondus re&longs;pondeat lineæ CB: <lb/><figure id="id.017.01.601.1.jpg" xlink:href="017/01/601/1.jpg"/><lb/>funis RSDTV &longs;it alligatus in R, & Poten­<lb/>tia &longs;it in V, quæ funem trahens intelligitur <lb/>con&longs;tituta in T, & oppo&longs;itum diametri <lb/>punctum S cen&longs;etur hypomochlium; at­<lb/>que adeò momentum Potentiæ ad momen­<lb/>tum Ponderis e&longs;t ut TS ad CS. <!-- KEEP S--></s> <s id="s.004359">Ex quo fit, <lb/>&longs;i reciprocè vis potentiæ ad gravitatem <lb/>ponderis &longs;it ut CS ad TS, ab huju&longs;modi <lb/>potentiâ &longs;u&longs;tineri pondus, & potentia &longs;i augeatur, etiam mo­<lb/>veri, orbiculo circà &longs;uum centrum revoluto, & ver&longs;us poten-<pb pagenum="586" xlink:href="017/01/602.jpg"/>tiam attracto. </s> <s id="s.004360">In conver&longs;ione autem orbiculi, prout aliæ atque <lb/>aliæ &longs;unt diametri, quas contingunt funis ductus RS, & VT, <lb/>alios &longs;ubinde, atque alios vectes e&longs;&longs;e commini&longs;cuntur. </s> </p> <p type="main"> <s id="s.004361">Verùm huju&longs;modi ratiocinationi nunquam aquie&longs;cere po­<lb/>tui; mihi enim per&longs;pectum e&longs;t, &longs;i orbiculus non fuerit ver&longs;atilis, <lb/>&longs;ed omnino fixus in &longs;uo loculamento, adhuc potentiam V faci­<lb/>liùs attollere pondus, quod in B intelligitur &longs;u&longs;pen&longs;um, quàm <lb/>illud directè, & immediatè attolleret; & tamen diameter ea­<lb/>dem TS &longs;emper maneret horizonti parallela (nam CB &longs;emper <lb/>e&longs;t in perpendiculo) nullumque haberet motum conver&longs;ionis <lb/>circà punctum, quod vocant, hypomochlij S, quo referret mo­<lb/>tum Vectis proprium. </s> <s id="s.004362">Adde orbiculum in &longs;uo loculamento <lb/>fixum perinde e&longs;&longs;e, atque &longs;i annulus ponderi adnectatur, & fu­<lb/>nis alligatus in R in&longs;eratur annulo, atque potentia in V funem <lb/>trahat; potentia enim duplo velociùs movetur, quàm annulus <lb/>& pondus: hìc autem in annulo, quem nullatenus convolvi cer­<lb/>tum e&longs;t, quomodo Vectis ve&longs;tigium deprehendes? </s> <s id="s.004363">Illud qui­<lb/>dem incommodi in annulo, & in orbiculo non ver&longs;atili, accide­<lb/>ret, quod funis ob &longs;uam a&longs;peritatem cum orbiculi orbitá, & cum <lb/>annulo confligeret; ex quo tritu non levis movendi difficultas <lb/>oriretur: propterea, ad vitandum huju&longs;modi incommodum <lb/>adhibentur orbiculi circa &longs;uum axem ver&longs;atiles; axis enim poli­<lb/>tus, aut etiam addito unguine lubricus, ferè nullam creat orbi­<lb/>culi rotationi difficultatem, funis verò non atterit eju&longs;dem or­<lb/>biculi orbitam, quâ revolutâ ille explicatur. </s> <s id="s.004364">Cæterum quod ad <lb/>Rationem motuum potentiæ & ponderis &longs;pectat, eadem e&longs;t Ra­<lb/>tio dupla, &longs;ive orbiculus ver&longs;atilis &longs;it, &longs;ive fixus, &longs;ive annulus <lb/>ponderi adnectatur, &longs;ive etiam ponderi in&longs;eratur funis, ità ut <lb/>pondus ip&longs;um excurrere queat. </s> <s id="s.004365">Hoc &longs;cilicet unicè pendet ex <lb/><figure id="id.017.01.602.1.jpg" xlink:href="017/01/602/1.jpg"/><lb/>ipsâ funis inflexione: nam &longs;i funis AB ita flectatur, <lb/>ut ad extremitatem extremitas accedat, & B veniat <lb/>in C propè A; utique non ni&longs;i media pars BE mo­<lb/>vetur; adeò ut, &longs;i annulus in&longs;eratur funi in B, & per <lb/>longitudinem funis, qui complicatur, excurrat, ve­<lb/>niat ex B in E interea, dum extremitas B, & poten­<lb/>tiam illam adducens, venit in C: quo in motu &longs;in­<lb/>gulæ funis particulæ inter B & E percurrunt &longs;pa­<lb/>tium duplum di&longs;tantiæ &longs;ingularum à medio, ante-<pb pagenum="587" xlink:href="017/01/603.jpg"/>quam complicarentur: & &longs;i potentia ex C ulteriùs progredia­<lb/>tur, &longs;ingulæ funis particulæ inter medium & caput A inter­<lb/>ceptæ perficiunt &longs;patium duplum di&longs;tantiæ &longs;ingularum à capi­<lb/>te A, ubi funis religatur. </s> </p> <p type="main"> <s id="s.004366">Jam verò &longs;tatue ampliorem aliquem, & &longs;atis gravem cylin­<lb/>drum DEFG, qui rotatu pro­<lb/><figure id="id.017.01.603.1.jpg" xlink:href="017/01/603/1.jpg"/><lb/>movendus &longs;it, aut in plano hori­<lb/>zontali, aut in &longs;uperiorem plani <lb/>inclinati locum: applicentur au­<lb/>tem homines in D & G, & quot­<lb/>quot nece&longs;&longs;arij fuerint juxta cylin­<lb/>dri longitudinem, qui illum im­<lb/>pellant. </s> <s id="s.004367">Quæro, an ibi ulla Vectis <lb/>ratio intercedat, ita ut &longs;it qua&longs;i <lb/>vectis DE, hypomochlium in E, <lb/>& pondus in puncto I, quod <lb/>re&longs;pondet centro gravitatis, at­<lb/>què adeò in cylindri conver&longs;ione <lb/>&longs;ubinde mutetur vectis, & locus <lb/>tùm potentiæ, tùm hypomochlij, <lb/>prout aliis atque aliis perimetri punctis applicatur potentia im­<lb/>pellens, quibus ex diametro opponuntur alia atque alia puncta, <lb/>in quibus à &longs;ubjecto plano cylinder tangitur. </s> <s id="s.004368">Vix, puto, au­<lb/>debis Vectem ibi agno&longs;cere, ubi demum Potentiam impellen­<lb/>tem, & Pondus, quod in centro gravitatis, &longs;cilicet in Axe cy­<lb/>lindri, con&longs;titutum intelligitur, æqualem motus lineam per­<lb/>curri&longs;&longs;e deprehenderis, ut manife&longs;tum e&longs;t in huju&longs;modi rotun­<lb/>dorum corporum revolutione, in qua æqualem lineam percur­<lb/>runt centrum, & punctum in peripheriâ notatum. </s> <s id="s.004369">Igitur duo­<lb/>rum funium capita firmiter alliga in M & H, ipsó&longs;que funes <lb/>cylindro &longs;ubjice, & in &longs;uperiorem partem reductos ita di&longs;po­<lb/>ne, ut cylindrum complectantur, atque à duabus potentiis, quæ <lb/>priùs in D & G impellebant, trahantur capita L & P. <!-- KEEP S--></s> <s id="s.004370">Certi&longs;&longs;i­<lb/>mo con&longs;tat experimento longè faciliùs cylindrum huju&longs;modi <lb/>funibus convolvi, quàm impul&longs;ione potentiarum illi proximè <lb/>applicitarum. </s> <s id="s.004371">Si nulla Vectis Ratio agno&longs;cenda e&longs;t in diame­<lb/>tro DE, utique facilitas illa movendi non habetur à vecte, qui <lb/>nullus e&longs;t: Sin autem Vectem ibi e&longs;&longs;e con&longs;tanter affirmes, igi-<pb pagenum="588" xlink:href="017/01/604.jpg"/>tur perindè e&longs;t &longs;i Potentia proximè, & immediatè applicetur <lb/>puncto D, aut H, ad impellendum, atque &longs;i medio fune MHL <lb/>applicetur puncto H trahens funis caput L: atqui longè majo­<lb/>ra momenta habet funem LH trahens, quàm impellens in H; <lb/>cum igitur utrobique idem Vectis; eadem &longs;cilicet cylindri dia­<lb/>meter, habeatur, &longs;ed non idem momentum, non ex rationibus <lb/>Vectis, &longs;ed aliundè petenda e&longs;t hæc momenti acce&longs;&longs;io: Quia <lb/>videlicet fune &longs;ic di&longs;po&longs;ito, potentia duplo velociùs movetur <lb/>quàm pondus, nullâ habitâ vectis ratione. </s> <s id="s.004372">Finge jam funem <lb/>laxiorem circumplecti cylindrum, & in nodum colligi in X: <lb/>utique &longs;i in X adderetur pondus aliquod raptandum unà cum <lb/>cylindro promoto; facilius raptaretur cylindro huju&longs;modi fu­<lb/>nibus revoluto, quàm &longs;i cylindrus impul&longs;ione potentiæ proxi­<lb/>mè applicatæ promoveretur; & tamen major hæc facilitas ex <lb/>nullo vecte addito oriretur. </s> <s id="s.004373">An non ergo cylindrus trochleæ <lb/>orbiculum refert, & funis X orbiculi loculamentum, cui pon­<lb/>dus adnectitur? </s> <s id="s.004374">manife&longs;to igitur experimento habetur non ex <lb/>Vectis rationibus ducendam e&longs;&longs;e majorem movendi facilitatem, <lb/>quæ ex &longs;implici trochleâ habetur, quando illi adnectitur <lb/>pondus. </s> </p> <p type="main"> <s id="s.004375">Sed præ&longs;tat examinare, quæ præterea dicuntur, quando ei­<lb/>dem &longs;implici Trochleæ, cui pondus M adnectitur, etiam funis <lb/>caput alligatur; tunc enim potentiæ momentum triplex e&longs;t, <lb/>adeò ut ad attollendum pondus M &longs;ufficiat potentia &longs;ubtripla <lb/><figure id="id.017.01.604.1.jpg" xlink:href="017/01/604/1.jpg"/><lb/>illius potentiæ, quæ ab&longs;que machinâ attol­<lb/>leret idem pondus. </s> <s id="s.004376">Sic igitur ratiocinantur <lb/>apud P. <!-- REMOVE S-->Schott in Magia mechanica Syn­<lb/>tagm. <!-- KEEP S-->4. cap. 2. prop.5. Si fuerit Vectis DE, <lb/>in cujus medio C &longs;it pondus, fuerit autem <lb/>quædam potentia in C &longs;u&longs;tinens, & alia po­<lb/>tentia illi æqualis &longs;u&longs;tinens in E, hypomo­<lb/>chlium verò in D, unaquæque potentia e&longs;t <lb/>&longs;ubtripla ponderis &longs;u&longs;tentati. </s> <s id="s.004378">Quia enim <lb/>potentia C di&longs;tat ab hypomochlio D æqua­<lb/>liter ac pondus in C con&longs;titutum, &longs;u&longs;tinet <lb/>pondus æquale &longs;uis viribus; potentia autem <lb/>E, quia e&longs;t in duplo majore di&longs;tantiâ quàm <lb/>Pondus C, &longs;u&longs;tinet pondus duplum &longs;uarum <pb pagenum="589" xlink:href="017/01/605.jpg"/>virium. </s> <s id="s.004379">Quoniam ergo Potentiæ ex hypothe&longs;i &longs;unt æquale, <lb/>& totius ponderis duæ partes &longs;u&longs;tinentur à Potentia E, & una <lb/>à Potentia C, illa autem e&longs;t &longs;ubdupla ponderis à &longs;e &longs;u&longs;tentati, <lb/>unaquæque e&longs;t eju&longs;dem totius ponderis &longs;ubtripla quo ad vires <lb/>&longs;u&longs;tentandi. </s> <s id="s.004380">Cum igitur in propo&longs;itis Trochleis &longs;it potentia F <lb/>&longs;u&longs;tinens in medio, & potentia G in alterâ extremitate &longs;u&longs;ti­<lb/>nens, unaquæque e&longs;t &longs;ubtripla ponderis M &longs;u&longs;tinendi, ac <lb/>propterea Potentia G &longs;i &longs;it paulo major quàm &longs;ubtripla, erit <lb/>etiam apta ad movendum pondus. </s> </p> <p type="main"> <s id="s.004381">His pariter a&longs;&longs;entiri nequeo, quæ de ponderis &longs;u&longs;tentatione <lb/>dicuntur; nec &longs;atis video, an Vecti &longs;ecundi generis congruant; <lb/>neque enim &longs;olùm Potentiæ in medio atque in altera extremita­<lb/>te applicatæ, verùm etiam hypomochlium ip&longs;um exercet vim <lb/>&longs;u&longs;tinendi: ex hoc &longs;iquidem quod addatur potentia in medio, <lb/>ubi e&longs;t pondus, non tollitur omnino pre&longs;&longs;io, quâ hypomo­<lb/>chlium à pondere urgetur. </s> <s id="s.004382">Quare non tota vis &longs;u&longs;tentandi di­<lb/>videnda e&longs;t inter duas illas potentias, &longs;ed etiam admittendum <lb/>e&longs;t hypomochlij con&longs;ortium. </s> </p> <p type="main"> <s id="s.004383">Dic autem, quænam e&longs;t potentia in F retinens pondus? </s> <lb/> <s id="s.004384">nonne &longs;tatim ac potentia in G remi&longs;&longs;iorem conatum adhibet, <lb/>etiam F cum pondere de&longs;cendit? </s> <s id="s.004385">ip&longs;a quippe Potentia G dum <lb/>intentum funem FI retinet, &longs;u&longs;tinet etiam pondus; atque adeò <lb/>non duæ &longs;unt potentiæ &longs;u&longs;tinentes, &longs;ed unica. </s> <s id="s.004386">Et quidem, &longs;i <lb/>res &longs;incerè exponatur, pondus &longs;u&longs;tinetur & à potentiâ G &longs;ur­<lb/>&longs;um conante, & à clavo S, cui &longs;uperior trochlea adnectitur, <lb/>mediis funibus HD, IF retinente, ità ut centrum gravitatis <lb/>ponderis &longs;it in lineâ Directionis tran&longs;eunte per ip&longs;um clavum S, <lb/>&longs;i funis GE &longs;it ad perpendiculum, nec in latus retrahat tro­<lb/>chleam C: eo autem ip&longs;o, quòd Potentia G &longs;uo conatu prohi­<lb/>bet, ne funis excurrat, retinet pondus ex eodem clavo S &longs;u&longs;­<lb/>pen&longs;um. </s> <s id="s.004387">Quapropter eju&longs;dem potentiæ G e&longs;t vis illa, quæ & <lb/>in F, hoc e&longs;t in C & in E retinet. </s> <s id="s.004388">Quandò verò &longs;ur&longs;um attolli­<lb/>tur pondus, eadem e&longs;t potentia G, quæ &longs;ur&longs;um trahit F, cui <lb/>non minùs applicatur medio fune IF, quàm applicetur ip&longs;i E <lb/>medio fune GE; neque enim in F e&longs;t alia potentia &longs;ponte &longs;ur­<lb/>&longs;um a&longs;cendens, & &longs;ecum rapiens pondus. </s> </p> <p type="main"> <s id="s.004389">Sed quid fru&longs;trà confugiamus ad vim &longs;u&longs;tentandi pondus ex <lb/>trochleis dependens? </s> <s id="s.004390">&longs;i pondus fuerit in plano horizontali tra-<pb pagenum="590" xlink:href="017/01/606.jpg"/>hendum, nihil in trochleis reperitur, à quo &longs;u&longs;tineatur pon­<lb/>dus omnino incumbens &longs;ubjecto plano, & tamen potentia G <lb/>e&longs;t &longs;ubtripla potentiæ, quæ &longs;ine machinâ in eodem plano trahe­<lb/>ret idem pondus: ratione vectis ED &longs;olùm e&longs;&longs;e pote&longs;t &longs;ubdu­<lb/>pla; in F nulla e&longs;t potentia trahens; unde ergo ratione vectis <lb/>potentia ad trahendum pondus habet momenti incrementum? </s> <lb/> <s id="s.004391">Quod &longs;i dixeris eandem potentiam, quæ in G trahit, etiam tra­<lb/>here in F; igitur conatum non adhibet &longs;ubtriplum, &longs;ed &longs;ub&longs;e&longs;­<lb/>quialterum; nam conatur & in extremitate E, & in vectis me­<lb/>dio C, ut tu quidem ais, ita ut utrobique &longs;it &longs;ubtripla vis mo­<lb/>vendi: fatendum e&longs;t ergo potentiam trahentem conari ut (2/33) <lb/>cum tamen reip&longs;a adhibeat &longs;olum conatum ut 1/3. </s> </p> <p type="main"> <s id="s.004392">Con&longs;ideremus demum Trochleas pluribus in &longs;tructas orbicu­<lb/>lis, & videamus, quid ex Vecte &longs;perari po&longs;&longs;it. </s> <s id="s.004393">Statuunt Autho­<lb/>res cum eodem P. <!-- REMOVE S-->Schott ibid. </s> <s id="s.004394">prop.7. &longs;i fuerint duo vectes <lb/>BA, & DC, ex quorum me­<lb/><figure id="id.017.01.606.1.jpg" xlink:href="017/01/606/1.jpg"/><lb/>dio E & F dependeat pondus G, <lb/>duas potentias æquales in B & <lb/>D con&longs;titutas, &longs;imúlque æqua­<lb/>liter in &longs;u&longs;tinendo pondere la­<lb/>borantes, &longs;ingulas e&longs;&longs;e &longs;ubqua­<lb/>druplas ponderis. </s> <s id="s.004395">Nam &longs;i &longs;ola <lb/>potentia D &longs;u&longs;tineret, e&longs;&longs;et pon­<lb/>deris &longs;ubdupla, &longs;cilicet ut FC <lb/>ad DC; & &longs;i &longs;ola potentia B <lb/>&longs;u&longs;tineret, e&longs;&longs;et ip&longs;a pariter &longs;ub­<lb/>dupla, nimirum ut EA ad BA. <!-- KEEP S--></s> <lb/> <s id="s.004396">Cum igitur ambæ æquales &longs;int, <lb/>& æqualiter conentur, unicui­<lb/>que re&longs;pondebit &longs;ubduplum <lb/>&longs;ubdupli, hoc e&longs;t quarta pars <lb/>ponderis. </s> <s id="s.004397">Atqui in Trochleis <lb/>binos orbiculos habentibus &longs;unt duo vectes HI, & PO in me­<lb/>dio &longs;u&longs;tinentes pondus, hypomochlia in I & O, atque Po­<lb/>tentiæ in H & P. <!-- KEEP S--></s> <s id="s.004398">Igitur potentia &longs;u&longs;tinens e&longs;t ponderis &longs;ub­<lb/>quadrupla, & movens paulò major &longs;ubquadruplâ. </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"/>eatenus admitto, quatenus &longs;ingulas potentias D & B &longs;u&longs;tinen­<lb/>tes &longs;ubquadruplas e&longs;&longs;e ponderis definiunt; nam perinde &longs;e ha­<lb/>bent, atque &longs;i utraque potentia in unius, eju&longs;démque vectis ex­<lb/>tremitate &longs;imul &longs;u&longs;tinerent, unicámque potentiam, con&longs;titue­<lb/>rent, quæ &longs;ubdupla e&longs;t ponderis: & quia &longs;ingulæ potentiæ &longs;unt <lb/>ad totam & integram potentiam &longs;ubduplæ, &longs;ingulæ &longs;unt ponde­<lb/>ris &longs;ubquadruplæ. </s> <s id="s.004400">Cæterum ex hoc quod ambæ potentiæ æqua­<lb/>les &longs;int, & &longs;ingulæ &longs;olitariæ e&longs;&longs;ent &longs;ubduplæ arguere, quod uni­<lb/>cuique re&longs;pondeat &longs;ubduplum &longs;ubdupli, materialiter quidem <lb/>verum e&longs;t, non autem formaliter ex modo argumentandi; alio­<lb/>quin &longs;i addatur tertius vectis, &longs;ervatâ eadem argumentandi for­<lb/>mâ, tres e&longs;&longs;ent potentiæ, & unicuique re&longs;pondebit &longs;ubduplum <lb/>&longs;ubdupli, hoc e&longs;t octava pars ponderis; id quod e&longs;t fal&longs;um. </s> <lb/> <s id="s.004401">Neque enim ex hoc quod potentia D &longs;u&longs;tineat pondus in F, <lb/>facit illud e&longs;&longs;e minùs grave, qua&longs;i transferatur in E factum <lb/>gravitatis &longs;ubduplæ, & potentia B &longs;ubduplam gravitatem pon­<lb/>deris &longs;u&longs;tineret &longs;ubduplo conatu, hoc e&longs;t &longs;ubquadruplo ejus, <lb/>qui requiritur ad &longs;u&longs;tinendum totum pondus; alioquin addito <lb/>tertio vecte in illius medium transferretur gravitas &longs;ubquadru­<lb/>pla ponderis, quæ &longs;u&longs;tineretur à potentiâ illius &longs;ubduplâ, ac <lb/>proinde &longs;uboctuplâ totius ponderis; cum tamen in tribus vecti­<lb/>bus &longs;ic di&longs;po&longs;itis tres potentiæ &longs;u&longs;tinentes &longs;ingulæ &longs;int &longs;olum <lb/>&longs;ub&longs;extuplæ. </s> <s id="s.004402">Quod &longs;i pondus alligetur medio primi vectis in F, <lb/>tum extremitas D alligetur medio &longs;ecundi vectis in E, & dein­<lb/>ceps extremitas B alligetur medio tertij vectis, optimè con­<lb/>cluditur potentiam in F &longs;u&longs;tinere &longs;ubduplum &longs;ubdupli, & po­<lb/>tentiam applicatam tertio vecti &longs;u&longs;tinere &longs;ubduplum &longs;ubdupli <lb/>&longs;ubdupli, ac proinde illam e&longs;&longs;e &longs;ubquadruplam, hanc verò &longs;ub­<lb/>octuplam. </s> <s id="s.004403">Sed hæc di&longs;po&longs;itio nil juvaret ad explicandum Tro­<lb/>chlearum momentum. </s> </p> <p type="main"> <s id="s.004404">Verùm in Trochleâ duas illas potentias in H & P non vi­<lb/>deo; nam unica potentia in X medio fune XSH applicatur <lb/>quidem puncto H, &longs;uóque conatu prohibet ne pondus &longs;uâ gra­<lb/>vitate deor&longs;um trahat ip&longs;am Trochleam: at in P quænam alia <lb/>Potentia hoc idem efficit? </s> <s id="s.004405">An non eadem Potentia X medio <lb/>fune XSHILMP applicatur vecti PO in P? igitur eadem <lb/>potentia exhibet conatum duarum potentiarum &longs;ubquadrupla­<lb/>rum: igitur Potentia non e&longs;t &longs;ubquadrupla, &longs;ed &longs;olum &longs;ubdu-<pb pagenum="592" xlink:href="017/01/608.jpg"/>pla; quemadmodum &longs;i duos &longs;imul vectes in D & B idem &longs;u&longs;ti­<lb/>neret, utique tantumdem virium impenderet in utroque &longs;imul <lb/>&longs;u&longs;tinendo, quantum &longs;i unicus e&longs;&longs;et vectis. </s> </p> <p type="main"> <s id="s.004406">Neque dixeris &longs;u&longs;tineri pondus à funibus inferiores orbicu­<lb/>los complectentibus: Hoc enim ad propo&longs;itam quæ&longs;tionem <lb/>nihil e&longs;t, tum quia nulla e&longs;t &longs;u&longs;tentatio, &longs;i pondus raptandum <lb/>&longs;it in plano horizontali, & tamen vis Trochleæ exercetur in <lb/>motu; tum quia ad pondus retinendum funes vim eandem <lb/>exercerent, &longs;i tam ampla e&longs;&longs;et unius orbiculi orbita, ut funem <lb/>utrumque caperet, vel unicus e&longs;&longs;et funis tam validus, ut utri­<lb/>que illi funi, quibus duo inferiores orbiculi in&longs;i&longs;tunt, æquiva­<lb/>leret; tum quia vero propius e&longs;t dicere, pondus &longs;u&longs;tineri à cla­<lb/>vo, ex quo &longs;uperior trochlea pendet, quàm à funibus, quemad­<lb/>modum ip&longs;a potentia &longs;u&longs;tinet; non autem vis &longs;u&longs;tinendi tribui­<lb/>tur funi illi, quem potentia arripit, & quo medio &longs;u&longs;tinet: <lb/>Clavus autem in huju&longs;modi trochleis, quando potentia trahens <lb/>proximè applicatur trochleæ clavo adnexæ, perinde &longs;u&longs;tinet <lb/>totam atque integram ponderis gravitatem, &longs;i plures fuerint or­<lb/>biculi, ac &longs;i unicus e&longs;&longs;et orbiculus; quamquam potentia minùs <lb/>reluctans in pluribus orbiculis, minore impetu conetur adver­<lb/>sùs pondus, ac proinde illa clavum minùs premat: quando verò <lb/>potentia proximè applicatur trochleæ inferiori, atque &longs;ur&longs;um <lb/>trahit, clavus nec urgetur ab impetu potentiæ, quem nullum <lb/>recipit, nec ip&longs;e &longs;u&longs;tinet totum pondus. </s> <s id="s.004407">Quod &longs;i pondus traha­<lb/>tur in plano horizontali, &longs;ola potentia e&longs;t, quæ adversùs cla­<lb/>vum &longs;uam vim exercet &longs;uperando re&longs;i&longs;tentiam ponderis, quod <lb/>nihil agit adversùs clavum, &longs;ed &longs;uâ gravitate urget &longs;ubjectum <lb/>planum. </s> </p> <p type="main"> <s id="s.004408">Ut autem manife&longs;tè deprehendas nihil e&longs;&longs;e Trochleis cum <lb/>Vecte commercij, duo ligna accipe, cuju&longs;cumque tandem fi­<lb/>guræ: &longs;ingulis tria in&longs;int foramina, quoad ejus fieri poterit, ex­<lb/>qui&longs;itè polita, ut minore conflictu funis excurrere po&longs;&longs;it: de­<lb/>inde funis alterno ab uno in alterum lignum ductu per forami­<lb/>na trajiciatur: Nam &longs;i alterum lignorum huju&longs;modi certo in lo­<lb/>co firmetur, alteri adnectatur pondus, tum funis extremitatem <lb/>arripiens trahas, idem planè præ&longs;tabis, quod adhibitis orbicu­<lb/>lis in communibus Trochleis: & tamen nullum hîc vectis ve&longs;ti­<lb/>gium apparet. </s> <s id="s.004409">Certè in majoribus navigiis malus hinc & hinc <pb pagenum="593" xlink:href="017/01/609.jpg"/>navis lateribus alligatur, ut rectam po&longs;itionem &longs;ervet: quia au­<lb/>tem rudentes aliquando remittuntur, ut in majore æ&longs;tu, illó&longs;­<lb/>que intendi oportet, propterea duo huju&longs;modi ligna in Ellip&longs;im <lb/>ferè deformata (vel potiùs in &longs;phæroides Hyperbolium factâ <lb/>conver&longs;ione non circa Axem, &longs;ed circa ordinatim Applicatam) <lb/>alterum navis lateri, alterum rudenti adnectunt nautæ, & fu­<lb/>nem non adeò cra&longs;&longs;um per foramina alterno ductu trajiciunt, <lb/>quem etiam axungiâ, aut aliâ pinguedine inficiunt, ut faciliùs <lb/>excurrat. </s> <s id="s.004410">Cum autem remi&longs;&longs;ior factus fuerit rudens, funis illius <lb/>caput &longs;olvunt, & trahentes cogunt ligna illa fieri propiora, ex <lb/>quo rudens intenditur exiguo trahentis conatu, &longs;i animadver­<lb/>tas quàm opero&longs;um & incommodum e&longs;&longs;et alio artificio ruden­<lb/>tem remi&longs;&longs;um intendere. </s> <s id="s.004411">Argumentum hoc, quod olim ante <lb/>annos vigintiquinque in Collegio Romano meis Auditoribus <lb/>in&longs;inuavi, conatus e&longs;t P. <!-- REMOVE S-->Schott ubi &longs;upra cap.3. eludere dicens <lb/><emph type="italics"/>ligna illa nullo modo habere rationem trochlearum, quia malus, qui <lb/>e&longs;t re&longs;i&longs;titivum, & debet trahi versùs latera navis, e&longs;t appen&longs;us uni <lb/>extremo illorum mediante fune, & potentia trahens e&longs;t applicata <lb/>alteri extremo eorumdem, & nihil dependet intermedium. </s> <s id="s.004412">Mirum <lb/>ergo non e&longs;t, &longs;i non habeat Vectis rationem.<emph.end type="italics"/></s> <s id="s.004413"> Verùm, tanti viri pa­<lb/>ce dixerim, ligna illa ita habent rationem Trochlearum, ut &longs;i <lb/>illorum loco communes Trochleas &longs;ub&longs;tituas, idem planè & <lb/>eodem modo efficias, trochleâ alterâ adnexa navis alteri, alterâ <lb/>rudenti intendendo: Neque enim malus e&longs;t re&longs;i&longs;titivum, quod <lb/>ponderis loco &longs;uccedit, neque ille ad navis latus trahendus e&longs;t, <lb/>aut inclinandus, &longs;ed rudentis caput trahendum e&longs;t, ut malo <lb/>immoto ad navim accedat, adeóque intendatur: Quare rudens <lb/>ip&longs;e intendendus vicem &longs;ubit ponderis, quatenus intentioni <lb/>repugnat, & potentia e&longs;t applicata funi per lignorum foramina <lb/>trajecto, &longs;icut applicaretur funi ductario trochlearum orbiculos <lb/>complexo. </s> <s id="s.004414">Quod &longs;i ligna illa non habent rationem Trochlea­<lb/>rum, & tamen trahendi facilitatem præ&longs;tant, ad quam Facul­<lb/>tatem Mechanicam &longs;pectant? </s> <s id="s.004415">Non ad Vectem, ut ille quoquè <lb/>admittit; non ad Axem, neque ad Cuneum, neque ad Co­<lb/>chleam, ut manife&longs;tum e&longs;t, pertinent: igitur vel novam Facul­<lb/>tatem con&longs;tituunt, vel omnino Trochleæ &longs;unt. <pb pagenum="594" xlink:href="017/01/610.jpg"/></s> </p> <p type="main"> <s id="s.004416"><emph type="center"/>CAPUT III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004417"><emph type="center"/><emph type="italics"/>An orbiculi magnitudo quicquam conferat.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004418">QUamquam Trochleæ Vires haberi etiam &longs;inè orbiculis &longs;u­<lb/>periùs dictum &longs;it, communiter tamen rotulas &longs;uis thecis <lb/>inclu&longs;as, & ver&longs;atiles adhibemus. </s> <s id="s.004419">Quæritur autem, an rotu­<lb/>larum huju&longs;modi magnitudo quicquam conferat ad faciliorem <lb/>motum: an verò indi&longs;criminatim rotulis &longs;ive majoribus, &longs;ive <lb/>minoribus uti po&longs;&longs;imus, citra virium notabile di&longs;pendium. </s> <lb/> <s id="s.004420">Quæ&longs;tioni huic locum fecit Ari&longs;toteles Mechan. quæ&longs;t. </s> <s id="s.004421">9. ubi <lb/>inquirit, <emph type="italics"/>Cur ea, quæ per majores circulos tolluntur, & trahuntur, <lb/>faciliùs & citiùs moveri contingit, veluti majoribus trochleis, quam <lb/>minoribus?<emph.end type="italics"/> & re&longs;pondet, <emph type="italics"/>An quoniam quanto major fuerit illa, quæ <lb/>à centro e&longs;t, in æquali tempore majus movetur &longs;patium? </s> <s id="s.004422">Quamobrem <lb/>æquali inexi&longs;tente onere idem faciet, quemadmodum diximus, & <lb/>majores libras minoribus exactiores e&longs;&longs;e; &longs;partum enim in illis cen­<lb/>trum e&longs;t.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.004423">Non de&longs;unt, qui negent faciliùs attolli pondus, ex. </s> <s id="s.004424">gr. <lb/><!-- REMOVE S-->&longs;itulam aquâ plenam è puteo, &longs;i funis in&longs;i&longs;tat orbiculo majo­<lb/>ri, quàm &longs;i minorem complectatur, ac proptereà ab Ari&longs;totele <lb/>fru&longs;trà quæri cau&longs;am facilitatis, quæ nulla &longs;it. </s> <s id="s.004425">Si enim diame­<lb/>ter orbiculi &longs;umatur ut Vectis primi generis hypomochlium <lb/>habens in centro, potentia & pondus in diametri extremitati­<lb/>bus æqualiter di&longs;tant ab hypomochlio, ac proinde &longs;ive major <lb/>&longs;it, &longs;ive minor diameter, eadem &longs;emper manet Ratio æqualita­<lb/>tis momentorum, quatenus ex po&longs;itione pendent; adeóque <lb/>nullum e&longs;t facilitatis in movendo di&longs;crimen. </s> <s id="s.004426">Sin autem nullus <lb/>agno&longs;catur Vectis, &longs;ed potentiæ motus cum motu ponderis <lb/>comparetur, hos &longs;emper æquales e&longs;&longs;e manife&longs;tum e&longs;t, &longs;ive ma­<lb/>jor, &longs;ive minor rotula adhibeatur: atque hinc nullum infert mo­<lb/>mentorum di&longs;crimen magnitudo, aut parvitas rotulæ. </s> </p> <p type="main"> <s id="s.004427">Ego tamen, Ari&longs;totelem omnino temerè majorem hanc mo­<lb/>vendi facilitatem per majores orbiculos a&longs;&longs;ump&longs;i&longs;&longs;e, affirmare <pb pagenum="595" xlink:href="017/01/611.jpg"/>non au&longs;im; neque enim carere potuit experimento aliquo, quo <lb/>id &longs;uaderetur. </s> <s id="s.004428">Difficultas potiùs &longs;uboriri pote&longs;t, an veram ille <lb/>afferat cau&longs;am majoris huju&longs;ce facilitatis: Nam quod innuit de <lb/>libris majoribus, quæ exqui&longs;itiores &longs;unt minoribus, quo pacto <lb/>intelligendum &longs;it, dictum e&longs;t lib. 3. cap. 6: illud autem hìc lo­<lb/>cum non habere manife&longs;tum e&longs;t. </s> <s id="s.004429">Nemo negat in majoribus <lb/>circulis, quorum major e&longs;t Radius, ab extremitate Radij ma­<lb/>jorem arcum de&longs;cribi, quàm à Radio minore, &longs;i tempore eodem <lb/>&longs;imilem arcum de&longs;cribant; &longs;unt &longs;cilicet arcus &longs;imiles in Ratione <lb/>Radiorum; &longs;ed quando rotulæ inæquales commune centrum <lb/>non habent, neque Radij omnino &longs;imul moventur, qua&longs;i minor <lb/>&longs;it pars majoris, quid prohibet eodem tempore arcus quídem <lb/>æquales, &longs;ed di&longs;&longs;imiles, de&longs;cribi? </s> <s id="s.004430">Nam &longs;i potentia trahens de­<lb/>&longs;cendat per &longs;patium palmare, &longs;ive rotula major &longs;it, &longs;ive minor, <lb/>pondus a&longs;cendit per palmum, & punctum in orbitâ rotulæ tam <lb/>majoris quàm minoris notatum de&longs;cribit arcum palmarem: hoc <lb/>autem tantummodo differunt, quod in univer&longs;o ponderis ele­<lb/>vati motu rotula minor &longs;æpiùs convertitur quàm major, & con­<lb/>ver&longs;ionum numeri &longs;unt reciprocè in Ratione Radiorum: &longs;ic &longs;i <lb/>Radius minor ad majorem &longs;it ut 4 ad 9, novem conver&longs;iones <lb/>minoris eodem tempore fiunt, ac quatuor conver&longs;iones majoris <lb/>rotulæ, &longs;i à Potentiâ æqualiter moveantur. </s> <s id="s.004431">Quare æquali tem­<lb/>pore major Radius non movetur per majus &longs;patium; movetur <lb/>&longs;iquidem æqualiter cum potentiâ trahente & pondere a&longs;cen­<lb/>dente, quemadmodum & minor Radius. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004432">Ut igitur Ari&longs;totelis dicto veritatem aliquam conciliemus, <lb/>quæ tamen experimentis re&longs;pondeat, illud ob&longs;ervandum e&longs;t, <lb/>quod &longs;uperiùs innui, videlicet eo con&longs;ilio excogitatos e&longs;&longs;e orbi­<lb/>culos, ut impedimentum ex funis attritu &longs;ubmoveatur, qui &longs;anè <lb/>tantus e&longs;&longs;et cum corpore, cui funis in&longs;i&longs;tit, quanta e&longs;t funis lon­<lb/>gitudo æqualis motui ponderis, quod trahitur. </s> <s id="s.004433">At in orbiculo <lb/>ver&longs;atili &longs;olus axis teritur à cavâ foraminis &longs;uperficie axis &longs;uper­<lb/>ficiei congruente, quæ eò minor e&longs;t, quò minor e&longs;t axis diame­<lb/>ter diametro ip&longs;ius orbiculi: perimetri enim &longs;unt in Ratione <lb/>diametrorum. </s> <s id="s.004434">Quare &longs;i Axis diameter ad orbiculi diametrum &longs;it <lb/>ex. </s> <s id="s.004435">gr. <!-- REMOVE S-->&longs;ubquadrupla, conflictus axis cum orbiculo e&longs;t &longs;ubqua­<lb/>druplus ejus, qui e&longs;&longs;et orbitæ orbiculi &longs;tabilis cum fune mobili; <lb/>immò adhuc minor e&longs;t quàm &longs;ubquadruplus, funis enim multò <pb pagenum="596" xlink:href="017/01/612.jpg"/>a&longs;perior e&longs;t quàm &longs;uperficies axis & foraminis &longs;ibi congruentes. </s> <lb/> <s id="s.004436">Quoniam verò axis &longs;oliditas definitur ex pondere, quod ab eo <lb/>&longs;u&longs;tinendum e&longs;t, idem e&longs;&longs;e pote&longs;t axis cum majore, & cum mi­<lb/>nore orbiculo. </s> <s id="s.004437">Si ergo eidem axi major orbiculus in&longs;eratur, <lb/>manife&longs;tum e&longs;t minorem fieri attritionem datâ motûs æqualita­<lb/>te. </s> <s id="s.004438">Nam orbiculorum orbitæ ex hypothe&longs;i &longs;int in Ratione du­<lb/>plâ, minoris autem orbiculi peripheria ad axis ambitum &longs;it in <lb/>Ratione quadrupla, jam orbita majoris orbiculi ad ambitum axis <lb/>e&longs;t in Ratione octupla: ponamus ambitum axis e&longs;&longs;e digitorum <lb/>4, orbita minor e&longs;t digitorum 16, orbita major digit. </s> <s id="s.004439">32: igitur &longs;i <lb/>adhibeatur minor orbiculus, dum potentia & pondus pariter <lb/>moventur per digitos 16, tritus cum axe e&longs;t per digitos 4 (pono <lb/>&longs;cilicet axem & foramen &longs;e invicem terere in puncto, in quo <lb/>exercetur &longs;u&longs;tentatio) adhibito autem majore orbiculo, dum <lb/>potentia & pondus per digitos 16 moventur, tritus cum axe e&longs;t <lb/>&longs;olum per digitos 2, &longs;emi&longs;&longs;em ambitûs foraminis. </s> <s id="s.004440">Ubi autem e&longs;t <lb/>minus movendi impedimentum, facilior e&longs;t motus; igitur ma­<lb/>jore orbiculo faciliùs movetur pondus. </s> </p> <p type="main"> <s id="s.004441">Sed ut rem ip&longs;am penitiùs intro&longs;piciamus, animadvertendum <lb/>e&longs;t conflictum orbiculi cum Axe non fieri in centro motûs, <lb/><figure id="id.017.01.612.1.jpg" xlink:href="017/01/612/1.jpg"/><lb/>quod idem e&longs;t cum centro <lb/>Axis, &longs;ed in ip&longs;ius axis &longs;uper­<lb/>ficie: quapropter hinc pon­<lb/>dus repugnans, hinc poten­<lb/>tia contranitens &longs;uas exer­<lb/>cent vires in axem non per <lb/>lineam ad ejus centrum <lb/>ductam, &longs;ed per lineam à <lb/>punctis potentiæ & ponde­<lb/>ris contingentem eju&longs;dem <lb/>axis &longs;uperficiem. </s> <s id="s.004442">Sic po&longs;ito <lb/>Axe, cujus centrum C, &longs;e­<lb/>midiameter ad perpendicu­<lb/>lum CA, &longs;i in extremitatibus diametri orbiculi DB &longs;it in B po­<lb/>tentia, in D pondus, illa &longs;uas vires in Axem exercet per lineam <lb/>contingentem BE, hoc verò per lineam DF. <!-- KEEP S--></s> <s id="s.004443">Similiter &longs;i po­<lb/>tentia &longs;it in G, & pondus in I extremitatibus diametri orbiculi <lb/>majoris circa eundem Axem, illa vires exercet per contingen-<pb pagenum="597" xlink:href="017/01/613.jpg"/>tem GH, hoc per IL. <!-- KEEP S--></s> <s id="s.004444">Potentia igitur B trahens, punctum F <lb/>orbiculi minoris cogit a&longs;cendere in A, & potentia G cogit <lb/>punctum L orbiculi majoris a&longs;cendere pariter in A. <!-- KEEP S--></s> <s id="s.004445">Porrò <lb/>punctum L propius e&longs;&longs;e puncto A, quàm punctum F, e&longs;t ma­<lb/>nife&longs;tum, quia minor Secans CD & minor Tangens DF com­<lb/>prehendunt arcum SF minorem, quàm &longs;it arcus SL compre­<lb/>hen&longs;us à majore Secante CI & majore Tangente IL. <!-- KEEP S--></s> <s id="s.004446">Hinc ex <lb/>Doctrinâ Sinuum con&longs;tat in Radio CA minorem particulam <lb/>re&longs;pondere arcui LA, quàm &longs;it particula re&longs;pondens æquali ar­<lb/>cui incipienti ab F versùs A. <!-- KEEP S--></s> <s id="s.004447">Igitur datâ motûs æqualitate, po­<lb/>tentiæ &longs;cilicet trahentis tantum funem, quantus e&longs;t arcus LA, <lb/>minùs re&longs;i&longs;tit a&longs;cen&longs;ui punctum L, quàm punctum F, & citiùs L <lb/>venit in A per breviorem arcum LA, quàm veniat F per lon­<lb/>giorem arcum FA. <!-- KEEP S--></s> <s id="s.004448">Potentia itaque in G faciliùs, hoc e&longs;t mi­<lb/>nore labore, cæteris paribus movebit pondus in I po&longs;itum, quàm <lb/>potentia eadem minori orbiculo in B applicata moveat idem <lb/>pondus in D. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004449">Neque dixeris ex æquali funis tractione pondus in &longs;uâ perpen­<lb/>diculari lineâ Directionis æqualiter a&longs;cendere. </s> <s id="s.004450">&longs;ive fuerit in D, <lb/>&longs;ive in I, ac propterea nullum inveniri facilitatis di&longs;crimen in <lb/>illo attollendo. </s> <s id="s.004451">Quia adhuc con&longs;iderandum e&longs;t pondus, quate­<lb/>nus e&longs;t <expan abbr="applicatũ">applicatum</expan> Axi medio orbiculo, in quo axe dum a&longs;cendit, <lb/>a&longs;cendit pariter in &longs;uâ <expan abbr="perp&etilde;diculari">perpendiculari</expan> lineâ Directionis; & quam­<lb/>vis in hac æqualiter &longs;e habeat, non tamen e&longs;t æqualiter <expan abbr="applicatũ">applicatum</expan> <lb/>axi: lineæ autem GH, IL productæ concurrerent in angulum <lb/>magis obtu&longs;um, quàm lineæ BE, DF; quapropter &longs;ibi invicem <lb/>minùs adver&longs;antur, quò propiùs accedunt ad rectitudinem. </s> </p> <p type="main"> <s id="s.004452">Verùm facilitas i&longs;ta non e&longs;t cum additamento momenti, quod <lb/>à machinâ efficitur; Machina enim tribuens movendi facilita­<lb/>tem e&longs;t pariter cau&longs;a tarditatis motûs; at hìc <emph type="italics"/>faciliùs & citiùs<emph.end type="italics"/> per <lb/>majores circulos moveri pondus docet Ari&longs;toteles; quatenus vi­<lb/>delicet &longs;ublatâ impedimenti particulâ, quæ ex tritu oriretur, <lb/>potentia faciliùs & citiùs movetur, cum qua pariter æquali pla­<lb/>nè motu etiam pondus movetur, quod tamen per machinam <lb/>tardiùs moveretur, quàm potentia. </s> </p> <p type="main"> <s id="s.004453">Quod &longs;i cylindricam axis &longs;uperficiem non admittas omnino <lb/>congruere cavæ &longs;uperficiei foraminis, jam contactus Axis e&longs;t &longs;o­<lb/>lùm ad punctum A utriu&longs;que orbiculi tam majoris, quàm mino-<pb pagenum="598" xlink:href="017/01/614.jpg"/>ris, & tunc refert quandam libræ &longs;imilitudinem, cujus jugum &longs;it <lb/>aut GI, aut BD, & &longs;partum in loco &longs;uperiore A. <!-- KEEP S--></s> <s id="s.004454">Sed non eadem <lb/>hìc militat ratio, quæ in libra: nam in brevioribus libræ brachiis, <lb/>quando pondera &longs;unt inæqualis gravitatis, extremitas brachij de­<lb/>&longs;cendentis in &longs;uo motu deflectens à lineâ rectâ, etiam deflectit à <lb/>perpendiculo eò magis, quò minor e&longs;t &longs;emidiameter circuli, cu­<lb/>jus arcum de&longs;cribit; at in longioribus brachiis majorem arcum <lb/>de&longs;cribentibus &longs;imilem minori, minùs deflectit à perpendiculo; <lb/>ac proinde in de&longs;cen&longs;u pauciora deteruntur gravitatis momenta, <lb/>cum magis ob&longs;ecundet naturali gravitatis propen&longs;ioni, quæ niti­<lb/>tur ad perpendiculum. </s> <s id="s.004455">At hìc in orbiculis, &longs;i Potentia movens &longs;it <lb/>gravitas aliqua major pondere attollendo, non cogitur deflectere <lb/>à perpendiculo, &longs;ive major, &longs;ive minor fuerit orbiculus. </s> <s id="s.004456">Quapro­<lb/>pter non ex Rationibus libræ philo&longs;ophandum e&longs;t, &longs;ed con&longs;ide­<lb/>randa e&longs;t pre&longs;&longs;io &longs;uperanda, quæ fit in A, tùm pondere, tùm po­<lb/>tentiâ deor&longs;um, ex hypothe&longs;i, conantibus; vel &longs;i pondus in pla­<lb/>no, cui in&longs;i&longs;tit, raptandum &longs;it, pre&longs;&longs;io fit vi potentiæ trahentis <lb/>pondus re&longs;i&longs;tens. </s> <s id="s.004457">Avellenda e&longs;t igitur ab Axe pars orbiculi illum <lb/>tangens in A: &longs;ed po&longs;ito æquali motu potentiæ tùm in B, tùm in <lb/>G, minor motus & tardior particulæ A efficitur, &longs;i potentia mo­<lb/>veat in G, quàm &longs;i moveat in B: facilius igitur illa movet præ i&longs;tâ. </s> <lb/> <s id="s.004458">Minorem autem & tardiorem e&longs;&longs;e motum in A, ubi vincenda e&longs;t <lb/>vis pre&longs;&longs;ionis, <expan abbr="cõ&longs;tat">con&longs;tat</expan>, quia, ut &longs;emel foramen orbiculi minoris per­<lb/>currat axem in A, totus ille convertendus e&longs;t; at orbiculi majo­<lb/>ris punctum in orbitâ de&longs;ignatum &longs;i moveatur æquali motu ac <lb/>punctum minoris orbitæ, non ab&longs;olvit integram revolutionem; <lb/>atque adeò orbiculus major æquale habens foramen cum orbi­<lb/>culo minore, &longs;ed multo majorem orbitam, motu æquali non per­<lb/>currit Axem in A, ni&longs;i juxta partem, quæ re&longs;pondeat revolutio­<lb/>ni orbitæ, quam con&longs;tat non e&longs;&longs;e integram. </s> </p> <figure id="id.017.01.614.1.jpg" xlink:href="017/01/614/1.jpg"/> <p type="main"> <s id="s.004459">Hinc conjicere licet po&longs;&longs;e orbiculo con­<lb/>&longs;trui &longs;atis exactam libram. </s> <s id="s.004460">Fiat ex ligno aut <lb/>ex materiâ metallicâ di&longs;cus RST, cujus <lb/>centrum V, eju&longs;que orbita ad tornum mo­<lb/>dicè excavetur, ut illi in&longs;i&longs;tere po&longs;&longs;int fu­<lb/>niculi lancium. </s> <s id="s.004461">Tum in V centro fiat fora­<lb/>men exqui&longs;itè rotundum atque politum, <lb/>cui indatur Axis pariter politus & lævis: <pb pagenum="599" xlink:href="017/01/615.jpg"/>axis <expan abbr="aut&etilde;">autem</expan> extremitatibus hinc atque hinc eminentibus alligen­<lb/>tur fila, inter quæ interceptus di&longs;cus po&longs;&longs;it &longs;u&longs;pendi. </s> <s id="s.004462">Si gravitas <lb/>fuerit per univer&longs;am laminam æquabiliter diffu&longs;a, <expan abbr="cõ&longs;i&longs;tet">con&longs;i&longs;tet</expan> di&longs;cus <lb/>in quacumque po&longs;itione; &longs;in autem partes fuerint &longs;ecundùm <lb/>gravitatem inæquales, ita &longs;ponte convertetur di&longs;cus, ut pars <lb/>gravior inferiorem occupatura locum u&longs;que eò de&longs;cendat, dum <lb/>centrum gravitatis &longs;it in lineâ directionis perpendiculari tran­<lb/>&longs;eunte per punctum &longs;u&longs;pen&longs;ionis, & punctum contactus orbi­<lb/>culi cum axe. </s> <s id="s.004463">Hanc perpendicularem lineam refert, atque de­<lb/>&longs;ignat filum, ex quo &longs;u&longs;penditur. </s> <s id="s.004464">Notato igitur diligenti&longs;&longs;imè <lb/>puncto S, in quo fila &longs;u&longs;pendentia tangunt extremam orbitam, <lb/>ibi e&longs;t locus apponendæ lingulæ, atque ibi firmandus e&longs;t uter­<lb/>que funiculus SR & ST. <!-- KEEP S--></s> <s id="s.004465">Amotis igitur funiculis, &longs;eu filis, ex <lb/>quibus prius &longs;u&longs;pendebatur orbiculus, atque adjectâ opportu­<lb/>nâ lingulâ, apponatur an&longs;a VM, quæ includet lingulam, &longs;i hæc <lb/>fuerit ritè collocata. </s> <s id="s.004466">Demum pendentibus funiculis adnectan­<lb/>tur lances ita, ut æquilibrium con&longs;tituant, quod à lingula indi­<lb/>cabitur. </s> <s id="s.004467">Sic parata erit, ut opinor, exacti&longs;&longs;ima libra, de qua du­<lb/>bitari non po&longs;&longs;it, an centrum motûs verè re&longs;pondeat lineæ, in <lb/>qua e&longs;t centrum gravitatis: æqualitas brachiorum VR, VT e&longs;t <lb/>manife&longs;ta propter faciliorem circuli con&longs;tructionem, quàm bra­<lb/>chiorum rectorum æqualitatem æquabili & æquali gravitate <lb/>præditam: pondera autem &longs;i inæqualia lancibus imponantur, <lb/>&longs;emper in eodem perpendiculo con&longs;i&longs;tunt, &longs;ive de&longs;cendant, &longs;ive <lb/>a&longs;cendant: lingula verò quia &longs;atis longa e&longs;t, quippe quæ incipit <lb/>ab V, quamvis additamentum factum &longs;it in S, vel modici&longs;&longs;imam <lb/>inclinationem in alterutram partem indicabit. </s> </p> <p type="main"> <s id="s.004468">Cum itaque negari non po&longs;&longs;it in &longs;implici orbiculo aliquam <lb/>demum movendi facilitatem aquiri, &longs;i ille major fuerit, quàm <lb/>&longs;i minor, hoc pariter in Trochleis contingere po&longs;&longs;e non nega­<lb/>rem adhibitis majoribus orbiculis potiùs quàm minoribus. </s> <s id="s.004469">Ve­<lb/>rùm attendendum e&longs;t, an &longs;it operæ pretium tam ingentes tro­<lb/>chleas movendis ponderibus adhibere; illa enim & majore di&longs;­<lb/>pendio con&longs;truerentur, & e&longs;&longs;ent valde graves, & ægre trans­<lb/>ferri po&longs;&longs;ent, &longs;i notabili aliquâ magnitudine præditæ e&longs;&longs;ent. </s> <lb/> <s id="s.004470">Quare nemini author e&longs;&longs;em, ut rejectis minoribus orbiculis ma­<lb/>jores quæreret; communiter enim valde mediocribus trochleis <lb/>utuntur artifices, & &longs;atis commodè perficitur motus, &longs;i orbicu-<pb pagenum="600" xlink:href="017/01/616.jpg"/>li facilè convolvantur: commodum verò, quod accederet ex <lb/>aliquatenus diminuto orbiculorum cum &longs;uis axibus conflictu, <lb/>non tantum e&longs;t, ut majore incommodo parandum &longs;it. <lb/></s> </p> <p type="main"> <s id="s.004471"><emph type="center"/>CAPUT IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004472"><emph type="center"/><emph type="italics"/>Qua Ratione Trochlearum vires augeantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004473">EX dictis cap.1. &longs;atis notum e&longs;t Trochlearum vires augeri <lb/>pro multitudine orbiculorum: &longs;ed quoniam non præ&longs;tat in­<lb/>gentes Trochleas con&longs;truere, proptereà &longs;atius e&longs;t Trochleas <lb/>cum aliâ quapiam Facultate componere, & poti&longs;&longs;imum cum <lb/>Axe in Peritrochio, &longs;ive Sucula &longs;it, &longs;ive Ergata, &longs;ive Tympa­<lb/>num, quorum Axi dum in conver&longs;ione circumducitur funis <lb/>ductarius, Trochleæ evadunt propiores, & adducitur pondus. <lb/><figure id="id.017.01.616.1.jpg" xlink:href="017/01/616/1.jpg"/><lb/>Ejus rei meminit Lucret. <!-- REMOVE S-->lib.4: Multaque per Trochleas & <lb/>Tympana pondere magno commovet, atque levi &longs;u&longs;tollit ma­<lb/>china ni&longs;u. </s> <s id="s.004474">Et quidem &longs;uperiore loco, uti de Tympano ageba­<lb/>tur, indicata e&longs;t methodus geminandi vires Tympani ABCS, <pb pagenum="601" xlink:href="017/01/617.jpg"/>&longs;i nimirum trabis exporrectæ extremitati D alligetur funis <lb/>ductarius, qui primùm tran&longs;eat per orbiculum E ponderi attol­<lb/>lendo adnexum, deinde per orbiculos F & G trabi adhæ­<lb/>rentes, per quos demum venit ad Axem H, cui circumducen­<lb/>dus e&longs;t. </s> <s id="s.004475">Quia enim potentia in F duplo velociùs movetur quàm <lb/>E, & Potentia in S premens Tympanum movetur velociùs <lb/>quam F, in Ratione partis &longs;emidiametri tympani ad &longs;emidia­<lb/>metrum Axis, hoc e&longs;t in Ratione AV ad AH, manife&longs;tum e&longs;t <lb/>geminari momenta tympani &longs;olitariè accepti. </s> <s id="s.004476">Quod &longs;i tam ex­<lb/>tremitati D, quàm Ponderi adnecterentur Trochleæ, adhuc <lb/>major e&longs;&longs;et vis Tympani aucta per Trochleas, & vici&longs;&longs;im ma­<lb/>jor Trochlearum vis aucta per Tympanum. <!-- KEEP S--></s> <s id="s.004477">Hinc &longs;i e&longs;&longs;ent duæ <lb/>trochleæ binis orbiculis in&longs;tructæ, & funis caput inferiori tro­<lb/>chleæ adjungeretur, quintuplex fieret tympani momentum, & <lb/>vici&longs;&longs;im trochlearum momentum acciperet incrementum in ra­<lb/>tione AH ad AV. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004478">Di&longs;tinguenda &longs;unt autem onera, quorum alia &longs;unt mediocria <lb/>(nam minora facilè &longs;olis trochleis attolluntur, arreptâ ab ho­<lb/>minibus funis ductarij extremitate) alia majora, & ingentia, <lb/>quæ à Vitruvio, ut aliàs innui, Colo&longs;&longs;icotera dicuntur. </s> <s id="s.004479">Pro <lb/>mediocribus ponderibus ad operarum numerum minuendum <lb/>Trochleis adjungi pote&longs;t Sucula, cui circumducatur funis <lb/>ductarius: compo&longs;itis enim Rationibus Suculæ & Trochlea­<lb/>rum, habetur Ratio momenti potentiæ ad Pondus. <!-- KEEP S--></s> <s id="s.004480">Si non ad <lb/>multam altitudinem attollendum &longs;it Pondus, neque proximo <lb/>parieti trabem infigi expediat, cui Trochlea adjungatur, & ex <lb/>qua onus dependeat, ex Vetruvij præ&longs;cripto lib.10. cap.2. tigna <lb/>tria parantur longitudine & &longs;oliditate re&longs;pondentia oneris <lb/>magnitudini & gravitati; hæc à capite fibulâ aut funibus con­<lb/>juncta, in imo divaricata eriguntur, qua&longs;i in pyramidis trian­<lb/>gularis &longs;peciem. </s> <s id="s.004481">Quod &longs;i timeatur, ne in hanc aut illam par­<lb/>tem machina inclinetur, funibus in capitibus collocatis, & <lb/>circa di&longs;po&longs;itis in adver&longs;as plagas, atque firmatis, erecta reti­<lb/>netur. </s> <s id="s.004482">In &longs;ummo, ubi tigna coëunt, alligatur Trochlea; & <lb/>inferiùs, ubi commodè applicari po&longs;&longs;it Potentia, exteriori duo­<lb/>rum tignorum divaricatorum faciei firmiter affiguntur Chelo­<lb/>nia, hoc e&longs;t fulcra quædam rotundum foramen habentia, in <lb/>quæ conjiciuntur Suculæ capita; ut Axis facile ver&longs;etur. </s> <s id="s.004483">Sucu-<pb pagenum="602" xlink:href="017/01/618.jpg"/>la autem proximè capita aut habet infixos Radios, aut &longs;altem <lb/>bina foramina ita temperata & di&longs;po&longs;ita, ut vectes in ea immit­<lb/>ti po&longs;&longs;int variæ longitudinis pro opportunitate atque nece&longs;&longs;ita­<lb/>te, habitâ ratione loci & ponderis. </s> <s id="s.004484">Altera Trochlea adnecti­<lb/>tur ponderi, prout commodius acciderit, & funis ductarij ex­<lb/>tremitas &longs;uperiori trochleæ adnectitur, eju&longs;que per Trochlea­<lb/>rum orbiculos trajecti caput ad Suculam religatur, cujus con­<lb/>ver&longs;ione attollitur pondus. </s> <s id="s.004485">Machinam hanc aliqui artifices <lb/><emph type="italics"/>Capram<emph.end type="italics"/> vocant. </s> </p> <p type="main"> <s id="s.004486">Ex his, &longs;i data fuerit ponderis gravitas, & nota Potentiæ <lb/>virtus, definies trochlearum orbiculos, aut &longs;altem vectium lon­<lb/>gitudinem, qui faciliùs parari po&longs;&longs;unt, & commutari pro re <lb/>natâ, quàm aliæ trochleæ inveniri. </s> <s id="s.004487">Sint itaque trochleæ bi­<lb/>nis orbiculis in&longs;tructæ; harum forma & po&longs;itio non ni&longs;i qua­<lb/>druplum potentiæ motum determinat, &longs;i cum motu ponderis <lb/>comparetur. </s> <s id="s.004488">At potentia univer&longs;a &longs;int duo homines, &longs;inguli <lb/>valentes attollere libras 25; atque adeò potentia e&longs;t lib. 50; <lb/>quæ &longs;i proximè applicetur funi ductario trochlearum, poterit <lb/>&longs;olùm attollere gravitatem quadruplam, hoc e&longs;t lib.200. Quon<lb/>niam verò oblatum pondus e&longs;t ex hypothe&longs;i lib.1000, hoc e&longs;t <lb/>quintuplum librarum 200, addenda e&longs;t trochleis Ratio quin­<lb/>tupla Succulæ, cujus Radij aut Vectes &longs;int quintupli &longs;emidia­<lb/>metri Axis eju&longs;dem Suculæ. </s> <s id="s.004489">Nam Potentia Vectibus aut Ra­<lb/>diis applicata quintuplo velociùs movetur, quàm extremitas <lb/>funis ductarij Axem complexi; hæc autem quadruplo velociùs <lb/>quàm pondus; atque idcircò potentia vigecuplo velociùs mo­<lb/>vetur quàm pondus, poteritque movere pondus vigecuplum <lb/>librarum 50, hoc e&longs;t lib.1000. </s> </p> <p type="main"> <s id="s.004490">Quod &longs;i ad in&longs;ignem aliquam altitudinem evehendum &longs;it <lb/>pondus, non e&longs;t opus tria huju&longs;modi tigna compingere, &longs;ed <lb/>ut &longs;umptibus & labori parcatur, &longs;atis e&longs;t non procul à pondere <lb/>longiorem trabem, etiam ex pluribus apte & firmiter con­<lb/>junctis compo&longs;itam, erigere, atque funibus in oppo&longs;itas vento­<lb/>rum plagas di&longs;po&longs;itis ita eju&longs;dem caput firmare, ut nullam in <lb/>partem vi &longs;u&longs;pen&longs;i ponderis inclinetur. </s> <s id="s.004491">Verùm quidem e&longs;t tra­<lb/>bem huju&longs;modi (Antennam aliqui dicunt) non omnino ad <lb/>perpendiculum erigi, &longs;ed modicè inclinatam &longs;tatui, ut à &longs;um­<lb/>mo vertice pendens ad perpendiculum &longs;arcina, quæ attollitur, <pb pagenum="603" xlink:href="017/01/619.jpg"/>non incurrat in trabem. </s> <s id="s.004492">Modicè, inquam, inclinata &longs;tatuitur <lb/>trabs i&longs;ta (ni&longs;i fortè illa altiùs defodiatur, & circùm fi&longs;tucatio­<lb/>ne &longs;olidetur, tunc enim poterit magis inclinata &longs;tatui) quia <lb/>propter notabilem longitudinem ita pote&longs;t inclinari, ut linea <lb/>directionis ab illius centro gravitatis ducta cadat intrà (vel cer­<lb/>tè non admodum ultrà) ba&longs;im &longs;u&longs;tentationis, atque perpen­<lb/>diculum à &longs;ummo vertice de&longs;cendens & illi lineæ parallelum <lb/>tanto ab&longs;it intervallo, quod &longs;atis &longs;it ad elevandum pondus citra <lb/>periculum colli&longs;ionis cum trabe: cui periculo occurri non po­<lb/>te&longs;t in tigno breviore, quo valde inclinato ad vitandum huju&longs;­<lb/>modi periculum colli&longs;ionis, linea directionis ab ejus centro <lb/>gravitatis cadens multo notabiliùs recederet à ba&longs;i &longs;u&longs;tentatio­<lb/>nis: propterea ubi brevioribus trabibus fuerit utendum, tres <lb/>modo &longs;uperiùs dicto compinguntur, ut &longs;e invicem fulcientes <lb/>&longs;ponte con&longs;i&longs;tant, & pondus non contingant. </s> <s id="s.004493">Capiti igitur <lb/>erectæ trabis longioris altera trochlea alligatur, altera oneri; <lb/>&longs;ed ad trabis pedem orbiculus unus firmiter adnectitur, per <lb/>quem funis ductarius juxta trabis longitudinem de&longs;cendens <lb/>trajicitur, & ad Ergatæ axem adducitur, ut ex ejus revolutio­<lb/>ne funis trahatur: orbiculum hunc Græci <foreign lang="greek">e)pago/nta</foreign> Latini Ar­<lb/>temonem vocant, ex Vitruvio lib. 10 cap.5. Hic tamen infi­<lb/>mus orbiculus cum nihil immutet aut potentiæ velocitatem, <lb/>aut ponderis tarditatem, nihil addit momenti ip&longs;i potentiæ ad <lb/>onus attollendum, &longs;ed ideò poti&longs;&longs;imùm adhibetur, ut funis <lb/>commodiùs Ergatæ circumducatur. </s> </p> <p type="main"> <s id="s.004494">Quare Potentiæ momenta componuntur ex momentis Tro­<lb/>chlearum & Ergatæ; quæ &longs;i innote&longs;cant, & data &longs;it potentiæ <lb/>virtus movendi, manife&longs;tum erit pondus, quod illa Ergatæ ap­<lb/>plicata movere poterit. </s> <s id="s.004495">Sic &longs;i Trochleæ binos habeant orbicu­<lb/>los, qui dant Rationem quadruplum, Vectis autem Ergatæ &longs;it <lb/>ad eju&longs;dem Axis &longs;emidiametrum ut 20 ad 1, Ratio, quæ ex <lb/>quadruplâ, & vigecuplâ componitur, e&longs;t octuagecupla; ac <lb/>proindè potentia extremo Vecti applicata poterit movere pon­<lb/>dus octuagecuplum ejus, quod &longs;inè machinâ movere pote&longs;t. </s> </p> <p type="main"> <s id="s.004496">Hinc &longs;i potentia movere valeat libras 50, huic machinæ ap­<lb/>plicata movebit pondus lib. 4000. Illud autem commodi ha­<lb/>bet Ergata, quod in illâ convolvendâ uti po&longs;&longs;umus jumentis <lb/>extremo vecti applicatis: & experimento didicimus trochleis <pb pagenum="604" xlink:href="017/01/620.jpg"/>binorum orbiculorum, & Ergatâ attolli à duobus equis pondus <lb/>librarum non minùs quàm triginta millium. </s> <s id="s.004497">Cum enim &longs;int <lb/>duo equi, unu&longs;qui&longs;que movet libras 15000; &longs;ed quia Trochleæ <lb/>dant Rationem quadruplam, accipe librarum 15000 quadran­<lb/>tem 3750, & ope trochearum, &longs;i &longs;olæ e&longs;&longs;ent & ab Ergatâ &longs;e­<lb/>junctæ, unicuique equo adhibendus e&longs;&longs;et ni&longs;us &longs;ubquadruplus, <lb/>videlicet conatus &longs;ufficiens ad <expan abbr="mov&etilde;das">movendas</expan> ab&longs;que trochleis libras <lb/>3750: quoniam demum Ergatæ Vectis ad &longs;emidiametrum axis <lb/>e&longs;t ex. </s> <s id="s.004498">gr. <!-- REMOVE S-->decuplus, &longs;inguli equi adhibent conatum adhuc <lb/>&longs;ubdecuplum, quo &longs;cilicer moverent libras 375: e&longs;t nimirum, <lb/>ex hypothe&longs;i harum trochlearum, & hujus Ergatæ, motus po­<lb/>tentiæ ad motum ponderis quadragecuplus; ac proindè poten­<lb/>tia adhibet conatum, quo moveret ab&longs;que machina gravitatem <lb/>dati ponderis &longs;ubquadragecuplam. </s> </p> <p type="main"> <s id="s.004499">At verò &longs;i pondera attollenda &longs;int omnino ingentia & colo&longs;­<lb/>&longs;icotera, non &longs;atis fuerit trabem erigere, &longs;ed ex pluribus trabi­<lb/>bus invicem compactis &longs;ive funibus, &longs;ive ferreis retinaculis, & <lb/>clavis qua&longs;i cra&longs;&longs;iores columnas erigere, eá&longs;que tran&longs;ver&longs;is aliis <lb/>trabibus inter &longs;e colligare, aut etiam obliquis fulcire oportet, <lb/>& circa pondus componere validi&longs;&longs;imum ca&longs;tellum, quod nul­<lb/>lam in partem inclinari queat: ut deinde pluribus Trochlea­<lb/>rum paribus cum &longs;uis Ergatis ritè collocatis machinator tutò <lb/>aggredi po&longs;&longs;it opus. </s> <s id="s.004500">Hìc autem multo commodius accidit plu­<lb/>res communes Trochleas & Ergatas adhibere, quàm paucio­<lb/>res trochleas plurimorum orbiculorum con&longs;truere, quæ lon­<lb/>gi&longs;&longs;imum funem ductarium exigerent, aut ingentes Ergatas &longs;ta­<lb/>tuere, quarum vectis valde longus non ni&longs;i in amplo &longs;patio <lb/>circumagi po&longs;&longs;et. </s> <s id="s.004501">Illud Machinatoris &longs;olertiæ relinquitur, <lb/>quod Ergatas &longs;ingulas atque Trochleas tam aptè di&longs;ponat, ut <lb/>&longs;ibi invicem impedimento non &longs;int. </s> <s id="s.004502">Quod &longs;i, dum moles ip&longs;a <lb/>elevatur, fulcra &longs;ubinde opportuno loco &longs;ubjicias, quibus illa <lb/>innitatur, multo certiùs, nec &longs;ine laboris compendio, rem to­<lb/>tam perficies. </s> </p> <p type="main"> <s id="s.004503">Quod demum ad funes attinet, in ponderum ingentium ele­<lb/>vatione duplex periculum præcavendum e&longs;t; alterum, quod <lb/>plures Trochleæ diver&longs;is ponderis partibus applicantur, ne &longs;ci­<lb/>licet funes aliquarum Trochlearum vi ponderis plus ju&longs;to <lb/>di&longs;tendantur, & longiores fiant, quàm par &longs;it, ut pondus u&longs;que <pb pagenum="605" xlink:href="017/01/621.jpg"/>in de&longs;tinatum evehatur locum, & aptè collocetur; una enim <lb/>parte jam ferè &longs;uum in locum deductâ, reliqua pars adhuc di&longs;ta­<lb/>ret, nec potentia trochleis illis applicata &longs;ola ad perficiendum <lb/>motum &longs;ufficeret. </s> <s id="s.004504">Alterum e&longs;t, ne ex motu, & vehementi <lb/>funium cum orbiculis, aut orbiculorum cum axibus tritu, ni­<lb/>mis incale&longs;cant, atque ignem concipiant. </s> <s id="s.004505">Sed utrique pericu­<lb/>lo occurritur, &longs;i aquam in promptu habeas, qua funes aut tro­<lb/>chleæ madefiant; illa enim non &longs;olum incen&longs;ionis periculum <lb/>&longs;ubmovet, verùm etiam funes contrahit. </s> </p> <p type="main"> <s id="s.004506">In plano autem horizontali aut inclinato longè facilior e&longs;t <lb/>motus, potentia quippe caret labore retinendi onus, quod in­<lb/>nitur plano; & quamvis hoc &longs;it inclinatum (non tamen lubri­<lb/>cum, neque pondus incumbat &longs;cytalis, &longs;eu cylindris) ita pon­<lb/>dus &longs;uâ gravitate premit &longs;ubjectum planum, ut etiam dimi&longs;&longs;um <lb/>non facile prolabatur: ut tamen in huju&longs;modi planis faciliùs <lb/>trahatur, expedit cylindros &longs;upponere, aut rotas addere, aut <lb/>illud trahæ imponere. </s> <s id="s.004507">Hìc pariter ad trahendum juvari pote&longs;t <lb/>potentia, &longs;i funis ductarij per Trochleas trajecti caput ad Axem <lb/>Ergatæ, aut Suculæ, aut Tympani referatur; prout majora aut <lb/>mediocria fuerint pondera. </s> <s id="s.004508">In minoribus autem ponderibus <lb/>raptandis, etiam &longs;implici Vecte, & quidem expediti&longs;&longs;imè, au­<lb/>geri po&longs;&longs;unt momenta Trochlearum; &longs;i nimirum vecti circa <lb/>medium alligetur caput funis ductarij, & inclinati vectis caput <lb/>&longs;ubinde transferatur. </s> <s id="s.004509">Sit enim ductarij funis extremitas A; <lb/>hæc in A religetur vecti BC, qui <lb/><figure id="id.017.01.621.1.jpg" xlink:href="017/01/621/1.jpg"/><lb/>terram premat in C, quod e&longs;t hy­<lb/>pomochlium, & potentia movens <lb/>&longs;it in B; quæ, manente extremi­<lb/>tate C, dum promovetur in D, fu­<lb/>nis caput venit ex A in E: tunc <lb/>iterum inclinetur vectis CD, ut <lb/>habeat po&longs;itionem FG, & poten­<lb/>tia &longs;imiliter circa punctum G ma­<lb/>nens moveatur ex F versùs D, at­<lb/>que ulteriùs adducatur funis ca­<lb/>put E, & &longs;ic deinceps. </s> <s id="s.004510">Quod &longs;i <lb/>progredi nolueris, &longs;ed eodem in loco con&longs;i&longs;tere, ubi vectis po­<lb/>&longs;itionem CD nactus fuerit, & A venerit in E, retrahe D ite-<pb pagenum="606" xlink:href="017/01/622.jpg"/>rum in B, atque particulam funis AE ita vecti convolve, ut <lb/>excurrere nequeat; nam iterato vectis motu trahetur funis, & <lb/>cum trochleâ pondus; motúque huju&longs;modi continuato de&longs;tina­<lb/>tumin locum adducetur pondus. </s> <s id="s.004511">Quantum verò &longs;it hoc compen. </s> <lb/> <s id="s.004512">dium, illicò innote&longs;cet, &longs;i ob&longs;ervaveris ut minimum geminari mo­<lb/>menta potentiæ, &longs;i videlicet punctum A præcisè medium fuerit <lb/>æqualiter ab extremitatibus B & C di&longs;tans: quod &longs;i AC &longs;it triens <lb/>totius BC, momentum potentiæ triplicatur, & e&longs;t Ratio com­<lb/>po&longs;ita ex Ratione Trochlearum, & Ratione Vectis. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004513">Ut autem manife&longs;to experimento deprehendas, quàm tenuis <lb/>Potentia Trochleis cum Axe in Peritrochio compo&longs;itis non leve <lb/>pondus trahat, in extremitate tabulæ non ruditer dolatæ duos <lb/>perpendiculares tigillulos erige intervallo digitorum quatuor, <lb/>illó&longs;que junge tran&longs;ver&longs;ario &longs;imilis cra&longs;&longs;itiei, non tamen à plano <lb/>ab&longs;it ni&longs;i quatuor digitos: Deinde &longs;upra tran&longs;ver&longs;arium interval­<lb/>lo &longs;altem digitorum octo &longs;tatue axem in tigillis facillimè ver&longs;ati­<lb/>lem, cujus diameter vix digitalis &longs;it: alteri autem hujus axis capiti <lb/>rotam circumpone, cujus diametrum digiti duodecim <expan abbr="metiãtur">metiantur</expan>: <lb/>quæ rota ut levis &longs;it, bino radiorum ordine con&longs;tet aptè colliga­<lb/>torum, & circa perimetrum emineant palmulæ, ut in molendino­<lb/>rum aquaticorum rotis, ex levi materiâ, cuju&longs;modi e&longs;&longs;et cra&longs;&longs;ior <lb/>charta, aut membrana, aut quid &longs;imile valens flatum excipere. </s> <lb/> <s id="s.004514">Tum parvulæ trochleæ duæ binis orbiculis in&longs;tructæ firmentur, <lb/>altera quidem in tran&longs;ver&longs;ario tigillorum, altera in extremi­<lb/>tate a&longs;&longs;erculi, cui onus aliquod e&longs;t imponendum, eique aut <lb/>rotæ, aut cylindruli &longs;ubjiciantur, ut facilè mobilis &longs;it; & tro­<lb/>chleæ mobili adnectatur extremitas funiculi &longs;erici, qui per or­<lb/>biculos trochlearum trajectus demum ad Axem referatur. </s> <lb/> <s id="s.004515">Nam &longs;i in rotæ palmulas vehementiùs in&longs;uffles, rota convol­<lb/>vetur, & cum illâ Axis, atque adeò funiculum involutum &longs;e­<lb/>queretur trochlea cum pondere ferè &longs;exagecuplo ejus, quod <lb/>flatu eodem ex&longs;ufflare po&longs;&longs;es. </s> <s id="s.004516">Aut potius Æolipilam aptè col­<lb/>loca, ut flatus ex illâ exiens in palmulas incurrat, & ex ponde­<lb/>ris, quod a&longs;&longs;erculo impo&longs;itum movetur, gravitate cogno&longs;tes <lb/>impetum, quo flatus ex Æolipilâ erumpit, &longs;i Ratio Trochlea­<lb/>rum, quæ e&longs;t quintupla, componatur cum Ratione diametri ro­<lb/>tæ ad diametrum axis, quæ, ex con&longs;tructione, e&longs;t duodecupla: <lb/>cum enim &longs;it Ratio, &longs;exagecupla, fiat ut 60 ad 1, ita gravitas <pb pagenum="607" xlink:href="017/01/623.jpg"/>ponderis, quod per huju&longs;modi machinulam trahitur, ad pon­<lb/>dus, quod flatu illo impelli po&longs;&longs;et &longs;ine machinâ. <lb/></s> </p> <p type="main"> <s id="s.004517"><emph type="center"/>CAPUT V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004518"><emph type="center"/><emph type="italics"/>Trochleæ Trochleis additæ plurimùm áugent <lb/>momenta Potentiæ.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004519">QUodlibet oblatum pondus datâ Potentiæ virtute movere <lb/>adhibitis Trochleis omnes norunt, &longs;i binas trochleas tot <lb/>in&longs;truant orbiculis, quot exigit Ratio ponderis ad potentiam, <lb/>aut plura communium trochlearum paria cum pluribus Ergatis <lb/>adhibeant: quo in opere quàm immanes trochleas e&longs;&longs;e oporte­<lb/>ret, &longs;i centenos aliquot, aut millenos orbiculos &longs;ingulæ conti­<lb/>nerent, aut quot Ergatæ, quantóque di&longs;pendio &longs;tatuendæ e&longs;­<lb/>&longs;ent, ex methodo &longs;uperiori capite traditâ, nemo non videt. </s> <s id="s.004520">Hìc <lb/>ergo &longs;olis Trochleis rem facillimè perfici po&longs;&longs;e me demon&longs;tratu­<lb/>rum confido, prout in <emph type="italics"/>Terrâ machinis motâ<emph.end type="italics"/> differt. </s> <s id="s.004521">1. indicavi. </s> </p> <p type="main"> <s id="s.004522">Et primò quidem &longs;implici orbiculo &longs;tabili elevari pote&longs;t pon­<lb/>dus cum incremento momentorum Potentiæ deor&longs;um trahentis <lb/>(id quod multo facilius accidit, quàm &longs;ur­<lb/><figure id="id.017.01.623.1.jpg" xlink:href="017/01/623/1.jpg"/><lb/>&longs;um trahere) &longs;i extremitati funis loco Po­<lb/>tentiæ adnectatur alius orbiculus, cujus fu­<lb/>nis in loco inferiore alligatus fuerit. </s> <s id="s.004523">Sit <lb/>Pondus P, orbiculo &longs;tabili A elevandum: <lb/>utique Potentia funi ductario in B appli­<lb/>cata non attollet pondus, ni&longs;i ejus gravitas, <lb/>aut virtus movendi major fuerit gravitate <lb/>ponderis P. <!-- KEEP S--></s> <s id="s.004524">Adnectatur in B orbiculus ver­<lb/>&longs;atilis E: & paxillo in C firmato alligetur <lb/>caput funis per orbiculum E tran&longs;euntis: <lb/>Nam potentia in F funem trahens duplo <lb/>velociùs movetur quàm orbiculus E, hoć <lb/>e&longs;t B extremitas funis ductarij, quæ cum <lb/>pondere P æqualiter movetur: ac proinde <pb pagenum="608" xlink:href="017/01/624.jpg"/>potentia, quæ duplo velociùs movetur quàm pondus, &longs;atis e&longs;t &longs;i <lb/>fuerit paulo major, quàm &longs;ubdupla ponderis. </s> </p> <p type="main"> <s id="s.004525">Ex hoc qua&longs;i rudimento continuò &longs;e &longs;e offert methodus com­<lb/>ponendi trochleas conjugatas; &longs;i nimirum duabus trochleis aptè <lb/>di&longs;po&longs;itis ad trahendum pondus, funis ductarij extremitati non <lb/>applicetur potentia, &longs;ed alia trochlea adnectatur, qua&longs;i ibi e&longs;&longs;et <lb/>pondus, & harum trochlearum &longs;ecundò po&longs;itarum funi applice­<lb/>tur potentia, cujus momenta ex Trochlearum rationibus com­<lb/><figure id="id.017.01.624.1.jpg" xlink:href="017/01/624/1.jpg"/><lb/>ponuntur. </s> <s id="s.004526">Sint duæ Trochleæ A & B bi­<lb/>nis orbiculis in&longs;tructæ; pondus in M ad­<lb/>nexum &longs;it trochleæ A; & trochlea B fir­<lb/>metur in C: funis autem ductarius in D <lb/>alligetur trochleæ A mobili: utique Po­<lb/>tentia in F habet momentum quintuplum, <lb/>quia quintuplo velociùs movetur, quàm <lb/>pondus in M adnexum. </s> <s id="s.004527">Sed iterum duæ <lb/>aliæ Trochleæ H & I binos orbiculos ha­<lb/>bentes parentur, & trochlea H mobilis <lb/>jungatur extremitati funis F, trochlea ve­<lb/>rò I &longs;tabilis &longs;it. </s> <s id="s.004528">Trochlearum H, I funis <lb/>ductarius alligetur in G trochleæ mobili; <lb/>& potentia in K applicata quinquies velo­<lb/>ciùs movetur quàm F; ergo & vige&longs;ies <lb/>quinquies velociùs quàm pondus adne­<lb/>xum in M. </s> <s id="s.004529">Illud igitur pondus, quod quin­<lb/>que homines applicati in F traherent, ab <lb/>unico homine in K applicato atque tra­<lb/>hente adducitur, qui unicus æquivalet vi­<lb/>gintiquinque hominibus pondus ab&longs;que <lb/>trochleis trahere conantibus. </s> </p> <p type="main"> <s id="s.004530">Cum itaque communes Trochleæ in promptu &longs;int, manife&longs;tum <lb/>e&longs;t, quàm facilè multiplicari valeant momenta potentiæ quæ cæ­<lb/>teroqui in exemplo propo&longs;ito duas trochleas &longs;ingulas duodenûm <lb/>orbiculorum exigeret, ut unus homo præ&longs;taret idem, quod vi­<lb/>ginti quinque Quod &longs;i adhuc duas &longs;imiles trochleas adhiberes, <lb/>& alteram &longs;imiliter in K adnecteres, unicus homo æquivaleret <lb/>hominibus 125 trahentibus: hinc &longs;i unicus ille homo tanto co­<lb/>natu trahat, quanto traheret libras 50, omninò &longs;olus tribus his <lb/>trochlearum paribus movebit pondus librarum 6250. </s> </p> <pb pagenum="609" xlink:href="017/01/625.jpg"/> <p type="main"> <s id="s.004531">Sed &longs;i quem admiratio capiat duodecim orbiculis in &longs;ex tro­<lb/>chleas di&longs;tributis tantum pondus moveri, admiretur adhuc am­<lb/>pliùs <expan abbr="ii&longs;d&etilde;">ii&longs;dem</expan> duodecim orbiculis in duodecim &longs;implices trochleas <lb/>di&longs;tinctis, quæ binæ & binæ conjungentur, longè majus pondus <lb/>po&longs;&longs;e trahi. </s> <s id="s.004532">Nam &longs;i trochleæ mobili, cui alligatur pondus, etiam <lb/>funis extremitas adnectatur, jam &longs;ingulæ Trochlearum conjuga­<lb/>tiones dant rationem triplam; &longs;unt igitur &longs;ex Rationes triplæ <lb/>compo&longs;itæ; ac propterea prima conjugatio dat Rationem 3 ad 1; <lb/>&longs;ecunda 9 ad 1; tertia 27 ad 1; quarta 81 ad 1; quinta 243 ad 1; <lb/>&longs;exta 729 ad 1: & hæc e&longs;t Ratio motûs po­<lb/><figure id="id.017.01.625.1.jpg" xlink:href="017/01/625/1.jpg"/><lb/>tentiæ ad motum ponderis. </s> <s id="s.004533">Quare &longs;i po­<lb/>tentia conetur ut 50, ducatur 729 per 50, <lb/>& potentia trahere valebit pondus libra­<lb/>rum 36450. At verò &longs;i duodecim illæ &longs;im­<lb/>plices trochleæ <expan abbr="nõ">non</expan> fuerint conjugatæ, &longs;ed <lb/>&longs;ingulæ &longs;eor&longs;im &longs;uos habeant funes, ita ut <lb/>primæ adnectatur <expan abbr="põdus">pondus</expan>, & &longs;ecundæ jun­<lb/>gatur extremitas funis ductarij primæ, at­<lb/>que ita deinceps, jam multo majus erit <lb/>momentum potentiæ; erunt &longs;cilicet duo­<lb/>decim Rationes duplæ <expan abbr="cõpo&longs;itæ">compo&longs;itæ</expan>. </s> <s id="s.004534">Sit enim <lb/>trochleæ X adnexum pondus, funis illius <lb/>alligatus in V, & eju&longs;dem capiti ad nexa &longs;it <lb/>&longs;ecunda Trochlea T; cujus pariter funis <lb/>alligatus in S reliquâ extremitate conjun­<lb/>gatur cum tertia Trochleâ R, eju&longs;que fu­<lb/>nis &longs;imiliter firmatus in Q veniat ad P, cui <lb/>deinceps quarta trochlea adjungatur, & <lb/>&longs;ic de cæteris con&longs;equentibus. </s> <s id="s.004535">Certum e&longs;t <lb/>T moveri duplo velociùs quàm X, & R <lb/>duplo velociùs quàm T, & P duplo velo­<lb/>ciùs quàm R; ac proinde P moveri octu­<lb/>plo velociùs quàm pondus in X. <!-- KEEP S--></s> <s id="s.004536">Si igitur <lb/>duodecim rationes duplæ componantur, <lb/>erit demum Ratio 4096 ad 1. Quaprop­<lb/>ter potentia in extremitate funis trochleæ <lb/>duodecimæ &longs;imiliter conata ut 50, move­<lb/>bit pondus librarum 204800. </s> </p> <pb pagenum="610" xlink:href="017/01/626.jpg"/> <p type="main"> <s id="s.004537">Ex his vides po&longs;terioribus trochleis minùs repugnare pondus <lb/>quàm prioribus, atque propterea funes ductarios po&longs;teriorum <lb/>trochlearum po&longs;&longs;e exiliores e&longs;&longs;e, quamvis longiores; eóque de­<lb/>veniri po&longs;&longs;e, ut potentia &longs;ubtili&longs;&longs;imo funiculo applicetur, & &longs;e­<lb/>curè trahat valde magnum pondus. </s> <s id="s.004538">Semper autem tractionis <lb/>mentionem feci, non elevationis, quia in illa faciliùs quàm in <lb/>hac uti po&longs;&longs;umus huju&longs;modi trochlearum complexione: quam­<lb/>quam etiam in elevatione ad mediocrem altitudinem, di&longs;po&longs;itis <lb/>duabus trochleis, qua&longs;i illas tantum adhibere oporteret, po&longs;&longs;u­<lb/>mus extremitati funis ductarij adjicere trochleam, cujus com­<lb/>parem paxillo in terram firmiter depacto alligemus; aut etiam, <lb/>&longs;i altitudo &longs;uppetat longè major ea, ad quam attollendum e&longs;t <lb/>pondus, in &longs;upremo loco &longs;tatuere po&longs;&longs;umus trochleam &longs;tabilem <lb/>&longs;ecundæ conjugationis, & mobili trochleæ adnectere extremi­<lb/>tatem funis ductarij priorum trochlearum, in quibus propterea <lb/>caput funis adnectendum e&longs;t trochleæ mobili, cui adhæret pon­<lb/>dus evehendum. </s> </p> <p type="main"> <s id="s.004539">Non e&longs;t autem di&longs;&longs;imulandum incommodum, quod ex hac <lb/>trochlearum di&longs;po&longs;itione atque complexione oritur, &longs;cilicet <lb/>magnam funium longitudinem requiri, nec non ingens &longs;pa­<lb/>tium, in quo di&longs;ponantur duo illa Trochlearum pariæ, quibus <lb/>vigequintupla fiunt Potentiæ momenta. </s> <s id="s.004540">Quia enim in Tro­<lb/>chleis adnexam &longs;arcinam adducentibus &longs;unt quatuor funis <lb/>ductus æquales trochlearum intervallo, utique, &longs;i eidem tro­<lb/>chleæ pondus ac funis alligatur, totus explicatur ultra termi­<lb/>num, cui trochlea &longs;tabilis adnectitur: quare trochleam mobi­<lb/>lem &longs;ecundæ conjugationis adnexam extremitati funis priorum <lb/>trochlearum con&longs;tituere oportet di&longs;tantem à &longs;uâ trochleâ &longs;tabi­<lb/>li non minùs quàm intervallo quintuplo di&longs;tantiæ priorum: ac <lb/>propterea harum po&longs;teriorum funis explicatus excurrit ultra <lb/>terminum, cui affigitur compar trochlea &longs;tabilis &longs;patio illius <lb/>quintupli intervalli quadruplo, hoc e&longs;t vigecuplo intervalli <lb/>priorum trochlearum; cui &longs;i addatur di&longs;tantia po&longs;teriorum <lb/>quintupla di&longs;tantiæ priorum, Potentia trochleæ &longs;ecundæ mo­<lb/>bili applicata funem trahens movetur vigequintuplo velociùs <lb/>quàm pondus, & exigit &longs;patium vigequintuplum di&longs;tantiæ prio­<lb/>rum trochlearum, &longs;i illa velit progredi, quantum fert longitudo <lb/>funis explicati; id quod nece&longs;&longs;e e&longs;t, &longs;i funis à jumentis trahatur, <pb pagenum="611" xlink:href="017/01/627.jpg"/>nec circumducatur Ergatæ; tunc enim non tantum &longs;patij re­<lb/>quiritur, & momentum Ratione Ergatæ augetur. </s> <s id="s.004541">At &longs;i Poten­<lb/>tia trahens &longs;int homines, &longs;atis e&longs;t &longs;i propè &longs;ecundam trochleam <lb/>&longs;tabilem con&longs;i&longs;tant. </s> <s id="s.004542">Quare &longs;i quis voluerit huju&longs;modi quatuor <lb/>trochlearum complexione uti, ut potentia obtineat momentum <lb/>vigequintuplum, requiritur &longs;patij longitudo quintupla &longs;patij, <lb/>per quod deducendum e&longs;t pondus. </s> <s id="s.004543">Quod igitur ad funium <lb/>longitudinem &longs;pectat, longitudo funis priorum trochlearum e&longs;t <lb/>quadrupla &longs;patij percurrendi à pondere, & longitudo funis <lb/>po&longs;teriorum e&longs;t eju&longs;dem &longs;patij vigecupla; hic tamen po&longs;terior <lb/>funis pote&longs;t e&longs;&longs;e priore tenuior atque exilior, ut dictum e&longs;t. </s> </p> <p type="main"> <s id="s.004544">Dixerit forta&longs;&longs;e aliquis, rem minùs attentè con&longs;iderans, po&longs;&longs;e <lb/>po&longs;teriores trochleas habere funem non longiorem fune prio­<lb/>rum; &longs;ed quia, ubi ille totus explicatus fuerit, pondus non e&longs;t <lb/>adductum ni&longs;i ad quintam partem &longs;patij, po&longs;&longs;e trochleas illas <lb/>po&longs;teriores ita invicem disjungi, ut ea, quæ e&longs;t mobilis, adjun­<lb/>gatur funi ductario propè trochleam priorem mobilem; nam <lb/>potentia iterum trahens adducet pondus: id quod &longs;æpius ite­<lb/>rari pote&longs;t. </s> </p> <p type="main"> <s id="s.004545">Verùm hoc fieri omnino non po&longs;&longs;e deprehendes, &longs;i ob&longs;erva­<lb/>veris, nunquam hoc pacto adduci pondus ni&longs;i per quintam par­<lb/>tem reliqui &longs;patij; quare aliquid &longs;emper relinquitur, quin ad <lb/>de&longs;tinatum locum pondus perveniat. </s> <s id="s.004546">Si placuerit tamen hunc <lb/>laborem a&longs;&longs;umere in disjungendis po&longs;terioribus trochleis, prio­<lb/>res trochleas ita invicem disjunctas initio colloca, ut earum in­<lb/>tervallum &longs;it &longs;altem &longs;e&longs;quialterum &longs;patij, per quod pondus mo­<lb/>veri oportet; &longs;ic enim repetito quinquies trahendi labore obti­<lb/>nebis propo&longs;itum motum: primâ videlicet tractione deducitur <lb/>pondus per totius intervalli 1/5; in &longs;ecunda per eju&longs;dem inter­<lb/>valli (4/25); in tertiâ per (16/125); in quartâ per (64/625); in quintâ per (256/1125); <lb/>quæ partes &longs;i in &longs;ummam redigantur, dant (2101/3125), hoc e&longs;t paulo <lb/>amplius quàm 2/3 propo&longs;iti intervalli, quantum &longs;atis e&longs;t ad perfi­<lb/>ciendum de&longs;tinatum &longs;patium. </s> <s id="s.004547">Ubi vides; &longs;i intervallum a&longs;­<lb/>&longs;umptum fui&longs;&longs;et paulo majus quàm duplum de&longs;tinati &longs;patij, ter­<lb/>tiâ tractione ab&longs;olvi propo&longs;itum motum; nam 1/5, (1/25), (16/125) &longs;i colli­<lb/>gantur in &longs;ummam, dant (61/125), hoc e&longs;t ferè 1/2. At &longs;i duobus &longs;im­<lb/>plicibus orbiculis utaris, quibus compo&longs;itis potentia habet mo-<pb pagenum="612" xlink:href="017/01/628.jpg"/>mentum quadruplum, etiam&longs;i &longs;ecundi orbiculi funem &longs;tatuas <lb/>æqualem funi prioris orbiculi, cui adnectitur pondus, facilli­<lb/>mum e&longs;t orbiculum &longs;ecundum retrahere ad orbiculum primum, <lb/>po&longs;tquam hic primâ tractione ab&longs;olvit &longs;emi&longs;&longs;em &longs;patij inter pon­<lb/>dus & paxillum, cui alligatur funis; & &longs;ecundâ tractione ab­<lb/>&longs;olvit quadrantem totius intervalli initio con&longs;tituti: Quare &longs;a­<lb/>tis fuerit funem prioris orbiculi æquari intervallo &longs;e&longs;quitertio <lb/>longitudinis &longs;patij, per quod deducendum e&longs;t pondus. </s> </p> <p type="main"> <s id="s.004548">Et quoniam hìc mentio incidit orbiculorum &longs;implicium, ob­<lb/>&longs;erva, quanto faciliùs duobus orbiculis perficiamus id, quod <lb/>duabus trochleis binos orbiculos habentibus præ&longs;taremus in <lb/>trahendo pondere, quando funis ductarius e&longs;t alligatus tro­<lb/>chleæ &longs;tabili; tunc enim potentia &longs;olùm habet momentum <lb/>quadruplum, quod pariter obtinet duobus orbiculis. </s> <s id="s.004549">Sit enim <lb/><figure id="id.017.01.628.1.jpg" xlink:href="017/01/628/1.jpg"/><lb/>AB di&longs;tantia &longs;e&longs;quitertia &longs;patij AI, per quod <lb/>trahendum e&longs;t pondus in P adnexum orbicu­<lb/>lo A: funis in B alligetur, & ejus caput C con­<lb/>nectatur cum orbiculo E, cujus pariter funis <lb/>in B alligetur, atque illius extremitas à Poten­<lb/>tiâ F trahatur. </s> <s id="s.004550">Quando potentia F adduxerit <lb/>orbiculum E prope B, erit orbiculus A in H: <lb/>retrahatur orbiculus E ex B, & propè H ad­<lb/>nectatur funi orbiculi A; factâ enim &longs;ecundâ <lb/>tractione, quando orbiculus E fuerit iterum <lb/>prope B, orbiculus A erit in I; e&longs;t autem ex <lb/>hypothe&longs;i di&longs;tantia AI æqualis &longs;patio, per <lb/>quod trahendum erat pondus, &longs;ub&longs;e&longs;quitertio <lb/>intervalli AB. <!