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<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Casati, Paolo</author> <title>Mechanica</title> <date>1684</date> <place>Lyons</place> <editor></editor> <publisher></publisher> <translator></translator> <lang>la</lang> <chunk unit="page*">page</chunk> <locator>0000000017</locator> </info> <text> <front> </front> <body> <chap> <pb/><p type="main"> <s><emph type="center"/>R. P. PAULI <lb/>CASATI <lb/>MECHANICA.<emph.end type="center"/></s></p><pb/><p type="main"> <s><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="main"> <s><emph type="center"/>principio Vectis vires Phy&longs;icè explicantur & Geometricè <lb/>demon&longs;trantur,<emph.end type="center"/></s></p><p type="main"> <s><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></figure><p type="main"> <s><emph type="center"/>LUGDUNI, <lb/>Apud ANISSONIOS, JOAN, POSUEL <lb/>& CLAUDIUM RIGAUD.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>M. </s> <s>D C. LXXXIV.<emph.end type="italics"/><lb/>CUM PRIVILEGIO REGIS.<emph.end type="center"/></s></p><pb/><figure></figure><p type="main"> <s><emph type="center"/>CHRISTIANISSIMO <lb/>GALLIARUM <lb/>ET NAVARRÆ REGI <lb/>LUDOVICO <lb/>MAGNO.<emph.end type="center"/></s></p><p type="main"> <s><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>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/><emph type="italics"/>citate quid me paterer excludi? </s> <s>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>Me verò Natu­<lb/>ræ atque Artis mutuam &longs;ocietatem coëuntium in <lb/>Machinis, ferè dixerim, miracula contemplari <lb/>a&longs;&longs;uetum rapuere admirabundum, 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>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>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/><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> <s><lb/>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> <s><lb/>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="main"> <s><emph type="center"/><emph type="italics"/>MAJESTATIS Tuæ<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Parmæ Kal, Maij 1683. </s></p><p type="main"> <s>Humillimus atque Ob&longs;equenti&longs;&longs;imus <lb/>Servus <lb/>PAULUS CASATUS è SOC. JESU. <pb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Facultas R. P. </s> <s>Provincialis Societatis Je&longs;u <lb/>in Provincia Veneta.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>R. P. N. </s> <s>Præpo&longs;ito Generali Jo. </s> <s>Paulo Oliva, faculta­<lb/>tem facio, ut Opus in&longs;criptum, <emph type="italics"/>Mcchanichorum Libri octo, <lb/>Authore P. </s> <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>Cujus rei <lb/>gratiâ has litteras meâ manu &longs;ub&longs;criptas, & &longs;igillo officij mei <lb/>munitas dedi. </s> <s>Parmæ 23. Februarij 1681. </s></p><p type="main"> <s>OCTAVIUS RUBEUS. <lb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Summa Privilegiy 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>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 Libriocto, Authore R. P. </s> <s>Paulo Ca&longs;ato Soc. </s> <s>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> <s><lb/>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>Dabatur Ver&longs;aliis die vige&longs;ima prima Januarij anno Dom. </s> <s>1684. </s></p><p type="main"> <s><emph type="italics"/>Ex mandato Regis.<emph.end type="italics"/></s></p><p type="main"> <s>JUNQUIERES. </s></p><p type="main"> <s>MECHA </s></p><pb/><figure></figure><p type="main"> <s><emph type="center"/>AD LECTOREM.<emph.end type="center"/></s></p><p type="main"> <s>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>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>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>Licuit autem, <lb/>præter &longs;pem, toties dimi&longs;&longs;um calamum re&longs;umere, ut tan-<pb/>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> <s><lb/>Quapropter alienæ utilitati &longs;erviendum potiùs fuit, quàm <lb/>meæ voluntati. </s></p><p type="main"> <s>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>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>Methodum ne culpes, quòod 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>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>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. </s> <s>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/>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>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> <s><lb/>Quare alia Geometricè, alia Phy&longs;icè tractata æquo animo <lb/>patere. </s></p><p type="main"> <s>Stylum autem quid excu&longs;em? </s> <s>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>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>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>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> <s><lb/>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/>&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>Nec te pluribus volo, Amice Lector. </s> <s>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>In iis verò, in quibus à me per impru­<lb/>dentiam peccatum fuerit, à tuâ Sapientiâ facilè patiar me <lb/>dedoceri. </s> <s>Vale. </s></p><figure></figure><p type="main"> <s>ELENCHUS </s></p><pb/><figure></figure><p type="main"> <s><emph type="center"/>ELENCHUS CAPITUM.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/>LIBER PRIMUS. </s> <s>De Centro Gravitatis.<emph.end type="center"/></s></p><p type="table"> <s>TABELLE WAR HIER</s></p><p type="table"> <s>TABELLE WAR HIER</s></p><p type="table"> <s>TABELLE WAR HIER</s></p><pb pagenum="54"/><p type="main"> <s>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>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 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>Et ut res i&longs;ta plani&longs;&longs;imè o&longs;tendatur, <lb/><figure id="fig1"></figure><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. </s> <s>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>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. </s> <s>Quid autem aliud e&longs;t &longs;olidum KD, quam tur­<lb/>ris inclinata? </s></p><p type="main"> <s>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>Quare litteris ad P. </s> <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"/><p type="main"> <s>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. </s> <s>Bonon. </s> <s>17. <lb/>unc. </s> <s>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>Areæ demum vacuæ ad ba­<lb/>&longs;im latus unum e&longs;t ped. </s> <s>6. alterum ped.6. unc.1. </s></p><p type="main"> <s>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>Sit itaque <lb/>turris inclinata DC, &longs;uperioris autem podij <lb/><figure id="fig2"></figure><lb/>A&longs;inellæ altitudo EB ped.234 1/2, unde ob&longs;er­<lb/>vatus e&longs;t angulus CEB gr. </s> <s>18. 40′. </s> <s>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. </s> <s>51. 51′. </s> <s>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>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. </s> <s>234 1/2, remanet altitudo GB, hoc e&longs;t CH, <lb/>ped. (123 1/12). </s></p><pb pagenum="56"/><p type="main"> <s>Demum ad inve&longs;tigandam turris inclinationem, applicito <lb/>ad punctum I perpendiculo ob&longs;ervatus e&longs;t angulus DIL <lb/>gr. </s> <s>3. 10′.: cùm autem IL parallela &longs;it perpendiculari CH, erit <lb/>pariter angulus DCH gr.3.10′. </s> <s>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>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>E&longs;t igitur EB ped. </s> <s>6. ac <lb/>propterea RB ped. </s> <s>300″; & quia HC e&longs;t ped. </s> <s>5, VC e&longs;t <lb/>ped.2. 50″. </s> <s>BD autem e&longs;t ped. </s> <s>6. unc.10, hoc e&longs;t ped.(6 10/12). </s></p><figure></figure><p type="main"> <s>In Triangulo BDC rectangulo datis BD <lb/>ped. </s> <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″. </s> <s><lb/>Fiat igitur ut CB ped. </s> <s>123. 27″, ad BD <lb/>ped. </s> <s>6. 83″. </s> <s>ita Radius ad &longs;inum anguli BCD <lb/>gr. </s> <s>3. 10′ 34″. </s> <s>Quare angulus reliquus CBD <lb/>gr. </s> <s>86. 49′. </s> <s>26″, cui æqualis e&longs;t alternus VCB <lb/>inter parallelas VC, RD; angulus autem, <lb/>qui e&longs;t deinceps, CBR gr. </s> <s>93. 10′. </s> <s>34′. </s> <s>In <lb/>triangulo VCB datis lateribus VC ped.2-50″, <lb/>CB ped. </s> <s>123. 27″, & angulo verticali VCB <lb/>gr. </s> <s>86. 49′. </s> <s>26″, reperitur CVB gr. </s> <s>92. 0′. </s> <s>36″, <lb/>& VBC. gr. </s> <s>1. 9′, 58″. </s> <s>Ex his verò invenitur <lb/>VB ped. </s> <s>122. 76″. </s></p><p type="main"> <s>Jam verò in Triangulo VBR, notus e&longs;t <lb/>angulus RBV æqualis alterno CVB gr.92. <lb/>0′. </s> <s>36′. </s> <s>& nota &longs;unt latera RB ped. </s> <s>300″, & <lb/>VB ped. </s> <s>122. 76″. </s> <s>Quare invenitur angulus <lb/>VRB gr. </s> <s>86. 35′ 43″. </s> <s>BVR gr. </s> <s>1. 23′. </s> <s>41″, & ba&longs;is VR <lb/>ped. </s> <s>123. 17″. </s></p><p type="main"> <s>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″. </s> <s>Ductâ igitur ex S centro gra-<pb pagenum="57"/>vitatis perpendiculari lineâ directionis SX, ex datis latere SR <lb/>ped. </s> <s>59. 72″, & angulo VRX gr. </s> <s>86, 35′, 43″, innote&longs;cit RX <lb/>ped. </s> <s>3. 54″. </s> <s>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>Ex his di&longs;cuties timorem corum, 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>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>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. </s> <s>0. 34. 22″. </s> <s>at &longs;i KE &longs;it ped. </s> <s>50, & KC iterum ped. </s> <s>1. <lb/>angulus KEC e&longs;t grad. </s> <s>11. 32′. </s> <s>13″. </s></p><p type="main"> <s>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>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"/>deinde que madmodum 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>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 var us, magis re&longs;i&longs;tit, & <lb/>datâ æqualitate motûs minùs declinat à perpendiculo. <lb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT X.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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 fummo 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>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"/>Sit terræ &longs;emidiameter CS, altitudo au­<lb/><figure id="fig3"></figure><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. </s> <s>6. duplicatam Rationem laterum <lb/>RI, SL, hoc e&longs;t, per 4. lib. </s> <s>6. duplica­<lb/>tam Rationis, quam habet CR ad CS. </s> <s><lb/>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. </s> <s>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>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>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>Nam &longs;i &longs;uper clivo AB <lb/>&longs;it &longs;tructura AL, cujus parietes per­<lb/><figure id="fig4"></figure><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> <s><lb/>Non igitur plures aut ampliores &longs;tructuras recipit clivus collis, <lb/>quàm &longs;ubjectum planum horizontale. </s> <s>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"/>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>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>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>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. </s></p><p type="main"> <s>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"/>AD; cujus arcus dividatur in par­<lb/><figure id="fig5"></figure><lb/>tes AB, BC, CD æquales, ita ut <lb/>&longs;inguli arcus pro rectiâ 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>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>Centro autem H, <lb/>intervallis HE & HF de&longs;cribantur arcus EI, & FK. </s> <s>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. </s> <s>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>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. </s> <s>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. </s> <s>Eademque e&longs;to de cæteris <lb/>ratio. </s> <s>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"/>intrà illud &longs;patium, ac in eá altitudine, ubi perpendiculorum <lb/>convergentia adeò exigua e&longs;t, ut evane&longs;cat. </s> <s>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>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>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. </s> <s>In &longs;patio igitur, quo <lb/>&longs;uperficies EF excedit &longs;uperficiem AE, poterit alia præterea <lb/>&longs;tructura excitari. <lb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT XI.<emph.end type="center"/></s></p><p type="main"> <s><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="main"> <s>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>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 au­<lb/>tem a&longs;&longs;ecuta e&longs;t motus ita di&longs;ponendo, ut linea directionis nun-<pb pagenum="63"/>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>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> <s><lb/>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>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>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> <s><lb/>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;uitentanti. </s> <s>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"/>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>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 adcò 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>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>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>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>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"/>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>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>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>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>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"/>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>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>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>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>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>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>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. </s> <s><lb/>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"/>autem peripheriæ inter &longs;e ut &longs;emi­<lb/><figure id="fig6"></figure><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>Fiat ita­<lb/>que ut TB &longs;emidiameter terræ mil­<lb/>liar. </s> <s>Rom. </s> <s>ant.4128.pa&longs;&longs;.635. ad BF <lb/>altitudinem hominis ex. </s> <s>gr. </s> <s>ped. </s> <s>Rom. </s> <s>ant. </s> <s>6. ita BC iter pe­<lb/>dum mill. </s> <s>500, ad IE exce&longs;&longs;um itineris capitis qui e&longs;t (726632/1000000) <lb/>unius pedis. </s> <s>Quòd &longs;i fiat ut terræ &longs;emidiameter ad hominis al­<lb/>titudinem, ita circulus terræ maximus mill. </s> <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. </s> <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>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>Incidi in Magiam Naturalem P. </s> <s>Ga&longs;paris <lb/>Schotti part.3.lib.1. pag. </s> <s>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"/>&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>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>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>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>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>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>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. </s> <s>5. hæc peripheria æqualis e&longs;t illi exce&longs;&longs;ui periphe­<lb/>riæ majoris. </s> <s>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. </s> <s>6. ad ped. </s> <s>37. <lb/>unc.8: qui numerus con&longs;entit cùm &longs;uperiore. <pb pagenum="69"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT XII.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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>Ex quo nece&longs;&longs;ariò colligitur mutatio centri gravitatis. </s></p><p type="main"> <s>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>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>Terram univer&longs;am & &longs;ingulas cjus 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"/>cætera omnia inde excludit. </s> <s>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>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>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>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>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 tantummedò repugnant, ne &longs;ur&longs;um in medium leve <lb/>transferantur. </s></p><p type="main"> <s>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 aöre <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>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 aöris 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"/>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>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/>Aör &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>Terræ itaque globus nihil habet, in quod <lb/>gravitatis vires exerceat deor&longs;um conando. </s></p><p type="main"> <s>Quæ cum ita &longs;int, nulla unquam continget in terrâ mutatio <lb/>atque gravium tran&longs;latio, quæ efficiat motum trepidationis. </s> <s><lb/>Sit enim terræ globus AB, cujus cen­<lb/><figure id="fig7"></figure><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>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. </s> <s>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>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. </s> <s>Nondum igitur terra movetur. </s></p><p type="main"> <s>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>Aio ne dum factam <lb/>e&longs;&longs;e mutationem, quæ ad motum telluri conciliandum &longs;ufficiat. </s> <s><lb/>Quamvis enim mons ille FHG, quippe quem ambit aër le-<pb pagenum="72"/>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. </s> <s>4.) <lb/>non poterit &longs;ubjectam terram loco dimovere. </s> <s>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>Quanta igitur <lb/>gravitate præditum e&longs;&longs;e montem oporteret, qui tantam re­<lb/>&longs;i &longs;tentiam &longs;uperare valeret? </s> <s>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>Quare neque hanc gravium tran&longs;lationem motus ul­<lb/>lus con&longs;equitur, quo tellus trepidare dicatur. </s></p><p type="main"> <s>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>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>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 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>Quod &longs;i ex librâ &longs;imilitudinem ducere placeat, petenda po-<pb pagenum="73"/>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/>bu; 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>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>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>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>Sed unum adhuc &longs;upere&longs;t, quod per di&longs;&longs;imulantiam præ­<lb/>tereundum non videtur. </s> <s>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"/>quis po&longs;&longs;it. </s> <s>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>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>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;ot explicandum. </s> <s>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>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>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"/>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>Illud potius hìc attendendum videtur, quod montis altitu­<lb/>do, atque magnitudo ad totius telluris molem Rationem habet <lb/>&longs;atis exiguam. </s> <s>Cum enim terræ ambitus probabiliter &longs;tatuatur, <lb/>ut aliàs o&longs;tendi, milliarium Rom. </s> <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;phætica (ut pote quadrupla maximi circuli ex demon&longs;tratis <lb/>ab Archimede) e&longs;t mill. </s> <s>quadratorum 297. 