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<?xml version="1.0"?><!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" > <archimedes> <info> <author>Monte, Guidobaldo del</author><title>Mechanicorum Liber</title> <date>1577</date><place>Pisauri</place><translator></translator><lang>LA</lang><cvs_file>monte_mecha_02_la_1577</cvs_file><cvs_version>2635.10</cvs_version><locator>036.xml</locator></info> <text> <front> <section> <pb id="p.0001" xlink:href="036/01/001.jpg"/> <p id="id.2.1.1.1.0.0.0" type="head"> <s id="id.2.1.1.1.2.1.0">GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS <lb/>MECHANICORVM <lb/>LIBER. </s> </p> <figure id="id.036.01.001.1.jpg" xlink:href="036/01/001/1.jpg"> </figure> <p id="id.2.1.1.1.4.1.0" type="head"> <s id="id.2.1.1.1.6.1.0">PISAVRI <lb/>Apud Hieronymum Concordiam. </s> <lb/> <s id="id.2.1.1.1.8.1.0">M. D. LXXVII. </s> <lb/> <s id="id.2.1.1.1.10.1.0">Cum Licentia Superiorum. </s> </p> <pb xlink:href="036/01/002.jpg"/> <p id="id.2.1.1.3.0.0.0" type="head"> <s id="id.2.1.1.3.1.1.0">PRAESENTI OPERE <lb/>CONTENTA. </s> </p> <p id="id.2.1.1.4.0.0.0" type="main"> <s id="id.2.1.1.4.1.1.0">De Libra. </s> </p> <p id="id.2.1.1.5.0.0.0" type="main"> <s id="id.2.1.1.5.1.1.0">De Vecte. </s> </p> <p id="id.2.1.1.6.0.0.0" type="main"> <s id="id.2.1.1.6.1.1.0">De Trochlea. </s> </p> <p id="id.2.1.1.7.0.0.0" type="main"> <s id="id.2.1.1.7.1.1.0">De Axe in peritrochio. </s> </p> <p id="id.2.1.1.8.0.0.0" type="main"> <s id="id.2.1.1.8.1.1.0">De Cuneo. </s> </p> <p id="id.2.1.1.9.0.0.0" type="main"> <s id="id.2.1.1.9.1.1.0">De Cochlea. </s> </p> <pb xlink:href="036/01/003.jpg"/> <p id="id.2.1.1.10.0.0.0" type="head"> <s id="id.2.1.1.11.1.1.0">AD FRANCISCVM <lb/>MARIAM II <lb/>VRBINATVM <lb/>AMPLISSIMVM DVCEM <lb/>GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS </s> </p> <p type="head"> <s id="id.2.1.1.11.3.1.0">PRAEFATIO. </s> </p> <p id="id.2.1.1.12.0.0.0" type="main"> <s id="id.2.1.1.12.1.1.0">DVAE res (AMPLISSIME PRIN­<lb/>CEPS) quæ ad conciliandas homi<lb/>nibus facultates, vtilitas nempè, & <lb/>nobilitas, plurimùm valere con&longs;ue<lb/>uerunt. </s> <s id="id.2.1.1.12.1.2.0">illæ ad exornandam mecha<lb/>nicam facultatem, & eam præ om­<lb/>nibus alijs appetibilem reddendam con&longs;pira&longs;&longs;e <lb/>mihi videntur: nam &longs;i nobilitatem (quod pleriq; <lb/>modò faciunt) ortu ip&longs;o metimur, occurret hinc <lb/>Geometria, illinc verò Phi&longs;ica; quorum gemina<lb/>to complexu nobili&longs;&longs;ima artium prodit mechani­<lb/>ca. </s> <s id="id.2.1.1.12.1.3.0">&longs;i enim nobilitatem magis, tùm &longs;tratæ materiæ, <lb/>tùm argumentorum nece&longs;&longs;itati (quod Ari&longs;tote­<lb/>les fatetur aliquandò) relatam volumus, omnium <lb/>procul dubiò nobili&longs;&longs;imam per&longs;piciemus. </s> <s id="id.2.1.1.12.1.4.0">quæ <pb xlink:href="036/01/004.jpg"/>quidem non &longs;olum geometriam (vt Pappus te&longs;ta<lb/>tur) ab&longs;oluit, & perficit; verùm etiam & phi&longs;ica­<lb/>rum rerum imperium habet: quandoquidem <lb/>quodcunq; Fabris, Architectis, Baiulis, Agricolis, <lb/>Nautis, & quàm plurimis alijs (repugnantibus na­<lb/>turæ legibus) opitulatur; id omne mechanicum <lb/>e&longs;t imperium. </s> <s id="id.2.1.1.12.1.5.0">quippè quod aduer&longs;us naturam <lb/>vel eiu&longs;dem emulata leges exercet; &longs;umma id <lb/>certè admiratione dignum; veri&longs;&longs;imum tamen, <lb/>& à quocunque liberaliter admi&longs;&longs;um, qui pri­<lb/>us ab Ari&longs;totele didicerit, omnia mechanica, <lb/>tùm problemata, tùm theoremata ad rotundam <lb/>machinam reduci, atq; ideo illo niti principio, <lb/><expan abbr="nõ">non</expan> minus &longs;en&longs;ui, quàm rationi noto. </s> <s id="id.2.1.1.12.1.6.0">Rotunda ma<lb/>china e&longs;t mouenti&longs;&longs;ima, & quò maior, eò mouen­<lb/>tior. </s> <s id="id.2.1.1.12.1.7.0">Verùm huic nobilitati adnexa e&longs;t &longs;umma re <lb/>rum ad vitam pertinentium vtilitas, quæ propte­<lb/>rea omnes alias à diuer&longs;is artibus propagatas an­<lb/>tecellit; quòd aliæ facultates po&longs;t mundi gene&longs;im <lb/>longa temporis intercapedine &longs;uos explicarunt <lb/>v&longs;us; i&longs;ta verò & in ip&longs;is mundi primordijs ita fuit <lb/>hominibus nece&longs;&longs;aria, vt ea &longs;ublata Sol de mun­<lb/>do &longs;ublatus videretur. </s> <s id="id.2.1.1.12.1.8.0">nam quacunq; nece&longs;&longs;ita­<lb/>te Adæ vita degeretur; & quamuis etiam ca&longs;is <lb/>contectis &longs;tramine, & angu&longs;tis tugurijs, ac gurgu­<lb/>&longs;tijs cœli de fenderet iniurias; &longs;ic & in corporis ve<lb/>&longs;titu, licet ip&longs;e nihil aliud &longs;pectaret, ni&longs;i vt imbres, <pb xlink:href="036/01/005.jpg"/>vt niues, vt ventos; vt Solem, vt frigus arceret; <lb/>quodcunque tamen id fuit, omne mechanicum <lb/>fuit. </s> <s id="id.2.1.1.12.1.9.0">neq; tamen huic facultati contingit, quod <lb/>ventis &longs;olet, qui cùm vndè oriuntur, ibi vehe­<lb/>menti&longs;&longs;imi &longs;int, ad longinqua tamen fracti, de­<lb/>bilitatiquè perueniunt: &longs;ed quod magnis flumini­<lb/>bus crebriu&longs; accidit, quæ cùm in ip&longs;o ortu parua <lb/>&longs;int, perpetuò tamen aucta, eò ampliori ferun<lb/>tur alueo, quò à fontibus &longs;uis longius rece&longs;&longs;e­<lb/>runt. </s> <s id="id.2.1.1.12.1.10.0">Nam & temporis progre&longs;&longs;u mechanica fa <lb/>cultas &longs;ub iugo æquum arationis laborem di­<lb/>&longs;pen&longs;are, atque aratrum agris circumagere cæ­<lb/>pit. </s> <s id="id.2.1.1.12.1.11.0">deinceps bigis, & quadrigis docuit comea<lb/>tus, merces, onera quælibet vehere, è finibus <lb/>no&longs;tri&longs; ad finitimos populos exportare, & ex il<lb/>lis contra importare ad nos. </s> <s id="id.2.1.1.12.1.12.0">præterea cùm iam <lb/>res non tantùm nece&longs;&longs;itate, verùm etiam orna­<lb/>tu, & commoditate metirentur, mechanicæ <lb/>fuit &longs;ubtilitatis, quòd nauigia remo impellere­<lb/>mus; quòd gubernaculo exiguo in extrema pup<lb/>pi collocato ingentes triremium moles inflecte­<lb/>remus; quòd vnius &longs;æpè manu pro multis fabro­<lb/>rum manibus modò pondera lapidum, & tra­<lb/>bium Fabris, & Architectis &longs;ubleuaremus; mo­<lb/>dò tollenonis &longs;pecie aquas è puteis olitoribus e­<lb/>xhauriremus. </s> <s id="id.2.1.1.12.1.13.0">hinc etiam è liquidorum prælis vi<lb/>na, olea, vnguenta expre&longs;&longs;a, & quicquid liquo­<pb xlink:href="036/01/006.jpg"/>ris habent, per&longs;oluere domino compul&longs;a. </s> <s id="id.2.1.1.12.1.14.0">hinc <lb/>magnas <expan abbr="arborũ">arborum</expan>, & marmorum moles duobus in <lb/>contrarias partes <expan abbr="di&longs;trah&etilde;tibus">di&longs;trahentibus</expan> vectibus diremp­<lb/>&longs;imus; hinc militiæ in aggeribus extruendis, in <lb/>con&longs;erenda manu, in opugnando, propugnan­<lb/>doq; loca infinitæ ferè redundarunt vtilitates; <lb/>hinc demum Lignatores, Lapicidæ, Marmorarij <lb/>Vinitores, Olearij, Vnguentarij, Ferrarij, Auri<lb/>fices, Metallici, Chirurgi, Ton&longs;ores, Pi&longs;tores, Sar<lb/>tores, omnes deniq; opifices beneficiarij, tot, tan<lb/>taq; vitæ humanæ &longs;uppeditarunt commoda. </s> <s id="id.2.1.1.12.1.15.0">Eant <lb/>nunc noui logodedali quidam mechanicorum <lb/>contemptores, perfricent frontem, &longs;i quam ha­<lb/>bent, & ignobilitatem, atquè inutilitatem fal&longs;ò <lb/>criminari de&longs;inant: quòd &longs;i & adhuc id minimè <lb/>velint, eos quæ&longs;o in in&longs;citia &longs;ua relinquamus: <lb/>Ari&longs;totelemquè potius philo&longs;ophorum cory­<lb/>phæum imitemur, cuius mechanici amoris ardo<lb/>rem acuti&longs;&longs;imæ illæ mechanicæ quæ&longs;tiones po&longs;te <lb/>ris traditæ &longs;atis declarant: qua quidem laude <lb/>Platonem magnificè &longs;uperauit; qui (vt te&longs;tatur <lb/>Plutarcus) Architam, & Eudoxum mechanicæ <lb/>vtilitatem impen&longs;ius colentes ab in&longs;tituto deter<lb/>ruit; quòd nobili&longs;&longs;imam philo&longs;ophorum po&longs;&longs;e&longs;­<lb/>&longs;ionem in vulgus indicarent, ac publicarent; & <lb/>velut arcana philo&longs;ophiæ my&longs;teria proderent. </s> <s id="id.2.1.1.12.1.16.0"><lb/>res &longs;anè meo quidem iudicio pro&longs;us vituperan­<pb xlink:href="036/01/007.jpg"/>da, ni&longs;i fortè velimus tam nobilis di&longs;ciplinæ con<lb/>templationem quidem ocio&longs;am laudare; fructum <lb/>verò, & v&longs;um, arti&longs;q; finem improbare. </s> <s id="id.2.1.1.12.1.17.0">&longs;ed præ <lb/>omnibus mathematicis vnus Archimedes ore <lb/>laudandus e&longs;t pleniore, quem voluit Deus in me­<lb/>chanicis velut ideam &longs;ingularem e&longs;&longs;e, quam om­<lb/>nes earum &longs;tudio&longs;i ad imitandum &longs;ibi propone­<lb/>rent. </s> <s id="id.2.1.1.12.1.18.0">is enim Cœle&longs;tem globum exiguo admo­<lb/>dum, fragili què vitreo orbe conclu&longs;um ita efin­<lb/>xit, &longs;imulatis a&longs;tris viuum naturæ opus, ac iura <lb/>poli motibus certis adeò præ &longs;e ferentibus; vt <lb/>æmula naturæ manus tale de &longs;e encomium &longs;it <lb/>promerita: &longs;ic manus naturam, vt natura ma­<lb/>num ip&longs;a immitata putetur. </s> <s id="id.2.1.1.12.1.19.0">is poli&longs;pa&longs;tu manu <lb/>leua, & &longs;ola, quinquies millenum modiorum <lb/>pondus attraxit. </s> <s id="id.2.1.1.12.1.20.0">nauem in &longs;iccum litus eductam, <lb/>ac grauius oneratam &longs;olus machinis &longs;uis ad &longs;e <lb/>perindè pertraxit, ac &longs;i in mari remis, veli&longs;uè <lb/>impul&longs;a moueretur, <expan abbr="quã">quam</expan> & po&longs;tea in litore (quod <lb/>omnes Siciliæ vires non potuerunt) in mare de­<lb/>duxit. </s> <s id="id.2.1.1.12.1.21.0">ab i&longs;to etiam ea extiterunt bellica tor­<lb/>menta, quibus Syracu&longs;æ aduer&longs;us Marcellum <lb/>ita defen&longs;æ &longs;unt, vt pa&longs;&longs;im eorum machinator <lb/>Briareus, & centimanus à Romanis appellare­<lb/>tur. </s> <s id="id.2.1.1.12.1.22.0">demum hac arte confi&longs;us eò proce&longs;&longs;it au­<lb/>daciæ, vt eam vocem naturæ legibus adeò re­<lb/>pugnantem protulerit. </s> <s id="id.2.1.1.12.1.23.0">Da mihi, vbi &longs;i&longs;tam, ter<pb xlink:href="036/01/008.jpg"/>ramq; mouebo. </s> <s id="id.2.1.1.12.1.24.0">quod tamen non modò nos <lb/>vecte tantùm fieri potui&longs;&longs;e in præ&longs;enti libro doce<lb/>mus; verùm etiam, & omnis antiquitas (quod <lb/>multis forta&longs;&longs;è mirabile videbitur) id penitus <lb/>credidi&longs;&longs;e mihi videtur; quæ Neptuno tri­<lb/>dentem tanquam vectem attribuit; cuius ope <lb/>terræ concu&longs;&longs;or vbiq; nuncupatur à poetis. </s> <s id="id.2.1.1.12.1.25.0">ad <lb/>quod etiam a&longs;piciens celeberrimus no&longs;ter poeta <lb/>Neptunum inducit i&longs;ta machina &longs;yrtes, quò ma­<lb/>gis apparerent Troianis, &longs;ubleuantem. </s> </p> <p id="id.2.1.1.13.0.0.0" type="main"> <s id="id.2.1.1.13.1.1.0">“Leuat ip&longs;e tridenti <lb/>& va&longs;tas aperit &longs;yrtes.” </s> </p> <p id="id.2.1.1.14.0.0.0" type="main"> <s id="id.2.1.1.14.1.1.0">Mechanici præterea fuerunt Heron, Cte&longs;ibius, <lb/>& Pappus, qui licet ad mechanicæ apicem, perin­<lb/>de atq; Archimedes, euecti forta&longs;&longs;è minimè &longs;int; <lb/>mechanicam tamen facultatem egregiè percal­<lb/>luerunt; tale&longs;q; fuerunt, & præ&longs;ertim Pappus, vt <lb/>eum me ducem &longs;equentem nemo (vt opinor) cul<lb/>pauerit. </s> <s id="id.2.1.1.14.1.2.0">quod & propterea libentius feci, quòd <lb/>nè latum quidem vnguem ab Archimedeis prin­<lb/>cipijs Pappus recedat. </s> <s id="id.2.1.1.14.1.3.0">ego enim in hac præ&longs;ertim <lb/>facultate Archimedis ve&longs;tigijs hærere &longs;emper vo <lb/>lui: & licet eius lucubrationes ad <expan abbr="mechanicã">mechanicam</expan> per­<pb xlink:href="036/01/009.jpg"/>tinentes multis ab hinc annis pa&longs;&longs;im &longs;oleant do­<lb/>ctis de&longs;iderari: eruditi&longs;&longs;imus tamen libellus de æ­<lb/>queponderantibus præ manibus <expan abbr="hominũ">hominum</expan> adhuc <lb/>ver&longs;atur, in quò tanquam in copio&longs;i&longs;&longs;ima pœnu <lb/>omnia ferè mechanica dogmata repo&longs;ita mihi vi­<lb/>dentur; quem &longs;anè libellum, &longs;i ætatis no&longs;træ mathe<lb/>matici &longs;ibi magis familiarem adhibui&longs;&longs;ent; reperi&longs;<lb/>&longs;ent &longs;anè <expan abbr="&longs;ent&etilde;tias">&longs;ententias</expan> multas, quas modó ip&longs;i firmas, <lb/>& ratas e&longs;&longs;e docent; &longs;ubtili&longs;&longs;imè, atquè veri&longs;­<lb/>&longs;imè conuul&longs;as, & labefactatas. </s> <s id="id.2.1.1.14.1.4.0">&longs;ed hoc vi­<lb/>derint ip&longs;i. </s> <s id="id.2.1.1.14.1.5.0">ego enim ad Pappum redeo, qui <lb/>ad v&longs;um mathematicarum vberiorem, emulu­<lb/>mentorumquè acce&longs;&longs;iones amplificandas peni­<lb/>tus conuer&longs;us, de quinque principibus machi­<lb/>nis, Vecte nempè, Trochlea, Axe in peri­<lb/>trochio, Cuneo, & Cochlea, multa egre­<lb/>giè philo&longs;ophatus e&longs;t; demon&longs;trauit què quicquid <lb/>in machinis, aut cogitari peritè, aut acutè <lb/>definiri, aut certò &longs;tatui pote&longs;t, id omne quin­<lb/>què illis infinita vi præditis machinis referen­<lb/>dum e&longs;&longs;e. </s> <s id="id.2.1.1.14.1.6.0">atquè vtinam iniuria temporis ni­<lb/>hil è tanti viri &longs;criptis abra&longs;i&longs;&longs;et: nec enim tam <lb/>den&longs;a in&longs;citiæ caligo vniuer&longs;um propè terra­<lb/>rum orbem obtexi&longs;&longs;et, neque tanta mechani<lb/>cæ facultatis e&longs;&longs;et ignoratio con&longs;ecuta, vt ma­<lb/>thematicarum proceres exi&longs;timarentur illi, qui <lb/>modò inepti&longs;&longs;ima quadam di&longs;tinctione, diffi­<pb xlink:href="036/01/010.jpg"/>cultates nonnullas, nec illas tamen &longs;atis ar­<lb/>duas, & ob&longs;curas è medio tollunt. </s> <s id="id.2.1.1.14.1.7.0">reperiun­<lb/>tur enim aliqui, no&longs;traq; ætate emunctæ naris <lb/>mathematici, qui mechanicam, tùm mathe­<lb/>maticè &longs;eor&longs;um, tùm phi&longs;icè con&longs;iderari po&longs;­<lb/>&longs;e affirmant; ac &longs;i aliquando, vel &longs;ine demon<lb/>&longs;trationibus geometricis, vel &longs;ine vero motu <lb/>res mechanicæ con&longs;iderari po&longs;&longs;int: qua &longs;anè di­<lb/>&longs;tinctione (vt leuius cum illis agam) nihil aliud mi­<lb/>hi commini&longs;ci videntur, quàm vt dum &longs;e, tùm <lb/>phi&longs;icos, tùm mathematicos proferant, vtra­<lb/>que (quod aiunt) &longs;ella excludantur. </s> <s id="id.2.1.1.14.1.8.0">nequè <lb/>enim amplius mechanica, &longs;i à machinis ab&longs;tra<lb/>hatur, & &longs;eiungatur, mechanica pote&longs;t appel<lb/>lari. </s> <s id="id.2.1.1.14.1.9.0">Emicuit tamen inter i&longs;tas tenebras (quam­<lb/>uis alij quoquè nonnulli fuerint præclari&longs;&longs;imi) <lb/>Solis in&longs;tar Federicus Commandinus, qui multis <lb/>docti&longs;&longs;imis elucubrationibus ami&longs;&longs;um mathema<lb/>ticarum patrimonium non modò re&longs;taurauit, <lb/>verùm etiam auctiùs, & locupletiùs effecit. </s> <s id="id.2.1.1.14.1.10.0"><lb/>erat enim &longs;ummus i&longs;te vir omnibus adeò facul­<lb/>tatibus mathematicis ornatus, vt in eo Archi­<lb/>tas, Eudoxus, Heron, Euclides, Theon, Ari­<lb/>&longs;tarcus, Diophantus, Theodo&longs;ius, Ptolemæus <lb/>Apollonius, Serenus, Pappus, quin & ip­<lb/>&longs;emet Archimedes (&longs;iquidem ip&longs;ius in Archi­<lb/>medem &longs;cripta Archimedis olent lucernam) re <pb xlink:href="036/01/011.jpg"/>uixi&longs;&longs;e viderentur. </s> <s id="id.2.1.1.14.1.11.0">& ecce repentè è tenebris (vt <lb/>confidimus) ac vinculis corporis in lucem, li­<lb/>bertatem què productus mathematicas alieni&longs;­<lb/>&longs;imo tempore optimo, & præ&longs;tanti&longs;&longs;imo patre <lb/>orbatas, nos verò ita con&longs;ternatos reliquit, vt e­<lb/>ius de&longs;iderium vix longo &longs;ermone mitigare <lb/>po&longs;&longs;e videamur. </s> <s id="id.2.1.1.14.1.12.0">Ille tamen perpetuò in alia­<lb/>rum mathematicarum explicationem ver&longs;ans, <lb/>mechanicam facultatem, aut penitus præter­<lb/>mi&longs;it, aut modicè attigit. </s> <s id="id.2.1.1.14.1.13.0">Quapropter in hoc <lb/>&longs;tudium ardentiùs ego incumbere cæpi, nec me <lb/>vnquam per omne mathematum genus vagan<lb/>tem ea &longs;olicitudo de&longs;eruit; ecquid ex vno <lb/>quoquè decerpi, ac delibari po&longs;&longs;it; quo ad me<lb/>chanicam expoliendam, & exornandam acco­<lb/>modatior e&longs;&longs;e po&longs;&longs;em. </s> <s id="id.2.1.1.14.1.14.0">Nunc verò cùm mihi <lb/>videar, noni ea quidem omnia, quæ ad mecha<lb/>nicam pertinent, perfeci&longs;&longs;e; &longs;ed eò v&longs;q; tamen <lb/>progre&longs;&longs;us, vt ijs, qui ex Pappo, ex Vitruuio, <lb/>& ex alijs didicerint, quid &longs;it Vectis, quid Tro­<lb/>chlea, quid Axis in peritrochio, quid Cuneus, <lb/>quid Cochlea; quomodoq; vt pondera moueri <lb/>po&longs;&longs;int, aptari debeant; adhuc tamen acciden­<lb/>tia permulta, quæ inter potentiam, & pondus <lb/>vectis virtute illis in&longs;unt in&longs;trumentis, perdi&longs;ce­<lb/>re cupiunt, opis aliquid adferre po&longs;&longs;im; putaui <lb/>tempus iam po&longs;tulare, vt prodirem; & nauatæ <pb xlink:href="036/01/012.jpg"/>in hoc genere operæ &longs;pecimen aliquod darem. </s> <s id="id.2.1.1.14.1.15.0"><lb/>Verùm quò facilius totius operis &longs;ub&longs;tructio <lb/>ad fa&longs;tigium &longs;uum per duceretur, nonnulla quo­<lb/>què de libra fuerunt pertractanda, & præ&longs;er­<lb/>tim dum vnico pondere alterum &longs;olum ip&longs;ius <lb/>brachium penitus deprimitur: que in re mi­<lb/>rum e&longs;t quantas fecerint ruinas Iordanus (qui <lb/>inter recentiores maximæ fuit auctoritatis) & <lb/>alij; qui hanc rem &longs;ibi di&longs;cutiendam propo&longs;ue<lb/>runt. </s> <s id="id.2.1.1.14.1.16.0">opus &longs;anè arduum, & for&longs;an viribus no­<lb/>&longs;tris impar aggre&longs;si &longs;umus; in eo tamen digni, vt <lb/>no&longs;tros conatus, & indu&longs;triam ad præclara ten<lb/>dentem bonorum omnium perpetuus applau­<lb/>&longs;us, approbatioq; comitetur; quòd ad &longs;tudium <lb/>tàm illu&longs;tre, tam magnificum, tam laudabile <lb/>contulimus quicquid habuimus virium. </s> <s id="id.2.1.1.14.1.17.0">quod <lb/>&longs;anè qualecunq; &longs;it, tibi celeberrime PRINCEPS <lb/>nuncupandum cen&longs;uimus; cuius &longs;anè con&longs;ilij, <lb/>atq; in&longs;tituti no&longs;tri rationes multas reddere in <lb/>promptu e&longs;t: & primùm hæreditaria tibi in fa­<lb/>miliam no&longs;tram promerita, quibus nos ita de­<lb/>uictos habes; vt facilè intelligamus ad fortunas <lb/>non modò no&longs;tras, verùm & ad &longs;anguinem, & <lb/>vitam quoq; pro tua dignitate propendendam <lb/>parati&longs;&longs;imos e&longs;&longs;e debere. </s> <s id="id.2.1.1.14.1.18.0">Præterea illud non <lb/>parui quoq; ponderis accedit, quòd à pueri­<lb/>tia literarum omnium, &longs;ed præcipuè mathe­<pb xlink:href="036/01/013.jpg"/>maticarum de&longs;iderio ita fueris incen&longs;us, vt ni­<lb/>&longs;i illis adeptis vitam tibi acerbam, atq; in&longs;ua­<lb/>uem &longs;tatueres. </s> <s id="id.2.1.1.14.1.19.0">proinde in earum &longs;tudio infi­<lb/>xus primam ætatis partem in illis percipiendis <lb/>exegi&longs;ti, eamquè &longs;æpius verè principe dignam <lb/>vocem protuli&longs;ti, te propterea mathematicis <lb/>præ&longs;ertim delectari, quòd i&longs;tæ maximè ex do­<lb/>me&longs;tico illo, & vmbratili vitæ genere in Solem <lb/>(quod dicitur) & puluerem prodire po&longs;sint: cu<lb/>ius &longs;anè rei tuum flagranti&longs;simum ab ineunte æta <lb/>te peritiæ militaris de&longs;iderium, exploratum in­<lb/>dicium poterat e&longs;&longs;e, ni&longs;i nimis emendicatæ men­<lb/>tis e&longs;&longs;et ea proponere, quæ à te &longs;perari po&longs;&longs;ent; <lb/>quando tu penitus adole&longs;cens, egregia multa fa<lb/>cinora proficere matura&longs;ti. </s> <s id="id.2.1.1.14.1.20.0">Tu enim cùm iam <lb/>à &longs;ancti&longs;&longs;imo Pontifice Pio V &longs;aluberrimæ Prin­<lb/>cipum Chri&longs;tianorum coniunctionis fundamen­<lb/>ta iacta e&longs;&longs;ent, alacer admodum ad debellan­<lb/>dos Chri&longs;ti ho&longs;tes profectus, &longs;olidi&longs;&longs;imam, ac ve­<lb/>ri&longs;&longs;imam gloriam tibi compara&longs;ti. </s> <s id="id.2.1.1.14.1.21.0">Tu quoties de <lb/>&longs;umma rerum deliberatum e&longs;t, eas &longs;ententias <lb/>dixi&longs;ti, quæ &longs;ummam prudentiam cùm &longs;umma <lb/>animi excel&longs;itate coniunctam indicarent. </s> <s id="id.2.1.1.14.1.22.0">ommit­<lb/>tam interim pleraq; alia illis temporibus egre­<lb/>giè, viriliter què à te ge&longs;ta, ne tibi ip&longs;i ea, quæ <lb/>omnibus &longs;unt manife&longs;ta, palàm facere videar: <pb xlink:href="036/01/014.jpg"/>quæ cùm omnia magna, & præclara &longs;int; mul­<lb/>tò tamen à te maiora, & præclara expectant <lb/>adhuc homines. </s> <s id="id.2.1.1.14.1.23.0">Vale interim præ&longs;tanti&longs;&longs;imum <lb/>orbis decus, & &longs;i quando aliquid otij nactus <lb/>fueris has meas vigiliolas a&longs;picere ne dedi­<lb/>gneris. </s> </p> <pb n="1" xlink:href="036/01/015.jpg"/> <p id="id.2.1.1.15.0.0.0" type="head"> <s id="id.2.1.1.16.1.1.0">GVIDIVBALDI <lb/>E MARCHIONIBVS <lb/>MONTIS. </s> </p> <p type="head"> <s id="id.2.1.1.16.3.1.0">MECHANICORVM <lb/>LIBER. </s> </p> </section> </front> <body> <chap> <p id="id.id.2.1.1.16.5.1.0.a"> <s id="id.2.1.1.16.7.1.0">DEFINITIONES. </s> </p> <p id="id.2.1.1.17.0.0.0" type="main"> <s id="id.2.1.1.17.1.1.0">Centrvm grauitatis vniu&longs;cu­<lb/>iu&longs;q; corporis e&longs;t punctum quod­<lb/>dam intra po&longs;itum, à quo &longs;i gra­<lb/>ue appen&longs;um mente concipiatur, <lb/>dum fertur, quie&longs;cit; & &longs;eruat eam, <lb/>quam in principio habebat po&longs;i­<lb/>tionem: neq; in ip&longs;a latione circumuertitur. </s> </p> <p id="id.2.1.1.18.0.0.0" type="main"> <s id="id.2.1.1.18.1.1.0">Hanc centri grauitatis definitionem Pappus Alexandrinus in <lb/>octauo Mathematicarum collectionum libro tradidit. </s> <s id="id.2.1.1.18.1.2.0">Federicus <lb/>verò Commandinus in libro de centro grauitatis &longs;olidorum idem <lb/>centrum de&longs;cribendo ita explicauit. </s> </p> <p id="id.2.1.1.19.0.0.0" type="main"> <s id="id.2.1.1.19.1.1.0">Centrum grauitatis vniu&longs;cuiu&longs;q; &longs;olidæ figu­<lb/>ræ e&longs;t punctum illud intra po&longs;itum, circa quod <lb/>vndiq; partes æqualium momentorum con&longs;i­<lb/>&longs;tunt. </s> <s id="id.2.1.1.19.1.2.0">&longs;i enim per tale centrum ducatur planum <lb/>figuram quomodocunq; &longs;ecans &longs;emper in par­<lb/>tes æqueponderantes ip&longs;am diuidet. </s> </p> <pb xlink:href="036/01/016.jpg"/> <p id="id.2.1.1.21.0.0.0" type="head"> <s id="id.2.1.1.21.1.1.0">COMMVNES NOTIONES. </s> </p> <p type="head"> <s id="id.2.1.1.21.3.1.0">I </s> </p> <p id="id.2.1.1.22.0.0.0" type="main"> <s id="id.2.1.1.22.1.1.0">Si ab æqueponderantibus æqueponderantia au­<lb/>ferantur, reliqua æqueponderabunt. </s> </p> <p id="id.2.1.1.23.0.0.0" type="head"> <s id="id.2.1.1.23.1.1.0">II </s> </p> <p id="id.2.1.1.24.0.0.0" type="main"> <s id="id.2.1.1.24.1.1.0">Si æqueponderantibus æqueponderantia adii­<lb/>ciantur, tota &longs;imul æqueponderabunt. </s> </p> <p id="id.2.1.1.25.0.0.0" type="head"> <s id="id.2.1.1.25.1.1.0">III </s> </p> <p id="id.2.1.1.26.0.0.0" type="main"> <s id="id.2.1.1.26.1.1.0">Quæ eidem æqueponderant, inter &longs;e æquè &longs;unt <lb/>grauia. </s> </p> <p id="id.2.1.1.27.0.0.0" type="head"> <s id="id.2.1.1.27.1.1.0">SVPPOSITIONES. </s> </p> <p type="head"> <s id="id.2.1.1.27.3.1.0">I </s> </p> <p id="id.2.1.1.28.0.0.0" type="main"> <s id="id.2.1.1.28.1.1.0">Vnius corporis vnum tantùm e&longs;t centrum gra­<lb/>uitatis. </s> </p> <p id="id.2.1.1.29.0.0.0" type="head"> <s id="id.2.1.1.29.1.1.0">II </s> </p> <p id="id.2.1.1.30.0.0.0" type="main"> <s id="id.2.1.1.30.1.1.0">Vnius corporis centrum grauitatis &longs;emper in <lb/>eodem e&longs;t &longs;itu re&longs;pectu &longs;ui corporis. </s> </p> <p id="id.2.1.1.31.0.0.0" type="head"> <s id="id.2.1.1.31.1.1.0">III </s> </p> <p id="id.2.1.1.32.0.0.0" type="main"> <s id="id.2.1.1.32.1.1.0">Secundùm grauitatis centrum pondera deor­<lb/>&longs;um feruntur. </s> </p> </chap> <pb n="2" xlink:href="036/01/017.jpg"/> <chap> <p id="id.2.1.1.33.0.0.0" type="head"> <s id="id.2.1.1.34.1.1.0">DE LIBRA. </s> </p> <p id="id.2.1.1.35.0.0.0" type="main"> <s id="id.2.1.1.35.1.1.0">Anteqvam de libra &longs;ermo ha<lb/>beatur, vtres clarior eluce&longs;cat, &longs;it <lb/>libra AB recta linea; CD verò <lb/>trutina, quæ &longs;ecundum commu­<lb/>nem con&longs;uetudinem horizonti <lb/>&longs;emper e&longs;t perpendicularis. </s> <s id="id.2.1.1.35.1.2.0">pun­<lb/>ctum autem C immobile, circa quod vertitur li­<lb/>bra, centrum libræ <lb/>vocetur. </s> <s id="id.2.1.1.35.1.3.0">itidemque <lb/>(quamuis tamen im­<lb/>proprie) &longs;iue &longs;upra, <lb/>&longs;iue infra libram fue<lb/>rit con&longs;titutum. </s> <s id="id.2.1.1.35.1.4.0">CA <lb/>verò, & CB, tum di<lb/>&longs;tantiæ, tum libræ <lb/>brachia nuncupen­<lb/>tur. </s> <s id="id.2.1.1.35.1.5.0">& &longs;i à centro li­<lb/>bræ &longs;upra, vel infra <lb/><figure id="id.036.01.017.1.jpg" xlink:href="036/01/017/1.jpg"></figure><lb/>libram con&longs;tituto ip&longs;i AB perpendicularis duca­<lb/>tur, hæc perpendiculum vocetur, quæ libram AB <lb/>&longs;ub&longs;tinebit; & quocunque modo moueatur libra, <lb/>ip&longs;i &longs;emper perpendicularis exi&longs;tet. </s> </p> <pb xlink:href="036/01/018.jpg"/> <p id="id.2.1.1.37.0.0.0" type="head"> <s id="id.2.1.1.37.1.1.0">LEMMA. </s> </p> <p id="id.2.1.1.38.0.0.0" type="main"> <s id="id.2.1.1.38.1.1.0">Sit linea AB horizonti perpendicularis, & dia <lb/>metro AB circulus de&longs;cribatur AEBD, cuius <lb/>centrum C. </s> <s id="id.2.1.1.38.1.1.0.a">Dico punctum B infimum e&longs;&longs;e lo­<lb/>cum circumferentiæ circuli AEBD; punctum <lb/>verò A &longs;ublimiorem; & quælibet puncta, vt DE <lb/>æqualiter à puncto A di&longs;tantia æqualiter e&longs;&longs;e <lb/>deor&longs;um; quæ verò propius &longs;unt ip&longs;i A eis, quæ <lb/>magis di&longs;tant, &longs;ublimiora e&longs;&longs;e. </s> </p> <p id="id.2.1.1.39.0.0.0" type="main"> <s id="id.2.1.1.39.1.1.0">Producatur AB v&longs;q; ad mundi cen­<lb/>trum, quod &longs;it F; deinde in circuli circum­<lb/><arrow.to.target n="note1"></arrow.to.target>ferentia quoduis accipiatur punctum G; <lb/>connectanturq; FG FD FE. </s> <s id="id.2.1.1.39.1.2.0">Quoniam <lb/>n. BF minima e&longs;t omnium, quæ à puncto <lb/>F ad circumferentiam AEBD ducun­<lb/>tur; erit BF ip&longs;a FG minor. </s> <s id="id.2.1.1.39.1.3.0">quare punctum <lb/>B propius erit puncto F, quàm G. </s> <s id="id.2.1.1.39.1.3.0.a">hacq; <lb/>ratione o&longs;tendetur punctum B quouis alio <lb/>puncto circumferentiæ circuli AEDB <lb/>mundi centro propius e&longs;&longs;e. </s> <s id="id.2.1.1.39.1.4.0">erit igitur pun­<lb/>ctum B circumferentiæ circuli AEBD <lb/>infimus locus. </s> <s id="id.2.1.1.39.1.5.0">Deinde quoniam AF per <lb/>centrum ducta maior e&longs;t ip&longs;a GF; erit <lb/>punctum A non <expan abbr="&longs;olũ">&longs;olum</expan> ip&longs;o G, verum etiam <lb/>quouis alio puncto circumferentiæ circuli <lb/>AEBD &longs;ublimius. </s> <s id="id.2.1.1.39.1.6.0">Præterea quoniam DF <lb/>FE &longs;unt æquales; puncta DE æqualiter <lb/><figure id="id.036.01.018.1.jpg" xlink:href="036/01/018/1.jpg"></figure><lb/>mundi centro di&longs;tabunt. </s> <s id="id.2.1.1.39.1.7.0">& cum DF maior &longs;it FG; erit pun­<lb/>ctum D ip&longs;i A propius puncto G &longs;ublimius. </s> <s id="id.2.1.1.39.1.8.0">quæ omnia demon­<lb/>&longs;trare oportebat. </s> </p> <p id="id.2.1.2.1.0.0.0" type="margin"> <s id="id.2.1.2.1.1.1.0"><margin.target id="note1"></margin.target>8. <emph type="italics"/>Tertil.<emph.end type="italics"/></s> </p> <pb n="3" xlink:href="036/01/019.jpg"/> <p id="id.2.1.3.1.0.0.0" type="head"> <s id="id.2.1.3.1.2.1.0">PROPOSITIO I. </s> </p> <p id="id.2.1.3.2.0.0.0" type="main"> <s id="id.2.1.3.2.1.1.0">Si Pondus in eius centro grauitatis a recta &longs;u­<lb/>&longs;tineatur linea, nunquam manebit, ni&longs;i eadem li­<lb/>nea horizonti fuerit perpendicularis. </s> </p> <p id="id.2.1.3.3.0.0.0" type="main"> <s id="id.2.1.3.3.1.1.0">Sit pondus A, cuius centrum gra<lb/>uitatis B, quod à linea CE &longs;u&longs;ti­<lb/>neatur. </s> <s id="id.2.1.3.3.1.2.0">Dico pondus nunquam <lb/>perman&longs;urum, ni&longs;i CB horizonti <lb/>perpendicularis exi&longs;tat. </s> <s id="id.2.1.3.3.1.3.0">&longs;it pun­<lb/>ctum C immobile, quod vt pon<lb/>dus &longs;u&longs;tineatur, nece&longs;&longs;e e&longs;t. </s> <s id="id.2.1.3.3.1.4.0">& cum <lb/>punctum C &longs;it immobile, &longs;i pon­<lb/>dus A mouebitur, punctum B cir<lb/>culi circumferentiam de&longs;cribet, <lb/>cuius &longs;emidiameter erit CB. qua<lb/>re centro C, &longs;patio verò BC, cir­<lb/>culus de&longs;cribatur BFDE. </s> <s id="id.2.1.3.3.1.4.0.a">&longs;itq; <lb/><figure id="id.036.01.019.1.jpg" xlink:href="036/01/019/1.jpg"></figure><lb/>primum BC horizonti perpendicularís, quæ v&longs;q; ad D produca­<lb/>tur; atq; punctum C &longs;it infra punctum B. </s> <s id="id.2.1.3.3.1.4.0.b">Quoniam enim pondus <arrow.to.target n="note2"></arrow.to.target><lb/>A &longs;ecundum grauitatis centrum B deor&longs;um mouetur; punctum <lb/>B deor&longs;um in centrum mundi, quò naturaliter tendit, per re­<lb/>ctam lineam BD mouebitur: totum ergo pondus A eius cen­<lb/>tro grauitatis B &longs;uper rectam lineam BC graue&longs;cet. </s> <s id="id.2.1.3.3.1.5.0">cum au­<lb/>tem pondus à linea CB &longs;u&longs;tineatur, linea CB totum &longs;u&longs;ti­<lb/>nebit pondus A; &longs;uper quam deor&longs;um moueri non pote&longs;t, cum <lb/>ab ip&longs;a prohibeatur: per definitionem igitur centri grauitatis pun<lb/>ctum B, pondu&longs;q; A in hoc &longs;itu manebunt. </s> <s id="id.2.1.3.3.1.6.0">& quamquam B quo­<lb/>cunq; alio puncto circuli &longs;it &longs;ublimius, ab hoc tamen &longs;itu deor&longs;um <lb/>per circuli circumferentiam nequaquam mouebitur non enim ver­<lb/>&longs;us F magis, quàm ver&longs;us E inclinabitur, cum ex vtraq; parte æqua­<lb/>lis &longs;it de&longs;cen&longs;us; neq; pondus A in vnam magis, quàm in alteram <lb/>partem propen&longs;ionem habeat: quod non accidit in quouis alio <lb/>puncto circumferentiæ circuli (præter D) &longs;it ponderis eiu&longs;dem <pb xlink:href="036/01/020.jpg"/>centrum grauitatis, vt in F; cum ex <lb/>puncto F ver&longs;us D &longs;it de&longs;cen&longs;us, at <lb/>verò ver&longs;us B a&longs;cen&longs;us. </s> <s id="id.2.1.3.3.1.7.0">quare pun­<lb/>ctum F deor&longs;um mouebitur. </s> <s id="id.2.1.3.3.1.8.0">& quo<lb/>niam per rectam lineam in centrum <lb/>mundi moueri non pote&longs;t, cum à <lb/>puncto C immobili propter lineam <lb/>CF prohibeatur; deor&longs;um tamen <lb/>&longs;icuti eius natura po&longs;tulat, &longs;emper <lb/>mouebitur. </s> <s id="id.2.1.3.3.1.9.0">& cum infimus locus &longs;it <lb/>D, per <expan abbr="circumferentiã">circumferentiam</expan> FD mouebi<lb/>tur, donec in D perueniat, in quo <lb/>&longs;itu manebit, <expan abbr="põdu&longs;q">pondu&longs;q</expan>; immobile exi <lb/><figure id="id.036.01.020.1.jpg" xlink:href="036/01/020/1.jpg"></figure><lb/>&longs;tet. </s> <s id="id.2.1.3.3.1.10.0">tum quia deor&longs;um amplius moueri non pote&longs;t, cum ex pun­<lb/>cto C &longs;it appen&longs;um; tum etiam, quia in eius centro grauitatis &longs;u&longs;ti<lb/>netur. </s> <s id="id.2.1.3.3.1.11.0">Quando autem F erit in D, erit quoq; linea FC in DC, <lb/>&longs;imulq; horizonti perpendicularis. </s> <s id="id.2.1.3.3.1.12.0">pondus ergo nunquam mane<lb/>bit, donec linea CF horizonti perpendicularis non exi&longs;tat. quod <lb/>o&longs;tendere oportebat. </s> <s id="id.2.1.3.3.1.13.0">quod <lb/>o&longs;tendere oportebat. </s> </p> <p id="id.2.1.4.1.0.0.0" type="margin"> <s id="id.2.1.4.1.1.1.0"><margin.target id="note2"></margin.target><emph type="italics"/>Supp.<emph.end type="italics"/> 3. <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.5.1.0.0.0" type="main"> <s id="id.2.1.5.1.1.1.0">Ex hoc elici pote&longs;t, pondus quocunq; modo <lb/>in dato puncto &longs;u&longs;tineatur, nunquam manere; ni <lb/>&longs;i quando a centro grauitatis ponderis ad id pun<lb/>ctum ducta linea horizonti &longs;it perpendicularis. </s> </p> <p id="id.2.1.5.2.0.0.0" type="main"> <s id="id.2.1.5.2.1.1.0">Vt ii&longs;dem po&longs;itis, &longs;u&longs;tineatur <lb/>pondus à lineis CG CH. </s> <s id="id.2.1.5.2.1.1.0.a">Dico <lb/>&longs;i ducta BC horizonti &longs;it perpen­<lb/>dicularis, pondus A manere. </s> <s id="id.2.1.5.2.1.2.0">&longs;i verò <lb/>ducta CF non &longs;it horizonti per­<lb/>pendicularis, punctum F deor&longs;um <lb/>v&longs;q; ad D moueri; in quo &longs;itu pon­<lb/>dus manebit, ductaq; CD horizon<lb/>ti perpendicularis exi&longs;tet. </s> <s id="id.2.1.5.2.1.3.0">quæ om­<lb/>nia eadem ratione o&longs;tendentur. <figure id="id.036.01.020.2.jpg" xlink:href="036/01/020/2.jpg"></figure></s> <pb n="4" xlink:href="036/01/021.jpg"/> <s id="id.2.1.5.2.3.1.0">PROPOSITIO II. </s> </p> <p id="id.2.1.5.3.0.0.0" type="main"> <s id="id.2.1.5.3.1.1.0">Libra horizonti æquidi&longs;tans, cuius centrum <lb/>&longs;it &longs;upra libram, æqualia in extremitatibus, æqua <lb/>literq; à perpendiculo di&longs;tantia habens pondera, <lb/>&longs;i ab eiu&longs;modi moueatur &longs;itu, in eundem rur&longs;us <lb/>relicta, redibit; ibíq; manebit. </s> </p> <p id="id.2.1.5.4.0.0.0" type="main"> <s id="id.2.1.5.4.1.1.0">Sit libra AB recta li­<lb/>nea horizonti æquidi­<lb/>&longs;tans, cuius centrum C <lb/>&longs;it &longs;upra libram; &longs;itq; CD <lb/><expan abbr="perpendiculũ">perpendiculum</expan>, quod ho­<lb/>rizonti perpendiculare <lb/>erit: atq; di&longs;tantia DA &longs;it <lb/>di&longs;tantiæ DB æqualis; <lb/>&longs;intq; in AB pondera æ­<lb/>qualia, <expan abbr="quorũ">quorum</expan> grauitatis <lb/>centra &longs;int in AB <expan abbr="pũctis">punctis</expan>. </s> <s id="id.2.1.5.4.1.2.0"><lb/>Moueatur AB libra ab <lb/><figure id="id.036.01.021.1.jpg" xlink:href="036/01/021/1.jpg"></figure><lb/>hoc &longs;itu, putá in EF, deinde relinquatur. </s> <s id="id.2.1.5.4.1.3.0">dico libram EF in AB ho<lb/>rizonti æquidi&longs;tantem redire, ibíq; manere. </s> <s id="id.2.1.5.4.1.4.0">Quoniam autem pun<lb/>ctum C e&longs;t immobile, dum libra mouetur, punctum D circuli cir­<lb/>cumferentiam de&longs;cribet, cuius &longs;emidiameter erit CD. quare cen­<lb/>tro C, &longs;patio verò CD, circulus de&longs;cribatur DGH. </s> <s id="id.2.1.5.4.1.4.0.a">Quoniam <lb/>enim CD ip&longs;i libræ &longs;emper e&longs;t perpendicularis, dum libra erit in <lb/>EF, linea CD erit in CG, ita vt CG &longs;it ip&longs;i EF perpendicula­<lb/>ris. </s> <s id="id.2.1.5.4.1.5.0">Cùm autem AB bifariam à puncto D diuidatur, & pondera <lb/>in AB &longs;int æqualia; erit magnitudinis ex ip&longs;is AB compo&longs;itæ cen<arrow.to.target n="note3"></arrow.to.target><lb/>trum grauitatis in medio, hoc e&longs;t in D. & <expan abbr="quãdo">quando</expan> libra vná cum pon<lb/>deribus erit in EF; erit magnitudinis ex vtri&longs;q; EF compo&longs;itæ cen<lb/>trum grauitatis G. </s> <s id="id.2.1.5.4.1.5.0.a">& quoniam CG horizonti non e&longs;t perpendi­<lb/>cularis; <arrow.to.target n="note4"></arrow.to.target>magnitudo ex ponderibus EF compo&longs;ita in hoc &longs;itu <expan abbr="mi­nimè">mi­<lb/>nime</expan> per&longs;i&longs;tet, &longs;ed deor&longs;um <expan abbr="&longs;ecũdùm">&longs;ecundum</expan> eius centrum grauitatis G per <lb/>circumferentiam GD mouebitur; donec CG horizonti fiat per­<pb xlink:href="036/01/022.jpg"/>pendicularis, &longs;cilicet do­<lb/>nec CG in CD redeat. </s> <s id="id.2.1.5.4.1.6.0"><lb/>Quando autem CG erit <lb/>in CD, linea EF, cùm <lb/>ip&longs;i CG &longs;emper ad rectos <lb/>&longs;it angulos, erit in AB; in <lb/><arrow.to.target n="note5"></arrow.to.target>quo &longs;itu quoq; manebit. </s> <s id="id.2.1.5.4.1.7.0">li<lb/>bra ergo EF in AB hori­<lb/>zonti <expan abbr="æquidi&longs;tãtem">æquidi&longs;tantem</expan> redi<lb/>bit, ibíq; manebit. </s> <s id="id.2.1.5.4.1.8.0">quod <lb/>demon&longs;trare oportebat. </s> </p> <p id="id.2.1.6.1.0.0.0" type="margin"> <s id="id.2.1.6.1.1.1.0"><margin.target id="note3"></margin.target>4. <emph type="italics"/>primi Archi<lb/>medis de <lb/>æqueponde­<lb/>rantibus.<emph.end type="italics"/></s> <s id="id.2.1.6.1.1.2.0"><margin.target id="note4"></margin.target>1. <emph type="italics"/>Huius<emph.end type="italics"/></s> <s id="id.2.1.6.1.1.3.0"><margin.target id="note5"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <figure id="id.036.01.022.1.jpg" xlink:href="036/01/022/1.jpg"> </figure> <p id="id.2.1.7.1.1.1.0" type="head"> <s id="id.2.1.7.1.3.1.0">PROPOSITIO III. </s> </p> <p id="id.2.1.7.2.0.0.0" type="main"> <s id="id.2.1.7.2.1.1.0">Libra horizonti æquidi&longs;tans æqualia in extre­<lb/>mitatibus, æqualiterq; à perpendiculo di&longs;tan­<lb/>tia habens pondera, centro infernè collocato, in <lb/>hoc &longs;itu manebit. </s> <s id="id.2.1.7.2.1.2.0">&longs;i verò inde moueatur, deor­<lb/>&longs;um relicta, &longs;ecundùm partem decliuiorem mo­<lb/>uebitur. <figure id="id.036.01.022.2.jpg" xlink:href="036/01/022/2.jpg"></figure></s> </p> <p id="id.2.1.7.3.0.0.0" type="main"> <s id="id.2.1.7.3.1.1.0">Sit libra AB rectá li­<lb/>nea horizonti æquidi­<lb/>&longs;tans, cuius centrum C <lb/>&longs;it infra libram; perpen­<lb/>diculumq; &longs;it CD, quod <lb/>horizonti perpendiculare <lb/>erit; & di&longs;tantia AD &longs;it <lb/>di&longs;tantiæ DB æqualis; <lb/>&longs;intq; in AB pondera <lb/>æqualia, quorum grauita­<lb/>tis centra &longs;int in punctis <lb/>AB. </s> <s id="id.2.1.7.3.1.1.0.a">Dico primùm libram AB in hoc &longs;itu manere. </s> <s id="id.2.1.7.3.1.2.0">Quoniam <lb/>enim AB bifariam diuiditur à puncto D, & pondera in AB &longs;unt <lb/>æqualia; erit punctum D centrum grauitatis magnitudinis ex <pb n="5" xlink:href="036/01/023.jpg"/>vtri&longs;q; AB ponderibus compo&longs;itæ. </s> <s id="id.2.1.7.3.1.3.0">& CD libram &longs;u&longs;tinens ho­<lb/>rizonti <arrow.to.target n="note6"></arrow.to.target>e&longs;t perpendicularis, libra ergo AB in hoc &longs;itu manebit. <arrow.to.target n="note7"></arrow.to.target><lb/>moueatur autem libra AB ab hoc &longs;itu, putà in EF, deinde relinqua<lb/>tur. </s> <s id="id.2.1.7.3.1.4.0">dico libram EF ex parte F moueri. </s> <s id="id.2.1.7.3.1.5.0">Quoniam igitur CD <lb/>ip&longs;i libræ &longs;emper e&longs;t perpendicularis, dum libra erit in EF, erit <lb/>CD in CG ip&longs;i EF perpendicularis. </s> <s id="id.2.1.7.3.1.6.0">& punctum G magnitudi­<lb/>nis ex EF compo&longs;itæ centrum grauitatis erit; quod dum moue­<lb/>tur, circuli circumferentiam de&longs;cribet DGH, cuius &longs;emidiameter <lb/>CD, & centrum C. </s> <s id="id.2.1.7.3.1.6.0.a">Quoniam autem CG horizonti non e&longs;t per­<lb/>pendicularis, magnitudo ex EF ponderibus compo&longs;ita in hoc &longs;i­<lb/>tu minimè manebit; &longs;ed &longs;ecundùm eius grauitatis centrum G deor<lb/>&longs;um per circumferentiam GH mouebitur. </s> <s id="id.2.1.7.3.1.7.0">libra ergo EF ex par <lb/>te F deor&longs;um mouebitur, quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.8.1.0.0.0" type="margin"> <s id="id.2.1.8.1.1.1.0"><margin.target id="note6"></margin.target>4. <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.8.1.1.3.0"><margin.target id="note7"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.9.1.0.0.0" type="head"> <s id="id.2.1.9.1.1.1.0">PROPOSITIO IIII. </s> </p> <p id="id.2.1.9.2.0.0.0" type="main"> <s id="id.2.1.9.2.1.1.0">Libra horizonti æquidi&longs;tans æqualia in ex­<lb/>tremitatibus, æqualiterq; à centro in ip&longs;a libra <lb/>collocato, di&longs;tantia habens pondera; &longs;iue inde <lb/>moueatur, &longs;iue minus; vbicunq; relicta, manebit. <figure id="id.036.01.023.1.jpg" xlink:href="036/01/023/1.jpg"></figure></s> </p> <p id="id.2.1.9.3.0.0.0" type="main"> <s id="id.2.1.9.3.1.1.0">Sit libra recta linea A <lb/>B horizonti æquidi&longs;tans, <lb/>cuius centrum C in ea­<lb/>dem &longs;it linea AB; di&longs;tan<lb/>tia verò CA &longs;it di&longs;tantiæ <lb/>CB æqualis: &longs;intq; pon­<lb/>dera in AB æqualia, quo­<lb/>rum centra grauitatis &longs;int <lb/>in <expan abbr="puntis">punctis</expan> AB. </s> <s id="id.2.1.9.3.1.1.0.a">Moueatur <lb/>libra, vt in DE, ibiquè <lb/>relinquatur. </s> <s id="id.2.1.9.3.1.2.0">Dico primùm libram DE non moueri, in eoquè &longs;itu <lb/>manere. </s> <s id="id.2.1.9.3.1.3.0">Quoniam enim pondera AB &longs;unt æqualia; erit magni­<lb/>tudinis ex vtroq; pondere, videlicet A, & B compo&longs;itæ centrum <lb/>grauitatis C. quare idem punctum C, & centrum libræ, & <expan abbr="centrũ">centrum</expan><lb/> grauitatis totius ponderis erit. </s> <s id="id.2.1.9.3.1.4.0">Quoniam autem centrum libræ <pb xlink:href="036/01/024.jpg"/>C, dum libra AB vnà <lb/>cum ponderibus in DE <lb/>mouetur, immobile re­<lb/>manet, centrum quoq; <lb/>grauitatis, quod e&longs;t idem <lb/>C, non mouebitur. </s> <s id="id.2.1.9.3.1.5.0">nec <lb/>igitur libra DE mouebi<lb/>tur, per definitionem <lb/>centri grauitatis, cum in <lb/>ip&longs;o &longs;u&longs;pendatur. </s> <s id="id.2.1.9.3.1.6.0">Idip­<lb/><figure id="id.036.01.024.1.jpg" xlink:href="036/01/024/1.jpg"></figure><lb/>&longs;um quoq; contingit libra in AB horizonti æquidi&longs;tante, vel in <lb/>quocunq; alio &longs;itu exi&longs;tente. </s> <s id="id.2.1.9.3.1.7.0">Manebit ergo libra, vbi relinque­<lb/>tur. </s> <s id="id.2.1.9.3.1.8.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.9.4.0.0.0" type="main"> <s id="id.2.1.9.4.1.1.0">Cum verò in iis, quæ dicta &longs;unt, grauitatis tantùm magnitudi<lb/>num, quæ in extremitatibus libræ po&longs;itæ &longs;unt æquales, ab&longs;q; <expan abbr="lí­bræ">li­<lb/>bræ</expan> grauitate con&longs;iderauerimus; quoniam tamen adhuc libræ bra­<lb/>chia &longs;unt æqualia, idcirco idem libræ, eius grauitate con&longs;iderata, <lb/>vnà cum ponderibus, vel &longs;ine ponderibus eueniet. </s> <s id="id.2.1.9.4.1.2.0">idem enim cen<lb/>trum grauitatis fine ponderibus libræ tantùm grauitatis centrum <lb/>erit. </s> <s id="id.2.1.9.4.1.3.0">Similiter &longs;i pondera in libræ extremitatibus appendantur, vt <lb/>fieri &longs;olet, idem eueniet; dummodo ex &longs;u&longs;pen&longs;ionum punctis ad <lb/>centra grauitatum ponderum ductæ lineæ (quocunq; modo mo­<lb/>ueatur libra) &longs;i protrahantur, in centrum mundi concurrant. </s> <s id="id.2.1.9.4.1.4.0">vbi <lb/>enim pondera hoc modo &longs;unt appen&longs;a, ibi graue&longs;cunt, ac &longs;i in ii&longs;­<lb/>dem punctis centra grauitatum haberent. </s> <s id="id.2.1.9.4.1.5.0">præterea, quæ &longs;equun­<lb/>tur, eodem pror&longs;us modo con&longs;iderare poterimus. </s> </p> <p id="id.2.1.9.5.0.0.0" type="main"> <s id="id.2.1.9.5.1.1.0"><arrow.to.target n="note8"></arrow.to.target>Quoniam autem huic determinationi vltimæ multa à nonnullis <lb/>aliter &longs;entientibus dicta officere videntur; idcirco in hac parte ali­<lb/><arrow.to.target n="note9"></arrow.to.target>quantulum immorari oportebit; & pro viribus, non &longs;olum pro­<lb/>priam &longs;ententiam, &longs;ed Archimedem ip&longs;um, qui in hac eadem e&longs;&longs;e <lb/><arrow.to.target n="note10"></arrow.to.target>&longs;ententia videtur, defendere conabor. <pb n="6" xlink:href="036/01/025.jpg"/><figure id="id.036.01.025.1.jpg" xlink:href="036/01/025/1.jpg"></figure></s> </p> <p id="id.2.1.9.6.0.0.0" type="main"> <s id="id.2.1.9.6.1.1.0">Ii&longs;dem po&longs;itis, duca­<lb/>tur FCG ip&longs;i AB, & <lb/>horizonti perpendicula­<lb/>ris; & centro C, <expan abbr="&longs;patio­què">&longs;patio­<lb/>que</expan> CA, circulus de&longs;cri<lb/>batur ADFBEG. erunt <lb/>puncta ADBE in circu<lb/>li circumferentia; cum li­<lb/>bræ brachia &longs;int æqualia. </s> <s id="id.2.1.9.6.1.2.0"><lb/>& quoniam in vnam con<lb/>ueniunt &longs;ententiam, a&longs;&longs;e­<lb/>rentes &longs;cilicet libram DE <lb/>neq; in FG moueri, ne­<lb/>que in DE manere, &longs;ed in AB horizonti æquidi&longs;tantem rediré. </s> <s id="id.2.1.9.6.1.3.0"><lb/>hanc eorum &longs;ententiam nullo modo con&longs;i&longs;tere po&longs;&longs;e o&longs;tendam. </s> <s id="id.2.1.9.6.1.4.0"><lb/>Non enim, &longs;ed &longs;i quod aiunt, euenerit, vel ideo erit, quia pondus <lb/>D pondere E grauius fuerit, vel &longs;i pondera &longs;unt æqualia, di&longs;tantiæ, <lb/>quibus &longs;unt po&longs;ita, non erunt æquales, hoc e&longs;t CD ip&longs;i CE non erit <lb/>æqualis, &longs;ed maior. </s> <s id="id.2.1.9.6.1.5.0">Quòd autem pondera in DE &longs;int æqualia, & <lb/>di&longs;tantia CD &longs;it æqualis di&longs;tantiæ CE: hæc ex &longs;uppo&longs;itione pa­<lb/>tent. </s> <s id="id.2.1.9.6.1.6.0">Sed quoniam dicunt pondus in D in eo &longs;itu pondere in E <lb/>grauius e&longs;&longs;e in altero &longs;itu deor&longs;um: dum pondera &longs;unt in DE, pun­<lb/>ctum C non erit amplius centrum grauitatis, nam non manent, &longs;i <lb/>ex C &longs;u&longs;pendantur; &longs;ed erit in linea CD, ex tertia primi Archi­<lb/>medis de æqueponderantibus. </s> <s id="id.2.1.9.6.1.7.0">non autem erit in linea CE, cum pon<lb/>dus D grauius &longs;it pondere E. &longs;it igitur in H, in quo &longs;i &longs;u&longs;pendan­<lb/>tur, manebunt. </s> <s id="id.2.1.9.6.1.8.0">Quoniam autem centrum grauitatis ponderum <lb/>in AB connexorum e&longs;t punctum C; ponderum verò in DE e&longs;t <lb/>punctum H: dum igitur pondera AB mouentur in DE, centrum <lb/>grauitatis C ver&longs;us D mouebitur, & ad D propius accedet; quod <lb/>e&longs;t impo&longs;sibile: cum pondera eandem inter &longs;e &longs;e &longs;eruent di&longs;tantiam. </s> <s id="id.2.1.9.6.1.9.0"><lb/>Vniu&longs;cuiu&longs;q; enim corporis centrum grauitatis in eodem &longs;emper <arrow.to.target n="note11"></arrow.to.target><lb/>e&longs;t &longs;itu re&longs;pectu &longs;ui corporis. </s> <s id="id.2.1.9.6.1.10.0">& quamquam punctum C &longs;it duo­<lb/>rum corporum AB centrum grauitatis, quia tamen inter &longs;e &longs;e ita à <lb/>libra connexa &longs;unt, vt &longs;emper eodem modo &longs;e &longs;e habeant; Ideo <lb/>punctum C ita eorum erit centrum grauitatis, ac &longs;i vna tantum <pb xlink:href="036/01/026.jpg"/><arrow.to.target n="note12"></arrow.to.target>e&longs;&longs;et magnitudo. </s> <s id="id.2.1.9.6.1.11.0">libra <lb/>enim vna cum ponderi­<lb/>bus vnum tantum conti<lb/>nuum efficit, cuius cen­<lb/>trum grauitatis erit &longs;em­<lb/>per in medio. </s> <s id="id.2.1.9.6.1.12.0">non igitur <lb/>pondus in D pondere in <lb/>E e&longs;t grauius. </s> <s id="id.2.1.9.6.1.13.0">Si autem <lb/>dicerent centrum graui­<lb/>tatis non in linea CD, <lb/>&longs;ed in CE e&longs;&longs;e debere; <lb/>idem eueniet ab&longs;urdum. <figure id="id.036.01.026.1.jpg" xlink:href="036/01/026/1.jpg"></figure></s> </p> <p id="id.2.1.9.7.0.0.0" type="main"> <s id="id.2.1.9.7.1.1.0">Amplius &longs;i pondus D <lb/>deor&longs;um mouebitur, pondus E &longs;ur&longs;um mouebit. </s> <s id="id.2.1.9.7.1.2.0">pondus igitur gra­<lb/>uius, quàm &longs;it E, in eodemmet &longs;itu ponderi D æqueponderabit, & <lb/>grauia inæqualia æquali di&longs;tantia po&longs;ita æqueponderabunt. </s> <s id="id.2.1.9.7.1.3.0">Adii­<lb/>ciatur ergo ponderi E aliquod graue, ita vt ip&longs;i D contraponde­<lb/>ret, &longs;i ex C &longs;u&longs;pendantur. </s> <s id="id.2.1.9.7.1.4.0">&longs;ed cum &longs;upra o&longs;ten&longs;um &longs;it punctum C <lb/>centrum e&longs;&longs;e grauitatis æqualium ponderum in DE; &longs;i igitur pon­<lb/><arrow.to.target n="note13"></arrow.to.target>dus E grauius fuerit pondere D, erit centrum grauitatis in linea <lb/>CE. &longs;itq; hoc centrum K. at per definitionem centri grauitatis, &longs;i <lb/>pondera &longs;u&longs;pendantur ex K, manebunt. </s> <s id="id.2.1.9.7.1.5.0">ergo &longs;i &longs;u&longs;pendantur ex <lb/>C, non manebunt, quod e&longs;t contra hypote&longs;im: &longs;ed pondus E deor<lb/>&longs;um mouebitur. </s> <s id="id.2.1.9.7.1.6.0">quòd &longs;i ex C quoque &longs;u&longs;pen&longs;a æqueponderarent; <lb/><arrow.to.target n="note14"></arrow.to.target>vnius magnitudinis duo e&longs;&longs;ent centra grauitatis; quod e&longs;t impo&longs;si<lb/>bile. </s> <s id="id.2.1.9.7.1.7.0">Non igitur pondus in E grauius eo, quod e&longs;t in D, ip&longs;i D æque­<lb/>ponderabit, cum ex puncto C fiat &longs;u&longs;pen&longs;io. </s> <s id="id.2.1.9.7.1.8.0">Pondera ergo in DE <lb/>æqualia ex eorum grauitatis centro C &longs;u&longs;pen&longs;a, æqueponderabunt, <lb/>manebuntquè. </s> <s id="id.2.1.9.7.1.9.0">quod demon&longs;trare fuerat propo&longs;itum. </s> </p> <p id="id.2.1.10.1.0.0.0" type="margin"> <s id="id.2.1.10.1.1.1.0"><margin.target id="note8"></margin.target><emph type="italics"/>Iordanus de Ponderibus. <emph.end type="italics"/></s> <s id="id.2.1.10.1.1.2.0"><margin.target id="note9"></margin.target><emph type="italics"/>Hyerommus Cardanus de &longs;ubtilitate. <emph.end type="italics"/></s> <s id="id.2.1.10.1.1.3.0"><margin.target id="note10"></margin.target><emph type="italics"/>Nicolaus Tartalea de quæ&longs;itis, ac inuentionibus. <emph.end type="italics"/></s> <s id="id.2.1.10.1.1.4.0"><margin.target id="note11"></margin.target>2. <emph type="italics"/>Sup. huius. <emph.end type="italics"/></s> <s id="id.2.1.10.1.1.6.0"><margin.target id="note12"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 4. <emph type="italics"/>primi Archim de Aequep.<emph.end type="italics"/></s> <s id="id.2.1.10.1.1.7.0"><margin.target id="note13"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 3. <emph type="italics"/>primi Archim de Aequep.<emph.end type="italics"/></s> <s id="id.2.1.10.1.1.8.0"><margin.target id="note14"></margin.target>1. <emph type="italics"/>Suppo&longs;. huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.11.1.0.0.0" type="main"> <s id="id.2.1.11.1.1.1.0"><arrow.to.target n="note15"></arrow.to.target>Huic autem po&longs;tremo inconuenienti occurrunt dicentes, im­<lb/>po&longs;sibile e&longs;&longs;e addere ip&longs;i E pondus adeo minimum, quin adhuc &longs;i <lb/>ex C &longs;u&longs;pendantur, pondus E &longs;emper deor&longs;um ver&longs;us G moueatur. </s> <s id="id.2.1.11.1.1.2.0"><lb/>quod nos fieri po&longs;&longs;e &longs;uppo&longs;uimus, atque fieri po&longs;&longs;e credebamus. </s> <s id="id.2.1.11.1.1.3.0">ex­<lb/>ce&longs;&longs;um enim ponderis D &longs;upra pondus E, cum quantitatis ratio­<lb/>nem habeat, non &longs;olum minimum e&longs;&longs;e, verum in infinitum diuidi <lb/>po&longs;&longs;e immaginabamur, quod quidem ip&longs;i, non &longs;olum minimum, <pb n="7" xlink:href="036/01/027.jpg"/>&longs;ed ne minimum quidem e&longs;&longs;e, cum reperiri non po&longs;sit, hoc mo­<lb/>do demon&longs;trare nituntur. <figure id="id.036.01.027.1.jpg" xlink:href="036/01/027/1.jpg"></figure></s> </p> <p id="id.2.1.11.2.0.0.0" type="main"> <s id="id.2.1.11.2.1.1.0">Exponantur eadem. </s> <s id="id.2.1.11.2.1.2.0"><lb/>à puncti&longs;què DE hori­<lb/>zonti <expan abbr="perp&etilde;diculares">perpendiculares</expan> du<lb/><expan abbr="cãtur">cantur</expan> DHEK, atq; alius <lb/>&longs;it circulus LDM, cu­<lb/>ius <expan abbr="centrũ">centrum</expan> N, qui FDG <lb/>in puncto D contingat, <lb/>ip&longs;iq; FDG &longs;it æqualis: <lb/>erit NC recta linea. </s> <s id="id.2.1.11.2.1.3.0">& <arrow.to.target n="note16"></arrow.to.target><lb/>quoniam angulus KEC <lb/>angulo HDN e&longs;t æqua <arrow.to.target n="note17"></arrow.to.target><lb/>lis, angulusq; CEG an­<lb/>gulo NDM e&longs;t etiam <lb/>æqualis; cum à &longs;emidiametris, æqualibusq; circumferentiis conti­<lb/>neatur; erit reliquus mixtu&longs;què angulus KEG reliquo mixtoquè <lb/>HDM æqualis. </s> <s id="id.2.1.11.2.1.4.0">& quia &longs;upponunt, quò minor e&longs;t angulus linea <lb/>horizonti perpendiculari, & circumferentia contentus, eò pondus <lb/>in eo &longs;itu grauius e&longs;&longs;e. </s> <s id="id.2.1.11.2.1.5.0">vt quò minor e&longs;t angulus HD, & circumfe<lb/>rentia DG contentus angulo KEG, hoc e&longs;t angulo HDM; ita &longs;e<lb/>cundum hanc proportionem pondus in D grauius e&longs;&longs;e pondere in <lb/>E. </s> <s id="id.2.1.11.2.1.5.0.a">Proportio autem anguli MDH ad angulum HDG minor e&longs;t <lb/>qualibet proportione, quæ &longs;it inter maiorem, & minorem quanti<lb/>tatem: ergo proportio ponderum DE omnium proportionum mi<lb/>nima erit. </s> <s id="id.2.1.11.2.1.6.0">immo neq; erit ferè proportio, cum &longs;it omnium pro <lb/>portionum minima. </s> <s id="id.2.1.11.2.1.7.0">quòd autem proportio MDH ad HDG &longs;it <lb/>omnium minima, ex hac nece&longs;sitate o&longs;tendunt; quia MDH exce<lb/>dit HDG angulo curuilineo MDG, qui quidem angulus omnium <lb/>angulorum rectilineorum minimus exi&longs;tit: ergo cum non po&longs;sit da <lb/>ri angulus minor MDG, erit proportio MDH ad HDG <expan abbr="omniũ">omnium</expan> <lb/>proportionum minima. </s> <s id="id.2.1.11.2.1.8.0">quæ ratio inutilis valde videtur e&longs;&longs;e; quia <lb/>quamquam angulus MDG &longs;it omnibus rectilineis angulis minor, <lb/>non idcirco &longs;equitur, ab&longs;olutè, &longs;impliciterq; omnium e&longs;&longs;e <expan abbr="angulorũ">angulorum</expan> <lb/>minimum: nam ducatur à puncto D linea DO ip&longs;i NC perpendicu<lb/>laris, hæc vtra&longs;q; tanget circumferentias LDM FDG in puncto <arrow.to.target n="note18"></arrow.to.target><pb xlink:href="036/01/028.jpg"/>D. quia verò circumfe<lb/>rentiæ &longs;unt æquales, erit <lb/>angulus MDO mixtus <lb/>angulo ODG mixto <lb/>æqualis; alter ergo an<lb/>gulus, vt ODG minor <lb/>erit MDG, hoc e&longs;t mi <lb/>nor minimo. </s> <s id="id.2.1.11.2.1.9.0">angulus <lb/>deinde OGH minor <lb/>erit angulo MDH; qua <lb/>re ODH ad angulum <lb/><arrow.to.target n="note19"></arrow.to.target>HDG minorem habe<lb/>bit <expan abbr="proportion&etilde;">proportionem</expan>, quàm <lb/><figure id="id.036.01.028.1.jpg" xlink:href="036/01/028/1.jpg"></figure><lb/>MDH ad eundem HDG. dabitur ergo quoquè proportio mi­<lb/>nor minima, quam in infinitum adhuc minorem ita o&longs;tende­<lb/>mus. </s> <s id="id.2.1.11.2.1.10.0">De&longs;cribatur circulus DR, cuius centrum E, & &longs;emidiame­<lb/><arrow.to.target n="note20"></arrow.to.target>ter ED. continget circumferentia DR circumferentiam DG in <lb/><arrow.to.target n="note21"></arrow.to.target>puncto D, lineamquè DO in puncto D; quare minor erit angu­<lb/>lus RDG angulo ODG. &longs;imiliter & angulus RDH angulo <lb/>ODH. </s> <s id="id.2.1.11.2.1.10.0.a">minorem igitur proportionem habebit RDH ad HDG, <lb/>quàm ODH ad HDG. </s> <s id="id.2.1.11.2.1.10.0.b">Accipiatur deinde inter EC vtcun­<lb/>que punctum P, ex quo in di&longs;tantia PD alia de&longs;cribatur circum­<lb/>ferentia DQ, quæ circumferentiam DR, circumferentiamquè <lb/>DG in puncto D continget; & angulus QDH minor erit <lb/>angulo RDH: ergo QDH ad HDG minorem habebit propor<lb/>tionem, quàm RDH ad HDG. eodemquè pror&longs;us modo, &longs;i <lb/>inter PC aliud accipiatur punctum, & inter hoc &C aliud, & &longs;ic <lb/>deinceps, infinitæ de&longs;cribentur circumferentiæ inter DO, & cir<lb/>cumferentiam DG; ex quibus proportionem in infinitum &longs;emper <lb/>minorem inueniemus. </s> <s id="id.2.1.11.2.1.11.0">atque ideo proportionem ponderis in D <lb/>ad pondus in E non adeo minorem e&longs;&longs;e &longs;equitur, quin ad infini <lb/>tum ip&longs;a &longs;emper minorem reperiri po&longs;sit. </s> <s id="id.2.1.11.2.1.12.0">& quia angulus MDG <lb/>in infinitum diuidi pote&longs;t; exce&longs;&longs;us quoque grauitatis D &longs;upra E <lb/>diuidi ad infinitum poterit. </s> </p> <p id="id.2.1.12.1.0.0.0" type="margin"> <s id="id.2.1.12.1.1.1.0"><margin.target id="note15"></margin.target><emph type="italics"/>Tartalea &longs;exta propo&longs;itione octaui libri.<emph.end type="italics"/></s> <s id="id.2.1.12.1.1.2.0"><margin.target id="note16"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 12. <emph type="italics"/>tertii.<emph.end type="italics"/></s> <s id="id.2.1.12.1.1.3.0"><margin.target id="note17"></margin.target>29. <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.12.1.1.4.0"><margin.target id="note18"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 18. <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.12.1.1.5.0"><margin.target id="note19"></margin.target>8. <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.12.1.1.6.0"><margin.target id="note20"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 11. <emph type="italics"/>tertit.<emph.end type="italics"/></s> <s id="id.2.1.12.1.1.7.0"><margin.target id="note21"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 18. <emph type="italics"/>tertii.