-- KEEP S--></s> <s id="s.004551">Ecce igitur Potentia habet <lb/>momentum quadruplum, & duorum funium <lb/>longitudines &longs;imul &longs;umptæ non dant longitu­<lb/>dinem triplam &longs;patij, per quod deducendum <lb/>e&longs;t pondus. </s> <s id="s.004552">At &longs;i e&longs;&longs;ent duæ Trochleæ cum <lb/>binis orbiculis, exigerent unicum funem qua­<lb/>druplum longitudinis &longs;patij, per quod in&longs;ti­<lb/>tuendus e&longs;t motus. </s> </p> <p type="main"> <s id="s.004553">Sed & illud addendum videtur, quod duobus &longs;implicibus or­<lb/>biculis etiam ad longiora &longs;patia adduci pote&longs;t pondus, ita ut <lb/>quilibet trahentium habeat momentum quadruplum. </s> <s id="s.004554">Expe-<pb pagenum="613" xlink:href="017/01/629.jpg"/>dit autem trahentium numerum geminari, ut alternâ quiete <lb/>faciliùs & citiùs onus trahant. </s> <s id="s.004555">Sit orbiculus M adnectendus <lb/>ponderi, & &longs;it <lb/><figure id="id.017.01.629.1.jpg" xlink:href="017/01/629/1.jpg"/><lb/>datus funis du­<lb/>ctarius SR, cu­<lb/>jus extremitati­<lb/>bus R & S re­<lb/>plicatis qua&longs;i in <lb/>laqueum, &longs;eu <lb/>an&longs;am facillimè immitti po&longs;&longs;int & paxillus R, & alterius tro­<lb/>chleæ uncus S. <!-- KEEP S--></s> <s id="s.004556">Funis alius paretur NT prioris duplus extre­<lb/>mitates &longs;imiliter replicatas habens, ut in V immitti po&longs;&longs;it tra­<lb/>hentis manus, & in T paxillus. </s> <s id="s.004557">Quare tantumdem paxillus R <lb/>di&longs;tat à Trochleâ M, quantum à paxillo T, & hic tantumdem à <lb/>paxillo X. <!-- KEEP S--></s> <s id="s.004558">Cum igitur toto fune VNT explicato orbiculus N <lb/>fuerit in T, orbiculus M erit in R, & funis extremitas S erit in <lb/>T. <!-- KEEP S--></s> <s id="s.004559">Itaque ex paxillo T auferatur funis explicatus, & ejus loco <lb/>injiciatur extremitas S. <!-- KEEP S--></s> <s id="s.004560">Eximatur tunc ex paxillo R extremitas <lb/>funis, & adnectatur alteri Trochleæ funem habenti æqualem <lb/>funi VNT, cujus extremitas alligata fuerit paxillo X, & ad T <lb/>adducetur trochlea M unà cum pondere. </s> <s id="s.004561">Atque ita alternâ ope­<lb/>rá adducetur pondus ad quancumque di&longs;tantiam; interea enim, <lb/>dum orbiculus M ex R trahitur ad T, is qui traxerat funem V, <lb/>alligat illum paxillo, ad quem progrediendo pervenitur, & extre­<lb/>mitatem T exemptam è paxillo trahet, ubi trochleam N eò jam <lb/>deductam iterum junxerit extremitati S in T exi&longs;tenti. </s> <s id="s.004562">Sunt <lb/>itaque pangendi in terram paxilli æqualibus intervallis. </s> </p> <p type="main"> <s id="s.004563">Monendus e&longs;t autem Lector ad hoc caput non pertinere illam <lb/>Trochlearum additionem, quæ non facit rationum Compo&longs;itio­<lb/>nem; quando &longs;cilicet plures trochleæ uno loculamento ita in­<lb/>cluduntur, ut &longs;ingulæ trochleæ tam &longs;uperior, quam inferior plu­<lb/>res habeant orbiculorum ordines in latitudinem collocatos, at­<lb/>que adeo tot funes ductarios, quot &longs;unt ordines illi orbiculorum, <lb/>exigunt; perinde enim e&longs;t atque &longs;i duæ aut tres trochleæ diver­<lb/>&longs;is loculamentis di&longs;tinctæ adhiberentur. </s> <s id="s.004564">Cum autem plures &longs;int <lb/>funes ductarij, qui uno eodémque tempore adducendi &longs;unt, di­<lb/>ligenter animum advertere oportet, ut operæ omnes æqualiter <lb/>trahant. <pb pagenum="614" xlink:href="017/01/630.jpg"/></s> </p> <p type="main"> <s id="s.004565"><emph type="center"/>CAPUT VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004566"><emph type="center"/><emph type="italics"/>Trochlearum ope moveri potest pondus velociter.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004567">HActenus Trochlearum in facilè movendis oneribus vires <lb/>expendimus, ubi quò majora momenta ope hujus Faculta­<lb/>tis adduntur Potentiæ, eò etiam tardior e&longs;t motus ponderis, po­<lb/>tentiæ autem velocior: quod &longs;i velociter movendum &longs;it pondus, <lb/>nece&longs;&longs;ariò augeri debet potentia. </s> <s id="s.004568">Verùm quia non rarò contin­<lb/>gere pote&longs;t, ut potentia quidem ip&longs;a per &longs;e viribus abundet, il­<lb/>lam tamen tardè moveri oporteat, aut contra in trahendo onere <lb/>fe&longs;tinato &longs;it opus, propterea hìc indicandum e&longs;t, qua methodo <lb/>uti po&longs;&longs;imus, ut hinc plenior hujus Facultatis notitia habeatur. </s> </p> <p type="main"> <s id="s.004569">Opus &longs;it in turrim, vel in urbis mœnia commeatum transfer­<lb/>re velociter: operarum &longs;uppetat &longs;atis, at non item temporis. <lb/><figure id="id.017.01.630.1.jpg" xlink:href="017/01/630/1.jpg"/><lb/>Statuatur in &longs;umma turri, <lb/>aut certè in loco opportu­<lb/>no, Sucula BC cum manu­<lb/>briis CEF, & BDI, qui­<lb/>bus plures operæ applicari <lb/>po&longs;&longs;int pro gravitate oneris <lb/>attollendi; immò etiam ha­<lb/>beat infixos radios, ut adhuc <lb/>plures recipiat, qui illam <lb/>ver&longs;are po&longs;&longs;int. </s> <s id="s.004570">Circà Axem <lb/>involutus &longs;it funis paulo <lb/>longior &longs;emi&longs;&longs;e altitudinis, <lb/>& extremitati &longs;it adnexus <lb/>girgillus G, cui in&longs;ertus &longs;it <lb/>funis ductarius. </s> <s id="s.004571">HGL <lb/>æqualis altitudini, ad quam <lb/>evehendum e&longs;t onus; alte­<lb/>ri hujus funis extremitati <lb/>cohæreat in H validus un­<lb/>cus, quo onus &longs;u&longs;pendatur, <lb/>alteram verò <expan abbr="extremitat&etilde;">extremitatem</expan> L <pb pagenum="615" xlink:href="017/01/631.jpg"/>firmet clavus, aut quid &longs;imile ad pedem turris. </s> <s id="s.004572">Nam &longs;i conver­<lb/>tatur Sucula, devenient pariter in A tum girgillus G, tum <lb/>onus unco H &longs;u&longs;pen&longs;um; quod &longs;anè duplo velociùs movetur, <lb/>quàm &longs;i adnexum funi ductario AK traheretur &longs;ur&longs;um ope <lb/>&longs;implicis &longs;uculæ. </s> <s id="s.004573">Quare momenta &longs;uculæ non ni&longs;i dimidiata <lb/>computanda &longs;unt, adeò ut &longs;i duo homines &longs;uculam BC cir­<lb/>cumagentes valerent attollere libras 400, eodem conatu, & la­<lb/>bore po&longs;&longs;int &longs;olum libras 200 attollere: at quia facilè multipli­<lb/>cari po&longs;&longs;unt homines &longs;uculam ver&longs;antes, geminetur eorum nu­<lb/>merus, & attollent libras 400, &longs;ed breviori tempore. </s> <s id="s.004574">In con­<lb/>trarium autem revoluta &longs;ucula demittet girgillum G, & &longs;uo pon­<lb/>dere duplo velociùs de&longs;cendet uncus H. <!-- KEEP S--></s> <s id="s.004575">Ex quo habetur quæ­<lb/>&longs;itum temporis compendium. </s> </p> <p type="main"> <s id="s.004576">At &longs;i duabus trochlei, &longs;implicibus &longs;ingulos orbiculos haben­<lb/>tibus res perficienda e&longs;&longs;et, ita ut uni trochleæ adnecteretur Po­<lb/>tentia, alteri Pondus, funis autem extremitas alicubi clavo re­<lb/>ligata e&longs;&longs;et, attentè di&longs;piciendum e&longs;t, utri trochleæ adnecta­<lb/>tur reliqua funis trochleas jungentis extremitas. </s> <s id="s.004577">Nam &longs;i tro­<lb/>chleæ A, quam trahit potentia N, <lb/><figure id="id.017.01.631.1.jpg" xlink:href="017/01/631/1.jpg"/><lb/>adnectatur in C funis per orbicu­<lb/>los trajectus, trochleæ verò B <lb/>pondus M, & funis religatus fue­<lb/>rit in D, intelligitur motus inci­<lb/>pere, quando trochleæ adhuc in­<lb/>vicem ab&longs;unt, ità ut in motu tro­<lb/>chlea ponderis ad trochleam po­<lb/>tentiæ accedat, ce&longs;&longs;are autem, <lb/>cùm illæ proximæ factæ fuerint in <lb/>maximâ di&longs;tantiâ à clavo D, ubi <lb/>funis extremitas alligatur. </s> <s id="s.004578">Contrà <lb/>verò accidit trochleis G & H, &longs;i <lb/>trochleæ H adnectatur pondus B, <lb/>atque in I funis ductarij caput: <lb/>nam trahente potentia S, quæ ini­<lb/>tio propiores erant trochleæ, à &longs;e <lb/>invicem recedunt, trochleâ po­<lb/>tentiæ &longs;ecedente à trochleâ pon­<lb/>deris; & demum ab&longs;olvitur mo-<pb pagenum="616" xlink:href="017/01/632.jpg"/>tus, cùm trochlea H ponderis acce&longs;&longs;erit ad R extremitatem <lb/>funis religati. </s> <s id="s.004579">Cum itaque in utroque ca&longs;u & potentia, & <lb/>pondus versùs eandem partem moveantur, in primo tamen <lb/>pondus, quod à potentiâ di&longs;tabat, ad illam accedat, & in <lb/>&longs;ecundo potentia vicina ponderi ab illo recedat, manife&longs;to <lb/>indicio e&longs;t in primo ca&longs;u pondus, in &longs;ecundo potentiam ve­<lb/>lociùs moveri: quare ibi potentia augenda e&longs;t, ut valeat mo­<lb/>vere pondus, hìc fieri pote&longs;t additamentum ponderi, ut po­<lb/>tentiæ virtuti re&longs;pondeat. </s> <s id="s.004580">E&longs;t autem motuum Ratio &longs;e&longs;qui­<lb/>altera, ut palàm faciunt funium ductus, eorúmque explica­<lb/>tio: Nam in primo ca&longs;u maxima trochlearum di&longs;tantia e&longs;t, <lb/>quando trochlea A e&longs;t clavo D proxima; igitur potentia <lb/>movetur per &longs;patium, cujus longitudinem metitur funis ex­<lb/>plicatus, qui e&longs;t duplus di&longs;tantiæ trochlearum, & pondus <lb/>accedens ad potentiam in&longs;uper percurrit &longs;patium, quo tro­<lb/>chleæ di&longs;tabant; igitur motus ponderis e&longs;t ut 3, & poten­<lb/>tia ut 2. In &longs;ecundo verò ca&longs;u, Trochleæ G & H cùm <lb/>proximæ &longs;unt, di&longs;tant à clavo R juxta longitudinem funis <lb/>explicati, cùm autem maximè invicem ab&longs;unt, & potentia <lb/>tran&longs;gre&longs;&longs;a e&longs;t clavum R, totus funis di&longs;tributus e&longs;t in duos <lb/>ductus, & trochlearum intervallum e&longs;t medietas longitudi­<lb/>nis funis; quare ponderis motus e&longs;t ut 1, & motus poten­<lb/>tiæ ut 1 1/2. </s> </p> <p type="main"> <s id="s.004581">Simili ratione philo&longs;ophandum erit, &longs;i trochleæ inæquales <lb/>proponantur, ut &longs;i altera &longs;it duorum orbiculorum, altera <lb/>unius orbiculi: Utique funis per orbiculos trajectus adnecten­<lb/>dus e&longs;t &longs;implici trochleæ, ejú&longs;que altera extremitas alicubi <lb/>firmanda. </s> <s id="s.004582">Non igitur indi&longs;criminatim &longs;ivè huic, &longs;ivè illi <lb/>trochleæ adjungenda e&longs;t potentia, &longs;ed priùs &longs;tatuendum ti­<lb/>bi e&longs;t, utrùm velis pondus movere facilè, an velociter; &longs;i <lb/>facilè, tardior &longs;it ponderis motus, quàm potentiæ; &longs;i velo­<lb/>citer, tardior &longs;it potentia. </s> <s id="s.004583">Facilè movebis pondus, &longs;i potentia <lb/>trahat &longs;implicem orbiculum, & pondus cohæreat trochleæ <lb/>duorum orbiculorum: Velociter autem movebitur pondus, &longs;i <lb/>illud adnectatur &longs;implici orbiculo, potentia verò trahat tro­<lb/>chleam duorum orbiculorum. </s> <s id="s.004584">Nam in primo ca&longs;u funis expli­<lb/>catus replicatur, & potentia recedit à pondere; in &longs;ecundo fu­<lb/>nis replićatus explicatur, & pondus accedit ad potentiam. </s> </p> <pb pagenum="617" xlink:href="017/01/633.jpg"/> <p type="main"> <s id="s.004585">Sit Trochlea MO, & orbiculus I; huic in L adnectitur fu­<lb/>nis, cujus altera extremitas religatur in A, <lb/><figure id="id.017.01.633.1.jpg" xlink:href="017/01/633/1.jpg"/><lb/>quò demum devenire pote&longs;t trochlea MO <lb/>cum pondere T adjecto. </s> <s id="s.004586">Vice versá Tro­<lb/>chleæ GC adhibeatur potentia, & pon­<lb/>dus S adjiciatur orbiculo E: huic in B ad­<lb/>nectitur funis, qui per orbiculos trajectus <lb/>de&longs;init in F, ubi ille religatur, & trochlea <lb/>GO maximè di&longs;tat ab orbiculo E. <!-- KEEP S--></s> <s id="s.004587">In&longs;ti­<lb/>tuto motu, Potentia D &longs;emper magis re­<lb/>cedit à pondere T; at pondus S &longs;emper <lb/>magis accedit ad Potentiam H: ibi ergo <lb/>potentia celerior e&longs;t pondere, hìc pondus <lb/>velocius e&longs;t Potentiâ; motuum autem Ra­<lb/>tio e&longs;t &longs;e&longs;quitertia: Nam explicato fune <lb/>toto, qui religatur in A, potentia proxi­<lb/>ma e&longs;t ponderi, & di&longs;tant ab A pro funis <lb/>longitudine; potentiâ trahente accedunt <lb/>ad A, &longs;ed potentia ulteriùs progreditur, atque ab&longs;oluto motu <lb/>replicatus e&longs;t funis in tres ductus, & Potentia di&longs;tat à pondere <lb/>tertiâ parte ip&longs;ius funis, ita ut pondus quidem &longs;it clavo A proxi­<lb/>mum, potentia verò tran&longs;gre&longs;&longs;a &longs;it clavum A intervallo OL: <lb/>igitur motus ponderi, quem longitudo funis metitur, e&longs;t ut 1, <lb/>potentiæ ut 1 1/3. Ex adver&longs;o Potentia applicata trochleæ GC <lb/>proxima e&longs;t clavo F, cum ab illâ pondus maximè abe&longs;t inter­<lb/>vallo tertiæ partis ip&longs;ius funis in tres ductus replicati: inito mo­<lb/>tu pondus accedit ad Potentiam, cui demum proximum e&longs;t, <lb/>quando jam totus funis e&longs;t explicatus; igitur motum potentiæ <lb/>metitur funis explicatus, motum autem ponderis adhuc tertia <lb/>pars, &longs;cilicet intervallum BC: adeóque ponderis motus ad mo­<lb/>tum potentiæ e&longs;t ut 1 1/3 ad 1. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004588">Quæ autem de his trochleis dicta &longs;unt, &longs;i attentè con&longs;ide­<lb/>rentur, etiam cæteris trochleis conjugatis, &longs;ed di&longs;pari orbicu­<lb/>lorum numero in&longs;tructis, conveniunt. </s> <s id="s.004589">Non po&longs;&longs;e verò orbicu­<lb/>lorum numeros differre ni&longs;i unitate, &longs;atis manife&longs;tum e&longs;t: nam <lb/>&longs;i differrent binario aut ternario, eorum aliquis aut plures pla­<lb/>nè otio&longs;i e&longs;&longs;ent, quippe qui recipere nequirent funem ducta-<pb pagenum="618" xlink:href="017/01/634.jpg"/>rium jam per reliquos orbiculos trajectum. </s> <s id="s.004590">Quare &longs;i altera tro­<lb/>chlea minor duos habeat orbiculos, altera major non ni&longs;i tres <lb/>habere pote&longs;t, aut &longs;i minor tres habeat, major non ni&longs;i quatuor <lb/>habere poterit. </s> <s id="s.004591">Attendendum e&longs;t igitur, utri trochlearum tro­<lb/>chlearum potentia applicetur; &longs;i enim illa trahendam arripiat <lb/>trochleam plures habentem orbiculos, tardiùs movetur, quàm <lb/>pondus in Ratione &longs;ub&longs;uperparticulari denominatâ à numero <lb/>omnium &longs;imul <expan abbr="orbiculorũ">orbiculorum</expan>: ut &longs;i potentia trochleæ trium, pondus <lb/>verò trochleæ duorum orbiculorum applicetur, motus potentiæ <lb/>e&longs;t ad motum ponderis in Ratione &longs;ub&longs;e&longs;quiquintâ, quia illa mo­<lb/>vetur ut 5, pondus ut 6: & &longs;i potentia trochleæ quatuor orbicu­<lb/>lorum applicetur, pondus autem trochleæ trium, Ratio e&longs;t &longs;ub­<lb/>&longs;e&longs;qui&longs;eptima, quia illa movetur ut 7, hoc ut 8. Quare augetur <lb/>motus ponderis, & in eadem Ratione difficultas potentiæ. </s> <s id="s.004592">Con­<lb/>trà autem &longs;i pondus alligetur majori trochleæ, etiam potentiæ <lb/>motus major, e&longs;t motu ponderis in Ratione &longs;uperparticulari de­<lb/>nominatâ à numero omnium &longs;imul orbiculolum: &longs;ic erit Ratio <lb/>&longs;e&longs;quiquinta, &longs;i potentia duobus, pondus tribus orbiculis allige­<lb/>tur, nam motus potentiæ e&longs;t ut 6, & motus ponderis ut 5: &longs;imi­<lb/>liter erit Ratio &longs;e&longs;qui&longs;eptima, quando pondus alligatum tro­<lb/>chleæ quatuor orbiculorum movetur ut 7, dum potentia appli­<lb/>cata trochleæ trium orbiculorum movetur ut 8. </s> </p> <p type="main"> <s id="s.004593">Quod &longs;i pari orbiculorum numero con&longs;tet utraque trochlea, <lb/>& utraque moveatur, &longs;imiliter motus erunt in Ratione &longs;uper­<lb/>particulari denominatâ à numero omnium orbiculorum &longs;imul: <lb/>hoc tamen erit di&longs;crimen, quod illud tardius movebitur, quod <lb/>applicabitur trochleæ, cui extremitas funis per orbiculos tra­<lb/>jecti adnectitur. </s> <s id="s.004594">Sic &longs;i trochleæ ambæ binos habeant orbiculos, <lb/>Ratio e&longs;t &longs;e&longs;quiquarta, &longs;i ternos &longs;e&longs;qui&longs;exta: &longs;i trochleam, cui <lb/>funis ductarij extremitas adnectitur, potentia trahat, illa move­<lb/>tur ut 4, aut ut 6, pondus verò movetur ut 5, aut ut 7: &longs;ed &longs;i tro­<lb/>chleæ, cui funis adnectitur, alligetur pondus, potentia movetur <lb/>ut 5, aut ut 7, pondus autem ip&longs;um ut 4, aut ut 6. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004595">Ex his itaque duplex trochlearum u&longs;us innote&longs;cit, alter com­<lb/>munis quo potentia applicatur extremitati funis ductarij (alterâ <lb/>trochlearum manente &longs;tabili) quem trahens attrahit pariter <lb/>pondus, & motus potentiæ e&longs;t in Ratione aliqua multiplici ad <lb/>motum ponderis. </s> <s id="s.004596">Alter verò e&longs;t, quando extremitas funis <pb pagenum="619" xlink:href="017/01/635.jpg"/>ductarij non trahitur, &longs;ed alicubi firmatur, potentia autem <lb/>trahit alteram trochleam, ad cujus motum etiam reliqua tro­<lb/>chlea cum pondere illor&longs;um movetur, quor&longs;um potentia tendit: <lb/>&longs;i in hoc motu trochleæ disjunguntur, & potentia recedit à <lb/>Pondere, Ratio motûs potentiæ ad motum ponderis e&longs;t &longs;uper­<lb/>particularis, & potentia con&longs;equitur aliquam movendi facilita­<lb/>tem: &longs;in autem pondus ad potentiam accedit, & trochleæ, quæ <lb/>disjunctæ erat, fiunt proximæ, Ratio motûs potentiæ ad motum <lb/>ponderis e&longs;t &longs;ub&longs;uperparticularis, & potentiam plus adhibere <lb/>conatûs oportet, quàm &longs;i illud ab&longs;que trochleis traheret; quia <lb/>pondus velociùs movetur quàm potentia. <lb/></s> </p> <p type="main"> <s id="s.004597"><emph type="center"/>CAPUT VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004598"><emph type="center"/><emph type="italics"/>Quàm validum e&longs;&longs;e oporteat trochlearum <lb/>retinaculum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004599">IN Trochlearum u&longs;u communi alteram &longs;tabilem e&longs;&longs;e ac fir­<lb/>mam, alteram mobilem (&longs;i enim plures e&longs;&longs;ent omnino &longs;tabi­<lb/>les, quantumvis multæ, non augerent motum potentiæ) illam <lb/>autem ab aliquo corpore, cui alligata e&longs;t, retineri, &longs;atis per &longs;e <lb/>patet; propterea corpus hoc adeò validum e&longs;&longs;e oportet, ut ne­<lb/>que gravitati ponderis, neque conatui potentiæ cedat, &longs;ed ita <lb/>immotum per&longs;i&longs;tat, ut univer&longs;us potentiæ impetus ad vincen­<lb/>dam ponderis re&longs;i&longs;tentiam referatur. </s> <s id="s.004600">Hinc à veritate non ad­<lb/>modum rece&longs;&longs;i&longs;&longs;e videntur, qui in Mechanicis motionibus qua­<lb/>&longs;i duplex munus di&longs;tinguunt, alterum, quo pondus retinetur, ne <lb/>vi &longs;uæ gravitatis labatur, alterum, quo gravitas ip&longs;a &longs;uperatur, <lb/>& cogitur inire motum &longs;uæ propen&longs;ioni adver&longs;antem: po&longs;te­<lb/>rius hoc &longs;oli potentiæ tribuendum, prius illud non uni poten­<lb/>tiæ, &longs;ed etiam corpori, cui machina innititur, ad&longs;cribendum <lb/>cen&longs;ent, & in illud maximam oneris partem rejici a&longs;&longs;erunt. </s> <s id="s.004601">Et <lb/>&longs;anè quid prode&longs;&longs;et trochleam &longs;uperiorem aut fune, qui laxari <lb/>nequiret, aut ferreo unco, quem revellere nulla gravitas po&longs;­<lb/>&longs;et, connecti cum tigno parieti infixo; &longs;i timendum e&longs;&longs;et, ne <pb pagenum="620" xlink:href="017/01/636.jpg"/>tignum ip&longs;um imbecillum, vímque gravitatis &longs;u&longs;pen&longs;æ ferre <lb/>non valens, frangeretur? </s> <s id="s.004602">Quare ne magnum in di&longs;crimen res <lb/>adducatur, & ad periculum omne &longs;ubmovendum, ne in&longs;titu­<lb/>tus motus repentina retinaculi abruptione intercidatur, atque <lb/>ut certiùs eligi po&longs;&longs;it, cuinam poti&longs;&longs;imùm corpori (tigno ne pa­<lb/>rieti infixo? </s> <s id="s.004603">an antennæ erectæ?) concredenda &longs;it oneris <lb/>&longs;u&longs;tentatio, machinatori attente di&longs;piciendum e&longs;t, quantam <lb/>vim tùm oneris gravitas, tùm potentiæ conatus exerceat adver­<lb/>sùs huju&longs;modi retinaculum. </s> <s id="s.004604">Propterea vim i&longs;tam placuit hoc <lb/>capite examinare, ut cætera &longs;ecurè definiri valeant. </s> <s id="s.004605">Ut verò <lb/>brevitati & per&longs;picuitati con&longs;ulatur, retinaculum hoc ponamus <lb/>e&longs;&longs;e clavum, ex quo trochleæ cum onere &longs;u&longs;pen&longs;o dependeant; <lb/>quæ enim de huju&longs;modi clavo dicentur, facilè ad cætera tradu­<lb/>ci poterunt. </s> </p> <p type="main"> <s id="s.004606">Et primò &longs;i trochlearum funis per orbiculos ritè trajectus de­<lb/>mum &longs;uâ extremitate in nodum colligatur, ne excurrere valeat, <lb/>totam atque integram oneris gravitatem (trochleas & funem à <lb/>&longs;uâ in&longs;itâ gravitate nunc quidem mente &longs;ecernamus) à clavo, <lb/>ex quo trochleæ &longs;u&longs;penduntur, retineri dubium e&longs;&longs;e non pote&longs;t; <lb/>nihil aliud quippe ade&longs;t, adversùs quod ponderis gravitas de­<lb/>or&longs;um &longs;e ip&longs;a urgens connitatur. </s> <s id="s.004607">Deinde &longs;i funis ductarij caput, <lb/>quod potentia trahere &longs;olita e&longs;t, alligetur &longs;olo, aut ingenti &longs;axo <lb/>longi&longs;&longs;imè graviori, quàm pondus &longs;u&longs;pen&longs;um, utique neque <lb/>&longs;axum illud &longs;ubjectæ telluri incumbens, neque tellus ip&longs;a, quip­<lb/>piam virium exercent adversùs pondus, cui &longs;olùm &longs;uâ longè <lb/>majori gravitate re&longs;i&longs;tunt Formaliter, non verò Activè; quia ni­<lb/>mirum nullum efficiunt impetum, quo de&longs;cen&longs;um moliantur; <lb/>ac proinde à clavo &longs;olo pondus trochleis adnexum &longs;u&longs;tinetur, <lb/>& &longs;olum pondus clavum deor&longs;um trahere conatur. </s> </p> <p type="main"> <s id="s.004608">At verò &longs;i funis ductarij extremitati adnectatur alia gravitas <lb/>pro trochlearum Ratione re&longs;pondens ponderis gravitati, ita ut <lb/>æqualibus momentis certantes ambæ &longs;u&longs;pen&longs;æ con&longs;i&longs;tant, utra­<lb/>que gravitas collatis viribus clavum trahere conatur, utraque <lb/>enim deor&longs;um connititur: & ideò tam validum &longs;tatui clavum <lb/>oportet, ut utriu&longs;que gravitatis conatum ferre valeat. </s> <s id="s.004609">Id quod <lb/>multo magis ob&longs;ervandum e&longs;t, quando gravitas adnexa præpon­<lb/>derans vim infert oneri, illúdque &longs;ur&longs;um trahit; ip&longs;a &longs;cilicet gra­<lb/>vitas plus conatur in motu, quàm in æquilibrio; ac propterea & <pb pagenum="621" xlink:href="017/01/637.jpg"/>potentiæ deor&longs;um connitentis in motu impetum, & oneris mo­<lb/>tui &longs;ur&longs;um repugnantis gravitatem fert clavus utrique re&longs;i&longs;tens <lb/>&longs;uâ &longs;oliditate. </s> <s id="s.004610">Sicut igitur gravitas inanimata ex trochlearum <lb/>fune pendens &longs;u&longs;pendit pondus, aut attollit; ita potentia vivens <lb/>funem retinendo &longs;uo impetu virtutem eju&longs;dem gravitatis æquat, <lb/>ac &longs;imilem vim exercet in clavum; funem verò trahendo virtu­<lb/>tem illam gravitatis &longs;uperat, atque impre&longs;&longs;o impetu quodam­<lb/>modo attenuat, ita tamen, ut quod videtur gravitati demptum, <lb/>intelligatur additum conatui potentiæ prævalentis. </s> </p> <p type="main"> <s id="s.004611">Mihi autem (quid fru&longs;tra di&longs;&longs;imulem?) non levis injicitur <lb/>&longs;crupulus & dubitatio, an vis illata clavo, ex quo trochleæ cum <lb/>onere dependent, men&longs;uram præcisè recipiat ex ab&longs;olutâ gra­<lb/>vitate oneris, quando abe&longs;t conatus potentiæ illud attollentis <lb/>aut &longs;u&longs;pendentis. </s> <s id="s.004612">Dubitandi an&longs;am offert quædam munerum <lb/>commutatio inter Potentiam, Pondus, & Clavum, &longs;i ad effectio­<lb/>nes diver&longs;as referantur. </s> <s id="s.004613">Si enim oneris &longs;u&longs;pen&longs;io aut elevatio vi <lb/>potentiæ ex adver&longs;o nitentis con&longs;ideretur, Clavus exercet mu­<lb/>nus Retinaculi: at &longs;i vim clavo illatam, ejú&longs;que inflexionem, aut <lb/>revul&longs;ionem intueamur, efficientia vim huju&longs;modi inferens tri­<lb/>buenda e&longs;t aut gravitati oneris, aut impetui potentiæ trahentis: <lb/>quapropter &longs;oliditas clavi inflexionem re&longs;puentis, aut ejus firma <lb/>cohæ&longs;io cum pariete aut ligno, cui infixus e&longs;t, vicem &longs;ubit Pon­<lb/>deris ope trochlearum movendi cum alterâ trochleâ connexi; <lb/>Sarcina autem ex reliquâ trochleâ dependens aut retinaculi <lb/>munus obtinet, &longs;i attollatur, aut Potentiæ vices &longs;ubit, &longs;i deor­<lb/>&longs;um moveatur. </s> </p> <p type="main"> <s id="s.004614">Sit clavo A adnexa &longs;implex Trochlea B, ejú&longs;que funis <lb/>ductarius CDE: adnectatur in C &longs;a­<lb/><figure id="id.017.01.637.1.jpg" xlink:href="017/01/637/1.jpg"/><lb/>xum P, & à Potentia G elevatum &longs;u&longs;­<lb/>pendatur religato funis capite in E. <!-- KEEP S--></s> <s id="s.004615">Si <lb/>&longs;axum P accipiatur, quatenus elevatur, <lb/>ip&longs;um e&longs;t Pondus, Clavus A e&longs;t Retina­<lb/>culum, & Potentia e&longs;t G, &longs;ive illa &longs;it <lb/>inanima &longs;ua majore gravitate contrani­<lb/>tens, &longs;ive &longs;it vivens &longs;uo impetu &longs;ur&longs;um <lb/>trahens, & po&longs;tmodum remi&longs;&longs;iore impe­<lb/>tu, & nervorum contentione impediens, <lb/>ne &longs;axum elevatum relabatur. </s> <s id="s.004616">At &longs;i ip&longs;ius <pb pagenum="622" xlink:href="017/01/638.jpg"/>clavi A pre&longs;&longs;io, &longs;ive inflexio con&longs;ideretur, jam vis efficien­<lb/>di pre&longs;&longs;ionem hanc, &longs;eu inflexionem, tota tribuenda e&longs;t &longs;axo <lb/>P, quod propterea inducit rationem Potentiæ, & retinacu­<lb/>lum e&longs;t paxillus E in terram firmiter depactus, qui nihil agit, <lb/>&longs;ed funem duntaxat retinet. </s> <s id="s.004617">Verùm in hac po&longs;itione mo­<lb/>mentum &longs;axi adversùs clavum non e&longs;t &longs;implicis gravitatis ab­<lb/>&longs;olutè acceptæ, perinde atque &longs;i funis FG infra orbiculum <lb/>reflexus colligeretur in nodum cum fune DC: tunc enim, <lb/>collecto in nodum fune, orbiculus e&longs;&longs;et planè otio&longs;us, & ni­<lb/>hil conferret ad momentorum varietatem, &longs;ed idem accide­<lb/>ret, ac &longs;i funis &longs;implex proximè & immediatè clavo ad­<lb/>necteretur &longs;epo&longs;itâ quacumque trochleâ. </s> <s id="s.004618">Sed fune in E re­<lb/>ligato (qua&longs;i duplex pondus ex &longs;implici fune penderet) ge­<lb/>minatur &longs;axi P momentum adversùs clavum, qui nequit vel <lb/>minimum flecti, quin duplo motu &longs;axum ip&longs;um moveatur, <lb/>neque enim, quod ad geminandum momentum &longs;pectat, dif­<lb/>fert &longs;axum à potentiâ vivente, quæ utique in C applicata <lb/>funi, & trochleam trahens, adversùs pondus trochleæ ad­<lb/>nexum habet momentum duplum ejus, quod obtineret, &longs;i <lb/>funem &longs;implicem traheret: e&longs;t autem trochleæ adnexus <lb/>clavus. </s> </p> <p type="main"> <s id="s.004619">Quod &longs;i in G contra &longs;axum P aut gravitas inanimata, aut <lb/>potentia vivens nitatur, &longs;i quidem æqualibus conatibus hinc <lb/>& hinc certetur, atque &longs;u&longs;pen&longs;um con&longs;i&longs;tat &longs;axum, aut clavus <lb/>&longs;imiliter premitur atque libræ agina, cùm jugum à duobus <lb/>æqualibus ponderibus in æquilibrio retinetur, aut alterutri <lb/>munus Potentiæ, & alteri Retinaculi ad&longs;cribendum e&longs;t, & <lb/>Potentia &longs;imiliter geminato momento clavum trahit deor­<lb/>&longs;um. </s> <s id="s.004620">Sin autem aut &longs;axum P, aut virtus movendi in G, <lb/>&longs;uperat, huic Potentiæ ratio tribuatur, oppo&longs;ito munus re­<lb/>tinaculi; &longs;ed Potentiæ ab&longs;olutè acceptæ momenta non ge­<lb/>minantur, quia retinaculum &longs;tabile non e&longs;t, &longs;ed cedit; <lb/>adeóque impetus à Potentia productus duos motus efficit, <lb/>alterum trahendo retinaculum, alterum inflectendo clavum, <lb/>qui propterea minùs flectitur, quò magis oppo&longs;itum retina­<lb/>culum movetur. </s> </p> <p type="main"> <s id="s.004621">Neque hæc quicquam habent admirationis: Nam &longs;i Vectis <lb/>&longs;it CD, habens in C hypomochlium; in medio autem <pb pagenum="623" xlink:href="017/01/639.jpg"/>puncto E adnexus &longs;it funiculus, qui incumbens orbiculo F <lb/>ver&longs;atili adnexum habeat pondus S <lb/><figure id="id.017.01.639.1.jpg" xlink:href="017/01/639/1.jpg"/><lb/>innixum plano &longs;ubjecto; utique in ex­<lb/>tremitate D pondus V paulo majus <lb/>quàm &longs;ubduplum ponderis S illud ele­<lb/>vabit, atque præcisè &longs;ubduplum non <lb/>elevabit quidem illud, &longs;ed adversùs <lb/>orbiculum F conatur momento duplo <lb/>ejus, quod obtineret, &longs;i ex E pende­<lb/>ret ip&longs;um pondus V, cui reluctaretur <lb/>pondus S gravius innixum plano. </s> <s id="s.004622">Ve­<lb/>rùm ad vectem retinendum in po&longs;itio­<lb/>ne horizontali CD nihil intere&longs;t. </s> <lb/> <s id="s.004623">utrùm in C aliquid &longs;uperius &longs;it prohibens, ne illa extremitas <lb/>vi ponderis V attollatur, an verò inferiùs funiculo connectatur <lb/>cum tellure, aut ex C pendeat onus H (&longs;ed plano &longs;ubjecto in­<lb/>nixum) vel æquale ip&longs;i V, vel eo majus; &longs;emper enim pondus <lb/>V eadem obtinet momenta. </s> <s id="s.004624">Quare &longs;i, amoto orbiculo F & pon­<lb/>dere S, manu retineas funiculum IE, percipies ad &longs;ervandum <lb/>vectem horizontalem, quantâ virium acce&longs;&longs;ione tibi opus &longs;it, <lb/>&longs;upra quàm exigeret &longs;implex gravitas ponderis V, &longs;i ex E pen­<lb/>deret, ubi nulla Vectis ratio intercederet. </s> </p> <p type="main"> <s id="s.004625">Cum itaque hæc in Vecte pariter ratione po&longs;itionis pon­<lb/>deris contingant, quæ trochleæ accidere diximus ratione <lb/>connexionis ponderis vel cum trochleâ, vel cum paxillo <lb/>telluri infixo, nil mirum &longs;i alia atque alia &longs;int eju&longs;dem pon­<lb/>deris momenta adversùs clavum. </s> <s id="s.004626">Sicut autem quando tam ab <lb/>hypomochlio quàm à potentiâ &longs;u&longs;tinetur onus in medio vecte <lb/>&longs;u&longs;pen&longs;um, hypomochlium à pondere non premitur ni&longs;i juxta <lb/>&longs;emi&longs;&longs;em gravitatis ponderis; ita quoque cum funis ductarius <lb/>alterâ extremitate adnexus e&longs;t clavo, alterâ retinetur à poten­<lb/>tia, pondus ex trochleâ &longs;implici pendens partim à Potentiâ, <lb/>partim à clavo &longs;u&longs;tinetur, adversùs quem minus virium exercet <lb/>ejus gravitas, ut con&longs;tabit, &longs;i clavo orbiculum ver&longs;atilem in­<lb/>figas, & funiculo per orbiculi orbitam excavatam tran&longs;eunti <lb/>aliud pondus adnectas, quod &longs;atis erit, &longs;i fuerit &longs;ubduplum <lb/>ponderis ex trochleâ pendentis; hoc enim &longs;u&longs;tinebitur à <lb/>duplici virtute &longs;ubduplâ gravitatis illius. </s> <s id="s.004627">Non igitur plus <pb pagenum="624" xlink:href="017/01/640.jpg"/>re&longs;i&longs;tentiæ requiritur in clavo, quàm in pondere illo &longs;ub­<lb/>duplo. </s> </p> <p type="main"> <s id="s.004628">His ita in unicâ &longs;implici trochleâ con&longs;titutis, examinandæ <lb/>&longs;unt trochleæ conjugatæ; nec difficile erit ex dictis &longs;uperiore <lb/>capite inve&longs;tigare momenta ponderis adversùs clavum, cui al­<lb/>tera trochlea adnectitur. </s> <s id="s.004629">Ibi enim alteri trochleæ potentiam <lb/>&longs;ur&longs;um trahentem, alteri pondus dependens adnecti po&longs;uimus, <lb/>funis verò extremitatem clavo alligari: Hìc loco clavi illius re­<lb/>tinentis extremitatem funis intelligendum e&longs;t retinaculum, <lb/>quodcumque tandem illud &longs;it, &longs;ive manus hominis, &longs;ive etiam <lb/>alius clavus: &longs;ed loco Potentiæ &longs;uperiorem trochleam &longs;ur&longs;um <lb/>trahentis &longs;it clavus, ex quo trochleæ fune ductario connexæ <lb/>unà cum pondere dependent; gravitas autem illa &longs;u&longs;pen&longs;a ex <lb/>inferiore trochleâ exercet munus potentiæ adversùs clavum, <lb/>qui &longs;ubit vicem ponderis movendi, quatenus aliquantulum <lb/>flectitur, aut inflexioni repugnat. </s> <s id="s.004630">Sicut ergo ibi o&longs;ten&longs;um e&longs;t <lb/>in duabus &longs;implicibus trochleis &longs;ingulos orbiculos habentibus, <lb/>&longs;i funis ductarij caput alligatum &longs;it &longs;uperiori trochleæ, motum <lb/>trochleæ &longs;uperioris ad motum inferioris e&longs;&longs;e ut 2 ad 3; &longs;i verò <lb/>funis caput alligatum fuerit inferiori trochleæ, motum &longs;uperio­<lb/>ris ad motum inferioris e&longs;&longs;e ut 3 ad 2: Ita hìc dicendum e&longs;t <lb/>(ponamus clavum flecti aliquantulum) in primo ca&longs;u motum <lb/>flexionis clavi ad motum de&longs;censûs ponderis e&longs;&longs;e ut 2 ad 3, in <lb/>&longs;ecundo autem ca&longs;u ut 3 ad 2. Ex quo fit in primo ca&longs;u pon­<lb/>dus habere adversùs clavum majus momentum quàm in &longs;ecun­<lb/>do ca&longs;u; & in primo ca&longs;u validiùs deor&longs;um trahere, quàm &longs;i <lb/>&longs;implici funiculo dependeret, & motus e&longs;&longs;ent æquales; major <lb/>&longs;iquidem e&longs;t Ratio 3 ad 2, quàm 2 ad 2; in &longs;ecundo verò ca&longs;u <lb/>debiliùs deor&longs;um trahere, quàm &longs;i nullæ e&longs;&longs;ent trochleæ, adeó­<lb/>que motus æquales e&longs;&longs;ent; minor quippe e&longs;t Ratio 2 ad 3, quam <lb/>1 ad 1, aut 3 ad 3. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004631">Simili planè methodo philo&longs;ophandum e&longs;t in reliquis tro­<lb/>chleis conjugatis: &longs;i enim duabus trochleis di&longs;par in&longs;it orbicu­<lb/>lorum numerus, ut altera major &longs;it, altera minor, ob&longs;ervandum <lb/>e&longs;t, an major trochlea alligetur clavo, an verò minor: Si tro­<lb/>chlea plures habens orbiculos clavo adnectatur, motus flexionis <lb/>clavi minor e&longs;t motu de&longs;censûs ponderis in Ràtione &longs;ub&longs;uper­<lb/>particulari denominatâ à numero omnium &longs;imul orbiculorum; <pb pagenum="625" xlink:href="017/01/641.jpg"/>ac proinde pondus habet momentum majus, quàm &longs;i nullæ in­<lb/>tercederent trochleæ: contra verò &longs;i clavo adnectatur trochlea <lb/>minor, motus flexionis clavi major e&longs;t motu de&longs;censûs ponde­<lb/>ris in Ratione &longs;uperparticulari denominatâ à numero omnium <lb/>&longs;imul orbiculorum; atque ideo pondus adversùs clavum minus <lb/>habet momenti, quàm &longs;i ex illo &longs;implici fune penderet. </s> <s id="s.004632">At &longs;i <lb/>utriu&longs;que trochleæ par &longs;it orbiculorum numerus, & pariter ra­<lb/>tio &longs;uperparticularis, aut &longs;ub&longs;uperparticularis denominata à <lb/>numero omnium &longs;imul orbiculorum; & &longs;i quidem trochleæ &longs;u­<lb/>periori adnectatur funis caput, pondus adversùs clavum habet <lb/>momentum majus, quàm &longs;i amotis trochleis ex &longs;implici fune <lb/>penderet; &longs;in autem inferiori trochleæ alligetur extremitas fu­<lb/>nis ductarij, ponderis momentum adversùs clavum minùs <lb/>e&longs;t, quàm &longs;i idem pondus ex eodem clavo &longs;implici fune &longs;u&longs;­<lb/>penderetur. </s> </p> <p type="main"> <s id="s.004633">Ex his &longs;atis apparet clavo eandem vim inferri, &longs;i pondus de­<lb/>pendens ex trochleis &longs;u&longs;pen&longs;um maneat, &longs;ive quia funis extre­<lb/>mitas religetur paxillo, &longs;ive quia ex eâdem funis extremitate <lb/>dependeat onus &longs;ubmultiplex ponderis ex inferiore trochleâ <lb/>pendentis, &longs;ecundùm Rationem, quam inferunt ip&longs;i orbiculi. </s> <lb/> <s id="s.004634">Sic ex trochleis binos orbiculos habentibus dependeat pondus, <lb/>& funis extremitas religetur paxillo: ex dictis, &longs;uperioris tro­<lb/>chleæ clavo adnexæ, & funis ductarij caput habentis, motus, <lb/>ad motum inferioris trochleæ & ponderis e&longs;t &longs;ub&longs;e&longs;quiquartus; <lb/>ac proinde pondus trochleis connexum cum clavo ad vim illi <lb/>inferendam perinde &longs;e habet, atque &longs;i ex eodem clavo ab&longs;que <lb/>trochleis &longs;implici fune appenderetur pondus aliud dati ponde­<lb/>ris Se&longs;quiquartum. </s> <s id="s.004635">At &longs;i extremitati funis adderetur pondus va­<lb/>lens &longs;u&longs;pendere onus adnexum trochleæ inferiori, e&longs;&longs;et ex dictis <lb/>cap.1. dati oneris &longs;ubquadruplum. </s> <s id="s.004636">Igitur duorum horum pon­<lb/>derum &longs;umma ad datum pondus e&longs;&longs;et ut 5 ad 4, cuju&longs;modi erat <lb/>Ratio motuum, ex quibus momentum de&longs;umitur. </s> <s id="s.004637">An non &longs;i <lb/>onus in plano horizontali raptandum &longs;implici trochleæ adne­<lb/>xum proponatur, & duo homines pariter utramque funis ex­<lb/>tremitatem arripiant, atque trahant, &longs;inguli medietatem ne­<lb/>ce&longs;&longs;arij conatûs adhibent? </s> <s id="s.004638">&longs;i verò alter trahentium deficiat, & <lb/>illa funis extremitas alligetur paxillo, nonne qui reliquus e&longs;t <lb/>eodem conatu trahens &longs;olus adducet idem pondus? </s> <s id="s.004639">non ni&longs;i <pb pagenum="626" xlink:href="017/01/642.jpg"/>quia, cùm ambo trahebant, pondus & potentia æqualiter mo­<lb/>vebantur; cùm alter tantùm trahit, ille movetur duplo velo­<lb/>cius, quàm pondus, quod ad &longs;ubduplam velocitatem &longs;atis ha­<lb/>bet impetum &longs;ubduplum impetûs nece&longs;&longs;arij ad velocitatem <lb/>æqualem. </s> <s id="s.004640">Eadem igitur militat ratio in clavo, cui vis infertur <lb/>à duobus ponderibus &longs;u&longs;pen&longs;is ex trochleis in æquilibrio, quæ <lb/>&longs;imul deor&longs;um trahentia motum habent æqualem cum motu <lb/>clavi, qui flectitur, aut revellitur; &longs;ed funis capite religato fir­<lb/>miter ad paxillum, pondus inferiori trochleæ adnexum motum <lb/>habet velociorem comparatum cum eju&longs;dem clavi motu, ac <lb/>propterea majus momentum habet. </s> </p> <p type="main"> <s id="s.004641">Hinc præterea inferendum e&longs;t non &longs;atis utiliter eos opera­<lb/>ri, qui pondus ex &longs;uperiore loco fune &longs;u&longs;pen&longs;um, &longs;ive orbicu­<lb/>lus intercedat, &longs;ive non, putant firmius &longs;u&longs;tineri, &longs;i funis ca­<lb/>put in inferiore loco religetur: &longs;i enim funis excurrere ne­<lb/>queat, inferiùs hoc retinaculum pror&longs;us inutile accidit, &longs;in au­<lb/>tem excurrere valeat, &longs;uperius illud retinaculum geminatam <lb/>vim &longs;u&longs;cipit, qua&longs;i duplex pondus ab illo &longs;u&longs;tineretur. <lb/></s> </p> <p type="main"> <s id="s.004642"><emph type="center"/>CAPUT VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004643"><emph type="center"/><emph type="italics"/>Aliqui Trochlearum u&longs;us indicantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004644">PRo more in &longs;uperioribus libris &longs;ervato, hìc pariter indican­<lb/>di &longs;unt aliqui Trochlearum u&longs;us, qui facilè ad &longs;imilia <lb/>traduci poterunt, &longs;pectato motu, qui exhibendus proponitur, <lb/>ut ei trochleæ re&longs;pondeant, & aptè collocentur, neque plu­<lb/>ribus, quàm opus &longs;it, orbiculis in&longs;truantur, ne dum poten­<lb/>tiæ facilitatem con&longs;ectaris, nimis tardè moveas pondus, aut <lb/>ex adver&longs;o, dum ponderi velocitatem concilias, nimio labore <lb/>potentiam opprimas. </s> </p> <pb pagenum="627" xlink:href="017/01/643.jpg"/> <p type="main"> <s id="s.004645"><emph type="center"/>PROPOSITIO I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004646"><emph type="center"/><emph type="italics"/>Auram in Conclavi excitare.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004647">QUæritur &longs;æpè æ&longs;tivo tempore aliquod ex aëris motu re­<lb/>frigerium; &longs;ed manuali flabello auram excitare aliquan­<lb/>do incommodum e&longs;t, &longs;i aliud agendo di&longs;tinearis: propterea <lb/>ventilabrum in conclavis angulo &longs;tatuere po&longs;&longs;umus, quod ali­<lb/>quandiu moveatur, aërémque agitet: ideóque illud ad angu­<lb/>lum &longs;tatuendum propo&longs;ui, ut commotus aër in proximos hinc <lb/>atque hinc parietes impactus reflectatur, & faciliùs reliquum <lb/>conclavis aërem exagitet. </s> </p> <p type="main"> <s id="s.004648">Excitetur angulo congruens turricula haud ab&longs;imilis iis, <lb/>quibus horologia reconduntur; in &longs;upremâ turriculæ parte <lb/>ab angulo ad oppo&longs;itum ex diametro angulum Axis horizon­<lb/>ti parallelus &longs;tatuatur facilè ver&longs;atilis, cujus tamen pars ex­<lb/>tra turriculam promineat tantæ longitudinis, quanta flabel­<lb/>lis latitudo de&longs;tinatur. </s> <s id="s.004649">Pars tamen hæc Axis extima nul­<lb/>lam exigit certam figuram, nihilque refert &longs;ive cylindrica <lb/>&longs;it, &longs;ive quadrata, &longs;ive quæcumque alia; modò ea &longs;it, ut illi <lb/>facilè flabella firmiter infigi, atque eximi pro opportunitate <lb/>po&longs;&longs;int, ii&longs;que exemptis aptari valeat manubrium, quo faciliùs <lb/>& citiùs ab homine convolvatur Axis. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004650">Parentur duæ trochleæ ternis orbiculis in&longs;tructæ, altera <lb/>in &longs;uperiore turriculæ loco firmetur, altera ad turriculæ pe­<lb/>dem con&longs;tituatur adnexam habens plumbeam ma&longs;&longs;am motui <lb/>perficiendo congruentem: huic eidem trochleæ adnectatur <lb/>extremitas funis ductarij, qui per omnes trochlearum orbicu­<lb/>los trajectus demum ad Axem referatur, ibíque alligetur. </s> <s id="s.004651">Tum <lb/>appo&longs;ito manubrio convolutum Axem circumplectetur funis <lb/>ductarius, & plumbea ma&longs;&longs;a in &longs;upremam turriculæ partem <lb/>Axi proximam deducetur. </s> <s id="s.004652">Infigantur Axi ventilabra, & amo­<lb/>to manubrio plumbea ma&longs;&longs;a &longs;ibi relicta lenti&longs;&longs;imo motu de&longs;cen­<lb/>det, convolvén&longs;que axem cum flabellis tandiu aërem commo­<lb/>vebit, quandiu illa de&longs;cendet. </s> </p> <p type="main"> <s id="s.004653">Hìc commutatas vices inter potentiam & pondus ob&longs;ervare <lb/>quilibet pote&longs;t; potentia &longs;iquidem movens e&longs;t plumbea ma&longs;-<pb pagenum="628" xlink:href="017/01/644.jpg"/>&longs;a, quæ &longs;eptuplo tardiùs movetur quàm extremitas funis <lb/>ductarij, quem aliàs trahere &longs;olita e&longs;t potentia. </s> <s id="s.004654">Loco autem <lb/>ponderis e&longs;t aër, qui à flabellis impellitur; ac proinde quò <lb/>ampliora &longs;unt flabella, eò major e&longs;t re&longs;i&longs;tentia aëris com­<lb/>moti, ratione cujus etiam retardatur motus potentiæ. </s> <s id="s.004655">Ubi <lb/>quoquè attendenda e&longs;t Ratio longitudinis flabellorum ad &longs;e­<lb/>midiametrum axis convoluti: nam &longs;i hæc Ratio componatur <lb/>cum Ratione &longs;eptuplâ, quam Trochleæ inferunt, habebitur <lb/>Ratio motûs extremi flabelli ad motum ma&longs;&longs;æ plumbeæ: <lb/>quamquam non ita computanda e&longs;t aëris re&longs;i&longs;tentia, qua&longs;i to­<lb/>ta in flabelli extremitate exerceretur; hæc &longs;cilicet per univer­<lb/>&longs;am flabelli longitudinem diffunditur inæqualiter di&longs;tributa <lb/>pro Ratione di&longs;tantiæ à centro motûs, aër quippe pro diversâ <lb/>impellentis velocitate inæqualiter re&longs;i&longs;tit. </s> </p> <p type="main"> <s id="s.004656">Quod &longs;i magis arrideret non continua convolutione flabel­<lb/>la circumagi, &longs;ed alternâ quadam modò in dextram, modò <lb/>in &longs;ini&longs;tram inflexione agitari; Axi, quem funis ductarius <lb/>complectitur, infige rotam dentatam, cujus dentes incurrant <lb/>in pinnulas fu&longs;i perpendicularis flabella &longs;u&longs;tinentes, quemad­<lb/>modum in Tempore horologij: &longs;imili enim ratione, ac Tem­<lb/>pus, ultrò citróque remeabunt flabella, & aërem in oppo&longs;itas <lb/>partes commovebunt. </s> <s id="s.004657">Cùm verò antè motum appo&longs;ito manu­<lb/>brio convolvendus erit Axis, ut funem ductarium recipiat, at­<lb/>que trochlea inferior cum pondere attollatur, ita fu&longs;um pau­<lb/>li&longs;per elevare oportebit, ut ejus pinnulæ non occurrant denti­<lb/>bus rotæ, ni&longs;i cùm iterum fu&longs;us &longs;uum in locum re&longs;tituetur. </s> <lb/> <s id="s.004658">Hac alternatione diuturnior erit motus. </s> </p> <p type="main"> <s id="s.004659"><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004660"><emph type="center"/><emph type="italics"/>Corpus aliquod in gyrum celeriter volvere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004661">IN rebus &longs;cenicis locum habere non infrequentem pote&longs;t <lb/>hæc propo&longs;itio: aliquando &longs;cilicet &longs;olis di&longs;cum in &longs;cenam <lb/>producimus, quem licet auro obductum, ac multis facibus <lb/>illu&longs;tratum, quas &longs;pectatorum oculis ex arte &longs;ubducimus, <lb/>non tamen radios ejaculantem mentimur, ni&longs;i ille circa <pb pagenum="629" xlink:href="017/01/645.jpg"/>&longs;uum centrum velociter circumagatur. </s> <s id="s.004662">Id quod variá qui­<lb/>dem methodo præ&longs;tari pote&longs;t infixum di&longs;ci centro cylindrum <lb/>convolvendo, &longs;ive ope rotæ dentatæ Vertebram &longs;triatam cy­<lb/>lindro circumpo&longs;itam moventis; &longs;ive fune cylindrum bis aut <lb/>ter arctè complexo, & in &longs;e&longs;e redeunte, ubi majoris alicu­<lb/>jus tympani orbitam pariter complexus fuerit; &longs;ive pondere <lb/>funem cylindro involutum explicante: &longs;ed po&longs;tremus hic <lb/>modus non ni&longs;i breve temporis &longs;patium exigit; duo priores, <lb/>&longs;i paulo longior futurus &longs;it motus, non ni&longs;i à potentiá vi­<lb/>vente commodè exhiberi po&longs;&longs;unt. </s> <s id="s.004663">Quare &longs;atius fuerit tro­<lb/>chleas, ut in &longs;uperiore propo&longs;itione, di&longs;po&longs;itas adhibere, at­<lb/>que loco flabellorum &longs;olis di&longs;cum Axi adnectere; &longs;ic enim <lb/>fiet, ut & celeriter in gyrum agatur, & diu per&longs;everet <lb/>motus. </s> </p> <p type="main"> <s id="s.004664">Similiter ad fingendum mare, & undarum motum vehe­<lb/>mentiorem, &longs;tatuuntur horizonti & invicem paralleli ali­<lb/>quot axes, quos ambiunt &longs;piræ profundiùs excavatæ colore <lb/>marinam undam imitantes: dum enim huju&longs;modi axes con­<lb/>volvuntur, marini æ&longs;tûs cur&longs;um &longs;pectatoribus repræ&longs;entant. </s> <lb/> <s id="s.004665">Ut autem axes illi citra cuju&longs;quam laborem volvantur tro­<lb/>chleas duas binis, aut ternis orbiculis in&longs;tructas (prout diu­<lb/>turnior motus requiritur) compone, & proximas &longs;tatue, al­<lb/>teram firmans in &longs;uperiore loco: Tum funis ductarius per <lb/>omnes Trochlearum orbiculos trajectus &longs;ingulorum axium ca­<lb/>pita ex ordine ambiat unâ &longs;altem aut alterâ &longs;pirâ, & demum <lb/>ad peculiarem alium axem deveniat, quem totus plures in <lb/>&longs;piras complicatus circumplectatur, ita tamen, ut facilè <lb/>evolvi queat. </s> <s id="s.004666">Ubi igitur tempus advenerit, inferiori tro­<lb/>chleæ congruum pondus adnecte; hoc enim licèt lentè de­<lb/>&longs;cendat, velociter tamen axes convolvit funem evolvens. </s> <lb/> <s id="s.004667">Procellam verò mite&longs;cere aut exa&longs;perari mentieris, factâ <lb/>ponderis aliquâ detractione aut acce&longs;&longs;ione, id quod difficile <lb/>non fuerit. </s> </p> <pb pagenum="630" xlink:href="017/01/646.jpg"/> <p type="main"> <s id="s.004668"><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004669"><emph type="center"/><emph type="italics"/>Se ip&longs;um ope trochlearum in altum evehere, aut promovere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004670">SElla paretur hinc & hinc habens fulcra, quibus brachia <lb/>innituntur, & in huju&longs;modi fulcrorum extremitate ante­<lb/>riore aptetur Sucula manubriata, quam &longs;edens commodè ver­<lb/>&longs;are valeat: &longs;ella autem quatuor funibus in nodum cum an­<lb/>nulo coëuntibus &longs;u&longs;pendatur ita, ut inferioris trochleæ uncus <lb/>annulo indatur, & funis ductarius per cunctos trochlearum <lb/>orbiculos trajectus demum &longs;uculæ alligetur. </s> <s id="s.004671">Nam in &longs;ellâ &longs;e­<lb/>dens, & &longs;uculæ manubria convertens, funem ductarium <lb/>trahit, atque ip&longs;e &longs;e in altum evehit eâ facilitate, quam in­<lb/>fert Ratio compo&longs;ita ex Rationibus trochlearum, & Suculæ: <lb/>e&longs;t &longs;iquidem Potentia ip&longs;a virtus animalis mu&longs;culorum con­<lb/>tentione ver&longs;ans manubria, pondus autem e&longs;t in&longs;ita corpori <lb/>gravitas, quæ eò minor apparet, quo majores &longs;unt, hoc e&longs;t <lb/>pluribus in&longs;tructæ orbiculis, trochleæ, & major e&longs;t Ratio <lb/>manubriorum ad &longs;emidiametrum Axis, qui fune obvolvitur. </s> <lb/> <s id="s.004672">Sit enim ex. </s> <s id="s.004673">gr. <!-- REMOVE S-->inferior trochlea, cui pondus movendum ad­<lb/>nectitur, & funis ductarij extremitas alligatur, orbiculorum <lb/>duorum; &longs;uperior autem trochlea, quæ &longs;tabilis manet, tres <lb/>habeat orbiculos: utique Ratio motûs potentiæ ad motum <lb/>ponderis e&longs;t quintupla: manubria autem Suculæ &longs;int quadru­<lb/>pla &longs;emidiametri Axis: Ratio compo&longs;ita ex quadruplâ & quin­<lb/>tuplâ e&longs;t vigecuplâ; igitur conatus Potentiæ manubria ver&longs;an­<lb/>tis &longs;atis e&longs;t, &longs;i re&longs;pondeat vige&longs;imæ parti ponderis. </s> </p> <p type="main"> <s id="s.004674">Similiter &longs;i cymba adver&longs;o flumine non procul à ripâ dedu­<lb/>cenda &longs;it, & qui in eâ &longs;unt nautæ, ita pauci &longs;int, ut non va­<lb/>leant eam adversùs vim profluentis remo agere, aut è ripá fu­<lb/>ne nautico trahere; &longs;ub&longs;idium ex trochleis petere poterunt; <lb/>ex&longs;cen&longs;u &longs;cilicet in terram facto, atque defixo in ripâ paxil­<lb/>lo alligatur trochlea una, altera adnectitur proræ cymbæ, <lb/>in qua nautæ duo funem ductarium trahentes illam adver&longs;o <lb/>flumine promovent perinde, atque &longs;i e&longs;&longs;ent octo aut duode­<lb/>cim homines, &longs;i trochleæ binos aut ternos habuerint orbicu-<pb pagenum="631" xlink:href="017/01/647.jpg"/>los. </s> <s id="s.004675">Quod &longs;i trochleis illi careant, utantur artificio &longs;equentis <lb/>propo&longs;itionis, unà cum iis, quæ cap. 5. dicta &longs;unt. </s> </p> <p type="main"> <s id="s.004676"><emph type="center"/>PROPOSITIO IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004677"><emph type="center"/><emph type="italics"/>Trochlearum defectum &longs;upplere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004678">EX his, quæ cap. 2. hujus libri indicata &longs;unt, &longs;atis con&longs;tat <lb/>etiam &longs;inè orbiculis haberi po&longs;&longs;e momentum Trochlea­<lb/>rum: quare his deficientibus annulos &longs;ufficere facile erit. <lb/><figure id="id.017.01.647.1.jpg" xlink:href="017/01/647/1.jpg"/><lb/>Et primo quidem &longs;ingulis annulis uti po&longs;&longs;umus: nam cùm <lb/>cymba communiter adnexum proræ annulum habeat, ut <lb/>medio fune, aut catenâ ad ripam religetur, funis ducta­<lb/>rius unus AB adnectatur paxillo A in ripâ defixo, & per <lb/>cymbæ annulum B trajiciatur; illius alteri extremitati C an­<lb/>nulus alius adnectatur, per quem alter ductarius funis DEF <lb/>trajectus & paxillo D alligatus &longs;i à Potentiâ in F con&longs;titutâ <lb/>trahatur, illa habebit momentum quadruplum, perinde at­<lb/>que de orbiculi, &longs;uperiùs dictum e&longs;t cap. 5. At &longs;i con&longs;i&longs;tentes <lb/>in cymbâ trahere illam velint nautæ, adjiciatur paxillo D an­<lb/>nulus G &longs;tabilis, per quem productus funis EF tran&longs;eat, & <lb/>veniat in H ad nautarum manus in cymbâ; nam illum trahen­<lb/>do cymbæ prora ex B accedet ad A. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004679">Porrò annuli nomine notatum volo quicquid eju&longs;modi e&longs;t, <lb/>ut funis per illud trajici po&longs;&longs;it, & liberè excurrere, &longs;ive &longs;it <lb/>ligni fru&longs;tum foramen habens politum & &longs;atis amplum, ut <lb/>per illud funis facilè moveri valeat, &longs;ive etiam &longs;it flexilis <pb pagenum="632" xlink:href="017/01/648.jpg"/>bacilli particula in arcum vel modicè &longs;inuata; modò illa non <lb/>&longs;it fractioni obnoxia. </s> <s id="s.004680">Illud autem in annullis ob&longs;ervandum <lb/>e&longs;t, quod faciliùs excurrit funis, &longs;i illi cra&longs;&longs;iores fuerint & po­<lb/>liti, quàm &longs;i exiles & a&longs;peri. </s> </p> <figure id="id.017.01.648.1.jpg" xlink:href="017/01/648/1.jpg"/> <p type="main"> <s id="s.004681">Quod &longs;i annulis Trochleas propiùs æmulari placuerit, duos <lb/>annulos R & S alliga paxillo V, duó&longs;que alios H & G ad­<lb/>necte in M ponderi trahendo: Tum funem ductarium eidem <lb/>paxillo V alligatum trajice primùm per annulum G, deinde <lb/>per annulum S, hinc per annulum H, demum per annulum R. <!-- KEEP S--></s> <lb/> <s id="s.004682">Nam &longs;i extremitati I potentia trahens applicetur, movebitur <lb/>quadruplo velociùs, quàm pondus in M, adeóque etiam ha­<lb/>bebit momentum quadruplum. </s> <s id="s.004683">Ne autem funes ob nimiam <lb/>propinquitatem &longs;ibi invicem impedimento &longs;int &longs;e mutuo con­<lb/>flictu atterentes, annulos tran&longs;ver&longs;is bacillis ON, & LP dis­<lb/>junge. </s> </p> <p type="main"> <s id="s.004684"><emph type="center"/>PROPOSITIO V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004685"><emph type="center"/><emph type="italics"/>Re&longs;i&longs;tentiam ex axium cum orbiculis conflictu in Trochleis <lb/>examinare.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004686">QUoniam variarum Trochlearum u&longs;um indicavimus, mo­<lb/>dò trochleâ alterâ manente atque &longs;tabili, modò utra­<lb/>que commotâ, placet hìc examinare propo&longs;itas duas tro­<lb/>chleas, an aliquid impedimenti afferant ex conflictu axium <lb/>cum orbiculis, aut etiam trochleas comparare cum annulis <pb pagenum="633" xlink:href="017/01/649.jpg"/>earum loco adhibitis, quantum videlicet præ trochleis afferat <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/>examine habens in C; <lb/><figure id="id.017.01.649.1.jpg" xlink:href="017/01/649/1.jpg"/><lb/>adnectatur in A tro­<lb/>chlea &longs;uperior, ex qua <lb/>cum inferiore trochleâ <lb/>pendeat &longs;axum F notæ <lb/>gravitatis, & funis <lb/>ductarij extremitas re­<lb/>ligetur clavo in E. <!-- KEEP S--></s> <s id="s.004688">In­<lb/>note&longs;cat autem tùm tro­<lb/>chlearum &longs;ingularum, <lb/>tùm funis ductarij gra­<lb/>vitas, ut congruum pon­<lb/>dus parari po&longs;&longs;it in B ap­<lb/>pendendum. </s> <s id="s.004689">Clavus igi­<lb/>tur E &longs;u&longs;tinet inferioris <lb/>trochleæ & adnexi &longs;axi <lb/>gravitatis partem quin­<lb/>tam, reliquas quatuor <lb/>quintas partes, & præ­<lb/>terea trochleæ &longs;uperio­<lb/>ris, atque quatuor duc­<lb/>tuum funis gravitatem <lb/>&longs;u&longs;tinet brachium libræ <lb/>in A. <!-- KEEP S--></s> <s id="s.004690">Quare in B tantum ponderis apponendum e&longs;t, quan­<lb/>tum &longs;ufficiat ad æquilibrium; proinde &longs;en&longs;im augendum e&longs;t <lb/>pondus in D, donec examen in C æqualitatem momentorum <lb/>indicet. </s> <s id="s.004691">Hoc peracto adde adhuc ponderi D aliam atque aliam <lb/>gravitatem, u&longs;que dum brachium B deor&longs;um inclinetur: hu­<lb/>ju&longs;modi enim additamentum indicabit re&longs;i&longs;tentiam ortam ex <lb/>conflictu axium cum orbiculis. </s> </p> <p type="main"> <s id="s.004692">Jam &longs;i trochlearum loco annulos &longs;ub&longs;tituas, eadémque me­<lb/>thodo invento primùm æquilibrio, deinde factâ in D ponderis <lb/>acce&longs;&longs;ione præponderantiam quæras, deprehendes, quanto <lb/>major re&longs;i&longs;tentia ex funis ductarij cum annulis affrictu oriatur, <lb/>quàm ex axium cum &longs;uis orbiculis conflictu in trochleis. </s> </p> <pb pagenum="634" xlink:href="017/01/650.jpg"/> <p type="main"> <s id="s.004693">Simili ratione &longs;i in A &longs;it clavus, cui &longs;uperior trochlea &longs;tabi­<lb/>lis permanens affigatur; extremitas verò funis ductarij M ad­<lb/>nectatur brachio libræ ad æquilibrium con&longs;tituendum &longs;ufficit <lb/>in N quinta pars gravitatis trochleæ inferioris unà cum &longs;axo F, <lb/>& ductu funis MI. </s> <s id="s.004694">Facto igitur in H additamento gravitatis, <lb/>ut tollatur æquilibrium, indicabitur quanta re&longs;i&longs;tentiæ acce&longs;&longs;io <lb/>fiat ponderi F ex axium cum orbiculis conflictu: atque &longs;imili­<lb/>ter repo&longs;itis loco trochlearum annulis, po&longs;t æquilibrium aucto <lb/>pondere N donec deprimatur, innote&longs;cet re&longs;i&longs;tentia orta ex fu­<lb/>nis cum annulis affrictu. </s> </p> <p type="main"> <s id="s.004695">Hinc apparet primò &longs;atius e&longs;&longs;e hoc po&longs;teriore modo ope­<lb/>rari, quia longè minus pondus requiritur in N, quàm in D. <!-- KEEP S--></s> <lb/> <s id="s.004696">Secundò ad tollendum pondus F cum trochlea inferiore, &longs;i <lb/>&longs;uperior fixa maneat, tantam vim in potentiâ requiri, quan­<lb/>tâ opus e&longs;&longs;et ad attollendum ab&longs;que ullâ machinâ pondus N <lb/>præponderans; ad attollendum verò idem &longs;axum F cum <lb/>utrâque trochleâ, trahendo &longs;cilicet &longs;ur&longs;um trochleam A, <lb/>tantam vim exigi in potentia, quanta requiritur ad attol­<lb/>lendum pondus D præponderans. </s> <s id="s.004697">Tertiò, retentis ii&longs;dem tro­<lb/>chleis, &longs;ed mutato pondere F, examinari po&longs;&longs;e, an, & quan­<lb/>to major re&longs;i&longs;tentia oriatur ex majore pre&longs;&longs;ione axium, <lb/>quando pondus e&longs;t majus. </s> <s id="s.004698">Quartò. <!-- KEEP S--></s> <s id="s.004699">mutatis trochleis, & pon­<lb/>dere eodem retento, di&longs;paritatem aliquam inveniri, quia non <lb/>omnium trochlearum axes &longs;unt æquè teretes, ac politi, <lb/>& &longs;uorum orbiculorum foramini congruentes. </s> <s id="s.004700">Quod &longs;i, exa­<lb/>mine huju&longs;modi &longs;emel in&longs;tituto, orbiculos manu pauli&longs;per <lb/>convertas, & iterum idem examen in&longs;tituas, neque æqua­<lb/>lis inveniatur re&longs;i&longs;tentia, indicium erit foramen orbiculi, <lb/>aut forta&longs;&longs;e etiam axem, non e&longs;&longs;e, exqui&longs;itè rotundum. </s> <lb/> <s id="s.004701">Quintò. <!-- KEEP S--></s> <s id="s.004702">&longs;imili examine in annulis inito deprehendi po&longs;&longs;e, <lb/>an faciliùs &longs;uccedat tractio fune cra&longs;&longs;iore, an verò te­<lb/>nuiore. </s> </p> <p type="main"> <s id="s.004703">Non ita tamen nece&longs;&longs;e e&longs;t indicatâ methodo uti, ut, &longs;i <lb/>non placeat jugum libræ æqualium brachiorum adhibere, <lb/>nequeas loco libræ &longs;tateram applicare ut trochleæ A, aut <lb/>funis extremitati M: primùm enim indicabitur æquili­<lb/>brium: deinde longiùs reducto &longs;acomate, u&longs;que dum appa­<lb/>rere incipiat præponderatio, innote&longs;cet quantitas impedimen-<pb pagenum="635" xlink:href="017/01/651.jpg"/>ti, quin opus &longs;it gravitatem aliam atque aliam addere, ut <lb/>in librâ. </s> </p> <p type="main"> <s id="s.004704"><emph type="center"/>PROPOSITIO VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004705"><emph type="center"/><emph type="italics"/>Vim Retinaculi Trochlearum augere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004706">SÆpè contingit infixo parieti tigillo alligari &longs;uperiorem <lb/>trochleam; & ni&longs;i paries valdè firmus ac &longs;olidus fuerit, <lb/>cuju&longs;modi &longs;unt antiqui parietes, non leve periculum immi­<lb/>net, ne ponderis vi labefactetur ip&longs;e paries, maximè &longs;i recens <lb/>fuerit, & tigillus non admodum procul à &longs;ummitate infigatur; <lb/>ut &longs;i recentis parietis BC <lb/><figure id="id.017.01.651.1.jpg" xlink:href="017/01/651/1.jpg"/><lb/>foramini immittatur tigillus <lb/>brevior AH, ex quo in A <lb/>dependet trochlea, & ex illâ <lb/>pondus cum reliquâ tro­<lb/>chleâ: fieri enim pote&longs;t, ut <lb/>tigillus ip&longs;e qua&longs;i Vectis à <lb/>pondere adnexo depre&longs;&longs;us at­<lb/>tollat lateres impo&longs;itos, & <lb/>&longs;uperioris parietis compa­<lb/>gem di&longs;&longs;olvat. </s> <s id="s.004707">Foramen igi­<lb/>tur ita fiat, ut paries pervius <lb/>&longs;it, illúmque pervadat lon­<lb/>gior tigillus AF, cujus ca­<lb/>put F fune FI connectatur <lb/>cum annulo in I parieti in­<lb/>fixo: &longs;ic enim ponderis gra­<lb/>vitas nullam inferre poterit <lb/>parieti labem quamvis re­<lb/>centi; & quò longior fuerit <lb/>tigilli pars HF &longs;upra partem HA, eò validius retinebitur tro­<lb/>chlea in A, tigillo rationem Vectis habente. </s> </p> <pb pagenum="636" xlink:href="017/01/652.jpg"/> <p type="main"> <s id="s.004708"><emph type="center"/>PROPOSITIO VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004709"><emph type="center"/><emph type="italics"/>Trochleis vim Vectis augere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004710">QUamvis ex dictis obvium &longs;it Trochleas cum aliis Faculta­<lb/>tibus componere, placet tamen hìc eas cum Vecte com­<lb/>ponendas indicare. </s> <s id="s.004711">Sit prælum in torculari, &longs;ed fortè di&longs;&longs;ipa­<lb/>ta fuerit Cochlea, qua illud deor&longs;um trahebatur, aut certè ad <lb/>&longs;ubitum u&longs;um properato prælo utendum &longs;it: trabem &longs;tatue <lb/>tran&longs;ver&longs;am, quæ altero capite retineatur objecto repagulo, ne <lb/>&longs;ur&longs;um attollatur. </s> <s id="s.004712">Vectis e&longs;t &longs;ecundi generis in medio habens <lb/>pondus premendum. </s> <s id="s.004713">Alteri trabis extremitati adnectatur tro­<lb/>chlea, ejú&longs;que compar in inferiore loco firmetur: nam funem <lb/>ductarium trahentes momentum habebunt, quod ex Ratione <lb/>Vectis, & ex Ratione Trochlearum componitur. </s> </p> <p type="main"> <s id="s.004714">Simili methodo utendum e&longs;t, &longs;i Vecte &longs;ecundi generis at­<lb/>tollendum &longs;it pondus: loco enim potentiæ de&longs;tinato, hoc e&longs;t <lb/>Vectis extremitati attollendæ, adnectatur Trochlea, ejú&longs;que <lb/>compar in &longs;uperiore loco firmetur: hìc enim pariter Vectis at­<lb/>que Trochlearum Rationes componuntur. </s> <s id="s.004715">Quòd &longs;i funem Su­<lb/>culâ traxeris, aut Ergatâ, tres erunt Rationes compo&longs;itæ; dua­<lb/>bus quippe illis addenda e&longs;t Ratio Suculæ aut Ergatæ. <lb/><figure id="id.017.01.652.1.jpg" xlink:href="017/01/652/1.jpg"/></s> </p> <pb pagenum="637" xlink:href="017/01/653.jpg"/> <figure id="id.017.01.653.1.jpg" xlink:href="017/01/653/1.jpg"/> <p type="main"> <s id="s.004716"><emph type="center"/>MECHANICORUM <emph.end type="center"/><!-- REMOVE S--><emph type="center"/>LIBER SEPTIMUS.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.004717"><emph type="center"/><emph type="italics"/>De Cuneo & Percu&longs;sionibus.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004718">MECHANICARUM Facultatum Quarta &longs;pecies, <lb/>Cuneus, præ&longs;entem di&longs;putationem exigit: neque <lb/>enim &longs;olam corporum gravitatem, ob id ip&longs;um <lb/>quia gravitas e&longs;t, vincere oportet, ac loco dimo­<lb/>vere, quemadmodum Vecte, Axe, & Trochleis, <lb/>&longs;ed etiam &longs;æpè conjunctas corporum partes, aut cohærentia <lb/>proximè corpora &longs;ejungere atque divellere: id quod Cuneo <lb/>poti&longs;&longs;imùm perficimus, & iis, quæ ad Cunei rationem &longs;pecta­<lb/>re videntur. </s> <s id="s.004719">Quoniam verò in iis, in quibus præcipuè Cu­<lb/>nei vis elucet, percu&longs;&longs;ione utimur, quæ &longs;anè in paulò lon­<lb/>giorem &longs;ermonem nos vocat, non erit abs re aliquanto latiùs <lb/>Percu&longs;&longs;ionis naturam explicare, ut potentiæ Cuneo applicatæ <lb/>virtus manife&longs;ta fiat. </s> <s id="s.004720">Quamquam non &longs;emper percu&longs;&longs;ione <lb/>indigeat Cuneus, &longs;ed non rarò impul&longs;ione contentus &longs;it, ne­<lb/>que &longs;emper ad divellenda ea, quæ conjuncta &longs;unt, illo uta­<lb/>mur, &longs;ed aliquando etiam ad deprimendum, aut attollendum <lb/>corpus aliquod, ut ex &longs;equentibus patebit. </s> <s id="s.004721">Ea verò, quæ <lb/>Cunei figuram imitantur, quia in apicem de&longs;inunt, ut trian­<lb/>gula, ideóque Cunei nomine &longs;unt indicata &longs;æpiùs à Vitruvio <lb/>in Architecturæ libris, ad præ&longs;entem di&longs;putationem non atti­<lb/>nent; quemadmodum neque &longs;ub&longs;cudes, &longs;eu &longs;ecuriclæ, quibus <lb/>arctè duæ tabulæ compinguntur; licèt enim cunei formam <lb/>imitentur, non tamen &longs;imilem, &longs;ed cuneo oppo&longs;itam effectio­<lb/>nem habent, & inter retinacula connumerandæ &longs;unt. <pb pagenum="638" xlink:href="017/01/654.jpg"/></s> </p> <p type="main"> <s id="s.004722"><emph type="center"/>CAPUT I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004723"><emph type="center"/><emph type="italics"/>Cunei forma, & vires explicantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004724">CUneus, quo ad ligna findenda communiter utimur, &longs;atis <lb/>notum e&longs;t in&longs;trumentum, quod ex ampliori ba&longs;e fa&longs;tigia­<lb/>tur in acumen de&longs;inens; duas &longs;cilicet facies quadrangulas in li­<lb/>neam coëuntes, & duobus triangulis hinc atque hinc connexas <lb/>&longs;uper quadrilateram ba&longs;im erigit, adeò ut &longs;cindendo corpori <lb/>acies ip&longs;a applicetur, percu&longs;&longs;ionem excipiat ba&longs;is. </s> <s id="s.004725">Nihil tamen <lb/>prohibet aliam cuneo figuram tribui: nam cum aliquando com­<lb/>planandus e&longs;&longs;et colliculus, cujus creta arenæ permixta in lapi­<lb/>deam quandam materiem concreverat, & ita obduruerat, ut li­<lb/>gonibus ægerrimè cederet, parari ju&longs;&longs;i longiu&longs;culos ferreos cy­<lb/>lindrulos digitorum duûm cra&longs;&longs;itudine extremitate alterâ capi­<lb/>tatos ad excipiendum mallei ictum, altera in planas duas &longs;uper­<lb/>ficies compre&longs;&longs;os atque exacutos, qui juxta lapidis intervenia ap­<lb/>plicati, ac tudite adacti lapidem penetrabant; qui demum rimas <lb/>agens di&longs;&longs;iliebat in fru&longs;ta &longs;atis con&longs;picua non pœnitendo labore. </s> <lb/> <s id="s.004726">Sed quæcumque demum figura cuneo &longs;tatuatur, illud omnibus <lb/>commune e&longs;t, quod ex minori latitudine in majorem procedant, <lb/>ut quibus corporibus Cuneus in&longs;eritur, magis atque magis alte­<lb/>rum ab altero &longs;ecedat, dum ille penitiùs adigitur. </s> </p> <p type="main"> <s id="s.004727">Nunc verò &longs;ci&longs;&longs;ionem tanti&longs;per &longs;eponamus, & &longs;olum motum <lb/>corporis gravis ex cunei impul&longs;ione con&longs;ideremus, &longs;icuti &longs;i duro <lb/>plano incumbentem marmoreum cubum, cui vectis &longs;ubjici ne­<lb/>quiret, ut attolleretur, addacto cuneo disjungeremus à &longs;ub­<lb/>jecto plano: id quod faciliùs acutiore cuneo præ&longs;tatur, ut omnes <lb/><figure id="id.017.01.654.1.jpg" xlink:href="017/01/654/1.jpg"/><lb/>nôrunt, quàm &longs;i ille in mi­<lb/>nus acutum angulum de­<lb/>&longs;ineret. </s> <s id="s.004728">Sit enim cubus H <lb/>marmoreus plano AB in­<lb/>cumbens; & applicatus <lb/>cuneus DE, atque tudite <lb/>in D validè percu&longs;&longs;us ita <lb/>cubo &longs;ubjiciatur, ut hic <pb pagenum="639" xlink:href="017/01/655.jpg"/>elevetur ad altitudinem FI. <!-- KEEP S--></s> <s id="s.004729">Cùm itaque citrà omnem dubi­<lb/>tationem is, qui tuditem movet, eóque cuneum percutit, illo <lb/>eodem conatu nequeat cubum attollere &longs;inè cuneo, quæritur, <lb/>unde vis tanta illi accedat, &longs;pectatâ præcisè cunei figurâ, nihil <lb/>interim ad percu&longs;&longs;ionem re&longs;piciendo. </s> </p> <p type="main"> <s id="s.004730">Qui Ari&longs;toteli Mechan. quæ&longs;t. </s> <s id="s.004731">17. in lateribus cunei ligno <lb/>&longs;cindendo interjecti duplicem Vectem agno&longs;centi adhærent, <lb/>illicò ad Vectis Rationes confugient, quas in latere DE re­<lb/>cogno&longs;cere conabuntur. </s> <s id="s.004732">Verùm quærenti, primi ne? </s> <s id="s.004733">an &longs;e­<lb/>cundi generis Vectis &longs;it DE? vix &longs;uppetet, quid re&longs;pondeant. </s> <lb/> <s id="s.004734">Si primi generis, ut volui&longs;&longs;e videtur Ari&longs;toteles; cum tria &longs;int <lb/>puncta D, & I, & E, atque primum D procul dubio potentiæ <lb/>a&longs;cribatur; reliquum e&longs;t medium punctum I e&longs;&longs;e hypomochlij, <lb/>extremum E ponderis. </s> <s id="s.004735">Atqui nihil pror&longs;us apparet, quod cu­<lb/>bum H applicet puncto E, à quo neque &longs;u&longs;tinetur, neque tan­<lb/>gitur: igitur vectis non e&longs;t primi generis. </s> <s id="s.004736">Quod &longs;i in E pon­<lb/>dus e&longs;&longs;e ultrò conce&longs;&longs;erim, illud certè ex hoc efficitur, atque <lb/>con&longs;equens e&longs;t, quod cuneo novâ percu&longs;&longs;ione ulteriùs adacto, <lb/>& Potentia ad hypomochlium, &longs;cilicet D ad I, accedat, & pon­<lb/>dus ab hypomochlio, &longs;cilicet E ab I, remotius fiat; igitur & <lb/>potentiæ momenta decre&longs;cerent, & ponderis momenta auge­<lb/>rentur; ac proinde hanc momentorum dece&longs;&longs;ionem, & acce&longs;­<lb/>&longs;ionem, major movendi difficultas, & quidem notabilis atque <lb/>con&longs;picua, con&longs;equeretur; quæ tamen cum experimentis non <lb/>con&longs;entit. </s> <s id="s.004737">Adde in Vecte primi generis potentiam & pondus <lb/>oppo&longs;itis motibus circa hypomochlium, qua&longs;i circà centrum, <lb/>circulariter moveri; at hìc neque ullus intercedit circa <lb/>punctum I motus circularis, neque potentia de&longs;cendit pondere <lb/>a&longs;cendente. </s> </p> <p type="main"> <s id="s.004738">At forta&longs;&longs;e, quod aliis magis placet, vectem ais e&longs;&longs;e &longs;ecundi <lb/>generis; pondus quippe in I &longs;u&longs;tinetur, & e&longs;t potentiæ in D <lb/>exi&longs;tenti proximum; quare hypomochlio relinquitur extre­<lb/>mum punctum E. <!-- KEEP S--></s> <s id="s.004739">Id quidem aliquantò magis appo&longs;itè dictum <lb/>videretur, &longs;i, quæ vectis Rationibus conveniunt, hìc quoquè <lb/>in Cuneo locum habere po&longs;&longs;ent; Vectis &longs;iquidem quò longior <lb/>e&longs;t, & potentia magis abe&longs;t ab hypomochlio, cæteris paribus, <lb/>plus momenti tribuit potentiæ: at Cunei DE longitudo &longs;i au­<lb/>geatur, manente eodem angulo IEF, eádemque di&longs;tantiâ IE, <pb pagenum="640" xlink:href="017/01/656.jpg"/>nullam facit momentorum acce&longs;&longs;ionem: nulla igitur ibi Vectis <lb/>Ratio intercedit. </s> <s id="s.004740">Contrà verò manente eadem cunei DE lon­<lb/>gitudine, eademque di&longs;tantiâ IE, diminuto autem, &longs;ive aucto <lb/>angulo ad E eædem per&longs;everarent vectis Rationes; ergo & ea­<lb/>dem momenta movendi: id tamen longi&longs;&longs;imè à vero abe&longs;&longs;e <lb/>manife&longs;tis docemur experimentis, nam major angulus ad E <lb/>movendi difficultatem auget, minor minuit. </s> <s id="s.004741">Cùm verò Cu­<lb/>neus, ex hypothe&longs;i, &longs;emper &longs;ubjecto plano incumbat, utique <lb/>potentia in D ab eo &longs;emper æqualiter di&longs;tat, & nunquam altiùs <lb/>a&longs;&longs;urgit, pondere tamen altiùs &longs;ublato, quo magis illi cuneus <lb/>&longs;ubjicitur: in Vecte autem &longs;ecundi generis potentia a&longs;cendit, &longs;i <lb/>pondus attollitur. </s> <s id="s.004742">Non igitur cuneus habet rationem vectis &longs;e­<lb/>cundi generis; &longs;i maximè nullum hìc haberi circa hypomo­<lb/>chlium, tanquam circa centrum, motum circularem potentiæ <lb/>animadvertas. </s> </p> <p type="main"> <s id="s.004743">Cùm itaque à Cuneo ab&longs;int Rationes Vectis, illius vires pe­<lb/>tendæ &longs;unt ex eo, quod olim con&longs;titutum e&longs;t, Facultatibus om­<lb/>nibus Mechanicis communi principio: videlicet, quia Cunei <lb/>forma ea e&longs;t, ut majore motu moveatur Potentia Cuneum im­<lb/>pellens, quàm pondus à Cuneo repul&longs;um ad latus; hoc minùs <lb/>re&longs;i&longs;tit, quàm &longs;i æquali motu cum potentiâ moveretur; atque <lb/>adeò impetus motum in potentiâ efficiens tantæ velocitatis, & <lb/>valens pari velocitate movere certum pondus, cui ine&longs;&longs;et æquè <lb/>inten&longs;us ac in potentiâ, poterit in majori pondere entitativè <lb/>æqualis, &longs;ed minùs inten&longs;us efficere motum tardiorem pro Ra­<lb/>tione minoris inten&longs;ionis, ita tamen, ut, quæ Ratio e&longs;&longs;et inten­<lb/>&longs;ionis majoris in minori pondere, ad inten&longs;ionem minorem in <lb/>majori pondere, ea pariter &longs;it Ratio majoris gravitatis ad mi­<lb/>norem gravitatem; &longs;ic enim contingit æqualem e&longs;&longs;e entitativè <lb/>motum tardiorem majoris ponderis, atque motum velociorem <lb/>minoris ponderis; quemadmodum aliàs in Vecte & in Tro­<lb/>chleâ explicatum e&longs;t. </s> <s id="s.004744">Quoniam igitur cuneus, vi impetûs im­<lb/>pre&longs;&longs;i à potentia, dum promovetur &longs;ub pondus juxta lineam EF, <lb/>repellit pondus juxta lineam FI, linea EF motum potentiæ me­<lb/>titur, linea autem FI motum ponderis. </s> <s id="s.004745">Atqui Cunei confor­<lb/>matio hoc habet, ut in triangulo EFI minimus angulus &longs;it ad <lb/>apicem E; igitur per 19. lib. 1 minimum latus e&longs;t FI, atque <lb/>proinde minùs movetur pondus per FI, quàm potentia per EF. <pb pagenum="641" xlink:href="017/01/657.jpg"/>Quanto igitur major e&longs;t EF quàm FI, tanto majus e&longs;&longs;e pote&longs;t <lb/>gravitatis momentum in I vincendum, quàm e&longs;&longs;et momentum <lb/>gravitatis propellendæ in E juxta directionem FE motûs po­<lb/>tentiæ. </s> </p> <p type="main"> <s id="s.004746">Hinc plani&longs;&longs;imè con&longs;tat, cur acutiores cunei majora habeant <lb/>movendi momenta, cæteris paribus. </s> <s id="s.004747">Fac enim angulum E e&longs;&longs;e <lb/>adhuc minorem, utique oppo&longs;itum latus minus erit quàm FI; <lb/>eadem igitur linea EF ad lineam breviorem, quàm FI, habet <lb/>majorem Rationem, quàm ad eandem lineam FI; ac propterea <lb/>pondus adhuc multo tardiùs movetur quàm potentia, & pote­<lb/>rit e&longs;&longs;e majus; vel, &longs;i majus non fuerit, potentia indigebit mino­<lb/>re conatu, & faciliùs movebit. </s> </p> <p type="main"> <s id="s.004748">Ob&longs;erva autem (quantum quidem ex Cunei Ratione e&longs;t) ut <lb/>&longs;e initium dederit, eandem &longs;emper e&longs;&longs;e facilitatem in proce&longs;&longs;u <lb/>motûs, quia eadem permanet Ratio motuum potentiæ cuneo <lb/>applicatæ, & ponderis: nam ex 4. lib. 6. ut EF ad FI, ita EG <lb/>ad CO, & IS, hoc e&longs;t FC, ad SO, propter triangulorum &longs;i­<lb/>militudinem, cùm &longs;it IS parallela ip&longs;i EC. Quantum, inquam, <lb/>e&longs;t ex Cunei Ratione; quandoquidem cubi H, dum manente <lb/>extremitate A elevatur ex I, momenta &longs;ubinde variari, &longs;uo lo­<lb/>co, &longs;uperiùs indicatum e&longs;t. </s> <s id="s.004749">In &longs;cindendis autem corporibus, <lb/>prout variè contingit &longs;ci&longs;&longs;io, aliquando peculiaris intercedere <lb/>pote&longs;t cau&longs;a faciliorem vel difficiliorem in proce&longs;&longs;u &longs;ci&longs;&longs;ionem <lb/>reddens. </s> </p> <p type="main"> <s id="s.004750">Et quidem in &longs;ci&longs;&longs;ione corporum vi cunei faciendâ non e&longs;t <lb/>ita proclivè Geometricas leges per&longs;equi, ad explicandam eo­<lb/>rum re&longs;i&longs;tentiam: neque enim &longs;icut gravitas loco dimovenda <lb/>facilè innote&longs;cit, certámque &longs;ub men&longs;uram cadit, ita corpo­<lb/>rum re&longs;i&longs;tentia, ne findantur, fieri pote&longs;t manife&longs;ta: E&longs;t &longs;iqui­<lb/>dem &longs;ci&longs;&longs;io partium conjunctarum &longs;eparatio; earum autem con­<lb/>junctionem adeò variam e&longs;&longs;e contingit, ut certam legem &longs;ubi­<lb/>re nequeat. </s> <s id="s.004751">Nam quemadmodum inter lapides, ut monet Vi­<lb/>truvius lib. 2. cap. 7. alij ita molles &longs;unt, ut etiam &longs;erra dentatâ, <lb/>qua&longs;i ligna, &longs;ecentur, immò &longs;ecundùm oras maritimas ab &longs;al­<lb/>&longs;ugine exe&longs;a diffluant, & in locis patentibus atque apertis, prui­<lb/>ná & gelu frientur, ac di&longs;&longs;olvantur; alij duriores, &longs;ed qui inter­<lb/>veniorum vacuitates habeant, quapropter ab igne tuti non &longs;int, <lb/>quin rare&longs;cente aëre vacuitatibus illis interjecto di&longs;&longs;iliant & di&longs;-<pb pagenum="642" xlink:href="017/01/658.jpg"/>&longs;ipentur; alii ita &longs;pi&longs;&longs;is compactionibus &longs;olidati, ut neque ab <lb/>tempe&longs;tatibus, neque ab ignis vehementiâ timeant: Ita pariter <lb/>inter ligna alia aliis &longs;olidiora &longs;unt, & unum præ alio faciliùs e&longs;t <lb/>fi&longs;&longs;ile, prout particulæ componentes cra&longs;&longs;iores, aut tenuiores, <lb/>&longs;unt magis aut minus exqui&longs;ite permi&longs;tæ, atque nimio, &longs;ive <lb/>modico, &longs;ive temperato humore concretæ, & prout juxta &longs;ta­<lb/>minum ductum, aut illa oblique &longs;ecando, in&longs;tituitur &longs;ci&longs;&longs;io. </s> <lb/> <s id="s.004752">Sunt autem corpora illa (quantum quidem ad præ&longs;entem <lb/>tractationem &longs;pectat) partium &longs;eparationi difficiliùs obnoxia, <lb/>quorum materia ita probè &longs;ubacta e&longs;t, ut eorum elementa in <lb/>minuti&longs;&longs;imas particulas conci&longs;a, & qua&longs;i individua corpu&longs;cula <lb/>in unam naturam inob&longs;ervabili permi&longs;tione temperata coalue­<lb/>rint eâ tantùm humoris copiâ, quæ &longs;atis fuerit ad illa firmiter <lb/>agglutinanda. </s> <s id="s.004753">Ex quo fit, ut huju&longs;modi corpora &longs;olidiora &longs;int, <lb/>minú&longs;que con&longs;picuas inanitates admittant, atque proinde, &longs;i <lb/>expoliantur, &longs;uperficiem induant lævem & undique æquabi­<lb/>lem: id quod cæteris non accidit, quorum particulæ frequenti­<lb/>bus hiatibus interci&longs;æ, cùm aliæ emineant, aliæ &longs;uperentur, <lb/>&longs;emper aliquid habeant a&longs;peritatis; quemadmodum animadver­<lb/>tere poterit, qui&longs;quis lapides cum marmoribus comparaverit. </s> </p> <p type="main"> <s id="s.004754">Si igitur ex &longs;olido corpore avellenda e&longs;t particula aliqua, hæc <lb/>i&longs;táque disjungenda e&longs;t à circum&longs;tantibus particulis, quibus <lb/>conjungitur, neque fieri pote&longs;t, ut illa moveatur, quin proxi­<lb/>marum particularum aliæ motui oppo&longs;itæ impellantur, aliæ <lb/>di&longs;trahantur; omnes autem ægrè à &longs;tatu &longs;ibi &longs;ecundùm natu­<lb/>ram debito recedentes repugnant: quò verò plures particulæ <lb/>vim &longs;ubire coguntur, eò major e&longs;t re&longs;i&longs;tentia plurium qua&longs;i col­<lb/>latis viribus &longs;imul repugnantium. </s> <s id="s.004755">Hinc &longs;i duriora ligna &longs;ecan­<lb/>da offerantur, potior e&longs;t u&longs;us &longs;ubtilioris &longs;erræ minutos denticu­<lb/>los habentis; quia videlicet, quò exilior atque &longs;ubtilior e&longs;t den­<lb/>ticulus, minorem particulam obviam habet, quam impellat, & <lb/>pauciores particulæ, à quibus &longs;eparetur, illam circun&longs;tant; <lb/>ideóque ab iis faciliùs avellitur, quàm particula major, quæ à <lb/>pluribus disjungenda e&longs;&longs;et. </s> <s id="s.004756">Contra verò quorum particulæ le­<lb/>vi impul&longs;u divelluntur, quia non adeò dura &longs;unt, cra&longs;&longs;iore &longs;er­<lb/>râ facilè &longs;ecantur, quæ in durioribus majorem re&longs;i&longs;tentiam in­<lb/>veniens parùm utilis accideret. </s> <s id="s.004757">Hæc autem in limá pariter ob­<lb/>&longs;ervari po&longs;&longs;unt; quàm enim di&longs;&longs;ipari a&longs;peritate opus e&longs;t in limâ, <pb pagenum="643" xlink:href="017/01/659.jpg"/>qua chalybs, aut qua lignum terendo expolitur? </s> <s id="s.004758">Sed & in mar­<lb/>morum &longs;ectione mirantur aliqui &longs;erras adhiberi nullis dentibus, <lb/>&longs;altem con&longs;picuis, a&longs;peras, non &longs;atis animum advertentes ad <lb/>arenas aquâ a&longs;per&longs;as, quæ huju&longs;modi in opere interveniunt; ut <lb/>enim loquitur Plinius lib. 36. cap. 6. <emph type="italics"/>arená hoc fit, & ferro vide­<lb/>tur fieri, &longs;errá in prætenui lineâ premente arenas, ver&longs;andóque tractu <lb/>ip&longs;o &longs;ecante.