987800 proximè. </s> <s><lb/>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. </s> <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>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 promincat, 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 &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>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>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"/>validioribus atque incitatioribus, &longs;ed temperatis etiam atque <lb/>mediocribus ventis flantibus. </s> <s>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>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<gap/>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>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>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>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> <s><lb/>Nam &longs;i ab ortu in occa&longs;um ex. </s> <s>gr. </s> <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"/>bus, quibus procedebant, di&longs;crimine, &longs;ed quod demùm modi­<lb/>cis illis acce&longs;&longs;ionibus in integrum diem excreverat. </s> <s>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 numeru ac­<lb/>cre&longs;ceret. </s> <s>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>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>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>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>Negabis <lb/>utique, quamvis enim utraque lacui &longs;tagnanti innate<gap/>, 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>Ita <lb/>planè e&longs;t. </s> <s>Sed jam animum transfer ad in&longs;titutam di&longs;putatio­<lb/>nem, ut di&longs;picias, undè irrep&longs;erit dubitatio hæc de relluris <pb pagenum="78"/>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>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> <s><lb/>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>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>At in telluris circa <lb/>&longs;uum centrum, vel axem, conver&longs;ione nihil ade&longs;&longs;et, quod Pe­<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>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>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> <s><lb/>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>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"/>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT XIII.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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="fig8"></figure><lb/>perficie conferatur. </s> <s>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>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>Sunt autem anguli inclinationis ABC, <lb/>ACD. </s> <s>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>Sin autem ita à <pb pagenum="80"/>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>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. </s> <s>1. & &longs;imiliter AFD maior e&longs;t angulo ACD. </s> <s>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>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>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>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>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>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>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"/>deri autem A &longs;ubjiciantur <lb/><figure id="fig9"></figure><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. </s> <s>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>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>Con&longs;tituantur itaque duo <lb/><figure id="fig10"></figure><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>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. </s> <s>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>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"/>con&longs;tituto, in E elevato de&longs;umenda e&longs;&longs;et ex differentiâ inter <lb/>KC & OE. </s> <s>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. </s> <s>5. ut CE <lb/>ad DG. </s> <s>Sed ut CE ad DG, ita EB ad GB, per 2. lib. </s> <s>6. & <lb/>ut BE, ad BG ita BC ad BD, per 4. lib. </s> <s>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. </s></p><p type="main"> <s>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>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>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>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>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"/>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> <s><lb/>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. </s></p><p type="main"> <s>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>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. </s></p><p type="main"> <s>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>gr. </s> <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>Quare &longs;ublatâ, aut potiùs impeditâ, <lb/>gravitatione corpus quie&longs;cit in plano horizontali. </s></p><p type="main"> <s>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>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"/><p type="main"> <s>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. </s> <s>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>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>Re autem ipsâ &longs;emper contingit angulum BCA e&longs;&longs;e obtu&longs;um <lb/>vel non minorem recto. </s> <s>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. </s> <s>0. 3′. </s> <s>26′; atque adeò DCA <lb/>gr. </s> <s>89. 56′. </s> <s>34″. </s> <s>Jam verò &longs;it CD ad DB ut 100 ad 87; erit <lb/>angulus BCD gr.4.1. 1′. </s> <s>23″: quare totus BCA gr.130. 57′. </s> <s>57′. </s> <s><lb/>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. </s> <s>0. 3′. </s> <s>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 inte c pta 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>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"/><p type="main"> <s>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>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>Sed <lb/>hoc communiter non accidit. </s></p><p type="main"> <s>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. </s> <s>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. </s></p><p type="main"> <s>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> <s><lb/>Sit ad horizontalem, SC per­<lb/><figure id="fig11"></figure><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. </s> <s><lb/>Dico ponderum M, O, N, gravitationes in &longs;uis planis e&longs;&longs;e <lb/>æquales. </s> <s><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"/>pendiculari BD, ad gravitationcm eju&longs;dem M in inclinatâ <lb/>BC, e&longs;t pariter ut BC ad BD; igitur per 11. lib. </s> <s>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. </s> <s>5. gravitatio O in per­<lb/>pendiculari BD æqualis e&longs;t gravitationi M in inclinatâ BC. </s> <s><lb/>Eâdem methodo o&longs;tenditur æqualem e&longs;&longs;e gravitationem N in <lb/>inclinatâ BS, gravitationi O in perpendiculari BD. </s> <s>Quare <lb/>gravitationes M & N æquales inter &longs;e &longs;unt, cum æquales &longs;int <lb/>gravitationi O. </s></p><p type="main"> <s>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>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>De­<lb/>tur O lib. </s> <s>15. & angulus DBC gr. </s> <s>36. Fiat ut radius 10000000 <lb/>ad &longs;ecantem 12360680, ita lib. </s> <s>15. ad lib. </s> <s>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>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. </s> <s>15. ad pondus M datum <lb/>lib. </s> <s>50, ita Radius 10000000 ad 333.33333.&longs;ecantem anguli in­<lb/>clinationis DBC gr. </s> <s>72. 32′. </s> <s>32″. </s> <s>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>Sit enim DBC <lb/>gr. </s> <s>36, & M lib. </s> <s>50. Erit ut Secans 12360680 ad Radium <lb/>10000000, ita M lib. </s> <s>50 ad pondus O fcrè lib.40 1/2, quod po&longs;&longs;it <lb/>à potentia in aöre libero &longs;u&longs;tineri. </s> <s>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 quod intelligitur ex vi præcisè <lb/>gravitationis; quicquid inferat di&longs;criminis partium conflictus. <pb pagenum="87"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT XIV.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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>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>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>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>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>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>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"/>dum, ut etiam in illud, à quo impediuntur, gravitent, il­<lb/>ludque urgeant. </s></p><p type="main"> <s>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>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>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 verticlai, quod & perpendiculare, ea e&longs;t men&longs;ura im­<lb/>pedimenti, quod à &longs;ubjecto plano infertur motui; atque <lb/><figure id="fig12"></figure><lb/>adeò gravitationis corporis in planum. </s></p><p type="main"> <s>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. </s> <s>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. </s> <s>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 <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>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"/>&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. </s></p><p type="main"> <s>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. </s> <s>50. ii&longs;dem <lb/>viribus, quibus in perpendiculari &longs;u&longs;penderentur lib. </s> <s>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>Ex his fieri pote&longs;t &longs;atis quæ­<lb/><figure id="fig13"></figure><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. </s> <s>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>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/>cò augetur magis conatus potentiæ in R detinentis columnam, <lb/>ne juxta plani inclinationem de&longs;cendat. </s> <s>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"/>laboris. </s> <s>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>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>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>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>Sit vectis, aut <lb/><figure id="fig14"></figure><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>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. </s></p><p type="main"> <s>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>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. </s> <s><lb/>Neque hic liceat ad æqualitatem potentiarum confugere, ut <lb/>&longs;icut per 47. lib. </s> <s>1. Eucl. </s> <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"/>Secans anguli inclinationis exprimat, gravitationem in plano <lb/>inclinato Radius, Tangens verò gravitationem in planum in­<lb/>clinatum. </s> <s>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"> <s>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> <s><lb/>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. </s> <s>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>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. </s></p><p type="main"> <s>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>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. </s> <s>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"/>FDC ad Radium; qui enim &longs;u&longs;tinet grave, dum de&longs;cendit in­<lb/>clinatum, habet rationem plani inclinati. </s> <s>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>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="fig15"></figure><lb/>jectum funem. </s> <s>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>Sic po&longs;itâ BD ped. </s> <s>150, & DC ped. </s> <s>200, BC e&longs;t <lb/>ped. </s> <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"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT XV.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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>Pendeat ex clavo C ad perpen­<lb/><figure id="fig16"></figure><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>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>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>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"/>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>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>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>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>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>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 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"/>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>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="fig17"></figure><lb/>riùs. </s> <s>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>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>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. </s> <s>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>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. </s> <s>13. hujus lib. </s> <s><lb/>demon&longs;travimus. </s></p><p type="main"> <s>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"/>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. </s> <s>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. </s> <s>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. </s></p><p type="main"> <s>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>Sit enim jam non in­<lb/><figure id="fig18"></figure><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>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>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>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. </s></p><p type="main"> <s>Ex quo apertiùs liquet, quàm ut in eo explicando diutiùs <pb pagenum="97"/>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>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> <s><lb/>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>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. </s> <s>5. e&longs;t nimirum KB major, <lb/>& GF minor. </s></p><p type="main"> <s>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 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. </s></p><p type="main"> <s>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>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. </s> <s>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. </s> <s>Plus igitur momenti ad gravitandum habet glo-<pb pagenum="98"/>bus F, &longs;i ex breviore funiculo MF pendeat, quàm &longs;i ex <lb/>longiore AF. </s></p><p type="main"> <s>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>Sit corpus grave ubi A, quod at­<lb/><figure id="fig19"></figure><lb/>tollere oporteat, & in &longs;uperiorem <lb/>locum RS transferre. </s> <s>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. </s> <s>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. </s> <s>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. </s> <s>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>Porrò angulus DAS <lb/>major e&longs;t angulo CAR; ergo & reliquus DSA major reliquo <lb/>CRA. </s> <s>Cum itaque tres anguli utriu&longs;que trianguli &longs;int æquales <lb/>duobus Rectis per 32. lib. </s> <s>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> <s><lb/>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"/>quàm cùm ex C, pendebit; quia declinationis angulus XCR <lb/>major e&longs;t angulo XDR per 16. lib. </s> <s>1. Verùm qua Ratione, in­<lb/>quis, vires, quas in utroque ca&longs;u retinens exerit, di&longs;criminan­<lb/>tur? </s> <s>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 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>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>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> <s><lb/>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. </s></p><p type="main"> <s>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"/>dam rationem daram 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. </s></p><p type="main"> <s>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>Sit enim globus A tùm <lb/><figure id="fig20"></figure><lb/>ex B, tùm ex C &longs;u&longs;pen&longs;us fu­<lb/>niculis BA & CA. </s> <s>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>Verùm ni&longs;i mea <lb/>mihi nimiùm blanditur opi­<lb/>nio, ex dictis facilis videtur <lb/>explicatio. </s> <s>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>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. </s> <s>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>Sunt igitur duæ Ra­<lb/>tiones DA ad AB, & EA ad AC. </s></p><p type="main"> <s>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. </s> <s>6. ut EA ad AC, ita DA ad <pb pagenum="101"/>AF. </s> <s>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>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>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. </s> <s>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>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>Si enim angulus declinatio­<lb/>nis DBA &longs;it gr. </s> <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. </s> <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"/>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>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>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. </s> <s>Sunt igitur duæ Rationes cundem 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>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>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>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"/>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>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 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>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>Si enim mobile &longs;it <lb/>ubi A, impetum verò cum tali <lb/><figure id="fig21"></figure><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>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"/>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. </s> <s>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>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. </s> <s><lb/>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>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. </s></p><p type="main"> <s>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. </s> <s>Porrò ex hypothe&longs;i &longs;unt angulus DBA gr. </s> <s>22. <lb/>12′, & angulus ECA gr. </s> <s>54. 35′: jungantur &longs;imul, & eorum <lb/>&longs;umma gr. </s> <s>76. 47′ auferatur ex gr. </s> <s>180, ut re&longs;iduum gr. </s> <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. </s> <s>76. 47′ quæ e&longs;t &longs;umma angulorum decli­<lb/>nationis. </s> <s>Sunt igitur in triangulo AGN nota latera AG, <lb/>GN (e&longs;t enim ex 34. lib. </s> <s>1. GN oppo&longs;ito lateri AR æquale) <pb pagenum="105"/>unâ cum angulo G comprehen&longs;o, & ex Trigonometriâ inno­<lb/>te&longs;cit tertium latus AN. </s> <s>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. </s> <s>51. 36 1/2 Tangens 126205 ad 46249 Tangentem gr. </s> <s>24. 49′ 2/5 <lb/>differentiæ infra, vel &longs;upra eandem &longs;emi&longs;ummam. </s> <s>E&longs;t igitur <lb/>angulus GAN gr. </s> <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. </s> <s>26. 47′ (3/10) Sinus 45070 ad anguli G <lb/>gr. </s> <s>76. 47′ Sinum 97351, ita latus GN 37784 ad latus AN <lb/>81613. </s></p><p type="main"> <s>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>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> <s><lb/>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>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>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>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"/><figure id="fig22"></figure><lb/>clinationis funiculorum, de­<lb/>mantur ex Radio. </s> <s>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. </s> <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. </s> <s>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>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>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. </s> <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. </s> <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′. </s> <s>Fiat demùm ut anguli <lb/>FAI gr.15.10′. </s> <s>Sinus 26163 ad anguli AFI gr. </s> <s>103. 13′. </s> <s>hoc e&longs;t <lb/>ad &longs;upplementi gr.76.47′. </s> <s>Sinum 97351, ita latus FI 18504 <lb/>ad ba&longs;im AI 68852. </s></p><p type="main"> <s>Inventa itaque momenta compo&longs;ita tùm in planis inclinatis, <lb/>tùm in plana inclinata, dividantur juxta Rationem momento-<pb pagenum="107"/>rum &longs;implicium, ut innote&longs;cat, quid demum cuique fi<gap/><lb/>tribuendum &longs;it in pondere retinendo. </s> <s>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 corum 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 corum 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>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>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>Orbiculo circà &longs;uum axem C ver&longs;atili, <pb pagenum="108"/><figure id="fig23"></figure><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>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. </s> <s>Jam verò &longs;it funiculus AFD, <lb/>qui in D religetur, & ponderi A &longs;umatur æquale pondus E, <lb/>vel potiù; ip&longs;um B transferatur in E, & funiculo AFD ad­<lb/>nectatur in H; ut &longs;int qua&longs;i duo funiculi DH, FH. </s> <s>Quæro <lb/>quantum ad deicendendum 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 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. </s> <s>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>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"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT XVI.