<emph.end type="italics"/></s> </p> <pb n="8" xlink:href="036/01/029.jpg"/> <p id="id.2.1.13.1.0.0.0" type="main"> <s id="id.2.1.13.1.2.1.0">Sed neque prætereundum <lb/>e&longs;t, ip&longs;os in demon&longs;tratio­<lb/>ne angulum KEG maiorem <lb/>e&longs;&longs;e angulo HDG, tanquam <lb/>notum accepi&longs;&longs;e. </s> <s id="id.2.1.13.1.2.2.0">quod e&longs;t <lb/>quidem verum, &longs;i DHEK <lb/>inter &longs;e &longs;e &longs;int æquidi&longs;tan­<lb/>tes. </s> <s id="id.2.1.13.1.2.3.0">Quoniam autem (vt <lb/>ip&longs;i quoque &longs;upponunt) li­<lb/>neæ DHEK in centrum <lb/>mundi conueniunt; lineæ <lb/>DHEK æquidi&longs;tantes nun<lb/>quam erunt, & angulus KEG <lb/>angulo HDG non &longs;olum <lb/>maior erit, &longs;ed minor. </s> <s id="id.2.1.13.1.2.4.0">vt <lb/>exempli gratia, producatur <lb/>FG v&longs;que ad centrum mun<lb/>di, quod &longs;it S; <expan abbr="connectan­turqué">connectan­<lb/>turque</expan> DSES. o&longs;tenden­<lb/>dum e&longs;t angulum SEG mi<lb/>norem e&longs;&longs;e angulo SDG. </s> <s id="id.2.1.13.1.2.4.0.a">du<lb/><figure id="id.036.01.029.1.jpg" xlink:href="036/01/029/1.jpg"></figure><lb/>catur à puncto E linea ET circulum DGEF contingens, ab eo <lb/>demqué puncto ip&longs;i DS æquidi&longs;tans ducatur EV. </s> <s id="id.2.1.13.1.2.4.0.b">Quoniam igi<lb/>tur EVDS inter &longs;e &longs;e &longs;unt æquidi&longs;tantes: &longs;imiliter ETDO æqui <lb/>di&longs;tantes: erit angulus VET angulo SDO æqualis. </s> <s id="id.2.1.13.1.2.5.0">& angulus <lb/>TEG angulo ODM e&longs;t æqualis; cum à lineis contingentibus, <lb/>circumferentii&longs;qué æqualibus contineatur: totus ergo angulus <lb/>VEG angulo SDM æqualis erit. </s> <s id="id.2.1.13.1.2.6.0">Auferatur ab angulo SDM <lb/>angulus curuilineus MDG; ab angulo autem VEG angulus au­<lb/>feratur VES; & angulus VES rectilineus maior e&longs;t curuilineo <lb/>MDG; erit reliquus angulus SEG minor angulo SDG. </s> <s id="id.2.1.13.1.2.6.0.a"><lb/>Quare ex ip&longs;orum &longs;uppo&longs;itionibus non &longs;olum pondus in D gra­<lb/>uius erit pondere in E; verùm è conuer&longs;o, pondus in E ip&longs;o D <lb/>grauius exi&longs;tet. </s> </p> <pb xlink:href="036/01/030.jpg"/> <p id="id.2.1.13.3.0.0.0" type="main"> <s id="id.2.1.13.3.1.1.0">Rationes tamen af<lb/>ferunt, quibus demon<lb/>&longs;trare nituntur, libram <lb/>DE in AB horizon­<lb/>ti æquidi&longs;tantem ex <lb/>nece&longs;sitate redire. </s> <s id="id.2.1.13.3.1.2.0"><expan abbr="Pri­mùm">Pri­<lb/>mum</expan> quidem o&longs;ten­<lb/>dunt, idem pondus <lb/>grauius e&longs;&longs;e in A, <lb/>quàm in alio &longs;itu, quem <lb/>æqualitatis &longs;itum no­<lb/>minant, cum linea <lb/>AB &longs;it horizonti æ­<lb/><figure id="id.036.01.030.1.jpg" xlink:href="036/01/030/1.jpg"></figure><lb/>quidi&longs;tans. </s> <s id="id.2.1.13.3.1.3.0">deinde quò propius e&longs;t ip&longs;i A, quouis alio remotiori <lb/>grauius e&longs;&longs;e. </s> <s id="id.2.1.13.3.1.4.0">Vt pondus in A grauius e&longs;&longs;e, quàm in D; & in D, <lb/>quàm in L. &longs;imiliter in A grauius, quam in N; & in N grauius, <lb/>quàm in M. </s> <s id="id.2.1.13.3.1.4.0.a">Vnum tantùm con&longs;iderando pondus in altero libræ <lb/><arrow.to.target n="note22"></arrow.to.target>brachio &longs;ur&longs;um deor&longs;umq; moto. </s> <s id="id.2.1.13.3.1.5.0">Quia (inquiunt) po&longs;ita trutina <lb/>in CF, pondus in A longius e&longs;t à trutina, quàm in D: & in D <lb/>longius, quàm in L. ductis enim DO LP ip&longs;i CF perpendicula­<lb/><arrow.to.target n="note23"></arrow.to.target>ribus, linea AC maior e&longs;t, quàm DO, & DO ip&longs;a LP. quod <lb/><arrow.to.target n="note24"></arrow.to.target>idem euenit in punctis NM. </s> <s id="id.2.1.13.3.1.5.0.a">deinde ex quo loco (aiunt) pon<lb/>dus velocius mouetur, ibi grauius e&longs;t; velocius autem ex A, quàm <lb/>ab alio &longs;itu mouetur; ergo in A grauius e&longs;t. </s> <s id="id.2.1.13.3.1.6.0">&longs;imili modo, quò <lb/>propius e&longs;t ip&longs;i A, velocius quoque mouetur; ergo in D gra­<lb/><arrow.to.target n="note25"></arrow.to.target>uius erit, quàm in L. </s> <s id="id.2.1.13.3.1.6.0.a">Altera deinde cau&longs;a, quam ex rectiori, & obli<lb/><arrow.to.target n="note26"></arrow.to.target>quiori motu deducunt, e&longs;t; quò pondus in arcubus æqualibus re­<lb/>ctius de&longs;cendit, grauius e&longs;&longs;e videtur; cum pondus liberum, atq; <lb/><arrow.to.target n="note27"></arrow.to.target>&longs;olutum &longs;uaptè natura rectè moueatur; &longs;ed in A rectius de&longs;cen<lb/>dit; ergo in A grauius erit. </s> <s id="id.2.1.13.3.1.7.0">hocq; o&longs;tendunt accipiendo arcum <lb/>AN arcui LD æqualem; à puncti&longs;q; NL lineæ FG (quam <lb/>etiam directionis vocant) æquidi&longs;tantes ducantur NRLQ, quæ <lb/>lineas AB DO &longs;ecent in QR; & à puncto N ip&longs;i FG perpen<lb/>dicularis ducatur NT. rectèq; demon&longs;trant LQ ip&longs;i PO æqua<lb/>lem e&longs;&longs;e, & NR ip&longs;i CT; lineamq; NR ip&longs;a LQ maiorem e&longs;&longs;e. </s> <s id="id.2.1.13.3.1.8.0"><lb/>Quoniam autem de&longs;cen&longs;u; ponderis ex A v&longs;q; ad N per circum­<pb n="9" xlink:href="036/01/031.jpg"/>ferentiam AN maiorem portionem lineæ FG pertran&longs;it (quod <lb/>ip&longs;i vocant capere de directo) quàm de&longs;cen&longs;us ex L in D per cir<lb/>cumferentiam LD; cùm de&longs;cen&longs;us AN lineam CT pertran&longs;eat, <lb/>de&longs;cen&longs;us verò LD lineam PO; & CT maior e&longs;t PO; rectior erit <lb/>de&longs;cen&longs;us AN, quám de&longs;cen&longs;us LD. </s> <s id="id.2.1.13.3.1.8.0.a">grauius ergo erit pondus <lb/>in A, quàm in L, & in quouis alio &longs;itu. </s> <s id="id.2.1.13.3.1.9.0">eodemq; pror&longs;us <lb/>modo o&longs;tendunt, quò propius e&longs;t ip&longs;i A, grauius e&longs;&longs;e. </s> <s id="id.2.1.13.3.1.10.0"><lb/>Vt &longs;int circumferentiæ LD DA inter &longs;e &longs;e æquales, & à puncto <lb/>D ip&longs;i AB perpendicularis ducatur DR; erit DR ip&longs;i CO æqua <arrow.to.target n="note28"></arrow.to.target><lb/>lis. </s> <s id="id.2.1.13.3.1.11.0">lineam deinde DR ip&longs;a LQ maiorem e&longs;&longs;e demon&longs;trant. </s> <s id="id.2.1.13.3.1.12.0">di­<lb/>cuntq; de&longs;cen&longs;um DA magis capere de directo de&longs;cen&longs;u LD, ma<lb/>ior enim e&longs;t linea CO, quàm OP; quare pondus grauius erit <lb/>in D, quàm in L. quod ip&longs;um euenit in punctis NM. </s> <s id="id.2.1.13.3.1.12.0.a">Suppo­<lb/>&longs;itionem itaq;, qua libram DE in AB redire demon&longs;trant, vt <arrow.to.target n="note29"></arrow.to.target><lb/>notam, manife&longs;tamq; proferunt. </s> <s id="id.2.1.13.3.1.13.0">Nempè Secundùm &longs;itum pon<lb/>dus grauius e&longs;&longs;e, quanto in eodem &longs;itu minus obliquus e&longs;t de&longs;cen<lb/>&longs;us. </s> <s id="id.2.1.13.3.1.14.0">huiu&longs;q; reditus cau&longs;am eam e&longs;&longs;e dicunt; Quoniam &longs;cilicet <arrow.to.target n="note30"></arrow.to.target><lb/>de&longs;cen&longs;us ponderis in D rectior e&longs;t de&longs;cen&longs;u ponderis in E, cùm <lb/>minus capiat de directo pondus in E de&longs;cendendo, quàm pon<arrow.to.target n="note31"></arrow.to.target><lb/>dus in D &longs;im liter de&longs;cendendo. </s> <s id="id.2.1.13.3.1.15.0">Vt &longs;i arcus EV &longs;it ip&longs;i DA <lb/>æqualis, ducanturq; VH ET ip&longs;i FG perpendiculares; maior <lb/>erit DR, quàm TH. quare per &longs;uppo&longs;itionem pondus in D ra<lb/>tione &longs;itus grauius erit pondere in E. </s> <s id="id.2.1.13.3.1.15.0.a">pondus ergo in D, cùm &longs;it <lb/>grauius, deor&longs;um mouebitur; pondus verò in E &longs;ur&longs;um, donec li<lb/>bra DE in AB redeat. </s> </p> <p id="id.2.1.14.1.0.0.0" type="margin"> <s id="id.2.1.14.1.1.1.0"><margin.target id="note22"></margin.target><emph type="italics"/>Cardanus primo de &longs;ubtilitate. <emph.end type="italics"/></s> <s id="id.2.1.14.1.1.2.0"><margin.target id="note23"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15. <emph type="italics"/>tertii.<emph.end type="italics"/></s> <s id="id.2.1.14.1.1.3.0"><margin.target id="note24"></margin.target><emph type="italics"/>Cardanus. <emph.end type="italics"/></s> <s id="id.2.1.14.1.1.4.0"><margin.target id="note25"></margin.target><emph type="italics"/>Cardanus. <emph.end type="italics"/></s> <s id="id.2.1.14.1.1.5.0"><margin.target id="note26"></margin.target><emph type="italics"/>Iordanus propo&longs;itio ne<emph.end type="italics"/> 4. </s> <s id="id.2.1.14.1.1.6.0"><margin.target id="note27"></margin.target><emph type="italics"/>Tartalea propo&longs;itione<emph.end type="italics"/> 5. </s> <s id="id.2.1.14.1.1.7.0"><margin.target id="note28"></margin.target>34 <emph type="italics"/>Primi. <emph.end type="italics"/></s> <s id="id.2.1.14.1.1.8.0"><margin.target id="note29"></margin.target><emph type="italics"/>Iordanus &longs;uppo&longs;itione<emph.end type="italics"/> 4. </s> <s id="id.2.1.14.1.1.9.0"><margin.target id="note30"></margin.target><emph type="italics"/>Iordanus propo&longs;itio ne<emph.end type="italics"/> 3. </s> <s id="id.2.1.14.1.1.10.0"><margin.target id="note31"></margin.target><emph type="italics"/>Tartalea propo&longs;itio ne<emph.end type="italics"/> 5. </s> </p> <p id="id.2.1.15.1.0.0.0" type="main"> <s id="id.2.1.15.1.1.1.0">Altera huius quoq; reditus ratio e&longs;t, cùm trutina &longs;upra libram <arrow.to.target n="note32"></arrow.to.target><lb/>e&longs;t in CF; linea CG e&longs;t meta. </s> <s id="id.2.1.15.1.1.2.0">& quoniam angulus GCD ma<lb/>ior e&longs;t angulo GCE, & maior à meta angulus grauius reddit <lb/>pondus; trutina igitur &longs;uperius exi&longs;tente, grauius erit pondus in <lb/>D, quàm in E. idcirco D in A, & E in B redibit. </s> </p> <p id="id.2.1.16.1.0.0.0" type="margin"> <s id="id.2.1.16.1.1.1.0"><margin.target id="note32"></margin.target><emph type="italics"/>Cardanus.<emph.end type="italics"/></s> </p> <p id="id.2.1.17.1.0.0.0" type="main"> <s id="id.2.1.17.1.1.1.0">His itaq; rationibus conantur o&longs;tendere libram DE in AB re<lb/>dire; quæ meo quidem iuditio facile &longs;olui po&longs;&longs;unt. </s> </p> <pb xlink:href="036/01/032.jpg"/> <p id="id.2.1.17.3.0.0.0" type="main"> <s id="id.2.1.17.3.1.1.0">Primùm itaq; quan<lb/>tum attinet ad ratio­<lb/>nes pondus in A gra<lb/>uius e&longs;&longs;e, quàm in a­<lb/>lio &longs;itu o&longs;tendentes, <lb/>quas ex longiori, & <lb/>propinquiori <expan abbr="di&longs;tãtia">di&longs;tantia</expan> à <lb/>linea FG, & ex velo­<lb/>ciori, & rectiori mo <lb/>tu à puncto A dedu­<lb/>cunt; primùm quidem <lb/>non demon&longs;trant, cur <lb/>pondus ex A velocius <lb/><figure id="id.036.01.032.1.jpg" xlink:href="036/01/032/1.jpg"></figure><lb/>moueatur, quàm ex alio &longs;itu. </s> <s id="id.2.1.17.3.1.2.0">nec quia CA e&longs;t DO maior, <lb/>& DO ip&longs;a LP, propterea &longs;equitur tanquam ex vera cau&longs;a, pon<lb/>dus in A grauius e&longs;&longs;e, quàm in D; & in D, quàm in L. </s> <s id="id.2.1.17.3.1.2.0.a">neq; <lb/>enim intellectus quie&longs;cit, ni&longs;i alia huius o&longs;tendatur cau&longs;a; cùm po<lb/>tius &longs;ignum, quàm vera cau&longs;a e&longs;&longs;e videatur. </s> <s id="id.2.1.17.3.1.3.0">id ip&longs;um quoq; al­<lb/>teri rationi contintingit, quam ex rectiori & obliquiori motu de­<lb/>ducunt. </s> <s id="id.2.1.17.3.1.4.0">Præterea quæcunq; ex velociori, & rectiori motu per­<lb/>&longs;uadent pondus in A grauius e&longs;&longs;e, quàm in D; non ideo de­<lb/>mon&longs;trant pondus in A, quatenus e&longs;t in A, grauius e&longs;&longs;e pon<lb/>dere in D, quatenus e&longs;t in D; &longs;ed quatenus à punctis DA rece<lb/>dit. </s> <s id="id.2.1.17.3.1.5.0">Idcirco antequàm vlterius progrediar, o&longs;tendam primùm <lb/>pondus, quò propius e&longs;t ip&longs;is FG, minus grauitare; tum qua­<lb/>tenus in eo &longs;itu, in quo reperitur, manet: tum quatenus ab eo <lb/>recedit. </s> <s id="id.2.1.17.3.1.6.0">&longs;imulq; fal&longs;um e&longs;&longs;e, pondus in A grauius e&longs;&longs;e, quàm in <lb/>alio &longs;itu. </s> </p> <pb n="10" xlink:href="036/01/033.jpg"/> <p id="id.2.1.17.5.0.0.0" type="main"> <s id="id.2.1.17.5.1.1.0">Producatur FG v&longs;q; ad mundi cen<lb/>trum, quod &longs;it S. & à puncto S circu<lb/>lum AFBG contingens ducatur. </s> <s id="id.2.1.17.5.1.2.0">neq; <lb/>enim linea à puncto S circulum con­<lb/>tingere pote&longs;t in A; nam ducta AS <lb/>triangulum ACS duos haberet angu<lb/>los rectos, nempè SAC ACS, quod <arrow.to.target n="note33"></arrow.to.target><lb/>e&longs;t impo&longs;sibile. </s> <s id="id.2.1.17.5.1.3.0">neq; &longs;upra punctum A <lb/>in circumferentia AF continget; cir<lb/>culum enim &longs;ecaret. </s> <s id="id.2.1.17.5.1.4.0">tanget igitur in­<lb/>fra, &longs;itq; SO. connectantur deinde SD <lb/>SL, quæ circumferentiam AOG in <lb/>punctis KH &longs;ecent. </s> <s id="id.2.1.17.5.1.5.0">& Ck CH con<lb/>iungantur. </s> <s id="id.2.1.17.5.1.6.0">Et quoniam pondus, quanto <lb/>propius e&longs;t ip&longs;i F, magis quoque inni­<lb/>titur centro; vt pondus in D magis ver­<lb/>&longs;ionis puncto C innititur tanquam <lb/>centro; hoc e&longs;t in D magis &longs;upra li­<lb/>neam CD grauitat, quàm &longs;i e&longs;&longs;et in A <lb/>&longs;upra lineam CA; & adhuc magis in <lb/>L &longs;upra lineam CL; Nam cùm tres <lb/>anguli cuiu&longs;cunq; trianguli duobus re­<lb/><figure id="id.036.01.033.1.jpg" xlink:href="036/01/033/1.jpg"></figure><lb/>ctis &longs;int æquales, & trianguli DCk æquicruris angulus DCk <lb/>minor &longs;it angulo LCH æquicruris trianguli LCH: erunt reli­<lb/>qui ad ba&longs;im &longs;cilicet CDk CkD &longs;imul &longs;umpti reliquis CLH <lb/>CHL maiores. </s> <s id="id.2.1.17.5.1.7.0">& horum dimidii; hoc e&longs;t angulus CDS angu<lb/>lo CLS maior erit. </s> <s id="id.2.1.17.5.1.8.0">cùm itaq; CLS &longs;it minor, linea CL ma<lb/>gis adhærebit motui naturali ponderis in L pror&longs;us &longs;oluti. </s> <s id="id.2.1.17.5.1.9.0">hoc <lb/>e&longs;t lineæ LS, quàm CD motui DS. </s> <s id="id.2.1.17.5.1.9.0.a">pondus enim in L <expan abbr="libe">li</expan>­<lb/>berum, atq; &longs;olutum in centrum mundi per LS moueretur, pon­<lb/>dusq; in D per DS. </s> <s id="id.2.1.17.5.1.9.0.b">quoniam verò pondus in L totum &longs;uper LS <lb/>grauitat, in D verò &longs;uper DS: pondus in L magis &longs;upra lineam <lb/>CL grauitabit, quàm exi&longs;tens in D &longs;upra lineam DC. ergo <lb/>linea CL pondus magis &longs;u&longs;tentabit, quàm linea CD. </s> <s id="id.2.1.17.5.1.9.0.c"><expan abbr="Eodem­qué">Eodem­<lb/>que</expan> modo, quò pondus propius fuerit ip&longs;i F, magis ob hanc cau­<lb/>&longs;am à linea CL &longs;u&longs;tineri o&longs;tendetur; &longs;emper enim angulus CLS <pb xlink:href="036/01/034.jpg"/>minor e&longs;&longs;et. </s> <s id="id.2.1.17.5.1.10.0">quod etiam patet; quia &longs;i <lb/>lineæ CL, & LS in vnam coinciderent <lb/>lineam, quod euenit in FCS; tunc linea <lb/>CF totum &longs;u&longs;tineret pondus in F, im­<lb/>mobilemq; redderet: neq; vllam pror­<lb/>&longs;us grauitatem in circumferentia circu­<lb/>li haberet. </s> <s id="id.2.1.17.5.1.11.0">Idem ergo pondus propter <lb/>&longs;ituum diuer&longs;itatem grauius, leuiu&longs;q; erit. </s> <s id="id.2.1.17.5.1.12.0"><lb/>non autem quia ratione &longs;itus interdum <lb/>maiorem re vera acquirat grauitatem, <lb/>interdum verò amittat, cùm eiu&longs;dem &longs;it <lb/>&longs;emper grauitatis, vbicunque reperiatur; <lb/>&longs;ed quia magis, minu&longs;uè in circumferen­<lb/>tia grauitat, vt in D magis &longs;upra circum<lb/>ferentiam DA grauitat, quàm in L &longs;upra <lb/>circumferentiam LD. </s> <s id="id.2.1.17.5.1.12.0.a">hoc e&longs;t, &longs;i pon<lb/>dus à circumferentiis, recti&longs;q; lineis &longs;u<lb/>&longs;tineatur; circumferentia AD magis &longs;u<lb/>&longs;tinebit pondus in D, quàm circumfe<lb/>rentia DL pondere exi&longs;tente in <emph type="italics"/>L.<emph.end type="italics"/> mi<lb/>nus enim coadiuuat CD, quàm CL. </s> <s id="id.2.1.17.5.1.12.0.b"><lb/>Præterea quando pondus e&longs;t in L, &longs;i e&longs;­<lb/><figure id="id.036.01.034.1.jpg" xlink:href="036/01/034/1.jpg"></figure><lb/>&longs;et omnino liberum, penitu&longs;q; &longs;olutum, deor&longs;um per LS moueretur; <lb/>ni&longs;i à linea CL prohiberetur, quæ pondus in L vltra lineam LS per <lb/><expan abbr="circumferentiã">circumferentiam</expan> LD moueri cogit; ip&longs;umq; quodammodo impellit, <lb/>impellendoq; pondus partim &longs;u&longs;tentabit. </s> <s id="id.2.1.17.5.1.13.0">ni&longs;i enim &longs;u&longs;tineret, ip&longs;iq; <lb/>reniteretur, deor&longs;um per lineam LS moueretur, non autem per <lb/>circumferentiam LD. &longs;imiliter CD ponderi in D renititur, cùm <lb/>illud per circumferentiam DA moueri cogat. </s> <s id="id.2.1.17.5.1.14.0">eodemq; modo <lb/>exi&longs;tente pondere in A, linea CA pondus vltra lineam AS per <lb/>circumferentiam AO moueri compellet. </s> <s id="id.2.1.17.5.1.15.0">e&longs;t enim angulus CAS <lb/>acutus; cùm angulus ACS &longs;it rectus. </s> <s id="id.2.1.17.5.1.16.0">lineæ igitur CA CD ali<lb/>qua ex parte, non tamen ex æquo ponderi renituntur. </s> <s id="id.2.1.17.5.1.17.0">& quotie&longs; <lb/>cunque angulus in circumferentia circuli à lineis à centro <lb/>mundi S, & centro C prodeuntibus, fuerit acutus; idem eue­<lb/>nire &longs;imiliter o&longs;tendemus. </s> <s id="id.2.1.17.5.1.18.0">Quoniam autem mixtus angulus CLD <pb n="11" xlink:href="036/01/035.jpg"/>æqualis e&longs;t angulo CDA, cùm à &longs;emidiametris, eademq; circumfe<lb/>rentia contineantur; & angulus C<emph type="italics"/>L<emph.end type="italics"/>S angulo CDS e&longs;t minor; <lb/>erit reliquus <emph type="italics"/>S<emph.end type="italics"/>LD reliquo SDA maior. </s> <s id="id.2.1.17.5.1.19.0">quare circumferentia <lb/>DA, hoc e&longs;t de&longs;cen&longs;us ponderis in D propior erit motui natu­<lb/>rali ponderis in D &longs;oluti, lineæ &longs;cilicet DS, quàm circumferen<lb/>tia LD lineæ LS. </s> <s id="id.2.1.17.5.1.19.0.a">minus igitur linea CD ponderi in D reniti­<lb/>tur, quàm linea CL ponderi in L. </s> <s id="id.2.1.17.5.1.19.0.b">linea ideo CD minus &longs;u&longs;tinet, <lb/>quàm CL; pondu&longs;q; magis liberum erit in D, quàm in L: <lb/>cùm pondus naturaliter magis per DA moueatur, quàm per LD. <lb/>quare grauius erit in D, quàm in L. &longs;imiliter o&longs;tendemus CA <lb/>minus &longs;u&longs;tinere, quàm CD: pondu&longs;q; magis in A, quàm in D li­<lb/>berum, grauiu&longs;q, e&longs;&longs;e. </s> <s id="id.2.1.17.5.1.20.0">Ex parte deinde inferiori ob ea&longs;dem cau&longs;as, <lb/>quò pondus propius fuerit ip&longs;i G, magis detinebitur, vt in H ma<lb/>gis à linea CH, quàm in K à linea CK. nam cùm angulus CHS <lb/>maior &longs;it angulo CkS, ad rectitudinem magis appropinquabunt <arrow.to.target n="note34"></arrow.to.target><lb/>&longs;e &longs;e lineæ CH HS, quàm Ck kS; atq; ob id pondus magis deti­<lb/>nebitur à CH, quàm à Ck &longs;i enim CH HS in vnam conuenirent <lb/>lineam vt euenit pondere exi&longs;tente in G; tunc linea CG totum &longs;u<lb/>&longs;tineret' pondus in G, ita vt immobilis per&longs;i&longs;teret. </s> <s id="id.2.1.17.5.1.21.0">quò igitur <lb/>minor erit angulus linea CH, & de&longs;cen&longs;u ponderis &longs;oluti, &longs;cilicet <lb/>HS contentus, eò minus quoq; eiu&longs;modi linea pondus detinebit. </s> <s id="id.2.1.17.5.1.22.0"><lb/>& vbi minus detinebitur, ibi magis liberum, grauiu&longs;q; exi&longs;tet. </s> <s id="id.2.1.17.5.1.23.0"><lb/>Præterea &longs;i pondus in k liberum e&longs;&longs;et, atq; &longs;olutum, per lineam <lb/>k S moueretur; à linea verò Ck prohibetur, quæ cogit pondus <lb/>citrà lineam k S per circumferentiam k H moueri. </s> <s id="id.2.1.17.5.1.24.0">ip&longs;um enim <lb/>quodammodo retrahit, retrahendoq; &longs;u&longs;tinet. </s> <s id="id.2.1.17.5.1.25.0">ni&longs;i enim &longs;u&longs;tineret. </s> <s id="id.2.1.17.5.1.26.0"><lb/>pondus deor&longs;um per rectam k S moueretur, non autem per cir<lb/>cumferentiam k H. &longs;imiliter CH pondus retinet, cùm per circum<lb/><expan abbr="ferentiã">ferentiam</expan> HG moueri compellat. </s> <s id="id.2.1.17.5.1.27.0"><expan abbr="Quoniã">Quoniam</expan> autem angulus CHS ma­<lb/>ior e&longs;t angulo CKS, <expan abbr="d&etilde;ptis">demptis</expan> æqualibus angulis CHG CkH; erit <lb/>reliquus SHG reliquo SKH maior. </s> <s id="id.2.1.17.5.1.28.0">circumferentia igitur k H, hoc <lb/>e&longs;t de&longs;cen&longs;us ponderis in k, propior erit motui naturali ponderis in <lb/>k &longs;oluti, hoc e&longs;t lineæ k S, quàm circumferentia HG lineæ HS. mi <lb/>nus idcirco detinet linea Ck, quàm CH: cùm pondus naturali­<lb/>ter magis moueatur per k H, quàm per HG. </s> <s id="id.2.1.17.5.1.28.0.a">&longs;imili ratione o&longs;ten­<lb/>detur, quò minor erit angulus SkH, lineam Ck minus &longs;u&longs;tinere. </s> <s id="id.2.1.17.5.1.29.0"><pb xlink:href="036/01/036.jpg"/>exi&longs;tente igitur pondere in O, quia angu<lb/>lus SOC non &longs;olum minor e&longs;t angulo <lb/>CKS, verùm etiam omnium angulorum <lb/>à punctis CS prodeuntium, verticemq; <lb/>in circumferuntia OkG habentium mi­<lb/>nimus; erit <expan abbr="anglus">angulus</expan> SOK, & angulo SkH, <lb/>& eiu&longs;modi omnium minimus. </s> <s id="id.2.1.17.5.1.30.0">ergo de­<lb/>&longs;cen&longs;us ponderis in O propior erit motui <lb/>naturali ip&longs;ius in O &longs;oluti, quàm in alio <lb/>&longs;itu circumferentiæ OkG. </s> <s>lineaq; CO <lb/>minus pondus &longs;u&longs;tinebit, quàm &longs;i pon­<lb/>dus in quouis alio fuerit &longs;itu eiu&longs;dem cir<lb/>cumferentiæ OG. </s> <s id="id.2.1.17.5.1.30.0.a">&longs;imiliter quoniam con<lb/>tingentiæ angulus SOk, & angulo SDA, <lb/>& SAO, ac quibu&longs;cunq; &longs;imilibus e&longs;t mi <lb/>nor; erit de&longs;cen&longs;us ponderis in O motui <lb/>naturali ip&longs;ius ponderis in O &longs;oluti pro­<lb/>pior, quàm in alio &longs;itu circumferentiæ <lb/>ODF. </s> <s id="id.2.1.17.5.1.30.0.b">Præterea quoniam linea GO pon<lb/>dus in O dum deor&longs;um mouetur, impelle­<lb/>re non pote&longs;t, ita vt vltra lineam OS mo<lb/>ueatur; cùm linea OS circulum non &longs;ecet, <lb/><figure id="id.036.01.036.1.jpg" xlink:href="036/01/036/1.jpg"></figure><lb/>&longs;ed contingat; angulu&longs;q; SOC &longs;it rectus, & non acutus; pondus <lb/>in O nihil &longs;upra lineam CO grauitabit. </s> <s id="id.2.1.17.5.1.31.0">neq; centro innitetur. </s> <s id="id.2.1.17.5.1.32.0">quem <lb/>admodum in quouis alio puncto &longs;upra O accideret. </s> <s id="id.2.1.17.5.1.33.0">erit igitur pon<lb/>dus in O magis ob has cau&longs;as liberum, atq; &longs;olutum in hoc &longs;itu, <lb/>quàm in quouis alio circumferentiæ FOG. </s> <s>ac idcirco in hoc <lb/>grauius erit, hoc e&longs;t magis grauitabit, quàm in alio &longs;itu. </s> <s id="id.2.1.17.5.1.34.0">& quò <lb/>propius fuerit ip&longs;i O remotiori grauius erit. </s> <s id="id.2.1.17.5.1.35.0">lineaq; CO horizonti <lb/>æquidi&longs;tans erit. </s> <s id="id.2.1.17.5.1.36.0">non tamen puncti C horizonti (vt ip&longs;i exi&longs;ti­<lb/>mant) &longs;ed ponderis in O con&longs;tituti, cùm ex centro grauitatis <lb/>ponderis &longs;ummendus &longs;it horizon. </s> <s id="id.2.1.17.5.1.37.0">quæ omnia demon&longs;trare opor­<lb/>tebat. </s> </p> <p id="id.2.1.18.1.0.0.0" type="margin"> <s id="id.2.1.18.1.1.1.0"><margin.target id="note33"></margin.target>18 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.18.1.1.2.0"><margin.target id="note34"></margin.target>21 <emph type="italics"/>primi.<emph.end type="italics"/></s> </p> <pb n="12" xlink:href="036/01/037.jpg"/> <p id="id.2.1.19.1.0.0.0" type="main"> <s id="id.2.1.19.1.2.1.0">Si autem libræ brachium ip&longs;o CO <lb/>fuerit maius, putá quantitate CD; erit <lb/>quoq; pondus in O grauius. </s> <s id="id.2.1.19.1.2.2.0">circulus de­<lb/>&longs;cribatur OH, cuius centrum &longs;it D, &longs;e<arrow.to.target n="note35"></arrow.to.target><lb/>midiameterq; DO. </s> <s>tanget circulus OH <lb/>circulum FOG in puncto O, lineamq; <arrow.to.target n="note36"></arrow.to.target><lb/>OS, quæ ponderis in O rectus, natura­<lb/>li&longs;q; e&longs;t de&longs;cen&longs;us, in eodem puncto con<lb/>tinget. </s> <s id="id.2.1.19.1.2.3.0">& quoniam angulus SOH mi­<lb/>nor e&longs;t angulo SOG, erit de&longs;cen&longs;us <lb/>ponderis in O per circumferentiam OH <lb/>motui naturali OS propior, quàm per <lb/>circumferentiam OG. </s> <s id="id.2.1.19.1.2.3.0.a">magis ergo li­<lb/>berum, atq; &longs;olutum, ac per con&longs;equens <lb/>grauius erit in O, centro libræ exi&longs;ten<lb/>te in D, quàm in C. &longs;imiliter o&longs;ten­<lb/>detur, quò maius fuerit brachium DO, <lb/>pondus in O adhuc grauius e&longs;&longs;e. <figure id="id.036.01.037.1.jpg" xlink:href="036/01/037/1.jpg"></figure></s> </p> <pb xlink:href="036/01/038.jpg"/> <p id="id.2.1.19.3.0.0.0" type="main"> <s id="id.2.1.19.3.1.1.0">Si verò idem circulus AFBG, <lb/>cuius centrum &longs;it R, propius fuerit <lb/>mundi centro S; circulumqué à pun­<lb/>cto S ducatur contingens ST; punctum <lb/>T (vbi grauius e&longs;t pondus) magis <lb/>à puncto A di&longs;tabit, quàm punctum <lb/>O. ducantur enim à punctis OT ip&longs;i <lb/>CS perpendiculares OMTN; conne<lb/>ctanturq; RT; &longs;itq; centrum R in li­<lb/>nea CS; lineaq; ARB ip&longs;i ACB æqui <lb/><arrow.to.target n="note37"></arrow.to.target>di&longs;tans. </s> <s id="id.2.1.19.3.1.2.0">Quoniam igitur triangula COS <lb/>RTS &longs;unt rectangula; erit SC ad CO, <lb/>vt CO ad CM. </s> <s>&longs;imiliter SR ad RT, <lb/>vt RT ad RN. </s> <s>cùm itaq; &longs;it RT ip­<lb/><arrow.to.target n="note38"></arrow.to.target>&longs;i CO æqualis, & SC ip&longs;a SR maior: <lb/>maiorem habebit proportionem SC <lb/>ad CO, quàm SR ad RT. </s> <s>quare ma<lb/>iorem quoq; proportionem habebit <lb/>CO ad CM, quàm RT ad RN. </s> <s id="id.2.1.19.3.1.2.0.a">mi<lb/><arrow.to.target n="note39"></arrow.to.target>nor ergo erit CM, quàm RN. </s> <s>&longs;ecetur <lb/>igitur RN in P, ita vt RP &longs;it ip&longs;i <lb/><figure id="id.036.01.038.1.jpg" xlink:href="036/01/038/1.jpg"></figure><lb/>CM æqualis; & à puncto P ip&longs;is MONT æquidi&longs;tans ducatur <lb/>PQ, quæ circumferentiam AT &longs;ecet in Q: deniq; connectatur <lb/>RQ. </s> <s>quoniam enim duæ CO CM duabus RQRP &longs;unt æqua<lb/><arrow.to.target n="note40"></arrow.to.target>les, & angulus CMO angulo RPQ e&longs;t æqualis; erit & angu­<lb/>lus MCO angulo PRQ æqualis. </s> <s id="id.2.1.19.3.1.3.0">angulus autem MCA rectus <lb/><arrow.to.target n="note41"></arrow.to.target>recto PRA e&longs;t æqualis; ergo reliquus OCA reliquo QRA <lb/>æqualis, & circumferentia OA circumferentiæ QA æqualis quo­<lb/>que erit. </s> <s id="id.2.1.19.3.1.4.0">punctum idcirco T, quia magis à puncto A di&longs;tat, <lb/>quàm Q; magis quoq; à puncto A di&longs;tabit, quàm punctum O. <lb/></s> <s>&longs;imiliter o&longs;tendetur, quò propius fuerit circulus mundi centro, eun­<lb/>dem magis di&longs;tare. </s> <s id="id.2.1.19.3.1.5.0">atq; ita vt prius demon&longs;trabitur pondus in cir<lb/>cumferentia TAF centro R inniti, in circumferentia verò TG <lb/>à linea detineri; atq; in puncto T grauius e&longs;&longs;e. </s> </p> <p id="id.2.1.20.1.0.0.0" type="margin"> <s id="id.2.1.20.1.1.1.0"><margin.target id="note35"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 11 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.20.1.1.2.0"><margin.target id="note36"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 18 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.20.1.1.3.0"><margin.target id="note37"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 8 <emph type="italics"/>&longs;exti<emph.end type="italics"/></s> <s id="id.2.1.20.1.1.4.0"><margin.target id="note38"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 8 <emph type="italics"/>quinti<emph.end type="italics"/></s> <s id="id.2.1.20.1.1.5.0"><margin.target id="note39"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 10 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.20.1.1.6.0"><margin.target id="note40"></margin.target>7 <emph type="italics"/>Sexti.<emph.end type="italics"/></s> <s id="id.2.1.20.1.1.7.0"><margin.target id="note41"></margin.target>26 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> </p> <pb n="13" xlink:href="036/01/039.jpg"/> <p id="id.2.1.21.1.0.0.0" type="main"> <s id="id.2.1.21.1.2.1.0">Si autem punctum G e&longs;&longs;et <lb/>in centro mundi; tunc quò <lb/>pondus propius fuerit ip&longs;i G, <lb/>grauius erit: & vbicunq; po<lb/>natur pondus præterquàm in <lb/>ip&longs;o G, &longs;emper centro C inni<lb/>tetur, vt in K. </s> <s>nam ducta <lb/>G k, efficiet hæc (&longs;ecun­<lb/>dùm quam fit ponderis natu<lb/>ralis motus) vná cum libræ <lb/>brachio k C angulum acu­<lb/>tum. </s> <s id="id.2.1.21.1.2.2.0">æquicruris enim trian­<lb/>guli CkG ad ba&longs;im anguli <lb/>ad k, & G &longs;unt &longs;emper acuti. </s> <s id="id.2.1.21.1.2.3.0"><lb/><figure id="id.036.01.039.1.jpg" xlink:href="036/01/039/1.jpg"></figure><lb/>Conferantur autem inuicem hæc duo, pondus videlicet in k, & <lb/>pondus in D: erit pondus in k grauius, quàm in D. nam iuncta <lb/>DG, cùm tres anguli cuiu&longs;cunque trianguli duobus &longs;int rectis <lb/>æquales, & trianguli CDG æquicruris angulus DCG maior &longs;it <lb/>angulo kCG æquicruris trianguli CkG: erunt reliqui ad ba&longs;im an<lb/>guli DGC GDC &longs;imul &longs;umpti reliquis KGCGkC &longs;imul &longs;umptis <lb/>minores. </s> <s id="id.2.1.21.1.2.4.0">horumq; dimidii; angulus &longs;cilicet CDG angulo CKG <lb/>minor erit. </s> <s id="id.2.1.21.1.2.5.0">quare cùm pondus in k &longs;olutum naturaliter per <lb/>KG moueatur, pondusq; in D per DG, tanquam per &longs;patia, <lb/>quibus in centrum mundi feruntur; linea CD, hoc e&longs;t libræ <lb/>brachium magis adhærebit motui naturali ponderis in D pror­<lb/>&longs;us &longs;oluti, lineæ &longs;cilicet DG; quàm Ck motui &longs;ecundùm kG <lb/>effecto. </s> <s id="id.2.1.21.1.2.6.0">magis igitur &longs;u&longs;tinebit linea CD, quàm Ck. </s> <s id="id.2.1.21.1.2.7.0">ac pro­<lb/>pterea pondus in k ex &longs;uperius dictis grauius erit, quàm in D. </s> <s id="id.2.1.21.1.2.7.0.a"><lb/>Præterea quoniam pondus in K &longs;i e&longs;&longs;et omnino liberum, pror&longs;u&longs;q; <lb/>&longs;olutum, deor&longs;um per k G moueretur; ni&longs;i à linea C k prohibere<lb/>tur, quæ pondus vltra lineam KG per circumferentiam KH mo­<lb/>ueri cogit; linea C k pondus partim &longs;u&longs;tinebit, ip&longs;iq; renitetur; <lb/>cùm illud per circumferentiam k H moueri compellat. </s> <s id="id.2.1.21.1.2.8.0">& <lb/>quoniam angulus CDG minor e&longs;t angulo CkG, & angulus CDk <lb/>angulo CkH e&longs;t æqualis; erit reliquus GDk reliquo G k H maior. </s> <s id="id.2.1.21.1.2.9.0"><lb/>circumferentia igitur k H motui naturali ponderis in k &longs;oluti, li­<pb xlink:href="036/01/040.jpg"/>neæ &longs;cilicet KG propior erit, <lb/>quàm circumferentia Dk li­<lb/>neæ DG. quare linea CD <lb/>ponderi in D magis renititur, <lb/>quàm linea C k ip&longs;i ponde­<lb/>ri in K. </s> <s id="id.2.1.21.1.2.9.0.a">ergo pondus in k <lb/>grauius erit, quàm in D. </s> <s id="id.2.1.21.1.2.9.0.b"><lb/>Similiter o&longs;tendetur pondus, <lb/>quò fuerit ip&longs;i F propius, vt <lb/>in L, minus grauitare: pro­<lb/>pius verò ip&longs;i G, vt in H, <lb/>grauius e&longs;&longs;e. <figure id="id.036.01.040.1.jpg" xlink:href="036/01/040/1.jpg"></figure></s> </p> <p id="id.2.1.21.2.0.0.0" type="main"> <s id="id.2.1.21.2.1.1.0">Si verò centrum mundi <lb/>S e&longs;&longs;et inter puncta CG; <lb/>primùm quidem &longs;imili­<lb/>ter o&longs;tendetur pondus vbi <lb/>cunq; po&longs;itum centro C <lb/>initi, vt in H. </s> <s>ductis enim <lb/>HG HS, angulus ad <lb/>ba&longs;im GHC æquicruris tri <lb/>anguli CHG e&longs;t &longs;emper <lb/>acutus: quare & SHC ip<lb/>&longs;o minor erit quoq; &longs;em<lb/>per acutus. </s> <s id="id.2.1.21.2.1.2.0">ducatur au­<lb/>tem à puncto S ip&longs;i CS <lb/>perpendicularis Sk. </s> <s id="id.2.1.21.2.1.3.0">di­<lb/><figure id="id.036.01.040.2.jpg" xlink:href="036/01/040/2.jpg"></figure><lb/>co pondus grauius e&longs;&longs;e in k, quàm in alio &longs;itu circumferentiæ FKG. <lb/>& quò propius fuerit ip&longs;i F, vel G, minus grauitare. </s> <s id="id.2.1.21.2.1.4.0">Accipiantur <lb/>ver&longs;us F puncta DL, connectanturq; LC LS DC DS, produ­<lb/>canturq; LS DS k SHS v&longs;q; ad circuli circumferentiam in EM <lb/>NO; connectanturq; CE, CM, CN, CO. </s> <s id="id.2.1.21.2.1.4.0.a">Quoniam enim <lb/><arrow.to.target n="note42"></arrow.to.target>LE DM &longs;e inuicem &longs;ecant in S; erit rectangulum LSE rectan­<lb/><arrow.to.target n="note43"></arrow.to.target>gulo DSM æquale. </s> <s id="id.2.1.21.2.1.5.0">quare vt LS ad DS ita erit SM <lb/><arrow.to.target n="note44"></arrow.to.target>ad SE. </s> <s id="id.2.1.21.2.1.5.0.a">maior autem e&longs;t LS, quàm DS; & SM ip&longs;a SE. </s> <s id="id.2.1.21.2.1.5.0.b"><pb n="14" xlink:href="036/01/041.jpg"/>ergo LS SE &longs;imul &longs;umptæ ip&longs;is DS SM maiores erunt. </s> <s id="id.2.1.21.2.1.6.0">eademq; <arrow.to.target n="note45"></arrow.to.target><lb/>ratione kN minorem e&longs;&longs;e DM o&longs;tendetur. </s> <s id="id.2.1.21.2.1.7.0">rur&longs;us quoniam re<lb/>ctangulum OSH æquale e&longs;t rectangulo kSN; ob eandem cau&longs;am <lb/>HO maior erit kN. </s> <s>eodemq; pror&longs;us modo kN omnibus a­<lb/>liis per punctum S tran&longs;euntibus minorem e&longs;&longs;e demon&longs;trabitur. </s> <s id="id.2.1.21.2.1.8.0"><lb/>& quoniam æquicrurium triangulorum CLE DCM latera LC <lb/>CE lateribus DC CM &longs;unt æqualia; ba&longs;is verò LE maior e&longs;t <lb/>DM: erit angulus LCE angulo DCM maior. </s> <s id="id.2.1.21.2.1.9.0">quare ad ba&longs;im <arrow.to.target n="note46"></arrow.to.target><lb/>anguli C<emph type="italics"/>L<emph.end type="italics"/>E CEL &longs;imul &longs;umpti angulis CDM CMD mi­<lb/>nores erunt. </s> <s id="id.2.1.21.2.1.10.0">& horum dimidii, angulus &longs;cilicet CLS angulo CDS <lb/>minor erit. </s> <s id="id.2.1.21.2.1.11.0">ergo pondus in <emph type="italics"/>L<emph.end type="italics"/> magis &longs;upra lineam LC, quàm <lb/>in D &longs;upra DC grauitabit. </s> <s id="id.2.1.21.2.1.11.0.a">magisqué centro innitetur in L, quàm <lb/>in D. </s> <s id="id.2.1.21.2.1.11.0.b">&longs;imiliter o&longs;tendetur in D magis <expan abbr="c&etilde;tro">centro</expan> C inniti, quàm in k. </s> <s id="id.2.1.21.2.1.12.0">ergo <lb/><expan abbr="ponds">pondus</expan> in k grauius erit, quàm in D; & in D, quàm in L. eademq; pror<lb/>&longs;us ratione quoniam kN minor e&longs;t HO, erit angulus CKS an­<lb/>gulo CHS maior. </s> <s id="id.2.1.21.2.1.13.0">quare pondus in H magis centro C innite­<lb/>tur, quàm in k. </s> <s id="id.2.1.21.2.1.14.0">& hoc modo o&longs;tendetur, vbicunq; in circum­<lb/>ferentia FDG fuerit pondus, minus in K centro C inniti, quàm <lb/>in alio &longs;itu: & quò propius fuerit ip&longs;i F, vel G, magis inniti. </s> <s id="id.2.1.21.2.1.15.0">dein­<lb/>de quoniam angulus CkS maior e&longs;t CDS, & CDk æqualis <lb/>e&longs;t CkH: erit reliquus SkH reliquo SDk minor. </s> <s id="id.2.1.21.2.1.16.0">quare cir­<lb/>cumferentia k H propior erit motui naturali recto ponderis in K <lb/>&longs;oluti, lineæ &longs;cilicet k S, quàm circumferentia D k motui DS. </s> <s id="id.2.1.21.2.1.16.0.a">& <lb/>ideo linea CD magis ip&longs;i ponderi in D renititur, quàm CK <lb/>ponderi in k con&longs;tituto. </s> <s id="id.2.1.21.2.1.17.0">hacq; ratione o&longs;tendetur angulum <lb/>SHG maiorem e&longs;&longs;e SkH: & per con&longs;equens lineam CH magis <lb/>ponderi in H reniti, quàm CK ponderi in K. </s> <s>&longs;imiliter demon­<lb/>&longs;trabitur lineam C<emph type="italics"/>L<emph.end type="italics"/> magis pondus &longs;u&longs;tinere, quàm CD: ob <lb/>ea&longs;demq; cau&longs;as o&longs;tendetur pondus in K minus &longs;upra lineam Ck <lb/>grauitare, quàm in quouis alio &longs;itu fuerit circumferentiæ FDG. <lb/></s> <s>& quò propius fuerit ip&longs;i F, vel G, minus grauitare. </s> <s id="id.2.1.21.2.1.18.0">grauius ergo <lb/>erit in k, quàm in alio &longs;itu: minu&longs;q; graue erit, quò propius fue­<lb/>rit ip&longs;i F, vel G. <pb xlink:href="036/01/042.jpg"/></s> </p> <p id="id.2.1.22.1.0.0.0" type="margin"> <s id="id.2.1.22.1.1.1.0"><margin.target id="note42"></margin.target>35 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.22.1.1.2.0"><margin.target id="note43"></margin.target>16 <emph type="italics"/>Sexti.<emph.end type="italics"/></s> <s id="id.2.1.22.1.1.3.0"><margin.target id="note44"></margin.target>7 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.22.1.1.4.0"><margin.target id="note45"></margin.target>25 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.22.1.1.5.0"><margin.target id="note46"></margin.target>25 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.23.1.0.0.0" type="main"> <s id="id.2.1.23.1.1.1.0">Si deniq; centrum C <lb/>e&longs;&longs;et in centro mundi, <lb/>pondus vbicunque con­<lb/>&longs;titutum manere mani­<lb/>fe&longs;tum e&longs;t. </s> <s id="id.2.1.23.1.1.2.0">vt po&longs;ito pon<lb/>dere in D, linea CD to­<lb/>tum &longs;u&longs;tinebit pondus; <lb/>cùm ip&longs;ius ponderis in D <lb/>horizonti &longs;it perpendicu<lb/><arrow.to.target n="note47"></arrow.to.target>laris. </s> <s id="id.2.1.23.1.1.3.0">pondus ergo ma <lb/>nebit. <figure id="id.036.01.042.1.jpg" xlink:href="036/01/042/1.jpg"></figure></s> </p> <p id="id.2.1.23.2.0.0.0" type="main"> <s id="id.2.1.23.2.1.1.0">Quoniam autem in his hactenus demon&longs;tratis, nullam de gra<lb/>uitate brachii libræ mentionem fecimus, idcirco &longs;i brachii quoq; <lb/>grauitatem con&longs;iderare voluerimus, centrum grauitatis magnitu<lb/>dinis ex pondere, brachioq; compo&longs;itæ inueniri poterit, circulo<lb/>rumq; circumferentiæ &longs;ecundum di&longs;tantiam à centro libræ ad <lb/>hoc ip&longs;um grauitatis centrum de&longs;cribentur, ac &longs;i in ip&longs;o (vt re ue<lb/>ra e&longs;t) pondus con&longs;titutum fuerit; omnia, &longs;icuti ab&longs;q; libræ bra<lb/>chii grauitate con&longs;iderata inuenimus; hoc quoq; modo eius con&longs;i<lb/>derata grauitate reperiemus. </s> </p> <p id="id.2.1.24.1.0.0.0" type="margin"> <s id="id.2.1.24.1.1.1.0"><margin.target id="note47"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="15" xlink:href="036/01/043.jpg"/> <p id="id.2.1.25.1.0.0.0" type="main"> <s id="id.2.1.25.1.2.1.0">Ex dictis igitur, con&longs;iderando li­<lb/>bram, vt longè à mundi centro a­<lb/>be&longs;t, quemadmodum ip&longs;i fecere, &longs;i­<lb/>cuti etiam actu e&longs;t, apparet fal&longs;itas <lb/>dicentium pondus in A grauius e&longs;&longs;e, <lb/>quàm in alio &longs;itu. </s> <s id="id.2.1.25.1.2.2.0">&longs;imulq; fal&longs;um e&longs;&longs;e, <lb/>quò pondus à linea FG magis di&longs;tat <lb/><expan abbr="grauiuis">grauius</expan> e&longs;&longs;e. </s> <s id="id.2.1.25.1.2.3.0">nam punctum O pro­<lb/>pius e&longs;t ip&longs;i FG, quàm punctum A. <lb/>e&longs;t enim linea à puncto O ip&longs;i FG <arrow.to.target n="note48"></arrow.to.target><lb/>perpendicularis ip&longs;a CA minor. </s> <s id="id.2.1.25.1.2.4.0">de­<lb/>inde ex puncto A pondus velocius mo <lb/>ueri, quàm ab alio &longs;itu, e&longs;t quoque <lb/>fal&longs;um. </s> <s id="id.2.1.25.1.2.5.0">ex puncto enim O pondus ve­<lb/>locius mouebitur, quàm ex puncto <lb/>A; cùm in O &longs;it magis liberum, atq; <lb/>&longs;olutum, quàm in alio &longs;itu: de&longs;cen&longs;us <lb/>qué ex puncto O propior &longs;it motui na­<lb/>turali recto, quàm quilibet alius de­<lb/>&longs;cen&longs;us. <figure id="id.036.01.043.1.jpg" xlink:href="036/01/043/1.jpg"></figure></s> </p> <p id="id.2.1.25.2.0.0.0" type="main"> <s id="id.2.1.25.2.1.1.0">Præterea cùm ex re­<lb/>ctiori, & obliquiori <expan abbr="de&longs;c&etilde;­&longs;u">de&longs;cen<lb/>&longs;u</expan> o&longs;tendunt, pondus in <lb/>A <expan abbr="grauiur">grauior</expan> e&longs;&longs;e, quàm in <lb/>D; & in D, quàm in <lb/>L; primùm quidem fal<lb/>&longs;um exi&longs;timant, &longs;i pon<lb/>dus aliquod collocatum <lb/>fuerit in quocunq; &longs;itu <lb/>circunferentiæ, vt in D, <lb/>rectum eius de&longs;cen&longs;um <lb/>per rectam lineam DR <lb/>ip&longs;i FG parallelam, tam <lb/>quàm &longs;ecundùm mo­<figure id="id.036.01.043.2.jpg" xlink:href="036/01/043/2.jpg"></figure><pb xlink:href="036/01/044.jpg"/>tum naturalem fieri de­<lb/>bere; &longs;icuti prius dictum <lb/>e&longs;t. </s> <s id="id.2.1.25.2.1.2.0">In quocunq; enim <lb/>&longs;itu pondus aliquod con<lb/>&longs;tituatur, &longs;i naturalem <lb/>eius ad propium locum <lb/>motionem &longs;pectemus, <lb/>cùm rectá ad eum &longs;ua­<lb/>ptè natura moueatur, &longs;up<lb/>po&longs;ita totius vniuer&longs;i figu<lb/>ra, eiu&longs;modi erit; vt <lb/>&longs;emper <expan abbr="&longs;patiũ">&longs;patium</expan>, per quod <lb/>naturaliter mouetur, ra­<lb/>tionem habere videatur <lb/><figure id="id.036.01.044.1.jpg" xlink:href="036/01/044/1.jpg"></figure><lb/>lineæ à circumferentia ad centrum productæ. </s> <s id="id.2.1.25.2.1.3.0">non igitur natura<lb/>les de&longs;cen&longs;us recti cuiuslibet &longs;oluti ponderis per lineas fieri po&longs;<lb/>&longs;unt inter &longs;e &longs;e parallelas; cùm omnes in centrum mundi conue­<lb/>niant. </s> <s id="id.2.1.25.2.1.4.0">&longs;upponunt deinde ponderis ex D in A per rectam lineam <lb/>ver&longs;us centrum mundi motum eiu&longs;dem e&longs;&longs;e quantitatis, ac &longs;i fui&longs;<lb/>&longs;et ex O in C: ita vt punctum A æqualiter à centro mundi &longs;it <lb/>di&longs;tans, vt C. </s> <s>quod e&longs;t etiam fal&longs;um; nam punctum A magis <lb/>à centro mundi di&longs;tat, quàm C: maior enim e&longs;t linea à cen­<lb/><arrow.to.target n="note49"></arrow.to.target>tro mundi v&longs;q; ad A, quàm à centro mundi v&longs;q; ad C: cùm li­<lb/>nea à centro mundi v&longs;q; ad A rectum &longs;ubtendat angulum à li­<lb/>neis AC, & à puncto C ad centrum mundi contentum. </s> <s id="id.2.1.25.2.1.5.0">ex qui­<lb/>bus non &longs;olum &longs;uppo&longs;itio illa, qua libram DE in AB redire demon<lb/>&longs;trant, verùm etiam omnes ferè ip&longs;orum demon&longs;trationes ruunt. </s> <s id="id.2.1.25.2.1.6.0"><lb/>ni&longs;i forta&longs;&longs;e dixerint, hæc omnia propter maximam à centro mun<lb/>di v&longs;q; ad nos di&longs;tantiam adeo in&longs;en&longs;ibilia e&longs;&longs;e, vt propter in&longs;en<lb/>&longs;ibilitatem tanquam vera &longs;upponi po&longs;sint: cùm omnes <expan abbr="quid&etilde;">quidem</expan> alii, qui <lb/>hæc tractauerunt, tanquam nota &longs;uppo&longs;uerint. </s> <s id="id.2.1.25.2.1.7.0">præ&longs;ertim quia <lb/>&longs;en&longs;ibilitas illa non efficit, quin de&longs;cen&longs;us ponderis ex L in D <lb/>(vt eorum verbis vtar) minus capiat de directo, quàm de&longs;cen­<lb/>&longs;us DA. </s> <s>&longs;imiliter arcus DA magis de directo capiet, quàm cir<lb/>cumferentia EV. </s> <s>quocirca vera erit &longs;uppo&longs;itio; aliæq; demon­<lb/>&longs;trationes in &longs;uo robore permanebunt. </s> <s id="id.2.1.25.2.1.8.0">Concedamus etiam pon<pb n="16" xlink:href="036/01/045.jpg"/>dus in A grauius e&longs;&longs;e, quàm in alio &longs;itu; rectumq; ponderis de­<lb/>&longs;cen&longs;um per rectam lineam ip&longs;i FG parallelam fieri debere; & <lb/>quælibet puncta in lineis horizonti æquidi&longs;tantibus accepta æ­<lb/>qualiter à centro mundi di&longs;tare: non tamen propterea &longs;equetur, <lb/>veram e&longs;&longs;e demon&longs;trationem, qua inferunt pondus in A grauius <lb/>e&longs;&longs;e, quàm in alio &longs;itu, vt in L. </s> <s>&longs;i enim verum e&longs;&longs;et, quò pon<lb/>dus hoc modo rectius de&longs;cendit, ibi grauius e&longs;&longs;e; &longs;equeretur etiam, <lb/>quò idem pondus in æqualibus arcubus æqualiter rectè de&longs;cende<lb/>ret, vt in ii&longs;dem locis æqualem haberet grauitatem, quod fal<lb/>&longs;um e&longs;&longs;e ita demon&longs;tratur. </s> </p> <p id="id.2.1.26.1.0.0.0" type="margin"> <s id="id.2.1.26.1.1.1.0"><margin.target id="note48"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.26.1.1.2.0"><margin.target id="note49"></margin.target>18 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.27.1.0.0.0" type="main"> <s id="id.2.1.27.1.1.1.0">Sint circumferentiæ AL AM inter &longs;e &longs;e æquales; & conne<lb/>ctatur LM, quæ AB &longs;ecet in X: erit LM ip&longs;i FG æquidi&longs;tans, <lb/>ip&longs;iq; AB perpendicularis. </s> <s id="id.2.1.27.1.1.2.0">& XM ip&longs;i XL æqualis erit. </s> <s id="id.2.1.27.1.1.3.0">&longs;i igi<arrow.to.target n="note50"></arrow.to.target><lb/>tur pondus ex L moueatur in A per circumferentiam LA, rectus <lb/>eius motus erit &longs;ecundùm lineam LX. </s> <s id="id.2.1.27.1.1.3.0.a">&longs;i verò moueatur ex A <lb/>in M per circumferentiam AM, &longs;ecundùm rectam eius motus <lb/>erit XM. </s> <s id="id.2.1.27.1.1.3.0.b">quare de&longs;cen&longs;us ex L in A æqualis erit de&longs;cen&longs;ui ex A <lb/>in M; tum ob circumferentias æquales, tum propter rectas li <lb/>neas ip&longs;i AB perpendiculares æquales. </s> <s id="id.2.1.27.1.1.4.0">ergo idem pondus in L <lb/>æquè graue erit, vt in A, quod e&longs;t fal&longs;um. </s> <s id="id.2.1.27.1.1.5.0">cum longé grauius &longs;it <lb/>in A, quàm in L. </s> </p> <p id="id.2.1.28.1.0.0.0" type="margin"> <s id="id.2.1.28.1.1.1.0"><margin.target id="note50"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 3 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> </p> <p id="id.2.1.29.1.0.0.0" type="main"> <s id="id.2.1.29.1.1.1.0">Quamuis autem AMLA æqualiter &longs;ecundùm ip&longs;os de directo <lb/>capiant; dicent forta&longs;&longs;e, quia tamen principium de&longs;cen&longs;us ex L <lb/>&longs;cilicet LD minus de directo capit, quàm principium de&longs;cen&longs;us <lb/>ex A, &longs;cilicet AN; pondus in A grauius erit, quàm in L. </s> <s id="id.2.1.29.1.1.1.0.a">nam <lb/>cùm circumferentia AN &longs;it ip&longs;i LD (vt &longs;upra po&longs;itum e&longs;t) <lb/>æqualis, quæ &longs;ecundùm ip&longs;os de directo capit CT; LD verò <lb/>de directo capit PO. </s> <s id="id.2.1.29.1.1.1.0.b">ideo pondus grauius erit in A, quàm in L. <lb/></s> <s id="id.2.1.29.1.1.1.0.c">quod &longs;i verum e&longs;&longs;et, &longs;equeretur idem pondus in eodem &longs;itu diuer<lb/>&longs;o duntaxat modo con&longs;ideratum in habitudine ad eundem &longs;itum, <lb/>tum grauius, tum leuius e&longs;&longs;e. </s> <s id="id.2.1.29.1.1.2.0">quod e&longs;t impo&longs;sibile. </s> <s id="id.2.1.29.1.1.3.0">hoc e&longs;t, &longs;i <lb/>de&longs;cen&longs;um con&longs;ideremus ponderis in L, quatenus ex L in A de­<lb/>&longs;cendit, grauius erit, quàm &longs;i eiu&longs;dem ponderis de&longs;cen&longs;um con­<lb/>&longs;ideremus ex L in D tantùm. </s> <s id="id.2.1.29.1.1.4.0">neq; enim negare po&longs;&longs;unt ex ei&longs;­<lb/>demmet dictis, quin de&longs;cen&longs;us ponderis ex L in A de directo ca<lb/>piat LX, &longs;iue PC. </s> <s>de&longs;cen&longs;us verò AM, quin &longs;imiliter de directo <pb xlink:href="036/01/046.jpg"/>capiat XM: cùm ip&longs;i <lb/>quoq; hoc modo acci­<lb/>piant, atq; ita accipe­<lb/>re &longs;it nece&longs;&longs;e. </s> <s id="id.2.1.29.1.1.5.0">&longs;i enim li­<lb/>bram DE in AB redire <lb/>demon&longs;trare volunt, com<lb/>parando de&longs;cen&longs;us pon­<lb/>deris in D cum de&longs;cen­<lb/>&longs;u ponderis in E, nece&longs;&longs;e <lb/>e&longs;t, vt o&longs;tendant rectum <lb/>de&longs;cen&longs;um OC corre­<lb/>&longs;pondentem circumferen<lb/>tiæ DA maiorem e&longs;&longs;e re<lb/>cto de&longs;cen&longs;u TH circum<lb/><figure id="id.036.01.046.1.jpg" xlink:href="036/01/046/1.jpg"></figure><lb/>ferentiæ EV corre&longs;pondente. </s> <s id="id.2.1.29.1.1.6.0">&longs;i enim partem tantùm totius de­<lb/>&longs;cen&longs;us ex D in A acciperent, vt D k; o&longs;tenderentq; magis cape­<lb/>re de directo de&longs;cen&longs;um Dk, quàm æqualis portio de&longs;cen&longs;us ex <lb/>puncto E. </s> <s>&longs;equetur pondus in D &longs;ecundùm ip&longs;os grauius e&longs;&longs;e pon<lb/>dere in E; & v&longs;q; ad k tantùm deor&longs;um moueri: ita vt libra mo<lb/>ta &longs;it in kI. </s> <s>&longs;imiliter &longs;i libram KI in AB redire demon&longs;trare vo<lb/>lunt accipiendo portionem de&longs;cen&longs;us ex k in A; hoc e&longs;t k S; <lb/>o&longs;tenderentq; k S magis de directo capere, quàm ex aduer&longs;o æ­<lb/>qualis de&longs;cen&longs;us ex puncto I: &longs;imili modo &longs;equetur pondus in k <lb/>grauius e&longs;&longs;e, quàm in I; & v&longs;q; ad S tantùm moueri. </s> <s id="id.2.1.29.1.1.7.0">& &longs;i rur&longs;us <lb/>o&longs;tenderent portionem de&longs;cen&longs;us ex S in A, atq; ita deinceps, re<lb/>ctiorem e&longs;&longs;e æquali de&longs;cen&longs;u ponderis oppo&longs;iti; &longs;emper &longs;equetur <lb/>libram SI ad AB propius accedere, nunquam tamen in AB per­<lb/>uenire demon&longs;trabunt. </s> <s id="id.2.1.29.1.1.8.0">&longs;i igitur libram DE in AB redire demon<lb/>&longs;trare volunt, nece&longs;&longs;e e&longs;t, vt de&longs;cen&longs;um ponderis ex D in A de di<lb/>recro capere quantitatem lineæ ex puncto D ip&longs;i AB ad rectos <lb/>angulos ductæ accipiant. </s> <s id="id.2.1.29.1.1.9.0">atq; ita, &longs;i æquales de&longs;cen&longs;us DA AN <lb/>inuicem comparemus, qui æqualiter de directo capient OC CT, <lb/>eueniet idem pondus in D æquè graue e&longs;&longs;e, vt in A. </s> <s>&longs;i verò por<lb/>tiones tantum ex D A accipiamus; grauius erit in A, quàm <lb/>in D. </s> <s>ergo ex diuer&longs;itate tantùm modi con&longs;iderandi, idem pon<lb/>dus, & grauius, & leuius e&longs;&longs;e continget. </s> <s id="id.2.1.29.1.1.10.0">non autem ex ip&longs;a na­<pb n="17" xlink:href="036/01/047.jpg"/>tura rei. </s> <s id="id.2.1.29.1.1.11.0">In&longs;uper ip&longs;orum &longs;uppo&longs;itio non a&longs;&longs;erit, pondus &longs;ecun<lb/>dùm &longs;itum grauius e&longs;&longs;e, quantò in eodem &longs;itu minus obliquum <lb/>e&longs;t principium ip&longs;ius de&longs;cen&longs;us. </s> <s id="id.2.1.29.1.1.12.0">Suppo&longs;itio igitur &longs;uperius alla<lb/>ta, hoc e&longs;t, &longs;ecundùm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo <lb/>dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; non &longs;olum ex his, quæ <lb/>diximus, vllo modo concedi pote&longs;t; &longs;ed quoniam huius oppo&longs;i<lb/>tum o&longs;tendere quoq; non e&longs;t difficile: &longs;cilicet idem pondus in <lb/>æqualibus circumferentiis, quò minus obliquus e&longs;t de&longs;cen&longs;us, ibi <lb/>minus grauitare. </s> </p> <p id="id.2.1.29.2.0.0.0" type="main"> <s id="id.2.1.29.2.1.1.0">Sint enim vt prius cir<lb/><expan abbr="cumferentræ">cumferentiae</expan> AL AM <lb/>inter &longs;e &longs;e æquales; &longs;itq; <lb/>punctum L propè F. </s> <s>& <lb/>connectatur LM, quæ <lb/>ip&longs;i AB perpendicularis <lb/>erit. </s> <s id="id.2.1.29.2.1.2.0">& LX ip&longs;i XM <lb/>æqualis. </s> <s id="id.2.1.29.2.1.3.0">deinde propè <lb/>M inter MG quoduis <lb/>accipiatur punctum P. <lb/>fiatq; circumferentia PO <lb/>circumferentiæ AM æ­<lb/>qualis. </s> <s id="id.2.1.29.2.1.4.0">erit punctum O <lb/><figure id="id.036.01.047.1.jpg" xlink:href="036/01/047/1.jpg"></figure><expan abbr="propè"><lb/>prope</expan> A. </s> <s>connectanturq; CL, CO, CM, CP, OP. </s> <s>& à <lb/>puncto P ip&longs;i OC perpendicularis ducatur PN. </s> <s id="id.2.1.29.2.1.4.0.a">& quoniam cir<lb/>cumferentia AM circumferentiæ OP e&longs;t æqualis: erit angu­<lb/>lus <arrow.to.target n="note51"></arrow.to.target>ACM æqualis angulo OCP; & angulus CXM rectus re­<lb/>cto CNP e&longs;t æqualis: erit quoq; reliquus XMC trianguli MCX <arrow.to.target n="note52"></arrow.to.target><lb/>reliquo NPC trianguli PCN æqualis. </s> <s id="id.2.1.29.2.1.5.0">&longs;ed & latus CM lateri <arrow.to.target n="note53"></arrow.to.target><lb/>CP e&longs;t æquale: ergo triangulum MCX triangulo PCN æquale <lb/>erit. </s> <s id="id.2.1.29.2.1.6.0">latu&longs;q; MX lateri NP æquale. </s> <s id="id.2.1.29.2.1.7.0">quare linea PN ip&longs;i LX æqua<lb/>lis erit. </s> <s id="id.2.1.29.2.1.8.0">ducatur præterea à puncto O linea OT ip&longs;i AC æqui<lb/>di&longs;tans, quæ NP &longs;ecet in V. </s> <s>atq; ip&longs;i OT à puncto P perpendi<lb/>cularis ducatur, quæ quidem inter OV cadere non pote&longs;t; nam <lb/>cùm angulus ONV &longs;it rectus; erit OVN acutus. </s> <s id="id.2.1.29.2.1.9.0">quare OVP <arrow.to.target n="note54"></arrow.to.target><lb/>obtu&longs;us erit. </s> <s id="id.2.1.29.2.1.10.0">non igitur linea à puncto P ip&longs;i OT intra OV <pb xlink:href="036/01/048.jpg"/>perpendicularis cadet. </s> <s id="id.2.1.29.2.1.11.0"><lb/>duo enim anguli vnius <lb/>trianguli, vnus quidem <lb/>rectus, alter verò ob­<lb/>tu&longs;us e&longs;&longs;et. </s> <s id="id.2.1.29.2.1.12.0">quod e&longs;t im<lb/>po&longs;sibile. </s> <s id="id.2.1.29.2.1.13.0">cadet ergo in <lb/>linea OT in parte VT. <lb/></s> <s>&longs;itq; PT. erit PT &longs;ecun<lb/>dùm ip&longs;os rectus circum<lb/>ferentiæ OP de&longs;cen&longs;us. </s> <s id="id.2.1.29.2.1.14.0"><lb/>Quoniam igitur angulus <lb/>ONV e&longs;t rectus; erit <lb/><arrow.to.target n="note55"></arrow.to.target>linea OV ip&longs;a ON ma<lb/>ior. </s> <s id="id.2.1.29.2.1.15.0">quare OT ip&longs;a <lb/><figure id="id.036.01.048.1.jpg" xlink:href="036/01/048/1.jpg"></figure><lb/>quoq; ON maior exi&longs;tet. </s> <s id="id.2.1.29.2.1.16.0">Cùm itaq; linèa OP angulos &longs;ubten­<lb/>dat rectos ONP OTP; erit quadratum ex OP quadratis ex <lb/><arrow.to.target n="note56"></arrow.to.target>ON NP &longs;imul &longs;umptis æquale. </s> <s id="id.2.1.29.2.1.17.0">&longs;imiliter quadratis ex OT TP <lb/>&longs;imul æquale. </s> <s id="id.2.1.29.2.1.18.0">quare quadrata &longs;imul ex ON NP quadratis ex <lb/>OT TP &longs;imul æqualia erunt. </s> <s id="id.2.1.29.2.1.19.0">quadratum autem ex OT maius <lb/>e&longs;t quadrato ex ON; cum linea OT &longs;it ip&longs;a ON maior. </s> <s id="id.2.1.29.2.1.20.0">ergo qua<lb/>dratum ex NP maius erit quadrato ex TP. </s> <s>ac propterea linea <lb/>TP minor erit linea PN, & linea LX. </s> <s>minus obliquus igitur e&longs;t <lb/>de&longs;cen&longs;us arcus LA, quàm arcus OP. </s> <s id="id.2.1.29.2.1.20.0.a">ergo pondus in L, ex ip<lb/>&longs;orum dictis, grauius erit, quàm in O. quod ex iis, quæ &longs;upra di<lb/>ximus e&longs;t manife&longs;tè fal&longs;um, cùm pondus in O grauius &longs;it, quàm <lb/>in L. </s> <s id="id.2.1.29.2.1.20.0.b">non igitur ex rectiori, & obliquiori motu ita accepto col­<lb/>ligi pote&longs;t, &longs;ecundùm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo<lb/>dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us. </s> <s id="id.2.1.29.2.1.21.0">Atq; hinc oritur omnis <lb/>fermé ip&longs;orum error in hac re, atq; deceptio: nam quamuis per <lb/>accidens interdum ex fal&longs;is &longs;equatur verum, per &longs;e tamen ex fal<lb/>&longs;is fal&longs;um &longs;equitur, quemadmodum ex veris &longs;emper verum, nil <lb/>idcirco mirum, &longs;i dum fal&longs;a accipiunt; illi&longs;q; tanquam veri&longs;si­<lb/>mis innituntur; fal&longs;i&longs;sima omninò colligunt, atq; concludunt. </s> <s id="id.2.1.29.2.1.22.0"><lb/>decipiuntur quinetiam, dùm libræ contemplationem mathemati<lb/>cè &longs;impliciter a&longs;&longs;ummunt; cùm eius con&longs;ideratio &longs;it pror&longs;us me­<lb/>chanica: nec vllo modo ab&longs;q; vero motu, ac ponderibus (en­<pb n="18" xlink:href="036/01/049.jpg"/>tibus omninò naturalibus) de ip&longs;a &longs;ermo haberi po&longs;sit: &longs;ine qui­<lb/>bus eorum, quæ libræ accidunt, veræ caulæ reperiri nullo mo <lb/>do po&longs;sint. </s> </p> <p id="id.2.1.30.1.0.0.0" type="margin"> <s id="id.2.1.30.1.1.1.0"><margin.target id="note51"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 27 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.30.1.1.2.0"><margin.target id="note52"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 32 <emph type="italics"/>primi.<emph.end type="italics"/></s> <s id="id.2.1.30.1.1.3.0"><margin.target id="note53"></margin.target>26 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.30.1.1.4.0"><margin.target id="note54"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 13 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.30.1.1.5.0"><margin.target id="note55"></margin.target>19 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.30.1.1.6.0"><margin.target id="note56"></margin.target>47 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.31.1.0.0.0" type="main"> <s id="id.2.1.31.1.1.1.0">Præterea &longs;i adhuc &longs;up<lb/>po&longs;itionem conceda­<lb/>mus; à con&longs;ideratione <lb/>libræ longè recedunt; <lb/>dum eo pacto, vt libra <lb/>DE in AB redire de­<lb/>beat, di&longs;currunt. </s> <s id="id.2.1.31.1.1.2.0">&longs;emper <lb/>enim alterum pondus <lb/>&longs;eor&longs;um accipiunt, putá <lb/>D, vel E; ac &longs;i modò <expan abbr="vnũ">vnum</expan> <lb/>modò alterum in libra <lb/>con&longs;titutum e&longs;&longs;et, nec <lb/>vllo modo ambo con­<lb/><figure id="id.036.01.049.1.jpg" xlink:href="036/01/049/1.jpg"></figure><lb/>nexa; cuius tamen oppo&longs;itum omninò fieri oportet; neq; alterum <lb/>&longs;ine altero rectè con&longs;iderari pote&longs;t; cùm de ip&longs;is in libra con&longs;ti­<lb/>tutis &longs;ermo habeatur. </s> <s id="id.2.1.31.1.1.3.0">cùm enim dicunt, de&longs;cen&longs;um ponderis in <lb/>D minus obliquum e&longs;&longs;e de&longs;cen&longs;u ponderis in E; erit pondus in <lb/>D per &longs;uppo&longs;itionem grauius pondere in E: quare cùm &longs;it graui­<lb/>us, nece&longs;&longs;e e&longs;t deor&longs;um moueri, libramq; DE in AB redire: di<lb/>&longs;cur&longs;us i&longs;te nullius pror&longs;us momenti e&longs;t. </s> <s id="id.2.1.31.1.1.4.0">Primùm quidem &longs;em­<lb/>per argumentantur, ac &longs;i pondera in DE de&longs;cendere debeant, <lb/>vnius tantùm &longs;ine alterius connexione con&longs;iderando de&longs;cen&longs;um. </s> <s id="id.2.1.31.1.1.5.0"><lb/>po&longs;tremò tamen ob ponderum de&longs;cen&longs;uum comparationem colli­<lb/>gentes inferunt, pondus in D deor&longs;um moueri, & pondus in E <lb/>&longs;ur&longs;um, vtraq; &longs;imul in libra inuicem connexa accipientes. </s> <s id="id.2.1.31.1.1.6.0"><expan abbr="ve­rùm">ve­<lb/>rum</expan> ex ii&longs;demmet, quibus vtuntur, principiis, ac demon&longs;tratio<lb/>nibus, oppo&longs;itum eius, quod defendere conantur, facillimè col­<lb/>ligi pote&longs;t. </s> <s id="id.2.1.31.1.1.7.0">Nam &longs;i comparetur de&longs;cen&longs;us ponderis in D cum a­<lb/>&longs;cen&longs;u ponderis in E, vt ductis EK DH ip&longs;i AB perpendicula­<lb/>ribus; cùm angulus DCH &longs;it æqualis angulo ECk; & angulus <arrow.to.target n="note57"></arrow.to.target><lb/>DHC rectus æqualis e&longs;t recto E k C; & latus DC lateri CE æqua <lb/>le: erit triangulum CDH triangulo CEk æquale, & latus DH la-<arrow.to.target n="note58"></arrow.to.target><pb xlink:href="036/01/050.jpg"/>teri Ek æquale. </s> <s id="id.2.1.31.1.1.8.0">cùm <lb/>autem angulus DCA <lb/>&longs;it angulo ECB æqua­<lb/>lis: erit quoq; circum­<lb/>ferentia DA <expan abbr="cirferen">circumferen</expan>­<lb/>tiæ BE æqualis. </s> <s id="id.2.1.31.1.1.9.0">dum <lb/>itaq; pondus in D de­<lb/>&longs;cendit per circumfe­<lb/>rentiam DA, pondus <lb/>in E per circumferen­<lb/>tiam EB ip&longs;i DA æ­<lb/>qualem a&longs;cendit. </s> <s id="id.2.1.31.1.1.10.0">& de­<lb/>&longs;cen&longs;us <expan abbr="põderis">ponderis</expan> in D de <lb/>directo (more <expan abbr="ip&longs;orũ">ip&longs;orum</expan>) <lb/><figure id="id.036.01.050.1.jpg" xlink:href="036/01/050/1.jpg"></figure><lb/>capiet DH; a&longs;cen&longs;us verò ponderis in E de directo capiet Ek ip<lb/>&longs;i DH æqualem: erit itaq; de&longs;cen&longs;us ponderis in D a&longs;cen&longs;ui pon<lb/>deris in E æqualis, & qualis erit propen&longs;io vnius ad motum deor<lb/>sum, talis etiam erit re&longs;i&longs;tentia alterius ad motum &longs;ur&longs;um. </s> <s id="id.2.1.31.1.1.11.0">re­<lb/>&longs;i&longs;tentia &longs;cilicet violentiæ ponderis in E in a&longs;cen&longs;u naturali po­<lb/>tentiæ ponderis in D in de&longs;cen&longs;u contrà nitendo apponitur; cùm <lb/>&longs;it ip&longs;i æqualis. </s> <s id="id.2.1.31.1.1.12.0">quò enim pondus in D naturali potentia deor<lb/>&longs;um velocius de&longs;cendit, eò tardius pondus in E violenter a&longs;cendit. </s> <s id="id.2.1.31.1.1.13.0"><lb/>quare neutrum ip&longs;orum alteri præponderabit, cùm ab æquali non <lb/>proueniat actio. </s> <s id="id.2.1.31.1.1.14.0">Non igitur pondus in D pondus in E &longs;ur&longs;um <lb/>mouebit. </s> <s id="id.2.1.31.1.1.15.0">&longs;i enim moueret; nece&longs;&longs;e e&longs;&longs;et, pondus in D maiorem <lb/>habere virtutem de&longs;cendendo, quàm pondus in E a&longs;cendendo; <lb/>&longs;ed hæc &longs;unt æqualia: ergo pondera manebunt. </s> <s id="id.2.1.31.1.1.16.0">& grauitas pon­<lb/>deris in D grauitati ponderis in E æqualis erit. </s> <s id="id.2.1.31.1.1.17.0">Præterea quoniam <lb/>&longs;upponunt, quò pondus à linea directionis FG magis di&longs;tat, eò <lb/>grauius e&longs;&longs;e: Idcirco ductis quoq; à punctis DE ip&longs;i FG perpen<lb/>dicularibus DO EI; &longs;imili modo demon&longs;trabitur, triangulum <lb/>CDO triangulo CEI æqualem e&longs;&longs;e: & lineam DO ip&longs;i EI æqua<lb/>lem. </s> <s id="id.2.1.31.1.1.18.0">tam igitur di&longs;tat à linea FG pondus in D, quàm pondus in <lb/>E. </s> <s>ex ip&longs;orum igitur rationibus, atq; &longs;uppo&longs;itionibus, pondera <lb/>in DE æquè grauia erunt. </s> <s id="id.2.1.31.1.1.19.0">Amplius quid prohibet, quin libram <lb/>DE ex nece&longs;sitate in FG moueri &longs;imili ratione o&longs;tendatur? </s> <s id="id.2.1.31.1.1.20.0">Pri­<pb n="19" xlink:href="036/01/051.jpg"/>mùm quidem ex eorummet demon&longs;trationibus colligi pote&longs;t, a­<lb/>&longs;cen&longs;um ponderis in E ver&longs;us B rectiorem e&longs;&longs;e a&longs;cen&longs;u ponderis <lb/>in D ver&longs;us F; hoc e&longs;t minus capere de directo a&longs;cen&longs;um pon­<lb/>deris in D in arcubus æqualibus a&longs;cen&longs;u ponderis in E. </s> <s id="id.2.1.31.1.1.20.0.a">&longs;uppona<lb/>tur ergo &longs;ecundùm &longs;itum pondus leuius e&longs;&longs;e, quantò in eodem &longs;i­<lb/>tu minus rectus e&longs;t a&longs;cen&longs;us: quæ quidem &longs;uppo&longs;itio, adeò ma­<lb/>nife&longs;ta e&longs;&longs;e videtur, veluti ip&longs;orum altera. </s> <s id="id.2.1.31.1.1.21.0">Quoniam igitur a&longs;cen­<lb/>&longs;us ponderis in E rectior e&longs;t a&longs;cen&longs;u ponderis in D; per &longs;uppo&longs;i­<lb/>tionem pondus in D leuius erit pondere in E. ergo pondus in <lb/>D &longs;ur&longs;um à pondere in E mouebitur, ita vt libra in FG perue<lb/>niat. </s> <s id="id.2.1.31.1.1.22.0">atq; ita demon&longs;trari poterit, libram DE in FG moueri.