<emph.end type="italics"/></s> <s id="s.004759"> Expedit autem &longs;ubtiliore arenâ uti, <emph type="italics"/>cra&longs;&longs;ior enim are­<lb/>na laxioribus &longs;egmentis terit, & plus erodit marmoris, majú&longs;que <lb/>opus &longs;cabritiâ polituræ relinquit:<emph.end type="italics"/> Sunt &longs;cilicet arenæ granula tam <lb/>quæ premuntur, quàm quæ &longs;erræ adhærent, qua&longs;i denticuli <lb/>mobiles mordacis limæ eodem ductu tùm cru&longs;tarum faciem le­<lb/>viter expolientis, tùm &longs;ubjectum marmor &longs;ecantis. </s> </p> <p type="main"> <s id="s.004760">E&longs;t autem manife&longs;tum non eâdem vi, qua &longs;uper lignum &longs;er­<lb/>ram adducentes & reducentes &longs;ci&longs;&longs;ionem inchoamus, idem pa­<lb/>riter obtineri, &longs;i cuneo lignum premamus citrà percu&longs;&longs;ionem: <lb/>quia nimirum cuneo prementes urgemus in directum &longs;ubjectas <lb/>ligni partes, quæ conjunctim re&longs;i&longs;tunt, ne comprimantur, at­<lb/>que à lateribus cohærentes particulæ repugnant, ne di&longs;trahan­<lb/>tur: &longs;erram verò ducentes obliquè urgemus ligni particulas <lb/>dentibus re&longs;pondentes, ac proinde pauculæ illæ tantùm, quæ <lb/>urgentur, re&longs;i&longs;tunt compre&longs;&longs;ioni, & illis attiguæ di&longs;tractioni. </s> <lb/> <s id="s.004761">Hinc fit cultro faciliùs aliquid &longs;cindi, &longs;i illius aciem quamvis <lb/>hebetem & obtu&longs;am adversùs corpus &longs;ci&longs;&longs;ile urgeas &longs;imul, atque <lb/>tran&longs;ver&longs;am agas; quia particulæ à cultro pre&longs;&longs;æ minorem in­<lb/>veniunt re&longs;i&longs;tentiam in tran&longs;ver&longs;o motu, ubi anteriores eâdem <lb/>cum po&longs;terioribus directione moventur, nec &longs;ibi adver&longs;antur, <lb/>quàm &longs;i &longs;olo pre&longs;&longs;u extimæ urgerent interiores, quæ comprimi <lb/>renuunt. </s> <s id="s.004762">Huc pariter referenda e&longs;t cau&longs;a, cur adeò valida con­<lb/>tingat &longs;ci&longs;&longs;io, &longs;i Harpe ictus infligatur: &longs;ic huju&longs;modi genere <lb/>en&longs;is in &longs;ummitate falcati, & in exteriore latere exacuti u&longs;um <lb/>Per&longs;eum in amputando Medu&longs;æ capite, & Mercurium in occi­<lb/>dendo Argo centoculo refert Ovidius lib. 5. Metam. <emph type="italics"/>Vertit in <lb/>hunc Harpem madefactam cæde Medu&longs;æ:<emph.end type="italics"/> id enim non ex &longs;olâ <lb/>Harpes gravitate, &longs;ed ex ip&longs;o poti&longs;&longs;imùm flexu oritur, qui ef­<lb/>ficit, ut dum vi impetùs de&longs;cendit, acies etiam tran&longs;ver&longs;o mo­<lb/>tu ducatur &longs;upra partes corporis &longs;ci&longs;&longs;ilis; ex quo & facilior &longs;ci&longs;­<lb/>&longs;io. </s> <s id="s.004763">Sic quidam vulgari en&longs;e, quo equitantes viatores non in <lb/>&longs;peciem, &longs;ed ad u&longs;um, præcingi &longs;olent, vituli caput uno ictu <pb pagenum="644" xlink:href="017/01/660.jpg"/>amputabat; aver&longs;o &longs;cilicet ictu percutiens gladius circulum <lb/>de&longs;cribebat, adeóque non &longs;olùm premendo &longs;ecabat, verùm <lb/>etiam motu tran&longs;ver&longs;o: quo in negotio dexteritate potiùs, <lb/>quàm viribus opus e&longs;t. </s> <s id="s.004764">Quòd autem alij re&longs;imum Harpes la­<lb/>tus in canaliculum excavant, eíque aliquid argenti vivi in­<lb/>dunt, quod à capulo ad cu&longs;pidem excurrat, id faciunt ad per­<lb/>cu&longs;&longs;ionem augendam, quia tran&longs;lata ad cu&longs;pidem gravitate <lb/>Mercurij, etiam percu&longs;&longs;ionis centrum transfertur longiùs à ca­<lb/>pulo, ideóque ictus fit validior, accedente præ&longs;ertim impetu, <lb/>quem Mercurius de&longs;cendens concipit. </s> </p> <p type="main"> <s id="s.004765">Non e&longs;t itaque comparandus gladius motu tran&longs;ver&longs;o &longs;cin­<lb/>dens cum &longs;errâ &longs;ecante; hæc enim obvias particulas in &longs;cabem <lb/>abeuntes &longs;en&longs;im à ligno divellens illud demum &longs;ublatis omni­<lb/>bus intermediis particulis bifariam divi&longs;um relinquit: ille ve­<lb/>rò &longs;uâ acie premens atque penetrans, &longs;ed nihil abradens, inter­<lb/>ponitur partibus, quæ invicem &longs;eparantur. </s> <s id="s.004766">Motus &longs;erræ motui <lb/>particularum ab&longs;ci&longs;&longs;arum planè æqualis e&longs;t; dens quippe parti­<lb/>culam, in quam incurrit, tangens impellit, &longs;uóque impul&longs;u vin­<lb/>cens nexum, quo particula &longs;ibi cohærentibus jungebatur, illam <lb/>avellit. </s> <s id="s.004767">Tran&longs;ver&longs;us verò gladij motus &longs;i comparetur cum mo­<lb/>tu particularum compre&longs;&longs;arum, atque invicem divul&longs;arum, <lb/>multò major e&longs;t illo; nam culter manûs moventis motui ob&longs;e­<lb/>cundat; & particulæ compre&longs;&longs;æ in latus recedunt; ac proinde <lb/>multo velociùs movetur potentia gladium adducens aut redu­<lb/>cens, quàm id, cui hoc motu vis infertur: atque idcircò gladij <lb/>vis &longs;cindendi hoc motu refertur ad cuneum. </s> <s id="s.004768">Quòd &longs;i gladius <lb/>non motu tran&longs;ver&longs;o ducatur &longs;uper id, quod &longs;cinditur, &longs;ed om­<lb/>nino motu recto pre&longs;&longs;ionis, &longs;it autem ferri cra&longs;&longs;ities &longs;en&longs;im exte­<lb/>nuata in aciem, quemadmodum cuneis omnibus commune e&longs;t, <lb/>idem planè dicendum erit, quod de vulgari cuneo, cui nomen <lb/>hoc præcipuè inditum e&longs;t. </s> </p> <p type="main"> <s id="s.004769">Quare Cuneus in corpus &longs;cindendum adactus con&longs;iderandus <lb/>e&longs;t ratione habitâ ip&longs;ius corporis, quod tenerum ac molle e&longs;&longs;e <lb/>pote&longs;t, atque ita flexibile, ut &longs;equatur quocunque torqueas, aut <lb/>etiam durum, & minimè tractabile. </s> <s id="s.004770">Si molle illud &longs;it, immi&longs;­<lb/>&longs;um cuneum recipit, séque illi accommodat, & comprimun­<lb/>tur tùm &longs;ubjectæ particulæ cunei aciem tangentes, tùm quæ à <lb/>lateribus cunei faciei congruunt: illæ omnino æqualiter pro-<pb pagenum="645" xlink:href="017/01/661.jpg"/>moventur, ac adigitur cuneus, neque motus earum à cuneo <lb/>pendet, quâ cuneus e&longs;t, &longs;ed quâ corpus e&longs;t &longs;uâ mole objectam <lb/>molem trudens: hæ verò ad latus &longs;ecedentes magis & magis, <lb/>prout cunei cra&longs;&longs;itudo excre&longs;cit, minori motu moventur, quàm <lb/>promoveatur immi&longs;&longs;us Cuneus; quandoquidem major e&longs;t cunei <lb/>longitudo eju&longs;dem motum metiens, quàm cra&longs;&longs;ities dextras at­<lb/>que &longs;ini&longs;tras partes impellens. </s> <s id="s.004771">Sin autem durum &longs;it corpus, cui <lb/>cuneus in&longs;eritur, illud quidem vix excogitari pote&longs;t ex adeò <lb/>con&longs;tipatis partibus inter &longs;e quàm apti&longs;&longs;imè cohærentibus con­<lb/>&longs;tare, ut nullius pror&longs;us compre&longs;&longs;ionis &longs;it capax, neque vel te­<lb/>nui&longs;&longs;imum cunei apicem admittat. </s> <s id="s.004772">Comprimuntur igitur ini­<lb/>tio partes proximè cuneo &longs;ubjectæ multò magis, quàm quæ la­<lb/>teri adjacent; &longs;ed propter duritiem certum compre&longs;&longs;ionis mo­<lb/>dum natura finivit, extra quem &longs;ubjectas partes diffindi potiùs <lb/>patiatur, atque à &longs;e mutuò divelli. </s> <s id="s.004773">Qua in fi&longs;&longs;ione ip&longs;æ etiam <lb/>&longs;uperiores partes majori compre&longs;&longs;ioni &longs;emper validiùs re­<lb/>pugnantes, quò penitiùs adigitur cuneus, plurimum habent <lb/>momenti, ut cunei vis, quâ cuneus e&longs;t, exerceatur: Quia vide­<lb/>licet &longs;uperiores partes cum inferioribus connexæ, neque flexi­<lb/>biles, dum ad latera cuneo urgente &longs;ecedunt, cogunt pariter in­<lb/>feriores ad dexteram & ad &longs;ini&longs;tram recedere; atque propterea <lb/>quæ adhuc connexæ erant, di&longs;trahuntur ita, ut demum di­<lb/>vellantur. </s> </p> <p type="main"> <s id="s.004774">Sit cuneus HI in &longs;ubjectum lignum immi&longs;&longs;us inter B & C; <lb/>dum percu&longs;&longs;ione urgetur <lb/><figure id="id.017.01.661.1.jpg" xlink:href="017/01/661/1.jpg"/><lb/>intror&longs;um, partes B versùs <lb/>A, & partes C ver&longs;us D re­<lb/>cedunt, & cum illis pariter <lb/>inferiores BM, atque CM: <lb/>ex quo fit partes in M con­<lb/>nexas di&longs;trahi atque invi­<lb/>cem divelli, & &longs;ci&longs;&longs;ionem <lb/>longiùs promoveri. </s> <s id="s.004775">Cum <lb/>igitur hoc &longs;it propo&longs;itum cuneo &longs;cindere &longs;ubjectum corpus, non <lb/>attendendus e&longs;t &longs;impliciter motus in B & C, &longs;ed etiam qui in M <lb/>efficitur, ibi quippe &longs;ci&longs;&longs;io contingit, &longs;emperque longiùs di&longs;tat <lb/>à punctis B, & C, locus &longs;ci&longs;&longs;ionis, quò magis intror&longs;um urge­<lb/>tur cuneus. </s> <s id="s.004776">Ex quo fit attentè di&longs;tinguendam e&longs;&longs;e facilitatem <pb pagenum="646" xlink:href="017/01/662.jpg"/>adigendi cunei, à facilitate &longs;cindendi: quandiu enim partes <lb/>faciem cunei tangentes non admodum repugnant compre&longs;&longs;io­<lb/>ni, facilè cedunt adacto cuneo; ubi verò ulteriùs comprimi <lb/>renuunt, tota vis exercenda e&longs;t in di&longs;tractione partium &longs;ejun­<lb/>gendarum; quam di&longs;tractionem quò majorem e&longs;&longs;e contingit, <lb/>augetur &longs;anè & cunei adigendi, & &longs;cindendi difficultas. </s> <s id="s.004777">Hæc <lb/>tamen alio ex capite minuitur, quia, quò magis punctum M, in <lb/>quo di&longs;trahendæ &longs;unt partes conjunctæ, abe&longs;t à punctis B & C, <lb/>faciliùs con&longs;equitur &longs;ci&longs;&longs;io, nam, cæteris paribus, motus particu­<lb/>larum, quæ &longs;ejunguntur, minorem habet Rationem ad motum <lb/>punctorum B & C. <!-- KEEP S--></s> <s id="s.004778">Sic fu&longs;tem cra&longs;&longs;iorem ab alterâ extremitate <lb/>fi&longs;&longs;um juxta notabilem longitudinem ulteriùs findimus etiam <lb/>&longs;olis manibus eò faciliùs, quò longior fuerit prior &longs;ci&longs;&longs;io: plu­<lb/>rimum &longs;iquidem intere&longs;t in lignis, quorum textura certum <lb/>quendam & rectum &longs;taminum ordinem habet, utrùm juxta eo­<lb/>rumdem &longs;taminum ductum in&longs;tituatur &longs;ci&longs;&longs;io, an hæc obliquè <lb/>&longs;ecentur; quemadmodum & in lapidibus præ&longs;tat cuneum in­<lb/>terveniis applicare, ut faciliùs &longs;cindantur: propterea nodo&longs;is <lb/>arborum partibus applicatus Cuneus ægrè illas findit, quia no­<lb/>dorum &longs;tamina non recto tramite, &longs;ed per anfractus & tortuosè <lb/>procedunt. </s> <s id="s.004779">In hac autem ligni &longs;ci&longs;&longs;ione &longs;i placeat tum particu­<lb/>las, quæ in M di&longs;trahuntur, tum illis &longs;ubjectas atque adhuc im­<lb/>motas, puta in N con&longs;iderare, atque ve&longs;tigium aliquod dupli­<lb/>cis Vectis &longs;ecundi generis recogno&longs;cere, itaut commune hypo­<lb/>mochlium &longs;it in N, longitudines vectium BN, & CN, poten­<lb/>tia medio cuneo applicata in B & C, atque re&longs;i&longs;tentia vincen­<lb/>da in M; non me difficilem præbebo: &longs;ed & illud &longs;tatim ad­<lb/>dam, non e&longs;&longs;e hunc duplicem illum Vectem, quem alij in Cu­<lb/>neo quærunt; cum potiùs &longs;int duo vectes in diver&longs;a impul&longs;i ab <lb/>interjecto cuneo. </s> </p> <p type="main"> <s id="s.004780">Quapropter ex his; quæ latiùs explicare placuit, illud confi­<lb/>citur, quod Cuneo aliquando re&longs;i&longs;tit gravitas, ut cùm ille cor­<lb/>pori gravi elevando &longs;upponitur, aut cùm disjunctorum quidem <lb/>corporum gravium, &longs;ed proximorum, &longs;altem alterum remove­<lb/>tur; aliquando re&longs;i&longs;tit partium nexus in M, qui ni&longs;i &longs;olvatur, <lb/>propelli nequeunt partes B & C cunei faciem tangentes: vis &longs;i­<lb/>quidem cunei proximè exercetur adversùs B & C, & propter <lb/>partium connexionem etiam adversùs M, quamvis hoc po&longs;tre-<pb pagenum="647" xlink:href="017/01/663.jpg"/>mum &longs;it &longs;copus &longs;cindentis: quò autem validiùs partes in M <lb/>conjunguntur, etiam difficiliùs urgentur partes B & C: ligna <lb/>verò adhuc viridia, & lento humore plena, quia particulæ ma­<lb/>jorem di&longs;tractionem ferunt, nec facilè di&longs;&longs;iliunt, difficiliùs <lb/>&longs;cinduntur, quàm ligna arida. </s> </p> <p type="main"> <s id="s.004781">Quæ itaque de Cuneo vulgari dicta &longs;unt, facilè innote&longs;cit <lb/>ea pariter convenire forficibus, cultris en&longs;ibus, novaculis, &longs;cal­<lb/>pris, dentibus hominis anterioribus, & &longs;imilibus, quibus ad <lb/>&longs;cindendum utimur; &longs;unt enim cunei diver&longs;imodè juxta varios <lb/>u&longs;us conformati; neque egent percu&longs;&longs;ione, quia res &longs;cindendæ <lb/>non admodum re&longs;i&longs;tunt, & &longs;olus impul&longs;us &longs;æpè &longs;ufficit. </s> <s id="s.004782">Forfi­<lb/>ces autem &longs;unt quidem cunei, &longs;ed vectem conjunctum haben­<lb/>tes, adeò ut potentia momenti augmentum acquirat ex Ratio­<lb/>nibus vectis juxta di&longs;tantias tùm potentiæ, tùm corporis &longs;cin­<lb/>dendi, à clavo, ubi decu&longs;&longs;antur. </s> <s id="s.004783">An verò etiam &longs;calpra, qui­<lb/>bus Marmorarij a&longs;&longs;ulas ex operibus dejiciunt; Cuneorum ratio­<lb/>nem habeant, non admodum curo; videntur &longs;iquidem non in­<lb/>cidere marmor, &longs;ed partes &longs;uperfluas decutere: quod &longs;i per­<lb/>cu&longs;&longs;i &longs;calpri mucro penetrat, & dividit marmor, cuneus perin­<lb/>de e&longs;t atque &longs;calpra, quibus lignum cælatur. </s> </p> <p type="main"> <s id="s.004784">Similiter acus, &longs;ubulæ, aculei, clavi ad cuneum referun­<lb/>tur: & quidem huju&longs;modi corpora quò &longs;ubtiliora &longs;unt, eò fa­<lb/>ciliùs penetrant, quia immi&longs;&longs;a, & juxta &longs;uam longitudinem <lb/>progredientia valdè moventur interea, dum corporis perforan­<lb/>di particulæ di&longs;trahendæ atque comprimendæ exiguo motu in <lb/>latera &longs;ecedunt. </s> <s id="s.004785">Hinc con&longs;tat, cur terebellâ paulo minore ape­<lb/>riendum &longs;it foramen, cui immittatur clavus cra&longs;&longs;iu&longs;culus, &longs;i <lb/>præ&longs;ertim lignum tenue &longs;it; ne videlicet immi&longs;&longs;o clavo tot par­<lb/>tes adeò invicem comprimantur, ut ulteriorem compre&longs;&longs;ionem <lb/>recu&longs;antes cogant alias di&longs;trahi, ac demum rimâ factâ lignum <lb/>di&longs;&longs;iliat: &longs;ublatis autem terebrâ particulis aliquot, reliquæ com­<lb/>primendæ ut clavum arctè complectantur, cum pauciores &longs;int, <lb/>faciliùs compre&longs;&longs;ionem ferunt citrà periculum fractionis aut <lb/>&longs;ci&longs;&longs;ionis ligni. </s> </p> <p type="main"> <s id="s.004786">Demum &longs;ecuris, & gladius cæ&longs;im feriens, cuneus e&longs;t, cui quo­<lb/>dammodo junctus e&longs;t tudes; illo &longs;iquidem percutimus: quid enim <lb/>intere&longs;t, quod cuneum manentem tudite percutiamus, &longs;ive cu­<lb/>neo velociter moto percutiatur corpus &longs;cindendum? <pb pagenum="648" xlink:href="017/01/664.jpg"/></s> </p> <p type="main"> <s id="s.004787"><emph type="center"/>CAPUT II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004788"><emph type="center"/><emph type="italics"/>Cunei inflexi u&longs;us ad movendum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004789">PRæter vulgarem Cunei formam, quæ planis extremitatibus <lb/>circum&longs;cribitur, &longs;i non ad &longs;cindendum, &longs;ed ad movendum <lb/>adhibeatur, utilis e&longs;&longs;e pote&longs;t Cuneus inflexus, ita ut, qua &longs;al­<lb/>tem parte movendo corpori applicatur, faciem habeat non pla­<lb/>nam, &longs;ed inflexam, moveatur autem non circa eju&longs;dem circu­<lb/>laris curvitatis centrum. </s> <s id="s.004790">In cra&longs;&longs;iore tabulâ a&longs;&longs;umpto A puncto <lb/><figure id="id.017.01.664.1.jpg" xlink:href="017/01/664/1.jpg"/><lb/>tanquam centro de&longs;cribatur pla­<lb/>cito intervallo AB pars arcûs <lb/>circuli BC, &longs;ive Quadrans, &longs;ive <lb/>Quadrante minor, &longs;ive major <lb/>fuerit. </s> <s id="s.004791">Tum centro alio a&longs;&longs;ump­<lb/>to, & majore aliquo intervallo, <lb/>alia circuli pars de&longs;cribatur DC <lb/>occurrens priori arcui BC in <lb/>puncto C, &longs;ive ibi &longs;e contingant, <lb/>&longs;ive &longs;ecent arcus, prout tibi <lb/>commodius acciderit. </s> <s id="s.004792">Re&longs;ecatis igitur &longs;upervacuis tabulæ par­<lb/>tibus, & retentâ parte curvilineâ, habetur cuneus BCD in­<lb/>flexus, qui &longs;uam vim exerceat non motu recto, ut cæteri cunei, <lb/>&longs;ed curvo: propterea illi ad firmitatem addantur tran&longs;ver&longs;aria <lb/>ab extremitatibus cunei exeuntia, & in unum punctum coëun­<lb/>tia, circa quod, tanquam centrum, moveri po&longs;&longs;it cuneus. </s> <s id="s.004793">Ad <lb/>firmitatem, inquam, quia, ad motum, &longs;atis e&longs;&longs;et cra&longs;&longs;iori ex­<lb/>tremitati BD addere appendicem BM, in qua a&longs;&longs;umi po&longs;&longs;it <lb/>punctum, circa quod moveatur, quodcumque illud &longs;it, modò <lb/>non &longs;it centrum arcûs DC, &longs;i arcus ille impellat corpus moven­<lb/>dum, neque punctum D minùs di&longs;tet ab huju&longs;modi centro mo­<lb/>tûs, quàm punctum aliud extremum C. <!-- KEEP S--></s> <s id="s.004794">Contra verò &longs;i cunei <lb/>conatus exercendus &longs;it trahendo, & corpori applicetur arcus <lb/>BC, oportet centrum motûs minùs abe&longs;&longs;e ab extremitate B, <pb pagenum="649" xlink:href="017/01/665.jpg"/>quàm ab apice cunei C. <!-- KEEP S--></s> <s id="s.004795">Hùc &longs;cilicet &longs;pectare videntur ferrei <lb/>uncini, quibus duo corpora fibulantur, aut refibulantur, ut <lb/>cum armaria, fene&longs;træ, aut cap&longs;ulæ clauduntur & recludun­<lb/>tur. </s> <s id="s.004796">Quare intellectâ rectâ DM tanquam parte diametri cir­<lb/>culi, cujus arcus exerceat vim cunei, &longs;i hic &longs;it arcus DC, <lb/>oportet ejus centrum inter extremitatem D, & punctum M <lb/>centrum motûs, interjacere; &longs;in autem &longs;it arcus BC, inter A <lb/>centrum circuli, & extremitatem B, oportet interjici centrum <lb/>motûs H: ab illo quippe centro M in primo ca&longs;u removeri <lb/>oportet corpus movendum, & ad hoc centrum H accedere <lb/>oportet corpus trahendum. </s> <s id="s.004797">Quod &longs;i recta DM non fuerit pars <lb/>diametri tran&longs;euntis per centra arcûs & motûs, &longs;altem oportet <lb/>illam huju&longs;modi diametro propiorem e&longs;&longs;e, quàm &longs;it recta ex <lb/>centro motûs ad C ducta; id quod ex 7.lib.3. manife&longs;tum e&longs;t. </s> </p> <p type="main"> <s id="s.004798">Firmato itaque pro loci opportunitate centro motûs, & appli­<lb/>catum ad impellendum pondus cuneum BCD urgens poten­<lb/>tia in D, de&longs;cribit circa M centrum motûs arcum circularem <lb/>DE, &longs;ed pondus non propellitur ni&longs;i juxta differentiam linea­<lb/>rum à punctis arcûs DC ad M centrum motûs ductarum. </s> <s id="s.004799">Ex <lb/>quo fit movendi facilitatem <lb/>&longs;ubinde augeri, cæteris pa­<lb/><figure id="id.017.01.665.1.jpg" xlink:href="017/01/665/1.jpg"/><lb/>ribus. </s> <s id="s.004800">Hoc autem ut pla­<lb/>niùs explicetur, &longs;it arcus <lb/>LQ in quinque æquales <lb/>partes divi&longs;us, & &longs;ingulæ <lb/>&longs;int gr. <!-- REMOVE S-->3. linea HL tran­<lb/>&longs;eat per centrum I, & ex H <lb/>ducantur rectæ HM, HN, <lb/>HO &c; Certum e&longs;t lineas <lb/>ha&longs;ce omnes ex Q ad L <lb/>&longs;emper majores e&longs;&longs;e, & ma­<lb/>ximam e&longs;&longs;e HL ex 7. lib. 3. atque a&longs;&longs;umptâ HF æquali ip&longs;i <lb/>HQ, differentiam totam e&longs;&longs;e FL. </s> <s id="s.004801">Quare &longs;i LQ &longs;it latus cu­<lb/>nei inflexi, potentia de&longs;cribens arcum æqualem arcui LQ ha­<lb/>beret motum, qui ad motum ponderis e&longs;&longs;et ut arcus LQ ad <lb/>rectam FL, &longs;eu QS. </s> <s id="s.004802">Sed quoniam centrum motûs e&longs;t H, po­<lb/>tentia circa illud de&longs;cribit arcum LS, cujus quantitas inno­<lb/>te&longs;cit, &longs;i dato IL, hoc e&longs;t IQ, Radio, atque di&longs;tantiâ IH, <pb pagenum="650" xlink:href="017/01/666.jpg"/>cùm ex hypothe&longs;i notus &longs;it angulus LIQ, inve&longs;tigetur angu­<lb/>lus IHQ, quem metitur arcus LS; hujus autem quantitas <lb/>prodit ex datis partibus Radij HL. <!-- KEEP S--></s> <s id="s.004803">Quare angulus LIQ &longs;it <lb/>gr. <!-- REMOVE S-->15. IL partium 10000, IH partium 5000. Igitur in trian­<lb/>gulo HIQ angulus IHQ e&longs;t gr. <!-- REMOVE S-->10. 0′. </s> <s id="s.004804">45″: atque &longs;i ex Cy­<lb/>clometricis ineatur Ratio men&longs;uræ arcûs LS, invenietur in <lb/>partibus Radij IL 10000 ferè par arcui <expan abbr="Lq;">LQ</expan> hic e&longs;t partium <lb/>2618, ille 2621. Quapropter in tam exiguâ circuli portione <lb/>perinde e&longs;t arcum LQ, atque arcum LS con&longs;iderare. </s> <s id="s.004805">Singu­<lb/>læ itaque partes quintæ arcûs de&longs;cripti &longs;unt particularum 524. </s> </p> <p type="main"> <s id="s.004806">Jam verò per Trigonometriam, ex datis lateribus HI 5000, <lb/>& IM 10000, atque angulo comprehen&longs;o HIM, ut pote &longs;up­<lb/>plemento ad duos rectos noti anguli LIM ex hypothe&longs;i gr. <!-- REMOVE S-->3. <lb/>(&longs;imiliter in reliquis triangulis eadem &longs;unt latera, & angulus <lb/>comprehen&longs;us &longs;en&longs;im per gr. <!-- REMOVE S-->3. minuitur) inveniatur linearum <lb/>longitudo; & e&longs;t in ii&longs;dem Radij IL partibus 10000 linea <lb/>HQ 14885, HP 14927, HO 14959, HN 14981, HM 14995, <lb/>atque demum HL 15000. Sunt igitur linearum ex Q ad L in­<lb/>crementa inæqualia, videlicet 42, 32, 22, 14, 5, quibus re&longs;pon­<lb/>det motus ponderis cuneo propul&longs;i, qui &longs;emper decre&longs;cit, dum <lb/>potentia motus æquales perficit. </s> <s id="s.004807">Ratio proinde motûs potentiæ <lb/>ad motum ponderis initio, dum cunei pars QP &longs;ubinde ponde­<lb/>ri applicatur, atque Potentia venit ex L in M, e&longs;t ut 524 ad 42, <lb/>deinde in PO ut 524 ad 32, in ON ut 524 ad 22, in NM ut <lb/>524 ad 14, in ML ut 524 ad 5. Cum itaque &longs;emper major fiat <lb/>Ratio motuum, augetur movendi facilitas; atque perinde fit, <lb/>ac &longs;i acutior &longs;emper atque acutior cuneus adhiberetur. </s> </p> <p type="main"> <s id="s.004808">Hìc tamen ob&longs;ervandum e&longs;t ita temperandum e&longs;&longs;e movendi <lb/>facilitatem cum ip&longs;o ponderis motu, ut illam con&longs;ectando hoc <lb/>minùs moveri non contingat, quàm par fuerit: quò enim <lb/>punctum, quod e&longs;t centrum motûs, minùs abe&longs;t ab I centro <lb/>arcûs LQ, eò quidem faciliùs movetur pondus, quia ad hujus <lb/>motum potentiæ motus majorem habet Rationem, &longs;ed à pon­<lb/>dere minus &longs;patium percurritur. </s> <s id="s.004809">Nam &longs;i centrum motûs &longs;it G, <lb/>& IG partium 2500, quarum IQ e&longs;t 10000, linea GQ e&longs;t <lb/>12432, GP 12456, GO 12475, GN 12489, GM 12497, <lb/>GL 12500: atque adeò linearum incrementa &longs;unt 24, 19, 14, 8, 3; <lb/>cum tamen quinta pars arcûs intervallo GL de&longs;cripti à poten-<pb pagenum="651" xlink:href="017/01/667.jpg"/>tiâ &longs;it proximè 524; Major e&longs;t autem Ratio 524 ad &longs;ingula hæc <lb/>linearum incrementa, quàm cum motûs centrum e&longs;t H. <!-- KEEP S--></s> <s id="s.004810">Cum <lb/>hac tamen movendi facilitate connectitur exiguus ponderis mo­<lb/>tus; nam inter 12432 & 12500, quæ &longs;unt extremæ lineæ GQ <lb/>& GL, differentia 68 minor e&longs;t quàm differentia 115 inter HQ <lb/>14885 & HL 15000, quæ differentia inter extremas lineas me­<lb/>titur ponderis motum: e&longs;t &longs;iquidem differentia inter aggrega­<lb/>tum laterum & ba&longs;im trianguli HIQ, aut GIQ, men&longs;ura, juxta <lb/>quam pondus promovetur impul&longs;u cunei. </s> <s id="s.004811">Cum verò IQ & IL, <lb/>utpote &longs;emidiametri, æquales &longs;int, ba&longs;is autem HQ major &longs;it ba&longs;i <lb/>GQ (nam in triangulo HGQ amblygonio ba&longs;is HQ opponi­<lb/>tur majori angulo) fieri non pote&longs;t, ut eadem &longs;it motûs ponderis <lb/><expan abbr="m&etilde;&longs;ura">men&longs;ura</expan> æqualis ip&longs;is FL aut QS, ni&longs;i ab a&longs;&longs;umpto motûs centro <lb/>de&longs;criptus arcus (intervallo u&longs;que ad Q punctum illi centro pro­<lb/>ximum) tran&longs;eat per extremitates ea&longs;dem Q & F, per quas tran­<lb/>&longs;iret arcus ex H intervallo HQ de&longs;criptus. </s> <s id="s.004812">Quare duo circuli <lb/>&longs;e in duobus punctis &longs;ecantes communem haberent rectam li­<lb/>neam QF, ad quam bifariam &longs;ectam in V perpendicularis VH <lb/>tran&longs;iret per utriu&longs;que circuli centrum, ex 3. lib. 3. ac proinde, <lb/>cum ex. </s> <s id="s.004813">5. lib.3. non habeant idem centrum H, alterius circuli <lb/>centrum e&longs;&longs;et extra rectam HI, puta in T. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004814">Utrùm autem dato eodem motûs centro, & datâ pari arcûs <lb/>portione, præ&longs;tet arcum e&longs;&longs;e majoris, an minoris, circuli partem, <lb/>vix e&longs;t dubitandi locus. </s> <s id="s.004815">Quando enim duo circuli idem planum <lb/>in eodem puncto contingunt, peripheria majoris interjicitur in­<lb/>ter planum datum & peripheriam minoris circuli; atque adeò <lb/>ab eodem motûs centro lineæ ad illam majoris circuli periphe­<lb/>riam ductæ omnes &longs;ecant peripheriam minoris, ac propterea, ut­<lb/>pote longiores, majorem efficiunt pre&longs;&longs;ionem, longiú&longs;que pro­<lb/>pellunt corpus, quod impellitur. </s> <s id="s.004816">Hinc &longs;i viribus potentia abun­<lb/>det, & ad majus &longs;patium protrudere oporteat pondus, adhiben­<lb/>da e&longs;t peripheria majoris circuli; contra verò minore utendum <lb/>e&longs;t, &longs;i parum movendum &longs;it, & 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/>admodum intere&longs;t, plana-ne? </s> <s id="s.004819">an inflexa? </s> <s id="s.004820">&longs;it ejus facies, modò <lb/>ex ejus interjectu duo disjuncta corpora magis invicem remo­<lb/>veantur, &longs;ive utrumque &longs;imul in diver&longs;as partes abeant, &longs;ive al­<lb/>tero manente, alterum tantummodo moveatur. </s> <s id="s.004821">Hìc autem im-<pb pagenum="652" xlink:href="017/01/668.jpg"/>motum manet centrum, circa quod vertitur portio circuli ex­<lb/>centrici, quæ &longs;ive &longs;implici impul&longs;ione, &longs;ive etiam percu&longs;&longs;ione, <lb/>adacta urget corpus, quod contingit, neque aliter quàm &longs;i inter <lb/>validum &longs;tipitem humi defixum, atque pondus interjiceretur <lb/>vulgaris cuneus planus. </s> </p> <p type="main"> <s id="s.004822">Verùm quamvis hactenus potentiam in ip&longs;a cunei inflexi ex­<lb/>tremitate po&longs;uerimus ad explicandum ejus motum, nihil tamen <lb/>refert: nam &longs;i etiam circa medium cuneum fuerit an&longs;a, qua ar­<lb/>reptâ ille valeat circumduci, perinde e&longs;t; motus &longs;iquidem po­<lb/>tentiæ ad ponderis motum eandem &longs;ervat Rationem. <!-- KEEP S--></s> <s id="s.004823">Ex quo <lb/>manife&longs;tò deprehenditur nullam e&longs;&longs;e in Cuneo Vectis umbram; <lb/>in Vecte &longs;iquidem certus e&longs;t potentiæ locus, quo mutato etiam <lb/>momenta variantur: at in huju&longs;modi Cuneo non contingit mo­<lb/>mentorum mutatio, cuju&longs;cumque tandem cunei parti applice­<lb/>tur potentia, dummodo ea &longs;it di&longs;po&longs;itio, ut vires &longs;uas æquè exer­<lb/>cere valeat, &longs;ive in hac, &longs;ive in illâ arcûs extremitate, hoc e&longs;t ad <lb/>L aut Q, &longs;ive circa medium ad O, aut N, con&longs;tituatur. </s> <s id="s.004824">Cave ta­<lb/>men putes æquè liberum e&longs;&longs;e in majore aut minore di&longs;tantiâ à <lb/>centro motûs H aut G potentiam collocare: id enim &longs;anè per­<lb/>peram fieret; pro Ratione &longs;iquidem di&longs;tantiæ à centro motûs <lb/>majorem aut minorem arcum potentia &longs;uo motu de&longs;criberet: <lb/>e&longs;to nihil inter&longs;it, cuinam parti applicetur, &longs;ervatâ eâdem à præ­<lb/>dicto motûs centro di&longs;tantiâ. </s> <s id="s.004825">Propterea &longs;i ad Q applicetur po­<lb/>tentia cuneum trahens, an&longs;a eju&longs;modi apponenda e&longs;t &longs;ur&longs;um re­<lb/>curva, cui applicata potentia non minorem arcum de&longs;cribat, <lb/>quàm &longs;i illa applicaretur puncto L impellens cuneum: non e&longs;t <lb/>&longs;cilicet par potentiæ motus, qui fit intervallo HQ, ac interval­<lb/>lo HL. <!-- KEEP S--></s> <s id="s.004826">Similiter autem Cuneo plano uti licebit, cujus latera &longs;i <lb/>ferreo paxillo hinc atque hinc extante trajeceris, ut arreptis <lb/>utrâque manu paxilli extremitatibus cuneum adducere valëas, <lb/>aut impellere, duo corpora, quibus cuneus interjicitur, disjun­<lb/>ges: immò &longs;i fi&longs;&longs;ili ligno bicubitali juxta &longs;taminum ductum cu­<lb/>neum eumdem ita per vim immi&longs;eris, ut cuneum elevatum &longs;e­<lb/>quatur pariter & lignum, tùm ligni calce &longs;axum percu&longs;&longs;eris, <lb/>cuneus &longs;ci&longs;&longs;ionem promovebit, quocumque tandem in loco &longs;ive <lb/>juxta ip&longs;ius cunei apicem, &longs;ive juxta ba&longs;im immi&longs;&longs;us fuerit pa­<lb/>xillus ille, cui potentia applicatur. <pb pagenum="653" xlink:href="017/01/669.jpg"/></s> </p> <p type="main"> <s id="s.004827"><emph type="center"/>CAPUT III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004828"><emph type="center"/><emph type="italics"/>Cuneus perpetuus circulo excentrico effingitur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004829">CUneum Perpetuum voco non eum, qui perpetuâ, hoc e&longs;t <lb/>majore &longs;emper atque majore impul&longs;ione corpus propellat <lb/>longiùs atque longiùs; id enim, ut &longs;atis clarum e&longs;t, infinitam <lb/>exigeret cunei longitudinem: &longs;ed eatenus dico <emph type="italics"/>perpetuum,<emph.end type="italics"/> qua­<lb/>tenus potentia illi &longs;emel applicata in&longs;titutum motum juxta ean­<lb/>dem directionem perpetuare pote&longs;t: id quod neutiquam con­<lb/>tingit &longs;ecundùm rectam lineam, quæ &longs;patium requireret infini­<lb/>tum perpetuo illo motu percurrendum; &longs;ed potentia in orbem <lb/>progrediens, & cuneum contorquens, motus alternos perficit. </s> </p> <p type="main"> <s id="s.004830">Et primùm quidem id fieri pote&longs;t circulo, cujus motûs cen­<lb/>trum ab&longs;it ab eju&longs;dem circuli centro; vis enim cunei erit ad <lb/>propellendum corpus intervallo duplo intervalli centrorum; <lb/>momentum verò potentiæ de&longs;umetur ex &longs;emiperipheriâ circuli, <lb/>cujus Radius æqualis &longs;it dati circuli &longs;emidiametro auctæ cen­<lb/>trorum illorum intervallo; &longs;i tamen extremitati à centro motûs <lb/>maximè di&longs;tanti ip&longs;a potentia applicetur. </s> </p> <p type="main"> <s id="s.004831">Sit datus circulus BED, cujus centrum A: fiat motûs cen­<lb/>trum C, circa quod in gyrum <lb/><figure id="id.017.01.669.1.jpg" xlink:href="017/01/669/1.jpg"/><lb/>agatur con&longs;titutus circulus. </s> <lb/> <s id="s.004832">In hoc motu duo circuli con­<lb/>centrici de&longs;cribuntur; alter <lb/>quidem à puncto D, Radio <lb/>CD, alter verò à puncto B, <lb/>Radio CB. <!-- KEEP S--></s> <s id="s.004833">Quare dum po­<lb/>tentia ex B per G venit in H, <lb/>punctum D per K venit in L, <lb/>& corpus, quod puncto D ap­<lb/>plicitum erat, à peripheriâ <lb/>circuli BED &longs;en&longs;im propel­<lb/>litur, donec veniat ex D in H. <!-- KEEP S--></s> <lb/> <s id="s.004834">E&longs;t autem DH æqualis ip&longs;i <lb/>BL, quia ex æqualibus CH & CB auferuntur æquales CD <pb pagenum="654" xlink:href="017/01/670.jpg"/>& CL: inter diametros verò BD & LD differentia e&longs;t BL: <lb/>igitur quia diametrorum differentia dupla e&longs;t differentiæ &longs;emi­<lb/>diametrorum, BL e&longs;t dupla ip&longs;ius AC differentiæ &longs;emidiame­<lb/>trorum AD & CD. <!-- KEEP S--></s> <s id="s.004835">Quapropter etiam DH &longs;patium, quod à <lb/>pondere propul&longs;o percurritur, duplum e&longs;t intervalli centrorum <lb/>AC Demùm potentia in B, ex hypothe&longs;i, applicata momentum <lb/>habet juxta Rationem &longs;emiperipheriæ BGH ad &longs;patium DH <lb/>duplum intervalli centrorum AC: hæc &longs;iquidem e&longs;t Ratio mo­<lb/>tuum potentiæ & ponderis. </s> <s id="s.004836">Hinc &longs;i ponatur dati circuli Ra­<lb/>dius AB 100, & centrorum di&longs;tantia AC 13, erit DH 26: At <lb/>&longs;emiperipheria BGH ad &longs;uum Radium BC e&longs;t ut 355 ad 113; <lb/>igitur BGH ad DH e&longs;t ut 355 ad 26. Quare, cæteris paribus, <lb/>quò majus e&longs;t centrorum intervallum, eò majores requiruntur <lb/>in potentiâ vires; quia hujus intervalli duplum e&longs;t &longs;patium, per <lb/>quod impellitur pondus, manente eodem potentiæ motu. </s> </p> <p type="main"> <s id="s.004837">E&longs;t <expan abbr="aut&etilde;">autem</expan> attentè <expan abbr="con&longs;iderandũ">con&longs;iderandum</expan>, utrùm præ&longs;tet, cæteris paribus, <lb/>majore circulo uti: Cæteris, inquam, paribus, ut &longs;cilicet idem &longs;it <lb/>centrorum intervallum, & eadem potentiæ à <expan abbr="c&etilde;tro">centro</expan> motûs di&longs;tan­<lb/>tia. </s> <s id="s.004838">Et primò <expan abbr="ob&longs;ervandũ">ob&longs;ervandum</expan> e&longs;t cuneum e&longs;&longs;e LBED, cujus vertex <lb/>e&longs;t angulus <expan abbr="cõtingentiæ">contingentiæ</expan> factus à peripheriâ dati circuli, & à pe­<lb/>ripheriâ circuli, quem circa centrum C in motu de&longs;cribit extre­<lb/>mitas D. <!-- KEEP S--></s> <s id="s.004839">Deinde &longs;emiperipheria BGH à potentiâ de&longs;cripta in <lb/>motu (& e&longs;t ex hypothe&longs;i <expan abbr="partiũ">partium</expan> 355, <expan abbr="quarũ">quarum</expan> Radius CB e&longs;t 113) <lb/>dividatur in partes æquales duodecim, ita ut &longs;ingulæ re&longs;pon­<lb/>deant gradibus 15, & &longs;ingulis competant partes 29 1/2. <!--neuer Satz-->Similiter <lb/>&longs;emiperipheria DOL in 12 æquales partes &longs;ingulas gr.15. divi­<lb/>datur: adeò ut cùm linea BD circa punctum C circumacta an­<lb/>gulum gr. <!-- REMOVE S-->15 de&longs;crip&longs;erit, potentia &longs;it progre&longs;&longs;a per partes 29 1/2. <lb/>Examinandum e&longs;t, quanto &longs;patio interim propellatur pondus, <lb/>quod erat in D, versùs H. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004840">Sit angulus DCO, hoc e&longs;t arcus DO, gr. <!-- REMOVE S-->15: ducta intelli­<lb/>gatur recta CO u&longs;que in N peripheriam dati circuli; e&longs;t igitur <lb/>ON motus ponderis ex D ver&longs;us H. <!-- KEEP S--></s> <s id="s.004841">Quapropter inve&longs;tiganda <lb/>e&longs;t ip&longs;ius CN longitudo, ut appareat eju&longs;dem exce&longs;&longs;us &longs;upra <lb/>CD. <!-- KEEP S--></s> <s id="s.004842">Ducatur dati circuli Radius AN notus partium 100; da­<lb/>tur item intervallum AC partium 13; notus e&longs;t angulus ACN <lb/>gr. <!-- REMOVE S-->165: ergo per Trigonometriam innote&longs;cit primò angu­<lb/>lus CNA gr.1.55′. <!-- REMOVE S-->41″; atque ex eo reliquus angulus NAC <pb pagenum="655" xlink:href="017/01/671.jpg"/>gr.13.4′. <!-- REMOVE S-->19″, hoc e&longs;t arcus ND: deinde habetur longitudo <lb/>CN partium (87 38/100), quarum CO, hoc e&longs;t CD e&longs;t 87: igitur <lb/>ON e&longs;t (38/100). Quod &longs;i arcus DO ponatur gr. <!-- REMOVE S-->30, in triangulo <lb/>ACN dantur <expan abbr="ead&etilde;">eadem</expan> latera AN 100, & AC 13, & angulus ACN <lb/>gr. <!-- REMOVE S-->150: igitur invenitur CNA gr.3. 43′. </s> <s id="s.004843">37″; atque angulus <lb/>NAC, hoc e&longs;t arcus ND gr. <!-- REMOVE S-->26. 16′. </s> <s id="s.004844">23″, & linea CN par­<lb/>tium (88 52/100): igitur ON e&longs;t part. (1 52/100). Atarcus DO &longs;it gr. <!-- REMOVE S-->45; <lb/>e&longs;t angulus ACN gr. <!-- REMOVE S-->135: datis ii&longs;dem lateribus AN 100, & <lb/>AC 13, invenitur angulus CNA gr. <!-- REMOVE S-->5. 16′. </s> <s id="s.004845">27″, atque angu­<lb/>lus NAC, hoc e&longs;t arcus ND gr. <!-- REMOVE S-->39. 43′. </s> <s id="s.004846">33″. <!-- REMOVE S-->& linea CN <lb/>part. (90 38/100): igitur ON part. (3 38/100). Similiter &longs;i DO &longs;it gr. <!-- REMOVE S-->60: <lb/>invenitur ON part. (5 86/100); &longs;i verò fuerit gr. <!-- REMOVE S-->75, e&longs;t ON (8 84/100): &longs;i <lb/>gr. <!-- REMOVE S-->90, e&longs;t ON (12 15/100). Mutetur jam hypothe&longs;is, & circuli dati <lb/>Radius &longs;it duplex, &longs;cilicet AD, hoc e&longs;t AN, partium 200, <lb/>quarum AC e&longs;t 13. Sit arcus DO gr. <!-- REMOVE S-->15: invenitur angulus <lb/>CNA gr. <!-- REMOVE S-->0. 57′. </s> <s id="s.004847">51″: atque angulus NAC gr. <!-- REMOVE S-->14. 2′. </s> <s id="s.004848">9″: ac <lb/>proinde linea GN partium (187 40/100), quarum CD, hoc e&longs;t CO, <lb/>e&longs;t 187; quare ON e&longs;t (40/100). Sit deinde arcus DO gr. <!-- REMOVE S-->30: in­<lb/>venitur angulus CNA gr. <!-- REMOVE S-->1.5 1′.45″, & angulus NAC, hoc e&longs;t <lb/>arcus DN, gr. <!-- REMOVE S-->28. 8′. </s> <s id="s.004849">15″, atque demum linea CN part. (188 63/100): <lb/>igitur ON part. (1 63/100). Denique arcus DO &longs;it gr. <!-- REMOVE S-->45: deprehen­<lb/>ditur angulus CNA gr. <!-- REMOVE S-->2. 38′. </s> <s id="s.004850">4″. <!-- REMOVE S-->angulus NAC, hoc e&longs;t ar­<lb/>cus DN, gr. <!-- REMOVE S-->42, 21′. </s> <s id="s.004851">56″; & linea CN part. (190 59/100); atque <lb/>adeò ON part. (3 59/100). Si DO &longs;it gr. <!-- REMOVE S-->60, ON e&longs;t part. (6 18/100); &longs;i <lb/>DO &longs;it gr. <!-- REMOVE S-->75, ON e&longs;t (9 35/100). Si &longs;it gr. <!-- REMOVE S-->90, ON e&longs;t (12 57/100). </s> </p> <p type="main"> <s id="s.004852">Ex his manife&longs;tò con&longs;tat initio motûs in primo quadrante à <lb/>circulo majore paulò ampliùs propelli pondus ex D versùs H, <lb/>quàm à circulo minore, datâ angulorum motûs ad centrum C <lb/>paritate. </s> <s id="s.004853">Verùm in circulo majoris diametri non &longs;olùm pari <lb/>graduum numero re&longs;pondet longior arcus pro Ratione diame­<lb/>trorum, &longs;ed etiam, ut ex &longs;uperioribus calculis con&longs;tat, major <lb/>circulus plures gradus ponderi coaptat, quàm minor. </s> <s id="s.004854">Sic in <lb/>motu ad centrum C gr. <!-- REMOVE S-->15, circulo minori, cujus Radius 100, <lb/>competunt gr. <!-- REMOVE S-->13. 4′. </s> <s id="s.004855">19″; at circulo majori, cujus Radius 200, <lb/>competunt gr. <!-- REMOVE S-->14. 2′. </s> <s id="s.004856">9″. <!-- KEEP S--></s> <s id="s.004857">Quare præterquam quod duplex e&longs;t <pb pagenum="656" xlink:href="017/01/672.jpg"/>longitudo arcus majoris, quia duplex e&longs;t Radius, adhuc &longs;u­<lb/>pere&longs;t longitudo gr. <!-- REMOVE S-->0. 57′. </s> <s id="s.004858">50″: cum tamen motus ponderis in <lb/>minore &longs;it (38/100), in majore (40/100); quod di&longs;crimen (2/100) longè minus e&longs;t <lb/>illo exce&longs;&longs;u arcûs. </s> </p> <p type="main"> <s id="s.004859">Quapropter &longs;i Potentia peripheriæ dati circuli partibus &longs;ub­<lb/>inde applicetur (ut &longs;i extarent ad orbitam paxilli perpendicula­<lb/>res) patet in majore circulo haberi majora momenta; multo <lb/>magis, &longs;i applicetur juxta maximam à motûs centro di&longs;tantiam; <lb/>id quod fieri expedit, &longs;i nihil ob&longs;it: At &longs;i Potentia à centro mo­<lb/>tûs æquè ab&longs;it in majore atque in minore circulo, non habetur <lb/>hoc momentorum compendium, quod ex di&longs;tantia facilè obti­<lb/>neri po&longs;&longs;et. </s> </p> <p type="main"> <s id="s.004860">Si itaque corpus ex D in H impul&longs;um aut vi ela&longs;ticâ re&longs;ti­<lb/>tuere &longs;e po&longs;&longs;it ex H in D, aut illud &longs;ublevatum (&longs;i circulus <lb/>fuerit in plano Verticali) &longs;uâ gravitate de&longs;cendere valeat ex H <lb/>in D, paulatim in priorem locum redibit, cum potentia tran&longs;­<lb/>gre&longs;&longs;a punctum H per I &longs;e re&longs;tituet in B: potentia igitur perpe­<lb/>tuò in gyrum circumactâ, circulum &longs;imiliter ver&longs;ando, corpus <lb/>illud in motu reciprocando &longs;ervabit con&longs;tantiam. </s> <s id="s.004861">Quod &longs;i vir­<lb/>tute ela&longs;ticâ præditum &longs;it corpus impul&longs;um ex D in H, illa pa­<lb/>riter, præter in&longs;itam corpori gravitatem, movendi difficulta­<lb/>tem augebit, quippe cui vis inferenda e&longs;t, quam deinde excu­<lb/>tere valeat. </s> <s id="s.004862">Quare &longs;atius fuerit omnem virtutem ela&longs;ticam amo­<lb/>vere (&longs;i id quidem fieri po&longs;&longs;it) ut &longs;ola gravitatis re&longs;i&longs;tentia &longs;u­<lb/>peranda &longs;it. </s> </p> <p type="main"> <s id="s.004863">Ut autem corpus ultro citróque remeare po&longs;&longs;it ex D in H, & <lb/>vici&longs;&longs;im ex H in D, regula &longs;tatuatur in Verticali plano erecta, <lb/>&longs;ed ver&longs;atilis circa axem, aut annulum alteri eju&longs;dem regulæ <lb/>extremitati infixum, & reliqua regulæ extremitas mobilis oc­<lb/>currat circulo in B: Tum funiculo longitudine diametrum BD <lb/>æquante connectatur regula cum pondere movendo; &longs;ic enim <lb/>fiet ut regulæ extremitas devenerit in L, quando pondus fue­<lb/>rit in H; atque propellendo regulam ex L in B, pondus ex H <lb/>trahetur in D, & perpetua vici&longs;&longs;itudine tum regula, tùm pon­<lb/>dus à circumacto cuneo impellentur: &longs;emper verò re&longs;i&longs;tentia <lb/>orietur ex ponderis modò impul&longs;i, modò attracti gravitate; re­<lb/>gula &longs;iquidem per &longs;e nihil ob&longs;i&longs;tit, &longs;ed quatenus cum pondere <lb/>trahendo conjungitur. </s> <s id="s.004864">Verùm qua po&longs;itione collocandus &longs;it fu-<pb pagenum="657" xlink:href="017/01/673.jpg"/>niculus circuli diametro BD re&longs;pondens, an &longs;upra, an infra cir­<lb/>culum BED, quid opus e&longs;t explicare? </s> <s id="s.004865">&longs;atis enim cuique mani­<lb/>fe&longs;tum e&longs;t attendendum e&longs;&longs;e, qua ratione ip&longs;i circulo applicetur <lb/>potentia movens; nam &longs;i illum Potentia agitet paxillo in &longs;upe­<lb/>riori aut inferiori facie extremæ orbitæ infixo, patet funiculum <lb/>adver&longs;æ faciei re&longs;pondere, ne in illum paxillus incurrat. </s> <s id="s.004866">Sin <lb/>autem potentia circulo non proximè adhæreat, nec illum tan­<lb/>gat, quia ex centro C exeunti axi additum e&longs;t manubrium cir­<lb/>culo parallelum, aut Vectis, aut circulus alius eidem parallelus, <lb/>liberum erit funiculum alterutri circuli faciei re&longs;pondentem <lb/>collocare, ex neutrâ &longs;cilicet parte impedimento e&longs;&longs;e pote&longs;t <lb/>potentiæ &longs;e in gyrum contorquenti. </s> </p> <p type="main"> <s id="s.004867">Ne me verò carpendum puta, quòd integrum circulum <lb/>FBED propo&longs;uerim, cum &longs;atis e&longs;&longs;e po&longs;&longs;it &longs;egmentum paulo <lb/>majus &longs;emicirculo BED (ut &longs;cilicet &longs;it locus axi infigendo in C <lb/>motûs centro) cui tota vis impellendi &longs;ive pondus, &longs;ive regu­<lb/>lam, tribuenda e&longs;t. </s> <s id="s.004868">Eo con&longs;ilio integrum circulum FBED <lb/>propo&longs;ui, ut liberum potentiæ &longs;it &longs;ive in dexteram, &longs;ive in &longs;i­<lb/>ni&longs;tram motum in&longs;tituere, atque promi&longs;cuè uti modò cuneo <lb/>inflexo BED, modò BFD, prout commodius acciderit. </s> <s id="s.004869">Dein­<lb/>de &longs;i &longs;emicirculo tantùm BED utamur, & vis ela&longs;tica interve­<lb/>niat, aut gravitas &longs;ublevata recidat, ubi potentia venerit in H, <lb/>& &longs;emicirculi BED &longs;it facta po&longs;itio HML, fieri non pote&longs;t, ut <lb/>potentia versùs I procedat, quin illicò & qua&longs;i momento pon­<lb/>dus redeat ad D; huju&longs;modi verò motus adeò velox vix contin­<lb/>gere &longs;æpiùs pote&longs;t citra aliquod detrimentrum; cui periculo <lb/>occurritur, &longs;i integer fuerit circulus FBED, &longs;en&longs;im enim fit <lb/>regre&longs;&longs;us ex H in D. <lb/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004870"><emph type="center"/>CAPUT IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004871"><emph type="center"/><emph type="italics"/>Ex Cylindro construi potest Cuneus perpetuus.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004872">ALtera &longs;pecies Cunei perpetui de&longs;umi poterit ex Cylindro <lb/>Recto obliquè &longs;ecto, erit &longs;iquidem &longs;ectio Ellip&longs;is; & poti&longs;-<pb pagenum="658" xlink:href="017/01/674.jpg"/>&longs;imùm in&longs;erviet ad deprimendum, &longs;i cylindrus fuerit horizon­<lb/>ti perpendicularis, atque addito vecte circumagatur Ergatæ in <lb/>morem: Sin autem cylindrus &longs;it horizonti parallelus, in&longs;erviet <lb/>ad propellendum pondus, & Radios admittet quemadmodum <lb/>Sucula. </s> <s id="s.004873">Sit cylindrus BC Rectus, <lb/><figure id="id.017.01.674.1.jpg" xlink:href="017/01/674/1.jpg"/><lb/>cujus &longs;cilicet Axis e&longs;t ad ba&longs;im <lb/>perpendicularis, & obliquè &longs;ece­<lb/>tur plano per DC; fit enim &longs;ectio <lb/>DECF ellip&longs;is. </s> <s id="s.004874">Quod &longs;i huju&longs;­<lb/>modi Ellip&longs;is planities officere po&longs;­<lb/>&longs;it motui, interiores partes aliquan­<lb/>tulùm excavari oportebit, quando­<lb/>quidem &longs;ufficit limbus perimetri, <lb/>modò &longs;it &longs;olidus, & &longs;atis validus; <lb/>quem & ferreâ laminâ non ruditer <lb/>politâ munire operæ pretium fue­<lb/>rit. </s> <s id="s.004875">Ut autem huju&longs;modi &longs;ectionis obliquitas major aut minor <lb/>opportunè fiat, &longs;tatuenda e&longs;t primùm men&longs;ura depre&longs;&longs;ionis aut <lb/>impul&longs;ionis, qua movendum e&longs;t pondus, & &longs;it ex. </s> <s id="s.004876">gr. <!-- REMOVE S-->CA, cui <lb/>æqualis &longs;umatur ID: tùm ex D in C fiat &longs;ectio, & erit con&longs;ti­<lb/>tutus cuneus ADFC, cujus latus unum e&longs;t cylindri &longs;emipe­<lb/>ripheria AD, aliud DFC &longs;emiperimeter Ellip&longs;is, cujus partes <lb/>ponderi in D con&longs;tituto &longs;ubinde applicantur ex convolutione <lb/>cylindri, & quatenus ab AD recedunt, pondus deprimunt, aut <lb/>impellunt, donec demum à puncto C attingatur pondus propul­<lb/>&longs;um ex D in I. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004877">Quoniam verò Ellipticum limbum ferreâ laminâ muniendum <lb/>dixi, non omninò abs re fuerit indicare, qua methodo illius <lb/>perimetrum indagare po&longs;&longs;imus, ut laminæ in ellipticam figu­<lb/>ram inflectendæ longitudo innote&longs;cat. </s> <s id="s.004878">In circulo quidem nota <lb/>e&longs;t aliqua Ratio diametri ad peripheriam, &longs;ive ut 7 ad 22, &longs;i­<lb/>ve ut 71 ad 223, &longs;ive ut 113 ad 355, &longs;ive quæcumque alia ma­<lb/>gis arrideat majoribus numeris explicata: at in Ellip&longs;i nulla hu­<lb/>ju&longs;modi Ratio perimetri ad alterutrum Axem (quod quidem <lb/>&longs;ciam) deprehen&longs;a adhuc e&longs;t: quapropter ad inve&longs;tigandam <lb/>ejus perimetrum ex datis Axibus variæ tentatæ &longs;unt viæ, quas <lb/>inire nunc non e&longs;t operæ pretium. </s> <s id="s.004879">Mihi hanc, utpote perbre­<lb/>vem, nec à veritatis formâ, quantum res Phy&longs;ica patitur, re-<pb pagenum="659" xlink:href="017/01/675.jpg"/>cedentem ineundam &longs;u&longs;cepi. </s> <s id="s.004880">Illud verò tanquam certum & <lb/>demon&longs;tratum pono, quod Ellip&longs;is, quæ ex Coni &longs;ectione, ab <lb/>ea quæ ex Cylindri &longs;ectione oritur, non differt, ut quidam mi­<lb/>nùs attenti perperam exi&longs;timârunt; proinde quælibet oblata <lb/>Ellip&longs;is ad aliquem Cylindrum &longs;pectare pote&longs;t. </s> <s id="s.004881">Ad quem au­<lb/>tem cylindrum illa pertineat, facile e&longs;t determinare ex ip&longs;ius <lb/>Ellip&longs;is Axe minori, qui e&longs;t æqualis diametro Cylindri. </s> <s id="s.004882">Datis <lb/>igitur, Axibus, Majore, & Minore, invenire oportet, quan­<lb/>ta &longs;it in huju&longs;modi Cylindro obliquitas &longs;ectionis Ellip&longs;im <lb/>con&longs;tituens: id quod obtinetur, &longs;i ex quadrato Axis Majoris <lb/>auferatur quadratum Axis Minoris; re&longs;idui enim Radix qua­<lb/>drata dabit in Cylindri latere longitudinem, qua di&longs;tant inter <lb/>&longs;e duo plana ba&longs;i parallela, inter quæ intercipitur &longs;ectio <lb/>obliqua. </s> </p> <p type="main"> <s id="s.004883">Sit data Ellip&longs;is DECF, cujus major Axis DC 25, Minor <lb/>FE 20. Axi FE æqualis e&longs;t cylindri diameter CI. <!-- KEEP S--></s> <s id="s.004884">Igitur pla­<lb/>num per axem cylindri ductum habet cum plano obliquè &longs;e­<lb/>cante communem &longs;ectionem DC Axem Ellip&longs;is, & cum cy­<lb/>lindri ba&longs;e &longs;ectionem facit CI, atque in &longs;uperficie dat latus DI. <!-- KEEP S--></s> <lb/> <s id="s.004885">Quare e&longs;t triangulum rectangulum CID, cujus datur hypo­<lb/>thenu&longs;a DC 25, & ba&longs;is CI 20: ex ip&longs;ius DC quadrato 625 <lb/>ablatum quadratum ex CI 400, relinquit 225, quadratum per­<lb/>pendiculi DI, quod propterea e&longs;t 15. Plana igitur CI, & AD <lb/>parallela di&longs;tant intervallo CI 15. Cum itaque ex ba&longs;is dia­<lb/>metro CI 20 innote&longs;cat eju&longs;dem cylindricæ ba&longs;is periphe­<lb/>ria (62 84/100) proximè, &longs;uperficies cylindrica ACID manife&longs;ta e&longs;t <lb/>(942 60/100), cujus &longs;emi&longs;&longs;em (471 30/100) dividit bifariam &longs;emiperimeter <lb/>Ellip&longs;is DEC. </s> <s id="s.004886">E&longs;t igitur triangulum rectangulum, cujus late­<lb/>ra circa rectum &longs;unt latus DI 15, & cylindricæ ba&longs;is &longs;emiperi­<lb/>pheria CHI (31 42/100): horum quadrata 225, & (987 2164/10000) in &longs;um­<lb/>mam colligantur, & quadrati (1212 2164/10000) Radix (34 82/100) ferè, e&longs;t <lb/>Ellip&longs;is &longs;emiperimeter DEC, integra verò DECF erit (69 64/100). </s> </p> <p type="main"> <s id="s.004887">Quapropter cum plana per AD & CI ex hypothe&longs;i &longs;int pa­<lb/>rallela, etiam DI, & AC æqualia &longs;unt latera. </s> <s id="s.004888">Igitur cum cy­<lb/>lindri dati nota &longs;it diameter 20, atque adeò &longs;emiperipheria <lb/>AD (31 42/100), &longs;it data obliquitas, quam metitur AC 15, ex &longs;um­<lb/>mâ quadratorum rectæ AC, & &longs;emiperipheriæ AD, cruatur <pb pagenum="660" xlink:href="017/01/676.jpg"/>Radix quadrata, & dabit &longs;emiperimetrum Ellip&longs;is DFC proxt­<lb/>mam veræ, quæ ex jam datis e&longs;t illa eadem, quam paulo antè <lb/>invenimus (34 82/100). </s> </p> <p type="main"> <s id="s.004889">Hinc igitur innote&longs;cunt momenta huju&longs;modi cunei, com­<lb/>paratis inter &longs;e lineis, quæ definiunt motum ponderis atque <lb/>potentiæ; pondus enim movetur juxta lineam AC, potentia <lb/>autem juxta &longs;emiperipheriam cylindri, quatenus videlicet præ­<lb/>cisè atque &longs;impliciter ratione ip&longs;ius cunei motus illi convenit. </s> <lb/> <s id="s.004890">Verùm quia non facilè potentia applicatur proximè &longs;uperficiei <lb/>cylindri, & &longs;æpius expedit potentiæ momenta augere; propterea <lb/>Cylindro infigitur Vectis MN, cujus longitudo de&longs;umitur à <lb/>puncto, ubi ille concurrit cum Axe Cylindri, u&longs;que ad extre­<lb/>mitatem N, cui potentia applicatur. </s> <s id="s.004891">Hæc autem longitudo, <lb/>manente eodem cuneo, varia omnino e&longs;&longs;e pote&longs;t, atque adeò <lb/>potentiæ momenta repræ&longs;entabit &longs;emiperipheria circuli ab ex­<lb/>tremitate N de&longs;cripti, quæ comparanda erit cum motu ip&longs;ius <lb/>ponderis ab obliquitate &longs;ectionis definito, ut dictum e&longs;t. </s> </p> <p type="main"> <s id="s.004892">At &longs;ubdubitare contingit, utrùm cra&longs;&longs;iore, an graciliore cy­<lb/>lindro uti expediat, manente eâdem obliquitatis men&longs;urâ, at­<lb/>que eâdem vectis longitudine; manet &longs;iquidem eadem mo­<lb/>tuum Ratio; &longs;ed augeri videtur corporum conflictus ex mutuo <lb/>tritu, nam in cra&longs;&longs;iore cylindro major e&longs;t elliptica &longs;emiperime­<lb/>ter, quàm in tenuiore, ut manife&longs;tum e&longs;t, &longs;i methodo paulò <lb/>antè indicatâ res ad calculos revocetur: quamobrem ex majore <lb/>hoc tritu augeri videtur difficultas movendi, cum maneat ea­<lb/>dem corporis gravitas, eadem potentiæ virtus, eadem motuum <lb/>Ratio. <!-- KEEP S--></s> <s id="s.004893">Longè tamen aliter &longs;e res habet; quandoquidem duo­<lb/>rum corporum &longs;e &longs;e invicem in motu contingentium conflictus, <lb/>qui ex &longs;uperficiei a&longs;peritate oritur (hìc corporis unius conatum <lb/>adversùs aliud vi &longs;uæ gravitatis mente &longs;ecernimus à conatu, <lb/>quo illud repellit præcisè vi &longs;uæ molis tanquam objectum im­<lb/>pedimentum, etiam&longs;i adversùs illud non gravitet) con&longs;ide­<lb/>randus e&longs;t, quatenus corpus impul&longs;um adver&longs;atur directioni <lb/>motûs corporis impellentis. </s> <s id="s.004894">Hinc e&longs;t minimum e&longs;&longs;e conflictum, <lb/>&longs;i ambæ facies &longs;e in plano Verticali contingant, & alterutrum <lb/>corpus in eodem plano Verticali moveatur; nam reliquum cor­<lb/>pus non repellitur, quia in illud non incurrit linea directionis <lb/>motûs alterius corporis, &longs;ed &longs;olùm prominulæ utriu&longs;que corpo-<pb pagenum="661" xlink:href="017/01/677.jpg"/>ris particulæ, quatenus aliæ in alias incurrunt, impediunt mo­<lb/>tum pro earum magnitudine & numero: quoad impedimentum <lb/>maximâ ex parte tollitur, &longs;i pingui aliquo humore delibutæ fa­<lb/>cies lubricæ fiant; replentur &longs;cilicet inanitates inter prominulas <lb/>particulas interjectæ, quas intercapedines &longs;ubire non tam faci­<lb/>lè po&longs;&longs;unt corporis proximi particulæ. </s> <s id="s.004895">Sic &longs;i integer e&longs;&longs;et cylin­<lb/>drus, &longs;uâ ba&longs;i aut limbo CHI contingens &longs;ubjectum corpus, <lb/>minimo tritu cum illo confligeret in motu circà &longs;uum Axem, <lb/>quia huju&longs;modi motui non opponitur corpus illud in I po&longs;itum. </s> <lb/> <s id="s.004896">At verò major e&longs;t conflictus, quando directioni motûs illud ad­<lb/>ver&longs;atur, ut cùm prope D e&longs;&longs;e intelligitur aliquâ &longs;ui parte &longs;ub­<lb/>jectum cylindro, qui obliquè &longs;ectus circumagi non pote&longs;t, quin <lb/>urgeat illud ex D versùs I. <!-- KEEP S--></s> <s id="s.004897">Quò autem majore angulo planum <lb/>Ellipticum CEDF inclinatur ad ba&longs;is planum CHI, eò magis <lb/>conver&longs;ioni cylindri adver&longs;atur objectum corpus, adeóque ma­<lb/>jor invenitur difficultas. </s> </p> <p type="main"> <s id="s.004898">Cùm itaque in majore cylindro, datâ æquali obliquitatis <lb/>men&longs;urâ AC (æqualem obliquitatem non dico) planum obli­<lb/>què &longs;ecans minorem angulum cum plano ba&longs;is cylindri con&longs;ti­<lb/>tuat, magi&longs;que ad ip&longs;am ba&longs;im accedat, minus habet re&longs;i&longs;ten­<lb/>tiæ ab objecto corpore, &longs;i particulæ &longs;ingulæ con&longs;iderentur, <lb/>quamvis cunctæ re&longs;i&longs;tentiæ &longs;imul collectæ demùm in æqualem <lb/>&longs;ummam à men&longs;ura AC definitam coëant. </s> <s id="s.004899">Sit enim minoris cy­<lb/>lindri &longs;emiperipheria RS, men­<lb/>&longs;ura obliquitatis RO, &longs;emiperi­<lb/><figure id="id.017.01.677.1.jpg" xlink:href="017/01/677/1.jpg"/><lb/>meter Ellip&longs;is SO: Manente au­<lb/>tem eâdem RO, &longs;it majoris cy­<lb/>lindri &longs;emiperipheria RT, & el­<lb/>lip&longs;is &longs;emiperimeter TO; utique <lb/>angulus RSO, utpote externus, <lb/>major e&longs;t interno oppo&longs;ito RTO; ac proinde alternus SOX <lb/>major e&longs;t alterno TOX, quibus angulis repræ&longs;entatur plani <lb/>obliquè &longs;ecantis inclinatio ad ba&longs;im cylindri. </s> <s id="s.004900">Quamvis igitur <lb/>ex cylindri convolutione &longs;emiperimeter Ellip&longs;is SO impellat <lb/>pondus juxta men&longs;uram RO, & juxta eandem men&longs;uram RO <lb/>impellatur pondus à &longs;emiperimetro Ellip&longs;is TO; item ab illius <lb/>quadrante SP, atque hujus quadrante TQ, æqualiter impella­<lb/>tur juxta men&longs;uram MP, & NQ, quæ æquales &longs;unt (utraque <pb pagenum="662" xlink:href="017/01/678.jpg"/>&longs;cilicet e&longs;t quadrans ip&longs;ius RO, &longs;iquidem propter triangulo­<lb/>rum &longs;imilitudinem, ut OS ad PS, ita RO ad MP, & ut OT <lb/>ad QT, ita RO ad NQ; &longs;ed ex hypothe&longs;i OS e&longs;t quadrupla <lb/>ip&longs;ius PS, &longs;icut OT e&longs;t quadrupla ip&longs;ius QT; igitur RO e&longs;t <lb/>quadrupla ip&longs;ius MP, & ip&longs;ius NQ, quæ propterea &longs;unt æqua­<lb/>les) quia tamen QT major e&longs;t quàm PS, qua Ratione RT ma­<lb/>jor e&longs;t quàm RS, & OT major quàm OS; idcircò eadem re­<lb/>&longs;i&longs;tentia di&longs;tributa per plures particulas longioris QT, &longs;eu OT, <lb/>minor e&longs;t in &longs;ingulis particulis, quàm cùm di&longs;tribuitur per pau­<lb/>ciores particulas brevioris PS, &longs;eu OS. </s> <s id="s.004901">Minùs igitur TO &longs;uis <lb/>particulis &longs;ubinde contingens corpus, quod impellit, cum eo <lb/>confligit, quàm confligat SO pertinens ad minorem cy­<lb/>lindrum. </s> </p> <p type="main"> <s id="s.004902">Huju&longs;modi Cuneum inflexum ex cylindro obliquè &longs;ecto per­<lb/><figure id="id.017.01.678.1.jpg" xlink:href="017/01/678/1.jpg"/><lb/>petuum e&longs;&longs;e in reciprocando motu, &longs;atis <lb/>manife&longs;tum e&longs;t, &longs;i po&longs;tquam pondus ex <lb/>D propul&longs;um e&longs;t in I, iterum redeat ad <lb/>D; ab&longs;olutâ enim cylindri conver&longs;ione <lb/>iterum pondus ex D ad I propellitur. </s> <s id="s.004903">Id <lb/>autem ut fiat, &longs;tatuatur jugum KL cir­<lb/>ca axem in V ver&longs;atile, ita tamen ut V <lb/>re&longs;pondeat axi cylindri: tùm in L ad­<lb/>nectatur pondus, quod ex D impelletur <lb/>in I, dum Cylindri dimidia revolutio ex <lb/>D per F in C perficietur: quia autem in reliquâ dimidiâ cylin­<lb/>dri revolutione jugi extremitas K jam repul&longs;a &longs;ur&longs;um versùs A, <lb/>impelletur iterum ad C, pondus re&longs;tituetur ex I in D, atque ita <lb/>deinceps reciprocando impul&longs;ionem tum ponderis adnexi in L, <lb/>tùm extremitatis K. <!-- KEEP S--></s> <s id="s.004904">Hinc &longs;i ex laqueari pendeat &longs;tatua ventum <lb/>referens, & in &longs;peciem volantis ingentes das expandens junctas <lb/>jugo KL, atque in &longs;uperiore conclavi ad&longs;it qui cylindrum cir­<lb/>cumagat, alis reciprocantibus commovebitur aër, & aura exci­<lb/>tabitur ad refrigerandum. </s> </p> <p type="main"> <s id="s.004905">Pro variis demum u&longs;ibus &longs;tatuetur cylindrus modò horizon­<lb/>ti, tanquam Ergata, perpendicularis, modò velut Sucula, paral­<lb/>lelus: Erítque expediti&longs;&longs;ima ejus conver&longs;io, &longs;i centrum Ellip&longs;is <lb/>nulli polo innitatur, &longs;ed cylindrus ip&longs;e congruo loculamento <lb/>ita in&longs;eratur, ut in eo &longs;it ver&longs;atilis, &, quam primò dederis, po-<pb pagenum="663" xlink:href="017/01/679.jpg"/>&longs;itionem deinceps &longs;ervet, ac neutram in partem nutet. <!--neuer Satz-->Id qui­<lb/>dem paulò longiorem cylindrum exigit; non tamen e&longs;t nece&longs;­<lb/>&longs;e unius perpetuæ cra&longs;&longs;itiei e&longs;&longs;e cylindrum; &longs;æpè enim nimis <lb/>cra&longs;&longs;um atque incommodum e&longs;&longs;e contingeret; &longs;ed fru&longs;to cy­<lb/>lindrico cra&longs;&longs;iori obliquè &longs;ecto firmiter in&longs;eri poterit gracilior <lb/>cylindrus, ita ut axis axi conveniat, & rectam lineam con&longs;ti­<lb/>tuant, atque hic in foramen immi&longs;&longs;us, in quo ver&longs;atilis e&longs;t, dum <lb/>contorquetur, cra&longs;&longs;iorem cylindrum pariter convolvit. <lb/> </s> </p> <p type="main"> <s id="s.004906"><emph type="center"/>CAPUT V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004907"><emph type="center"/><emph type="italics"/>Cuneum perpetuum Circulus inclinatus imitatur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004908">TErtiam hanc cunei perpetui &longs;peciem duabus &longs;uperioribus <lb/>adjicere non inutile fuerit, ut legenti hujus libri cap. ult. </s> <lb/> <s id="s.004909">prop. 6. con&longs;tabit, quanquam forta&longs;&longs;is alicui à &longs;uperiore parùm <lb/>di&longs;tare videatur; ibi enim Ellip&longs;im ex cylindri recti &longs;ectione <lb/>obliquâ, hìc circulum &longs;uo quidem centro in&longs;i&longs;tentem, &longs;ed in­<lb/>clinatum proponimus. </s> <s id="s.004910">Ut autem res <lb/>clariùs exponatur, concipiamus circu­<lb/><figure id="id.017.01.679.1.jpg" xlink:href="017/01/679/1.jpg"/><lb/>lum à plano horizontali RS &longs;ectum <lb/>per centrum A, ita ut eorum commu­<lb/>nis &longs;ectio &longs;it diameter BC, &longs;emicircu­<lb/>lus autem &longs;uperior ad horizontem in­<lb/>clinatus &longs;it BDC; qui per AD bifa­<lb/>riam &longs;ecetur plano Verticali ad &longs;ub­<lb/>jectum planum RS horizontale recto; <lb/>sítque horum planorum communis <lb/>&longs;ectio recta AE. <!-- KEEP S--></s> <s id="s.004911">Tum ex D Quadran­<lb/>tis extremitate demittatur per 11.lib.11. perpendicularis ad &longs;ub­<lb/>jectum planum recta DE; quæ propterea ex defin. </s> <s id="s.004912">3. lib.11. fa­<lb/>cit cum rectà AE angulum rectum. </s> <s id="s.004913">Accipiatur arcus DF, & <lb/>per F in circuli plano ductâ FG parallelâ ip&longs;i AD, per eam <lb/>ductum intelligatur planum parallelum plano tran&longs;eunti per <lb/>AD: Igitur planum horizontale plana illa parallela &longs;ecans, per <lb/>16. lib. 11. facit &longs;ectiones AE & GH parallelas, ac proinde, <lb/>cum duæ rectæ AD & AE duabus rectis GF & GH &longs;int pa-<pb pagenum="664" xlink:href="017/01/680.jpg"/>rallelæ, etiam per 10. lib. 11 anguli DAE, & FGH (cum illæ <lb/>&longs;int &longs;imiliter po&longs;itæ) &longs;unt æquales. </s> <s id="s.004914">Jam ex F demittatur in &longs;ub­<lb/>jectum planum perpendicularis FH, quæ cum rectâ GH <lb/>con&longs;tituit angulum rectum. </s> <s id="s.004915">Cum itaque duo triangula AED <lb/>& GHF rectangula habeant angulum DAE angulo FGH <lb/>æqualem, & reliquus reliquo æqualis e&longs;t, atque &longs;imilia &longs;unt <lb/>triangula: Quapropter ut AD ad DE, ita GF ad FH, & per­<lb/>mutando ut AD ad GF, ita DE ad FH. <!-- KEEP S--></s> <s id="s.004916">Eadem erit ratioci­<lb/>natio in triangulo KLI &longs;imiliter facto, quod erit reliquis &longs;imi­<lb/>le, & ut AD ad KI, ita DE ad IL: atque ita deinceps de cæ­<lb/>teris omnibus triangulis, quæ efformari po&longs;&longs;unt à lineis paral­<lb/>lelis Radio AD, tanquam hypothenu&longs;is, & à perpendiculari­<lb/>bus cadentibus in &longs;ubjectum planum ex peripheriâ circuli in­<lb/>clinati, & à rectis, quæ jungunt punctum, in quod cadit per­<lb/>pendiculum, cum extremitate altera hypothenu&longs;æ. </s> <s id="s.004917">Quoniam <lb/>verò omnes lineæ parallelæ Radio AD &longs;unt Sinus arcuum à <lb/>puncto C incipientium (&longs;ic IK e&longs;t Sinus arcûs IC, FG e&longs;t Si­<lb/>nus arcûs FC) omnium illarum Ratio manife&longs;ta e&longs;t ex Canone <lb/>Sinuum, &longs;i arcuum quantitas in gradibus data &longs;it, vel nota; <lb/>quare etiam nota e&longs;t Ratio perpendiculorum DE, FH, IL. <!-- KEEP S--></s> <lb/> <s id="s.004918">Hæc autem, quæ de Quadrante DAC dicta &longs;unt, etiam de <lb/>reliquo Quadrante DAB intelliguntur; & quæ de hoc &longs;upe­<lb/>riore &longs;emicirculo demon&longs;trata &longs;unt, etiam de inferiore &longs;emicir­<lb/>culo vera &longs;unt, quatenus ille ad hoc idem planum horizontale <lb/>RS refertur, à quo circulus bifariam &longs;ecatur. </s> </p> <p type="main"> <s id="s.004919">Jam verò integer circulus cum alio plano horizontali non <lb/>Secante, &longs;ed Tangente circulum inclinatum in puncto infimo, <lb/>comparetur: &longs;unt autem duo hæc plana horizontalia invicem <lb/>parallela; & perpendiculum à puncto D cadens in planum ho­<lb/>rizontale Tangens, e&longs;t duplum perpendiculi DE, quemadmo­<lb/>dum totius circuli diameter e&longs;t dupla Radij AD. <!-- KEEP S--></s> <s id="s.004920">Quapropter <lb/>cum nota &longs;it dati circuli diameter &longs;ecundùm certam men&longs;uram, <lb/>& data &longs;it circuli inclinatio, &longs;ive perpendiculi longitudo, qui <lb/>e&longs;t Sinus anguli inclinationis, facile e&longs;t invenire &longs;ingularum <lb/>perpendicularium quantitatem. </s> <s id="s.004921">Nam &longs;i datur angulus inclina­<lb/>tionis circuli ad planum Tangens (cùm hoc &longs;it parallelum pla­<lb/>no Secanti) angulus ille æqualis e&longs;t angulo DAE: quare &longs;icut <lb/>in triangulo DAE rectangulo datur hypothenu&longs;a AD Radius <pb pagenum="665" xlink:href="017/01/681.jpg"/>circuli, & angulus acutus adjacens DAE, ex quibus inveni­<lb/>tur latus DE, ita manife&longs;tum fit perpendiculum à &longs;ummo cir­<lb/>culi inclinati puncto D in planum horizontale Tangens, quod <lb/>e&longs;t duplum lateris DE inventi. </s> </p> <p type="main"> <s id="s.004922">Sivè igitur detur puncti D à plano horizontali Tangente <lb/>di&longs;tantia, &longs;ivè inveniatur, di&longs;tantia hæc bipartitò dividatur, <lb/>ejú&longs;que medietas tribuatur perpendiculari DE. <!-- KEEP S--></s> <s id="s.004923">Tum Qua­<lb/>drans DC in quotlibet partes æquales divi&longs;us intelligatur, puta <lb/>in decem, & ex Canone accipiantur &longs;ingulorum arcuum Sinus <lb/>gr. <!-- REMOVE S-->81.72. 63.54. 45.36.27.18.9: deinde fiat ut Radius ad &longs;ingu­<lb/>los Sinus, ita notum perpendiculum DE ad aliud, & proveniet <lb/>&longs;ingulorum perpendiculorum in planum Secans cadentium <lb/>men&longs;ura: quibus &longs;ingillatim addenda e&longs;t quantitas ip&longs;ius DE, <lb/>hoc e&longs;t dimidia altitudo &longs;umma, ut habeatur &longs;ingulorum alti­<lb/>tudo &longs;uprà planum horizontale Tangens. <!-- KEEP S--></s> <s id="s.004924">Ponatur &longs;umma cir­<lb/>culi elevatio à po&longs;itione horizontali, palmi unius: igitur DE <lb/>e&longs;t &longs;emipalmus, qui intelligatur di&longs;tinctus in particulas 100.000, <lb/>adeóque totus palmus in part. </s> <s id="s.004925">200.000. Ideò in &longs;uperiori <lb/>Quadrante &longs;ingulis perpendiculis addito &longs;emipalmo perpen­<lb/>diculorum altitudo ea e&longs;t, quam adjecta tabella exhibet. </s> </p> <p type="table"> <s id="s.004926">TABELLE WAR HIER<!-- KEEP S--></s> </p> <p type="main"> <s id="s.004927">Pro inferiori autem Quadrante ponendo gr. <!-- REMOVE S-->90. in puncto con­<lb/>tactûs circuli cum plano Tangente, lineæ perpendiculares ad <pb pagenum="666" xlink:href="017/01/682.jpg"/>planum Secans auferendæ &longs;unt à &longs;emi&longs;&longs;e datæ elevationis, hoc <lb/>e&longs;t ex &longs;emipalmo part. </s> <s id="s.004928">100000, & re&longs;iduum e&longs;t di&longs;tantia per­<lb/>pendicularis à &longs;ubjecto plano Tangente &longs;ingulis partium <lb/>punctis re&longs;pondens; quemadmodum adjectæ tabellæ pars alte­<lb/>ra o&longs;tendit. </s> </p> <p type="main"> <s id="s.004929">His fundamentis po&longs;itis innititur &longs;pecies hæc cunei petita ex <lb/>circulo inclinato, qui eandem &longs;ervans inclinationem circa <lb/>&longs;uum centrum convertitur. </s> <s id="s.004930">Sit enim circu­<lb/><figure id="id.017.01.682.1.jpg" xlink:href="017/01/682/1.jpg"/><lb/>lus BFED, cujus centrum C, ad horizon­<lb/>tem, &longs;ive ad planum Verticale inclinatus, & <lb/>&longs;it in D pondus impellendum. </s> <s id="s.004931">Potentia in B <lb/>exi&longs;tens, circulumque retinens in eâdem in­<lb/>clinatione, &longs;i illum circa &longs;uum centrum C <lb/>circumagat, paulatim pondus impellit, prout <lb/>illud tangitur modò à puncto E, modò à <lb/>puncto F, donec demum à puncto B infimo <lb/>ad extremum motûs terminum deducatur. </s> <s id="s.004932">Quare potentiæ <lb/>motus definitur à &longs;emicirculi peripheriâ BFED, motus verò <lb/>ponderis à rectâ DH. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.004933">Ut autem circulus in conver&longs;ione eandem &longs;emper inclinatio­<lb/>nem &longs;ervet, fru&longs;tum ligni G obliquè in parte &longs;uperiori &longs;ectum, <lb/>inferiùs affigatur circulo (aut contrà in parte inferiori &longs;ectum <lb/>&longs;uperiùs affigatur, prout commodius acciderit) atque in ligno <lb/>foramen fiat re&longs;pondens circuli centro C, per quod foramen <lb/>tran&longs;eat polus, cui centrum in&longs;i&longs;tit. </s> <s id="s.004934">Cum enim foramen illud <lb/>perpendiculare maneat ad horizontem, circulus eandem reti­<lb/>net in conver&longs;ione inclinationem. </s> </p> <p type="main"> <s id="s.004935">Verùm quia priùs &longs;tatuendum e&longs;t &longs;patium DH, per quod <lb/>ponderi commeandum e&longs;t, quàm circuli amplitudo definiatur, <lb/>non &longs;olùm ut Potentiæ motus ad ponderis motum Rationem <lb/>habeat majorem pro circuli &longs;emiperipheriæ longitudine, &longs;ed <lb/>etiam ut quàm minimum fieri po&longs;&longs;it, inclinatio ip&longs;a recedat à <lb/>paralleli&longs;mo cum plano, ad quod inclinari dicitur, &longs;ive illud <lb/>horizontale &longs;it, &longs;ive Verticale, quò enim minùs directioni mo­<lb/>tûs potentiæ opponitur pondus, eò minùs re&longs;i&longs;tit: Propterea <lb/>data linea DH &longs;tatuatur ut Sinus anguli inclinationis, & Ra­<lb/>dio re&longs;pondebit diameter circuli opportuni. </s> <s id="s.004936">Sic &longs;i recta DH &longs;it <lb/>linea palmaris, & angulus inclinationis ponatur gr. <!-- REMOVE S-->10: fiat ut <pb pagenum="667" xlink:href="017/01/683.jpg"/>gr.10 Sinus 17365 ad Radium 100000, ita 1 palmus ad pal­<lb/>mos (5 76/100) ferè, quæ e&longs;&longs;et diameter circuli eam inclinationem <lb/>habentis; atque adeò motus potentiæ cum circulo in gyrum <lb/>actæ e&longs;&longs;et ad motum ponderis &longs;altem noncuplus. </s> <s id="s.004937">Ex quibus &longs;a­<lb/>tis apertum e&longs;t ampliorem circulum præ minore utiliorem e&longs;&longs;e, <lb/>cæteris paribus. </s> </p> <p type="main"> <s id="s.004938">At circulum ip&longs;um convolvere aut non placet, aut non licet, <lb/>quia forta&longs;&longs;e pondus illius peripheriæ adnexum e&longs;t, atque id­<lb/>circò non ni&longs;i in gyrum pariter cum circulo ageretur. </s> <s id="s.004939">Idem <lb/>planè a&longs;&longs;equemur, &longs;i circulum horizonti, aut plano Verticali, <lb/>con&longs;titutum parallelum potentia urgeat in B; tùm puncto D <lb/>applicetur pondus; deinde potentia pergens in F & E percurrat <lb/>circuli ambitum illum deprimendo & inclinando; quandoqui­<lb/>dem utroque modo mutatur di&longs;tantia ponderis ab infimo <lb/>puncto circuli. </s> <s id="s.004940">Ponatur enim punctum F æquè di&longs;tans à puncto <lb/>B, atque punctum E di&longs;tat à puncto D. <!-- KEEP S--></s> <s id="s.004941">Si potentia manens ap­<lb/>plicata eidem puncto B convertat circulum ita, ut ip&longs;a poten­<lb/>tia di&longs;tet à pondere arcu BE, pondus non adnexum circulo <lb/>impellitur pro Ratione, quam arcus ille exigit: at verò &longs;i ma­<lb/>nente pondere applicato ad punctum D, cui adnectitur, poten­<lb/>tia pergat ex B in F, &longs;imiliter di&longs;tat à pondere arcu FD, qui <lb/>e&longs;t æqualis arcui BE; atque proinde æqualiter deprimitur, pro­<lb/>ut idem arcus exigit, juxta &longs;uperius explicata, & in tabellâ ex­<lb/>po&longs;ita; atque ita deinceps, donec potentia veniat in D: &longs;ingulas <lb/>autem impul&longs;iones metitur differentia perpendicularium. </s> </p> <p type="main"> <s id="s.004942">Hìc igitur ubi pondus circulo adnexum ponitur, manife&longs;ta <lb/>e&longs;t motûs reciprocatio: potentia &longs;iquidem ubi per F & E vene­<lb/>rit in circuli punctum D, & impulerit pondus u&longs;que in H, <lb/>percurrendo reliquum &longs;emicirculum DIB iterum retrahit pon­<lb/>dus ex H in D. <!