<emph.end type="center"/></s></p><p type="main"> <s><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>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>Et primùm quidem &longs;uper plano <lb/><figure id="fig24"></figure><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. </s> <s>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>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. </s> <s>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> <s><lb/>Quare cum pondus B obliquè trahitur &longs;uper planum inclina-<pb pagenum="110"/>tum, minorem &longs;ubit violentiam, quàm cum ab illo perpendi­<lb/>culari elevatione avellitur. </s></p><p type="main"> <s>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. </s> <s>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>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. </s> <s>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>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. </s></p><p type="main"> <s>Ex his manife&longs;tum e&longs;t plus virium requiri ad trahendum <pb pagenum="111"/>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>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>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>Fiat igitur ip&longs;i BPO æqualis angulus <lb/>BFK, ac proinde K cadit inter puncta I & G. </s> <s>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. </s> <s>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. </s> <s>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>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>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"/>BR minorem e&longs;&longs;e quàm BO. </s> <s>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="fig25"></figure><lb/>lud adversùs planum trahat. </s> <s><lb/>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. </s> <s>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. </s> <s>Dico CH motum <lb/>globi majorem e&longs;&longs;e &longs;upra CG <lb/>motum potentiæ trahentis. </s> <s><lb/>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. </s> <s>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>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>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>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"/>&longs;im I&longs;o&longs;celis MDL. </s> <s>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 extermus 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>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>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>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">internno</expan> HCG; <lb/>igitur reliquus angulus LMH vel e&longs;t æqualis angulo GHC, <lb/>vel illo minor, vel illo major. </s> <s>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>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>Cum autem angu-<pb pagenum="114"/>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>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>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>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>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>Similiter in navium tractione, quæ <pb pagenum="115"/>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 confideraverimus, 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>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>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>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. </s> <s>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="fig26"></figure><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. </s> <s>Con&longs;tat trian-<pb pagenum="116"/>gulum DGC fimile 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;tractione 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. </s> <s>6. ut <lb/>BA ad AS, ita DG ad GC. </s> <s>Quoniam itaque, &longs;i pondus in <lb/>plano inclinato ad pondus in perpendiculari &longs;it ut inclinata BA <lb/>ad perpendicularem AS, corum 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. </s> <s>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>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>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. </s> <s>1. qui utique CID minor e&longs;t externo CGD per 16. <lb/>lib. </s> <s>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> <s><lb/>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>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>Atqui pondera &longs;uper BA & AP, quæ &longs;int <pb pagenum="117"/>ut BA ad AP, æquiponderant ex dictis cap. </s> <s>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. </s></p><p type="main"> <s>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>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>Hic autem tria ob&longs;ervanda occurrunt. </s> <s>Primum e&longs;t, quòd <lb/>Rationes prædictæ momentorum potentiæ retinentis compara­<lb/>tæ ad pondus idem, quamvis pro diversâ obliquitate aliis atque <lb/>aliiis 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>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>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>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>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"/><figure id="fig27"></figure><lb/>CF perpendicularis horizonti; & Rati<gap/><lb/>ponderis ad vires retinentes erunt ut CF <lb/>ad FD. </s> <s>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. </s> <s>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. </s></p><p type="main"> <s>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>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> <s><lb/>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"/>intere&longs;t, quanto conatu &longs;ubjectam tellurem urgeat modò certi <lb/>&longs;imus de fulcri ip&longs;ius firmitate. </s> <s>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>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>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>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>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"/><figure></figure><p type="main"> <s>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>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. </s> <s>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>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. </s> <s><lb/>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"/>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>Ne quid verò dubitationis <lb/><figure id="fig28"></figure><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 can­<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. </s> <s>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>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. </s> <s>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>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/><gap/>i&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 cidem ponderi G. </s> <s>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"/><figure></figure><p type="main"> <s>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. </s> <s>Ducatur per B perpendicularis ad ho­<lb/>rizontem FH, & ex H ad BE perpendicularis HO. </s> <s>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. </s> <s>Vires ergo retinentes in D ad vires retinentes in F &longs;unt <lb/>ut DB ad BF. </s></p><p type="main"> <s>Retineatur pri&longs;ma &longs;ecundùm obliquam GB, quæ producta <lb/>u&longs;que ad Horizontalem concurrat in L. </s> <s>Iterum ex L ad DE <lb/>cadat ad angulos rectos LC, quæ perpendicularem FH &longs;ecabiz <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. </s> <s>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. </s></p><p type="main"> <s>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/>d cularem 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"/><p type="main"> <s>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. </s> <s>Hinc &longs;i inclinatio­<lb/>nis angulus &longs;it gr. </s> <s>36. DB ad BF erit ut Radius ad Secantem <lb/>gr. </s> <s>36. vel ut Sinus gr.54. complementi gr.36. ad Radium. </s> <s>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. </s> <s>29. ad Sinum <lb/>gr.126, hoc e&longs;t &longs;upplementi gr.54. </s></p><p type="main"> <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. </s></p><p type="main"> <s>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"/>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. </s> <s>Ex quo <gap/>riàs <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;erimen, quod trahentis linea initio applicata, <lb/>ut angulum faciat acutum cum axe pri&longs;matis, in ipsâ t<gap/>ione <lb/>&longs;emper majorem facit cum ip&longs;o axe angulum, donee venrat 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>Quare in primâ <lb/>illâ tractione minuitur conatus, in hac &longs;ecunda augetur. <lb/><figure id="fig29"></figure></s></p><pb pagenum="125"/><figure></figure><p type="main"> <s><emph type="center"/>MECHANICORUM<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/>LIBER SECUNDUS.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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, anlapidea. </s> <s>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>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>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 qui<gap/><lb/>quam in reliquo opere &longs;uper&longs;it difficultatis. <pb pagenum="126"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT I.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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> <s><lb/>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 Poöta apud Ari&longs;totelem in quæ&longs;t.Mechan. <lb/></s> <s>&longs;ub initium, <foreign lang="greek">te/xnh| kratou=men, w)_n fu/s<gap/> nikw/meqa. </foreign></s> <s>Sed hic plani&longs;­<lb/>&longs;imè philo&longs;ophandi locus e&longs;t, non gloriandi in&longs;olentiùs. </s> <s>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>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>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>Quare &longs;icut laberetur turpiter, qui machinam &longs;axo ab <lb/>imo mariad &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"/>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 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. </s> <s>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>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>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 aöre 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 fiuctuante &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> <s><lb/>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>Quoniam verò obtineri nequit, ut corporum &longs;e contingen­<lb/>tium &longs;uperficies &longs;int continuo lævore lubricæ, earum autem <pb pagenum="128"/>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>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>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>Hinc Am&longs;telodami 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>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;iuli&longs;&longs;e, & loco Pa­<lb/>langum, per quos ducerentur, tergoribus animalium recens cæ&longs;orum <lb/>olco inunctis u&longs;um,<emph.end type="italics"/> Et Silius Ital. </s> <s>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/>Æquorcam rota ducebat per gramina puppim.<emph.end type="italics"/></s></p><p type="main"> <s>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>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"/>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/>lusaxem &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>Huc &longs;pectat id, quod non &longs;ine voluptate ob&longs;ervare aliouan­<lb/>do contigit Bononiæ. </s> <s>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>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>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-n<gap/> &longs;it? </s> <s><lb/>an erectum? </s> <s>læve, an a&longs;perum? </s> <s>amplâ, an tenui &longs;uperficie <lb/>contingat? </s> <s>hinc &longs;i quidem varia re&longs;i&longs;tentiæ momenta exur­<lb/>gunt. </s> <s>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>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>Satis igitur fuerit <pb pagenum="130"/>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>Hîc autem fortè nec à præ&longs;enti in&longs;tituto alienum, nec lect<gap/>­<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, ennarrare omninò non <lb/>poterat. </s> <s>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="fig30"></figure><lb/>tuo fulcri, & axis tritu. </s> <s>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>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>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"/>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>Verum non e&longs;t nece&longs;&longs;e rotulis B & C tàm &longs;olidos axes dare; <lb/>nam &longs;iaxis 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>Cum verò cylindro­<lb/>rum re&longs;i&longs;tentiæ, ne frangantur, &longs;int in triplicatâ Raticne &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>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>Quâ <lb/>autem Ratione minor &longs;it con&longs;tat. </s> <s>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. </s> <s>1. etiam AD <lb/>innote&longs;cit, cujus &longs;cilicet quadratum habetur, &longs;i ex BA quadra­<lb/>to dematur quadraturm BD. </s></p><p type="main"> <s>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> <s><lb/>dix 2. 23″. </s> <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>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″. </s> <s>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>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>Sunt igitur quatuor numeri 600. 431 1/<gap/>. <pb pagenum="132"/>31<gap/>. </s> <s>223 continuè proportionales proximè, &longs;pretis fractiuncu­<lb/>lis. </s> <s>Quare &longs;i &longs;iat ut 60<gap/>′ 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>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>Atqui octans peripheriæ ex Radio BE æqua­<lb/>lis e&longs;t quadranti ex Radio. </s> <s>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/>affiictus cum &longs;uo axe; ac proinde &longs;ubjectis rotulis, quarum dia­<lb/>meter &longs;it tantum dupla diametri axis. </s> <s>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>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>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>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> <s><lb/>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>Quamquam negandum non &longs;it hæc pari-<pb pagenum="133"/>ter ad mechanicam contemplationem pertmere; quippe quæ <lb/>machinis, præcipuo nimirum mechanices &longs;copo. </s> <s>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>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/>cos 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>Hinc &longs;i quis <lb/><figure id="fig31"></figure><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 pregravata 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>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"/>modò contento, modò remi&longs;&longs;o conatu opus urgere. </s> <s>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>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>Cæterùm &longs;i totius motûs, qui Ar&longs;i pariter con&longs;tat ac <lb/>The&longs;i, habeatur ratio, inficiari nemo poterit, minus multo la­<lb/>boris impendi, quàm &longs;i hæc omnia &longs;ublata intelligantur. </s> <s>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>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>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"/><p type="main"> <s>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>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>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> <s><lb/>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>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>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"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT II.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Impetùs motum proximè e&longs;&longs;icientis natura <lb/>explicatur.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>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>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>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>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>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"/>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>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>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;<gap/>t 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>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 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>At in cæteris, quibus &longs;e movendi <lb/>principium ine&longs;t, neme jure negaverit aut in motu impetum <lb/>acquiti, 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>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>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>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"/>buitur, quia id illa gignit, quod proximè motus con&longs;equitur, <lb/>& ad motum efficiendum natura de&longs;tinavit. </s> <s>Quid? </s> <s>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>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>Non igitur impetum motus, <lb/>&longs;ed motum impetus efficit. </s></p><p type="main"> <s>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>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>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>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>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>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"/>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>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>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>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 quod in <lb/><figure id="fig32"></figure><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 afcen&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 oporter. </s> <s>At <pb pagenum="140"/>&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>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 aqu<gap/> 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>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>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>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>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>Si quis tamen animum diligentiùs adverrat, manife&longs;tò de-<pb pagenum="141"/>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>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> <s><lb/>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>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>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>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>utique, &longs;i cætera omnia paria &longs;int, triplo <lb/>majorem conatum adhibendum concedes, inten&longs;ione exten­<lb/>&longs;ionom 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>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>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"/><p type="main"> <s>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>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>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>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>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>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>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"/>ex quibus re&longs;i&longs;tentia componitur, re&longs;i&longs;tentia ip&longs;a innote&longs;cere <lb/>poterit. </s> <s>Quare & impetus à facultate movendi principium ha­<lb/>bente productus major &longs;it nece&longs;&longs;e e&longs;t, quàm dimoti corperis <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 confiigerent vires, à neutrâ parte &longs;taret victoria. </s></p><p type="main"> <s>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>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>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>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>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/>ctiam tandem aliquando corrupto particularum nexu disjicit. </s></p><pb pagenum="144"/><p type="main"> <s>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>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>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>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 ficri, 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>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>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>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>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"/>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 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>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>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>Motus autem non pro impetûs tantum, &longs;ed & pro re­<lb/>&longs;i&longs;tentiæ modo con&longs;equitur. <lb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT III.