<lb/></s> <s id="id.2.1.31.1.1.23.0">quæ quidem demon&longs;tratio inutilis e&longs;t pror&longs;us, ea&longs;demq; patitur <lb/>difficultates. </s> <s id="id.2.1.31.1.1.24.0">licet enim tanquàm verum admittatur pondus in E <lb/>a&longs;cendendo grauius e&longs;&longs;e pondere in D &longs;imiliter a&longs;cendendo, <lb/>non tamen ex hoc &longs;equitur, pondus in E de&longs;cendendo grauius <lb/>e&longs;&longs;e pondere in D a&longs;cendendo. </s> <s id="id.2.1.31.1.1.25.0">Neutra igitur harum demon­<lb/>&longs;trationum libram DE, vel in AB redire, vel in FG moue­<lb/>ri, o&longs;tendentium, vera e&longs;t. </s> </p> <p id="id.2.1.32.1.0.0.0" type="margin"> <s id="id.2.1.32.1.1.1.0"><margin.target id="note57"></margin.target>15 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.32.1.1.2.0"><margin.target id="note58"></margin.target>26 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.33.1.0.0.0" type="main"> <s id="id.2.1.33.1.1.1.0">Præterea &longs;i ip&longs;orum &longs;uppo&longs;itionem, eorumq; verborum vim <lb/>rectè perpendamus; alium certè habere &longs;en&longs;um con&longs;piciemus. </s> <s id="id.2.1.33.1.1.2.0">nam <lb/>cùm &longs;emper &longs;patium, per quod naturaliter pondus mouetur, à cen<lb/>tro grauitatis ip&longs;ius ponderis ad centrum mundi, in&longs;tar rectæ li­<lb/>neæ à centro grauitatis ad centrum mundi productæ, &longs;it &longs;umendum; <lb/>tantò huiusmodi ponderis de&longs;cen&longs;us, magis, minusuè obliquus <lb/>dicetur; quantò &longs;ecundùm &longs;patium in&longs;tar prædictæ lineæ de&longs;igna <lb/>tum, magis, aut minus (naturalem tamen locum petens, &longs;emperq; <lb/>magis ip&longs;i appropinquans) mouebitur; ita vt tantò obliquior de­<lb/>&longs;cen&longs;us dicatur, quantò recedit ab eiu&longs;modi &longs;patio: rectior verò, <lb/>quantò ad idem accedit. </s> <s id="id.2.1.33.1.1.3.0">& in hoc &longs;en&longs;u &longs;uppo&longs;itio illa nemini <lb/>difficultatem parere debet, adeò enim veritas eius con&longs;picua e&longs;t; <lb/>rationiq; con&longs;entanea: vt nulla pro&longs;us manife&longs;tatione egere vi­<lb/>deatur. </s> </p> <pb xlink:href="036/01/052.jpg"/> <p id="id.2.1.33.3.0.0.0" type="main"> <s id="id.2.1.33.3.1.1.0">Si itaq; pondus &longs;olutum in &longs;itu D <lb/>collocatum ad propium locum mo­<lb/>ueri debeat; proculdubio po&longs;ito cen­<lb/>tro mundi S, per lineam DS moue­<lb/>bitur. </s> <s id="id.2.1.33.3.1.2.0">&longs;imiliter pondus in E &longs;olutum <lb/>per lineam ES mouebitur. </s> <s id="id.2.1.33.3.1.3.0">quare &longs;i <lb/>(vt rei veritas e&longs;t) ponderis de&longs;cen­<lb/>&longs;us magis, minu&longs;uè obliquus dicetur <lb/>&longs;ecundùm rece&longs;&longs;um, & acce&longs;&longs;um ad <lb/>&longs;patia per lineas DSES de&longs;ignata, <lb/>iuxta naturales ip&longs;orum ad propria lo <lb/>ca lationes; con&longs;picuum e&longs;t, minus <lb/>obliquum e&longs;&longs;e de&longs;cen&longs;um ip&longs;ius E <lb/>per EG, quàm ip&longs;ius D per DA: <lb/>cùm angulum SEG angulo SDA <lb/>minorem e&longs;&longs;e &longs;upra o&longs;ten&longs;um &longs;it. </s> <s id="id.2.1.33.3.1.4.0">qua <lb/>re in E pondus magis grauitabit, <lb/>quàm in D. quod e&longs;t penitus oppo­<lb/>&longs;itum eius, quod ip&longs;i o&longs;tendere cona<lb/>ti &longs;unt. </s> <s id="id.2.1.33.3.1.5.0">In&longs;urgent autem forta&longs;&longs;e <lb/>contrarios, &longs;i igitur (dicent) pondus <lb/>in E grauius e&longs;t pondere in D, libra <lb/><figure id="id.036.01.052.1.jpg" xlink:href="036/01/052/1.jpg"></figure><lb/>DE in hoc &longs;itu minimè per&longs;i&longs;tet, quod <expan abbr="equid&etilde;">equidem</expan> tueri propo&longs;uimus: <lb/>&longs;ed in FG mouebitur. </s> <s id="id.2.1.33.3.1.6.0">quibus re&longs;pondemus, plurimum referre, &longs;iue <lb/>con&longs;ideremus pondera, quatenus &longs;unt inuicem di&longs;iuncta, &longs;iue quate <lb/>nus &longs;unt &longs;ibi inuicem connexa. </s> <s id="id.2.1.33.3.1.7.0">alia e&longs;t enim ratio ponderis in E &longs;ine <lb/>connexione ponderis in D, alia verò eiu&longs;dem alteri ponderi con<lb/>nexi; ita vt alterum &longs;ine altero moueri non po&longs;sit. </s> <s id="id.2.1.33.3.1.8.0">nam ponde<lb/>ris in E, quatenus e&longs;t &longs;ine alterius ponderis connexione, rectus <lb/>naturalis de&longs;cen&longs;us e&longs;t per lineam ES; quatenus verò connexum <lb/>e&longs;t ponderi in D, eius naturalis de&longs;cen&longs;us non erit amplius per <lb/>lineam ES, &longs;ed per lineam ip&longs;i CS parallelam. </s> <s id="id.2.1.33.3.1.9.0">magnitudo enim <lb/>ex ponderibus ED, & libra DE compo&longs;ita, cuius grauitatis cen­<lb/>trum e&longs;t C, &longs;i nullibi &longs;u&longs;tineatur, deor&longs;um eo modo, quo reperi<lb/>tur, &longs;ecundùm grauitatis centrum per rectam à centro grauita<lb/>tis C ad centrum mundi S ductam naturaliter mouebitur, donec <pb n="20" xlink:href="036/01/053.jpg"/>centrum C in centrum S perueniat. </s> <s id="id.2.1.33.3.1.10.0">libra igitur DE vná cum pon<lb/>deribus eo modo, quo reperitur, deor&longs;um mouebitur, ita vt pun­<lb/>ctum C per lineam CS moueatur, donec C in S, libraq; DE in <lb/>Hk perueniat; habeatq; libra in Hk eandem, quam prius habe­<lb/>bat po&longs;itionem; hoc e&longs;t Hk &longs;it ip&longs;i DE æquidi&longs;tans. </s> <s id="id.2.1.33.3.1.11.0">connectantur <lb/>igitur DH Ek. </s> <s id="id.2.1.33.3.1.12.0">manife&longs;tum e&longs;t, dum libra DE in Hk mouetur pun<lb/>cta DE per lineas DH Ek moueri, quippe exi&longs;tentibus inter &longs;e <arrow.to.target n="note59"></arrow.to.target><lb/>&longs;e, ip&longs;iq; CS æqualibus, & æquidi&longs;tantibus. </s> <s id="id.2.1.33.3.1.13.0">Quare pondera in <lb/>DE, quatenus &longs;unt &longs;ibi inuicem connexa, &longs;i ip&longs;orum naturalem mo <lb/>tum &longs;pectemus, non &longs;ecundùm lineas DS ES, &longs;ed &longs;ecundùm <lb/>LDH MEk ip&longs;i CS æquidi&longs;tantes mouebuntur. </s> <s id="id.2.1.33.3.1.14.0">ponderis <expan abbr="ve­rò">ve­<lb/>ro</expan> in E liberi, ac &longs;oluti, naturalis propen&longs;io erit per ES: ponderis <lb/>autem in D &longs;imiliter &longs;oluti erit per DS. ac propterea non e&longs;t incon­<lb/>ueniens idem pondus modò in E, modò in D, grauius e&longs;&longs;e in E, <lb/>quàm in D. </s> <s id="id.2.1.33.3.1.14.0.a">&longs;i verò pondera in ED &longs;ibi inuicem connexa, quate­<lb/>nusq; &longs;unt connexa con&longs;iderauerimus; erit ponderis in E natura­<lb/>lis propen&longs;io per lineam MEK: grauitas enim alterius ponde­<lb/>ris in D efficit, nè pondus in E per lineam ES grauitet, &longs;ed per <lb/>Ek. </s> <s id="id.2.1.33.3.1.15.0">quod ip&longs;um quoq; grauitas ponderis in E efficit, nè &longs;cilicet <lb/>pondus in D per rectam DS degrauet; &longs;ed &longs;ecundùm DH: vtra­<lb/>que enim &longs;e impediunt, nè ad propria loca <expan abbr="permeent">permeant</expan>. </s> <s id="id.2.1.33.3.1.16.0">Cùm igi<lb/>tur naturalis de&longs;cen&longs;us rectus ponderum in DE &longs;it &longs;ecundùm <lb/>LDH MEK: erit <expan abbr="&longs;imliter">similiter</expan> rectus eorum a&longs;cen&longs;us &longs;ecundùm ea&longs;<lb/>dem lineas HDL KEM. atq; a&longs;cen&longs;us ponderis in E magis, mi<lb/>nu&longs;uè obliquus dicetur; quantò &longs;ecundùm &longs;patium magis, mi­<lb/>nu&longs;uè iuxta lineam Mk mouebitur. </s> <s id="id.2.1.33.3.1.17.0">hocq; pror&longs;us modo iuxta li<lb/>neam LH &longs;ummendus e&longs;t, tùm de&longs;cen&longs;us, tùm a&longs;cen&longs;us ponde­<lb/>ris in D. </s> <s>&longs;i itaq; pondus in E deor&longs;um per EG moueretur; pon<lb/>dus in D &longs;ur&longs;um per DF moueret. </s> <s id="id.2.1.33.3.1.18.0">& quoniam angulus CEK <arrow.to.target n="note60"></arrow.to.target><lb/>æqualis e&longs;t angulo CDL, & angulus CEG angulo CDF æqua­<lb/>lis; erit reliquus GEK reliquo LDF æqualis. </s> <s id="id.2.1.33.3.1.19.0">cùm autem &longs;up­<lb/>po&longs;itio illa, quæ ait, &longs;ecundúm &longs;itum pondus grauius e&longs;&longs;e, <expan abbr="quan­tò">quan­<lb/>to</expan> in eodem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us; tanquam clara, <lb/>atq; con&longs;picua admittatur; proculdubio hæc quoq; accipienda <lb/>erit; nempè, &longs;ecundúm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo­<lb/>dem &longs;itu minus obliquus e&longs;t a&longs;cen&longs;us. </s> <s id="id.2.1.33.3.1.20.0">cùm non minus manife&longs;ta, <pb xlink:href="036/01/054.jpg"/>rationiq; &longs;it con&longs;entanea. </s> <s id="id.2.1.33.3.1.21.0">æqualis <lb/>igitur erit de&longs;cen&longs;us ponderis in E <lb/>a&longs;cen&longs;ui ponderis in D. </s> <s>eandem <lb/>enim obliquitatem habet de&longs;cen&longs;us <lb/>ponderis in E, quam habet a&longs;cen­<lb/>&longs;us ponderis in D; & qualis erit <lb/>propen&longs;io vnius ad motum deor&longs;um, <lb/>talis quoq; erit re&longs;i&longs;tentia alterius ad <lb/>motum &longs;ur&longs;um. </s> <s id="id.2.1.33.3.1.22.0"><expan abbr="nõ">non</expan> ergo pondus in E <lb/>pondus in D &longs;ur&longs;um mouebit. </s> <s id="id.2.1.33.3.1.23.0">neq; <lb/>pondus in D deor&longs;um mouebitur, ita <lb/>vt &longs;ur&longs;um moueat pondus in E. nam <lb/><expan abbr="cũ">cum</expan> angulus CEB &longs;it ip&longs;i CDA æqua­<lb/><arrow.to.target n="note61"></arrow.to.target>lis, & Angulus CEM &longs;it angulo <lb/>CDH æqualis; erit reliquus MEB <lb/>reliquo HDA æqualis. </s> <s id="id.2.1.33.3.1.24.0">de&longs;cen&longs;us <lb/>igitur ponderis in D a&longs;cen&longs;ui ponde<lb/>ris in E æqualis erit. </s> <s id="id.2.1.33.3.1.25.0">non ergo pon<lb/>dus in D pondus in E &longs;ur&longs;um moue<lb/>bit. </s> <s id="id.2.1.33.3.1.26.0">ex quibus &longs;equitur pondera in <lb/>DE, quatenus &longs;unt &longs;ibi inuicem con<lb/>nexa, æquè grauia e&longs;&longs;e. <figure id="id.036.01.054.1.jpg" xlink:href="036/01/054/1.jpg"></figure></s> </p> <p id="id.2.1.33.4.0.0.0" type="main"> <s id="id.2.1.33.4.1.1.0">Alia deinde ratio, li­<lb/>bram &longs;imiliter DE in AB <lb/>redire o&longs;tendens, cùm in­<lb/>quiunt, exi&longs;tente trutina in <lb/>CF meta e&longs;t CG. </s> <s id="id.2.1.33.4.1.1.0.a">& quo­<lb/>niam angulus DCG maior <lb/>e&longs;t angulo ECG; pondus <lb/>in D grauius erit pondere <lb/>in E; ergo libra DE in AB <lb/>redibit: nihil meo iudicio <lb/>concludit. </s> <s id="id.2.1.33.4.1.2.0">figmentumq; <lb/>hoc de trutina, & meta po­<lb/>tius omittendum, ac &longs;ilen­<figure id="id.036.01.054.2.jpg" xlink:href="036/01/054/2.jpg"></figure><pb n="21" xlink:href="036/01/055.jpg"/>tio <expan abbr="prætereundũ">prætereundum</expan> e&longs;&longs;et, quàm <expan abbr="verbũ">verbum</expan> <expan abbr="vllũ">vllum</expan> in eius confutatione &longs;umen<lb/>dum; cùm &longs;it pror&longs;us voluntarium. </s> <s id="id.2.1.33.4.1.3.0">nece&longs;sitas enim cur pondus <lb/>in D ex maiore angulo &longs;it grauius; curq; maior angulus maioris <lb/>&longs;it cau&longs;a grauitatis; nu&longs;quam apparet. </s> <s id="id.2.1.33.4.1.4.0">&longs;i autem comparentur in­<lb/>uicem anguli, cùm angulus GCD &longs;it æqualis angulo FCE; &longs;i angu<lb/>lus GCD e&longs;t cau&longs;a grauitatis; quare angulus FCE &longs;imiliter gra­<lb/>uitatis non e&longs;t cau&longs;a? </s> <s id="id.2.1.33.4.1.5.0">Huius autem rei eam in medium rationem <lb/>afferre videntur, quoniam CG e&longs;t meta, & CF trutina. </s> <s id="id.2.1.33.4.1.6.0">&longs;i (inquiunt) <lb/>CG e&longs;&longs;et trutina, & CF meta, tunc angulus FCE grauitatis e&longs;&longs;et <lb/>cau&longs;a; non autem DCG ip&longs;i æqualis. </s> <s id="id.2.1.33.4.1.7.0">quæ quidem ratio imma­<lb/>ginaria pror&longs;us, ac voluntaria e&longs;&longs;e videtur. </s> <s id="id.2.1.33.4.1.8.0">quid enim refert, &longs;iue tru<lb/>tina &longs;it in CF, &longs;iue in CG, cùm libra DE in eodem &longs;emper pun­<lb/>cto C &longs;u&longs;tineatur? </s> <s id="id.2.1.33.4.1.9.0">Vt autem eorum deceptio clarius appa­<lb/>reat. </s> </p> <p id="id.2.1.34.1.0.0.0" type="margin"> <s id="id.2.1.34.1.1.1.0"><margin.target id="note59"></margin.target>33 <emph type="italics"/>Prmi.<emph.end type="italics"/></s> <s id="id.2.1.34.1.1.2.0"><margin.target id="note60"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.34.1.1.3.0"><margin.target id="note61"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.35.1.0.0.0" type="main"> <s id="id.2.1.35.1.1.1.0">Sit eadem libra AB, cu­<lb/>ius medium C. &longs;it deinde <lb/>tota FG trutina. </s> <s id="id.2.1.35.1.1.2.0">eaq; im<lb/>mobilis exi&longs;tat; quæ libram <lb/>AB in puncto C &longs;u&longs;tineat. </s> <s id="id.2.1.35.1.1.3.0"><lb/>moueaturq; libra in DE. </s> <s>& <lb/>quoniam trutina e&longs;t, & &longs;u­<lb/>pra, & infra libram, quis <lb/>nam angulus erit cau&longs;a gra­<lb/>uitatis, cùm libra DE in <lb/><figure id="id.036.01.055.1.jpg" xlink:href="036/01/055/1.jpg"></figure><expan abbr="eod&etilde;"><lb/>eodem</expan> &longs;emper puncto &longs;u&longs;tineatur? </s> <s id="id.2.1.35.1.1.4.0">dicent for&longs;an, &longs;i trutina à potentia <lb/>in F &longs;u&longs;titeneatur, tunc CG erit tanquam meta, & angulus <lb/>DCG grauitatis erit cau&longs;a. </s> <s id="id.2.1.35.1.1.5.0">&longs;i verò &longs;u&longs;tineatur in G, tunc FCE <lb/>erit cau&longs;a grauitatis, CF verò tanquam meta erit. </s> <s id="id.2.1.35.1.1.6.0">cuius quidem <lb/>rei nulla videtur e&longs;&longs;e cau&longs;a, ni&longs;i immaginaria. </s> <s id="id.2.1.35.1.1.7.0">meta enim (quod <lb/>aiunt) nullam pror&longs;us vim attractiuam, quandoq; ex maioris an­<lb/>guli parte, quandoq; ex parte minoris habere videtur. </s> <s id="id.2.1.35.1.1.8.0">Verùm à dua<lb/>bus potentiis &longs;u&longs;tineatur trutina, in F &longs;cilicet, & in G, quod præ ne<lb/>ce&longs;sitate fieri pote&longs;t, veluti &longs;i potentia in F &longs;it adeò debilis, vt ex &longs;e <lb/>ip&longs;a medietatem tantùm ponderis &longs;u&longs;tinere quæat: &longs;itq; potentia in <lb/>G ip&longs;i potentiæ in F æqualis, vtræq; <expan abbr="aut&etilde;">autem</expan> &longs;imul libram vná cum pon<lb/>deribus &longs;u&longs;tineant. </s> <s id="id.2.1.35.1.1.9.0">tunc quis nam angulus erit cau&longs;a grauitatis? </s> <s id="id.2.1.35.1.1.10.0">non <pb xlink:href="036/01/056.jpg"/>FCE, quia trutina e&longs;t in <lb/>CF, & in F &longs;u&longs;tinetur. </s> <s id="id.2.1.35.1.1.11.0">neq; <lb/>DCG, cùm trutina &longs;it in <lb/>CG, & in G quoq; &longs;u&longs;ti<lb/>neatur; non igitur anguli <lb/>grauitatis cau&longs;a erunt. </s> <s id="id.2.1.35.1.1.12.0">ergo <lb/>neq; libra DE ab hoc &longs;itu <lb/>ob hanc cau&longs;am mouebi­<lb/><arrow.to.target n="note62"></arrow.to.target>tur. </s> <s id="id.2.1.35.1.1.13.0">Hanc autem eorum <lb/>&longs;ententiam dupliciter con­<lb/><figure id="id.036.01.056.1.jpg" xlink:href="036/01/056/1.jpg"></figure><lb/>firmare videntur. </s> <s id="id.2.1.35.1.1.14.0">primùm quidem a&longs;&longs;erunt Ari&longs;totelem in quæ&longs;tio<lb/>nibus mechanicis has duas tantùm quæ&longs;tiones propo&longs;ui&longs;&longs;e; eiu&longs;q; <lb/>demon&longs;trationes, tum maiori, & minori angulo, tùm trutinæ po&longs;i<lb/>tioni inniti. </s> <s id="id.2.1.35.1.1.15.0">Affirmant deinde experientiam hoc idem docere; <lb/>hoc e&longs;t libram DE trutina exi&longs;tente in CF, in AB horizonti <lb/>æquidi&longs;tantem redire. </s> <s id="id.2.1.35.1.1.16.0">quando autem trutina e&longs;t in CG, in FG <lb/>moueri. </s> <s id="id.2.1.35.1.1.17.0">Verùm neq; Ari&longs;toteles, neq; experientia huic eorum <lb/>opinioni fauent, quin potius aduer&longs;antur. </s> <s id="id.2.1.35.1.1.18.0">quantùm enim atti­<lb/>net ad experientiam decipiuntur, ip&longs;a quidem experientia ma­<lb/>nife&longs;tum e&longs;t hoc accidere, quando libræ quoq; centrum, vel &longs;u­<lb/>pra, vel infra libram fuerit collocatum: non autem trutina dun<lb/>taxat &longs;upra, vel infra exi&longs;tente, id contingere. </s> </p> <p id="id.2.1.36.1.0.0.0" type="margin"> <s id="id.2.1.36.1.1.1.0"><margin.target id="note62"></margin.target><emph type="italics"/>Cardanus.<emph.end type="italics"/></s> </p> <pb n="22" xlink:href="036/01/057.jpg"/> <p id="id.2.1.37.1.0.0.0" type="main"> <s id="id.2.1.37.1.2.1.0">Nam &longs;i libra AB habeat <lb/>centrum C &longs;upra libram; <lb/>&longs;itq; trutina CD infra li­<lb/>bram; moueaturq; libra in <lb/>EF; tunc EF rur&longs;us in AB <lb/>horizonti æquidi&longs;tantem <arrow.to.target n="note63"></arrow.to.target><lb/>redibit. </s> <s id="id.2.1.37.1.2.2.0">&longs;imiliter &longs;i libra <lb/>centrum C habeat infra li<lb/>bram, &longs;itq; trutina CD &longs;u<lb/>pra libram, & moueatur <lb/>libra in EF; patet libram <arrow.to.target n="note64"></arrow.to.target><lb/>ex parte F deor&longs;um moue <lb/>ri, trutina &longs;upra libram e­<lb/>xi&longs;tente. </s> <s id="id.2.1.37.1.2.3.0">& in quocunq; a­<lb/>lio &longs;itu fuerit trutina, idem <lb/>&longs;emper eueniet. </s> <s id="id.2.1.37.1.2.4.0">non igitur <lb/>trutina, &longs;ed centrum libræ <lb/>harum diuer&longs;itatum cau­<lb/>&longs;a erit. <figure id="id.036.01.057.1.jpg" xlink:href="036/01/057/1.jpg"></figure></s> </p> <p id="id.2.1.37.2.0.0.0" type="main"> <s id="id.2.1.37.2.1.1.0">Animaduertendum e&longs;t <lb/>itaq; in hac parte difficulter materialem libram con&longs;titui po&longs;&longs;e, <lb/>quæ in vno tantùm puncto &longs;u&longs;tineatur; quemadmodum mente <lb/>concipimus. </s> <s id="id.2.1.37.2.1.2.0">brachiaq; ab eiu&longs;modi centro adeò æqualia habeat, <lb/>non &longs;olum in longitudine, verùm etiam in latitudine, & profun<lb/>ditate, vt omnes partes hinc indé ad vnguem æqueponderent. </s> <s id="id.2.1.37.2.1.3.0"><lb/>hoc enim materia difficilimè patitur. </s> <s id="id.2.1.37.2.1.4.0">quocirca &longs;i centrum in ip&longs;a <lb/>libra e&longs;&longs;e con&longs;iderauerimus, ad &longs;en&longs;um confugiendum non e&longs;t: <lb/>cùm artificilia ad &longs;ummum illud perfectionis gradum ab artifice <lb/>deduci minimè po&longs;sint. </s> <s id="id.2.1.37.2.1.5.0">In aliis verò experientia quidem appa­<lb/>rentia docere poterit; propterea quod, quamquam centrum libræ <lb/>&longs;it &longs;emper punctum, quando tamen &longs;upra libram fuerit, parùm re­<lb/>fert, &longs;i libra in eo puncto adamu&longs;&longs;im minimè &longs;u&longs;tineatur; quia cùm <lb/>&longs;it &longs;emper &longs;upra libram, idem &longs;emper eueniet. </s> <s id="id.2.1.37.2.1.6.0">&longs;imili quoq; modo <lb/>quando e&longs;t infra libram: quod tamen non accidit centro in ip&longs;a li­<lb/>bra exi&longs;tente. </s> <s id="id.2.1.37.2.1.7.0">&longs;i enim ad vnguem &longs;emper in illo medio non &longs;u­<lb/>&longs;tineatur, diuer&longs;itatem efficiet; cùm facillimum &longs;it, centrum il­<pb xlink:href="036/01/058.jpg"/>lud, dùm libra mouetur, proprium mutare &longs;itum. </s> </p> <p id="id.2.1.38.1.0.0.0" type="margin"> <s id="id.2.1.38.1.1.1.0"><margin.target id="note63"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.38.1.1.2.0"><margin.target id="note64"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.39.1.0.0.0" type="main"> <s id="id.2.1.39.1.1.1.0">Quòd autem Ari&longs;toteles duas tantùm quæ&longs;tiones propo­<lb/>&longs;uerit, cur &longs;cilicet trutina &longs;uperius exi&longs;tente, &longs;i libra non &longs;it <lb/>horizonti æquidi&longs;tans in æquilibrium, hoc e&longs;t horizonti æqui <lb/>di&longs;tans redit: &longs;i autem trutina deor&longs;um fuerit con&longs;tituta, non <lb/>redit; &longs;ed adhuc &longs;ecundùm partem depre&longs;&longs;am mouetur: verum <lb/>quidem e&longs;t. </s> <s id="id.2.1.39.1.1.2.0">non tamen eius demon&longs;trationes maiori, & mino <lb/>ri angulo, po&longs;itioniqué trutinæ (vt ip&longs;i dicunt) innituntur. </s> <s id="id.2.1.39.1.1.3.0">In <lb/>hoc enim mentem philo&longs;ophi a&longs;ignantis rationem diuer&longs;itatis <lb/>motuum libræ minimè attingunt. </s> <s id="id.2.1.39.1.1.4.0">tantùm enim abe&longs;t philo&longs;o­<lb/>phum has diuer&longs;itates in angulos referre, vt potius in cau&longs;a e&longs;&longs;e <lb/>dicat magnitudinis alterius brachii libræ exce&longs;&longs;um à perpendiculo, <lb/>modò ex vna, modò ex altera parte contingentem. </s> </p> <p id="id.2.1.39.2.0.0.0" type="main"> <s id="id.2.1.39.2.1.1.0">Vt trutina &longs;uperius in <lb/>CF exi&longs;tente, perpendicu<lb/>lum erit FCG, quod <expan abbr="&longs;e­cundùm">&longs;e­<lb/>cundum</expan> ip&longs;um in centrum <lb/>mundi &longs;emper vergit; <lb/>quod quidem libram mo­<lb/>tam in DE in partes di­<lb/>uidit inæquales; & maior <lb/>pars e&longs;t ver&longs;us D: id au­<lb/>tem, quod plus e&longs;t, deor<lb/>&longs;um fertur; ergo ex par­<lb/>te D deor&longs;um libra moue<lb/>bitur, donec in AB re­<lb/>deat. </s> <s id="id.2.1.39.2.1.2.0">&longs;i verò trutina &longs;it <lb/><figure id="id.036.01.058.1.jpg" xlink:href="036/01/058/1.jpg"></figure><lb/>in CG deor&longs;um, erit GCF perpendiculum, quod libram DE <lb/>in partes inæquales &longs;imiliter diuidit: maior autem pars erit ver&longs;us <lb/>E; quare ex parte E deor&longs;um libra mouebitur. </s> <s id="id.2.1.39.2.1.3.0">quod vt rectè in­<lb/>telligatur, cùm trutina e&longs;t &longs;upra libram, libræ quoq; centrum &longs;u­<lb/>pra libram e&longs;&longs;e intelligendum e&longs;t; & &longs;i deor&longs;um, centrum quoque <lb/>deor&longs;um: vt infra patebit. </s> <s id="id.2.1.39.2.1.4.0">Aliter ip&longs;a Ari&longs;totelis demon&longs;tratio <lb/>nihil concluderet. </s> <s id="id.2.1.39.2.1.5.0">exi&longs;tente enim centro in ip&longs;a libra, vt in C; quo­<lb/>cunq; modo moueatur libra, nunquam perpendiculum FG libram, <pb n="23" xlink:href="036/01/059.jpg"/>ni&longs;i in puncto C, & in partes diuidet æquales. </s> <s id="id.2.1.39.2.1.6.0">quare Ari&longs;totelis <lb/>&longs;ententia ip&longs;is non &longs;olum non fauet, verùm etiam maximè aduer­<lb/>&longs;atur. </s> <s id="id.2.1.39.2.1.7.0">quòd non &longs;olum ex &longs;ecunda, & tertia huius liquet; verùm <lb/>quia exi&longs;tente centro &longs;upra libram pondus eleuatum maiorem <lb/>propter &longs;itum acquirit grauitatem. </s> <s id="id.2.1.39.2.1.8.0">ex quò contingit redditus li­<lb/>bræ ad æqualem horizonti di&longs;tantiam. </s> <s id="id.2.1.39.2.1.9.0">è contra verò, quando <lb/>centrum e&longs;t infra libram. </s> <s id="id.2.1.39.2.1.10.0">Quæ omnia hoc modo o&longs;tendentur; <lb/>&longs;upponendo ea, quæ &longs;upra declarata &longs;unt. </s> <s id="id.2.1.39.2.1.11.0">&longs;cilicet pondus ex quò <lb/>loco rectius de&longs;cendit, grauius fieri. </s> <s id="id.2.1.39.2.1.12.0">& ex quo rectius a&longs;cendit, gra<lb/>uius quoq; reddi. </s> </p> <p id="id.2.1.39.3.0.0.0" type="main"> <s id="id.2.1.39.3.1.1.0">Sit libra AB horizonti <lb/>æquidi&longs;tans, cuius centrum <lb/>C &longs;it &longs;upra libram, perpen­<lb/>diculumq; &longs;it CD. &longs;intq; in <lb/>AB ponderum æqualium <lb/>centra grauitatis po&longs;ita: mo<lb/>taq; &longs;it libra in EF. </s> <s id="id.2.1.39.3.1.1.0.a">Dico <lb/>pondus in E maiorem ha­<lb/>bere grauitatem, quàm pon<lb/>dus in F. </s> <s>& ob id libram <lb/>EF in AB redire. </s> <s id="id.2.1.39.3.1.2.0">Produ<lb/>catur primùm CD v&longs;q; ad <lb/>mundi <expan abbr="centrũ">centrum</expan>, quod &longs;it S. de <lb/>inde AC CB EC CF HS <lb/><expan abbr="cõnectantur">connectantur</expan>, à puncti&longs;q; EF <lb/>ip&longs;i HS æquidi&longs;tantes du<lb/>cantur Ek GFL. </s> <s id="id.2.1.39.3.1.2.0.a">Quoniam <lb/>igitur naturalis de&longs;cen&longs;us re<lb/>ctus totius magnitudinis, <lb/>libræ &longs;cilicet EF &longs;ic con&longs;ti­<lb/>tutæ vná cum ponderibus, <lb/>e&longs;t <expan abbr="&longs;cundùm">secundum</expan> grauitatis cen<lb/>trum H per rectam HS; erit <lb/><figure id="id.036.01.059.1.jpg" xlink:href="036/01/059/1.jpg"></figure><lb/>quoq; ponderum in EF ita po&longs;sitorum de&longs;cen&longs;us &longs;ecundùm re­<lb/>ctas Ek FL ip&longs;i HS parallelas; &longs;icuti &longs;upra demon&longs;trauimus. </s> <s id="id.2.1.39.3.1.3.0"><pb xlink:href="036/01/060.jpg"/>De&longs;cen&longs;us igitur, & a&longs;cen­<lb/>&longs;us ponderum in EF ma­<lb/>gis, minu&longs;uè obliquus di­<lb/>cetur &longs;ecundùm acce&longs;&longs;um, <lb/>& rece&longs;&longs;um iuxta lineas Ek <lb/>FL de&longs;ignatum. </s> <s id="id.2.1.39.3.1.4.0"><expan abbr="Quoniã">Quoniam</expan> <expan abbr="au­t&etilde;">au­<lb/>tem</expan> duo latera AD DC duo<lb/>bus lateribus BD DE &longs;unt <lb/>æqualia; anguliq; ad D &longs;unt <lb/><arrow.to.target n="note65"></arrow.to.target>recti; erit latus AC lateri <lb/>CB æquale. </s> <s id="id.2.1.39.3.1.5.0">& cùm pun­<lb/>ctum C &longs;it immobile; dum <lb/>puncta AB mouentur, cir<lb/>culi circumferentiam de&longs;cri<lb/>bent, cuius &longs;emidiameter <lb/>erit AC. quare centro C, <lb/>circulus de&longs;cribatur AEBF. <lb/>puncta AB EF in circuli <lb/>circumferentia erunt. </s> <s id="id.2.1.39.3.1.6.0">&longs;ed <lb/>cùm EF &longs;it ip&longs;i AB æqua <lb/><arrow.to.target n="note66"></arrow.to.target>lis; erit circumferentia <lb/>EAF circumferentiæ AFB <lb/>æqualis. </s> <s id="id.2.1.39.3.1.7.0">quare dempta <lb/><figure id="id.036.01.060.1.jpg" xlink:href="036/01/060/1.jpg"></figure><lb/>communi AF, erit circumferentia EA circumferentiæ FB æqua<lb/>lis. </s> <s id="id.2.1.39.3.1.8.0">Quoniam autem mixtus angulus CEA e&longs;t æqualis mixto <lb/>CFB; & HFB ip&longs;o CFB e&longs;t maior; angulus verò HEA ip&longs;o <lb/>CEA minor; erit angulus HFB angulo HEA maior. </s> <s id="id.2.1.39.3.1.9.0">à quibus <lb/><arrow.to.target n="note67"></arrow.to.target>&longs;i auferantur anguli HFG HEk æquales; erit angulus GFB an <lb/>gulo kEA maior. </s> <s id="id.2.1.39.3.1.10.0">ergo de&longs;cen&longs;us ponderis in E minus obliquus <lb/>erit a&longs;cen&longs;u ponderis in F. </s> <s>& quamquam pondus in E de&longs;cen<lb/>dendo, & pondus in F a&longs;cendendo per circumferentias mouean<lb/>tur æquales; quia tamen pondus in E ex hoc loco rectius de&longs;cen<lb/>dit, quàm pondus in F a&longs;cendit: idcirco naturalis potentia pon<lb/>deris in E re&longs;i&longs;tentiam violentiæ ponderis F &longs;uperabit. </s> <s id="id.2.1.39.3.1.11.0">quare <lb/>maiorem grauitatem habebit pondus in E, quàm pondus in F. </s> <s id="id.2.1.39.3.1.11.0.a"><lb/>ergo pondus in E deor&longs;um, pondus verò in F &longs;ur&longs;um mouebitur: <pb n="24" xlink:href="036/01/061.jpg"/>donec libra EF in AB redeat. </s> <s id="id.2.1.39.3.1.12.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.40.1.0.0.0" type="margin"> <s id="id.2.1.40.1.1.1.0"><margin.target id="note65"></margin.target>4 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.40.1.1.2.0"><margin.target id="note66"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 28 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.40.1.1.3.0"><margin.target id="note67"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.41.1.0.0.0" type="main"> <s id="id.2.1.41.1.1.1.0">Huius autem effectus ratio ab Ari&longs;totele po&longs;ita, hic manife&longs;ta in <arrow.to.target n="note68"></arrow.to.target><lb/>tueri pote&longs;t. </s> <s id="id.2.1.41.1.1.2.0">&longs;it enim punctum N vbi CS EF &longs;e inuicem &longs;ecant. </s> <s id="id.2.1.41.1.1.3.0"><lb/>& quoniam HE e&longs;t ip&longs;i HF æqualis; erit NE maior NF. </s> <s>li­<lb/>nea ergo CS, quam perpendiculum vocat, libram EF in partes di<lb/>uidet inæquales. </s> <s id="id.2.1.41.1.1.4.0">cùm itaq; pars libræ NE &longs;it maior NF; atq; id, <lb/>quod plus e&longs;t, nece&longs;&longs;e e&longs;t, deor&longs;um ferri: libra ergo EF ex parte E <lb/>deor&longs;um mouebitur, donec in AB redeat. </s> </p> <p id="id.2.1.42.1.0.0.0" type="margin"> <s id="id.2.1.42.1.1.1.0"><margin.target id="note68"></margin.target><emph type="italics"/>Ari&longs;totelis ratio.<emph.end type="italics"/></s> </p> <p id="id.2.1.43.1.0.0.0" type="main"> <s id="id.2.1.43.1.1.1.0">Ex iis præterea, quæ ha<lb/>ctenus dicta &longs;unt inferre li<lb/>cet, libram EF velocius ab <lb/>eo &longs;itu in AB moueri; vndè <lb/>linea EF in directum pro­<lb/>tracta in centrum mundi <lb/>perueniat. </s> <s id="id.2.1.43.1.1.2.0">vt &longs;it EFS recta <lb/>linea. </s> <s id="id.2.1.43.1.1.3.0">& quoniam CD <lb/>CH, &longs;unt inter &longs;e &longs;e æqua<lb/>les. </s> <s id="id.2.1.43.1.1.4.0">&longs;i igitur centro C, &longs;pa<lb/>tioq; CD, circulus de&longs;cri­<lb/>batur DHM; erunt pun­<lb/>cta DH in circuli circum­<lb/>ferentia. </s> <s id="id.2.1.43.1.1.5.0">Quoniam au­<lb/>tem CH ip&longs;i EF e&longs;t per­<lb/>pendicularis; continget li­<lb/>nea EHS circulum DHM <lb/>in puncto H. </s> <s id="id.2.1.43.1.1.5.0.a">pondus igi­<lb/>tur in H (&longs;icuti &longs;upra de­<lb/>mon&longs;trauimus) grauius <lb/><figure id="id.036.01.061.1.jpg" xlink:href="036/01/061/1.jpg"></figure><lb/>erit, quàm in alio &longs;itu circuli DHM. </s> <s id="id.2.1.43.1.1.5.0.b">ergo magnitudo ex EF <lb/>ponderibus, & libra EF compo&longs;ita, cuius centrum grauitatis e&longs;t <lb/>in H, in hoc &longs;itu magis grauitabit, quàm in quocunq; alio &longs;itu <pb xlink:href="036/01/062.jpg"/>circuli fuerit punctum H. <lb/></s> <s>ab hoc igitur &longs;itu velo­<lb/>cius, quàm à quocunq; <lb/>alio mouebitur. </s> <s id="id.2.1.43.1.1.6.0">& &longs;i H <lb/>propius fuerit ip&longs;i D mi <lb/>nus grauitabit, minu&longs;q; <lb/>ab eo &longs;itu mouebitur. </s> <s id="id.2.1.43.1.1.7.0"><lb/>&longs;emper enim de&longs;cen&longs;us <lb/>obliquior e&longs;t, & minus re<lb/>ctus. </s> <s id="id.2.1.43.1.1.8.0">libra ergo EF velo<lb/>cius ab hoc &longs;itu mouebi­<lb/>tur, quàm ab alio &longs;itu. </s> <s id="id.2.1.43.1.1.9.0">& <lb/>&longs;i propius ad AB acce­<lb/>det, inde minus mouebi<lb/>tur. </s> <s id="id.2.1.43.1.1.10.0">Deinde quò longius <lb/>punctum H à puncto C <lb/>di&longs;tabit, velocius moue­<lb/>bitur; quod <expan abbr="nõ">non</expan> <expan abbr="&longs;olũ">&longs;olum</expan> ex Ari<lb/>&longs;totele in principio quæ&longs;t­<lb/>io num mechanicarum, & <lb/><figure id="id.036.01.062.1.jpg" xlink:href="036/01/062/1.jpg"></figure><lb/>ex &longs;uperius dictis patet; verùm etiam ex iis, quæ infra in &longs;exta <lb/>propo&longs;itione dicemus, manife&longs;tum erit. </s> <s id="id.2.1.43.1.1.11.0">libra igitur EF, quò ma<lb/>gis ab eius centro di&longs;tabit, adhuc velocius mouebitur. </s> </p> <pb n="25" xlink:href="036/01/063.jpg"/> <p id="id.2.1.43.3.0.0.0" type="main"> <s id="id.2.1.43.3.1.1.0">Sit deinde libra AB, <lb/>cuius centrum C &longs;it infra li<lb/>bram; &longs;intq; in AB pon<lb/>dera æqualia; libraq; &longs;it <lb/>mota in EF. </s> <s id="id.2.1.43.3.1.1.0.a">Dico maio­<lb/>rem habere grauitatem <lb/>pondus in F, quàm pondus <lb/>in E. atq; ideo libram EF <lb/>deor&longs;um ex parte F moue­<lb/>ri. </s> <s id="id.2.1.43.3.1.2.0">Producatur DC ex <lb/>vtraq; parte v&longs;q; ad mun­<lb/>di centrum S, & v&longs;q; ad <lb/>O, lineaq; HS ducatur, <lb/>cui à punctis EF æquidi­<lb/>&longs;tantes ducantur GEk FL; <lb/>connectanturq; CE CF: <lb/>atq; centro C, &longs;patioq; CE <lb/>circulus de&longs;cribatur AEO <lb/>BF. </s> <s id="id.2.1.43.3.1.2.0.a">&longs;imiliter demon&longs;tra­<lb/>bitur puncta ABEF in <lb/>circuli circumferentia e&longs;&longs;e; <lb/>de&longs;cen&longs;umq; libræ EF vná <lb/>cum ponderibus rectum &longs;e<lb/>cundùm lineam HS fieri; <lb/>ponderumq; in EF &longs;ecun<lb/><figure id="id.036.01.063.1.jpg" xlink:href="036/01/063/1.jpg"></figure><expan abbr="dùm"><lb/>dum</expan> lineas GK FL ip&longs;i HS æquidi&longs;tantes. </s> <s id="id.2.1.43.3.1.3.0">Quoniam autem an<lb/>gulus CFP æqualis e&longs;t angulo CEO: erit angulus HFP angulo <lb/>HEO maior. </s> <s id="id.2.1.43.3.1.4.0">angulus verò HFL æqualis e&longs;t angulo HEG. à <arrow.to.target n="note69"></arrow.to.target><lb/>quibus igitur &longs;i demantur anguli HFP HEO, erit angulus <lb/>LFP angulo GEO minor. </s> <s id="id.2.1.43.3.1.5.0">quare de&longs;cen&longs;us ponderis in F rectior <lb/>erit a&longs;cen&longs;u ponderis in E. ergo naturalis potentia ponderis in <lb/>F re&longs;i&longs;tentiam violentiæ ponderis in E &longs;uperabit. </s> <s id="id.2.1.43.3.1.6.0">& ideo ma­<lb/>iorem habebit grauitatem pondus in F, quàm pondus in E. </s> <s id="id.2.1.43.3.1.6.0.a"><lb/>Pondus igitur in F deor&longs;um, pondus verò in E &longs;ur&longs;um mo­<lb/>uebitur. </s> </p> <p id="id.2.1.44.1.0.0.0" type="margin"> <s id="id.2.1.44.1.1.1.0"><margin.target id="note69"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.45.1.0.0.0" type="main"> <s id="id.2.1.45.1.1.1.0">Ari&longs;totelis quoq; ratio hic per&longs;picua erit. </s> <s id="id.2.1.45.1.1.2.0">&longs;it enim punctum <arrow.to.target n="note70"></arrow.to.target><pb xlink:href="036/01/064.jpg"/>N vbi CO EF &longs;e inuicem <lb/>&longs;ecant; erit NF maior <lb/>NE. </s> <s id="id.2.1.45.1.1.2.0.a">& quoniam CO per<lb/>pendiculum (&longs;ecundùm <lb/>ip&longs;um) libram EF in par<lb/>tes inæquales diuidit, & <lb/>maior pars e&longs;t ver&longs;us F, hoc <lb/>e&longs;t NF; libra EF ex par<lb/>te F deor&longs;um mouebitur: <lb/>cùm id, quod plus e&longs;t, deor<lb/>&longs;um feratur. </s> </p> <p id="id.2.1.46.1.0.0.0" type="margin"> <s id="id.2.1.46.1.1.1.0"><margin.target id="note70"></margin.target><emph type="italics"/>Ari&longs;totelis ratio.<emph.end type="italics"/></s> </p> <p id="id.2.1.47.1.0.0.0" type="main"> <s id="id.2.1.47.1.1.1.0">Similiter, éx dictis <lb/>quoq; eliciemus libram EF <lb/>centrum habens infra li­<lb/>bram, quò magis à &longs;itu <lb/>AB di&longs;tabit, velocius mo <lb/>ueri. </s> <s id="id.2.1.47.1.1.2.0">centrum enim graui<lb/>tatis H, quò magis á pun­<lb/>cto D di&longs;tat, eò volecius <lb/>pondus ex EF ponderibus, <lb/>libraq; EF compo&longs;itum <lb/>mouebitur, donec angulus <lb/>CHS rectus euadat. </s> <s id="id.2.1.47.1.1.3.0">ad­<lb/>huc in&longs;uper velocius moue<lb/>bitur, quò libram à centro <lb/>C magis di&longs;tabit. <figure id="id.036.01.064.1.jpg" xlink:href="036/01/064/1.jpg"></figure></s> </p> <p id="id.2.1.47.2.0.0.0" type="main"> <s id="id.2.1.47.2.1.1.0">Ex ip&longs;orum quinetiam rationibus, ac fal&longs;is &longs;upo&longs;itionibus iam <lb/>declaratos libræ effectus, ac motus deducere, ac manife&longs;tare libet; <lb/>vt quanta &longs;it veritatis efficacia appareat, quippè ex fal&longs;is etiam <lb/>eluce&longs;cere contendit. </s> </p> <pb n="26" xlink:href="036/01/065.jpg"/> <p id="id.2.1.47.4.0.0.0" type="main"> <s id="id.2.1.47.4.1.1.0">Exponantur eadem, &longs;ci <lb/>licet &longs;it circulus AEBF; <lb/>libraqué AB, cuius cen­<lb/>trum C &longs;it &longs;upra libram, <lb/>moueatur in EF. </s> <s id="id.2.1.47.4.1.1.0.a">dico <lb/>pondus in E maiorem ibi <lb/>habere grauitatem, quàm <lb/>pondus in F; libramq; EF <lb/>in AB redire. </s> <s id="id.2.1.47.4.1.2.0">Ducantur <lb/>à punctis EF ip&longs;i AB <lb/>perpendiculares EL FM, <lb/>quæ inter &longs;e æquidi&longs;tan­<lb/>tes <arrow.to.target n="note71"></arrow.to.target><figure id="id.036.01.065.1.jpg" xlink:href="036/01/065/1.jpg"></figure>erunt; &longs;itq; punctum N, vbi AB EF &longs;e inuicem &longs;ecant. </s> <s id="id.2.1.47.4.1.3.0"><lb/>Quoniam igitur angulus FNM e&longs;t æqualis angulo ENL, & an­<lb/>gulus <arrow.to.target n="note72"></arrow.to.target>F MN rectus recto ELN æqualis, ac reliquus NFM reli­<lb/>quo <arrow.to.target n="note73"></arrow.to.target>NEL e&longs;t etiam æqualis; erit triangulum NLE triangu<lb/>lo NMF &longs;imile. </s> <s id="id.2.1.47.4.1.4.0">vt igitur NE ad EL, ita NF ad FM; & per <arrow.to.target n="note74"></arrow.to.target><lb/>mutando vt EN ad NF, ita EL ad FM. &longs;ed cùm &longs;it HE ip&longs;i <arrow.to.target n="note75"></arrow.to.target><lb/>HF æqualis, erit EN maior NF; quare & EL maior erit FM. </s> <s id="id.2.1.47.4.1.4.0.a"><lb/>& quoniam dum pondus in E per <expan abbr="circumferentiiam">circumferentiam</expan> EA de&longs;cendit, <lb/>pondus in F per circumferentiam FB ip&longs;i circumferentiæ EA <lb/>æqualem a&longs;cendit; de&longs;cen&longs;u&longs;q; ponderis in E de directo (vt ip­<lb/>&longs;i dicunt) capit EL: a&longs;cen&longs;us verò ponderis in F de directo ca­<lb/>pit FM; minus de directo capiet a&longs;cen&longs;us ponderis in F, quàm <lb/>de&longs;cen&longs;us ponderis in E. maiorem igitur grauitatem habebit pon<lb/>dus in E, quàm pondus in F. </s> </p> <p id="id.2.1.48.1.0.0.0" type="margin"> <s id="id.2.1.48.1.1.1.0"><margin.target id="note71"></margin.target>28 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.48.1.1.2.0"><margin.target id="note72"></margin.target>15 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.48.1.1.3.0"><margin.target id="note73"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.48.1.1.4.0"><margin.target id="note74"></margin.target>4 <emph type="italics"/>Sexti.<emph.end type="italics"/></s> <s id="id.2.1.48.1.1.5.0"><margin.target id="note75"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.49.1.0.0.0" type="main"> <s id="id.2.1.49.1.1.1.0">Producatur CD ex vtraq; parte in OP, quæ lineam EF in <lb/>puncto S &longs;ecet. </s> <s id="id.2.1.49.1.1.2.0">& quoniam (vt aiunt) quò magis pondus à li­<lb/>nea directionis OP di&longs;tat, eò fit grauius; idcirco hoc quoq; me <lb/>dio pondus in E maiorem habere <expan abbr="grauitauitatem">grauitatem</expan> pondere in F o­<lb/>&longs;tendetur. </s> <s id="id.2.1.49.1.1.3.0">Ducantur à punctis EF ip&longs;i OP perpendiculares EQ <lb/>FR. &longs;imili ratione o&longs;tendetur, triangulum QES triangulo RFS <lb/>&longs;imile e&longs;&longs;e; lineamq; EQ ip&longs;a RF maiorem e&longs;&longs;e. </s> <s id="id.2.1.49.1.1.4.0">pondus itaq; <lb/>in E magis à linea OP di&longs;tabit, quàm pondus in F; ac propterea <lb/>pondus in E maiorem habebit grauitatem pondere in F. ex quibus <lb/>reditus libræ EF in AB manife&longs;tus apparet. </s> </p> <pb xlink:href="036/01/066.jpg"/> <p id="id.2.1.49.3.0.0.0" type="main"> <s id="id.2.1.49.3.1.1.0">Si autem centrum libræ <lb/>&longs;it infra libram, tunc pon­<lb/>dus depre&longs;&longs;um maiorem <lb/>habere grauitatem eleuato <lb/>ii&longs;dem mediis o&longs;tendetur. </s> <s id="id.2.1.49.3.1.2.0"><lb/>ducantur à punctis EF ip­<lb/>&longs;i AB perpendiculares EL <lb/>FM. </s> <s>&longs;imiliter demon&longs;tra<lb/>bitur EL maiorem e&longs;&longs;e <lb/>FM; & ob id de&longs;cen&longs;us <lb/>ponderis in F minus de di <lb/>recto capiet, quàm a&longs;cen­<lb/><figure id="id.036.01.066.1.jpg" xlink:href="036/01/066/1.jpg"></figure><lb/>&longs;us ponderis in E: quocirca re&longs;i&longs;tentia violentiæ ponderis in E &longs;u<lb/>perabit naturalem propen&longs;ionem ponderis in F. </s> <s>ergo pondus in E <lb/>pondere in F grauius erit. </s> </p> <p id="id.2.1.49.4.0.0.0" type="main"> <s id="id.2.1.49.4.1.1.0">Producatur etiam CD ex vtraq; parte in OP; ip&longs;iq; à punctis <lb/>EF perpendiculares ducantur EQ FR. </s> <s>eodem pror&longs;us modo <lb/>o&longs;tendetur, lineam EQ maiorem e&longs;&longs;e FR. </s> <s>pondus ideò in E ma<lb/>gis à linea directionis OP di&longs;tabit, quàm pondus in F. </s> <s>maio­<lb/>rem igitur grauitatem habebit pondus in E, quàm pondus in F. <lb/></s> <s>ex quibus &longs;equitur, libram EF ex parte E deor&longs;um moueri. </s> </p> <p id="id.2.1.49.5.0.0.0" type="main"> <s id="id.2.1.49.5.1.1.0">Ari&longs;toteles itaq; has duas tantùm quæ&longs;tiones propo&longs;uit, ter­<lb/>tiamq; reliquit; &longs;cilicet cùm centrum libræ in ip&longs;a e&longs;t libra: hanc <lb/>autem ommi&longs;sit, vt notam, quemadmodum res valde notas præ­<lb/>termittere &longs;olet. </s> <s id="id.2.1.49.5.1.2.0">nam cui dubium, &longs;i pondus in eius centro gra<lb/>uitatis &longs;u&longs;tineatur, quin maneat? </s> <s id="id.2.1.49.5.1.3.0">Ea verò, quæ ex ip&longs;ius &longs;enten<lb/>tia attulimus, aliquis reprehendere po&longs;&longs;et, nos integram eius &longs;enten<lb/>tiam minimè protuli&longs;&longs;e <expan abbr="affimans">affirmans</expan>. </s> <s id="id.2.1.49.5.1.4.0">nam cùm in &longs;ecunda parte &longs;e<lb/>cundæ quæ&longs;tionis proponit, cur libra, trutina deor&longs;um con&longs;tituta, <lb/>quando deor&longs;um lato pondere qui&longs;piam id amouet, non a&longs;cen<lb/>dit, &longs;ed manet? </s> <s id="id.2.1.49.5.1.5.0">non a&longs;&longs;erit adhuc libram deor&longs;um moueri; &longs;ed <lb/>manere. </s> <s id="id.2.1.49.5.1.6.0">quod in vltima quoq; conclu&longs;ione colligi&longs;&longs;e videtur. </s> <s id="id.2.1.49.5.1.7.0">Ve <lb/>rùm hoc non &longs;olum nobis non repugnat, &longs;ed &longs;i rectè intelligitur, <lb/>maximè &longs;uffragatur. </s> </p> <pb n="27" xlink:href="036/01/067.jpg"/> <p id="id.2.1.49.7.0.0.0" type="main"> <s id="id.2.1.49.7.1.1.0">Sit enim libra AB <lb/>horizonti æquidi&longs;tans, <lb/>cuius centrum E &longs;it <lb/>infra libram. </s> <s id="id.2.1.49.7.1.2.0">quia ve <lb/>rò Ari&longs;toteles libram, <lb/>&longs;icuti actu e&longs;t, con&longs;ide<lb/>rat; ideò nece&longs;&longs;e e&longs;t <lb/>trutinam, vel aliquid <lb/>aliud infra centrum E <lb/>collocare, vt EF <lb/>(quod quidem truti­<lb/>na erit) ita vt centrum <lb/>E &longs;u&longs;tineat. </s> <s id="id.2.1.49.7.1.3.0">&longs;itq; per­<lb/><figure id="id.036.01.067.1.jpg" xlink:href="036/01/067/1.jpg"></figure><lb/>pendiculum ECD. </s> <s>& vt libra AB ab hoc moueatur &longs;itu; dicit <lb/>Ari&longs;toteles, ponatur pondus in B, quod cùm &longs;it graue, libram ex <lb/>parte B deor&longs;um mouebit; putá in G. </s> <s>ita vt propter impedimen<lb/>tum deor&longs;um amplius moueri non poterit. </s> <s id="id.2.1.49.7.1.4.0">non enim dicit Ari<lb/>&longs;toteles, moueatur libra ex parte B deor&longs;um, quou&longs;q; libuerit; dein <lb/>de relinquatur, vt nos diximus: &longs;ed præcipit, vt in ip&longs;o B po­<lb/>natur pondus, quod ex ip&longs;ius natura deor&longs;um &longs;emper mouebi­<lb/>tur; donec libra trutinæ, &longs;iue alicui alii adhæreat. </s> <s id="id.2.1.49.7.1.5.0">& quando B erit <lb/>in G, erit libra in GH; in quo &longs;itu, ablato pondere, manebit: <lb/>cùm maior pars libræ à perpendiculo &longs;it ver&longs;us G, quæ e&longs;t DG, <lb/>quàm DH. </s> <s id="id.2.1.49.7.1.5.0.a">nec deor&longs;um amplius mouebitur; nam libra, vel <lb/>trutinæ, vel alteri cuipiam, quod centrum libræ &longs;u&longs;tineat, incum<lb/>bet. </s> <s id="id.2.1.49.7.1.6.0">&longs;i enim huic non adhæreret, libra ex parte G deor&longs;um ex <lb/>ip&longs;ius &longs;ententia moueretur; cùm id, quod plus e&longs;t, &longs;cilicet DG, <lb/>deor&longs;um ferri &longs;it nece&longs;&longs;e. </s> </p> <p id="id.2.1.49.8.0.0.0" type="main"> <s id="id.2.1.49.8.1.1.0">Cæterum quis adhuc dicere poterit, &longs;i paruum imponatur pon<lb/>dus in B, mouebitur quidem libra deor&longs;um, non autem v&longs;q; ad <lb/>G. </s> <s>in quò &longs;itu &longs;ecundùm Ari&longs;totelem, ablato pondere, mane­<lb/>re deberet. </s> <s id="id.2.1.49.8.1.2.0">quod experimento patet; cùm in vna tantùm libræ <lb/>extremitate, impo&longs;ito onere, hocq; vel maiore, vel minore, libra <lb/>plus, minu&longs;uè inclinetur. </s> <s id="id.2.1.49.8.1.3.0">Quod e&longs;t quidem veri&longs;&longs;imum, centro &longs;upra <lb/>libram, non autem infra, neq; in ip&longs;a libra collocato. </s> <s id="id.2.1.49.8.1.4.0">Vt exempli <lb/>gratia. </s> </p> <pb xlink:href="036/01/068.jpg"/> <p id="id.2.1.49.10.0.0.0" type="main"> <s id="id.2.1.49.10.1.1.0">Sit libra horizonti æ­<lb/>quidi&longs;tans AB, cuius cen<lb/>trum C &longs;it &longs;upra libram, <lb/>perpendiculumq; CD ho<lb/>rizonti perpendiculare, <lb/>quod ex parte D produca<lb/>tur in H. </s> <s id="id.2.1.49.10.1.1.0.a">Quoniam enim <lb/>con&longs;iderata libræ grauita­<lb/>te, erit punctum D libræ <lb/>centrum grauitatis. </s> <s id="id.2.1.49.10.1.2.0">&longs;i ergo <lb/>in B paruum imponatur <lb/>pondus, cuius centrum <lb/><figure id="id.036.01.068.1.jpg" xlink:href="036/01/068/1.jpg"></figure><lb/>grauitatis &longs;it in puncto B; magnitudinis ex libra AB, & pondere <lb/>in B compo&longs;itæ non erit amplius centrum grauitatis D; &longs;ed erit in <lb/><arrow.to.target n="note76"></arrow.to.target>linea DB, vt in E: ita vt DE ad EB &longs;it, vt pondus in B ad gra­<lb/>uitatem libræ AB. </s> <s>Connectatur CE. </s> <s id="id.2.1.49.10.1.2.0.a">Quoniam autem pun­<lb/>ctum C e&longs;t immobile, dum libra mouetur, punctum E circuli cir<lb/>cumferentiam EFG de&longs;cribet, cuius &longs;emidiameter CE, & cen­<lb/>trum C. </s> <s>quia verò CD horizonti e&longs;t perpendicularis, linea CE <lb/>horizonti perpendicularis nequaquam erit. </s> <s id="id.2.1.49.10.1.3.0">quare magnitudo ex <lb/>AB, & pondere in B compo&longs;ita minimè in hoc &longs;itu manebit; &longs;ed <lb/><arrow.to.target n="note77"></arrow.to.target>deor&longs;um &longs;ecundùm eius grauitatis centrum E per circumferen­<lb/>tiam EFG mouebitur; donec CE horizonti perpendicularis eua<lb/>dat; hoc e&longs;t, donec CE in CDF perueniat. </s> <s id="id.2.1.49.10.1.4.0">atq; tunc libra AB <lb/>mota erit in kL, in quo &longs;itu libra vná cum pondere manebit. </s> <s id="id.2.1.49.10.1.5.0">nec <lb/>deor&longs;um amplius mouebitur. </s> <s id="id.2.1.49.10.1.6.0">Si verò in B ponatur pondus graui­<lb/>us; centrum grauitatis totius magnitudinis erit ip&longs;i B propius, vt in <lb/>M. </s> <s>& tunc libra deor&longs;um, donec iuncta CM in linea CDH per <lb/>ueniat, mouebitur. </s> <s id="id.2.1.49.10.1.7.0">Ex maiore igitur, & minore pondere in B po<lb/>&longs;ito, libra plus, minu&longs;uè inclinabitur. </s> <s id="id.2.1.49.10.1.8.0">ex quo &longs;equitur pondus B <lb/>quarta circuli parte minorem &longs;emper circumferentiam de&longs;cribe­<lb/>re, cùm angulus FCE &longs;it &longs;emper acutus. </s> <s id="id.2.1.49.10.1.9.0">nunquam enim punctum <lb/>B v&longs;q; ad lineam CH perueniet, cùm centrum grauitatis ponde­<lb/>ris, & libræ &longs;imul &longs;emper inter DB exi&longs;tat. </s> <s id="id.2.1.49.10.1.10.0">quò tamen pondus <lb/>in B grauius fuerit, maiorem quoq; circumferentiam de&longs;cribet. </s> <s id="id.2.1.49.10.1.11.0"><lb/>eò enim magis punctum B ad lineam CH accedet. </s> </p> <p id="id.2.1.50.1.0.0.0" type="margin"> <s id="id.2.1.50.1.1.1.0"><margin.target id="note76"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.50.1.1.3.0"><margin.target id="note77"></margin.target>1. <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="28" xlink:href="036/01/069.jpg"/> <p id="id.2.1.51.1.0.0.0" type="main"> <s id="id.2.1.51.1.2.1.0">Habeat autem libra AB <lb/>centrum C in ip&longs;a libra, atq; <lb/>in eius medio: erit C libræ <lb/>centrum quoq; grauitatis; <lb/>à quo ip&longs;i AB, horizontiq; <lb/>perpendicularis ducatur FC <lb/>G. </s> <s>ponatur deinde in B <lb/>quoduis pondus; erit totius <lb/>magnitudinis centrum gra­<lb/>uitatis putá in E; ita vt CE <lb/><figure id="id.036.01.069.1.jpg" xlink:href="036/01/069/1.jpg"></figure><lb/>ad EB &longs;it, vt pondus in B ad libræ grauitatem. </s> <s id="id.2.1.51.1.2.2.0">& quoniam CE <lb/>non e&longs;t horizonti perpendicularis, libra AB, atq; pondus in B <lb/>in hoc &longs;itu nunquam manebunt; &longs;ed deor&longs;um ex parte B mouebun<lb/>tur, donec CE horizonti fiat perpendicularis. </s> <s id="id.2.1.51.1.2.3.0">hoc e&longs;t donec li­<lb/>bra AB in FG perueniat. </s> <s id="id.2.1.51.1.2.4.0">ex quo patet, quolibet pondus in B <lb/>circuli quartam &longs;emper de&longs;cribere. </s> </p> <p id="id.2.1.51.2.0.0.0" type="main"> <s id="id.2.1.51.2.1.1.0">Sit autem centrum C in­<lb/>fra libram AB. </s> <s>&longs;itq; DCE <lb/>perpendiculum. </s> <s id="id.2.1.51.2.1.2.0">&longs;imiliter <lb/>po&longs;ito in B pondere, cen­<lb/>trum grauitatis magnitudi<lb/>nis ex AB libra, & ponde<lb/>re in B compo&longs;itæ in linea <lb/>DB erit; vt in F; ita vt DF <lb/>ad FB &longs;it, vt pondus in B <lb/><figure id="id.036.01.069.2.jpg" xlink:href="036/01/069/2.jpg"></figure><lb/>ad libræ pondus. </s> <s id="id.2.1.51.2.1.3.0">Iungatur CF. </s> <s>& quoniam CD horizonti e&longs;t <lb/>perpendicularis; linea CF horizonti nequaquam perpendicula­<lb/>ris exi&longs;tet. </s> <s id="id.2.1.51.2.1.4.0">quare magnitudo ex AB libra, ac pondere in B com<lb/>po&longs;ita in hoc &longs;itu nunquam per&longs;i&longs;tet; &longs;ed deor&longs;um, ni&longs;i aliquid <lb/>impediat, mouebitur; donec CF in DCE perueniat: in quo &longs;itu <lb/>libra vná cum pondere manebit. </s> <s id="id.2.1.51.2.1.5.0">& punctum B erit vt in G, atq; <lb/>punctum A in H, libraq; GH non amplius centrum infra, &longs;ed &longs;u<lb/>pra ip&longs;am habebit. </s> <s id="id.2.1.51.2.1.6.0">quod idem &longs;emper eueniet; quamuis mini­<lb/>mum imponatur pondus in B. </s> <s>ergo priu&longs;quam B perueniat ad <lb/>G; nece&longs;&longs;e e&longs;t libram, &longs;iue trutinæ deor&longs;um po&longs;itæ, vel alicui <pb xlink:href="036/01/070.jpg"/>alteri, quod centrum C &longs;u­<lb/>&longs;tineat, occurrere; ibiq; ad­<lb/>hærere. </s> <s id="id.2.1.51.2.1.7.0">ex hoc &longs;equitur, pon<lb/>dus in B vltra lineam Dk <lb/>&longs;emper moueri; ac circuli <lb/>quarta maiorem &longs;emper cir<lb/><expan abbr="cumfer&etilde;tiam">cumferentiam</expan> de&longs;cribere: e&longs;t <lb/>enim angulus FCE &longs;emper <lb/>obtu&longs;us, cùm angulus DCF <lb/>&longs;emper &longs;it acutus. </s> <s id="id.2.1.51.2.1.8.0">quò au­<lb/><figure id="id.036.01.070.1.jpg" xlink:href="036/01/070/1.jpg"></figure><lb/>tem pondus in B fuerit leuius, maiorem tamen adhuc circumfe­<lb/>rentiam de&longs;cribet. </s> <s id="id.2.1.51.2.1.9.0">nam quò pondus in G leuius fuerit, eò ma­<lb/>gis pondus in G eleuabitur; libraq; GH ad &longs;itum horizonti æqui<lb/>di&longs;tantem propius accedet. </s> <s id="id.2.1.51.2.1.10.0">quæ omnia ex iis, quæ &longs;upra dixi­<lb/>mus, manife&longs;ta &longs;unt. </s> </p> <p id="id.2.1.51.3.0.0.0" type="main"> <s id="id.2.1.51.3.1.1.0">His demon&longs;tratis. </s> <s id="id.2.1.51.3.1.2.0">Manife&longs;tum e&longs;t, centrum libræ cau&longs;am e&longs;&longs;e <lb/>diuer&longs;itatis effectuum in libra. </s> <s id="id.2.1.51.3.1.3.0">atq; patet omnes Archimedis de <lb/>æqueponderantibus propo&longs;itiones ad hoc pertinentes in omni &longs;itu <lb/>veras e&longs;&longs;e. </s> <s id="id.2.1.51.3.1.4.0">hoc e&longs;t &longs;iue libra &longs;it horizonti æquidi&longs;tans, &longs;iue non: <lb/>dummodo centrum libræ in ip&longs;a &longs;it libra; quemadmodum ip&longs;e <lb/>con&longs;iderat. </s> <s id="id.2.1.51.3.1.5.0">& quamquam libra brachia habeat inæqualia, idem eue<lb/>niet; eodemq; pro&longs;us modo o&longs;tendetur, centrum libræ diuer&longs;imo<lb/>dè collocatum varios producere effectus. </s> </p> <p id="id.2.1.51.4.0.0.0" type="main"> <s id="id.2.1.51.4.1.1.0">Sit enim libra AB hori­<lb/>zonti æquidi&longs;tans; & in AB <lb/>&longs;int pondera inæqualia, quo <lb/>rum grauitatis centrum &longs;it <lb/>C: &longs;u&longs;pendaturq; libra in <lb/>eodem puncto C. </s> <s>& mo­<lb/>ueatur libra in DE. </s> <s id="id.2.1.51.4.1.1.0.a">mani<lb/><arrow.to.target n="note78"></arrow.to.target>fe&longs;tum e&longs;t libram non &longs;o­<lb/>lum in DE, &longs;ed in quouis <lb/>alio &longs;itu manere. <figure id="id.036.01.070.2.jpg" xlink:href="036/01/070/2.jpg"></figure></s> </p> <pb n="29" xlink:href="036/01/071.jpg"/> <p id="id.2.1.51.6.0.0.0" type="main"> <s id="id.2.1.51.6.1.1.0">Sit autem centrum libræ <lb/>AB &longs;upra C in F; &longs;itq; <lb/>FC ip&longs;i AB, & horizonti <lb/>perpendicularis: & &longs;i mo­<lb/>ueatur libra in DE, linea <lb/>CF mota erit in FG; quæ <lb/>cùm non &longs;it horizonti per­<lb/>pendicularis, libra DE <arrow.to.target n="note79"></arrow.to.target><lb/>deor&longs;um ex parte D moue<lb/>bitur, donec FG in FC <lb/>redeat: atq; tunc libra DE <lb/>in AB erit, in quò &longs;itu <lb/>quoq; manebit. <figure id="id.036.01.071.1.jpg" xlink:href="036/01/071/1.jpg"></figure></s> </p> <p id="id.2.1.51.7.0.0.0" type="main"> <s id="id.2.1.51.7.1.1.0">Et &longs;i centrum libræ F <lb/>&longs;it infra libram; &longs;itq; mota <lb/>libra in DE; primùm qui<lb/>dem manife&longs;tum e&longs;t li­<lb/>bram in AB manere; in <arrow.to.target n="note80"></arrow.to.target><lb/>DE verò deor&longs;um ex par <lb/>te E moueri: cùm linea <lb/>FG non &longs;it horizonti per­<lb/>pendicularis. <figure id="id.036.01.071.2.jpg" xlink:href="036/01/071/2.jpg"></figure></s> </p> <p id="id.2.1.51.8.0.0.0" type="main"> <s id="id.2.1.51.8.1.1.0">Ex his determinatis &longs;i libra &longs;it <lb/>arcuata, vel libræ brachia angulum <lb/>con&longs;tituant; centrumq; diuer&longs;imo<lb/>dè collocetur (quamquam hæc pro<lb/>priè non &longs;it libra) varios tamen <lb/>huius quoq; effectus o&longs;tendere pote<lb/>rimus. </s> <s id="id.2.1.51.8.1.2.0">Vt &longs;it libra ACB, cuius <lb/>centrum, circa quod vertitur, &longs;it C. <lb/></s> <s>ductaq; AB, &longs;it arcus &longs;iue angulus <lb/><figure id="id.036.01.071.3.jpg" xlink:href="036/01/071/3.jpg"></figure><lb/>ACB &longs;upra lineam AB; & in AB grauitatis centra ponderum <lb/>ponantur, quæ in hoc &longs;itu maneant. </s> <s id="id.2.1.51.8.1.3.0">moueatur deinde libra ab <pb xlink:href="036/01/072.jpg"/>hoc &longs;itu, putá in ECF. </s> <s id="id.2.1.51.8.1.3.0.a">Dico li­<lb/>bram ECF in ACB redire. </s> <s id="id.2.1.51.8.1.4.0">to­<lb/>tius magnitudinis centrum grauita<lb/>tis inueniatur D. </s> <s>& CD iunga­<lb/>tur. </s> <s id="id.2.1.51.8.1.5.0">Quoniam enim pondera AB <lb/><arrow.to.target n="note81"></arrow.to.target>manent, linea CD horizonti per­<lb/>pendicularis erit. </s> <s id="id.2.1.51.8.1.6.0">quando igitur <lb/>libra erit in ECF, linea CD erit <lb/>putá in CG; quæ cùm non &longs;it ho<lb/><figure id="id.036.01.072.1.jpg" xlink:href="036/01/072/1.jpg"></figure><lb/>rizonti perpendicularis; libra ECF in ACB redibit. </s> <s id="id.2.1.51.8.1.7.0">quod idem <lb/>eueniet, &longs;i centrum C &longs;upra libram con&longs;tituatur, vt in H. </s> </p> <p id="id.2.1.52.1.0.0.0" type="margin"> <s id="id.2.1.52.1.1.1.0"><margin.target id="note78"></margin.target><emph type="italics"/>Per def. <expan abbr="c&etilde;tri">centri</expan> grauitatis. <emph.end type="italics"/></s> <s id="id.2.1.52.1.1.2.0"><margin.target id="note79"></margin.target>1 <emph type="italics"/>Huius. <emph.end type="italics"/></s> <s id="id.2.1.52.1.1.3.0"><margin.target id="note80"></margin.target>1. <emph type="italics"/>Huius. <emph.end type="italics"/></s> <s id="id.2.1.52.1.1.4.0"><margin.target id="note81"></margin.target>1 <emph type="italics"/>Huius. <emph.end type="italics"/></s> </p> <p id="id.2.1.53.1.0.0.0" type="main"> <s id="id.2.1.53.1.1.1.0">Si verò arcus, &longs;iue angulus <lb/>ACB, &longs;it infra lineam AB; eo <lb/>dem modo libram ECF, cuius <lb/>centrum, &longs;iue &longs;it in C, &longs;iue in H, <lb/>deor&longs;um ex parte F moueri o­<lb/>&longs;tendemus. <figure