-- KEEP S--></s> <s id="s.004943">At quando pondus non connectitur cum circu­<lb/>lo, & circulus ip&longs;e convertitur, tunc opus e&longs;t aliquo artificio, <lb/>ut pondus ex H remeet in D, quemadmodum indicatum e&longs;t <lb/>capite &longs;uperiori. </s> </p> <p type="main"> <s id="s.004944">Porrò circulus i&longs;te non convolutus, &longs;ed à potentiâ ejus am­<lb/>bitum percurrente &longs;ecundùm alias atque alias partes inclinatus, <lb/>non e&longs;t à Ratione Cunei excludendus; quandoquidem parùm <lb/>intere&longs;t utrùm &longs;imili motu potentia atque organum moveantur, <lb/>an verò di&longs;&longs;imili motu. </s> <s id="s.004945">Quando potentia in eodem puncto B <pb pagenum="668" xlink:href="017/01/684.jpg"/>&longs;emper applicata in gyrum pergit, &longs;imili motu cum circulo in <lb/>gyrum acto movetur: quando verò potentia quidem circulari­<lb/>ter movetur, &longs;ed non &longs;ecum rapit circulum, quem &longs;olummodò <lb/>inclinat, e&longs;t quidem diver&longs;us potentiæ motus à motu organi, <lb/>&longs;ed ponderis motus idem planè efficitur in utroque ca&longs;u, & <lb/>æqualis e&longs;t potentiæ ip&longs;ius motus determinatus à Rationibus <lb/>Cunei, quamvis hic non promoveatur, &longs;ed &longs;olùm impellatur. <lb/></s> </p> <p type="main"> <s id="s.004946"><emph type="center"/>CAPUT VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.004947"><emph type="center"/><emph type="italics"/>Vnde oriatur vis Percu&longs;sionis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.004948">CUnei vires, quatenus ex ejus formâ proveniunt, hactenus <lb/>con&longs;ideravimus; nunc ad id, quod poti&longs;&longs;imum in hac <lb/>tractatione videtur, tran&longs;eundum e&longs;t, videlicet ad percu&longs;&longs;io­<lb/>nem, qua dum adigitur Cuneus, multo faciliùs con&longs;equitur mo­<lb/>tus (&longs;ive &longs;ci&longs;&longs;io &longs;it, &longs;ive &longs;implex impul&longs;io, citrà corporis divi&longs;io­<lb/>nem) quàm &longs;i onere impo&longs;ito prægravaretur, aut Vecte &longs;eu aliâ <lb/>qualibet Facultate augerentur Potentiæ momenta. </s> <s id="s.004949">Certè <lb/>Ari&longs;toteles Mechan. quæ&longs;t. </s> <s id="s.004950">19. quærit, <emph type="italics"/>Cur &longs;i quis &longs;uper lignum <lb/>magnam imponat &longs;ecurim, de&longs;upérque illi magnum adjiciat pondus, <lb/>ligni quippiam, quod curandum &longs;it, non dividit: Si verò &longs;ecurim ex­<lb/>tollens percutiat, illud &longs;cindit; cum alioquin multo minus habeas <lb/>ponderis id, quod percutit, quàm id quod &longs;uperjacet, & premit?<emph.end type="italics"/> Id <lb/>quod in cæteris quoquè percu&longs;&longs;ionibus, ubi nulla intervenit <lb/>Cunei Ratio, manife&longs;tum e&longs;t; quemadmodum in &longs;implici com­<lb/>pre&longs;&longs;ione, ut cùm lamella aurea in &longs;ubtili&longs;&longs;imam bracteolam di­<lb/>ducitur repetitâ mallei percu&longs;&longs;ione; quod enim, licèt immen­<lb/>&longs;um, pondus vi &longs;uæ gravitatis tantumdem præ&longs;tare po&longs;&longs;et? </s> <s id="s.004951">Per­<lb/>cu&longs;&longs;ionis igitur natura inve&longs;tiganda e&longs;t, ut ejus vires in Cuneo <lb/>innote&longs;cant. </s> </p> <p type="main"> <s id="s.004952">Certum autem e&longs;&longs;e debet, & extra omnem controver&longs;iam <lb/>po&longs;itum nihil e&longs;&longs;e in hac rerum univer&longs;itate, quod vacet cor­<lb/>pore, &longs;ed corporibus omnem ob&longs;ideri locum, nullúmque e&longs;&longs;e <lb/>inane, in quod &longs;e recipere valeant, ac propterea corpora om-<pb pagenum="669" xlink:href="017/01/685.jpg"/>nia ita &longs;ibi vici&longs;&longs;im &longs;uâ mole ob&longs;i&longs;tere, ut nullum moveri va­<lb/>leat, quin alterius in locum &longs;uccedat; quod proinde loco pelli <lb/>nece&longs;&longs;e e&longs;t, quantum &longs;atis fuerit, ut &longs;ubeunti corpori &longs;patium <lb/>concedat; &longs;ive id contingat, quia ob&longs;i&longs;tens corpus inter an­<lb/>gu&longs;tias deprehen&longs;um &longs;e comprimi patiatur, &longs;ivè quia divi&longs;um <lb/>in latera &longs;ecedat, &longs;ive quia circumfu&longs;a corpora circumpellat, <lb/>quæ abeuntis ve&longs;tigia &longs;equantur. </s> <s id="s.004953">Cum itaque nec omnia pla­<lb/>nè corpora perpetuò quie&longs;cant, nec omnia æquali pror&longs;us agi­<lb/>tatione commoveantur, fieri non pote&longs;t, quin aliquibus vis <lb/>aliqua &longs;altem aliquando inferatur, &longs;eu quia non licet diu juxta <lb/>naturæ in&longs;titutum quieta con&longs;i&longs;tere, &longs;eu quia externo pul&longs;u ad <lb/>velociorem motum incitantur. </s> <s id="s.004954">Quare nullius corporis ex loco <lb/>in locum migratio excogitari pote&longs;t, cui nullum aliud corpus <lb/>adver&longs;etur & repugnet, vel ut &longs;uo &longs;e tutetur in loco juxta præ­<lb/>&longs;criptum à natura ordinem, vel ut partium nexum, & natura­<lb/>lem earum po&longs;itionem &longs;ervet citrà divi&longs;ionem, aut compre&longs;&longs;io­<lb/>nem, aut di&longs;tractionem. </s> <s id="s.004955">Ex quo & illud con&longs;equens e&longs;t, quod <lb/>nullum reip&longs;a (quicquid animo finxeris) quie&longs;cit corpus ad <lb/>omnem omninò motum adeò indifferens, ut nihil pror&longs;us re­<lb/>tundat impetûs ab alio corpore commoto &longs;ponte concepti, aut <lb/>extrin&longs;ecùs impre&longs;&longs;i: nullum quippe e&longs;t, quod neque quicquam <lb/>habeat proni, neque &longs;ur&longs;um &longs;ubvolare contendat, &longs;i di&longs;paris &longs;e­<lb/>cundùm &longs;peciem gravitatis corpori permeabili proximum <lb/>con&longs;i&longs;tat, ubi fortè ordinem perturbari contigerit: ac propterea <lb/>ad motum indifferens cen&longs;endum non e&longs;t, ni&longs;i ut exqui&longs;itam <lb/>circuli peripheriam circa centrum gravium percurrat externâ <lb/>vi impellente: id quod animo fingere facile e&longs;t, opere exequi, <lb/>ut miti&longs;&longs;imè loquar, difficillimum; certè &longs;emper incertum. </s> </p> <p type="main"> <s id="s.004956">Hoc verò di&longs;crimen e&longs;t inter corpora (quantum quidem ad <lb/>præ&longs;entem di&longs;putationem attinet) quod aliqua ita liberè fluunt, <lb/>ut nu&longs;quam adhære&longs;cere videantur, quemadmodum aër, & ex­<lb/>tenuatus vapor: Alia liquida & fu&longs;a manant, atque labuntur, <lb/>ut aqua cæteríque humores, per quos tran&longs;ire & permeare li­<lb/>cet, dirempti enim iterùm coëunt. </s> <s id="s.004957">Alia partibus con&longs;tant, quæ <lb/>junctione aliquâ tenentur, & &longs;ub certâ quidem conformatione, <lb/>atque figurâ con&longs;i&longs;tunt, quandiu nullo impellente urgentur; <lb/>quia tamen facilè comprimi queunt, in aliam figuram transfe­<lb/>runtur; cuju&longs;modi &longs;unt lutum, cera, & reliqua mollia ac tene-<pb pagenum="670" xlink:href="017/01/686.jpg"/>ra, quæ aut ita tractabilia &longs;unt, ut quamcumque in formam <lb/>fingantur, aut ita flexibilia, ut &longs;equantur quocumque tor­<lb/>queas: Alia demum &longs;olida & dura &longs;unt, quæ figuræ terminos, <lb/>quibus circum&longs;cribuntur, non facilè mutant, & &longs;i fortè &longs;e ali­<lb/>quatenus comprimi patiantur, pri&longs;tinam formam &longs;ibi reparant: <lb/>Ex hi&longs;ce quatuor corporum generibus priora rationem medij <lb/>&longs;ubire po&longs;&longs;unt, in quo reliquorum corporum motus exercean­<lb/>tur, ut ex alio in alium locum commigrent; po&longs;teriora, &longs;i unum <lb/>in aliud incurrat, aut &longs;i &longs;ibi invicem occurrant, ea &longs;unt, per <lb/>quæ tran&longs;itus non pateat, &longs;ed aliorum corporum motui tanti&longs;­<lb/>per mole &longs;uâ unumquodque obluctatur, dum pul&longs;u externo re­<lb/>moveatur. </s> </p> <p type="main"> <s id="s.004958">Porrò Impul&longs;ionem à Percu&longs;&longs;ione di&longs;tinguere opus e&longs;t, ni&longs;i <lb/>vocabulis abuti velimus; quamvis enim utraque objecti corpo­<lb/>ris re&longs;i&longs;tentiam inveniat, nemo tamen dixerit idem e&longs;&longs;e, ap­<lb/>prehen&longs;um manu Vectem impellendo, atque illum percutien­<lb/>do deprimere, innatans aquæ lignum conto propellere, atque <lb/>inflicto ictu illud à ripâ longiùs ab&longs;trahere, etiam&longs;i æquè & <lb/>Vectis deprimatur, & lignum promoveatur. </s> <s id="s.004959">Simplex nimi­<lb/>rum Impul&longs;io nullum per &longs;e antecedentem corporis impellentis <lb/>motum exigit: at Percu&longs;&longs;io ob idip&longs;um, quia Percu&longs;&longs;io e&longs;t, cor­<lb/>poris percutientis motum requirit, qui ip&longs;orum corporum col­<lb/>li&longs;ionem præcedat. </s> <s id="s.004960">Quare in Percu&longs;&longs;ione intervenit in&longs;tituti <lb/>jam & inchoati motûs interruptio ex novâ objecti corporis re­<lb/>&longs;i&longs;tentiâ. </s> <s id="s.004961">Hinc Ari&longs;toteles lib. 4. Meteor. <!-- REMOVE S-->&longs;umma 3. cap. 2. ait, <lb/><emph type="italics"/>E&longs;t autem Pul&longs;io, motus à movente, qui fit à tactu; Percu&longs;&longs;io autem; <lb/>cum à latione.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.004962">Hæc omnia conjunctim Percu&longs;&longs;io po&longs;tulat: Primò inchoa­<lb/>tum e&longs;&longs;e jam & in&longs;titutum motum oportet: quamvis etenim <lb/>impul&longs;io omnis vincat corporis urgendi aut &longs;cindendi re&longs;i&longs;ten­<lb/>tiam etiam primo motûs momento; quia tamen præcedens cor­<lb/>poris impellentis motus, quo antè acce&longs;&longs;it ad corporis impul&longs;i <lb/>contactum, quàm illud urgere incipiat, omnino præter Impul­<lb/>&longs;ionis naturam accidit (hæc &longs;i quidem eadem &longs;equetur, etiam <lb/>&longs;i priùs in mutuo contactu diuti&longs;&longs;imè quie&longs;cant) propterea non <lb/>&longs;atis e&longs;t re&longs;i&longs;tentiam invenire, &longs;ed hanc in&longs;tituto jam motui in­<lb/>tervenire nece&longs;&longs;e e&longs;t, ut &longs;it Percu&longs;&longs;io. </s> <s id="s.004963">Deinde, licèt præce&longs;&longs;e­<lb/>rit motus, atque adhuc continuatus novam inveniat re&longs;i&longs;ten-<pb pagenum="671" xlink:href="017/01/687.jpg"/>tiam, quia alias medij eju&longs;dem &longs;cindendi partes offendit; &longs;i ta­<lb/>men æquabilis per&longs;everet pri&longs;tinam velocitatem aut tarditatem <lb/>nulla ex parte imminutam continenter &longs;ervans, non e&longs;t cen&longs;en­<lb/>da nova re&longs;i&longs;tentia; &longs;ed quemadmodum continuus e&longs;t idem <lb/>motus aliis atque aliis partibus &longs;ibi &longs;uccedentibus, ita conti­<lb/>nuatur eadem re&longs;i&longs;tentia, nec po&longs;teriores medij partes percuti <lb/>dicuntur, &longs;ed, ut priùs, præcisè impelli, aut &longs;cindi: quia vide­<lb/>licet præcedens motus nihil confert ad novam hanc impul&longs;io­<lb/>nem prioribus omnino &longs;imilem. </s> <s id="s.004964">Quod &longs;i corpus in motu ob id­<lb/>ip&longs;um quia movetur, majorem atque majorem adhiberet cele­<lb/>ritatem, adeò ut in medio, hoc e&longs;t aëre ip&longs;o, re&longs;i&longs;tentiæ mo­<lb/>dus augeretur, facilè acquie&longs;cam contendenti aërem verberari <lb/>& percuti: &longs;ed quia nimis facilem &longs;e præbet aër ad hoc, ut <lb/>&longs;cindatur, non de huju&longs;modi percu&longs;&longs;ione medij, in quo fit mo­<lb/>tus, mihi hìc e&longs;t &longs;ermo, &longs;ed potiùs de percu&longs;&longs;ione corporis, ad <lb/>quod per medium accedit corpus percutiens. </s> <s id="s.004965">Hinc aquam per­<lb/>cuti non negaverim, quando en&longs;is bonitatem examinaturi, <lb/>utrùm &longs;cilicet ritè & æquabiliter in chalybem temperatum &longs;it <lb/>ferrum, horizontalem aquæ &longs;tagnantis &longs;uperficiem plano gla­<lb/>dio vehementer percutimus; per aërem &longs;cilicet, tanquam per <lb/>medium antecedentis motûs, ad aquam devenit gladius; quic­<lb/>quid &longs;it, quod & ip&longs;a aqua ad ulteriorem motum, quo en&longs;is <lb/>profundiùs immergatur, medij rationem habere po&longs;&longs;it. </s> <s id="s.004966">Simi­<lb/>liter aërem ip&longs;um percuti à corpore, quod ex aquâ emergit, <lb/>haud ægrè conce&longs;&longs;erim, &longs;i id quidem ex vi præcedentis motûs <lb/>contingat: e&longs;to, minùs ob&longs;i&longs;tat aër, quàm aqua, ob&longs;i&longs;tit tamen, <lb/>&longs;i à quiete dimoveatur, aut velociùs moveri cogatur, quàm mo­<lb/>veretur, &longs;i huju&longs;modi nova impul&longs;io vi præcedentis motûs non <lb/>accideret: Neque aër, cum primùm emergens corpus in eum <lb/>incurrit, habet rationem medij, &longs;ed perinde &longs;e habet primo il­<lb/>lo momento, atque &longs;i tabella &longs;u&longs;pen&longs;a horizonti parallela fa­<lb/>ciem aquæ proximè contingeret, & in eam ex aqua emergens <lb/>corpus incurreret; quanquam ab hac majorem, quàm ab aere, <lb/>re&longs;i&longs;tentiam &longs;ubiret, atque adeo validiorem huic ictum infli­<lb/>geret. </s> </p> <p type="main"> <s id="s.004967">Sed quid vocabula in quæ&longs;tionem fru&longs;tra vocamus? </s> <s id="s.004968">De his <lb/>loquere, ut libet: per me &longs;anè licebit, quando corpus ab uno <lb/>fluido, per quod inchoatus e&longs;t motus, ad aliud fluidum tran&longs;it <pb pagenum="672" xlink:href="017/01/688.jpg"/>(&longs;ive hoc magis, &longs;ive minùs cra&longs;&longs;um atque concretum fuerit) <lb/>hujus po&longs;terioris fluidi primum contactum cum impul&longs;ione <lb/>Percu&longs;&longs;ionem æquè appellare, atque &longs;i non fluidum e&longs;&longs;et, &longs;ed <lb/>durum; validiùs &longs;cilicet impetitur vi antecedentis motûs, quàm <lb/>&longs;i corpus incurrens tunc primùm à quiete recederet. </s> <s id="s.004969">Hìc &longs;oli­<lb/>dorum atque con&longs;i&longs;tentium corporum percu&longs;&longs;iones per&longs;equi­<lb/>mur, quarum vim inquirimus, & modum recipiunt à re&longs;i&longs;ten­<lb/>tiâ corporis percu&longs;&longs;i, quæ quò major e&longs;t, validior quoquè cæte­<lb/>ris paribus efficitur percu&longs;&longs;io. </s> <s id="s.004970">Sic &longs;i quis velit alteri alapam in­<lb/>fligere, nullus erit ictus, &longs;i æquè velociter ad ea&longs;dem partes <lb/>moveantur tum percutientis manus, tum is, cui de&longs;tinata e&longs;t <lb/>alapa; quia nulla e&longs;t re&longs;i&longs;tentia motum manûs impediens, aut <lb/>retardans: erit verò ictus genere ip&longs;o validi&longs;&longs;imus, &longs;i &longs;ibi oc­<lb/>currant, & quò majore impetu atque velocitate occurrent, eò <lb/>validior; quia nullum e&longs;t majus re&longs;i&longs;tentiæ genus, quàm &longs;i duo <lb/>oppo&longs;iti motus &longs;e invicem retundant. </s> <s id="s.004971">Quod &longs;i demum percu­<lb/>tientis manus moveatur velociùs, quàm is, qui percutitur, <lb/>quamvis ad ea&longs;dem partes moveantur, ictus infligetur validus <lb/>pro Ratione exce&longs;sûs velocitatis, cui motus tardior re&longs;i&longs;tit, qua­<lb/>tenus corpus tardum tandem à velociore deprehenditur, atque <lb/>urgetur: antè ictum verò &longs;i corpus percu&longs;&longs;um quie&longs;cat, quo ve­<lb/>locior erit percutientis motus, validior quoquè erit ictus; ad <lb/>eandem enim re&longs;i&longs;tentiam major motus habet majorem Ratio­<lb/>nem, quàm minor. </s> </p> <p type="main"> <s id="s.004972">Vim igitur percu&longs;&longs;ionis ex antecedenti motu originem duce­<lb/>re manife&longs;tum videtur; non quidem quâ motus e&longs;t ex loco in <lb/>locum tran&longs;itus, hic enim ante corporum contactum ictum <lb/>nullum infligere pote&longs;t, in ictu autem ip&longs;o motus omnis præce­<lb/>dens evanuit, nec jam extinctus quicquam efficere pote&longs;t, <lb/>etiam&longs;i motui præ&longs;enti vis aliqua efficiendi tribueretur. </s> <s id="s.004973">Sed <lb/>quia cum motu illo antecedente acqui&longs;itus e&longs;t impetus, qui <lb/>adhuc durans ip&longs;o percu&longs;&longs;ionis momento longè plus habet vi­<lb/>rium, quàm &longs;i tunc omnino inciperet motus cum impul&longs;ione; <lb/>augetur &longs;iquidem in motu impetus ab eâdem cau&longs;a movente <lb/>productus &longs;ingulis momentis, &longs;emper enim ad agendum cau&longs;a <lb/>nece&longs;&longs;aria applicata e&longs;t, atque, &longs;i maneat, utilitate non caret <lb/>impetus, quem &longs;ub&longs;equi pote&longs;t motus. </s> <s id="s.004974">Quid nimirum cau&longs;æ <lb/>e&longs;t, quare ligneus globus leniter aquæ impo&longs;itus innataret, &longs;i <pb pagenum="673" xlink:href="017/01/689.jpg"/>verò ex editâ turri in &longs;ubjectam fo&longs;&longs;am dimittatur, aquam al­<lb/>tiùs penetrat? </s> <s id="s.004975">ni&longs;i quia impetum in motu globus acqui&longs;ivit, <lb/>quo per&longs;everante terminos &longs;uæ gravitati à Naturâ præ&longs;criptos <lb/>tran&longs;ilit, eóque demum langue&longs;cente, aut illum aqua &longs;ur&longs;um <lb/>extrudet, aut vi &longs;uæ levitatis &longs;ponte a&longs;cendet. </s> <s id="s.004976">Sic citrà nota­<lb/>bilem doloris &longs;en&longs;um &longs;u&longs;tinemus capiti impo&longs;itum lapidem for­<lb/>tè bipedalem, at non item &longs;crupuli duorum digitorum ex alti­<lb/>tudine centum cubitorum decidentis ictum ferre po&longs;&longs;umus ci­<lb/>trà incommodum non &longs;anè leve: id quod ex acqui&longs;ito impetu <lb/>contingere palàm e&longs;t, nulla quippe alia præter impetum in <lb/>promptu e&longs;t cau&longs;a, cui vis hæc efficiendi commodè, atque pro­<lb/>babili conjecturâ, tribuenda &longs;it. </s> </p> <p type="main"> <s id="s.004977">Hunc impetum in motu acqui&longs;itum <emph type="italics"/>Gravitatis<emph.end type="italics"/> nomine indi­<lb/>gitare placuit Ari&longs;toteli, cùm propo&longs;itæ quæ&longs;tioni 19. &longs;atisfa­<lb/>cere contendens ait, <emph type="italics"/>An quia omnia cum motu fiunt, & grave <lb/>ip&longs;um gravitatis magis a&longs;&longs;umit motum, dum movetur, quàm dum <lb/>quie&longs;cit? </s> <s id="s.004978">Incumbens igitur connatam gravi motionem non movetur; <lb/>motum verò & &longs;ecundùm hanc movetur, & &longs;ecundùm eam, quæ e&longs;t <lb/>percutientis.<emph.end type="italics"/></s> <s id="s.004979"> Neque enim adeò in rebus Phy&longs;icis Ari&longs;totelem, <lb/>ejú&longs;que peritiores a&longs;&longs;eclas cæcutii&longs;&longs;e dixerim, ut gravitatem <lb/>corpori in&longs;itam, quæ prima radix atque origo e&longs;t, cui motus <lb/>debeatur, in ip&longs;o motu revera augeri exi&longs;timaverint (quamvis <lb/>nullâ factâ naturæ, &longs;altem con&longs;tipatis partibus, mutatione) <lb/>haud &longs;ecus, ac calori calor addatur. </s> <s id="s.004980">Sed idcirco plus gravitatis <lb/>a&longs;&longs;umi dicitur à corpore gravi dum movetur, quàm dum quie&longs;­<lb/>cit, quia in motu vi ac pote&longs;tate &longs;e movendi æquiparat corpora <lb/>graviora, atque adeò plus habet gravitatis non Formaliter, &longs;ed <lb/>Virtualiter & Æquivalenter, ut ip&longs;orum Peripateticorum vo­<lb/>cabulis utar. </s> <s id="s.004981">Cæterùm <emph type="italics"/>gravitatis<emph.end type="italics"/> nomine non ip&longs;um pondus <lb/>intelligi ab Ari&longs;totele &longs;uadet ip&longs;a loquendi formula, qua gravi­<lb/>tatem a&longs;&longs;umptam dum movetur, confert cum gravitate a&longs;&longs;ump­<lb/>tâ dum quie&longs;cit, ut hæc illâ minor cen&longs;eatur: videtur enim <lb/>Ari&longs;toteles in corpore gravi ad motum prono agno&longs;cere a&longs;­<lb/>&longs;umptum impetum, quo fieret connata ip&longs;i corpori gravi mo­<lb/>tio, ni&longs;i impediretur, dum quie&longs;cit, & præter hunc impetum, <lb/>alium in motu acqui&longs;itum, adeò ut demum utroque impetu <lb/>moveatur, ac proinde dicatur in motu plus a&longs;&longs;umere gravitatis. </s> </p> <p type="main"> <s id="s.004982">Quòd &longs;i hæc philo&longs;ophandi ratio placeat, qua corpori gravi <pb pagenum="674" xlink:href="017/01/690.jpg"/>idcircò &longs;altem ad &longs;peciem quie&longs;centi, quòd removere non va­<lb/>leat ea, quæ ob&longs;tant, & motum, qui &longs;ub &longs;en&longs;um cadat, impe­<lb/>diunt, conceditur impetus Innatus, qui &longs;it ip&longs;a actualis gravi­<lb/>tatio in&longs;itæ gravitati addita, reip&longs;a connitens aut adversùs &longs;ub­<lb/>jectum corpus, aut contra vim &longs;u&longs;pendentem: Cùm gravitas <lb/>in motu alium atque alium adhibeat novum conatum ad <lb/>de&longs;cendendum, perinde videtur contingere, ac &longs;i toties mul­<lb/>tiplicata fui&longs;&longs;et eadem gravitas, quoties multiplicatus fuit co­<lb/>natus priori illi æqualis. </s> <s id="s.004983">Hac autem ratione non ineptè dixit <lb/>Ari&longs;toteles grave ip&longs;um a&longs;&longs;umere plus gravitatis in motu, quia <lb/>&longs;ublato motûs impedimento & impetus Innatus &longs;uas omnes vi­<lb/>res exerit, & augetur Acqui&longs;ito: ac propterea minor gravitas <lb/>&longs;ic æquivalenter multiplicata longè plus efficit, quàm &longs;i major <lb/>gravitas incumberet, cujus impetus Innatus impediretur, ne <lb/>motum efficeret ullum neque compre&longs;&longs;ionis corporis &longs;ubjecti, <lb/>neque di&longs;tentionis corporis &longs;u&longs;pendentis. </s> <s id="s.004984">Quando autem nul­<lb/>lus omnino motus, &longs;ivè qui &longs;ub &longs;en&longs;um cadat, &longs;ive qui aciem <lb/>omnem fugiat, tribuitur corpori gravi, nullus quoquè &longs;uperad­<lb/>ditus Impetus ip&longs;i connatæ gravitati re&longs;pondens concedendus <lb/>e&longs;t; neque enim deor&longs;um reip&longs;a connititur; quamvis in &longs;e ha­<lb/>beat principium & originem gravitandi, &longs;i impedimentum &longs;al­<lb/>tem ex parte removeatur. </s> </p> <p type="main"> <s id="s.004985">Cùm itaque omne id, quod percu&longs;&longs;ionis ictum con&longs;equitur, <lb/>ab impetu oriatur, neque impetum gravitas, aut ulla moven­<lb/>di facultas, concipere valeat, quin aliquo &longs;altem motu ip&longs;a mo­<lb/>veatur; nil mirum, &longs;i ingens moles prohibita, ne prorsùs mo­<lb/>veatur, nullam labem inferat &longs;ubjecto lapidi, quem minor gra­<lb/>vitas cadens, atque percutiens in fru&longs;ta comminuit: minor &longs;ci­<lb/>licet gravitas liberè de&longs;cendens multum concipit impetum, <lb/>quem lapidi percu&longs;&longs;o communicans cogit eum in fru&longs;ta di&longs;&longs;ili­<lb/>re, &longs;i vis impetûs &longs;uperet partium nexum, aut &longs;altem cum con­<lb/>cutit. </s> <s id="s.004986">Nullum autem effectum impetûs ab ingenti mole prorsùs <lb/>quie&longs;cente expectare po&longs;&longs;umus, quippe quæ nullum imprime­<lb/>re pote&longs;t impetum &longs;ubjecto corpori. </s> </p> <p type="main"> <s id="s.004987">Hinc mirari ce&longs;&longs;ent, qui plumbeum globulum primo mallei <lb/>ictu certam compre&longs;&longs;ionem pati ob&longs;ervant, &longs;ecundo verò ictu <lb/>priori omninò æquali adhuc magis comprimi, quamvis minore <lb/>compre&longs;&longs;ione, quia particulæ jam per vim con&longs;tipatæ validiùs <pb pagenum="675" xlink:href="017/01/691.jpg"/>rejiciunt majorem violentiam. </s> <s id="s.004988">At &longs;i globulum &longs;imilem &longs;ubji­<lb/>ciant ponderi, quod illum æquè comprimat, ac prior ictus mal­<lb/>lei, addito adhuc æquali pondere non &longs;equitur compre&longs;&longs;io glo­<lb/>buli tanta, quæ re&longs;pondeat &longs;ecundo ictui mallei: ex quo &longs;atis <lb/>con&longs;tat duplicis percu&longs;&longs;ionis vires non æquari à duplici gravita­<lb/>te. </s> <s id="s.004989">Si enim animum attentè advertant, videbunt mallei mo­<lb/>tus tam in primâ quàm in &longs;ecunda percu&longs;&longs;ione planè æquales <lb/>e&longs;&longs;e, tùm ratione velocitatis, tùm ratione &longs;patij, ac proinde <lb/>æquali impetu malleum percutere: At factâ jam primâ &longs;ubjecti <lb/>globuli compre&longs;&longs;ione, in qua gravitas incumbens motum ha­<lb/>buit illi compre&longs;&longs;ioni re&longs;pondentem, manife&longs;tum e&longs;t, propter <lb/>majorem globuli jam compre&longs;&longs;i re&longs;i&longs;tentiam, non po&longs;&longs;e &longs;ecun­<lb/>dam gravitatem priori æqualem additam æquali motu, nec <lb/>æquali velocitate moveri, atque propterea neque po&longs;&longs;e æqua­<lb/>lem impetum concipere, quo po&longs;&longs;it effectum &longs;ecundo mallei <lb/>ictui &longs;imilem producere. </s> <s id="s.004990">Adde quod &longs;ecundum pondus addi­<lb/>tum priori, atque illi impo&longs;itum, &longs;uum habet gravitatis cen­<lb/>trum, & commune totius ponderis globulo incumbentis cen­<lb/>trum gravitatis transfertur in aliud molis compo&longs;itæ punctum; <lb/>ideóque linea directionis non &longs;imiliter incurrit in &longs;ubjectum <lb/>globulum, adversùm quem &longs;imiles exhibeat vires. </s> <s id="s.004991">Neque mihi <lb/>facilè per&longs;uadebis tam accuratè &longs;ecundum pondus adjectum <lb/>priori, ut po&longs;terius gravitatis centrum in eadem &longs;it lineâ di­<lb/>rectionis, nec ab illâ quicquam deflectat. </s> <s id="s.004992">Quare vel po&longs;terior <lb/>hæc gravitas addita priori, jam quie&longs;centi, ubi facta e&longs;t vis <lb/>comprimendi par virtuti re&longs;i&longs;tendi, omni pror&longs;us motu caret, & <lb/>nihil impetûs pote&longs;t concipere, aut imprimere; vel &longs;olùm te­<lb/>nui&longs;&longs;imum, & qui vix po&longs;t multum tempus con&longs;picuus fiat, mo­<lb/>tum habet, & non ni&longs;i levem impetum imprimit, quo &longs;ubjectus <lb/>globulus demum aliquantulo compre&longs;&longs;ior appareat: ideò, ut <lb/>compre&longs;&longs;io &longs;imilis illi, quæ fit à &longs;ecundo mallei ictu, habeatur, <lb/>nece&longs;&longs;e e&longs;t gravitatem additam e&longs;&longs;e adhuc majorem, ut gravi­<lb/>tas tota compo&longs;ita impetum efficere valeat, quem con&longs;equa­<lb/>tur motus æqualis &longs;ecundæ illi compre&longs;&longs;ioni à malleo factæ. </s> </p> <p type="main"> <s id="s.004993">Cur itaque &longs;ecuris ligno incumbens, quamvis ingenti præ­<lb/>gravata pondere, vix levem fi&longs;&longs;ionem inferat &longs;ubjecto ligno, <lb/>quod tamen altiùs penetratur ab eâdem &longs;ecuri cadente & per­<lb/>cutiente, in promptu cau&longs;a e&longs;t: quia videlicet compre&longs;&longs;io, quæ <pb pagenum="676" xlink:href="017/01/692.jpg"/>& impul&longs;io e&longs;t, cum motu quidem fit, &longs;ed ip&longs;o &longs;tatim initio & <lb/>in progre&longs;&longs;u ade&longs;t re&longs;i&longs;tentia, ne producatur totus impetus, <lb/>quem vis motiva po&longs;&longs;et efficere, & motus non e&longs;t, ni&longs;i quan­<lb/>tum impellitur objectum corpus; ideóque &longs;ecuris vi ponderis <lb/>incumbentis non valet in huju&longs;modi motu alium impetum con­<lb/>cipere præter illum, quem fert præ&longs;ens motus, qui valde exi­<lb/>guus e&longs;t: At percu&longs;&longs;io ea e&longs;t, ut cùm primùm &longs;ecuris cadens <lb/>applicatur ligno, jam multum habeat concepti impetus in aëre <lb/>libero, & nihil adhuc re&longs;i&longs;tente ligno, ac propterea po&longs;&longs;it ve­<lb/>lociùs moveri comprimendo & dividendo &longs;ubjectum lignum. </s> <lb/> <s id="s.004994">Ex quo fit onus &longs;ecuri impo&longs;itum tantæ gravitatis e&longs;&longs;e oportere, <lb/>ut quæ Ratio e&longs;t &longs;patij à &longs;ecuri cadente decur&longs;i ad &longs;patium, quo <lb/>illa penetrat lignum, ea &longs;altem &longs;it Ratio gravitatis conflatæ ex <lb/>&longs;ecuri & addito pondere ad <expan abbr="gravitat&etilde;">gravitatem</expan> &longs;implicis &longs;ecuris, ut fieret <lb/>æqualis &longs;ci&longs;&longs;io ab eâdem &longs;ecuri: ut videlicet tantumdem im­<lb/>petûs concipiatur à magnâ gravitate in exiguo motu præ&longs;ente <lb/>re&longs;i&longs;tentiâ, quantum impetûs concipitur à &longs;ecuri in anteceden­<lb/>te motu longiore ab&longs;que re&longs;i&longs;tentiâ ullâ, præterquam medij. </s> </p> <p type="main"> <s id="s.004995">Similiter nullum adhiberi po&longs;&longs;e pondus, quo aureæ lamellæ <lb/>impo&longs;ito hæc diduci po&longs;&longs;it in &longs;ubtili&longs;&longs;imam bracteolam, quem­<lb/>admodum vi mallei percutientis, ex ii&longs;dem principiis con&longs;tat. </s> <lb/> <s id="s.004996">Attende enim, quanto motu moveri po&longs;&longs;it illud pondus com­<lb/>primens; utique non ni&longs;i quantum e&longs;t altitudinis di&longs;crimen in­<lb/>ter lamellam & bracteolam: at tantillum &longs;patium, in quo exer­<lb/>cendus e&longs;&longs;et motus, quam Rationem habet ad toties multipli­<lb/>catum &longs;patium, in quo iteratis &longs;æpiùs ictibus liberè movetur <lb/>malleus? </s> <s id="s.004997">Cùm itaque minimus motus, aut etiam forta&longs;sè nul­<lb/>lus, po&longs;t tenui&longs;&longs;imam auri compre&longs;&longs;ionem ingenti illi oneri con­<lb/>veniat, nil mirum &longs;i exiguo impetu ferè nihil efficiat, cùm ta­<lb/>men malleus novo &longs;emper impetu &longs;ingulis ictibus concepto ali­<lb/>quam, licèt &longs;emper minorem atque minorem, compre&longs;&longs;ionem <lb/>efficiat. </s> </p> <p type="main"> <s id="s.004998">Ut autem res hæc pleniùs innote&longs;cat, ob&longs;erva impul&longs;ionem, <lb/>qua corpus urgetur, opponi tractioni, & compre&longs;&longs;ionem par­<lb/>tium di&longs;tractioni, atque &longs;icut corporis, quod urgetur, particu­<lb/>læ aliquando comprimuntur, ita corporis, quod trahitur, par­<lb/>ticulas aliquando di&longs;trahi, aut divelli, neque di&longs;&longs;imilem e&longs;&longs;e <lb/>re&longs;i&longs;tentiam corporum vi &longs;uæ gravitatis, ne impellantur, aut <pb pagenum="677" xlink:href="017/01/693.jpg"/>ratione po&longs;itionis partium, ne comprimantur, ac ne trahantur, <lb/>aut particularum nexus di&longs;&longs;olvatur. </s> <s id="s.004999">Quapropter ubi primùm <lb/>incipit impul&longs;io aut tractio, &longs;ive compre&longs;&longs;io aut di&longs;tractio, in­<lb/>cipit etiam re&longs;i&longs;tentia, quæ eò major evadit, quò majorem vio­<lb/>lentiam &longs;ubit corpus. </s> <s id="s.005000">Hinc e&longs;t potentiam impellentem aut tra­<lb/>hentem &longs;emper minore impetu ferri, quàm &longs;i liberè moveretur, <lb/>dum nulla ade&longs;&longs;et re&longs;i&longs;tentia. </s> <s id="s.005001">Sic &longs;i quis funiculum, quem re­<lb/>tinet clavus parieti infixus, arripiat, atque jam extentum <lb/>trahat, illum quidem multo ni&longs;u intendit, &longs;ed nec illum di&longs;­<lb/>rumpere valet, nec clavum revellere: &longs;ed &longs;i eodem conatu fu­<lb/>niculum languidum nec dum extentum trahat, celeriter mo­<lb/>vetur manus, antequam funiculus extendatur, & facilè aut hic <lb/>abrumpitur, aut ille revellitur. </s> <s id="s.005002">Quia nimirum extenti jam fu­<lb/>niculi re&longs;i&longs;tentia, ne intendatur, impedit, ne potentia pro Ra­<lb/>tione &longs;ui conatûs moveatur, multo impetu ab&longs;umpto in vincen­<lb/>dâ illâ re&longs;i&longs;tentia; neque movetur potentia ni&longs;i cunctabunda, <lb/>& per brevi&longs;&longs;imum &longs;patium, quantum vi inten&longs;ionis funiculus <lb/>magis extenditur: At ubi languidus e&longs;t funiculus, potentia <lb/>ab&longs;que ullo retinente per aliquantum &longs;patij liberè movetur, & <lb/>totum impetum &longs;uo conatui re&longs;pondentem in efficiendo celeri <lb/>motu impendit, quem jam notabiliter auctum invenit funicu­<lb/>lus, cum primùm e&longs;t extentus, & adhuc magis augetur per&longs;e­<lb/>verante eodem conatu. </s> <s id="s.005003">Quare cùm multò major &longs;it impetus, <lb/>&longs;atis e&longs;&longs;e pote&longs;t non &longs;olùm ad intendendum funiculum, verùm <lb/>etiam ad illum di&longs;rumpendum, aut, &longs;i, hujus particulæ validio­<lb/>re nexu jungantur, ad revellendum clavum. </s> </p> <p type="main"> <s id="s.005004">Ex his habes, quid re&longs;pondeat docti&longs;&longs;imis viris vim percu&longs;­<lb/>&longs;ionis inve&longs;tigantibus. </s> <s id="s.005005">Ut apparet, quantâ vi plumbeus globu­<lb/>lus unciarum duarum ex cubitali altitudine cadens percuteret <lb/>&longs;ubjectum corpus, exi&longs;timârunt &longs;atis innote&longs;cere, &longs;i globulus <lb/>ille funiculo cubitali adnecteretur chordæ arcûs medio loco in­<lb/>ter extremitates. </s> <s id="s.005006">Tum &longs;ublatus globulus u&longs;que ad chordam <lb/>ip&longs;am, dimi&longs;&longs;us e&longs;t, atque ob&longs;ervatum e&longs;t punctum, ad quod <lb/>adducta e&longs;t chorda: proclive enim erat arguere, globulum <lb/>tantâ vi percu&longs;&longs;urum &longs;ubjectum corpus, quantâ vi inflectebat <lb/>bali&longs;tæ arcum. </s> <s id="s.005007">Quare tentando varia pondera addiderunt <lb/>chordæ arcûs, donec demum pondus decem librarum chordam <lb/>ad idem punctum adduxit, ad quod adducta fuerat à globo ca-<pb pagenum="678" xlink:href="017/01/694.jpg"/>dente, atque in eodem flexionis &longs;tatu chordam & arcum deti­<lb/>nuit. </s> <s id="s.005008">Arguebant igitur percu&longs;&longs;ionem globi plumbei duarum <lb/>unciarum ex cubitali altitudine cadentis æquiparari pre&longs;&longs;ioni <lb/>decem librarum. </s> <s id="s.005009">Ulteriùs autem progrediendo, adhibita e&longs;t ba­<lb/>li&longs;ta alia validior, cujus arcus ob duriorem ferri temperationem <lb/>minùs erat flexibilis: quapropter cum eju&longs;dem potentiæ eadem <lb/>&longs;it vis, eju&longs;dem globuli ex eâdem altitudine &longs;imiliter cadentis <lb/>non ni&longs;i eædem e&longs;&longs;e poterant vires ad vincendam æqualem re­<lb/>&longs;i&longs;tentiam: atque adeò durioris arcûs minor flexio æquè re­<lb/>&longs;i&longs;tens, ac major flexio arcûs mollioris, breviore termino de­<lb/>finivit de&longs;cen&longs;um globuli plumbei, & ad propius punctum ad­<lb/>ducta e&longs;t chorda. </s> <s id="s.005010">Verùm, ut in eodem flexionis &longs;tatu fortior <lb/>hic arcus retineretur, non &longs;atis fuit decem libras appendere, <lb/>&longs;ed viginti librarum pondere opus fuit. </s> <s id="s.005011">Hinc inferebant ean­<lb/>dem eju&longs;dem globuli duarum unciarum percu&longs;&longs;ionem æquare <lb/>non &longs;olùm vires librarum decem, &longs;ed & viginti: atque u&longs;que <lb/>eò argumentationem deducebant, ut a&longs;&longs;umpto robu&longs;tiore ali­<lb/>quo arcu concluderent, ne pondus quidem librarum mille &longs;atis <lb/>e&longs;&longs;e ad arcum illum in eâ po&longs;itione retinendum, ad quam fui&longs;­<lb/>&longs;et adductus à globo duarum unciarum cadente: id quod vim <lb/>quandam percu&longs;&longs;ionis infinitam indicare videbatur. </s> </p> <p type="main"> <s id="s.005012">Verùm quamvis hos ingenio&longs;orum hominum conatus non <lb/>modò non improbem, &longs;ed multâ commendatione dignos exi&longs;ti­<lb/>mem, liceat tamen mihi argumentationis infirmitatem expo­<lb/>nere; tam enim non e&longs;t vis percu&longs;&longs;ionis duarum unciarum in­<lb/>finita, quàm infinita non e&longs;t vis pre&longs;&longs;ionis decem librarum. </s> <lb/> <s id="s.005013">Quando enim vi ponderis adnexi flectitur arcus, utique pon­<lb/>dus de&longs;cendit, & &longs;uâ gravitate &longs;uperat rigidi chalybis vires, <lb/>donec demum æqualitas quædam intercedat inter vim arcûs <lb/>ela&longs;ticam, & gravitatis conatum ad de&longs;cendendum; tunc &longs;cili­<lb/>cet fit con&longs;i&longs;tentia. </s> <s id="s.005014">Prout igitur robu&longs;tiores &longs;unt arcus, minùs <lb/>permittunt de&longs;cendere pondus chordæ appen&longs;um, &longs;i omnia &longs;int <lb/>paria: Nam &longs;i brevior &longs;it arcus mollis & languidus, longior ve­<lb/>rò arcus durioris temperationis, fieri pote&longs;t, ut idem pondus <lb/>æqualiter adducat longiorem chordam atque breviorem, &longs;imi­<lb/>li planè ratione ac de ponderibus fune &longs;u&longs;pen&longs;is præponderan­<lb/>tibus atque æquilibribus dictum e&longs;t lib.3. cap.12: ideò ponendi <lb/>&longs;unt arcus ita &longs;imiles & æquales, ut &longs;olâ ferri temperatione di&longs;-<pb pagenum="679" xlink:href="017/01/695.jpg"/>crepent. </s> <s id="s.005015">Si igitur validioris arcus repugnantia, ut flectatur ad <lb/>duos digitos, tanta e&longs;t, quanta repugnantia mollioris arcûs, ut <lb/>flectatur ad &longs;ex digitos, patet non e&longs;&longs;e eumdem impetum de­<lb/>cem librarum de&longs;cendentium &longs;olùm per duos priores digitos, <lb/>atque per &longs;ex: ac proinde cùm decem libræ applicatæ arcui va­<lb/>lidiori &longs;olùm po&longs;&longs;unt per duos digitos (& quidem lentiùs <lb/>propter majorem re&longs;i&longs;tentiam) moveri, minus po&longs;&longs;ent, quàm <lb/>per impetum conceptum in motu &longs;ex digitorum; & propterea <lb/>neque po&longs;&longs;ent illius robu&longs;tioris arcûs chordam adducere ad <lb/>duos digitos; &longs;ed neque adductam ab aliâ potentiâ po&longs;&longs;ent reti­<lb/>nere in eo &longs;tatu ac po&longs;itione: quia etiam &longs;i vis ela&longs;tica arcûs ro­<lb/>bu&longs;tioris inflexi ad duos digitos par e&longs;&longs;et virtuti ela&longs;ticæ arcûs <lb/>imbecillioris inflexi ad &longs;ex digitos, cui reluctantur decem li­<lb/>bræ; hæ minùs repugnant, ne ad duos digitos, quàm ne ad &longs;ex <lb/>attollantur; igitur decem libræ minùs re&longs;i&longs;tunt virtuti ela&longs;ticæ <lb/>arcûs fortioris, adeóque nec po&longs;&longs;unt in eo flexionis &longs;tatu reti­<lb/>nere arcum fortiorem & chordam: &longs;i enim pares &longs;unt vires <lb/>ela&longs;ticæ arcûs inflexi ad duos digitos, & arcûs inflexi ad &longs;ex di­<lb/>gitos, pari impetu &longs;e re&longs;tituunt, ut parem violentiam excu­<lb/>tiant; at pondus par utrique chordæ adnexum non pari veloci­<lb/>tate movetur, &longs;i ad duos ac &longs;i ad &longs;ex digitos attollatur; igitur <lb/>minùs re&longs;i&longs;tunt decem libræ motui duorum, quàm motui &longs;ex <lb/>digitorum. </s> </p> <p type="main"> <s id="s.005016">Porrò vis globi cadentis non e&longs;t comparanda cum pondere <lb/>quatenus retinente chordam in eadem flexione, &longs;ed quatenus <lb/>illam adducente & flectente, ut motus cum motu, non verò <lb/>motus cum quiete comparetur. </s> <s id="s.005017">In eo autem motu ponderis ad­<lb/>ducentis chordam, & arcum inflectentis, quò major e&longs;t re­<lb/>&longs;i&longs;tentia, eò minor e&longs;t impetus & velocitas, qua pondus illud <lb/>movetur: igitur idem pondus non parem vim habere pote&longs;t, <lb/>ubi di&longs;pari impetu & velocitate movetur. </s> <s id="s.005018">At globus cadens <lb/>antequam incipiat trahere chordam, nullum pror&longs;us habet im­<lb/>pedimentum, &longs;ed &longs;ivè fortior, &longs;ivè mollior &longs;it arcus, eodem im­<lb/>petu & velocitate movetur; ubi verò re&longs;i&longs;tentiam invenit, &longs;o­<lb/>lùm de&longs;cendit ulteriùs pro ratione repugnantiæ; & factä de­<lb/>mum æqualitate inter vim de&longs;cendendi à globulo acqui&longs;itam, <lb/>& vim ela&longs;ticam in arcu, ce&longs;&longs;at de&longs;cen&longs;us, atque extincto im­<lb/>petu acqui&longs;ito, vi ela&longs;ticâ vincente globuli gravitatem, hic &longs;ur-<pb pagenum="680" xlink:href="017/01/696.jpg"/>&longs;um trahitur. </s> <s id="s.005019">Cùm itaque quicquid vi extrin&longs;ecùs a&longs;&longs;umptâ <lb/>movetur, moveatur juxta exce&longs;&longs;um virtutis motivæ &longs;upra re­<lb/>&longs;i&longs;tentiam; &longs;i æqualis re&longs;i&longs;tentiæ men&longs;ura, quæ ex di&longs;&longs;imilium <lb/>arcuum majori aut minori flexione de&longs;umitur, eumdem exce&longs;­<lb/>&longs;um virtutis motivæ exigat, ut vincatur, & hunc exce&longs;&longs;um ha­<lb/>beat globulus cadens, nil mirum, &longs;i idem globulus cadens id <lb/>præ&longs;tare po&longs;&longs;it, quod &longs;uperat vires alicujus ponderis, cujus vis <lb/>movendi non eumdem &longs;emper exce&longs;&longs;um habet &longs;upra illam re­<lb/>&longs;i&longs;tentiam priori re&longs;i&longs;tentiæ æqualem; quia videlicet non æqua­<lb/>li impetûs inten&longs;ione aggreditur motum, ubi ip&longs;o &longs;tatim initio <lb/>major invenitur difficultas, & tardior e&longs;t motus. </s> <s id="s.005020">Non e&longs;t igitur <lb/>vis infinita globuli duarum unciarum nullo impedimento prohi­<lb/>biti, quin ad trahendam cuju&longs;cumque arcûs chordam &longs;emper <lb/>afferat, exempli gratiâ, centum gradus impetûs in motu acqui­<lb/>&longs;itos, quando pondera majora & majora tractionem incipientia <lb/>à quiete non parem habent impetûs exce&longs;&longs;um, &longs;ed minorem & <lb/>minorem pro duriore arcûs temperatione. </s> <s id="s.005021">An infinitam dixe­<lb/>ris equi virtutem, qui &longs;olus in liberâ planitie currum trahat, ad <lb/>quem trahendum in eâdem planitie altioribus atque altioribus <lb/>nivibus ob&longs;itâ requiruntur plures & plures equi? </s> <s id="s.005022">igitur nec in­<lb/>finita e&longs;t vis decem librarum, qua flectitur arcus mollis, quia <lb/>ad flectendos arcus fortiores majus & majus pondus requiritur: <lb/>huic autem virtuti decem librarum æqualis e&longs;t vis globuli ca­<lb/>dentis; hæc igitur & ip&longs;a finita e&longs;t. </s> <s id="s.005023">Nimirum aucta re&longs;i&longs;tentia <lb/>quodammodo imminuit virtutem agendi; ac propterea non &longs;a­<lb/>tis aptè comparantur decem libræ cum viginti libris perinde, <lb/>atque &longs;i utræque e&longs;&longs;ent omnino liberæ; &longs;ed unumquodque pon­<lb/>dus componi debet cum &longs;uâ re&longs;i&longs;tentiâ, ut demum habeatur <lb/>exce&longs;&longs;us virtutis motivæ &longs;upra re&longs;i&longs;tentiam. </s> </p> <p type="main"> <s id="s.005024">At, inquis, arcus fortior retinetur à libris viginti, & infir­<lb/>mior à libris decem. </s> <s id="s.005025">Ita planè e&longs;t: &longs;ed hìc pondera propriè non <lb/>habent rationem efficientis, &longs;ed potiùs re&longs;i&longs;tentis, quatenus <lb/>impediunt arcuum vim ela&longs;ticam, ne &longs;e re&longs;tituant: cùm verò <lb/>virtutes ela&longs;ticæ ex genere &longs;uo propter di&longs;parem temperatio­<lb/>nem inæquales &longs;int, nil mirum, &longs;i ab inæqualibus re&longs;i&longs;tentiis <lb/>impediendæ &longs;int, ne agant. </s> <s id="s.005026">Hinc autem non e&longs;t de&longs;umenda <lb/>ulla comparatio cum virtute globuli cadentis, quippe qui ac­<lb/>qui&longs;itum impetum amittens non habet vim retinendi arcum in <pb pagenum="681" xlink:href="017/01/697.jpg"/>eo &longs;tatu; ad quem illum adduxit: at ponderis adnexi gravitas <lb/>manet, & ibi retinet arcum, quò eum adduxit; ni&longs;i fortè ali­<lb/>quem impetum acqui&longs;ierit in de&longs;cen&longs;u, quo pereunte, aliquan­<lb/>tulum præpolleat vis ela&longs;tica, & &longs;ur&longs;um retrahat appen&longs;um <lb/>pondus. </s> <s id="s.005027">Licet igitur globulo cadenti æqualiter re&longs;i&longs;tere dican­<lb/>tur arcus fortior qui minùs flectitur, & mollior qui magis flecti­<lb/>tur; po&longs;tquam tamen jam per vim inflexi &longs;unt arcus, naturali­<lb/>ter partes minùs flexibiles validiùs conantur &longs;e re&longs;tituere, quàm <lb/>flexibiliores: quemadmodum gravitas ut quatuor, & gravitas <lb/>ut duo, &longs;i moveantur per vim motu reciprocè &longs;ubduplo, æqua­<lb/>liter re&longs;i&longs;tunt moventi; &longs;ed &longs;i utraque &longs;u&longs;pendatur, inæquali­<lb/>ter conantur &longs;uos motus naturales. <lb/></s> </p> <p type="main"> <s id="s.005028"><emph type="center"/>CAPUT VII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005029"><emph type="center"/><emph type="italics"/>Quàm di&longs;pares ex motûs velocitate &longs;int <lb/>percu&longs;siones.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005030">PErcu&longs;&longs;ionem ex ea parte, quatenus à &longs;implici Impul&longs;ione <lb/>di&longs;tinguitur, motum exigere antecedentem, quo impetus <lb/>acquiratur, &longs;uperiori capite definitum e&longs;t. </s> <s id="s.005031">Nunc verò, quia pro <lb/>motuum velocitate diversâ di&longs;pares &longs;unt percu&longs;&longs;ionum vires, <lb/>quærendum e&longs;t, unde di&longs;&longs;imilitudo i&longs;ta procreetur, & quænam <lb/>&longs;ervari Ratio videatur, &longs;ivè inæquales eju&longs;dem corporis, &longs;ive <lb/>diver&longs;orum corporum percu&longs;&longs;iones inter &longs;e comparentur. </s> <s id="s.005032">E&longs;t <lb/>autem con&longs;iderandum in Impetu, qui e&longs;t proximè efficiens mo­<lb/>tum, aliud e&longs;&longs;e ejus quantitatem, &longs;ivè entitatem accipere, aliud <lb/>in eju&longs;dem Inten&longs;ione con&longs;i&longs;tere: inten&longs;ionem con&longs;equitur ve­<lb/>locitas motûs, at ex entitate ipsâ magis extensâ, quamvis mi­<lb/>nùs intensâ, ac proinde ex motu tardiore, oriri pote&longs;t validior <lb/>ictus, de quo in &longs;equentibus. </s> <s id="s.005033">Ex motûs autem velocitate, quæ <lb/>corpori vi præcedentis motûs congrueret, &longs;i nihil ob&longs;taret, per­<lb/>cu&longs;&longs;ionem fieri majorem tam certis experimentis con&longs;tat, ut vel <lb/>cæci, &longs;i quando præfidentes concitatiùs ambulando caput ad <lb/>objectum parietem allidunt, id abundè te&longs;tari valeant; corpus <pb pagenum="682" xlink:href="017/01/698.jpg"/>&longs;iquidem, quod motui re&longs;i&longs;tit, majori & velociori motui ma­<lb/>gis re&longs;i&longs;tit, quare & percu&longs;&longs;io fit validior. </s> </p> <p type="main"> <s id="s.005034">Ubi verò de motûs velocitate &longs;ermo e&longs;t, non videtur di&longs;&longs;imu­<lb/>landa medij &longs;cindendi re&longs;i&longs;tentia; hoc quippe tardiori motui <lb/>minùs, velociori magis ob&longs;tat. </s> <s id="s.005035">Si enim ex lento & flexili vir­<lb/>gulto ab&longs;tractam virgam per aërem molli brachio huc, illuc, <lb/>&longs;ur&longs;um, deor&longs;um duxeris, hæc, aëre tenui&longs;&longs;imam aut ferè nul­<lb/>lam compre&longs;&longs;ionem &longs;ubeunte, vix, aut ne vix quidem, tantu­<lb/>lum à directâ &longs;uarum partium po&longs;itione deflectet: at &longs;i eam ve­<lb/>hementius agitaveris, aëre tantam particularum compre&longs;&longs;io­<lb/>nem renuente, manife&longs;tè inflexam videbis, & illatam &longs;ibi vim <lb/>aër acuto &longs;ibilo prodet. </s> <s id="s.005036">Sic baculo aquam &longs;en&longs;im ac leniter di­<lb/>videns non admodum repugnantem experiris; at velociùs con­<lb/>citanti illa validè re&longs;i&longs;tit, eóque validiùs, quò cra&longs;&longs;ior fuerit <lb/>baculus. </s> <s id="s.005037">Ex his liquidò conficitur de percu&longs;&longs;ione philo&longs;ophan­<lb/>tem fru&longs;tra medij re&longs;i&longs;tentiam mente ab&longs;trahere: Nam &longs;i nulla <lb/>e&longs;t &longs;ine motu percu&longs;&longs;io, nullus motus ni&longs;i per medium, neque <lb/>&longs;inè certa velocitatis aut tarditatis men&longs;urâ, cui medium inæ­<lb/>qualiter re&longs;i&longs;tit; utique & motum à percu&longs;&longs;ione ita mente pari­<lb/>ter &longs;ejungere poteris, ut nihil pror&longs;us de motu cogites, &longs;i nul­<lb/>lam cum medio rationem habendam exi&longs;timas: At motum, ejú&longs;­<lb/>que velocitatem attendendam e&longs;&longs;e in percu&longs;&longs;ione nemo negat; <lb/>igitur neque medij re&longs;i&longs;tentiam, quæ velocitati modum ali­<lb/>quem &longs;tatuit, omnino contemnere oportet. </s> </p> <p type="main"> <s id="s.005038">Hinc duæ ferè ex diametro oppo&longs;itæ &longs;ententiæ cavendæ <lb/>&longs;unt, quarum altera gravium inæqualium motum &longs;tatuit ip&longs;o­<lb/>rum gravitatibus analogum, ut decuplò velociùs moveatur il­<lb/>lud, quod e&longs;t decuplò gravius: altera æqualem omnibus velo­<lb/>citatem tribuit. </s> <s id="s.005039">Utramque manife&longs;ta experimenta fal&longs;itatis re­<lb/>darguunt, &longs;i ex congruâ altitudine in&longs;tituantur: Si enim ex val­<lb/>dè editâ turri inæqualia corpora, aut eju&longs;dem, aut diver&longs;æ &longs;e­<lb/>cundùm &longs;peciem gravitatis dimittas, illud, quod gravius e&longs;t, <lb/>terram citiùs attingere ob&longs;ervabis, &longs;ed tam brevi momentorum <lb/>di&longs;crimine, ut nulla &longs;ube&longs;&longs;e po&longs;&longs;it &longs;u&longs;picio &longs;ervatæ velocitatum <lb/>cum gravitatibus analogiæ, neque tamen de velocitatum inæ­<lb/>qualitati dubitari queat. </s> <s id="s.005040">Meis &longs;cilicet auribus & oculis fidem <lb/>abrogare nequeo, quicquid obtrudant aliqui in contrarium &longs;ua <lb/>aut aliorum experimenta afferentes ex nimis brevi altitudine. <pb pagenum="683" xlink:href="017/01/699.jpg"/>Nam & &longs;æpiùs in profundi&longs;&longs;imum puteum inæquales lapides <lb/>dimi&longs;i &longs;imul, & ictuum &longs;onitum alium alio priorem &longs;emper au­<lb/>divi; id quod &longs;atis e&longs;t ad illam velocitatum omnimodam æqua­<lb/>litatem rejiciendam, quamvis uter prior aquam attigerit, certò <lb/>digno&longs;cere non valeret auris; quòd &longs;i alter in &longs;ubjectam peluim <lb/>æneam, alter in vas ligneum decidi&longs;&longs;et, potui&longs;&longs;et auris dijudi­<lb/>care ex &longs;onitu: Et ex alti&longs;&longs;imâ turri Bononien&longs;i dimi&longs;&longs;a pondera <lb/>inæqualia ob&longs;ervavi initio qua&longs;i æqualiter de&longs;cendere ita, ut <lb/>oculus nullam velocitatum di&longs;&longs;imilitudinem adhuc digno&longs;ce­<lb/>ret; deinde procedente de&longs;cen&longs;u paulatim gravitas major præ­<lb/>currere notabiliter incipiebat, &longs;empérque magis augebatur ve­<lb/>locitas, adeò ut aliquando gravitas major terram attigerit, <lb/>quando minor adhuc aberat intervallo pedum quadraginta, <lb/>quemadmodum ex notâ in turris latere dimetiri licuit. </s> <s id="s.005041">Verum <lb/>quidem e&longs;t brevi&longs;&longs;imâ temporis men&longs;urâ & hanc minorem in <lb/>terram decidi&longs;&longs;e. </s> <s id="s.005042">Id quod forta&longs;&longs;e fucum fecit non animadver­<lb/>tentibus magnæ velocitati multum re&longs;pondere &longs;patij, quod <lb/>qua&longs;i momento percurritur; ac propterea æqualitatem veloci­<lb/>tatum utrique gravitati tribuendam cen&longs;uerunt, quia exiguum <lb/>erat temporum di&longs;crimen. </s> <s id="s.005043">An vellus, quantum pugno com­<lb/>prehenditur, æquè velociter ac prægrande &longs;axum de&longs;cen&longs;urum <lb/>exi&longs;timas? </s> <s id="s.005044">Figuræ dices tribuendum plurimum, non enim ab <lb/>omnibus corporibus æquè facilè dividitur aër nunquam non <lb/>fluctuans. </s> <s id="s.005045">Ita &longs;ane: igitur &longs;i aër inæqualiter re&longs;i&longs;tit, inæqua­<lb/>liter moveri po&longs;&longs;unt corpora cadentia. </s> <s id="s.005046">Adde non di&longs;&longs;imiliter <lb/>gravia & levia ad &longs;uos motus à naturâ incitari; atque adeò &longs;i, <lb/>ubi plus e&longs;t levitatis, velociorem motum &longs;ur&longs;um ob&longs;ervamus, <lb/>etiam, ubi plus e&longs;t gravitatis concitatiorem motum deor&longs;um <lb/>arguere debemus. </s> <s id="s.005047">Aquam in longiore fi&longs;tulâ vitreá aliquan­<lb/>diu agita, ut aër fi&longs;tulæ inclu&longs;us aquæ admi&longs;ceatur: ubi ab agi­<lb/>tatione ce&longs;&longs;atum fuerit, majores aëris particulas citò a&longs;cenden­<lb/>tes videbis, dum minores cunctabundæ paulatim moventur: id <lb/>quod clariùs con&longs;tabit, &longs;i aquæ loco hydrargyrum in fi&longs;tulam <lb/>admi&longs;eris. </s> <s id="s.005048">Quidni igitur gravia pariter deor&longs;um di&longs;pari velo­<lb/>citate moveantur, &longs;i inæqualia fuerint? </s> </p> <p type="main"> <s id="s.005049">Non tamen &longs;ervandam e&longs;&longs;e gravitatum analogiam hinc <lb/>apertè con&longs;tat, quod in corporibus eju&longs;dem &longs;peciei Ratio gra­<lb/>vitatum eadem e&longs;t ac magnitudinum: magnitudines autem <pb pagenum="684" xlink:href="017/01/700.jpg"/>&longs;unt in triplicatâ Ratione homologorum laterum: At impedi­<lb/>mentum, quod ex medio &longs;cindendo oritur, & velocitati mo­<lb/>dum &longs;tatuit, non e&longs;t ip&longs;is magnitudinibus analogum, &longs;ed ad <lb/>&longs;ummum ea e&longs;&longs;e pote&longs;t Ratio, quæ inter corporum &longs;uperficies <lb/>intercedit; hæ autem tantùm &longs;unt in duplicatâ Ratione late­<lb/>rum homologorum. </s> <s id="s.005050">Non igitur velocitates, quatenus ab im­<lb/>pedimento temperantur, &longs;unt directè gravitatibus analogæ. </s> <lb/> <s id="s.005051">Ubi autem corpora non eju&longs;dem &longs;ecundùm &longs;peciem gravitatis, <lb/>habuerint gravitates magnitudinibus reciprocè analogas, atque <lb/>adeò æquali gravitate ab&longs;olutâ, &longs;eu pondere, prædita fuerint, <lb/>adhuc inæquales e&longs;&longs;e aëris re&longs;i&longs;tentias, &longs;i figuræ &longs;imiles &longs;int, <lb/>&longs;atis probabiliter concedimus, plus &longs;iquidem majori repugnat, <lb/>quàm minori: &longs;i verò & di&longs;&longs;imiles figuræ, & inæquales gravi­<lb/>tates ponantur, ex omnibus &longs;imul compo&longs;itis quodammodo <lb/>conflari re&longs;i&longs;tentiarum Rationem facile e&longs;t opinari; &longs;ed Ratio­<lb/>nis terminos temerè definire non au&longs;im. </s> </p> <p type="main"> <s id="s.005052">Porrò certam legem, qua in re&longs;i&longs;tendo aër contineatur, om­<lb/>ninò afferre non po&longs;&longs;umus, &longs;i quemadmodum ille re&longs;i&longs;tat, per­<lb/>pendamus. </s> <s id="s.005053">Non eadem e&longs;t aquæ & aëris re&longs;i&longs;tendi Ratio, ne <lb/>dividatur; aqua enim nondum in vaporem extenuata, & fu&longs;a, <lb/>con&longs;tipari &longs;e, & in angu&longs;tiora &longs;patia coarctari non &longs;init; &longs;ed ubi <lb/>locum de&longs;cendenti ex aëre corpori concedere cogitur, &longs;uper­<lb/>ficiem intrà vas, quo continetur, attollens tantumdem aëri, <lb/>quem impellit, &longs;urripit &longs;patij, quantum immer&longs;o corpori per­<lb/>mittit. </s> <s id="s.005054">Aut &longs;i non de&longs;cendat corpus, &longs;ed obliquè feratur (ut &longs;i <lb/>baculum partim aquæ immi&longs;&longs;um, partim extantem tran&longs;ver&longs;um <lb/>agas, aut navis prora illam findat) tunc quæ motui opponitur, <lb/>aqua cri&longs;patur, & baculus &longs;ivè navis foveam ponè relinquit, in <lb/>quam deinde aqua refluat; quò autem cra&longs;&longs;ior baculus aut am­<lb/>plior prora, & vehementior atque concitatior fuerit motus, <lb/>aqua impul&longs;a altiùs a&longs;&longs;urgit, & magis depre&longs;&longs;a fovea apparet. </s> <lb/> <s id="s.005055">Ex quo &longs;atis apertè con&longs;tat à corpore, quod movetur, proximas <lb/>aquæ oppo&longs;itæ particulas impelli, & per has interjectas etiam <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/>perimentis novimus, dum locum corpori commoto concedit, <lb/>neque opus e&longs;t, ut &longs;upremi ætheris regionem invadat locum <lb/>&longs;ibi quærens, neque in foveam excavatus vel ad momentum <pb pagenum="685" xlink:href="017/01/701.jpg"/>hiat; &longs;ed tanti&longs;per dum ejus particulæ circumpul&longs;æ in abeun­<lb/>tis corporis relictum &longs;patium &longs;uccedant, quæ antè &longs;unt, com­<lb/>primuntur, quæ ponè, dilatantur; compre&longs;&longs;æ autem &longs;e expli­<lb/>cantes aërem lateribus adhærentem repellunt, quem dilatatæ <lb/>attrahunt &longs;e contrahentes. </s> <s id="s.005057">Si tardus &longs;it motus, exiguâ aëris <lb/>con&longs;tipatione aut di&longs;tractione opus e&longs;t; at &longs;i velocior, oppo&longs;itæ <lb/>aëris particulæ magis comprimuntur, &longs;equentes magis dilatan­<lb/>tur; quæ proinde &longs;e re&longs;tituere vehementiùs conantes, etiam <lb/>velociorem efficiunt reliquarum particularum circumpul&longs;io­<lb/>nem. </s> <s id="s.005058">Verùm quia &longs;apienti&longs;&longs;imo Naturæ in&longs;tituto ita compara­<lb/>tum e&longs;t, ut quàm minimum ejus ordo perturbetur, & mini­<lb/>mam, quoad fieri po&longs;&longs;it, corpora &longs;ingula patiantur violentiam, <lb/>hanc pluribus potiùs di&longs;pertiendam cen&longs;uit, quàm uni &longs;ubeun­<lb/>dam: propterea &longs;i digitale &longs;patium multo aëri &longs;urripiendum e&longs;t, <lb/>exigua contingit &longs;ingulis particulis naturalis &longs;patij jactura, <lb/>quam di&longs;&longs;imulanter ferunt, nec admodum repugnant; contra <lb/>verò &longs;i modicus &longs;it aër, & tantumdem de ejus &longs;patio demendum <lb/>&longs;it, reluctatur acriùs, ut pro viribus naturæ jura tueatur. </s> <s id="s.005059">Hinc <lb/>&longs;i corpus, quod movetur, brevi intervallo ab&longs;it à corpore &longs;oli­<lb/>do & duro, quod ejus motum ob&longs;i&longs;tendo compe&longs;cet, atque <lb/>adeò etiam aërem impul&longs;um remoratur, hunc inter angu&longs;tias <lb/>deprehen&longs;um magis con&longs;tipari nece&longs;&longs;e e&longs;t, magí&longs;que re&longs;i&longs;tere. </s> <lb/> <s id="s.005060">Non e&longs;t tamen aëri denegandum, quod cæteris corporibus ul­<lb/>tro concedimus; nam & ip&longs;e jam commotus ex concepto per <lb/>impul&longs;ionem externam, aut ex vi &longs;uâ ela&longs;ticâ, impetu faciliùs <lb/>pergit in&longs;titutum iter conficere, quàm &longs;i tunc primùm à quie­<lb/>te recederet: Ex quo fit in motu corporis accelerato, licèt ra­<lb/>tione habitâ velocitatis augenda e&longs;&longs;et re&longs;i&longs;tentia aëris, hanc ta­<lb/>men non augeri ni&longs;i pro exce&longs;&longs;u velocitatis illius &longs;upra motum, <lb/>quo aër moveretur ad ea&longs;dem partes, ni&longs;i acriùs ab ip&longs;o corpo­<lb/>re urgeretur. </s> </p> <p type="main"> <s id="s.005061">Hanc aëris re&longs;i&longs;tentiam paulò explicatiùs commemorare pla­<lb/>cuit eo con&longs;ilio, ut mihi ip&longs;e per&longs;uadeam non modò ip&longs;um <lb/>nihil omnino non officere motui, verùm etiam tam fieri non <lb/>po&longs;&longs;e, ut percu&longs;&longs;ionibus certi&longs;&longs;imam legem &longs;tatuamus, quàm <lb/>evidens e&longs;t adeò incon&longs;tantem & variam e&longs;&longs;e aëris re&longs;i&longs;ten­<lb/>tiam, ut ad calculos &longs;ubtiliter & exqui&longs;itè revocari nequeat, <lb/>quippe quæ ex tam variis cau&longs;is pendet: quemadmodum enim <pb pagenum="686" xlink:href="017/01/702.jpg"/>aquæ, cæterorúmque liquorum di&longs;&longs;imilium re&longs;i&longs;tentia inæqua­<lb/>lis con&longs;picua e&longs;t, ita purum ac tenuem aërem non æquè re&longs;i&longs;te­<lb/>re atque cra&longs;&longs;um & concretum ratio &longs;uadet: quis autem &longs;ynce­<lb/>rum aërem ab aëre cum terræ expirationibus permi&longs;to di&longs;cer­<lb/>nat? </s> <s id="s.005062">Quid &longs;i corpus in motu aërem aliò directum, aut in con­<lb/>trarias partes reflexum, aut turbine aliquo perver&longs;um atque <lb/>adhuc agitatum offendat? </s> <s id="s.005063">an non aliquis velocitatis gradus im­<lb/>minuitur? </s> <s id="s.005064">Sed quis certum habeat, utrùm quie&longs;cat aër, neque <lb/>corporis impetum frangat, aut reprimat aliena impre&longs;&longs;ione ad­<lb/>ver&longs;us, an verò ad ea&longs;dem partes delatus motui ob&longs;ecundet, & <lb/>velocitati faveat? </s> <s id="s.005065">Quare laudandi quidem quicumque percu&longs;­<lb/>&longs;ionum naturam ve&longs;tigantes, & ea, quibus in ejus notitiam <lb/>deduci po&longs;&longs;ent, conjecturá pro&longs;picientes, in in&longs;tituendi expe­<lb/>rimentis &longs;edulò &longs;e exercuerunt; parùm tamen mihi de veritate <lb/>blandiri me po&longs;&longs;e arbitrarer, &longs;i hæc qua&longs;i Apodixes temerè reci­<lb/>perem; &longs;ed neque ore tam duro fuerim, ut ea prorsùs rejiciam. </s> <lb/> <s id="s.005066">Confirmatis igitur experimentis me duci &longs;inam, quatenus ad <lb/>veritatis &longs;imilitudinem me proximè acce&longs;&longs;urum &longs;pero. </s> </p> <p type="main"> <s id="s.005067">Percu&longs;&longs;io itaque, &longs;i motum naturalem ex gravitate ortum &longs;ub­<lb/>&longs;equatur, certam aliquam Rationem ob idip&longs;um &longs;ortiri videtur, <lb/>quia cum velocitate con&longs;entit impetus: velocitas autem ex &longs;pa­<lb/>tio deprehenditur æqualibus temporibus re&longs;pondente; &longs;patia <lb/>verò cum temporibus comparata certis Rationibus definita vi­<lb/>deri, iterata experimenta docuerunt, quæ vix qui&longs;quam &longs;anus <lb/>neget; in iis &longs;iquidem tot docti&longs;&longs;imi viri po&longs;t Galilæum ver&longs;ati <lb/>&longs;unt pari exitu, & &longs;ummo con&longs;en&longs;u, ut in his omnibus in&longs;it <lb/>quidam, &longs;ine ullo fuco veritatis color. </s> <s id="s.005068">Hujus rei &longs;pecimen <lb/>exhibeamus in globo argillaceo unciarum octo, qui &longs;patio unius <lb/>&longs;crupuli &longs;ecundi (quantus ferè e&longs;t pul&longs;us arteriæ hominis &longs;ani) <lb/>ob&longs;ervatus e&longs;t percurrere pedes Romanos 15; duplo autem <lb/>tempore incipiendo à quiete, hoc e&longs;t &longs;crupulis &longs;ecundis duo­<lb/>bus, pedes 60: quare &longs;i priori &longs;crupulo &longs;ecundo re&longs;pondent pe­<lb/>des 15, po&longs;teriori tribuendi &longs;unt pedes 45: igitur motus e&longs;t ce­<lb/>lerior, cùm majus &longs;patium pari tempore confecerit. </s> <s id="s.005069">Plura hu­<lb/>ju&longs;modi experimenta (&longs;i te à tentando ab&longs;terreat labor) &longs;uppe­<lb/>ditabit Ricciolius tom.1. Almag. <!-- REMOVE S-->lib.9. &longs;ect.4. cap.16. ex quibus <lb/>demum infertur velocitatis incrementa fieri juxta incremen­<lb/>tum progre&longs;&longs;ionis Arithmeticæ numerorum imparium ab unita-<pb pagenum="687" xlink:href="017/01/703.jpg"/>te incipientis 1.3.5.7.9.11.13, &c. </s> <s id="s.005070">adeò ut, &longs;i quod &longs;patium pri­<lb/>mo momento percurritur, &longs;tatuatur ut 1, triplo velociùs movea­<lb/>tur corpus grave de&longs;cendens in &longs;ecundo momento, quintuplo <lb/>velocius in tertio, &longs;eptuplo velociùs in quarto, atque ita dein­<lb/>ceps. </s> <s id="s.005071">Quoniam verò numerorum imparium &longs;eries ab unitate <lb/>incipiens hoc habet, quòd, &longs;i colligantur in &longs;ummam, nume­<lb/>ros quadratos con&longs;tituant; hinc e&longs;t, quòd collectis in &longs;ummam <lb/>omnibus incrementis velocitatis (hoc e&longs;t omnibus &longs;patiis, ex <lb/>his quippe digno&longs;citur velocitas) habeatur numerus quadratus <lb/>temporis, quod duravit motus. </s> <s id="s.005072">Collatis igitur invicem duobus <lb/>motibus naturalibus eju&longs;dem corporis gravis, &longs;ed non i&longs;ochro­<lb/>nis, erunt ut quadrata temporum ita & &longs;patia, atque è conver­<lb/>&longs;o ut &longs;patia inter &longs;e, ita & temporum quadrata. </s> </p> <p type="main"> <s id="s.005073">Hinc cognito &longs;patio, quod à dato corpore gravi percurritur <lb/>dato tempore, &longs;tatim innote&longs;cet, quantum &longs;patij conficere va­<lb/>leat alio tempore dato, vel quanto tempore aliud datum &longs;pa­<lb/>tium. </s> <s id="s.005074">Quæratur enim, quantum &longs;patium de&longs;cendendo percur­<lb/>ret uno horæ quadrante globus idem argillaceus, qui uno mi­<lb/>nuto &longs;ecundo Romanos pedes 15 percurrit? </s> <s id="s.005075">Datum tempus, <lb/>&longs;cilicet horæ quadran, &longs;crupula Secunda 900 continet, cujus <lb/>numeri quadratum e&longs;t 810000. Fiat igitur ut 1 ad 810000, ita <lb/>pedes 15 ad 12150000: qui pedum numerus in milliaria Itali­<lb/>ca re&longs;olutus dat milliaria 2430, quæ uno horæ quadrante con­<lb/>ficeret. </s> <s id="s.005076">Vici&longs;&longs;im quæratur quantum temporis idem globus in­<lb/>&longs;umeret in primo milliari percurrendo, hoc e&longs;t ped. <!-- REMOVE S-->5000. Fiat <lb/>ut 15 ad 5000, ita 1 quadratum dati temporis, &longs;cilicet unius <lb/>&longs;crupuli &longs;ecundi, ad 333 1/3 quadratum quæ&longs;iti temporis; cujus <lb/>quadrati Radix inve&longs;tiganda e&longs;t, & demum invenitur Scrup. <lb/><!-- KEEP S--></s> <s id="s.005077">&longs;ec. <!-- REMOVE S-->18 1/4 & paulo ampliûs; nam huic tempori præcisè re&longs;pon­<lb/>dent &longs;olùm pedes (4995 15/16). </s> </p> <p type="main"> <s id="s.005078">Incrementa hæc velocitatis ex concepti impetûs incremento <lb/>de&longs;umenda e&longs;&longs;e nullus dubito; &longs;ed opero&longs;um videri po&longs;&longs;et au­<lb/>ge&longs;centis impetûs cau&longs;am exponere. </s> <s id="s.005079">Cùm junior Ari&longs;totelem <lb/>interpretarer, & primas curas huju&longs;modi rerum contemplatio­<lb/>ni impenderem, hanc excogitavi hypothe&longs;im; videlicet impe­<lb/>tûs producti diuturnitatem maximam duobus tantùm momen­<lb/>tis circum&longs;cribebam, ita ut primo momento oriretur, &longs;ecundo <pb pagenum="688" xlink:href="017/01/704.jpg"/>æqualem &longs;ibi impetum gigneret, in quo &longs;uper&longs;tes e&longs;&longs;et, tertio <lb/>periret: gravitati autem &longs;ingulis momentis vim producendi <lb/>certum impetûs gradum &longs;ibi congruentem tribuebam. </s> <s id="s.005080">Hinc <lb/>corpus de&longs;cendens primo momento primum habebat impetûs <lb/>gradum à gravitate productum; &longs;ecundo momento primus ille <lb/>gradus alium gradum gignebat præter eum, qui à gravitate <lb/>tunc oriebatur; quare duo novi gradus cum uno antiquo tres <lb/>gradus con&longs;tituebant. </s> <s id="s.005081">Tertio momento primus gradus peribat, <lb/>duo &longs;ecundi gradus duos pariter producebant, & gravitas &longs;uum <lb/>tertium gradum; quare quinque gradus erant. </s> <s id="s.005082">Quarto mo­<lb/>mento duobus &longs;ecundis gradibus pereuntibus, tres gradus ter­<lb/>tio momento producti reliqui erant, & &longs;ibi tres alios gradus <lb/>addebant, quos producebant, atque gravitas &longs;uum quartum <lb/>gradum efficiebat, ut in univer&longs;um e&longs;&longs;ent &longs;eptem gradus. </s> <s id="s.005083">Ex <lb/>his &longs;eptem quinto momento peribant tres tertij gradus; qua­<lb/>tuor reliqui item alios quatuor adjiciebant quinto gradui à gra­<lb/>vitate profici&longs;centi, & erant novem. </s> <s id="s.005084">Atque ita deinceps, tot <lb/>pereuntibus gradibus, quotum erat momentum uno interjecto <lb/>præcedens, & tot productis, quotum erat ip&longs;ius motûs momen­<lb/>tum. </s> <s id="s.005085">Sic momento vige&longs;imo nono peribant gradus 27 pro­<lb/>ducti momento vige&longs;imo &longs;eptimo, remanentibus gradibus 28 <lb/>productis momento vige&longs;imo octavo, à quibus totidem produ­<lb/>cebantur unâ cum gradu proprio gravitatis, hoc e&longs;t gradus 29, <lb/>& tunc erat impetûs inten&longs;io graduum 57 momento vige&longs;imo <lb/>nono. </s> <s id="s.005086">Quemlibet verò terminum in &longs;erie numerorum impa­<lb/>rium facilè invenies, &longs;i illi duplicato demas unitatem: &longs;ic quæ­<lb/>rens octavum terminum ex denominatore termini duplicato, <lb/>&longs;cilicet bis 8, hoc e&longs;t 16, deme unitatem, & 15 e&longs;t octavus ter­<lb/>minus: &longs;ic terminus &longs;eptuage&longs;imus habetur demptâ unitate ex <lb/>140, & e&longs;t 139. </s> </p> <p type="main"> <s id="s.005087">Huic hypothe&longs;i cum Phenomeno optimè conveniebat, & ea <lb/>&longs;tatuebatur impetûs inten&longs;io, quæ velocitati efficiendæ par e&longs;­<lb/>&longs;et, &longs;ervatâ incrementorum Ratione, quæ ex iteratis experimen­<lb/>tis innotuerat. </s> <s id="s.005088">Verùm commentitia, & fabulæ proxima vide­<lb/>batur tàm brevis impetûs vita, quam non ni&longs;i duo momenta <lb/>metirentur: in iis &longs;anè, quæ vi externâ moventur, & longiùs <lb/>projiciuntur, aut in gyrum aguntur, licet extinctâ effectrice <lb/>causâ impre&longs;&longs;us impetus diutiùs permanet; quidni & impetus <pb pagenum="689" xlink:href="017/01/705.jpg"/>&longs;ponte &longs;uâ conceptus, &longs;uæque origini cohærens aliquandiu per­<lb/>&longs;everet? </s> <s id="s.005089">quippe qui aut eju&longs;dem, aut &longs;altem non deterioris na­<lb/>turæ cen&longs;endus e&longs;t. </s> <s id="s.005090">Adde nimis incertum e&longs;&longs;e, an impetus im­<lb/>petum producere valeat in eodem corpore, cui ine&longs;t, quamvis <lb/>impetum in alienis corporibus percu&longs;&longs;is efficiendi vis illi conce­<lb/>datur: nam & calor, & cæteræ qualitates effectrices, quas de­<lb/>perditas &longs;ibi forma &longs;ub&longs;tantialis reparare incipit, non alios &longs;imi­<lb/>les gradus &longs;ibi addunt, licèt eos in proximo corpore efficere va­<lb/>leant. </s> <s id="s.005091">Præterquam quod, cur illo ip&longs;o momento, quò primùm <lb/>exi&longs;tit impetus, &longs;imilem gradum non producit? </s> <s id="s.005092">nihil &longs;cilicet il­<lb/>li dee&longs;t, nullo impedimento prohibetur, neque cau&longs;am ætate <lb/>præcedere effectum, &longs;ed origine, nece&longs;&longs;e e&longs;t. </s> <s id="s.005093">Si autem primo <lb/>momento & oritur impetus, & impetum efficit, hic pariter &longs;uam <lb/>vim primo eodem momento exerens alium impetum producit, <lb/>& infinita graduum impetûs æqualium multitudo con&longs;urgit; <lb/>cujus ne ve&longs;tigium quidem apparere pote&longs;t, cùm in cau&longs;arum <lb/>& effectuum &longs;erie &longs;emper ab infinitate natura di&longs;cedat. </s> </p> <p type="main"> <s id="s.005094">Quare impetum à gravitate de&longs;cendente productum, ex tam <lb/>expedito interitu vendicandum, & virtute &longs;e novo impetu au­<lb/>gendi &longs;poliandum, longè probabiliore conjecturâ cen&longs;ui. </s> <s id="s.005095">Im­<lb/>petum igitur certâ quadam men&longs;urâ gravitati corporis con­<lb/>gruente, &longs;tatim ac in motum erumpere pote&longs;t, produci exi&longs;timo, <lb/>& quandiu motus per&longs;everat, permanere; eadem enim gravi­<lb/>tas, quæ primo momento illum effecit, reliquis con&longs;equentibus <lb/>momentis con&longs;ervare valet; finis, quò refertur, & cujus causâ <lb/>productus e&longs;t, adhuc obtineri pote&longs;t, videlicet motus; liberè <lb/>de&longs;cendenti corpori nullum objicitur impedimentum; nihil <lb/>ade&longs;t, quod ip&longs;ius concepti impetûs interitum exigat: ergo im­<lb/>petum à gravibus de&longs;cendentibus conceptum non perire in mo­<lb/>tu &longs;i dixerimus, &longs;imilitudinem veri nos con&longs;ecutos arbitror. </s> <lb/> <s id="s.005096">Quoniam verò gravitas inter eas cau&longs;as enumeratur, in quibus <lb/>ine&longs;t efficiendi nece&longs;&longs;itas, & quandiu opus e&longs;t juxta naturæ pro­<lb/>po&longs;itum, quantum po&longs;&longs;unt, efficiunt; &longs;ingulis momentis, qui­<lb/>bus pote&longs;t de&longs;cendere, &longs;ingulos impetûs gradus æquales priori­<lb/>bus adjicit, adeò ut, quot momenta motum metiuntur, tot gra­<lb/>dus impetûs po&longs;tremo momento inten&longs;ionem con&longs;tituant, cui <lb/>motûs velocitas re&longs;pondeat. </s> <s id="s.005097">Velocitatum igitur incrementa <lb/>fiunt juxta naturalem numerorum progre&longs;&longs;ionem 1.2.3.4.5, &c. <pb pagenum="690" xlink:href="017/01/706.jpg"/>nam juxta hanc eandem &longs;eriem impetûs, velocitatis cau&longs;a, au­<lb/>getur, &longs;ingulis gradibus in &longs;ingula momenta additis. </s> </p> <p type="main"> <s id="s.005098">Ab omni tamen infinitatis &longs;u&longs;picione recedendum e&longs;t hìc, <lb/>ubi momentorum vocabulum u&longs;urpo, qua&longs;i infinita puncta <lb/>temporis agno&longs;cerem, & ad vim percu&longs;&longs;ionis infinitam ad&longs;truen­<lb/>dam, infinitis momentis &longs;ingulis impetûs gradum tribuerem. </s> <lb/> <s id="s.005099">Quemadmodum enim corpora punctis prorsùs individuis non <lb/>con&longs;tare &longs;uadetur multiplici argumento præ&longs;ertim ex A&longs;ympto­<lb/>tis lineis de&longs;umpto, ita motui atque tempori puncta omnibus <lb/>omnino partibus carentia nunquam concedenda cen&longs;ui. </s> <s id="s.005100">Sed <lb/>huic verbo, cum momentum dico, &longs;ubjecta notio e&longs;t, minima <lb/>temporis particula Phy&longs;ica, quæ licèt particulas alias adhuc mi­<lb/>nores contineat &longs;ibi ordine &longs;uccedentes, ex quibus illa con&longs;ti­<lb/>tuitur, tota tamen ad primi impetûs effectionem ita requiritur, <lb/>ut juxta naturæ leges nihil effici po&longs;&longs;et motûs, ni&longs;i integra illa <lb/>temporis particula &longs;uppeteret. </s> <s id="s.005101">Huju&longs;modi autem non indivi­<lb/>duas particulas minimas certas atque æquales in tempore, aut <lb/>motu, finito non e&longs;&longs;e ni&longs;i certo numero definitas manife&longs;tum <lb/>e&longs;t: Quapropter &longs;icut momentorum, ita & graduum impetûs <lb/>æqualium multitudo finita e&longs;t. </s> </p> <p type="main"> <s id="s.005102">Verùm nemo temerè hanc momentorum multitudinem ad <lb/>calculos revocare in&longs;tituat; res enim planè incerta e&longs;t. </s> <s id="s.005103">Utique <lb/>tardi&longs;&longs;imos reperiri & languidi&longs;&longs;imos motus aliquos novimus, <lb/>qui diu latent, nec ni&longs;i po&longs;t tempus benè con&longs;picuum demum <lb/>innote&longs;cunt: Ex quo deprehendimus in tempore aut motu, qui <lb/>&longs;en&longs;ibus percipi po&longs;&longs;it, multas numerari huju&longs;modi momento­<lb/>rum myriadas: &longs;i enim in uno aliquo motu exiguo particulæ il­<lb/>lius &longs;ibi ex ordine &longs;uccedentes re&longs;pondent motui longi&longs;&longs;imè ma­<lb/>jori (cuju&longs;modi e&longs;t cælorum motus) ex quo definitur tempus, <lb/>& in hoc plurimæ partes notabiles, & &longs;ub Phy&longs;icam men&longs;uram <lb/>cadentes numerantur, utique & in illo plurimæ particulæ om­<lb/>nem &longs;en&longs;us aciem fugientes inveniuntur. </s> <s id="s.005104">Et quidem &longs;i cum il­<lb/>lis <expan abbr="A&longs;tronõmis">A&longs;tronomis</expan> philo&longs;ophemur, qui cæle&longs;tium graduum minu­<lb/>ta u&longs;que eò in &longs;exage&longs;imas partiuntur, ut demum in &longs;crupulis <lb/>Decimis con&longs;i&longs;tant, cùm Æquatoris gradus quindecim in Pri­<lb/>mo mòbili uni horæ re&longs;pondeant, &longs;atis con&longs;tat, quantus &longs;it hu­<lb/>ju&longs;modi &longs;crupulorum Decimorum numerus, in quorum fluxum <lb/>unica hora re&longs;olvatur: ac proinde in uno horæ minuto Secun-<pb pagenum="691" xlink:href="017/01/707.jpg"/>do, hoc e&longs;t in pul&longs;u arteriæ, continentur plu&longs;quam decies mil­<lb/>lies millena millia myriadum huju&longs;modi &longs;crupulorum Decimo­<lb/>rum, quæ momenta appellari po&longs;&longs;unt. </s> <s id="s.005105">Quapropter illud uni­<lb/>cum generatim &longs;tatuere po&longs;&longs;umus, in quolibet tempore Phy&longs;i­<lb/>cè notabili plurima e&longs;&longs;e momenta, quamvis eorum certum nu­<lb/>merum explicare nequeamus: ideóque cùm incrementa velo­<lb/>citatum men&longs;uram de&longs;umant ex momentorum numero, qui <lb/>&longs;emper unitatis additione augetur, inten&longs;io autem impetûs ha­<lb/>beat graduum numerum parem numero momentorum; quàm <lb/>difficiles explicatus habet momentorum multitudo, tam ob&longs;cu­<lb/>ra e&longs;t impetûs inten&longs;io; &longs;i minutam &longs;ubtilitatem per&longs;equamur. </s> <lb/> <s id="s.005106">Sed &longs;i datum tempus in aliquot particulas no&longs;tro arbitratu <lb/>di&longs;tinguamus, quo plures fuerint huju&longs;modi particulæ, eò pro­<lb/>piùs accedemus ad id, quod experimentis deprehen&longs;um e&longs;t; vi­<lb/>delicet, etiam &longs;i &longs;patia in motu decur&longs;a juxta &longs;eriem naturalem <lb/>numerorum augeantur in motu, demum eorum collectiones in­<lb/>cipiendo à quiete habere inter &longs;e duplicatam Rationem tempo­<lb/>rum inveniemus. </s> </p> <p type="main"> <s id="s.005107">Comparatis igitur invicem motibus alicujus corporis gravis <lb/>de&longs;cendentis, cujus motus unus jam innotuerit, quantum &longs;cili­<lb/>cet &longs;patij dato tempore confecerit, innote&longs;cet, quantus futurus <lb/>&longs;it alio tempore motus, &longs;i fiant ut quadrata datorum temporum, <lb/>ita & &longs;patia; vel quanto tempore percurrendum &longs;it &longs;patium de­<lb/>finitum, &longs;i fiant ut Radices quadratæ datorum &longs;patiorum, ita & <lb/>tempora. </s> <s id="s.005108">Quia nimirum po&longs;ita illa incrementa impetûs, & ve­<lb/>locitatum, atque &longs;patiorum juxta &longs;eriem naturalem numero­<lb/>rum ab unitate incipientem con&longs;tituunt collectiones habentes <lb/>inter &longs;e proximè Rationem duplicatam temporum. </s> <s id="s.005109">Habemus <lb/>experimento globum, argillaceum unciarum octo percurrere <lb/>uno minuto Secundo horæ pedes 15, & duobus Secundis pe­<lb/>des 60, hoc e&longs;t &longs;patium quadruplum, & quia tempora &longs;unt ut 1 <lb/>ad 2, &longs;patia &longs;unt ut quadrata, &longs;cilicet ut 1 ad 4. Ponamus in uno <lb/>Secundo e&longs;&longs;e momenta 10000; &longs;unt igitur ultimo momento <lb/>10000 gradus velocitatis &longs;imiles & æquales primo gradui primi <lb/>momenti, & &longs;patium ultimo hoc momento decur&longs;um, ad &longs;pa­<lb/>tium primi momenti e&longs;t ut 10000 ad 1. Coge igitur in &longs;um­<lb/>mam omnia &longs;patia incipiendo ab unitate u&longs;que ad 10000, vi­<lb/>delicet ultimi termini dimidiato quadrato adde eju&longs;dem ultimi <pb pagenum="692" xlink:href="017/01/708.jpg"/>termini &longs;emi&longs;&longs;em, & prodibit omnium &longs;patiorum &longs;umma. </s> <s id="s.005110">Ultimi <lb/>termini 10000 quadratum e&longs;t 100000000, cui adde ip&longs;um ul­<lb/>timum terminum; & hujus &longs;ummæ medietas 50005000 e&longs;t <lb/>&longs;umma minimorum &longs;patiorum, quibus conflantur pedes 15. In <lb/>duobus Secundis erunt momenta 20000, & &longs;imiliter invenitur <lb/>&longs;umma 200010000 minimorum &longs;patiorum, quibus con&longs;tant <lb/>pedes 60. Non &longs;unt quidem duæ huju&longs;modi &longs;ummæ hìc <lb/>inventæ 50005000 & 200010000, omnino ut 1 ad 4, &longs;ed ut 1 <lb/>ad (3 49995/50005): verùm tantula differentia (10/50005) quid officit allato ex­<lb/>perimento? </s> <s id="s.005111">an potuit ob&longs;ervari? </s> <s id="s.005112">Si 4. &longs;unt pedes 60, quid <lb/>&longs;unt (3 49995/50005)? utique pedes (59 49855/50005); dee&longs;t igitur pedis particula <lb/>(150/50005), hoc e&longs;t unciæ qua&longs;i pars vige&longs;ima &longs;eptima. </s> <s id="s.005113">Quis autem <lb/>tam minutæ &longs;ubtilitati locus &longs;it in ob&longs;ervando motu? </s> </p> <p type="main"> <s id="s.005114">Ut autem per&longs;picuè appareat hanc hypothe&longs;im incrementi <lb/>juxta &longs;eriem naturalem numerorum con&longs;entire cum experi­<lb/>mentis, & &longs;patia &longs;e habere ut quadrata temporum, &longs;tatuamus <lb/>eadem &longs;patia, ut primum &longs;it ad &longs;ecundum in Ratione 50005000 <lb/>ad 200010000. Radix primi &longs;patij e&longs;t (7071 5959/14142), Radix autem <lb/>&longs;ecundi &longs;patij e&longs;t (14142 6918/14142); quæ &longs;unt ut 1 ad 2, &longs;i fractiones <lb/>contemnantur; nec repugnat experimentum; nam tantula <lb/>differentia temporum, ne &longs;it Ratio præcisè dupla, di&longs;cerni <lb/>non potuit: quarum enim partium (7071 5959/14142) e&longs;t unum minu­<lb/>tum Secundum horæ, dee&longs;t unius partis (5000/14142), ut &longs;int duo mi­<lb/>nuta Secunda, hoc e&longs;t unius pulsûs alteriæ pars una vicies mil­<lb/>le&longs;ima de&longs;ideratur, ut &longs;int planè duo Secunda. </s> <s id="s.005115">Quære argu­<lb/>menta, &longs;i qua potes; an experimento revinces e&longs;&longs;e plani&longs;&longs;imè <lb/>duo minuta Secunda, nec vel unicum momentum defui&longs;&longs;e? </s> </p> <p type="main"> <s id="s.005116">Hoc idem, quod exempli causâ in Ratione duplâ temporum <lb/>& quadruplâ &longs;patiorum explicatum e&longs;t, in cæteris pariter de­<lb/>prehendes. </s> <s id="s.005117">Fac enim e&longs;&longs;e tempus quadruplum, hoc e&longs;t Secun­<lb/>dorum 4, hoc e&longs;t minimorum temporis 40000. Tota collectio <lb/>&longs;patiorum erit 800020000. Quare 50005000 ad 800020000 <lb/>e&longs;t ut 1 ad (15 49925/50005), qua&longs;i ut 1 ad 16; e&longs;t autem defectus (60/50005). <lb/>Spatium igitur uno Secundo decur&longs;um cum &longs;it ped. <!