<emph.end type="center"/></s></p><p type="main"> <s><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>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"/><p type="main"> <s>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>Sed neque temporis mo­<lb/>mento circum&longs;cribi impetum &longs;ivè in naturali motu acqui&longs;itum, <lb/>&longs;ive in violento imprelium, plura &longs;unt, quæ palam faciunt: ut <lb/>enim reliqua &longs;ileam nullæ e&longs;&longs;ent funependulorum ofcillatio­<lb/>nes, nullus emi&longs;&longs;æ &longs;agittæ motus, &longs;i conceptus impetus illicò <lb/>periret. </s></p><p type="main"> <s>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 conctur 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 cá fa­<lb/>cultate &longs;it præditum, quæ in contrariam partem conniti valeat <lb/>illicò, ac vis inferri cæperit. </s> <s>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 aöre circumfuia motum expetit, quo <lb/>&longs;uum repetat locum repugnans &longs;u&longs;tinenti. </s> <s>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>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"/>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>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<gap/>one po&longs;­<lb/>&longs;et augeri. </s></p><p type="main"> <s>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>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>Periit ergo innatus impetus, quo aqua &longs;itulam <lb/>replens de&longs;cen&longs;um moliebatur. </s></p><p type="main"> <s>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;eat. </s> <s>Quæ <lb/>&longs;anè immer&longs;io, ni&longs;i Acqui&longs;itus impetus adhuc duraret, omninò <lb/>non contingeret. </s> <s>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"/>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>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>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>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>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>Præ&longs;tat autem hîc funependuli <lb/><figure id="fig33"></figure><lb/>motum paulò attentiùs con&longs;iderare. </s> <s><lb/>Sit plumbeus globulus B filo AB <lb/>connexus clavo in A. </s> <s>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. </s> <s>1. cap. </s> <s>15 explicatum e&longs;t; hæc autem <lb/>&longs;unt pro Ratione Sinuum angulorum declinationis à perpendi­<lb/>culo AK. </s> <s>Quare totum momentum, quod in B e&longs;&longs;et ut AB, <lb/>fingulis 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"/>e&longs;t ut HI, &longs;ed ut HG. </s> <s>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> <s><lb/>Q<gap/>ropter 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>Ex quo fit quod, <lb/>cùm arcus BK ad Radium AB, hoc e&longs;t ad BC æqualc<gap/><lb/>proximè ut 11 ad 7, ex Cyclometricis, multò plus t<gap/><lb/>percurrendo arcu BK, quàm in rectâ BC, in&longs;umit<gap/><lb/>&longs;cilicet movetur quàm in perpendiculari, quæ ad BC<gap/> ut <lb/>11 ad 7. manente itaque, quamdiu corpus naturá urg<gap/><lb/>vetur, impetu acqui&longs;ito, qui re&longs;i&longs;tentiam exced<gap/><lb/>de&longs;censûs in K totus impetus e&longs;t ut aggregatum om<gap/><lb/>nuum Quadrantis: at in perpendiculari BC in fin<gap/><lb/>in C e&longs;&longs;et ut aggregatum omnium parallelarum ip&longs;i AB<gap/><lb/>Quadrato AC; ac propterea (in re Phy&longs;icâ &longs;i liceat <gap/><lb/>metrizantibus per Indivi&longs;ibilia ratiocinari) erit impetus pe<gap/><lb/>cum BK acqui&longs;itus ad impetum per rectam BC acqui&longs;i<gap/><lb/>Quadrans ABK ad Quadratum AC, hoc e&longs;t ut 11 ad <gap/><lb/>iis quæ in Cyclometriâ demon&longs;trantur. </s></p><p type="main"> <s>Quoniam verò ubi ad perpendiculum AK globulus de&longs;cen­<lb/>dens venerit, nihil objicitur, quod motum pror 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>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>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>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"/>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 glo<gap/>us <lb/>iterum de&longs;cendat. </s> <s>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>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>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;eit, &longs;ed <lb/>pergit moveri, aut deor&longs;um comprimendo corpus molle, aut <lb/>illicò &longs;ursùm reflexum à duro. </s> <s>Ita autem à corpore molli ex-<pb pagenum="151"/>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>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>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>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 aliqueties 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>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>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>&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;tinco. </s></p><p type="main"> <s>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 cli&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>Sic & <lb/>pilá in &longs;phæri&longs;terio ludentes &longs;atis nôrunt eam validiùs re&longs;lecti <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"/>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>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>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>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>Sic Atlas Sinicus pag. </s> <s>123. in XI. </s> <s><lb/>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 tamct&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"/> 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 pramittunt nautæ aliquot &longs;tra­<lb/>minis &longs;o&longs;ces, ad quos navis leviùs impingat, ac tran&longs;eat.<emph.end type="italics"/></s></p><p type="main"> <s>Jam verò ad impetum extrin&longs;ecùs impre&longs;&longs;um mentem ocu­<lb/><gap/>e intendente, non illum &longs;emper momento perire animad­<lb/><gap/> aut illicò, ac externus agitator ce&longs;&longs;at. </s></p><p type="main"> <s><gap/> nim tit, ut concitato navigio, cùm vela nautæ con­<lb/><gap/>ut remiges inhibuerunt, retineat tamen ip&longs;a navis <lb/><gap/>ur&longs;um &longs;uum, intermi&longs;&longs;o ventorum incur&longs;u, pulsúve <pb pagenum="153"/>remorum? </s> <s>ni&longs;i quia navis, etiam nullo impellente, vi impre&longs;sâ <lb/>urgetur. </s> <s>Quid rhedam cur&longs;u procedente faciliùs quàm initiò <lb/>promovet, equis licet languidius connitentibus? </s> <s>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>ni&longs;i quia reliquus ex priore motu impetus <lb/>adhuc per&longs;everans po&longs;teriorem motum juvat. </s> <s>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>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>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>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 ctiam pror­<lb/>sùs perire. </s></p><p type="main"> <s>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"/>id genus) illa e&longs;t externo impul&longs;ui peculiaris repugnantia, <lb/>quæ ex inhærente corpori gra vitate oritur, &longs;ive illi innatus im­<lb/>petus, &longs;ive acqui&longs;itus modum &longs;tatuat. </s> <s>Neque id &longs;impiiciter <lb/>tantùm, &longs;ed comparatè con&longs;iderandum e&longs;t, quam &longs;cilicet in <lb/>plagam impul&longs;us motum dirigat, & quatenu gravitatis pro­<lb/>pen&longs;ioni opponatur. </s> <s>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> <s><lb/>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>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 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 quasal­<lb/>li&longs;us paululùm inclinetur, aut aliò reflectatur. </s></p><p type="main"> <s>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"/>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>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>Con&longs;tat enim, &longs;i æquabili & æquali cona­<lb/>ru 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>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>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>Hîc tamen advertendus animus e&longs;t, & ob&longs;ervare oporter <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> <s><lb/>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>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"/>tur. </s> <s>Quare &longs;i corpus cò deveniat, ut nullo pror&longs;us pacto agi­<lb/>tari queat, aut interiore motu cieri, quo momento impeditur <lb/>motus, ne &longs;it, co momento impetus perit, ce&longs;&longs;ante videlicet <lb/>causa effectiva al ejus con&longs;ervatione co ip&longs;o quod ce&longs;&longs;at finis, <lb/>propter quem impetus e&longs;t. </s> <s>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT IV.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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>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>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>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>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"/>trariam plagam directum efficit, cujus vi motum impedit, ac <lb/>proptereà Activè re&longs;i&longs;tit. </s> <s>Huic autem verbo, cum <emph type="italics"/>Re&longs;i, lere<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>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>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/>resomnes, 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>Quid igitur mirum, &longs;i di&longs;tractis viribus languidior <lb/>&longs;equatur metus? </s> <s>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>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>quis enim ambigat, an, &longs;i pares <lb/>illæ fuerint, nullus futurus &longs;it motus? </s> <s>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>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>At &longs;i corpus, cui motus debetur, <pb pagenum="158"/>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>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>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>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>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>Hæc tamen nune mente &longs;ecernamus, per­<lb/>inde atque &longs;i nihil officerent motui. </s></p><p type="main"> <s>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>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"/>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. </s> <s>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>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 e&longs;&longs;iciens 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>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>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>At quòd im-<pb pagenum="160"/>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>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>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>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>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>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>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>Quod &longs;i lapis ta­<lb/>bellæ non impo&longs;itus, &longs;ed &longs;uppo&longs;itus, arctè tamen connexus <pb pagenum="161"/>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>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>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>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>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 mereurio ut 2, gra­<lb/>vitat in aëre ut 11; igitur plumbeæ ma&longs;&longs;æ partes 11 levitantes <lb/>fingulæ 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>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"/><p type="main"> <s>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;irum corpus. </s> <s>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;euju&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>Fieri autem hane mutuam impetûs communicationem hinc <lb/>apparet, quòd &longs;i duo corpora, quorum virtus movendi ut AB <lb/><figure id="fig34"></figure><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. </s> <s>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>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>At &longs;i propter virium æqualitatem percur-<pb pagenum="163"/>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;eriberent latus Quadrati. </s></p><p type="main"> <s>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>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 anguh, po&longs;ito Radio AI. </s></p><p type="main"> <s>Firge 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. </s> <s>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. </s></p><p type="main"> <s>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>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>Primum verò, <pb pagenum="164"/>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. </s> <s>Ego autem libentiùs pro&longs;iteor 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>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>Hinc fit corpus habens di­<lb/>rectionem & velocitatem AC minorem velocitate AB promo­<lb/>veri ultrà punctum F in linea mixti motûs AD. </s></p><p type="main"> <s>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>contrarium &longs;anè vi­<lb/>detur experientia evincere. </s> <s>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>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>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>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"/>& alterum alterius vires extenuat impediens, ne juxtà propriæ <lb/>virtutis men&longs;uram motum concipiat. </s></p><p type="main"> <s>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>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>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>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>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"/>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>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>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;quialter, 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>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>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>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"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT V.<emph.end type="center"/></s></p><p type="main"> <s><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>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> <s><lb/>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>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>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;a&longs;tentanda laborem potiùs <lb/>creabit, quàm ut præ &longs;imili ligneo vecte motum hunc facilio­<lb/>rem reddat. </s> <s>Quare præter Mechanicæ facnltatis 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"/>tis additamento; &longs;ed aliam potentiam, quæ conjunctis cum illi <lb/>viribus agat, con&longs;ociando. </s></p><p type="main"> <s>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>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>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>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>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"/>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>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>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>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>gr. </s> <s>&longs;ub­<lb/>quintuplo ejus quo illa movetur quintuplo velociùs ac ip&longs;um <lb/>Pondus. </s> <s>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>Quinque igitur im­<lb/>petus huic æquales po&longs;&longs;unt totam re&longs;i&longs;tentiam &longs;uperare. </s> <s>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>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, cettum 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"/><figure id="fig35"></figure><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>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> <s><lb/>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>Ne tamen <lb/>motui globi D tribue men&longs;uram arcûs AB &longs;ed rectæ AB. </s></p><p type="main"> <s>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>Sit ex. </s> <s>gr. </s> <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"/>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>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>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>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. </s> <s>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>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>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"/>hujus quidem per vim &longs;ursùm; illius verò juxtà naturam deor­<lb/>&longs;um. </s> <s>Sit adhuc eadem Potentia; &longs;ed offeratur Onus, cujus <lb/>particulæ gravitatis &longs;int non jam 20; &longs;ed 50: Potentiæ virtuse&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; cò 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>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 cju&longs;dem Po­<lb/>tentiæ requirunt particulas 100 impetûs. </s> <s>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>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>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>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"/>tuti vim movendi, etiam&longs;i minorem, habenti pro ratâ portio­<lb/>ne re&longs;pondeat. </s> <s>Satis igitur erit, ubi &longs;ingularum machmarum <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 quiequid 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>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/>m<gap/>raculorum omnium e&longs;&longs;e prmcipium. </s> <s>Ea igitur quæ circà libram fiunt, <lb/>ad circulum referuntur, quæ verò circa ve<gap/>em, ad ip&longs;am libram; <lb/>alia autem ferè omnia, quæ circa mechanicas &longs;unt motiones, ad <lb/>vectem.<emph.end type="italics"/> 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>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>Cavum <lb/>&longs;i quidem à convexo, quæ recto interjecto di&longs;eriminantur, 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>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>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>Sed quid hæc in circulo potiùs &longs;u&longs;­<lb/>piciamus, quàm in Helice, cui gene&longs;is haud di&longs;par contingit? </s> <s><lb/>Quòd &longs;i circulo primas ideò deferendas exi&longs;timemus, quòd <pb pagenum="174"/>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>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 ettiam (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>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>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>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>Quare &longs;apienti&longs;&longs;imo con&longs;ilio factum, ut corum, <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>Quæ verò ad explicandum, cur ea, quæ centro propiora <lb/>&longs;unt, tardiùs in gyrum contorqueantur, Author illius libri <lb/>Quæ&longs;t. </s> <s>mechan. </s> <s>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. </s> <s>1. de Cælo. </s> <s>&longs;umma 4. circularem motum & &longs;implicem, & <pb pagenum="175"/>naturalem, & priorem recto di&longs;erti&longs;&longs;imè pronunciat; <emph type="italics"/>Perfectum <lb/>enim,<emph.end type="italics"/> inquit text. </s> <s>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"/> 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>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"/> 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/>unumquod que &longs;ui motus centrum con&longs;tituat in eâ finitimi o&longs;&longs;is <lb/>parte, cui &longs;ivè <foreign lang="greek">*kaq) e)na/r<gap/>rwsin</foreign>, &longs;ive <foreign lang="greek">kata/ dia)rqrwsin</foreign> flexili com­<lb/>page in&longs;eritur. </s> <s>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>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>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>An quia gravia &longs;uis nutibus ad <lb/>terræ centrum rectâ feruntur? </s> <s>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>Verùm mirifica &longs;it circuli natura; quid hæc ad explicandam <lb/>Mechanicarum motionum cau&longs;am? </s> <s>an ut hanc ignotam fatea­<lb/>mur, quia admirandam prædicamus? </s> <s>&longs;ed unico argumento, <lb/>commenta huju&longs;modi disjiciamus. </s> <s>Si minor potentia majori <pb pagenum="176"/>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>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></figure><p type="main"> <s>Sed age raptandus &longs;it in plano horizontali, aut inclinato, aut <lb/>etiam elevandus &longs;it ad perpendiculum cylindrus A. </s> <s>Experire <lb/>primùm quanto labore id præ&longs;tes illum trahens illigato fune <lb/>in C, & arreptâ extremitate funis B. </s> <s>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, atreptáque alterâ funis extremitate D iterum trahe cy­<lb/>lindrum, & quantò minori labore id perficias, tu te ip&longs;e doce­<lb/>bis. </s> <s>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>Facilitatis autem di&longs;erimen 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"/>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. </s></p><p type="main"> <s>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="fig36"></figure><lb/>bentes a&longs;peritatis infige. </s> <s><lb/>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>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. </s> <s>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 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. </s> <s><lb/>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"/>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>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 cò 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>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> <s><lb/>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>Hæc autem clariùs patebunt lib.4. cap.1. <lb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT VI.