id="id.036.01.072.2.jpg" xlink:href="036/01/072/2.jpg"></figure><figure id="id.036.01.072.3.jpg" xlink:href="036/01/072/3.jpg"></figure></s> </p> <p id="id.2.1.53.2.0.0.0" type="main"> <s id="id.2.1.53.2.1.1.0">Sit autem angulus ACB &longs;upra lineam AB; ac libræ centrum <lb/>&longs;it H; lineaq; CH libram &longs;u&longs;tineat; & moueatur libra in EKF: <lb/>libra EkF in ACB redibit. </s> </p> <pb n="30" xlink:href="036/01/073.jpg"/> <p id="id.2.1.53.4.0.0.0" type="main"> <s id="id.2.1.53.4.1.1.0">Si verò centrum libræ &longs;it D, quocunq; modo moueatur libra; <lb/>vbi relinquetur, manebit. </s> </p> <p id="id.2.1.53.5.0.0.0" type="main"> <s id="id.2.1.53.5.1.1.0">Si deinde punctum H &longs;it infra lineam AB; tunc libra EkF <lb/>deor&longs;um ex parte F mouebitur. </s> </p> <p id="id.2.1.53.6.0.0.0" type="main"> <s id="id.2.1.53.6.1.1.0">Similiq; pror&longs;us ratione, &longs;i an<lb/>gulus ACB &longs;it infra lineam AB; <lb/>&longs;itq; libræ centrum H; &longs;u&longs;tineaturq; <lb/>libra linea CH; &longs;i libra ab hoc mo<lb/>ueatur &longs;itu, deor&longs;um ex parte pon­<lb/>deris inferioris mouebitur. </s> <s id="id.2.1.53.6.1.2.0">& &longs;i cen<lb/>trum libræ &longs;it D; vbi relinquetur, <lb/>manebit. </s> <s id="id.2.1.53.6.1.3.0">&longs;i verò &longs;it in K; &longs;i ab eiu&longs; <lb/><figure id="id.036.01.073.1.jpg" xlink:href="036/01/073/1.jpg"></figure><lb/>modi moueatur &longs;itu, in eundem pro&longs;us redibit. </s> <s id="id.2.1.53.6.1.4.0">quæ omnia ex iis, <lb/>quæ in principio diximus, &longs;unt manife&longs;ta. </s> <s id="id.2.1.53.6.1.5.0">&longs;imiliter &longs;i centrum li<lb/>bræ, vel in altero brachiorum, vel intra, vel extra vtcunq; po<lb/>natur; eadem inueniemus. </s> </p> <pb xlink:href="036/01/074.jpg"/> <p id="id.2.1.53.8.0.0.0" type="head"> <s id="id.2.1.53.8.1.1.0">PROPOSITIO. V. </s> </p> <p id="id.2.1.53.9.0.0.0" type="main"> <s id="id.2.1.53.9.1.1.0">Duo pondera in libra appen&longs;a, &longs;i libra inter <lb/>hæc ita diuidatur, vt partes ponderibus per­<lb/>mutatim re&longs;pondeant; tàm in punctis appen&longs;is <lb/>ponderabunt, quàm &longs;i vtraq; ex diui&longs;ionis pun­<lb/>cto &longs;u&longs;pendantur. <figure id="id.036.01.074.1.jpg" xlink:href="036/01/074/1.jpg"></figure></s> </p> <p id="id.2.1.53.10.0.0.0" type="main"> <s id="id.2.1.53.10.1.1.0">Sit AB libra, cuius centrum C; &longs;intq; duo pondera EF ex pun<lb/>ctis BG &longs;u&longs;pen&longs;a: diuidaturq; BG in H, ita vt BH ad HG <lb/>eandem habeat proportionem, quam pondus E ad pondus F. </s> <s id="id.2.1.53.10.1.1.0.a"><lb/>Dico pondera EF tàm in BG ponderare, quàm &longs;i vtraq; ex pun<lb/>cto H &longs;u&longs;pendantur. </s> <s id="id.2.1.53.10.1.2.0">fiat AC ip&longs;i CH æqualis. </s> <s id="id.2.1.53.10.1.3.0">& vt AC ad <lb/>CG, ita fiat pondus E ad pondus L. </s> <s>&longs;imiliter vt AC ad CB, <lb/>ita fiat pondus F ad pondus M. </s> <s>ponderaq; LM ex puncto A &longs;u<lb/>&longs;pendantur. </s> <s id="id.2.1.53.10.1.4.0">Quoniam enim AC e&longs;t æqualis CH, erit BC ad <lb/>CH vt pondus M ad pondus F. </s> <s id="id.2.1.53.10.1.4.0.a">& quoniam maior e&longs;t BC, <lb/>quàm CH; erit & pondus M ip&longs;o F maius. </s> <s id="id.2.1.53.10.1.5.0">diuidatur igitur pon<lb/>dus M in duas partes QR, &longs;itq; pars Q ip&longs;i F æqualis; erit BC <lb/><arrow.to.target n="note82"></arrow.to.target>ad CH, vt RQ ad Q: & diuidendo, vt BH ad HC, ita R ad q. <lb/><arrow.to.target n="note83"></arrow.to.target>deinde conuertendo, vt CH ad HB, ita Q ad R. </s> <s id="id.2.1.53.10.1.5.0.a">Præterea quo­<lb/>niam CH e&longs;t æqualis ip&longs;i CA, erit HC ad CG, vt pondus <lb/>E ad pondus L: maior autem e&longs;t HC, quàm CG; erit & pon­<pb n="31" xlink:href="036/01/075.jpg"/>dus E pondere L maius. </s> <s id="id.2.1.53.10.1.6.0">diuidatur itaq; pondus E in duas partes <lb/>NO ita, vt pars O &longs;it ip&longs;i L æqualis, erit HC ad CG, vt to­<lb/>tum NO ad O; & diuidendo, vt HG ad GC, ita N ad O: <arrow.to.target n="note84"></arrow.to.target><lb/>conuertendoq; vt CG ad GH, ita O ad N. </s> <s>& iterum com­<lb/>ponendo, vt CH ad HG, ita ON ad N. </s> <s>vt autem GH <arrow.to.target n="note85"></arrow.to.target><lb/>ad HB, ita e&longs;t F ad ON. </s> <s>quare ex æquali, vt CH ad HB, ita F <arrow.to.target n="note86"></arrow.to.target><lb/>ad N. &longs;ed vt CH ad HB ita e&longs;t Q ad R: erit igitur Q ad R, vt <arrow.to.target n="note87"></arrow.to.target><lb/>F ad N; & permutando, vt Q ad F, ita R ad N. </s> <s>e&longs;t autem pars <arrow.to.target n="note88"></arrow.to.target><lb/>Q ip&longs;i F æqualis; quare & pars R ip&longs;i N æqualis erit. </s> <s id="id.2.1.53.10.1.7.0">Itaq; cùm <lb/>pondus L &longs;it ip&longs;i O æquale, & pondus F ip&longs;i Q etiam æquale, atq; <lb/>pars R ip&longs;i N æqualis; erunt pondera LM ip&longs;is EF ponderibus <lb/>æqualia. </s> <s id="id.2.1.53.10.1.8.0">& quoniam e&longs;t, vt AC ad CG, ita pondus E ad pon­<lb/>dus L; pondera EL æqueponderabunt. </s> <s id="id.2.1.53.10.1.9.0">&longs;imiliter quoniam e&longs;t, vt <arrow.to.target n="note89"></arrow.to.target><lb/>AC ad CB, ita <expan abbr="pundus">pondus</expan> F ad pondus M; pondera quoq; FM <lb/>æqueponderabunt. </s> <s id="id.2.1.53.10.1.10.0">Pondera igitur LM ponderibus EF in BG <arrow.to.target n="note90"></arrow.to.target><lb/>appen&longs;is æqueponderabunt. </s> <s id="id.2.1.53.10.1.11.0">cùm autem di&longs;tantia CA æqualis &longs;it <lb/>di&longs;tantiæ CH; &longs;i igitur vtraq; pondera EF in H appendantur, <lb/>pondera LM ip&longs;is EF ponderibus in H appen&longs;is æquepondera­<lb/>bunt. </s> <s id="id.2.1.53.10.1.12.0">&longs;ed LM ip&longs;is EF in GB quoq; æqueponderant: æquè <arrow.to.target n="note91"></arrow.to.target><lb/>igitur grauia erunt pondera EF in GB, vt in H appen&longs;a. </s> <s id="id.2.1.53.10.1.13.0">tàm igi<lb/>tur ponderabunt in BG, quàm in H appen&longs;a. <figure id="id.036.01.075.1.jpg" xlink:href="036/01/075/1.jpg"></figure></s> </p> <p id="id.2.1.53.11.0.0.0" type="main"> <s id="id.2.1.53.11.1.1.0">Sint autem pondera EF in CB appen&longs;a; &longs;itq; C libræ centrum; <lb/>& diuidatur CB in H, ita vt CH ad HB &longs;it, vt pondus in F ad <lb/>E. </s> <s id="id.2.1.53.11.1.1.0.a">Dico pondera EF tàm in CB ponderare, quàm in puncto H. </s> <s id="id.2.1.53.11.1.1.0.b"><lb/>fiat CA ip&longs;i CH æqualis, & vt CA ad CB, ita fiat pondus F ad <lb/>aliud D, quod appendatur in A. </s> <s id="id.2.1.53.11.1.1.0.c">Quoniam enim CH e&longs;t æqua­<pb xlink:href="036/01/076.jpg"/><figure id="id.036.01.076.1.jpg" xlink:href="036/01/076/1.jpg"></figure><lb/>lis CA, erit CH ad CB, vt F ad D; & maior quidem e&longs;t CB, <lb/>quàm CH; idcirco D pondere F maius erit. </s> <s id="id.2.1.53.11.1.2.0">Diuidatur ergo D <lb/>in duas partes Gk, &longs;itq; G ip&longs;i F æqualis; erit vt BC ad CH, <lb/>vt Gk ad G; & diuidendo, vt BH ad HC, ita K ad G; & conuer<lb/><arrow.to.target n="note92"></arrow.to.target>tendo, vt CH ad HB, ita G ad k. </s> <s id="id.2.1.53.11.1.3.0">Vt autem CH ad HB, ita e&longs;t <lb/><arrow.to.target n="note93"></arrow.to.target>F ad E. </s> <s>vt igitur G ad k, ita e&longs;t F ad E; & permutando vt G <lb/><arrow.to.target n="note94"></arrow.to.target>ad F, ita k ad E. </s> <s>&longs;unt autem GF æqualia; erunt & kE inter &longs;e <lb/>&longs;e æqualia. </s> <s id="id.2.1.53.11.1.4.0">cùm itaq; pars G &longs;it ip&longs;i F æqualis, & K ip&longs;i E; erit <lb/>totum C k ip&longs;is EF ponderibus æquale. </s> <s id="id.2.1.53.11.1.5.0">& quoniam AC e&longs;t ip­<lb/>&longs;i CH æqualis; &longs;i igitur pondera EF ex puncto H &longs;u&longs;pendantur, <lb/>pondus D ip&longs;is EF in H appen&longs;is æqueponderabit. </s> <s id="id.2.1.53.11.1.6.0">&longs;ed & ip&longs;is <lb/>æqueponderat in CB, hoc e&longs;t F in B, & E in C; cùm &longs;it vt AC <lb/>ad CB, ita F ad. D. </s> <s id="id.2.1.53.11.1.7.0">pondus enim E ex centro libræ C &longs;u&longs;pen­<lb/>&longs;um non efficit, vt libra in alterutram moueatur partem. </s> <s id="id.2.1.53.11.1.8.0">tàm igi­<lb/>tur grauia erunt pondera EF in CB, quàm in H appen&longs;a. <pb n="32" xlink:href="036/01/077.jpg"/><figure id="id.036.01.077.1.jpg" xlink:href="036/01/077/1.jpg"></figure></s> </p> <p id="id.2.1.53.12.0.0.0" type="main"> <s id="id.2.1.53.12.1.1.0">Sit deniq; libra AB, & ex punctis AB &longs;u&longs;pen&longs;a &longs;int pondera <lb/>EF; &longs;itq; centrum libræ C intra pondera; diuidaturq; AB in <lb/>D, ita vt AD ad DB &longs;it, vt pondus F ad pondus E. </s> <s id="id.2.1.53.12.1.1.0.a">Dico pon<lb/>dera EF tàm in AB ponderare, quám &longs;i vtraq; ex puncto D &longs;u&longs;pen<lb/>dantur. </s> <s id="id.2.1.53.12.1.2.0">fiat CG æqualis ip&longs;i CD; & vt DC ad CA, ita fiat <lb/>pondus E ad aliud H; quod appendatur in D. vt autem GC ad <lb/>CB, ita fiat pondus F ad aliud K; appendaturq; k in G. </s> <s id="id.2.1.53.12.1.2.0.a"><expan abbr="Quoniã">Quoniam</expan> enim <lb/>e&longs;t, vt BC ad CG, hoc e&longs;t ad CD, ita pondus k ad F; erit K ma <lb/>ius pondere F. </s> <s>quare diuidatur pondus k in L, & MN; fiatq; <lb/>pars L ip&longs;i F æqualis; erit vt BC ad CD, vt totum LMN ad <lb/>L; & diuidendo, vt BD ad DC, ita pars MN ad partem L. </s> <s>vt <arrow.to.target n="note95"></arrow.to.target><lb/>igitur BD ad DC, ita pars MN ad F. </s> <s>vt autem AD ad DB, <lb/>ita F ad E: quare ex æquali, vt AD ad DC, ita MN ad E. </s> <s>cùm <arrow.to.target n="note96"></arrow.to.target><expan abbr="verò"><lb/>vero</expan> AD &longs;it ip&longs;a CD maior; erit & pars MN pondere E <lb/>maior: diuidatur ergo MN in duas partes MN, &longs;itq; M æqua <lb/>lis ip&longs;i E. </s> <s>erit vt AD ad DC, vt NM ad M; & diuidendo, vt <arrow.to.target n="note97"></arrow.to.target><lb/>AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M <lb/>ad N. </s> <s>vt autem DC ad CA, ita e&longs;t E ad H; erit igitur M ad N <arrow.to.target n="note98"></arrow.to.target><lb/>vt E ad H; & permutando, vt M ad E, ita N ad H. </s> <s>&longs;ed ME <arrow.to.target n="note99"></arrow.to.target><lb/>&longs;unt inter &longs;e æqualia, erunt NH inter &longs;e&longs;e quoq; æqualia. </s> <s id="id.2.1.53.12.1.3.0">& quo­<lb/>niam ita e&longs;t AC ad CD, vt H ad E: pondera HE æqueponde­<lb/>rabunt. <arrow.to.target n="note100"></arrow.to.target></s> <s id="id.2.1.53.12.1.4.0">&longs;imiliter quoniam e&longs;t vt GC ad CB, ita F ad k, ponde­<pb xlink:href="036/01/078.jpg"/><figure id="id.036.01.078.1.jpg" xlink:href="036/01/078/1.jpg"></figure><lb/><arrow.to.target n="note101"></arrow.to.target>ra etiam kF æqueponderabunt. </s> <s id="id.2.1.53.12.1.5.0">pondera igitur Ek HF in li­<lb/>bra AB, cuius centrum C, æqueponderabunt. </s> <s id="id.2.1.53.12.1.6.0">cùm autem GC <lb/>ip&longs;i CD &longs;it æqualis, & pondus H &longs;it ip&longs;i N æquale; pondera NH <lb/>æqueponderabunt. </s> <s id="id.2.1.53.12.1.7.0">& quoniam omnia æqueponderant, demptis <lb/><arrow.to.target n="note102"></arrow.to.target>HN ponderibus, quæ æqueponderant, reliqua æqueponderabunt; <lb/>hoc e&longs;t pondera EF & pondus LM ex centro libræ C &longs;u&longs;pen&longs;a. </s> <s id="id.2.1.53.12.1.8.0"><lb/>quia verò pars L ip&longs;i F e&longs;t æqualis, & pars M ip&longs;i E æqualis; erit <lb/>totum LM ip&longs;is FE ponderibus &longs;imul &longs;umptis æquale. </s> <s id="id.2.1.53.12.1.9.0">& cùm <lb/>&longs;it CG ip&longs;i CD æqualis, &longs;i igitur pondera EF ex puncto D &longs;u&longs;pen­<lb/>dantur, pondera EF in D appen&longs;a ip&longs;i LM æqueponderabunt. </s> <s id="id.2.1.53.12.1.10.0">quare <lb/>LM tàm ip&longs;is EF in AB appen&longs;is æqueponderat, quàm in pun<lb/>cto D appen&longs;is. </s> <s id="id.2.1.53.12.1.11.0">libra enim &longs;emper eodem modo manet. </s> <s id="id.2.1.53.12.1.12.0">Ponde­<lb/><arrow.to.target n="note103"></arrow.to.target>ra ergo EF tàm in AB ponderabunt, quàm in puncto D. </s> <s id="id.2.1.53.12.1.9.0.a">quod <lb/><expan abbr="demon&longs;tre">demonstrare</expan> oportebat. </s> </p> <p id="id.2.1.54.1.0.0.0" type="margin"> <s id="id.2.1.54.1.1.1.0"><margin.target id="note82"></margin.target>17 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.2.0"><margin.target id="note83"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.3.0"><margin.target id="note84"></margin.target>17 <emph type="italics"/>Quinti. </s> <s id="id.2.1.54.1.1.4.0">Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.5.0"><margin.target id="note85"></margin.target>18 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.6.0"><margin.target id="note86"></margin.target>23 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.7.0"><margin.target id="note87"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.8.0"><margin.target id="note88"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.9.0"><margin.target id="note89"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.11.0"><margin.target id="note90"></margin.target>2 <emph type="italics"/>Com. not. huius.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.14.0"><margin.target id="note91"></margin.target>3 <emph type="italics"/>Com. not. huius.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.16.0"><margin.target id="note92"></margin.target>17 <emph type="italics"/>Quinti. </s> <s id="id.2.1.54.1.1.17.0">Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.18.0"><margin.target id="note93"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.19.0"><margin.target id="note94"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.20.0"><margin.target id="note95"></margin.target>17 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.21.0"><margin.target id="note96"></margin.target>23 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.22.0"><margin.target id="note97"></margin.target>17 <emph type="italics"/>Quinti. </s> <s id="id.2.1.54.1.1.23.0">Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.24.0"><margin.target id="note98"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.25.0"><margin.target id="note99"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.26.0"><margin.target id="note100"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.28.0"><margin.target id="note101"></margin.target>2 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.30.0"><margin.target id="note102"></margin.target>1 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/></s> <s id="id.2.1.54.1.1.32.0"><margin.target id="note103"></margin.target>3 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.55.1.0.0.0" type="main"> <s id="id.2.1.55.1.1.1.0">Hæc autem omnia (mechanicè tamen ma­<lb/>gis) aliter o&longs;tendemus. <pb n="33" xlink:href="036/01/079.jpg"/><figure id="id.036.01.079.1.jpg" xlink:href="036/01/079/1.jpg"></figure></s> </p> <p id="id.2.1.55.2.0.0.0" type="main"> <s id="id.2.1.55.2.1.1.0">Sit libra AB, cuius centrum C; &longs;intq; vt in primo ca&longs;u duo pon<lb/>dera EF ex punctis BG &longs;u&longs;pen&longs;a: &longs;itq; GH ad HB, vt pondus <lb/>F ad pondus E. </s> <s id="id.2.1.55.2.1.1.0.a">Dico pondera EF tàm in GB ponderare, quàm <lb/>&longs;i vtraq; ex diui&longs;ionis puncto H &longs;u&longs;pendantur. </s> <s id="id.2.1.55.2.1.2.0">Con&longs;truantur ea <lb/>dem, hoc e&longs;t fiat AC ip&longs;i CH æqualis, & ex puncto A duo ap­<lb/>pendantur pondera LM, ita vt pondus E ad pondus L, &longs;it vt <lb/>CA ad CG; vt autem CB ad CA, ita &longs;it pondus M ad pondus <lb/>F. </s> <s id="id.2.1.55.2.1.2.0.a">pondera LM ip&longs;is EF in GB appen&longs;is (vt &longs;upra dictum e&longs;t) <lb/>æqueponderabunt. </s> <s id="id.2.1.55.2.1.3.0">Sint deinde puncta NO centra grauitatis pon<lb/>derum EF; connectanturq; GN BO; iungaturq; NO, quæ tan­<lb/>quam libra erit; quæ etiam efficiat lineas GN BO inter &longs;e &longs;e æqui­<lb/>di&longs;tantes e&longs;&longs;e; à punctoq; H horizonti perpendicularis ducatur <lb/>HP, quæ NO &longs;ecet in P, atq; ip&longs;is GN BO &longs;it æquidi&longs;tans. <lb/></s> <s id="id.2.1.55.2.1.3.0.a">deniq; connectatur GO, quæ HP &longs;ecet in R. </s> <s id="id.2.1.55.2.1.4.0">Quoniam igitur <lb/>HR e&longs;t lateri BO trianguli GBO æquidi&longs;tans; erit GH ad HB, <lb/>vt GR ad RO. </s> <s>&longs;imiliter quoniam RP e&longs;t lateri GN trianguli <arrow.to.target n="note104"></arrow.to.target><lb/>OGN æquidi&longs;tans; erit GR ad RO, vt NP ad PO. </s> <s>quare <lb/>vt GH ad HB, ita e&longs;t NP ad PO. </s> <s>vt autem GH ad HB, ita <arrow.to.target n="note105"></arrow.to.target><lb/>e&longs;t pondus F ad pondus E; vt igitur NP ad PO, ita e&longs;t pondus <lb/>F ad pondus E. </s> <s id="id.2.1.55.2.1.4.0.a">punctum ergo P centrum erit grauitatis magni­<lb/>tudinis ex vtri&longs;q; EF ponderibus compo&longs;itæ. </s> <s id="id.2.1.55.2.1.5.0">Intelligantur itaq; <arrow.to.target n="note106"></arrow.to.target><lb/>pondera EF ita e&longs;&longs;e à libra NO connexa, ac &longs;i vna tantùm e&longs;&longs;et <lb/>magnitudo ex vtri&longs;q; EF compo&longs;ita, in puncti&longs;q; BG appen&longs;a. </s> <s id="id.2.1.55.2.1.6.0">&longs;i <lb/>igitur ponderum &longs;u&longs;pen&longs;iones BG &longs;oluantur, manebunt pondera <arrow.to.target n="note107"></arrow.to.target><lb/>EF ex HP &longs;u&longs;pen&longs;a; &longs;icuti in GB prius manebant. </s> <s id="id.2.1.55.2.1.7.0">pondera verò EF <lb/>in GB appen&longs;a ip&longs;is LM ponderibus æqueponderant, & pondera <pb xlink:href="036/01/080.jpg"/><figure id="id.036.01.080.1.jpg" xlink:href="036/01/080/1.jpg"></figure><lb/>EF ex puncto H &longs;u&longs;pen&longs;a, eandem habent con&longs;titutionem ad li­<lb/>bram AB, quam in BG appen&longs;a: eadem ergo pondera EF ex <lb/>H &longs;u&longs;pen&longs;a ei&longs;dem ponderibus LM æqueponderabunt. </s> <s id="id.2.1.55.2.1.8.0">æquè igi­<lb/>tur &longs;unt grauia pondera EF in GB, vt in H appen&longs;a. <figure id="id.036.01.080.2.jpg" xlink:href="036/01/080/2.jpg"></figure></s> </p> <p id="id.2.1.55.3.0.0.0" type="main"> <s id="id.2.1.55.3.1.1.0">Similiter demon&longs;trabitur, pondera EF in quibu&longs;cunq; aliis pun<lb/>ctis appen&longs;a tàm <expan abbr="põderare">ponderare</expan>, quàm &longs;i vtraq; ex diui&longs;ionis puncto H &longs;u<lb/>&longs;pendantur. </s> <s id="id.2.1.55.3.1.2.0">&longs;i enim (vt &longs;upra docuimus) in libra pondera inue­<lb/>niantur, quibus pondera EF æqueponderent; eadem pondera EF <lb/>ex H &longs;u&longs;pen&longs;a ei&longs;dem inuentis ponderibus æqueponderabunt; cùm <lb/>punctum P &longs;it &longs;emper eorum centrum grauitatis; & HP horizon <lb/>ri perpendicularis. </s> </p> <p id="id.2.1.56.1.0.0.0" type="margin"> <s id="id.2.1.56.1.1.1.0"><margin.target id="note104"></margin.target>2 <emph type="italics"/>Sexti.<emph.end type="italics"/></s> <s id="id.2.1.56.1.1.2.0"><margin.target id="note105"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.56.1.1.3.0"><margin.target id="note106"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.56.1.1.5.0"><margin.target id="note107"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="34" xlink:href="036/01/081.jpg"/> <p id="id.2.1.57.1.0.0.0" type="head"> <s id="id.2.1.57.1.2.1.0">PROPOSITIO. VI. </s> </p> <p id="id.2.1.57.2.0.0.0" type="main"> <s id="id.2.1.57.2.1.1.0">Pondera æqualia in libra appen&longs;a eam in gra<lb/>uitate proportionem habent; quam di&longs;tantiæ, ex <lb/>quibus appenduntur. <figure id="id.036.01.081.1.jpg" xlink:href="036/01/081/1.jpg"></figure></s> </p> <p id="id.2.1.57.3.0.0.0" type="main"> <s id="id.2.1.57.3.1.1.0">Sit libra BAC &longs;u&longs;pen&longs;a ex puncto A; & &longs;ecetur AC vtcunq; <lb/>in D: ex punctis autem DC appendantur æqualia pondera EF. <lb/></s> <s id="id.2.1.57.3.1.1.0.a">Dico pondus F ad pondus E eam in grauitate proportionem ha­<lb/>bere, quam habet di&longs;tantia CA ad di&longs;tantiam AD. </s> <s id="id.2.1.57.3.1.1.0.b">fiat enim vt <lb/>CA ad AD, ita pondus F ad aliud pondus, quod &longs;it G. </s> <s id="id.2.1.57.3.1.1.0.c">Dico pri<lb/>múm pondera GF ex puncto C &longs;u&longs;pen&longs;a tantùm ponderare, quan<lb/>tùm pondera EF ex punctis DC. </s> <s id="id.2.1.57.3.1.1.0.d">Secetur DC bifariam in H, & <lb/>ex H appendantur vtraq; pondera EF. </s> <s>ponderabunt EF &longs;imul <lb/>&longs;umpta in eo &longs;itu, quantùm ponderant in DC. ponatur BA <arrow.to.target n="note108"></arrow.to.target><lb/>æqualis AH, &longs;eceturq; BA in K, ita vt &longs;it KA æqualis AD: <lb/>deinde ex puncto B appendatur pondus L duplum ponderis F, <lb/>hoc e&longs;t æquale duobus ponderibus EF, quod quidem æqueponde<lb/>rabit ponderibus EF in H appen&longs;is, hoc e&longs;t appen&longs;is in DC. </s> <s id="id.2.1.57.3.1.1.0.e"><expan abbr="Quoniã">Quoniam</expan> <lb/>igitur, vt CA ad AD, ita e&longs;t pondus F ad pondus G; erit compo<lb/>nendo vt CA AD ad AD, hoc e&longs;t vt Ck ad AD, ita ponde­<lb/>ra <arrow.to.target n="note109"></arrow.to.target>FG ad pondus G. </s> <s>&longs;ed cùm &longs;it, vt CA ad AD, ita F pon­<lb/>dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <arrow.to.target n="note110"></arrow.to.target><lb/>G ad pondus F; & con&longs;equentium dupla, vt DA ad duplam ip&longs;ius <lb/>AC, ita pondus G ad duplum ponderis F, hoc e&longs;t ad pondus <lb/>L. </s> <s id="id.2.1.57.3.1.1.0.f">Quare vt Ck ad DA, ita pondera EF ad pondus G; & vt <pb xlink:href="036/01/082.jpg"/><figure id="id.036.01.082.1.jpg" xlink:href="036/01/082/1.jpg"></figure><lb/><arrow.to.target n="note111"></arrow.to.target>AD ad <expan abbr="duplã">duplam</expan> ip&longs;ius AC, ita pondus G ad pondus L; ergo ex æquali, <lb/>vt Ck ad <expan abbr="duplã">duplam</expan> ip&longs;ius AC, ita pondera FG ad pondus L. </s> <s>&longs;ed vt Ck <lb/>ad duplam AC, ita dimidia CK, videlicet AH, hoc e&longs;t BA, ad <lb/>AC. </s> <s id="id.2.1.57.3.1.1.0.g">Vt igitur BA ad AC, ita FG pondera ad pondus L. </s> <s id="id.2.1.57.3.1.1.0.h">Qua<lb/>re ex &longs;exta eiu&longs;dem primi Archimedis, duo pondera FG ex pun<lb/>cto C &longs;u&longs;pen&longs;a tantùm ponderabunt, quantùm pondus L ex B; <lb/>hoc e&longs;t quantùm pondera EF ex punctis DC &longs;u&longs;pen&longs;a. </s> <s id="id.2.1.57.3.1.2.0">Itaq; quo<lb/>niam pondera FG tantùm ponderant, quantum pondera EF; &longs;u­<lb/>blato communi pondere F, tàm ponderabit pondus G in C ap­<lb/>pen&longs;um, quàm pondus E in D. </s> <s id="id.2.1.57.3.1.2.0.a">ac propterea pondus F ad pon­<lb/><arrow.to.target n="note112"></arrow.to.target>dus E eam in grauitate proportionem habet, quam habet ad pon<lb/>dus G. </s> <s>&longs;ed pondus F ad G erat, vt CA ad AD: ergo & F pon­<lb/>dus ad pondus E eam in grauitate proportionem habebit, quam ha<lb/>bet CA ad AD. </s> <s>quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.58.1.0.0.0" type="margin"> <s id="id.2.1.58.1.1.1.0"><margin.target id="note108"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.58.1.1.2.0"><margin.target id="note109"></margin.target>18 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.58.1.1.3.0"><margin.target id="note110"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.58.1.1.4.0"><margin.target id="note111"></margin.target>22 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.58.1.1.5.0"><margin.target id="note112"></margin.target>7 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.59.1.0.0.0" type="main"> <s id="id.2.1.59.1.1.1.0">Si verò in libra <lb/>BAC pondera EF <lb/>æqualia ex punctis <lb/>BC &longs;u&longs;pendantur; &longs;i­<lb/>militer dico pondus <lb/>E ad pondus F eam <lb/><figure id="id.036.01.082.2.jpg" xlink:href="036/01/082/2.jpg"></figure><lb/>in grauitate proportionem habere, quàm habet di&longs;tantia CA ad di<lb/>&longs;tantiam AB. </s> <s id="id.2.1.59.1.1.1.0.a">fiat AD ip&longs;i AB æqualis, & ex puncto D &longs;u&longs;pen­<lb/>datur pondus G æquale ponderi F; quod etiam ip&longs;i E erit æquale. </s> <s id="id.2.1.59.1.1.2.0"><lb/>& quoniam AD e&longs;t æqualis ip&longs;i AB; pondera FG æqueponde<lb/>rabunt, eandemq; habebunt grauitatem. </s> <s id="id.2.1.59.1.1.3.0">cùm autem grauitas pon<lb/>deris E ad grauitatem ponderis G &longs;it, vt CA ad AD; erit graui<lb/>tas ponderis E ad grauitatem ponderis F, vt CA ad AD, hoc e&longs;t <lb/>CA ad AB. quod erat quoq; o&longs;tendendum. </s> </p> <pb n="35" xlink:href="036/01/083.jpg"/> <p id="id.2.1.59.2.0.0.0" type="head"> <s id="id.2.1.59.3.1.1.0">ALITER. </s> </p> <p id="id.2.1.59.4.0.0.0" type="main"> <s id="id.2.1.59.4.1.1.0">Sit libra BAC, cu­<lb/>ius centrum A; in pun­<lb/>ctis verò BC pondera <lb/>appendantur æqualia G <lb/>F: &longs;itq; primùm cen­<lb/>trum A vtcunque inter <lb/>BC. </s> <s id="id.2.1.59.4.1.1.0.a">Dico pondus F ad <lb/>pondus G eam in graui<lb/><figure id="id.036.01.083.1.jpg" xlink:href="036/01/083/1.jpg"></figure><lb/>tate proportionem habere, quam habet di&longs;tantia CA ad di&longs;tan­<lb/>tiam AB. </s> <s id="id.2.1.59.4.1.1.0.b">fiat vt BA ad AC, ita pondus F ad aliud H, quod ap<lb/>pendatur in B: pondera HF ex A æqueponderabunt. </s> <s id="id.2.1.59.4.1.2.0">&longs;ed cùm <arrow.to.target n="note113"></arrow.to.target><lb/>pondera FG &longs;int æqualia, habebit pondus H ad pondus G ean­<lb/>dem proportionem, quam habet ad F. </s> <s>vt igitur CA ad AB, ita <arrow.to.target n="note114"></arrow.to.target><lb/>e&longs;t H ad G. </s> <s>vt autem H ad G, ita e&longs;t grauitas ip&longs;ius H ad graui<lb/>tatem ip&longs;ius G; cùm in eodem puncto B &longs;int appen&longs;a. </s> <s id="id.2.1.59.4.1.3.0">quare vt CA <lb/>ad AB, ita grauitas ponderis H ad grauitatem ponderis G. </s> <s>cùm au<lb/>tem grauitas ponderis F in C appen&longs;i &longs;it æqualis grauitati ponderis <lb/>H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA <lb/>ad AB, videlicet vt di&longs;tantia ad di&longs;tantiam. </s> <s id="id.2.1.59.4.1.4.0">quod demon&longs;trare <lb/>oportebat. </s> </p> <p id="id.2.1.60.1.0.0.0" type="margin"> <s id="id.2.1.60.1.1.1.0"><margin.target id="note113"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.60.1.1.3.0"><margin.target id="note114"></margin.target>7 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.61.1.0.0.0" type="main"> <s id="id.2.1.61.1.1.1.0">Si verò libra B <lb/>AC &longs;ecetur vtcunq; <lb/>in D, & in DC ap­<lb/>pendantur pondera <lb/>æqualia EF. </s> <s id="id.2.1.61.1.1.1.0.a">Dico <lb/>&longs;imiliter ita e&longs;&longs;e gra­<lb/><figure id="id.036.01.083.2.jpg" xlink:href="036/01/083/2.jpg"></figure><lb/>uitatem ponderis F ad grauitatem ponderis E, vt di&longs;tantia CA ad <lb/>di&longs;tantiam AD. </s> <s id="id.2.1.61.1.1.1.0.b">fiat AB æqualis ip&longs;i AD, & in B appendatur <lb/>pondus G æquale ponderi E, & ponderi F. </s> <s id="id.2.1.61.1.1.1.0.c">Quoniam enim AB e&longs;t <lb/>æqualis AD; pondera GE æqueponderabunt. </s> <s id="id.2.1.61.1.1.2.0">&longs;ed cùm grauitas <lb/>ponderis F ad grauitatem ponderis G &longs;it, vt CA ad AB, & graui<lb/>tas ponderis E &longs;it æqualis grauitati ponderis G; erit grauitas pon-<lb/>deris F ad grauitatem ponderis E, vt CA ad AB, hoc e&longs;t vt CA <lb/>ad AD. </s> <s>quod demon&longs;trare oportebat. </s> </p> <pb xlink:href="036/01/084.jpg"/> <p id="id.2.1.61.2.0.0.0" type="head"> <s id="id.2.1.61.3.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.61.4.0.0.0" type="main"> <s id="id.2.1.61.4.1.1.0">Ex hoc manife&longs;tum e&longs;t, quò pondus à centro <lb/>libræ magis di&longs;tat, eò grauius e&longs;&longs;e; & per con&longs;e­<lb/>quens velocius moueri. </s> </p> <p id="id.2.1.61.5.0.0.0" type="main"> <s id="id.2.1.61.5.1.1.0"><arrow.to.target n="note115"></arrow.to.target>Hinc præterea &longs;tateræ quoq; ratio facilè o&longs;ten<lb/>detur. </s> </p> <p id="id.2.1.62.1.0.0.0" type="margin"> <s id="id.2.1.62.1.1.1.0"><margin.target id="note115"></margin.target><emph type="italics"/>Stateræ ratio.<emph.end type="italics"/></s> </p> <p id="id.2.1.63.1.0.0.0" type="main"> <s id="id.2.1.63.1.1.1.0">Sit enim &longs;tate<lb/>ræ &longs;capus AB, cu<lb/>ius trutina &longs;it in <lb/>C; &longs;itq; &longs;tateræ <lb/>appendiculum E. <lb/></s> <s>appendatur in A <lb/>pondus D, quod <lb/>æqueponderet ap<lb/>pendiculo E in F <lb/><figure id="id.036.01.084.1.jpg" xlink:href="036/01/084/1.jpg"></figure><lb/>appen&longs;o. </s> <s id="id.2.1.63.1.1.2.0">aliud quoq; appendatur pondus G in A, quod etiam <lb/>appendiculo E in B appen&longs;o æqueponderet. </s> <s id="id.2.1.63.1.1.3.0">Dico grauitatem <lb/>ponderis D ad grauitatem ponderis G ita e&longs;&longs;e, vt CF ad CB. </s> <s id="id.2.1.63.1.1.3.0.a"><lb/>Quoniam enim grauitas ponderis D e&longs;t æqualis grauitati ponde­<lb/>ris E in F appen&longs;i, & grauitas ponderis G e&longs;t æqualis grauitati pon<lb/>deris E in B; erit grauitas ponderis D ad grauitatem ponderis E in <lb/>F, vt grauitas ponderis G ad grauitatem ponderis E in B: & permu<lb/><arrow.to.target n="note116"></arrow.to.target>tando, vt grauitas ponderis D ad grauitatem ponderis G, ita graui<lb/>tas ip&longs;ius E in F, ad grauitatem ip&longs;ius E in B; grauitas autem pon<lb/><arrow.to.target n="note117"></arrow.to.target>deris E in F ad grauitatem ponderis E in B e&longs;t, vt CF ad CB; vt <lb/>igitur grauitas ponderis D ad grauitatem ponderis G, ita e&longs;t CF <lb/>ad CB &longs;i ergo pars &longs;capi CB in partes diuidatur æquales, &longs;olo <lb/>pondere E, & propius, & longius à puncto C po&longs;ito; ponderum <lb/>grauitates, quæ ex puncto A &longs;u&longs;penduntur inter &longs;e &longs;e notæ erunt. </s> <s id="id.2.1.63.1.1.4.0"><pb n="36" xlink:href="036/01/085.jpg"/>Vt &longs;i di&longs;tantia CB tripla &longs;it di&longs;tantiæ CF, erit quoq; grauitas ip­<lb/>&longs;ius G grauitatis ip&longs;ius D tripla, quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.64.1.0.0.0" type="margin"> <s id="id.2.1.64.1.1.1.0"><margin.target id="note116"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.64.1.1.2.0"><margin.target id="note117"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.65.1.0.0.0" type="main"> <s id="id.2.1.65.1.1.1.0">Alio quoq; modo &longs;tatera vti po&longs;&longs;umus, vt <lb/>ponderum grauitates notæ reddantur. </s> </p> <p id="id.2.1.65.2.0.0.0" type="main"> <s id="id.2.1.65.2.1.1.0">Sit &longs;capus AB, cuius tru­<lb/>tina &longs;it in C; &longs;itq; &longs;tateræ ap<lb/>pendiculum E, quod appen­<lb/>datur in A; &longs;intqué pon­<lb/>dera DG inæqualia, quorum <lb/>inter &longs;e &longs;e grauitatum propor­<lb/>tiones quærimus: appenda­<lb/>tur pondus D in B, ita vt ip&longs;i <lb/><figure id="id.036.01.085.1.jpg" xlink:href="036/01/085/1.jpg"></figure><lb/>E æqueponderet. </s> <s id="id.2.1.65.2.1.2.0">&longs;imiliter pondus G appendatur in F, quod ei­<lb/>dem ponderi E æqueponderet. </s> <s id="id.2.1.65.2.1.3.0">dico D ad G ita e&longs;&longs;e, vt CF ad <lb/>CB. </s> <s id="id.2.1.65.2.1.3.0.a">Quoniam enim pondera DE æqueponderant, erit D ad E, <arrow.to.target n="note118"></arrow.to.target><lb/>vt CA ad CB. </s> <s>cùm autem pondera quoque GE æquepon­<lb/>derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua <lb/>li pondus D ad pondus G ita erit, vt CF ad CB. </s> <s>quod o&longs;tende<arrow.to.target n="note119"></arrow.to.target><lb/>re quoq; oportebat. </s> </p> <p id="id.2.1.66.1.0.0.0" type="margin"> <s id="id.2.1.66.1.1.1.0"><margin.target id="note118"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.66.1.1.3.0"><margin.target id="note119"></margin.target>23 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/086.jpg"/> <p id="id.2.1.67.1.0.0.0" type="head"> <s id="id.2.1.67.1.2.1.0">PROPOSITIO VII. </s> </p> <p type="head"> <s id="id.2.1.67.1.4.1.0">PROBLEMA. </s> </p> <p id="id.2.1.67.2.0.0.0" type="main"> <s id="id.2.1.67.2.1.1.0">Quotcunque datis in libra ponderibus <lb/>vbicunque appen&longs;is, centrum libræ inuenire, <lb/>ex quo &longs;i &longs;u&longs;pendatur libra, data pondera ma­<lb/>neant. <figure id="id.036.01.086.1.jpg" xlink:href="036/01/086/1.jpg"></figure></s> </p> <p id="id.2.1.67.3.0.0.0" type="main"> <s id="id.2.1.67.3.1.1.0">Sit libra AB, &longs;intq; data quotcunque pondera CDEFG. <lb/></s> <s id="id.2.1.67.3.1.1.0.a">accipiantur in libra vtcunque puncta AHkLB, ex quibus <lb/>data pondera <expan abbr="&longs;pu&longs;pendantur">suspendantur</expan>. </s> <s id="id.2.1.67.3.1.2.0">Centrum libræ inuenire oportet, <lb/>ex quo &longs;i fiat &longs;u&longs;pen&longs;io, data pondera maneant. </s> <s id="id.2.1.67.3.1.3.0">Diuidatur <pb n="37" xlink:href="036/01/087.jpg"/><figure id="id.036.01.087.1.jpg" xlink:href="036/01/087/1.jpg"></figure><lb/>AH in M, ita vt HM ad MA, &longs;it vt grauitas ponderis <lb/>C ad grauitatem ponderis D. </s> <s id="id.2.1.67.3.1.3.0.a">deinde diuidatur BL in N, ita <lb/>vt LN ad NB, &longs;it vt grauitas ponderis G ad grauitatem pon<lb/>deris F. </s> <s>diuidaturq; MN in O, ita vt MO ad ON &longs;it, vt <lb/>grauitas ponderum FG ad grauitatem ponderum CD. </s> <s id="id.2.1.67.3.1.3.0.b">tandem­<lb/>què diuidatur kO in P, ita vt kP ad PO, &longs;it vt grauitas pon<lb/>derum CDFG ad grauitatem ponderis E. </s> <s id="id.2.1.67.3.1.3.0.c">Quoniam igitur pon<lb/>dera CDFG tàm ponderant in O, quàm CD in M, & FG in N; <arrow.to.target n="note120"></arrow.to.target><lb/>æqueponderabunt pondera CD in M, & FG in N, & pondus E <lb/>in K, &longs;i ex puncto P &longs;u&longs;pendantur. </s> <s id="id.2.1.67.3.1.4.0">cùm verò pondera CD tan<lb/>tùm ponderent in M, quantùm in AH, & FG in N, quantùm <lb/>in LB; pondera CDFG ex AHLB punctis &longs;u&longs;pen&longs;a, & pon­<lb/>dus E ex k, &longs;i ex P &longs;u&longs;pendantur, æqueponderabunt, atq; mane­<lb/>bunt. </s> <s id="id.2.1.67.3.1.5.0">Inuentum e&longs;t ergo centrum libræ P, ex quo data pondera <lb/>manent. </s> <s id="id.2.1.67.3.1.6.0">quod facere oportebat. </s> </p> <p id="id.2.1.68.1.0.0.0" type="margin"> <s id="id.2.1.68.1.1.1.0"><margin.target id="note120"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/088.jpg"/> <p id="id.2.1.69.1.0.0.0" type="head"> <s id="id.2.1.69.1.2.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.69.2.0.0.0" type="main"> <s id="id.2.1.69.2.1.1.0">Ex hoc manife&longs;tum e&longs;t, &longs;i ponderum CDEFG <lb/>centra grauitatis e&longs;&longs;ent in AHKLB punctis; e&longs;­<lb/>&longs;et punctum P magnitudinis ex omnibus CD <lb/>EFG ponderibus compo&longs;itæ centrum graui­<lb/>tatis. <figure id="id.036.01.088.1.jpg" xlink:href="036/01/088/1.jpg"></figure></s> </p> <p id="id.2.1.69.3.0.0.0" type="main"> <s id="id.2.1.69.3.1.1.0">Hoc enim ex definitione centri grauitatis patet, cùm ponde­<lb/>ra, &longs;i ex puncto P &longs;u&longs;pendantur, maneant. </s> </p> </chap> <pb n="38" xlink:href="036/01/089.jpg"/> <chap> <p id="id.2.1.69.4.0.0.0" type="head"> <s id="id.2.1.69.5.1.1.0">DE VECTE. </s> </p> <p type="head"> <s id="id.2.1.69.5.3.1.0">LEMMA. </s> </p> <p id="id.2.1.69.6.0.0.0" type="main"> <s id="id.2.1.69.6.1.1.0">Sint quatuor magnitudines A <lb/>BCD; &longs;itq; A maior B, & C ma<lb/>ior D. </s> <s id="id.2.1.69.6.1.1.0.a">Dico A ad D maiorem <lb/>habere proportionem; quàm <lb/>habet B ad C. </s> </p> <p id="id.2.1.69.7.0.0.0" type="main"> <s id="id.2.1.69.7.1.1.0">Quoniam enim A ad C maiorem habet pro­<lb/>portionem, quàm B ad C; & A ad D maio­<lb/>rem <arrow.to.target n="note121"></arrow.to.target>quoq; habet proportionem, quam habet <lb/>ad C: A igitur ad D maiorem habebit, quam B <lb/>ad C. quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.70.1.0.0.0" type="margin"> <s id="id.2.1.70.1.1.1.0"><margin.target id="note121"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <figure id="id.036.01.089.1.jpg" xlink:href="036/01/089/1.jpg"> </figure> <p id="id.2.1.71.1.1.1.0" type="head"> <s id="id.2.1.71.1.3.1.0">PROPOSITIO I. </s> </p> <p id="id.2.1.71.2.0.0.0" type="main"> <s id="id.2.1.71.2.1.1.0">Potentia &longs;u&longs;tinens pondus vecti appen&longs;um; <lb/>eandem ad ip&longs;um pondus proportionem habe­<lb/>bit, quam vectis di&longs;tantia inter fulcimentum, ac <lb/>ponderis &longs;u&longs;pen&longs;ionem ad di&longs;tantiam à fulcimen<lb/>to ad potentiam interiectam. <pb xlink:href="036/01/090.jpg"/><figure id="id.036.01.090.1.jpg" xlink:href="036/01/090/1.jpg"></figure></s> </p> <p id="id.2.1.71.3.0.0.0" type="main"> <s id="id.2.1.71.3.1.1.0">Sit vectis AB, cuius fulcimentum C; &longs;itq; pondus D ex A &longs;u­<lb/>&longs;pen&longs;um AH, ita vt AH &longs;it &longs;emper horizonti perpendicularis: <lb/>&longs;itq; potentia &longs;u&longs;tinens pondus in B. </s> <s id="id.2.1.71.3.1.1.0.a">Dico potentiam in B ad pon<lb/>dus D ita e&longs;&longs;e, vt CA ad CB. </s> <s id="id.2.1.71.3.1.1.0.b">fiat vt BC ad CA, ita pondus D <lb/><arrow.to.target n="note122"></arrow.to.target>ad aliud pondus E, quippè quod &longs;i in B appendatur; ip&longs;i D æque <lb/>ponderabit, exi&longs;tente C amborum grauitatis centro. </s> <s id="id.2.1.71.3.1.2.0">quare poten<lb/>tia æqualis ip&longs;i E ibidem con&longs;tituta ip&longs;i D æqueponderabit, vecte <lb/>AB, eius fulcimento in C collocato, hoc e&longs;t prohibebit, ne pon<lb/>dus D deor&longs;um vergat, quemadmodum prohibet pondus E. </s> <s id="id.2.1.71.3.1.2.0.a">Po<lb/><arrow.to.target n="note123"></arrow.to.target>tentia verò in B ad pondus D eandem habet proportionem, quam <lb/>pondus E ad idem pondus D: ergo potentia in B ad pondus D <lb/>erit, vt CA ad CB; hoc e&longs;t vectis di&longs;tantia à fulcimento ad pon<lb/>deris &longs;u&longs;pendium ad di&longs;tantiam à fulcimento ad potentiam. </s> <s id="id.2.1.71.3.1.3.0">quod <lb/>demon&longs;trare oportebat. </s> </p> <p id="id.2.1.72.1.0.0.0" type="margin"> <s id="id.2.1.72.1.1.1.0"><margin.target id="note122"></margin.target>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s> <s id="id.2.1.72.1.1.3.0"><margin.target id="note123"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.73.1.0.0.0" type="main"> <s id="id.2.1.73.1.1.1.0">Hinc facilè o&longs;tendi pote&longs;t, fulcimentum quò <lb/>ponderi fuerit propius, minorem ad idem pon­<lb/>dus &longs;u&longs;tinendum requiri potentiam. </s> </p> <p id="id.2.1.73.2.0.0.0" type="main"> <s id="id.2.1.73.2.1.1.0">Ii&longs;dem po&longs;i­<lb/>tis, &longs;it fulcimen <lb/>tum in F ip&longs;i A <lb/>propius, quàm <lb/>C; fiatq; vt BF <lb/>ad FA, ita pon<lb/>dus D ad aliud <lb/><figure id="id.036.01.090.2.jpg" xlink:href="036/01/090/2.jpg"></figure><lb/>G, quod &longs;i appendatur in B, pondera DG ex fulcimento E <lb/><arrow.to.target n="note124"></arrow.to.target>æqueponderabunt. </s> <s id="id.2.1.73.2.1.2.0">quoniam autem BF maior e&longs;t BC, & CA <lb/><arrow.to.target n="note125"></arrow.to.target>maior AC; maior erit proportio BF ad FA, quàm BC ad CA: <pb n="39" xlink:href="036/01/091.jpg"/>& ideo maior quoq; erit proportio ponderis D ad pondus G, <lb/>quàm idem D ad E: pondus igitur G minus erit pondere E. cùm <arrow.to.target n="note126"></arrow.to.target><lb/>autem potentia in B ip&longs;i G æqualis ponderi D æqueponderet, mi­<lb/>nor potentia, quàm ea, quæ ponderi E e&longs;t æqualis, pondus D &longs;u<lb/>&longs;tinebit; exi&longs;tente vecte AB, eius verò fulcimento vbi F, quàm &longs;i <lb/>fuerit vbi C. &longs;imiliter quoq; o&longs;tendetur, quò propius erit fulci­<lb/>mentum ponderi D, adhuc &longs;emper minorem requiri potentiam <lb/>ad &longs;u&longs;tinendum pondus D. </s> </p> <p id="id.2.1.74.1.0.0.0" type="margin"> <s id="id.2.1.74.1.1.1.0"><margin.target id="note124"></margin.target><emph type="italics"/>Ex eadem Sexta.<emph.end type="italics"/></s> <s id="id.2.1.74.1.1.2.0"><margin.target id="note125"></margin.target><emph type="italics"/>Lemma.<emph.end type="italics"/></s> <s id="id.2.1.74.1.1.3.0"><margin.target id="note126"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.75.1.0.0.0" type="head"> <s id="id.2.1.75.1.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.75.2.0.0.0" type="main"> <s id="id.2.1.75.2.1.1.0">Vnde palàm colligere licet, exi&longs;tente AF ip&longs;a <lb/>FB minore, minorem quoq; requiri potentiam <lb/>in ip&longs;o B pondere D &longs;u&longs;tinendo. </s> <s id="id.2.1.75.2.1.2.0">æquali verò <lb/>æqualem. maiore verò maiorem. </s> </p> <p id="id.2.1.75.3.0.0.0" type="head"> <s id="id.2.1.75.3.1.1.0">PROPOSITIO II. </s> </p> <p id="id.2.1.75.4.0.0.0" type="main"> <s id="id.2.1.75.4.1.1.0">Alio modo vecte vti po&longs;sumus. </s> </p> <p id="id.2.1.75.5.0.0.0" type="main"> <s id="id.2.1.75.5.1.1.0">Sit vectis AB, cuius <lb/>fulcimentum &longs;it B, & <lb/>pondus C vtcunq; in <lb/>D inter AB appen­<lb/>&longs;um; &longs;itq; potentia in <lb/>A &longs;u&longs;tinens pondus C. </s> <s id="id.2.1.75.5.1.1.0.a"><lb/>Dico vt BD ad BA, <lb/><figure id="id.036.01.091.1.jpg" xlink:href="036/01/091/1.jpg"></figure><lb/>ita e&longs;&longs;e potentiam in A ad pondus C. appendatur in A pondus <lb/>E æquale ip&longs;i C; & vt AB ad BD, ita fiat pondus E ad aliud F. <lb/>& quoniam pondera CE &longs;unt inter &longs;e &longs;e æqualia, erit pondus C <lb/>ad pondus F, vt AB ad BD. appendatur quoq; pondus F in A. <lb/>& quoniam pondus E ad pondus F e&longs;t, vt grauitas ip&longs;ius E ad gra­<lb/>uitatem <arrow.to.target n="note127"></arrow.to.target>ip&longs;ius F; & pondus E ad F e&longs;t, vt AB ad BD; vt igitur <lb/>grauitas ponderis E ad grauitatem ponderis F, ita e&longs;t AB ab BD. <lb/>vt autem AB ad BD, ita e&longs;t grauitas ponderis E ad grauitatem <arrow.to.target n="note128"></arrow.to.target><pb xlink:href="036/01/092.jpg"/>ponderis C: quare gra<lb/>uitas ponderis E ad <lb/>grauitatem ponderis <lb/>F ita erit, vt grauitas <lb/>ponderis E ad gra­<lb/>uitatem ponderis C. </s> <s id="id.2.1.75.5.1.1.0.b"><lb/>Pondera igitur CF <lb/><figure id="id.036.01.092.1.jpg" xlink:href="036/01/092/1.jpg"></figure><lb/><arrow.to.target n="note129"></arrow.to.target>eandem habent grauitatem. </s> <s id="id.2.1.75.5.1.2.0">Ponatur itaq; potentia in A &longs;u&longs;tinens <lb/>pondus F; erit potentia in A æqualis ip&longs;i ponderi F. </s> <s id="id.2.1.75.5.1.2.0.a">& quoniam <lb/>pondus F in A appen&longs;um æquè graue e&longs;t, vt pondus C in D ap­<lb/>pen&longs;um; eandem proportionem habebit potentia in A ad grauita­<lb/><arrow.to.target n="note130"></arrow.to.target>tem ponderis F in A appen&longs;i, quam habet ad grauitatem ponde­<lb/>ris C in D appen&longs;i. </s> <s id="id.2.1.75.5.1.3.0">Potentia verò in A ip&longs;i F æqualis &longs;u&longs;tinet <lb/>pondus F, ergo potentia in A pondus quoq; C &longs;u&longs;tinebit. </s> <s id="id.2.1.75.5.1.4.0">Itaq; <lb/>cùm potentia in A &longs;it æqualis ponderi F, & pondus C ad pon­<lb/>dus F &longs;it, vt AB ad BD; erit pondus C ad potentiam in A, vt <lb/><arrow.to.target n="note131"></arrow.to.target>AB ad BD. & è conuer&longs;o, vt BD ad BA, ita potentia in A ad <lb/>pondus C. potentia ergo ad pondus ita erit, vt di&longs;tantia fulci­<lb/>mento, ac ponderis &longs;u&longs;pen&longs;ioni intercepta ad di&longs;tantiam à fulci <lb/>mento ad potentiam. </s> <s id="id.2.1.75.5.1.5.0">quod oportebat demon&longs;trare. </s> </p> <p id="id.2.1.76.1.0.0.0" type="margin"> <s id="id.2.1.76.1.1.1.0"><margin.target id="note127"></margin.target><emph type="italics"/>In &longs;exta huius de libra Ex<emph.end type="italics"/> 11 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.76.1.1.2.0"><margin.target id="note128"></margin.target>6 <emph type="italics"/>Huius. de libra.<emph.end type="italics"/></s> <s id="id.2.1.76.1.1.4.0"><margin.target id="note129"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.76.1.1.5.0"><margin.target id="note130"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.76.1.1.6.0"><margin.target id="note131"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.77.1.0.0.0" type="head"> <s id="id.2.1.77.1.1.1.0">ALITER. </s> </p> <figure id="id.036.01.092.2.jpg" xlink:href="036/01/092/2.jpg"> </figure> <p id="id.2.1.77.2.0.0.0" type="main"> <s id="id.2.1.77.2.1.1.0">Sit vectis AB, cuius fulcimentum &longs;it B, & pondus E ex puncto <lb/>C &longs;u&longs;pen&longs;um; &longs;itq; vis in A &longs;u&longs;tinens pondus E. </s> <s id="id.2.1.77.2.1.1.0.a">Dico vt BC ad BA, <lb/>ita e&longs;&longs;e potentiam in A ad pondus E. </s> <s id="id.2.1.77.2.1.1.0.b">Producatur AB in C, & <lb/>fiat BD æqualis BC; & ex puncto D appendatur pondus F æqua <lb/>le ponderi E; itemq; ex puncto A &longs;u&longs;pendatur pondus G ita, vt <lb/>pondus F ad pondus G eandem habeat proportionem, quam AB <pb n="40" xlink:href="036/01/093.jpg"/>ad BA. pondera FG æqueponderabunt. </s> <s id="id.2.1.77.2.1.2.0">cùm autem &longs;it CB æqua <lb/>lis BD, pondera quoq; FE æqualia æqueponderabunt. </s> <s id="id.2.1.77.2.1.3.0">pondera <lb/>verò FEG in libra, &longs;eu vecte DBA appen&longs;a, cuius fulcimentum <lb/>e&longs;t B, non æqueponderabunt; &longs;ed ex parte A deor&longs;um tendent. </s> <s id="id.2.1.77.2.1.4.0">po<lb/>natur itaq; in A tanta vis, vt pondera FEG æqueponderent; erit <lb/>potentia in A æqualis ponderi G. pondera enim FE <expan abbr="æqueponderãt">æqueponderant</expan>, <lb/>& vis in A nihil aliud efficere debet, ni&longs;i &longs;u&longs;tinere <expan abbr="põdus">pondus</expan> G, ne de&longs;cen<lb/>dat. </s> <s id="id.2.1.77.2.1.5.0">& quoniam pondera FEG, & potentia in A æqueponderant, <lb/>demptis igitur FG ponderibus, quæ æqueponderant, reliqua æque <lb/>ponderabunt; &longs;cilicet potentia in A ponderi E, hoc e&longs;t potentia <lb/>in A pondus E &longs;u&longs;tinebit, ita vt vectis AB maneat, vt prius erat. </s> <s id="id.2.1.77.2.1.6.0"><lb/>Cùm autem potentia in A &longs;it æqualis ponderi G, & pondus E pon<lb/>deri F æquale; habebit potentia in A ad pondus E eandem pro­<lb/>portionem, quam habet BD, hoc e&longs;t BC ad BA. quod demon­<lb/>&longs;trare oportebat. </s> </p> <p id="id.2.1.77.3.0.0.0" type="head"> <s id="id.2.1.77.3.1.1.0">COROLLARIVM I. </s> </p> <p id="id.2.1.77.4.0.0.0" type="main"> <s id="id.2.1.77.4.1.1.0">Ex hoc etiam (vt prius) manife&longs;tum e&longs;&longs;e po­<lb/>te&longs;t, &longs;i ponatur pondus E propius fulcimento B, <lb/>vt in H; minorem potentiam in A &longs;u&longs;tinere po&longs;­<lb/>&longs;e ip&longs;um pondus. </s> </p> <p id="id.2.1.77.5.0.0.0" type="main"> <s id="id.2.1.77.5.1.1.0">Minorem enim proportionem habet HB ad BA, quam CB ad <arrow.to.target n="note132"></arrow.to.target><lb/>BA. & quò propius pondus erit fulcimento, adhuc &longs;emper mino <lb/>rem po&longs;&longs;e potentiam &longs;u&longs;tinere pondus E &longs;imiliter o&longs;tendetur. </s> </p> <p id="id.2.1.78.1.0.0.0" type="margin"> <s id="id.2.1.78.1.1.1.0"><margin.target id="note132"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.79.1.0.0.0" type="head"> <s id="id.2.1.79.1.1.1.0">COROLLARIVM II. </s> </p> <p id="id.2.1.79.2.0.0.0" type="main"> <s id="id.2.1.79.2.1.1.0">Sequitur etiam potentiam in A &longs;emper mino <lb/>rem e&longs;&longs;e pondere E. </s> </p> <p id="id.2.1.79.3.0.0.0" type="main"> <s id="id.2.1.79.3.1.1.0">Sumatur enim inter AB quoduis punctum C, &longs;emper BC <lb/>minor erit BA. </s> </p> <pb xlink:href="036/01/094.jpg"/> <p id="id.2.1.79.5.0.0.0" type="head"> <s id="id.2.1.79.5.1.1.0">COROLLARIVM III. </s> </p> <p id="id.2.1.79.6.0.0.0" type="main"> <s id="id.2.1.79.6.1.1.0">Ex hoc quoq; elici pote&longs;t, &longs;i duæ fuerint poten<lb/>tiæ, vna in A, altera in B, & vtraq; &longs;u&longs;tentet <lb/>pondus E; potentiam in A ad potentiam in B e&longs;­<lb/>&longs;e, vt BC ad CA. </s> </p> <p id="id.2.1.79.7.0.0.0" type="main"> <s id="id.2.1.79.7.1.1.0">Vectis enim BA fungi­<lb/>tur officio duorum <expan abbr="vectiũ">vectium</expan>; <lb/>& AB &longs;unt tanquam duo <lb/>fulcimenta, hoc e&longs;t quan­<lb/>do AB e&longs;t vectis, & poten<lb/>tia &longs;u&longs;tinens in A; erit eius <lb/><figure id="id.036.01.094.1.jpg" xlink:href="036/01/094/1.jpg"></figure><lb/>fulcimentum B. </s> <s id="id.2.1.79.7.1.1.0.a">Quando verò BA e&longs;t vectis, & potentia in B; <lb/>erit A fulcimentum: & pondus &longs;emper ex puncto C remanet &longs;u­<lb/>&longs;pen&longs;um. </s> <s id="id.2.1.79.7.1.2.0">& quoniam potentia in A ad pondus E e&longs;t, vt BC ad <lb/>BA; vt autem pondus E ad potentiam, quæ e&longs;t in B, ita e&longs;t <lb/><arrow.to.target n="note133"></arrow.to.target>BA ad AC; erit ex æquali, potentia in A ad potentiam in B, vt <lb/>BC ad CA. & hoc modo facilè etiam proportionem, quæ in <lb/>Quæ&longs;tionibus Mechanicis quæ&longs;tione vige&longs;ima nona ab Ari&longs;totele <lb/>ponitur, noui&longs;&longs;e poterimus. </s> </p> <p id="id.2.1.80.1.0.0.0" type="margin"> <s id="id.2.1.80.1.1.1.0"><margin.target id="note133"></margin.target>22 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.81.1.0.0.0" type="head"> <s id="id.2.1.81.1.1.1.0">COROLLARIVM IIII. </s> </p> <p id="id.2.1.81.2.0.0.0" type="main"> <s id="id.2.1.81.2.1.1.0">E&longs;t etiam manife&longs;tum, vtra&longs;q; potentias in A, <lb/>& B &longs;imul &longs;umptas æquales e&longs;&longs;e ponderi E. </s> </p> <p id="id.2.1.81.3.0.0.0" type="main"> <s id="id.2.1.81.3.1.1.0">Pondus enim E ad potentiam in A e&longs;t, vt BA ad BC; & idem <lb/>pondus E ad potentiam in B e&longs;t, vt BA ad AC; quare pondus <lb/>E ad vtra&longs;q; potentias in A, & B &longs;imul &longs;umptas e&longs;t, vt AB ad BC <lb/>CA &longs;imul, hoc e&longs;t ad BA. pondus igitur E vtri&longs;q; potentiis &longs;imul <lb/>&longs;umptis æquale erit. </s> </p> <pb n="41" xlink:href="036/01/095.jpg"/> <p id="id.2.1.81.4.0.0.0" type="head"> <s id="id.2.1.81.5.1.1.0">PROPOSITIO III. </s> </p> <p id="id.2.1.81.6.0.0.0" type="main"> <s id="id.2.1.81.6.1.1.0">Alio quoq; modo vecte vti po&longs;sumus. </s> </p> <p id="id.2.1.81.7.0.0.0" type="main"> <s id="id.2.1.81.7.1.1.0">Sit Vectis AB, <lb/>cuius fulcimentum <lb/>B; &longs;itq; ex puncto <lb/>A pondus C appen­<lb/>&longs;um; &longs;itq; potentia <lb/>in D vtcunq; inter <lb/>AB &longs;u&longs;tinens pon­<lb/>dus C. </s> <s id="id.2.1.81.7.1.1.0.a">Dico vt AB <lb/><figure id="id.036.01.095.1.jpg" xlink:href="036/01/095/1.jpg"></figure><lb/>ad BD, ita e&longs;&longs;e potentiam in D ad pondus C. </s> <s id="id.2.1.81.7.1.1.0.b">Appendatur ex <lb/>puncto D pondus E æquale ip&longs;i C; & vt BD ad BA, ita fiat pon<lb/>dus E ad aliud F. & cùm pondera CE &longs;int inter &longs;e &longs;e æqualia; erit <lb/>pondus C ad pondus F, vt BD ad BA. </s> <s id="id.2.1.81.7.1.1.0.c">appendatur pondus <lb/>F quoq; in D. </s> <s id="id.2.1.81.7.1.1.0.d">& quoniam pondus E ad ip&longs;um F e&longs;t, vt grauitas <lb/>ponderis E ad grauitatem ponderis F; & pondus E ad pondus F <arrow.to.target n="note134"></arrow.to.target><lb/>e&longs;t, vt BD ad BA: vt igitur grauitas ponderis E ad grauitatem <lb/>ponderis F, ita e&longs;t BD ad BA. vt autem BD ad BA, ita e&longs;t gra<arrow.to.target n="note135"></arrow.to.target><lb/>uitas ponderis E ad grauitatem ponderis C; quare grauitas ponde­<lb/>ris E ad grauitatem ponderis F eandem habet proportionem, <lb/>quam habet ad grauitatem ponderis C. pondera ergo CF eandem <arrow.to.target n="note136"></arrow.to.target><lb/>habent grauitatem. </s> <s id="id.2.1.81.7.1.2.0">&longs;it igitur potentia in D &longs;u&longs;tinens pondus F, <lb/>erit potentia in D ip&longs;i ponderi F æqualis. </s> <s id="id.2.1.81.7.1.3.0">& quoniam pondus F <lb/>in D æquè graue e&longs;t, vt pondus C in A; habebit potentia in D <lb/>eandem proportionem ad grauitatem ponderis F, quam habet ad <arrow.to.target n="note137"></arrow.to.target><lb/>grauitatem ponderis C. </s> <s id="id.2.1.81.7.1.3.0.a">&longs;ed potentia in D pondus F &longs;u&longs;tinet; po­<lb/>tentia igitur in D pondus quoq; C &longs;u&longs;tinebit: & pondus C ad po­<lb/>tentiam in D ita erit, vt pondus C ad pondus F; & C ad F e&longs;t, vt <lb/>BD ad BA; erit igitur pondus C ad potentiam in D, vt BD ad <lb/>BA: & conuertendo, vt AB ad BD, ita potentia in D ad pondus <lb/>C. </s> <s id="id.2.1.81.7.1.3.0.b">potentia ergo ad pondus e&longs;t, vt di&longs;tantia à fulcimento ad pon<lb/>deris &longs;u&longs;pendium ad di&longs;tantiam à fulcimento ad potentiam. </s> <s id="id.2.1.81.7.1.4.0">quod <lb/>demon&longs;trare oportebat. </s> </p> <p id="id.2.1.82.1.0.0.0" type="margin"> <s id="id.2.1.82.1.1.1.0"><margin.target id="note134"></margin.target><emph type="italics"/>In &longs;exta huius de libra.<emph.end type="italics"/></s> <s id="id.2.1.82.1.1.2.0"><margin.target id="note135"></margin.target>6 <emph type="italics"/>Huius de libra.<emph.end type="italics"/></s> <s id="id.2.1.82.1.1.3.0"><margin.target id="note136"></margin.target>9 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.82.1.1.4.0"><margin.target id="note137"></margin.target>7 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/096.jpg"/> <p id="id.2.1.83.1.0.0.0" type="head"> <s id="id.2.1.83.1.2.1.0">ALITER. </s> </p> <figure id="id.036.01.096.1.jpg" xlink:href="036/01/096/1.jpg"> </figure> <p id="id.2.1.83.2.0.0.0" type="main"> <s id="id.2.1.83.2.1.1.0">Sit vectis AB, cuius fulcimentum B; & ex puncto A &longs;it pon­<lb/>dus C &longs;u&longs;pen&longs;um; &longs;itq; potentia in D &longs;u&longs;tinens pondus C. </s> <s id="id.2.1.83.2.1.1.0.a">Dico <lb/>vt AB ad BD, ita e&longs;&longs;e potentiam in D ad pondus C. </s> <s id="id.2.1.83.2.1.1.0.b">Produca<lb/>tur AB in E, fiatq; BE æqualis ip&longs;i BA; & ex puncto E appen<lb/>datur pondus F æquale ponderi C; & vt BD ad BE, ita fiat pon<lb/>dus F ad aliud G, quod ex puncto D &longs;u&longs;pendatur. </s> <s id="id.2.1.83.2.1.2.0">pondera FG <lb/>æqueponderabunt. </s> <s id="id.2.1.83.2.1.3.0">& quoniam AB e&longs;t æqualis BE, & pondera <lb/>FC æqualia; &longs;imiliter pondera FC æqueponderabunt. </s> <s id="id.2.1.83.2.1.4.0">Pondera <lb/>verò FGC &longs;u&longs;pen&longs;a in vecte EBA, cuius fulcimentum e&longs;t B, non <lb/>æqueponderabunt; &longs;ed ex parte A deor&longs;um tendent. </s> <s id="id.2.1.83.2.1.5.0">Ponatur igi<lb/>tur in D tanta vis, vt pondera FGC æqueponderent; erit po­<lb/>tentia in D æqualis ponderi G: pondera enim FC æqueponde­<lb/>rant, & potentia in D nil aliud efficere debet, ni&longs;i &longs;u&longs;tinere pon­<lb/>dus G ne de&longs;cendat. </s> <s id="id.2.1.83.2.1.6.0">& quoniam pondera FGC, & potentia in <lb/>D æqueponderant, demptis igitur FG ponderibus, quæ æquepon<lb/>derant; reliqua æqueponderabunt, &longs;cilicet potentia in D ponderi C. <lb/>hoc e&longs;t potentia in D pondus C &longs;u&longs;tinebit, ita vt vectis AB ma­<lb/>neat, vt prius. </s> <s id="id.2.1.83.2.1.7.0">& cùm potentia in D &longs;it æqualis ponderi G, & pon­<lb/>dus C æquale ponderi F; habebit potentia in D ad pondus C ean<lb/>dem proportionem, quam EB, hoc e&longs;t AB ad BD. quod de­<lb/>mon&longs;trare oportebat. </s> </p> <p id="id.2.1.83.3.0.0.0" type="head"> <s id="id.2.1.83.3.1.1.0">COROLLARIVM I. </s> </p> <p id="id.2.1.83.4.0.0.0" type="main"> <s id="id.2.1.83.4.1.1.0">Ex hoc etiam pàtet, vt prius, &longs;i coftituatur pon<lb/>dus fulcimento B propius, vt in H; à minori po­<lb/>tentia pondus ip&longs;um &longs;ub&longs;tineri debere. </s> </p> <pb n="42" xlink:href="036/01/097.jpg"/> <p id="id.2.1.83.6.0.0.0" type="main"> <s id="id.2.1.83.6.1.1.0">Minorem enim proportionem habet HB ad BD, quàm AB ad <arrow.to.target n="note138"></arrow.to.target><lb/>BD. & quò propius erit fulcimento, adhuc &longs;emper minorem re­<lb/>quiri potentiam. </s> </p> <p id="id.2.1.84.1.0.0.0" type="margin"> <s id="id.2.1.84.1.1.1.0"><margin.target id="note138"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.85.1.0.0.0" type="head"> <s id="id.2.1.85.1.1.1.0">COROLLARIVM II. </s> </p> <p id="id.2.1.85.2.0.0.0" type="main"> <s id="id.2.1.85.2.1.1.0">Manife&longs;tum quoq; e&longs;t, potentiam in D &longs;emper <lb/>maiorem e&longs;&longs;e pondere C. </s> </p> <p id="id.2.1.85.3.0.0.0" type="main"> <s id="id.2.1.85.3.1.1.0">Si enim inter AB &longs;umatur quoduis punctum D, &longs;emper AB <lb/>maior erit BD. </s> </p> <p id="id.2.1.85.4.0.0.0" type="main"> <s id="id.2.1.85.4.1.1.0">Et aduertendum e&longs;t ha&longs;ce, quas attulimus demon&longs;trationes <lb/>non &longs;olum vectibus horizonti æquidi&longs;tantibus, verùm etiam ve­<lb/>ctibus horizonti inclinatis ad hæc omnia o&longs;tendenda commodè <lb/>aptari po&longs;&longs;e. </s> <s id="id.2.1.85.4.1.2.0">quod ex iis, quæ de libra diximus, patet. </s> </p> <p id="id.2.1.85.5.0.0.0" type="head"> <s id="id.2.1.85.5.1.1.0">PROPOSITIO IIII. </s> </p> <p id="id.2.1.85.6.0.0.0" type="main"> <s id="id.2.1.85.6.1.1.0">Si potentia pondus in vecte appen&longs;um mo­<lb/>ueat; erit &longs;patium potentiæ motæ ad &longs;patium <lb/>moti ponderis, vt di&longs;tantia à fulcimento ad po­<lb/>tentiam ad di&longs;tantiam ab eodem ad ponderis &longs;u<lb/>&longs;pen&longs;ionem. </s> </p> <pb xlink:href="036/01/098.jpg"/> <p id="id.2.1.85.8.0.0.0" type="main"> <s id="id.2.1.85.8.1.1.0">Sit vectis AB, cuius ful­<lb/>cimentum C; & ex puncto B <lb/>&longs;it pondus D &longs;u&longs;pen&longs;um; &longs;itq; <lb/>potentia in A mouens pon­<lb/>dus D vecte AB. </s> <s id="id.2.1.85.8.1.1.0.a">Dico &longs;pa­<lb/>tium potentiæ in A ad &longs;pa­<lb/>tium ponderis ita e&longs;&longs;e, vt CA <lb/>ad CB. </s> <s id="id.2.1.85.8.1.1.0.b">Moueatur vectis AB, <lb/>& vt pondus D &longs;ur&longs;um mo­<lb/>ueatur, oportet B &longs;ur&longs;um mo <lb/>ueri, A verò deor&longs;um. </s> <s id="id.2.1.85.8.1.2.0">& quo­<lb/>niam C e&longs;t punctum immobi<lb/>le; idcirco dum A, & B mo­<lb/>uentur, <expan abbr="circulorũ">circulorum</expan> circumferen<lb/>tias de&longs;cribent. </s> <s id="id.2.1.85.8.1.3.0">Moueatur igi­<lb/>tur AB in EF; erunt AE <lb/><figure id="id.036.01.098.1.jpg" xlink:href="036/01/098/1.jpg"></figure><lb/>BF circulorum circumferentiæ, quorum &longs;emidiametri &longs;unt CA <lb/>CB. tota compleatur circumferentia AGE, & tota BHF; &longs;intq; <lb/>KH puncta, vbi AB, & EF circulum BHF &longs;ecant. </s> <s id="id.2.1.85.8.1.4.0">Quoniam e­<lb/><arrow.to.target n="note139"></arrow.to.target>nim angulus BCF e&longs;t æqualis angulo HCk; erit circumferentia <lb/><arrow.to.target n="note140"></arrow.to.target>kH circumferentiæ BF æqualis. </s> <s id="id.2.1.85.8.1.5.0">cùm autem circumferentiæ AE <lb/>kH &longs;int &longs;ub eodem angulo ACE, & circumferentia AE ad to­<lb/>tam circumferentiam AGE &longs;it, vt angulus ACE ad quatuor re­<lb/>ctos; vt autem idem angulus HCk ad quatuor rectos, ita quoq; <lb/>e&longs;t circumferentia HK ad totam circumferentiam HBK; erit cir<lb/>cumferentia AE ad totam circumferentiam AGE, vt circumfe­<lb/><arrow.to.target n="note141"></arrow.to.target>rentia kH ad totam kFH. & permutando, vt circumferentia <lb/>AE ad circumferentiam kH, hoc e&longs;t BF, ita tota circumferen­<lb/>tia AGE ad totam circumferentiam BHF. </s> <s id="id.2.1.85.8.1.5.0.a">tota verò circumfe<lb/>rentia AGE ita &longs;e habet ad totam BHF, vt diameter circuli AEG <lb/><arrow.to.target n="note142"></arrow.to.target>ad diametrum circuli BHF. </s> <s id="id.2.1.85.8.1.5.0.b">Vt igitur circumferentia AE ad cir<lb/><arrow.to.target n="note143"></arrow.to.target>cumferentiam BF, ita diameter circuli AGE ad diametrum cir<lb/>culi BHF: vt autem diameter ad diametrum, ita &longs;emidiameter <lb/>ad &longs;emidiametrum, hoc e&longs;t CA ad CB: quare vt circumferen­<lb/>tia AE ad circumferentiam BF, ita CA ad CF. circumferentia <lb/>verò AE &longs;patium e&longs;t potentiæ motæ, & circumferentia BF e&longs;t <pb n="43" xlink:href="036/01/099.jpg"/>æqualis &longs;patio ponderis D moti. </s> <s id="id.2.1.85.8.1.6.0">&longs;patium enim motus ponderis <lb/>D &longs;emper æquale e&longs;t &longs;patio motus puncti B, cùm in B &longs;it appen<lb/>&longs;um: &longs;patium ergo potentiæ motæ ad &longs;patium moti ponderis e&longs;t, <lb/>vt CA ad CB; hoc e&longs;t vt di&longs;tantia à fulcimento ad potentiam <lb/>ad di&longs;tantiam ab eodem ad ponderis &longs;u&longs;pen&longs;ionem. </s> <s id="id.2.1.85.8.1.7.0">quod demon<lb/>&longs;trare oportebat. </s> </p> <p id="id.2.1.86.1.0.0.0" type="margin"> <s id="id.2.1.86.1.1.1.0"><margin.target id="note139"></margin.target>15 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.86.1.1.2.0"><margin.target id="note140"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 26 <emph type="italics"/>tertii.