-- REMOVE S-->15, qua­<lb/>tuor Secundis erit ped. <!-- REMOVE S-->240 minùs una ferè &longs;exage&longs;ima nona <lb/>particulâ unciæ. </s> <s id="s.005118">Vici&longs;&longs;im ut tempora invenias in &longs;ubduplicatâ <pb pagenum="693" xlink:href="017/01/709.jpg"/>Ratione &longs;patiorum, quære illorum tanquam quadratorum Ra­<lb/>dices; & primi quidem Radix e&longs;t, ut priùs fuit inventa <lb/>(7071 5959/14142), &longs;ecundi Radix e&longs;t (28284 8836/14142), quarum Ratio e&longs;t <lb/>quadrupla, &longs;i fractiones &longs;pernantur; at aliquid dee&longs;t, ut &longs;int <lb/>integra quatuor Secunda minuta horæ; qui defectus demum <lb/>vix major e&longs;t quàm (1/6667) unius pulsûs arteriæ. </s> </p> <p type="main"> <s id="s.005119">Cùm itaque con&longs;tituta hypothe&longs;is incrementi &longs;patiorum, <lb/>velocitatis, atque impetûs juxta &longs;eriem naturalem numerorum <lb/>&longs;it naturæ con&longs;entanea, nequè Phy&longs;icè repugnet experimentis, <lb/>non debemus e&longs;&longs;e &longs;olliciti, ut aliam quæramus hypothe&longs;im ad <lb/>&longs;tatuenda incrementa exqui&longs;itè juxta numeros impares; cùm <lb/>maximè in aquâ ob majorem re&longs;i&longs;tentiam, quam in illâ divi­<lb/>denda inveniunt corpora gravia de&longs;cendentia, non exactè &longs;er­<lb/>vari eandem Rationem incrementorum, quæ in aëre apparet, <lb/>experimenta iterata declarent, quamvis ad illam Rationem <lb/>proximè accedant, ut apud Ricciolium tom. </s> <s id="s.005120">1. Almag. <!-- REMOVE S-->lib. 9. <lb/>&longs;ect.4. cap. 16. n. </s> <s id="s.005121">16. varia experimenta afferentem legi pote&longs;t. </s> <lb/> <s id="s.005122">Præterquam quod &longs;i tam in aëre quàm in aquâ adhibeantur in <lb/>experimentum corpora &longs;ecundùm gravitatem &longs;pecificam ab il­<lb/>lis minimum di&longs;crepantia, &longs;tatim apparebit non &longs;ervari illam <lb/>temporum atque &longs;patiorum Analogiam: id quod pariter ob&longs;er­<lb/>vabitur, &longs;i in diver&longs;is liquoribus eadem gravia corpora dimit­<lb/>tantur: varia &longs;cilicet e&longs;t re&longs;i&longs;tentia; hæc autem latet, quando <lb/>grave dimi&longs;&longs;um valde differt à gravitate, aut levitate medij. </s> </p> <p type="main"> <s id="s.005123">His ita con&longs;titutis, percu&longs;&longs;ionis vires, quatenus ex velocita­<lb/>te oriuntur, proximè definire poterimus, &longs;i innote&longs;cant &longs;patia, <lb/>per quæ idem corpus grave liberè de&longs;cendit; nam hinc inno­<lb/>te&longs;cet Ratio inten&longs;ionum impetûs ultimo de&longs;censûs momento, <lb/>quo contingit percu&longs;&longs;io. </s> <s id="s.005124">E&longs;t &longs;iquidem numerus graduum im­<lb/>petus in motu naturali libero concepti par numero momento­<lb/>rum motûs; at momenta, quibus con&longs;tant tempora motuum <lb/>inæqualium, &longs;unt in Ratione &longs;ubduplicatâ Rationis &longs;patiorum: <lb/>igitur &longs;icut Radices quadratæ &longs;patiorum <expan abbr="indicãt">indicant</expan> Rationem tem­<lb/>porum, ita pariter eædem indicant Rationem inten&longs;ionum im­<lb/>petûs. </s> <s id="s.005125">Quare &longs;i alicujus gravis ex datâ altitudine cadentis per­<lb/>cu&longs;&longs;io manife&longs;ta fuerit, facilè inferemus, quanta proximè &longs;it <lb/>futura eju&longs;dem percu&longs;&longs;io ex majori, aut minori altitudine, &longs;i fiat <pb pagenum="694" xlink:href="017/01/710.jpg"/>ut Radix datæ altitudinis prioris ad Radicem po&longs;terioris altitu­<lb/>dinis, ita nota percu&longs;&longs;io ad quæ&longs;itam percu&longs;&longs;ionem. </s> </p> <p type="main"> <s id="s.005126">Neque hìc conabor dicta confirmare experimentis tùm à <lb/>Ricciolio loc. </s> <s id="s.005127">cit. </s> <s id="s.005128">cap.16. n.12, tùm à Mer&longs;enno tom.3. in Re­<lb/>flexionibus Phy&longs;ico-Mathemat. <!-- REMOVE S-->cap. 8. allatis, ex quibus proxi. </s> <lb/> <s id="s.005129">mé infertur hæc Ratio &longs;ubduplicata &longs;patiorum. </s> <s id="s.005130">Nam cùm adhi­<lb/>bita &longs;it libra, ut in alteram lancem cadens pondus ex diver&longs;is <lb/>altitudinibus dimi&longs;&longs;um elevaret pondera inæqualia oppo&longs;itæ <lb/>lanci impo&longs;ita, res e&longs;t anceps & incetta. </s> <s id="s.005131">Quandoquidem, ut <lb/>ob&longs;ervavi jam tum ab anno 54 labentis &longs;æculi &longs;criptis Romæ de <lb/>hoc eodem argumento publicè traditis, & funiculi, ex quibus <lb/>lanx percu&longs;&longs;a pendet, di&longs;trahuntur, & libræ jugum flectitur, <lb/>immò & lanx ip&longs;a ictum cadentis ponderis excipiens flexilis <lb/>e&longs;t; ac propterea impetûs vim retundunt, cùm maximè vi <lb/>ela&longs;ticâ &longs;e re&longs;tituentes conantur &longs;ur&longs;um. </s> <s id="s.005132">Præterquam quod, &longs;i <lb/>pondus cadens non exactè incidat in lancis centrum re&longs;pon­<lb/>dens extremitati jugi, plurimum intere&longs;t ad varianda momen­<lb/>ta, prout libræ brachium aut decurtatum aut productum intel­<lb/>ligitur. </s> <s id="s.005133">Quò autem majus e&longs;t pondus in oppo&longs;itâ lance attollen­<lb/>dum ex vi depre&longs;&longs;ionis lancis percu&longs;&longs;æ, magis re&longs;i&longs;tit, ac proin­<lb/>de locus e&longs;t flexioni majori ip&longs;ius libræ, aut funiculorum <lb/>di&longs;tractioni, præ&longs;ertim &longs;i lanx fuerit concava, & cadens pon­<lb/>dus illam tangens prolabatur in depre&longs;&longs;iorem lancis locum. </s> <s id="s.005134">Ex <lb/>quo accidit, ut docuit experientia, aucto pondere elevando non <lb/>&longs;atis e&longs;&longs;e dimittere pondus cadens ex altitudine, quæ &longs;it ad prio­<lb/>rem altitudinem ut quadratum ponderis majoris ad quadratum <lb/>ponderis minoris initio elevati; percu&longs;&longs;io enim contingit in lan­<lb/>ce, cujus re&longs;i&longs;tentiam ad hoc, ut vi ponderis cadentis deprima­<lb/>tur, metitur re&longs;i&longs;tentia ponderis elevandi in oppo&longs;itâ lance: at <lb/>hæc &longs;i fuerit major quàm re&longs;i&longs;tentia jugi, aut lancis, ne flecta­<lb/>tur, aut funiculorum ne di&longs;trahantur, in hac flexione aut <lb/>di&longs;tractione in&longs;umitur vis percu&longs;&longs;ionis, quin oppo&longs;ita lanx attol­<lb/>latur. </s> <s id="s.005135">Quapropter ex altitudine adhuc majori dimittendum e&longs;t <lb/>pondus cadens; nam adhuc majore impetu concepto tam validè <lb/>percutiet lancem, ut re&longs;i&longs;tentia jugi & lancis ad flexionem ul­<lb/>teriorem, atque funiculorum ad longiorem di&longs;tractionem, ma­<lb/>jor &longs;it quàm re&longs;i&longs;tentia ponderis oppo&longs;itæ lancis: atque adeò <lb/>lanx percu&longs;&longs;a non deprimetur &longs;olùm, quantum funiculorum <pb pagenum="695" xlink:href="017/01/711.jpg"/>di&longs;tractio & ip&longs;a flexio permittit, &longs;ed adhuc ulteriùs; atque op­<lb/>po&longs;ita lanx attolletur, quatenus impetûs vires excedunt oppo­<lb/>&longs;itæ gravitatis re&longs;i&longs;tentiam. </s> </p> <p type="main"> <s id="s.005136">Ex his, quæ de Percu&longs;&longs;ionibus corporum gravium naturali­<lb/>ter de&longs;cendentium hactenus dicta &longs;unt, conjectura &longs;umenda e&longs;t <lb/>de reliquis percu&longs;&longs;ionibus, quæ motûs originem non à &longs;olâ gra­<lb/>vitate ducunt: nam in his pariter ex motûs velocitate, qua in­<lb/>dicatur inten&longs;io impetûs, oritur validior percu&longs;&longs;io: nam &longs;ive <lb/>conceptus à potentiâ vivente impetus, &longs;ive extrin&longs;ecùs im­<lb/>pre&longs;&longs;us (ni&longs;i novam offendat re&longs;i&longs;tentiam, quæ illum retundat <lb/>ac minuat) augetur novo impetu, quem potentia &longs;imiliter ap­<lb/>plicata, ii&longs;démque viribus prædita, nec ob&longs;taculo ullo aut reti­<lb/>naculo impedita, &longs;equentibus momentis efficere pote&longs;t: ideó­<lb/>que quò major e&longs;t potentiæ percutientis motus quoad &longs;patium, <lb/>cæteris paribus, validiùs percutit. </s> <s id="s.005137">Cæteris, inquam, paribus; <lb/>nam &longs;i po&longs;terioribus momentis motûs, minor impetus addatur à <lb/>potentiâ movente, quàm deperdatur ex priore impetu impre&longs;­<lb/>&longs;o, quem natura repugnans excutit, langue&longs;cit motus, & mi­<lb/>nuitur impetus: aut &longs;i tantumdem acquiratur impetûs, quan­<lb/>tum deperditur, motus e&longs;t æquabilis, nec ad validiorem per­<lb/>cu&longs;&longs;ionem quicquam confert diuturna potentiæ moventis appli­<lb/>catio, cum eadem &longs;it impetûs inten&longs;io in fine horæ, atque in <lb/>primo momento. </s> <s id="s.005138">Quia tamen frequentiùs (etiam &longs;i fortè ali­<lb/>quid antiquioris impetûs ex novâ re&longs;i&longs;tentiâ deteratur) plus ad­<lb/>ditur, velocitas incrementum &longs;umit, &longs;i potentia maneat diutiùs <lb/>applicata. </s> </p> <p type="main"> <s id="s.005139">Hinc patet, cur, cum quis ho&longs;tem &longs;ari&longs;sâ confodere tentat, <lb/>aut po&longs;tes cra&longs;&longs;iore fu&longs;te arietare, brachium retrahat quantum <lb/>pote&longs;t; ut nimirum potentia diutiùs applicata maneat in motu, <lb/>&longs;empérque novum impetum gignens &longs;ari&longs;&longs;æ aut fu&longs;ti imprimat. </s> <lb/> <s id="s.005140">Sic duobus digladiantibus, &longs;i alter alteri &longs;ini&longs;trum latus obver­<lb/>tat, & tam longo en&longs;e utatur, ut protecto corpore po&longs;&longs;it manum <lb/>valde retrahere, hic validi&longs;&longs;imum ictum infliget extento dex­<lb/>tro brachio, tùm quia ex celerrimâ corporis totius conver&longs;ione <lb/>impetus aliquis brachio communicatur præter impetum, quem <lb/>conferunt mu&longs;culi movendo brachio de&longs;tinati, tùm quia diu­<lb/>tiùs movetur gladius à manu per majus &longs;patium. </s> <s id="s.005141">Quod &longs;i eo <lb/>ictu, quem Itali <emph type="italics"/>Quartam<emph.end type="italics"/> vocamus, ho&longs;tem impetat, adhuc va-<pb pagenum="696" xlink:href="017/01/712.jpg"/>dior erit ictus, quia longior motus; nam in conver&longs;ione corpo­<lb/>ris obvertitur ho&longs;ti dextrum latus præcisè ita, ut brachium to­<lb/>tum extendi queat; & præterea pars exterior manûs deor&longs;um <lb/>convertitur, ex quo propter conformationem juncturarum cubi­<lb/>ti & manus ictus evadit duos ferè digitos longior, quàm &longs;i non <lb/>fieret huju&longs;modi manûs conver&longs;io: demum cùm &longs;ini&longs;trum bra­<lb/>chium in po&longs;teriora projiciatur extentum, fit, ut corpori major <lb/>impetus in anteriora po&longs;&longs;it imprimi citrà periculum cadendi; <lb/>brachium &longs;iquidem eo pacto in po&longs;teriora projectum, tran&longs;lato <lb/>gravitatis, ut gravitationis, centro &longs;ervat totius corporis æqui­<lb/>librium. </s> </p> <p type="main"> <s id="s.005142">Nec di&longs;&longs;imili ratione manife&longs;tum fit, cur pugnum validiùs <lb/>impingant, qui brachium magis contrahunt, ideóque validio­<lb/>res ictus ab iis proveniant, qui longiora habentes brachia ea <lb/>magis contrahunt; &longs;icut calce fortiùs impetunt, qui longiora <lb/>habent crura; quia videlicet diutiùs moventur, magi&longs;que im­<lb/>petum augent & velocitatem, antè ictum. </s> <s id="s.005143">Sic vides ab equis <lb/>calcitronibus pedem in anteriora retrahi, & ab irato tauro col­<lb/>lum depre&longs;&longs;um in po&longs;teriora inflecti, corpore pariter curvato, <lb/>& qua&longs;i in po&longs;teriora retracto, ut longiore motu validiùs impe­<lb/>tant: hinc qui propior e&longs;t equo calcitranti, minùs læditur, quia <lb/>nondum tantum impetum concepit, quantum longiore motu <lb/>concepi&longs;&longs;et. </s> </p> <p type="main"> <s id="s.005144">Huju&longs;modi percu&longs;&longs;ionibus à potentiâ vivente, quæ &longs;uos mo­<lb/>tus ex arbitrio temperat, provenientibus certam legem &longs;tatui <lb/>non po&longs;&longs;e nemo non videt, &longs;ive corpus percutiens impetu ex­<lb/>trin&longs;ecùs a&longs;&longs;umpto feratur naturâ omnino repugnante, &longs;ive im­<lb/>petus impre&longs;&longs;us cum impetu vi interiore acqui&longs;ito in eumdem <lb/>motum con&longs;pirent. </s> <s id="s.005145">Hoc unum tanquam manife&longs;tò comper­<lb/>tum atque deprehen&longs;um tenemus, quod in longiore motu factâ <lb/>novi impetûs acce&longs;&longs;ione velocitas augetur, & vis percu&longs;&longs;ionis <lb/>e&longs;t major. </s> <s id="s.005146">Hinc &longs;icut quando fi&longs;tucâ cadente pali in terram <lb/>adiguntur, initio illa modicum attollitur, quia exigua &longs;uperan­<lb/>da e&longs;t re&longs;i&longs;tentia, hac autem cre&longs;cente quò altiùs adacti fuerint, <lb/>magis illa attollitur; &longs;ic lignarios fabros clavum in tabulam in­<lb/>figentes, initio quidem breviore mallei motu uti videmus, quem <lb/>deinceps augent, donec demum totâ brachij exten&longs;ione con­<lb/>nitantur, prout re&longs;i&longs;tentiæ incrementa validiore percu&longs;&longs;ione <pb pagenum="697" xlink:href="017/01/713.jpg"/>vinci oportet. </s> <s id="s.005147">Sic ru&longs;ticos ligna findentes altiùs elevare &longs;ecu­<lb/>rim, aut tuditem, quo cuneum percutiant, ob&longs;ervamus, quò <lb/>major e&longs;t &longs;cindendi difficultas, ut auctus impetus velociorem <lb/>motum efficiat, quem percu&longs;&longs;io con&longs;equitur. </s> </p> <p type="main"> <s id="s.005148">Cum itaque duæ velocitates inter &longs;e comparatæ conferri po&longs;­<lb/>&longs;int vel ratione temporis, vel ratione &longs;patij, ita ut vel æqualia <lb/>&longs;patia inæqualibus temporibus, vel inæqualia &longs;patia tempore <lb/>eodem conficiant; illa utique erit major velocitas, quando in <lb/>motu temporis brevitas, & &longs;patij amplitudo con&longs;en&longs;erint. </s> <s id="s.005149">Hinc <lb/>percu&longs;&longs;io contingit validior à corpore, quod multum &longs;patij bre­<lb/>vi tempore decurrat, quàm à corpore conficiente minus &longs;patij <lb/>longiore tempore. </s> <s id="s.005150">Propterea quæ de velocitate dicta &longs;unt, & <lb/>percu&longs;&longs;ionum viribus, &longs;pectatâ diuturnitate motûs, ita intelli­<lb/>genda &longs;unt, ut corpus percutiens vel eodem tempore plus &longs;pa­<lb/>tij, vel breviore tempore æquale &longs;patium, vel breviore tempore <lb/>plus &longs;patij decurrat: hoc enim ex majore impetûs inten&longs;ione <lb/>oritur. <lb/></s> </p> <p type="main"> <s id="s.005151"><emph type="center"/>CAPUT VIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005152"><emph type="center"/><emph type="italics"/>An validior &longs;it ictus Mallei à &longs;itu Verticali ad <lb/>Horizontalem, an verò ab Horizontali ad <lb/>Verticalem de&longs;cendentis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005153">CErtum e&longs;t percu&longs;&longs;iones fieri validiores, quando cum impe­<lb/>tu ab extraneâ potentiâ impre&longs;&longs;o con&longs;entit vis intrin&longs;eca <lb/>ip&longs;ius corporis percutientis motu naturali de&longs;cendentis; ip&longs;um <lb/>enim &longs;uum pariter impetum concipit, quem addit impre&longs;&longs;o: &longs;ic <lb/>&longs;axum, quod ex editâ turri deor&longs;um rectâ projicis, validiùs per­<lb/>cutit, quàm &longs;i illud dimitteres &longs;ponte &longs;ua ca&longs;urum. </s> <s id="s.005154">Hinc qui <lb/>malleo deor&longs;um percutit aliquod corpus, ad motum eundem, <lb/>cum percutientis impul&longs;u con&longs;pirantem invenit mallei gravita­<lb/>tem, & citrà omnem controver&longs;iam majorem ictum infligit, <lb/>quàm &longs;i &longs;ur&longs;um, aut in latus urgeret malleum, gravitate aut re­<lb/>pugnante, aut &longs;altem nihil juvante. </s> </p> <pb pagenum="698" xlink:href="017/01/714.jpg"/> <p type="main"> <s id="s.005155">Porrò cùm circa juncturam brachij cum humero, tanquam <lb/>circa centrum, de&longs;cribatur &longs;emicirculus de&longs;cendens, in quo <lb/>&longs;unt duo Quadrantes, alter cùm brachium &longs;ummè elevatum in <lb/>perpendiculo exi&longs;tens de&longs;cendit, ut fiat horizonti parallelum, <lb/>alterùm brachium à po&longs;itione horizonti parallelâ ad imum de­<lb/>primitur, ut iterum in perpendiculo &longs;tatuatur; dubitari pote&longs;t, <lb/>an mallei ictus juxta duas ha&longs;ce po&longs;itiones &longs;int omnino æquales, <lb/>an verò inæquales; & hoc quidem non ex vi impul&longs;ionis exter­<lb/>næ, quam æquabilem ponimus, homine æqualiter connitente, <lb/>brachio æqualiter extento, & datâ eâdem manubrij longitudi­<lb/>ne, &longs;ed ratione ip&longs;ius gravitatis mallei naturaliter de&longs;cendentis. </s> <lb/> <s id="s.005156">Prioris motus, quo in circulo Verticali malleus deprimitur u&longs;­<lb/>que ad planum horizonti parallelum, in quo fit percu&longs;&longs;io, exem­<lb/>plum præbent tùm fabri ferrarij incudem tundentes, tùm ligna­<lb/>rij clavos a&longs;&longs;eribus in plano horizontali con&longs;titutis infigentes. </s> <lb/> <s id="s.005157">Po&longs;terioris autem motûs, quo malleum horizonti parallelum <lb/>u&longs;que eò deprimimus, ut fiat perpendicularis, &longs;pecimen habe­<lb/>mus, cum corpus in pavimento jacens, aut non procul ab illo, <lb/>ita percutimus, ut percu&longs;&longs;um moveatur horizonti ferè paralle­<lb/>lum; cuju&longs;modi e&longs;&longs;et, quando cuneum jacenti &longs;axo &longs;ubjicere <lb/>conamur, aut ligneam pilam ludentes malleo excutimus. </s> </p> <p type="main"> <s id="s.005158">Ut autem dilucidè propo&longs;ita quæ&longs;tio exponatur, &longs;ecernendus <lb/><figure id="id.017.01.714.1.jpg" xlink:href="017/01/714/1.jpg"/><lb/>e&longs;t mallei motus naturalis ab <lb/>ea parte, quam externa im­<lb/>pul&longs;io addit; & perinde con­<lb/>&longs;iderandus e&longs;t malleus, atque <lb/>&longs;i manubrij extremitas axi in­<lb/>fixa e&longs;&longs;et circa eum ver&longs;atilis, <lb/>adeò ut &longs;ibi relictus malleus <lb/>arcum de&longs;cendendo de&longs;cri­<lb/>beret. </s> <s id="s.005159">Sit malleus AB; & <lb/>manubrij extremitas A &longs;it <lb/>circa axem in A ver&longs;atilis; <lb/>centrum autem gravitatis <lb/>mallei intelligatur in B: quod <lb/>quandiu in perpendiculo im­<lb/>minet axi A, totam &longs;uam vim <lb/>in illum exerens &longs;u&longs;tinetur, <pb pagenum="699" xlink:href="017/01/715.jpg"/>nec motum inchoat, ni&longs;i à perpendiculo BA removeatur; hoc <lb/>verò ubi tran&longs;gre&longs;&longs;us fuerit malleus, de&longs;cen&longs;um molitur: &longs;ed <lb/>quia rigido manubrio connectitur cum Axe A, cogitur in latus <lb/>&longs;ecedere, & de&longs;cribere arcum BC, cui motui re&longs;pondet &longs;olùm <lb/>de&longs;cen&longs;us BD Sinus Ver&longs;us anguli BAC; & de&longs;cripto integro <lb/>Quadrante BE, de&longs;cen&longs;um metitur Radius BA. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005160">Verùm quamvis motus huju&longs;modi per arcum BE &longs;it con&longs;en­<lb/>taneus propen&longs;ioni gravitatis, quæ proinde &longs;ingulis momentis <lb/>novam impetûs particulam concipiens motum, quantum pote&longs;t, <lb/>accelerat, non tamen illa Ratio &longs;ervatur, de qua &longs;uperiori Capi­<lb/>te dictum e&longs;t, ut Ratio &longs;patiorum &longs;it in duplicatâ Ratione tem­<lb/>porum; neque enim hìc liberè de&longs;cendit malleus, &longs;ed in &longs;ingu­<lb/>lis punctis Quadrantis alia atque alia habet momenta de&longs;cen­<lb/>dendi, &longs;ingula minora momento, quod haberet idem malleus <lb/>nullo impedimento prohibitus; quod momentum integrum ille <lb/>obtinet tantummodo in Quadrantis extremitate E, ubi nullâ <lb/>ratione &longs;u&longs;tinetur aut retinetur ab axe A manubrium &longs;u&longs;tinen­<lb/>te, aut retinente. </s> <s id="s.005161">E&longs;t autem momentum gravitatis in unoquo­<lb/>que arcûs puncto, qua&longs;i illa e&longs;&longs;et in plano ibi circulum tangen­<lb/>te, ac propterea inclinato: idcirco, ut lib.1. cap.13. dictum e&longs;t, <lb/>eju&longs;dem gravitatis momentum in plano inclinato, ad momen­<lb/>tum in lineâ perpendiculari eam Rationem habet, quam in <lb/>triangulo rectangulo, cujus angulus Verticalis &longs;it æqualis an­<lb/>gulo inclinationis plani, habet Perpendiculum ad Hypothe­<lb/>nu&longs;am. </s> <s id="s.005162">Quare in puncto C malleus habet momentum, ac &longs;i e&longs;­<lb/>&longs;et in plano inclinato FC, & ad momentum liberum ita &longs;e ha­<lb/>bet, ut DF ad FC, hoc e&longs;t per 8.lib.6. ut DC ad CA, & &longs;i­<lb/>militer in puncto G ut IG ad GA. <!-- KEEP S--></s> <s id="s.005163">Ex quo patet momentorum <lb/>incrementa analoga e&longs;&longs;e incrementis Sinuum Rectorum arcu­<lb/>bus &longs;ubinde majoribus convenientium. </s> </p> <p type="main"> <s id="s.005164">At hìc, ubi de gravitatis momentis &longs;ermo in&longs;tituitur, caven­<lb/>dum e&longs;t, ne quem fortè in errorem inducat ambiguitas nomi­<lb/>nis. </s> <s id="s.005165">Nam quando in C momentum dicimus e&longs;&longs;e ut DC, & in <lb/>G momentum e&longs;&longs;e ut IG, hoc intelligendum e&longs;t præcisè ra­<lb/>tione po&longs;itionis, quatenus in hoc aut illo puncto con&longs;tituta gra­<lb/>vitas concipitur, nullâ habitâ ratione antecedentis motûs aut <lb/>quietis: & &longs;ub voce momenti Gravitatis hæc &longs;ubjecta e&longs;t &longs;en­<lb/>tentia, ut gravitas mallei, quæ non impedita &longs;ingulis punctis <pb pagenum="700" xlink:href="017/01/716.jpg"/>temporis conciperet novum impetum ut AC, AG &c. </s> <s id="s.005166">quia à <lb/>rigido manubrio modò magis, modò minùs &longs;u&longs;tinetur, quando <lb/>e&longs;t in C, impeditur, ne concipiat impetum ni&longs;i ut DC, & in G <lb/>ut IG, atque ita de cæteris, donec in E concipiat impetum ut <lb/>AE. <!-- KEEP S--></s> <s id="s.005167">Cæterùm quia in præcedentibus temporis punctis acqui­<lb/>&longs;itæ &longs;unt particulæ impetûs re&longs;pondentes. </s> <s id="s.005168">Sinubus Rectis præ­<lb/>cedentium arcuum, ea fit impetûs inten&longs;io, ac proinde motûs <lb/>velocitas, quæ omnium illorum Sinuum aggregato ferè re&longs;pon­<lb/>deat: Et in fine Quadrantis in E vis e&longs;t complectens omnes <lb/>impetus, quibus additio facta e&longs;t &longs;emper non tamen æqualis pri­<lb/>mo impetui, qui valdè languidus fuit, &longs;ed &longs;emper major atque <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/>drans con&longs;iderandus e&longs;t, in quo Axis A retinet malleum, ne ex <lb/>E recto tramite de&longs;cendat, &longs;ed eum cogit deflectere, & arcum <lb/>ES de&longs;cribere, in cujus &longs;ingulis punctis momentum perinde e&longs;t <lb/>atque in plano inclinato. </s> <s id="s.005171">Quare in L intelligitur de&longs;cendens <lb/>in plano inclinato KL, & ibi ejus momentum ad momentum <lb/>liberum e&longs;t ut RK ad KL, hoc e&longs;t ut ML, ad LK, hoc e&longs;t per <lb/>8. lib.6. ut MA ad AL, hoc e&longs;t ut Sinus Complementi arcûs <lb/>EL ad Radium. <!-- KEEP S--></s> <s id="s.005172">Similiter in N e&longs;t ut PA ad AN; & &longs;ic de <lb/>cæteris. </s> <s id="s.005173">E&longs;t autem manife&longs;tum huju&longs;modi Sinus Complemen­<lb/>torum eo&longs;dem planè e&longs;&longs;e cum Sinubus Rectis Superioris Qua­<lb/>drantis, &longs;ed ordine præpo&longs;tero acceptis, atque adeò horum ag­<lb/>gregatum e&longs;&longs;e illorum &longs;ummæ æquale. </s> </p> <p type="main"> <s id="s.005174">Non tamen hinc &longs;tatim conficitur eandem e&longs;&longs;e in S vim mal­<lb/>lei de&longs;cendentis ex E, atque e&longs;t in E vis eju&longs;dem de&longs;cendentis <lb/>ex B. </s> <s id="s.005175">Non inficior æqualem in utroque Quadrante produci <lb/>impetûs entitatem, &longs;i in &longs;ummam referantur omnes impetûs <lb/>particulæ, quæ momentis &longs;ingulis efficiuntur; &longs;ed an æqualem <lb/>demum conflent inten&longs;ionem, ex qua vis percu&longs;&longs;ionis oritur, <lb/>non omninò temerè, ut mihi quidem videor, &longs;ubdubito. </s> <s id="s.005176">Cùm <lb/>enim acqui&longs;itus impetus interveniente re&longs;i&longs;tentiâ imminuatur, <lb/>ac debilitetur, & quidem eò magis, &longs;i à rectâ &longs;ecundùm natu­<lb/>ram lineâ magis declinare cogitur; utique in&longs;titutâ Superioris <lb/>cum Inferiori Quadrante comparatio o&longs;tendit in illo quidem <lb/>re&longs;i&longs;tentiam &longs;emper decre&longs;cere, in hoc &longs;emper augeri, ac proin­<lb/>de impetum prioribus momentis acqui&longs;itum, licèt aliquid in <pb pagenum="701" xlink:href="017/01/717.jpg"/>con&longs;equentibus amittat, tamen hujus decrementi men&longs;urâ ma­<lb/>gis ac magis extenuatâ, non adeò in Superiore Quadrante lan­<lb/>gue&longs;cere, &longs;icut in inferiore, ubi re&longs;i&longs;tentia &longs;emper augetur, & <lb/>impetus magis ac magis deteritur. </s> <s id="s.005177">Adde impetum de novo <lb/>productum in po&longs;terioribus momentis, in &longs;uperiore quidem <lb/>Quadrante e&longs;&longs;e majorem, in inferiore verò minorem. </s> <s id="s.005178">Quare <lb/>cùm in po&longs;tremis motûs momentis in &longs;uperiore Quadrante <lb/>multus producatur impetus, & ferè nulla &longs;it re&longs;i&longs;tentia, in in­<lb/>feriore autem Quadrante multa inveniatur re&longs;i&longs;tentia, & valdè <lb/>exiguus impetus producatur, &longs;atis probabili conjecturâ ali­<lb/>quam &longs;tatuemus ictuum inæqualitatem, ita ut aliquanto vali­<lb/>dior &longs;it ex B in E, quàm ex E in S. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005179">Neque ob&longs;tat, quod in E maximus producatur impetus, qui <lb/>deinde in L & N, atque con&longs;equentibus punctis additamen­<lb/>tum accipiat, licèt &longs;emper minus ac minus, quo ita augetur, ut <lb/>&longs;emper incitetur motus. </s> <s id="s.005180">Hoc enim non facit, quin impetus in E <lb/>conceptus majoribus &longs;emper decrementis imminuatur u&longs;que <lb/>in S, & impetus in L conceptus &longs;imiliter magis langue&longs;cat, at­<lb/>que ita de cæteris. </s> <s id="s.005181">Finge &longs;cilicet nullum impetum novum con­<lb/>cipi in L, aut nullum in N, adhuc impetus in E conceptus deor­<lb/>&longs;um tenderet, &longs;ed retinaculo illo debilitatus languidiùs adduce­<lb/>ret malleum in S: idem dic de quolibet impetu &longs;ingulis mo­<lb/>mentis concepto, qui cre&longs;cente re&longs;i&longs;tentiâ majoribus decre­<lb/>mentis imminueretur, & languidè veniret in S. <!-- KEEP S--></s> <s id="s.005182">At quoniam <lb/>plurima &longs;unt momenta in brevi&longs;&longs;imi temporis particulâ, tot <lb/>&longs;unt reliqui impetus, ut &longs;imul con&longs;tituant notabilem inten­<lb/>&longs;ionem. </s> </p> <p type="main"> <s id="s.005183">Quod autem re&longs;i&longs;tentia in &longs;uperiore Quadrante minuatur, <lb/>argumento non e&longs;t opus; manife&longs;tum quippe e&longs;t gravitatem in <lb/>arcu BE de&longs;cendentem &longs;ubinde transferri à plano magis incli­<lb/>nato in minùs inclinatum, & magis accedens ad perpendicu­<lb/>lare: quis autem neget de&longs;cendenti gravitati eò minùs ob&longs;tare <lb/>planum &longs;ubjectum, quò fuerit minùs inclinatum? </s> <s id="s.005184">Atqui an­<lb/>gulus CAF minor e&longs;t angulo GAH, anguli autem ad C & G <lb/>&longs;unt recti; igitur angulus AFC major e&longs;t angulo AHG; at­<lb/>que propterea planum FC e&longs;t magis inclinatum, quàm pla­<lb/>num HG, & cætera plana con&longs;equentia u&longs;que ad planum per­<lb/>pendiculare in E. <!-- KEEP S--></s> <s id="s.005185">Contra verò in inferiore Quadrante re&longs;i&longs;ten-<pb pagenum="702" xlink:href="017/01/718.jpg"/>tiam &longs;emper augeri ex eo con&longs;tat, quòd à plano perpendicula­<lb/>ri ad inclinatum, immò ad &longs;emper magis atque magis inclina­<lb/>tum, fit tran&longs;itus, donec demum de&longs;cendens gravitas veniat ad <lb/>planum horizontale. </s> <s id="s.005186">Angulus videlicet inclinationis plani e&longs;t <lb/>æqualis angulo, quem denotat arcus in inferiore Quadrante <lb/>decur&longs;us. </s> <s id="s.005187">Nam in triangulo ALK, cujus in ba&longs;im AK cadit <lb/>perpendicularis LM, ex 8.lib.6. angulo KAL æqualis e&longs;t an­<lb/>gulus KLM, & propter paralleli&longs;mum linearum ML & KR, <lb/>angulo KLM æqualis e&longs;t alternus LKR angulus inclinationis <lb/>plani KL; igitur hic angulus inclinationis e&longs;t æqualis angulo, <lb/>quem denotat arcus EL. <!-- KEEP S--></s> <s id="s.005188">Idem dic de angulo EAN & cæte­<lb/>ris, qui &longs;emper majores indicant gravitatem tran&longs;ire ad plana <lb/>magis & magis inclinata. <lb/></s> </p> <p type="main"> <s id="s.005189"><emph type="center"/>CAPUT IX.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005190"><emph type="center"/><emph type="italics"/>Quomodo percu&longs;siones ex mole pendeant.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005191">QUamvis ad validiorem ictum infligendum corporis percu­<lb/>tientis velocitas, impetûs inten&longs;ionem indicans, pluri­<lb/>mum conferat, ut dictum e&longs;t; non ad hanc tamen velut ad uni­<lb/>cam cau&longs;am referenda e&longs;t vis percu&longs;&longs;ionis; &longs;ed & corporis eju&longs;­<lb/>dem percutientis moles attendenda e&longs;t: videmus &longs;cilicet ex <lb/>mole ipsâ percu&longs;&longs;iones augeri, cæteris paribus; perinde enim <lb/>e&longs;t, atque &longs;i tot corpora percutientia e&longs;&longs;ent, quàm multiplex <lb/>e&longs;t moles major collata cum minore. </s> <s id="s.005192">Nam &longs;i nota e&longs;t vis per­<lb/>cutiendi, quæ ine&longs;t globulo duarum unciarum cadenti ex cer­<lb/>tâ quadam altitudine utique probabilis conjectura & ratio &longs;ua­<lb/>det &longs;extuplam e&longs;&longs;e vim globi ex &longs;imili materiâ unciarum duo­<lb/>decim ex altitudine eâdem cadentis: gravitas &longs;iquidem &longs;exies <lb/>multiplicata etiam impetum efficere pote&longs;t &longs;extuplum, non qui­<lb/>dem inten&longs;ivè, &longs;ed entitativè; neque enim pro Ratione molis <lb/>augetur velocitas, quippe quæ requireret &longs;extuplam inten&longs;io­<lb/>nem. </s> <s id="s.005193">Hanc tamen majorem vim Ratione molis ita intelligi <lb/>velim, ut medij re&longs;i&longs;tentia di&longs;&longs;imuletur: nam eo ip&longs;o, quod mo-<pb pagenum="703" xlink:href="017/01/719.jpg"/>les eju&longs;dem &longs;ecundùm &longs;peciem gravitatis augetur, etiam &longs;uper­<lb/>ficiem augeri nece&longs;&longs;e e&longs;t, quæ non eâdem facilitate medium di­<lb/>vidit. </s> <s id="s.005194">Sed quoniam ita augeri pote&longs;t moles, ut non &longs;imilem <lb/>&longs;ervet figuram, &longs;ed alias atque alias induat figuras manente <lb/>æquali gravitate, ac propterea valde incerta e&longs;t re&longs;i&longs;tentiæ <lb/>men&longs;ura, quæ ex medij divi&longs;ione oritur, prout hanc aut illam <lb/>faciem medio dividendo obvertit ip&longs;a moles; hinc e&longs;t, quod il­<lb/>lam re&longs;i&longs;tentiam tanti&longs;per di&longs;&longs;imulare licet, dum reliquas per­<lb/>cu&longs;&longs;ionis cau&longs;as ve&longs;tigamus. </s> <s id="s.005195">Cæterùm illius quoque ratio e&longs;t <lb/>habenda, ut aliquid virium detractum intelligatur percu&longs;&longs;ioni, <lb/>ubi & moles & &longs;patium con&longs;ideratur; quatenus velocitas, auctâ <lb/>mole, hoc e&longs;t aucto ex medij re&longs;i&longs;tentia impedimento, aliquan­<lb/>tulum imminuitur, ita tamen ut pariter plures majoris molis <lb/>partes, collatis viribus medium urgentes, aliquid afferant faci­<lb/>litatis in dividendo medio, adeóque etiam velocitatis. </s> </p> <p type="main"> <s id="s.005196">Ex his habetur percu&longs;&longs;ionis vires componi ex mole & ex ve­<lb/>locitate corporis percutientis: Moles &longs;iquidem determinat en­<lb/>titativè men&longs;uram impetûs &longs;ingulis momentis producti, veloci­<lb/>tas indicat inten&longs;ionem, hoc e&longs;t &longs;ummam impetuum in motu <lb/>acqui&longs;itorum, hoc e&longs;t eju&longs;dem impetûs gravitati, aut potentiæ <lb/>virtuti, primo momento re&longs;pondentis multiplicationem. </s> <s id="s.005197">Qua­<lb/>re &longs;i duorum corporum ictus comparentur, percu&longs;&longs;ionum Ratio <lb/>erit compo&longs;ita ex Rationibus velocitatum, & gravitatum, &longs;eu <lb/>potentiarum, quibus vis movendi tribuitur. </s> <s id="s.005198">Velocitatem au­<lb/>tem illam intelligo, quæ ratione impetus acqui&longs;iti conveniret <lb/>corpori eo momento, quo percutit, ni&longs;i inveniret re&longs;i&longs;tentiam: <lb/>Huju&longs;modi verò impetûs eo momento inten&longs;ionem indicat &longs;pa­<lb/>tium in antecedenti motu decur&longs;um, ex quo, prout dictum e&longs;t <lb/>cap.7. cogno&longs;citur Ratio temporum, quibus e&longs;t analoga inten­<lb/>&longs;io impetûs. </s> <s id="s.005199">Hinc &longs;i cadat ex altitudine 100 palmorum globus <lb/>unciarum duarum, deinde ex altitudine decem palmorum glo­<lb/>bus unciarum 12, &longs;unt duæ Rationes, altera velocitatum, hoc e&longs;t <lb/>inten&longs;ionum impetûs in &longs;ubduplicatâ Ratione altitudinum, vi­<lb/>delicet ut 10 ad (3 16/100), altera gravitatum ut 1 ad 6; quæ compo&longs;i­<lb/>tæ dant Rationem ut 10 ad (18 96/100); ac proinde percu&longs;&longs;io globi <lb/>minoris e&longs;timari poterit proximè ut 10, majoris ut (18 96/100): Nam <lb/>duæ unciæ majoris de&longs;cendendo per decem palmos haberent <pb pagenum="704" xlink:href="017/01/720.jpg"/>impetum ut (3 16/100): ergo &longs;exies duæ unciæ habent entitativè im­<lb/>petum ut (18 96/100). Quare &longs;i gravitates fuerint reciprocè ut velo­<lb/>citates, hoc e&longs;t impetûs inten&longs;iones, erunt æquales percu&longs;&longs;io­<lb/>nes, ut e&longs;t manife&longs;tum. </s> </p> <p type="main"> <s id="s.005200">Hoc autem, quod de gravitate motu naturali de&longs;cendente <lb/>dictum e&longs;t, de potentiis pariter, &longs;ervatâ analogiâ, e&longs;t intelligen­<lb/>dum (a&longs;&longs;umendo &longs;cilicet loco gravitatis vim ip&longs;am potentiæ, <lb/>quæ &longs;imiliter conata per&longs;everet in motu antecedente percu&longs;&longs;io­<lb/>nem) &longs;i innote&longs;cat, quantum potentiæ inæqualiter conentur, <lb/>& per inæqualia &longs;patia moveant idem corpus, quo ictum infli­<lb/>gunt; nam ex Rationibus conatuum, & velocitatum componi­<lb/>tur Ratio percu&longs;&longs;ionum. </s> <s id="s.005201">At &longs;i potentiæ duæ inæqualiter co­<lb/>nantes per inæqualia &longs;patia moveant inæqualia corpora, qui­<lb/>bus alteri corpori ictus infligatur, attendendum e&longs;t, an &longs;olùm <lb/>tam diuturnus &longs;it motus ictum præcedens, ut nihil impre&longs;&longs;i im­<lb/>petûs deteratur: nam &longs;i alternis quibu&longs;dam incrementis & de­<lb/>crementis modò augeatur, modò minuatur, non e&longs;t habenda <lb/>ratio totius temporis, aut &longs;patij, in quo factus e&longs;t motus: quis <lb/>enim exi&longs;timet aptè computari po&longs;&longs;e, utrùm navis percurrerit <lb/>&longs;ex, aut octo milliaria, ut ejus ictus, quo cymbam percutit, <lb/>cogno&longs;catur? </s> <s id="s.005202">Quare &longs;atius erit in huju&longs;modi motibus ab impe­<lb/>tu extrin&longs;ecùs impre&longs;&longs;o provenientibus, qui ut plurimum re­<lb/>pugnantem habet ip&longs;ius corporis naturam, nec totus permanet <lb/>quemadmodum impetus acqui&longs;itus, ip&longs;am velocitatem con&longs;i­<lb/>derare, quatenus apparet non multo tempore antè ictum: tunc <lb/>enim, quia potentia movens eum impetum imprimit, qui &longs;atis <lb/>&longs;it ad molem illam movendam tantâ velocitate, ut impetus hu­<lb/>ju&longs;modi innote&longs;cat, & molis & velocitatis ratio habenda e&longs;t; <lb/>atque idcircò ad comparandas invicem percu&longs;&longs;iones compo­<lb/>nenda e&longs;t Ratio ex Rationibus molium, & velocitatum. </s> <s id="s.005203">Hinc <lb/>&longs;i navis oneraria lentè moveatur velocitate ut duo, & navis alia <lb/>&longs;extuplo minor moveatur velocitate ut decem (quia videlicet <lb/>paulò antè ictum ob&longs;ervatum e&longs;t, quo tempore illa procedebat <lb/>duos pa&longs;&longs;us, hanc percurri&longs;&longs;e decem pa&longs;&longs;us) ictus majoris ad <lb/>ictum minoris erit ut 12 ad 10, compo&longs;itis &longs;cilicet Ratione mo­<lb/>lium 6 ad 1, & Ratione velocitatum 2 ad 10. </s> </p> <p type="main"> <s id="s.005204">Hìc autem ubi Molis nomen u&longs;urpamus, cavendus e&longs;t in <pb pagenum="705" xlink:href="017/01/721.jpg"/>vocabuli ambiguitate lap&longs;us: neque enim corporis tantummo­<lb/>do amplitudinem, quatenus &longs;ub Geometricam dimen&longs;ionem <lb/>cadens &longs;patium occupat, intelligere oportet, verùm etiam na­<lb/>turam ip&longs;am atque &longs;ub&longs;tantiam: ea &longs;cilicet, quæ minore &longs;ecun­<lb/>dùm &longs;peciem gravitate prædita &longs;unt, &longs;ub magnis dimen&longs;ionibus <lb/>parum habent &longs;ub&longs;tantiæ atque materiæ, ideóque & tenuem <lb/>movendi ac impetum producendi virtutem; neque propterea <lb/>quod molem magnam præ&longs;eferant, validiora in percutiendo <lb/>cen&longs;enda &longs;unt, qua&longs;i à globo ligneo librarum duarum, quia fe­<lb/>rè decuplo major e&longs;t globo plumbeo eju&longs;dem ponderis, expecta­<lb/>ri po&longs;&longs;et validior ictus, &longs;i ex eâdem altitudine dimittantur: nam <lb/>vis producendi impetum connata e&longs;t &longs;ub&longs;tantiæ, quâ &longs;ub&longs;tantia <lb/>talis e&longs;t, non quantitati, prout exten&longs;io e&longs;t. </s> <s id="s.005205">Quare ubi molis <lb/>habendam e&longs;&longs;e rationem diximus, ut fiat Rationum Compo&longs;i­<lb/>tio, ex qua percu&longs;&longs;ionum incrementa aut decrementa inno­<lb/>te&longs;cant, ip&longs;am poti&longs;&longs;imùm &longs;ub&longs;tantiam intelligimus, quam, non <lb/>ni&longs;i intra idem genus corporis, exten&longs;io major aut minor con&longs;e­<lb/>qui &longs;olet: propterea &longs;i ferrei cylindri ictus, atque lignei, confer­<lb/>re invicem volueris, non ip&longs;os cylindros, quatenus cylindri &longs;unt <lb/>&longs;ub tantâ ba&longs;i & altitudine, dimetiri oportet, &longs;ed potiùs eorum <lb/>gravitatem, ut quanta &longs;it moles virtutis ip&longs;i naturæ atque &longs;ub­<lb/>&longs;tantiæ re&longs;pondens, ex gravitate inferatur. </s> </p> <p type="main"> <s id="s.005206">Non tamen idcircò exten&longs;ionis atque figuræ animadver&longs;io <lb/>otio&longs;a e&longs;t, aut contemnenda, in percu&longs;&longs;ionibus; quinimmò non <lb/>o&longs;citanter con&longs;ideranda, ut deprehendatur, qua &longs;ui corporis <lb/>parte validi&longs;&longs;imum ictum infligat in&longs;trumentum percutiens. </s> <lb/> <s id="s.005207">Hoc autem tripliciter poti&longs;&longs;imùm movetur, videlicet, primò ad <lb/>perpendiculum de&longs;cendendo motu naturali; deinde horizonta­<lb/>liter, &longs;eu obliquè, cùm à dextrâ in &longs;ini&longs;tram, aut vici&longs;&longs;im à lævâ <lb/>in dexteram, aut in anteriora extrin&longs;ecùs motu recto impelli­<lb/>tur; demum in orbem, circuli arcum de&longs;cribendo. </s> </p> <p type="main"> <s id="s.005208">Et quidem corpus &longs;ponte &longs;uâ de&longs;cendens, quodcumque tan­<lb/>dem illud &longs;it, &longs;uam habet Directionis lineam, per quam in mo­<lb/>tu Centrum gravitatis progreditur. </s> <s id="s.005209">In infimâ igitur corporis <lb/>parte ip&longs;a Directionis linea definit punctum, in quo &longs;i fiat cor­<lb/>poris percutientis contactus, ille erit validi&longs;&longs;imus ictus, quem <lb/>huju&longs;modi corpus ex datâ altitudine de&longs;cendens infligere po­<lb/>te&longs;t, ibi quippe maximam reperit re&longs;i&longs;tentiam, cum æquales <pb pagenum="706" xlink:href="017/01/722.jpg"/>vires hinc atque hinc con&longs;i&longs;tentes ibi con&longs;pirent, & obicem <lb/>motui directè oppo&longs;itum offendant, adeò ut neque in hanc, <lb/>neque in illam partem Centrum gravitatis dirigatur. </s> <s id="s.005210">Quod &longs;i <lb/>punctum contactûs corporum colli&longs;orum non &longs;it in lineâ Di­<lb/>rectionis corporis cadentis, &longs;ed à latere; eò validior erit ictus, <lb/>quo minore intervallo punctum contactûs ab huju&longs;modi lineâ <lb/>Directionis aberit; magis videlicet opponitur motui directo, <lb/>quàm &longs;i ab eâ longiùs abe&longs;&longs;et: quando enim contactus procul <lb/>e&longs;t à lineâ Directionis, ab hac minùs deflectere cogitur cen­<lb/>trum gravitatis, quod multò magis repellendum e&longs;&longs;et à contactu <lb/><figure id="id.017.01.722.1.jpg" xlink:href="017/01/722/1.jpg"/><lb/>propiore. </s> <s id="s.005211">Sic globus, cujus centrum gra­<lb/>vitatis &longs;it C, de&longs;cendens per lineam Di­<lb/>rectionis CD, &longs;i percutiat puncto D, om­<lb/>nium validi&longs;&longs;imum ictum infligit, quia <lb/>corpus percu&longs;&longs;um omninò opponitur mo­<lb/>tui CD, nec centro C relinquit locum <lb/>&longs;altem obliquè de&longs;cendendi: at verò &longs;i <lb/>contactus fiat in E, impeditur quidem <lb/>de&longs;cen&longs;us globi per rectam CD ulteriùs <lb/>productam, pote&longs;t tamen centrum gravitatis de&longs;cendere de&longs;cri­<lb/>bendo circa punctum E manens arcum CF; quapropter in E <lb/>minorem invenit re&longs;i&longs;tentiam quàm in D, ubi nihil de&longs;cende­<lb/>re pote&longs;t, &longs;i &longs;ubjectum corpus loco non cedat. </s> <s id="s.005212">Similiter &longs;i con­<lb/>tactus fiat in G, adhuc impeditur motus directus per CD, atta­<lb/>men centrum gravitatis C pote&longs;t obliquè de&longs;cendere de&longs;cri­<lb/>bendo arcum CH. <!-- KEEP S--></s> <s id="s.005213">Sed quoniam per arcum CF magis decli­<lb/>nat à perpendiculo, & minùs de&longs;cendit, quàm per arcum CH <lb/>(quamvis arcus illi æquales ponantur paribus ℞adiis EC, & <lb/>GC de&longs;cripti) propterea magis impeditur motus in contactu E <lb/>propiori lineæ Directionis, quàm in G remotiori. </s> <s id="s.005214">Cum itaque <lb/>eò validiorem ictum infligant corpora percutientia, quò ma­<lb/>jorem inchoato motui re&longs;i&longs;tentiam offendunt, manife&longs;tum e&longs;t <lb/>in corporibus naturali motu de&longs;cendentibus validi&longs;&longs;imum e&longs;&longs;e <lb/>ictum in puncto, quod lineæ directionis motûs re&longs;pondet, &longs;em­<lb/>pérque imbecilliores e&longs;&longs;e ictus, quò magis puncta contactûs ab­<lb/>&longs;unt à lineâ Directionis. </s> </p> <p type="main"> <s id="s.005215">Hoc idem, quod de lineâ Directionis gravium &longs;ponte &longs;uâ <lb/>de&longs;cendentium dictum e&longs;t, analogiâ &longs;ervatâ, traducendum e&longs;t <pb pagenum="707" xlink:href="017/01/723.jpg"/>ad ea corpora, quæ externo impul&longs;u agitata motu recto &longs;ivè <lb/>Horizonti parallelo, &longs;ivè ad Horizontem aut obliquè, aut ad <lb/>perpendiculum, inclinato moventur. </s> <s id="s.005216">Cum enim, ex hypo­<lb/>the&longs;i, partes omnes huju&longs;modi corporis impul&longs;i æquali veloci­<lb/>tate per æqualia &longs;patia moveantur, æqualem impetum &longs;ingulæ <lb/>recipiunt, à movente impre&longs;&longs;um. </s> <s id="s.005217">Similiter igitur in corpore <lb/>illo concipiendum e&longs;t punctum, quod <emph type="italics"/>Centrum Impetûs<emph.end type="italics"/> vocari <lb/>pote&longs;t, quia illud æquales hinc & hinc Impetus circum&longs;tant, <lb/>quemadmodum Centrum Gravitatis dicitur, circa quod æqua­<lb/>lia gravitatis momenta di&longs;po&longs;ita intelliguntur. </s> <s id="s.005218">Hinc &longs;i corpo­<lb/>ris particulæ fuerint omnino homogeneæ, adeóque æquè capa­<lb/>ces impetûs recipiendi, illud idem erit Centrum Impetûs, quod <lb/>e&longs;t centrum molis, &longs;eu magnitudinis; nam eadem plana, quæ <lb/>molem æqualiter dividunt, etiam æqualiter dividunt Impetum <lb/>per &longs;ingulas particulas æquabiliter diffu&longs;um. </s> <s id="s.005219">At &longs;i non eju&longs;dem <lb/>generis fuerint partes corpus illud componentes, &longs;ed raræ aliæ, <lb/>aliæ den&longs;æ, hoc e&longs;t ex materiâ partim tenui, partim con&longs;tipatâ, <lb/>&longs;icut non e&longs;&longs;et idem Centrum Gravitatis, atque Centrum <lb/>Magnitudinis, ita neque idem e&longs;t cum Molis centro Centrum <lb/>Impetûs impre&longs;&longs;i; quia, ut ex Projectis con&longs;tat, ea quæ &longs;ecun­<lb/>dùm &longs;peciem leviora &longs;unt, cæteris paribus, minorem impetum <lb/>concipiunt (& globuli ex argillâ efficti, quos bali&longs;tæ evibrant, <lb/>majorem ictum infligunt, quàm pares globuli lignei, qui &longs;unt <lb/>argillâ leviores) ac proinde Centrum Impetûs impre&longs;&longs;i a&longs;&longs;umi <lb/>pote&longs;t idem, ac punctum illud, quod in motu naturali e&longs;&longs;et <lb/>Centrum gravitatis ip&longs;i corpori inexi&longs;tens. </s> </p> <p type="main"> <s id="s.005220">Quare in motibus corporum externâ vi impul&longs;orum atten­<lb/>denda e&longs;t pariter linea, &longs;ecundùm quam dirigitur motus huju&longs;­<lb/>modi Centri Impetûs: & punctum illud in corporis percutien­<lb/>tis &longs;uperficie, quod linea directionis motûs à Centro Impetûs <lb/>ducta de&longs;ignat, ip&longs;um e&longs;t, in quo corpus percutiens vim &longs;uam <lb/>validi&longs;&longs;imè exercet. </s> <s id="s.005221">Cum enim omnia plana per hanc Di­<lb/>rectionis motûs lineam tran&longs;euntia (quorum illa e&longs;t communis <lb/>&longs;ectio) dividant univer&longs;um Impetum in partes hinc & hinc <lb/>æquales, quippe quæ etiam per Centrum Impetûs tran&longs;eunt, <lb/>ita ex percu&longs;&longs;ione in puncto illo impeditur motus, ut neque ad <lb/>hanc, neque ad illam partem deflectere po&longs;&longs;it corpus impactum <lb/>in obicem, qui re&longs;i&longs;tit. </s> <s id="s.005222">Quod &longs;i punctum contactûs fuerit ex-<pb pagenum="708" xlink:href="017/01/724.jpg"/>tra lineam Directionis motûs, inæquales &longs;unt impetus, & majo­<lb/>re præpollente, corpus pergit in motu, quamvis ad latus in­<lb/>flectatur &longs;ivè magis, &longs;ivè minùs, prout majus aut minus fuerit <lb/>intervallum inter punctum contactûs, & lineam Directionis <lb/>motûs. </s> </p> <p type="main"> <s id="s.005223">Sit corpus AB, quod tran&longs;latum à potentiâ impellente ha­<lb/>beat Centrum Impetûs C, & linea, per quam dirigitur motus, <lb/><figure id="id.017.01.724.1.jpg" xlink:href="017/01/724/1.jpg"/><lb/>&longs;it CD, cui parallelæ <lb/>&longs;unt lineæ à &longs;ingulis par­<lb/>tibus in motu de&longs;criptæ. </s> <lb/> <s id="s.005224">Si ergo in obicem incur­<lb/>rat punctum D, ita im­<lb/>peditur motus, ut ulte­<lb/>riùs promoveri nequeat <lb/>corpus, ni&longs;i obex loco cedat; quia nimirum impetus in DA <lb/>æqualis e&longs;t impetui in DB, ideò neutra pars æquali impetu af­<lb/>fecta promoveri pote&longs;t: e&longs;t igitur maxima re&longs;i&longs;tentia, & ictus <lb/>validi&longs;&longs;imus. </s> <s id="s.005225">Sin autem non puncto. </s> <s id="s.005226">D, &longs;ed puncto E fiat per­<lb/>cu&longs;&longs;io &longs;ecundùm eandem directionem GE, jam impetus &longs;unt <lb/>inæquales, & minor impetus e&longs;t in EA, quàm in EB; proin­<lb/>de pars EB validior pergens in motu inflectitur circa obicem <lb/>in puncto E, tanquam circa centrum, & re&longs;i&longs;tentia e&longs;t minor, <lb/>quàm ad punctum D. <!-- KEEP S--></s> <s id="s.005227">Simile quid contingit, fi fiat percu&longs;&longs;o <lb/>in puncto F, multo enim major impetuum inæqualitas interce­<lb/>dit inter FA, & FB, quàm inter EA, & EB, atque faciliùs <lb/>fit conver&longs;io & inflexio motûs circa obicem in puncto F, quàm <lb/>in puncto E: arcus &longs;iquidem majore Radio FD de&longs;criptus mi­<lb/>nùs de&longs;ci&longs;cit à rectitudine lineæ, per quam dirigitur motus, <lb/>quàm arcus minore Radio ED de&longs;criptus. </s> <s id="s.005228">Quò igitur magis <lb/>punctum contactûs in percu&longs;&longs;ione abe&longs;t à puncto D, eò infir<lb/>mior e&longs;t ictus, minorem quippe invenit re&longs;i&longs;tentiam. </s> </p> <p type="main"> <s id="s.005229">At &longs;i corpus idem AB ita impellatur, ut linea directionis <lb/>motûs ducta ex C centro impetûs &longs;it CA, &longs;imiliter con&longs;tat va­<lb/>lidi&longs;&longs;imum ictum fieri in A, imbecilliorem verò in extremis an­<lb/>gulis eju&longs;dem &longs;uperficiei. </s> <s id="s.005230">Hinc vides, cur ex vetere di&longs;ciplinâ <lb/>Poliorceticâ ad murorum, aut po&longs;tium expugnationem, arie­<lb/>tes, quibus concutiebantur, non planâ facie, &longs;ed convexâ com­<lb/>muniter, aut acutâ con&longs;truerentur: quia &longs;cilicet trabem ferro <pb pagenum="709" xlink:href="017/01/725.jpg"/>in capite armatam funibus &longs;u&longs;pen&longs;am (ne &longs;u&longs;tinendi laborem <lb/>&longs;ubirent, &longs;ed vires omnes in motu impenderent) retro ducen­<lb/>tes, ac deinde propellentes, non planè horizontaliter, &longs;ed qua­<lb/>&longs;i circulariter movebant; planum autem &longs;i fui&longs;&longs;et trabis caput, <lb/>ictus inflictus fui&longs;&longs;et ab extremo illius &longs;uperficiei latere, non ve­<lb/>rò à partibus circa medium exi&longs;tentibus, à quibus multò vali­<lb/>dior ictus expectari potui&longs;&longs;et; quemadmodum certiùs contingit <lb/>facie convexâ, aut in apicem de&longs;inente. </s> </p> <p type="main"> <s id="s.005231">Si demum linea directionis motûs e&longs;&longs;et CF, utique in F e&longs;­<lb/>&longs;et validi&longs;&longs;imus ictus, quia planum FH bifariam divideret <lb/>æqualiter univer&longs;um impetum, & impetus FAH æqualis e&longs;&longs;et <lb/>impetui FBH. </s> <s id="s.005232">E&longs;&longs;et autem infirmior ictus, quem infligeret <lb/>punctum D, cujus directio DI parallela directioni Centri FH <lb/>inæqualiter divideret impetum, & pars impetûs DBI minor <lb/>e&longs;&longs;et parte DAHI; quapropter hæc circa obicem in D mo­<lb/>veri po&longs;&longs;et, & minorem inveniret re&longs;i&longs;tentiam quàm in F. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005233">Ubi ob&longs;ervandum e&longs;t non aptè quæri, quonam in puncto va­<lb/>lidi&longs;&longs;imus fiat ictus, ni&longs;i pariter &longs;tatuatur, quænam &longs;it linea Di­<lb/>rectionis motûs: Nam in eodem puncto D validi&longs;&longs;imus e&longs;t ictus, <lb/>&longs;i directio fuerit CD, quia tunc e&longs;t maxima re&longs;i&longs;tentia; nullus <lb/>e&longs;t ictus in directione CA, quia nihil illi opponitur; imbecillis <lb/>e&longs;t ictus in Directione CF, quia mediocrem offendit re&longs;i&longs;ten­<lb/>tiam. </s> <s id="s.005234">Præterea comparatis invicem Directionibus CD & CA, <lb/>validior e&longs;t ictus in A quàm in D; plures &longs;iquidem partes in <lb/>eandem longitudinis lineam directè con&longs;pirantes plus obtinent <lb/>virium, quàm pauciores in lineâ latitudinis: præterquam quod <lb/>partium ad latera adjacentium lineæ, quæ propiores &longs;unt lineæ <lb/>Directionis Centri Impetûs, qua&longs;i in unam Phy&longs;icè coale&longs;cunt; <lb/>id quod non contingit partibus notabili intervallo disjunctis ab <lb/>illâ Directionis lineâ: quæ eatenus &longs;olùm in percu&longs;&longs;ionem con­<lb/>&longs;entiunt, quatenus cum intermediis conjunctæ nexu non facilè <lb/>di&longs;&longs;olubili eas pariter juvant; nam &longs;i e&longs;&longs;et corpus percutiens in <lb/>plures partes, ceu virgulas, di&longs;&longs;ectum, iis, quæ obicem con­<lb/>tingerent, manentibus, reliquæ &longs;inè ictu excurrerent: propterea <lb/>etiam conjunctæ faciliùs à directâ po&longs;itione deflectentes corpo­<lb/>ris longitudinem inflectunt: ut in tenui & gracili ligno accide­<lb/>re pote&longs;t extremis partibus A & B, quæ ex impul&longs;u, quo pro­<lb/>moventur, po&longs;&longs;unt circa punctum D inflecti; ex quo infirmior <pb pagenum="710" xlink:href="017/01/726.jpg"/>percu&longs;&longs;io, quàm cùm tanta e&longs;t corporis cra&longs;&longs;ities, ut pro longi­<lb/>tudine breviore non valeat flecti. </s> <s id="s.005235">At &longs;i cum his directionibus <lb/>CD & CA, ad perpendiculum incidentibus in faciem corpo­<lb/>ris percutientis, comparetur Directionis linea CF obliquè in­<lb/>cidens, licèt longior &longs;it linea CF, quàm CD, non idcirco va­<lb/>lidior e&longs;t in F ictus Directionis CF, quàm in D ictus Directio­<lb/>nis CD: id quod oritur ex minore re&longs;i&longs;tentiâ ratione obliqui­<lb/>tatis; faciliùs quippe pote&longs;t ulteriùs excurrere corpus obliquè <lb/>percutiens, quàm &longs;i directè percuteret. </s> </p> <p type="main"> <s id="s.005236">Porrò non levis error obreperet minùs accuratè perpenden­<lb/>tibus ea, quæ hactenus de Centro Impetûs di&longs;putata &longs;unt, &longs;i hoc <lb/>Centrum ab&longs;olutè in eo in&longs;trumento, quo ad percutiendum <lb/>utimur, quærendum e&longs;&longs;e exi&longs;timarent: non enim rarò etiam <lb/>ejus, à quo in&longs;trumentum impellitur, con&longs;iderandus e&longs;t impe­<lb/>tus & motus. </s> <s id="s.005237">Sic quando duo lanceis concurrunt, non e&longs;t æ&longs;ti­<lb/>manda percu&longs;&longs;io ex &longs;olo impetu lanceæ impre&longs;&longs;o, verùm etiam <lb/>ex eo, quem militis corpori imprimit equus, cui currenti in&longs;i­<lb/>det, immò & ip&longs;ius equi impetus, quem virtute &longs;uâ animali <lb/>concipit: univer&longs;um quippe hunc impetum retundi oportet <lb/>ab eo, qui ictum recipit: hinc &longs;i miles minùs robu&longs;tus fuerit, <lb/>infirmior e&longs;t ictus, quia ip&longs;o ictûs momento ille cedit, & perin­<lb/>de e&longs;t, atque &longs;i lancea ip&longs;a cederet, aut flecteretur. </s> <s id="s.005238">Non e&longs;t igi­<lb/>tur Centrum impetûs in lanceâ ipsâ, &longs;ed potiùs in corpore mili­<lb/>tis non procul ab equo; ac proinde inclinata lancea, ut, quam <lb/>minimùm fieri po&longs;&longs;it, recedat à po&longs;itione parallelâ lineæ Di­<lb/>rectionis motûs, & ab hac lineâ non longè ab&longs;it, validi&longs;&longs;imum <lb/>ictum infliget: hoc autem quia faciliùs obtinetur longiore lan­<lb/>ceâ, quàm breviore, ideò, cæteris paribus, præ&longs;tat longiore <lb/>lanceâ uti. </s> </p> <p type="main"> <s id="s.005239">Res autem aliter &longs;e habet, quando percu&longs;&longs;io contingit in&longs;tru­<lb/>mento non ampliùs cohærente ip&longs;i cau&longs;æ, à qua impetum re­<lb/>cipit, &longs;ed jam ab eâ disjuncto; in eo enim præcisè e&longs;t Cen­<lb/>trum Impetûs, & attendenda e&longs;t linea Directionis motûs ab <lb/>huju&longs;modi centro ducta, ut vis percu&longs;&longs;ionis maxima innote&longs;cat. </s> </p> <p type="main"> <s id="s.005240">At hìc quæris; &longs;i ha&longs;tam manu &longs;tringentes impetum illi im­<lb/>primimus, & brachium pariter impetum concipit, atque ex <lb/>utroque impetu æ&longs;timandus e&longs;t ictus; cur validiùs ha&longs;tam ean­<lb/>dem intorquemus jaculantes, quàm manu tenentes? </s> <s id="s.005241">Sic anti-<pb pagenum="711" xlink:href="017/01/727.jpg"/>quis, ad acriùs feriendum, placuit ha&longs;tis amentatis uti, ut po&longs;t <lb/>ejaculationem ha&longs;tam loris ligatam retraherent, iterúmque <lb/>evibrarent: E&longs;&longs;e autem validiorem ictum hinc cogno&longs;ces, quòd <lb/>ha&longs;tæ evibratæ mucro altiùs infigitur objectæ tabulæ, quàm <lb/>cùm illam manu retinentes &longs;imiliter tabulam cu&longs;pide percuti­<lb/>mus. </s> <s id="s.005242">Ex multiplici causâ id petendum videtur. </s> <s id="s.005243">Et primò qui­<lb/>dem, quia cùm ha&longs;tam manu &longs;tringimus, caro, quæ e&longs;t in volâ <lb/>manûs, illicò cedit, ac obicem ha&longs;tæ re&longs;i&longs;tentem offendit, ex <lb/>qua ce&longs;&longs;ione minuitur impetûs; qui & multo magis debilitatur, <lb/>&longs;i brachium pariter in po&longs;teriora modicè revocemus timentes, <lb/>ne ex præconcepto impetu, & corporis percu&longs;&longs;i re&longs;i&longs;tentiâ <lb/>oriatur nimia aliqua partium convul&longs;io, aut dolor: hoc autem <lb/>incommodum vitatur in ha&longs;tâ jam emi&longs;sâ. </s> <s id="s.005244">Deinde quando ali­<lb/>quid jaculamur, ultimo momento, quo illud tenemus, bra­<lb/>chium validi&longs;&longs;imo conatu in anteriora movemus, &longs;tatímque re­<lb/>trahimus dimittentes mi&longs;&longs;ile, cui propterea plurimus impetus <lb/>imprimitur: con&longs;tat autem non po&longs;&longs;e à nobis ha&longs;tam retinenti­<lb/>bus (alia &longs;cilicet e&longs;t mu&longs;culorum contentio & motio) moveri <lb/>brachium motu adeò concitato. </s> <s id="s.005245">Demum impetus brevi&longs;&longs;imo <lb/>illo motu tantâ vi productus in mi&longs;&longs;ili &longs;uam retinet directionem <lb/>(quicquid &longs;it, an gravitas in&longs;ita aliquid officiat) quæ in lon­<lb/>giore motu brachij &longs;i non dimittatur, aliquantulum labefacta­<lb/>tur, eo quod plures motus circa diver&longs;a centra, videlicet circa <lb/>os humeri, & os cubiti, mi&longs;ceantur; atque ex diver&longs;a illâ di­<lb/>rectione vis impetûs minuitur. </s> <s id="s.005246">Cùm itaque ex omnibus hi&longs;ce <lb/>cau&longs;is major inveniatur impetus in ha&longs;tâ evibratâ, quo momen­<lb/>to illa percutit, majorem quoquè ictum ab eâ infligi con&longs;e­<lb/>quens e&longs;t. </s> </p> <p type="main"> <s id="s.005247">Ad hoc percu&longs;&longs;ionum horizontalium genus &longs;pectat illa per­<lb/>cu&longs;&longs;io, qua in ludo minoris tudiculæ globus unus tudiculâ im­<lb/>pellente emi&longs;&longs;us alium globulum percutit. </s> <s id="s.005248">Si enim in eâdem <lb/>directionis motûs lineâ reperiantur centra utriu&longs;que globi, per­<lb/>cutientis &longs;cilicet & percu&longs;&longs;i, maximus ictus infligitur, quia <lb/>maximam invenit re&longs;i&longs;tentiam, cum totus globulus percu&longs;&longs;us <lb/>toti percutienti opponatur, cujus &longs;ingularum partium lineæ di­<lb/>rectionis motûs &longs;i producantur, occurrunt globulo percu&longs;&longs;o <lb/>(æquales &longs;unt globuli ex hypothe&longs;i) quamvis &longs;ola linea directio­<lb/>nis Centri illum contingat. </s> <s id="s.005249">Sin autem globus emi&longs;&longs;us ita alium <pb pagenum="712" xlink:href="017/01/728.jpg"/>quie&longs;centem tangat, ut recta linea per contactûs punctum ducta <lb/>&longs;it utriu&longs;que globi Tangens; & lineæ directionis motûs paralle­<lb/>la, nullus e&longs;t ictus, quia nullum motui impedimentum infertur. </s> <lb/> <s id="s.005250">Demum &longs;i linea, per quam dirigitur centrum globuli emi&longs;&longs;i, non <lb/>occurrat centro globi percu&longs;&longs;i, ita tamen &longs;e habeat, ut lineæ <lb/>utrumque globulum Tangenti occurrat extra punctum con­<lb/>tactûs, tunc major aut minor erit ictus pro ratione impedimen­<lb/>ti & re&longs;i&longs;tentiæ, prout majori aut minori parti globuli percu­<lb/>tientis opponitur globulus percu&longs;&longs;us: Ex quo fit eò majorem <lb/>e&longs;&longs;e re&longs;i&longs;tentiam, quò linea directionis motûs Centri percu­<lb/>tientis propiùs ad punctum contactûs occurrit lineæ Tangenti. </s> </p> <p type="main"> <s id="s.005251">Sit globus B emi&longs;&longs;us adversùs globum A quie&longs;centem, & <lb/><figure id="id.017.01.728.1.jpg" xlink:href="017/01/728/1.jpg"/><lb/>linea, per quam dirigitur motus <lb/>centri B, &longs;it BC occurrens <lb/>puncto contactûs C, atque adeò, <lb/>ut colligitur ex 12. lib. 3. etiam <lb/>centro A, &longs;i producta intelliga­<lb/>tur. </s> <s id="s.005252">Hìc invenit maximam re­<lb/>&longs;i&longs;tentiam globus percutiens, ne­<lb/>que ad hanc, neque ad illam <lb/>partem deflectere pote&longs;t à priori <lb/>directione; linea &longs;iquidem Di­<lb/>rectionis BC ad angulos rectos <lb/>incidit in Tangentem DE, & omnes &longs;ingularum partium di­<lb/>rectiones occurrerent globulo A, &longs;i productæ intelligantur ex­<lb/>tra globulum B. </s> <s id="s.005253">Quòd &longs;i linea directionis motûs fui&longs;&longs;et BF pa­<lb/>rallela Tangenti DE, nullum planè inferretur impedimentum <lb/>motui à globo quie&longs;cente A; nam partium globi impul&longs;i Di­<lb/>rectiones parallelæ lineæ BF, eæ e&longs;&longs;ent, ut earum nulla incur­<lb/>reret in globum A, ideóque nullus e&longs;&longs;et ictus, ubi nulla e&longs;t re­<lb/>&longs;i&longs;tentia. </s> <s id="s.005254">At &longs;i globus B habeat Directionem BE, aut BG, & <lb/>quie&longs;centem globum tangeret in C, etiam&longs;i neque BE, neque <lb/>BG directiones centri incurrerent in globum A, tamen par­<lb/>tium aliquarum eju&longs;dem globi B Directiones parallelæ Directio­<lb/>ni BE, aut BG, &longs;i productæ intelligantur, incurrerent in glo­<lb/>bum A, atque invenirent ex eo impedimentum. </s> <s id="s.005255">Sit enim Di­<lb/>rectio BE, & in globo B linea LO ip&longs;i BE paralella, quæ pro­<lb/>ducta contingeret in K globum A: utique omnes partes &longs;eg-<pb pagenum="713" xlink:href="017/01/729.jpg"/>menti OLS habentes Directionem parallelam Directioni BE, <lb/>quæ e&longs;t directio Centri, inveniunt re&longs;i&longs;tentiam, cum earum di­<lb/>rectiones incurrant in oppo&longs;itum globum A. <!-- KEEP S--></s> <s id="s.005256">Similiter &longs;i Di­<lb/>rectio Centri &longs;it BG, parallela Directio HI producta tangit <lb/>in P globum A, qui proinde opponitur Directionibus omnium <lb/>partium &longs;egmenti IHS. </s> <s id="s.005257">Cùm autem &longs;egmentum IHS majus <lb/>&longs;it &longs;egmento OLS, etiam major e&longs;t ictus, quando Directio BG <lb/>ea e&longs;t, ut Tangenti lineæ DE occurrat in puncto G non ita <lb/>procul à contactu C, quàm &longs;i directio BE ea e&longs;&longs;et, quæ in <lb/>puncto E remotiore à contactu C occurreret eidem Tangenti <lb/>DE: Maximam &longs;iquidem habet veritatis &longs;peciem, in huju&longs;mo­<lb/>di ictibus impetum in globo percu&longs;&longs;o eâ inten&longs;ione imprimi, <lb/>quæ proportione re&longs;pondeat partibus, quæ directè &longs;ecundùm <lb/>illam Directionis lineam impediuntur. </s> <s id="s.005258">Quoniam verò impe­<lb/>tus globo percu&longs;&longs;o impre&longs;&longs;us illum afficit æquabiliter, hinc e&longs;t, <lb/>quod ille non pote&longs;t à percu&longs;&longs;ione determinari, ni&longs;i ut moveatur <lb/>per lineam Directionis, quæ conjungat punctum contactûs C <lb/>cum centro A: quandoquidem æquales &longs;unt globi partes circa <lb/>centrum, adeóque & æquales impetus. </s> </p> <p type="main"> <s id="s.005259">Ob&longs;erva hìc à me a&longs;&longs;umptos e&longs;&longs;e circulos pro globis, & lineas <lb/>vice planorum &longs;ecantium globos, ut res faciliùs explicaretur: <lb/>cæterùm quæ de lineis dicta &longs;unt, &longs;i de planis per lineas illas <lb/>tran&longs;euntibus intelligantur, rem &longs;imiliter ob oculos ponent, &longs;i <lb/>ip&longs;a parallela fuerint, aut inclinata, prout de lineis con&longs;tituta <lb/>e&longs;t hypothe&longs;is. </s> <s id="s.005260">Supere&longs;t tertius percutientis motus, videlicet <lb/>in arcûs circularis &longs;peciem ductus, quando, altera extremitate <lb/>manente, corpus in gyrum movetur. </s> <s id="s.005261">Experimentis autem do­<lb/>cemur validi&longs;&longs;imum ictum non &longs;emper fieri ab extremitate, <lb/>quamvis hæc veloci&longs;&longs;imè moveatur præ cæteris punctis alte­<lb/>ri extremitati manenti propioribus. </s> <s id="s.005262">Ut igitur inveniatur <lb/>punctum, in quo corpus percutiens maximam habeat re­<lb/>&longs;i&longs;tentiam, ponendum e&longs;t illud e&longs;&longs;e æquabiliter ductum, & <lb/>ex materiâ homogeneâ æqualiter capaci impetûs; atque <lb/>ibi &longs;anè maxima erit re&longs;i&longs;tentia, ubi impetûs momenta <lb/>æqualiter dividuntur: id quod contingit in puncto ita re­<lb/>moto à motûs centro, ut cadat inter be&longs;&longs;em, & dodrantem <lb/>totius longitudinis, quæ habet rationem Radij, quo arcus <lb/>de&longs;cribitur. </s> </p> <pb pagenum="714" xlink:href="017/01/730.jpg"/> <p type="main"> <s id="s.005263">Sit corporis percutientis longitudo AB. <!-- KEEP S--></s> <s id="s.005264">Si motu naturali <lb/><figure id="id.017.01.730.1.jpg" xlink:href="017/01/730/1.jpg"/><lb/>&longs;ponte &longs;uâ de&longs;cenderet, & <lb/>in motu po&longs;itionem hori­<lb/>zonti parallelam &longs;ervaret, <lb/>utique validi&longs;&longs;imus e&longs;&longs;et <lb/>ictus in D puncto, quod <lb/>re&longs;pondet centro gravita­<lb/>tis C; e&longs;&longs;ent enim hinc atque hinc æquales gravitates, & æqua­<lb/>lia impetûs momenta, ut &longs;uperiùs dictum e&longs;t. </s> <s id="s.005265">At manente ex­<lb/>tremitate A, tanquam centro motûs, & corpore ip&longs;o vi &longs;uæ gra­<lb/>vitatis de&longs;cendente, licèt &longs;ingulæ particulæ, utpote naturæ <lb/>eju&longs;dem, paribus viribus &longs;int præditæ, non tamen æquali mo­<lb/>mento feruntur; &longs;ed cum in A retineantur, quæ puncto A pro­<lb/>piores &longs;unt, magis detorquentur à directione naturalis gravita­<lb/>tis, adeóque plus momenti habent partes inter DB, quàm in­<lb/>ter AD con&longs;titutæ. </s> <s id="s.005266">Porrò momenta &longs;unt in Ratione Di&longs;tan­<lb/>tiarum: Momentum &longs;iquidem e&longs;t Exce&longs;&longs;us virtutis moventis <lb/>&longs;upra re&longs;i&longs;tentiam, qua impedimentum prohibet, ne &longs;equatur <lb/>motus juxta naturalem propen&longs;ionem: quare &longs;ingularum par­<lb/>tium momenta ex earum motu digno&longs;cuntur: moventur autem <lb/>per circulorum arcus &longs;imiles, quorum etiam &longs;imiles &longs;unt Sinus <lb/>de&longs;cen&longs;um metientes, qui &longs;unt in Ratione Radiorum, hoc e&longs;t <lb/>di&longs;tantiarum ab A communi centro. </s> <s id="s.005267">Sic momentum puncti <lb/>D e&longs;t ut AD, puncti G ut AG, puncti H ut AH, atque ita <lb/>de cæteris. </s> <s id="s.005268">Hinc e&longs;t omnium momentorum &longs;ummam conflari <lb/>ex illorum aggregato, quà&longs;i ex aggregato arcuum quos de&longs;cri­<lb/>bunt, aut Sinuum arcubus &longs;imilibus re&longs;pondentium, quorum <lb/>Ratio eadem e&longs;t cum aggregato Radiorum, ex quibus de&longs;cri­<lb/>buntur arcus. </s> <s id="s.005269">Cum autem univer&longs;a longitudo AB in particu­<lb/>las æquales divi&longs;a intelligatur, manife&longs;tum e&longs;t di&longs;tantias à cen­<lb/>tro A con&longs;tituere Progre&longs;&longs;ionem Arithmeticam juxta &longs;eriem <lb/>naturalem numerorum, ac proinde punctum, quod vocari po­<lb/>te&longs;t <emph type="italics"/>Centrum Momentorum Impetûs,<emph.end type="italics"/> illud e&longs;&longs;e, in quo momenta <lb/>illa bifariam æqualiter dividuntur. </s> </p> <p type="main"> <s id="s.005270">Hoc verò punctum e&longs;&longs;e ultra be&longs;&longs;em totius longitudinis hinc <lb/>apparet, quòd, &longs;i longitudo AB in tres æquales partes di&longs;tincta <lb/>intelligatur, prima centro A proxima habet momentum ut 1, &longs;e­<lb/>cunda ut 2, tertia ut 3: igitur po&longs;t finem &longs;ecundæ, hoc e&longs;t in G, <pb pagenum="715" xlink:href="017/01/731.jpg"/>videtur e&longs;&longs;e æqualitas momentorum; nam AG habet ut 3, <lb/>& GB item ut 3. Sed non e&longs;&longs;e in G momentorum æqualitatem <lb/>con&longs;tat, &longs;i adhuc plures in partes AB di&longs;tincta intelligatur, & <lb/>eadem &longs;it Ratio dupla AG ad GB. <!-- KEEP S--></s> <s id="s.005271">Sit AB partium 6; AG <lb/>e&longs;t 4, GB 2: Momenta AG &longs;unt 10, GB 11; &longs;unt &longs;cilicet <lb/>ip&longs;ius AG momenta quatuor di&longs;tantiarum 1. 2. 3. 4. hoc e&longs;t 10, <lb/>GB verò momenta quintæ & &longs;extæ di&longs;tantiæ 5 & 6, hoc e&longs;t 11. <lb/>Quod &longs;i ponatur AB partium 9, & AG 6, GB 3; momenta <lb/>AG &longs;unt 21, GB &longs;unt 24. Similiter &longs;tatuamus longitudinem <lb/>AB partium 12, &longs;cilicet AG 8, GB 4, momenta AG &longs;unt 36, <lb/>GB 42. Item AB &longs;it partium 15; AG 10, GB 5: momenta <lb/>AG &longs;unt 55, GB 65. Demum AB &longs;it partium 18; AG 12, <lb/>GB 6: momenta AG &longs;unt 78, GB 93. Non igitur momen­<lb/>torum æqualitas e&longs;t præcisè in G ad be&longs;&longs;em longitudinis AB, <lb/>&longs;ed e&longs;t ultra G versùs B. </s> </p> <p type="main"> <s id="s.005272">Verùm, &longs;i AH &longs;it dodrans longitudinis AB, non in H, &longs;ed <lb/>citrà H, inter G & H e&longs;t quæ&longs;itum punctum, in quo momen­<lb/>torum æqualitas invenitur. </s> <s id="s.005273">Nam &longs;i AB &longs;it partium 4, atque <lb/>AH &longs;it 3, HB verò &longs;it 1; momenta AH &longs;unt 6, & HB 4: <lb/>Si AB &longs;it part. </s> <s id="s.005274">8: & AH 6, HB 2; momenta AH &longs;unt 21, <lb/>HB 15; & &longs;ic de cæteris &longs;ervatâ eádem Ratione triplâ AH <lb/>ad HB. <!-- KEEP S--></s> <s id="s.005275">Cum itaque momenta AG minora &longs;int quàm momen­<lb/>ta GB, contrà verò momenta AH majora &longs;int momentis HB, <lb/>con&longs;tat æqualitatem momentorum e&longs;&longs;e inter G & H, hoc e&longs;t <lb/>inter be&longs;&longs;em & dodrantem. </s> <s id="s.005276">Ubi autem proximè &longs;it huju&longs;modi <lb/>punctum, deprehendes, &longs;i totam AB &longs;tatus partium 576, <lb/>& AI partium 407: Cum enim momenta omnia totius AB <lb/>&longs;int 166176, & momenta AI &longs;int 83028, remanent momen­<lb/>ta IB 83148, proximè æqualia momentis AI. <!-- KEEP S--></s> <s id="s.005277">E&longs;t autem Ra­<lb/>tio 407 ad 576 minor Ratione 3 ad 4: & Ratio AI ad IB mi­<lb/>nor e&longs;t Ratione AH ad HB, hoc e&longs;t minor Ratione Dodran­<lb/>tis ad A&longs;&longs;em: item Ratio 407 ad 576 major e&longs;t Ratione 2 ad 3, <lb/>hoc e&longs;t Ratione Be&longs;&longs;is ad A&longs;&longs;em. <!-- KEEP S--></s> <s id="s.005278">Quæ de lineâ AB hactenus <lb/>dicta &longs;unt, in reliquis pariter illi parallelis vera e&longs;&longs;e deprehen­<lb/>duntur &longs;imili ratiocinatione; ac propterea à linea IL omnes in <lb/>eâdem Ratione &longs;ecantur. </s> <s id="s.005279">Maxima igitur percu&longs;&longs;io à corpore <lb/>AE circa lineam AF in gyrum acto fiet in lineâ IL; in qua <lb/>medium punctum K denotat locum validi&longs;&longs;imi ictûs; in eo &longs;ci-<pb pagenum="716" xlink:href="017/01/732.jpg"/>licet omnia momenta impetûs æqualiter dividuntur tam jux­<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/>bili ductu partes omnes di&longs;po&longs;itas habere, &longs;icut hactenus hypo­<lb/>the&longs;im in cylindro aut pri&longs;mate con&longs;tituimus, idcircò frequen­<lb/>ti&longs;&longs;imè centrum hoc momentorum Impetûs, ex quo ictus vehe­<lb/>mentia pendet, aut magis ad centrum motûs accedit, aut ab <lb/>hoc magis recedit, prout ad hanc aut illam extremitatem plu­<lb/>res &longs;unt partes majoris impetûs, aut majorum momentorum ca­<lb/>paces: fieri &longs;iquidem pote&longs;t, ut plures partes centro motûs pro­<lb/>ximæ tenuioribus momentis &longs;int præditæ, pauciores autem par­<lb/>tes ab huju&longs;modi motûs centro remotæ majora obtineant mo­<lb/>menta, adeò ut inæqualitas partium reciprocâ quadam inæqua­<lb/>litate momentorum compen&longs;etur; & fieri pote&longs;t, ut plures par­<lb/>tes cum majore di&longs;tantiâ componantur, adeò ut centrum mo­<lb/>mentorum impetûs proximum &longs;it extremitati, quæ veloci&longs;&longs;imè <lb/>movetur. </s> <s id="s.005281">Hinc quia malleo & &longs;ecuri infligendus e&longs;t ictus, in <lb/>illorum manubriis &longs;tatuendis cavendum e&longs;t, ne nimis gravia <lb/>&longs;int, ne fortè extra malleum aut &longs;ecurim, quibus fit percu&longs;&longs;io, &longs;it <lb/>centrum momentorum impetûs. </s> <s id="s.005282">Centrum hoc momentorum &longs;i <lb/>appellare libeat <emph type="italics"/>Centrum Percu&longs;&longs;ionis,<emph.end type="italics"/> per me licet; neque enim <lb/>hæreo in vocabulis. </s> </p> <p type="main"> <s id="s.005283">Ut autem oblato quocumque corpore ad percutiendum apto, <lb/>quo utendum &longs;it motu circulari, cuju&longs;modi e&longs;t malleus, clava, <lb/>&longs;ecuris, & &longs;imilia, ejus centrum Momentorum impetûs Phy&longs;icè <lb/>& Mechanicè habeamus, hæc methodus forta&longs;&longs;e non inutilis <lb/>accidat. </s> <s id="s.005284">Extremam illam partem, quæ manu apprehendi &longs;olet, <lb/>ex clavo immobili ita &longs;u&longs;pende, ut circa illum liberè moveri <lb/>valeat: tum &longs;u&longs;pen&longs;am clavam à perpendiculo remove, & in <lb/>hanc atque illam partem vibrari permitte. </s> <s id="s.005285">Interim ex &longs;ubtili&longs;&longs;i­<lb/>mo filo æreus, aut plumbeus, globulus pendeat, qui pariter vi­<lb/>bretur: & hujus perpendiculi vibrationes cum clavæ &longs;u&longs;pen&longs;æ <lb/>vibrationibus compara, an videlicet &longs;ingulæ &longs;ingulis i&longs;ochronæ <lb/>&longs;int, hoc e&longs;t æqualis durationis, an verò inæqualis; &longs;i una per­<lb/>pendiculi vibratio diuturnior &longs;it, quàm una clavæ vibratio, de­<lb/>curtandum e&longs;t filum, &longs;i brevior, producendum u&longs;que eò, dum <lb/>perpendiculi vibrationes &longs;ingulæ &longs;ingulis clavæ vibrationibus <lb/>i&longs;ochronæ fuerint. </s> <s id="s.005286">Hoc ubi con&longs;ecutus fueris, haud temerè <pb pagenum="717" xlink:href="017/01/733.jpg"/>pronunciabis quæ&longs;itum Centrum momentorum Impetûs clavæ <lb/>tanto intervallo abe&longs;&longs;e à puncto &longs;u&longs;pen&longs;ionis, quanta e&longs;t per­<lb/>pendiculi longitudo, non quidem exacti&longs;&longs;imè & Geometricè, <lb/>&longs;ed quantum &longs;atis e&longs;t ad Phy&longs;icum opus. </s> <s id="s.005287">Cur ita argumentari <lb/>liceat, &longs;i rationem repo&longs;cas, hæc &longs;atis probabilis afferri pote&longs;t; <lb/>quia &longs;cilicet Centrum momentorum Impetûs e&longs;t punctum illud, <lb/>in quo cum &longs;it æqualitas momentorum, omnes ip&longs;ius clavæ par­<lb/>tes &longs;uam vim exercent ad motum illius o&longs;cillationis; quemad­<lb/>modum in centro globuli ex filo pendentis (filum ex hypothe­<lb/>&longs;i nullum habet notabile momentum ad motum adnexi globuli <lb/>variandum) e&longs;t æqualitas momentorum eju&longs;dem de&longs;cendentis, <lb/>& ad po&longs;itionem perpendicularem &longs;e re&longs;tituentis. </s> <s id="s.005288">E &longs;t igitur cla­<lb/>va qua&longs;i perpendiculum tantæ longitudinis, quantum e&longs;t inter­<lb/>vallum inter Centrum motûs atque Centrum momentorum Im­<lb/>petûs. </s> <s id="s.005289">At perpendicula æqualis longitudinis &longs;unt i&longs;ochrona: <lb/>igitur invento perpendiculo i&longs;ochrono cum o&longs;cillationibus cla­<lb/>væ, nota erit ex hujus longitudine etiam longitudo rigidi illius <lb/>perpendiculi, quod concipitur in clavâ, videlicet di&longs;tantia Cen­<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/>exactè punctum Centri momentorum in clavâ, &longs;ed illud adhuc <lb/>paulò longiùs abe&longs;&longs;e; quia nimirum perpendiculorum omnino <lb/>æqualium, præterquam in gravitate ponderis appen&longs;i, illud, <lb/>quod gravius e&longs;t, plures vibrationes eodem tempore perficit; <lb/>atque perpendiculorum omnino æqualium, præterquam in lon­<lb/>gitudine, illud, quod longius e&longs;t, paucioribus vibrationibus <lb/>eodem tempore agitatur. </s> <s id="s.005291">Cum autem clava &longs;it perpendiculum <lb/>gravius globulo, qui ex filo pendet, po&longs;itâ æquali longitudine, <lb/>clava velociùs moveretur: Si igitur motus clavæ e&longs;t i&longs;ochronus <lb/>cum motu globuli ex filo &longs;u&longs;pen&longs;i, nece&longs;&longs;e e&longs;t longitudine gra­<lb/>vioris inferente vibrationum raritatem compen&longs;ari ejus gravita­<lb/>tem, quæ crebriores efficeret vibrationes. </s> <s id="s.005292">Quare hoc certum <lb/>habebis, quæ&longs;itum Centrum Momentorum e&longs;&longs;e ultra punctum <lb/>illud inventum ex longitudine perpendiculi adhibiti. </s> </p> <p type="main"> <s id="s.005293">Sed & illud præterea ob&longs;ervandum e&longs;t, motus i&longs;tos circula­<lb/>res corporum percutientium communiter non habere pro &longs;ui <lb/>motûs Centro alteram extremitatem, ni&longs;i fortè, quando ad &longs;olius <lb/>manûs motum moventur, cubito ac brachio immotis: cæterùm <pb pagenum="718" xlink:href="017/01/734.jpg"/>pro centro motûs habent aut cubiti, aut humeri juncturam, <lb/>prout cubitus, vel totum brachium movetur: & tunc Centrum <lb/>momentorum transferri contingit, nec opus e&longs;t adeò longa e&longs;&longs;e <lb/>manubria; ut vides malleos, quibus ad contundendos libros <lb/>utuntur bibliopægi, brevioris e&longs;&longs;e manubrij, quia extento bra­<lb/>chio percutiunt, quod fungitur vice valde longi manubrij: <lb/>contra verò fabrorum ferrariorum mallei, bipennes, & cætera <lb/>in&longs;trumenta, quæ utrâque manu apprehen&longs;a tractamus, lon­<lb/>giora habent manubria, tunc enim brachium adeò extendere <lb/>nequimus. <lb/></s> </p> <p type="main"> <s id="s.005294"><emph type="center"/>CAPUT X.