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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>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"/>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>Sit exempli gratiâ pondus A, quod &longs;it trochleâ attollendum <lb/>in D. </s> <s>Poterit id duplici ratione fieri; primùm raptando illud in <lb/>plano Horizontali ita, ut ex B <lb/><figure id="fig37"></figure><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>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>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>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. </s> <s>1. cap. </s> <s>15, ac <lb/>propterea etiam minùs dubitari pote&longs;t de trochleæ firmitate. </s></p><p type="main"> <s>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"/>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>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>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="fig38"></figure><lb/>trochlea, altera quidem in A imminens pon­<lb/>deri ad perpendiculum, altera verò in B. </s> <s><lb/>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>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. </s> <s>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>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>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"/>alioquin par non e&longs;&longs;et utriu&longs;que trahentis labor. </s> <s>Cæterùm <lb/>&longs;atius e&longs;t A & B parùm di&longs;tare. </s></p><p type="main"> <s>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>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. </s> <s>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. </s> <s>63 1/4 ferè <lb/>e&longs;t Perpendiculum AS. </s> <s>Igitur elevatio CS e&longs;t ped. </s> <s>6 3/4. </s></p><p type="main"> <s>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. </s> <s>63 1/4, & Ba&longs;is EV e&longs;t ped. </s> <s>30: <lb/>Quare inter &longs;ummam ped. </s> <s>93 1/4, ac differentiam ped. </s> <s>33 1/4 media <lb/>proportionalis ped. </s> <s>55. 67″. </s> <s>e&longs;t Perpendiculum BV; atque <lb/>adeò elevatio DV e&longs;t ped. </s> <s>7. 58″. </s> <s>major quàm CS. </s> <s>Et &longs;ic de <lb/>reliquis. </s></p><p type="main"> <s>At &longs;tatue di&longs;tantiam AB &longs;olùm ped. </s> <s>20: reperies perpendi­<lb/>culum AS vix excedere ped. </s> <s>67; quare elevatio CS crit ped. </s> <s>3 <lb/>ferè; ac propterea etiam Perpendiculum BV erit paulò majus <lb/>ped. </s> <s>63. 94″; & elevatio DV ped. </s> <s>3. 06″; & &longs;ic de cæteris. </s></p><p type="main"> <s>Potentiæ verò elevantis motum metitur differentia, quæ <lb/>inter lineas BC & BD intercedit: quando autem di&longs;tantia <lb/>AB e&longs;t ped. </s> <s>30, linea BC e&longs;t ped. </s> <s>76. 15″; at cum e&longs;t ped. </s> <s>20, <lb/>BC e&longs;t ped. </s> <s>72 4/5. Cum igitur in primo ca&longs;u BD &longs;it ped. </s> <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. </s> <s>67; linea autem BC &longs;it ped. </s> <s>72 4/5, motus potentiæ e&longs;t <lb/>ped. </s> <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. </s> <s>5. Majorem igi­<lb/>tur Rationem habet motus Potentiæ ad motum ponderis, quan-<pb pagenum="182"/>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>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="fig39"></figure><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>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>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>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 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>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>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>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"/>BC ita cubum &longs;u&longs;tentat, ut <lb/><figure id="fig40"></figure><lb/>etiam reliquo latere cubus <lb/>idem &longs;ubjecto plano incumbat. </s></p><p type="main"> <s>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>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 populcum: 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>an obliquè? </s> <s>&longs;ectio facienda &longs;it. </s> <s>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>In metallis <lb/>pariter eorum natura con&longs;ideranda e&longs;t, molli&longs;ne illa &longs;it, ac <lb/>flexibilis? </s> <s>an verò dura? </s> <s>ut eam, quam &longs;emel induit figu­<lb/>ram, con&longs;tanter retineat. </s> <s>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>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>Dubitare videlicet nemo pote&longs;t à <lb/>duplici partium dividendarum numero duplicem oriri re&longs;i&longs;ten­<lb/>tiam. </s> <s>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"/>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>Hinc &longs;i fuerint duo parallelepipeda quadrata A & B, quorum <lb/>latera &longs;int in Ratione quadruplâ, altitudines verò AC, & BD <lb/><figure id="fig41"></figure><lb/>æquales; con&longs;tat ex 32. <lb/>l. </s> <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. </s> <s><lb/>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 quod manife&longs;tum fiat, &longs;i virgam ex falicto <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>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>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"/>læ vi nexûs, quo unaquæque cum &longs;ibi proximè cohærentibus <lb/>particulis colligatur. </s> <s>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>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="fig42"></figure><lb/>que ita vici&longs;&longs;im con­<lb/>gruentes, ut earum ex­<lb/>tremitates con&longs;tituant <lb/>planum HI. </s> <s>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>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>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>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>Hinc <lb/>oritur major &longs;olidi, quàm laminarum, re&longs;i&longs;tentia, ne fran­<lb/>gatur. </s> <s>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"/>cere occœperit, periculum e&longs;t, ne malum &longs;erpat juxta vena­<lb/>rum aut fibrarum ductum. </s> <s>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>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>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 fint 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>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>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. </s></p><p type="main"> <s>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"/> 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>Ut &longs;i fuerint duo &longs;oli-<pb pagenum="187"/>da CD & EF æqualis longitu­<lb/><figure id="fig43"></figure><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>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. </s> <s>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>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 eruntin 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>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>Nam &longs;i lon­<lb/>gitudines frangendorum corporum comparemus, quæ &longs;uâ va­<lb/>rietate mutant frangendi difficultatem, aut facilitatem, re-<pb pagenum="188"/>&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="fig44"></figure><lb/>manife&longs;tum e&longs;t. </s> <s>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. </s> <s>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. </s> <s>5. minor e&longs;t Ratio FG <lb/>ad CD, quàm &longs;it Ratio BE ad eandem CD. </s> <s>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;ccundùm Reciprocam Ra­<lb/>tionem longitudinum. </s></p><p type="main"> <s>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;tititivam, 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"/>quâ particulæ &longs;olidi invicem connexæ repugnant, ne di&longs;tra­<lb/>hantur, aut comprimantur. </s></p><p type="main"> <s>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>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>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>Quare ad tollenda <lb/>in &longs;uperiores ædificiorum partes ingentia &longs;axa multo cau­<lb/>tiùs atque tutiùs ij operantur, qui longam trabe<gap/>, 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"/>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>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>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>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>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/>tem 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 1<gap/>. </s></p><p type="main"> <s>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>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"/>magis ad B accedit; <lb/><figure id="fig45"></figure><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>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>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="fig46"></figure><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>Quæ omnia <lb/>ita ex dictis aperta &longs;unt, ut ulte­<lb/>riori explicatione non egeant. </s></p><p type="main"> <s>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"/>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>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>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>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 quod etiam <lb/>hinc con&longs;tat, quia cùm &longs;ingula illius longirudinis 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>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>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"/>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>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>Si enim fuerit <lb/>vectis AB in­<lb/><figure id="fig47"></figure><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>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"/><figure id="fig48"></figure><lb/>&longs;olùm pondus S, &longs;ed & Poten­<lb/>tiam, quæ e&longs;t in R. </s> <s>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>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>Potentia enim R major <lb/>e&longs;t pondere S. </s></p><p type="main"> <s>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> <s><lb/>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>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"/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT VII.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Præ&longs;tet-ne Machinam augere? </s> <s>an componere.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>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 Pondu 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. </s> <s>Sit enim ex. </s> <s>gr. </s> <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>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>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>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"/>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>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>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 Anteccdentis <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>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>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"/>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>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>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>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. </s> <s>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>aut Cochleam adhibere, cujus &longs;piras intervallum capilla­<lb/>ceum &longs;ecernat? </s></p><p type="main"> <s>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>an verò in compo&longs;itione? </s> <s>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 folum 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>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 Coehlea, Trochleis Throchleæ adjunguntur; aut diver­<lb/>&longs;i generis, ut cum facultates ip&longs;æ permi&longs;centur, vecti trochleas, <pb pagenum="198"/>Cochleæ vectem, Trochleis Cochleam, & deinceps, adjun­<lb/>gendo. </s> <s>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>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>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>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>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. </s> <s>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>Sic dato Vecte palmorum &longs;ex, quo potentia mo-<pb pagenum="199"/>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>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>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>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>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> <s><lb/>quinque perficimus, &longs;ervatâ &longs;emper eâdem Ponderis ab hypo­<lb/>mochlio di&longs;tantiâ, nimirum palmi unius. </s> <s>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"/>&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>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. </s> <s>Idem planè de cæteris dicendum Faculta­<lb/>tibus. </s></p><p type="main"> <s>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>In co 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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT VIII.<emph.end type="center"/></s></p><p type="main"> <s><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>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"/>multò faciliùs à potentia moveatur, quàm &longs;i ea illud &longs;u&longs;tinere <lb/>pariter atque transferre cogeretur. </s> <s>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 criretur; 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>Hinc Ci­<lb/>&longs;ia, Sarraca, Vehes, Carri & genus omne plau&longs;trorum. </s> <s>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>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>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"/>vehiculum onu&longs;tum, quàm minore, quia & longiore vecte fa­<lb/>ciliùs pondera moventur, quàm breviore. </s> <s>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>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>Igitur in Rotâ ratio Vectis <lb/>non invenitur, ideóque neque major Rota accipienda e&longs;t qua­<lb/>&longs;i longior Vectis. </s> <s>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>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/>joribu, 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>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. </s> <s>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>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;er, & axis depre&longs;&longs;ior: quò autem minor e&longs;t tractionis <pb pagenum="203"/>obliquitas, minorem quoque e&longs;&longs;e trahendi difficultatem loco <lb/>citato explicatum e&longs;t. </s></p><p type="main"> <s>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. </s> <s>5. prop. </s> <s>8.) idem majorem habet Rationem ad minus, <lb/>quàm ad majus. </s> <s>Nam &longs;i <lb/><figure id="fig49"></figure><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>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>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> <s><lb/>Porrò lineam HF majorem e&longs;&longs;e quàm IG &longs;ic o&longs;tenditur. </s> <s>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. </s> <s>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. </s> <s>1. &longs;unt autem triangula IGA & HFE <lb/>rectangula ad G & F ex con&longs;tructione; &longs;unt igitur &longs;imilia, & <pb pagenum="204"/>per 4. lib. </s> <s>6. ut. </s> <s>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>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. </s> <s>In triangulo igitur CEO angulus internus <lb/>E, per 16. lib. </s> <s>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>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>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. </s> <s><lb/>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. </s> <s>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"/>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>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>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>Quòd &longs;i ad calculos rem hanc curiosiùs revocare libeat, &longs;ic <lb/>ex gr. </s> <s>Rotæ minoris &longs;emidiameter CA pedum duorum, &longs;cilicet <lb/>digitorum 32, offendiculi verò DE <lb/><figure id="fig50"></figure><lb/>altitudo digitorum 4. Cum igitur <lb/>FD & CA parallelæ &longs;int, &longs;icut & <lb/>FC ac DA per 34. lib. </s> <s>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> <s><lb/>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. </s> <s>28. 57′ 18″. </s> <s>Jam verò po&longs;itâ &longs;emidiametro <lb/>CA digitorum 32, fiat ut 113 ad 355, ita data &longs;emidiameter <lb/>digit. </s> <s>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>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"/>pariter digit. </s> <s>4. Quare HI e&longs;t digit. </s> <s>44 Sinus anguli IBH, <lb/>ex quo innote&longs;cet arcus complementi HA. </s> <s>Fiat ut BH 48 ad <lb/>HI 44, &longs;eu ut 12 ad 11, ita Radius 100000 ad 91666 Sinum <lb/>arcûs gr. </s> <s>66. 26′. </s> <s>33″; & e&longs;t quæ&longs;itus arcus HA gr.23.33′.27″. </s> <s><lb/>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. </s></p><p type="main"> <s>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 <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>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 ipfum 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>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"/>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>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>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>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>Sic &longs;i Rotæ <lb/>fuerinc A & B, pla­<lb/><figure id="fig51"></figure><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. </s> <s>Si enim in utro­<lb/>que plano pondus fuerit &longs;imiliter po&longs;itum (puta circà me-<pb pagenum="208"/>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. </s> <s>14. <lb/>§. <emph type="italics"/>Ex his fieri pote&longs;t.<emph.end type="italics"/> 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>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>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT IX.