<emph.end type="italics"/></s> <s id="id.2.1.86.1.1.3.0"><margin.target id="note141"></margin.target>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.86.1.1.4.0"><margin.target id="note142"></margin.target>23 <emph type="italics"/>Octaui Pappi.<emph.end type="italics"/></s> <s id="id.2.1.86.1.1.5.0"><margin.target id="note143"></margin.target>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.87.1.0.0.0" type="main"> <s id="id.2.1.87.1.1.1.0">Sit autem vectis AB, cu­<lb/>ius fulcimentum B; <expan abbr="potentia­qué">potentia­<lb/>que</expan> mouens in A; & pondus <lb/>in C. </s> <s id="id.2.1.87.1.1.1.0.a">dico &longs;patium potentiæ <lb/>translatæ ad &longs;patium transla<lb/>ti ponderis ita e&longs;&longs;e, vt BA ad <lb/>BC. </s> <s id="id.2.1.87.1.1.1.0.b">Moueatur vectis, & vt <lb/>pondus sursum attollatur, ne­<lb/>ce&longs;&longs;e e&longs;t puncta C A &longs;ur&longs;um <lb/>moueri. </s> <s id="id.2.1.87.1.1.2.0">Moueatur igitur A <lb/>&longs;ur&longs;um v&longs;q; ad D; &longs;itq; ve­<lb/>ctis motus BD. eodemq; <lb/>modo (vt prius dictum e&longs;t) <lb/>o&longs;tendemus puncta CA cir­<lb/>culorum circumferentias de­<lb/><figure id="id.036.01.099.1.jpg" xlink:href="036/01/099/1.jpg"></figure><lb/>&longs;cribere, <expan abbr="quorũ">quorum</expan> &longs;emidiametri &longs;unt BA BC. &longs;imiliterq; o&longs;tendemus <lb/>ita e&longs;&longs;e AD ad CE, vt &longs;emidiameter AB ad &longs;emidiametrum BC. </s> </p> <p id="id.2.1.87.2.0.0.0" type="main"> <s id="id.2.1.87.2.1.1.0">Eademq; ratione, &longs;i potentia e&longs;&longs;et in C, & pondus in A, <lb/>o&longs;tendetur ita e&longs;&longs;e CE ad AD, vt BC ad BA; hoc e&longs;t di&longs;tan<lb/>tia à fulcimento ad potentiam ad di&longs;tantiam ab eodem ad ponde<lb/>ris &longs;u&longs;pen&longs;ionem. </s> <s id="id.2.1.87.2.1.2.0">quod oportebat demon&longs;trare. </s> </p> <p id="id.2.1.87.3.0.0.0" type="head"> <s id="id.2.1.87.3.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.87.4.0.0.0" type="main"> <s id="id.2.1.87.4.1.1.0">Ex his manife&longs;tum e&longs;t maiorem habere pro­<lb/>portionem &longs;patium potentiæ mouentis ad &longs;pa­<lb/>tium ponderis moti, quàm pondus ad eandem <lb/>potentiam. </s> </p> <p id="id.2.1.87.5.0.0.0" type="main"> <s id="id.2.1.87.5.1.1.0">Spatium enim potentiæ ad &longs;patium ponderis eandem habet, <pb xlink:href="036/01/100.jpg"/>quam pondus ad potentiam pondus &longs;u&longs;tinentem; potentia <expan abbr="ve­rò">ve­<lb/>ro</expan> &longs;u&longs;tinens minor e&longs;t potentia mouente, quare minorem habebit <lb/><arrow.to.target n="note144"></arrow.to.target>proportionem pondus ad potentiam ip&longs;um mouentem, quàm ad <lb/>potentiam ip&longs;um &longs;u&longs;tinentem. </s> <s id="id.2.1.87.5.1.2.0">&longs;patium igitur potentiæ mouentis <lb/>ad &longs;patium ponderis maiorem habebit proportionem, quàm pon­<lb/>dus ad eandem potentiam. </s> </p> <p id="id.2.1.88.1.0.0.0" type="margin"> <s id="id.2.1.88.1.1.1.0"><margin.target id="note144"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.89.1.0.0.0" type="head"> <s id="id.2.1.89.1.1.1.0">PROPOSITIO V. </s> </p> <p id="id.2.1.89.2.0.0.0" type="main"> <s id="id.2.1.89.2.1.1.0">Potentia quomodocunq; vecte pondus &longs;u&longs;ti­<lb/>nens ad ip&longs;um pondus eandem habebit propor­<lb/>tionem, quam di&longs;tantia à fulcimento ad punctum, <lb/>vbi à centro grauitatis ponderis horizonti ducta <lb/>perpendicularis vectem &longs;ecat, intercepta, ad <lb/>di&longs;tantiam inter fulcimentum, & potentiam. </s> </p> <p id="id.2.1.89.3.0.0.0" type="main"> <s id="id.2.1.89.3.1.1.0">Sit vectis AB <lb/>horizonti æqui­<lb/>di&longs;tans, cuius ful<lb/>cimentum N; &longs;it <lb/>deinde pondus <lb/>AC, cuius cen­<lb/>trum grauitatis <lb/>&longs;it D, quod pri<lb/>mùm &longs;it infra ve<lb/>ctem; pondus ve<lb/>rò &longs;it ex punctis <lb/>AO &longs;u&longs;pen&longs;um; <lb/><figure id="id.036.01.100.1.jpg" xlink:href="036/01/100/1.jpg"></figure><lb/>& à puncto D horizonti, & ip&longs;i AB perpendicularis ducatur DE. </s> <s id="id.2.1.89.3.1.1.0.a"><lb/>&longs;i verò alii &longs;int quoq; vectes AF AG, quorum fulcimenta &longs;int <lb/>HK; pondu&longs;q; AC in vecte AG ex punctis AQ &longs;it appen&longs;um; <lb/>in vecte autem AF in punctis AP: lineaq; DE producta &longs;ecet <lb/>AF in L, & AG in M. </s> <s id="id.2.1.89.3.1.1.0.b">dico potentiam in F pondus AC &longs;u&longs;tinen<lb/>tem ad ip&longs;um pondus eam habere proportionem, quam habet kL <pb n="44" xlink:href="036/01/101.jpg"/>ad kF; & potentiam in B ad pondus eam habere, quam NE ad <lb/>NB; & potentiam in G ad pondus eam, quam HM ad HG. </s> <s id="id.2.1.89.3.1.1.0.c"><lb/>Quoniam enim DL horizonti e&longs;t perpendicularis, pondus AC <lb/>vbicunq; in linea DL fuerit appen&longs;um, eodem modo, quo reperi­<lb/>tur, manebit. </s> <s id="id.2.1.89.3.1.2.0">quare in vecte AB &longs;i &longs;u&longs;pen&longs;iones, quæ &longs;unt ad AO <lb/>&longs;oluantur, pondus AC in E appen&longs;um eodem modo manebit, &longs;i­<lb/>cuti nunc manet; hoc e&longs;t &longs;ublato puncto A, & linea QO, codem <lb/>modo pondus in E appen&longs;um manebit, vt ab ip&longs;is AO pun­<lb/>ctis &longs;u&longs;tinebatur; ex commentario Federici Commandini in &longs;extam <lb/>Archimedis <expan abbr="propo&longs;ion&etilde;">propo&longs;itionem</expan> de quadratura parabolæ, & ex prima huius <lb/>de libra. </s> <s id="id.2.1.89.3.1.3.0">Itaq; quoniam pondus AC eandem ad libram habet con&longs;ti<lb/>tutionem, &longs;iue in AO &longs;u&longs;tineatur, &longs;iue ex puncto E &longs;it appen&longs;um; <lb/>eadem potentia in B idem pondus AC, &longs;iue in E, &longs;iue in AO <lb/>&longs;u&longs;pen&longs;um &longs;u&longs;tinebit. </s> <s id="id.2.1.89.3.1.4.0">potentia verò in B &longs;u&longs;tinens pondus AC <lb/>in E appen&longs;um ad ip&longs;um pondus ita &longs;e habet, vt NE ad NB; po­<lb/>tentia <arrow.to.target n="note145"></arrow.to.target>igitur in B &longs;u&longs;tinens pondus AC ex punctis AO &longs;u&longs;pen<lb/>&longs;um ad ip&longs;um pondus ita erit, vt NE ad NB. </s> <s id="id.2.1.89.3.1.4.0.a">Non aliter o&longs;ten <lb/>detur pondus AC ex puncto L &longs;u&longs;pen&longs;um manere, &longs;icuti à pun<lb/>ctis AP &longs;u&longs;tinetur; potentiamq; in F ad ip&longs;um pondus ita e&longs;&longs;e, vt kL <lb/>ad KF. </s> <s id="id.2.1.89.3.1.4.0.b">In vecte verò AG pondus AC in M appen&longs;um ita mane <lb/>re, vt à punctis AQ &longs;u&longs;tinetur; potentiamq; in G ad pondus <lb/>AC ita e&longs;&longs;e, vt HM ad HG; hoc e&longs;t vt di&longs;tantia à fulcimento <lb/>ad punctum, vbi à centro grauitatis ponderis horizonti ducta <lb/>perpendicularis vectem &longs;ecat, ad di&longs;tantiam à fulcimento ad poten<lb/>tiam. </s> <s id="id.2.1.89.3.1.5.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.90.1.0.0.0" type="margin"> <s id="id.2.1.90.1.1.1.0"><margin.target id="note145"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.91.1.0.0.0" type="main"> <s id="id.2.1.91.1.1.1.0">Si autem FBG e&longs;&longs;ent vectium fulcimenta, potentiæq; e&longs;&longs;ent <lb/>in KNH pondus &longs;u&longs;tinentes, &longs;imili modo o&longs;tendetur ita e&longs;&longs;e po<lb/>tentiam in H ad pondus, vt GM ad GH; & potentiam in N ad <lb/>pondus, vt BE ad BN; ac potentiam in k ad pondus, vt FL <lb/>ad Fk. </s> </p> <pb xlink:href="036/01/102.jpg"/> <p id="id.2.1.91.3.0.0.0" type="main"> <s id="id.2.1.91.3.1.1.0">Et &longs;i vectes AB <lb/>AF AG habeant <lb/>fulcimenta in A, <lb/>& pondus &longs;it NO; <lb/>deinde ab eius <lb/>centro grauitatis <lb/>D ducatur ip&longs;i A <lb/>B, & horizonti <lb/><expan abbr="perp&etilde;dicularis">perpendicularis</expan> D <lb/>MEL; &longs;intq; po<lb/>tentiæ in FBG: <lb/>&longs;imiliter o&longs;tende­<lb/>tur ita e&longs;&longs;e poten­<lb/><figure id="id.036.01.102.1.jpg" xlink:href="036/01/102/1.jpg"></figure><lb/>tiam in G pondus NO &longs;u&longs;tinentem ad ip&longs;um pondus, vt AM <lb/>ad AG; ac potentiam in B, vt AE ad AB; & potentiam in F, <lb/>vt AL ad AF. </s> </p> <p id="id.2.1.91.4.0.0.0" type="main"> <s id="id.2.1.91.4.1.1.0">Sit deinde <lb/>vectis AB ho<lb/>rizonti æqui­<lb/>di&longs;tans, cuius <lb/>fulcimentum <lb/>D; & &longs;it BE <lb/>pondus, cuius <lb/>centrum <expan abbr="gaui">graui</expan><lb/>tatis &longs;it F &longs;u­<lb/>pra vectem: à <lb/>punctoq; F ho<lb/>rizonti, & ip&longs;i <lb/>AB ducatur <lb/><figure id="id.036.01.102.2.jpg" xlink:href="036/01/102/2.jpg"></figure><lb/>FH; pondu&longs;q; à puncto B, & PQ &longs;u&longs;tineatur. </s> <s id="id.2.1.91.4.1.2.0">Sint deinde alii ve­<lb/>ctes BL BM, quorum fulcimenta &longs;int NO; lineaq; FH producta &longs;e­<lb/>cet BM in k, & BL in G; pondus autem in vecte BL in pun­<lb/>ctis BP &longs;u&longs;tineatur; in vecte autem BM à puncto B, & PR. </s> <s id="id.2.1.91.4.1.2.0.a">Di­<lb/>co potentiam in L pondus BE vecte BL &longs;u&longs;tinentem ad ip&longs;um <lb/>pondus eam habere proportionem, quam NG ad NL; & po­<pb n="45" xlink:href="036/01/103.jpg"/>tentiam in A ad pondus eam habere, quam DH ad DA; poten<lb/>tiamq; in M ad pondus eam, quam Ok ad OM. </s> <s id="id.2.1.91.4.1.2.0.b">Quoniam e­<lb/>nim à centro grauitatis F ducta e&longs;t kF horizonti perpendicularis, <lb/>ex quocunq; puncto lineæ kF &longs;u&longs;tineatur pondus, manebit; vt <arrow.to.target n="note146"></arrow.to.target><lb/>nunc &longs;e habet. </s> <s id="id.2.1.91.4.1.3.0">&longs;i igitur &longs;u&longs;tineatur in H, manebit vt prius; &longs;cili­<lb/>cet &longs;ublato puncto B, & PQ, quæ pondus &longs;u&longs;tinent, pondus BE <lb/>manebit, &longs;icuti ab ip&longs;is &longs;u&longs;tinebatur. </s> <s id="id.2.1.91.4.1.4.0">quare in vecte AB graue&longs;cet <lb/>in H, & ad vectem eandem habebit con&longs;titutionem, quam prius; <lb/>idcirco erit, ac &longs;i in H e&longs;&longs;et appen&longs;um. </s> <s id="id.2.1.91.4.1.5.0">eadem igitur potentia ìdem <lb/>pondus BE, &longs;iue in H, &longs;iue in B, & Q &longs;uffultum, &longs;u&longs;tinebit. </s> <s id="id.2.1.91.4.1.6.0">Potentia ve <arrow.to.target n="note147"></arrow.to.target><expan abbr="rò"><lb/>ro</expan> in A &longs;u&longs;tinens pondus BE vecte AB in H appen&longs;um ad ip&longs;um <lb/>pondus eandem habet proportionem, quam DH ad DA; eadem <lb/>ergo potentia in A &longs;u&longs;tinens pondus BE in punctis BQ &longs;u&longs;tenta <lb/>tum ad ip&longs;um pondus erit, vt DH ad DA. </s> <s id="id.2.1.91.4.1.6.0.a">Similiter o&longs;tende­<lb/>tur pondus BE &longs;i in G &longs;u&longs;tineatur, manere; &longs;icuti à punctis BP <lb/>&longs;u&longs;tinebatur: & in puncto k, vt à punctis BR. quare potentia in <lb/>L &longs;u&longs;tinens pondus BE ad ip&longs;um pondus ita erit, vt NG ad NL. <lb/>potentia verò in M ad pondus, vt OK ad OM; hoc e&longs;t vt di&longs;tan<lb/>tia à fulcimento ad punctum, vbi à centro grauitatis ponderis ho<lb/>rizonti ducta perpendicularis vectem &longs;ecat, ad di&longs;tantiam à fulci­<lb/>mento ad potentiam. </s> <s id="id.2.1.91.4.1.7.0">quod demon&longs;trare quoq; oportebat. </s> </p> <p id="id.2.1.92.1.0.0.0" type="margin"> <s id="id.2.1.92.1.1.1.0"><margin.target id="note146"></margin.target>1 <emph type="italics"/>Huius de libra.<emph.end type="italics"/></s> <s id="id.2.1.92.1.1.2.0"><margin.target id="note147"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.93.1.0.0.0" type="main"> <s id="id.2.1.93.1.1.1.0">Si verò LAM e&longs;&longs;ent fulcimenta, & potentiæ in NDO; &longs;imi <lb/>liter o&longs;tendetur ita e&longs;&longs;e potentiam in N ad pondus, vt LG ad L <lb/>N; & potentiam in D, vt AH ad AD; & potentiam in O, vt <lb/>Mk ad MO. <pb xlink:href="036/01/104.jpg"/></s> </p> <p id="id.2.1.93.2.0.0.0" type="main"> <s id="id.2.1.93.2.1.1.0">Et &longs;i vectes BA <lb/>BL BM habeant <lb/>fulcimenta in B, & <lb/>pondus &longs;upra <expan abbr="vect&etilde;">vectem</expan> <lb/>&longs;it NO; & ab eius <lb/>centro grauitatis F <lb/>ducatur ip&longs;i AB, & <lb/>horizonti perpendi<lb/>cularis FDEG; &longs;int <lb/>qué potentiæ in L <lb/>AM; &longs;imiliter o­<lb/>&longs;tendetur ita e&longs;&longs;e po<lb/>tentiam in L pon­<lb/><figure id="id.036.01.104.1.jpg" xlink:href="036/01/104/1.jpg"></figure><lb/>dus &longs;u&longs;tinentem ad ip&longs;um pondus, vt BD ad BL; & potentiam <lb/>in A ad pondus, vt BE ad BA, atq; potentiam in M, vt BG <lb/>ad BM. </s> </p> <p id="id.2.1.93.3.0.0.0" type="main"> <s id="id.2.1.93.3.1.1.0">Sit deniq; <lb/>vectis AB ho<lb/>rizonti æqui­<lb/>di&longs;tans, cuius <lb/>fulcimentum <lb/>C, & pondus <lb/>DE habeat <expan abbr="c&etilde;">cen</expan><lb/>trum grauita­<lb/>tis F in ip&longs;o <lb/>vecte AB; <lb/>&longs;intq; deniq; <lb/>alii vectes G <lb/>H kL, quo­<lb/><figure id="id.036.01.104.2.jpg" xlink:href="036/01/104/2.jpg"></figure><lb/>rum fulcimenta &longs;int MN; pondusq; in vecte GH &longs;u&longs;tineatur à <lb/>punctis GO; in vecte autem AB à punctis AP; & in uecte KL <lb/>à punctis KQ; & centrum grauitatis F &longs;it quoq; in utroq; uecte <lb/>GH kL; &longs;intq; potentiæ in HBL. </s> <s id="id.2.1.93.3.1.1.0.a">Dico potentiam in H ad <lb/>pondus ita e&longs;&longs;e, ut NF ad NH; & potentiam in B ad pondus, ut <lb/>CF ad CB; ac potentiam in L ad pondus, ut MF ad ML. </s> <s id="id.2.1.93.3.1.1.0.b">Quo­<lb/>niam enim F centrum e&longs;t grauitatis ponderis DE, &longs;i igitur in F <pb n="46" xlink:href="036/01/105.jpg"/>&longs;u&longs;tineatur, pondus DE manebit &longs;icut prius, per deffinitionem cen<lb/>tri grauitatis; eritq; ac &longs;i in F e&longs;&longs;et appen&longs;um; atq; in vecte eodem <lb/>modo manebit, &longs;iue à punctis AP, &longs;iue à puncto F &longs;u&longs;tineatur. </s> <s id="id.2.1.93.3.1.2.0"><lb/>quod idem in vectibus GH kL eueniet; &longs;cilicet pondus eodem mo <lb/>do manere, &longs;iue in F, &longs;iue in GO, vel in kQ &longs;u&longs;tineatur. </s> <s id="id.2.1.93.3.1.3.0">eadem <lb/>igitur potentia in B idem pondus DE, vel in F, vel in AP appen&longs;um <lb/>&longs;u&longs;tinebit: & quando appen&longs;um e&longs;t in F ad ip&longs;um pon­<lb/>dus e&longs;t, vt CF ad CB, ergo potentia &longs;u&longs;tinens pondus DE in <lb/>AP appen&longs;um ad ip&longs;um pondus erit, vt CF ad CB. eodemq; mo <lb/>do potentia in H ad pondus in GO appen&longs;um ita erit, vt NF ad <lb/>NH. potentiaq; in L ad pondus in kQ appen&longs;um erit, vt MF <lb/>ad ML. quod o&longs;tendere quoq; oportebat. </s> </p> <p id="id.2.1.93.4.0.0.0" type="main"> <s id="id.2.1.93.4.1.1.0">Si verò HBL e&longs;&longs;ent fulcimenta, & potentiæ e&longs;&longs;ent in NCM; &longs;i­<lb/>militer o&longs;tendetur potentiam in N ad pondus ita e&longs;&longs;e, vt HF ad <lb/>HN; & potentiam in C, vt BF ad BC, & potentiam in M, vt <lb/>LF ad LM. </s> </p> <p id="id.2.1.93.5.0.0.0" type="main"> <s id="id.2.1.93.5.1.1.0">Et &longs;i vectes BA <lb/>BC BD <expan abbr="habeãt">habeant</expan> ful<lb/>cimenta in B, &longs;intq; <lb/>pondera in EF GH <lb/>kL, ita vt eorum <lb/>centra MNO gra­<lb/>uitatis &longs;int in vecti<lb/>bus; &longs;intq; poten­<lb/>tiæ in CAD: &longs;imi <lb/>liter o&longs;tendetur po<lb/>tentiam in C ad <lb/>pondus EF ita e&longs;&longs;e, <lb/><figure id="id.036.01.105.1.jpg" xlink:href="036/01/105/1.jpg"></figure><lb/>vt BM ad BC, & potentiam in A ad pondus GH, vt BN ad <lb/>BA, potentiamq; in D ad pondus KL, vt BO ad BD. </s> </p> <pb xlink:href="036/01/106.jpg"/> <p id="id.2.1.93.7.0.0.0" type="head"> <s id="id.2.1.93.7.1.1.0">PROPOSITIO VI. </s> </p> <p id="id.2.1.93.8.0.0.0" type="main"> <s id="id.2.1.93.8.1.1.0">Sit AB recta linea, cui ad angulos &longs;it rectos <lb/>AD, quæ ex parte A producatur vtcunq; v&longs;q; <lb/>ad C; connectaturq; CB, quæ ex parte B quoq; <lb/>producatur v&longs;q; ad E. ducantur deinde à pun­<lb/>cto B vtcunq; inter AB BE lineæ BF BG ip&longs;i <lb/>AB æquales; à puncti&longs;q; FG ip&longs;is perpendicula­<lb/>res ducantur FH GK, quæ & inter &longs;e &longs;e, & ip&longs;i <lb/>AD con&longs;tituantur æ­<lb/>quales, ac &longs;i BA AD <lb/>motæ &longs;int in BF FH, <lb/>& in BG GK; con­<lb/>nectanturq; CH CK, <lb/>quæ lineas BF BG <lb/>in punctis MN &longs;e­<lb/>cent. </s> <s id="id.2.1.93.8.1.2.0">Dico BN mi­<lb/>norem e&longs;&longs;e BM, & <lb/>BM ip&longs;a BA. <lb/><figure id="id.036.01.106.1.jpg" xlink:href="036/01/106/1.jpg"></figure></s> </p> <p id="id.2.1.93.9.0.0.0" type="main"> <s id="id.2.1.93.9.1.1.0">Connectantur BD BH <lb/>BK. & quoniam duæ lineæ <lb/>DA AB duabus HF FB <lb/>&longs;unt æquales, & angulus <lb/>DAB rectus recto HFB e&longs;t <lb/><arrow.to.target n="note148"></arrow.to.target>etiam æqualis; erunt reliqui <lb/>anguli reliquis angulis æqua­<lb/>les, & HB ip&longs;i DB æqualis. </s> <s id="id.2.1.93.9.1.2.0"><lb/>&longs;imiliter o&longs;tendetur triangu­<lb/>lum BkG triangulo BHF æqualem e&longs;&longs;e. </s> <s id="id.2.1.93.9.1.3.0">quare centro B, inter­<pb n="47" xlink:href="036/01/107.jpg"/>uallo quidem vna ip&longs;arum circulus de&longs;cribatur DH kE, qui li­<lb/>neas CH CK &longs;ecet in punctis OP; connectanturq; OB PB. </s> <s id="id.2.1.93.9.1.3.0.a"><lb/>Quoniam igitur punctum k propius e&longs;t ip&longs;i E, quàm H; erit linea <arrow.to.target n="note149"></arrow.to.target><lb/>Ck maior ip&longs;a CH, & CP ip&longs;a CO minor: ergo PK ip&longs;a OH <lb/>maior erit. </s> <s id="id.2.1.93.9.1.4.0">Quoniam autem triangulum BkP æquicrure latera <lb/>Bk BP lateribus BH BO trianguli BHO æquicruris æqualia ha<lb/>bet, ba&longs;im verò KP ba&longs;i HO maiorem, erit angulus kBP an­<lb/>gulo <arrow.to.target n="note150"></arrow.to.target>HBO maior. </s> <s id="id.2.1.93.9.1.5.0">ergo reliqui ad ba&longs;im anguli, hoc e&longs;t kPB <lb/>PkB &longs;imul &longs;umpti, qui inter &longs;e &longs;unt æquales, reliquis ad ba&longs;im an­<lb/>gulis, nempè OHB HOB, qui etiam inter &longs;e &longs;unt æquales, mino­<lb/>res <arrow.to.target n="note151"></arrow.to.target>erunt: cùm omnes anguli cuiu&longs;cunq; trianguli duobus &longs;int rectis <lb/>æquales. </s> <s id="id.2.1.93.9.1.6.0">quare & horum dimidii, &longs;cilicet NkB minor MHB. </s> <s id="id.2.1.93.9.1.6.0.a"><lb/>Cùm autem angulus BkG æqualis &longs;it angulo BHF, erit NkG <lb/>ip&longs;o MHF maior. </s> <s id="id.2.1.93.9.1.7.0">&longs;i igitur à puncto k con&longs;tituatur angulus GKQ <lb/>ip&longs;i FHM æqualis, fiet triangulum GkQ triangulo FHM æqua <lb/>le; nam duo anguli ad FH vnius duobus ad Gk alterius &longs;unt <lb/>æquales, & latus FH lateri Gk e&longs;t æquale, erit GQ ip&longs;i FM æ­<lb/>quale. <arrow.to.target n="note152"></arrow.to.target></s> <s id="id.2.1.93.9.1.8.0">ergo GN maior erit ip&longs;a FM. </s> <s id="id.2.1.93.9.1.8.0.a">Cùm itaq; BG ip&longs;i BF &longs;it æqua <lb/>lis, erit BN minor ip&longs;a BM. </s> <s id="id.2.1.93.9.1.8.0.b">Quòd autem BM &longs;it ip&longs;a BA minor, <lb/>e&longs;t manife&longs;tum; cùm BM ip&longs;a BF, quæ ip&longs;i BA e&longs;t æqualis, &longs;it <lb/>minor. </s> <s id="id.2.1.93.9.1.9.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.94.1.0.0.0" type="margin"> <s id="id.2.1.94.1.1.1.0"><margin.target id="note148"></margin.target>4 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.94.1.1.2.0"><margin.target id="note149"></margin.target>8 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.94.1.1.3.0"><margin.target id="note150"></margin.target>25 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.94.1.1.4.0"><margin.target id="note151"></margin.target>5 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.94.1.1.5.0"><margin.target id="note152"></margin.target>26 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.95.1.0.0.0" type="main"> <s id="id.2.1.95.1.1.1.0">In&longs;uper &longs;i intra BG BE alia vtcunq; ducatur linea ip&longs;i BG æ­<lb/>qualis; fiatq; operatio, quemadmodum &longs;upra dictum e&longs;t; &longs;imili­<lb/>ter o&longs;tendetur lineam BR minorem e&longs;&longs;e BN. & quò propius fue<lb/>rit ip&longs;i BE, adhuc minorem &longs;emper e&longs;&longs;e. </s> </p> <pb xlink:href="036/01/108.jpg"/> <p id="id.2.1.95.3.0.0.0" type="main"> <s id="id.2.1.95.3.1.1.0">Si verò æqualia triangula BFH BGK &longs;int <lb/>deor&longs;um inter BC BA con&longs;tituta; connectan­<lb/>turq; HC KC, quæ lineas BF BG ex parte <lb/>FG productas in punctis MN &longs;ecent erit BN <lb/>maior BM, & BM ip&longs;a BA. </s> </p> <p id="id.2.1.95.4.0.0.0" type="main"> <s id="id.2.1.95.4.1.1.0">Nam producatur CH <lb/>Ck v&longs;q; ad circumferentiam <lb/>in OP, Connectanturq; BO <lb/>BP; &longs;imili modo o&longs;tende­<lb/>tur lineam Pk maiorem e&longs; <lb/>&longs;e OH, angulumq; PkB mi<lb/>norem e&longs;&longs;e angulo OHB. </s> <s id="id.2.1.95.4.1.1.0.a">& <lb/>quoniam angulus BHF e&longs;t <lb/>æqualis angulo BkG; erit to<lb/>tus PKG angulus angulo <lb/>OHF minor: quare reliquus <lb/>GKN reliquo FHM maior <lb/>erit. </s> <s id="id.2.1.95.4.1.2.0">&longs;i it aq; con&longs;tituatur angu<lb/>lus GkQ ip&longs;i FHM æqua <lb/>lis, linea KQ ip&longs;am GN ita <lb/>&longs;ecabit, vt GQ ip&longs;i FM æqua <lb/>lis euadat: quare maior. </s> <s id="id.2.1.95.4.1.3.0">erit <lb/>GN, quàm FM; quibus &longs;i <lb/>æquales adiiciantur BF BG, <lb/>erit BN ip&longs;a BM maior. </s> <s id="id.2.1.95.4.1.4.0">& <lb/>cùm BM &longs;it ip&longs;a FB maior, <lb/>erit quoq; ip&longs;a BA maior. </s> <s id="id.2.1.95.4.1.5.0">&longs;i <lb/>militer o&longs;tendetur, quò pro <lb/>pius fuerit BG ip&longs;i BC, li­<lb/>neam BN &longs;emper maiorem <lb/>e&longs;&longs;e. <figure id="id.036.01.108.1.jpg" xlink:href="036/01/108/1.jpg"></figure></s> <pb n="48" xlink:href="036/01/109.jpg"/> <s id="id.2.1.95.4.3.1.0">PROPOSITIO VII. </s> </p> <p id="id.2.1.95.5.0.0.0" type="main"> <s id="id.2.1.95.5.1.1.0">Sit recta linea AB, cuì perpendicularis exi­<lb/>&longs;tat AD, quæ ex parte D producatur vtcunq; v&longs;q; <lb/>ad C; connectaturq; CB, quæ producatur e­<lb/>tiam v&longs;q; ad E; & inter AB BE lineæ &longs;imiliter <lb/>vtcunq; ducantur BF BG ip&longs;i AB æquales; à <lb/>punctisq; FG lineæ FH GK ip&longs;i AB æquales, <lb/>ip&longs;is verò BF BG <expan abbr="per­p&etilde;diculares">per­<lb/>pendiculares</expan> ducantur; <lb/>ac &longs;i BA AD motæ <lb/>&longs;int in BF FH BG <lb/>GK: Connectanturq; <lb/>CH CK, quæ lineas <lb/>BF BG productas &longs;e­<lb/>cent in punctis MN. </s> <s id="id.2.1.95.5.1.1.0.a"><lb/>Dico BN maiorem e&longs; <lb/>&longs;e BM, & BM ip&longs;a BA. <lb/><figure id="id.036.01.109.1.jpg" xlink:href="036/01/109/1.jpg"></figure></s> </p> <p id="id.2.1.95.6.0.0.0" type="main"> <s id="id.2.1.95.6.1.1.0">Connectantur BD BH Bk, <lb/>& centro B, interuallo quidem <lb/>BD, circulus de&longs;cribatur. </s> <s id="id.2.1.95.6.1.2.0">&longs;imi <lb/>liter vt in præcedenti demon­<lb/>&longs;trabimus puncta kHDOP in <lb/>circuli circumferentia e&longs;&longs;e, trian<lb/>gulaq; ABD FBH GBk in­<lb/>ter &longs;e &longs;e æqualia e&longs;&longs;e, atq; lineam <lb/>Pk maiorem OH, angulumq; <lb/>PKB minorem e&longs;&longs;e angulo O <lb/>HB. </s> <s id="id.2.1.95.6.1.2.0.a">Quoniam igitur angulus BHF æqualis e&longs;t angulo BkG, <pb xlink:href="036/01/110.jpg"/>erit totus angulus PkG angu­<lb/>lo OHF minor: quare reliquus <lb/>GkN reliquo FHM maior <lb/>erit. </s> <s id="id.2.1.95.6.1.3.0">&longs;i igitur fiat angulus GK <lb/>Q ip&longs;i FHM æqualis, erit trian<lb/>gulum GKQ triangulo FHM <lb/>æquale, & latus GQ lateri FM <lb/>æquale; ergo maior erit GN ip<lb/>&longs;a FM; ac propterea BN ma­<lb/>ior erit BM. </s> <s id="id.2.1.95.6.1.3.0.a">BM autem ma­<lb/>ior erit BA; nam BM maior e&longs;t <lb/>ip&longs;a BF. quod demon&longs;trare <lb/>oportebat. <figure id="id.036.01.110.1.jpg" xlink:href="036/01/110/1.jpg"></figure></s> </p> <p id="id.2.1.95.7.0.0.0" type="main"> <s id="id.2.1.95.7.1.1.0">Eodemq; pror&longs;us modo, quo <lb/>propius fuerit BG ip&longs;i BE, li­<lb/>neam BN &longs;emper maiorem e&longs;&longs;e <lb/>o&longs;tendetur. </s> </p> <p id="id.2.1.95.8.0.0.0" type="main"> <s id="id.2.1.95.8.1.1.0">Si autem triangula BFH BGK deor&longs;um in­<lb/>ter AB BC con&longs;tituantur, ducanturq; CHO <lb/>CKP, quæ lineas BF BG &longs;ecent in punctis M <lb/>N; erit linea BN minor ip&longs;a BM, & BM <lb/>ip&longs;a BA. </s> </p> <pb n="49" xlink:href="036/01/111.jpg"/> <p id="id.2.1.95.10.0.0.0" type="main"> <s id="id.2.1.95.10.1.1.0">Connectantur enim BO BP, <lb/>&longs;imiliter o&longs;tendetur angulum <lb/>PKB minorem e&longs;&longs;e OHB. </s> <s id="id.2.1.95.10.1.1.0.a">& <lb/>quoniam angulus FHB æqua­<lb/>lis e&longs;t angulo GkB; erit angu<lb/>lus GkN angulo FHM ma­<lb/>ior: quare & linea GN ma­<lb/>ior erit ip&longs;a FM. ideoq; linea <lb/><expan abbr="nea"></expan> BN minor erit linea BM. </s> <s id="id.2.1.95.10.1.1.0.b"><lb/>Cùm autem maior &longs;it BF ip&longs;a <lb/>BM; erit BM ip&longs;a BA minor. </s> <s id="id.2.1.95.10.1.2.0">Si­<lb/>miliq; modo o&longs;tendetur, quò <lb/>propius fuerit BG ip&longs;i BC, li­<lb/>neam BN &longs;emper minorem <lb/>e&longs;&longs;e. </s> </p> <figure id="id.036.01.111.1.jpg" xlink:href="036/01/111/1.jpg"> </figure> <p id="id.2.1.95.10.2.1.0" type="head"> <s id="id.2.1.95.10.4.1.0">PROPOSITIO VIII. </s> </p> <p id="id.2.1.95.11.0.0.0" type="main"> <s id="id.2.1.95.11.1.1.0">Potentia pondus &longs;u&longs;tinens centrum grauitatis <lb/>&longs;upra vectem horizonti æquidi&longs;tantem habens, <lb/>quò magis pondus ab hoc &longs;itu vecte eleuabitur; <lb/>minori &longs;emper, vt &longs;u&longs;tineatur, egebit potentia: <lb/>&longs;i verò deprimetur, maiori. <pb xlink:href="036/01/112.jpg"/><figure id="id.036.01.112.1.jpg" xlink:href="036/01/112/1.jpg"></figure></s> </p> <p id="id.2.1.95.12.0.0.0" type="main"> <s id="id.2.1.95.12.1.1.0">Sit vectis AB horizonti æquidi&longs;tans, cuius fulcimentum C; <lb/>pondus autem BD, eiu&longs;dem verò grauitatis centrum &longs;it &longs;upra ve<lb/>ctem vbi H: &longs;itq; potentia &longs;u&longs;tinens in A. </s> <s id="id.2.1.95.12.1.1.0.a">moueatur deinde ve<lb/>ctis AB in EF, &longs;itq; pondus motum in FG. </s> <s id="id.2.1.95.12.1.1.0.b">Dico primùm mino <lb/>rem <expan abbr="potentiã">potentiam</expan> in E &longs;u&longs;tinere pondus FG vecte EF, quàm <expan abbr="pot&etilde;tia">potentia</expan> in <lb/>A pondus BD vecte AB. </s> <s id="id.2.1.95.12.1.1.0.c">&longs;it k centrum grauitatis ponderis FG; <lb/>deinde tùm ex H, tùm ex K ducantur HL kM ip&longs;orum horizon<lb/>tibus perpendiculares, quæ in <expan abbr="centrũ">centrum</expan> mundi conuenient; &longs;itq; HL ip<lb/>&longs;i quoq; AB perpendicularis. </s> <s id="id.2.1.95.12.1.2.0">ducatur deinde kN ip&longs;i EF perpen­<lb/>dicularis, quæ ip&longs;i HL æqualis erit, & CN ip&longs;i CL æqualis. </s> <s id="id.2.1.95.12.1.3.0">Quo­<lb/><arrow.to.target n="note153"></arrow.to.target>niam enim HL horizonti e&longs;t perpendicularis, potentia in A &longs;u<lb/>&longs;tinens pondus BD ad ip&longs;um pondus eam habebit proportionem, <lb/>quam CL ad CA. </s> <s id="id.2.1.95.12.1.3.0.a">rur&longs;us quoniam kM horizonti e&longs;t perpendicu<lb/>laris, potentia in E pondus FG &longs;u&longs;tinens ita erit ad pondus, vt <lb/>CM ad CE. </s> <s id="id.2.1.95.12.1.3.0.b">Cùm autem CN NK ip&longs;is CL LH &longs;int æquales, <lb/><arrow.to.target n="note154"></arrow.to.target>angulosq; rectos contineant; erit CM minor ip&longs;a CL; ergo CM <lb/><arrow.to.target n="note155"></arrow.to.target>ad CA minorem habebit proportionem, quam CL ad CA; & <pb n="45" xlink:href="036/01/113.jpg"/>CA ip&longs;i CE e&longs;t æqualis, minorem igitur proportionem habebit <lb/>CM ad CE. quàm CL ad CA: & cùm pondera BD FG &longs;int <lb/>æqualia, e&longs;t enim idem pondus; ergo minor erit proportio po<lb/>tentiæ in E pondus FG &longs;u&longs;tinentis ad ip&longs;um pondus, quàm po<lb/>tentiæ in A pondus BD &longs;u&longs;tinentis ad ip&longs;um pondus. </s> <s id="id.2.1.95.12.1.4.0">Quare <arrow.to.target n="note156"></arrow.to.target><lb/>minor potentia in E &longs;u&longs;tinebit pondus FG, quàm potentia in A <lb/>pondus BD. & quò pondus magis eleuabitur; &longs;emper o&longs;tendetur <lb/>minorem adhuc potentiam pondus &longs;u&longs;tinere; cùm linea PC mi <arrow.to.target n="note157"></arrow.to.target><lb/>nor &longs;it linea CM. </s> <s id="id.2.1.95.12.1.4.0.a">&longs;it deinde vectis in QR, & pondus in QS, <lb/>cuius <expan abbr="centrũ">centrum</expan> grauitatis &longs;it O. </s> <s id="id.2.1.95.12.1.4.0.b">dico maiorem requiri potentiam in R <lb/>ad <expan abbr="&longs;u&longs;tinendũ">&longs;u&longs;tinendum</expan> pondus QS, quàm in A ad pondus BD. ducatur à cen<lb/>tro grauitatis O linea OT horizonti perpendicularis. </s> <s id="id.2.1.95.12.1.5.0">& quo­<lb/>niam HL OT, &longs;i ex parte L, atq; T producantur, in centrum <lb/>mundi conuenient; erit CT maior CL: e&longs;t autem CA ip&longs;i CR <arrow.to.target n="note158"></arrow.to.target><lb/>æqualis, habebit ergo TC ad CR maiorem proportionem, quàm <lb/>LC ad CA. </s> <s id="id.2.1.95.12.1.5.0.a">Maior igitur erit potentia in R &longs;u&longs;tinens pondus <arrow.to.target n="note159"></arrow.to.target><lb/>QS, quàm in A &longs;u&longs;tinens BD. &longs;imiliter o&longs;tendetur; quò vectis <lb/>RQ magis à vecte AB di&longs;tabit deor&longs;um vergens, &longs;emper maio­<lb/>rem potentiam requiri ad &longs;u&longs;tinendum pondus: di&longs;tantia enim CV <arrow.to.target n="note160"></arrow.to.target><lb/>longior e&longs;t CT. </s> <s id="id.2.1.95.12.1.5.0.b">Quò igitur pondus à &longs;itu horizonti æquidi&longs;tan<lb/>te magis eleuabitur à minori &longs;emper potentia pondus &longs;u&longs;tinebitur; <lb/>quò verò magis deprimetur, maiori, vt &longs;u&longs;tineatur, egebit potentia. <lb/></s> <s id="id.2.1.95.12.1.6.0"><lb/>quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.96.1.0.0.0" type="margin"> <s id="id.2.1.96.1.1.1.0"><margin.target id="note153"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.96.1.1.2.0"><margin.target id="note154"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.96.1.1.3.0"><margin.target id="note155"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.96.1.1.4.0"><margin.target id="note156"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.96.1.1.5.0"><margin.target id="note157"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.96.1.1.6.0"><margin.target id="note158"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.96.1.1.7.0"><margin.target id="note159"></margin.target>8 <emph type="italics"/>Quinti. </s> <s id="id.2.1.96.1.1.8.0">Ex<emph.end type="italics"/> 10 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.96.1.1.9.0"><margin.target id="note160"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.97.1.0.0.0" type="main"> <s id="id.2.1.97.1.1.1.0">Hinc facile elicitur potentiam in A ad poten­<lb/>tiam in E ita e&longs;&longs;e, vt CL ad CM. </s> </p> <p id="id.2.1.97.2.0.0.0" type="main"> <s id="id.2.1.97.2.1.1.0">Nam ita e&longs;t LC ad CA, vt potentia in A ad pondus; vt au­<lb/>tem CA, hoc e&longs;t CE ad CM, ita e&longs;t pondus ad potentiam in E; <lb/>quare ex æquali potentia in A ad potentiam in E ita erit, vt CL <arrow.to.target n="note161"></arrow.to.target><lb/>ad CM. </s> </p> <p id="id.2.1.98.1.0.0.0" type="margin"> <s id="id.2.1.98.1.1.1.0"><margin.target id="note161"></margin.target>22 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.99.1.0.0.0" type="main"> <s id="id.2.1.99.1.1.1.0">Similiq; ratione non &longs;olum o&longs;tendetur, potentiam in A ad po­<lb/>tentiam in R ita e&longs;&longs;e, vt CL ad CT; &longs;ed & potentiam quoq; in E <lb/>ad potentiam in R ita e&longs;&longs;e, vt CM ad CT. & ita in reliquis. <pb xlink:href="036/01/114.jpg"/><figure id="id.036.01.114.1.jpg" xlink:href="036/01/114/1.jpg"></figure></s> </p> <p id="id.2.1.99.2.0.0.0" type="main"> <s id="id.2.1.99.2.1.1.0">Sit deinde vectis AB horizonti æquidi&longs;tans, cuius fulcimen­<lb/>tum B; & centrum grauitatis H ponderis CD &longs;it &longs;upra vectem; <lb/>moueaturq; vectis in BE, pondu&longs;q; in FG. </s> <s id="id.2.1.99.2.1.1.0.a">dico minorem po­<lb/>tentiam in E &longs;u&longs;tinere pondus FG vecte EB, quàm potentia in <lb/>A pondus CD vecte AB. </s> <s id="id.2.1.99.2.1.1.0.b">&longs;it k centrum grauitatis ponderis FG, <lb/>& à centris grauitatum Hk ip&longs;orum horizontibus perpendicu­<lb/><arrow.to.target n="note162"></arrow.to.target>lares ducantur HL kM. </s> <s id="id.2.1.99.2.1.1.0.c">Quoniam enim (ex &longs;upra demon&longs;tratis) <lb/><arrow.to.target n="note163"></arrow.to.target>BM minor e&longs;t BL, & BE ip&longs;i BA æqualis; minorem habebit <lb/><arrow.to.target n="note164"></arrow.to.target>proportionem BM ad BE, quàm BL ad BA. &longs;ed vt BM ad <lb/>BE, ita potentia in E &longs;u&longs;tinens pondus FG ad ip&longs;um pondus; & <lb/>vt BL ad BA, ita potentia in A ad pondus CD; minorem <lb/>habebit proportionem potentia in E ad pondus FG, quàm poten<lb/><arrow.to.target n="note165"></arrow.to.target>tia in A ad pondus CD. </s> <s id="id.2.1.99.2.1.1.0.d">Ergo potentia in E minor erit poten­<lb/>tia in A. &longs;imiliter o&longs;tendetur, quò magis pondus eleuabitur, &longs;em­<lb/>per minorem potentiam pondus &longs;u&longs;tinere. </s> <s id="id.2.1.99.2.1.2.0">Sit autem vectis in <lb/>BO, & pondus in PQ, cuius centrum grauitatis &longs;it R. </s> <s id="id.2.1.99.2.1.2.0.a">dico maio<lb/>rem potentiam in O requiri ad &longs;u&longs;tinendum pondus PQ vecte BO, <lb/>quàm pondus CD vecte BA. </s> <s id="id.2.1.99.2.1.2.0.b">ducatur à puncto R horizonti per­<lb/><arrow.to.target n="note166"></arrow.to.target>pendicularis RS. </s> <s id="id.2.1.99.2.1.2.0.c">& quoniam BS maior e&longs;t BL, habebit BS ad <lb/>BO maiorem proportionem, quàm BL ad BA; quare maior erit <lb/>potentia in O &longs;u&longs;tinens pondus PQ, quàm potentia in A &longs;u&longs;ti<lb/>nens pondus CD. & hoc modo o&longs;tendetur' quò vectis BO ma<lb/>gis à vecte AB deor&longs;um tendens di&longs;tabit, &longs;emper maiorem ponderi <pb n="51" xlink:href="036/01/115.jpg"/>&longs;u&longs;tinendo requiri potentiam. </s> </p> <p id="id.2.1.100.1.0.0.0" type="margin"> <s id="id.2.1.100.1.1.1.0"><margin.target id="note162"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.100.1.1.2.0"><margin.target id="note163"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.100.1.1.3.0"><margin.target id="note164"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.100.1.1.4.0"><margin.target id="note165"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.100.1.1.5.0"><margin.target id="note166"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.101.1.0.0.0" type="main"> <s id="id.2.1.101.1.1.1.0">Hinc quoq; vt &longs;upra patet pontentiam in A ad potentiam in E e&longs; <lb/>&longs;e, vt BL ad BM; potentiamq; in A ad potentiam in O, vt BL <lb/>ad BS. atque potentiam in E ad potentiam in O, vt BM <lb/>ad BS. </s> </p> <p id="id.2.1.101.2.0.0.0" type="main"> <s id="id.2.1.101.2.1.1.0">Præterea &longs;i in B alia intelligatur potentia, ita vt duæ &longs;int poten<lb/>tiæ pondus &longs;u&longs;tinentes; minor erit potentia in B &longs;u&longs;tinens pon­<lb/>dus PQ vecte BO, quàm pondus CD vecte BA aduer&longs;o au<lb/>tem maior requiritur potentia in B ad &longs;u&longs;tinendum pondus FG ve<lb/>cte BE, quàm pondus CD vecte AB. ducta enim kN ip&longs;i EB <lb/>perpendicularis, erit EN ip&longs;i AL æqualis: quare EM ip&longs;a LA <lb/>maior erit. </s> <s id="id.2.1.101.2.1.2.0">ergo maiorem habebit proportionem EM ad E<emph type="italics"/>B<emph.end type="italics"/>, <arrow.to.target n="note167"></arrow.to.target><expan abbr="quàm"><lb/>quam</expan> LA ad A<emph type="italics"/>B<emph.end type="italics"/>; & LA ad A<emph type="italics"/>B<emph.end type="italics"/> maiorem, quàm SO ad O<emph type="italics"/>B<emph.end type="italics"/>; <arrow.to.target n="note168"></arrow.to.target><lb/>quæ &longs;unt proportiones potentiæ ad pondus. </s> </p> <p id="id.2.1.102.1.0.0.0" type="margin"> <s id="id.2.1.102.1.1.1.0"><margin.target id="note167"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.102.1.1.2.0"><margin.target id="note168"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.103.1.0.0.0" type="main"> <s id="id.2.1.103.1.1.1.0">Similiter o&longs;tendetur potentiam in <emph type="italics"/>B<emph.end type="italics"/> pondus vecte A<emph type="italics"/>B<emph.end type="italics"/> &longs;u&longs;ti­<lb/>nentem ad potentiam in eodem puncto <emph type="italics"/>B<emph.end type="italics"/> vecte E<emph type="italics"/>B<emph.end type="italics"/> &longs;u&longs;tinentem <lb/>e&longs;&longs;e, vt LA ad EM; ad potentiam autem in B pondus vecte O<emph type="italics"/>B<emph.end type="italics"/> <lb/>&longs;u&longs;tinentem ita e&longs;&longs;e, vt AL ad OS. quæ verò vectibus E<emph type="italics"/>B<emph.end type="italics"/> OB <lb/>&longs;u&longs;tinent inter &longs;e &longs;e e&longs;&longs;e, vt EM ad OS. </s> </p> <p id="id.2.1.103.2.0.0.0" type="main"> <s id="id.2.1.103.2.1.1.0">Deinde vt in iis, quæ &longs;uperius dicta &longs;unt, demon&longs;trabimus po­<lb/>tentiam in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in E eam habere proportionem, quam <arrow.to.target n="note169"></arrow.to.target><lb/>EM ad M<emph type="italics"/>B<emph.end type="italics"/>; & potentiam in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in A ita e&longs;&longs;e, vt AL ad <arrow.to.target n="note170"></arrow.to.target><lb/>L<emph type="italics"/>B<emph.end type="italics"/>, potentiamq; in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in O, vt OS ad S<emph type="italics"/>B.<emph.end type="italics"/></s> </p> <p id="id.2.1.104.1.0.0.0" type="margin"> <s id="id.2.1.104.1.1.1.0"><margin.target id="note169"></margin.target>3 <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.104.1.1.2.0"><margin.target id="note170"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.105.1.0.0.0" type="main"> <s id="id.2.1.105.1.1.1.0">Sit autem vectis A<emph type="italics"/>B<emph.end type="italics"/> <lb/>horizonti æquidi&longs;tans, <lb/>cuius fulcimentum <emph type="italics"/>B<emph.end type="italics"/>, <lb/>grauitati&longs;q; centrum H <lb/>ponderis AC &longs;it &longs;upra <lb/>vectem: moueaturq; ve<lb/>ctis in <emph type="italics"/>B<emph.end type="italics"/>E, ac pondus <lb/>in EF, potentiaq; in G. <lb/>&longs;imiliter vt &longs;upra o&longs;ten­<lb/>detur potentiam in G <lb/>pondus EF <expan abbr="&longs;ui&longs;tinen">sustinen</expan>­<lb/><figure id="id.036.01.115.1.jpg" xlink:href="036/01/115/1.jpg"></figure><lb/>tem minorem e&longs;&longs;e potentia in D pondus AC &longs;u&longs;tinente. </s> <s id="id.2.1.105.1.1.2.0">cùm <pb xlink:href="036/01/116.jpg"/>enim minor &longs;it BM ip&longs;a <lb/>BL, minorem habebit <lb/>proportionem MB ad <lb/>BG, quàm LB ad BD. <lb/>atq; hoc modo o&longs;ten­<lb/>detur, quò pondus ve­<lb/>cte magis eleuabitur, mi<lb/>norem &longs;emper. ad pon­<lb/>dus &longs;u&longs;tinendum requi­<lb/>ri potentiam. </s> <s id="id.2.1.105.1.1.4.0">Simili­<lb/>ter &longs;i moueatur vectis <lb/>in BO, potentiaq; &longs;u­<lb/><figure id="id.036.01.116.1.jpg" xlink:href="036/01/116/1.jpg"></figure><lb/>&longs;tinens in N, o&longs;tendetur potentiam in N maiorem e&longs;&longs;e potentia in <lb/>D. maiorem enim habet proportionem SB ad BN, quàm LB <lb/>ad BD. o&longs;tendetur etiam, quò magis pondus deprimetur; ma­<lb/>iorem &longs;emper (vt &longs;u&longs;tineatur) requiri potentiam. quod demon<lb/>&longs;trare oportebat. </s> <s id="id.2.1.105.1.1.5.0">quod demon<lb/>&longs;trare oportebat. </s> </p> <p id="id.2.1.105.2.0.0.0" type="main"> <s id="id.2.1.105.2.1.1.0">Hinc quoq; liquet potentias in GDN inter &longs;e &longs;e ita e&longs;&longs;e, vt <lb/>BM ad BL, atq; vt BL ad BS, deniq; vt BM ad BS. </s> </p> <p id="id.2.1.105.3.0.0.0" type="head"> <s id="id.2.1.105.3.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.105.4.0.0.0" type="main"> <s id="id.2.1.105.4.1.1.0">Ex his manife&longs;tum e&longs;t; &longs;i potentia vecte &longs;ur­<lb/>&longs;um moueat pondus, cuius centrum grauitatis <lb/>&longs;it &longs;upra vectem, quò magis pondus eleuabitur; <lb/>&longs;emper minorem potentiam requiri vt pondus <lb/>moueatur. </s> </p> <p id="id.2.1.105.5.0.0.0" type="main"> <s id="id.2.1.105.5.1.1.0">Vbi enim potentia pondus &longs;u&longs;tinens e&longs;t &longs;emper minor, erit <lb/>quoq; potentia ip&longs;um mouens &longs;emper minor. <pb n="52" xlink:href="036/01/117.jpg"/><figure id="id.036.01.117.1.jpg" xlink:href="036/01/117/1.jpg"></figure></s> </p> <p id="id.2.1.105.6.0.0.0" type="main"> <s id="id.2.1.105.6.1.1.0">Ex iis etiam demon&longs;trabitur, &longs;i centrum grauitatis eiu&longs;dem pon<lb/>deris, &longs;iue propinquius, &longs;iue remotius fuerit à vecte AB horizon­<lb/>ti æquidi&longs;tante, eandem potentiam in A pondus nihilominus <lb/>&longs;u&longs;tinere: vt &longs;i centrum grauitatis H ponderis BD longius ab&longs;it <lb/>à vecte BA, quàm centrum grauitatis N ponderis PV, dum­<lb/>modo ducta à puncto H perpendicularis HL horizonti, vectiq; <lb/>AB tran&longs;eat per N; &longs;itq; pondus PV ponderi BD æquale; <lb/>erit tùm pondus BD, tùm pondus PV, ac &longs;i ambo in L e&longs;­<lb/>&longs;ent appen&longs;a; atque &longs;unt æqualia, cùm loco vnius ponderis ac­<lb/>cipiantur, eadem igitur potentia in A &longs;u&longs;tinens pondus BD, <lb/>pondus quoq; PV &longs;u&longs;tinebit. </s> <s id="id.2.1.105.6.1.2.0">Vecte autem EF, quò centrum <lb/>grauitatis longius fuerit à vecte, eò facilius potentia idem pon­<lb/>dus &longs;u&longs;tinebit: vt &longs;i centrum grauitatis k ponderis FG longius <lb/>&longs;it à vecte EF, quàm centrum grauitatis X ponderis YZ; ita ta<lb/>men vt ducta à puncto k vecti FE perpendicularis tran&longs;eat per <lb/>X; &longs;itq; pondus FG ponderi YZ æquale; & à punctis kX ip­<lb/>&longs;orum horizontibus perpendiculares ducantur KM X9; erit C9 <lb/>maior CM; ac propterea pondus FG in vecte erit, ac &longs;i in M e&longs; <lb/>&longs;et appen&longs;um, & pondus YZ, ac &longs;i in 9 e&longs;&longs;et appen&longs;um. </s> <s id="id.2.1.105.6.1.3.0">quo<pb xlink:href="036/01/118.jpg"/><figure id="id.036.01.118.1.jpg" xlink:href="036/01/118/1.jpg"></figure><lb/><arrow.to.target n="note171"></arrow.to.target>niam autem maiorem habet proportionem C9 ad CE, quàm <lb/>CM ad CE, maior potentia in E &longs;u&longs;tinebit pondus YZ, quàm <lb/>FG. </s> <s id="id.2.1.105.6.1.3.0.a">In vecte autem QR è conuer&longs;o demon&longs;trabitur, &longs;cilicet <lb/>quò centrum grauitatis eiu&longs;dem ponderis &longs;it longius à vecte, eò <lb/>maiorem e&longs;&longs;e potentiam pondus &longs;u&longs;tinentem. </s> <s id="id.2.1.105.6.1.4.0">maior enim e&longs;t <lb/>CT, quàm CI; & ob id maiorem habebit proportionem CT <lb/>ad CR, quàm CI ad CR. </s> <s id="id.2.1.105.6.1.4.0.a">Similiter demon&longs;trabitur, &longs;i pondus <lb/>intra potentiam, & fulcimentum fuerit collocatum; vel poten­<lb/>tia intra fulcimentum, & pondus. </s> <s id="id.2.1.105.6.1.5.0">Quod idem etiam potentiæ <lb/>eueniet mouenti. </s> <s id="id.2.1.105.6.1.6.0">vbi enim minor potentia &longs;u&longs;tinet pondus, ibi <lb/>minor potentia mouebit; & vbi maior in &longs;u&longs;tinendo, ibi maior <lb/>quoq; in mouendo requiretur. </s> </p> <p id="id.2.1.106.1.0.0.0" type="margin"> <s id="id.2.1.106.1.1.1.0"><margin.target id="note171"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.107.1.0.0.0" type="head"> <s id="id.2.1.107.1.1.1.0">PROPOSITIO VIIII. </s> </p> <p id="id.2.1.107.2.0.0.0" type="main"> <s id="id.2.1.107.2.1.1.0">Potentia pondus &longs;u&longs;tinens infra vectem ho­<lb/>rizonti æquidi&longs;tantem ip&longs;ius centrum grauitatis <pb n="53" xlink:href="036/01/119.jpg"/>habens, quò magis ab hoc &longs;itu vecte pondus ele<lb/>uabitur maiori &longs;emper potentia, vt &longs;u&longs;tineatur, <lb/>egebit. </s> <s id="id.2.1.107.2.1.2.0">&longs;i verò deprimetur, minori. <figure id="id.036.01.119.1.jpg" xlink:href="036/01/119/1.jpg"></figure></s> </p> <p id="id.2.1.107.3.0.0.0" type="main"> <s id="id.2.1.107.3.1.1.0">Sit vectis AB horizonti æquidi&longs;tans, cuius fulcimentum C; <lb/>&longs;itq; pondus AD, cuius centrum grauitatis L &longs;it infra vectem; <lb/>&longs;itq; potentia in B &longs;u&longs;tinens pondus AD: moueatur deinde ve­<lb/>ctis in FG, & pondus in FH. </s> <s id="id.2.1.107.3.1.1.0.a">Dico primum maiorem requiri <lb/>potentiam in G ad &longs;u&longs;tinendum pondus FH vecte FG, quàm <lb/>&longs;it potentia in B pondere exi&longs;tente AD vecte autem AB. </s> <s id="id.2.1.107.3.1.1.0.b">&longs;it M <lb/>grauitatis centrum ponderis FH, & à punctis LM ip&longs;orum ho­<lb/>rizontibus perpendiculares ducantur Lk MN: ip&longs;i verò FG per­<lb/>pendicularis ducatur MS, quæ æqualis erit LK, & CK ip&longs;i CS <lb/>erit etiam æqualis. </s> <s id="id.2.1.107.3.1.2.0">Quoniam igitur CN maior e&longs;t Ck, habe­<lb/>bit <arrow.to.target n="note172"></arrow.to.target>NC ad CG maiorem proportionem, quàm Ck ad CB; po<arrow.to.target n="note173"></arrow.to.target><lb/>tentia uerò in B ad pondus AD eandem habet, quam kC ad CB: <arrow.to.target n="note174"></arrow.to.target><lb/>& vt potentia in G ad pondus FH, ita e&longs;t NC ad CG; ergo <lb/>maiorem habebit proportionem potentia in G ad pondus FH, <lb/>quàm potentia in B ad pondus AD. </s> <s id="id.2.1.107.3.1.2.0.a">maior igitur e&longs;t potentia <arrow.to.target n="note175"></arrow.to.target><lb/>in G ip&longs;a potentia in B. &longs;i verò vectis &longs;it in OP, & pondus in <lb/>OQ; erit potentia in B maior, quàm in P. eodem enim mo­<lb/>do o&longs;tendetur CR minorem e&longs;&longs;e Ck, & CR ad CP minorem <arrow.to.target n="note176"></arrow.to.target><pb xlink:href="036/01/120.jpg"/><figure id="id.036.01.120.1.jpg" xlink:href="036/01/120/1.jpg"></figure><lb/>habere proportionem, quàm Ck ad CB; & ob id potentiam in <lb/>B maiorem e&longs;&longs;e potentia in P. & hoc modo o&longs;tendetur, quò ma­<lb/>gis à &longs;itu AB pondus eleuabitur, &longs;emper maiorem potentiam ad <lb/>pondus &longs;u&longs;tinendum requiri. è </s> <s id="id.2.1.107.3.1.3.0">è contra verò &longs;i deprimetur. </s> <s id="id.2.1.107.3.1.4.0">quod <lb/>demon&longs;trare oportebat. </s> </p> <p id="id.2.1.108.1.0.0.0" type="margin"> <s id="id.2.1.108.1.1.1.0"><margin.target id="note172"></margin.target>7 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.108.1.1.2.0"><margin.target id="note173"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.108.1.1.3.0"><margin.target id="note174"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.108.1.1.4.0"><margin.target id="note175"></margin.target>10 <emph type="italics"/>Quinti<emph.end type="italics"/></s> <s id="id.2.1.108.1.1.5.0"><margin.target id="note176"></margin.target>7 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.109.1.0.0.0" type="main"> <s id="id.2.1.109.1.1.1.0">Hinc quoq; facilè elici pote&longs;t potentias in PBG inter &longs;e &longs;e ita <lb/>e&longs;&longs;e, vt CR ad Ck; & vt Ck ad CN; atq; vt CN ad CR. <lb/><figure id="id.036.01.120.2.jpg" xlink:href="036/01/120/2.jpg"></figure></s> </p> <p id="id.2.1.109.2.0.0.0" type="main"> <s id="id.2.1.109.2.1.1.0">Sit deinde vectis AB horizonti æquidi&longs;tans, cuius fulcimentum <lb/>B; pondu&longs;q; CD habeat centrum grauitatis O infra vectem; &longs;itq; <lb/>potentia in A &longs;u&longs;tinens pondus CD. </s> <s id="id.2.1.109.2.1.1.0.a">Moueatur deinde vectis in <pb n="54" xlink:href="036/01/121.jpg"/>BE BF, pondu&longs;q; transferatur in GH kL. </s> <s id="id.2.1.109.2.1.1.0.b">Dico maiorem re­<lb/>quiri potentiam in E, vt pondus &longs;u&longs;tineatur, quàm in A; & ma<lb/>iorem in A, quàm in F. ducantur à centris grauitatum horizon­<lb/>tibus perpendiculares NM OP QR, quæ ex parte NOQ <lb/>protractæ in centrum mundi conuenient. </s> <s id="id.2.1.109.2.1.2.0">&longs;imiliter vt &longs;upra o&longs;ten <lb/>detur BM <expan abbr="maior&etilde;">maiorem</expan> e&longs;&longs;e BP, & <emph type="italics"/>B<emph.end type="italics"/>P maiorem BR; & BM ad BE ma­<lb/>iorem <arrow.to.target n="note177"></arrow.to.target>habere proportionem, <expan abbr="qaàm">quam</expan> BP ad BA; & BP ad BA ma­<lb/>iorem, quàm BR ad BF: & propter hoc potentiam in E maio­<lb/>rem e&longs;&longs;e potentia in A; & potentiam in A maiorem potentia in <lb/>F. & quò vectis magis à &longs;itu AB eleuabitur, &longs;emper o&longs;tendetur, <lb/>maiorem requiri potentiam ponderi &longs;u&longs;tinendo. &longs;i verò depri­<lb/>metur, minorem. </s> </p> <p id="id.2.1.110.1.0.0.0" type="margin"> <s id="id.2.1.110.1.1.1.0"><margin.target id="note177"></margin.target>7 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.111.1.0.0.0" type="main"> <s id="id.2.1.111.1.1.1.0">Hinc patet etiam potentias in EAF inter &longs;e &longs;e ita e&longs;&longs;e, vt BM ad <lb/>BP; & vt BP ad BR; ac vt BM ad BR. </s> </p> <p id="id.2.1.111.2.0.0.0" type="main"> <s id="id.2.1.111.2.1.1.0">In&longs;uper &longs;i in B altera &longs;it potentia, ita vt duæ &longs;int potentiæ pondus <lb/>&longs;u&longs;tinentes, maiore opus e&longs;t potentia in B pondus kL &longs;u&longs;tinente <lb/>vecte BF, quàm pondus CD vecte AB. & adhuc maiore vecte <lb/>AB, quàm vecte BE. </s> <s id="id.2.1.111.2.1.1.0.a">maiorem enim habet proportionem RF <lb/>ad FB, quàm PA ad AB; & PA ad AB maiorem habet, quàm <lb/>EM ad EB. </s> </p> <p id="id.2.1.111.3.0.0.0" type="main"> <s id="id.2.1.111.3.1.1.0">Similiterq; o&longs;tendetur potentias in B pondus vectibus &longs;u&longs;tinen­<lb/>tes inter &longs;e &longs;e ita e&longs;&longs;e, vt EM ad AP; & ut <lb/>AP ad FR; atque ut <lb/>EM ad FR. </s> </p> <p id="id.2.1.111.4.0.0.0" type="main"> <s id="id.2.1.111.4.1.1.0">Præterea potentia in B ad potentiam in F ita erit, ut RF ad <arrow.to.target n="note178"></arrow.to.target><lb/>RB; & potentia in B ad potentiam in A, ut PA ad PB, & po­<lb/>tentia <arrow.to.target n="note179"></arrow.to.target>in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in E, ut EM ad M<emph type="italics"/>B.<emph.end type="italics"/></s> </p> <p id="id.2.1.112.1.0.0.0" type="margin"> <s id="id.2.1.112.1.1.1.0"><margin.target id="note178"></margin.target>3 <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.112.1.1.2.0"><margin.target id="note179"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/122.jpg"/> <p id="id.2.1.113.1.0.0.0" type="main"> <s id="id.2.1.113.1.2.1.0">Sit autem vectis <lb/>AB horizonti æqui­<lb/>di&longs;tans, cuius fulci­<lb/>mentum B; & pon­<lb/>dus AC, cuius cen­<lb/>trum grauitatis &longs;it in­<lb/>fra vectem: &longs;itq; po­<lb/>tentia in D pondus <lb/><expan abbr="&longs;u&longs;tin&etilde;s">&longs;u&longs;tinens</expan>; moueaturq; <lb/>vectis in BE BF, & <lb/>potentia in GH: &longs;i­<lb/>militer o&longs;tendetur po<lb/><figure id="id.036.01.122.1.jpg" xlink:href="036/01/122/1.jpg"></figure><lb/>tentiam in G maiorem e&longs;&longs;e debere potentia in D; & potentiam in <lb/>D maiorem potentia in H. </s> <s id="id.2.1.113.1.2.1.0.a">maiorem enim proportionem habet <lb/>KB ad BG, quàm BL ad BD; & BL ad BD maiorem, quàm <lb/>MB ad BH. </s> <s id="id.2.1.113.1.2.1.0.b">& hoc modo o&longs;tendetur, quò vectis magis à &longs;itu <lb/>AB eleuabitur, adhuc &longs;emper maiorem e&longs;&longs;e debere potentiam pon<lb/>dus &longs;u&longs;tinentem. </s> <s id="id.2.1.113.1.2.2.0">quò autem magis deprimetur; minorem. </s> <s id="id.2.1.113.1.2.3.0">quod <lb/>demon&longs;trare oportebat. </s> </p> <p id="id.2.1.113.2.0.0.0" type="main"> <s id="id.2.1.113.2.1.1.0">Similiter in his potentiæ in GDH inter &longs;e &longs;e ita. erunt, vt BK <lb/>ad BL; & vt BL ad BM; deniq; vt Bk ad BM. </s> </p> <p id="id.2.1.113.3.0.0.0" type="head"> <s id="id.2.1.113.3.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.113.4.0.0.0" type="main"> <s id="id.2.1.113.4.1.1.0">Ex his patet etiam, &longs;i potentia vecte &longs;ur&longs;um <lb/>moueat pondus, cuius centrum grauitatis &longs;it in­<lb/>fra vectem; quò magis pondus eleuabitur, &longs;em<lb/>per maiorem requiri potentiam, vt pondus mo<lb/>ueatur. </s> </p> <p id="id.2.1.113.5.0.0.0" type="main"> <s id="id.2.1.113.5.1.1.0">Nam &longs;i potentia pondus &longs;u&longs;tinens &longs;emper e&longs;t maior: erit quoq; <lb/>potentia mouens &longs;emper maior. <pb n="55" xlink:href="036/01/123.jpg"/><figure id="id.036.01.123.1.jpg" xlink:href="036/01/123/1.jpg"></figure></s> </p> <p id="id.2.1.113.6.0.0.0" type="main"> <s id="id.2.1.113.6.1.1.0">Et his etiam facilè elicietur, &longs;i centrum grauitatis eiu&longs;dem pon­<lb/>deris, &longs;iue propius, &longs;iue remotius fuerit à vecte AB horizonti æ­<lb/>quidi&longs;tante; eandem potentiam in B pondus &longs;u&longs;tinere. </s> <s id="id.2.1.113.6.1.2.0">vt &longs;i cen­<lb/>trum grauitatis L ponderis AD &longs;it remotius à vecte BA, quàm <lb/>centrum grauitatis N ponderis PV; dummodo ducta à puncto L <lb/>perpendicularis LK horizonti, vectiq; AB tran&longs;eat per N: &longs;imili­<lb/>ter vt in præcedenti o&longs;tendetur, eandem potentiam in B, & pondus <lb/>AD, & pondus PV &longs;u&longs;tinere. </s> <s id="id.2.1.113.6.1.3.0">In vecte auté EF, quò <expan abbr="centrũ">centrum</expan> grauitatis <lb/>longius aberit à vecte, eò maiori opus erit potentia ponderi &longs;u&longs;ti­<lb/>nendo. </s> <s id="id.2.1.113.6.1.4.0">vt centrum grauitatis M ponderis FH remotius &longs;it à ue<lb/>cte EF, quàm S centrum grauitatis ponderis XZ; ducantur à pun<lb/>ctis MS horizontibus perpendiculares MI SG; erit CI maior <lb/>CG: ac propterea maior e&longs;&longs;e debet potentia in E pondus FH &longs;u<lb/>&longs;tinens, quàm pondus XZ. </s> <s id="id.2.1.113.6.1.4.0.a">Contra uerò in uecte OR o&longs;tende<lb/>tur, quò &longs;cilicet centrum grauitatis eiu&longs;dem ponderis longius ab <lb/>&longs;it à uecte, à minori potentia pondus &longs;u&longs;tineri. </s> <s id="id.2.1.113.6.1.5.0">minor enim e&longs;t <lb/>CY, quàm CT. </s> <s id="id.2.1.113.6.1.5.0.a">Simili quoq; modo demon&longs;trabitur, &longs;i pondus <lb/>&longs;it intra potentiam, & fulcimentum; uel potentia intra fulci­<lb/>mentum, & pondus. </s> <s id="id.2.1.113.6.1.6.0">Quod idem potentiæ eueniet mouenti: <pb xlink:href="036/01/124.jpg"/>vbi enim minor potentia &longs;u&longs;tinet pondus, ibi minor potentia mo­<lb/>uebit. </s> <s id="id.2.1.113.6.1.7.0">& vbi maior potentia in &longs;u&longs;tinendo; ibi quoq; maior in mo<lb/>uendo aderit. </s> </p> <p id="id.2.1.113.7.0.0.0" type="head"> <s id="id.2.1.113.7.1.1.0">PROPOSITIO X. </s> </p> <p id="id.2.1.113.8.0.0.0" type="main"> <s id="id.2.1.113.8.1.1.0">Potentia pondus &longs;u&longs;tinens in ip&longs;o vecte cen­<lb/>trum grauitatis habens, quomodocunq; vecte <lb/>transferatur pondus; eadem &longs;emper, vt &longs;u&longs;tinea­<lb/>tur, potentia opus erit. <figure id="id.036.01.124.1.jpg" xlink:href="036/01/124/1.jpg"></figure></s> </p> <p id="id.2.1.113.9.0.0.0" type="main"> <s id="id.2.1.113.9.1.1.0">Sit vectis AB horizonti æquidi&longs;tàns, cuius fulcimentum C. <lb/>E verò centrum grauitatis ponderis in ip&longs;o &longs;it vecte. </s> <s id="id.2.1.113.9.1.2.0">Moueatur <lb/>deinde uectis in FG, Hk; & centrum grauitatis in LM. </s> <s id="id.2.1.113.9.1.2.0.a">dico ean<lb/>dem potentiam in kBG idemmet &longs;emper &longs;u&longs;tinere pondus. </s> <s id="id.2.1.113.9.1.3.0"><lb/>Quoniam enim pondus in uecte AB perinde &longs;e habet, ac &longs;i e&longs;&longs;et <lb/><arrow.to.target n="note180"></arrow.to.target>appen&longs;um in E; & in uecte GF, ac &longs;i e&longs;&longs;et appen&longs;um in L; & in <lb/>uecte Hk. </s> <s id="id.2.1.113.9.1.4.0">ac &longs;i in M e&longs;&longs;et appen&longs;um; di&longs;tantiæ uerò CL CE <lb/>CM &longs;unt inter &longs;e &longs;e æquales; nec non CK CB CG inter &longs;e æ­<lb/>quales; erit potentia in B ad pondus, ut CE ad CB; atque poten<pb n="56" xlink:href="036/01/125.jpg"/>tia in k ad pondus, ut CM ad Ck; & potentia in G ad pondus, <lb/>vt CL ad CG. eadem igitur potentia in k<emph type="italics"/>B<emph.end type="italics"/>G idem translatum <lb/>pondus &longs;u&longs;tinebit. </s> <s id="id.2.1.113.9.1.5.