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005295"><emph type="center"/><emph type="italics"/>Quid conferat re&longs;istentia corporis percu&longs;si.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005296">EAtenus ictum corpori percu&longs;&longs;o infligi, quatenus hoc motui <lb/>corporis percutientis opponitur, illíque ob&longs;i&longs;tit, dictum e&longs;t <lb/>cap.6. eóque vehementiorem e&longs;&longs;e percu&longs;&longs;ionem, quò major e&longs;t <lb/>re&longs;i&longs;tentia. </s> <s id="s.005297">Hæc autem re&longs;i&longs;tentia originem ducit ex ipsâ cor­<lb/>porum naturâ; omne &longs;iquidem corpus, quâ corpus e&longs;t, nulli <lb/>corpori penetrabile e&longs;t, neque fieri pote&longs;t, citrà Divinam vir­<lb/>tutem longiùs, quàm naturæ termini po&longs;tulant, excurrentem, <lb/>ut uno eodémque in &longs;patio duo corpora collocentur, quemad­<lb/>modum duæ &longs;ub&longs;tantiæ ab omni pror&longs;us &longs;en&longs;u disjunctæ, &longs;ed <lb/>quæ &longs;olâ ratione, & intelligentiâ comprehenduntur, &longs;e vici&longs;&longs;im <lb/>eodem in loco facilè patiuntur. </s> <s id="s.005298">Corpus igitur percu&longs;&longs;um tùm <lb/>ex &longs;uâ con&longs;titutione, & temperie, tum ex recedendi difficulta­<lb/>te, tùm ex po&longs;itione, &longs;ecundùm quam ictum excipit, habet, ut <lb/>modum percu&longs;&longs;ioni &longs;tatuat; ex triplici enim hoc capite re&longs;i&longs;ten­<lb/>tiarum varietas petenda e&longs;t, qua corpus percu&longs;&longs;um reluctatur, <lb/>ne loco cedat. </s> </p> <p type="main"> <s id="s.005299">Et ad primum quidem quod attinet, corpora dura magis re­<lb/>&longs;i&longs;tere, quàm mollia, manife&longs;tum e&longs;t ex ipsá Duri & Mollis <lb/>notione. <emph type="italics"/>E&longs;t autem Durum,<emph.end type="italics"/> ut ait Ari&longs;toteles lib. 4. Meteor. <lb/><!-- REMOVE S-->&longs;umma 2. cap. 1. <emph type="italics"/>quod non cedit in &longs;eip&longs;um &longs;ecundùm &longs;uperficiem:<emph.end type="italics"/><pb pagenum="719" xlink:href="017/01/735.jpg"/><emph type="italics"/>Molle autem, quod cedit, non circumob&longs;i&longs;tendo; aqua enim non Mol­<lb/>lis, non enim cedit compre&longs;&longs;ione &longs;uperficies in profundum, &longs;ed circum­<lb/>ob&longs;i&longs;tit:<emph.end type="italics"/> aqua &longs;cilicet, & humores urgenti quidem cedunt, at non <lb/>induendo &longs;uperficiem, quæ maneat, (ut ceræ ac luto accidit) <lb/>&longs;ed ita &longs;ecedendo, ut, urgente remoto, ad &longs;uperficiem antiquæ <lb/>&longs;uperficiei &longs;imilem confluant particulæ, quæ &longs;ece&longs;&longs;erant. </s> <s id="s.005300">Qua­<lb/>propter ad mollium genus, in rem præ&longs;entem &longs;pectare cen&longs;en­<lb/>da &longs;unt, quæcumque &longs;e premi patiuntur, hoc e&longs;t, externo im­<lb/>pul&longs;u in &longs;e ip&longs;a coëunt, cùm in profundum &longs;uperficies permu­<lb/>tatur, nec dividitur; &longs;ivè imprimi & formari po&longs;&longs;int, pul&longs;u tan­<lb/>tùm, ut cera & argilla, aut percu&longs;&longs;ione, ut plumbum; &longs;ivè im­<lb/>pre&longs;&longs;ionem & formam rejiciant, ut lana, & &longs;pongiæ. </s> <s id="s.005301">Ea autem, <lb/>quæ dura &longs;unt, &longs;ed ductilia, quia <emph type="italics"/>eâdem percußione po&longs;&longs;unt &longs;imulin <lb/>latus, & in profundum &longs;ecundùm &longs;uperficiem transferri &longs;ecundùm <lb/>partem,<emph.end type="italics"/> ut ferro candenti, alií&longs;que metallis &longs;ub fabri malleo <lb/>contingit, aliquatenus ad mollia pertinere videntur, &longs;altem <lb/>comparatè, quia videlicet cedunt percutienti, quod propterea <lb/>durius cen&longs;etur. </s> <s id="s.005302">Sic in arce Antuerpien&longs;i memini me vidi&longs;&longs;e <lb/>ænea aliquot ingentia tormenta bellica, olim ex Sckenckianâ <lb/>munitione, cum in Hi&longs;panorum pote&longs;tatem venit, a&longs;portata, <lb/>in quorum tubis non mediocres contu&longs;ionum notæ ab ho&longs;tili­<lb/>bus globis impre&longs;&longs;æ apparebant. </s> <s id="s.005303">Quæ verò corpora dura &longs;unt, <lb/>neque &longs;e ita comprimi patiuntur, ut &longs;uperficies depre&longs;&longs;a cra&longs;&longs;i­<lb/>tiem minuat, &longs;ed &longs;olùm, &longs;ervatâ longitudine, flexibilia &longs;unt eâ <lb/>ratione, ut à rectitudine ad curvitatem, aut vici&longs;&longs;im à curvitate <lb/>ad rectitudinem torqueantur, cedunt quidem, &longs;ed inter mollia, <lb/>ex hoc quidem capite, recen&longs;enda non &longs;unt. </s> <s id="s.005304">Quod &longs;i vehe­<lb/>mentiore percu&longs;&longs;ione non &longs;olùm flectantur, &longs;ed etiam frangan­<lb/>tur (quemadmodum contingit cra&longs;&longs;iu&longs;culo baculo, cujus ex­<lb/>tremitates in acumen de&longs;inentes innituntur duobus vitreis cya­<lb/>this; qui circa medium valido fu&longs;te percu&longs;&longs;us flectitur, & in­<lb/>flexione declinans vitra, iis integris frangitur) in magnas par­<lb/>tes dividuntur, & &longs;eparantur: at &longs;i in partes plures di&longs;&longs;iliant ex <lb/>unicâ percu&longs;&longs;ione, friantur, ut vitrum, lapis, fictile; id quod ex <lb/>duritie oritur. </s> </p> <p type="main"> <s id="s.005305">Hæc eadem corporis habitudo, quæ particularum compo­<lb/>nentium complexionem re&longs;picit, æquè in percutiente, ac in <lb/>percu&longs;&longs;o attendenda e&longs;t; quandoquidem &longs;i di&longs;par fuerit eorum <pb pagenum="720" xlink:href="017/01/736.jpg"/>durities, fieri pote&longs;t, ut ex ictu labefactetur potiùs percutiens, <lb/>quàm percu&longs;&longs;um: &longs;ic globus plumbeus ex editâ turri decidens <lb/>in &longs;ubjectum &longs;ilicem ex ictu contunditur, rotundâ &longs;uperficie in <lb/>planam mutatâ, qua parte fuit contactus; & vitrum ad &longs;axa <lb/>alli&longs;um friatur; & follis lu&longs;orius in parietem impactus compri­<lb/>mitur. </s> <s id="s.005306">Hinc tamen non fit, quò minus corpus illud, in quod <lb/>plumbeus globus, aut vitrum, aut follis incurrit, percu&longs;&longs;um di­<lb/>catur; ex ictu enim &longs;altem concutitur, & contremi&longs;cit. </s> <s id="s.005307">Neque <lb/>tremorem huju&longs;modi temerè confictum &longs;u&longs;picabitur, qui&longs;quis <lb/>longi&longs;&longs;imæ trabis extremitati aurem admoverit, ut alterâ extre­<lb/>mitate quamvis levi&longs;&longs;imè digito percu&longs;sâ &longs;onitum audiat, aut <lb/>noctu &longs;crobiculo in terrâ facto aurem immi&longs;erit, ut adventan­<lb/>tis alicujus adhuc procul po&longs;iti pa&longs;&longs;us percipiat: nullum verò <lb/>&longs;onum fieri &longs;ine tremore & motu, extra controver&longs;iam po&longs;uit <lb/>experientia. </s> </p> <p type="main"> <s id="s.005308">Porrò &longs;pectatâ corporum temperie, percu&longs;&longs;ionum vehemen­<lb/>tiâ æ&longs;timatur ex iis, quæ con&longs;equuntur re&longs;i&longs;tentiam ortam ex <lb/>corporum colli&longs;orum duritie &longs;eu mollitudine majori au minori, <lb/>tùm ab&longs;olutè, tùm comparatè. </s> <s id="s.005309">Cum <emph type="italics"/>Ab&longs;olutè<emph.end type="italics"/> dico, alterutrius <lb/>&longs;olùm duritiem &longs;eu mollitudinem con&longs;idero ita, ut aut corpora <lb/>percu&longs;&longs;a inter &longs;e, aut corpora percutientia &longs;imiliter inter &longs;e <lb/>conferantur: <emph type="italics"/>Comparatè<emph.end type="italics"/> autem, quando percutiens cum per­<lb/>cu&longs;&longs;o comparatur, prout duritie &longs;e excedunt. </s> <s id="s.005310">Si corpus percu­<lb/>tiens valdè durum ponatur, & corpus percu&longs;&longs;um molle fuerit, <lb/>hoc cedendo retundit ictum; ex levi enim illâ re&longs;i&longs;tentiâ tan­<lb/>diu durante, quandiu fit partium compre&longs;&longs;io, minuitur in per­<lb/>cutiente impetus, & quod corpori molli &longs;ubjectum e&longs;t corpus, <lb/>levi&longs;&longs;imam impre&longs;&longs;ionem ex ictu recipit. </s> <s id="s.005311">Sic apud Sinas, ut in <lb/>Atlante Sinico pag. </s> <s id="s.005312">127. <emph type="italics"/>In flumine, per quod ad Ienping navi­<lb/>gatur, Catadupæ aquarum multæ &longs;unt, & periculo&longs;i&longs;&longs;ima Syrtibus <lb/>loca, duo præ&longs;ertim propè Cinglieu, unus Kieulung, alter Changcung <lb/>dictus. </s> <s id="s.005313">Cum naves tran&longs;eunt, ne cum aquâ decidentes fractionis in­<lb/>currant periculum, &longs;citè præmittunt aliquot &longs;traminis fa&longs;ces, ad quos <lb/>navis leviùs impingat, ac tran&longs;eat.<emph.end type="italics"/></s> <s id="s.005314"> Sic ferreis tormentorum glo­<lb/>bis objecti &longs;acci lanâ aut terrâ repleti illorum vim elidunt, ne <lb/>diruant muros huju&longs;modi &longs;accis protecto: &longs;ic farti go&longs;&longs;ypio <lb/>thoraces non levi munimento &longs;unt digladiantibus. </s> <s id="s.005315">Quò autem <lb/>mollius fuerit corpus percu&longs;&longs;um, quia magis cedit, minùs læ-<pb pagenum="721" xlink:href="017/01/737.jpg"/>ditur à percutiente; & vici&longs;&longs;im quò durius illud fuerit, magis <lb/>ab eodem percutiente læditur, cujus impul&longs;um excipit. </s> </p> <p type="main"> <s id="s.005316">Hinc vides cur ferreos militum thoraces, & galeas no&longs;tro <lb/>hoc ævo aliter temperare oporteat, ac antiquis temporibus, <lb/>quando gladiorum, ha&longs;tarum, &longs;agittarum ictus tantummodo <lb/>repellere opus erat; tunc enim durâ temperatione &longs;olidandum <lb/>erat ferrum, ne pror&longs;us cederet, huju&longs;modi armorum mucro­<lb/>nem admittendo: nunc verò ut innoxiè excipiantur ictus glo­<lb/>borum à Sclopis emi&longs;&longs;orum, ferrum molle e&longs;&longs;e expedit, ut con­<lb/>tu&longs;um flectatur, & aliquantulum cedens ita imminuat globi <lb/>ejaculati vires, ut penetrare ulteriùs non valeat. </s> <s id="s.005317">Quod &longs;i in <lb/>chalybem temperatus e&longs;&longs;et thorax militaris, nec admodum <lb/>cra&longs;&longs;us e&longs;&longs;et, ne gravitate nimiâ incommodus, aut inutilis ac­<lb/>cideret, facilè chalybs ex globi ictu di&longs;&longs;iliret, & vulneri locum <lb/>aperiret. </s> <s id="s.005318">Sed quia globi plumbei &longs;unt, & &longs;e comprimi patiun­<lb/>tur, ex huju&longs;modi percu&longs;&longs;ione compre&longs;&longs;io qua&longs;i di&longs;tribuitur in­<lb/>ter plumbeum globum explo&longs;um, atque ferreum thoracem, qui <lb/>multo magis contunderetur (aut fortè etiam perforaretur à glo­<lb/>bulo ferreo; globulus autem plumbeus, &longs;i thorax aut ip&longs;a galea <lb/>nihil cederet, magis comprimeretur, quemadmodum cùm in <lb/>marmor exploditur. </s> </p> <p type="main"> <s id="s.005319">At ubi corpus percu&longs;&longs;um non cedit in &longs;eip&longs;um &longs;ecundùm &longs;u­<lb/>perficiem, flectitur tamen, adhuc minùs re&longs;i&longs;tit, quàm corpo­<lb/>ra rigida, nec flexioni notabili obnoxia. </s> <s id="s.005320">Notabili, inquam, ne <lb/>in quæ&longs;tionem vocemus, utrùm flecti dicenda &longs;int illa corpora, <lb/>quæ ex ictu tremorem concipiunt, ut æri campano, cum pul&longs;a­<lb/>tur, accidit: nam vix excogitari pote&longs;t corpus aliquod, cui ex <lb/>vi percu&longs;&longs;ionis accidere nequeat tremor; cum & terram ip&longs;am <lb/>licèt altiùs defo&longs;&longs;am in cuniculis concuti & contremi&longs;cere <lb/>o&longs;tendant lapilli, & fabæ in tympani militaris planâ facie &longs;ub&longs;i­<lb/>lientes ex profundo illo ligonis ictu. </s> <s id="s.005321">Certè, &longs;i Atlanti Sinico <lb/>pag.57. credimus, ubi in IV Provinciâ Xantung mentionem <lb/>facit de monte, cui nomen Mingxe, hoc e&longs;t Sonorum lapis; <lb/><emph type="italics"/>in hujus montis vertice cippus erectus &longs;tat centum altus perticas <emph.end type="italics"/><lb/>(Pertica apud Sinas e&longs;t decem cubitorum) <emph type="italics"/>qui vel leviter digito <lb/>percu&longs;&longs;us ad tympani modum &longs;onum edere dicitur, à quo monti nomen<emph.end type="italics"/>; <lb/>nullus autem &longs;onus ab&longs;que corporis &longs;onori tremore efficitur. </s> <lb/> <s id="s.005322">Quod &longs;i non ni&longs;i levi&longs;&longs;imè flecti queat corpus percu&longs;&longs;um, &longs;ed ci-<pb pagenum="722" xlink:href="017/01/738.jpg"/>tra tremorem frangatur, aut frietur, indicium e&longs;t majoris re­<lb/>&longs;i&longs;tentiæ, ac proinde, cæteris paribus, vehementiorem futu­<lb/>ram percu&longs;&longs;ionem, quàm &longs;i con&longs;picuam flexionem admitteret. </s> <lb/> <s id="s.005323">Cæterùm re&longs;i&longs;tentia ferè maxima eorum corporum e&longs;t, quæ & <lb/>partes nexu ægrè di&longs;&longs;olubili copulatas habent, & congruá cra&longs;­<lb/>&longs;itudine prædita non ni&longs;i creberrimo & minuti&longs;&longs;imo tremore <lb/>concuti po&longs;&longs;unt, &longs;i percutiantur. </s> <s id="s.005324">Nam omnium re&longs;i&longs;tentiarum <lb/>ab&longs;olutè maxima e&longs;t, cùm pror&longs;us immotum à percu&longs;&longs;ione ma­<lb/>net corpus. </s> </p> <p type="main"> <s id="s.005325">Hinc &longs;i percu&longs;&longs;i corporis durities major fuerit, quàm percu­<lb/>tientis, fieri pote&longs;t, ut impetus qui corpori percu&longs;&longs;o imprimi <lb/>non pote&longs;t, disjiciat ip&longs;ius percutientis partes, aut in latus im­<lb/>pellat ita, ut vel contundatur, vel frangatur, vel frietur, sítque <lb/>percutientis conditio deterior, quàm percu&longs;&longs;i. </s> <s id="s.005326">Huju&longs;modi e&longs;&longs;et <lb/>apud nos conditio gladij, quo marmor percuteremus; neque <lb/>enim no&longs;trates en&longs;es comparandi &longs;unt cum illo, de quo Atlas <lb/>Sinicus pag.159.in XV Provincia Junnam ad urbem Chinkiang, <lb/>ubi hæc habet. <emph type="italics"/>Ad urbis Borealem partem ad hæc u&longs;que tempora in­<lb/>gens con&longs;picitur lapis, ubi Mung Rex Sinulo alterius Regis legatos <lb/>excipiens, cum illi minimè &longs;atisfacerent, extracto gladio rapidens <lb/>ita percu&longs;&longs;it, ut ictus ad tres cubitos penetraret, verbis in&longs;uper mi­<lb/>nacibus legatos alloquens; Ite, & Regi ve&longs;tro renunciare, quales apud <lb/>me gladij &longs;int.<emph.end type="italics"/></s> </p> <p type="main"> <s id="s.005327">Altera re&longs;i&longs;tentiæ origo habetur ex difficultate recedendi; <lb/>quando videlicet corpus percu&longs;&longs;um &longs;ivè ratione figuræ, &longs;ivè ra­<lb/>tione molis & gravitatis, &longs;ivè ratione ob&longs;taculi alicujus, aut <lb/>retinaculi, &longs;ivè ratione motûs oppo&longs;iti, nequit ob&longs;ecundare <lb/>motui percutientis, &longs;ed potiùs illum aut cohibet, aut retardat, <lb/>aut reflectit; hæc enim tria accidere po&longs;&longs;unt motui percutientis <lb/>ex percu&longs;&longs;i re&longs;i&longs;tentiâ. </s> <s id="s.005328">Primum &longs;iquidem &longs;i corpus percu&longs;&longs;um <lb/>volubile non fuerit, & in orbem incitari nequeat, &longs;ed planâ fa­<lb/>cie incumbat &longs;olo, præ&longs;ertim &longs;alebro&longs;o, quò ampliori facie fit <lb/>contactus, eò difficiliùs impelli pote&longs;t. </s> <s id="s.005329">Deinde etiam&longs;i rotun­<lb/>dum fuerit corpus, & facilis motionis principium habeat &longs;pecta­<lb/>tâ figurâ, &longs;i tamen ingens fuerit globus marmoreus, aut æreus, <lb/>tanta e&longs;&longs;e pote&longs;t gravitas, ut vix, aut ne vix quidem, loco dimo­<lb/>veri queat. </s> <s id="s.005330">Non tamen &longs;emper faciliùs moventur ex percu&longs;&longs;io­<lb/>ne, quæ leviora &longs;unt; nam &longs;i quis ex cupre&longs;&longs;u galbulum, aut <pb pagenum="723" xlink:href="017/01/739.jpg"/>ex quercu gallam decerpat, & malleo percutiat, ob nimiam <lb/>galbuli, & gallæ levitatem multo infirmior erit ictus, quàm &longs;i <lb/>æqualem globulum eburneum percuteret. </s> <s id="s.005331">Ad hæc, forma <lb/>quidem apta e&longs;&longs;e pote&longs;t, nec gravitas aut moles nimia, &longs;ed quia <lb/>corpus percu&longs;&longs;um nequit percutienti cedere, ni&longs;i corpus aliud <lb/>proximum repellatur, propterea augeri pote&longs;t re&longs;i&longs;tentia: &longs;ic <lb/>&longs;ublicas acuminatas in terram adigimus fi&longs;tucâ &longs;ivè directas ad <lb/>perpendiculum, &longs;ivè pronas; &longs;ed ea pote&longs;t e&longs;&longs;e telluris den&longs;itas <lb/>compre&longs;&longs;ionem re&longs;puens, ut &longs;æpiùs cadens fi&longs;tuca parum profi­<lb/>ciat. </s> <s id="s.005332">Demum &longs;i corpus percu&longs;&longs;um antè ictum non quie&longs;cat, &longs;ed <lb/>oppo&longs;ito motu occurrat percutienti, quò velociùs movetur, & <lb/>directione magis oppo&longs;itâ, etiam magis re&longs;i&longs;tit, & utrumque <lb/>vici&longs;&longs;im e&longs;t percutiens & percu&longs;&longs;um; ni&longs;i quod percutientis vo­<lb/>cabulum validiori conceditur. </s> <s id="s.005333">Contra verò languidior accidit <lb/>percu&longs;&longs;io, &longs;i corpus percutiens a&longs;&longs;equatur aliud, quod ad ea&longs;­<lb/>dem partes tardiùs movetur; eóque minor e&longs;t re&longs;i&longs;tentia, quò <lb/>minor e&longs;t in velocitate motuum differentia. </s> <s id="s.005334">Sic decidentis ex <lb/>altitudine non modicâ lapidis ictum manu citrà læ&longs;ionem exci­<lb/>pimus, &longs;i illius motui, ubi manum attigerit, exiguo minoris <lb/>velocitatis di&longs;crimine ob&longs;ecundemus: hæc &longs;iquidem exigua re­<lb/>&longs;i&longs;tentia modicum quid impetûs deterit, & quia aliquot mo­<lb/>menta durat, ita &longs;en&longs;im extenuatur impetus, ut demum qui re­<lb/>liquus e&longs;t nocere non valeat. </s> </p> <p type="main"> <s id="s.005335">Hoc artificio procul dubio utebatur quidam, qui ante ali­<lb/>quot annos, ut ex viro fide digno tanquam rem noti&longs;&longs;imam ac­<lb/>cepi, Mutinæ en&longs;em eâ dexteritate in altum projiciebat, ut per­<lb/>pendicularis recideret mucrone deor&longs;um conver&longs;o, quem ca­<lb/>dentem nuda manûs vola innoxiè excipiebat; &longs;ed cum aliquan­<lb/>do invitus cogeretur, ut id noctu experiretur in conclavi mul­<lb/>tis facibus illu&longs;trato, cùm (deficiente con&longs;tanti & clari&longs;&longs;imâ <lb/>diurnâ luce, quam æmulari non pote&longs;t tremula & incon&longs;tans <lb/>facium, quamvis multarum, flamma) non ita exactè a&longs;&longs;eque­<lb/>retur de&longs;cendentis gladij motum & velocitatem, finem fecit lu­<lb/>do manum trajectam referens. </s> </p> <p type="main"> <s id="s.005336">Po&longs;tremum caput, ex quo re&longs;i&longs;tentiæ modus de&longs;umitur in <lb/>percu&longs;&longs;ionibus, e&longs;t ip&longs;a po&longs;itio corporis percu&longs;&longs;i, prout directè, <lb/>aut obliquè, ictum excipit, hoc e&longs;t quatenus linea directionis <lb/>motûs, quo fertur corpus percutiens, incurrit in corporis per-<pb pagenum="724" xlink:href="017/01/740.jpg"/>cu&longs;&longs;i &longs;uperficiem ad angulos æquales, aut inæquales. </s> <s id="s.005337">Si enim <lb/>ad angulos æquales opponatur Directioni motûs, cum ad neu­<lb/>tram partem corpus percutiens declinare po&longs;&longs;it, tota vis ictus <lb/>excipitur à corpore percu&longs;&longs;o. </s> <s id="s.005338">Sin autem obliquè, & ad angu­<lb/>los inæquales, à corpore percu&longs;&longs;o excipiatur percutientis ictus, <lb/>quò major erit angulorum inæqualitas, eò languidior erit per­<lb/>cu&longs;&longs;io, minùs quippe motûs Directioni opponitur &longs;uperficies <lb/>percu&longs;&longs;a, quò fuerit angulus Incidentiæ magis acutus. </s> <s id="s.005339">Ut au­<lb/>tem horum ictuum Ratio aliqua innote&longs;cat, nulla mihi con­<lb/>gruentior methodus occurrit, quàm &longs;i philo&longs;ophemur &longs;imili <lb/>planè ratiocinatione, ac cùm lib. 1. cap. 14. expendimus gravi­<lb/>tationem corporis in planum inclinatum; &longs;icut enim ibi gravi­<lb/>tatem cum &longs;uâ directione deor&longs;um ad centrum gravium con&longs;i­<lb/>deravimus, ita hìc in percu&longs;&longs;ione impetum corporis percutien­<lb/>tis, & ejus directionem accipere oportet: & quemadmodum in <lb/>plano inclinato gravia obtinent momenta de&longs;cendendi majora, <lb/>aut minora, prout angulus inclinationis plani cum perpendi­<lb/>culo minor e&longs;t, aut major; &longs;imiliter in percu&longs;&longs;ione momentum <lb/>progrediendi juxta conceptam aut impre&longs;&longs;am directionem mo­<lb/>tûs ii&longs;dem tenetur legibus, juxta plani percu&longs;&longs;i obliquitatem; <lb/>ac proinde minor invenitur re&longs;i&longs;tentia, ubi majus e&longs;t progre­<lb/>diendi momentum. </s> <s id="s.005340">Quare hìc &longs;atis erit recolere, quæ dicta <lb/>&longs;unt lib. 1. cap. 13. & 14. de gravitatione in plano inclinato, & <lb/>in planum inclinatum, eáque percu&longs;&longs;ionibus &longs;ervatâ analogiâ <lb/>applicare. </s> </p> <p type="main"> <s id="s.005341">Cum itaque in percutiente con&longs;ideranda &longs;it & moles, & mo­<lb/>tûs velocitas, & Directio motûs, & durities; in corpore autem <lb/>percu&longs;&longs;o & naturæ temperatio, & recedendi difficultas, & po­<lb/>&longs;itio, &longs;ecundùm quam excipitur ictus, &longs;pectanda &longs;it; mani­<lb/>fe&longs;tum e&longs;t ex his omnibus ictuum vim temperari; atque adeò &longs;i <lb/>duo ictus comparandi &longs;int, a&longs;&longs;umendæ &longs;unt in corporibus per­<lb/>cutientibus invicem comparatis Rationes omnes & molis ad <lb/>molem (hoc e&longs;t gravitatis ad gravitatem, aut virtutis moventis <lb/>ad virtutem moventem) & velocitatis ad velocitatem, & di­<lb/>rectionis ad directionem, & duritiei ad duritiem; & &longs;imiliter <lb/>in corporibus percu&longs;&longs;is Rationes eorum, quæ in illis con&longs;ide­<lb/>rantur: atque demum facta Rationum compo&longs;itio indicabit Ra­<lb/>tionem ictuum. </s> <s id="s.005342">Hinc vides quàm multæ fieri po&longs;&longs;int huju&longs;modi <pb pagenum="725" xlink:href="017/01/741.jpg"/>Rationum complexiones; quas &longs;i juxta earum varietatem in <lb/>Propo&longs;itiones digerere otium e&longs;&longs;et, in molem non exiguam hæc <lb/>&longs;criptio excre&longs;ceret, &longs;ed non majore fructu, quàm &longs;i tu ip&longs;e Ra­<lb/>tiones, ut indicatum e&longs;t, componas. <lb/></s> </p> <p type="main"> <s id="s.005343"><emph type="center"/>CAPUT XI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005344"><emph type="center"/><emph type="italics"/>Quomodo ex Percu&longs;sionibus determinentur <lb/>Reflexiones.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005345">UT Percu&longs;&longs;ionis natura plenè perfectéque innote&longs;cat, di&longs;­<lb/>piciendum &longs;upere&longs;t, quomodo ex illâ determinetur Re­<lb/>flexio. </s> <s id="s.005346">Motus &longs;iquidem, qui propriè e&longs;t reflexus, percu&longs;&longs;ionem <lb/>con&longs;equitur, quatenus id, quod motu directo ferebatur, inve­<lb/>nit obicem, ne ulteriùs juxta eandem Directionem progredia­<lb/>tur: &longs;ed quia adhuc acqui&longs;itus, &longs;eu impre&longs;&longs;us, impetus &longs;upere&longs;t, <lb/>aliam inire viam cogitur; eóque magis reflectitur, quò majo­<lb/>rem invenit re&longs;i&longs;tentiam ortam ex utriu&longs;que corporis impene­<lb/>trabilitate, atque duritie. </s> <s id="s.005347">Quòd &longs;i utrique corpori, percutien­<lb/>ti videlicet atque percu&longs;&longs;o, &longs;umma durities ine&longs;&longs;e poneretur, <lb/>ita ut in neutro ex vi percu&longs;&longs;ionis ulla &longs;equeretur partium com­<lb/>pre&longs;&longs;io, aut depre&longs;&longs;io, aut attritio &longs;eu divi&longs;io, perfecta quoquè <lb/>intelligeretur reflexio, in qua corpus percutiens non ni&longs;i in <lb/>tran&longs;itu, citrà omnem vel brevi&longs;&longs;imam morulam, contingeret <lb/>corpus, à quo reflectitur; & nulla fieret impetûs acqui&longs;iti, &longs;ive <lb/>impre&longs;&longs;i, diminutio præter eam, quam &longs;ecum trahit nova re­<lb/>flectentis determinatio oppo&longs;ita lineæ directionis, &longs;ecundùm <lb/>quam priùs movebatur. </s> <s id="s.005348">Nemini autem dubium e&longs;&longs;e debet, an <lb/>corpus reflexum pergat moveri ex vi impetûs adhuc re&longs;idui <lb/>po&longs;t motum directum: nam corpus reflectens pror&longs;us immo­<lb/>tum & quie&longs;cens non pote&longs;t impetum illi communicare; cum <lb/>perpetuis experimentis doceamur nihil moveri ab alio <lb/>quie&longs;cente. </s> </p> <p type="main"> <s id="s.005349">Sæpiùs tamen contingit (&longs;i quis dixerit <emph type="italics"/>&longs;emper,<emph.end type="italics"/> quibus argu­<lb/>mentis eum coarguerem manife&longs;tæ fal&longs;itatis, me non habere <pb pagenum="726" xlink:href="017/01/742.jpg"/>candidè profiteor) non fieri puram reflexionem ex merâ re­<lb/>&longs;i&longs;tentiâ; &longs;ed in alterutro &longs;altem corporum colli&longs;orum, ex per­<lb/>cu&longs;&longs;ione &longs;equitur aliqua partium violenta compre&longs;&longs;io, aut <lb/>di&longs;tractio; hanc autem naturæ repugnantem partium po&longs;itio­<lb/>nem excutere dum nititur, séque in pri&longs;tinum &longs;tatum re&longs;titue­<lb/>re, novum impetum concipit, quem & pote&longs;t reflectens re­<lb/>flexo imprimere, atque in eo diminuti ex re&longs;i&longs;tentiâ impetûs <lb/>jacturam, aliquâ &longs;altem ex parte, re&longs;arcire. </s> <s id="s.005350">Hinc, in rem præ­<lb/>&longs;entem di&longs;tinguere oportet, quid inter Compre&longs;&longs;ionem & De­<lb/>pre&longs;&longs;ionem inter&longs;it: quæ enim deprimuntur, ut plumbum, cera, <lb/>argilla, non re&longs;iliunt &longs;ecundùm &longs;uperficiem, ut pri&longs;tinam figu­<lb/>ram induant; ideóque quando huju&longs;modi corpora in aliud im­<lb/>pinguntur, vel aliud in illa impingitur, valdè debilitatur re­<lb/>flexio, &longs;i modò aliqua contingere pote&longs;t. </s> <s id="s.005351">Quæ verò compri­<lb/>muntur, externâ vi deficiente &longs;e in pri&longs;tinam figuram ab&longs;que <lb/>cunctatione re&longs;tituunt concepto novo impetu. </s> <s id="s.005352">Exemplum ex <lb/>folle pugillatorio peti pote&longs;t, ut res in apertum deducatur. </s> <s id="s.005353">Ca­<lb/>dens in &longs;ubjectum pavimentum follis lu&longs;orius, ritè inflatus, im­<lb/>peditur, ne ulteriùs procedat; &longs;ed quia inclu&longs;i aëris particulæ <lb/>eæ &longs;unt, quæ per vim con&longs;tipari ampliùs po&longs;&longs;int, ideò ex illis <lb/>anteriores hinc urgentur à po&longs;terioribus, quæ vi acqui&longs;iti im­<lb/>petûs inchoatum iter pro&longs;equuntur, hinc tellure re&longs;i&longs;tente, <lb/>hinc alutâ continente, inter angu&longs;tias deprehen&longs;æ comprimun­<lb/>tur: id quod cùm motum exigat, certam aliquam brevi&longs;&longs;imi <lb/>temporis men&longs;uram requirit, quo fluente, terra à folle tangi­<lb/>tur, motú&longs;que aliquatenus impeditur (nunquam tamen ita, ut <lb/>ce&longs;&longs;et omnino motus illarum &longs;altem partium, à quibus anterio­<lb/>res urgentur ac premuntur) & quò diutiùs huju&longs;modi com­<lb/>pre&longs;&longs;io durat, eò magis impeditur motus totius follis, atque adeò <lb/>plus impetûs deperditur. </s> <s id="s.005354">Sed quoniam &longs;tatus ille majoris com­<lb/>pre&longs;&longs;ionis aëri intrà follem con&longs;tipato contra naturam accidit, <lb/>ubi primùm, debilitato impetu urgente, re&longs;tituere &longs;e pote&longs;t <lb/>aër, impetum &longs;ibi imprimit, quo moveatur ad ampliorem lo­<lb/>cum occupandum, &longs;i facta fuerit conden&longs;atio, vel certè ad par­<lb/>tes in pri&longs;tino & naturali &longs;tatu con&longs;tituendas (quemadmodum <lb/>alutæ contingit, cujus partes aliæ compre&longs;&longs;æ, aliæ di&longs;tractæ &longs;e­<lb/>&longs;e re&longs;tituunt) cúmque id præ&longs;tare nequeat motu ad terram di­<lb/>recto, quippe quæ re&longs;i&longs;tit, in oppo&longs;itam partem motum dirigit; <pb pagenum="727" xlink:href="017/01/743.jpg"/>novóque hoc impetu &longs;i non æquatur, qui re&longs;i&longs;tentiâ diminutus <lb/>fuerat, &longs;altem incrementi alicujus compen&longs;atione lenitur in­<lb/>commodum detrimenti, & major fit motus, quàm pro Ratione <lb/>re&longs;idui impetûs ante percu&longs;&longs;ionem & compre&longs;&longs;ionem concepti. </s> </p> <p type="main"> <s id="s.005355">Hæc eadem proportione dicenda &longs;unt, quando non corpus <lb/>percutiens, ut follis in terram decidens, &longs;ed percu&longs;&longs;um com­<lb/>primitur aut di&longs;trahitur, & virtute ela&longs;ticâ &longs;e re&longs;tituit; impe­<lb/>tum enim concipiens, quo ami&longs;&longs;am figuram recuperet, etiam <lb/>percutienti impetum imprimit, quo repellitur. </s> <s id="s.005356">Sic in &longs;phæ­<lb/>ri&longs;terio immi&longs;&longs;æ pilæ &longs;i reticulum ex contortis animalium in­<lb/>te&longs;tinis in plagas di&longs;tinctum objeceris, validiùs reflectitur pila, <lb/>quàm objecto batillo ligneo, intenti enim nervi illi, ex impetu <lb/>pilæ inflexi, validi&longs;&longs;imè &longs;e re&longs;tituunt, id quod ligno non con­<lb/>tingit, quippe quod vix ela&longs;ticam hanc virtutem exercet, &longs;i ta­<lb/>men à pilâ impacta quicquam inflexionis recipit, quæ compre&longs;­<lb/>&longs;io &longs;it potiùs, quàm depre&longs;&longs;io. </s> <s id="s.005357">Quæ &longs;cilicet corpora eam par­<lb/>tium texturam habent, ut minùs ferant &longs;e à priore po&longs;itione & <lb/>figurâ dimoveri, illa &longs;e&longs;e majore impetu re&longs;tituunt. </s> </p> <p type="main"> <s id="s.005358">Ex his con&longs;tat, cur partium depre&longs;&longs;io officiat reflexioni cor­<lb/>poris percutientis: quia nimirum à po&longs;terioribus illius partibus <lb/>urgentur anteriores contactui proximæ, quæ interim vel quie&longs;­<lb/>cunt, vel multò tardiùs moventur, vel ad latus &longs;ecedunt, & <lb/>idcircò vel totum, vel ferè totum, &longs;uum impetum deperdunt: <lb/>po&longs;teriores verò dum urgent ac premunt, moventur quidem, <lb/>&longs;ed reperiunt re&longs;i&longs;tentiam &longs;ub&longs;identium partium anteriorum, <lb/>atque adeò in illis pariter minuitur impetus; &longs;æpiú&longs;que tanta fit <lb/>impetûs diminutio, ut, depre&longs;&longs;ione ab&longs;olutâ, partes illæ po&longs;te­<lb/>riores reliquum non habeant tantum impetûs, qui vincere va­<lb/>leat gravitatem, & reflexionem efficere; neque enim aliquid <lb/>ami&longs;&longs;i impetûs compen&longs;atur ab impetu novo partium &longs;e re&longs;ti­<lb/>tuentium, quemadmodum fieri diximus in compre&longs;&longs;ione. </s> <s id="s.005359">Quan­<lb/>do autem depre&longs;&longs;io partium accidit corpori, ad quod alliditur <lb/>corpus percutiens, ut cùm in arenam &longs;iccam ac pulverulentam, <lb/>aut in limo&longs;am terram decidit globus, tunc multum impetûs <lb/>deperditur, ut dictum e&longs;t &longs;uperiùs de ictu, qui eò infirmior e&longs;t, <lb/>quò mollius e&longs;t corpus percu&longs;&longs;um; reflexio autem eò major e&longs;t, <lb/>quò validiore ictu percutitur corpus reflectens. </s> <s id="s.005360">Quòd &longs;i <lb/>utrumque corpus, tam percutiens, quàm percu&longs;&longs;um, patiatur <pb pagenum="728" xlink:href="017/01/744.jpg"/>compre&longs;&longs;ionem aut depre&longs;&longs;ionem, aut partium attritum, tunc <lb/>multò minor e&longs;t reflexio, quia dum invicem cedunt, aliquo <lb/>tempore durat re&longs;i&longs;tentia, multóque magis minuitur impetus. </s> <lb/> <s id="s.005361">id quod adhuc magis contingit, &longs;i &longs;e invicem conterant, & <lb/>particulæ aliquæ majores re&longs;iliant. </s> </p> <p type="main"> <s id="s.005362">Quapropter cum incerta &longs;emper, & varia &longs;it complexio hu­<lb/>ju&longs;modi re&longs;i&longs;tentiarum & ce&longs;&longs;ionum, juxta variam corporum <lb/>temperationem; ut reflexionis certæ regulæ &longs;tatuantur, &longs;emo­<lb/>tis iis, quæ percu&longs;&longs;ionis accidunt, con&longs;ideranda & a&longs;&longs;umenda <lb/>e&longs;t re&longs;i&longs;tentia ab&longs;que ullâ ce&longs;&longs;ione, perinde atque &longs;i duri&longs;&longs;imo­<lb/>rum corporum colli&longs;io fieret. </s> </p> <p type="main"> <s id="s.005363">Cum itaque in reflexione, corporis duri in aliud decidentis, <lb/>aut impacti, motus ad novam lineam dirigatur, nova hæc di­<lb/>rectio oritur ex lineâ directionis prioris motûs quatenus compa­<lb/>ratâ cum plano reflectente, videlicet quatenus ad illud incli­<lb/>natur, & cum eo angulum con&longs;tituit in puncto contactûs. </s> <lb/> <s id="s.005364">Quando autem &longs;uperficies corporis reflectentis eo loco, ubi <lb/>percutitur, plana non e&longs;t, &longs;ed convexa (&longs;imile quid dicendum, <lb/>&longs;i cava fuerit) &longs;ivè &longs;phærica &longs;it, &longs;ivè Elliptica, &longs;ivè Conica, re­<lb/>verâ nullum ibi e&longs;t planum reflectens (ni&longs;i fortè huju&longs;modi <lb/>convexas &longs;uperficies ex plurimis planis minimis con&longs;titui fin­<lb/>gas, quemadmodum circuli peripheriam ex infinitis lineolis <lb/>rectis, quarum rectitudo &longs;en&longs;um omnem fugiat, componi opi­<lb/>nantur aliqui) &longs;ed communiter mente concipiunt planum, <lb/>quod in puncto percu&longs;&longs;ionis tangeret &longs;uperficiem con­<lb/>vexam; & ex illo angulos tùm Incidentiæ, tùm Reflexionis <lb/>definiunt. </s> </p> <p type="main"> <s id="s.005365">Porrò planum reflectens (quod quidem &longs;pectat ad novam <lb/>directionem motûs &longs;tatuendam corpori percutienti, quem po­<lb/>namus e&longs;&longs;e globum) ita &longs;e habere videtur, ac &longs;i in globum <lb/>quie&longs;centem motu parallelo impingeretur ip&longs;um planum tanto <lb/>impetu, quanto impetu fertur globus adversùs planum: &longs;i enim <lb/>ex duobus colli&longs;is alterum quie&longs;cit, alterum movetur, ad ra­<lb/>tionem ictûs nil refert, utrum illorum quie&longs;cat, aut moveatur, <lb/>modò cætera omnia paria fuerint; ad rationem verò reflexio­<lb/>nis, quà reflexio e&longs;t, attenditur poti&longs;&longs;imum ordinatio novæ li­<lb/>neæ motûs, quæ ex ob&longs;taculi po&longs;itione de&longs;umitur, adeò ut no­<lb/>va linea directionis, quatenus à plano reflectente pendet, & à <pb pagenum="729" xlink:href="017/01/745.jpg"/>centro gravitatis de&longs;cendentis, aut à centro Impetûs corpo­<lb/>ris impacti, certâ Ratione re&longs;piciat priorem lineam directio­<lb/>nis. </s> <s id="s.005366">Eo igitur ip&longs;o quod concipimus planum reflectens mo­<lb/>veri motu parallelo, hoc e&longs;t &longs;ervatâ po&longs;itione priori po&longs;i­<lb/>tioni parallelâ, adversùs globum quie&longs;centem, manife&longs;tum <lb/>e&longs;t novam determinationem ex illo ortam e&longs;&longs;e versùs lineam <lb/>plano perpendicularem, ex puncto contactûs erectam: nam <lb/>impetus, qui ex illo plani motu imprimeretur globo <lb/>quie&longs;centi, hunc deferret per lineam jungentem punctum <lb/>contactûs cum centro globi: hæc autem linea ex centro <lb/>&longs;phæræ ducta ad punctum contactûs plani e&longs;t ip&longs;i plano per­<lb/>pendicularis, ut ex Sphæricis con&longs;tat. </s> <s id="s.005367">Quamvis igitur res <lb/>contrario modo &longs;e habeat, &longs;cilicet planum quie&longs;cat, & glo­<lb/>bus moveatur, directio tamen, quatenus orta ex re&longs;i&longs;tentiâ <lb/>plani, eodem modo &longs;e habet, & e&longs;t versùs perpendicula­<lb/>rem ex puncto contactûs. </s> <s id="s.005368">Sed quia cum impetu globi ut <lb/>plurimùm manet adhuc prior directio, ex his duabus mo­<lb/>tuum ordinationibus oritur tertia mixta; ita ut neque ad <lb/>perpendiculum reflectatur, ni&longs;i incidentiæ linea perpendi­<lb/>cularis fuerit, neque recta in&longs;titutum iter pro&longs;equatur. </s> </p> <p type="main"> <s id="s.005369">Quoniam igitur ex puncto contactûs innumeræ lineæ exi­<lb/>re po&longs;&longs;unt cùm variâ inclinatione ad planum reflectens, <lb/>nec ulla peculiaris e&longs;t cau&longs;a, cur ad hos potiùs, quàm ad <lb/>illos angulos, reflectatur corpus percutiens, qui majores <lb/>&longs;int aut minores angulo incidentiæ, quem linea directio­<lb/>nis motûs con&longs;tituit cum eodem plano reflectente; reli­<lb/>quum e&longs;t, ut angulo incidentiæ æqualis &longs;it angulus refle­<lb/>xionis; hæc &longs;iquidem linea ad angulum priori æqualem re­<lb/>flexa unica e&longs;t, quæ inter innumeras alias lineas magis aut <lb/>minùs inclinatas potiori quodam jure exigitur à naturâ <lb/>prioris directionis leges, quoad fieri pote&longs;t, retinente. </s> <lb/> <s id="s.005370">Non e&longs;t autem nece&longs;&longs;e tyronem monere, duas lineas, di­<lb/>rectam & reflexam in puncto reflexionis concurrentes e&longs;&longs;e <lb/>in uno & eodem plano, ut con&longs;tat ex. </s> <s id="s.005371">2. lib. 11. ab hoc <lb/>autem plano &longs;ecari planum reflectens, ac proinde ad lineam, <lb/>quæ e&longs;t duorum planorum communis &longs;ectio, referendam e&longs;&longs;e <lb/>linearum illarum inclinationem. </s> </p> <p type="main"> <s id="s.005372">Quare &longs;it plani reflectentis, & plani, in quo fit motus, <pb pagenum="730" xlink:href="017/01/746.jpg"/>communis &longs;ectio linea AB, & &longs;uper planum ad rectos angulos <lb/><figure id="id.017.01.746.1.jpg" xlink:href="017/01/746/1.jpg"/><lb/>cadat linea directionis prioris DC, <lb/>per quam movetur globus tanto <lb/>impetu, ut ni&longs;i planum ob&longs;taret, <lb/>ulteriùs procederet rectà versùs E: <lb/>Verùm quoniam à plano ob&longs;i&longs;ten­<lb/>te repellitur per lineam perpendi­<lb/>cularem CD, nova hæc determi­<lb/>natio ad motum e&longs;t omninò & <lb/>adæquatè oppo&longs;ita priori directioni DC, ideóque ictus e&longs;t va­<lb/>lidi&longs;&longs;imus propter maximam re&longs;i&longs;tentiam. </s> <s id="s.005373">Hinc quia ex re­<lb/>&longs;i&longs;tentiâ oritur reflexio, maxima e&longs;t reflexio, quæ fit per li­<lb/>neam perpendicularem, nihil enim remanet de priori directio­<lb/>ne: in hoc quippe comparantur invicem reflexiones, ut illa <lb/>major dicatur, in qua nova motûs ordinatio magis minuit prio­<lb/>rem directionem, ut &longs;cilicet minùs pergat ad eam partem, ad <lb/>quam ferebatur motu directo corpus percutiens. </s> <s id="s.005374">In reflexione <lb/>autem perpendiculari ita tollitur prior directio, ut nullo pacto <lb/>globus, qui ex D per DC movebatur, ampliùs versùs E ten­<lb/>dat. </s> <s id="s.005375">Cùm ergo nova ordinatio &longs;it per perpendicularem CD <lb/>ad angulos rectos, manife&longs;tò con&longs;tat, angulum reflexionis e&longs;&longs;e <lb/>æqualem angulo incidentiæ; nam omnes anguli recti &longs;unt <lb/>æquales. </s> </p> <p type="main"> <s id="s.005376">At moveatur corpus per lineam FC, & fiat incidentiæ an­<lb/>gulus FCB acutus: ni&longs;i planum re&longs;i&longs;teret, progrederetur cor­<lb/>pus juxta eandem directionem ultra C in G; quo motu rece­<lb/>dens à puncto C partim tenderet à C versùs A, partim à C ver­<lb/>sùs E, ita ut à lineâ CA di&longs;taret intervallo AG, à lineâ au­<lb/>tem CE intervallo EG; e&longs;&longs;et enim directio CG æquivalens <lb/>directioni mixtæ ex CA, & CE. <!-- KEEP S--></s> <s id="s.005377">Verùm nova motûs ordina­<lb/>tio à plano reflectente, quatenus opponitur ulteriori motui, e&longs;t <lb/>per lineam perpendicularem CD; hæc autem priori directio­<lb/>ni FCG adver&longs;atur &longs;olùm, prout æquivalet Directioni CE <lb/>(nam quatenus æquivalet directioni CA, non illi opponitur; <lb/>globo &longs;cilicet, qui per CA moveretur, planum non re&longs;i&longs;teret, <lb/>nec illum reflecteret) ac propterea dat oppo&longs;itam directionem <lb/>CD, cujus longitudinem ponamus æqualem ip&longs;i CE. <!-- KEEP S--></s> <s id="s.005378">Manen­<lb/>te igitur directione per CA, & directione CE mutatâ in CD, <pb pagenum="731" xlink:href="017/01/747.jpg"/>e&longs;t ex utrâque mixta directio CH, &longs;ecundùm quam movetur <lb/>corpus reflexum. </s> <s id="s.005379">Quoniam itaque directionum &longs;ingularum <lb/>men&longs;uræ &longs;unt CA, & CE, per A ducatur parallela ip&longs;i DE; & <lb/>per E, atque per D, ducantur EG & DH ip&longs;i CA parallelæ. </s> <lb/> <s id="s.005380">E&longs;t ergo rectangulum HE; & quia CD a&longs;&longs;umpta e&longs;t æqualis <lb/>ip&longs;i CE, etiam AH & AG &longs;unt illis æquales. </s> <s id="s.005381">Quapropter cum in <lb/>triangulis CAH, CAG rectangulis, latera AC & AG æqualia <lb/>&longs;int lateribus AC & AH, atque angulus comprehen&longs;us ad A <lb/>&longs;it rectus, per 4. lib. 1. angulus ACH (qui e&longs;t angulus Re­<lb/>flexionis) e&longs;t æqualis angulo ACG: at angulo ACG æqualis <lb/>e&longs;t ad verticem angulus incidentiæ FCB, per 15. lib. 1: ergo <lb/>angulo FCB incidentiæ æqualis e&longs;t ACH angulus re­<lb/>flexionis. </s> </p> <p type="main"> <s id="s.005382">Eâdem methodo, &longs;i angulus incidentiæ fuerit ICB, often­<lb/>demus angulum reflexionis KCA e&longs;&longs;e illi æqualem; quando­<lb/>quidem directio CM mutatur in CN, & manet directio CA; ac <lb/>propterea directio mixta ex CN, & CA, e&longs;t CK. <!-- KEEP S--></s> <s id="s.005383">Atque ita de <lb/>cæteris. </s> </p> <p type="main"> <s id="s.005384">Ex quibus ob&longs;ervabis, quò acutior fuerit angulus inciden­<lb/>tiæ, in reflexione ita mi&longs;ceri novam directionem cum anti­<lb/>quâ, ut magis prævaleat antiqua; nova &longs;iquidem ad anti­<lb/>quam, &longs;ecundùm id, quod de illâ remanet, &longs;e habet ut <lb/>Sinus Rectus anguli incidentiæ ad Sinum Complementi. </s> <lb/> <s id="s.005385">Nam &longs;i incidentiæ angulus &longs;it FCB, & illi æqualis <lb/>HCA, nova directio CD, hoc e&longs;t AH ad antiquæ <lb/>re&longs;iduum CA, &longs;e habet ut HA ad AC: Sin autem in­<lb/>cidentiæ angulus fuerit ICB, hoc e&longs;t illi æqualis angu­<lb/>lus reflexionis KCA, nova directio ad id, quod de anti­<lb/>quâ remanet, e&longs;t ut KA ad CA. <!-- KEEP S--></s> <s id="s.005386">E&longs;t autem major Ratio <lb/>AC ad AK minorem, quàm eju&longs;dem AC ad AH majo­<lb/>rem per 8. lib. 5. Quare quandiu angulus incidentiæ mi­<lb/>nor e&longs;t &longs;emirecto, majus e&longs;t re&longs;iduum antiquæ directionis <lb/>(attentè ob&longs;erva me de &longs;olâ directione loqui) quâm nova <lb/>ordinatio: ubi fuerit angulus &longs;emirectus, &longs;unt æquales; <lb/>&longs;i angulus incidentiæ fuerit &longs;emirecto major, nova ordina­<lb/>tio major e&longs;t eo, quod remanet de antiquâ directione: ubi <lb/>demum fuerit angulus rectus in perpendiculari inciden­<lb/>tiâ, nova directio ad priorem &longs;e habet ut Radius ad <pb pagenum="732" xlink:href="017/01/748.jpg"/>nihil, motus enim reflexus nihil retinet de priori di­<lb/>rectione. </s> </p> <p type="main"> <s id="s.005387">Antè tamen quàm in hac di&longs;putatione procedamus, mentis <lb/><figure id="id.017.01.748.1.jpg" xlink:href="017/01/748/1.jpg"/><lb/>oculos tanti&longs;per in globum <lb/>C, à quo percutitur planum <lb/>AB, convertamus; hacte­<lb/>nus enim univer&longs;a contem­<lb/>platio in meris lineis ver&longs;a­<lb/>ta e&longs;t. </s> <s id="s.005388">Et quidem &longs;i directio­<lb/>nis linea &longs;it RS perpendi­<lb/>cularis tran&longs;iens per globi <lb/>centrum C, & punctum <lb/>contactûs S, nulla e&longs;&longs;e po­<lb/>te&longs;t difficultas, quin per <lb/>eandem lineam SR re&longs;iliat <lb/>ad angulos rectos. </s> <s id="s.005389">Sed &longs;i in <lb/>planum obliquè incidat li­<lb/>nea directionis globi per <lb/>centrum C deducta, & &longs;it MN; certum e&longs;t in plano AB <lb/>punctum N, in quod directionis linea MC producta incurrit, <lb/>non e&longs;&longs;e punctum contactûs; alioquin linea à globi centro <lb/>ducta ad punctum contactûs, caderet ad angulos inæquales ex <lb/>hypothe&longs;i, cum tamen angulos rectos con&longs;tituere demon&longs;tretur <lb/>in Sphæricis. </s> <s id="s.005390">E&longs;t igitur contactus in puncto S, extra lineam <lb/>directionis centri, ideóque angulus reflexionis non e&longs;t BNQ <lb/>æqualis angulo ANM. </s> <s id="s.005391">Propterea in globo attendendum e&longs;t <lb/>punctum S, à quo reipsâ percutitur planum; & &longs;icuti in circu­<lb/>lo globum bifariam dividente punctum I delatum e&longs;t per li­<lb/>neam MI, ita punctum S per lineam OS ip&longs;i MI parallelam <lb/>(pono hìc globum non rotari dum movetur, &longs;ed recto itinere <lb/>deduci) venit ad contactum & percu&longs;&longs;ionem plani. </s> <s id="s.005392">Cum igi­<lb/>tur OS & MN &longs;int parallelæ, anguli OSA, & MNA &longs;unt <lb/>æquales: & &longs;icuti &longs;i punctum I &longs;olitarium e&longs;&longs;et, atque juxta <lb/>&longs;uam directionem veniret in N, reflecteretur per NQ, ut re­<lb/>flexionis angulus QNB e&longs;&longs;et æqualis angulo Incidentiæ <lb/>MNA; ita punctum S globi reflectitur per SP, & angulus <lb/>reflexionis PSB æqualis e&longs;t incidentiæ angulo OSA; ac pro­<lb/>inde anguli PSB, & QNB &longs;unt æquales inter &longs;e. </s> <s id="s.005393">Centrum <pb pagenum="733" xlink:href="017/01/749.jpg"/>igitur C cùm in directione MN haberet directionem mixtam <lb/>ex directione CS versùs planum, & directione SB, eum im­<lb/>pediatur à globi &longs;oliditate ne ad planum ulteriùs accedat, mu­<lb/>tatâ directione CS in CR, atque retentâ priore directione SB, <lb/>habet directionem mixtam CH omnino &longs;imilem directioni <lb/>puncti S; atque propterea CH e&longs;t parallela ip&longs;i SP. <!-- KEEP S--></s> <s id="s.005394">Quod &longs;i <lb/>globus rotari intelligatur, loco linearum, de quibus hactenus <lb/>fuit &longs;ermo, concipe plana, in quibus puncta illa &longs;uas periodos <lb/>de&longs;criberent in motu rotationis, & plana illa e&longs;&longs;ent ad planum <lb/>reflectens &longs;imiliter inclinata, ut de lineis dictum e&longs;t. </s> </p> <p type="main"> <s id="s.005395">In cæteris verò corporibus non rotundis idem de eorum re­<lb/>flexione dicendum e&longs;t, &longs;ervatâ analogiâ, quantum ferre pote&longs;t <lb/>anomala eorum figura, & di&longs;par partium po&longs;itio circa centrum <lb/>gravitatis aut magnitudinis: in multis enim huju&longs;modi æqua­<lb/>litas angulorum incidentiæ & reflexionis non exactè &longs;ervatur. </s> <lb/> <s id="s.005396">Sic ha&longs;tam &longs;i obliquè contorqueas in rupem, non modò inæ­<lb/>qualitatem angulorum deprehendes, &longs;ed vix reflexionem fieri <lb/>admittes; quia videlicet extremo ha&longs;tæ calce rupem tangente, <lb/>reliquæ partes habentes circa centrum gravitatis inæqualia mo­<lb/>menta, valdè turbant motum: &longs;olùm autem quando ha&longs;ta in <lb/>planum impingitur, aut cadit, ad perpendiculum, &longs;ervata re­<lb/>flexionis regulâ ad angulos rectos re&longs;ilit; quia tunc partes om­<lb/>nes circa centrum gravitatis paria habent momenta. </s> <s id="s.005397">Hæc au­<lb/>tem momentorum diver&longs;itas in globo non reperitur, ni&longs;i fortè <lb/>aut deficiat à perfectâ rotunditate, aut centrum magnitudinis <lb/>non &longs;it idem cum centro gravitatis, adeò ut linea punctum <lb/>contactûs cum centro gravitatis, jungens non &longs;it plano re­<lb/>flectenti perpendicularis; tunc enim perturbaretur globi <lb/>reflexio. </s> </p> <p type="main"> <s id="s.005398">Ex dictis &longs;atis apertè con&longs;tat reflexionem non ex impetu de­<lb/>&longs;umendam e&longs;&longs;e, &longs;ed ex directione motus, cui opponitur corpus <lb/>reflectens, juxta hu us po&longs;itionem perpendicularem aut obli­<lb/>quam: multus enim impetus aliquando officere pote&longs;t æquali­<lb/>tati angulorum, &longs;i ex colli&longs;ione corporis impacti cum corpore <lb/>reflectente, aut alterutrum, aut utrumque notabiliter cedat, <lb/>adeò ut non contingat &longs;incera reflexio. </s> <s id="s.005399">Cæterùm cum &longs;emper <lb/>in reflexione &longs;it nova directio priori directioni oppo&longs;ita, ali­<lb/>quid impetûs perit pro Ratione oppo&longs;itionis. </s> <s id="s.005400">Ex quo fit in re-<pb pagenum="734" xlink:href="017/01/750.jpg"/>flexione ad angulos magis acutos impetum minori decremento <lb/>minui, quia nova directio minùs opponitur antiquæ, & minùs <lb/>impeditur motus; idcirco globus ad angulum valde acutum re­<lb/>flexus, &longs;i offendat in motu reflexo aliquem obicem, multò va­<lb/>lidiùs illum percutit, quàm &longs;i ad angulum minùs acutum re­<lb/>flecteretur, quia, cæteris paribus, majore impetu &longs;uper&longs;tite <lb/>ictum infligit. </s> </p> <p type="main"> <s id="s.005401">Quapropter obvium e&longs;t cuique rationem reddere omnium, <lb/>quæ in pilæ ludo contingunt circa &longs;altus in pavimento, in quod <lb/>pila emi&longs;&longs;a decidit, & reflexiones ad parietem, in quem illa <lb/>impingitur. </s> <s id="s.005402">Duo tamen poti&longs;&longs;imùm ob&longs;ervare placet. </s> <s id="s.005403">Primùm, <lb/>quando pila cadit obliqua in pavimentum non procul à pariete, <lb/>&longs;æpè fit duplex reflexio, altera &longs;cilicet à pavimento, altera à <lb/>pariete: ex quo fit, ut pila aliquando longè altiorem &longs;altum <lb/>edat, &longs;i multum habeat impetus; quia videlicet à pavimento <lb/>re&longs;iliens, &longs;i in parietem non incurreret, lineam curvam in re­<lb/>flexione de&longs;cribens vi &longs;uæ gravitatis impetum extrin&longs;ecùs im­<lb/>pre&longs;&longs;um temperantis, citiùs deprimeretur, & magis à recto tra­<lb/>mite deor&longs;um deflecteret: at quia proximus ponitur e&longs;&longs;e paries, <lb/>linea primò reflexa nondum differt notabiliter à lineâ rectâ; <lb/>atque proinde in &longs;ecundâ reflexione altiùs pila a&longs;&longs;urgit, quàm <lb/>à pavimento di&longs;taret apex lineæ curvæ, quæ ex primâ reflexio­<lb/>ne de&longs;criberetur; nam directio illa &longs;ecunda magis elevata &longs;upra <lb/>horizontem minùs permittit pilam à rectâ lineâ declinare; ut <lb/>in bali&longs;tarum & bombardarum globis cum majori elevatione <lb/>emi&longs;&longs;is con&longs;tat. </s> <s id="s.005404">Deinde quando reticulis luditur, non rarò re­<lb/>ticulum movetur in plano aliquo horizontali, aut valde incli­<lb/>nato (nos Itali dicimus <emph type="italics"/>Tagliare, ò Trinciare una palla<emph.end type="italics"/>) ita ut, <lb/>dum pilam rectâ expellit, illi etiam motum quendam imprimat, <lb/>quo ip&longs;a circa &longs;uum centrum movetur: unde fit, ut, ni&longs;i pilam <lb/>excipias, repellá&longs;que antè, quàm pavimentum attingat, fru&longs;tra <lb/>deinde &longs;altum illius expectes juxta regulas reflexionis, quia ni­<lb/>mirum pila terram tangens, dum pergit moveri circa &longs;uum <lb/>centrum motu orbiculari, nequit à plano impediente recipere <lb/>directionem illam, cujus e&longs;&longs;et capax, &longs;i &longs;olùm &longs;implici motu <lb/>centri mota fui&longs;&longs;et; motus enim peripheriæ globi contrarius e&longs;t <lb/>motui centri. </s> <s id="s.005405">Idem accidit quando pila leviore affrictu funem <lb/>per&longs;tringit; tunc &longs;cilicet concipit motum circularem, adeóque <pb pagenum="735" xlink:href="017/01/751.jpg"/>&longs;altus fallit. </s> <s id="s.005406">Quantum autem in motu valeat directiones com­<lb/>mi&longs;cere, alteram centri rectam, alteram peripheriæ circularem <lb/>&longs;ed oppo&longs;itam, &longs;atis norûnt, qui minoribus orbiculis ludentes <lb/>globum qua&longs;i pendentem ex manu tenent, dúmque illum pro­<lb/>jiciunt, manu ei motum circularem communicant; unde oritur, <lb/>quod, ubi terram globus attigerit, vel &longs;i&longs;tit &longs;e, &longs;i directio peri­<lb/>pheriæ ad motum circularem e&longs;t æqualis directioni centri ad <lb/>motum rectum; vel tardiùs promovetur, quàm &longs;i &longs;olam centri <lb/>directionem haberet, prout directio centri major e&longs;t directione <lb/>peripheriæ, quæ cum primùm terram attingit, apta e&longs;t &longs;uá con­<lb/>ver&longs;ione retrahere centrum versùs projicientem. </s> </p> <p type="main"> <s id="s.005407">Quandoquidem verò in ludicris philo&longs;ophamur, liceat hìc <lb/>vulgarem errorem retegere; quando &longs;cilicet rotundum verticil­<lb/>lum impre&longs;&longs;us impetus in gyrum agit, &longs;i verticillus corruat, mo­<lb/>vetur, quoad impetus extinguatur, &longs;ed ita ut videatur omnino <lb/>in contrarias partes agi, ac priùs: id quod communiter tribuunt <lb/>reflexioni, quia in pavimentum recidit. </s> <s id="s.005408">Nullam hìc reflexio­<lb/>nem intercedere, & eandem permanere motûs directionem, <lb/>memini me aliquando a&longs;&longs;erentem vi&longs;um fui&longs;&longs;e pluribus, qui <lb/>aderant, paradoxum loqui: &longs;ed ubi inter nos convenit eundem <lb/>motum e&longs;&longs;e, quandiu circà axem ita fit convolutio, ut quæ par­<lb/>tes peripheriæ verticilli præcedebant axe in&longs;i&longs;tente, eædem axe <lb/>inclinato & procumbente præcedant; ju&longs;&longs;i aliquas notas peri­<lb/>pheriæ imprimi, ut priores à po&longs;terioribus di&longs;cerni po&longs;&longs;ent; de­<lb/>inde yerticillo corruente, & procumbente ob&longs;ervatum e&longs;t par­<lb/>tes, quæ priores erant in circuitione, ea&longs;dem &longs;ubinde altiùs at­<lb/>tolli à pavimento, atque circa axem eandem fieri conver&longs;io­<lb/>nem: & quia verticilli pes qua&longs;i centrum retinet inclinatam <lb/>peripheriam, illa eadem conver&longs;io circa axem facit, ut peri­<lb/>pheria &longs;ecundùm po&longs;teriores partes &longs;ubinde attingat &longs;ubjectum <lb/>alveolum, adeóque ratione habitâ alveoli videatur in contrarias <lb/>partes ferri ac priùs. </s> <s id="s.005409">Quare cùm nulla &longs;it nova motûs directio <lb/>ex plani oppo&longs;itione, nulla quoque e&longs;t reflexio. </s> </p> <p type="main"> <s id="s.005410">At &longs;i duo corpora &longs;ibi invicem occurrant, &longs;ibi mutuo ob­<lb/>&longs;i&longs;tunt, & diminuto ex re&longs;i&longs;tentiâ impetu, &longs;i quid adhuc re&longs;i­<lb/>duum fuerit impetûs, qui excedat in&longs;itam repugnantiam ex <lb/>gravitate ortam, fit reflexio, aut alterius tantùm, &longs;i in reliquo <lb/>impetus obtundatur, aut utriu&longs;que, &longs;i fuerint &longs;ibi invicem per-<pb pagenum="736" xlink:href="017/01/752.jpg"/>cutiens & percu&longs;&longs;um: ut cùm duo globi &longs;ibi in motu occur­<lb/>runt aut æquali, aut non immodicè inæquali impetu acti. <lb/><emph type="italics"/>Æquali<emph.end type="italics"/> inquam, <emph type="italics"/>impetu,<emph.end type="italics"/> non <emph type="italics"/>æquali velocitate<emph.end type="italics"/>; &longs;i enim inæqua­<lb/>les fuerint globi, fieri pote&longs;t, ut eorum velocitates &longs;int in Re­<lb/>ciprocâ Ratione gravitatum; tunc &longs;cilicet impetus æquales &longs;unt; <lb/>contingere &longs;iquidem pote&longs;t majorem globum tardè quidem <lb/>moveri, &longs;ed multo impetu re&longs;pondente ejus moli, adeò ut ex­<lb/>cedat minoris globi impetum, qui proptereà non præcisè re­<lb/>flectatur, &longs;ed à majore globo & impetum recipiat, & directio­<lb/>nem non ex &longs;olâ re&longs;i&longs;tentiâ definitam, &longs;ed etiam ex ip&longs;ius ma­<lb/>joris globi motu. </s> <s id="s.005411">Id quod &longs;i contingat, minor quidem reflecti­<lb/>tur, &longs;ed qui majore impetu ferebatur, modicam inveniens re­<lb/>&longs;i&longs;tentiam non reflectitur, quia dum minor globus cedit, plu­<lb/>rimum impetûs deperditur à majore; & ubi re&longs;i&longs;tentia minor <lb/>e&longs;t ce&longs;&longs;ione, e&longs;&longs;e nequit reflexio. </s> </p> <p type="main"> <s id="s.005412">Ponamus itaque globos duos tanto impetu actos, ut po&longs;&longs;it <lb/>uterque reflecti. </s> <s id="s.005413">Non placet inter illos ad punctum contactûs <lb/>interjicere planum, ut ex angulis determinetur reflexio; hoc <lb/>enim planum cogitatione nobis ip&longs;i fingimus; &longs;ed, licèt in <lb/>idem res recidat, tamen ad veritatem &longs;inceriùs me acce&longs;&longs;urum <lb/>&longs;pero, &longs;i rem ex ip&longs;is motuum directionibus & re&longs;i&longs;tentiis de&longs;i­<lb/><figure id="id.017.01.752.1.jpg" xlink:href="017/01/752/1.jpg"/><lb/>niero. </s> <s id="s.005414">Quare occurrant &longs;ibi <lb/>globi in puncto A; & illo­<lb/>rum directiones primò eæ <lb/>&longs;int, quæ &longs;ibi maximè ad­<lb/>ver&longs;antes in rectam lineam <lb/>BC coëant: haud dubium <lb/>quin globus V per rectam <lb/>AB, & globus R per rectam <lb/>AC re&longs;iliat, uterque per li­<lb/>neam, quâ venit, jungentem <lb/>cum puncto contactûs Centrum impetûs: &longs;implici enim di­<lb/>rectione alter adversùs alterum fertur, & &longs;ibi toto conatu re­<lb/>pugnant. </s> <s id="s.005415">Deinde obliquus &longs;it morus, & globi R directio &longs;it DE: <lb/>quapropter RE directio quædam e&longs;t mixta ex lineâ maximæ re­<lb/>&longs;i&longs;tentiæ RA, & ex lineâ nullius re&longs;i&longs;tentiæ RI, ita ut partim <lb/>versùs A, partim versùs I tendat: priorem directionem versùs A <lb/>metitur linea RF, po&longs;teriorem versùs I metitur linea FE. <!-- KEEP S--></s> <s id="s.005416">Cum <pb pagenum="737" xlink:href="017/01/753.jpg"/>igitur globus V &longs;olùm priori directioni RF opponatur, hæc <lb/>mutatur in oppo&longs;itam RG, manet autem directio versùs I æqua­<lb/>lis ip&longs;i FE, & e&longs;t GH: propterea motus centri R e&longs;t RH <lb/>parallelus motui puncti A, quod per lineam KA incidens in <lb/>Tangentem MN, reflecteretur ad angulos æquales KAN & <lb/>LAM percurrendo lineam AL. <!-- KEEP S--></s> <s id="s.005417">Simili ratione, &longs;i globi V di­<lb/>rectio &longs;it PO, à globo R occurrente in A reflectitur centrum <lb/>per rectam VS. <lb/></s> </p> <p type="main"> <s id="s.005418"><emph type="center"/>CAPUT XII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005419"><emph type="center"/><emph type="italics"/>Quomodo impetus in percu&longs;sione communicetur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005420">ANtè &longs;atisfaciendum e&longs;t Phy&longs;icis, quàm percu&longs;&longs;ionum con­<lb/>templationem dimittamus. </s> <s id="s.005421">Quoniam percu&longs;&longs;io omnis mo­<lb/>tum antecedentem exigit; motus non habetur ab&longs;que impetu <lb/>concepto aut impre&longs;&longs;o; ex impetu pendet ictus, quo corporis <lb/>percu&longs;&longs;i re&longs;i&longs;tentia aliqua vincitur, &longs;ivè illud totum impellatur, <lb/>&longs;ivè expellatur, &longs;ivè concutiatur, &longs;ivè flectatur, &longs;ivè compri­<lb/>matur, &longs;ivè deprimatur, &longs;ivè di&longs;&longs;iliat in partes earum unione <lb/>&longs;olutâ, &longs;ivè quamcumque aliam vim &longs;ubeat; corporis percu&longs;&longs;i <lb/>partes, vel omnes, vel aliquæ &longs;altem, moveantur, & impetum <lb/>recipiant nece&longs;&longs;e e&longs;t, à quo motus ip&longs;e efficiatur impre&longs;&longs;i impe­<lb/>tûs inten&longs;ioni re&longs;pondens. </s> <s id="s.005422">Quærat autem Phy&longs;icus, cuinam <lb/>tribuenda &longs;it virtus efficiendi impetum corpori percu&longs;&longs;o im­<lb/>pre&longs;&longs;um. </s> </p> <p type="main"> <s id="s.005423">Exi&longs;timabit forta&longs;&longs;e non nemo à virtute eâdem, quæ in cor­<lb/>pore percutiente in&longs;idet, ut &longs;eip&longs;um moveat, effici novum im­<lb/>petum, quo corpus percu&longs;&longs;um impellatur, aut agitetur. </s> <s id="s.005424">Sed <lb/>quid? </s> <s id="s.005425">&longs;i percutiens neque animans &longs;it, cujus in pote&longs;tate po&longs;ita <lb/>&longs;it motio, neque juxta in&longs;itæ gravitatis directionem &longs;eip&longs;um <lb/>agat. </s> <s id="s.005426">Huic certè inhærens facultas &longs;e movendi planè otio&longs;a e&longs;t, <lb/>quippe quæ pror&longs;us immota con&longs;i&longs;teret, ni&longs;i impetum extra­<lb/>neum reciperet. </s> <s id="s.005427">Aliunde igitur quàm ex hac &longs;e movendi facul­<lb/>tate originem ducit impetus corpori percu&longs;&longs;o impre&longs;&longs;us. </s> <s id="s.005428">Dein-<pb pagenum="738" xlink:href="017/01/754.jpg"/>de certum e&longs;t corporis percutientis naturam non priùs imprime­<lb/>re po&longs;&longs;e percu&longs;&longs;o impetum; quàm illud attingat: at in ip&longs;o per­<lb/>cutientis appul&longs;u ea e&longs;t percu&longs;&longs;i re&longs;i&longs;tentia, ut eju&longs;dem percu­<lb/>tientis motum ex ipsâ naturâ provenientem imminuat: cùm <lb/>igitur natura percutientis vix &longs;eip&longs;a movere valeat, quàm te­<lb/>nues habet vires ad vincendam obicis re&longs;i&longs;tentiam? </s> <s id="s.005429">Præterea, <lb/>ni&longs;i facta fuerit notabilis in longiore motu naturali acqui&longs;iti im­<lb/>petûs acce&longs;&longs;io, manife&longs;tò apparet valdè languida & enervata <lb/>percu&longs;&longs;io; &, quamvis &longs;ivè longior, &longs;ivè exiguus motus præ­<lb/>ce&longs;&longs;erit, eadem manens virtus movendi, nec &longs;ibi di&longs;&longs;imilis, va­<lb/>rietatem in &longs;e habet nullam: cum tamen ex di&longs;paribus incre­<lb/>mentis impetûs in motu acqui&longs;iti di&longs;&longs;imiles fiant percu&longs;&longs;iones: <lb/>Non igitur à &longs;olâ in&longs;itâ vi movendi producitur in percu&longs;&longs;o im­<lb/>petus. </s> </p> <p type="main"> <s id="s.005430">Propterea, ut una atque eadem in percu&longs;&longs;ionibus omnibus <lb/>a&longs;&longs;ignetur producti impetûs cau&longs;a, &longs;ivè percutiens &longs;ponte &longs;uâ, <lb/>&longs;ivè per vim &longs;ibi illatam moveatur, percutientis impetum plu­<lb/>res cen&longs;ent dicendum e&longs;&longs;e principium & cau&longs;am effectricem <lb/>impetûs percu&longs;&longs;o impre&longs;&longs;i; ab illo enim, prout major fuerit, aut <lb/>minor, hujus men&longs;uram pendere &longs;atis innotui&longs;&longs;e videtur ex <lb/>quotidianis experimentis. </s> </p> <p type="main"> <s id="s.005431">Verùm, ne raptim in hanc &longs;ententiam pedarius Philo&longs;ophus <lb/>curram, illud me remoratur, quod, &longs;icuti eam e&longs;&longs;e con&longs;tat im­<lb/>petûs naturam, ut illico pror&longs;us pereat, ac motus ce&longs;&longs;at omni­<lb/>no illius corporis, in quo priùs inerat motum efficiens, ita pari­<lb/>ter eodem momento impetum minui nece&longs;&longs;e e&longs;t, eáque Ratio­<lb/>ne, quo momento, & qua Ratione illius eju&longs;dem corporis mo­<lb/>tus ex parte impeditur. </s> <s id="s.005432">Quò igitur magis impeditur percutien­<lb/>tis motus, eò magis eju&longs;dem impetum minui con&longs;equens e&longs;t: <lb/>propterea, quo momento à percutiente attingitur corpus per­<lb/>cu&longs;&longs;um, extenuatur in illo impetus, quia tunc illius motus im­<lb/>peditur; eóque minor evadit in percutiente impetus, quò ma­<lb/>jus invenit impedimentum motûs. </s> <s id="s.005433">Cùm autem effectui tenui­<lb/>tatem importet cau&longs;æ imbecillitas, exiguum utique impetum <lb/>in corpore percu&longs;&longs;o efficere valeret attenuatus percutientis im­<lb/>petus, quo momento accidit appul&longs;us atque alli&longs;io; eóque mi­<lb/>norem impetum reciperet corpus percu&longs;&longs;um, quò magis re­<lb/>&longs;i&longs;tens plus inferret impedimenti motui percutientis, quippe <pb pagenum="739" xlink:href="017/01/755.jpg"/>cujus impetus fieret languidior; neque enim quicquam juvat <lb/>antiqua virtus, &longs;i nunc e&longs;t effœta. </s> <s id="s.005434">Quò igitur magis re&longs;i&longs;tit <lb/>corpus percu&longs;&longs;um, languidiorem ictum exciperet, cum levior <lb/>infirmiórque impetus in eo efficeretur à tenuiore & languidio­<lb/>re percutientis impetu. </s> <s id="s.005435">Sed cum manife&longs;ta refragetur expe­<lb/>rientia validiores ictus à majore re&longs;i&longs;tentia ortos demon&longs;trans, <lb/>quæ&longs;o à Philo&longs;ophis, ut in hac causâ mihi dent hanc veniam, <lb/>ut patiantur me ab eorum placitis aliquantulum di&longs;cedere, nec <lb/>percutientis impetui tribuere facultatem effectricem impetûs <lb/>in corpore percu&longs;&longs;o, lyceo quamvis reclamante; cui &longs;ilentium <lb/>&longs;i tanti&longs;per indicere po&longs;&longs;em, dum me audiret po&longs;tulantem id, <lb/>quod æqui&longs;&longs;imum e&longs;t, ut ne quid huc præjudicati afferat, meam <lb/>forta&longs;&longs;e in &longs;ententiam volens deduceretur. </s> </p> <p type="main"> <s id="s.005436">Cùm itaque nec à virtute movendi, quæ corpori percutien­<lb/>ti inhæret, nec ab impetu eju&longs;dem percutientis effici novum <lb/>impetum in corpore percu&longs;&longs;o, &longs;atis probabili conjecturâ dicen­<lb/>dum videatur, quænam demum erit cau&longs;a impetûs, & eorum, <lb/>quæ impetum con&longs;equuntur, in corpore percu&longs;&longs;o? </s> <s id="s.005437">Ut quæ&longs;tio­<lb/>nibus &longs;atisfiat, quas percu&longs;&longs;iones excitant, nihil &longs;e mihi offert <lb/>vero propius, quàm &longs;i dicamus ex percutiente in corpus per­<lb/>cu&longs;&longs;um migrare impetum, aut totum, aut ex parte, prout alicu­<lb/>jus motûs capax fuerit corpus, quod motui percutientis re&longs;i&longs;tit. </s> <lb/> <s id="s.005438">Si totus impetus à percutiente recedat, hoc neque reflectitur <lb/>ab obice percu&longs;&longs;o, neque quicquam procedit in motu: Si quid <lb/>impetûs in percutiente remaneat, hoc aut juxta in&longs;titutam di­<lb/>rectionem pergit moveri unà cum corpore percu&longs;&longs;o, &longs;ive lentiùs <lb/>illud &longs;equitur, aut aliò reflectitur, pro re&longs;idui impetûs inten­<lb/>&longs;ione, aut vibratur, & concutitur. </s> </p> <p type="main"> <s id="s.005439">Hinc quia gravi&longs;&longs;ima &longs;imul & duri&longs;&longs;ima corpora tantum im­<lb/>petûs obtinere à percutiente nequeunt, quanto opus e&longs;&longs;et, ut <lb/>motum aliquem con&longs;picuum ex percu&longs;&longs;ione reciperent, pro­<lb/>pterea validi&longs;&longs;imè re&longs;i&longs;tunt, & reflectunt, cùm univer&longs;us ferè <lb/>impetus in percutiente remaneat: in corpus enim percu&longs;&longs;um <lb/>non migrat ni&longs;i impetus, qui re&longs;pondeat motui, cujus illud <lb/>tunc e&longs;t capax. </s> <s id="s.005440">Contra verò à corporibus, quæ leviter re­<lb/>&longs;i&longs;tunt, & facilè moventur aliquo motu, aut nihil, aut langui­<lb/>dè reflectitur percutiens; quia illa plurimum impetûs reci­<lb/>piunt, & exiguus impetus in percutiente reliquus e&longs;t. </s> <s id="s.005441">Hinc <pb pagenum="740" xlink:href="017/01/756.jpg"/>pariter globus æqualem in mole & gravitate globum percu­<lb/>tiens eâ directione, quæ per utriu&longs;que globi centra tran&longs;eat, <lb/>con&longs;i&longs;tit in loco, ubi percutit, & percu&longs;&longs;um globum vehemen­<lb/>ter excutit; quia videlicet globus æqualis &longs;atis re&longs;i&longs;tit, & capax <lb/>e&longs;t totius impetûs eum æquali inten&longs;ione afficientis, hic de&longs;ti­<lb/>tuens globum percutientem æquè velocem motum percu&longs;&longs;o <lb/>conciliat, & percutiens omni de&longs;titutus impetu con&longs;i&longs;tit. </s> <s id="s.005442">Sin <lb/>autem percutiatur globus major & gravior; hic quidem (ni&longs;i <lb/>nimia &longs;it gravitatis aut molis differentia) loco cedit; &longs;ed quia ad <lb/>motum æquè velocem plus requirit impetus, quàm illi impri­<lb/>mere valeat globus minor, propterea minore inten&longs;ione af­<lb/>fectus tardiùs movetur, & minorem globum aliquando reflectit. </s> <lb/> <s id="s.005443">Si demum globus major minorem & leviorem percutiat, hic <lb/>languidiùs re&longs;i&longs;tens impetum recipit velociori motui congruum; <lb/>& quia in globo majore adhuc aliquid &longs;upere&longs;t impetûs, ille <lb/>pariter pergit moveri, &longs;ed tardiùs. </s> </p> <p type="main"> <s id="s.005444">At, inquis, impetus ex eo genere e&longs;t, quod Accidentia tan­<lb/>quam partes complectitur: Accidentia autem ex &longs;ubjecto in <lb/>&longs;ubjectum non tran&longs;ire, ip&longs;i &longs;cholarum parietes clamant. </s> <s id="s.005445">Mul­<lb/>ta i&longs;tiu&longs;modi, non diffiteor, dicuntur in &longs;cholis: verùm an &longs;atis <lb/>examinata, momentóque &longs;uo ponderata fuerint, ignoro: non <lb/>pauca quippe habemus de manu, ut aiunt, in manum tradita, <lb/>non ad aurificis &longs;tateram revocata, &longs;ed populari trutinæ per­<lb/>mi&longs;&longs;a. </s> <s id="s.005446">In illis certè Accidentium generibus, quæ po&longs;tremis no­<lb/>vem Categoriis comprehenduntur, &longs;i &longs;ex demas, Relationem, <lb/>Actionem, Pa&longs;&longs;ionem, Ubi, Quando, Situm, quos alij (libera­<lb/>liter-ne? </s> <s id="s.005447">dicam, an prodigè?) <emph type="italics"/>Modos<emph.end type="italics"/> certæ naturæ, à qua avel­<lb/>li nequeunt, affixos appellant, alij minimo contenti, & parciùs <lb/>philo&longs;ophantes, nihil e&longs;&longs;e præter mera nomina, aut ab&longs;tractas <lb/>à rebus inter &longs;e comparatis intelligentias exi&longs;timant; vix tria <lb/>reliqua genera Quantitas, Qualitas, Habitus con&longs;tituere con­<lb/>trover&longs;iam po&longs;&longs;unt. </s> <s id="s.005448">Et quidem de Habitu nullus videtur re­<lb/>lictus ambigendi locus; quis enim neget potui&longs;&longs;e Ther&longs;item eâ­<lb/>dem Achillis galeâ, eodémque thorace armari, & regiâ chla­<lb/>myde &longs;ervum indui? </s> <s id="s.005449">mutatâ &longs;cilicet armorum aut indumento­<lb/>rum Ubicatione comparatâ cum hominis, qui armatus dicitur, <lb/>aut ve&longs;titus, Ubicatione & po&longs;itione. </s> <s id="s.005450">Quantitatem verò, qua <lb/>locus ob&longs;idetur (nam de Numero, qui præter individua cogita-<pb pagenum="741" xlink:href="017/01/757.jpg"/>tioni ea complectenti &longs;ubjecta nihil e&longs;t, non attinet dicere) <lb/>quam multi à materiâ non dividunt? </s> <s id="s.005451">quot Philo&longs;ophi &longs;uam &longs;in­<lb/>gulis corporeis rebus tribuunt quantitatem? </s> <s id="s.005452">De &longs;olâ igitur <lb/>Qualitate oriri pote&longs;t quæ&longs;tio: cujus tamen &longs;pecies aliæ me­<lb/>ram membrorum aut terminorum corporis collocationem & <lb/>conformationem dicunt, ut Forma & Figura; aliæ particula­<lb/>rum in extimâ &longs;uperficie po&longs;itionem, ut a&longs;peritas & lævor; aliæ <lb/>earumdem toto corpore diffu&longs;arum complexionem, ut molli­<lb/>tudo & durities, raritas & den&longs;itas; aliæ non ni&longs;i intelligentiâ <lb/>&longs;ecretæ accidere dicuntur Naturæ, cuju&longs;modi non paucæ Na­<lb/>turales Potentiæ & Impotentiæ; aliæ Patibiles Qualitates aut <lb/>Pa&longs;&longs;iones immi&longs;&longs;ione corpu&longs;culorum effluentium communican­<lb/>tur, quemadmodum Odores & Sapores, & fortè etiam quas Pri­<lb/>mas Qualitates vocant. </s> </p> <p type="main"> <s id="s.005453">Sed quicquid tandem de huju&longs;modi Accidentibus a&longs;&longs;erere <lb/>placeat (neque enim hìc de iis philo&longs;ophandi e&longs;t locus) ultro <lb/>demus ea e&longs;&longs;e, quæ licèt à &longs;ub&longs;tantiâ di&longs;tinguantur, per &longs;e ta­<lb/>men &longs;tare nequeant, & nece&longs;&longs;ariò &longs;ubjectam aliquam naturam <lb/>afficiant, in qua inhæreant: verùm Qualitates omnes (ni&longs;i ex <lb/>earum genere &longs;int, quos Modos appellant, quia Actuales De­<lb/>terminationes, cuju&longs;modi &longs;unt cogitationes, appetitiones, & <lb/>motus, quibus actio vitæ continetur) quid prohibet nunc huic, <lb/>mox illi &longs;ubjecto inhærere, quemadmodum in locum pereun­<lb/>tis Cau&longs;æ Effectricis, cujus virtute hactenus con&longs;ervabantur, <lb/>aliam &longs;ub&longs;titui cau&longs;am, cujus vi adhuc permaneant, omnes fa­<lb/>temur? </s> <s id="s.005454">Nonne causâ effectrice magis indigent Accidentia, <lb/>quàm Materiali & Subjectivâ? </s> <s id="s.005455">Divinâ &longs;iquidem vi accidentia <lb/>à Subjecto avul&longs;a permanere po&longs;&longs;e docemur ex My&longs;teriis Eu­<lb/>chari&longs;ticis; at &longs;inè ullâ causâ effectrice con&longs;i&longs;tere nullatenus <lb/>po&longs;&longs;unt: hanc &longs;ubinde permutant citrà Naturæ incommodum; <lb/>quidni & &longs;ubjectum? </s> <s id="s.005456">Nihil igitur extra modum ab&longs;onum & ab­<lb/>&longs;urdum loquatur, qui impetum migrantem ex percutiente in <lb/>percu&longs;&longs;um ita &longs;ubjectum mutare dixerit, quemadmodum om­<lb/>nes novum impetum à percutiente malleo produci in percu&longs;&longs;o <lb/>& excu&longs;&longs;o globo opinantes, aliam eju&longs;dem impetûs, quandiu <lb/>durat, cau&longs;am, à qua con&longs;ervetur, ultro admittunt. </s> </p> <p type="main"> <s id="s.005457">Quamvis autem hoc cæteris qualitatibus ratum ac firmum <lb/>e&longs;&longs;et, quod ita &longs;ubjecto, cui &longs;emel inhæ&longs;erint, affigantur, ut <pb pagenum="742" xlink:href="017/01/758.jpg"/>aut in illo in&longs;idere nece&longs;&longs;e &longs;it, aut interire; impetui tamen <lb/>privatam legem à Naturâ irrogatam fui&longs;&longs;e non e&longs;t incon­<lb/>gruum, quippe qui motui efficiendo, & locorum commutatio­<lb/>ni, tamquam proxima cau&longs;a, de&longs;tinatus e&longs;t; &longs;i enim illi corpo­<lb/>rum tran&longs;latio tribuenda e&longs;t, quidni & ip&longs;e à corpore, quod <lb/>jam commovere nequit ob re&longs;i&longs;tentiam, in aliud corpus proxi­<lb/>mum faciliùs mobile tran&longs;mittat, ut &longs;ubmoveatur impedimen­<lb/>tum? </s> <s id="s.005458">Neque mihi videor temerè in hanc &longs;ententiam di&longs;ce&longs;&longs;i&longs;&longs;e: <lb/>ob&longs;ervavi &longs;cilicet quàm multum inter&longs;it in vehementi brachij <lb/>projectione, &longs;i verè lapidem manu longiùs excutias, ac &longs;i tan­<lb/>tummodo, eâdem quidem contentione, &longs;ed manu vacuâ, te la­<lb/>pidem jactare mentiaris: hoc enim po&longs;tremum &longs;inè dolore non <lb/>accidit, quia impetus à brachio in lapidem jactandum transfe­<lb/>rendus &longs;i in brachio permaneat, hoc &longs;ecum rapit, & nexum <lb/>di&longs;trahit, quo tenetur cum humero colligatum. </s> <s id="s.005459">At ne fortè <lb/>me potiùs opinionis commento, quàm re ductum &longs;u&longs;piceris <lb/>(quamquam & alij hunc eundem brachij dolorem experientes <lb/>non &longs;emel probârunt) bali&longs;tæ arcum chalybeum intento nervo <lb/>inflecte, ac &longs;æpiùs, nullo adjecto globo aut telo, quod explodat <lb/>& ejiciat, &longs;ubmoto nervi adducti retinaculo dimitte: an diutiùs <lb/>inani hoc ludo uti licebit? </s> <s id="s.005460">&longs;excenties utique & millies bali&longs;tâ <lb/>hac globos argillaceos ejaculaberis citrà arcûs detrimentum; <lb/>&longs;ed non item &longs;ine incommodo &longs;æpiùs vacuum nervum dimittes, <lb/>quin arcus ip&longs;e in periculum ac di&longs;crimen vocetur, ne facilè <lb/>di&longs;rumpatur: impetus &longs;iquidem, quem mi&longs;&longs;ili imprimere opor­<lb/>tuit, in arcu, dum &longs;e&longs;e vi ela&longs;ticâ re&longs;tituit, permanens illum va­<lb/>lidiùs concutit, ac &longs;æpiùs labefactans demum diffindit. </s> </p> <p type="main"> <s id="s.005461">Quapropter, cùm ex projectionibus &longs;atis habeamus argumen­<lb/>ti, po&longs;&longs;e impetum ex projiciente migrare in projectum, quo mo­<lb/>mento projicitur; cur non item poterit impetus ex percutiente <lb/>in percu&longs;&longs;um tran&longs;ire, quo momento percutitur, prout hoc mo­<lb/>tum aliquem concipere pote&longs;t pro impetûs Ratione? </s> </p> <p type="main"> <s id="s.005462">Neque ut percu&longs;&longs;i impetum à percutientis virtute tunc pri­<lb/>mò productum ad&longs;truas, conferenda e&longs;t Percu&longs;&longs;io cum Impul­<lb/>&longs;ione; non enim par e&longs;t in Percu&longs;&longs;ione aut Projectione, atque in <lb/>Simplici Impul&longs;ione aut Tractione philo&longs;ophandi ratio: Poten­<lb/>tia enim corpori impul&longs;o aut raptato applicata quandiu cum <lb/>illo nectitur, & &longs;e, & illud movet qua&longs;i corpus unum ex utro-<pb pagenum="743" xlink:href="017/01/759.jpg"/>que conflatum: propterea &longs;icut mu&longs;culi in animante o&longs;&longs;a &longs;ibi <lb/>cohærentia attollentes & &longs;e movent, & o&longs;&longs;a; ita potentia Vecti <lb/>applicata & &longs;e movet, & vectem, & pondus, atque equi cur­<lb/>rui adjuncti non modò &longs;eip&longs;i, &longs;ed & currum, trahentes movent. </s> <lb/> <s id="s.005463">At Percu&longs;&longs;io &longs;æpè corpus percu&longs;&longs;um procul à percutiente ejicit, <lb/>quemadmodum & Projectio. </s> <s id="s.005464">Quod &longs;i cum Percu&longs;&longs;ione junga­<lb/>tur Impul&longs;io (quæ &longs;emper Projectionem præcedit) impetus in <lb/>Impul&longs;ione producitur à potentiâ impellente; &longs;ed &longs;icut momen­<lb/>to Projectionis qui erat in projiciente impetus, migrat in pro­<lb/>jectum, quod di&longs;cedit; ita in Percu&longs;&longs;ione primo Percu&longs;&longs;ionis mo­<lb/>mento tran&longs;it impetus in corpus percu&longs;&longs;um pro ejus capacitate: <lb/>quod &longs;i præterea impellatur à corpore percutiente, cujus motus <lb/>juxta &longs;uam directionem procedat, & urgeat partes corporis per­<lb/>cu&longs;&longs;i (ut in iis, quæ deprimuntur, aut comprimuntur contin­<lb/>git & cùm &longs;ublicas, dum panguntur, fi&longs;tuca ex ca&longs;u non re&longs;i­<lb/>liens impellit) impetum aliquem habet ab impellente pro­<lb/>ductum præter impetum ab eodem tamquam percutiente, ip&longs;o <lb/>percu&longs;&longs;ionis momento communicatum: &longs;ed qui ab impellente <lb/>efficitur, non admodum multus e&longs;t, &longs;i cum eo componatur, qui <lb/>ex percu&longs;&longs;ione habetur. </s> </p> <p type="main"> <s id="s.005465">Simile quid Impul&longs;ioni, quæ Percu&longs;&longs;ionem &longs;equitur, habe­<lb/>tur in Tractione, quam Excur&longs;us præce&longs;&longs;it, in quo acqui&longs;itus <lb/>e&longs;t impetus: quo enim momento Excur&longs;us ce&longs;&longs;at, & incipit <lb/>Tractio, tran&longs;it impetus, & minuitur in trahente; ut &longs;i lapis in <lb/>pavimento jacens fune jungatur alteri lapidi paulò minori, fu­<lb/>nis autem orbiculo ver&longs;atili in&longs;ideat, & lapis ille minor cadens, <lb/>donec funem intendat, impetum ex motu acquirat; &longs;tatim ac <lb/>intentus e&longs;t funis, & lapis jacens de&longs;cendentis lapidis motui re­<lb/>&longs;i&longs;tit impetus acqui&longs;itus migrat ad vincendam jacentis lapidis <lb/>re&longs;i&longs;tentiam, atque acceptâ à trahentis motu directione cogi­<lb/>tur a&longs;cendere, quandiu alter de&longs;cendit, & hunc aliquantulum <lb/>trahit; &longs;ed impetu impre&longs;&longs;o langue&longs;cente in lapide graviore hic <lb/>de&longs;cendit, & &longs;ur&longs;um vici&longs;&longs;im rapit eum, à quo vim pa&longs;&longs;us fue­<lb/>rat. </s> <s id="s.005466">Sic potentia velociter languidum funem intendens mul­<lb/>tum concipit impetum, quem ponderi adnexo imprimit, <lb/>dum illo de&longs;tituitur, cum primùm re&longs;i&longs;tentiam patitur, &longs;ed & <lb/>aliam impetûs particulam trahendo producit atque efficit in <lb/>pondere. </s> </p> <pb pagenum="744" xlink:href="017/01/760.jpg"/> <p type="main"> <s id="s.005467">Cum igitur duplex &longs;it in motu &longs;ubmovendorum impedimen­<lb/>torum genus, alia, videlicet, quæ inchoatum motum ab­<lb/>rumpunt, alia quæ ob&longs;i&longs;tunt, ne fiat motus; illa tollenda <lb/>&longs;unt per impetum, quo motus continuandus fui&longs;&longs;et, ni&longs;i im­<lb/>pedimentum occurri&longs;&longs;et; hæc verò &longs;uperat pars impetûs pro­<lb/>ducta à potentiâ, quæ &longs;e tardiùs movet, quia vires dividit, <lb/>partem impetûs &longs;ibi re&longs;ervans, partem impertiens ob&longs;taculo, <lb/>quod removet impellendo aut trahendo. </s> <s id="s.005468">Quare nil mirum, &longs;i <lb/>impetus, qui periturus e&longs;&longs;et in percutiente, cujus motus im­<lb/>peditur, tran&longs;eat in obicem percu&longs;&longs;um, quem &longs;ubmovendo <lb/>locum relinquit ulteriori motui, &longs;i facultas &longs;e movendi &longs;uppe­<lb/>tat corpori percutienti. <lb/></s> </p> <p type="main"> <s id="s.005469"><emph type="center"/>CAPUT XIII.