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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. </s> <s>7. cap. </s> <s>37. cum Vi­<lb/>truvio lib.10. cap. </s> <s>6, quem tamen idem Plinius lib.36. cap.14. <pb pagenum="209"/>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/>a<gap/>ò 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>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>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>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>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>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"/><p type="main"> <s>Methagenis artificium imitatus Paconius, te&longs;te Vitruvio <lb/>lib. </s> <s>10. cap. </s> <s>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> <s><lb/>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>Tùm circà fu&longs;os fu­<lb/>nem involvit, qui bobus trahentibus explicabatur, & con­<lb/>vertebantur Rotæ. </s> <s>Verùm quia funis circumvoluti &longs;piræ ad <lb/>unam, aut ad alteram partem &longs;pectabant, non poterat <gap/><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>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>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>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>Neque aliud <lb/>quidpiam cen&longs;endus e&longs;t Cæ&longs;ar intellexi&longs;&longs;e, ubi lib. </s> <s>3. Belli <lb/>Civil. </s> <s>&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>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"/> Sunt autem &longs;cytalæ ut apud Sui­<lb/>dam, rotunda & polita ligna: aliquid tamen peculiare. </s> <s>ad­<lb/>dit Ari&longs;toteles in Mechan. </s> <s>quæ&longs;t. </s> <s>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>Scytalis nimirum pu-<pb pagenum="211"/>&longs;illas rotas adjectas intelligit, <lb/><figure id="fig52"></figure><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>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>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>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>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>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>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"/>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>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>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>Nam &longs;i fue­<lb/>rit ingens lapis AB cylin­<lb/><figure id="fig53"></figure><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> <s><lb/>modetur, citrà omnem &longs;axi & cylindri affrictum. </s></p><p type="main"> <s>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"/>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>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>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. </s> <s>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>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>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"/>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>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 quod, <lb/>per &longs;e quidem &longs;atis clarum, dilucidiùs explicabitur, &longs;i ex mi­<lb/><figure id="fig54"></figure><lb/>nore circulo extet particula, cu­<lb/>jus altitudo &longs;it ON, ex majore <lb/>autem circulo æqualis altitudo <lb/>emineat IM. </s> <s>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. E&longs;t igl­<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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT X.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Circulorum Concentricorum motus explicatur.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>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"/>cem &longs;uccedentes commeant, ut circulus totus &longs;patium, in quo <lb/>volvitur, omninò non mutet. </s> <s>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>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>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>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>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>Sit <lb/>explicandi gratiâ, <lb/><figure id="fig55"></figure><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. </s> <s>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"/>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. </s> <s>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. </s></p><p type="main"> <s>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>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>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>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. </s> <s>In po&longs;teriori verò Quadrante &longs;upe-<pb pagenum="217"/>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>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>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. </s></p><p type="main"> <s>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. </s> <s>quæ&longs;t. </s> <s>24. de circulorum concentricorum <lb/>motu, quando alter ad alterius motum promoto communi cen­<lb/>tro movetur. </s> <s>Sit <lb/><figure id="fig56"></figure><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>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"/>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>Major igitur Quadrans percurrit &longs;patium BF, & mi­<lb/>nor &longs;patium ST. </s> <s>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. </s> <s>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 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> <s><lb/>Quid igitur probabilius dicendum videatur, paucis expono. </s></p><p type="main"> <s>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>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>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"/><p type="main"> <s>Quod &longs;i minor circulus in&longs;i&longs;tat &longs;ubjecto &longs;ibi plano, <gap/>n­<lb/>que det motui centri; quia minor peripheria de&longs;ignat <gap/>n <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> <s><lb/>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. </s> <s>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>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>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>cit. <lb/><emph type="italics"/>Quando hic quidem movet, ille verò movetur ab i&longs;<gap/>o, quantum uti­<lb/>que moverit alter, tantum alter movebitur.<emph.end type="italics"/></s></p><p type="main"> <s>Cur igitur parem lineam rectam de&longs;ignat in plano utraque <lb/>orbita major & minor? </s> <s>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"/>ripheriæ de&longs;criptione, &longs;olum centri motum relinquunt. </s> <s>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>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>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>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>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>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"/>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>Res, clarita­<lb/><figure id="fig57"></figure><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. </s> <s>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. </s> <s>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>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>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"/>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. </s> <s>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>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>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>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>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>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>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>Si <pb pagenum="223"/>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>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>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>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>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"/>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>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>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>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="fig58"></figure></s></p><pb pagenum="225"/><figure></figure><p type="main"> <s><emph type="center"/>MECHANICORUM<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/>LIBER TERTIUS.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>De Libra.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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. </s> <s>Mathem. </s> <s>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> <s><lb/>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>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"/>momentis, ut infra in loco o&longs;tendetur. </s> <s>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>Nihil autem officit libræ dignitati, quod <lb/>Cain authorem agno&longs;cere videatur, qui, ut Jo&longs;ephus lib. </s> <s>1. <lb/><expan abbr="Antiq.">Antique</expan> Jud. </s> <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"/> Quid enim &longs;i quis præclaro artifi­<lb/>cio ex naturæ the&longs;auris deprompto abutatur? </s> <s>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>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. </s> <s>61. <emph type="italics"/>Mendaces filij hominum in &longs;tateris:<emph.end type="italics"/> ubi <lb/>S. </s> <s>Ba&longs;ilius hom. </s> <s>in P&longs;alm. </s> <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>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT I.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Libræ forma, & natura exponitur.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>In hoc <pb pagenum="227"/>in&longs;trumento con&longs;ideratur pri­<lb/><figure id="fig59"></figure><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;torelem in quæ&longs;t. </s> <s>Mechan. <emph type="italics"/>Spartum<emph.end type="italics"/> dicitur. </s> <s><lb/>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>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>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>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>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"/> 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>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>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"/>puncta jugi; ex quibus pondera dependent: horum etenim <lb/>di&longs;tantiam à centro omnino æqualem e&longs;&longs;e oportet. </s> <s>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>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>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>Finge enim pondus B de&longs;cendere in F, utique <lb/><figure id="fig60"></figure><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>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> <s><lb/>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>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"/>& 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>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. </s> <s>13. lib. </s> <s>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 &longs;i fiat facilè ab æquilibrio deflectet &longs;capus, &longs;i vel modi­<lb/>ca intercedat ponderum inæqualitas. </s> <s>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>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 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>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>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>Quod &longs;i axis non &longs;it an&longs;æ, &longs;ed &longs;capo, firmiter infixus, volua-<pb pagenum="230"/>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>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>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>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>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>&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>Subtilitas hæc relinquatur gemmariis, aurifici­<lb/>bus, auríque monetalis cu&longs;oribus, quibus damnum e&longs;&longs;et minu­<lb/>tias contemnere. </s> <s>Quamquam nec i&longs;tis author fuerim, ut &longs;in­<lb/><gap/>aribus granis uterentur, &longs;ed potiùs ponderibus, quæ plturi­<lb/><gap/>anis æquivalerent; &longs;i enim &longs;ingula grana à legitimo pon­<lb/>dere <gap/>iciunt per cente&longs;imam grani partem, quæ facilè &longs;ensûs <pb pagenum="231"/>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>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>Alterius Unciæ <lb/>octavam partem a&longs;&longs;umens habebis Draclimam. </s> <s>Drachmæ pars <lb/>tertia dabit &longs;crupulum. </s> <s>Scrupuli &longs;emi&longs;&longs;is e&longs;t obolus. </s> <s>Oboli <lb/>triens e&longs;t &longs;iliqua. </s> <s>Demùm &longs;iliquæ quadrans e&longs;t Granum. </s> <s>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 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>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>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>Sic dentur octo pon­<lb/>dera in Ratione duplâ incipiendo ab uncia 1; octavum e&longs;t <lb/>unc. </s> <s>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"/>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>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>TABELLE WAR HIER</s></p><pb pagenum="233"/><p type="table"> <s>TABELLE WAR HIER</s></p><p type="main"> <s>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>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>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"/>&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>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>TABELLE WAR HIER</s></p><p type="table"> <s>TABELLE WAR HIER</s></p><p type="main"> <s>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;t<gap/><pb pagenum="235"/>tur Galenus lib. </s> <s>6. cap. </s> <s>8. <emph type="italics"/>de compo&longs;itione medicam. </s> <s>per genera.<emph.end type="italics"/><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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT II.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Libra inæqualium brachiorum expenditur.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>Sit <lb/>libra AB, cujus centro C <lb/><figure id="fig61"></figure><lb/>dividatur jugum in brachia <lb/>inæqualia CA & CB. </s> <s><lb/>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>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. </s> <s>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"/>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>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>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>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. </s> <s>1. <lb/>&longs;unt æquiangula; igitur per 4 lib. </s> <s>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. </s> <s>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. </s></p><p type="main"> <s>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>Si <lb/>enim pondera e&longs;&longs;ent æqualia, ponderis A re&longs;i&longs;tentia ratione <pb pagenum="237"/>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>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 unaqu<gap/>que 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> <s><lb/>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>Quare <lb/>cum eadem &longs;it gravitatum Ratio ut CA ad CB, atque motuum <lb/>reciprocè ut FD ad GE, ex 16 lib. </s> <s>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. </s></p><p type="main"> <s>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>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"/>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>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>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>Sed quænam &longs;unt, & quanta utriu&longs;que brachij momenta? </s> <s><lb/>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>Sit igi­<lb/><figure id="fig62"></figure><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. </s> <s>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"/>fuum momentum juxtà di&longs;tantiam centri &longs;uæ gravitatis à <lb/>centro motûs. </s> <s>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>eadem quippe e&longs;t gravitas &longs;ingulis in­<lb/>&longs;ita, eadem &longs;ingularum à centro di&longs;tantia. </s> <s>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>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>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>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>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 quod adhuc ul­<lb/>teriùs &longs;ic explicari po&longs;&longs;e videtur. </s> <s>Sit libræ jugum M. N, & <lb/>motûs centrum O: intelligatur moveri, ut obtineat po&longs;itio-<pb pagenum="240"/>nem PR. </s> <s>Momentum brachij minoris OM referre videtur <lb/>&longs;ector MOP, momentum verò brachij majoris ON referre <lb/><figure id="fig63"></figure><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>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. </s></p><p type="main"> <s>At quæris. </s> <s>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. </s> <s>4, & longitudo SB e&longs;t quadrupla longitu­<lb/>dinis SA. </s></p><p type="main"> <s>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"/>unica libra partis C tantumdem habet momenti, quantum ha­<lb/>beret &longs;i totâ ex A penderet. </s></p><p type="main"> <s>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>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>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>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>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>Primùm igitur a&longs;&longs;umptus e&longs;t ligneus cylindrus, cujus dia­<lb/>meter CE unc. </s> <s>1. 06″ pedis Romani antiqui, & addito in A <pb pagenum="242"/>pondere D unciarum 40 1/2 collocatus e&longs;t in æquilibrio, quod <lb/>factum e&longs;t in B puncto. </s> <s>Fuit autem longitudo BA unciarum <lb/><figure id="fig64"></figure><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>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>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>Tùm &longs;ic <lb/>ratiocinatus &longs;um Algebricè; ut 136900 ad 4481689, ita mo­<lb/>mentum BA 1 ℞ ad 32.73″ ℞ momentum BC. </s> <s>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>40 1/2: & per An­<lb/>tithe&longs;im demptâ utrinque 1 ℞, æquatio e&longs;t inter 37. 73″ ℞ & <lb/>unc. </s> <s>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″. </s> <s>Quare perinde e&longs;t atque &longs;i gravitas <lb/>unc. </s> <s>1. 27″ poneretur in extremitate Alineæ Mathematicæ, ac <lb/>in extremitate C poneretur gravitas unc. </s> <s>41. 57″. </s> <s>At in A fuit <lb/>additum pondus unc. </s> <s>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>2 1/8, hoc e&longs;t in cente&longs;imis paulò ul­<lb/>tra 2. 12″. </s> <s>Si verò momentis brachij BA pondo unc. </s> <s>1.27″ ad­<lb/>datur gravitas D pondo unc. </s> <s>40. 50″, fit aggregatum 41.77″, <lb/>quod excedit inventum momentum brachij BC unc.41.57″. </s> <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>Quod &longs;i fiat ut longitudo BC 2117 ad <pb pagenum="243"/>longitudinem AB 370, ita pondus in A unc.41.77″ ad pon­<lb/>dus in B unc. </s> <s>7. 30″, con&longs;tat e&longs;&longs;e ferè &longs;emi&longs;&longs;em gravitatis <lb/>unc. </s> <s>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>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. </s> <s>1. unc.1 1/4. Alte­<lb/>ri extremitati additus e&longs;t <lb/><figure id="fig65"></figure><lb/>plumbeus cylindrus ad per­<lb/>pendiculum pendens, cujus <lb/>pondus unc. </s> <s>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>18 1/6, partes verò major fuit men&longs;urâ palm. </s> <s>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> <s><lb/>Quare CB pondo unc. </s> <s>1059, AC verò pondo unc. </s> <s>2. 66″. </s> <s><lb/>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>20.00″ & 1 ℞; & factâ Antithe&longs;i e&longs;t æquatio inter <lb/>unc. </s> <s>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>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>21. 343‴ planè <lb/>æqualis momento brachij BC. </s></p><p type="main"> <s>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>Cum igitur &longs;it CB pondo unc. </s> <s>10. 50″; & AC <lb/>pondo unc. </s> <s>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. </s> <s>Quoniam verò AC + D hoc e&longs;t (70756/1121481) ℞ <pb pagenum="244"/>+ unc. </s> <s>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>20.00″ & (10507<gap/>5/1121481) ℞, & factâ divi&longs;ione emer­<lb/>git pretium 1 ℞, hoc e&longs;t momentum BC pondo unc. </s> <s>21. 347‴. </s> <s><lb/>atque adeò momentum ip&longs;ius AC e&longs;t pondo unc. </s> <s>1. 347″; cui <lb/>&longs;i addatur cylindri D gravitas unc. </s> <s>20, totum momentum in A <lb/>e&longs;t unc. </s> <s>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 (<gap/>/1000). Hîc pariter brachij AC gravitas ab&longs;o­<lb/>luta pondo unc. </s> <s>2. 66″. </s> <s>habet momentum unc. </s> <s>1. 