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.114.1.0.0.0" type="margin"> <s id="id.2.1.114.1.1.1.0"><margin.target id="note180"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.115.1.0.0.0" type="main"> <s id="id.2.1.115.1.1.1.0">Similiter o&longs;tendetur, &longs;i pondus e&longs;&longs;et intra potentiam, & fulci­<lb/>mentum; vel potentia inter fulcimentum, & pondus. </s> <s id="id.2.1.115.1.1.2.0">quod idem <lb/>potentiæ mouenti eueniet. </s> </p> <p id="id.2.1.115.2.0.0.0" type="head"> <s id="id.2.1.115.2.1.1.0">PROPOSITIO XI. </s> </p> <p id="id.2.1.115.3.0.0.0" type="main"> <s id="id.2.1.115.3.1.1.0">Si vectis di&longs;tantia inter fulcimentum, & poten<lb/>tiam ad di&longs;tantiam fulcimento, punctoq;, vbi <lb/>à centro grauitatis ponderis horizonti ducta <lb/>perpendicularis vectem &longs;ecat, interiectam ma­<lb/>iorem habuerit proportionem, quàm pondus <lb/>ad potentiam; pondus vtiq; à potentia moue­<lb/>bitur. </s> </p> <p id="id.2.1.115.4.0.0.0" type="main"> <s id="id.2.1.115.4.1.1.0">Sit véctis AB, ex <lb/>punctoq; A &longs;u&longs;penda<lb/>tur pondus C; hoc e&longs;t <lb/>punctum A &longs;emper &longs;it <lb/>punctum, vbi perpen<lb/>dicularis à grauitatis <lb/>centro ponderis du­<lb/>cta vectem &longs;ecat; &longs;itq; <lb/><figure id="id.036.01.125.1.jpg" xlink:href="036/01/125/1.jpg"></figure><lb/>potentia in B, ac fulcimentum &longs;it D; & DB ad DA maiorem <lb/>habeat proportionem, quàm pondus C ad potentiam in B. </s> <s id="id.2.1.115.4.1.1.0.a">Di­<lb/>co pondus Cà potentia in B moueri. </s> <s id="id.2.1.115.4.1.2.0">fiat vt BD ad DA, ita <lb/>pondus E ad potentiam in B; atq; pondus E quoq; appendatur <lb/>in A: patet potentiam in B æqueponderare ip&longs;i E; hoc e&longs;t pon­<lb/>dus <arrow.to.target n="note181"></arrow.to.target>E &longs;u&longs;tinere. </s> <s id="id.2.1.115.4.1.3.0">& quoniam BD ad DA maiorem habet pro­<lb/>portionem, quàm Cad potentiam in B; & vt BD ad DA, ita <pb xlink:href="036/01/126.jpg"/>e&longs;t pondus E ad po­<lb/>tentiam: igitur E ad <lb/>potentiam maiorem <lb/>habebit proportio­<lb/>nem, quàm pondus <lb/>C ad eandem poten­<lb/><arrow.to.target n="note182"></arrow.to.target>tiam. </s> <s id="id.2.1.115.4.1.4.0">quare pondus <lb/>E maius erit ponde­<lb/><figure id="id.036.01.126.1.jpg" xlink:href="036/01/126/1.jpg"></figure><lb/>re C. & cùm potentia ip&longs;a E æqueponderet, potentia igitur ip&longs;i <lb/>C non æqueponderabit, &longs;ed &longs;ua ui deor&longs;um verget. </s> <s id="id.2.1.115.4.1.5.0">pondus igitur <lb/>C à potentia in B mouebitur vecte AB, cuius fulcimentum <lb/>e&longs;t D. </s> </p> <p id="id.2.1.116.1.0.0.0" type="margin"> <s id="id.2.1.116.1.1.1.0"><margin.target id="note181"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.116.1.1.2.0"><margin.target id="note182"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.117.1.0.0.0" type="main"> <s id="id.2.1.117.1.1.1.0">Si verò &longs;it vectis AB, & <lb/>fulcimentum A, pondu&longs;q; C <lb/>in D appen&longs;um, & potentia <lb/>in B; & BA ad AD maio­<lb/>rem habeat proportionem, <lb/>quàm pondus C ad poten­<lb/>tiam in B. </s> <s id="id.2.1.117.1.1.1.0.a">dico pondus C à <lb/><figure id="id.036.01.126.2.jpg" xlink:href="036/01/126/2.jpg"></figure><lb/>potentia in B moueri. </s> <s id="id.2.1.117.1.1.2.0">fiat vt BA ad AD; ita pondus E ad poten<lb/><arrow.to.target n="note183"></arrow.to.target>tiam in B: & &longs;i E appendatur in D, potentia in B pondus E &longs;u&longs;ti<lb/>nebit. </s> <s id="id.2.1.117.1.1.3.0">&longs;ed cùm BA ad AD maiorem habeat proportionem, <lb/>quàm pondus C ad potentiam in B; & vt BA ad AD, ita e&longs;t <lb/>pondus E ad potentiam in B: pondus igitur E ad potentiam, <lb/>quæ e&longs;t in B, maiorem habebit proportionem, quàm pondus C <lb/><arrow.to.target n="note184"></arrow.to.target>ad eandem potentiam. </s> <s id="id.2.1.117.1.1.4.0">& ideo pondus E maius erit pondere C. <lb/>potentia verò in B &longs;u&longs;tinet pondus E; ergo potentia in B pondus <lb/>C minus pondere E in D appen&longs;um mouebit vecte AB, cuius fulci <lb/>mentum e&longs;t A. </s> </p> <p id="id.2.1.118.1.0.0.0" type="margin"> <s id="id.2.1.118.1.1.1.0"><margin.target id="note183"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.118.1.1.2.0"><margin.target id="note184"></margin.target>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <pb n="57" xlink:href="036/01/127.jpg"/> <p id="id.2.1.119.1.0.0.0" type="main"> <s id="id.2.1.119.1.2.1.0">Sit rur&longs;us vectis <lb/>AB, cuius fulcimen <lb/><expan abbr="tũ">tum</expan> A; & pondus C in <lb/>B &longs;it appen&longs;um; &longs;itq; <lb/>potentia in D: & <lb/>DA ad AB maio­<lb/>rem habeat propor­<lb/>tionem, quàm pon­<lb/><figure id="id.036.01.127.1.jpg" xlink:href="036/01/127/1.jpg"></figure><lb/>dus C ad potentiam, quæ e&longs;t in D. </s> <s id="id.2.1.119.1.2.1.0.a">dico pondus C à potentia <lb/>in D moueri. </s> <s id="id.2.1.119.1.2.2.0">fiat vt DA ad AB, ita pondus E ad potentiam in <lb/>D; & &longs;it pondus E ex puncto B &longs;u&longs;pen&longs;um: potentia in D pondus <lb/>E &longs;u&longs;tinebit. </s> <s id="id.2.1.119.1.2.3.0">&longs;ed DA ad AB maiorem habet proportionem, <lb/>quàm C ad potentiam in D; & vt DA ad AB, ita e&longs;t pondus E <lb/>ad potentiam in D; pondus igitur E ad potentiam, quæ e&longs;t in D, <lb/>maiorem habebit proportionem, quàm pondus C ad eandem po<lb/>tentiam. </s> <s id="id.2.1.119.1.2.4.0">quare pondus E maius e&longs;t pondere C. & cùm poten­<lb/>tia in D pondus E &longs;u&longs;tineat, potentia igitur in D pondus C in B <lb/>appen&longs;um vecte AB, cuius fulcimentum e&longs;t A, mouebit. </s> <s id="id.2.1.119.1.2.5.0">quod <lb/>demon&longs;trare oportebat. </s> </p> <p id="id.2.1.119.2.0.0.0" type="head"> <s id="id.2.1.119.2.1.1.0">ALITER. </s> </p> <p id="id.2.1.119.3.0.0.0" type="main"> <s id="id.2.1.119.3.1.1.0">Sit vectis AB, & <lb/>pondus C in A ap­<lb/>pen&longs;um & poten­<lb/>tia in B; &longs;it qué fulci­<lb/>mentum D: & DB <lb/><figure id="id.036.01.127.2.jpg" xlink:href="036/01/127/2.jpg"></figure><lb/>ad DA maiorem habeat proportionem, quàm pondus C ad po<lb/>tentiam in B. </s> <s id="id.2.1.119.3.1.1.0.a">dico pondus C à potentia in B moueri. </s> <s id="id.2.1.119.3.1.2.0">fiat BE ad <lb/>EA, vt pondus C ad potentiam, erit punctum E inter BD. </s> <s id="id.2.1.119.3.1.2.0.a">opor<lb/>tet enim BE ad EA minorem habere proportionem, quàm DB <lb/>ad DA, & ideo BE minor erit BD. </s> <s id="id.2.1.119.3.1.2.0.b">& quoniam potentia in B &longs;u<arrow.to.target n="note185"></arrow.to.target><lb/>&longs;tinet pondus C in A appen&longs;um uecte AB, cuius <expan abbr="fulcimentũ">fulcimentum</expan> E; minor <lb/>igitur potentia in B, quàm data, idem pondus &longs;u&longs;tinebit fulcimen<lb/>to D. data ergo potentia in B pondus C mouebit uecte AB, cuius <lb/>fulcimentum e&longs;t D. <pb xlink:href="036/01/128.jpg"/></s> </p> <p id="id.2.1.120.1.0.0.0" type="margin"> <s id="id.2.1.120.1.1.1.0"><margin.target id="note185"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.121.1.0.0.0" type="main"> <s id="id.2.1.121.1.1.1.0">Sit deinde vectis AB, & fulci <lb/>mentum A, & pondus C in D <lb/>appen&longs;um, &longs;itq; potentia in B; & <lb/>AB ad AD maiorem habeat pro­<lb/>portionem, quàm pondus C ad <lb/>potentiam in B. </s> <s id="id.2.1.121.1.1.1.0.a">dico pondus C <lb/><figure id="id.036.01.128.1.jpg" xlink:href="036/01/128/1.jpg"></figure><expan abbr="à"><lb/>a</expan> potentia in B moueri. </s> <s id="id.2.1.121.1.1.2.0">Fiat AB ad AE, vt pondus C ad poten<lb/><arrow.to.target n="note186"></arrow.to.target>tiam; erit &longs;imiliter punctum E inter BD. nece&longs;&longs;e e&longs;t enim AE <lb/>maiorem e&longs;&longs;e AD. & &longs;i pondus C e&longs;&longs;et in E appen&longs;um, potentia <lb/><arrow.to.target n="note187"></arrow.to.target>in B illud &longs;u&longs;tineret. </s> <s id="id.2.1.121.1.1.3.0">minor autem potentia in B, quàm data, &longs;u&longs;ti­<lb/><arrow.to.target n="note188"></arrow.to.target>net pondus C in D appen&longs;um; data ergo potentia in B pondus C in <lb/><arrow.to.target n="note189"></arrow.to.target>D appen&longs;um vecte AB, cuius fulcimentum e&longs;t A, mouebit. </s> </p> <p id="id.2.1.122.1.0.0.0" type="margin"> <s id="id.2.1.122.1.1.1.0"><margin.target id="note186"></margin.target>8 <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.122.1.1.2.0"><margin.target id="note187"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.122.1.1.3.0"><margin.target id="note188"></margin.target>1 <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.122.1.1.4.0"><margin.target id="note189"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.123.1.0.0.0" type="main"> <s id="id.2.1.123.1.1.1.0">Sit rur&longs;us vectis AB, cu<lb/>ius fulcimentum A, & pon<lb/>dus C in B &longs;it appen&longs;um; <lb/>&longs;itq; potentia in D; & DA <lb/>ad AB maiorem habeat <lb/><figure id="id.036.01.128.2.jpg" xlink:href="036/01/128/2.jpg"></figure><lb/>proportionem, quàm pondus C ad potentiam in D. </s> <s id="id.2.1.123.1.1.1.0.a">dico pon­<lb/>dus C à potentia in D moueri. </s> <s id="id.2.1.123.1.1.2.0">fiat vt pondus C ad potentiam, <lb/><arrow.to.target n="note190"></arrow.to.target>ita DA ad AE; erit AE maior AB; cùm maior &longs;it proportio <lb/>DA ad AB, quàm DA ad AE. & &longs;i pondus C appendatur in <lb/><arrow.to.target n="note191"></arrow.to.target>E, patet potentiam in D &longs;u&longs;tinere pondus C in E appen&longs;um. </s> <s id="id.2.1.123.1.1.3.0">mi­<lb/><arrow.to.target n="note192"></arrow.to.target>nor autem potentia, quàm data, &longs;u&longs;tinet idem pondus C in B; <lb/><arrow.to.target n="note193"></arrow.to.target>data igitur potentia in D pondus C in B appen&longs;um mouebit ve­<lb/>cte AB, cuius fulcimentum e&longs;t A. quod oportebat demon­<lb/>&longs;trare. </s> </p> <p id="id.2.1.124.1.0.0.0" type="margin"> <s id="id.2.1.124.1.1.1.0"><margin.target id="note190"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.124.1.1.2.0"><margin.target id="note191"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.124.1.1.3.0"><margin.target id="note192"></margin.target>1 <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.124.1.1.4.0"><margin.target id="note193"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.125.1.0.0.0" type="head"> <s id="id.2.1.125.1.1.1.0">PROPOSITIO XII. </s> </p> <p type="head"> <s id="id.2.1.125.1.3.1.0">PROBLEMA. </s> </p> <p id="id.2.1.125.2.0.0.0" type="main"> <s id="id.2.1.125.2.1.1.0">Datum pondus à data potentia dato vecte <lb/>moueri. <pb n="58" xlink:href="036/01/129.jpg"/><figure id="id.036.01.129.1.jpg" xlink:href="036/01/129/1.jpg"></figure></s> </p> <p id="id.2.1.125.3.0.0.0" type="main"> <s id="id.2.1.125.3.1.1.0">Sit pondus A vt centum, potentia verò mouens &longs;it vt decem; <lb/>&longs;itq; datus vectis BC. </s> <s id="id.2.1.125.3.1.1.0.a">oportet potentiam, quæ e&longs;t decem pondus <lb/>A centum vecte BC mouere. </s> <s id="id.2.1.125.3.1.2.0">Diuidatur BC in D, ita vt CD <lb/>ad DB eandem habeat proportionem, quàm habet centum ad <lb/>decem, hoc e&longs;t decem ad vnum; etenim &longs;i D fieret fulcimentum, <lb/>con&longs;tat potentiam vt decem in C æqueponderare ponderi A in B <arrow.to.target n="note194"></arrow.to.target><lb/>appen&longs;o: hoc e&longs;t pondus A &longs;u&longs;tinere. </s> <s id="id.2.1.125.3.1.3.0">accipiatur inter BD quod <lb/>uis punctum E, & fiat E fulcimentum. </s> <s id="id.2.1.125.3.1.4.0">Quoniam enim maior <arrow.to.target n="note195"></arrow.to.target><lb/>e&longs;t proportio CE ad EB, quàm CD ad DB; maiorem habebit <lb/>proportionem CE ad EB, quàm pondus A ad potentiam decem <lb/>in C: potentia igitur decem in C pondus A centum in B appen­<lb/>&longs;um vecte BC, cuius fulcimentum &longs;it E, mouebit. <arrow.to.target n="note196"></arrow.to.target></s> </p> <p id="id.2.1.125.4.0.0.0" type="main"> <s id="id.2.1.125.4.1.1.0">Si verò &longs;it vectis <lb/>BC, & fulcimen­<lb/>tum B. diuidatur CB <lb/>in D, ita vt CB ad <lb/>BD eandem habeat <lb/>proportionem, <expan abbr="quã">quam</expan><lb/><figure id="id.036.01.129.2.jpg" xlink:href="036/01/129/2.jpg"></figure><lb/> habet centum ad decem: & &longs;i pondus A in D &longs;u&longs;pendatur, & po­<lb/>tentia in C, potentia vt decem in C pondus A in D appen&longs;um &longs;u<arrow.to.target n="note197"></arrow.to.target><lb/>&longs;tinebit. </s> <s id="id.2.1.125.4.1.2.0">accipiatur inter DB quoduis punctum E, ponaturq; pon<lb/>dus A in E; & cùm &longs;it maior proportio CB ad BE, quàm <arrow.to.target n="note198"></arrow.to.target><lb/>BC ad BD; maiorem habebit proportionem CB ad BE, quàm <lb/>pondus A centum ad potentiam decem. </s> <s id="id.2.1.125.4.1.3.0">potentia igitur decem <arrow.to.target n="note199"></arrow.to.target><lb/>in C pondus A centum in E appen&longs;um mouebit vecte BC, cu<lb/>ius fulcimentum e&longs;t B. quod facere oportebat. </s> </p> <p id="id.2.1.126.1.0.0.0" type="margin"> <s id="id.2.1.126.1.1.1.0"><margin.target id="note194"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.126.1.1.2.0"><margin.target id="note195"></margin.target><emph type="italics"/>Lemma huius.<emph.end type="italics"/></s> <s id="id.2.1.126.1.1.3.0"><margin.target id="note196"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.126.1.1.4.0"><margin.target id="note197"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.126.1.1.5.0"><margin.target id="note198"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.126.1.1.6.0"><margin.target id="note199"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/130.jpg"/> <p id="id.2.1.127.1.0.0.0" type="main"> <s id="id.2.1.127.1.2.1.0">Hoc autem fieri non po­<lb/>te&longs;t exi&longs;tente vecte BC, cuius <lb/>fulcimentum &longs;it B, & pondus <lb/>A centum in C appen&longs;um: po<lb/>natur enim potentia &longs;u&longs;tinens <lb/>pondus A vtcunq; inter BC, <lb/><arrow.to.target n="note200"></arrow.to.target>vt in D, &longs;emper potentia ma<lb/><arrow.to.target n="note201"></arrow.to.target>ior erit pondere A. quare opor<lb/><figure id="id.036.01.130.1.jpg" xlink:href="036/01/130/1.jpg"></figure><lb/>tet datam potentiam maiorem e&longs;&longs;e pondere A. &longs;it igitur poten­<lb/>tia data vt centum quinquaginta. </s> <s id="id.2.1.127.1.2.2.0">diuidatur BC in D, ita vt CB <lb/>ad BD &longs;it, vt centum quinquaginta ad centum; hoc e&longs;t tria ad duo: <lb/><arrow.to.target n="note202"></arrow.to.target>& &longs;i ponatur potentia in D, patet potentiam in D &longs;u&longs;tinere pon­<lb/>dus A in C <expan abbr="appep&longs;um">appensum</expan>. </s> <s id="id.2.1.127.1.2.3.0">accipiatur itaq; inter DC quoduis pun­<lb/><arrow.to.target n="note203"></arrow.to.target>ctum E, ponaturq; potentia mouens in E; & cùm maior &longs;it pro­<lb/>portio EB ad BC, quàm DB ad BC; habebit EB ad BC maio<lb/>rem proportionem, quàm pondus A ad potentiam in E. </s> <s id="id.2.1.127.1.2.3.0.a">poten<lb/><arrow.to.target n="note204"></arrow.to.target>tia igitur vt centum quinquaginta in E pondus A centum in C <lb/>appen&longs;um vecte BC, cuius fulcimentum e&longs;t B, mouebit. </s> <s id="id.2.1.127.1.2.4.0">quod <lb/>facere oportebat. </s> </p> <p id="id.2.1.128.1.0.0.0" type="margin"> <s id="id.2.1.128.1.1.1.0"><margin.target id="note200"></margin.target>2 <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.128.1.1.2.0"><margin.target id="note201"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.128.1.1.3.0"><margin.target id="note202"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.128.1.1.4.0"><margin.target id="note203"></margin.target>8 <emph type="italics"/>Quinti.<emph.end type="italics"/></s> <s id="id.2.1.128.1.1.5.0"><margin.target id="note204"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.129.1.0.0.0" type="head"> <s id="id.2.1.129.1.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.129.2.0.0.0" type="main"> <s id="id.2.1.129.2.1.1.0">Hinc manife&longs;tum e&longs;t &longs;i data potentia &longs;it dato <lb/>pondere maior; hoc fieri po&longs;&longs;e, &longs;iue ita exi&longs;ten<lb/>te vecte, vt eius fulcimentum &longs;it inter pondus, <lb/>& potentiam; &longs;iue pondus inter fulcimentum, <lb/>& potentiam habente; &longs;iue demum potentia in­<lb/>ter pondus, & fulcimentum con&longs;tituta. </s> </p> <p id="id.2.1.129.3.0.0.0" type="main"> <s id="id.2.1.129.3.1.1.0">Sin autem data potentia minor, vel æqualis <lb/>dato pondere fuerit; palam quoq; e&longs;t id ip&longs;um <lb/>dumtaxat a&longs;&longs;e qui po&longs;&longs;e vecte ita exi&longs;tente, vt eius <lb/>fulcimentum &longs;it inter pondus, & pontentiam; <pb n="59" xlink:href="036/01/131.jpg"/>vel pondus intra fulcimentum, & potentiam <lb/>habente. </s> </p> <p id="id.2.1.129.4.0.0.0" type="head"> <s id="id.2.1.129.4.1.1.0">PROPOSITIO XIII. </s> </p> <p type="head"> <s id="id.2.1.129.4.3.1.0">PROBLEMA. </s> </p> <p id="id.2.1.129.5.0.0.0" type="main"> <s id="id.2.1.129.5.1.1.0">Quotcunq; datis in vecte ponderibus <expan abbr="vbicun­què">vbicun­<lb/>que</expan> appen&longs;is, cuius fulcimentum &longs;it quoq; da­<lb/>tum, potentiam inuenire, quæ in dato puncto <lb/>data pondera &longs;u&longs;tineat. <figure id="id.036.01.131.1.jpg" xlink:href="036/01/131/1.jpg"></figure></s> </p> <p id="id.2.1.129.6.0.0.0" type="main"> <s id="id.2.1.129.6.1.1.0">Sint data pondera ABC in vecte DE, cuius fulcimentum F, <lb/>vbicunq; in punctis DGH appen&longs;a: collocandaq; &longs;it potentia in <lb/>puncto E. potentiam inuenire oportet, quæ in E data pondera <lb/>ABC vecte DE &longs;u&longs;tineat. </s> <s id="id.2.1.129.6.1.2.0">diuidatur DG in k, ita vt Dk ad KG <lb/>&longs;it, vt pondus B ad pondus A; deinde diuidatur kH in L, ita vt kL <lb/>ad LH, &longs;it vt pondus C ad pondera BA; atq; vt FE ad FL, ita <lb/>fiant pondera ABC &longs;imul ad potentiam, quæ ponatur in E. </s> <s id="id.2.1.129.6.1.2.0.a">di­<lb/>co potentiam in E data pondera ABC in DGH appen&longs;a vecte <lb/>DE, cuius fulcimentum e&longs;t F, &longs;u&longs;tinere. </s> <s id="id.2.1.129.6.1.3.0">Quoniam enim &longs;i ponde<lb/>ra ABC &longs;imul e&longs;&longs;ent in L appen&longs;a, potentia in E data pondera <arrow.to.target n="note205"></arrow.to.target><lb/>in L appen&longs;a &longs;u&longs;tineret; pondera verò ABC tàm in L ponderant, <arrow.to.target n="note206"></arrow.to.target><expan abbr="quàm"><lb/>quam</expan> &longs;i C in H, & BA &longs;imul in K e&longs;&longs;ent appen&longs;a; & AB in k tàm <pb xlink:href="036/01/132.jpg"/><figure id="id.036.01.132.1.jpg" xlink:href="036/01/132/1.jpg"></figure><lb/>ponderant, quàm &longs;i A in D, & B in G appen&longs;a e&longs;&longs;ent; ergo po­<lb/>tentia in E data pondera ABC in DGH appen&longs;a vecte DE, cu­<lb/>ius fulcimentum e&longs;t F, &longs;u&longs;tinebit. </s> <s id="id.2.1.129.6.1.4.0">Si autem potentia in quouis <lb/>alio puncto vectis DE (præterquàm in F) con&longs;tituenda e&longs;&longs;et, <lb/>vt in k; fiat vt Fk ad FL, ita pondera ABC ad potentiam: &longs;i­<lb/><arrow.to.target n="note207"></arrow.to.target>militer demon&longs;trabimus potentiam in k pondera ABC in pun­<lb/>ctis DGH appen&longs;a &longs;u&longs;tinere. </s> <s id="id.2.1.129.6.1.5.0">quod facere oportebat. <figure id="id.036.01.132.2.jpg" xlink:href="036/01/132/2.jpg"></figure></s> </p> <p id="id.2.1.129.7.0.0.0" type="main"> <s id="id.2.1.129.7.1.1.0">Ex hac, & ex quinta huius, &longs;i pondera ABC &longs;int in vecte <lb/>DE quomodocunq; po&longs;ita; oporteatq; potentiam inuenire, quæ <lb/>in E data pondera &longs;u&longs;tinere debeat: ducantur à centris grauita­<lb/>tum ponderum ABC horizontibus perpendiculares, quæ ve­<lb/>ctem DE in DGH punctis &longs;ecent; cæteraq; eodem modo fiant: <lb/>Manife&longs;tum e&longs;t, potentiam in E, vel in K data pondera &longs;u&longs;tinere. </s> <s id="id.2.1.129.7.1.2.0"><lb/>idem enim e&longs;t, ac &longs;i pondera in DGH e&longs;&longs;ent appen&longs;a. </s> </p> <p id="id.2.1.130.1.0.0.0" type="margin"> <s id="id.2.1.130.1.1.1.0"><margin.target id="note205"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.130.1.1.2.0"><margin.target id="note206"></margin.target>5 <emph type="italics"/>Huius. de libra.<emph.end type="italics"/></s> <s id="id.2.1.130.1.1.4.0"><margin.target id="note207"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="60" xlink:href="036/01/133.jpg"/> <p id="id.2.1.131.1.0.0.0" type="head"> <s id="id.2.1.131.1.2.1.0">PROPOSITIO XIIII. </s> </p> <p type="head"> <s id="id.2.1.131.1.4.1.0">PROBLEMA. </s> </p> <p id="id.2.1.131.2.0.0.0" type="main"> <s id="id.2.1.131.2.1.1.0">Data quotcunq; pondera in dato vecte vbi­<lb/>cunq; & quomodocunq; po&longs;ita à data potentia <lb/>moueri. <figure id="id.036.01.133.1.jpg" xlink:href="036/01/133/1.jpg"></figure></s> </p> <p id="id.2.1.131.3.0.0.0" type="main"> <s id="id.2.1.131.3.1.1.0">Sit datus vectis DE, & &longs;int data pondera vt in præcedenti co<lb/>rollario; &longs;itq; A vt centum, B vt quinquaginta, C vt triginta; <lb/>dataq; potentia &longs;it vt triginta. </s> <s id="id.2.1.131.3.1.2.0">exponantur eadem, inueniaturq; <lb/>punctum L; deinde diuidatur LE in F, ita vt FE ad FL &longs;it, vt <lb/>centum octoginta ad triginta, hoc e&longs;t &longs;ex ad vnum: & &longs;i F fieret <lb/>fulcimentum, potentia vt triginta in E &longs;u&longs;tineret pondera ABC. </s> <s id="id.2.1.131.3.1.2.0.a"><arrow.to.target n="note208"></arrow.to.target><lb/>accipiatur igitur inter LF quoduis punctum M, fiatq; M fulci­<lb/>mentum: manife&longs;tum e&longs;t potentiam in E vt triginta pondera <arrow.to.target n="note209"></arrow.to.target><lb/>ABC vt centum octoginta vecte DE mouere. </s> <s id="id.2.1.131.3.1.3.0">quod facere <lb/>oportebat. </s> </p> <p id="id.2.1.132.1.0.0.0" type="margin"> <s id="id.2.1.132.1.1.1.0"><margin.target id="note208"></margin.target>13 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.132.1.1.2.0"><margin.target id="note209"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.133.1.0.0.0" type="main"> <s id="id.2.1.133.1.1.1.0">Hoc autem vniuersè a&longs;&longs;equi minimè poterimus, &longs;i in extremita­<lb/>te vectis fulcimentum e&longs;&longs;et, vt in D; quia proportio DE, ad DL <lb/>hoc e&longs;t proportio ponderum ABC ad potentiam, quæ pondera <lb/>&longs;u&longs;tinere debeat, &longs;emper e&longs;t data. </s> <s id="id.2.1.133.1.1.2.0">quod multo quoq; minus fieri <lb/>po&longs;&longs;et, &longs;i ponenda e&longs;&longs;et potentia inter DL. </s> </p> <pb xlink:href="036/01/134.jpg"/> <p id="id.2.1.133.3.0.0.0" type="head"> <s id="id.2.1.133.3.1.1.0">PROPOSITIO XV. </s> </p> <p type="head"> <s id="id.2.1.133.3.3.1.0">PROBLEMA. </s> </p> <p id="id.2.1.133.4.0.0.0" type="main"> <s id="id.2.1.133.4.1.1.0">Quia verò dum pondera vecte mouentur, <lb/>vectis quoq; grauitatem habet, cuius nulla ha­<lb/>ctenus mentio facta e&longs;t: idcirco primùm quo­<lb/>modo inueniatur potentia, quæ in dato puncto <lb/>datum vectem, cuius fulcimentum &longs;it quoq; da­<lb/>tum, &longs;u&longs;tineat, o&longs;tendamus. <figure id="id.036.01.134.1.jpg" xlink:href="036/01/134/1.jpg"></figure></s> </p> <p id="id.2.1.133.5.0.0.0" type="main"> <s id="id.2.1.133.5.1.1.0">Sit datus vectis AB, cuius fulcimentum &longs;it datum C; &longs;itq; <lb/>punctum D, in quo collocanda &longs;it potentia, quæ vectem AB &longs;u<lb/>&longs;tinere debeat, ita vt immobilis per&longs;i&longs;tat. </s> <s id="id.2.1.133.5.1.2.0">ducatur à puncto C <lb/>linea CE horizonti perpendicularis, quæ vectem AB in duas di­<lb/>uidat partes AE EF, &longs;itq; partis AE centrum grauitatis G, & <lb/>partis EF centrum grauitatis H; à punctisqué GH horizon­<lb/>tibus perpendiculares ducantur Gk HL, quæ lineam AF <lb/>in punctis KL &longs;ecent. </s> <s id="id.2.1.133.5.1.3.0">quoniam enim vectis AB à linea CE in duas <lb/>diuiditur partes AE EF; ideo vectis AB nihil aliud erit, ni&longs;i <lb/>duo pondera AE EF in vecte, &longs;iue libra AF po&longs;ita; cuius &longs;u­<lb/>&longs;pen&longs;io, &longs;iue fulcimentum e&longs;t C. quare pondera AE EF ita erunt <lb/>po&longs;ita, ac &longs;i in kL e&longs;&longs;ent appen&longs;a. </s> <s id="id.2.1.133.5.1.4.0">diuidatur ergo kL in M, <lb/>ita vt kM ad ML, &longs;it vt grauitas partis EF ad grauitatem par­<lb/>tis AE; & vt CA ad CM, ita fiat grauitas totius vectis AB ad <lb/>potentiam, quæ &longs;i collocetur in D (dummodo DA horizonti <pb n="61" xlink:href="036/01/135.jpg"/>perpendicularis exi&longs;tat) vecti æqueponderabit; hoc e&longs;t vectem <arrow.to.target n="note210"></arrow.to.target><lb/>AB deor&longs;um premendo &longs;u&longs;tinebit. </s> <s id="id.2.1.133.5.1.5.0">quod inuenire oportebat. </s> </p> <p id="id.2.1.134.1.0.0.0" type="margin"> <s id="id.2.1.134.1.1.1.0"><margin.target id="note210"></margin.target>13 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.135.1.0.0.0" type="main"> <s id="id.2.1.135.1.1.1.0">Si verò potentia in puncto B ponenda e&longs;&longs;et. </s> <s id="id.2.1.135.1.1.2.0">fiat vt CF ad CM <lb/>ita pondus AB ad potentiam. </s> <s id="id.2.1.135.1.1.3.0">&longs;imili modo o&longs;tendetur poten­<lb/>tiam in B vectem AB &longs;u&longs;tinere. </s> <s id="id.2.1.135.1.1.4.0">&longs;imiliterq; demon&longs;trabitur in quo­<lb/>cunq; alio &longs;itu (præterquàm in e) ponenda fuerit potentia, vt in <lb/>N. fiat enim vt CO ad CM, ita AB ad potentiam; quæ &longs;i pona­<lb/>tur in N, vectem AB &longs;u&longs;tinebit. </s> </p> <p id="id.2.1.135.2.0.0.0" type="main"> <s id="id.2.1.135.2.1.1.0">Adiiciatur autem pondus in vecte appen&longs;um, <lb/>&longs;iue po&longs;itum; vt iisdem po&longs;itis &longs;it pondus P in <lb/>A appen&longs;um; potentiaq; &longs;it ponenda in B, ita <lb/>vt vectem AB vnà cum pondere P &longs;u&longs;tineat. <figure id="id.036.01.135.1.jpg" xlink:href="036/01/135/1.jpg"></figure></s> </p> <p id="id.2.1.135.3.0.0.0" type="main"> <s id="id.2.1.135.3.1.1.0">Diuidatur AM in Q, ita vt AQ ad QM &longs;it, ut grauitas ue­<lb/>ctis AB ad grauitatem ponderis P; deinde ut CF ad CQ, ita fat <lb/>grauitas AB, & P &longs;imul ad potentiam, quæ ponatur in B: patet <lb/>potentiam in B uectem AB unà cum pondere P &longs;u&longs;tinere. </s> <s id="id.2.1.135.3.1.2.0">Si ue­<arrow.to.target n="note211"></arrow.to.target><expan abbr="rò"><lb/>ro</expan> e&longs;&longs;et CA ad CM, vt AB ad P; e&longs;&longs;et punctum C eorum centrum <arrow.to.target n="note212"></arrow.to.target><lb/>grauitatis, & ideo vectis AB vná cum pondere P ab&longs;q; potentia in <arrow.to.target n="note213"></arrow.to.target><lb/>B manebit. </s> <s id="id.2.1.135.3.1.3.0">&longs;ed &longs;i ponderum grauitatis centrum e&longs;&longs;et inter CF, vt <lb/>in O; fiat vt CF ad CO, ita AB&P &longs;imul ad potentiam, quæ <lb/>in B, & vectem AB, & pondus P &longs;u&longs;tinebit. <pb xlink:href="036/01/136.jpg"/><figure id="id.036.01.136.1.jpg" xlink:href="036/01/136/1.jpg"></figure></s> </p> <p id="id.2.1.135.4.0.0.0" type="main"> <s id="id.2.1.135.4.1.1.0">Similiter o&longs;tendetur, &longs;i plura e&longs;&longs;ent pondera in vecte AB ubi­<lb/>cunq;, & quomodocunq; po&longs;ita. </s> </p> <p id="id.2.1.136.1.0.0.0" type="margin"> <s id="id.2.1.136.1.1.1.0"><margin.target id="note211"></margin.target>13 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.136.1.1.2.0"><margin.target id="note212"></margin.target><emph type="italics"/>Ex &longs;exta<emph.end type="italics"/></s> <s id="id.2.1.136.1.1.3.0"><margin.target id="note213"></margin.target>1 <emph type="italics"/>Arch. de æquep.<emph.end type="italics"/></s> </p> <p id="id.2.1.137.1.0.0.0" type="main"> <s id="id.2.1.137.1.1.1.0">In&longs;uper ex his non &longs;olum, ut in decimaquarta huius docuimus, <lb/>quomodo &longs;cilicet data pondera ubicunq; in uecte po&longs;ita data poten<lb/>tia dato uecte mouere po&longs;&longs;umus, eodem modo grauitate uectis <lb/>con&longs;iderata idem facere poterimus; uerùm etiam accidentia reli­<lb/>qua, quæ &longs;upra ab&longs;q; uectis grauitatis con&longs;ideratione demon&longs;tra­<lb/>ta &longs;unt; &longs;imili modo uectis grauitate con&longs;iderata vná cum ponde<lb/>ribus, uel &longs;ine ponderibus o&longs;tendentur. </s> </p> </chap> <pb n="62" xlink:href="036/01/137.jpg"/> <chap> <p id="id.2.1.137.2.0.0.0" type="head"> <s id="id.2.1.137.3.1.1.0">DE TROCHLEA. </s> </p> <p id="id.2.1.137.4.0.0.0" type="main"> <s id="id.2.1.137.4.1.1.0">Trochleae in&longs;trumento pon<lb/>dus multipliciter moueri pote&longs;t; <lb/>quia verò in omnibus e&longs;t eadem <lb/>ratio: ideo (vt res euidentior ap­<lb/>pareat) in iis, quæ dicenda &longs;unt, <lb/>intelligatur pondus &longs;ur&longs;um ad re<lb/>ctos horizontis plano angulos hoc modo &longs;em­<lb/>per moueri. </s> </p> <pb xlink:href="036/01/138.jpg"/> <p id="id.2.1.137.6.0.0.0" type="main"> <s id="id.2.1.137.6.1.1.0">Sit pondus A, quod ip&longs;i ho<lb/>rizontis plano &longs;ur&longs;um ad rectos <lb/>angulos &longs;it attollendum; & vt <lb/>fieri &longs;olet, trochlea duos habens <lb/>orbiculos, quorum axiculi &longs;int <lb/>in BC, &longs;upernè appendatur; <lb/>trochlea verò duos &longs;imiliter ha<lb/>bens orbiculos, quorum axicu­<lb/>li &longs;int in DE, ponderi alligetur: <lb/>ac per omnes vtriu&longs;q; trochleæ <lb/>orbiculos circunducatur ducta­<lb/>rius funis, quem in altero eius ex <lb/>tremo, putá in F, oportet e&longs;&longs;e <lb/>religatum. </s> <s id="id.2.1.137.6.1.2.0">potentia autem mo<lb/>uens ponatur in G, quæ dum <lb/>de&longs;cendit, pondus A &longs;ur&longs;um ex <lb/>aduer&longs;o attolletur; quemadmo<lb/>dum Pappus in octauo libro Ma<lb/>thematicarum collectionum a&longs;­<lb/>&longs;erit; nec non Vitruuius in deci <lb/>mo de Architectura, & alii. <figure id="id.036.01.138.1.jpg" xlink:href="036/01/138/1.jpg"></figure></s> </p> <p id="id.2.1.137.7.0.0.0" type="main"> <s id="id.2.1.137.7.1.1.0">Quomodo autem hoc trochleæ in&longs;trumen­<lb/>tum reducatur ad vectem; cur magnum pondus <lb/>ab exigua virtute, & quomodo, quantoq; in tem<lb/>pore moueatur; cur funis in vno capite debeat <lb/>e&longs;&longs;e religatus; quodq; &longs;uperioris, inferioris&qacute;ue <lb/>trochleæ fuerit officium; & quomodo omnis in <pb n="63" xlink:href="036/01/139.jpg"/>numeris data proportio inter potentiam, & pon<lb/>dus inueniri po&longs;sit; dicamus. </s> </p> <p id="id.2.1.137.8.0.0.0" type="head"> <s id="id.2.1.137.8.1.1.0">LEMMA. </s> </p> <p id="id.2.1.137.9.0.0.0" type="main"> <s id="id.2.1.137.9.1.1.0">Sint rectæ lineæ AB CD parallelæ, quæ in <lb/>punctis AC circulum ACE contingant, cuius <lb/>centrum F: & FA FC connectantur. </s> <s id="id.2.1.137.9.1.2.0">Dico <lb/>AFC rectam lineam e&longs;&longs;e. </s> </p> <p id="id.2.1.137.10.0.0.0" type="main"> <s id="id.2.1.137.10.1.1.0">Ducatur FE ip&longs;is AB CD æquidi&longs;tans. </s> <s id="id.2.1.137.10.1.2.0"><lb/>& quoniam AB, & FE &longs;unt parallelæ, & <lb/>angulus BAF e&longs;t rectus; erit & AFE re­<lb/>ctus. </s> <s id="id.2.1.137.10.1.3.0">eodemq; modo CFE rectus erit. </s> <s id="id.2.1.137.10.1.4.0">li­<lb/>nea igitur <arrow.to.target n="note214"></arrow.to.target>AFC recta e&longs;t. </s> <s id="id.2.1.137.10.1.5.0">quod erat de­<lb/>mon&longs;trandum. <arrow.to.target n="note215"></arrow.to.target><arrow.to.target n="note216"></arrow.to.target><lb/></s> </p> <figure id="id.036.01.139.1.jpg" xlink:href="036/01/139/1.jpg"> </figure> <p id="id.2.1.138.1.0.0.0" type="margin"> <s id="id.2.1.138.1.1.1.0"><margin.target id="note214"></margin.target>18 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.138.1.1.2.0"><margin.target id="note215"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.138.1.1.3.0"><margin.target id="note216"></margin.target>14 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.139.1.0.0.0" type="head"> <s id="id.2.1.139.1.2.1.0">PROPOSITIO I. </s> </p> <p id="id.2.1.139.2.0.0.0" type="main"> <s id="id.2.1.139.2.1.1.0">Si funis trochleæ &longs;upernè appen&longs;æ orbiculo <lb/>circunducatur, alterumq; eius extremum pon­<lb/>deri alligetur, altero interim à potentia pondus <lb/>&longs;u&longs;tinente apprehen&longs;o: erit potentia ponderi <lb/>æqualis. </s> </p> <pb xlink:href="036/01/140.jpg"/> <p id="id.2.1.139.4.0.0.0" type="main"> <s id="id.2.1.139.4.1.1.0">Sit pondus A, <lb/>cui alligatus &longs;it fu­<lb/>nis in B; trochleaq; <lb/>habens orbiculum C <lb/>EF, cuius centrum <lb/>D, &longs;ur&longs;um appenda­<lb/>tur; &longs;itq; D quoq; <lb/>centrum axiculi; & <lb/>circa orbiculum uo­<lb/>luatur funis BC EF <lb/>G; &longs;itq; potentia <lb/>in G &longs;u&longs;tinens pon­<lb/>dus A. </s> <s id="id.2.1.139.4.1.1.0.a">dico poten­<lb/>tiam in G ponderi A <lb/>æqualem e&longs;&longs;e. </s> <s id="id.2.1.139.4.1.2.0">Sit FG <lb/>æquidi&longs;tans CB. </s> <s id="id.2.1.139.4.1.2.0.a"><lb/>Quoniam igitur pon<lb/><arrow.to.target n="note217"></arrow.to.target>dus A manet; erit <lb/><figure id="id.036.01.140.1.jpg" xlink:href="036/01/140/1.jpg"></figure><lb/>CB horizonti plano perpendicularis <*> quare FG eidem plano per­<lb/><arrow.to.target n="note218"></arrow.to.target>pendicularis erit. </s> <s id="id.2.1.139.4.1.3.0">Sint CF <expan abbr="pũcta">puncta</expan> in orbiculo, à quibus funes CB FG <lb/>in horizontis <expan abbr="planũ">planum</expan> ad rectos angulos de&longs;cendunt; tangent BC FG <lb/><expan abbr="orbiculũ">orbiculum</expan> CEF in punctis CF. <expan abbr="orbiculũ">orbiculum</expan> enim &longs;ecare <expan abbr="nõ">non</expan> po&longs;&longs;unt. </s> <s id="id.2.1.139.4.1.4.0">con<lb/>nectantur DC DF; erit CF recta linea, & anguli DCB DFG recti. </s> <s id="id.2.1.139.4.1.5.0"><lb/><arrow.to.target n="note219"></arrow.to.target><expan abbr="Quoniã">Quoniam</expan> <expan abbr="aut&etilde;">autem</expan> BC tùm horizonti, tùm ip&longs;i CF e&longs;t perpendicularis; <lb/>erit linea CF horizonti æquidi&longs;tans. </s> <s id="id.2.1.139.4.1.6.0">cùm verò <expan abbr="põdus">pondus</expan> appen&longs;um &longs;it <lb/><arrow.to.target n="note220"></arrow.to.target>in BC, & potentia &longs;it in G; quod idem e&longs;t, ac &longs;i e&longs;&longs;et in F; erit <lb/>CF tanquam libra, &longs;iue vectis, cuius centrum, &longs;iue fulcimentum e&longs;t <lb/>D; nam in axiculo <expan abbr="orbuculus">orbiculus</expan> &longs;u&longs;tinetur; atq; punctum D, cùm &longs;it <lb/>centrum axiculi, & orbiculi, etiam vtri&longs;que circumuolutis <lb/>immobile remanet. </s> <s id="id.2.1.139.4.1.7.0">Itaq; cùm di&longs;tantia DC &longs;it æqualis di&longs;tantiæ <lb/>DF, potentiaq; in F ponderi A in C appen&longs;o æqueponderet, cùm <lb/><arrow.to.target n="note221"></arrow.to.target>pondus &longs;u&longs;tineat, ne deor&longs;um vergat; erit potentia in F, &longs;iue in G <lb/>(nam idem e&longs;t) con&longs;tituta ponderi A æqualis. </s> <s id="id.2.1.139.4.1.8.0">Idem enim effi­<lb/>cit potentia in G, ac &longs;i in G aliud e&longs;&longs;et appen&longs;um pondus æquale <lb/>ponderi A; quæ pondera in CF appen&longs;a æquæponderabunt. </s> <s id="id.2.1.139.4.1.9.0">Præ­<lb/>terea, cùm in neutram fiat motus partem, idem erit vnico exi­<pb n="64" xlink:href="036/01/141.jpg"/>&longs;tente fune BC EFG hoc modo orbiculo circumuoluto, ac &longs;i duo <lb/>e&longs;&longs;ent funes BC FG alligati in vecte, &longs;iue libra CF. </s> </p> <p id="id.2.1.140.1.0.0.0" type="margin"> <s id="id.2.1.140.1.1.1.0"><margin.target id="note217"></margin.target>1 <emph type="italics"/>Huius. de libra.<emph.end type="italics"/></s> <s id="id.2.1.140.1.1.3.0"><margin.target id="note218"></margin.target>8 <emph type="italics"/>Vndecimi.<emph.end type="italics"/></s> <s id="id.2.1.140.1.1.4.0"><margin.target id="note219"></margin.target>18 <emph type="italics"/>Tertii.<emph.end type="italics"/></s> <s id="id.2.1.140.1.1.5.0"><margin.target id="note220"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 28 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.140.1.1.6.0"><margin.target id="note221"></margin.target>1 <emph type="italics"/>Primi. Archim. de æquepond.<emph.end type="italics"/></s> </p> <p id="id.2.1.141.1.0.0.0" type="head"> <s id="id.2.1.141.1.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.141.2.0.0.0" type="main"> <s id="id.2.1.141.2.1.1.0">Ex hoc manife&longs;tum e&longs;&longs;e pote&longs;t, idem pon­<lb/>dus ab eadem potentia ab&longs;q; ullo huius tro­<lb/>chleæ auxilio nihilominus &longs;u&longs;tineri po&longs;&longs;e. </s> </p> <p id="id.2.1.141.3.0.0.0" type="main"> <s id="id.2.1.141.3.1.1.0">Sit enim pondus H æquale <lb/>ponderi A, cui alligatus &longs;it funis <lb/>kL; &longs;itq; potentia in L &longs;u&longs;tinens <lb/>pondus H. cùm autem pondus <lb/>ab&longs;q; vllo adminiculo &longs;u&longs;tinere <lb/>volentes tanta vi opus &longs;it, quanta <lb/>ponderi e&longs;t æqualis; erit potentia <lb/>in L ponderi H æqualis; pondus <lb/>verò H ip&longs;i ponderi A e&longs;t æquale, <lb/>cui potentia in G e&longs;t æqualis; erit <lb/>igitur potentia in G potentiæ in L <lb/>æqualis. </s> <s id="id.2.1.141.3.1.2.0">quod idem e&longs;t, ac &longs;i <expan abbr="ead&etilde;">eadem</expan> <lb/>potentia idem pondus &longs;u&longs;tineret. <figure id="id.036.01.141.1.jpg" xlink:href="036/01/141/1.jpg"></figure></s> </p> <p id="id.2.1.141.4.0.0.0" type="main"> <s id="id.2.1.141.4.1.1.0">Præterea &longs;i potentiæ in G, & <lb/>in L inuicem fuerint æquales, &longs;eor<lb/>&longs;um autem ponderibus minores; <lb/>patet potentias ponderibus &longs;u&longs;ti­<lb/>nendis non &longs;ufficere. </s> <s id="id.2.1.141.4.1.2.0">&longs;i verò maiores, manife&longs;tum e&longs;t pondera à <lb/>pontentiis moueri. </s> <s id="id.2.1.141.4.1.3.0">& &longs;ic in eadem e&longs;&longs;e proportione potentiam in <lb/>L. ad pondus H, veluti potentia in G ad pondus A. </s> </p> <p id="id.2.1.141.5.0.0.0" type="main"> <s id="id.2.1.141.5.1.1.0">Sed quoniam in demon&longs;tratione a&longs;&longs;umptum fuit axiculum cir­<lb/>cumuerti, qui vt plurimum immobilis manet; idcirco immobili <lb/>quoq; manente axiculo idem o&longs;tendatur. </s> </p> <pb xlink:href="036/01/142.jpg"/> <p id="id.2.1.141.7.0.0.0" type="main"> <s id="id.2.1.141.7.1.1.0">Sit orbiculus trochleæ CEF, cu<lb/>ius centrum D; &longs;itq; axiculus GHk, <lb/>cuius idem &longs;it centrum D. </s> <s id="id.2.1.141.7.1.1.0.a">Ducatur <lb/>CG DkF diameter horizonti æ­<lb/>quidi&longs;tans. </s> <s id="id.2.1.141.7.1.2.0">& quoniam dum orbi­<lb/>culus circumuertitur, circumferen­<lb/>tia circuli CEF &longs;emper e&longs;t æquidi­<lb/>&longs;tans circumferentiæ axiculi GHk; <lb/>circa enim axiculum circumuerti­<lb/>tur; & circulorum æquidi&longs;tantes cir<lb/>cumferentiæ idem habent centrum; <lb/>erit punctum D &longs;emper & orbiculi, <lb/><figure id="id.036.01.142.1.jpg" xlink:href="036/01/142/1.jpg"></figure><lb/>& axiculi centrum. </s> <s id="id.2.1.141.7.1.3.0">Itaq; cùm DC &longs;it æqualis DF, & DG ip&longs;i <lb/>Dk; erit GC ip&longs;i kF æqualis. </s> <s id="id.2.1.141.7.1.4.0">&longs;i igitur in vecte, &longs;iue libra CF <lb/>pondera appendantur æqualia, æqueponderabunt. </s> <s id="id.2.1.141.7.1.5.0">di&longs;tantia enim <lb/>CG æqualis e&longs;t di&longs;tantiæ kF; axiculu&longs;q; GHK immobilis gerit <lb/>vicem centri, &longs;iue fulcimenti. </s> <s id="id.2.1.141.7.1.6.0">immobili igitur manente axicu­<lb/>lo, &longs;i ponatur in F potentia &longs;u&longs;tinens pondus in C appen&longs;um; erit <lb/>potentia in F ip&longs;i ponderi æqualis. </s> <s id="id.2.1.141.7.1.7.0">quod erat o&longs;tendendum. </s> </p> <p id="id.2.1.141.8.0.0.0" type="main"> <s id="id.2.1.141.8.1.1.0">Et cùm idem pror&longs;us &longs;it, &longs;iue axiculus circumuertatur, &longs;iue mi­<lb/>nus; liceat propterea in iis, quæ dicenda &longs;unt, loco axiculi cen­<lb/>trum tantùm accipere. </s> </p> <p id="id.2.1.141.9.0.0.0" type="head"> <s id="id.2.1.141.9.1.1.0">PROPOSITIO II. </s> </p> <p id="id.2.1.141.10.0.0.0" type="main"> <s id="id.2.1.141.10.1.1.0">Si funis orbiculo trochleæ ponderi alligatæ <lb/>circumducatur, altero eius extremo alicubi reli­<lb/>gato, altero uerò à potentia pondus &longs;u&longs;tinente <lb/>apprehen&longs;o; erit potentia ponderis &longs;ubdupla. </s> </p> <pb n="65" xlink:href="036/01/143.jpg"/> <p id="id.2.1.141.12.0.0.0" type="main"> <s id="id.2.1.141.12.1.1.0">Si pondus A; &longs;it BCD <lb/>orbiculus trochleæ pon­<lb/>deri A alligate, cuius cen<lb/>trum E; funis deinde FB <lb/>CDG circa orbiculum <lb/>voluatur, qui religetur in <lb/>F; &longs;itq; potentia in G &longs;u<lb/>&longs;tinens pondus A. </s> <s id="id.2.1.141.12.1.1.0.a">dico <lb/>potentiam in G &longs;ubdu­<lb/>plam e&longs;&longs;e ponderis A. </s> <s id="id.2.1.141.12.1.1.0.b">&longs;int <lb/>funes FB GD puncti E <lb/>horizonti perpendicula­<lb/>res, qui inter &longs;e &longs;e æqui­<lb/>di&longs;tantes <arrow.to.target n="note222"></arrow.to.target>erunt; tangantq; <lb/>funes FB GD circulum <lb/>BCD in BD punctis. </s> <s id="id.2.1.141.12.1.2.0"><lb/>connectatur BD; erit BD <lb/>per centrum E ducta, <arrow.to.target n="note223"></arrow.to.target><lb/><figure id="id.036.01.143.1.jpg" xlink:href="036/01/143/1.jpg"></figure><expan abbr="ip&longs;iu&longs;qué"><lb/>ip&longs;iu&longs;que</expan> centri horizonti æquidi&longs;tans. </s> <s id="id.2.1.141.12.1.3.0">Cùm autem <expan abbr="potén­tia">poten­<lb/>tia</expan> in G trochlea pondus A &longs;u&longs;tinere debeat, funem ex altero ex­<lb/>tremo religatum e&longs;&longs;e oportet, puta in F; ita vt F æqualiter &longs;altem <lb/>potentiæ in G re&longs;i&longs;tat, alioquin potentia in G nullatenus pondus <lb/>&longs;u&longs;tinere po&longs;&longs;et. </s> <s id="id.2.1.141.12.1.4.0">Et quoniam potentia fune &longs;u&longs;tinet orbiculum, <lb/>qui reliquam trochleæ partem, cui appen&longs;um e&longs;t pondus, &longs;u&longs;tinet <lb/>axiculo; grauitabit hæc trochleæ pars in axiculo, hoc e&longs;t in centro <lb/>E. quare pondus A in eodem quoq; centro E ponderabit, ac &longs;i <lb/>in E e&longs;&longs;et appen&longs;um. </s> <s id="id.2.1.141.12.1.5.0">po&longs;ita igitur potentia, quæ in G, vbi D <lb/>(idem enim pror&longs;us e&longs;t) erit BD tanquam vectis, cuius fulci<lb/>mentum erit B, pondus in E appen&longs;um, & potentia in D. con<lb/>uenienter enim fulcimenti rationem ip&longs;um B &longs;ubire pote&longs;t, exi<lb/>&longs;tente fune FB immobili. </s> <s id="id.2.1.141.12.1.6.0">cæterum hoc po&longs;terius magis eluce&longs;cet. </s> <s id="id.2.1.141.12.1.7.0"><lb/>Quoniam autem potentia ad pondus eandem habet proportio­<lb/>nem, <arrow.to.target n="note224"></arrow.to.target>quàm BE ad BD; & BE in &longs;ubdupla e&longs;t proportione <lb/>ad BD: potentia igitur in G ponderis A &longs;ubdupla erit. </s> <s id="id.2.1.141.12.1.8.0">quod de­<lb/>mon&longs;trare oportebat. </s> </p> <p id="id.2.1.142.1.0.0.0" type="margin"> <s id="id.2.1.142.1.1.1.0"><margin.target id="note222"></margin.target>6 <emph type="italics"/>Vndecimi<emph.end type="italics"/></s> <s id="id.2.1.142.1.1.2.0"><margin.target id="note223"></margin.target><emph type="italics"/>Ex præcedenti.<emph.end type="italics"/></s> <s id="id.2.1.142.1.1.3.0"><margin.target id="note224"></margin.target>2 <emph type="italics"/>Huius de vecte.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/144.jpg"/> <p id="id.2.1.143.1.0.0.0" type="main"> <s id="id.2.1.143.1.2.1.0">Hoc igitur ita &longs;e ha­<lb/>bet vnico exi&longs;tente fune <lb/>FBC DG ip&longs;i orbi culo <lb/>circumducto, ac &longs;i duo e&longs;<lb/>&longs;ent funes BF GD ve­<lb/>cti BD alligati, cuius ful<lb/>cimentum erit B, pon­<lb/>dus in E appen&longs;um, & <lb/>potentia &longs;u&longs;tinens in D, <lb/>vel quod idem e&longs;t in G. </s> </p> <figure id="id.036.01.144.1.jpg" xlink:href="036/01/144/1.jpg"> </figure> <p id="id.2.1.143.1.3.1.0" type="head"> <s id="id.2.1.143.1.5.1.0">COROLLARIVM I. </s> </p> <p id="id.2.1.143.2.0.0.0" type="main"> <s id="id.2.1.143.2.1.1.0">Ex hoc itaq; manife&longs;tum e&longs;t, pondus hoc mo <lb/>do à minori in &longs;ubdupla proportione potentia <lb/>&longs;u&longs;tineri, quam &longs;ine vllo huiu&longs;modi trochleæ <lb/>auxilio. </s> </p> <pb n="66" xlink:href="036/01/145.jpg"/> <p id="id.2.1.143.4.0.0.0" type="main"> <s id="id.2.1.143.4.1.1.0">Veluti &longs;it pondus H ponderi A <lb/>æquale, cui religatus &longs;it funis kL; <lb/>potentiaq; in L &longs;u&longs;tineat pondus H; <lb/>erit potentia in L &longs;eor&longs;um ponderi <lb/>H, & ponderi A æqualis; &longs;ed poten<lb/>tia in G &longs;ubdupla e&longs;t ponderis A, <lb/>quare potentia in G &longs;ubdupla erit po<lb/>tentiæ, quæ e&longs;t in L. & hoc modo in <lb/>huiu&longs;cemodi reliquis omnibus pro <lb/>portio inueniri poterit. </s> </p> <figure id="id.036.01.145.1.jpg" xlink:href="036/01/145/1.jpg"> </figure> <p id="id.2.1.143.4.2.1.0" type="head"> <s id="id.2.1.143.4.4.1.0">COROLLARIVM. II. </s> </p> <p id="id.2.1.143.5.0.0.0" type="main"> <s id="id.2.1.143.5.1.1.0">Manife&longs;tum e&longs;t etiam; &longs;i duæ fuerint poten­<lb/>tiæ vna in G, altera in F, pondus A &longs;u&longs;tinentes; <lb/>vtra&longs;q; &longs;imul ponderi A æquales e&longs;&longs;e: & vnam <lb/>quamque &longs;u&longs;tinere dimidium ponderis A. </s> </p> <p id="id.2.1.143.6.0.0.0" type="main"> <s id="id.2.1.143.6.1.1.0">Hoc autem ex tertio, & quarto corollario &longs;ecundæ huius in <lb/>tractatu de vecte patet. </s> </p> <p id="id.2.1.143.7.0.0.0" type="head"> <s id="id.2.1.143.7.1.1.0">COROLLARIVM III. </s> </p> <p id="id.2.1.143.8.0.0.0" type="main"> <s id="id.2.1.143.8.1.1.0">Illud quoq; præterea innote&longs;cit, cur &longs;cilicet fu<lb/>nis ex altero religatus e&longs;&longs;e debeat extremo. </s> </p> <pb xlink:href="036/01/146.jpg"/> <p id="id.2.1.143.10.0.0.0" type="head"> <s id="id.2.1.143.10.1.1.0">PROPOSITIO III. </s> </p> <p id="id.2.1.143.11.0.0.0" type="main"> <s id="id.2.1.143.11.1.1.0">Si vtri&longs;q; duarum trochlearum &longs;ingulis or­<lb/>biculis, quarum altera &longs;upernè, altera verò <expan abbr="in­fernè">in­<lb/>ferne</expan> con&longs;tituta, ponderiq; alligata fuerit, cir<lb/>cunducatur funis; altero eius extremo alicubi <lb/>religato, altero verò à potentia pondus &longs;u&longs;ti­<lb/>nente detento; erit potentia ponderis &longs;ub du­<lb/>pla. </s> </p> <p id="id.2.1.143.12.0.0.0" type="main"> <s id="id.2.1.143.12.1.1.0">Sit pondus A; &longs;it BCD orbiculus trochleæ pon<lb/>deri A alligatæ, cuius centrum K; EFG verò <lb/>&longs;it trochleæ &longs;ur&longs;um appen&longs;æ, cuius centrum H. <lb/>deinde LBC DME FGN funis circa orbicu­<lb/>los ducatur, qui religetur in L; &longs;itq; potentia in <lb/>N &longs;u&longs;tinens pondus A. </s> <s id="id.2.1.143.12.1.1.0.a">dico potentiam in N <lb/>&longs;ubduplam e&longs;&longs;e ponderis A. &longs;i enim potentia &longs;u<lb/>&longs;tinens pondus A vbi M collocata foret, e&longs;&longs;et <lb/>vtiq; potentia in M &longs;ubdupla ponderis A. po­<lb/><arrow.to.target n="note225"></arrow.to.target>tentiæ verò in M æqualis e&longs;t vis in N. e&longs;t e­<lb/><arrow.to.target n="note226"></arrow.to.target>nim ac &longs;i potentia in M dimidium ponderis <lb/>A &longs;ine trochlea &longs;u&longs;tineret, cui æqueponderat <lb/>pondus in N ponderis A dimidio æquale. </s> <s id="id.2.1.143.12.1.2.0"><lb/>quare vis in N æqualis dimidio ponderis A <lb/>ip&longs;um A &longs;u&longs;tinebit. </s> <s id="id.2.1.143.12.1.3.0">Potentia igitur in N &longs;u&longs;ti<lb/>nens pondus A &longs;ubdupla e&longs;t ip&longs;ius A. quod <lb/>demon&longs;trare oportebat. <figure id="id.036.01.146.1.jpg" xlink:href="036/01/146/1.jpg"></figure></s> </p> <pb n="67" xlink:href="036/01/147.jpg"/> <p id="id.2.1.143.14.0.0.0" type="main"> <s id="id.2.1.143.14.1.1.0">Si verò vt in &longs;ecunda figura &longs;it fu<lb/>nis BC DEF GHkL orbiculis cir<lb/>cum uolutus, & religatus in B; poten<lb/>tiaq; in L pondus A &longs;u&longs;tineat: erit <lb/>potentia in L &longs;imiliter ponderis &longs;ubdu<lb/>pla. </s> <s id="id.2.1.143.14.1.2.0">orbiculus enim trochleæ &longs;upe­<lb/>rioris, ip&longs;aqué trochlea penitus &longs;unt <lb/>inutiles: & idem e&longs;t, ac &longs;i funis reli<lb/>gatus e&longs;&longs;et in F, & potentia in L &longs;u<lb/>&longs;tineret pondus &longs;ola trochlea ponderi <lb/>alligata, quæ potentia ponderis A o&longs;ten<lb/>&longs;a e&longs;t &longs;ubdupla. </s> </p> <p id="id.2.1.144.1.0.0.0" type="margin"> <s id="id.2.1.144.1.1.1.0"><margin.target id="note225"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.144.1.1.2.0"><margin.target id="note226"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <figure id="id.036.01.147.1.jpg" xlink:href="036/01/147/1.jpg"> </figure> <p id="id.2.1.145.1.1.1.0" type="head"> <s id="id.2.1.145.1.3.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.145.2.0.0.0" type="main"> <s id="id.2.1.145.2.1.1.0">Ex his &longs;equitur, &longs;i duæ &longs;int potentiæ in BL; <lb/>vtra&longs;q; inter &longs;e &longs;e æquales e&longs;&longs;e. </s> </p> <p id="id.2.1.145.3.0.0.0" type="main"> <s id="id.2.1.145.3.1.1.0">Vtraq; enim &longs;eor&longs;um e&longs;t ip&longs;ius A &longs;ubdupla. </s> </p> <pb xlink:href="036/01/148.jpg"/> <p id="id.2.1.145.4.0.0.0" type="head"> <s id="id.2.1.145.5.1.1.0">PROPOSITIO IIII. </s> </p> <p id="id.2.1.145.6.0.0.0" type="main"> <s id="id.2.1.145.6.1.1.0">Sit vectis AB, cuius fulcimentum &longs;it A; qui <lb/>bifariam diuidatur in D: &longs;itq; pondus C in D <lb/>appen&longs;um; duæq; &longs;int potentiæ æquales in BD <lb/>pondus C &longs;u&longs;tinentes. </s> <s id="id.2.1.145.6.1.2.0">Dico unamquamq; poten<lb/>tiam in BD ponderis C &longs;ubtriplam e&longs;&longs;e. </s> </p> <p id="id.2.1.145.7.0.0.0" type="main"> <s id="id.2.1.145.7.1.1.0">Quoniam enim altera <lb/>potentia e&longs;t in D colloca<lb/>ta, & pondus C in eodem <lb/>puncto D e&longs;t appen&longs;um; <lb/>potentia in D partem <lb/>ponderis C &longs;u&longs;tinebit ip­<lb/>&longs;i potentiæ D æqualem. </s> <s id="id.2.1.145.7.1.2.0"><lb/><figure id="id.036.01.148.1.jpg" xlink:href="036/01/148/1.jpg"></figure><lb/>quare potentia in B partem &longs;u&longs;tinebit reliquam, quæ pars dupla erit <lb/>ip&longs;ius potentiæ in B; cùm pondus ad potentiam eandem habeat <lb/>proportionem, quam AB ad AD: & potentiæ in BD &longs;unt æqua­<lb/>les; ergo potentia in B duplam &longs;u&longs;tinebit partem eius, quam &longs;u&longs;ti<lb/>net potentia in D. </s> <s id="id.2.1.145.7.1.2.0.a">diuidatur ergo pondus C in duas partes, qua <lb/>rum vna &longs;it reliquæ dupla; quod fiet, &longs;i in tres partes æquales EFG <lb/>diui&longs;erimus: tunc enim FG dupla erit ip&longs;ius E. </s> <s id="id.2.1.145.7.1.2.0.b">Itaq; potentia <lb/>in D partem E &longs;u&longs;tinebit, & potentiam in B reliquas FG. vtreq; <lb/>igitur inter &longs;e &longs;e æquales potentiæ in BD &longs;imul totum &longs;u&longs;tinebunt <lb/>pondus C. </s> <s id="id.2.1.145.7.1.2.0.c">& quoniam potentia in D partem E &longs;u&longs;tinet, quæ ter<lb/>tia e&longs;t pars ponderis C, ip&longs;iq; e&longs;t æqualis; erit potentia in D &longs;ub <lb/>tripla ponderis C. & cùm potentia in B &longs;u&longs;tineat partes FG, qua <lb/>rum potentia in B e&longs;t &longs;ubdupla; erit in B potentia vni partium FG, <lb/>putà G æqualis. </s> <s id="id.2.1.145.7.1.3.0">G verò tertia e&longs;t pars ponderis C; potentia <lb/>igitur in B &longs;ubtripla erit ponderis C. </s> <s id="id.2.1.145.7.1.3.0.a">Vnaquæq; ergo potentia in <lb/>BD &longs;ubtripla e&longs;t ponderis C. quod demon&longs;trare oportebat. <pb n="68" xlink:href="036/01/149.jpg"/><figure id="id.036.01.149.1.jpg" xlink:href="036/01/149/1.jpg"></figure></s> </p> <p id="id.2.1.145.8.0.0.0" type="main"> <s id="id.2.1.145.8.1.1.0">Et &longs;i duo e&longs;&longs;ent vectes AB EF bifariam in GD diui&longs;i, quorum <lb/>fulcimenta e&longs;&longs;ent AF, & pondus C in DG vtriq; vecti appen­<lb/>&longs;um, ita tamen vt in vtroq; æqualiter ponderet; duæq; e&longs;&longs;ent <lb/>æquales potentiæ in BG: eadem pror&longs;us ratione o&longs;tendetur, <lb/>vnamquamq; potentiam in B, & G ponderis C &longs;ubtriplam <lb/>e&longs;&longs;e. </s> </p> <p id="id.2.1.145.9.0.0.0" type="head"> <s id="id.2.1.145.9.1.1.0">PROPOSITIO V. </s> </p> <p id="id.2.1.145.10.0.0.0" type="main"> <s id="id.2.1.145.10.1.1.0">Si vtri&longs;q; duarum <expan abbr="trochlearũ">trochlearum</expan> &longs;ingulis orbiculis, <lb/>quarum altera &longs;upernè, altera verò infernè con&longs;ti<lb/>tuta, ponderiq; alligata fuerit, circumducatur fu<lb/>nis; altero eius extremo inferiori trochleæ reli­<lb/>gato, altero verò à potentia pondus &longs;u&longs;tinente <lb/>detento: erit potentia ponderis &longs;ubtripla. </s> </p> <pb xlink:href="036/01/150.jpg"/> <p id="id.2.1.145.12.0.0.0" type="main"> <s id="id.2.1.145.12.1.1.0">Sit pondus A; &longs;it BCD orbiculus tro­<lb/>chleæ ponderi A alligate, cuius centrum <lb/>E; & FGH trochleæ &longs;ur&longs;um appen&longs;æ, cu­<lb/>ius centrum k; & LFGHBCDM funis <lb/>orbiculis circumducatur, qui religetur in L <lb/>trochleæ inferiori; &longs;itq; potentia in M &longs;u­<lb/>&longs;tinens pondus A. </s> <s id="id.2.1.145.12.1.1.0.a">dico potentiam in M <lb/>&longs;ubtriplam e&longs;&longs;e ponderis A. </s> <s id="id.2.1.145.12.1.1.0.b">ducantur FH <lb/>BD per centra kE horizonti æquidi&longs;tan­<lb/>tes, &longs;icut in præcedentibus dictum e&longs;t Quo­<lb/>niam enim funis FL trochleam &longs;u&longs;tinet in­<lb/>feriorem, quæ &longs;u&longs;tinet orbiculum in eius <lb/>centro E; erit funis in L vt potentia &longs;u&longs;ti­<lb/>nens orbiculum, ac &longs;i in ip&longs;o E centro e&longs;&longs;et; <lb/>potentia verò in M e&longs;t, ac &longs;i e&longs;&longs;et in D; <lb/>efficietur igitur DB tanquam vectis, cuius <lb/><arrow.to.target n="note227"></arrow.to.target>fulcimentum erit B; pondus verò A (vt &longs;u<lb/>pra o&longs;ten&longs;um e&longs;t) ex E &longs;u&longs;pen&longs;um à dua­<lb/>bus potentiis altera in D, altera in E &longs;u&longs;ten<lb/>tatum. </s> <s id="id.2.1.145.12.1.2.0">Cùm autem in pondere &longs;u&longs;tinendo <lb/>vectes FH BD immobiles maneant, &longs;i in <lb/>funibus FL HB appendantur pondera, e­<lb/><arrow.to.target n="note228"></arrow.to.target>runt hæc ip&longs;a æqualia; cùm vectis FH ha­<lb/>beat fulcimentum in medio; alioquin ex al<lb/>tera parte deor&longs;um fieret motus, quod <expan abbr="tam&etilde;">tamen</expan> <lb/>non contingit. </s> <s id="id.2.1.145.12.1.3.0">tam igitur &longs;u&longs;tinet funis FL, <lb/>quàm HB. deinde quoniam ex medio ve­<lb/><figure id="id.036.01.150.1.jpg" xlink:href="036/01/150/1.jpg"></figure><lb/>cte BD pondus &longs;u&longs;penditur, idcirco &longs;i duæ fuerint potentiæ in BD <lb/><arrow.to.target n="note229"></arrow.to.target>pondus &longs;u&longs;tinentes, erunt inuicem æquales. </s> <s id="id.2.1.145.12.1.4.0">& quamquam funis <pb n="69" xlink:href="036/01/151.jpg"/>FL ip&longs;e quoq; pondus &longs;u&longs;tineat, cùm potentiæ in E <expan abbr="vic&etilde;">vicem</expan> gerat; quia <lb/>tamen ex eodemmet puncto &longs;u&longs;tinet, vbi appen&longs;um e&longs;t pondus, non <lb/>efficiet propterea, quin potentiæ in BD &longs;int inter &longs;e &longs;e æquales; <lb/>opitulatur enim tàm vni, quàm alteri. </s> <s id="id.2.1.145.12.1.5.0">potentiæ verò in BD eæ­<lb/>dem &longs;unt, ac &longs;i e&longs;&longs;ent in HM; quare tàm &longs;u&longs;tinebit funis MD, <lb/>quàm HB. </s> <s id="id.2.1.145.12.1.5.0.a">ita verò &longs;u&longs;tinet HB, atq; FL; funis igitur MD ita <lb/>&longs;u&longs;tinebit, &longs;icut FL, hoc e&longs;t, ac &longs;i in D, & L appen&longs;a e&longs;&longs;ent pon­<lb/>dera æqualia. </s> <s id="id.2.1.145.12.1.6.0">Cùm itaq; æqualia pondera à potentiis &longs;u&longs;tinean­<lb/>tur æqualibus, potentiæ in ML æquales erunt; quarum eadem pror<lb/>&longs;us e&longs;t ratio, ac &longs;i e&longs;&longs;ent ambæ in DE. </s> <s id="id.2.1.145.12.1.6.0.a">Itaq; cùm pondus A in <lb/>medio vectis BD &longs;it appen&longs;um, duæq; potentiæ &longs;int æquales in <lb/>DE pondus &longs;u&longs;tinentes; erit B fulcimentum, ac vnaquæq; potentia, <arrow.to.target n="note230"></arrow.to.target><lb/>&longs;iue in DE, &longs;iue in ML &longs;ubtripla ponderis A. ergo potentia in M <lb/>&longs;u&longs;tinens pondus &longs;ubtripla erit ponderis A. quod o&longs;tendere o­<lb/>portebat. </s> </p> <p id="id.2.1.146.1.0.0.0" type="margin"> <s id="id.2.1.146.1.1.1.0"><margin.target id="note227"></margin.target><emph type="italics"/>In<emph.end type="italics"/> 2 <emph type="italics"/>Huius<emph.end type="italics"/></s> <s id="id.2.1.146.1.1.2.0"><margin.target id="note228"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.146.1.1.3.0"><margin.target id="note229"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 3 <emph type="italics"/>Cor.<emph.end type="italics"/> 2 <emph type="italics"/>Huius vecte.<emph.end type="italics"/></s> <s id="id.2.1.146.1.1.4.0"><margin.target id="note230"></margin.target>4 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.147.1.0.0.0" type="head"> <s id="id.2.1.147.1.