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005470"><emph type="center"/><emph type="italics"/>Cunei u&longs;us promovetur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005471">NE quis fortè Cuneum &longs;olis ru&longs;ticis ad findenda ligna u&longs;ui <lb/>e&longs;&longs;e &longs;ibi per&longs;uadeat, fontes aliquos indicare placet ex <lb/>quibus non levis utilitas derivatur. </s> <s id="s.005472">Ad Machinarum &longs;cilicet <lb/>Rationem pertinet poti&longs;&longs;imùm motus corporis, cujus re&longs;i&longs;ten­<lb/>tia &longs;uperatur, &longs;ivè illa demum ex gravitate oriatur, &longs;ivè ex <lb/>nexu, quo colligatur cum proximo corpore; id quod iis con­<lb/>tingit, quæ in corpus unum coale&longs;cunt, & fi&longs;&longs;ione &longs;ejun­<lb/>guntur. </s> </p> <p type="main"> <s id="s.005473"><emph type="center"/>PROPOSITIO I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005474"><emph type="center"/><emph type="italics"/>Vectis vires Cuneo augere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005475">COntingit aliquando potentiam incommodè applicari <lb/>vecti, ut cum hominem valde curvari oportet ad <lb/>vectem &longs;ecundi generis ferè in &longs;olo jacentem attollendum; <lb/>tunc &longs;ub&longs;idium à Cuneo non incongruè peti pote&longs;t. </s> <s id="s.005476">Sit <lb/>Vectis AB &longs;ubjectus foribus DC &longs;uis è cardinibus avellen-<pb pagenum="745" xlink:href="017/01/761.jpg"/>dis, ut reficiantur: hypomochlium e&longs;t in A, & pondus in C. <!-- KEEP S--></s> <lb/> <s id="s.005477">At &longs;i potentiam adeò in­<lb/><figure id="id.017.01.761.1.jpg" xlink:href="017/01/761/1.jpg"/><lb/>clinari atque curvari opor­<lb/>teat, ut arripiat extremum <lb/>vectem B, &longs;atis manife&longs;tum <lb/>e&longs;t, quanto id incommodo <lb/>fiat. </s> <s id="s.005478">Subjiciatur vecti in B <lb/>(&longs;iquidem &longs;olum æquo <lb/>mollius fuerit) a&longs;&longs;eris aut <lb/>lapidis pars, quæ compre&longs;&longs;ioni re&longs;i&longs;tat, atque inter illam & <lb/>vectem apex Cunei E immittatur. </s> <s id="s.005479">Nam &longs;i tudite cuneum per­<lb/>cutias, vectem facilè attollet, ac proinde etiam valvas in C <lb/>incumbentes. </s> </p> <p type="main"> <s id="s.005480">Quod &longs;i vectis &longs;ecundi generis FG habens hypomochlium <lb/>in G ita fuerit altiùs col­<lb/><figure id="id.017.01.761.2.jpg" xlink:href="017/01/761/2.jpg"/><lb/>locatus, ut ægrè brachio­<lb/>rum contentione attolle­<lb/>re valeas pondus in K ad­<lb/>nexum, utere cuneo in­<lb/>flexo FH, quem &longs;olo in I <lb/>incumbentem, & vecti in <lb/>F &longs;ubjectum, &longs;i propellas <lb/>lateri HI, arrepto manu­<lb/>brio LM, applicatus, prout commodiùs acciderit, vectem cum <lb/>pondere eatenus elevabis, quoad latus IH longius, &longs;olo ad <lb/>pendiculum in&longs;i&longs;tat. </s> <s id="s.005481">Vectem autem, qua parte cuneum huju&longs;­<lb/>modi contingit, ita extenuatum e&longs;&longs;e oportere, ut cunei orbitæ ex­<lb/>cavatæ congruat, ne elabatur, res per &longs;e ip&longs;a loquitur. </s> </p> <p type="main"> <s id="s.005482">Maximè verò opportunum duxerim huju&longs;modi cuneo in­<lb/>flexo uti, ubi tertij generis Vectis <lb/><figure id="id.017.01.761.3.jpg" xlink:href="017/01/761/3.jpg"/><lb/>adhibendus fuerit RS, & pondus <lb/>adnexum ex S in V &longs;u&longs;tollendum: <lb/>Nam &longs;i loco Potentiæ de&longs;tinato <lb/>in T &longs;ubjicias cuneum inflexum <lb/>TP, &longs;olo in X incumbentem, & <lb/>hunc urgeas ex latere XP, aut <lb/>trahas ex latere TX, ubi P venerit <lb/>in Q, pondus ex S erit in V. <!-- KEEP S--></s> </p> <pb pagenum="746" xlink:href="017/01/762.jpg"/> <p type="main"> <s id="s.005483"><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005484"><emph type="center"/><emph type="italics"/>Vecte, aut Trochleâ, aut Succulâ, Cunei inflexi <lb/>vires augere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005485">IN&longs;trumenta pre&longs;&longs;oria varij generis excogitantur; &longs;ed ad &longs;ubi­<lb/>tum u&longs;um, & non &longs;inè compendio, uti aliquando po&longs;&longs;umus <lb/>Cuneo inflexo, cui, &longs;i validiùs premendum &longs;it, vectem adjice­<lb/><figure id="id.017.01.762.1.jpg" xlink:href="017/01/762/1.jpg"/><lb/>re licebit. </s> <s id="s.005486">Sit cra&longs;&longs;ior tabula AB, <lb/>cui &longs;upponatur id, quod premen­<lb/>dum proponitur. </s> <s id="s.005487">Paretur cuneus <lb/>inflexus DE, & pro illius motûs <lb/>centro &longs;tatuatur punctum C, cui <lb/>axis infigatur: Nam urgendo lon­<lb/>gius latus CE, aut trahendo bre­<lb/>vius CD, &longs;ubjectam tabulam AB <lb/>premes. </s> <s id="s.005488">Quod &longs;i validiore pre&longs;&longs;u <lb/>opus fuerit, lateri longiori CE <lb/>Vectem FG adhibe: potentia &longs;i­<lb/>quidem in G majorem arcum de&longs;­<lb/>cribens circa centrum motûs C, <lb/>majora obtinebit momenta, quàm <lb/>&longs;i proximè illa applicaretur Cuneo: illa tamen momenta poten­<lb/>tiæ in G &longs;en&longs;im minuuntur, prout cunei partes tabulam con­<lb/>tingentes propiores &longs;unt extremo puncto E. <!-- KEEP S--></s> <s id="s.005489">Ne verò ip&longs;a ea­<lb/>dem tabula AB impedimento &longs;it, &longs;i lateri CE proximè vectis <lb/>adhæreret, affigatur cuneo unum aut alterum Chelonion HF, <lb/>IK, in quæ conjectus vectis FG di&longs;tet à Cuneo citrà pericu­<lb/>lum incurrendi in &longs;ubjectam tabulam AB. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005490">Vel &longs;i Vectem cuneo affigere non placuerit, ip&longs;ius vectis ca­<lb/>put hypomochlio re&longs;pondens ita collocetur, ut vectis horizonti <lb/>ferè paralleli longitudo tran&longs;ver&longs;a cadat in latus cunei DE, <lb/>séque non procul ab E decu&longs;&longs;ent: hac enim ratione vecti &longs;ua <lb/>con&longs;tabunt momenta, quibus momenta cunei augeantur. </s> </p> <p type="main"> <s id="s.005491">At &longs;i fortè loci di&longs;po&longs;itio non ferat, ut vectis adhibeatur im­<lb/>pellendo cuneo, trahatur ille ex D, ubi aut annulus infigatur, <pb pagenum="747" xlink:href="017/01/763.jpg"/>aut foramini in&longs;eratur funis, cui deinde trochlea &longs;ive &longs;implex, <lb/>&longs;ive multiplex adnectatur, prout opus fuerit. </s> <s id="s.005492">Immò & Succula <lb/>addi poterit, ad quam funis caput religetur: erúntque momen­<lb/>ta potentiæ, quæ componuntur ex Rationibus Succulæ, Tro­<lb/>chleæ, & Cunei. </s> </p> <p type="main"> <s id="s.005493"><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005494"><emph type="center"/><emph type="italics"/>Cuneum inflexum validi&longs;simum con&longs;truere ad <lb/>Vectem tùm trahendum, tùm repellendum.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005495">AS&longs;umatur planum aliquod circulare circa axem per cen­<lb/>trum ductum ver&longs;atile, ita cra&longs;&longs;um & validum, ut in eo <lb/>in&longs;culpi po&longs;&longs;it profundiùs &longs;pira, quæ Vectis caput ferreo clavo <lb/>capitato, & in globum rotundato, armatum contineat, ne ela­<lb/>batur. </s> <s id="s.005496">Hinc enim fiet, ut in &longs;piræ cavitatem immi&longs;&longs;um Vectis <lb/>caput aut propellatur, aut attrahatur, prout plani illius motus <lb/>in hanc aut illam partem dirigitur: tantus &longs;cilicet erit Vectis <lb/>motus, quanta erit Radiorum à centro ad &longs;piræ ambitum ducto­<lb/>rum differentia. </s> <s id="s.005497">Ex quo orietur tractio, aut impul&longs;io; Radiis <lb/>enim decre&longs;centibus trahitur Vectis ad centrum, illis cre&longs;cen­<lb/>tibus propellitur à centro. </s> <s id="s.005498">Potentia igitur certæ plani illius cir­<lb/>cularis parti applicata integrum circulum de&longs;cribit, dum vectis <lb/>caput per unum &longs;piræ flexum excurrit, & tot circulos potentia <lb/>de&longs;cribit, quot &longs;piræ flexus vectis caput &longs;ubinde complectun­<lb/>tur. </s> <s id="s.005499">Quare comparanda e&longs;t di&longs;tantia à centro plani circumacti, <lb/>quam in motu caput Vectis mutavit, cum univer&longs;is circulis, <lb/>quos potentia interim de&longs;crip&longs;it, & &longs;tatim innote&longs;cet Ratio mo­<lb/>mentorum. </s> <s id="s.005500">Hinc &longs;i plano huju&longs;modi, in quo excavata e&longs;t &longs;pi­<lb/>ra, addideris circa extremam orbitam Radios, quemadmodum <lb/>Axi in Peritrochio, motus potentiæ &longs;atis amplos circulos <lb/>de&longs;cribet. </s> </p> <p type="main"> <s id="s.005501">Statuamus, exempli gratiâ, plani circularis a&longs;&longs;umpti diame­<lb/>trum cubitalem, hoc e&longs;t &longs;e&longs;quipedalem, &longs;eu digitorum 24; &longs;it <lb/>autem inter &longs;piræ excavatæ flexum & flexum intercapedo digi­<lb/>torum duûm, adeò ut, peractâ circulatione unâ, vectis caput <lb/>per &longs;piram excurrens digitos duos à primâ &longs;uâ &longs;ede dimotum <pb pagenum="748" xlink:href="017/01/764.jpg"/>fuerit: at potentia in extremâ orbitâ plani circularis con&longs;tituta <lb/>&longs;uo motu integram peripheriam circuli, cujus diameter digi­<lb/>torum 24, de&longs;crip&longs;erit, hoc e&longs;t digitorum 75. Motus igitur <lb/>potentiæ ad motum capitis vectis e&longs;t ut 75 ad 2: cui &longs;i addatur <lb/>Ratio ad ip&longs;um Vectem &longs;pectans, quatenus cum pondere com­<lb/>paratur, fiet Ratio Compo&longs;ita indicans Rationem motûs poten­<lb/>tiæ ad motum ponderis. </s> </p> <p type="main"> <s id="s.005502">Quapropter etiam&longs;i ad conciliandum ponderi motum paulò <lb/>velociorem, uteremur Vecte primi generis &longs;ed inver&longs;o, ita ut <lb/>ab hypomochlio plus di&longs;taret pondus, quàm potentia capiti <lb/>vectis applicata, adhuc haberetur non modicum momentorum <lb/>compendium. </s> <s id="s.005503">Sit enim di&longs;tantia potentiæ in capite vectis ab <lb/>hypomochlio ut 1, ponderis verò ut 5; atque adeò, dum Vectis <lb/>caput deprimitur digitos duos, pondus attollatur digitos de­<lb/>cem: Componantur duæ Rationes 75 ad 2, & 1 ad 5; erit Ra­<lb/>tio 15 ad 2, & potentia &longs;piræ applicata movebit pondus huju&longs;­<lb/>modi vecti adnexum lib.150, quo conatu ab&longs;que ullâ machinâ <lb/>moveret pondus lib. 20. </s> </p> <p type="main"> <s id="s.005504">Hanc propo&longs;itionem hìc potiùs afferre placuit, quàm in &longs;e­<lb/>quentem librum de Cochleâ re&longs;ervare, quia hìc caput vectis <lb/>excurrit per ip&longs;am &longs;piram, & proximè pertinere videtur hìc <lb/>motus ad motum &longs;uper faciem Cunei inflexi: in Cochleâ verò, <lb/>prout communiter illa u&longs;urpatur, pondus movetur ad motum <lb/>cylindri, cui in&longs;culpta e&longs;t Cochlea. <!-- KEEP S--></s> <s id="s.005505">Dixi, prout communiter <lb/>u&longs;urpatur, quia aliquid &longs;imile contingit Cochleæ infinitæ, ut <lb/>videbimus. </s> </p> <p type="main"> <s id="s.005506"><emph type="center"/>PROPOSITIO IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005507"><emph type="center"/><emph type="italics"/>Flatum vehementem non interruptum excitare <lb/>follibus adhibitis.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005508">GLebam metallicam ex fodinis erutam valido igne exco­<lb/>quere oportet, ut metallum fluat, atque id, quod utile e&longs;t, <lb/>ab inutili &longs;ecernatur. </s> <s id="s.005509">Ignis autem ut ex carbonibus excitetur <lb/>eâ vehementiâ, qua opus e&longs;t, etiam vehementem flatum, qui <lb/>ex follibus exprimatur adhibendum manife&longs;tum e&longs;t omnibus: <pb pagenum="749" xlink:href="017/01/765.jpg"/>neque enim ubique commodum reperiri pote&longs;t conclave hy­<lb/>pogæum, in quod præceps delap&longs;a aqua aërem vapori mi&longs;tum <lb/>per tubum in camini focum impellat. </s> <s id="s.005510">Quare præter vulgarem <lb/>& noti&longs;&longs;imam methodum folles alterno motu agitandi, ex his, <lb/>quæ hujus lib. cap. 3. dicta &longs;unt, rationem aliquam mire po&longs;­<lb/>&longs;umus, qua plurimum flatus in carbones accen&longs;os immit­<lb/>tamus. </s> </p> <p type="main"> <s id="s.005511">Quod ad folles ip&longs;os &longs;pectat, non illos &longs;implices vellem, &longs;ed <lb/>&longs;ingulos duplices, ita videlicet conformatos ut &longs;inguli ex binis <lb/>a&longs;&longs;eribus con&longs;tent invicem &longs;ecundùm alteram extremitatem in­<lb/>clinatis, qua&longs;i in angulum coituri e&longs;&longs;ent, qui omnino &longs;tabiles <lb/>permaneant. </s> <s id="s.005512">In ip&longs;o autem tigillo, cui firmiter infixa manent <lb/>a&longs;&longs;erum illorum capita, excavatus &longs;it congruè ductus, per quem <lb/>flatus exprimatur in tubum adnexum, quo ad focum defertur: <lb/>atque opportuno loco in &longs;ingulis a&longs;&longs;eribus, ut moris e&longs;t, fora­<lb/>men excipiendo aëri de&longs;tinatum a&longs;&longs;ario muniatur. </s> <s id="s.005513">Hos inter <lb/>immotos, planum aliud &longs;imile, ad extremitatem fibulâ ver&longs;atili <lb/>connexum cum tigillo illo communi, adjiciatur, & cum extre­<lb/>mis a&longs;&longs;eribus corio plicatili jungatur, adeò ut duo &longs;int conjuncti <lb/>folles, quorum alter clauditur, alter recluditur, cùm ex medio <lb/>hoc plano mobili exiens an&longs;a adducitur & reducitur: hoc enim <lb/>mobile planum e&longs;t diaphragma &longs;ejungens folles, ne ex altero in <lb/>alterum compre&longs;&longs;us aër effugiat, &longs;ed per infimum ductum in <lb/>tubum erumpat, per quem ad focum devehatur. </s> <s id="s.005514">Quatuor pa­<lb/>rentur huju&longs;modi folles duplices, quorum bini &longs;ibi ex diametro <lb/>oppo&longs;iti ita &longs;tatuantur, ut inter illos di&longs;cus circularis congruæ <lb/>magnitudinis interjectus eorum an&longs;as &longs;ubinde propellere va­<lb/>leat: bini autem oppo&longs;iti funiculo jungantur aut loro, aut ca­<lb/>tenulâ, an&longs;as connectente longitudinis æqualis diametro cir­<lb/>culi: Ex quo fiet, ut operâ eâdem follis unius an&longs;a propellatur, <lb/>oppo&longs;iti verò an&longs;a trahatur. </s> </p> <p type="main"> <s id="s.005515">Porrò attendendum e&longs;t, quantum &longs;patij percurrat &longs;ingulo­<lb/>rum follium an&longs;a ultro citróque remeando, quo loco illa tangi­<lb/>tur à circulo: hujus enim &longs;patij &longs;emi&longs;&longs;e definietur intervallum, <lb/>quo circuli centrum abe&longs;&longs;e oportet à centro, quod &longs;tatuendum <lb/>e&longs;t, ut circa illud fiat eju&longs;dem circuli convolutio. </s> <s id="s.005516">Huic motûs <lb/>centro infigendus e&longs;t firmiter axis, &longs;ivè ille &longs;it communis exte­<lb/>riori rotæ ab aquâ fluente convolutæ, &longs;ivè cui vectis opportu-<pb pagenum="750" xlink:href="017/01/766.jpg"/>næ longitudinis adjiciatur, ut ab homine, aut à jumento con­<lb/><figure id="id.017.01.766.1.jpg" xlink:href="017/01/766/1.jpg"/><lb/>torqueatur. </s> <s id="s.005517">Sic an&longs;æ motus <lb/>univer&longs;us. </s> <s id="s.005518">Sit ex. </s> <s id="s.005519">gr. <!-- REMOVE S-->AB <lb/>circuli centrum &longs;it C: acci­<lb/>piatur intervallum CD &longs;ub­<lb/>duplum ip&longs;ius AB; & erit <lb/>in D infigendus axis, ex cu­<lb/>jus convolutione circulus pa­<lb/>riter circumagatur, & fol­<lb/>lium an&longs;as in quatuor oppo­<lb/>&longs;itis punctis A, E, G, F &longs;ubinde tangat, eá&longs;que vici&longs;&longs;im propel­<lb/>lat, & trahat: Cùm &longs;cilicet incipit propelli follis an&longs;a, quæ e&longs;t <lb/>in E, propellitur pariter ea, quæ e&longs;t in G (&longs;i quidem conver­<lb/>&longs;io fiat ex E in F) atque ex adver&longs;o tantumdem trahitur quæ <lb/>e&longs;t in A, quantùm propellitur quæ e&longs;t in E; atque &longs;imiliter <lb/>tractio ejus, quæ e&longs;t in F, e&longs;t æqualis impul&longs;ioni an&longs;æ, quæ <lb/>e&longs;t in G. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005520">Si igitur circulus non &longs;it in plano Verticali, & axem in D in­<lb/>fixum non habeat communem cum rotâ, quæ ab aquâ volva­<lb/>tur, &longs;ed &longs;it in plano horizontali, axi infixo in D addatur vectis <lb/>DH, ut potentia in H vectem impellens aut trahens circum­<lb/>agat circulum. </s> <s id="s.005521">Quo autem loco &longs;tatuendus &longs;it vectis, pendet <lb/>ex loci po&longs;itione, prout vel in &longs;uperiori, vel in inferiori, vel in <lb/>eodem conclavi folles collocantur; axis &longs;iquidem certam non <lb/>exigit longitudinem, &longs;ed ea illi tribuenda e&longs;t, quæ commodior <lb/>acciderit. </s> <s id="s.005522">Vectis tamen longitudinem ita temperare oportet, <lb/>ut, dum potentiæ movendi facilitatem affectas, nimiam tardi­<lb/>tatem compre&longs;&longs;ionis follium effugias. </s> </p> <p type="main"> <s id="s.005523">Ex his &longs;atis apparet, quantùm aëris impellatur in prunas à <lb/>quatuor follibus, qui clauduntur, dum quatuor reliqui reclu­<lb/>duntur, perpetuú&longs;que e&longs;t flatus nunquam interruptus. </s> <s id="s.005524">Quòd <lb/>&longs;i potentia movens viribus abundet, & circulus fieri po&longs;&longs;it am­<lb/>plior ita, ut non quatuor &longs;olùm follibus duplicibus, &longs;ed etiam <lb/>&longs;ex aut octo &longs;imilibus in gyrum di&longs;ponendis commodus locus <lb/>&longs;uppetat, &longs;atis vides, quantus excitari po&longs;&longs;it flatus. </s> </p> <p type="main"> <s id="s.005525">Quoniam verò ex dictis infertur folles e&longs;&longs;e erigendos, ut in <lb/>eorum an&longs;as circulus horizonti parallelus incurrat, ob&longs;erva po&longs;­<lb/>&longs;e illos etiam jacentes (modò a&longs;&longs;arium &longs;uperioris a&longs;&longs;eris foramen <pb pagenum="751" xlink:href="017/01/767.jpg"/><expan abbr="claud&etilde;s">claudens</expan> exactè fungi po&longs;&longs;it &longs;uo munere) u&longs;ui e&longs;&longs;e po&longs;&longs;e, &longs;i <expan abbr="artificiũ">artificium</expan> <lb/>aliquod adhibeatur, quo an&longs;a attollatur, atque deprimatur. </s> <s id="s.005526">Fiat <lb/>inflexus vectis, &longs;eu qua&longs;i vec­<lb/><figure id="id.017.01.767.1.jpg" xlink:href="017/01/767/1.jpg"/><lb/>tis RSG, cujus angulo S <lb/>addatur axis, circa quem fa­<lb/>cilè converti po&longs;&longs;it, & ad ex­<lb/>tremitatem G &longs;it regula GH <lb/>follis an&longs;æ adnexa; & &longs;imilia <lb/>omnia ex adver&longs;o parentur, <lb/>atque funiculo MP conjun­<lb/>gantur. </s> <s id="s.005527">Nam circulus im­<lb/>pellens in R attollet extre­<lb/>mitatem G, ac proinde an&longs;am illi adnexam, trahendo autem <lb/>funiculum MP deprimet extremitatem O, & cum illâ an&longs;am <lb/>oppo&longs;iti follis: atque ita vici&longs;&longs;im impellendo N, attolletur O, <lb/>& deprimetur G. <!-- KEEP S--></s> <s id="s.005528">Hinc colliges hoc eodem artificio, &longs;i ha&longs;tulæ <lb/>GH non follis an&longs;am, &longs;ed antliæ embolum adjeceris, fieri po&longs;­<lb/>&longs;e machinam, qua, multiplicatis antiis, plurimum aquæ &longs;ur­<lb/>&longs;um impellere valeas. </s> <s id="s.005529">An autem di&longs;ci circularis exteriorem or­<lb/>bitam ferreo annulo polito & lævi munire, atque ferream lami­<lb/>nam pariter politàm an&longs;æ follis, aut vecti inflexo, apponere <lb/>præ&longs;tet, ut quàm minimo tritu inter &longs;e confligant, non e&longs;t opus <lb/>monere, &longs;i fuerit operæ pretium machinæ diuturnitati con&longs;u­<lb/>lere, & faciliorem motum exhibere. </s> </p> <p type="main"> <s id="s.005530">Quòd demum &longs;pectat ad ip&longs;ius circuli collocationem, quam­<lb/>quam cylindrus illi infixus po&longs;&longs;it inniti polis, circa quos ver&longs;e­<lb/>tur; ut tamen &longs;ubter circulum liberrimè trahi po&longs;&longs;int funiculi, <lb/>placeret potiùs illum omnino &longs;u&longs;pen&longs;um pendere ex cra&longs;&longs;iore <lb/>(&longs;ive &longs;implici, &longs;ive ex duobus compacto) tigno, cujus forami­<lb/>ni in&longs;eratur cylindrus, ferreo annulo munitus tam in &longs;uperiori, <lb/>quàm in inferiori parte, qua foramini re&longs;pondet, ita ut neque <lb/>&longs;ur&longs;um agi, neque deor&longs;um de&longs;cendere valeat, &longs;ed intrà fo­<lb/>ramen illud convertatur, quod pariter utrinque circulis fer­<lb/>reis muniatur re&longs;pondentibus &longs;uperiori & inferiori annulo <lb/>cylindri. </s> <s id="s.005531">Sit circularis di&longs;cus AB, in quo motûs centrum <lb/>&longs;it C, cui infigatur cylindrus CD convertendus à vecte, <lb/>&longs;ivè in I, &longs;ivè in H immittendo. </s> <s id="s.005532">Horizonti parallelum <pb pagenum="752" xlink:href="017/01/768.jpg"/>tignum KL &longs;ecundùm &longs;uas extremitates in pariete, aut ali­<lb/><figure id="id.017.01.768.1.jpg" xlink:href="017/01/768/1.jpg"/><lb/>ter, firmetur, in eóque &longs;it <lb/>foramen F capax cylindri: <lb/>foramen ferreo circulo, <lb/>quoad fieri po&longs;&longs;it, lævi at­<lb/>que polito muniatur, cui <lb/>æqualis annulus cylindrum <lb/>ve&longs;tiens, illíque affixus, <lb/>re&longs;pondeat. </s> <s id="s.005533">Manebit ex F <lb/>&longs;u&longs;pen&longs;us cylindrus DC uná <lb/>cum adjuncto circulari di&longs;­<lb/>co AB. <!-- KEEP S--></s> <s id="s.005534">In inferiore tigni <lb/>facie &longs;imiliter &longs;it circulus <lb/>ferreus, & annulus, ne &longs;ur­<lb/>&longs;um excurrere queat cylin­<lb/>drus. </s> <s id="s.005535">Si tignum quidem <lb/>valde di&longs;tet à circulo AB, <lb/>immitti poterit vectis in H; <lb/>at &longs;i exiguum fuerit inter­<lb/>valium inter tignum & cir­<lb/>culum AB, atque tignum exi&longs;tat infra planum, in quo ho­<lb/>mo aut jumentum vectem impellens aut trahens movetur, <lb/>vectis in I immittatur. </s> </p> <p type="main"> <s id="s.005536"><emph type="center"/>PROPOSITIO V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005537"><emph type="center"/><emph type="italics"/>Plures antlias duplices perpetuo ductu agitare.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005538">UBi jumentorum operâ uti oportet ad agitandas antlias, <lb/>quibus aqua in &longs;uperiorem locum aut attrahitur, aut im­<lb/>pellitur, illa in gyrum agere nece&longs;&longs;e e&longs;t; id quod multo <lb/>tempore eget; neque enim circuitus illos currendo efficere <lb/>po&longs;&longs;unt: quapropter rotarum dentatarum complexionem <lb/>con&longs;truere &longs;olemus, ut, dum &longs;emel jumentum &longs;uam con­<lb/>ver&longs;ionem ab&longs;olvit, &longs;æpiùs antliæ agitentur. </s> <s id="s.005539">Verùm minore <lb/>impendio ab&longs;que rotis idem forta&longs;&longs;e a&longs;&longs;equemur, &longs;i poti&longs;&longs;imùm <lb/>aqua in altum propellenda &longs;it. </s> </p> <pb pagenum="753" xlink:href="017/01/769.jpg"/> <p type="main"> <s id="s.005540">Ad perpendiculum erigatur tignum, quod imo puteo in­<lb/>nitatur, &longs;ive &longs;olidiori tigno <lb/><figure id="id.017.01.769.1.jpg" xlink:href="017/01/769/1.jpg"/><lb/>AB putei lateribus infixo <lb/>in&longs;i&longs;tat brevius tignum <lb/>CD, ita tamen obliquè <lb/>con&longs;titutum, ut hujus an­<lb/>guli latera illius re&longs;piciant, <lb/>quatenus ha&longs;tulæ ex hoc <lb/>exeuntes, medio jugo, <lb/>nullum recipiant à &longs;ub­<lb/>jecto tigno AB impedi­<lb/>mentum. </s> <s id="s.005541">Suprema pars <lb/>tigni CD ita &longs;ecetur, ut <lb/>circa axem in E infixum <lb/>liberè ver&longs;ati po&longs;&longs;it ju­<lb/>gum, cujus extremitatibus <lb/>adnexæ &longs;unt ha&longs;tulæ an­<lb/>tliarum embolos attollen­<lb/>tes atque deprimentes. </s> <lb/> <s id="s.005542">Paulò infra axem E ape­<lb/>riatur foramen F, cui pa­<lb/>riter immitti queat jugum <lb/>alterum ver&longs;atile circa <lb/>axem H infixum paulò <lb/>infrà crenam &longs;uperiori ju­<lb/>go &longs;ub&longs;ervientem. </s> </p> <p type="main"> <s id="s.005543">Porrò utriu&longs;que jugi non eadem e&longs;t forma: Nam &longs;upe­<lb/>rius jugum axi E infixum rectum e&longs;t GI, additamento <lb/>ad K auctum, ut paulo depre&longs;&longs;ius &longs;it foramen K ad reci­<lb/>piendum axem, quàm &longs;int axes ad G & I, quibus jun­<lb/>guntur cum jugo ha&longs;tulæ ad embolorum motum perficien­<lb/>dum de&longs;tinatæ. </s> <s id="s.005544">At verò jugum inferius non ni&longs;i extremi­<lb/>tates LM & NO rectas habet, cætera inflexum e&longs;t, & <lb/>ad mediam curvaturam habet in P foramen, quo innita­<lb/>tur axi in H infixo. </s> <s id="s.005545">Quantam autem e&longs;&longs;e oporteat hu­<lb/>ju&longs;modi inflexionem MPN, ex hoc definies, quod ubi <lb/>jugum GI in &longs;uo axe con&longs;i&longs;tens horizonti parallelum fue­<lb/>rit, etiam inferioris jugi in &longs;uo axe H con&longs;i&longs;tentis extre-<pb pagenum="754" xlink:href="017/01/770.jpg"/>mitates LM & NO in eodem horizontali plano cum GI <lb/>conveniant. </s> </p> <p type="main"> <s id="s.005546">His paratis fru&longs;tum cylindricum diametri (&longs;i id quidem <lb/>commodè fieri po&longs;&longs;it) non multo minoris, quàm &longs;it jugi GI <lb/>intercapedo inter ha&longs;tularum axes, con&longs;truatur. </s> <s id="s.005547">Quod &longs;i <lb/>tantæ cra&longs;&longs;itudinis lignum præ&longs;to non fuerit, plura aptè <lb/>compinge, atque ferreo circulo con&longs;tringe, ne di&longs;&longs;ilire va­<lb/>leant. </s> <s id="s.005548">Tum de&longs;tinatæ embolorum depre&longs;&longs;ioni atque eleva­<lb/>tioni æqualis &longs;altem pars QS toreutæ operâ rotundetur; <lb/>deinde &longs;errâ obliquè &longs;ecetur, ut fiat ellip&longs;is QT: cujus <lb/>limbus ferreâ lamellâ exactè planâ & politâ muniatur tùm <lb/>ad perpetuitatem, ne lignum atteratur, tùm ad faciliorem <lb/>motum, ut minor &longs;it cum &longs;ubjectis ligneis jugis conflictus: <lb/>interiores autem ellip&longs;is partes &longs;calpro eximi po&longs;&longs;unt, ut factâ <lb/>cavitate nullum motui impedimentum afferant tigni CD &longs;u­<lb/>premi anguli. </s> </p> <p type="main"> <s id="s.005549">Fru&longs;to huic cylindrico ad axem firmiter in&longs;eratur minor <lb/>cylindrus VR, cui immitti po&longs;&longs;it vectis XY à jumento in X <lb/>circumducendus. </s> <s id="s.005550">Minor huju&longs;modi cylindrus methodo &longs;u­<lb/>periùs indicatâ &longs;ub finem prop. 4. &longs;u&longs;pendatur eâ lege, ut <lb/>jugo GI maximè inclinato conveniat ellip&longs;is diameter QT, <lb/>minori autem ellip&longs;is Axi conveniant extremitates LM & <lb/>NO inferioris jugi, quæ erunt horizonti parallelæ. </s> <s id="s.005551">Hinc <lb/>fiet, ut conver&longs;o cylindro &longs;ubinde deveniant ad maximam <lb/>depre&longs;&longs;ionem, atque vici&longs;&longs;im ad maximam elevationem, &longs;in­<lb/>guli antliarum emboli. </s> </p> <p type="main"> <s id="s.005552">Quod &longs;i liceret in puteo, ubi aqua &longs;caturit, aut in va&longs;e, in <lb/>quod aqua influit, in altum elevanda vi antliæ propellentis, <lb/>non quadratum tantùm, &longs;ed hexagonum aut octogonum pri&longs;­<lb/>ma erigere ad perpendiculum, & in oppo&longs;itis faciebus fora­<lb/>mina excavare, quibus immitterentur inflexa juga, eâ in­<lb/>flexione, quæ &longs;atis e&longs;&longs;et, ut demum omnium extremitates in <lb/>eodem horizontali plano convenirent, &longs;atis manife&longs;tum e&longs;t <lb/>tribus aut quatuor jugis po&longs;&longs;e deinceps &longs;ex aut octo antlias <lb/>agitari. </s> </p> <pb pagenum="755" xlink:href="017/01/771.jpg"/> <p type="main"> <s id="s.005553"><emph type="center"/>PROPOSITIO VI.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005554"><emph type="center"/><emph type="italics"/>Alia ratione plures antlias componere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005555">EX iis, quæ hujus libri cap. 5. dicta &longs;unt, genus aliud ad <lb/>Cuneum pertinens excogitare po&longs;&longs;umus, quo &longs;imul plu­<lb/>res antlias agitare po&longs;&longs;it potentia, cui maximè virium copia <lb/>&longs;uppetat, & valde &longs;implex machina con&longs;truenda proponatur. </s> <lb/> <s id="s.005556">Ex &longs;olidis a&longs;&longs;eribus <lb/><figure id="id.017.01.771.1.jpg" xlink:href="017/01/771/1.jpg"/><lb/>compingatur cir­<lb/>culus: hic in octo <lb/>partes di&longs;tribuatur, <lb/>& duæ proximæ <lb/>confixum habeant <lb/>tigillum AB, cu­<lb/>jus extremitates ita <lb/>extra circulum pro­<lb/>mineant, ut inci&longs;is <lb/>crenis ha&longs;tulæ em­<lb/>bolo adnexæ circa <lb/>&longs;uum axem ver&longs;a­<lb/>tiles &longs;inè impedi­<lb/>mento moveri que­<lb/>ant. </s> <s id="s.005557">Tres &longs;imiles <lb/>tigilli tran&longs;ver&longs;arij <lb/>affigantur CD, EF, GH extremitatibus &longs;imiliter prominenti­<lb/>bus extra circuli ambitum, & excavatis in crenas ha&longs;tularum <lb/>capaces. </s> <s id="s.005558">Ha&longs;tularum verò formam &longs;uaderem, quæ prope em­<lb/>bolum e&longs;&longs;ent plicatiles in dextram atque &longs;ini&longs;tram, quemad­<lb/>modum in &longs;upremâ parte, ubi tran&longs;ver&longs;ariis cohærent, &longs;unt <lb/>circa axem flexiles in anteriorem atque in po&longs;teriorem partem: <lb/>ex hac enim flexibilitate in omnem partem facilior oritur mo­<lb/>tus. </s> <s id="s.005559">Duos autem tigillos CD & EF exi&longs;timo apponendos e&longs;&longs;e <lb/>tran&longs;ver&longs;arios, ad majorem circuli firmitatem: quamquam &longs;uf­<lb/>ficeret ad propo&longs;itum finem breviores apponere ad CE & DF, <lb/>omnino &longs;imiles & æquales ip&longs;is AB & GH. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005560">His paratis alius æqualis circulus &longs;uperponatur, firmitérque <pb pagenum="756" xlink:href="017/01/772.jpg"/>cum inferiore cohæreat. </s> <s id="s.005561">Tum validus &longs;tylus ferreus RT figu­<lb/>ræ primùm cylindricæ, deinde ad S &longs;phæricæ, demum in T <lb/>de&longs;inens in conum con&longs;truatur, & columnæ, cui univer&longs;a ma­<lb/>china inniti debet, ad perpendiculum infigatur. </s> <s id="s.005562">Ad centrum <lb/>verò circuli inferioris foramen fiat, per quod facilè globus S <lb/>immitti po&longs;&longs;it, & in centro circuli &longs;uperioris aliud pariter fo­<lb/>ramen aperiatur, &longs;ed tantùm capax coni ST, adeò ut machi­<lb/>na &longs;u&longs;tineatur à globo S, & in quancumque partem facilè in­<lb/>clinari queat: id quod etiam faciliùs continget, &longs;i foramen il­<lb/>lud &longs;uperioris circuli, qua parte globum S contingit, annulo, <lb/>&longs;eu limbo ferreo muniatur. </s> <s id="s.005563">Quod &longs;i circulus ille &longs;uperior cra&longs;­<lb/>&longs;ior fuerit, quàm ut facilè inclinari po&longs;&longs;it, ne &longs;uperior ora fo­<lb/>raminis incurrat in conum, abradi poterit, quantum &longs;atis fue­<lb/>rit, in calathoidem, ut magis pateat, atque liberam inclina­<lb/>tionem permittat. </s> </p> <p type="main"> <s id="s.005564">Circulari hac compage impo&longs;itâ &longs;tylo RS, ha&longs;tulæ embolo­<lb/>rum &longs;uis axibus adnectantur extremitatibus tigillorum promi­<lb/><figure id="id.017.01.772.1.jpg" xlink:href="017/01/772/1.jpg"/><lb/>nentibus. </s> <s id="s.005565">Tum ad conciliandum <lb/>motum machinæ, cylindrus IK <lb/>&longs;uo centro K innitatur apici &longs;ty­<lb/>li T, & in &longs;uperiore loco, axis I <lb/>congruo foramini immi&longs;&longs;us ier­<lb/>vet cylindri po&longs;itionem perpen­<lb/>dicularem. </s> <s id="s.005566">Sit autem in cylin­<lb/>dri latere profundiùs excavata <lb/>crena, cui in&longs;eri po&longs;&longs;it triangu­<lb/>lum OPN obtu&longs;angulum ad P, <lb/>quod validum &longs;it, & cum cylin­<lb/>dro firmi&longs;&longs;imè cohæreat: &longs;ic enim fiet, ut trianguli extremi­<lb/>tas O tangens circulum, illum à po&longs;itione horizonti parallelâ <lb/>removeat, & in eam partem inclinet, atque ex adversâ elevet. </s> <lb/> <s id="s.005567">Potentia verò vecti VX applicata, & cylindrum volvens, alam <lb/>pariter NOP circumducet; quæ aliis atque aliis &longs;ubjecti circu­<lb/>li partibus &longs;ubinde applicata illas deprimet, & ex diametro op­<lb/>po&longs;itas elevabit: intermediæ autem aliæ deprimentur, ad quas <lb/>&longs;cilicet extremitas O accedit, aliæ elevabuntur, à quibus ea­<lb/>dem extremitas O recedit. </s> </p> <p type="main"> <s id="s.005568">Quantum autem extremitas O infra ba&longs;im cylindri de&longs;cen-<pb pagenum="757" xlink:href="017/01/773.jpg"/>dere oporteat, definiendum e&longs;t primò ex motu, quem embolus <lb/>elevatus atque depre&longs;&longs;us perficit; cujus motús medietas acci­<lb/>pienda e&longs;t: deinde attendenda e&longs;t di&longs;tantia ba&longs;is cylindri à pla­<lb/>no circuli, &longs;i hoc con&longs;titueretur horizonti parallelum; hæc ve­<lb/>rò di&longs;tantia addenda e&longs;t &longs;emi&longs;&longs;i motûs emboli, ut innote&longs;cat, <lb/>quantum oporteat extremitatem O deprimi infra ba&longs;im cylin­<lb/>dri. </s> <s id="s.005569">Neque cuiquam dubium e&longs;&longs;e pote&longs;t; an &longs;ic definienda &longs;it <lb/>huju&longs;modi depre&longs;&longs;io extremitatis O; &longs;iquidem inclinato circu­<lb/>lo tantum extremitas altera diametri deprimitur infra planum <lb/>horizontale, quantum altera attollitur; hæc autem duplicata <lb/>differentia dat univer&longs;um motum emboli; igitur hujus motús <lb/>&longs;emi&longs;&longs;e definitur circuli depre&longs;&longs;io & inclinatio. </s> <s id="s.005570">Quia autem ad <lb/>faciliorem motum, tùm ne cylindri cra&longs;&longs;ities plano circuli in­<lb/>clinato occurrat, tùm ne latus PO circulum tangat præter­<lb/>quam extremitate O, ad vitandum tritum atque conflictum <lb/>partium, præ&longs;tat cylindrum non proximè adhærere circulo; <lb/>propterea di&longs;tantia ba&longs;is cylindri à centro &longs;ubjecti circuli com­<lb/>putanda e&longs;t. </s> </p> <p type="main"> <s id="s.005571">Porrò expedire extremitatem O munitam ferreâ laminâ <lb/>percurrere in &longs;ubjecto circulo laminam pariter ferream ex­<lb/>qui&longs;itè politam, non opus e&longs;t monere: &longs;atis quippe per &longs;e <lb/>patet. </s> <s id="s.005572">Illud cavendum e&longs;t, ut modum &longs;erves in alæ NOP <lb/>amplitudine; nam &longs;i nimis exigua &longs;it, paulò difficiliùs mo­<lb/>vet, quia nimis di&longs;tat ab ha&longs;tulis embolorum: &longs;in autem <lb/>æquo amplior fuerit, cùm maximam re&longs;i&longs;tentiæ partem illa <lb/>&longs;u&longs;tineat, &longs;ubit periculum luxationis. </s> <s id="s.005573">Cæterùm hoc pende­<lb/>bit ex circuli amplitudine, cujus diametrum con&longs;tituen­<lb/>dam e&longs;&longs;e habitâ ratione motûs embolo antliæ communican­<lb/>di, nemo ignorat; quemadmodum & in &longs;implici antliá ex <lb/>hoc eodem definitur di&longs;tantia ha&longs;tulæ à centro motûs. </s> <s id="s.005574">Quo­<lb/>niam enim motus ille depre&longs;&longs;ionis & elevationis emboli con­<lb/>nectitur cum motu circulari &longs;emidiametri circuli, cui ha&longs;tu­<lb/>læ adnectuntur, eum Radium circulo tribuere oportet, ut <lb/>arcus ab extremo puncto de&longs;criptus quàm minimùm differat <lb/>à lineâ rectâ; &longs;ic enim faciliùs movetur embolus. </s> <s id="s.005575">Quare ar­<lb/>cus eju&longs;modi de&longs;cribendus e&longs;t, ut illius medietas Sinum <lb/>Ver&longs;um habeat, quoad fieri poterit, minimum. </s> <s id="s.005576">Ponamus <lb/>univer&longs;um emboli motum e&longs;&longs;e unciarum 4, ejus &longs;emi&longs;&longs;em <pb pagenum="758" xlink:href="017/01/774.jpg"/>unciarum 2: Sit circuli Radius BD unciarum 8. Inveniatur <lb/><figure id="id.017.01.774.1.jpg" xlink:href="017/01/774/1.jpg"/><lb/>in Canone Sinuum arcus, cujus Si­<lb/>nus ad Radium &longs;it ut 2 ad 8, & e&longs;t <lb/>proximè gr. <!-- REMOVE S-->14. 28′ 40″. <!-- REMOVE S-->E&longs;t igitur <lb/>arcus ab extremâ &longs;emidiametro D <lb/>de&longs;cribendus CE gr. <!-- REMOVE S-->28. 57′. </s> <s id="s.005577">20″: <lb/>quo bifariam divi&longs;o in D e&longs;t arcus <lb/>CD gr.14. 28′. </s> <s id="s.005578">40″; cujus Sinus CI; <lb/>& Sinus Ver&longs;us ID e&longs;t totius Radij BD (3/100), hoc e&longs;t unius <lb/>unciæ 4/25; quæ deflexio arcûs CE à rectitudine non admodum <lb/>nocet. </s> <s id="s.005579">Satis igitur fuerit, &longs;i circuli diameter &longs;it unc.14. & ti­<lb/>gilli hinc atque hinc aliquantulum præter unam unciam pro­<lb/>mineant, ubi illis ha&longs;tulæ embolorum adnectuntur; &longs;ic enim <lb/>fiet, ut ha&longs;tulæ &longs;atis commodè moveantur, maximè &longs;i lon­<lb/>giores fuerint. </s> </p> <p type="main"> <s id="s.005580">Quod &longs;i ligneis tigillis uti nolueris, &longs;ed potius ferreis pri&longs;­<lb/>matibus inter utrumque ligneum circulum aptè con&longs;erendis, <lb/>adeò ut circuli plana &longs;ibi vici&longs;&longs;im adhæreant, non dubium, <lb/>quin multò firmior futura &longs;it machina: hoc te monitum volo, <lb/>quod circulos cra&longs;&longs;iu&longs;culos e&longs;&longs;e oportet, ut in illis opportunum <lb/>foramen excavetur, quo commodè machina in&longs;i&longs;tat &longs;tyli glo­<lb/>bulo, &, prout oportet, inclinetur. <lb/><figure id="id.017.01.774.2.jpg" xlink:href="017/01/774/2.jpg"/></s> </p> <pb pagenum="759" xlink:href="017/01/775.jpg"/> <figure id="id.017.01.775.1.jpg" xlink:href="017/01/775/1.jpg"/> <p type="main"> <s id="s.005581"><emph type="center"/>MECHANICORUM <emph.end type="center"/><!-- REMOVE S--><emph type="center"/>LIBER OCTAVUS.<emph.end type="center"/></s> </p> <p type="main"> <s id="s.005582"><emph type="center"/><emph type="italics"/>De Cochlea.<emph.end type="italics"/><emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005583">POSTREMO loco inter Mechanicas Facultates <lb/>numeratur Cochlea, non tamen po&longs;tremo loco <lb/>habenda, &longs;i ejus vires perpendantur; immò &longs;i <lb/>cum cæteris Facultatibus comparetur, omnium <lb/>efficaci&longs;&longs;ima cen&longs;enda erit, cæteris paribus, ut <lb/>ex iis, quæ hoc libro di&longs;putabuntur, mani­<lb/>fe&longs;tum fiet. </s> <s id="s.005584">Cur de Cochleâ po&longs;tremus habeatur &longs;ermo, &longs;i <lb/>quis inquirat, non pauci ex iis, qui inter Mechanicas faculta­<lb/>tes cognationis nexus quo&longs;dam perve&longs;tigant, ideò po&longs;t Cu­<lb/>neum numerari Cochleam autumabunt, quia Cochlea longior <lb/>quidam Cuneus cylindro convolutus cen&longs;eri pote&longs;t, cujus <lb/>propterea vires ad Cuneum revocare contendunt. </s> <s id="s.005585">Mihi tamen, <lb/>qui Facultates &longs;ingulas ita à reliquis ab&longs;olutas agno&longs;co, ut nul­<lb/>lo alio vinculo invicem copulentur, ni&longs;i quatenus omnes ab <lb/>uno eodémque principio ortum ducunt, ea tantummodo e&longs;&longs;e <lb/>videtur cau&longs;a, quod reliquæ Facultates &longs;implices &longs;int, ac faci­<lb/>liùs parabiles, quàm Cochlea, atque hæc &longs;i &longs;olitaria adhibea­<lb/>tur, nec cum ullâ reliquarum Facultatum componatur, licèt <lb/>validè urgeat, aut trahat, eâ tamen communiter non utamur <lb/>ad majores motus efficiendos, quos unâ aliquâ reliquarum Fa­<lb/>cultatum, minore operâ, con&longs;equimur. </s> </p> <p type="main"> <s id="s.005586">Hùc autem non &longs;pectat Archimedea Cochlea ad aquam in <lb/>altum evehendam in&longs;tituta: e&longs;t enim tubus in &longs;piram convolu­<lb/>tus circa &longs;uperficiem conicam aut cylindricam, &longs;eu in cono ip&longs;o <lb/>aut cylindro ita excavatus, ut aquam continere valeat, quam <lb/>extremum tubi o&longs;culum ex &longs;ubjectâ profluente hau&longs;it: dum &longs;ci-<pb pagenum="760" xlink:href="017/01/776.jpg"/>licet circa &longs;uum axem Conus aut Cylinder ad horizontem in­<lb/>clinatus convertitur, quæ ingre&longs;&longs;a fuerat aqua, per &longs;piras a&longs;cen­<lb/>dens ad alteram tubi extremitatem &longs;uperiorem demum effundi­<lb/>tur; atque hac ratione ad tantam altitudinem illa attollitur, <lb/>quantus e&longs;t Sinus anguli, quo ad horizontem inclinatur axis <lb/>eoni aut cylindri, po&longs;ito codem axe tanquam Radio. <!-- KEEP S--></s> <s id="s.005587">Hic, in­<lb/>quam, motus aquæ in tubo huju&longs;modi &longs;pirali a&longs;cendentis, non <lb/>e&longs;t præ&longs;entis di&longs;putationis, aqua &longs;iquidem non trahitur &longs;ur&longs;um, <lb/>&longs;ed &longs;emel ingre&longs;&longs;a in tubo &longs;pirali convoluto &longs;ponte de&longs;cendit, <lb/>donec ad &longs;upremum o&longs;culum provehatur; haud &longs;ecus ac plum­<lb/>beus globulus in eundem tubum immi&longs;&longs;us, &longs;i volvatur cylin­<lb/>der, non valens con&longs;i&longs;tere in eâ &longs;piræ parte, quæ priùs infima <lb/>& horizonti proxima, modò in conver&longs;ione removetur ab ho­<lb/>rizonte & attollitur, &longs;uâ autem gravitate repugnans a&longs;cen&longs;ui, <lb/>&longs;ponte de&longs;cendit per tubum tanquam per planum inclinatum, <lb/>atque ita deinceps, quoad ex &longs;upremo tubi o&longs;culo erumpat. </s> <lb/> <s id="s.005588">Idem planè contingit aquæ in huju&longs;modi tubo &longs;pirali vi &longs;uæ <lb/>gravitatis &longs;ubinde fluenti ac de&longs;cendenti in &longs;ingulis &longs;piris &longs;ta­<lb/>tim, ac modicum quid elevata e&longs;t in conver&longs;ione. </s> </p> <p type="main"> <s id="s.005589">Cochlea igitur, de qua hìc di&longs;putabitur, ea e&longs;t, quæ ad vim <lb/>gravitati inferendam, &longs;i repugnet, in&longs;tituta e&longs;t, adeò ut corpo­<lb/>ris vim pa&longs;&longs;i motus impul&longs;ui à Potentiâ per Cochleam commu­<lb/>nicato adæquatè tribuendus &longs;it; & &longs;i quid gravitas ip&longs;a confe­<lb/>rat, id planè contingens reputetur. </s> <s id="s.005590">Nomen autem Cochleæ <lb/>inditum e&longs;t ex &longs;imili quadam convolutione in te&longs;tâ limacis, <lb/>quæ in &longs;piras contorquetur, &longs;icut & Cochlides dicuntur &longs;calæ, <lb/>per quas in gyrum a&longs;cenditur. <lb/></s> </p> <p type="main"> <s id="s.005591"><emph type="center"/>CAPUT I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005592"><emph type="center"/><emph type="italics"/>Cochleæ forma & virtus de&longs;cribitur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005593">COchlea, quam explicandam &longs;u&longs;cipimus, ex limacis te&longs;tâ <lb/>eatenus &longs;olùm &longs;imilitudinem ducit, quatenus in &longs;piras du­<lb/>citur, cæterùm animalis illius &longs;piræ inæquales &longs;unt, & major <pb pagenum="761" xlink:href="017/01/777.jpg"/>&longs;pira minorem qua&longs;i complectitur, non quemadmodum helix <lb/>in plano de&longs;cripta, &longs;ed ferè &longs;icut &longs;pira in coni aut globi &longs;uper­<lb/>ficie deformata. </s> <s id="s.005594">Spira autem conicè ducta, aut &longs;phæricè, pa­<lb/>rum utilis accideret Machinatoris in&longs;tituto; cum enim, ut fir­<lb/>metur, in&longs;erenda &longs;it foramini &longs;imiliter in &longs;piram excavato, ma­<lb/>jores coni, aut globi, &longs;piræ non congruerent minoribus &longs;piris <lb/>foraminis conici aut &longs;phærici in modum &longs;caphij, nec per eas <lb/>promoveri po&longs;&longs;ent; atque minores coni, aut globi, &longs;piræ in am­<lb/>plioribus &longs;piris foraminis firmari nequirent. </s> <s id="s.005595">Oportet igitur &longs;pi­<lb/>ram omnino &longs;imilibus ductibus, atque æqualibus con&longs;tare; id <lb/>quod non ni&longs;i in cylindro obtinetur. </s> <s id="s.005596">Quapropter Cochlea, de <lb/>qua hìc agimus, e&longs;t &longs;olida &longs;pira in &longs;uperficie excavati cylindri <lb/>efformata; quæ vitium Capreolos arboris ramum complexos <lb/>imitata vulgari vocabulo <emph type="italics"/>Vitis<emph.end type="italics"/> (& forta&longs;&longs;e aptiùs) nominatur. </s> <lb/> <s id="s.005597">Receptaculum verò concavum, cui cylindrus in helicem de­<lb/>formatus immittitur, habétque &longs;pirales cavitates &longs;olidæ cylin­<lb/>dri &longs;piræ congruentes, <emph type="italics"/>Matrix<emph.end type="italics"/> dicitur, alij <emph type="italics"/>Tylum, Cochlidium <emph.end type="italics"/><lb/>alij, vocabulo ad hanc &longs;ignificationem detorto, vulgus <emph type="italics"/>Matrem <lb/>Vitis<emph.end type="italics"/> nuncupat. </s> </p> <p type="main"> <s id="s.005598">Ut autem &longs;piram cylindro æqualibus atque &longs;imilibus gyris <lb/>circumductam intelligas, concipe triangulum rectangulum, <lb/>cujus perpendiculum æquale &longs;it dato lateri aut Axi cylindri <lb/>Recti, ba&longs;is verò trianguli toties contineat perimetrum ba&longs;is <lb/>cylindri, quoties &longs;pira cylindrum ip&longs;um complecti debet; nam <lb/>huju&longs;modi trianguli hypothenu&longs;a lineam &longs;piralem omnino &longs;i­<lb/>militer ductam in cylindri &longs;uperficie de&longs;cribet, &longs;i triangulum <lb/>cylindro circumvolvatur. </s> </p> <p type="main"> <s id="s.005599">Sit cylindri altitudo AB, ejú&longs;que ba&longs;is circulari peripheriæ <lb/>&longs;it æqualis recta BC ad rectum an­<lb/><figure id="id.017.01.777.1.jpg" xlink:href="017/01/777/1.jpg"/><lb/>gulum CBA con&longs;tituta. </s> <s id="s.005600">Oporteat <lb/>autem &longs;piram quatuor gyris com­<lb/>plecti cylindrum; idcirco recta BC <lb/>producatur, ut tota BF &longs;it ip&longs;ius BG <lb/>quadrupla: ducta enim hypothenu­<lb/>&longs;a FA, &longs;i triangulum cylindro cir­<lb/>cumplicetur quadruplici convolutio­<lb/>ne, de&longs;ignabit in cylindri &longs;uperficie quatuor &longs;piras omnino &longs;i­<lb/>miles & æquales. </s> <s id="s.005601">Spirarum æqualitatem & &longs;imilitudinem fa-<pb pagenum="762" xlink:href="017/01/778.jpg"/>cilè demon&longs;trabis, &longs;i trianguli ba&longs;im BF, & altitudinem BA, <lb/>utramque in quatuor æquales partes di&longs;tinxeris; deinde ex &longs;in­<lb/>gulis divi&longs;ionum punctis rectas CM, DL, EK altitudini BA <lb/>parallelas, & rectas GK, HL, IM parallelas ba&longs;i BF excita­<lb/>veris; &longs;ibi enim occurrentes in punctis K, L, M, divident hy­<lb/>pothenu&longs;am in quatuor æquales partes, ut patet ex 2. lib. 6: Ni­<lb/>mirum ut FE ad ED, ita FK ad KL; & ut FD ad DC, ita FL <lb/>ad LM; & ut FC ad CB, ita FM ad MA: &longs;unt autem FE & <lb/>ED ex hypothe&longs;i æquales, igitur etiam FK & KL æquales: <lb/>FD po&longs;ita e&longs;t ip&longs;ius DC dupla, ergo FL ip&longs;ius LM dupla; <lb/>ergo LM æqualis e&longs;t ip&longs;i KL, aut FK: Demum FC ex con­<lb/>&longs;tructione e&longs;t ip&longs;ius CB tripla; igitur etiam FM e&longs;t tripla <lb/>ip&longs;ius MA; quare MA æqualis e&longs;t &longs;ingulis reliquis partibus <lb/>FK, KL, LM; & tota hypothenu&longs;a divifa e&longs;t in quatuor <lb/>æquales partes. </s> <s id="s.005602">Item in parallelogrammo KD, per 34. lib. 1. <lb/>æqualia &longs;unt oppo&longs;ita latera KN & ED, atque in parallelo­<lb/>grammo LC æqualia &longs;unt LO & DC, quemadmodum & in <lb/>parallelogrammo MB æqualia &longs;unt MI & CB: Sicut igitur <lb/>rectæ FE, ED, DC, CB ex hypothe&longs;i &longs;unt æquales, etiam <lb/>FE, KN, LO, MI &longs;unt inter &longs;e æquales. </s> <s id="s.005603">Similiter o&longs;tendes <lb/>&longs;icut æquales &longs;unt ex con&longs;tructione BG, GH, HI, IA, ita <lb/>æquales inter &longs;e e&longs;&longs;e EK, NL, OM, IA. </s> <s id="s.005604">Cum itaque trian­<lb/>gula FEK, KNL, LOM, MIA habeant tria latera &longs;ingula <lb/>&longs;ingulis æqualia, & &longs;imiliter po&longs;ita, ip&longs;a &longs;unt quoque æquian­<lb/>gula, ac proinde &longs;imiliter inclinatæ &longs;unt &longs;ingulæ &longs;piræ FK, KL, <lb/>LM, MA, quæ pariter demon&longs;tratæ &longs;unt æquales. </s> <s id="s.005605">Quam &longs;i­<lb/>milem inclinationem o&longs;tendit æqualitas angulorum ad F, K, <lb/>L, M, propter linearum paralleli&longs;mum. </s> <s id="s.005606">Triangulum igitur <lb/>ABF &longs;uâ hypothenusâ FA de&longs;ignat in cylindri &longs;uperficie qua­<lb/>tuor &longs;imiles & æquales &longs;piras. </s> </p> <p type="main"> <s id="s.005607">Verùm quid juvaret in exteriore cylindri &longs;uperficie &longs;piralem <lb/>lineam exqui&longs;itè de&longs;crip&longs;i&longs;&longs;e, ni&longs;i corpus ip&longs;um cylindricum in <lb/>&longs;olidam &longs;piram deformaretur? </s> <s id="s.005608">Quapropter nece&longs;&longs;ariò cylin­<lb/>drum circumplectuntur duæ &longs;piræ, cava altera & depre&longs;&longs;a, al­<lb/>tera convexa & prominens, quibus &longs;imiliter atque æqualiter <lb/>depre&longs;&longs;æ & prominentes duæ &longs;piræ in receptaculi &longs;eu Matricis <lb/>foramine cylindricè excavato requiruntur ita illis re&longs;ponden­<lb/>tes, ut depre&longs;&longs;am receptaculi &longs;piram &longs;ubeat prominens cylindri <pb pagenum="763" xlink:href="017/01/779.jpg"/>&longs;pira, & vici&longs;&longs;im prominentem receptaculi &longs;piram excipiat de­<lb/>pre&longs;&longs;a cylindri &longs;pira. </s> <s id="s.005609">Ex quo fit, ut convolutus circa &longs;uum <lb/>axem cylindrus attollatur aut deprimatur, adducatur aut redu­<lb/>catur, prout opus fuerit, atque cum eo corpus ba&longs;i illius proxi­<lb/>mum, &longs;eu adnexum urgeatur, aut trahatur, elevetur, aut pie­<lb/>matur. </s> </p> <p type="main"> <s id="s.005610">Vulgati&longs;&longs;imus autem & frequenti&longs;&longs;imus e&longs;t hujus Facultatis <lb/>u&longs;us, ubi poti&longs;&longs;imùm opus e&longs;t validâ pre&longs;&longs;ione, ut in prælis vi­<lb/>nariis ad exprimendum ex uvæ jam pre&longs;&longs;æ reliquiis tortivum <lb/>mu&longs;tum, apud typographos ad imprimendos &longs;ubjectæ chartæ <lb/>ex typis characteres, apud bibliopægos ad comprimendos li­<lb/>bros, jam compactos, apud fabros ferrarios ad firmandas fer­<lb/>reas laminas limâ expoliendas, atque apud alios artifices. </s> <lb/> <s id="s.005611">Quamquam & &longs;æpi&longs;&longs;imè clavorum loco, quibus ligna, aut me­<lb/>tallicæ laminæ configuntur citrà mallei percu&longs;&longs;ionem, cochleis <lb/>utimur, & quidem ad validiorem atque perennem firmitatem; <lb/>neque enim revelli pote&longs;t cochlea, aut excuti, quemadmodum <lb/>clavus. </s> <s id="s.005612">Sed tunc huju&longs;modi cochleæ non exercent vim facul­<lb/>tatis Mechanicæ; eatenus &longs;cilicet validiùs, quàm clavi, duo <lb/>corpora, quæ compinguntur, connectunt, quatenus multipli­<lb/>ces in cylindruli facie &longs;olidarum &longs;pirarum ductus pluribus cavis <lb/>foraminum &longs;piris implicantur ex cylindruli convolutione; qui <lb/>propterea eximi non pote&longs;t, ni&longs;i in contrarium revolvatur; <lb/>quandiu quidem incorruptum permanet lignum neque ex hu­<lb/>more putre&longs;cens, neque vermiculo erodente cario&longs;um, neque <lb/>calore nimio ita di&longs;cedens atque dehi&longs;cens, ut laxato foramine <lb/>jam non ampliùs &longs;olida cylindruli &longs;pira congruentibus &longs;triis <lb/>coërceatur. </s> </p> <p type="main"> <s id="s.005613">Hinc e&longs;t in &longs;u&longs;tentando pondere ex cochleâ &longs;u&longs;pen&longs;o pro­<lb/>priè non exerceri vim Mechanicam; nihil enim ampliùs co­<lb/>nante Potentiâ (quemadmodum in Vecte, aut Axe in Peritro­<lb/>chio, aut fune Trochlearum retinendo opus e&longs;t, quæ pondus <lb/>elevavit convoluto cylindro in cochleam deformato, &longs;ola &longs;pira­<lb/>rum cavæ atque convexæ complexio efficit, ut cylindrus cum <lb/>adnexo pondere retineatur, ne recidat, quatenus à &longs;ubjectâ lo­<lb/>culamenti &longs;pirâ &longs;olidâ &longs;u&longs;tinetur: quemadmodum & &longs;ub&longs;cudi­<lb/>bus compagem cohibentibus accidit, quatenus &longs;ecuricla ex mi­<lb/>nore in majorem amplitudinem explicata decre&longs;centis recepta-<pb pagenum="764" xlink:href="017/01/780.jpg"/>culi angu&longs;tiis coërcetur, ne excurrat; adeóque confixum hu­<lb/>ju&longs;imodi &longs;ub&longs;cude corpus grave inferius retinetur, ne à &longs;uperio­<lb/>re disjungatur, & cadat. </s> </p> <p type="main"> <s id="s.005614">Tota igitur vis Machinalis à Cochleâ exercetur in motu, <lb/>quem à potentiâ illam circumagente recipit. </s> <s id="s.005615">Et &longs;anè &longs;i poten­<lb/>tiæ cylindrum ver&longs;antis motum comparemus cum motu ponde­<lb/>ris, quod à cochleâ urgetur, aut trahitur; &longs;tatim apparebit po­<lb/>tentiam quidem circulum de&longs;cribere circa convoluti cylindri <lb/>axem, pondus verò recta moveri, prout promovetur, aut retra­<lb/>hitur cylindrus. </s> <s id="s.005616">Cum itaque in &longs;ingulis cylindri conver&longs;ioni­<lb/>bus ejus motum definiat &longs;piræ à &longs;pirâ intervallum; &longs;i hoc cum <lb/>circulari peripheriâ conferatur, innote&longs;cet motuum Ratio, & <lb/>Potentiæ momentum, quæ eò minorem in pondere re&longs;i&longs;ten­<lb/>tiam invenit, quò tardiùs hoc movetur. </s> <s id="s.005617">Hinc &longs;i cylindri alti­<lb/>tudo ad eju&longs;dem diametrum &longs;it ut 20 ad 1, numeratá&longs;que &longs;piras <lb/>cylindrum complectentes inveneris e&longs;&longs;e 35, rectè definies con­<lb/>volutionibus 35 re&longs;pondere totum cylindri motum, atque adeò <lb/>&longs;piræ à &longs;pirà intervallum e&longs;&longs;e ad cylindri diametrum ut 4. ad 7: <lb/>ex quo infertur circuli peripheriam ad &longs;pirarum di&longs;tantiam, <lb/>hoc e&longs;t potentiæ motum ad motum ponderis, e&longs;&longs;e proximè <lb/>ut 22 ad 4, atque potentiæ conatum ut 4. vincere po&longs;&longs;e quam­<lb/>libet re&longs;i&longs;tentiam minorem quàm ut 22, &longs;pectatâ Ratione, quam <lb/>infert cylindri cra&longs;&longs;ities, & &longs;pirarum obliquitas. </s> </p> <p type="main"> <s id="s.005618">Verùm quia non ni&longs;i parvulis cochleis, aut ubi levis conatus <lb/>requiritur, ita applicatur potentia, ut cylindri &longs;uperficiei ap­<lb/>plicata intelligatur, complanatâ &longs;cilicet eju&longs;dem cylindri extre­<lb/>mitate, quam &longs;ummis digitis apprehendere valeas, communi­<lb/>ter adhuc majus e&longs;t momentum Potentiæ, quàm ut ex circuli <lb/>peripheriâ ba&longs;im cylindri ambiente circum&longs;cribatur; additur <lb/>enim aut Radius cylindri Capiti quadrato infixus, aut aliquid <lb/>manubrij rationem habens, adeò ut potentia longè majorem <lb/>circulum de&longs;cribat, quàm &longs;it cylindri in &longs;piram deformati ba&longs;is: <lb/>ac proinde non ex cylindri cra&longs;&longs;itie, &longs;ed ex di&longs;tantiâ potentiæ <lb/>ab axe cylindri definiendus e&longs;t eju&longs;dem potentiæ circulum per­<lb/>ficientis motus, atque cum &longs;pirarum intervallo motum ponde­<lb/>ris metiente comparandus. </s> </p> <p type="main"> <s id="s.005619">Hinc ad imprimendas metallicæ laminæ ex argento, aut <lb/>auro, aut cupro imagines citrà percu&longs;&longs;ionem, &longs;uper &longs;olido pla-<pb pagenum="765" xlink:href="017/01/781.jpg"/>no erectis atque infixis ad perpendiculum duobus ferreis pedi­<lb/>bus ferreo pariter tran&longs;ver&longs;ario firmatis, in quo excavata co­<lb/>chleæ congruens Matrix, typus inter laminam & cylindrum <lb/>interjectus validè urgetur ex cylindri convolutione, & imagi­<lb/>nem exprimit: quia videlicet &longs;uperiori cylindri Capiti quadra­<lb/>to in&longs;eritur longior ferreus vectis hinc atque hinc productus, <lb/>ut duplici ejus extremitati duplex potentia, &longs;i opus fuerit, ap­<lb/>plicetur. </s> <s id="s.005620">Quapropter ab axe cylindri ad vectis huju&longs;modi ex­<lb/>tremitatem ducta linea e&longs;t Radius circuli potentiæ motum de­<lb/>terminantis; atque &longs;i huju&longs;modi Radij longitudo ad &longs;pirarum <lb/>intervallum fuerit ut 50 ad 1, circuli diameter e&longs;t 100, eju&longs;que <lb/>peripheria major quàm 314; & potentiæ motus ad motum typi <lb/>laminam prementis e&longs;t ut 314 ad 1: idcirco &longs;i in vectis extremi­<lb/>tatibus &longs;int &longs;inguli homines perinde conantes, ac &longs;i libras 50 <lb/>&longs;inguli moverent, premitur typus vi hujus cochleæ qua&longs;i à <lb/>pondere librarum 31400. </s> </p> <p type="main"> <s id="s.005621">Quod autem de pre&longs;&longs;ione dicitur, &longs;imili ratione intelligen­<lb/>dum e&longs;t de ponderis elevatione, &longs;i fortè aut inferiori cylindri <lb/>ba&longs;i adnexum fuerit, aut ejus capiti impo&longs;itum; &longs;icut enim cor­<lb/>pus prementi re&longs;i&longs;tit ratione particularum con&longs;tipatarum, ita <lb/>elevanti repugnat ratione &longs;uæ gravitatis: utrobique igitur &longs;imi­<lb/>lem virtutem habet potentia ad vincendam re&longs;i&longs;tentiam, quan­<lb/>do utrobique eadem invenitur Ratio motuum atque momento­<lb/>rum. </s> <s id="s.005622">Propterea in huju&longs;modi cochleis, quæ infixo Radio con­<lb/>volvuntur, non e&longs;t admodum anxiè procuranda cylindri cra&longs;­<lb/>&longs;ities, modò &longs;atis &longs;olidus &longs;it, nec fragilis: eadem quippe e&longs;t cir­<lb/>culi à potentiæ motu de&longs;cripti peripheria, &longs;ivè major &longs;it, &longs;ivè <lb/>minor cylindri cra&longs;&longs;itudo, quando eadem e&longs;t potentiæ di&longs;tan­<lb/>tia à cylindri axe, ac proinde idem e&longs;t momentum. </s> </p> <p type="main"> <s id="s.005623">Illud quidem ad rem facit maximè, quam obliquè inclinatus <lb/>&longs;it &longs;pirarum ductus; hinc enim oritur intervalli Ratio inter pro­<lb/>ximos &longs;pirarum circuitus, qui frequenti&longs;&longs;imi &longs;unt, ac brevi in­<lb/>tervallo disjuncti, &longs;i linea &longs;piralis &longs;it maximè inclinata, rari au­<lb/>tem atque notabiliter &longs;ejuncti, &longs;i illa fuerit ad majorem angu­<lb/>lum (acutum tamen) erecta: E&longs;t nimirum huju&longs;modi interval­<lb/>lum æquale Tangenti anguli inclinationis, po&longs;ito Radio am­<lb/>bitu ba&longs;is cylindri; ip&longs;a autem &longs;pira e&longs;t eju&longs;dem anguli Secans. </s> <lb/> <s id="s.005624">Quare datâ cylindri diametro, invenitur peripheria ba&longs;is; & <pb pagenum="766" xlink:href="017/01/782.jpg"/>dato &longs;pirarum intervallo, invenitur angulus huic intervallo tan­<lb/>quam Tangenti oppo&longs;itus, &longs;cilicet inclinatio &longs;piræ, & hypo­<lb/>thenu&longs;a tanquam eju&longs;dem anguli Secans dat ip&longs;ius lineæ &longs;pira­<lb/>lis longitudinem. </s> </p> <p type="main"> <s id="s.005625">Quod &longs;i totius lineæ &longs;piralis univer&longs;um cylindrum com­<lb/>plectentis lineam de&longs;ideras, toties peripheriam ba&longs;is multipli­<lb/>ca, quot &longs;unt &longs;pirarum circuitus, & habebis Radium; cylindri <lb/>altitudo dabit Tangentem, cui re&longs;pondens Secans indicabit to­<lb/>tius &longs;piræ integram longitudinem. </s> <s id="s.005626">Sit ex. </s> <s id="s.005627">gr. <!-- REMOVE S-->cylindri altitudo <lb/>ped. <!-- REMOVE S-->3. hoc e&longs;t unciarum 36. ejus diameter unciarum 7; ergo <lb/>ba&longs;is perimeter unc. </s> <s id="s.005628">22: Spirarum circuitus &longs;int 25: igitur <lb/>ductâ perimetro 22 in 25, habetur 550 tanquam Radius, & 36 <lb/>tanquam Tangens: igitur ut unciæ 550 ad uncias 36, ita Ra­<lb/>dius 100000 ad 6545 Tangentem gr. <!-- REMOVE S-->3. m. </s> <s id="s.005629">45. cui re&longs;pondet <lb/>Secans 100214: Quare ut 100000 ad 100214, ita unciæ 550 <lb/>ad uncias (551 177/1000) longitudinem totius lineæ &longs;piralis; quam, &longs;i <lb/>careas Canone Trigonometrico, etiam habebis ex 47. lib. 1. <lb/>addendo quadrata numerorum 550 & 36, erit enim horum <lb/>&longs;umma quadratum, cujus Radix dabit eandem quæ&longs;itam &longs;piræ <lb/>longitudinem. </s> </p> <p type="main"> <s id="s.005630">Cum autem hæc &longs;piræ longitudo, &longs;ive univer&longs;a, &longs;ive parti­<lb/>culatim a&longs;&longs;umatur, &longs;emper longior &longs;it, &longs;ivè multiplici, &longs;ivè &longs;in­<lb/>gulari perimetro circuli, qui e&longs;t ba&longs;is cylindri, utique motus <lb/>potentiæ, ejú&longs;que momentum, non ex hac &longs;pirali lineâ de&longs;u­<lb/>mendum e&longs;t; neque enim ip&longs;a e&longs;&longs;e pote&longs;t men&longs;ura motûs po­<lb/>tentiæ cylindro applicatæ ad ejus diametri extremitatem. </s> <s id="s.005631">Hinc <lb/>e&longs;t mihi non arridere eorum &longs;ententiam, qui cochleæ vires re­<lb/>ferunt ad planum inclinatum, quod ab ipsâ lineâ &longs;pirali repræ­<lb/>&longs;entetur. </s> <s id="s.005632">In plano &longs;iquidem inclinato momentum gravitatis, <lb/>ad eju&longs;dem gravitatis momentum in perpendiculo, &longs;e habet re­<lb/>ciprocè ut perpendiculum ad ip&longs;am lineam inclinatam; ac pro­<lb/>pterea eandem Rationem &longs;ervant conatus Potentiæ moventis <lb/>pondus aut in perpendiculo, aut in plano inclinato. </s> <s id="s.005633">At hìc po­<lb/>tentia non movetur juxta lineæ inclinatæ longitudinem, &longs;ed <lb/>breviore motu juxta ba&longs;im trianguli rectanguli, cujus hypothe­<lb/>nu&longs;a e&longs;t ip&longs;a linea inclinata: Igitur potentiæ momentum ali­<lb/>quanto minus cen&longs;endum e&longs;t, quam pro Ratione plani inclina­<lb/>ti. </s> <s id="s.005634">Adde momenta gravitatis ponderis alicujus tunc &longs;olùm <pb pagenum="767" xlink:href="017/01/783.jpg"/>fieri minora in plano inclinato, quando illi in&longs;i&longs;tit, & deor&longs;um <lb/>nititur premendo ip&longs;um planum: at &longs;i pondus idem incumbat <lb/>plano horizontali, verum quidem e&longs;t planum verticale, quod <lb/>adversùs pondus moveatur non &longs;ecundùm directionem, quæ <lb/>recta occurrat centro gravitatis eju&longs;dem ponderis, &longs;ed obliquè, <lb/>minus invenire re&longs;i&longs;tentiæ: &longs;ed pondus illud propriè non mo­<lb/>vetur &longs;uper plano, licèt ab eo obliquè repellatur; & potiùs pla­<lb/>num movetur juxta pondus: hìc verò &longs;i pondus in plano hori­<lb/>zontali jacens &longs;it adnexum cochleæ trahenti, aut oppo&longs;itum <lb/>cochleæ repellenti & urgenti, non movetur obliquè, &longs;ed motu <lb/>directo: non igitur movetur &longs;uper planum inclinatum. </s> </p> <p type="main"> <s id="s.005635">Porrò unum e&longs;t in Cochleâ quodammodo &longs;ingulare, quod <lb/>in nullam aliam Facultatem æquè convenire deprehenditur: <lb/>Cum enim requiratur & cylindrus in helicem inflexus, & Ma­<lb/>trix illi congruens, ita ut alteri quies, alteri motus debeatur; <lb/>perinde e&longs;t &longs;i matrice immotâ cylindrus convertatur, atque &longs;i <lb/>manente cylindro matrix ip&longs;a convolvatur, modò Potentia <lb/>æquali Radio utatur, &longs;ivè cylindri capiti, &longs;ivè Matrici infixo: <lb/>eadem &longs;iquidem &longs;unt potentiæ momenta, & æqualis motus pon­<lb/>deris; æqualiter enim promovetur matrix in cylindro &longs;tabili, at­<lb/>que cylindrus in Matrice immotâ. </s> <s id="s.005636">Id quod maxime locum <lb/>habet, ubi opus e&longs;t compre&longs;&longs;ione, & vulgati&longs;&longs;imus e&longs;t apud va­<lb/>rios artifices u&longs;us. </s> </p> <p type="main"> <s id="s.005637">Jam verò quod ad diuturnitatem &longs;pectat, diffitendum non e&longs;t <lb/>cochleam frequenti u&longs;u atteri, e&longs;t enim perpetuus illius cum <lb/>&longs;uâ Matrice conflictus, quamvis plurimùm juvet, &longs;i &longs;megmate, <lb/>aut pingui aliquo humore inungatur, quo lubrica fiat, ut faci­<lb/>liùs convolvatur, minú&longs;que atteratur. </s> <s id="s.005638">Deinde quamvis unica <lb/>&longs;pira matrici &longs;ufficiat, ut vel ip&longs;a, vel cylindrus promoveatur, <lb/>aut retrahatur, nihilominus faciliùs labem patitur, quàm &longs;i plu­<lb/>res in &longs;piras fuerit excavata: cum enim aut à gravitate ponde­<lb/>ris &longs;u&longs;tollendi, aut à partium con&longs;tipatione repugnantium com­<lb/>pre&longs;&longs;ioni, ip&longs;a unica re&longs;i&longs;tentiam inveniat, utique vel ponderis <lb/>gravitas ip&longs;i uni innititur, vel potentiæ conatus, reluctante cor­<lb/>pore comprimendo aut trahendo, in illam &longs;olam effunditur. </s> <lb/> <s id="s.005639">Propterea expedit alteram &longs;altem, aut tertiam &longs;piram addere, <lb/>ut divi&longs;o in plures conatu firmitati con&longs;ulatur. </s> </p> <p type="main"> <s id="s.005640">Eandem ob cau&longs;am aliquando cylindrum complectitur non <pb pagenum="768" xlink:href="017/01/784.jpg"/>unica &longs;pirarum &longs;eries, &longs;ed & alia illi parallela additur (nec quic­<lb/>quam prohibet, quin & plures duabus &longs;int huju&longs;modi paralle­<lb/>larum &longs;pirarum &longs;eries) ut multo validior ac firmior &longs;it cochlea, <lb/>ne facilè &longs;pira aliqua di&longs;&longs;ipetur, aut &longs;i qua labefactetur, nullum <lb/>&longs;equatur incommodum, alterâ &longs;pirâ parallelâ ejus vices &longs;up­<lb/><figure id="id.017.01.784.1.jpg" xlink:href="017/01/784/1.jpg"/><lb/>plente. </s> <s id="s.005641">Sic &longs;piræ ABCDEF paral­<lb/>lela &longs;tatuitur altera ab I incipiens, & <lb/>per KLMNOP &longs;imili lap&longs;u &longs;erpens. </s> <lb/> <s id="s.005642">Ex hac tamen multiplici &longs;pirâ non au­<lb/>getur momentum potentiæ applicatæ <lb/>Radio VS; neque enim circulus à po­<lb/>tentiâ in S applicatâ de&longs;criptus com­<lb/>parandus e&longs;t cum AK, &longs;ed cum AC; <lb/>unicâ &longs;iquidem cylindri convolutione <lb/>promovetur cylindrus non ex A in K, <lb/>&longs;ed ex A in C. <!-- KEEP S--></s> <s id="s.005643">Propterea oblato cy­<lb/>lyndro in Cochleam deformato dili­<lb/>genter attendendum e&longs;t, utrum plures &longs;int &longs;pirarum &longs;eries, an <lb/>unica; ne fortè ex brevi inter proximas &longs;piras intervallo perpe­<lb/>ram conjicias lineam &longs;piralem magis inclinatam, quàm reip&longs;a <lb/>&longs;it; prius enim dijudicandum e&longs;t, an illæ proximæ &longs;piræ ad ean­<lb/>dem Helicem &longs;pectent; attenditur &longs;cilicet intervallum &longs;pirarum <lb/>ad eandem &longs;eriem continuo ductu pertinentium. <lb/></s> </p> <p type="main"> <s id="s.005644"><emph type="center"/>CAPUT II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005645"><emph type="center"/><emph type="italics"/>An utilis &longs;it Cochlea duplex contraria.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005646">QUamvis ad &longs;uperandam modico labore re&longs;i&longs;tentiam non <lb/>modicam corporis, quod Cochlea urget, aut trahit, hu­<lb/>ju&longs;modi Facultas &longs;it poti&longs;&longs;imùm excogitata, &longs;æpi&longs;&longs;imè tamen <lb/>cochleam adhibemus non ad vincendam re&longs;i&longs;tentiam, quæ ali­<lb/>quando tenui&longs;&longs;ima e&longs;t, &longs;ed unicè ad motum ita temperandum, <lb/>ut pro opportunitate exiguus &longs;it; neque enim mu&longs;culorum mo­<lb/>tum ita attenuare pro arbitrio pote&longs;t homo, ut &longs;emper quàm <pb pagenum="769" xlink:href="017/01/785.jpg"/>minimus contingat: propterea deformato in cochleam cylin­<lb/>dro utimur, ut majori potentiæ motui minor motus in corpore <lb/>movendo re&longs;pondeat. </s> <s id="s.005647">Sic ad excipiendas objectorum corpo­<lb/>rum &longs;pecies opticas aut lumen, cum non eadem &longs;emper tu­<lb/>bo&longs;pecilli longitudo opportuna &longs;it pro variâ tum objecti di&longs;tan­<lb/>tiâ, tum oculi conformatione, prudenter ab aliquibus extre­<lb/>mus tubulus, cui lens ocularis in&longs;eritur, in &longs;piram contorque­<lb/>tur, ut faciliùs & citiùs ju&longs;tam longitudinem a&longs;&longs;equantur: id <lb/>quod ægrè obtinerent, &longs;i rectà tubulum illum adducerent, aut <lb/>reducerent, ut &longs;atis experientiâ con&longs;tat. </s> </p> <p type="main"> <s id="s.005648">Hinc aliquando contingit oppo&longs;itos motus conciliandos e&longs;&longs;e <lb/>duobus corporibus ita, ut aut ad &longs;e mutuò accedant, aut ma­<lb/>gis invicem disjungantur, &longs;ive illa motui valde repugnent, &longs;ive <lb/>&longs;ola motûs tarditas requiratur. </s> <s id="s.005649">Propterea eju&longs;dem cylindri lon­<lb/>gitudo in duas helices di&longs;tinguitur, quæ &longs;imili quidem ductu <lb/>cylindrum circumplectuntur, &longs;ed illis in diver&longs;a abeuntibus, <lb/>unius &longs;piræ non &longs;unt alterius &longs;piris parallelæ; quæ eatenus con­<lb/>trariæ vocari po&longs;&longs;unt, quatenus oppo&longs;itos motus efficiunt, ne­<lb/>que eæ &longs;unt, quæ in unam continuam &longs;piram coale&longs;cere queant. <lb/><figure id="id.017.01.785.1.jpg" xlink:href="017/01/785/1.jpg"/><lb/>Ex cylindri AB medio puncto C exeant duæ &longs;piræ ad ea&longs;dem <lb/>partes inclinatæ, hinc CD versùs extremitatem A procedens, <lb/>hinc verò CE versùs extremitatem B; utraque enim &longs;uam ma­<lb/>tricem habens, cui in&longs;eratur, dum convolvitur cylindrus, ma­<lb/>tricem longiùs à medio promovet, aut ad medium attrahit; at­<lb/>que cum matrice adnexa corpora &longs;imili & æquali motu moven­<lb/>tur. </s> <s id="s.005650">Hinc &longs;i utraque matrix proxima &longs;it medio puncto C, ex <lb/>primâ convolutione cylindri altera per CD removetur u&longs;que <lb/>in F, altera per CE in G; atque adeò &longs;icut matrices moven­<lb/>tur per CF, & CG, ita eadem men&longs;ura corporum adnexorum <pb pagenum="470" xlink:href="017/01/786.jpg"/>motum metitur, quæ invicem removentur intervallo FG; & <lb/>ita deinceps in cæteris cylindri convolutionibus. </s> </p> <p type="main"> <s id="s.005651">Quod &longs;i movendorum in oppo&longs;itas partes corporum re&longs;i&longs;ten­<lb/>tia exigua &longs;it, &longs;atis fuerit extremitatibus cylindri an&longs;ulas appo­<lb/>nere, quibus circumactis cylindrus ip&longs;e in cochleas deforma­<lb/>tus convertatur. </s> <s id="s.005652">Sic antè annos ferè quadraginta (cum non ar­<lb/>rideret vulgaris tunc apud artifices circinorum forma, qui in­<lb/>terjecto elatere crura divaricant; &longs;ed inflexam in arcum co­<lb/>chleam alteri crurum infixam, & per alterum trajectam, de­<lb/>currente matrice exteriùs appo&longs;itâ, dilatationem moderatur ar­<lb/>tifex) ju&longs;&longs;i mihi parari ab&longs;que ullo elatere circinum, quem ip&longs;e <lb/>dilatarem atque contraherem pro arbitrio, cochleam huju&longs;mo­<lb/>di duplicem in hanc atque in illam partem convertens. </s> <s id="s.005653">Ad <lb/>trientem totius longitudinis à nodo, &longs;ingula crura cylindricum <lb/>foramen habent, ut &longs;ingulis in&longs;erantur cylindruli congruentes <lb/>exqui&longs;itè politi, quorum &longs;uperiori extremitati &longs;unt adnexæ co­<lb/>chlearum matrices, inferior extremitas extra circini &longs;oliditatem <lb/>exiens in helicem de&longs;init, ut appo&longs;itâ matrice cylindrulus in­<lb/>tra foramen contineatur. </s> <s id="s.005654">Huju&longs;modi e&longs;t cylindrulus IS cra&longs;­<lb/>&longs;itiei circini re&longs;pondens, &longs;uperior pars e&longs;t matrix R, infima ex­<lb/>tra circini &longs;oliditatem in helicem deformata e&longs;t SV, cui addita <lb/>matrix X continet cylindrulum intrà foramen, cui inditus e&longs;t, <lb/>ita tamen, ut cylindruli ip&longs;ius opportunam convolutionem non <lb/>impediat. </s> <s id="s.005655">Duæ igitur matrices, cuju&longs;modi e&longs;t R, coaptantur <lb/>duplici cochleæ cujus deinde extremitatibus, ad facilem <lb/>conver&longs;ionem, an&longs;ulæ adduntur, adeò ut illæ matrices non <lb/>&longs;int exemptiles. </s> <s id="s.005656">Quare utrique crurum circini foramini, <lb/>utriu&longs;que matricis cylindrulus IS in&longs;eratur, & inferius ma­<lb/>trice X firmetur: Nam convertendo cylindrum in dupli­<lb/>cem cochleam deformatum, circini crura divaricabis, aut <lb/>adduces, ut libuerit. </s> <s id="s.005657">Neque quicquam officiet cylindri <lb/>rectitudo, quia matricum cylindruli IS pro opportunitate <lb/>volvuntur. </s> <s id="s.005658">Hinc circino eodem uti poteris ab&longs;que cochleâ, <lb/>deduci enim hæc pote&longs;t, exemptis matricibus è foramine, <lb/>cui in&longs;eruntur. </s> </p> <p type="main"> <s id="s.005659">At verò &longs;i validiore conatu opus fuerit, ad medium cy­<lb/>lindrum, ubi cochlearum &longs;piræ incipiunt, oportebit forami­<lb/>na excavare, quibus immitti queat vectis, ut potentiæ mo-<pb pagenum="771" xlink:href="017/01/787.jpg"/>tus ad ponderum motum habeat majorem Rationem. <!