347‴, cum <lb/>ejus &longs;emi&longs;&longs;is &longs;it unc. </s> <s>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>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>&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>21. 347‴ ad pondus in B unc. </s> <s>5. 354‴: erat au­<lb/>tem brachij BC gravitas ab&longs;oluta unc. </s> <s>10. 59″ cujus, &longs;emi&longs;&longs;is <lb/>5. 295‴. </s> <s>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>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>Quare eundem plumbeum cy­<lb/>lindrum, cujus longitudo erat palmi 1. unc. </s> <s>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"/>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. </s> <s><lb/>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>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>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>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>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"/>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> <s><lb/>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT III.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Quomodò corporum æquilibria explicentur.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>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>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>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>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"/>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>Sit propo&longs;itum Trapezium, cujus gravita­<lb/>tis centrum C puncto re&longs;pondeat, & <lb/><figure id="fig66"></figure><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. </s> <s>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>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>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"/>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>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>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>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>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>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>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>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 <gap/><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"/>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>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>gr. </s> <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>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>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"/>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> <s><lb/>Simili ratione explicanda &longs;unt, quæ idem Atlas Sinicus in XI <lb/>Provincia Fokien habet pag. </s> <s>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>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>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>Plau&longs;trum adhibent cum <gap/><lb/>rotâ ita con&longs;titutum, ut uni illius medium oceupandi, & 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"/> 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 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"/>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>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>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>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>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>Sic Trapezij, NPMQ in­<lb/><figure id="fig67"></figure><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"/>& e&longs;t D punctum æquilibrij, &longs;eu centrum gravitatis Trapezij <lb/>quæ&longs;itum. </s> <s>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>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>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> <s><lb/>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>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>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"/>proptereà huic occurrendum e&longs;t artificio, quo &longs;itum eumdem <lb/>perpetuò &longs;ervet. </s> <s>Rem exemplo declaro. </s> <s>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> <s><lb/>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>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>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>Sit pyxis <lb/>ip&longs;a ABCD, in qua venti de&longs;­<lb/><figure id="fig68"></figure><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>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"/>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>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>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>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="fig69"></figure><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"/>majora habent momenta, quibus deorium urgentibus corpus <lb/>revolvitur. </s> <s>Id quod multò magis contingit in Acrobarycis, quæ <lb/>nimirum gravitatem in &longs;ummitate habent, ut &longs;i corpori BC <gap/><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>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>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>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>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>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>Quare &longs;i <pb pagenum="256"/>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="fig70"></figure><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> <s><lb/>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>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>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>Sit ex. </s> <s>gr. </s> <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>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. </s> <s>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"/>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>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>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> <s><lb/>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT IV.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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>Certè Keplerus in A&longs;tronomiâ Opticâ cap.1. <pb pagenum="258"/>prop. </s> <s>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"/> 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>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>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>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>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>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="fig71"></figure><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>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? <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"/>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>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>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>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="fig72"></figure><lb/>vitati jugi, ut facere con­<lb/>&longs;ueverunt accuratiores <lb/>monetarij. </s> <s>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. </s> <s>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. </s> <s>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. </s></p><p type="main"> <s>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"/>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> <s><lb/>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. </s> <s>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>Nicolaus Tartalea lib. </s> <s>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> <s><lb/>Id quod hâc methodo conatur o&longs;tendere. </s> <s>Si ex C æqualiter <lb/><figure id="fig73"></figure><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. </s> <s>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. </s> <s>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. </s> <s>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. </s></p><p type="main"> <s>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"/>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>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>Cujus rei <lb/>cau&longs;am inquirens &longs;tatuit libræ jugum <lb/><figure id="fig74"></figure><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>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. </s> <s>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. </s> <s>Igitur <lb/>ut pondus C ad pondus D, ita FG ad GB. </s></p><p type="main"> <s>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">habentla­<lb/>tera</expan> AG & AK æqualia: ergo & re&longs;idua FG, DR æqualia &longs;unt. </s> <s><lb/>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>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>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"/><emph type="italics"/>revolvuntur ad æquilibrium; cum cnim æque ponderent, æquales c<gap/><lb/>in circulo fieri de&longs;cen&longs;us par e&longs;t.<emph.end type="italics"/></s></p><p type="main"> <s>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>an propterea con&longs;equens e&longs;t fieri æquilibrium <lb/>etiam in &longs;itu CAD, ni&longs;i aliunde probetur? </s> <s>&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> <s><lb/>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>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? 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>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. </s> <s>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>At, inquis, quid cau&longs;æ e&longs;t, <pb pagenum="263"/>cur &longs;imilem libram in quácumque po&longs;itione quie&longs;centem non <lb/>habemus? </s> <s>&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;it<gap/> convenire, hoc e&longs;t cen­<lb/>trum motus & centrum gravitatis, nimirùm punctum illud, <lb/>quod brachiorum longitudinem di&longs;eriminat. </s> <s>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, librac&longs;­<lb/>&longs;et inæqualium brachiorum, & tunc non adnexorum ponderum <lb/>æqualitas e&longs;&longs;et con&longs;ideranda, &longs;ed corum 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>Hinc e&longs;t de illâ duplici tantummedo libræ &longs;pecie locutum <lb/>fui&longs;&longs;e Ari&longs;totelem in Mechan. </s> <s><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>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>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>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>Quare hujus lineæ pars intercepta inter cen­<lb/>trum motûs, & lineam longitudinis brachiorum, &longs;ivè exigua <pb pagenum="264"/>&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>Sit libra æqualium brachiorum <lb/><figure id="fig75"></figure><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. </s> <s>Deprimatur extremitas A, <lb/>ut veniat in E, reliqua extremitas <lb/>B a&longs;cendit in F, & C venit in G. </s></p><p type="main"> <s>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. </s> <s>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. </s> <s>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. </s> <s>6. ut FO ad OE, ita FH <lb/>ad EI. </s> <s>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. </s></p><p type="main"> <s>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/>crunt in Rationc 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"/>pancto &longs;u&longs;pen&longs;ionis demi&longs;&longs;a cadet in centrum gravitatis compo­<lb/>&longs;itæ libræ & ponderum. </s> <s>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>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>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="fig76"></figure><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>Depri­<lb/>matur L in P, & elevabitur M in <lb/>Q, & N in V. </s></p><p type="main"> <s>Dico libram &longs;ummoto deprimen­<lb/>te, ad æquilibrium LM redituram. </s> <s><lb/>Ducantur perpendiculares PT & QR, productâ LM horizon­<lb/>tali, &longs;i opus fuerit. </s> <s>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> <s><lb/>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>Majorem autem e&longs;&longs;e <lb/>Rationem &longs;ic o&longs;tenditur. </s> <s>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. </s> <s>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>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 enimal-<pb pagenum="266"/>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>Quæ omnia licet per&longs;picua <lb/>&longs;int, &longs;i &longs;uperiores duæ figuræ invertantur, clarioris tamen ex­<lb/><figure id="fig77"></figure><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> <s><lb/>Deprimatur A in E, elevatur pariter B in F, & C deprimitur <lb/>in G. </s> <s>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>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. </s> <s>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 corum 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>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>De&longs;cendet itaque E, & nullo occur­<lb/>rente obice ea fiet totius libræ revolutio circà centrum D, ut <pb pagenum="267"/>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 libiæ. </s> <s>Ea­<lb/>dem dicta intelligantur de librâ brachiorum inæqualium, quæ <lb/>&longs;upervacaneum e&longs;t iterum inculcare. </s></p><p type="main"> <s>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>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT V.<emph.end type="center"/></s></p><p type="main"> <s><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>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"/>corporum movendorum di&longs;po&longs;itio, exigunt collocari ponde­<lb/>la, ut & libræ Rationes &longs;erventur, & tamen jugi rectitudo nul­<lb/>la appareat; non erit hic inutile libram curvam examinare, ut, <lb/>&longs;i quando eâ uti contigerit, innote&longs;cat, quænam &longs;int brachio­<lb/><gap/>m, & motuum Rationes. </s> <s>Libram autem curvam voco, quæ <lb/>a commun<gap/> deflectens latera habet non in directum po&longs;i­<lb/><gap/>, &longs;ed in angulum concurrentia, aut in arcum &longs;inuata, quo­<lb/><gap/>m extremitates &longs;ivè &longs;ur&longs;um, &longs;ivè deor&longs;um re&longs;piciunt: factâ <lb/><gap/> &longs;u&longs;pen&longs;ione &longs;ive ubi angulum latera con&longs;tituunt, &longs;ivè in <lb/>aliquo arcus puncto, ea fieri pote&longs;t hinc & hinc ponderum ad­<lb/>ditio, quam horizontale æquilibrium con&longs;equatur. </s> <s>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>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="fig78"></figure><lb/>rum longitudinibus; &longs;ed ex ipsâ extre­<lb/>mitatum B & C di&longs;tantiâ BC; quæ &longs;em­<lb/>per cadem 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>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>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>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? &longs;ed attendenda e&longs;t æqualitas aut in­<lb/>æqualitas linearum ex D ductarum ad extremitates B & C. </s></p><p type="main"> <s>Neque me arguas, quòd dixerim jugum e&longs;&longs;e BC, & attenden-<pb pagenum="269"/>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<gap/>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. </s> <s>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>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. </s> <s>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>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, corumque latera (&longs;cilicet lineæ MB & <lb/>MC) inæqualia erunt pronuncianda. </s></p><p type="main"> <s>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>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>Si enim &longs;it libræ &longs;ca­<lb/><figure id="fig79"></figure><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"/>quia &longs;cilicet centrum motûs C e&longs;t &longs;upra lincam DE jungentem <lb/>puncta contactuum, quibus pondera adnectuntur. </s> <s>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>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>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>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>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/>æoualia; &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="fig80"></figure><lb/>ra e&longs;&longs;ent æqualia, contra hy­<lb/>pothe&longs;im. </s> <s>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"/>rum DB & DC longitudinem. </s> <s>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>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>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. </s> <s>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. </s> <s>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>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. </s></p><p type="main"> <s>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. </s> <s>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 cidem <lb/>ponderi B, cui in C æquivalet pondus majus. </s> <s>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. </s></p><p type="main"> <s>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"/>e&longs;t ip&longs;ajugi &longs;egmenta inveniamus, quippe quod &longs;olá mente <lb/>concipitur ad laterum extremitates jungedas deductum. </s> <s>Hæc <lb/>autem e&longs;&longs;e poterit praxis. </s> <s>Laterum AB & AC longitudine <lb/>metire, tùm ex B ad C extentum funiculum ad &longs;imilem men­<lb/>&longs;uram revoca. </s> <s>His paratis certum e&longs;t hane 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/>fe<gap/>i pars, de quâ mox dicetur; debet &longs;cilicet excedere me­<lb/>diam proportionalem inter aggregatum laterum, & corum dif­<lb/>ferentiam. </s> <s>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>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>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>Hujus operationis ratio manife&longs;ta <lb/>e&longs;t ex corollario primo prop. </s> <s>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. </s> <s>E&longs;t igitur brachiorum (atque adeò etiam ponde­<lb/>rum reciprocè) Ratio ut 424 ad 105. </s></p><p type="main"> <s>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>At &longs;i pro ratione tui in&longs;tituti nimia &longs;it hujus perpendiculi <pb pagenum="273"/>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>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/<gap/>9) 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>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 latcribus, 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>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 tatione 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"/>clinatur latus longius faciens cum perpendiculo majorem an­<lb/>gulum: pro va<gap/>a autem inclinatione ip&longs;am eju&longs;dem lateris gra­<lb/>vitatem varia obtinere momenta manife&longs;tum videtur. </s> <s>Pona­<lb/><figure id="fig81"></figure><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> <s><lb/>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>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>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. </s> <s>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. </s> <s><lb/>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, hoe <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. </s> <s>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>Si igitur in lineá AC perpendiculari lamina AC <pb pagenum="275"/>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. </s></p><p type="main"> <s>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>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 remoriores 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. </s> <s>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. </s> <s>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>Hoc pacto habetur totum <lb/>momentum lateris AL; &longs;imiliterque habebitur momentum la­<lb/>teris oppo&longs;iti. </s> <s>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>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"/>Sinu complementi eju&longs;dem anguli CAL, nimirum ex li­<lb/>neâ IL. </s></p><p type="main"> <s>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>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. </s> <s>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>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"/>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>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>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>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>Nunc inquiramus, quantum ponderis addendum &longs;it lateri <lb/>minori, ut fiat æquilibrium cum &longs;olâ majoris lateris gravitate. </s> <s><lb/>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>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>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>24.755‴; & in B intelli­<lb/>gitur gravitas unc. (6 3/23), hoc e&longs;t in mille&longs;imis unc. </s> <s>6.130‴ ferè. </s> <s><lb/>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"/>tet exiguum hoc di&longs;crimen oriri ex neglectis fractiun­<lb/>culis. </s></p><p type="main"> <s>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT VI.