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.147.2.0.0.0" type="main"> <s id="id.2.1.147.2.1.1.0">Ex hoc manife&longs;tum e&longs;t, vnumquemq; funem <lb/>MD FL HB tertiam &longs;u&longs;tinere partem pon­<lb/>deris A. <pb xlink:href="036/01/152.jpg"/></s> </p> <p id="id.2.1.147.3.0.0.0" type="main"> <s id="id.2.1.147.3.1.1.0">Præterea, &longs;i funis ex M per a­<lb/>lium adhuc deferatur orbiculum &longs;u<lb/>periorem in trochlea &longs;ur&longs;um &longs;imi­<lb/>liter appen&longs;a con&longs;titutum, cuius <lb/>centrum N; ita vt perueniat in O; <lb/>ibiq; à potentia detineatur; erit po<lb/>tentia in O &longs;u&longs;tinens pondus A iti <lb/>dem &longs;ubtripla ip&longs;ius ponderis. </s> <s id="id.2.1.147.3.1.2.0">fu<lb/>nis enim MD tantùm ponderis &longs;u<lb/>&longs;tinet, ac &longs;i in D appen&longs;um e&longs;&longs;et <lb/>pondus æquale tertiæ parti ponde<lb/><arrow.to.target n="note231"></arrow.to.target>ris A, cui æquiualet potentia in <lb/>O ip&longs;i æqualis, hoc e&longs;t &longs;ubtripla <lb/>ponderis A. </s> <s id="id.2.1.147.3.1.2.0.a">Potentia igitur in O <lb/>&longs;ubtripla e&longs;t ponderis A. <lb/><figure id="id.036.01.152.1.jpg" xlink:href="036/01/152/1.jpg"></figure></s> </p> <p id="id.2.1.147.4.0.0.0" type="main"> <s id="id.2.1.147.4.1.1.0">Et ne idem &longs;æpius repetatur, no<lb/>ui&longs;&longs;e oportet potentiam in O &longs;em<lb/>per æqualem e&longs;&longs;e ei, quæ e&longs;t in M; <lb/>hoc e&longs;t &longs;i potentia in M e&longs;&longs;et &longs;ub <lb/>quadrupla, &longs;ubquintupla, vel huiu&longs; <lb/>modi aliter ip&longs;ius ponderis; poten<lb/>tia quoq; in O erit itidem &longs;ubqua<lb/>drupla, &longs;ubquintupla, atq; ita dein<lb/>ceps eiu&longs;demmet ponderis, quem<lb/>madmodum &longs;e habet potentia <lb/>in M. </s> </p> <p id="id.2.1.148.1.0.0.0" type="margin"> <s id="id.2.1.148.1.1.1.0"><margin.target id="note231"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="70" xlink:href="036/01/153.jpg"/> <p id="id.2.1.149.1.0.0.0" type="head"> <s id="id.2.1.149.1.2.1.0">PROPOSITIO VI. </s> </p> <p id="id.2.1.149.2.0.0.0" type="main"> <s id="id.2.1.149.2.1.1.0">Sint duo vectes AB CD bifariam diui&longs;i in <lb/>EF, quorum fulcimenta &longs;int. </s> <s id="id.2.1.149.2.1.2.0">in BD; &longs;itq; pon<lb/>dus G in EF vtriq; vecti appen&longs;um, ita ut ex <lb/>vtroq; æqualiter ponderet; duæq; &longs;int potentiæ <lb/>in AC æquales pondus &longs;u&longs;tinentes. </s> <s id="id.2.1.149.2.1.3.0">Dico unam <lb/>quamq; potentiam in AC &longs;ubquadruplam e&longs;­<lb/>&longs;e ponderis G. </s> </p> <p id="id.2.1.149.3.0.0.0" type="main"> <s id="id.2.1.149.3.1.1.0">Cùm enim potentiæ in <lb/>AC totum &longs;u&longs;tineant pon­<lb/>dus G, potentiaq; in A ad <lb/>partem ponderis, quod &longs;u&longs;ti<lb/>net, &longs;it vt BE ad BA; po­<lb/>tentia <arrow.to.target n="note232"></arrow.to.target>verò in C ad partem <lb/>ip&longs;ius G, quod &longs;u&longs;tinet, ita <lb/>&longs;it vt DF ad DC; & vt BE <lb/>ad BA, ita e&longs;t DF ad DC; <lb/><figure id="id.036.01.153.1.jpg" xlink:href="036/01/153/1.jpg"></figure><lb/>erit potentia in A ad partem ponderis, quod &longs;u&longs;tinet, vt poten­<lb/>tia in C ad ip&longs;ius ponderis, quod &longs;u&longs;tinet, partem; & potentiæ <lb/>in AC &longs;unt æquales; æquales igitur erunt partes ponderis G, <lb/>quæ à potentiis &longs;u&longs;tinentur. </s> <s id="id.2.1.149.3.1.2.0">quare vnaquæq; potentia in A C di­<lb/>midium &longs;u&longs;tinebit ponderis G. </s> <s id="id.2.1.149.3.1.2.0.a">Potentia verò in A &longs;ubdupla e&longs;t pon<lb/>deris, quod &longs;u&longs;tinet: ergo potentia in A dimidio dimidii, hoc <lb/>e&longs;t quartæ portioni ponderis G æqualis erit; ideoq; &longs;ubquadrupla <lb/>erit ponderis G. </s> <s id="id.2.1.149.3.1.2.0.b">neq; aliter demon&longs;trabitur potentiam in C &longs;ub-quadruplam <lb/>e&longs;&longs;e eiu&longs;dem ponderis G. quod demon&longs;trare opor­<lb/>tebat. </s> </p> <p id="id.2.1.150.1.0.0.0" type="margin"> <s id="id.2.1.150.1.1.1.0"><margin.target id="note232"></margin.target>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/154.jpg"/> <p id="id.2.1.151.1.0.0.0" type="main"> <s id="id.2.1.151.1.2.1.0">Si verò tres &longs;int vectes <lb/>AB CD EF bifariam di­<lb/>ui&longs;i in GHk, quorum fulci <lb/>menta &longs;int BDF; & pondus <lb/>L eodem modo in GHK <lb/>appen&longs;um; &longs;intq; tres poten<lb/>tiæ in ACE æquales pondus <lb/>&longs;u&longs;tinentes; &longs;imiliter o&longs;ten<lb/>detur vnamquamque po­<lb/>tentiam &longs;ub&longs;excuplam e&longs;&longs;e <lb/>ponderis L. atq; hoc ordi<lb/>ne &longs;i quatuor e&longs;&longs;ent vectes, <lb/>& quatuor potentiæ; erit vnaquæq; potentia &longs;uboctupla ponderis. </s> <lb/> <s id="id.2.1.151.1.2.2.0">atq; ita deinceps in infinitum. </s> </p> <figure id="id.036.01.154.1.jpg" xlink:href="036/01/154/1.jpg"> </figure> <p id="id.2.1.151.1.3.1.0" type="head"> <s id="id.2.1.151.1.5.1.0">PROPOSITIO VII. </s> </p> <p id="id.2.1.151.2.0.0.0" type="main"> <s id="id.2.1.151.2.1.1.0">Si tribus duarum trochlearum orbiculis, <expan abbr="quarũ">quarum</expan> <lb/>altera &longs;upernè vnico duntaxat, altera verò <expan abbr="infer­nè">infer­<lb/>ne</expan> duobus autem in&longs;ignita orbiculis, ponderiq; <lb/>alligata con&longs;tituta fuerit, funis circumponatur; al<lb/>tero eius extremo alicubi religato, altero verò à <lb/>potentia pondus &longs;u&longs;tinente retento; erit potentia <lb/>ponderis &longs;ubquadrupla. </s> </p> <pb n="71" xlink:href="036/01/155.jpg"/> <p id="id.2.1.151.4.0.0.0" type="main"> <s id="id.2.1.151.4.1.1.0">Sit pondus A; &longs;int tres orbiculi, quorum <lb/>centra BCD; orbiculu&longs;q;, cuius centrum D, <lb/>&longs;it trochleæ &longs;ur&longs;um appen&longs;æ; quorum verò <lb/>&longs;unt centra BC, &longs;int trochleæ ponderi A alli<lb/>gatæ; funi&longs;q; EFGHkLNOP per omnes <lb/>circumducatur orbiculos, qui religetur in E; <lb/>&longs;itq; vis in P &longs;u&longs;tinens pondus A. </s> <s id="id.2.1.151.4.1.1.0.a">dico po<lb/>tentiam in P &longs;ubquadruplam e&longs;&longs;e ponderis <lb/>A. </s> <s id="id.2.1.151.4.1.1.0.b">ducantur kL GF ON per rotularum <lb/>centra, & horizonti æquidi&longs;tantes, quæ (ex <lb/>iis, quæ dicta &longs;unt) tanquam vectes erunt. </s> <s id="id.2.1.151.4.1.2.0"><lb/>& quoniam propter vectem, &longs;iue libram kL, <lb/>cuius fulcimentum, &longs;iue centrum e&longs;t in me <lb/>dio, tàm &longs;u&longs;tinet funis kG, quàm LN, cùm <arrow.to.target n="note233"></arrow.to.target><lb/>in neutram partem fiat motus. </s> <s id="id.2.1.151.4.1.3.0">nec non <lb/>propter vectem GF, è cuius medio veluti &longs;u<lb/>&longs;pen&longs;um dependet onus; &longs;i duæ e&longs;&longs;ent in GF <lb/>potentiæ, &longs;eu in HE (e&longs;t enim par vtriu&longs;q; <lb/>&longs;itus ratio, vt iam &longs;epius dictum e&longs;t) e&longs;&longs;ent <arrow.to.target n="note234"></arrow.to.target><lb/>vtiq; huiu&longs;modi potentiæ inuicem æquales. </s> <s id="id.2.1.151.4.1.4.0"><lb/>quare ita &longs;u&longs;tinet funis HG, vt EF. &longs;imiliter <lb/>o&longs;ten detur funem PO tàm &longs;u&longs;tinere, quàm <lb/>LN: quare funes PO kG EF LN æqua<lb/>liter &longs;u&longs;tinent. </s> <s id="id.2.1.151.4.1.5.0">æqualiter igitur funis PO &longs;u<lb/>&longs;tinet, vt kG. &longs;i ergo duæ intelligantur e&longs; <lb/><figure id="id.036.01.155.1.jpg" xlink:href="036/01/155/1.jpg"></figure><lb/>&longs;e potentiæ in OG, &longs;eu in PH, quod idem e&longs;t, pondus nihilomi<lb/>nus &longs;u&longs;tinentes, quemadmodum funes &longs;u&longs;tinent, æquales vtiq; e&longs;<lb/>&longs;ent; & GF ON duorum vectium vires gerent; quorum fulci <lb/>menta erunt FN, & pondus A in BC medio vectium appen&longs;um. </s> <s id="id.2.1.151.4.1.6.0"><lb/>& quoniam omnes funes æqualiter &longs;u&longs;tinent, tàm &longs;u&longs;tinebunt <lb/>duo PO LN, quàm duo KGEF; tàm igitur &longs;u&longs;tinebit vectis <lb/>ON, quàm vectis GF. quare in vtroq; vecte ON GF æquali <lb/>ter pondus <expan abbr="põderabit">ponderabit</expan>. </s> <s id="id.2.1.151.4.1.7.0">erit ergo vnaquæq; potentia in PH &longs;ubquadru<arrow.to.target n="note235"></arrow.to.target><lb/>pla ponderis A. & cùm funis KG potentiæ loco &longs;umatur, quippè <lb/>qui haud &longs;ecus &longs;u&longs;tinet, quàm PO; erit potentia in P &longs;u&longs;tinens pon­<lb/>dus A ip&longs;ius ponderis &longs;ubquadrupla. </s> <s id="id.2.1.151.4.1.8.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.152.1.0.0.0" type="margin"> <s id="id.2.1.152.1.1.1.0"><margin.target id="note233"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.152.1.1.2.0"><margin.target id="note234"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/>2 <emph type="italics"/>Cor.<emph.end type="italics"/> 2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.152.1.1.3.0"><margin.target id="note235"></margin.target>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/156.jpg"/> <p id="id.2.1.153.1.0.0.0" type="head"> <s id="id.2.1.153.1.2.1.0">COROLLARIVM I. </s> </p> <p id="id.2.1.153.2.0.0.0" type="main"> <s id="id.2.1.153.2.1.1.0">Hinc manife&longs;tum e&longs;t vnumquemq; funem EF <lb/>GK LN OP quartam &longs;u&longs;tinere partem pon­<lb/>deris A. </s> </p> <p id="id.2.1.153.3.0.0.0" type="head"> <s id="id.2.1.153.3.1.1.0">COROLLARIVM II. </s> </p> <p id="id.2.1.153.4.0.0.0" type="main"> <s id="id.2.1.153.4.1.1.0">Patet etiam orbiculum, cuius centrum C, <lb/>non minus eo, cuius centrum e&longs;t B, &longs;u&longs;tinere. </s> </p> <p id="id.2.1.153.5.0.0.0" type="head"> <s id="id.2.1.153.5.1.1.0">ALITER. </s> </p> <p id="id.2.1.153.6.0.0.0" type="main"> <s id="id.2.1.153.6.1.1.0">Adhuc ii&longs;dem po&longs;itis, &longs;i duæ e&longs;&longs;ent poten<lb/>tiæ æquales pondus A &longs;u&longs;tinentes, vna in O <lb/><arrow.to.target n="note236"></arrow.to.target>altera in C; e&longs;&longs;et vnaquæq; dictarum poten<lb/>tiarum ponderis A &longs;ubtripla. </s> <s id="id.2.1.153.6.1.2.0">&longs;ed quoniam <lb/>vectis GF, cuius fulcimentum e&longs;t F bifariam <lb/>diui&longs;us e&longs;t in C; &longs;i igitur ponatur in G poten<lb/>tia idem pondus &longs;u&longs;tinens, vt potentia in C; <lb/>erit potentia in G &longs;ubdupla potentiæ, quæ e&longs; <lb/>&longs;et in C; nam &longs;i potentia in C &longs;e ip&longs;a pon­<lb/>dus in C appen&longs;um &longs;u&longs;tineret, e&longs;&longs;et vtiq; ip<lb/>&longs;i ponderi æqualis; & idem pondus, &longs;i à po<lb/><arrow.to.target n="note237"></arrow.to.target>tentia in G &longs;u&longs;tineretur, e&longs;&longs;et ip&longs;ius poten<lb/>tiæ in G duplum; potentia veró in C &longs;ubtri<lb/>pla e&longs;&longs;et ponderis A; ergo potentia in G <lb/>&longs;ub&longs;excupla e&longs;&longs;et ponderis A. </s> <s id="id.2.1.153.6.1.2.0.a">Cùm itaq; <lb/>potentia in O &longs;ubtripla &longs;it ponderis A, & <lb/>potentia in G &longs;ub&longs;excupla; erunt vtræq; &longs;i­<lb/>mul potentiæ in OG ip&longs;ius ponderis A &longs;ub <lb/>duplæ. </s> <s id="id.2.1.153.6.1.3.0">tertia enim pars cum &longs;exta dimi­<lb/>dium efficit. </s> <s id="id.2.1.153.6.1.4.0">quoniam autem potentiæ in <lb/>OG, &longs;iue in PH (vt prius dictum e&longs;t) <lb/>&longs;unt inter &longs;e æquales, ac vtræq; &longs;imul &longs;ubdu<lb/>plæ &longs;unt ponderis A. erit vnaquæq; poten<lb/><figure id="id.036.01.156.1.jpg" xlink:href="036/01/156/1.jpg"></figure><pb n="72" xlink:href="036/01/157.jpg"/>tia in P H ip&longs;ius A &longs;ubquadrupla. </s> <s id="id.2.1.153.6.1.5.0">Potentia igitur in P &longs;u&longs;tinens pon<lb/>dus A ip&longs;ius ponderis A &longs;ubquadrupla erit. </s> <s id="id.2.1.153.6.1.6.0">quod erat o&longs;ten­<lb/>dendum. </s> </p> <p id="id.2.1.154.1.0.0.0" type="margin"> <s id="id.2.1.154.1.1.1.0"><margin.target id="note236"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 4 <emph type="italics"/>Huius<emph.end type="italics"/></s> <s id="id.2.1.154.1.1.2.0"><margin.target id="note237"></margin.target>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/></s> </p> <p id="id.2.1.155.1.0.0.0" type="main"> <s id="id.2.1.155.1.1.1.0">Si verò funis religetur in E, <lb/>& &longs;ecundùm quatuor adhuc <lb/>circumuoluatur orbiculos, per <lb/>ueniatq; ad P. &longs;imiliter o&longs;ten <lb/>detur potentiam in P &longs;ubqua­<lb/>druplam e&longs;&longs;e ponderis A. <lb/>idem enim e&longs;t, ac &longs;i funis re­<lb/>ligatus e&longs;&longs;et in L, potentiaq; <lb/>&longs;u&longs;tineret pondus fune tribus <lb/>tantùm orbiculis circumdu­<lb/>cto, quorum centra e&longs;&longs;ent B <lb/>CQ. </s> <s>orbiculus enim cuius <lb/>centrum D e&longs;t pœnitus inu­<lb/>tilis. <figure id="id.036.01.157.1.jpg" xlink:href="036/01/157/1.jpg"></figure></s> <pb xlink:href="036/01/158.jpg"/> <s id="id.2.1.155.1.3.1.0">PROPOSITIO VIII. </s> </p> <p id="id.2.1.155.2.0.0.0" type="main"> <s id="id.2.1.155.2.1.1.0">Sint duo vetes AB CD bifariam diui&longs;i in EF, <lb/>quorum fulcimenta &longs;int AC, & pondus G in <lb/>punctis EF vtriq; vecti &longs;it appen&longs;um, ita vt ex <lb/>vtroq; æqualiter ponderet; tre&longs;q; &longs;int potentiæ <lb/>æquales in BDE pondus G &longs;u&longs;tinentes. </s> <s id="id.2.1.155.2.1.2.0">Dico <lb/>vnamquamq; &longs;eor&longs;um ex dictis potentiis &longs;ub­<lb/>quintuplam e&longs;&longs;e ponderis G. </s> </p> <p id="id.2.1.155.3.0.0.0" type="main"> <s id="id.2.1.155.3.1.1.0">Quoniam enim pondus G <lb/>appen&longs;um e&longs;t in EF, & tres <lb/>&longs;unt potentiæ in EBD æqua<lb/>les; ideo potentia in E partem <lb/>tantùm ponderis G &longs;u&longs;tinebit <lb/>ip&longs;i potentiæ in E æqualem; <lb/>potentiæ verò in BD partem <lb/>&longs;u&longs;tinebunt reliquam; & pars, <lb/><arrow.to.target n="note238"></arrow.to.target>quam &longs;u&longs;tinet B, erit ip&longs;ius <lb/>dupla; pars autem, quam &longs;u<lb/><figure id="id.036.01.158.1.jpg" xlink:href="036/01/158/1.jpg"></figure><lb/>&longs;tinet D, erit &longs;imiliter ip&longs;ius D dupla; propter proportionem <lb/>BA ad AE, & DC ad CF. </s> <s id="id.2.1.155.3.1.1.0.a">Cùm itaq; potentiæ in BD &longs;int æqua<lb/><arrow.to.target n="note239"></arrow.to.target>les, erunt (ex iis, quæ &longs;upra dictum e&longs;t) partes ponderis G, quæ <lb/>à potentiis BD &longs;u&longs;tinentur, inter &longs;e &longs;e æquales; & vnaquæq; du<lb/>pla eius partis, quæ à potentia in E &longs;u&longs;tinetur. </s> <s id="id.2.1.155.3.1.2.0">diuidatur er­<lb/>go pondus G in tres partes, quarum duæ &longs;int inter &longs;e &longs;e æquales, <lb/>nec non vnaquæq; &longs;eor&longs;um alterius tertiæ partis dupla. </s> <s id="id.2.1.155.3.1.3.0">quod <lb/>fiet, &longs;i in quinq; partes æquales HKLMN diuidatur; pars <lb/>enim compo&longs;ita ex duabus partibus kL dupla e&longs;t partis H; pars <lb/>quoq; MN eiu&longs;dem partis H e&longs;t &longs;imiliter dupla. </s> <s id="id.2.1.155.3.1.4.0">quare & pars <lb/>kL parti MN erit æqualis. </s> <s id="id.2.1.155.3.1.5.0">Su&longs;tineat autem potentia in E par<lb/>tem H; & potentia in B partes KL; potentia verò in D partes <pb n="73" xlink:href="036/01/159.jpg"/>MN: tres igitur potentiæ æquales in BDE totum &longs;u&longs;tinebunt pon<lb/>dus G; & vnaquæq; potentia in BD duplum &longs;u&longs;tinebit eius, quod <lb/>&longs;u&longs;tinet potentia in E. </s> <s id="id.2.1.155.3.1.5.0.a">Cùm itaq; potentia in E partem H &longs;u&longs;ti­<lb/>neat, quæ quinta e&longs;t pars ponderis G, ip&longs;iq; &longs;it æqualis; erit po<lb/>tentia in E &longs;ubquintupla ponderis G. </s> <s id="id.2.1.155.3.1.5.0.b">& quoniam potentia in B <lb/>partes kL &longs;u&longs;tinet, quæ quidem duplæ &longs;unt potentiæ B, & partis H; <lb/>erit quoq; potentia in B ip&longs;i H æqualis: quare &longs;ubquintupla erit <lb/>ponderis G. </s> <s id="id.2.1.155.3.1.5.0.c">Non aliter o&longs;tendetur potentiam in D &longs;ubquintu­<lb/>plam e&longs;&longs;e ponderis G. vnaquæq; igitur potentia in BDE &longs;ubquin­<lb/>tupla e&longs;t ponderis G. quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.156.1.0.0.0" type="margin"> <s id="id.2.1.156.1.1.1.0"><margin.target id="note238"></margin.target>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/></s> <s id="id.2.1.156.1.1.3.0"><margin.target id="note239"></margin.target><emph type="italics"/>In<emph.end type="italics"/> 6 <emph type="italics"/>Huius<emph.end type="italics"/></s> </p> <p id="id.2.1.157.1.0.0.0" type="main"> <s id="id.2.1.157.1.1.1.0">Si verò &longs;int tres vectes AB <lb/>CD EF bifariam diui&longs;i in <lb/>GHk, quorum fulcimenta <lb/>&longs;int ACE; & pondus L eo <lb/>dem modo in GHk &longs;it ap­<lb/>pen&longs;um; quatuorq; &longs;int po­<lb/>tentiæ æquales in BDFG <lb/>pondus L &longs;u&longs;tinentes; &longs;imili <lb/>modo o&longs;tendetur vnam­<lb/>quamq; potentiam in BD <lb/>FG &longs;ub&longs;eptuplam e&longs;&longs;e ponde<lb/>ris L. & &longs;i quatuor e&longs;&longs;ent vectes, & quinq; potentiæ æquales pon­<lb/>dus &longs;u&longs;tinentes; eodem quoq; modo o&longs;tendetur vnamquamq; <lb/>potentiam &longs;ubnonuplam e&longs;&longs;e ponderis. </s> <s id="id.2.1.157.1.1.2.0">atq; ita deinceps. </s> </p> <figure id="id.036.01.159.1.jpg" xlink:href="036/01/159/1.jpg"> </figure> <p id="id.2.1.157.1.2.1.0" type="head"> <s id="id.2.1.157.1.4.1.0">PROPOSITIO VIIII. </s> </p> <p id="id.2.1.157.2.0.0.0" type="main"> <s id="id.2.1.157.2.1.1.0">Si quatuor duarum trochlearum binis orbi­<lb/>culis, quarum altera &longs;upernè, altera vero <expan abbr="in­fernè">in­<lb/>ferne</expan>, ponderiq; alligata, di&longs;po&longs;ita fuerit, cir<lb/>cumducatur funis; altero eius extremo inferiori <pb xlink:href="036/01/160.jpg"/>trochleæ religato, altero verò à potentia pon­<lb/>dus &longs;u&longs;tinente retento: erit potentia ponderis <lb/>&longs;ubquintupla. </s> </p> <p id="id.2.1.157.3.0.0.0" type="main"> <s id="id.2.1.157.3.1.1.0">Sit pondus A, cui alligata &longs;it trochlea duos <lb/>habens orbiculos, quorum centra &longs;int BC; <lb/>&longs;itq; trochlea &longs;ur&longs;um appen&longs;a duos alios ha­<lb/>bens orbiculos, quorum centra &longs;int DE; funi&longs;q; <lb/>per omnes circumducatur orbiculos, qui tro­<lb/>chleæ inferiori religetur in F; &longs;it qué poten<lb/>tia in G &longs;u&longs;tinens pondus A. </s> <s id="id.2.1.157.3.1.1.0.a">dico poten­<lb/>tiam in G &longs;ubquintuplam e&longs;&longs;e ponderis A. <lb/>ducantur Hk LM per centra BC horizon­<lb/>ti æquidi&longs;tantes, quas eodem modo, quo &longs;u­<lb/>pra dictum e&longs;t, e&longs;&longs;e tanquam vectes o&longs;tende­<lb/>mus, quorum fulcimenta kM, & pondus A <lb/>ex medio vtriu&longs;q; vectis BC &longs;u&longs;pen&longs;um, & tres <lb/>potentiæ in LHC pondus &longs;u&longs;tinentes, quas <lb/>&longs;imili modo æquales e&longs;&longs;e demon&longs;trabimus; fu<lb/>nes enim idem efficiunt, ac &longs;i e&longs;&longs;ent potentiæ. </s> <s id="id.2.1.157.3.1.2.0"><lb/>& quoniam pondus æqualiter ex vtroq; ve­<lb/>cte HK LM ponderat, quod quidem o&longs;ten­<lb/>detur quoque, vt in præcedentibus demon­<lb/><arrow.to.target n="note240"></arrow.to.target>&longs;tratum e&longs;t: erit vnaquæq; potentia, tùm in <lb/>L, &longs;eu in G, quod idem e&longs;t; tùm in H, atq; <lb/>in C, hoc e&longs;t in F, &longs;ubquintupla ponderis A. </s> <s id="id.2.1.157.3.1.2.0.a"><lb/>Potentia ergo in G &longs;u&longs;tinens pondus A ip&longs;ius <lb/>A &longs;ubquintupla erit. </s> <s id="id.2.1.157.3.1.3.0">quod o&longs;tendere opor­<lb/>tebat. <figure id="id.036.01.160.1.jpg" xlink:href="036/01/160/1.jpg"></figure></s> </p> <pb n="74" xlink:href="036/01/161.jpg"/> <p id="id.2.1.157.5.0.0.0" type="main"> <s id="id.2.1.157.5.1.1.0">Si verò funis in F adhuc de­<lb/>feratur circa alium orbiculum, <lb/>cuius centrum N, qui religetur <lb/>in O; &longs;imiliter duplici medio <lb/>(vt in &longs;eptima huius) demon<lb/>&longs;trabitur potentiam in G pon­<lb/>dus A &longs;u&longs;tinentem &longs;ub&longs;excu<arrow.to.target n="note241"></arrow.to.target><lb/>plam e&longs;&longs;e ponderis A. </s> <s id="id.2.1.157.5.1.1.0.a">Primùm <lb/>quidem ex tribus vectibus LM <lb/>Hk FP, quorum fulcimenta <lb/>&longs;unt MkP, & pondus in me <lb/>dio vectium appen&longs;um; & tres <lb/>potentiæ in LHF æquales pon<lb/>dus &longs;u&longs;tinéres. </s> <s id="id.2.1.157.5.1.2.0">deinde ex poten<arrow.to.target n="note242"></arrow.to.target><lb/>tiis in LHN, quarum vnaquæq; <lb/>&longs;ubquintupla e&longs;&longs;et ponderis A. <lb/>e&longs;&longs;ent enim ambæ &longs;imul poten<lb/>tiæ in LH &longs;ubduplæ &longs;exquialte<lb/>ræ ip&longs;ius ponderis, <expan abbr="pot&etilde;tia">potentia</expan> verò <lb/>in F &longs;ubdecupla e&longs;&longs;et, cùm &longs;it ip<lb/>&longs;ius N &longs;ubdupla: &longs;ed duæ quin <lb/>tæ cùm decima dimidium ef<lb/>ficiunt, quòd &longs;i per terna diui <lb/>datur, &longs;exta pars ponderis re<lb/>&longs;pondebit vnicuiq; potentiæ in <lb/>LHF. ex quibus patet poten<lb/>tiam in G &longs;ub&longs;excuplam e&longs;&longs;e <lb/>ponderis A. &longs;imiliterq; demon<lb/>&longs;trabitur vnumquemque orbi<lb/>culum æqualem &longs;u&longs;tinere por­<lb/>tionem. <figure id="id.036.01.161.1.jpg" xlink:href="036/01/161/1.jpg"></figure></s> </p> <pb xlink:href="036/01/162.jpg"/> <p id="id.2.1.157.7.0.0.0" type="main"> <s id="id.2.1.157.7.1.1.0">Quòd &longs;i, vt in tertia figura <lb/>funis in O protrahatur; per <lb/>aliumq; circumducatur orbi­<lb/>culum, cuius centrum Q; qui <lb/>deinde in R trochleæ relige­<lb/>tur inferiori; erit potentia in <lb/><arrow.to.target n="note243"></arrow.to.target>G ponderis &longs;ub&longs;eptupla. </s> <s id="id.2.1.157.7.1.2.0">atq; <lb/>ita in infinitum procedendo <lb/>proportio potentiæ ad pon­<lb/>dus quotcunq; &longs;ubmulti­<lb/>plex inueniri poterit. </s> <s id="id.2.1.157.7.1.3.0">dein­<lb/>de &longs;emper o&longs;tendetur vt in <lb/>præcedentibus; &longs;i potentia <lb/>pondus &longs;u&longs;tinens fuerit, vel <lb/>&longs;ubquadrupla, vel &longs;ubquitu­<lb/>pla, vel quouis alio modo &longs;e <lb/>habebit ad pondus; &longs;imiliter <lb/>vnumquemque funem, vel <lb/>quartam, vel quintam, vel <lb/>quamuis aliam partem &longs;u&longs;ti­<lb/>nere ponderis, quemadmo­<lb/>dum potentia ip&longs;a; funes e­<lb/>nim idem efficiunt, ac &longs;i tot <lb/>e&longs;&longs;ent potentiæ: orbiculi ve <lb/>rò, ac &longs;i tot e&longs;&longs;ent vectes. </s> </p> <p id="id.2.1.158.1.0.0.0" type="margin"> <s id="id.2.1.158.1.1.1.0"><margin.target id="note240"></margin.target>8 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.158.1.1.2.0"><margin.target id="note241"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 6 <emph type="italics"/>huius<emph.end type="italics"/></s> <s id="id.2.1.158.1.1.3.0"><margin.target id="note242"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 8 <emph type="italics"/>huius<emph.end type="italics"/></s> <s id="id.2.1.158.1.1.4.0"><margin.target id="note243"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 8 <emph type="italics"/>Huius<emph.end type="italics"/></s> </p> <figure id="id.036.01.162.1.jpg" xlink:href="036/01/162/1.jpg"> </figure> <p id="id.2.1.159.1.1.1.0" type="head"> <s id="id.2.1.159.1.3.1.0">COROLLARIVM </s> </p> <p id="id.2.1.159.2.0.0.0" type="main"> <s id="id.2.1.159.2.1.1.0">Ex his manife&longs;tum e&longs;t orbiculos trochleæ, cui <lb/>e&longs;t alligatum pondus, efficere, vt pondus mino­<pb n="75" xlink:href="036/01/163.jpg"/>re &longs;u&longs;tineatur potentia, quàm &longs;it ip&longs;um pondus; <lb/>quod quidem trochleæ &longs;uperioris orbiculi non <lb/>efficiunt. </s> </p> <p id="id.2.1.159.3.0.0.0" type="main"> <s id="id.2.1.159.3.1.1.0">Noui&longs;&longs;e tamen oportet, quòd (vt fieri &longs;olet) inferioris tro<lb/>chleæ orbiculus, cuius centrum N, minor e&longs;&longs;e debet eo, cuius cen<lb/>trum C; hic autem minor adhuc eo, cuius centrum B; ac deniq; <lb/>&longs;i plures fuerint orbiculi in trochlea inferiori ponderi alligata, &longs;em<lb/>per cæteris maior e&longs;&longs;e debet, qui annexo ponderi e&longs;t propinquior. </s> <s id="id.2.1.159.3.1.2.0"><lb/>oppo&longs;ito autem modo di&longs;ponendi &longs;unt in trochlea &longs;uperiori. </s> <s id="id.2.1.159.3.1.3.0">quod <lb/>fieri con&longs;ueuit, ne funes inuicem complicentur; nam quantùm <lb/>ad orbiculos attinet, &longs;iue magni fuerint, &longs;iue parui, nihil refert; <lb/>cùm &longs;emper idem &longs;equatur. </s> </p> <p id="id.2.1.159.4.0.0.0" type="main"> <s id="id.2.1.159.4.1.1.0">Præterea notandum e&longs;t, quod etiam ex dictis facilè patet, &longs;i <lb/>funis, &longs;iue religetur in R trochleæ inferiori, &longs;iue in S, maximam <lb/>indè oriri differentiam inter potentiam, & pondus: nam &longs;i relige<lb/>tur in S, erit potentia in G ponderis &longs;ub&longs;excupla. </s> <s id="id.2.1.159.4.1.2.0">&longs;i verò in R, <lb/>&longs;ub&longs;eptupla. </s> <s id="id.2.1.159.4.1.3.0">quod trochleæ &longs;uperiori non contingit, quia &longs;iue <lb/>religetur funis (vt in præcedenti figura) in T, &longs;iue in O; &longs;em<lb/>per potentia in G &longs;ub&longs;excupla erit ip&longs;ius ponderis. </s> </p> <p id="id.2.1.159.5.0.0.0" type="main"> <s id="id.2.1.159.5.1.1.0">Po&longs;t hæc con&longs;iderandum e&longs;t, quonam modo vis moueat pon<lb/>dus; necnon potentiæ mouentis, ponderi&longs;q; moti &longs;patium, atque <lb/>tempus. </s> </p> <p id="id.2.1.159.6.0.0.0" type="head"> <s id="id.2.1.159.6.1.1.0">PROPOSITIO X. </s> </p> <p id="id.2.1.159.7.0.0.0" type="main"> <s id="id.2.1.159.7.1.1.0">Si funis orbiculo trochleæ &longs;ur&longs;um appen&longs;æ <lb/>fuerit circumuolutus, cuius altero extremo &longs;it al<lb/>ligatum pondus; alteri autem mouens collocata <lb/>&longs;it potentia: mouebit hæc vecte horizonti &longs;em­<lb/>per æquidi&longs;tante. </s> </p> <pb xlink:href="036/01/164.jpg"/> <p id="id.2.1.159.9.0.0.0" type="main"> <s id="id.2.1.159.9.1.1.0">Sit pondus A, &longs;it orbiculus trochleæ &longs;ur<lb/>&longs;um appen&longs;æ' cuius centrum K; &longs;it deinde <lb/>funis HBCDEF aligatus ponderi A in H, <lb/>orbiculoq; circumductus; &longs;itq; trochlea ita in <lb/>L appen&longs;a, & nullum alium habeat motum <lb/>præter liberam orbiculi circa axem ver&longs;ionem; <lb/>&longs;itq; potentia in F mouens pondus A. </s> <s id="id.2.1.159.9.1.1.0.a">Dico <lb/>potentiam in F &longs;emper mouere pondus A <lb/>vecte horizonti æquidi&longs;tante. </s> <s id="id.2.1.159.9.1.2.0">ducatur BKE <lb/>horizonti æquidi&longs;tans; &longs;intq; BE puncta, vbi <lb/>funes BH, & EF circulum tangunt; erit BkE <lb/><arrow.to.target n="note244"></arrow.to.target>vectis, cuius fulcimentum e&longs;t in eius medio <lb/>k. </s> <s id="id.2.1.159.9.1.3.0">&longs;icut &longs;upra o&longs;ten&longs;um e&longs;t. </s> <s id="id.2.1.159.9.1.4.0">dum itaq; vis <lb/>in F deor&longs;um tendit ver&longs;us M, vectis EB <lb/>mouebitur, cùm totus orbiculus moueatur, <lb/><figure id="id.036.01.164.1.jpg" xlink:href="036/01/164/1.jpg"></figure><lb/>hoc e&longs;t circumuertatur. </s> <s id="id.2.1.159.9.1.5.0">dum igitur F e&longs;t in M, &longs;it punctum E ve<lb/>ctis v&longs;q; ad I motum; B autem v&longs;q; ad C, ita vt vectis &longs;it in <lb/>CI. </s> <s id="id.2.1.159.9.1.5.0.a">fiat deinde NM æqualis ip&longs;i FE: & quando punctum E <lb/>erit in I, <expan abbr="tnnc">tunc</expan> funis punctum, quod erat in E, erit in N: quod au<lb/>tem erat in B erit in C; ita vt ducta CI per centrum K tran&longs;eat. </s> <s id="id.2.1.159.9.1.6.0"><lb/>dum autem B e&longs;t in C, &longs;it punctum H in G; eritq; BH ip&longs;i <lb/>CBG æqualis; cùm &longs;it idem funis. </s> <s id="id.2.1.159.9.1.7.0">& quoniam dum EF tendit <lb/>in NM, adhuc &longs;emper remanet EFM horizonti perpendicularis, <lb/>circulumq; tangens in puncto E; ita vt ducta à puncto E per cen<lb/>trum k, &longs;it &longs;emper horizonti æquidi&longs;tans. </s> <s id="id.2.1.159.9.1.8.0">quod idem euenit funi <lb/>BG, & puncto B. dum igitur circulus, &longs;iue orbiculus circumuer<lb/>titur, &longs;emper mouetur vectis EB, &longs;emperq; adhuc remanet alius <lb/>vectis in EB. </s> <s id="id.2.1.159.9.1.8.0.a">&longs;iquidem ex ip&longs;ius rotulæ natura, in qua &longs;emper <lb/>dum mouetur, remanet diameter ex B in E (quæ vectis vicem ge<lb/>rit) euenit, vt recedente vna, &longs;emper altera &longs;uccedat; eiu&longs;modi <lb/>durante circumductione: atq; ita fit, vt potentia &longs;emper moueat <lb/>pondus vecte EB horizonti æquidi&longs;tante. </s> <s id="id.2.1.159.9.1.9.0">quod demon&longs;trare opor­<lb/>tebat. </s> </p> <p id="id.2.1.160.1.0.0.0" type="margin"> <s id="id.2.1.160.1.1.1.0"><margin.target id="note244"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="76" xlink:href="036/01/165.jpg"/> <p id="id.2.1.161.1.0.0.0" type="main"> <s id="id.2.1.161.1.2.1.0">Ii&longs;dem po&longs;itis, &longs;patium potentiæ pondus <lb/>mouentis e&longs;t æquale &longs;patio eiu&longs;dem ponderis <lb/>moti. </s> </p> <p id="id.2.1.161.2.0.0.0" type="main"> <s id="id.2.1.161.2.1.1.0">Quoniam enim o&longs;ten&longs;um e&longs;t, dum F e&longs;t in M, pondus A, hoc <lb/>e&longs;t punctum H e&longs;&longs;e in G; & cùm funis HBCDEF &longs;it æqualis <lb/>GBCDENFM, e&longs;t enim idem funis; dempto igitur communi <lb/>GBCDENF, erit HG ip&longs;i FM æqualis. </s> <s id="id.2.1.161.2.1.2.0">&longs;imiliterq; o&longs;tende­<lb/>tur, de&longs;cen&longs;um F &longs;emper æqualem e&longs;&longs;e a&longs;cen&longs;ui H. ergo &longs;patium <lb/>potentiæ æquale e&longs;t &longs;patio ponderis. </s> <s id="id.2.1.161.2.1.3.0">quod erat demon&longs;tran­<lb/>dum. </s> </p> <p id="id.2.1.161.3.0.0.0" type="main"> <s id="id.2.1.161.3.1.1.0">Præterea potentia idem pondus per æquale <lb/>&longs;patium in æquali tempore mouet, tàm fune <lb/>hoc modo orbiculo trochleæ &longs;ur&longs;um appen&longs;æ <lb/>circumuoluto, quàm &longs;ine trochlea: dummo­<lb/>do ip&longs;ius potentiæ lationes in velocitate &longs;int æ­<lb/>quales. </s> </p> <pb xlink:href="036/01/166.jpg"/> <p id="id.2.1.161.5.0.0.0" type="main"> <s id="id.2.1.161.5.1.1.0">Ii&longs;dem po&longs;itis &longs;it aliud pondus P <lb/>æquale ponderi A, cui alligatus &longs;it <lb/>funis TQ <expan abbr="horizõti">horizonti</expan> <expan abbr="perp&etilde;dicularis">perpendicularis</expan>; <lb/>et &longs;it TQ ip&longs;i HB æqualis; moueat <lb/>qué <expan abbr="pot&etilde;tia">potentia</expan> in Q <expan abbr="põdus">pondus</expan> P &longs;ur&longs;um <lb/>ad rectos angulos horizonti, quem <lb/>admodum mouetur pondus A. </s> <s id="id.2.1.161.5.1.1.0.a">di<lb/>co per æquale &longs;patium in eodem <lb/>tempore potentiam in Q pondus <lb/>P, & potentiam in F pondus A <lb/>mouere. </s> <s id="id.2.1.161.5.1.2.0">quod idem e&longs;t, ac &longs;i e&longs;&longs;et <lb/>idem pondus in æquali tempore <lb/>motum; &longs;icut propo&longs;uimus. </s> <s id="id.2.1.161.5.1.3.0">Pro­<lb/>ducatur EF in S, & TQ in R; <lb/>fiantq; QR FS non &longs;olum inter <lb/>&longs;e &longs;e, verùm etiam ip&longs;i BH æqua<lb/>les. </s> <s id="id.2.1.161.5.1.4.0">Cùm autem TQ QR &longs;int <lb/>ip&longs;is HB FS æquales, & vis in Q <lb/>moueat pondus P per rectam T <lb/>QR; vis autem in F moueat A <lb/>per rectam HB, & velocitates <lb/><figure id="id.036.01.166.1.jpg" xlink:href="036/01/166/1.jpg"></figure><lb/>motuum vtriu&longs;q; potentiæ &longs;int æquales; tunc in eodem tempore <lb/>potentia in Q erit in R, & potentia in F erit in S; cùm &longs;patia &longs;int <lb/>æqualia. </s> <s id="id.2.1.161.5.1.5.0">&longs;ed dum potentia in Q e&longs;t in R, pondus P, hoc e&longs;t <lb/>punctum T erit in Q; cùm TQ &longs;it ip&longs;i QR æqualis. </s> <s id="id.2.1.161.5.1.6.0">& dum po<lb/>tentia in F e&longs;t in S, pondus A, hoc e&longs;t punctum H erit in B; &longs;ed <lb/>&longs;patium TQ æquale e&longs;t &longs;patio HB, potentiæ ergo in FQ æquali <lb/>ter motæ pondera PA æqualia per æqualia &longs;patia in eodem tempo<lb/>re mouebunt. </s> <s id="id.2.1.161.5.1.7.0">quod erat demon&longs;trandum </s> </p> <p id="id.2.1.161.6.0.0.0" type="head"> <s id="id.2.1.161.6.1.1.0">PROPOSITIO XI. </s> </p> <p id="id.2.1.161.7.0.0.0" type="main"> <s id="id.2.1.161.7.1.1.0">Si funis orbiculo trochleæ ponderi alligatæ <lb/>fuerit circumuolutus, qui in altero eius extre­<pb n="77" xlink:href="036/01/167.jpg"/>mo alicubi religetur, altero autem à potentia <lb/>mouente pondus appræhen&longs;o; vecte &longs;emper ho<lb/>rizonti æqui&longs;tante potentia mouebit. </s> </p> <p id="id.2.1.161.8.0.0.0" type="main"> <s id="id.2.1.161.8.1.1.0">Sit pondus A; Sit orbiculus. </s> <s id="id.2.1.161.8.1.2.0"><lb/>CED trochleæ ponderi A alli­<lb/>gatæ ex kH; &longs;itq; KH ad rectos <lb/>angulos horizonti, ita vt pon­<lb/>dus &longs;emper trochleæ motum, &longs;i­<lb/>ue &longs;ur&longs;um, &longs;iue deor&longs;um factum <lb/>&longs;equatur; &longs;itq; orbiculi centrum <lb/>K; & funis orbiculo circumuo­<lb/>lutus &longs;it BCDEF, qui relige­<lb/>tur in B, ita vt in B immobilis <lb/>maneat; & &longs;it potentia in F mo­<lb/>uens pondus A. </s> <s id="id.2.1.161.8.1.2.0.a">dico potentia m <lb/>in F &longs;emper mouere <expan abbr="põdus">pondus</expan> A ve<lb/>cte horizonti æquidi&longs;tante. </s> <s id="id.2.1.161.8.1.3.0">&longs;int <lb/>BC EF inter &longs;e &longs;e, ip&longs;iq; kH æ­<lb/>quidi&longs;tantes, & eiu&longs;dem kH ho<lb/>rizonti perpendiculares, tangen<lb/>te&longs;q; <expan abbr="circulũ">circulum</expan> CED in EC <expan abbr="pũctis">punctis</expan>; <lb/>et connectatur EC, quæ per cen<arrow.to.target n="note245"></arrow.to.target><lb/>trum k tran&longs;ibit, horizontiq; <lb/>æquidi&longs;tans erit; &longs;icuti prius di<lb/>ctum e&longs;t. </s> <s id="id.2.1.161.8.1.4.0">Quoniam enim or<lb/>biculus CED circa eius cen<lb/>trum K vertitur; ideo dum vis <lb/>in F trahit &longs;ur&longs;um punctum E, <lb/>deberet punctum C de&longs;cende <lb/>re, ac trahere deor&longs;um B; &longs;ed fu<lb/><figure id="id.036.01.167.1.jpg" xlink:href="036/01/167/1.jpg"></figure><lb/>nis in B e&longs;t immobilis, & BC <expan abbr="de&longs;cedere">descendere</expan> non pote&longs;t; quare dum <lb/>potentia in F trahit &longs;ur&longs;um E, totus orbiculus &longs;ur&longs;um mouebitur; <lb/>ac per con&longs;equens tota trochlea, & pondus; & EkC erit tanquam <arrow.to.target n="note246"></arrow.to.target><lb/>vectis, cuius fulcimentum erit C; e&longs;t enim punctum C propter BC <lb/>ferè immobile, potentia verò mouens vectem e&longs;t in F fune EF, <pb xlink:href="036/01/168.jpg"/>& pondus in k appen&longs;um. </s> <s id="id.2.1.161.8.1.5.0"><lb/>quòd &longs;i punctum C omnino fue<lb/>rit immobile, moueaturq; ve<lb/>ctis EC in NC; & diuidatur <lb/>NC bifariam in L: erunt CL <lb/>LN ip&longs;is Ck KE æquales. </s> <s id="id.2.1.161.8.1.6.0"><lb/>quare &longs;i vectis EC e&longs;&longs;et in CN, <lb/>punctum k e&longs;&longs;et in L; & &longs;i du<lb/>catur LM horizonti perpendi<lb/>cularis, quæ &longs;it etiam æqualis <lb/>kH; e&longs;&longs;et pondus A, hoc e&longs;t <lb/>punctum H in M. </s> <s id="id.2.1.161.8.1.6.0.a">&longs;ed quoniam <lb/>potentia in F dum tendit &longs;ur­<lb/>&longs;um mouendo orbiculum, &longs;em<lb/>per mouetur &longs;uper rectam EFG, <lb/>quæ &longs;emper e&longs;t quoq; æquidi<lb/>&longs;tans BC; nece&longs;&longs;e erit orbicu<lb/>lum trochleæ &longs;emper inter li­<lb/>neas EG BC e&longs;&longs;e: & centrum <lb/>k, cum &longs;it in medio, &longs;uper <lb/>rectam lineam HkT &longs;emper <lb/>moueri. </s> <s id="id.2.1.161.8.1.7.0">Itaq; ducatur per L li<lb/>nea PTLQ horizonti, & EC <lb/>æquidi&longs;tans, quæ &longs;ecet Hk pro­<lb/>ductam in T; & centro T, &longs;pa<lb/>tio verò TQ, circulus de&longs;criba<lb/><figure id="id.036.01.168.1.jpg" xlink:href="036/01/168/1.jpg"></figure><lb/>tur QRPS, qui æqualis erit circulo CED; & puncta PQ tangent fu<lb/><arrow.to.target n="note247"></arrow.to.target>nes FE BC in PQ punctis. </s> <s id="id.2.1.161.8.1.8.0">rectangulum enim e&longs;t PECQ, & <lb/>PT TQ ip&longs;is EK kC &longs;unt æquales. </s> <s id="id.2.1.161.8.1.9.0">deinde per T ducatur R <lb/>TS diameter circuli PQS æquidi&longs;tans ip&longs;i NC; fiatqué TO æqua <lb/>lis kH. </s> <s id="id.2.1.161.8.1.9.0.a">dum autem centrum k motum erit v&longs;q; ad lineam PQ, <lb/>tunc centrum k erit in T. o&longs;ten&longs;um e&longs;t enim centrum orbiculi &longs;u<lb/>per rectam HT &longs;emper moueri. </s> <s id="id.2.1.161.8.1.10.0">idcirco vt centrum k &longs;it in li<lb/>nea PQ ip&longs;i EC æquidi&longs;tante, nece&longs;&longs;e e&longs;t vt &longs;it in T. & vt vectis <lb/>EC eleuetur in angulo ECN, nece&longs;&longs;e e&longs;t, vt &longs;it in RS, non au­<lb/><arrow.to.target n="note248"></arrow.to.target>tem in CN: angulus enim RSE angulo NCE e&longs;t æqualis, & &longs;ic <pb n="78" xlink:href="036/01/169.jpg"/>fulcimentum C non e&longs;t penitus immobile. </s> <s id="id.2.1.161.8.1.11.0">cùm totus orbiculus &longs;ur<lb/>&longs;um moueatur, toru&longs;q; mutet totum locum; habet tamen C ratio <lb/>nem fulcimenti, quia minus mouetur C, quàm k, & E: punctum <lb/>enim E mouetur v&longs;q; ad R, & K v&longs;q; ad T, punctum verò C v&longs;q; <lb/>ad S tantùm. </s> <s id="id.2.1.161.8.1.12.0">quare dum centrum K e&longs;t in T, po&longs;itio orbiculi erit <lb/>QR PS: & pondus A. hoc e&longs;t punctum H erit in O; cùm TO <lb/>&longs;it æqualis kH; po&longs;itio verò EC, &longs;cilicet vectis moti, erit RS, po<lb/>tentiaq; in F mota erit &longs;ur&longs;um per rectam EFG. </s> <s id="id.2.1.161.8.1.12.0.a">eodem autem <lb/>tempore, quo k erit in T, &longs;it potentia in G: dum autem vectis EC <lb/>hoc modo mouetur, adhuc &longs;emper remanent GP BQ inter &longs;e &longs;e æ­<lb/>quidi&longs;tantes, atq; horizonti perpendiculares, ita vt vbi orbiculum <lb/>tangunt, vt in punctis PQ; &longs;emper linea PQ erit diameter orbi <lb/>culi, & tanquam vectis horizonti æquidi&longs;tans. </s> <s id="id.2.1.161.8.1.13.0">dum igitur orbi­<lb/>culus mouetur, & circumuertitur, &longs;emper etiam mouetur vectis <lb/>EC, & &longs;emper remanet alius vectis in orbiculo horizonti æqui&longs;tans, <lb/>vt PQ; ita vt potentia in F &longs;emper moueat pondus vecte hori<lb/>zonti æquidi&longs;tante, cuius fulcimentum erit &longs;emper in linea CB; & <lb/>pondus in medio vectis appen&longs;um; potentiaq; in linea EG. quod <lb/>erat o&longs;tendendum. </s> </p> <p id="id.2.1.162.1.0.0.0" type="margin"> <s id="id.2.1.162.1.1.1.0"><margin.target id="note245"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 1 <emph type="italics"/>huius<emph.end type="italics"/></s> <s id="id.2.1.162.1.1.2.0"><margin.target id="note246"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 2 <emph type="italics"/>huius<emph.end type="italics"/></s> <s id="id.2.1.162.1.1.3.0"><margin.target id="note247"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 34 <emph type="italics"/>primi.<emph.end type="italics"/></s> <s id="id.2.1.162.1.1.4.0"><margin.target id="note248"></margin.target>29 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.163.1.0.0.0" type="main"> <s id="id.2.1.163.1.1.1.0">Ii&longs;dem po&longs;itis, &longs;patium potentiæ pondus <lb/>mouentis duplum e&longs;t &longs;patii eiu&longs;dem ponderis <lb/>moti. </s> </p> <p id="id.2.1.163.2.0.0.0" type="main"> <s id="id.2.1.163.2.1.1.0">Cùm enim o&longs;ten&longs;um &longs;it, dum k e&longs;t in T, pondus A, hoc e&longs;t <lb/>punctum H e&longs;&longs;e in O, & in eodem etiam tempore potentiam in <lb/>F e&longs;&longs;e in G: & quoniam funis BCDEF e&longs;t æqualis funi BQS <lb/>PG; funis enim e&longs;t idem; & funis circa &longs;emicirculum CDE e&longs;t <lb/>æqualis funi circa &longs;emicirculum QSP; demptis igitur communi<lb/>bus BQ, & FP; erit reliquus FG ip&longs;is CQ, & EP &longs;imul &longs;umptis <lb/>æqualis. </s> <s id="id.2.1.163.2.1.2.0">&longs;ed EP ip&longs;i TK e&longs;t æqualis, & CQ ip&longs;i quoq; Tk æqualis, <lb/>&longs;unt enim Pk TC parallelogramma rectangula; quare lineæ EP <lb/>CQ &longs;imul ip&longs;ius Tk duplæ erunt. </s> <s id="id.2.1.163.2.1.3.0">funis igitur FC ip&longs;ius TK du<lb/>plus erit. </s> <s id="id.2.1.163.2.1.4.0">& quoniam kH e&longs;t æqualis TO, dempto communi kO, <lb/>erit kT ip&longs;i HO æqualis; quare funis FG ip&longs;ius HO duplus erit; <pb xlink:href="036/01/170.jpg"/>hoc e&longs;t &longs;patium potentiæ &longs;patii ponderis duplum. </s> <s id="id.2.1.163.2.1.5.0">quod erat <lb/>demon&longs;trandum. </s> </p> <p id="id.2.1.163.3.0.0.0" type="main"> <s id="id.2.1.163.3.1.1.0">Potentia deinde idem pondus in æquali tem­<lb/>pore per dimidium &longs;patium mouebit fune circa <lb/>orbiculum trochleæ ponderi alligatæ reuoluto, <lb/>quàm &longs;ine trochlea; dummodo ip&longs;ius potentiæ <lb/>velocitates motuum &longs;int æquales. </s> </p> <p id="id.2.1.163.4.0.0.0" type="main"> <s id="id.2.1.163.4.1.1.0">Sit enim (ii&longs;dem po&longs;i<lb/>tis) aliud pondus V æqua <lb/>le ponderi A, cui alligatus <lb/>&longs;it funis 9X; &longs;itq; poten<lb/>tia in X mouens pondus <lb/>V. </s> <s id="id.2.1.163.4.1.1.0.a">dico &longs;i vtriu&longs;q; poten<lb/>tiæ motuum velocitates <lb/>&longs;int æquales, in eodem <lb/>tempore potentiam in F <lb/>mouere pondus A per di<lb/>midium &longs;patium eius, per <lb/>quod à potentia in X mo<lb/>uetur pondus V; quod <lb/>idem e&longs;t, ac &longs;i e&longs;&longs;et idem <lb/>pondus in æquali tempo<lb/>re motum. </s> <s id="id.2.1.163.4.1.2.0">Moueat po<lb/>tentia in X pondus V, po<lb/>tentiaq; perueniat in Y; <lb/>&longs;itq; XY æqualis ip&longs;i FG; <lb/>& fiat YZ æqualis X9, ita <lb/>vt quando potentia in X <lb/>erit in Y, &longs;it pondus V, <lb/>hoc e&longs;t punctum 9 in Z. </s> <s id="id.2.1.163.4.1.2.0.a"><lb/>&longs;ed 9 Z e&longs;t æqualis FG, <lb/><figure id="id.036.01.170.1.jpg" xlink:href="036/01/170/1.jpg"></figure><pb n="79" xlink:href="036/01/171.jpg"/>cùm &longs;it æqualis XY; ergo 9 Z ip&longs;ius HO dupla erit. </s> <s id="id.2.1.163.4.1.3.0">Itaq; dum poten<lb/>tiæ erunt in GY, pondera AV erunt in OZ. in eodem autem <lb/>tempore erunt potentiæ in GY, ip&longs;arum enim velocitates mo <lb/>tuum &longs;unt æquales; quare vis in F pondus A in eodem tempore <lb/>mouebit per dimidium &longs;patium eius, per quod mouetur à poten<lb/>tia in X pondus V: & pondera &longs;unt æqualia; Potentia ergo idem <lb/>pondus in æquali tempore per dimidium &longs;patium mouebit fune, <lb/>trochleaq; hoc modo ponderi alligata, quàm &longs;ine trochlea; dum <lb/>modo potentiæ motuum velocitates &longs;int æquales. </s> <s id="id.2.1.163.4.1.4.0">quod erat de­<lb/>mon&longs;trandum. </s> </p> <p id="id.2.1.163.5.0.0.0" type="head"> <s id="id.2.1.163.5.1.1.0">PROPOSITIO XII. </s> </p> <p id="id.2.1.163.6.0.0.0" type="main"> <s id="id.2.1.163.6.1.1.0">Si funis circa plures reuoluatur orbiculos, al­<lb/>tero eius extremo alicubi religato, altero au­<lb/>tem à potentia pondus mouente detento; poten<lb/>tia vectibus horizonti &longs;emper æquidi&longs;tantibus <lb/>mouebit. </s> </p> <pb xlink:href="036/01/172.jpg"/> <p id="id.2.1.163.8.0.0.0" type="main"> <s id="id.2.1.163.8.1.1.0">Sit pondus A, &longs;it orbiculus CED tro­<lb/>chleæ ponderi alligatæ ex kS ad rectos an<lb/>gulos horizonti; ita vt pondus &longs;emper eius <lb/>motum &longs;ur&longs;um, ac deor&longs;um factum &longs;equa­<lb/>tur. </s> <s id="id.2.1.163.8.1.2.0">&longs;it deinde orbiculus circa centrum L <lb/>trochleæ &longs;ur&longs;um appen&longs;æ &longs;itq; funis circa <lb/>orbiculos reuolutus BCDEHMNO, <lb/>qui religatus &longs;it in B; &longs;itq; vis in O mouens <lb/>pondus A mouendo &longs;e deor&longs;um per OP. </s> <s id="id.2.1.163.8.1.2.0.a"><lb/>dico potentiam in O &longs;emper mouere pon­<lb/>dus A vectibus horizonti &longs;emper æquidi­<lb/>&longs;tantibus. </s> <s id="id.2.1.163.8.1.4.0">ducatur NH per centrum L ho<lb/><arrow.to.target n="note249"></arrow.to.target>rizonti æquidi&longs;tans, quæ erit vectis orbi­<lb/>culi, cuius centrum e&longs;t L. ducatur deinde <lb/>EC per centrum k &longs;imiliter horizonti æqui <lb/><arrow.to.target n="note250"></arrow.to.target>di&longs;tans, quæ etiam erit vectis orbiculi, cu­<lb/>ius centrum e&longs;t k. </s> <s id="id.2.1.163.8.1.5.0">Moueatur potentia in <lb/>O deor&longs;um, quæ dum deor&longs;um mouetur, ve<lb/>ctem NH mouebit; & dum vectis moue­<lb/><arrow.to.target n="note251"></arrow.to.target>tur, N deor&longs;um mouebitur, H verò &longs;ur­<lb/>&longs;um, vti &longs;upra dictum e&longs;t. </s> <s id="id.2.1.163.8.1.6.0">dum autem H <lb/>mouetur &longs;ur&longs;um, mouet etiam &longs;ur&longs;um E; & <lb/>vectem EC, cuius fulcimentum e&longs;t C, &longs;ed <lb/>fulcimentum C non pote&longs;t mouere deor­<lb/>&longs;um B; ideo orbiculus, cuius centrum K, &longs;ur<lb/><figure id="id.036.01.172.1.jpg" xlink:href="036/01/172/1.jpg"></figure><lb/>&longs;um mouebitur, & per con&longs;equens trochlea, & pondus A; vt in <lb/>præcedenti dictum e&longs;t. </s> <s id="id.2.1.163.8.1.7.0">& quoniam ob eandem cau&longs;am in præce­<lb/>dentibus a&longs;signatam in HN, & EC &longs;emper remanent vectes hori<lb/>zonti æquidi&longs;tantes; potentia ergo mouens pondus A &longs;emper <lb/>eum mouebit vectibus horizonti æquidi&longs;tantibus. </s> <s id="id.2.1.163.8.1.8.0">quod erat o­<lb/>&longs;tendendum. </s> </p> <p id="id.2.1.164.1.0.0.0" type="margin"> <s id="id.2.1.164.1.1.1.0"><margin.target id="note249"></margin.target>1, <emph type="italics"/>Et<emph.end type="italics"/> 10 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.164.1.1.2.0"><margin.target id="note250"></margin.target>11 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.164.1.1.3.0"><margin.target id="note251"></margin.target>10 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.165.1.0.0.0" type="main"> <s id="id.2.1.165.1.1.1.0">Et &longs;i funis circa plures &longs;it reuolutus orbiculos; &longs;imiliter o&longs;tende­<lb/>tur, potentiam mouere pondus vectibus horizonti &longs;emper æqui­<lb/>di&longs;tantibus: & vectes orbiculorum trochleæ &longs;uperioris &longs;emper <lb/>e&longs;&longs;e, vt HN, quorum fulcimenta erunt &longs;emper in medio: vectes au­<lb/>tem orbiculorum trochleæ inferioris &longs;emper exi&longs;tere, vt EC; quo­<pb n="80" xlink:href="036/01/173.jpg"/>rum fulcimenta erunt in extremitatibus vectium. </s> </p> <p id="id.2.1.165.2.0.0.0" type="main"> <s id="id.2.1.165.2.1.1.0">Ii&longs;dem po&longs;itis, &longs;patium potentiæ duplum e&longs;t <lb/>&longs;patii ponderis. </s> </p> <p id="id.2.1.165.3.0.0.0" type="main"> <s id="id.2.1.165.3.1.1.0">Sit motum centrum K v&longs;q; ad centrum R; & orbiculus &longs;it FTG. <lb/>deinde per centrum R ducatur GF ip&longs;i EC æquidi&longs;tans: tangent <lb/>funes EH CB orbiculum in GF punctis. </s> <s id="id.2.1.165.3.1.2.0">fiat deniq; RQ æqua <lb/>lis KS. dum igitur k erit in R; pondus A, &longs;cilicet punctum S erit <lb/>in q. & dum centrum orbiculi e&longs;t in R, &longs;it potentia in O mota <lb/>in P. </s> <s id="id.2.1.165.3.1.2.0.a">& quoniam funis BCDEHMNO e&longs;t æqualis funi BFT <lb/>GHMNP; e&longs;t enim idem funis; & FTG æqualis e&longs;t CDE; dem<lb/>ptis igitur communibus BF, & GHMNO, erit reliquus OP ip<lb/>&longs;is FCEG &longs;imul &longs;umptis æqualis: & per con&longs;equens duplus kR, <lb/>& QS & cùm OP &longs;it &longs;patium potentiæ motæ, & SQ &longs;patium pon<lb/>deris moti; erit &longs;patium potentiæ duplum &longs;patii ponderis. </s> <s id="id.2.1.165.3.1.3.0">quod <lb/>erat o&longs;tendendum. </s> </p> <p id="id.2.1.165.4.0.0.0" type="main"> <s id="id.2.1.165.4.1.1.0">Præterea potentia idem pondus in æquali <lb/>tempore per dimidium &longs;patium mouebit fune <lb/>circa duos orbiculos reuoluto, quorum vnus <lb/>&longs;it trochleæ &longs;uperioris, alter verò &longs;it trochleæ <lb/>ponderi alligatæ; quàm &longs;ine trochleis: dummo­<lb/>do ip&longs;ius potentiæ lationes &longs;int æqualiter ve­<lb/>loces </s> </p> <pb xlink:href="036/01/174.jpg"/> <p id="id.2.1.165.6.0.0.0" type="main"> <s id="id.2.1.165.6.1.1.0">Ii&longs;dem namq; po&longs;itis, &longs;it pon<lb/>dus V æquale ip&longs;i A, cui alliga­<lb/>tus &longs;it funis X9; &longs;itq; <expan abbr="pot&etilde;tia">potentia</expan> in X <lb/>mouens <expan abbr="põdus">pondus</expan> V; quæ dum pon<lb/>dus mouet, perueniat in Y: fiant <lb/>qué XY Z9 ip&longs;i OP æquales; <lb/>erit Z9 dupla QS. & &longs;i vtriu&longs;­<lb/>que potentiæ velocitates mo­<lb/>tuum &longs;int æquales; patet pon­<lb/>dus V duplum pertran&longs;ire &longs;pa­<lb/>tium in eodem tempore eìus, <lb/>quod pertran&longs;it pondus A. </s> <s id="id.2.1.165.6.1.1.0.a">in eo <lb/>dem enim tempore potentia in <lb/>X peruenit ad Y, & potentia in <lb/>O ad P; ponderaq; &longs;imiliter in <lb/>Z Q. quod erat demon&longs;tran­<lb/>dum. </s> </p> <figure id="id.036.01.174.1.jpg" xlink:href="036/01/174/1.jpg"> </figure> <p id="id.2.1.165.6.2.1.0" type="head"> <s id="id.2.1.165.6.4.1.0">PROPOSITIO XIII. </s> </p> <p id="id.2.1.165.7.0.0.0" type="main"> <s id="id.2.1.165.7.1.1.0">Fune circa &longs;ingulos duarum trochlearum <lb/>orbiculos, quarum altera &longs;upernè, altera verò <lb/>infernè, ponderiq; alligata fuerit, reuoluto; <lb/>altero etiam eius extremo inferiori trochleæ re­<pb n="81" xlink:href="036/01/175.jpg"/>ligata, altero autem à mouente potentia deten­<lb/>to: erit decur&longs;um trahentis potentiæ &longs;patium, mo<lb/>ti ponderis &longs;patii triplum. </s> </p> <p id="id.2.1.165.8.0.0.0" type="main"> <s id="id.2.1.165.8.1.1.0">Sit pondus A; &longs;it BCD orbiculus tro<lb/>chleæ ponderi A ex EQ &longs;u&longs;pen&longs;o alligatæ; <lb/>&longs;itq; orbiculi centrum E; &longs;it deinde FGH <lb/>orbiculus trochleæ &longs;ur&longs;um appen&longs;æ, cuius <lb/>centrum k; &longs;itq; funis LFGHDCBM <lb/>circa omnes reuolutus orbiculos, tro­<lb/>chleæq; inferiori in L religatus: &longs;itq; in <lb/>M potentia mouens. </s> <s id="id.2.1.165.8.1.2.0">dico &longs;patium de­<lb/>cur&longs;um à potentia in M, dum mouet pon<lb/>dus, triplum e&longs;&longs;e &longs;patii moti ponderis A. </s> <s id="id.2.1.165.8.1.2.0.a"><lb/>Moueatur potentia in M v&longs;q; ad N; & <lb/>centrum E &longs;it motum v&longs;q; ad O; & L v&longs;<lb/>que ad P; atq; pondus A, hoc e&longs;t pun­<lb/>ctum Q v&longs;q; ad R; orbiculu&longs;q; motus, &longs;it <lb/>TSV. ducantur per EO lineæ ST BD <lb/>horizonti æquidi&longs;tantes, quæ inter &longs;e &longs;e <lb/>quoq; æquidi&longs;tantes erunt. </s> <s id="id.2.1.165.8.1.3.0">quoniam au<lb/>tem dum E e&longs;t in O, punctum Q e&longs;t in <lb/>R; erit EQ æqualis OR, & EO ip&longs;i QR <lb/>æqualis; &longs;imiliter LQ æqualis erit PR, <lb/>& L P ip&longs;i QR æqualis. </s> <s id="id.2.1.165.8.1.4.0">tres igitur QR <lb/>EO LP inter &longs;e &longs;e æquales erunt; quibus <lb/>etiam &longs;unt æquales BS DT. </s> <s id="id.2.1.165.8.1.4.0.a">& quoniam fu<lb/>nis LFGHDCBM æqualis e&longs;t funi PF <lb/>GHTVSN, cùm &longs;it idem funis, & qui <lb/>circa &longs;emicirculum TVS e&longs;t æqualis funi <lb/>circa &longs;emicirculum BCD; demptis igi<lb/>tur communibus PFGHT' & SM; erit <lb/>reliquus MN tribus BS LP DT &longs;imul <lb/>&longs;umptis æqualis. </s> <s id="id.2.1.165.8.1.5.0">BS verò LP DT &longs;imul <lb/>tripli &longs;unt EO, & ex con&longs;equenti QR. <lb/><figure id="id.036.01.175.1.jpg" xlink:href="036/01/175/1.jpg"></figure><pb xlink:href="036/01/176.jpg"/>&longs;patium igitur MN translatæ potentiæ &longs;patii QR ponderis mo<lb/>ti triplum erit. </s> <s id="id.2.1.165.8.1.6.0">quod erat demon&longs;trandum. </s> </p> <p id="id.2.1.165.9.0.0.0" type="main"> <s id="id.2.1.165.9.1.1.0">Tempus quoq; huius motus manife&longs;tum e&longs;t, eadem enim po<lb/>tentia in æquali tempore &longs;patio &longs;ecundùm triplum ampliori &longs;ine <lb/>huiu&longs;modi trochleis idem pondus mouebit, quàm cum ei&longs;dem <lb/>hoc modo accomodatis. </s> <s id="id.2.1.165.9.1.2.0">&longs;patium ponderis &longs;ine trochleis moti <lb/>æquale e&longs;t &longs;patio potentiæ. </s> <s id="id.2.1.165.9.1.3.0">& hoc modo in omnibus inueniemus <lb/>tempus. </s> </p> <p id="id.2.1.165.10.0.0.0" type="head"> <s id="id.2.1.165.10.1.1.0">PROPOSITIO XIIII. </s> </p> <p id="id.2.1.165.11.0.0.0" type="main"> <s id="id.2.1.165.11.1.1.0">Fune circa tres duarum trochlearum orbicu<lb/>los, quarum altera &longs;upernè vnico dumtaxat, al <lb/>tera verò in&longs;ernè, duobus autem in&longs;ignita or­<lb/>biculis, ponderi&qacute;ue alligata fuerit, reuoluto; <lb/>altero eius e&longs;tremo alicubi religato, altero autem <lb/>à potentia pondus mouente detento: erit decur­<lb/>&longs;um trahentis potentiæ &longs;patium moti ponderis <lb/>&longs;patii quadruplum. </s> </p> <pb n="82" xlink:href="036/01/177.jpg"/> <p id="id.2.1.165.13.0.0.0" type="main"> <s id="id.2.1.165.13.1.1.0">Sit pondus A, &longs;int duo orbiculi, <expan abbr="quorũ">quorum</expan> <expan abbr="c&etilde;­tra;">cen­<lb/>tra</expan> k I trochleæ ponderi alligatæ k <foreign lang="greek">a</foreign>; ita vt <lb/>pondus motum trochleæ &longs;ur&longs;um, & deor&longs;um <lb/>&longs;emper &longs;equatur: &longs;it deinde orbiculus, cuius cen<lb/>trum L, trochleæ &longs;ur&longs;um appen&longs;æ in <35>; &longs;itq; <lb/>funis circa omnes orbiculos circumuolutus BC<lb/>DEFGHZMNO, religatu&longs;q; in B; &longs;itq; po<lb/>tentia in O mouens pondus A. </s> <s id="id.2.1.165.13.1.1.0.a">dico &longs;patium, <lb/>quod mouendo pertran&longs;it potentia in O, qua­<lb/>druplum e&longs;&longs;e &longs;patii moti ponderis A. </s> <s id="id.2.1.165.13.1.1.0.b">mouean<lb/>tur orbiculi trochleæ ponderi alligatæ; & dum <lb/>centrum k e&longs;t in R, centrum I &longs;it in S, & pon<lb/>dus A, hoc e&longs;t punctum <foreign lang="greek">a</foreign>in <foreign lang="greek">b</foreign>: erunt IS kR <lb/><foreign lang="greek">ab</foreign>inter &longs;e &longs;e æquales, itemq; k I ip&longs;i RS e­<lb/>rit æqualis. </s> <s id="id.2.1.165.13.1.2.0">orbiculi enim inter &longs;e &longs;e eandem <lb/>&longs;emper &longs;eruant di&longs;tantiam; & k <foreign lang="greek">a</foreign>ip&longs;i R <foreign lang="greek">b</foreign>æ­<lb/>qualis erit. </s> <s id="id.2.1.165.13.1.3.0">ducantur per orbiculorum centra <lb/>lineæ FH QT EC VX NZ horizonti æqui<lb/>di&longs;tantes, quæ tangent funes in FHQTEC <lb/>VX NZ punctis, & inter &longs;e &longs;e quoq; æquidi<lb/>&longs;tantes erunt: & EQ CT VN XZ non &longs;o<lb/>lum inter &longs;e &longs;e, &longs;ed etiam ip&longs;is IS KR <foreign lang="greek">ab</foreign>æqua<lb/>les erunt. </s> <s id="id.2.1.165.13.1.4.0">& dum centra kI &longs;unt in RS, po<lb/>tentia in O &longs;it mota in P. </s> <s id="id.2.1.165.13.1.4.0.a">& quoniam funis <lb/>BCDEFGHZMNO e&longs;t æqualis funi BT9 <lb/>QFGHXYVP, e&longs;t enim <expan abbr="id&etilde;">idem</expan> funis, & funes cir<lb/><figure id="id.036.01.177.1.jpg" xlink:href="036/01/177/1.jpg"></figure><lb/>ca T9Q XYV &longs;emicirculos &longs;unt æquales funibus, qui &longs;unt circa <lb/>CDE ZMN; Demptis igitur communibus BT, QF GHX, <lb/>& VO; erit OP æqualis ip&longs;is VN XZ CT QE &longs;imul &longs;umptis. </s> <s id="id.2.1.165.13.1.5.0"><lb/>quatuor verò VN ZX CT QE &longs;unt inter &longs;e &longs;e æquales, & &longs;imul <lb/>quadruplæ kR, & <foreign lang="greek">ab</foreign>; quare OP quadrupla erit ip&longs;ius <foreign lang="greek">ab</foreign>. </s> <s id="id.2.1.165.13.1.6.0">&longs;pa<lb/>tium igitur potentiæ quadruplum e&longs;t &longs;patii ponderis. </s> <s id="id.2.1.165.13.1.7.0">quod erat <lb/>o&longs;tendendum. </s> </p> <p id="id.2.1.165.14.0.0.0" type="main"> <s id="id.2.1.165.14.1.1.0">Et &longs;i funis in P circa alium adhuc reuoluatur orbiculum ver&longs;us <lb/><35>, potentiaqué mouendo &longs;e deor&longs;um moueat &longs;ur&longs;um pondus; &longs;imi <lb/>liter o&longs;tendetur &longs;patium potentiæ quadruplum e&longs;&longs;e &longs;patii ponderis. </s> </p> <pb xlink:href="036/01/178.jpg"/> <p id="id.2.1.165.16.0.0.0" type="main"> <s id="id.2.1.165.16.1.1.0">Si verò funis in B circumuoluatur al<lb/>teri orbiculo, qui deinde trochleæ in­<lb/><arrow.to.target n="note252"></arrow.to.target>feriori religetur; erit potentia in O <lb/>&longs;u&longs;tinens pondus A &longs;ubquintupla pon<lb/>deris. </s> <s id="id.2.1.165.16.1.2.0">& &longs;i in O &longs;it potentia mouens <lb/>pondus A; &longs;imiliter demon&longs;trabitur <lb/>&longs;patium potentiæ in O quintuplum e&longs; <lb/>&longs;e &longs;patii ponderis A. <lb/><figure id="id.036.01.178.1.jpg" xlink:href="036/01/178/1.jpg"></figure></s> </p> <p id="id.2.1.165.17.0.0.0" type="main"> <s id="id.2.1.165.17.1.1.0">Et &longs;i funis ita circa orbiculos apte­<lb/>tur, vt potentia in O &longs;u&longs;tinens pon­<lb/>dus &longs;it ponderis &longs;ub&longs;extupla; & loco <lb/>potentiæ &longs;u&longs;tinentis ponatur in O po­<lb/>tentia mouens pondus: eodem modo <lb/>o&longs;tendetur &longs;patium potentiæ &longs;extu­<lb/>plum e&longs;&longs;e &longs;patii ponderis moti. </s> <s id="id.2.1.165.17.1.2.0">& &longs;ic <lb/>procedendo in infinitum proportiones <lb/>&longs;patii potentiæ ad &longs;patium ponderis <lb/>moti quotcunq; multiplices inuenien­<lb/>tur. </s> </p> <p id="id.2.1.166.1.0.0.0" type="margin"> <s id="id.2.1.166.1.1.1.0"><margin.target id="note252"></margin.target>9 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.167.1.0.0.0" type="head"> <s id="id.2.1.167.1.1.1.0">COROLLARIVM I. </s> </p> <p id="id.2.1.167.2.0.0.0" type="main"> <s id="id.2.1.167.2.1.1.0">Ex his manife&longs;tum e&longs;t ita &longs;e habere pondus <lb/>ad potentiam ip&longs;um &longs;u&longs;tinentem, &longs;icuti &longs;patium <lb/>potentiæ mouentis ad &longs;patium ponderis moti. </s> </p> <p id="id.2.1.167.3.0.0.0" type="main"> <s id="id.2.1.167.3.1.1.0">Vt &longs;i pondus A quintuplum &longs;it potentiæ in O pondus A &longs;u&longs;ti­<lb/>nentis; erit & &longs;patium OP potentiæ pondus mouentis quintuplum <lb/>&longs;patii <foreign lang="greek">ab</foreign>ponderis moti. </s> </p> <pb n="83" xlink:href="036/01/179.jpg"/> <p id="id.2.1.167.4.0.0.0" type="head"> <s id="id.2.1.167.5.1.1.0">COROLLARIVM II. </s> </p> <p id="id.2.1.167.6.0.0.0" type="main"> <s id="id.2.1.167.6.1.1.0">Patet etiam per ea, quæ dicta &longs;unt, orbiculos <lb/>trochleæ, quæ ponderi e&longs;t alligata, efficere; vt à <lb/>moto pondere minus, quàm à trahente poten­<lb/>tia de&longs;cribatur &longs;patium; maioriq; tempore datum <lb/>æquale &longs;patium de&longs;cribi, quàm &longs;ine illis. </s> <s id="id.2.1.167.6.1.2.0">quod <lb/>quidem orbiculi trochleæ &longs;uperioris non effi­<lb/>ciunt. </s> </p> <p id="id.2.1.167.7.0.0.0" type="main"> <s id="id.2.1.167.7.1.1.0">Multiplici o&longs;ten&longs;a ponderis ad potentiam proportione, iam ex <lb/>aduer&longs;o potentiæ ad pondus proportio multiplex o&longs;tendatur. </s> </p> <p id="id.2.1.167.8.0.0.0" type="head"> <s id="id.2.1.167.8.1.1.0">PROPOSITIO XV. </s> </p> <p id="id.2.1.167.9.0.0.0" type="main"> <s id="id.2.1.167.9.1.1.0">Si funis orbiculo trochleæ à potentia &longs;ur&longs;um <lb/>detentæ fuerit circumuolutus; altero eius extre­<lb/>mo alicubi religato, alteri verò pondere appen<lb/>&longs;o; dupla erit ponderis potentia. </s> </p> <pb xlink:href="036/01/180.jpg"/> <p id="id.2.1.167.11.0.0.0" type="main"> <s id="id.2.1.167.11.1.1.0">Sit trochlea habens orbiculum, cuius <lb/>centrum A; & &longs;it pondus B alligatum fu<lb/>ni CDEFG, qui circa orbiculum &longs;it re­<lb/>uolutus, ac tandem religatus in G: &longs;itq; <lb/>potentia in H &longs;u&longs;tinens pondus. </s> <s id="id.2.1.167.11.1.2.0">dico po<lb/>tentiam in H duplam e&longs;&longs;e ponderis B. du<lb/>catur DF per <expan abbr="centrũ">centrum</expan> A horizonti æquidi<lb/>&longs;tans. </s> <s id="id.2.1.167.11.1.3.0"><expan abbr="quoniã">quoniam</expan> igitur potentia in H &longs;u&longs;tinet <lb/><expan abbr="trochleã">trochleam</expan>, quæ &longs;u&longs;tinet <expan abbr="orbiculũ">orbiculum</expan> in eius <expan abbr="c&etilde;tro">centro</expan> <lb/>A, qui pondus &longs;u&longs;tinet; erit potentia &longs;u&longs;ti<lb/>nens <expan abbr="orbiculũ">orbiculum</expan>, ac &longs;i in A <expan abbr="cõ&longs;tituta">con&longs;tituta</expan> e&longs;&longs;et; ip&longs;a <lb/>ergo in A exi&longs;tente, pondere verò in D <lb/>appen&longs;o, funiq; CD religato; erit DF <lb/>tanquam vectis, cuius fulcimentum erit <lb/>F, pondus in D, & potentia in A. </s> <s id="id.2.1.167.11.1.3.0.a">po­<lb/><arrow.to.target n="note253"></arrow.to.target>tentia verò ad pondus e&longs;t, vt DF ad <lb/>ad FA, & DF dupla e&longs;t ip&longs;ius FA; Po­<lb/><figure id="id.036.01.180.1.jpg" xlink:href="036/01/180/1.jpg"></figure><lb/>tentia igitur in A, &longs;iue in H, quod idem e&longs;t, ponderis B dupla erit. </s> <lb/> <s id="id.2.1.167.11.1.4.