-- KEEP S--></s> <s id="s.005660">Sic <lb/><figure id="id.017.01.787.1.jpg" xlink:href="017/01/787/1.jpg"/><lb/>duplici cochleâ cylindro circumductâ ad medium E &longs;int fora­<lb/>mina, quibus vectis BC &longs;ubinde inferri po&longs;&longs;it: duo autem <lb/>membra FD, & MN ex materiâ &longs;atis &longs;olidâ, qua extremita­<lb/>te re&longs;piciunt vectem, matricem habeant cochleæ congruen­<lb/>tem, ut ex vectis & cochleæ conver&longs;ione aut ad &longs;e invicem <lb/>accedant, aut &longs;ejungantur: reliqua extremitas exterior D <lb/>& N cava &longs;it, ut corpus repellendum comprehendatur, re­<lb/>flectatur verò qua&longs;i in uncos K & R, ut &longs;i duo corpora attra­<lb/>henda fuerint, iis apprehendantur, &longs;ivè proximè & immedia­<lb/>tè, &longs;ivè funibus adnexa. </s> </p> <p type="main"> <s id="s.005661">Quanta &longs;it hujus in&longs;trumenti vis, etiam ad frangenda aut <lb/>dilatanda ferrea clathra, hinc patet, quòd, longiore vecte ad­<lb/>dito, potentiæ momenta notabiliter augentur; quia potentia <lb/>percurrit peripheriam circuli, cujus Radius à cylindri centro <lb/>ad vectis extremitatem producitur, pondera verò non ni&longs;i pro <lb/>&longs;pirarum intervallo moventur. </s> <s id="s.005662">Ubi tamen advertendum e&longs;t, <lb/>utrum cylindrus ita &longs;it alicui loculamento in&longs;ertus, ut ejus <lb/>medium DE nec ad dexteram, nec ad &longs;ini&longs;tram declinare <lb/>queat, an verò liber omninó &longs;it. </s> <s id="s.005663">Si enim interjectum mo­<lb/>vendis corporibus in&longs;trumentum omnino liberum &longs;it, pon­<lb/>dera verò movenda inæqualiter re&longs;i&longs;tant comparatis, aut eo­<lb/>rum gravitatibus, aut momentis ratione planorum non uno <lb/>modo inclinatorum, aut ex di&longs;parili &longs;uperficierum a&longs;perita­<lb/>te, non &longs;equitur æqualis eorum motus, &longs;ed qua parte ma­<lb/>jor invenitur re&longs;i&longs;tentia, minor quoque e&longs;t motus; quamvis <lb/>utrumque æqualiter di&longs;ter à medio cylindri, quod repellitur <lb/>quodammodo ad eam partem, ubi levior e&longs;t re&longs;i&longs;tentia. </s> <s id="s.005664">Si <lb/>enim ad N &longs;it aliquid ob&longs;tans motui, ut paries, aut firmi-<pb pagenum="472" xlink:href="017/01/788.jpg"/>ter infixus paxillus, ad D verò corpus aliquod repellendum; <lb/>utique ex vectis BC conver&longs;ione etiam cochlea convolvitur, <lb/>& corpus in D po&longs;itum tantumdem promovetur, quanto in­<lb/>tervallo ab&longs;unt F & M ex cochleæ conver&longs;ione; nam propter <lb/>impedimentum in N exi&longs;tens, nullo pacto ip&longs;um M move­<lb/>tur. </s> <s id="s.005665">Sin autem corpus in N non omnino reluctetur motui, <lb/>&longs;ed tamen re&longs;i&longs;tat magis, quàm corpus in D, illud quidem <lb/>aliquantulum repellitur, &longs;ed multò magis repellitur corpus, <lb/>quod e&longs;t in D; & in hoc motu medium cylindri punctum E <lb/>ad eas partes accedit, ad quas movetur corpus in D repul­<lb/>&longs;um. </s> <s id="s.005666">Quod &longs;i medium E ita e&longs;&longs;et loculamento aliquo con­<lb/>clu&longs;um, ut po&longs;itionem mutare nequeat, &longs;ed &longs;olùm convolvi <lb/>po&longs;&longs;it, tunc utrumque corpus æqualiter repellitur, quia &longs;pi­<lb/>rarum intervalla in utráque cochleâ æqualia &longs;unt. </s> <s id="s.005667">Hinc pa­<lb/>tet po&longs;&longs;e fieri motus inæquales, &longs;i &longs;pirarum inclinationes non <lb/>fuerint æquales; minùs enim movetur illud, quod &longs;piris &longs;pi&longs;­<lb/>&longs;ioribus urgetur. </s> </p> <p type="main"> <s id="s.005668">Quamvis autem huju&longs;modi duplex cochlea ad duo corpo­<lb/>ra disjungenda aut attrahenda &longs;æpè utilis &longs;it, ubi tamen exi­<lb/>guus motus requiritur, &longs;ivè ad augenda potentiæ momenta, <lb/>&longs;ive ad affectandam tarditatem, præ&longs;tabit &longs;implicem co­<lb/>chleam adhibere. </s> <s id="s.005669">Nam in duplicis cochleæ conver&longs;ione <lb/>disjunguntur, aut ad &longs;e invicem accedunt matrices (ac pro­<lb/>inde & corpora, quæ moventur) quantum e&longs;t duplex inter­<lb/>vallum, quo &longs;pira abe&longs;t à &longs;pira; &longs;ingulis nimirum cochleis <lb/>&longs;uum re&longs;pondet intervallum: at in &longs;implicis cochleæ con­<lb/>ver&longs;ione motus re&longs;pondet &longs;implici duarum proximarum &longs;pi­<lb/>rarum intervallo; ad quod idem potentiæ motus majorem <lb/>habet Rationem, quàm ad duplex intervallum. </s> <s id="s.005670">Quare fa­<lb/>tis e&longs;t, &longs;i alterutrum membrum cochleam includens longius <lb/>&longs;it, & matricem habeat; reliquum membrum, ut MN, <lb/>brevius e&longs;&longs;e pote&longs;t, quantum opus fuerit ad recipiendum <lb/>cylindri caput extenuatum in minorem cylindrum, intrà <lb/>foramen cylindricum exqui&longs;itè politum, ut facillimè con­<lb/>verti po&longs;&longs;it: ita verò caput illud muniatur, ut ex locula­<lb/>mento extrahi nequeat, quando utendum fuerit in&longs;tru­<lb/>mento ad corpora attrahenda: nam ad illa disjungenda cùm <lb/>adhibetur, &longs;atis reluctatur major diameter cylindri in co-<pb pagenum="773" xlink:href="017/01/789.jpg"/>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/>cini crura, uti <lb/><figure id="id.017.01.789.1.jpg" xlink:href="017/01/789/1.jpg"/><lb/>poteram unicâ & <lb/>&longs;implici cochleâ <lb/>RS: cylindri il­<lb/>lius extremitas <lb/>extenuetur in mi­<lb/>norem cylindrum <lb/>exqui&longs;itè lævigatum, qui prominentis capitis O foramini <lb/>cylindrico pariter polito in&longs;eratur, & exteriore an&longs;ula V <lb/>converti pro arbitrio po&longs;&longs;it. </s> <s id="s.005672">Alterius clavi TX caput T <lb/>matricem habeat cochleæ congruentem: nam conver&longs;a an­<lb/>&longs;ula V adducet clavum T, & cum eo crus circini, ad O, <lb/>aut ab hoc illum removebit, & crura divaricabit: & qui­<lb/>dem faciliùs licebit minutam in accipiendis punctorum di­<lb/>&longs;tantiis &longs;ubtilitatem per&longs;equi; quandoquidem uni integræ <lb/>conver&longs;ioni cylindri re&longs;pondet unicum &longs;pirarum intervallum, <lb/>non autem duo intervalla huju&longs;modi, quemadmodum cùm <lb/>duplex e&longs;t cochlea. </s> </p> <p type="main"> <s id="s.005673">Quæ verò hì dicta &longs;unt, in pluribus aliis locum habe­<lb/>re po&longs;&longs;unt, in quibus pro opportunitate modò &longs;implicem, <lb/>modò duplicem cochleam prudens Machinator adhibebit: <lb/>Et quidem &longs;i duplex futura &longs;it cochlea, neque æquali mo­<lb/>tu movenda &longs;int in oppo&longs;itas partes corpora, cochleas ip&longs;as <lb/>non &longs;imili, &longs;ed inæquali, &longs;pirarum inclinatione formari ju­<lb/>bebit. </s> <s id="s.005674">Cochleam autem ip&longs;am opportuno loco &longs;tatuet: & <lb/>&longs;i fortè corporum ip&longs;orum motus paulò velocior aut major <lb/>requiratur, quàm ferat ip&longs;a cochleæ convolutio, duobus <lb/>vectibus decu&longs;&longs;atis, & circa axem in decu&longs;&longs;atione ver&longs;ati­<lb/>libus uti poterit, atque cochleam cum &longs;uis clavis & matri­<lb/>cibus (ut &longs;uperiùs de circino dictum e&longs;t) non procul à de­<lb/>cu&longs;&longs;atione collocabit; nam modica illius convolutio non <lb/>exiguum motum tribuet corporibus in vectium illorum ex­<lb/>tremitate po&longs;itis, quippe quæ à decu&longs;&longs;atione magis di&longs;tant, <lb/>quàm cochlea. </s> </p> <p type="main"> <s id="s.005675">At, inquies, hoc idem Ergatâ præ&longs;tari poterit: &longs;i enim <lb/>funes breviorum brachiorum extremitatibus C & E adnexi <pb pagenum="774" xlink:href="017/01/790.jpg"/>connectantur cum Ergatæ cylindro, ex hujus conver&longs;ione ac­<lb/><figure id="id.017.01.790.1.jpg" xlink:href="017/01/790/1.jpg"/><lb/>cedent ad &longs;e invicem ex­<lb/>tremitates C & E, ac pro­<lb/>pterea etiam velociùs cor­<lb/>pora in F & D movebun­<lb/>tur. </s> <s id="s.005676">Ita planè: non diffi­<lb/>teor: &longs;ed &longs;i extremitates C <lb/>& E proximas disjungere <lb/>oporteat, adeóne promptus <lb/>erit Ergatæ u&longs;us, quin alio artificio opus &longs;it, ut hujus ope <lb/>disjungantur? </s> <s id="s.005677">Præterquam, quod &longs;æpè multum &longs;patij ad col­<lb/>locandam Ergatam requiritur; &longs;i maximè opus &longs;it illi pegma <lb/>con&longs;truere. </s> <s id="s.005678">Quid verò &longs;i vectes ip&longs;i promovendi &longs;int, non <lb/>retrahendi, ut in rebus &longs;cenicis contingere pote&longs;t? </s> <s id="s.005679">Quid &longs;i in <lb/>&longs;ublimiore loco res perficienda &longs;it? </s> <s id="s.005680">quàm incommodè opportu­<lb/>na Ergata &longs;atis longo vecte in&longs;tructa ibi parabitur? </s> <s id="s.005681">Sed illud <lb/>poti&longs;&longs;imum attendendum e&longs;t, quod vis cochleæ longè major <lb/>e&longs;t; nam in Ergatâ Ratio motûs potentiæ ad motum ponderis <lb/>e&longs;t eadem cum Ratione peripheriæ ab extremitate vectis de­<lb/>&longs;criptæ ad cylindri ambitum, hoc e&longs;t longitudinis vectis u&longs;que <lb/>ad axem cylindri, ad ip&longs;ius cylindri &longs;emidiametrum: at in co­<lb/>chleâ peripheria ab extremo vecte de&longs;cripta non comparatur <lb/>cum ip&longs;ius cylindri perimetro, &longs;ed cum proximarum &longs;pirarum <lb/>intervallo, quod &longs;æpi&longs;&longs;imè minus e&longs;t (&longs;altem pote&longs;t e&longs;&longs;e minus, <lb/>&longs;i magis inclinatæ & &longs;pi&longs;&longs;æ &longs;int &longs;piræ) quàm cylindri diameter, <lb/>aut ejus perimeter: ac proinde major e&longs;t Ratio motûs potentiæ <lb/>ad motum ponderis. <lb/></s> </p> <p type="main"> <s id="s.005682"><emph type="center"/>CAPUT III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005683"><emph type="center"/><emph type="italics"/>Cochlea cum Vecte, atque cum Axe componitur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005684">COntingere pote&longs;t aliquando onus Vecte elevandum e&longs;&longs;e, <lb/>tantam verò illius gravitatem deprehendi, ut &longs;ufficiens <lb/>Vectis longitudo non &longs;uppetat pro ratione operarum, quas <pb pagenum="775" xlink:href="017/01/791.jpg"/>adhibere po&longs;&longs;umus, aut &longs;altem eæ &longs;int loci angu&longs;tiæ, ut neque <lb/>huju&longs;modi Vectis longitudinem, neque operarum multitudi­<lb/>nem capiat: plures autem Vectes componere omnino incom­<lb/>modum &longs;it, quia, ut in loco dictum e&longs;t, minimus fieret oneris <lb/>elevandi motus. </s> <s id="s.005685">Præ&longs;tabit igitur Cochleam Vecti addere, ubi <lb/>maximè frequens futura &longs;it huju&longs;modi ponderum elevatio, <lb/>quemadmodum propè telonia, ubi ingentes mercium &longs;arcinæ <lb/>attollendæ &longs;unt, ut plau&longs;tris avehendæ imponantur, aut ad­<lb/>vectæ ex iis deponantur. </s> </p> <p type="main"> <s id="s.005686">Erigatur tignum AB ad perpendiculum firmiter infixum <lb/>plano &longs;ubjecto, tanta verò &longs;it cra&longs;&longs;ities tigni, ut in eo excavari <lb/><figure id="id.017.01.791.1.jpg" xlink:href="017/01/791/1.jpg"/><lb/>po&longs;&longs;it crena CD, per quam tignum aliud EF immitti facilè <lb/>valeat adeò &longs;olidum, ut Vectis munere fungatur, ubi extremo <lb/>unco G funibus adnexum fuerit onus. </s> <s id="s.005687">Ut igitur facillimè one­<lb/>ris elevatio perficiatur, cochlea HI pri&longs;mati HL infixa (&longs;a­<lb/>tiùs fuerit, &longs;i eju&longs;dem ligni pars in pri&longs;ma, pars in cochleam <lb/>deformetur) ad perpendiculum erigatur in&longs;erta matrici NM: <lb/>&longs;it autem ita &longs;olida matrix, ut illi adnecti queat tignum EF <lb/>clavo E, circa quem facilè ver&longs;ari po&longs;&longs;it tignum ip&longs;um, quan­<lb/>do attollitur aut deprimitur. </s> <s id="s.005688">Demum &longs;ubjecto pri&longs;mati HL <pb pagenum="776" xlink:href="017/01/792.jpg"/>non &longs;olùm in quatuor faciebus in&longs;int foramina, quibus immit­<lb/>tatur Radius KO, verùm etiam in infimâ ba&longs;is parte, qua <lb/>re&longs;pondet axi cylindri in cochleam deformati, &longs;it polus, circa <lb/>quem convolvi po&longs;&longs;it: Hic tamen (ut &longs;atis manife&longs;tum e&longs;t) in­<lb/>trà ferream laminam ritè applumbatam lapidi in terrâ firmi&longs;&longs;i­<lb/>mè defixo, aut certè adeò gravi, ut longè omnem elevandarum <lb/>&longs;arcinarum gravitatem vincat, ita contineri debet, ut indè nul­<lb/>lo pacto eximi valeat, neque à pri&longs;mate avelli. </s> </p> <p type="main"> <s id="s.005689">Quod &longs;i non fuerit inter pavimentum & laqueare interval­<lb/>lum enorme, faciliùs erit congruam trabis partem in cochleam <lb/>deformare, illámque ba&longs;i imponere (cujus altitudo commodam <lb/>Radij KO conver&longs;ionem præ&longs;tet) atque circa duos polos, al­<lb/>terum eidem &longs;ubjectæ ba&longs;i, alterum lacunari infixum, convol­<lb/>vere. </s> <s id="s.005690">Aut &longs;altem proximo parieti infigatur tignum horizonta­<lb/>le, quod prominens excipiat &longs;uperiorem polum, atque prohi­<lb/>beat, ne vi ponderis ex unco G dependentis, in altum abripia­<lb/>tur cochlea. </s> </p> <p type="main"> <s id="s.005691">Hìc vides compo&longs;itam cum Vecte Cochleam, quæ Vectis vi­<lb/>res notabili incremento auget. </s> <s id="s.005692">E&longs;t autem vectis hypomochlium <lb/>in eâ inci&longs;æ aut in&longs;culptæ crenæ parte, quæ cochleam re&longs;picit, <lb/>quando pondus attollitur, & vectis caput F &longs;upra lineam hori­<lb/>zontalem elevatur; &longs;ecùs verò, quando deprimitur vectis infra <lb/>lineam horizontalem, tunc enim hypomochlium e&longs;t in D. <!-- KEEP S--></s> <s id="s.005693">Sit <lb/>igitur hypomochlium ad totius longitudinis EF be&longs;&longs;em; ac <lb/>proinde Ratio motûs potentiæ in E, ad motum ponderis in F, <lb/>e&longs;t dupla. </s> <s id="s.005694">Ponamus Radij KO extremitatem O ab axe cylin­<lb/>dri di&longs;tare intervallo &longs;altem decuplo intervalli &longs;pirarum co­<lb/>chleæ HV: ergo potentia de&longs;cribens peripheriam circuli, cu­<lb/>jus diameter e&longs;t 20, habet motum, qui e&longs;t, ut minimum, ut 62 <lb/>ad motum HV, hoc e&longs;t ad motum extremitatis E: hujus autem <lb/>motus e&longs;t duplus motûs ponderis in F: igitur motus potentiæ e&longs;t <lb/>ad motum ponderis ut 124 ad 1. Quare ut major &longs;it motus pon­<lb/>deris, poterit hypomochlium minùs abe&longs;&longs;e ab extremitate E; <lb/>vix enim tales &longs;unt &longs;arcinæ, quæ ut moveantur, citrà laborem <lb/>&longs;u&longs;tentandi, indigeant 60 hominibus. </s> </p> <p type="main"> <s id="s.005695">At &longs;i loci ratio ferat, &longs;uaderem potiùs Vectem &longs;ecundi gene­<lb/>ris, ita ut altera vectis extremitas in&longs;i&longs;teret tigno AB, & pon­<lb/>dus inter tignum & cochleam interciperetur: &longs;ic enim quò <pb pagenum="777" xlink:href="017/01/793.jpg"/>pondus e&longs;&longs;et propius cochleæ, ad majorem altitudinem attolle­<lb/>retur, licèt majore conatu; &longs;ed cochleæ vis abundare videtur, <lb/>& Vectis &longs;ecundi generis &longs;emper auget momenta. </s> </p> <p type="main"> <s id="s.005696">Nec di&longs;&longs;imili ratione Vectem manu tractabilem ita cochleâ <lb/>in&longs;truere po&longs;&longs;umus, ut ad ingentia pondera movenda &longs;atis &longs;it. </s> <lb/> <s id="s.005697">Finge &longs;iquidem re­<lb/><figure id="id.017.01.793.1.jpg" xlink:href="017/01/793/1.jpg"/><lb/>vellendas &longs;uis è car­<lb/>dinibus ingentes a­<lb/>licujus ba&longs;ilicæ val­<lb/>vas, ut reficiantur; <lb/>ferreus vectis AB <lb/>paretur extremita­<lb/>te A &longs;ubjiciendus <lb/>ponderi, & altera <lb/>extremitas B in ma­<lb/>tricem cochleæ ex­<lb/>cavetur: in hanc <lb/>immittatur cochlea CD, manubrium habens CE: atque ut <lb/>cochlea faciliùs convertatur, neque pavimentum atterat, pa­<lb/>ratam habeto ferream laminam H, quæ illi &longs;ubjiciatur. </s> <s id="s.005698">Nam <lb/>&longs;i ponderi &longs;upponatur vectis ex. </s> <s id="s.005699">gr. <!-- REMOVE S-->in F ad totius longitudinis <lb/>&longs;extantem, manubrij autem longitudo CE &longs;it &longs;altem decupla <lb/>intervalli &longs;pirarum cochleæ, utique peripheria de&longs;cripta à po­<lb/>tentiâ in E, ad elevationem extremitatis B, e&longs;t &longs;altem ut 62 <lb/>ad 1: elevatio autem ip&longs;ius B, ad elevationem ponderis in F, <lb/>e&longs;t ut 6 ad 1: igitur motus potentiæ ad motum ponderis e&longs;t ut <lb/>372 ad 1. Quapropter, et&longs;i initio parùm attollantur fores, & <lb/>&longs;ubjecto cuneo, ne recidant, atque revolutâ cochleâ deprimen­<lb/>dus, ac promovendus &longs;it vectis, ut pondus &longs;it in I, puta ad to­<lb/>tius longitudinis quadrantem aut trientem, adhuc potentiæ <lb/>momenta erunt ut 248, aut 186 ad 1: quæ &longs;anè exigua non &longs;unt <lb/>pro &longs;implici huju&longs;modi machinulâ. </s> </p> <p type="main"> <s id="s.005700">Huc pariter &longs;pectat præli genus in meâ patriâ vulgare (in lo­<lb/>cis poti&longs;&longs;imùm montanis, ubi faciliùs ingentes lapides non pro­<lb/>cul advehendi &longs;uppetunt) quo ex uvæ jam calcatæ reliquiis <lb/>tortivum mu&longs;tum exprimitur. </s> <s id="s.005701">Roboris, quoad fieri pote&longs;t, lon­<lb/>gi&longs;&longs;imum truncum unâ cum imo caudice a&longs;&longs;umunt, & ita ramis <lb/>omnibus &longs;poliant, ut tamen bifurcum relinquant, quatenus <pb pagenum="778" xlink:href="017/01/794.jpg"/>bifidæ illi extremitati inniti, atque connecti valeat matrix co­<lb/>chleæ, quæ convertatur circa polum ingenti &longs;ubjecto lapidi in­<lb/>&longs;i&longs;tentem, &longs;ed eâ ratione, ut demum etiam lapis attolli queat. </s> <lb/> <s id="s.005702">Truncum verò illum, qui præli munere fungi debet, præter <lb/>extremum caudicem cra&longs;&longs;i&longs;&longs;imum, dedolant, ut inter bina tigna <lb/>hinc atque hinc in alvei lateribus ad perpendiculum erecta in­<lb/>terjectum prælum attolli ac deprimi po&longs;&longs;it citrà impedimentum, <lb/>quod alioqui ip&longs;a rudis a&longs;peritas pareret. </s> <s id="s.005703">Porrò tigna illa bina <lb/>erecta, aut ex adver&longs;o rotundis aliquot foraminibus perforant, <lb/>quibus immitti po&longs;&longs;it cra&longs;&longs;iu&longs;culus cylindrus, &longs;eu ferreus vectis, <lb/>aut illa incidunt patente crenâ, cui in&longs;eri valeat repagulum; eo <lb/>con&longs;ilio, ut alterutra præli extremitas pro opportunitate prohi­<lb/>beatur, ne a&longs;cendat, aut de&longs;cendat. </s> <s id="s.005704">Quare convoluta cochlea <lb/>attollit matricem, & oppo&longs;ita præli extremitas amotis omnibus <lb/>&longs;ubjectis repagulis &longs;en&longs;im de&longs;cendit: ubi autem eò venerit, ut <lb/>non ab&longs;it ab altitudine eorum, quæ in torculari calcanda &longs;unt, <lb/>immittitur &longs;uperiùs repagulum, ne amplius attolli valeat; Tum <lb/>revoluta in contrarium cochleâ matricem cum præli extremita­<lb/>te deor&longs;um trahit: & quoniam reliqua extremitas attolli nequit <lb/>ob&longs;tante repagulo, premuntur uvæ, & in Lacum defluit <lb/>mu&longs;tum. </s> <s id="s.005705">Ubi demum adeò compre&longs;&longs;a fuerint vinacea, ut fa­<lb/>cilius &longs;it lapidem cochleæ adnexum attollere, quàm illa magis <lb/>comprimere, ex cochleæ conver&longs;ione attollitur lapis; quem ad <lb/>mediocrem altitudinem elevatum pendere diutiùs permittunt, <lb/>ut, lapidis gravitate deor&longs;um conante, à prælo exprimatur, <lb/>quantulumcumque mu&longs;ti adhuc vinaceis ine&longs;t. </s> <s id="s.005706">Duplex igitur <lb/>hìc con&longs;ideranda e&longs;t pre&longs;&longs;io: altera quidem vi potentiæ co­<lb/>chleam volventis; & hìc cochlea cum vecte &longs;ecundi generis <lb/>componitur; e&longs;t enim prælum vectis, cujus hypomochlium e&longs;t <lb/>in eâ extremitate, quæ repagulo prohibetur, ne attollatur; po­<lb/>tentiæ autem vectem huju&longs;modi deprimentis vices obit cochlea <lb/>claviculatim &longs;triata; quæ tamen motûs originem non habens <lb/>&longs;ibi in&longs;itam, potentiæ munus & nomen relinquit vectiariis il­<lb/>lam ver&longs;antibus. </s> <s id="s.005707">Altera pre&longs;&longs;io fit, ce&longs;&longs;ante convolutione co­<lb/>chleæ, vi gravitatis lapidis &longs;u&longs;pen&longs;i; & tunc non ni&longs;i Ratio <lb/>vectis intervenit, atque Potentia e&longs;t ip&longs;a gravitas. </s> </p> <p type="main"> <s id="s.005708">Sed quoniam non ubique reperiuntur aut tam ingentes lapi­<lb/>des, aut tam longæ arbores, communiter univer&longs;us premendi <pb pagenum="779" xlink:href="017/01/795.jpg"/>labor vectiariis incumbit cochleam unam, aut alteram ver&longs;an­<lb/>tibus. </s> <s id="s.005709">Si duæ &longs;int cochleæ ad oppo&longs;ita torcularis latera con&longs;ti­<lb/>tutæ, matricem habent in ip&longs;o prælo excavatam, quod &longs;uá con­<lb/>ver&longs;ione deor&longs;um trahunt, ut ex &longs;ubjectis vinaceis exprimatur <lb/>mu&longs;tum: & tunc nihil e&longs;t, quod Vectis momenta exerceat, <lb/>&longs;ed &longs;ola vis Cochleæ habetur. </s> <s id="s.005710">At &longs;i unica fuerit cochlea <lb/>(quemadmodum & in typographorum torculis) præli non e&longs;t <lb/>u&longs;us; &longs;ed tran&longs;ver&longs;æ trabi &longs;uperiori immotæ in&longs;eritur per exca­<lb/>vatas congruentes &longs;trias cochlea, quæ in conver&longs;ione depre&longs;&longs;a <lb/>calcat impo&longs;itum vinaceis planum ex &longs;olidis a&longs;&longs;eribus. </s> <s id="s.005711">Verùm <lb/>contingere pote&longs;t ut non &longs;it vectiario &longs;patium expeditum, <lb/>quando, po&longs;t modicam & faciliorem compre&longs;&longs;ionem breviore <lb/>vecte peractam, adhuc longiore Radio utendum e&longs;&longs;et ad con­<lb/>torquendam cochleam. </s> <s id="s.005712">Propterea ab Axe in Peritrochio &longs;ub­<lb/>&longs;idium facile peti pote&longs;t; &longs;i videlicet extra torcularis alveum <lb/>ligneus cylindrus ad perpendiculum erigatur circa &longs;uos polos, <lb/>alterum &longs;ubjecto plano, alterum exporrectæ è proximo pariete <lb/>trabi, infixos ver&longs;atilis: huic funem adnecte, qui extremo un­<lb/>co apprehendat annulum Radij, quo cochlea ver&longs;atur: cylin­<lb/>dro enim infixus Radius dum illum volvit, & funem illi cir­<lb/>cumducit, cochleæ vectem ad &longs;e rapit, & vehementiùs premi­<lb/>tur &longs;ubjectum cochleæ planum, quàm &longs;i eadem cochlea duplo <lb/>longiore vecte convolveretur. </s> </p> <p type="main"> <s id="s.005713">Unum tamen hìc ob&longs;ervandum, videlicet in huju&longs;modi con­<lb/>ver&longs;ione non eadem e&longs;&longs;e momenta, quandoquidem funis exten­<lb/>tus non eundem &longs;emper cum cochleæ vecte angulum con&longs;ti­<lb/>tuit; eò autem minora &longs;unt momenta, quò magis hic ab an­<lb/>gulo recto recedit, ut ex iis con&longs;tat, quæ lib.4. cap.7. dicta &longs;unt. </s> <lb/> <s id="s.005714">Quapropter expedit cylindrum illum ver&longs;atilem non longiùs à <lb/>torculari abe&longs;&longs;e, ut funis minùs acutum angulum cum vecte <lb/>con&longs;tituat, quando vectis extremitas incipit &longs;uæ peripheriæ ar­<lb/>cum de&longs;cribere: atque adeò ita &longs;tatuendus videtur cylindrus, <lb/>ut quando funis angulum rectum con&longs;tituet cum cochleæ vecte, <lb/>hic jam percurrerit arcum non majorem &longs;emirecto angulo, re&longs;­<lb/>pondentem gradibus 45: &longs;ic enim fiet, non nimis acutum e&longs;&longs;e <lb/>angulum initio tractionis, & progrediendo augeri, donec fiat <lb/>rectus: deinde, licèt momenta decre&longs;cant angulo in obtu&longs;um <lb/>tran&longs;eunte, ubi nimis obliquus factus fuerit angulus, poterit in <pb pagenum="780" xlink:href="017/01/796.jpg"/>aliud foramen immitti Radius, laxato priùs fune ex cylindri <lb/>revolutione. </s> <s id="s.005715">Quapropter ut potentiæ cylindrum volventis mo­<lb/>menta ad calculos revoces, maximum momentum e&longs;t fune ad <lb/>cochleæ Radium perpendiculari: quamvis autem non &longs;emper <lb/>progrediente motu con&longs;tituat angulum rectum, tunc tamen <lb/>perinde computandum e&longs;t momentum, atque &longs;i eandem po&longs;i­<lb/>tionem ad angulum rectum &longs;ervatura e&longs;&longs;et potentia per funem <lb/>Radio applicata; præterita &longs;iquidem atque futura applicatio <lb/>nihil minuit præ&longs;entis applicationis virtutem. </s> <s id="s.005716">In eâ verò appli­<lb/>catione perpendiculari, momenti Ratio de&longs;umenda e&longs;t ex circu­<lb/>li à Radio de&longs;cripti peripheriâ, atque ex intervallo &longs;pirarum co­<lb/>chleæ: quæ Ratio componenda e&longs;t cum Ratione Radij cylin­<lb/>drum volventis ad eju&longs;dem cylindri &longs;emidiametrum. </s> </p> <p type="main"> <s id="s.005717">Sed ut habeantur momenta aliarum po&longs;itionum, inquiren­<lb/>dus e&longs;t angulus applicationis funis ad eundem cochleæ vectem. <lb/><figure id="id.017.01.796.1.jpg" xlink:href="017/01/796/1.jpg"/><lb/>Et primò qui­<lb/>dem datur Ra­<lb/>dij cochleæ in­<lb/>fixi longitudo <lb/>AB, &longs;emidia­<lb/>meter cylindri <lb/>CD, & di&longs;tan­<lb/>tia AD. Inqui­<lb/>ratur, in quo <lb/>puncto accidat <lb/>po&longs;itio funis <lb/>perpendicularis <lb/>ad Radium co­<lb/>chleæ: hæc uti­<lb/>que non fit, ni&longs;i <lb/>fune tangente <lb/>utramque peri­<lb/>pheriam tum <lb/>circuli à Radio <lb/>AB de&longs;cripti, <lb/>tum cylindri; & erit BC. <!-- KEEP S--></s> <s id="s.005718">Quia igitur linea BC utrumque <lb/>circulum tangit, ductis &longs;emidiametris AB & CD, anguli ABC, <lb/>& DCB &longs;unt recti, ex 18. lib.3: igitur ex 27. lib.1. lineæ AB <pb pagenum="781" xlink:href="017/01/797.jpg"/>& DC &longs;unt parallelæ, & per 29. lib.1. anguli alterni BAD, <lb/>& CDA &longs;unt æquales: &longs;ed & anguli ad verticem E &longs;unt æqua­<lb/>les: ergo triangula BAE, CDE &longs;unt &longs;imilia; & per 4. lib. 6. <lb/>ut BA ad CD, ita AE ad DE; & componendo ut AB plus <lb/>CD ad CD, ita AD ad DE: innote&longs;cit itaque DE. <!-- KEEP S--></s> <s id="s.005719">Quare <lb/>ex quadrato ip&longs;ius DE auferatur quadratum lateris DC, & re­<lb/>&longs;idui Radix erit recta CE. <!-- KEEP S--></s> <s id="s.005720">Fiat ergo ut DC ad CE, ita AB <lb/>ad BE; atque additis BE & CE nota e&longs;t tota BC. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005721">Deinde a&longs;&longs;umatur po&longs;itio Radij AF, ita ut arcus FB non &longs;it <lb/>major gradibus 45. Dato igitur arcu illo, hoc e&longs;t angulo FAB, <lb/>noti &longs;unt anguli AFB, ABF trianguli i&longs;o&longs;celis ad ba&longs;im BF, <lb/>&longs;inguli enim habent &longs;emi&longs;&longs;em re&longs;idui ad duos rectos: atque <lb/>adeò, &longs;i recto CBA addatur notus ABF, innote&longs;cit anguli ob­<lb/>tu&longs;i CBF quantitas: Inventum jam e&longs;t latus CB, & latus BF <lb/>&longs;ubten&longs;a dati arcûs ex Canone Sinuum innote&longs;cit in partibus <lb/>Radij AB; quapropter ex Trigonometria inveniri pote&longs;t ba&longs;is <lb/>CF, & angulus BFC; qui &longs;i auferatur ex noto angulo BFA, <lb/>remanet quæ&longs;itus angulus CFA applicationis funis CF ad <lb/>vectem AF. <!-- KEEP S--></s> <s id="s.005722">Habetur itaque ex huju&longs;modi applicatione ad <lb/>vectem per angulum acutum CFA Ratio momenti comparati <lb/>cum momento applicationis ad angulum rectum CBA: e&longs;t <lb/>enim ex dictis lib. 4. cap.7. ut Sinus anguli acuti ad Radium. <!-- KEEP S--></s> <lb/> <s id="s.005723">Fiat igitur ut Radius ad Sinum anguli AFC, ita peripheria <lb/>de&longs;cripta à vecte AB ad aliud; & hoc inventum comparandum <lb/>e&longs;t cum intervallo &longs;pirarum cochleæ, ut habeatur Ratio mo­<lb/>menti potentiæ in C con&longs;titutæ, & applicatæ ad vectem AF <lb/>cum directione CF. <!-- KEEP S--></s> <s id="s.005724">Componenda deinde e&longs;t hæc Ratio cum <lb/>Ratione Radij DH cylindrum volventis, ad eju&longs;dem cylindri <lb/>&longs;emidiametrum DC, & habebitur adæquata Ratio momenti <lb/>potentiæ in H. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005725">Tertiò. <!-- KEEP S--></s> <s id="s.005726">Ex puncto C ad A ducatur recta CA: & cum in <lb/>triangulo ABC rectangulo nota jam &longs;int latera AB & BC cir­<lb/>ca rectum, invenitur hypothenu&longs;a AC, & angulus BAC. <!-- KEEP S--></s> <lb/> <s id="s.005727">Tum ex 33. lib.3. &longs;uper rectâ AC de&longs;cripto circuli &longs;egmento <lb/>capiente angulum obtu&longs;um æqualem Supplemento anguli <lb/>AFC ad duos rectos, in puncto I, ubi hujus &longs;egmenti arcus <lb/>&longs;ecat peripheriam à Cochleæ vecte de&longs;criptam, concurrant duæ <lb/>rectæ AI & CI. <!-- KEEP S--></s> <s id="s.005728">E&longs;t igitur triangulum CAI, in quo data &longs;unt <pb pagenum="782" xlink:href="017/01/798.jpg"/>latera CA & AI unâ cum angulo AIC noto, utpote ex con­<lb/>&longs;tructione æquali &longs;upplemento ad duos rectos anguli jam noti <lb/>AFC. </s> <s id="s.005729">Quapropter inveniatur angulus IAC, qui demptus ex <lb/>jam invento angulo BAC, relinquit angulum BAI; ac <lb/>propterea notus e&longs;t arcus BI, qui additus arcui BF dabit totum <lb/>arcum FI, in quo primùm momenta cre&longs;cunt ex F in B, dein­<lb/>de decre&longs;cunt ex B in I, ubi angulus obtu&longs;us tantumdem exce­<lb/>dit rectum, quantum à recto deficit acutus AFC; atque adeò <lb/>in F & I æqualia &longs;unt momenta. </s> </p> <p type="main"> <s id="s.005730">Antequam verò praxim hanc exemplo illu&longs;trem, ut &longs;ub­<lb/>ductis calculis noverit Machinator, quonam pacto omnia di&longs;­<lb/>ponenda &longs;int, monendus e&longs;t lector à me ideò &longs;emper idem <lb/>punctum C a&longs;&longs;umptum fui&longs;&longs;e, quia in re Phy&longs;icâ nullus &longs;ubre­<lb/>pere pote&longs;t error notabilis. </s> <s id="s.005731">Cæterùm &longs;i funem extentum con­<lb/>&longs;ideremus &longs;emper qua&longs;i lineam tangentem cylindri periphe­<lb/>riam, &longs;atis manife&longs;tum e&longs;t, &longs;i linea BC e&longs;t tangens in puncto <lb/>C, & angulus BCD e&longs;t rectus, non po&longs;&longs;e lineam à puncto F <lb/>productam ad contactum cadere in punctum C, &longs;ed ultrà illud, <lb/>ita ut demum veniat punctum contactûs in C, quando po&longs;itio <lb/>vectis fuerit AB, & iterum punctum contactûs recedat à C, <lb/>quando po&longs;itio vectis fiat AI. <!-- KEEP S--></s> <s id="s.005732">Verùm quia exiguum e&longs;t huju&longs;­<lb/>modi di&longs;crimen, propterea unum idémque punctum C a&longs;­<lb/>&longs;umptum e&longs;t, cum non &longs;equatur phy&longs;icè ullum incommodum <lb/>ex hoc Geometricæ accurationis contemptu. </s> </p> <p type="main"> <s id="s.005733">Sit igitur ex. </s> <s id="s.005734">gr. <!-- REMOVE S-->&longs;pirarum cochleæ intervallum unc. </s> <s id="s.005735">2; & <lb/>vectis longitudo AB cubitorum 3, hoc e&longs;t unc. </s> <s id="s.005736">36: quare inte­<lb/>gra peripheria hoc Radio e&longs;t unc. </s> <s id="s.005737">226; atque ideò in B, ubi <lb/>applicatio e&longs;t ad angulum rectum, Ratio motuum &longs;eu momen­<lb/>torum e&longs;t ut 226 ad 2, hoc e&longs;t 113 ad 1. Sit cylindri &longs;emidia­<lb/>meter DC unc. </s> <s id="s.005738">3, atque DH &longs;imiliter unc. </s> <s id="s.005739">36: e&longs;t igitur Ra­<lb/>tio motûs &longs;eu momenti potentiæ in H, ad motum &longs;eu momen­<lb/>tum in C ut 12 ad 1. Ratio itaque compo&longs;ita ex Rationibus 113 <lb/>ad 1, & 12 ad 1 e&longs;t Ratio 1356 ad 1: quæ longê major e&longs;t, <lb/>quàm &longs;i cochleæ adhiberi potui&longs;&longs;et Radius cubitorum 6, non <lb/>addito Axe in Peritrochio. <!-- KEEP S--></s> <s id="s.005740">Ut inveniatur longitudo BC, pri­<lb/>mùm fiat ut AB plus DC ad DC, hoc e&longs;t ut unc.39. ad unc.3. <lb/>ita di&longs;tantia AD data unc. </s> <s id="s.005741">65, ad ED unc. </s> <s id="s.005742">5. Igitur in trian­<lb/>gulo ECD rectangulo, cujus hypothenu&longs;a ED unc. </s> <s id="s.005743">5. la-<pb pagenum="783" xlink:href="017/01/799.jpg"/>tus DC unc. </s> <s id="s.005744">3. e&longs;t latus EC unc. </s> <s id="s.005745">4: atque adeò ut DC 3 ad <lb/>CE 4, ita AB 36 ad BE 48; cui addita CE 4 dat totam per­<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 &longs;ubten&longs;a 76536 partium, <lb/>quarum Radius AB unc.36 e&longs;t 100000, erit unc. </s> <s id="s.005748">27 1/2: anguli <lb/>verò AFB, ABF &longs;unt &longs;inguli gr. <!-- REMOVE S-->67. 30′. </s> <s id="s.005749">Quare in triangu­<lb/>lo FCB datur angulus CBF gr. <!-- REMOVE S-->157. 30′. </s> <s id="s.005750">comprehen&longs;us à la­<lb/>teribus CB unc. </s> <s id="s.005751">52, & BF unc. </s> <s id="s.005752">27 1/2. Invenitur ergo angulus <lb/>BFC gr. <!-- REMOVE S-->14. 45′, qui ex angulo BFA gr. <!-- REMOVE S-->67. 30. demptus relin­<lb/>quit angulum AFC gr. <!-- REMOVE S-->52. 45′; cujus Sinus e&longs;t particularum <lb/>79600. Igitur ut Radius 100000 ad 79600, ita momentum <lb/>Applicationis per angulum rectum, quod erat ut 113, ad 90 <lb/>proximè, momentum Applicationis per hunc angulum AFC <lb/>acutum. </s> <s id="s.005753">Compo&longs;itis itaque Rationibus 90 ad 1, & 12 ad 1, mo­<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/>& BC 52, reperitur hypothenu&longs;a AC 63 1/4, & angulus BAC <lb/>gr.55. 18′. </s> <s id="s.005755">Quapropter in triangulo AIC datur latus AC 63 1/4, <lb/>& angulus illi oppo&longs;itus AIC gr. <!-- REMOVE S-->127. 15′. </s> <s id="s.005756">& præterea latus AI <lb/>36: ex quibus invenitur huic oppo&longs;itus angulus ICA gr.26.56. <lb/>Igitur tertius angulus CAI e&longs;t gr. <!-- REMOVE S-->25. 49′: qui &longs;i auferatur ex <lb/>angulo BAC gr. <!-- REMOVE S-->55. 18, reliquus e&longs;t angulus BAI gr. <!-- REMOVE S-->29. 29′. </s> <lb/> <s id="s.005757">Igitur totus arcus FI e&longs;t gr.74. 29′. </s> </p> <p type="main"> <s id="s.005758">Ut autem appareat, quid conferat amplitudo arcûs BF, &longs;ta­<lb/>tuatur hic gr. <!-- REMOVE S-->60. & huic æquales &longs;unt anguli ad ba&longs;im BF; <lb/>quæ recta BF e&longs;t ip&longs;i AB æqualis, hoc e&longs;t unc.36. Quare in <lb/>triangulo FBC datur latus FB unc.36. & latus BC unc. </s> <s id="s.005759">52, & <lb/>angulus ab iis comprehen&longs;us gr. <!-- REMOVE S-->150: invenitur ergo angulus <lb/>BFC gr.17.47′; qui ablatus ex BFA gr. <!-- REMOVE S-->60. relinquit CFA <lb/>gr.42, 13′: cujus Sinus e&longs;t partium 67193. Igitur ut Radius <lb/>100000 ad 67193, ita 113 ad 76, quod e&longs;t momentum Applica­<lb/>tionis per hunc angulum acutum: atque compo&longs;itis Rationibus <lb/>76 ad 1, & 12 ad 1, momentum potentiæ in H e&longs;t ut 912. In trian­<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/>& Supplementum anguli AFC ad duos rectos e&longs;t angulus <lb/>AIC gr. <!-- REMOVE S-->137, 47″: ergo invenitur angulus ACI gr. <!-- REMOVE S-->22. 29′ <lb/>E&longs;t igitur angulus IAC gr. <!-- REMOVE S-->19. 44′: qui demptus ex angu-<pb pagenum="784" xlink:href="017/01/800.jpg"/>lo BAC gr.55. 18′. </s> <s id="s.005762">&longs;uperiùs invento, relinquit angulum IAB, <lb/>hoc e&longs;t arcum IB gr. <!-- REMOVE S-->35. 34′. </s> <s id="s.005763">Quare totus arcus FI e&longs;&longs;et <lb/>gr. <!-- REMOVE S-->95. 34′. </s> <s id="s.005764">Ex quo vides intra eo&longs;dem terminos æqualium mo­<lb/>mentorum, minora e&longs;&longs;e extrema momenta in F & I, &longs;ed per <lb/>majorem arcum, &longs;i incipias motum in majore di&longs;tantiá à puncto <lb/>Applicationis per angulum rectum: propterea &longs;atius videtur <lb/>majora obtinere momenta, & minorem arcum de&longs;cribere: ideò <lb/>dixi a&longs;&longs;umendum e&longs;&longs;e arcum BF non majorem gradibus 45. </s> </p> <p type="main"> <s id="s.005765">His &longs;imilia de Succulâ dicenda &longs;unt, quæ de Axe perpendi­<lb/>culari diximus, &longs;i &longs;ucculâ potiùs utendum loci & motûs quæ­<lb/>&longs;iti opportunitas &longs;uadeat: id quod ita per &longs;e clarum e&longs;t, ut in <lb/>his diutiùs immorari non &longs;it opus. <lb/></s> </p> <p type="main"> <s id="s.005766"><emph type="center"/>CAPUT IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005767"><emph type="center"/><emph type="italics"/>Cochleæ Infinitæ vires explicantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005768">VAlidi&longs;&longs;imam omnium Facultatum Cochleam e&longs;&longs;e ex &longs;upe­<lb/>rioribus manife&longs;tum e&longs;t: &longs;ed illud accidit incommodum, <lb/>quod nimis brevibus terminis coërcetur; quos nimirum ejus <lb/>longitudo definit; &longs;ivè illa circa &longs;uum axem convoluta intrà <lb/>Matricem immotam moveatur, &longs;ivè illa po&longs;itionem non mu­<lb/>tans ex convolutione attrahat aut repellat Matricem & pondus <lb/>ei adnexum. </s> <s id="s.005769">Propterea alius cochleæ u&longs;us excogitatus e&longs;t citrà <lb/>ullam Matricem, cui in&longs;eratur, atque eju&longs;modi, ut cochleæ <lb/>conver&longs;ioni nullus &longs;tatuatur finis, ea&longs;démque &longs;emper exerceat <lb/>vires. </s> <s id="s.005770">Hinc Cochleæ Infinitæ, aut Viti Perpetuæ nomen in­<lb/>ditum e&longs;t. </s> </p> <p type="main"> <s id="s.005771">Cylindrus circa &longs;uum axem, appo&longs;ito manubrio, ver&longs;atilis <lb/>in brevem cochleam deformatur unâ aut alterâ &longs;pirâ conten­<lb/>tus: ita autem ad tympani dentes accommodatur, ut eorum in­<lb/>tervallum &longs;it &longs;pirarum intervallo congruens; hoc e&longs;t initium <lb/>&longs;piræ apprehendat unum tympani dentem; dúmque ex Co­<lb/>chleæ convolutione dens primus tántum promovetur, quantum <lb/>exigit &longs;pirarum di&longs;tantia, unâ conver&longs;ione ab&longs;olutâ iterum ini-<pb pagenum="785" xlink:href="017/01/801.jpg"/>tium &longs;piræ apprehendat &longs;ecundum tympani dentem proximè <lb/>con&longs;equentem, ex tympani convolutione jam con&longs;titutum in <lb/>codem loco, in quo erat primus dens initio motûs: atque ita <lb/>deinceps omnes &longs;ubinde dentes apprehenduntur à cochleâ; <lb/>&longs;emelque revoluto tympano, iterum à primo dente incipit &longs;e­<lb/>cunda illius convolutio. </s> <s id="s.005772">Hinc quia cochleâ huju&longs;modi, quate­<lb/>nus ad &longs;e pertinet, nullum &longs;tatuit convolutionibus terminum, <lb/>etiam&longs;i definitum habet &longs;pirarum numerum, immò unicam ha­<lb/>beat &longs;piram, Infinita dicitur, nam & tympanum orbitam ha­<lb/>bens in &longs;e&longs;e redeuntem plurimis &longs;ine fine convolutionibus cir­<lb/>cumagi pote&longs;t. </s> <s id="s.005773">At &longs;i tympani loco rectam appo&longs;ueris laminam <lb/>denticulatam, quæ ex Cochleæ huju&longs;modi conver&longs;ione alium <lb/>atque &longs;ubinde alium dentem apprehendentis adduceretur, aut <lb/>repelleretur; an illa appellanda e&longs;&longs;et Cochlea Infinita, quia <lb/>longiorem atque longiorem &longs;inè fine laminam &longs;imiliter movere <lb/>po&longs;&longs;et; iis examinanda relinquatur quæ&longs;tio, quibus de vocabu­<lb/>lo di&longs;putandi otium e&longs;t. </s> </p> <p type="main"> <s id="s.005774">Tympano autem infixus e&longs;t Axis, &longs;ive ille &longs;implex &longs;it, cui <lb/>ductarius funis circumvolvatur, &longs;ivè &longs;triatus fuerit, qui aliud <lb/>tympanum convertat, prout &longs;uo loco, ubi de Axe in Peritro­<lb/>chio di&longs;putatum e&longs;t. </s> <s id="s.005775">Quapropter vis Cochleæ componitur cum <lb/>vi tympani, quod ab illà convertitur: idcirco huic Machinæ <lb/>Cochleæ Compo&longs;itæ aliqui nomen fecerunt. </s> <s id="s.005776">Cùm itaque &longs;in­<lb/>gulis cochleæ conver&longs;ionibus &longs;inguli dentes tympani promo­<lb/>veantur, toties convertitur cochlea, quot in tympani orbitâ <lb/>numerantur dentes. </s> <s id="s.005777">Potentiæ igitur motus, quo illa manubrium <lb/>ver&longs;ans de&longs;cribit circuli peripheriam, ducendus e&longs;t per den­<lb/>tium numerum, ut habeatur Ratio motûs Potentiæ, ad motum <lb/>orbitæ tympani. </s> <s id="s.005778">Cum verò data &longs;it Ratio tympani ad &longs;uum <lb/>Axem, data e&longs;t Ratio motûs orbitæ tympani ad motum ponde­<lb/>ris fune ductario attracti. </s> <s id="s.005779">Hæ duæ Rationes componantur, & <lb/>nota erit Ratio motû potentiæ ad motum ponderis. </s> <s id="s.005780">Sit co­<lb/>chleæ manubrium digitorum 7; igitur peripheria circuli à po­<lb/>tentiâ manubrio applicatâ de&longs;cripti e&longs;t ferè digit. </s> <s id="s.005781">44: tympani <lb/>&longs;emidiameter ad &longs;ui Axis &longs;emidiametrum &longs;it ut 4 ad 1: Sit au­<lb/>tem tympani orbita in dentes 24. di&longs;tincta; ac propterea dum &longs;e­<lb/>mel tympanum cum &longs;uo Axe volvitur, motus Potentiæ e&longs;t digi­<lb/>torum ferè 44 vicies & quater &longs;umptorum hoc e&longs;t digit. </s> <s id="s.005782">1056. <pb pagenum="786" xlink:href="017/01/802.jpg"/>Si igitur tympani &longs;emidiameter &longs;it digit. </s> <s id="s.005783">4, & Axis &longs;emidiame­<lb/>ter dig.1, illius peripheria e&longs;t &longs;altem digit. </s> <s id="s.005784">25, hujus verò pe­<lb/>ripheria &longs;altem digit. </s> <s id="s.005785">6 1/4, quantus e&longs;t ex unâ tympani conver­<lb/>&longs;ione motus ponderis. </s> <s id="s.005786">Itaque motus Potentiæ ad motum pon­<lb/>deris e&longs;t ut 1056 ad 6 1/4, hoc e&longs;t proximè ut 169 ad 1. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005787">Hinc &longs;i plura fuerint Compo&longs;ita Tympana, eorum Ratio, <lb/>quæ ex Rationibus diametrorum tympanorum ad &longs;uorum <lb/>Axium diametros componitur, a&longs;&longs;umenda e&longs;t, atque attenden­<lb/>dum quoties volvatur cochlea, ut primum tympanum cochleæ <lb/>proximum circumagatur: deinde per numerum dentium primi <lb/>tympani ducendus e&longs;t motus potentiæ manubrio cochleæ appli­<lb/>catæ; & ex Ratione tympanorum, atque ex Ratione Cochleæ, <lb/>componenda e&longs;t Ratio. <!-- KEEP S--></s> <s id="s.005788">Sit cochlea eadem, quæ priùs, eodém­<lb/>que manubrio in&longs;tructa, adeò ut potentia &longs;emel cochleam ver­<lb/>&longs;ans de&longs;cribat circuli peripheriam digitorum ferè 44; & pri­<lb/>mum tympanum habens peripheriam dig.25 in dentes 24 di&longs;tri­<lb/>butam, dum &longs;emel volvitur, potentia vicies & quater periphe­<lb/>riam dig.44 de&longs;cribens percurrit digitos 1056. <!--neuer Satz-->Sit idem pri­<lb/>mum tympanum ad &longs;uum Axem &longs;triatum ut 4 ad 1, &longs;ecundum <lb/>tympanum ad &longs;uum Axem fune ductario involutum &longs;it ut 3 ad 2: <lb/>Ratio compo&longs;ita horum duorum tympanorum e&longs;t ut 6 ad 1. <lb/><!--neuer Satz-->Cum verò motus potentiæ manubrio applicatæ ad integrum <lb/>motum peripheriæ tympani primi &longs;it ut 1056 ad 25 (nam &longs;ingu­<lb/>læ conver&longs;iones manubrij cochleæ ad motum unius dentis &longs;unt <lb/>ut 44 ad 25/24) componatur hæc Ratio cum Ratione 6 ad 1, & erit <lb/>motus potentiæ ad motum ponderis axi &longs;ecundi tympani per fu­<lb/>nem ductarium applicati, ut 6336 ad 25, hoc e&longs;t ferè ut 253 1/2 <lb/>ad 1: atque adeò quo conatu potentia moveret libras decem; <lb/>hac machinâ movebit libras 2535. </s> </p> <p type="main"> <s id="s.005789">Verùm adhuc augeri po&longs;&longs;unt vires Cochleæ Infinitæ non <lb/>multiplicatis tympanis dentatis, &longs;ed cum illo unico, quod à <lb/>Cochleâ movetur, componendo Trochleas: &longs;i videlicet alteri <lb/>Trochleæ adnectatur pondus, altera Trochlea alicubi firmetur: <lb/>tum funis ductarius, qui à Potentiâ arripiendus e&longs;&longs;et atque <lb/>trahendus, axi tympani alligetur. </s> <s id="s.005790">Nam &longs;i Ratio, quam Tro­<lb/>chleæ inferunt, componatur cum Ratione Axis in Peritrochio, <lb/>atque Ratione Cochleæ, fit Ratio ex tribus Rationibus trium <pb pagenum="787" xlink:href="017/01/803.jpg"/>Facultatum compo&longs;ita. </s> <s id="s.005791">Sic Ratio Cochleæ &longs;it, ut priùs, 44 ad 25/24, <lb/>Ratio Tympani ad Axem &longs;it 4 ad 1, Ratio Trochlearum, capite <lb/>funis ad trochleam ponderis alligato (&longs;int autem Trochleæ bi­<lb/>norum orbiculorum) &longs;it 5 ad 1: tres hæ Rationes Compo&longs;itæ <lb/>con&longs;tituunt Rationem 845 ad 1. Quare quo conatu moveres <lb/>libras decem, movebis libras 8450 tam facili & parabili ma­<lb/>chinâ. </s> </p> <p type="main"> <s id="s.005792">Ob&longs;ervanda &longs;unt autem tam commoda, quàm incommoda, <lb/>quæ hujus machinæ, &longs;cilicet Cochleæ Infinitæ u&longs;um comitan­<lb/>tur. </s> <s id="s.005793">Neque in po&longs;tremis illud numerandum e&longs;t, quod tantula <lb/>machinula facillimè transferri pote&longs;t, ad pondera &longs;atis magna <lb/>dimovenda; maximè &longs;i in plano raptanda &longs;int &longs;uppo&longs;itis &longs;cytalis <lb/>& trochleæ adhibeantur, quas non adeò cra&longs;&longs;o fune connecti <lb/>oportet, quemadmodum &longs;i in &longs;ublime attollendum e&longs;&longs;et pon­<lb/>dus, & fune ip&longs;o retinendum, ne relabatur. </s> </p> <p type="main"> <s id="s.005794">Adde non requiri ampliora &longs;patia, ut cochlea huju&longs;modi infi<lb/>nita circumagatur, & vel &longs;edentem hominem &longs;olâ, neque mul­<lb/>tâ, lacertorum manubrium ver&longs;antium contentione po&longs;&longs;e mo­<lb/>tum quæ&longs;itum perficere: atque &longs;i pondus attollatur, licet poten<lb/>tiæ, quandocumque libitum fuerit, ce&longs;&longs;are à motu, quin pon <lb/>dus &longs;u&longs;pen&longs;um recidat, etiam&longs;i neque illi fulcrum &longs;ubjiciatur <lb/>neque cochleæ manubrium retinaculo aliquo firmetur. </s> <s id="s.005795">Verùm <lb/>in attollendis ingentibus oneribus non expedit hac machinâ <lb/>uti, ni&longs;i tympanum dentatum &longs;atis magnum fuerit, ut Axem <lb/>cra&longs;&longs;iorem atque validiorem admittat, cui ductarius funis cir <lb/>cumduci queat; hic autem funis cum <expan abbr="teñuis">tenuis</expan> e&longs;&longs;e non po&longs;&longs;it, ne­<lb/>que exilem Axem exigit. </s> <s id="s.005796">Præterea di&longs;&longs;imulandum non e&longs;t per <lb/>culum, ne cochlea inutilis fiat; &longs;i videlicet vel unicus tympano <lb/>dens excutiatur: ubi enim in conver&longs;ione ad cam lacunam <lb/>ventum fuerit, illico ce&longs;&longs;at tympani conver&longs;io, cum nullus ejus <lb/>dens occurrat cochleæ. </s> <s id="s.005797">Propterea rem prudenter admini&longs;trare <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/>cipi pote&longs;t ad augendas communis Cochleæ vires &longs;ivè preme<lb/>tis, &longs;ivè etiam attrahentis. </s> <s id="s.005799">Eo videlicet loco, ubi aptandus e&longs;­<lb/>&longs;et Radius ad Cochleæ conver&longs;ionem, tympanum dentatum ad­<lb/>jiciatur, ex cujus centro exeat cylindrus in cochleam deforma­<lb/>tus, & Matrici in&longs;ertus: tympani verò dentes congruâ cochleæ <pb pagenum="788" xlink:href="017/01/804.jpg"/>infinitæ &longs;pirâ excipiantur: Manubrio enim ver&longs;ato cochlea in­<lb/>finita convertitur, & &longs;ingulis conver&longs;ionibus &longs;ingulos tympani <lb/>dentes, alios &longs;ubinde atque alios promovens, tympani covolu­<lb/>tionem efficit, atque cum eo pariter infixa cochlea ver&longs;atur. </s> <lb/> <s id="s.005800">Prudenti autem Machinatori non deerit methodus, qua huju&longs;­<lb/>modi Cochlea infinita applicetur, & &longs;imul cum tympano den­<lb/>tato deprimatur aut attollatur, &longs;i opus fuerit. </s> <s id="s.005801">Quapropter Ra­<lb/>tio peripheriæ tympani ad intervallum &longs;pirarum &longs;uæ cochleæ, <lb/>componenda e&longs;t cum Ratione peripheriæ à manubrio de&longs;criptæ <lb/>ad intervallum &longs;pirarum cochleæ infinitæ: ex hoc &longs;iquidem in­<lb/>tervallo pendet motus peripheriæ tympani, cujus dentes ap­<lb/>prehenduntur; quo enim pre&longs;&longs;ior e&longs;t cochleæ infinitæ &longs;pira, eò <lb/>tenuiores & frequentiores in&longs;unt tympano dentes. </s> <s id="s.005802">Sit ex. </s> <s id="s.005803">gr. <lb/><!-- REMOVE S-->&longs;pirarum cochleæ prementis intervallum &longs;ubtriplum &longs;emidia­<lb/>metri tympani, cui illa infixa e&longs;t: igitur Ratio perimetri tym­<lb/>pani ad intervallum &longs;pirarum e&longs;t ut 18 84/100 ad 1. <!--neuer Satz-->At Cochleæ in­<lb/>finitæ manubrium ad eju&longs;dem &longs;pirarum di&longs;tantiam &longs;it ut 10 ad 1: <lb/>Motus igitur potentiæ manubrium ver&longs;antis e&longs;t ut peripheria <lb/>de&longs;cripta 62 83/100 ad motum unius dentis tympani ut 1. <!--neuer Satz-->Ratio <lb/>itaque ex his duabus Rationibus Compo&longs;ita e&longs;t 1183 7/10 ad 1. <lb/><!--neuer Satz-->Ex quo &longs;atis innote&longs;cit, quanto virium incremento addatur co­<lb/>chleæ vulgari cochlea hæc infinita tam brevi manubrio in­<lb/>&longs;tructa, loco vectis admodum longi, quem &longs;patij angu&longs;tiæ non <lb/>caperent. </s> </p> <p type="main"> <s id="s.005804">Verùm non ad augendas tantummodo vires, &longs;eu, ut veriùs <lb/>dicam, ad momentorum potentiæ incrementum, adhiberi po­<lb/>te&longs;t cochlea infinita, &longs;ed ad motum quantumvis exiguum: <lb/>&longs;æpè enim motum extenuare opus e&longs;t. </s> <s id="s.005805">Sic in automatis horas <lb/>indicantibus vi laminæ ela&longs;ticæ longioris in &longs;piram convolutæ, <lb/>ad rotarum celeritatem aut tarditatem moderandam oportet <lb/>ip&longs;um elaterem modò intendere, modò remittere: quia verò <lb/>in vulgaribus horologiis id perficitur convolutione rotæ denta­<lb/>tæ (cujus axi intimum &longs;piræ ela&longs;ticæ caput adnectitur, atque <lb/>ne lamina per vim complicata &longs;e in laxiorem &longs;piram re&longs;tituat, <lb/>axem ip&longs;um & rotam dentatam revolvendo, obliquis rotæ eju&longs;­<lb/>dem dentibus, qua parte recti &longs;unt, objicitur virgula ela&longs;tica) <lb/>ut minimum dentem unum promovere aut retrahere nece&longs;&longs;e <pb pagenum="789" xlink:href="017/01/805.jpg"/>e&longs;t. </s> <s id="s.005806">At &longs;æpè contingere pote&longs;t, ut ela&longs;ticam laminam jam val­<lb/>de intentam amplius intendere, quantum fert integra dentis <lb/>unius conver&longs;io, celeriorem motum inferat, quam temporis ra­<lb/>tio po&longs;tularet; propterea &longs;cienti&longs;&longs;imi artifices, rejecta virgulâ <lb/>illâ ela&longs;ticâ, ita rotæ illius dentes conformant, ut cochleolæ <lb/>infinitæ congruant; hæc enim convoluta valde minutis pro­<lb/>gre&longs;&longs;ionibus laminam ela&longs;ticam intendit, aut remittit, & ubi­<lb/>cunque placuerit, &longs;i&longs;titur. </s> </p> <p type="main"> <s id="s.005807">Illud quoque non leve commodum (ut paulò &longs;uperius indi­<lb/>catum e&longs;t) in attollendis ponderibus animadver&longs;ione dignum <lb/>e&longs;t, quod &longs;ublato pondere atque &longs;u&longs;pen&longs;o, ce&longs;&longs;are pote&longs;t po­<lb/>tentia; & quamvis nec ab illâ, nec ab alio quolibet retinaculo <lb/>manubrium cochleæ infinitæ retineatur, neque pendenti oneri <lb/>fulcrum ullum &longs;ubjiciatur, ip&longs;a per &longs;e cochlea tympanum &longs;i&longs;tit, <lb/>& &longs;u&longs;pen&longs;um pondus impeditur, ne &longs;uâ vi recidat. </s> <s id="s.005808">Id quod in <lb/>tympanis dentatis, neque in Succulis, neque in Trochleis, ne­<lb/>que in Vecte obtinetur: quas Facultates &longs;i potentia dimi&longs;erit, <lb/>inchoato jam motu, neque illas aliquo retinaculo coërceat, <lb/>priorem laborem irritum facit gravitas &longs;ibi dimi&longs;&longs;a, ut &longs;atis aper­<lb/>tè con&longs;tat. </s> </p> <p type="main"> <s id="s.005809">Po&longs;tremò Cocheas infinitas cochleis pariter infinitis coag­<lb/>mentare &longs;i quis voluerit, is profectò momentis potentiæ immen­<lb/>&longs;am quandam acce&longs;&longs;ionem fecerit. </s> <s id="s.005810">Si enim primi tympani den­<lb/>tati Axem deformaveris in cochleam, quæ aliud tympanum <lb/>pariter dentatum moveat, & &longs;ecundi hujus tympani Axem item <lb/>in &longs;piralem &longs;triam excavaveris, quæ tertium tympanum con­<lb/>vertat unà cum Axe, cui ductarius funis circumducitur; ecce <lb/>quot Rationibus componitur Ratio motuum potentiæ & pon­<lb/>deris. </s> <s id="s.005811">Prima Ratio e&longs;t Peripheriæ à manubrio de&longs;criptæ ad di­<lb/>&longs;tantiam &longs;pirarum primæ cochleæ. </s> <s id="s.005812">Secunda Ratio e&longs;t periphe­<lb/>riæ primi tympani ad intervallum &longs;pirarum &longs;ecundæ cochleæ. </s> <lb/> <s id="s.005813">Secunda Ratio e&longs;t peripheriæ primi tympani ad intervallum <lb/>&longs;pirarum &longs;ecundæ cochleæ. </s> <s id="s.005814">Tertia Ratio e&longs;t peripheriæ &longs;ecun­<lb/>di tympani ad intervallum &longs;pirarum tertiæ cochleæ. </s> <s id="s.005815">Quarta <lb/>demum e&longs;t Ratio peripheriæ tertij tympani ad ambitum &longs;ui <lb/>Axis. <!-- KEEP S--></s> <s id="s.005816">Ponamus &longs;ingulas peripherias ad &longs;uæ cochleæ &longs;pirarum <lb/>intervallum e&longs;&longs;e ut 30 ad 1, & tertij tympani orbitam ad &longs;ui <lb/>Axis ambitum e&longs;&longs;e ut 5 ad 1; componendæ &longs;unt tres Rationes <pb pagenum="790" xlink:href="017/01/806.jpg"/>trigecuplæ cum unâ quintuplâ, & exurgit Ratio motûs poten­<lb/>tiæ manubrio applicatæ, ad motum ponderis ut 135000 ad 1. <lb/>Quo igitur conatu potentia moveret libras decem, hac trium <lb/>cochlearum infinitarum complexione movebit millies mille tre­<lb/>centas quinquaginta libras, &longs;eu, ut vulgari vocabulo utar, mil­<lb/>lionem & trecenta quinquaginta millia librarum. </s> <s id="s.005817">Quid autem, <lb/>&longs;i plura tympana cochleas infinitas habentia addantur? </s> <s id="s.005818">utique <lb/>&longs;i primæ cochleæ manubrio agitatæ quatuor con&longs;equentia tym­<lb/>pana cum &longs;uis cochleis addantur, eandem Rationem trigecu­<lb/>plam habentia, & quintum tympanum cum &longs;uo Axe Rationem <lb/>quintuplam habeat, demum potentia momentum obtinebit <lb/>ut 121. 500000: &, &longs;i ab&longs;que machinâ moveret libras decem, <lb/>hac machinâ ex quinque cochleis cum &longs;ibi congruentibus <lb/>tympanis movere poterit mille ducentos quindecim milliones <lb/>librarum. </s> </p> <p type="main"> <s id="s.005819">Neque &longs;ibi qui&longs;quam per&longs;uadeat opus e&longs;&longs;e ingentibus tym­<lb/>panis, ut validi&longs;&longs;imis cochleis re&longs;pondeant: Experimento enim <lb/>didicimus valde exiguas cochleas &longs;atis e&longs;&longs;e ad ingentia pondera <lb/>attollenda, modò axis funi ductario de&longs;tinatus &longs;atis firmus &longs;it ac <lb/>validus, & ferendo oneri par. </s> <s id="s.005820">Hic autem Axis (quemadmo­<lb/>dum & in Ergatâ) &longs;i plurimum funem excipere debeat, ne in <lb/>nimiam longitudinem protendatur, conformari pote&longs;t in Cy­<lb/>lindroides Hyperbolicum: nam ductarius funis illum aliquo­<lb/>ties complexus (quantum &longs;atis fuerit, ne excurrat) colligi po­<lb/>terit, & in convolutione &longs;e ad apicem Hyperbolæ continebit. </s> </p> <p type="main"> <s id="s.005821">At, inquis, huju&longs;modi motus ponderis nimis longa temporis <lb/>&longs;patia exigit. </s> <s id="s.005822">Ita planè: neque aliter contingere pote&longs;t, &longs;i qui­<lb/>dem tam ingens pondus movere volueris: an non præ&longs;tat tan­<lb/>tam molem demum loco ce&longs;&longs;i&longs;&longs;e, quam omnino immotam cui­<lb/>cumque conatui reluctari? </s> <s id="s.005823">Sed quid, &longs;i opportuni&longs;&longs;imum &longs;e <lb/>præbeat proximus rivulus perennis? </s> <s id="s.005824">primæ cochleæ apponatur <lb/>loco manubrij rota cum pinnis, in quas aqua incurrat; illa enim <lb/>circumacta cochleam & con&longs;equentia tympana ver&longs;abit, ac de­<lb/>mum vel dormientibus operis moles ab exiguâ aquâ dimo­<lb/>vebitur. </s> </p> <p type="main"> <s id="s.005825">Quod &longs;i ex pluribus cochleis infinitis compo&longs;itam machinam <lb/>tibi con&longs;truere volueris, ita tamen, ut modò majoribus, modò <lb/>minoribus ponderibus movendis &longs;it idonea citrà temporis di&longs;-<pb pagenum="791" xlink:href="017/01/807.jpg"/>pendium, ubi &longs;atis virium habetur in potentiâ; eâ ratione in <lb/>loculamento di&longs;pone &longs;ingulos axes in cochleam deformatos, ut <lb/>eorum poli ex loculamento promineant, atque pro re natâ pro­<lb/>pelli &longs;eu retrahi aliquanti&longs;per valeat hic aut ille axis, ne ejus <lb/>&longs;tria occurrat &longs;ubjecti tympani dentibus. </s> <s id="s.005826">Nam &longs;i alterius &longs;al­<lb/>tem poli extremitas in quadratam figuram de&longs;inat, quæ in&longs;eri <lb/>po&longs;&longs;it manubrio, hoc poterit huic aut illi axi aptari, quin &longs;upe­<lb/>riores cochleæ hujus tympani convolutionem impediant. </s> <s id="s.005827">Quod <lb/>&longs;i majora adhuc requirantur potentiæ momenta, proximè &longs;u­<lb/>perior axis &longs;uum in locum re&longs;tituatur, ut cochleæ &longs;tria in &longs;ub­<lb/>jecti tympani dentes incurrat. </s> <s id="s.005828">Quapropter ad minora pondera <lb/>movenda adhibeantur inferiores cochleæ, ad majora &longs;uperiores. <lb/></s> </p> <p type="main"> <s id="s.005829"><emph type="center"/>CAPUT V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005830"><emph type="center"/><emph type="italics"/>Cochleæ u&longs;us aliqui indicantur.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005831">ADeò frequens e&longs;t & vulgatus apud plero&longs;que artifices co­<lb/>chleæ u&longs;us, ut ex tam variâ ejus cum cæteris complexio­<lb/>ne unu&longs;qui&longs;que facilè colligere po&longs;&longs;it, quid facto &longs;it opus, ubi <lb/>eâ utendum nece&longs;&longs;itas aut utilitas &longs;ua&longs;erit. </s> <s id="s.005832">Ne tamen ab initâ <lb/>in antecedentibus libris con&longs;uetudine in hujus operis calce re­<lb/>cedam, pauca quædam indicare placuit, quæ in reliquis non <lb/>admodum di&longs;&longs;imilibus facem præferant. </s> </p> <p type="main"> <s id="s.005833"><emph type="center"/>PROPOSITIO I.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005834"><emph type="center"/><emph type="italics"/>Aërem validè comprimere, aut dilatare.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005835">FOllibus lu&longs;oriis aërem pyulco ingerentes majorem &longs;ubinde <lb/>atque majorem difficultatem percipiunt; quo enim magis <lb/>aër conclu&longs;us à naturali raritate recedere cogitur, etiam majo­<lb/>re ni&longs;u re&longs;i&longs;tit, neque &longs;olùm magis den&longs;ari renuit, &longs;ed & &longs;e la­<lb/>tiùs explicare molitur. </s> <s id="s.005836">Hinc didicimus & pneumaticos fontes <lb/>con&longs;truere, qui Spiritu interno urgente aquam in altum evi-<pb pagenum="792" xlink:href="017/01/808.jpg"/>brant, & plumbeas glandes fi&longs;tulis ejaculari, non pulvere ni­<lb/>trato ignem concipiente, &longs;ed aëre per vim den&longs;ato ad antiquas <lb/>dimen&longs;iones recuperandas erumpente. </s> <s id="s.005837">Quoniam verò inge&longs;ta <lb/>jam in conceptaculum non exigua aëris copia difficiliùs com­<lb/>primitur nová aëris acce&longs;&longs;ione, quàm ut manus valeat tru&longs;illum <lb/>rectâ impellere; idcirco tru&longs;illi ha&longs;tulam deformatam in heli­<lb/>cem, & &longs;uæ matrici in&longs;ertam, adhibere operæ pretium erit: <lb/>dum enim manubrio agitante contorquetur cochlea, &longs;en&longs;im de­<lb/>primitur embolus, aërémque ingerit. </s> <s id="s.005838">Ne autem morâ longio­<lb/>re opus &longs;it perpetuâ ver&longs;atione manubrij, ita cochleæ matrix <lb/>externam va&longs;is faciem contingat, ut illi adnecti, atque ab eo <lb/>disjungi valeat: initio enim, quando adhuc levis e&longs;t aeris mo­<lb/>dicè compre&longs;&longs;i re&longs;i&longs;tentia, lamella illa &longs;uo foramine interiùs <lb/>claviculatim &longs;triato cohærens ha&longs;tulæ emboli, &longs;i à va&longs;e dis­<lb/>juncta fuerit, unà cum ha&longs;tulâ movebitur: deinde verò, quan­<lb/>do jam tru&longs;illus ægrè impellitur, lamella illa cum va&longs;e con­<lb/>nectatur, & non ni&longs;i ver&longs;ato manubrio adduci atque reduci em­<lb/>bolus poterit; id quod &longs;atis lentè perficietur. </s> <s id="s.005839">Rem claritatis <lb/>gratia in fonte pneumatico explicemus. </s> </p> <p type="main"> <s id="s.005840">Sit vas AB ex materiâ metallicâ, in cujus &longs;uperiore parte la­<lb/><figure id="id.017.01.808.1.jpg" xlink:href="017/01/808/1.jpg"/><lb/>brum, ex quo per fo­<lb/>ramen A immittatur <lb/>in vas aqua, ita ta­<lb/>men, ut non implea­<lb/>tur; aqua enim in vas <lb/>modicè inclinatum <lb/>de&longs;cendens aërem ex­<lb/>pellet per tubulum <lb/>CD. <!-- KEEP S--></s> <s id="s.005841">Ubi &longs;atis aquæ <lb/>immi&longs;&longs;um fuerit, oc­<lb/>cludatur foramen A <lb/>diligenti&longs;&longs;imè co­<lb/>chleolâ congruente, <lb/>& convoluto epi&longs;to­<lb/>mio E, tubus DC &longs;it <lb/>aëri impervius ad va&longs;is latus &longs;tatuatur modiolus cum embolo <lb/>congruente HI, & emboli ha&longs;tula &longs;it connexa cum mobili va­<lb/>&longs;is ansâ HO. <!-- KEEP S--></s> </p> <pb pagenum="793" xlink:href="017/01/809.jpg"/> <p type="main"> <s id="s.005842">Porrò ha&longs;tula HK perforata &longs;it, & continuo ductu u&longs;que <lb/>ad emboli KS fundum pateat aëri ingredienti via HS; &longs;ed fo­<lb/>ramini S adjecta &longs;it valvula, quæ aëri regre&longs;&longs;um ob&longs;truat. </s> <s id="s.005843">Simi­<lb/>liter modioli fundo in I valvula exteriùs appo&longs;ita aperiatur in­<lb/>ge&longs;to aëri tran&longs;itum præbens, &longs;ed aëri intra vas compre&longs;&longs;o cum <lb/>nu&longs;quam exitus pateat, valvula ip&longs;a modioli foramen I occlu­<lb/>dit. </s> <s id="s.005844">Ha&longs;tulæ verò HK exterior facies &longs;it in helicem &longs;triata, & <lb/>lamellæ MN tanquam matrici congruat, quæ in M & N co­<lb/>chleolis adnecti queat exteriùs va&longs;i, qua&longs;i e&longs;&longs;et an&longs;æ fulcrum. </s> </p> <p type="main"> <s id="s.005845">Ubi immi&longs;&longs;um fuerit quantum &longs;atis e&longs;t aquæ, cochleolis M <lb/>& N revolutis disjungatur matrix à va&longs;e: tum attractâ ansâ <lb/>HO, unà cum lamellâ MN attrahitur embolus KS, & per <lb/>apertum ductum HS ingreditur aër modiolum implens. </s> <s id="s.005846">Im­<lb/>pul&longs;o deinde embolo, valvula ad S clauditur, & aër ex modio­<lb/>lo per patentem valvulam I ingeritur in vas; ex quo nequit exi­<lb/>re, neque aquam propellere, clau&longs;o &longs;cilicet epi&longs;tomio E, & fo­<lb/>ramine A: quapropter comprimitur, & den&longs;atur; ideóque at­<lb/>tracto denuo embolo KS inclu&longs;us va&longs;i aër &longs;e latiùs explicare <lb/>connitens valvulam I valide applicat foramini modioli, &longs;ibíque <lb/>exitum ob&longs;truit. </s> <s id="s.005847">Toties adducitur atque reducitur embolus, & <lb/>aër ingeritur, quoad magna premendi difficultas percipiatur; <lb/>ubi eò ventum fuerit, tunc lamella MN iterum va&longs;i adnecta­<lb/>tur &longs;uis cochleolis; nec jam embolus rectâ adduci pote&longs;t; &longs;ed <lb/>arreptum in O manubrium ver&longs;atur, & embolus intrà modio­<lb/>lum circumactus &longs;en&longs;im attollitur, qui deinde revoluto in con­<lb/>trarium manubrio deprimitur, & multâ vi aër in va&longs;e compri­<lb/>mitur. </s> <s id="s.005848">Laxato demum Epi&longs;tomio E, compre&longs;&longs;us in va&longs;e aër <lb/>aquam exprimit per tubum CD, primùm quidem vehemen­<lb/>tiùs, &longs;ubinde remi&longs;&longs;ius, prout aëris vis ela&longs;tica &longs;en&longs;im lan­<lb/>gue&longs;cit. </s> </p> <p type="main"> <s id="s.005849">Hoc idem quod de aëre intra vas comprimendo ad aquam <lb/>evibrandam commini&longs;ci placuit, &longs;ervatâ analogiâ dicendum <lb/>e&longs;t de aëre, tùm conatu manûs rectâ tru&longs;illum impellentis, tum <lb/>ope cochleæ &longs;imiliter conformatæ, intrà conceptaculum com­<lb/>primendo, ut ex fi&longs;tulâ deinde multâ vi emittatur plumbea <lb/>glans, ubi re&longs;eratus aëri exitus illum &longs;ubitò dilatari permi&longs;erit. </s> <lb/> <s id="s.005850">Quin & pneumatica huju&longs;modi tormenta citrà conceptaculum <lb/>aëris compre&longs;&longs;i con&longs;truere non inutile accidat, &longs;i, quemadmo-<pb pagenum="794" xlink:href="017/01/810.jpg"/>dum no&longs;trates pueri &longs;urculos &longs;ambuceos fungosâ medullâ <lb/>exhauriunt, & utráque tubuli extremitate papyraceis globulis <lb/>ob&longs;tructâ, alterum globulum congruo cylindro propellunt, at­<lb/>que inclu&longs;um aërem den&longs;ant, quoad aëris vim ela&longs;ticam, & im­<lb/>pellentis manûs conatum non ferens extremus alter globulus <lb/>edito &longs;cloppo expellatur; ita ferream fi&longs;tulam longiorem para­<lb/>veris, cujus alteri extremitati immittatur plumbea glans ob­<lb/>ducta papyro, aut &longs;imili materiâ, ut exqui&longs;itè tubi o&longs;culum <lb/>implens demum univer&longs;am aëris vim excipiat, alteram extre­<lb/>mitatem aliquot &longs;piris ambiat cava cochlea, quam impleat cy­<lb/>lindrus ferreus in congruentem cochleam deformatus: Si enim <lb/>huju&longs;modi cylindrus vix brevior fuerit, quàm fi&longs;tula, & apto <lb/>manubrio convolutus in fi&longs;tulam &longs;en&longs;im immittatur, totum aë­<lb/>rem, quo fi&longs;tula replebatur, ad exiguas &longs;patij angu&longs;tias adiget, <lb/>ex quibus magnâ vi demum, quâ data porta, erumpens ejacu­<lb/>labitur plumbeum globulum. </s> </p> <p type="main"> <s id="s.005851">Quod &longs;i aërem non comprimere, &longs;ed di&longs;trahere atque dila­<lb/>tare libitum fuerit, eâdem ratione parandus e&longs;t modiolus cum <lb/>embolo, ac ha&longs;tulâ in helicem &longs;triatâ, atque perforatâ, & co­<lb/>chleæ matrici in&longs;erta, ni&longs;i quod valvulæ contrariam po&longs;itionem <lb/>exigunt; nam modioli valvula I intrà ip&longs;um modiolum &longs;tatuen­<lb/>da e&longs;t, ut adducto embolo aperiatur, & ex va&longs;e aër in modio­<lb/>lum attrahatur: Emboli verò valvula non ad S, &longs;ed in H ap­<lb/>ponenda e&longs;t, ut reducto embolo, aër in modiolum admi&longs;&longs;us ex­<lb/>primatur per tubulum SH, &longs;ivè manu urgeatur tru&longs;illus, &longs;ive <lb/>cochlea convolvatur. </s> <s id="s.005852">Aërem autem, licèt valde compre&longs;&longs;um, <lb/>magis etiam convolutâ cochleâ den&longs;ari, aut valde rarum ma­<lb/>gis adhuc dilatari manife&longs;tum e&longs;t; id quod rectâ manûs impul­<lb/>&longs;ione aut attractione nequaquam fieri po&longs;&longs;et. </s> </p> <p type="main"> <s id="s.005853"><emph type="center"/>PROPOSITIO II.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005854"><emph type="center"/><emph type="italics"/>Forcipum vires cochleâ augere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005855">DUplicem exerceri à forcipibus vim con&longs;tat; altera e&longs;t con­<lb/>&longs;tringendo id, quod illis apprehenditur, & earum vis <lb/>major aut minor ex eo æ&longs;timatur, quod brachia longè à nodo, <lb/>aut prope illum, arripiantur: altera vis e&longs;t in extrahendo ali-<pb pagenum="795" xlink:href="017/01/811.jpg"/>quid, ut clavum tabulæ aut parieti infixum; cum enim curva <lb/>&longs;it forceps, qua parte clavum apprehendit, adnexum in ip&longs;o <lb/>flexu habet hypomochlium, & brachia inclinando, pro eorum <lb/>longitudine, vis extrahendi exercetur qua&longs;i per vectem. </s> <s id="s.005856">At <lb/>aliquando opus e&longs;t majore conatu, quàm ut &longs;olis forcipibus va­<lb/>leat potentia infixum clavum extrahere; momentum &longs;iquidem <lb/>potentiæ pendet ex Ratione, quam habet di&longs;tantia potentiæ ad <lb/>di&longs;tantiam clavi ab ip&longs;o flexu, qui fungitur munere hypomo­<lb/>chlij. </s> <s id="s.005857">Quare vis extrahendi major communicari pote&longs;t ope co­<lb/>chleæ, ita tamen, ut forceps non exerceat munus vectis. </s> </p> <p type="main"> <s id="s.005858">Paretur itaque valida & &longs;atis cra&longs;&longs;a lamina chalybea AB, <lb/>matricem cochleæ habens in C, & &longs;it <lb/><figure id="id.017.01.811.1.jpg" xlink:href="017/01/811/1.jpg"/><lb/>cochlea FE, manubrium habens ED. <!-- KEEP S--></s> <lb/> <s id="s.005859">Cochleæ verò extremitas in cylindrum <lb/>de&longs;inat, qui cra&longs;&longs;ioris laminæ HI fora­<lb/>mini exqui&longs;itè polito in&longs;eratur, & in eo <lb/>facillimè convolvi valeat. </s> <s id="s.005860">Cylindri ex­<lb/>tremitas infra laminam HI ita dilatetur, <lb/>ut eandem laminam HI &longs;u&longs;tineat, non <lb/>tamen convolutionem impediat. </s> <s id="s.005861">Porrò <lb/>laminæ HI adnexi &longs;int duo annuli ita <lb/>conformati, ut forcipis brachia exci­<lb/>piant: nam &longs;i brachia in huju&longs;modi an­<lb/>nulos immittantur, ut hi proximi &longs;int nodo forcipis maximè di­<lb/>latatæ, antequam apprehendat clavum extrahendum, po&longs;tmo­<lb/>dum con&longs;trictâ forcipe & clavum apprehendente, elevata la­<lb/>mina HI annulos &longs;ecum rapiet, qui per forcipis brachia diva­<lb/>ricata excurrentes demum validè illa con&longs;tringent, nec ulte­<lb/>riùs excurrere poterunt. </s> <s id="s.005862">His paratis utrique extremitati AB <lb/>&longs;ubjiciantur fulcra (&longs;ivè &longs;int tigillorum fru&longs;ta, &longs;ivè quæcum­<lb/>que alia) inter ip&longs;am laminam & planum, ex quo educendus <lb/>e&longs;t clavus, interjecta: Nam manubrio DE convoluta cochlea <lb/>ita matricem AB applicabit fulcris, ut firmi&longs;&longs;imè cohæreant <lb/>cum &longs;ubjecto plano. </s> <s id="s.005863">Jam &longs;i pergas cochleam contorquere, hæc <lb/>&longs;ecum rapiet laminam HI, & adjectos annulos cum forcipe, & <lb/>clavo, quem revellit. </s> </p> <p type="main"> <s id="s.005864">Quod &longs;i fortè placuerit forcipem habere peculiarem huic <lb/>in&longs;trumento aptandam, habeat in brachiorum extremitatibus <pb pagenum="796" xlink:href="017/01/812.jpg"/>uncos aut annulos annulis H & I in&longs;erendos aut connectendos, <lb/>eâ tamen ratione di&longs;po&longs;itos, ut dum lamina HI vi cochleæ tra­<lb/>hitur, brachia ip&longs;a ad &longs;e invicem accedendo forcipem con­<lb/>&longs;tringant. </s> </p> <p type="main"> <s id="s.005865">Unum præterea addendum, quod non levis e&longs;t momenti, & <lb/>aliàs quoque ob&longs;ervari poterit. </s> <s id="s.005866">Contingere pote&longs;t, ut omnibus <lb/>modo dicto paratis, potentia &longs;e infirmiorem &longs;entiat, quàm ut <lb/>valeat circumducto manubrio DE cochleam contorquere. </s> <s id="s.005867">Hoc <lb/>igitur tibi remedium compara: longiorem vectem validis funi­<lb/>culis colliga cum manubrio DE, & vecte illo qua&longs;i manubrio <lb/>utens experieris pro Ratione longitudinis aucta momenta; am­<lb/>plior &longs;iquidem peripheria, quæ tunc à potentiâ de&longs;cribitur, ad <lb/>&longs;pirarum cochleæ intervallum habet Majorem Rationem. <!-- KEEP S--></s> </p> <p type="main"> <s id="s.005868"><emph type="center"/>PROPOSITIO III.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005869"><emph type="center"/><emph type="italics"/>Numerum pa&longs;&longs;uum aut rotæ conver&longs;ionem metiri.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005870">HOc idem problema lib.5. cap.9. prop.2. propo&longs;itum e&longs;t, & <lb/>per rotulas dentatas &longs;ingulis prioris rotæ conver&longs;ionibus <lb/>excipientes impul&longs;ionem &longs;ingulorum dentium, in quos promi­<lb/>nens paxillus incurrat, perfici po&longs;&longs;e indicatum e&longs;t. </s> <s id="s.005871">Nunc <lb/>aliam methodum indicare placet ex iis, quæ &longs;uperiore capite <lb/>&longs;unt dicta de Cochleâ Infinitâ. </s> <s id="s.005872">Primam quidem rotulam, cui <lb/>motus origo ine&longs;t ex funiculi tractione, prout ibi dictum e&longs;t, <lb/>eandem &longs;tatue, & illius axis extremitas appo&longs;ito indice tot pa&longs;­<lb/>&longs;us, aut tot rotæ conver&longs;iones indicabit, quot in dentes ip&longs;a <lb/>prima rotula di&longs;tributa intelligitur. </s> <s id="s.005873">Hujus rotulæ axis in co­<lb/>chleam infinitam deformetur, cui &longs;ua rotula dentata congruat; <lb/>& &longs;ingulis primæ rotulæ conver&longs;ionibus &longs;inguli dentes &longs;ecundæ <lb/>promoventur: atque adeò quot dentes &longs;ecundæ huic rotulæ in­<lb/>&longs;unt, ut hæc integram conver&longs;ionem perficiat, tot requiruntur <lb/>prioris rotulæ conver&longs;iones. </s> <s id="s.005874">Similiter &longs;ecundæ rotulæ axis in <lb/>cochleam infinitam deformetur, & tertiam rotulam dentatam <lb/>convertat, cujus axis pariter tertiam cochleam infinitam con&longs;ti­<lb/>tuere pote&longs;t, & quartam rotulam cum &longs;uo axe & indice convol­<lb/>vere. </s> <s id="s.005875">Singulorum axium extremitates in facie loculamenti ad­<lb/>jecto indice ob oculos ponunt numerum revolutionum proxi-<pb pagenum="797" xlink:href="017/01/813.jpg"/>mè antecedentis rotulæ. </s> <s id="s.005876">Quapropter numerus à po&longs;tremâ ro­<lb/>tulâ indicatus multiplicandus e&longs;t per numerum omnium den­<lb/>tium penultimæ rotulæ, & productus per numerum dentium <lb/>antepenultimæ ducendus; atque iterum hunc productum per <lb/>numerum omnium dentium antecedentis rotulæ multiplicare <lb/>oportet, ut omnium pa&longs;&longs;uum, aut conver&longs;ionum rotæ currûs, nu­<lb/>merus innote&longs;cat. </s> <s id="s.005877">Quare artificis indu&longs;tria in hoc requiritur, <lb/>ut rotularum dentibus eos numeros &longs;tatuat, quorum rationem <lb/>inire non &longs;it nimis opero&longs;um. </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/>cochlearum infinitarum complexione contingit nece&longs;&longs;ariò, <lb/>quod axes &longs;unt in planis invicem rectis, ac proinde indices non <lb/>in eâdem loculamenti facie con&longs;titui po&longs;&longs;unt: cum enim unu&longs;­<lb/>qui&longs;que axis ad planum &longs;ui tympani dentati, cui infigitur, &longs;it <lb/>rectus, ip&longs;um verò tympanum &longs;it in eodem plano, in quo e&longs;t <lb/>cochlea infinita, à qua convertitur, manife&longs;tum e&longs;t plana ip&longs;a, <lb/>in quibus &longs;unt axes, e&longs;&longs;e invicem recta, atque idcirco non ad <lb/>eandem loculamenti faciem pertinere eorum indices. </s> </p> <p type="main"> <s id="s.005881"><emph type="center"/>PROPOSITIO IV.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005882"><emph type="center"/><emph type="italics"/>Lunæ motum & pha&longs;es in automato indicare.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005883">QUæ communiter parantur automata horas indicantia, in­<lb/>dicem habent horis duodecim perficientem integrum cir­<lb/>cuitum: quapropter lunæ motum, eju&longs;que ætatem ob oculos <lb/>ponere cupiens, &longs;atis erit, &longs;i axem, cui horarum index in&longs;eri­<lb/>tur, in cochleam infinitam deformaveris, quæ convertat tym­<lb/>panum in dentes 59 di&longs;tributum; axis enim tympani indicem <lb/>convertens ætatem lunæ common&longs;trabit in dexterâ, aut in &longs;i­<lb/>ni&longs;trâ facie loculamenti, cui automatum includitur. </s> <s id="s.005884">Cum <lb/>enim lunaris men&longs;is Synodicus complectatur dies 29 1/2, index <lb/>autem horarum &longs;emi&longs;&longs;em diei perficiat, erunt indicis hujus <lb/>conver&longs;iones 59, dum &longs;emel index lunæ &longs;uam conver&longs;ionem <lb/>ab&longs;olvit. </s> <s id="s.005885">Si igitur index lunæ &longs;it lamina rotundum habens fo­<lb/>ramen propè indicis lingulam, per quod appareat pictus in &longs;ub­<lb/>jectâ facie circulus centrum habens extra indicis centrum, adeò <lb/>ut primâ die lunæ nihil illius circuli appareat, & die decima-<pb pagenum="798" xlink:href="017/01/814.jpg"/>quintâ foramen integrum exhibeat eju&longs;dem circuli colorem, <lb/>lunæ Pha&longs;es à foramine, & ejus ætas à lingulâ indicabuntur. </s> </p> <p type="main"> <s id="s.005886">Quod &longs;i placuerit in eádem facie, in qua de&longs;criptæ &longs;unt ho­<lb/>ræ, etiam lunæ pha&longs;es & motum apparerere, oportebit axi in­<lb/>dicis horarum aptatam rotulam denticulos habere ad perpendi­<lb/>culum infixos, qui curriculum, &longs;eu Vertebram &longs;triatam con­<lb/>vertant, ita ut vertebræ huju&longs;modi una conver&longs;io planè i&longs;o­<lb/>chrona &longs;it uni conver&longs;ioni indicis horarum. </s> <s id="s.005887">Curriculi autem <lb/>axis in cochleam infinitam deformatus convertat tympanum in <lb/>dentes 59 di&longs;tinctum, quod collocetur faciei loculamenti pa­<lb/>rallelum; hujus &longs;iquidem conver&longs;io in eâdem loculamenti fa­<lb/>cie, in qua & horæ indicantur, repræ&longs;entabit lunæ pha&longs;es. </s> </p> <p type="main"> <s id="s.005888">At &longs;i forta&longs;&longs;e volueris in eâdem Automati facie ita apparere <lb/>horas & lunæ ætatem, ut proximè &longs;altem indicetur, quotâ ho­<lb/>râ accidat Novilunium aut Plenilunium, po&longs;tquam &longs;emel jux­<lb/>ta Ephemerides conciliaveris indices horarum & lunæ; non &longs;a­<lb/>tis erit in dentes 59 di&longs;tinxi&longs;&longs;e tympanum, cujus &longs;inguli dentes <lb/>horis 12 promoveantur; &longs;iquidem men&longs;is lunaris Synodicus <lb/>complectitur dies 29, horas 12, minuta 44, hoc e&longs;t ferè tres <lb/>horæ quadrantes; atque adeò po&longs;t duos men&longs;es index lunæ in­<lb/>dicaret Novilunium &longs;e&longs;quihorâ citiùs, quàm par fuerit, & po&longs;t <lb/>annum index anteverteret verum Novilunium novem horis. </s> <lb/> <s id="s.005889">Quare axi horas indicanti non e&longs;&longs;et copulandus axis cochleæ <lb/>infinitæ, cujus tympanum aliam exigeret dentium multitudi­<lb/>nem; &longs;ed peculiaris axis &longs;tatuendus e&longs;&longs;et, cujus conver&longs;io ita <lb/>temperaretur, ut horis undecim cum quadrante ab&longs;olveretur; <lb/>tympanum verò, ex cujus conver&longs;ione convolveretur index lu­<lb/>næ, di&longs;tribuendum e&longs;&longs;et in dentes 63; hujus enim unica con­<lb/>ver&longs;io re&longs;ponderet conver&longs;ionibus 63 axis, cujus &longs;ingulæ con­<lb/>ver&longs;iones perficerentur horis 11 1/4: quapropter index lunæ <lb/>&longs;uam conver&longs;ionem ab&longs;olveret horis 708 3/4, hoc e&longs;t diebus 29, <lb/>horis 12, minutis 45. E&longs;&longs;et igitur in &longs;ingulis lunationibus pau­<lb/>lò tardior non ni&longs;i uno minuto; &longs;ed demum ab&longs;olutis duode­<lb/>cim lunationibus exiguum e&longs;&longs;et di&longs;crimen. </s> <s id="s.005890">Quod &longs;i rotulæ ho­<lb/>ras indicantis faciem interiorem in partes 16 di&longs;tinxeris, & <lb/>denticulos ad perpendiculum erexeris, qui Curriculum con­<lb/>vertant, ita tamen, ut curriculus unâ conver&longs;ione excipiat &longs;o­<lb/>lùm quindecim denticulos, utique una curriculi conver&longs;io <pb pagenum="799" xlink:href="017/01/815.jpg"/>perficietur horis 11 1/4, hoc e&longs;t (15/16) horarum duodecim, &longs;eu hora­<lb/>rum quadrantibus 45; qui per 63 multiplicati dant horæ qua­<lb/>drantes 2835, quot una lunatio complectitur. </s> </p> <p type="main"> <s id="s.005891"><emph type="center"/>PROPOSITIO V.<emph.end type="center"/><!-- KEEP S--></s> </p> <p type="main"> <s id="s.005892"><emph type="center"/><emph type="italics"/>Pancratium ad onera Vecte attollenda opportunum con&longs;truere.<emph.end type="italics"/><emph.end type="center"/></s> </p> <p type="main"> <s id="s.005893">SÆpè contingit Vecte &longs;ecundi generis attollendum e&longs;&longs;e ali­<lb/>quod onus, cui impar &longs;it potentia: idcirco præ&longs;tò e&longs;&longs;e po­<lb/>te&longs;t in&longs;trumentum (cui Pancratio nomen fieri po&longs;&longs;e o&longs;tendit <lb/>vis &longs;atis magna) plures in alios u&longs;us accommodatum, quod & <lb/>facillimè quocumque in loco collocari valet, & quocumque <lb/>transferri. </s> <s id="s.005894">Cochlea infinita cum &longs;uo tympano dentato con­<lb/>gruente paretur: tympani axis &longs;it excavatus in tres aut quatuor <lb/>&longs;trias convenientes dentibus laminæ rectæ chalybeæ dentatæ <lb/>&longs;atis &longs;olidæ, cuju&longs;modi illa e&longs;t, quam lib. 5. cap. 6. exhibui. </s> <lb/> <s id="s.005895">Nam &longs;i hæc includantur cap&longs;ulæ paulò longiori, quàm &longs;it la­<lb/>mina illa dentata, & cochleæ axis extra loculamentum promi­<lb/>neat, ut ei aptari po&longs;&longs;it manubrium; ex Cochleæ conver&longs;ione <lb/>volvitur tympanum, & unà cum illo eju&longs;dem axis &longs;triatus, qui <lb/>dentes laminæ chalybeæ &longs;ubiens illam elevat. </s> <s id="s.005896">Et quoniam hu­<lb/>jus laminæ caput &longs;inuatum &longs;ubjicitur vecti, etiam vectis attol­<lb/>litur, & cum eo pondus. </s> <s id="s.005897">Quanta &longs;it cochleæ infinitæ cum &longs;uo <lb/>tympano & axe vis ad elevandam laminam, con&longs;tat ex dictis: <lb/>Componenda e&longs;t autem hæc Ratio cum Ratione Vectis, ut ha­<lb/>beatur momentum Potentiæ manubrio applicatæ comparatæ <lb/>cum onere. </s> </p> <p type="main"> <s id="s.005898"><emph type="center"/>FINIS.<emph.end type="center"/><!-- KEEP S--></s> </p> </chap> </body> <back/> </text> </archimedes>