<emph.end type="center"/></s></p><p type="main"> <s><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>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>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>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>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>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"/>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>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>Sit enim li­<lb/><figure id="fig82"></figure><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. </s> <s>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>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>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>& non potiùs lingulam CO con&longs;iderat, <lb/>an cum ansâ perpendiculari illa conveniat? </s> <s>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. </s> <s>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>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"/>tum pote&longs;t, de&longs;cendit. </s> <s>Locutus igitur videtur Ari&longs;toteles de <lb/>librâ &longs;partum habente in &longs;uperiore jugi loco extrà lineam, quz <lb/>jugi longitudinem definit. </s></p><p type="main"> <s>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="fig83"></figure><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>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> <s><lb/>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> <s><lb/>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>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. </s> <s>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>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"/>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>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>Dico itaque &longs;par­<lb/>tum lineæ jugi proximum utilius e&longs;&longs;e, quàm remotum. </s> <s>Sit <lb/>enim libra AB bifariam in C di­<lb/><figure id="fig84"></figure><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. </s> <s>Igitur DSR <lb/>e&longs;t linea, quæ facta perpendicularis con&longs;tituit cum lingulâ SI <lb/>angulum ISR. </s> <s>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. </s> <s>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>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> <s><lb/>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"/>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>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 ma ore 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>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>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>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<gap/>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, vidctur 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>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"/>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>Gravitatem autem motui im­<lb/>pedimento e&longs;&longs;e ex eo con&longs;tat, <emph type="italics"/>quòd faciliùs quanào &longs;ine pondere <lb/>e&longs;t, <gap/>r libra, quàm cum pondus habet,<emph.end type="italics"/> ut ob&longs;ervavit <lb/>Ari&longs;toreles 9. 10. Mechan. </s> <s>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 e<gap/> 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 contia 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, vei &longs;altem hanc imminui non per­<lb/>mitteret. </s> <s>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>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>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 ang latus 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>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>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"/>couver&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>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>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 eæte­<lb/>róqui fuerint ritè con&longs;tructæ. </s></p><p type="main"> <s>At quærat hîc qui&longs;piam. </s> <s>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>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>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>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> <s><lb/>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"/>propterea alterutri vacuæ libræ lanci imponendam unciam, ut <lb/>primò di&longs;po&longs;ita &longs;it libra ad recedendum ab æquilibrio. </s> <s>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>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 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>gr. </s> <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>Jam imponatur alteri lanci merx, quæ cum pondere le­<lb/>gitimo lib. </s> <s>2. faciat æquilibrium: aio non po&longs;&longs;e pronuncia­<lb/>ri mercem e&longs;&longs;e unc. </s> <s>25: nam &longs;i ponatur merx unc. </s> <s>25: ad­<lb/>ditâ gravitate lancis & brachij unc. </s> <s>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. </s> <s>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>27. Ex quo liquet, quò <lb/>majora pondera lancibus imponuntur, eò majorem e&longs;&longs;e diffe­<lb/>rentiam à pondere legitimo. </s> <s>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"/>æquilibrium con&longs;tituitur, merx erit unicarum 14, quia ut 12 <lb/>ad 13, ita unc. </s> <s>2<gap/> 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>4; unica autem ponde­<lb/>ratio dabat tantum uncias 3: quia videlicet &longs;ingulis vicibus ad­<lb/>ditui id, qued 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>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>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>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>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>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>Sic Ari&longs;tote­<lb/>les quærit quæ&longs;t. </s> <s>27. Mechan. <emph type="italics"/>Cur &longs;i valde procerum fucrit idem <lb/>pondus, difficiliùs &longs;uper humeros ge&longs;iatur, 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;mcdi e&longs;&longs;et longior ha&longs;ta aut bractea <lb/>ferrea mediocris cra&longs;&longs;itiei. </s> <s>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"/>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>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>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>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>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>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT VII.<emph.end type="center"/></s></p><p type="main"> <s><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>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"/>æqualitatem, vel quia ponderum æqualitatem non indicat ve­<lb/>to æquilibrio. </s> <s>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>Et primò quidem &longs;e offert dolus, cujus meminit Ari&longs;toteles <lb/>quæ&longs;t.1.Mechan. </s> <s>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>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>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>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"/>brium appareat, centrum motûs inæqualiter dividit longitudi­<lb/>nem jugi. </s> <s>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>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>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>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>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>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>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. </s> <s>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>Sint brachia, quæ brevitatis gratia <pb pagenum="290"/>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 & innoteicit 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. </s> <s>Sit itaque ex. </s> <s>gr. </s> <s>in primo æquilibrio <lb/>primum pondus legitimum unc. </s> <s>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>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>Tertiò. </s> <s>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>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>Sit &longs;capus libræ AB, ex quo perpendicularis a&longs;&longs;urgat lingula <lb/><figure id="fig85"></figure><lb/>CD, & ex D per O centrum mo­<lb/>tûs ducta recta linea occurrat li­<lb/>neæ SV jangenti extrema puncta, <lb/>ex quibus lances pendent, eam­<lb/>quc bifariam dividat in I: &longs;ed quo­<lb/>niam punctum S e&longs;t paulò altiùs <pb pagenum="291"/>quàm punctum V, fiat angulus SIO minor, & VIO major. </s> <s><lb/>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>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>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>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>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>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. </s> <s>4. hujus lib. </s> <s>3. </s></p><p type="main"> <s>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>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>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>Rarò tamen continget jugum SV <lb/>planè æqualiter dividi à lineâ lingulæ ad angulos obliquos in-<pb pagenum="292"/>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>Quartò. </s> <s>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. </s></p><p type="main"> <s>Quintò. </s> <s>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>Manife&longs;tum <lb/>e&longs;t autem eandem jugi lineam non po&longs;&longs;e in duobus punctis <lb/>æqualiter dividi. </s> <s>Tripliciter pote&longs;t hoc fieri. </s> <s>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>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="fig86"></figure><lb/>C & D. </s> <s>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>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"/><p type="main"> <s>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. </s> <s>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>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>Ad digno&longs;cendum, an libra hoc vitio laboret, uti poteris hac <lb/>methodo. </s> <s>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"/>dem in primâ lancium commutatione man&longs;it æquilibrium; & <lb/>e&longs;t primus ca&longs;us. </s> <s>Quòd &longs;i facto æquilibrio, alterutri lancium <lb/>addas pondus, & æquilibrium maneat, adde quantum &longs;atise&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>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>an inæqualia? </s> <s>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>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>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>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>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>Sextò. </s> <s>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>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"/>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>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>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>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>Quòd &longs;i libra huju&longs;modi <lb/>funiculis inæqualibus in&longs;tructa &longs;partum haberet in loco &longs;upc­<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>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"/>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;itz <lb/>lancis funiculum extendit. </s></p><p type="main"> <s>Septimò. </s> <s>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>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>Octavò. </s> <s>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>Hinc &longs;i pondus præter opinionem gravius aut le-<pb pagenum="297"/>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>Hoc <lb/>tamen intellige, quando ex huju&longs;modi po&longs;itione &longs;equeretur in­<lb/>æqualis velocitas motuum oppo&longs;itorum ponderum. <lb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT VIII.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Stateræ natura & forma explicatur.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>& 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>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>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>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"/>proinde etiam idem æquipondium variam habet Rationem ad <lb/>merces inæquales. </s></p><p type="main"> <s>Sunt autem &longs;tateræ partes Jugum, An&longs;a, Uncus aut lanx, <lb/>Æquipondium, quod aliis Sacoma, aliis Cur&longs;orium dicitur. </s> <s><lb/>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>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>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 oporter. <lb/></s> <s>Æquipondium e&longs;t certæ gravitatis pondus, ex quo oppo&longs;itæ <lb/>gravitatis Ratio innote&longs;cit. </s></p><p type="main"> <s>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="fig87"></figure><lb/>tur cum unco aut lance E, & CB <lb/>brachium majus, cujus longitu­<lb/>dinem pro opportunitate percurrit <lb/>æquipondium F. </s> <s>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>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>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>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"/>&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>Sic di&longs;tan­<lb/>tia AC tran&longs;lata in brachium CB ex. </s> <s>gr. </s> <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. </s></p><p type="main"> <s>Hîc animad vertendum 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>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. </s> <s>5. manife&longs;tum e&longs;t igitur majorem e&longs;&longs;e Rationem BS <lb/>ad SA, quàm BC ad CA. </s> <s>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>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"/>&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>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>Cùm itaque ex dictis cap.2. hujus lib. </s> <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>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>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>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 adddito &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>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>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>Sit tigillus AB, in quo intelliga-<pb pagenum="301"/>tur ip&longs;i AC brachio minori æqualis pars CH; e&longs;t igitur bra­<lb/>chiorum differentia HB. </s> <s>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>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>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> <s><lb/>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/>titotius jugi. </s> <s>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/>fic deinceps. </s></p><p type="main"> <s>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>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>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"/>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>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>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="fig88"></figure><lb/>extremitate poneretur. </s> <s>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. </s> <s>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 &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>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"/>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>gr. </s> <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> <s><lb/>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. </s> <s>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. </s> <s><lb/>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. </s></p><p type="main"> <s>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>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>gr. </s> <s>pri­<lb/>ma libra. </s> <s>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"/>&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>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 amplim­<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>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>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>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> <s><lb/>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>Quod &longs;i adhuc pondera infrà libram unam, hoc e&longs;t infra un-<pb pagenum="305"/>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>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>hæc enim vul­<lb/>garia &longs;unt. </s> <s>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>Hîc verò curiosiùs inquirenti, præ&longs;tantiorne dicenda &longs;it &longs;ta­<lb/>tera? </s> <s>an libra? </s> <s>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>Compen­<lb/>dium habet &longs;tatera unico &longs;acomate contenta; pluribus ponderi­<lb/>bus eget libra. </s> <s>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>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>cum ta-<pb pagenum="306"/>men à minore brachio expoliendo manum ab&longs;tinuerit; quippe <lb/>qui &longs;atis norat 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>Siverò hoc acciderit, occultum manet &longs;tate­<lb/>ræ vitium, nec ip&longs;a &longs;e prodit. </s></p><p type="main"> <s>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/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>CAPUT IX.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Antiquorum Statera examinatur.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>Dubitationi locum fecit Ari&longs;toteles in quæ&longs;t. </s> <s>20. Mechan. </s> <s><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;modili­<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>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>Plura hæc &longs;parta, quorum Ari&longs;toteles meminit, Blancano in <lb/>locis Mathem. </s> <s>Ari&longs;t. </s> <s>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"/>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>Quoniam verò quæ tunc Blancano oppo&longs;ui, video <lb/>placui&longs;&longs;e Authori Magiæ Naturalis P. </s> <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>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>In extremitate A <lb/><figure id="fig89"></figure><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>gr. </s> <s>unius libræ, aut duarum, aut trium, prout magis trutinandæ <lb/>merci idoneum erit, & hoc erit proprium æquipondij pondus. </s> <s>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. </s> <s>&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. </s> <s>Similiter &longs;i fieret æquilibrium ex trutin â 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> <s><lb/>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>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 apprchenderentur, nunquam <pb pagenum="308"/>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>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>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? 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>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>Aut <lb/><figure id="fig90"></figure><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>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"/><p type="main"> <s>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>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> <s><lb/>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>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>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"/>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>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>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>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>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>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>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. </s> <s><lb/>lib.4, remanent lib. </s> <s>8: igitur ut AE 3 ad EB 1, ita lib. </s> <s>8 ad <lb/>lib. </s> <s>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>At &longs;i lanx <pb pagenum="311"/>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. </s></p><p type="main"> <s>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>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>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>&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>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&a