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.168.1.0.0.0" type="margin"> <s id="id.2.1.168.1.1.1.0"><margin.target id="note253"></margin.target>3 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/></s> </p> <p id="id.2.1.169.1.0.0.0" type="main"> <s id="id.2.1.169.1.1.1.0">Præterea con&longs;iderandum occurrit, cùm hæc omnia maneant, <lb/>idem e&longs;&longs;e vnico exi&longs;tente fune CD EFG hoc modo orbiculo <expan abbr="cicum">circum</expan> <lb/>uoluto, ac &longs;i duo e&longs;&longs;ent funes CD FG in vecte &longs;iue libra DF al­<lb/>ligati. </s> </p> <p id="id.2.1.169.2.0.0.0" type="head"> <s id="id.2.1.169.2.1.1.0">ALITER. </s> </p> <p id="id.2.1.169.3.0.0.0" type="main"> <s id="id.2.1.169.3.1.1.0">Ii&longs;dem po&longs;itis, &longs;i in G appen&longs;um e&longs;&longs;et pondus k æquale pon­<lb/>deri B, pondera B k æqueponderabunt in libra DF, cuius centrum <lb/>A. </s> <s id="id.2.1.169.3.1.1.0.a">potentia verò in H &longs;u&longs;tinens pondera Bk e&longs;t ip&longs;is &longs;imul &longs;um<lb/>ptis æqualis, & pondera BK ip&longs;ius B &longs;unt dupla; potentia ergo in <lb/>H ponderis B dupla erit. </s> <s id="id.2.1.169.3.1.2.0">& quoniam funis religatus in G nihil a­<lb/>liud efficit, ni&longs;i quòd pondus B &longs;u&longs;tinet, ne de&longs;cendat; quod idem <lb/>efficit pondus k in G appen&longs;um: potentia igitur in H &longs;u&longs;tinens <lb/>pondus B, fune religato in G, dupla e&longs;t ponderis B. quod de­<lb/>mon&longs;trare oportebat. </s> </p> <pb n="84" xlink:href="036/01/181.jpg"/> <p id="id.2.1.169.4.0.0.0" type="head"> <s id="id.2.1.169.5.1.1.0">PROPOSITIO XVI. </s> </p> <p id="id.2.1.169.6.0.0.0" type="main"> <s id="id.2.1.169.6.1.1.0">Ii&longs;dem po&longs;itis &longs;i in H &longs;it potentia mouens pon<lb/>dus, mouebit hæc eadem vecte horizonti &longs;em­<lb/>per æquidi&longs;tante </s> </p> <p id="id.2.1.169.7.0.0.0" type="main"> <s id="id.2.1.169.7.1.1.0">Hoc etiam (&longs;icut in &longs;uperioribus dictum <lb/>e&longs;t) o&longs;tendetur. </s> <s id="id.2.1.169.7.1.2.0">moueatur enim orbiculus <lb/>&longs;ur&longs;um, po&longs;itionemq; habeat MNO, cuius <lb/>centrum L: & per L ducatur MLO ip&longs;i DF, <lb/>& horizonti æquidi&longs;tans. </s> <s id="id.2.1.169.7.1.3.0">& quoniam funes <lb/>tangunt circulum MON in punctis MO; <lb/>ideo cùm potentia in A, &longs;eu in H, quod <lb/>idem e&longs;t, moueat pondus B in D appen&longs;um <lb/>vecte DF, cuius fulcimentum e&longs;t F; &longs;emper <lb/>adhuc remanebit alius vectis, .vt MO hori<lb/>zonti æquidi&longs;tans, ita vt &longs;emper potentia <lb/>moueat pondus vecte horizonti æquidi&longs;tan<lb/>te, cuius fulcimentum e&longs;t &longs;emper in linea <lb/>OG, & pondus in MC, potentiaq; in cen<lb/>tro orbiculi. <figure id="id.036.01.181.1.jpg" xlink:href="036/01/181/1.jpg"></figure></s> </p> <p id="id.2.1.169.8.0.0.0" type="main"> <s id="id.2.1.169.8.1.1.0">Ii&longs;dem po&longs;itis, &longs;patium ponderis moti duplum <lb/>e&longs;t &longs;patii potentiæ mouentis. </s> </p> <pb xlink:href="036/01/182.jpg"/> <p id="id.2.1.169.10.0.0.0" type="main"> <s id="id.2.1.169.10.1.1.0">Sit motus orbiculus à centro A <lb/>v&longs;q; ad centrum L; & pondus B, <lb/>hoc e&longs;t punctum C, in eodem tem­<lb/>pore &longs;it motum in P; & potentia in <lb/>H v&longs;q; ad K; erit AH ip&longs;i LK æqua <lb/>lis, & AL ip&longs;i Hk. </s> <s id="id.2.1.169.10.1.2.0">& quoniam fu<lb/>nis CDEFG e&longs;t æqualis funi PM <lb/>NOG, idem enim e&longs;t funis, & fu <lb/>nis circa &longs;emicirculum MNO æ­<lb/>qualis e&longs;t funi circa &longs;emicirculum <lb/>DEF; demptis igitur communi­<lb/>bus DP FG, erit PC æqualis <lb/>DM FO &longs;imul &longs;umptis, qui funes <lb/>&longs;unt dupli ip&longs;ius AL, & con&longs;equen­<lb/>ter ip&longs;ius Hk. </s> <s id="id.2.1.169.10.1.3.0">&longs;patium ergo pon<lb/>deris moti CP duplum e&longs;t &longs;patii <lb/>Hk potentiæ. </s> <s id="id.2.1.169.10.1.4.0">quod oportebat de­<lb/>mon&longs;trare. </s> </p> <figure id="id.036.01.182.1.jpg" xlink:href="036/01/182/1.jpg"> </figure> <p id="id.2.1.169.10.2.1.0" type="head"> <s id="id.2.1.169.10.4.1.0">COROLLARIVM </s> </p> <p id="id.2.1.169.11.0.0.0" type="main"> <s id="id.2.1.169.11.1.1.0">Ex hoc manife&longs;tum e&longs;t, idem pondus trahi <lb/>ab eadem potentia in æquali tempore per du­<lb/>plum &longs;patium trochlea hoc modo accommoda<lb/>ta, quàm &longs;ine trochlea; dummodo ip&longs;ius poten<lb/>tiæ lationes in velocitate &longs;int æquales. </s> </p> <p id="id.2.1.169.12.0.0.0" type="main"> <s id="id.2.1.169.12.1.1.0">Spatium enim ponderis moti &longs;ine trochlea æquale e&longs;t &longs;patio <lb/>potentiæ. </s> </p> <pb n="85" xlink:href="036/01/183.jpg"/> <p id="id.2.1.169.14.0.0.0" type="main"> <s id="id.2.1.169.14.1.1.0">Si autem funis in G circa alium reuoluatur <lb/>orbiculum, cuius centrum k; &longs;itq; huiu&longs;mo<lb/>di orbiculi trochlea deor&longs;um affixa, quæ nul<lb/>lum alium habeat motum, ni&longs;i liberam orbi <lb/>culi circa axem reuolutionem; funi&longs;q; relige<lb/>tur in M; erit potentia in H &longs;u&longs;tinens pondus <lb/>B &longs;imiliter ip&longs;ius ponderis dupla. </s> <s id="id.2.1.169.14.1.2.0">quod qui <lb/>dem manife&longs;tum e&longs;t, cùm idem pror&longs;us &longs;it, <lb/>&longs;iue funis &longs;it religatus in M, &longs;iue in G. orbicu<lb/>lus enim, cuius centrum k, nihil efficit; penitu&longs; <lb/>qué inutilis e&longs;t. <figure id="id.036.01.183.1.jpg" xlink:href="036/01/183/1.jpg"></figure></s> </p> <p id="id.2.1.169.15.0.0.0" type="main"> <s id="id.2.1.169.15.1.1.0">Si verò &longs;it potentia in M &longs;u&longs;tinens pon<lb/>dus B, & trochlea &longs;uperior &longs;it &longs;ur&longs;um appen<lb/>&longs;a; erit potentia in M æqualis ponderi B. </s> </p> <p id="id.2.1.169.16.0.0.0" type="main"> <s id="id.2.1.169.16.1.1.0">Quoniam enim potentia in G &longs;u&longs;tinens <arrow.to.target n="note254"></arrow.to.target><lb/>pondus B æqualis e&longs;t ponderi B, & ip&longs;i po<lb/>tentiæ in G æqualis e&longs;t potentia in L; e&longs;t <lb/>enim GL vectis, cuius fulcimentum e&longs;t k; <lb/>& di&longs;tantia Gk di&longs;tantiæ kL e&longs;t æqualis; <lb/>erit igitur potentia in L, &longs;iue (quod idem e&longs;t) <lb/>in M, ponderi B æqualis. </s> </p> <p id="id.2.1.170.1.0.0.0" type="margin"> <s id="id.2.1.170.1.1.1.0"><margin.target id="note254"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.171.1.0.0.0" type="main"> <s id="id.2.1.171.1.1.1.0">Huiu&longs;modi autem motus fit vectibus DF LG, quorum fulci <lb/>menta &longs;unt kA, & pondus in D, & potentia in F. &longs;ed in vecte <lb/>LG potentia e&longs;t in L, pondus verò, ac &longs;i e&longs;&longs;et in G. </s> </p> <p id="id.2.1.171.2.0.0.0" type="main"> <s id="id.2.1.171.2.1.1.0">Si deinde in M &longs;it potentia mouens pondus, transferaturq; po<lb/>tentia in N, pondus autem motum fuerit v&longs;q; ad O; erit MN <lb/>&longs;patium potentiæ æquale &longs;patio CO ponderis. </s> <s id="id.2.1.171.2.1.2.0">Cùm enim funis <lb/>MLGFDC æqualis &longs;it funi NLGFDO. e&longs;t enim idem funis; <lb/>dempto communi MLGFDO; erit &longs;patium MN potentiæ æ­<lb/>quale &longs;patio CO ponderis. </s> </p> <p id="id.2.1.171.3.0.0.0" type="main"> <s id="id.2.1.171.3.1.1.0">Et &longs;i funis in M circa plures reuoluatur orbiculos, &longs;emper erit <lb/>potentia altero eius extremo pondus &longs;u&longs;tinens æqualis ip&longs;i ponderi. </s> <s id="id.2.1.171.3.1.2.0"><lb/>&longs;patiaq; ponderis, atq; potentiæ mouentis &longs;emper o&longs;tendentur <lb/>æqualia. </s> </p> <pb xlink:href="036/01/184.jpg"/> <p id="id.2.1.171.5.0.0.0" type="head"> <s id="id.2.1.171.5.1.1.0">PROPOSITIO XVII. </s> </p> <p id="id.2.1.171.6.0.0.0" type="main"> <s id="id.2.1.171.6.1.1.0">Si vtri&longs;q; duarum trochlearum &longs;ingulis orbicu<lb/>lis, quarum vna &longs;upernè à potentia &longs;u&longs;tineatur, <lb/>altera verò infernè, ibiq; affixa, con&longs;tituta fue­<lb/>rit, funis circumducatur; altero eius extremo &longs;u<lb/>periori trochleæ religato, alteri verò pondere <lb/>appen&longs;o; tripla erit ponderis potentia. </s> </p> <p id="id.2.1.171.7.0.0.0" type="main"> <s id="id.2.1.171.7.1.1.0">Sit orbiculus, cuius centrum A, tro­<lb/>chleæ infernè affixæ; & &longs;it funis BCD <lb/>EFG non &longs;olum huic orbiculo circumuo<lb/>lutus, verùm etiam orbiculo trochleæ &longs;u­<lb/>perioris, cuius centrum k; &longs;itq; funis in <lb/>B &longs;uperiori trochleæ religatus; & in G &longs;it ap<lb/>pen&longs;um pondus H; potentiaq; in L &longs;u&longs;ti<lb/>neat pondus H. </s> <s id="id.2.1.171.7.1.1.0.a">dico potentiam in L tri­<lb/>plam e&longs;&longs;e ponderis H. </s> <s id="id.2.1.171.7.1.1.0.b">&longs;i enim duæ e&longs;&longs;ent <lb/>potentiæ pondus H <expan abbr="&longs;u&longs;tidentes">sustinentes</expan>, vna in <lb/>K, altera in B, erunt vtræq; &longs;imul triplæ <lb/><arrow.to.target n="note255"></arrow.to.target>ponderis H potentia enim in k dupla e&longs;t <lb/>ponderis H, & potentia in B ip&longs;i ponderi <lb/>æqualis. </s> <s id="id.2.1.171.7.1.2.0">& quoniam &longs;ola potentia in L <lb/>vtri&longs;q; &longs;cilicet potentiæ in KB e&longs;t æqua­<lb/>lis. </s> <s id="id.2.1.171.7.1.3.0">&longs;u&longs;tinet enim potentia in L; tùm po­<lb/>tentiam in K, tùm potentiam in B; idem <lb/>qué efficit potentia in L, ac &longs;i duæ e&longs;&longs;ent <lb/>potentiæ, vna in k, altera in B: Tri­<lb/>pla igitur erit potentia in L ponderis H. <lb/>quod demon&longs;trare oportebat. <figure id="id.036.01.184.1.jpg" xlink:href="036/01/184/1.jpg"></figure></s> </p> <pb n="86" xlink:href="036/01/185.jpg"/> <p id="id.2.1.171.9.0.0.0" type="main"> <s id="id.2.1.171.9.1.1.0">Si autem in L &longs;it potentia mouens pondus. </s> <s id="id.2.1.171.9.1.2.0">di<lb/>co &longs;patium ponderis moti triplum e&longs;&longs;e &longs;patii po­<lb/>tentiæ motæ. </s> </p> <p id="id.2.1.172.1.0.0.0" type="margin"> <s id="id.2.1.172.1.1.1.0"><margin.target id="note255"></margin.target>15 <emph type="italics"/>Huius. </s> <s id="id.2.1.172.1.1.2.0">In præcedenti.<emph.end type="italics"/></s> </p> <p id="id.2.1.173.1.0.0.0" type="main"> <s id="id.2.1.173.1.1.1.0">Moueatur centrum or­<lb/>biculi K v&longs;q; ad M; cuius <lb/>quidem motus &longs;patium <lb/>motæ potentiæ &longs;patio e&longs;t <arrow.to.target n="note256"></arrow.to.target><lb/>æquale, &longs;icuti &longs;upra dictum <lb/>e&longs;t: & quando k erit in M, <lb/>B erit in N; & NB æqualis <lb/>erit M k; & dum k e&longs;t in M, <lb/>&longs;it pondus H, hoc e&longs;t pun<lb/>ctum G motum in O; & per <lb/>MK ducantur EF PQ ho<lb/>rizonti æquidi&longs;tantes; erit <lb/>vnaquæq; EP BN FQ ip<lb/>&longs;i KM æqualis. </s> <s id="id.2.1.173.1.1.2.0">& quoniam <lb/>funis BCDEFG æqualis <lb/>e&longs;t funi NCDPQO; <lb/>idem enim e&longs;t funis; & fu­<lb/>nis circa &longs;emicirculum ER <lb/>F æqualis e&longs;t funi circa &longs;e­<lb/>micirculum PSQ: dem­<lb/>ptis igitur communibus <lb/>BCDE, & FO, erit OG <lb/>tribus QF NB PE &longs;imul <lb/>&longs;umptis æqualis. </s> <s id="id.2.1.173.1.1.3.0">&longs;ed QF <lb/>NB PE &longs;imul triplæ &longs;unt <lb/>Mk, hoc e&longs;t &longs;patii poten­<lb/>tiæ motæ; &longs;patium ergo <lb/>GO ponderis H moti tri­<lb/><figure id="id.036.01.185.1.jpg" xlink:href="036/01/185/1.jpg"></figure><lb/>plum e&longs;t &longs;patii potentiæ motæ. </s> <s id="id.2.1.173.1.1.4.0">quod o&longs;tendere oportebat. </s> </p> <p id="id.2.1.174.1.0.0.0" type="margin"> <s id="id.2.1.174.1.1.1.0"><margin.target id="note256"></margin.target><emph type="italics"/>In præcedenti.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/186.jpg"/> <p id="id.2.1.175.1.0.0.0" type="head"> <s id="id.2.1.175.1.2.1.0">PROPOSITIO XVIII. </s> </p> <p id="id.2.1.175.2.0.0.0" type="main"> <s id="id.2.1.175.2.1.1.0">Si vtriu&longs;q; duarum trochlearum binis orbicu<lb/>lis, quarum altera &longs;upernè à potentia &longs;u&longs;tineatur, <lb/>altera verò infernè, ibiq; annexa, collocata fue­<lb/>rit, funis circumnectatur; altero eius extremo <lb/>alicubi, non autem &longs;uperiori trochleæ religato, <lb/>alteri verò pondere appen&longs;o; quadrupla erit <lb/>ponderis potentia. </s> </p> <p id="id.2.1.175.3.0.0.0" type="main"> <s id="id.2.1.175.3.1.1.0">Sit trochlea inferior, duos habens orbiculos, <lb/>quorum centra AB; &longs;it qué trochlea &longs;uperior <lb/>duos &longs;imiliter habens orbiculos, quorum cen­<lb/>tra CD; funi&longs;q; EFGHKLMNOP &longs;it cir­<lb/>ca omnes orbiculos reuolutus, qui &longs;it religatus <lb/>in E; & in P appendatur pondus Q; &longs;itq; po­<lb/>tentia in R. </s> <s id="id.2.1.175.3.1.1.0.a">dico potentiam in R quadruplam <lb/>e&longs;&longs;e ponderis q. Cùm enim &longs;i duæ intelligan<lb/>tur potentiæ, vna in k, altera in D, potentia <lb/><arrow.to.target n="note257"></arrow.to.target>in k &longs;u&longs;tinens pondus Q fune k LMNOP æ­<lb/>qualis erit ponderi; erunt duæ &longs;imul potentiæ, <lb/>vna in D, altera in k, pondus Q &longs;u&longs;tinentes, <lb/>triplæ eiu&longs;dem ponderis. </s> <s id="id.2.1.175.3.1.2.0">Potentia verò in C <lb/>dupla e&longs;t potentiæ in k, & per con&longs;equens pon<lb/>deris Q; idem enim e&longs;t, ac &longs;i in k appen&longs;um e&longs; <lb/><arrow.to.target n="note258"></arrow.to.target>&longs;et pondus æquale ponderi Q, cuius dupla e&longs;t <lb/>potentia in C; duæ igitur potentiæ in DC qua­<lb/>druplæ &longs;unt ponderis q. & cùm potentia in R <lb/>orbiculis &longs;u&longs;tineat pondus Q, erit <expan abbr="pot&etilde;tia">potentia</expan> in R, <lb/>ac &longs;i duæ e&longs;&longs;ent potentiæ, vna in D, altera in C, <lb/>& vtræq; &longs;imul pondus Q &longs;u&longs;tinerent. </s> <s id="id.2.1.175.3.1.3.0">ergo po­<lb/>tentia in R quadrupla e&longs;t ponderis q. quod <lb/>oportebat demon&longs;trare. <figure id="id.036.01.186.1.jpg" xlink:href="036/01/186/1.jpg"></figure></s> <pb n="87" xlink:href="036/01/187.jpg"/> <s id="id.2.1.175.3.3.1.0">COROLLARIVM </s> </p> <p id="id.2.1.176.1.0.0.0" type="margin"> <s id="id.2.1.176.1.1.1.0"><margin.target id="note257"></margin.target>16 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.176.1.1.2.0"><margin.target id="note258"></margin.target>15 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.177.1.0.0.0" type="main"> <s id="id.2.1.177.1.1.1.0">Ex quo patet, &longs;i funis fuerit religatus in G, & <lb/>circa orbiculos, quorum centra &longs;unt BCD reuo­<lb/>lutus; potentiam in R pondus &longs;u&longs;tinentem &longs;imili­<lb/>ter ponderis Q quadruplam e&longs;&longs;e. </s> <s id="id.2.1.177.1.1.2.0">orbiculus enim, <lb/>cuius centrum A, nihil efficit. </s> </p> <p id="id.2.1.177.2.0.0.0" type="main"> <s id="id.2.1.177.2.1.1.0">Si autem in R &longs;it potentia mouens pondus. </s> <s id="id.2.1.177.2.1.2.0">dico <lb/>&longs;patium ponderis moti quadruplum e&longs;&longs;e &longs;patii <lb/>potentiæ. </s> </p> <p id="id.2.1.177.3.0.0.0" type="main"> <s id="id.2.1.177.3.1.1.0">Moueantur centra CD orbiculorum v&longs;q; ad <lb/>ST; erunt ex &longs;uperius dictis CS DT &longs;patio <lb/>potentiæ æqualia; & per CSDT ducantur Hk <lb/>VX NO YZ horizonti æquidi&longs;tantes; & <expan abbr="dũ">dum</expan> <lb/>centra CD &longs;unt in ST, &longs;it pondus Q, hoc e&longs;t <lb/>punctum P motum in 9. & quoniam funis EF <lb/>GHKLMNOP æqualis e&longs;t funi EFGVX <lb/>LMYZ 9; cùm &longs;it idem funis: & funes circa <lb/>&longs;emicirculos NIO H <foreign lang="greek">a</foreign>k &longs;unt æquales funi­<lb/>bus, qui &longs;unt circa &longs;emicirculos Y<35>Z V<foreign lang="greek">b</foreign>X; <lb/>demptis igitur communibus EFGH kLMN <lb/>& O9; erit P9 ip&longs;is NY ZO VH <emph type="italics"/>X<emph.end type="italics"/>k &longs;i­<lb/>mul &longs;umptis æqualis. </s> <s id="id.2.1.177.3.1.2.0">quatuor autem NY ZO <lb/>VH Xk &longs;imul quadrupli &longs;unt DT, hoc e&longs;t <lb/>&longs;patii potentiæ; &longs;patium igitur P9 ponderis <lb/>quadruplum e&longs;t &longs;patii potentiæ quod demon<lb/>&longs;trandum fuerat. <figure id="id.036.01.187.1.jpg" xlink:href="036/01/187/1.jpg"></figure></s> </p> <pb xlink:href="036/01/188.jpg"/> <p id="id.2.1.177.5.0.0.0" type="main"> <s id="id.2.1.177.5.1.1.0">Si autem funis &longs;it re­<lb/>ligatus in E trochleæ &longs;u<lb/>periori, & potentia in R <lb/>&longs;u&longs;tineat pondus Q; e­<lb/>rit potentia in R ponde<lb/>ris Q quintupla. </s> <s id="id.2.1.177.5.1.2.0">& &longs;i in <lb/>R &longs;it potentia mouens <lb/>pondus; erit &longs;patium pon<lb/>deris moti quintuplum <lb/>&longs;patii potentiæ. </s> <s id="id.2.1.177.5.1.3.0">quæ om­<lb/>nia &longs;imili modo o&longs;ten­<lb/>dentur, &longs;icut in præce­<lb/>dentibus demon&longs;tra­<lb/>tum e&longs;t. <figure id="id.036.01.188.1.jpg" xlink:href="036/01/188/1.jpg"></figure></s> </p> <pb n="88" xlink:href="036/01/189.jpg"/> <p id="id.2.1.177.7.0.0.0" type="main"> <s id="id.2.1.177.7.1.1.0">Si verò potentia in R &longs;ub&longs;tineat pon­<lb/>dus Q trochlea tres orbiculos habente, <lb/>quorum centra &longs;int ABC; & &longs;it alia tro<lb/>chlea infernè affixa duos, vel tres orbicu­<lb/>los habens, quorum centra DEF; &longs;itq; <lb/>funis circa omnes orbiculos reuolutus, &longs;i­<lb/>ue in G, &longs;iue in H religatus; &longs;imiliter <lb/>o&longs;tendetur potentiam in R &longs;excuplam <lb/>e&longs;&longs;e ponderis q. Et &longs;i in R &longs;it potentia <lb/>mouens pondus, o&longs;tendetur &longs;patium pon<lb/>deris moti &longs;excuplum e&longs;&longs;e &longs;patii poten­<lb/>tiæ. <figure id="id.036.01.189.1.jpg" xlink:href="036/01/189/1.jpg"></figure></s> </p> <p id="id.2.1.177.8.0.0.0" type="main"> <s id="id.2.1.177.8.1.1.0">Et &longs;i funis &longs;it religatus in K trochleæ <lb/>&longs;uperiori, & in R &longs;it potentia pondus <lb/>&longs;u&longs;tinens; &longs;imili modo o&longs;tendetur poten<lb/>tiam in R &longs;eptuplam e&longs;&longs;e ponderis q. </s> </p> <p id="id.2.1.177.9.0.0.0" type="main"> <s id="id.2.1.177.9.1.1.0">Et &longs;i in R &longs;it potentia mouens, o&longs;ten <lb/>detur &longs;patium ponderis Q &longs;eptuplum e&longs;&longs;e <lb/>&longs;patii potentiæ. </s> <s id="id.2.1.177.9.1.2.0">atq; ita in infinitum <lb/>omnis potentiæ ad pondus multiplex <lb/>proportio inueniri poterit. </s> <s id="id.2.1.177.9.1.3.0">&longs;emperq; o­<lb/>&longs;tendetur, ita e&longs;&longs;e pondus ad potentiam <lb/>ip&longs;um &longs;u&longs;tinentem, &longs;icuti &longs;patium poten<lb/>tiæ pondus mouentis ad &longs;patium ponde­<lb/>ris moti. </s> </p> <p id="id.2.1.177.10.0.0.0" type="main"> <s id="id.2.1.177.10.1.1.0">Vectium autem ip&longs;orum orbiculorum <lb/>motus in his fit hoc modo, videlicet vectes <lb/>orbiculorum trochleæ &longs;uperioris mouen<lb/>tur, vti dictum e&longs;t in decima &longs;exta huius; <lb/>hoc e&longs;t habent fulcimentum in extremita<lb/>te, potentiam in medio, pondus in altera extremitate appen&longs;um. </s> <s id="id.2.1.177.10.1.2.0">ve<lb/>ctes verò trochleæ inferioris habent fulcimentum in medio, pon<lb/>dus, & potentiam in extremitatibus. </s> </p> <pb xlink:href="036/01/190.jpg"/> <p id="id.2.1.177.11.0.0.0" type="head"> <s id="id.2.1.177.12.1.1.0">COROLLARIVM </s> </p> <p id="id.2.1.177.13.0.0.0" type="main"> <s id="id.2.1.177.13.1.1.0">Manife&longs;tum e&longs;t in his, orbiculos trochleæ &longs;u<lb/>perioris efficere, vt pondus moueatur maiori <lb/>potentia, quàm &longs;it ip&longs;um pondus, & per maius <lb/>&longs;patium potentiæ &longs;patio, & per æquale tempo­<lb/>re minori; quod quidem orbiculi trochleæ in­<lb/>ferioris non efficiunt. </s> </p> <p id="id.2.1.177.14.0.0.0" type="main"> <s id="id.2.1.177.14.1.1.0">Alio quoq; modo hanc potentiæ ad pondus multiplicem propor<lb/>tionem inuenire po&longs;&longs;umus. </s> </p> <p id="id.2.1.177.15.0.0.0" type="head"> <s id="id.2.1.177.15.1.1.0">PROPOSITIO XVIIII. </s> </p> <p id="id.2.1.177.16.0.0.0" type="main"> <s id="id.2.1.177.16.1.1.0">Si vtriu&longs;q; duarum trochlearum &longs;ingulis orbi <lb/>culis, quarum altera &longs;upernè appen&longs;a, altera ve­<lb/>rò infernè à &longs;u&longs;tinente potentia <expan abbr="rententa">retenta</expan> fuerit, <lb/>funis circumuoluatur; altero eius extremo alicu<lb/>bi religato, alteri autem pondere appen&longs;o; du­<lb/>pla erit ponderis potentia. </s> </p> <pb n="89" xlink:href="036/01/191.jpg"/> <p id="id.2.1.177.18.0.0.0" type="main"> <s id="id.2.1.177.18.1.1.0">Sit orbiculus trochleæ &longs;upernè appen&longs;æ, cu<lb/>ius centrum &longs;it A; & BCD &longs;it trochleæ infe<lb/>rioris; &longs;it deinde funis EBC DFGHL reli­<lb/>gatus in E; & in L &longs;it appen&longs;um pondus M; <lb/>&longs;itq; potentia in N &longs;u&longs;tinens pondus M. </s> <s id="id.2.1.177.18.1.1.0.a"><lb/>dico potentiam in N duplam e&longs;&longs;e ponderis <lb/>M. </s> <s id="id.2.1.177.18.1.1.0.b">Cùm enim &longs;upra o&longs;ten&longs;um &longs;it potentiam <lb/>in L, quæ pondus, exempli gratia, O &longs;u&longs;ti­<lb/>neat <arrow.to.target n="note259"></arrow.to.target>in N appen&longs;um, &longs;ubduplam e&longs;&longs;e eiu&longs;dem <lb/>ponderis; potentia igitur in N ponderi O æ­<lb/>qualis pondus M potentiæ in L æquale &longs;u&longs;ti<lb/>nebit; ponderi&longs;q; M dupla erit. </s> <s id="id.2.1.177.18.1.2.0">quod demon<lb/>&longs;trare oportebat. </s> </p> <p id="id.2.1.178.1.0.0.0" type="margin"> <s id="id.2.1.178.1.1.1.0"><margin.target id="note259"></margin.target>3 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <figure id="id.036.01.191.1.jpg" xlink:href="036/01/191/1.jpg"> </figure> <p id="id.2.1.179.1.1.1.0" type="head"> <s id="id.2.1.179.1.3.1.0">ALITER. </s> </p> <p id="id.2.1.179.2.0.0.0" type="main"> <s id="id.2.1.179.2.1.1.0">Ii&longs;dem po&longs;itis. </s> <s id="id.2.1.179.2.1.2.0">Quoniam potentia in F, <arrow.to.target n="note260"></arrow.to.target><lb/>&longs;eu in D, quod idem e&longs;t, æqualis e&longs;t ponde<lb/>ri M; & BD e&longs;t vectis, cuius fulcimentum <lb/>e&longs;t B, & potentia in N e&longs;t, ac &longs;i e&longs;&longs;et in me­<lb/>dio vectis, & pondus æquale ip&longs;i M, ac &longs;i e&longs;­<lb/>&longs;et in D propter funem FD; quod idem <lb/>e&longs;t, ac &longs;i BCD e&longs;&longs;et orbiculus trochleæ &longs;upe<lb/>rioris, pondusq; appen&longs;um e&longs;&longs;et in fune DF, <lb/>&longs;icut in decimaquinta, & decima&longs;exta dictum e&longs;t; ergo potentia in <lb/>N dupla e&longs;t ponderis M. quod erat o&longs;tendendum. </s> </p> <p id="id.2.1.180.1.0.0.0" type="margin"> <s id="id.2.1.180.1.1.1.0"><margin.target id="note260"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.181.1.0.0.0" type="main"> <s id="id.2.1.181.1.1.1.0">Si autem in N &longs;it potentia mouens pondus M, erit &longs;patium <lb/>ponderis M duplum &longs;patii potentiæ in N. quod ex duodecima <lb/>huius manife&longs;tum e&longs;t; &longs;patium enim puncti L deor&longs;um ten­<lb/>dentis duplum e&longs;t &longs;patii N &longs;ur&longs;um; erit igitur è conuer&longs;o &longs;patium <lb/>potentiæ in N deor&longs;um tendentis dimidium <expan abbr="&longs;aptii">spatii</expan> ponderis M &longs;ur<lb/>&longs;um moti. </s> </p> <p id="id.2.1.181.2.0.0.0" type="main"> <s id="id.2.1.181.2.1.1.0">Sicut autem ex tertia, quinta, &longs;eptima huius, &c. </s> <s id="id.2.1.181.2.1.2.0">colligi po&longs;&longs;unt <lb/>ponderis O rationes quotcunq; multiplices ip&longs;ius potentiæ in L, <lb/><expan abbr="eod&etilde;">eodem</expan> quoq; modo o&longs;tendi poterunt potentiæ in N pondus &longs;u&longs;tinen<lb/>tis ponderis M quotcunq; multiplices. </s> <s id="id.2.1.181.2.1.3.0">Atq; ita ex decimatertia <pb xlink:href="036/01/192.jpg"/>decimaquarta rationes o&longs;ten <lb/>dentur quotcunq; multiplices <lb/>&longs;patii ponderis M ad &longs;patium <lb/>potentiæ mouentis in N con&longs;ti<lb/>tutæ. <figure id="id.036.01.192.1.jpg" xlink:href="036/01/192/1.jpg"></figure></s> </p> <p id="id.2.1.181.3.0.0.0" type="main"> <s id="id.2.1.181.3.1.1.0">Poterit quoq; ex decima&longs;e<lb/>ptima decimaoctaua huius mul<lb/>tiplex inueniri proportio, quam <lb/>habet potentia pondus &longs;u&longs;ti<lb/>nens ad ip&longs;um pondus; &longs;icut <lb/>proportio potentiæ in N ad pon<lb/>dus M ex decimaquinta, & deci<lb/>ma&longs;exta o&longs;tendebatur: inuenie<lb/>turq; ita e&longs;&longs;e pondus ad poten<lb/>tiam pondus &longs;u&longs;tinentem, vt &longs;pa<lb/>tium potentiæ mouentis ad &longs;pa<lb/>tium ponderis. </s> </p> <p id="id.2.1.181.4.0.0.0" type="main"> <s id="id.2.1.181.4.1.1.0">Vectium motus in his fit <lb/>hoc modo, videlicet vectes or<lb/>biculorum trochleæ inferioris <lb/>mouentur, vt vectis BD, quæ <lb/>mouetur, ac &longs;i B e&longs;&longs;et fulcimen <lb/>tum, & pondus in D, & poten<lb/>tia in medio. </s> <s id="id.2.1.181.4.1.2.0">Vectes verò or<lb/>biculorum trochleæ &longs;uperioris mouentur, vt FH, cuius fulcimen <lb/>tum e&longs;t in medio, pondus in H, & potentia in F. </s> </p> <p id="id.2.1.181.5.0.0.0" type="head"> <s id="id.2.1.181.5.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.181.6.0.0.0" type="main"> <s id="id.2.1.181.6.1.1.0">Ex hoc manife&longs;tum e&longs;t, orbiculos trochleæ <lb/>inferioris in his efficere, vt pondus maiori po­<pb n="90" xlink:href="036/01/193.jpg"/>tentia moueatur, quàm &longs;it ip&longs;um pondus, & <lb/>per maius &longs;patium &longs;patio potentiæ, & minori <lb/>tempore per æquale. </s> <s id="id.2.1.181.6.1.2.0">quod quidem orbiculi &longs;u<lb/>perioris trochleæ non efficiunt. </s> </p> <p id="id.2.1.181.7.0.0.0" type="main"> <s id="id.2.1.181.7.1.1.0">Cognitis proportionibus multiplicibus, iam ad &longs;uperparticu<lb/>lares accedendum e&longs;t. </s> </p> <p id="id.2.1.181.8.0.0.0" type="head"> <s id="id.2.1.181.8.1.1.0">PROPOSITIO XX. </s> </p> <p id="id.2.1.181.9.0.0.0" type="main"> <s id="id.2.1.181.9.1.1.0">Si vtriu&longs;q; duarum trochlearum &longs;ingulis or­<lb/>biculis, quarum altera &longs;upernè à potentia &longs;u&longs;ti­<lb/>neatur, altera verò infernè, ponderiq; alligata, <lb/><expan abbr="cõ&longs;tituta">con&longs;tituta</expan> fuerit, funis reuoluatur; altero eius extre<lb/>mo alicuibi, altero verò inferiori trochleæ reli<lb/>gato; pondus potentiæ &longs;e&longs;quialterum erit. </s> </p> <pb xlink:href="036/01/194.jpg"/> <p id="id.2.1.181.11.0.0.0" type="main"> <s id="id.2.1.181.11.1.1.0">Sit ABC orbiculus <lb/>trochleæ &longs;uperioris, & <lb/>DEF trochleæ inferio­<lb/>ris ponderi G alligatæ; <lb/>&longs;itq; funis HABCDE <lb/>Fk circa orbiculos re­<lb/>uolutus, qui &longs;it religatus <lb/>in K, & in H trochleæ <lb/>inferiori; &longs;itq; potentia <lb/>in L &longs;u&longs;tinens pondus <lb/>G. </s> <s id="id.2.1.181.11.1.1.0.a">dico pondus poten<lb/>tiæ &longs;e&longs;quialterum e&longs;&longs;e. </s> <s id="id.2.1.181.11.1.2.0"><lb/><arrow.to.target n="note261"></arrow.to.target>Quoniam enim vterque <lb/>funis CD AH tertiam <lb/>&longs;u&longs;tinet partem ponde­<lb/>ris G, erit vnaquæq; po<lb/>tentia in DH &longs;ubtripla <lb/>ponderis G; quibus &longs;i­<lb/>mul a&longs;&longs;umptis e&longs;t æqua­<lb/><figure id="id.036.01.194.1.jpg" xlink:href="036/01/194/1.jpg"></figure><lb/><arrow.to.target n="note262"></arrow.to.target>lis potentia in L: potentia enim in L dupla e&longs;t potentiæ in D, & <lb/>eius, quæ e&longs;t in H. quare potentia in L &longs;ub&longs;e&longs;quialtera e&longs;t ponde­<lb/>ris G. </s> <s id="id.2.1.181.11.1.2.0.a">pondus ergo G ad pontentiam in L e&longs;t, vt tria ad duo; <lb/>hoc e&longs;t &longs;e&longs;quialterum. </s> <s id="id.2.1.181.11.1.3.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.182.1.0.0.0" type="margin"> <s id="id.2.1.182.1.1.1.0"><margin.target id="note261"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 5 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.182.1.1.2.0"><margin.target id="note262"></margin.target><emph type="italics"/>Ex.<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <pb n="91" xlink:href="036/01/195.jpg"/> <p id="id.2.1.183.1.0.0.0" type="main"> <s id="id.2.1.183.1.2.1.0">Si autem in L &longs;it potentia mouens pondus. </s> <s id="id.2.1.183.1.2.2.0"><lb/>Dico &longs;patium potentiæ &longs;patii ponderis &longs;e&longs;quial­<lb/>terum e&longs;&longs;e. </s> </p> <p id="id.2.1.183.2.0.0.0" type="main"> <s id="id.2.1.183.2.1.1.0">Ii&longs;dem po&longs;itis, perueniat orbi­<lb/>culus ABC v&longs;q; ad MNO, & <lb/>DEF ad PQR; & H in S; & <lb/>pondus G v&longs;q; ad T. </s> <s id="id.2.1.183.2.1.1.0.a">Et quoniam <lb/>funis HABCDEFK e&longs;t æqualis <lb/>funi SMNOPQRk, cùm &longs;it <lb/>idem funis; & funes circa &longs;emicir<lb/>culos ABC MNO &longs;unt inter &longs;e <lb/>&longs;e æquales; qui verò &longs;unt circa <lb/>DEF PQR &longs;imiliter inter &longs;e æ­<lb/>quales; Demptis igitur AS CP <lb/>RK communibus, erunt duo CO <lb/>MA tribus DP HS FR æqua­<lb/>les. </s> <s id="id.2.1.183.2.1.2.0">&longs;ed vterq; CO AM &longs;eor&longs;um <lb/>e&longs;t æqualis &longs;patio potentiæ motæ. </s> <s id="id.2.1.183.2.1.3.0"><lb/>quare duo CO MA, &longs;imul &longs;patii <lb/>potentiæ dupli erunt: tre&longs;q; DP <lb/>HS FR &longs;imul &longs;imili modo &longs;patii <lb/>ponderis moti tripli erunt. </s> <s id="id.2.1.183.2.1.4.0">dimidia <lb/>verò pars, hoc e&longs;t &longs;patium poten<lb/>tiæ motæ ad tertiam, ad &longs;patium <lb/>&longs;cilicet ponderis moti ita &longs;e habet, <lb/>vt duplum dimidii ad duplum ter­<lb/>tii; hoc e&longs;t, vt totum ad duas ter<lb/><figure id="id.036.01.195.1.jpg" xlink:href="036/01/195/1.jpg"></figure><lb/>tias, quod e&longs;t vt tria ad duo. </s> <s id="id.2.1.183.2.1.5.0">&longs;patium ergo potentiæ in L &longs;pa­<lb/>tii ponderis G moti &longs;e&longs;quialterum e&longs;t. </s> <s id="id.2.1.183.2.1.6.0">quod o&longs;tendere opor­<lb/>tebat. </s> </p> <pb xlink:href="036/01/196.jpg"/> <p id="id.2.1.183.3.0.0.0" type="head"> <s id="id.2.1.183.4.1.1.0">PROPOSITIO XXI. </s> </p> <p id="id.2.1.183.5.0.0.0" type="main"> <s id="id.2.1.183.5.1.1.0">Si tribus duarum trochlearum orbiculis, qua <lb/>rum altera vnius tantùm orbiculi &longs;upernè à po­<lb/>tentia &longs;u&longs;tineatur, altera verò duorum infernè, <lb/>ponderiq; alligata, collocata fuerit, funis cir­<lb/>cumuoluatur; altero eius extremo alicubi, altero <lb/>autem &longs;uperiori trochleæ religato: pondus poten<lb/>tiæ &longs;e&longs;quitertium erit. </s> </p> <p id="id.2.1.183.6.0.0.0" type="main"> <s id="id.2.1.183.6.1.1.0">Sit pondus A trochleæ inferiori alliga­<lb/>tum, quæ duos habeat orbiculos, quorum <lb/>centra &longs;int BC; &longs;uperiorq; trochlea orbicu­<lb/>lum habeat, cuius centrum D; & &longs;it funis <lb/>EFGHkLMN circa omnes orbiculos re <lb/>uolutus, qui religatus &longs;it in N, & in E tro<lb/>chleæ &longs;uperiori; &longs;itqué potentia in O <lb/>&longs;u&longs;tinens pondus A. </s> <s id="id.2.1.183.6.1.1.0.a">dico pondus po­<lb/><arrow.to.target n="note263"></arrow.to.target>tentiæ &longs;e&longs;quitertium e&longs;&longs;e. </s> <s id="id.2.1.183.6.1.2.0">Quoniam enim <lb/>vnu&longs;qui&longs;q; funis NM HG EF KL quar­<lb/>tam &longs;u&longs;tinent partem ponderis A, & omnes <lb/>&longs;imul totum &longs;u&longs;tinent pondus; tres HG <lb/>EF kL &longs;imul tres &longs;u&longs;tinebunt partes pon­<lb/>deris A. quare pondus A ad hos omnes <lb/>&longs;imul erit, vt quatuor ad tria: & cùm po­<lb/>tentia in O idem efficiat, quod HG EF kL <lb/>&longs;imul efficiunt; omnes enim &longs;u&longs;tinet; erit po<lb/>tentia in O tribus &longs;imul HG EF kL æ­<lb/>qualis; & ob id pondus A ad potentiam <lb/>in O erit, vt quatuor ad tria; hoc e&longs;t &longs;e&longs;qui<lb/>tertium. </s> <s id="id.2.1.183.6.1.3.0">quod demon&longs;trare oportebat. <figure id="id.036.01.196.1.jpg" xlink:href="036/01/196/1.jpg"></figure></s> </p> <pb n="92" xlink:href="036/01/197.jpg"/> <p id="id.2.1.183.8.0.0.0" type="main"> <s id="id.2.1.183.8.1.1.0">Si vero in O &longs;it potentia mouens pondus A. </s> <s id="id.2.1.183.8.1.1.0.a"><lb/>Dico &longs;patium potentiæ in O decur&longs;um &longs;patii pon<lb/>deris A moti &longs;e&longs;quitertium e&longs;&longs;e. </s> </p> <p id="id.2.1.184.1.0.0.0" type="margin"> <s id="id.2.1.184.1.1.1.0"><margin.target id="note263"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 1 <emph type="italics"/>&longs;eptimebuius.<emph.end type="italics"/></s> </p> <p id="id.2.1.185.1.0.0.0" type="main"> <s id="id.2.1.185.1.1.1.0">Ii&longs;dem po&longs;itis, &longs;it centrum B motum <lb/>in P; &C v&longs;q; ad Q; & D in R; & E in <lb/>S eodem tempore: & per centra ducantur <lb/>ML 9Z FG TV Hk XY horizonti, <lb/>& inter &longs;e &longs;e æquidi&longs;tantes. </s> <s id="id.2.1.185.1.1.2.0">Similiter, vt in <lb/>præcedente o&longs;tendetur tres <emph type="italics"/>X<emph.end type="italics"/>H SE Yk <lb/>quatuor TG VF ZL 9M æquales e&longs;&longs;e. </s> <s id="id.2.1.185.1.1.3.0">& <lb/>quoniam tres XH SE Yk &longs;imul triplæ <lb/>&longs;unt &longs;patii potentiæ, quatuor verò TG VF <lb/>ZL 9M &longs;imul quadruplæ &longs;unt &longs;patii pon<lb/>deris moti; erit &longs;patium potentiæ ad &longs;pa­<lb/>tium ponderis, vt tertia pars ad quartam. </s> <s id="id.2.1.185.1.1.4.0"><lb/>&longs;ed tertia pars ad quartam e&longs;t, vt tres ter<lb/>tiæ ad tres quartas, hoc e&longs;t, vt totum ad <lb/>tres quartas; quod e&longs;t, vt quatuor ad tria. </s> <s id="id.2.1.185.1.1.5.0"><lb/>&longs;patium ergo potentiæ &longs;patii ponderis mo<lb/>ti &longs;e&longs;quitertium e&longs;t. </s> <s id="id.2.1.185.1.1.6.0">quod erat demon­<lb/>&longs;trandum. <figure id="id.036.01.197.1.jpg" xlink:href="036/01/197/1.jpg"></figure></s> </p> <p id="id.2.1.185.2.0.0.0" type="main"> <s id="id.2.1.185.2.1.1.0">Si verò funis in E per alium circumuol<lb/>uatur orbiculum, qui deinde trochleæ in <lb/>feriori religetur; &longs;imiliter o&longs;tendetur pro <lb/>portionem ponderis ad <expan abbr="potentiã">potentiam</expan> in O pon<lb/>dus &longs;u&longs;tinentem &longs;e&longs;quiquartam e&longs;&longs;e. </s> <s id="id.2.1.185.2.1.2.0">quòd <lb/>&longs;i in O &longs;it potentia mouens pondus, o&longs;ten <lb/>detur &longs;patium potentiæ &longs;patii ponderis &longs;e&longs;<lb/>quiquartum e&longs;&longs;e. </s> <s id="id.2.1.185.2.1.3.0">& &longs;ic in infinitum proce<lb/>dendo quamcunq; &longs;uperparticularem pro <lb/>portionem ponderis ad potentiam inuenie<lb/>mus; &longs;emperq; reperiemus, ita e&longs;&longs;e pondus <lb/>ad potentiam pondus &longs;u&longs;tinentem, vt &longs;pa­<lb/>tium potentiæ mouentis ad &longs;patium ponde­<lb/>ris moti. </s> </p> <pb xlink:href="036/01/198.jpg"/> <p id="id.2.1.185.4.0.0.0" type="main"> <s id="id.2.1.185.4.1.1.0">Motus verò vectium fit hoc mo <lb/>do, videlicet vectis ML fulci­<lb/>mentum e&longs;t M, cùm funis &longs;it re <lb/>ligatus in N, & pondus in me­<lb/>dio, & potentia in L. quia <expan abbr="ve­rò">ve­<lb/>ro</expan> punctum L tendit &longs;ur&longs;um, quod <lb/>à fune KL mouetur, idcirco K &longs;ur­<lb/>&longs;um mouebitur, & vectis HK ful<lb/>cimentum erit H, pondus ac &longs;i e&longs;<lb/>&longs;ent in k, & potentia in medio; <lb/>vectis autem FG fulcimentum <lb/>erit G, pondus in medio; & poten<lb/>tia in F. </s> <s id="id.2.1.185.4.1.1.0.a">punctum enim F &longs;ur&longs;um <lb/>mouetur à fune EF. </s> <s id="id.2.1.185.4.1.1.0.b">Præterea <lb/>G in orbiculo deor&longs;um tendit, <lb/>quia H quoque in eius orbiculo <lb/>deor&longs;um mouetur. <figure id="id.036.01.198.1.jpg" xlink:href="036/01/198/1.jpg"></figure></s> <pb n="93" xlink:href="036/01/199.jpg"/> <s id="id.2.1.185.4.3.1.0">PROPOSITIO XXII. </s> </p> <p id="id.2.1.185.5.0.0.0" type="main"> <s id="id.2.1.185.5.1.1.0">Si vtri&longs;que duarum trochlearum &longs;ingulis <lb/>orbiculis, quarum altera &longs;upernè à potentia <lb/>&longs;u&longs;tineatur, altera verò infernè, ponderiq; alli­<lb/>gata, collocata fuerit, circumducatur funis; al­<lb/>tero eius extremo alicubi, altero autem &longs;uperio<lb/>ri trochleæ religato. </s> <s id="id.2.1.185.5.1.2.0">erit potentia ponderis &longs;e&longs;<lb/>quialtera. </s> </p> <p id="id.2.1.185.6.0.0.0" type="main"> <s id="id.2.1.185.6.1.1.0">Sit orbiculus ABC trochleæ ponderi D al <lb/>ligatæ; & EFG trochleæ &longs;uperioris, cuius <lb/>centrum H; &longs;it deinde funis k ABCEFGL <lb/>circa orbiculos reuolutus, & religatus in L, & <lb/>in k trochleæ &longs;uperiori; &longs;itq; potentia in M <lb/>&longs;u&longs;tinens pondus D. </s> <s id="id.2.1.185.6.1.1.0.a">dico potentiam ponde<lb/>ris &longs;e&longs;quialteram e&longs;&longs;e. </s> <s id="id.2.1.185.6.1.2.0">Quoniam enim poten<arrow.to.target n="note264"></arrow.to.target><lb/>tia in E &longs;u&longs;tinens pondus D &longs;ubdupla e&longs;t pon<arrow.to.target n="note265"></arrow.to.target><lb/>deris D, potentiæ verò in E dupla e&longs;t poten<arrow.to.target n="note266"></arrow.to.target><lb/>tia in H; erit potentia in H ponderi D æqua <arrow.to.target n="note267"></arrow.to.target><lb/>lis; & cùm potentia in K &longs;ubdupla &longs;it ponde<lb/>ris D; erunt vtræq; &longs;imul potentiæ in H k &longs;e&longs;<lb/>quialteræ ponderis D. </s> <s id="id.2.1.185.6.1.2.0.a">Itaq; cùm potentia in <lb/>M duabus potentiis in Hk &longs;imul &longs;umptis &longs;it <lb/>æqualis, quemadmodum in &longs;uperioribus o­<lb/>&longs;ten&longs;um e&longs;t; erit potentia in M &longs;e&longs;quialtera <lb/>ponderis D. quod oportebat demon&longs;trare. </s> </p> <p id="id.2.1.186.1.0.0.0" type="margin"> <s id="id.2.1.186.1.1.1.0"><margin.target id="note264"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.186.1.1.2.0"><margin.target id="note265"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.186.1.1.3.0"><margin.target id="note266"></margin.target>2 <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.186.1.1.4.0"><margin.target id="note267"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.187.1.0.0.0" type="main"> <s id="id.2.1.187.1.1.1.0">Si verò in M &longs;it potentia mouens pondus, <lb/>&longs;imiliter vt in præcedentibus o&longs;tendetur, &longs;pa<lb/>tium ponderis &longs;patii potentiæ &longs;e&longs;quialterum <lb/>e&longs;&longs;e. <figure id="id.036.01.199.1.jpg" xlink:href="036/01/199/1.jpg"></figure></s> </p> <pb xlink:href="036/01/200.jpg"/> <p id="id.2.1.187.3.0.0.0" type="main"> <s id="id.2.1.187.3.1.1.0">Et &longs;i funis in K per alium circumuoluatur <lb/>orbiculum, cuius centrum &longs;it N; qui dein­<lb/>de trochleæ inferiori religetur in O; & po­<lb/>tentia in M &longs;u&longs;tineat pondus D. </s> <s id="id.2.1.187.3.1.1.0.a">dico pro­<lb/>portionem potentiæ ad pondus &longs;e&longs;quiter­<lb/>tiam e&longs;&longs;e. <figure id="id.036.01.200.1.jpg" xlink:href="036/01/200/1.jpg"></figure></s> </p> <p id="id.2.1.187.4.0.0.0" type="main"> <s id="id.2.1.187.4.1.1.0">Quoniam enim potentia in E &longs;u&longs;tinens <lb/><arrow.to.target n="note268"></arrow.to.target>pondus D fune ECB AKPO &longs;ubtripla e&longs;t <lb/><arrow.to.target n="note269"></arrow.to.target>ip&longs;ius D, ip&longs;ius autem E dupla e&longs;t potentia <lb/>in H; erit potentia in H &longs;ub&longs;e&longs;quialtera pon<lb/>deris D. &longs;imili quoq; modo quoniam po<lb/>tentia in O, quæ e&longs;t, ac &longs;i e&longs;&longs;et in centro or<lb/><arrow.to.target n="note270"></arrow.to.target>biculi ABC, &longs;ubtripla e&longs;t ponderis D; ip­<lb/>&longs;ius autem O dupla e&longs;t potentia in N; erit <lb/>quoq; potentia in N &longs;ub&longs;e&longs;quialtera ponde­<lb/>ris D. quare duæ &longs;imul potentiæ in HN pon<lb/>dus D &longs;uperant tertia parte, &longs;e &longs;e habentq; ad <lb/>D in ratione &longs;e&longs;quitertia: & cùm potentia <lb/>in M duabus &longs;it potentiis in HN &longs;imul &longs;um<lb/>ptis æqualis, &longs;uperabit itidem potentia in <lb/>M pondus D tertia parte. </s> <s id="id.2.1.187.4.1.2.0">ergo proportio <lb/>potentiæ in M ad pondus D &longs;e&longs;quitertia <lb/>e&longs;t. </s> <s id="id.2.1.187.4.1.3.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.188.1.0.0.0" type="margin"> <s id="id.2.1.188.1.1.1.0"><margin.target id="note268"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.188.1.1.2.0"><margin.target id="note269"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.188.1.1.3.0"><margin.target id="note270"></margin.target>3, 15,<emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.189.1.0.0.0" type="main"> <s id="id.2.1.189.1.1.1.0">Si autem in M &longs;it potentia mouens pon­<lb/>dus, &longs;imili modo o&longs;tendetur &longs;patium ponderis D &longs;patii potentiæ in <lb/>M &longs;e&longs;quitertium e&longs;&longs;e. </s> </p> <p id="id.2.1.189.2.0.0.0" type="main"> <s id="id.2.1.189.2.1.1.0">Et &longs;i funis in O per alium circumuoluatur orbiçulum, qui tro­<lb/>chleæ &longs;uperiori deinde religetur; eodem modo demon&longs;trabimus <lb/>proportionem potentiæ in M pondus &longs;u&longs;tinentis ad pondus &longs;e&longs;­<lb/>quiquartam e&longs;&longs;e. </s> <s id="id.2.1.189.2.1.2.0">& &longs;i in M &longs;it potentia mouens, &longs;imiliter o&longs;ten­<lb/>detur &longs;patium ponderis &longs;patii potentiæ &longs;e&longs;quiquartum e&longs;&longs;e. </s> <s id="id.2.1.189.2.1.3.0">pro­<lb/>cedendoq; hoc modo in infinitum quamcunq; proportionem <lb/>potentiæ ad pondus &longs;uperparticularem inueniemus; &longs;emperqué <pb n="94" xlink:href="036/01/201.jpg"/>o&longs;tendemus potentiam pondus &longs;u&longs;tinentem ita e&longs;&longs;e ad pondus, <lb/>vt &longs;patium ponderis ad &longs;patium potentiæ pondus mouentis. </s> </p> <p id="id.2.1.189.3.0.0.0" type="main"> <s id="id.2.1.189.3.1.1.0">Motus verò vectis EG e&longs;t, ac &longs;i G e&longs;&longs;et fulcimentum, cùm <lb/>funis &longs;it religatus in L; pondus ac &longs;i in E e&longs;&longs;et appen&longs;um, & po­<lb/>tentia in medio. </s> <s id="id.2.1.189.3.1.2.0">Vectis verò CA fulcimentum e&longs;t A pondus in <lb/>medio, & potentia in C. & K fulcimentum e&longs;t vectis Pk, pon­<lb/>dus in P, & potentia in medio. </s> <s id="id.2.1.189.3.1.3.0">quæ omnia &longs;icut in præceden­<lb/>ti o&longs;tendentur. </s> </p> <p id="id.2.1.189.4.0.0.0" type="head"> <s id="id.2.1.189.4.1.1.0">PROPOSITIO XXIII. </s> </p> <p id="id.2.1.189.5.0.0.0" type="main"> <s id="id.2.1.189.5.1.1.0">Si vtri&longs;q; duarum trochlearum &longs;ingulis or­<lb/>biculis, quarum altera &longs;upernè à potentia &longs;u&longs;ti­<lb/>neatur, altera verò infernè, ponderiq; alligata, <lb/><expan abbr="cõ&longs;tituta">con&longs;tituta</expan> fuerit, circumferatur funis; vtroq; eius <lb/>extremo alicuibi, non autem trochleis religato; <lb/>æqualis erit ponderi potentia. </s> </p> <pb xlink:href="036/01/202.jpg"/> <p id="id.2.1.189.7.0.0.0" type="main"> <s id="id.2.1.189.7.1.1.0">Sit orbiculus trochleæ &longs;uperioris <lb/>ABC, cuius centrum D; & EFG <lb/>trochleæ ponderi H alligatæ, cu­<lb/>ius centrum k; & &longs;it funis LEF <lb/>GABCM circa orbiculos reuo­<lb/>lutus, religatu&longs;q; in LM; &longs;itq; <lb/>potentia in N &longs;u&longs;tinens pondus <lb/>H. </s> <s id="id.2.1.189.7.1.1.0.a">dico potentiam in N æqua<lb/>lem e&longs;&longs;e ponderi H. </s> <s id="id.2.1.189.7.1.1.0.b">Accipiatur <lb/>quoduis punctum O in AG. </s> <s id="id.2.1.189.7.1.1.0.c">& <lb/>quoniam &longs;i in O e&longs;&longs;et potentia &longs;u<lb/><arrow.to.target n="note271"></arrow.to.target>&longs;tinens pondus H, &longs;ubdupla e&longs;&longs;et <lb/><arrow.to.target n="note272"></arrow.to.target>ponderis H, & potentiæ in O <lb/>dupla e&longs;t ea, quæ e&longs;t in D, &longs;iue <lb/>(quod idem e&longs;t) in N; erit po<lb/>tentia in N ponderi H æqualis. </s> <lb/> <s id="id.2.1.189.7.1.2.0">quod demon&longs;trare oportebat. <figure id="id.036.01.202.1.jpg" xlink:href="036/01/202/1.jpg"></figure></s> </p> <p id="id.2.1.189.8.0.0.0" type="main"> <s id="id.2.1.189.8.1.1.0">Et &longs;i in N &longs;it potentia mouens pondus. </s> <s id="id.2.1.189.8.1.2.0">Dico <lb/>&longs;patium potentiæ in N æqualem e&longs;&longs;e &longs;patio pon<lb/>deris H moti. </s> </p> <p id="id.2.1.190.1.0.0.0" type="margin"> <s id="id.2.1.190.1.1.1.0"><margin.target id="note271"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.190.1.1.2.0"><margin.target id="note272"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.191.1.0.0.0" type="main"> <s id="id.2.1.191.1.1.1.0">Quoniam enim &longs;patium puncti O moti, duplum e&longs;t, tùm &longs;patii <lb/><arrow.to.target n="note273"></arrow.to.target>ponderis H moti, tùm &longs;patii potentiæ in N motæ; erit &longs;patium <lb/><arrow.to.target n="note274"></arrow.to.target>potentiæ in N &longs;patio ponderis H æquale. </s> </p> <p id="id.2.1.192.1.0.0.0" type="margin"> <s id="id.2.1.192.1.1.1.0"><margin.target id="note273"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.192.1.1.2.0"><margin.target id="note274"></margin.target>16 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="95" xlink:href="036/01/203.jpg"/> <p id="id.2.1.193.1.0.0.0" type="head"> <s id="id.2.1.193.1.2.1.0">ALITER. </s> </p> <p id="id.2.1.193.2.0.0.0" type="main"> <s id="id.2.1.193.2.1.1.0">Ii&longs;dem po&longs;itis, transfera<lb/>tur centrum orbiculi ABC <lb/>v&longs;q; ad P; orbiculu&longs;q; po&longs;i<lb/>tionem habeat QRS; dein<lb/>de eodem tempore orbiculus <lb/>EFG &longs;it in TVX, cuius cen<lb/>trum &longs;it Y; & pondus perue<lb/>nerit in Z. ducantur per or<lb/>biculorum centra lineæ GE <lb/>TX AC QS horizonti æqui <lb/>di&longs;tantes. </s> <s id="id.2.1.193.2.1.2.0">& &longs;icut in aliis <lb/>demon&longs;tratum fuit, d uo fu­<lb/>nes AQ CS duobus XG <lb/>TE æquales erunt; &longs;ed AQ <lb/>CS &longs;imul dupli &longs;unt &longs;patii po<lb/>tentiæ motæ; & duo XG TE <lb/>&longs;imul &longs;unt &longs;imiliter dupli &longs;pa<lb/>tii ponderis; erit igitur <expan abbr="&longs;patiũ">&longs;patium</expan> <lb/>potentiæ &longs;patio ponderis æ­<lb/>quale. </s> <s id="id.2.1.193.2.1.3.0">quod demon&longs;trare o­<lb/>portebat. <figure id="id.036.01.203.1.jpg" xlink:href="036/01/203/1.jpg"></figure></s> </p> <pb xlink:href="036/01/204.jpg"/> <p id="id.2.1.193.4.0.0.0" type="main"> <s id="id.2.1.193.4.1.1.0">Quod etiam &longs;i vtraq; trochlea duos <lb/>habuerit orbiculos, quorum centra <lb/>&longs;int ABCD, funi&longs;q; per omnes cir<lb/>cumuoluatur, qui in LM religetur; <lb/>&longs;imiliter o&longs;tendetur potentiam in N <lb/>æqualem e&longs;&longs;e ponderi H. vnaquæq; <lb/>enim potentia in EF &longs;u&longs;tinens pon­<lb/>dus &longs;ubquadrupla e&longs;t ponderis; & po<lb/>tentiæ in CD duplæ &longs;unt earum, <lb/>quæ &longs;unt in EF; erit vnaquæq; po­<lb/>tentia in CD &longs;ubdupla ponderis H. <lb/>quare potentiæ in CD &longs;imul &longs;umptæ <lb/>ponderi H erunt æquales. </s> <s id="id.2.1.193.4.1.2.0">& quo­<lb/>niam potentia in N duabus in CD <lb/>pontentiis e&longs;t æqualis; erit potentia <lb/>in N ponderi H, æqualis. </s> </p> <p id="id.2.1.193.5.0.0.0" type="main"> <s id="id.2.1.193.5.1.1.0">Et &longs;i in N &longs;it potentia mouens, &longs;i <lb/>mili modo o&longs;tendetur, &longs;patium po­<lb/>tentiæ æquale e&longs;&longs;e &longs;patio ponderis. </s> </p> <p id="id.2.1.193.6.0.0.0" type="main"> <s id="id.2.1.193.6.1.1.0">Si autem vtraq; trochlea tres, vel <lb/>quatuor, vel quotcunq; habeat orbi­<lb/>culos; &longs;emper o&longs;tendetur <expan abbr="pot&etilde;tiam">potentiam</expan> in <lb/>N æqualem e&longs;&longs;e ponderi H; & &longs;pa<lb/>tium potentiæ pondus mouentis æ­<lb/>quale e&longs;&longs;e &longs;patio ponderis moti. <figure id="id.036.01.204.1.jpg" xlink:href="036/01/204/1.jpg"></figure></s> </p> <p id="id.2.1.193.7.0.0.0" type="main"> <s id="id.2.1.193.7.1.1.0">Vectium autem motus hoc pacto &longs;e habent; orbiculorum qui <lb/>dem trochleæ &longs;uperioris, veluti AC in præcedenti figura fulcimen <lb/>tum e&longs;t C, pondus verò in A appen&longs;um, & potentia in D medio. </s> <s id="id.2.1.193.7.1.2.0"><lb/>vectes autem orbiculorum trochleæ inferioris ita mouentur, vt ip<lb/>&longs;ius GE fulcimentum &longs;it E, pondus in medio appen&longs;um, & po<lb/>tentia in G. </s> </p> <pb n="96" xlink:href="036/01/205.jpg"/> <p id="id.2.1.193.9.0.0.0" type="head"> <s id="id.2.1.193.9.1.1.0">PROPOSITIO XXIIII. </s> </p> <p id="id.2.1.193.10.0.0.0" type="main"> <s id="id.2.1.193.10.1.1.0">Si tribus duarum trochlearum orbiculis, qua <lb/>rum altera vnius dumtaxat orbiculi &longs;upernè à <lb/>potentia &longs;u&longs;tineatur, altera verò duorum <expan abbr="infer­nè">infer­<lb/>ne</expan>, ponderiq; alligata fuerit con&longs;tituta, cir­<lb/>cundetur funis; vtroq; eius extremo alicubi, &longs;ed <lb/>non &longs;uperiori trochleæ religato: duplum erit <lb/>pondus potentiæ. </s> </p> <p id="id.2.1.193.11.0.0.0" type="main"> <s id="id.2.1.193.11.1.1.0">Sint AB centra orbiculorum <lb/>trochleæ ponderi C alligatæ; D ve<lb/>rò &longs;it centrum orbiculi trochleæ &longs;u<lb/>perioris; &longs;it deinde funis per om<lb/>nes orbiculos circumuolutus, reli<lb/>gatu&longs;q; in EF; & &longs;it potentia in <lb/>G &longs;u&longs;tinens pondus C. </s> <s id="id.2.1.193.11.1.1.0.a">dico pon<lb/>dus C duplum e&longs;&longs;e potentiæ in G. </s> <s id="id.2.1.193.11.1.1.0.b"><lb/>Quoniam enim &longs;i in H k duæ e&longs;­<lb/>&longs;ent potentiæ pondus &longs;u&longs;tinentes <lb/>duobus funibus orbiculis trochleæ <lb/>inferioris tantùm circumuolutis, e&longs;<lb/>&longs;et vtiq; vtraq; potentia in k H &longs;ub <arrow.to.target n="note275"></arrow.to.target><lb/>quadrupla ponderis C; &longs;ed poten­<lb/>tia in G æqualis e&longs;t potentiis in Hk <arrow.to.target n="note276"></arrow.to.target><lb/>&longs;imul &longs;umptis; vniu&longs;cuiu&longs;q; enim <lb/>potentiæ in H, & k dupla e&longs;t: erit <lb/>potentia in G &longs;ubdupla ponderis <lb/>C. pondus ergo potentiæ duplum <lb/>erit. </s> <s id="id.2.1.193.11.1.2.0">quod demon&longs;trare opor­<lb/>tebat. <figure id="id.036.01.205.1.jpg" xlink:href="036/01/205/1.jpg"></figure></s> </p> <pb xlink:href="036/01/206.jpg"/> <p id="id.2.1.193.13.0.0.0" type="main"> <s id="id.2.1.193.13.1.1.0">Et &longs;i in G &longs;it potentia mouens pondus. </s> <s id="id.2.1.193.13.1.2.0">Dico <lb/>&longs;patium potentiæ duplum e&longs;&longs;e &longs;patii ponderis. </s> </p> <p id="id.2.1.194.1.0.0.0" type="margin"> <s id="id.2.1.194.1.1.1.0"><margin.target id="note275"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>huius<emph.end type="italics"/></s> <s id="id.2.1.194.1.1.2.0"><margin.target id="note276"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.195.1.0.0.0" type="main"> <s id="id.2.1.195.1.1.1.0">Ii&longs;dem po&longs;itis, &longs;int <lb/>moti orbiculi, &longs;imiliter <lb/>demon&longs;trabitur ambos <lb/>illos LM NO æquales <lb/>e&longs;&longs;e quatuor PQ RS <lb/>TV XY. &longs;ed LM NO <lb/>&longs;imul dupli &longs;unt &longs;patii po<lb/>tentiæ in G motæ; & <lb/>quatuor PQ RS TV <lb/>XY &longs;imul quadrupli &longs;unt <lb/>&longs;patii ponderis moti. &longs;pa<lb/>tium igitur potentiæ ad <lb/>&longs;patium ponderis e&longs;t tan<lb/>quam &longs;ubduplum ad &longs;ub <lb/>quadruplum. </s> <s id="id.2.1.195.1.1.2.0">erit ergo <lb/>potentiæ &longs;patium pon­<lb/>deris &longs;patii duplum. <figure id="id.036.01.206.1.jpg" xlink:href="036/01/206/1.jpg"></figure></s> </p> <pb n="97" xlink:href="036/01/207.jpg"/> <p id="id.2.1.195.3.0.0.0" type="main"> <s id="id.2.1.195.3.1.1.0">Hinc autem con&longs;iderandum <lb/>e&longs;t quomodo fiat motus; quia, <lb/>cùm funis &longs;it religatur in F, vectis <lb/>NO in prima figura habebit ful­<lb/>cimentum O, pondus in medio, <lb/>& potentia in N. &longs;imiliter quo­<lb/>niam funis e&longs;t religatus in E, ve<lb/>ctis PQ habebit <expan abbr="fulcimentũ">fulcimentum</expan> P, & <lb/>pondus in medio, & potentia in <lb/>q. idcirco partes orbiculorum <lb/>in N, & Q &longs;ur&longs;um mouebuntur; <lb/>orbiculi ergo non in eandem, &longs;ed <lb/>in contrarias mouebuntur partes, <lb/>videlicet vnus <expan abbr="dextro&longs;um">dextrorsum</expan>, alter &longs;i­<lb/>ni&longs;tror&longs;um. </s> <s id="id.2.1.195.3.1.2.0">& quoniam potentiæ <lb/>in NQ eædem &longs;unt, quæ &longs;unt in <lb/>LM; potentiæ igitur in LM æ­<lb/>quales &longs;ur&longs;um mouebuntur. </s> <s id="id.2.1.195.3.1.3.0">ve<lb/>ctis igitur LM in neutram moue<lb/>bitur partem. </s> <s id="id.2.1.195.3.1.4.0">quare neq; orbicu<lb/>lus circumuertetur. </s> <s id="id.2.1.195.3.1.5.0">Itaq; LM <lb/>erit tanquam libra, cuius centrum <lb/>D, ponderaqué appen&longs;a in LM <lb/>æqualia quartæ parti ponderis C; <lb/>vnu&longs;qui&longs;q; enim funis LN MQ <lb/>quartam &longs;u&longs;tinet partem ponderis C. mouebitur ergo totus orbi <lb/>culus, cuius centrum D, &longs;ur&longs;um; &longs;ed non circumuertetur. <figure id="id.036.01.207.1.jpg" xlink:href="036/01/207/1.jpg"></figure></s> </p> <pb xlink:href="036/01/208.jpg"/> <p id="id.2.1.195.5.0.0.0" type="main"> <s id="id.2.1.195.5.1.1.0">Et &longs;i funis in F circa alios duos <lb/>voluatur orbiculos, quorum cen­<lb/>tra &longs;int HK, qui deinde religetur <lb/>in L; erit proportio ponderis ad <lb/>potentiam &longs;e&longs;quialtera. </s> </p> <p id="id.2.1.195.6.0.0.0" type="main"> <s id="id.2.1.195.6.1.1.0">Si enim quatuor e&longs;&longs;ent potentiæ <lb/><arrow.to.target n="note277"></arrow.to.target>in MNOI, e&longs;&longs;et vnaquæq; &longs;ub&longs;e&longs;­<lb/>cupla ponderis C, quare quatuor <lb/>&longs;imul potentiæ in MNOI qua­<lb/>tuor &longs;extæ erunt ponderis C. & <lb/>quoniam duæ &longs;imul potentiæ in <lb/>HD quatuor potentiis in MNOI <lb/>&longs;unt æquales; & potentia in G æ­<lb/>qualis e&longs;t potentiis in DH: erit <lb/>potentia in G quatuor &longs;imul po­<lb/>tentiis in MNOI æqualis; & ob <lb/>id quatuor &longs;extæ erit ponderis C. </s> <s id="id.2.1.195.6.1.1.0.a"><lb/>proportio igitur ponderis C ad po<lb/>tentiam in G &longs;e&longs;quialtera e&longs;t. </s> </p> <p id="id.2.1.196.1.0.0.0" type="margin"> <s id="id.2.1.196.1.1.1.0"><margin.target id="note277"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius<emph.end type="italics"/></s> </p> <p id="id.2.1.197.1.0.0.0" type="main"> <s id="id.2.1.197.1.1.1.0">Et &longs;i in G &longs;it potentia mouens, <lb/>&longs;imili modo o&longs;tendetur &longs;patium <lb/>potentiæ &longs;patii ponderis &longs;e&longs;quialte<lb/>rum e&longs;&longs;e. <figure id="id.036.01.208.1.jpg" xlink:href="036/01/208/1.jpg"></figure></s> </p> <p id="id.2.1.197.2.0.0.0" type="main"> <s id="id.2.1.197.2.1.1.0">Et &longs;i funis in L adhuc circa duos <lb/>alios orbiculos reuoluatur &longs;imi­<lb/>liter o&longs;tendetur proportionem <lb/>ponderis ad potentiam &longs;e&longs;qui­<lb/>tertiam e&longs;&longs;e. </s> <s id="id.2.1.197.2.1.2.0">quòd &longs;i in G &longs;it <lb/>potentia mouens, o&longs;tende­<lb/>tur &longs;patium potentiæ &longs;patii ponde<lb/>ris &longs;e&longs;quitertium e&longs;&longs;e, atq; ita dein­<lb/>ceps in infinitum procedendo, <lb/>quamcunq; proportionem ponderis ad potentiam &longs;uperparticula<lb/>rem inueniemus &longs;emperq; reperiemus ita e&longs;&longs;e pondus ad poten<lb/>tiam pondus &longs;u&longs;tinentem, vt &longs;patium potentiæ mouentis ad &longs;pa<lb/>tium ponderis à potentia moti. </s> </p> <pb n="98" xlink:href="036/01/209.jpg"/> <p id="id.2.1.197.4.0.0.0" type="main"> <s id="id.2.1.197.4.1.1.0">Motus vectium fit hoc modo, vectis YZ, cùm funis &longs;it religatus <lb/>in E, habet fulcimentum in Y, pondus in B medio appen&longs;um, & <lb/>potentia in Z. & vectis PQ habet fulcimentum in P potentia in <lb/>medio, & pondus in q. oportet enim orbiculos, quorum cen­<lb/>tra &longs;unt BD in eandem partem moueri, videlicet vt QZ &longs;ur­<lb/>&longs;um moueantur. </s> <s id="id.2.1.197.4.1.2.0">& quoniam funis religatus e&longs;t in L, erit T fulci <lb/>mentum vectis ST, qui pondus habet in medio, & potentia in <lb/>S. & quia S mouetur &longs;ur&longs;um, nece&longs;&longs;e e&longs;t etiam R &longs;ur&longs;um moue <lb/>ri; & ideo F erit fulcimentum vectis FR, & pondus erit in R, <lb/>& potentia in medio. </s> <s id="id.2.1.197.4.1.3.0">orbiculi igitur, quorum centra &longs;unt H k, <lb/>in contrariam mouentur partem eorum, quorum centra &longs;unt BD: <lb/>quare partes <expan abbr="orbiculorũ">orbiculorum</expan> PF in orbiculis deor&longs;um <expan abbr="tend&etilde;t">tendent</expan>; videlicet <lb/>ver&longs;us XV. </s> <s id="id.2.1.197.4.1.3.0.a">vectis igitur VX in neutram partem mouebitur, cùm <lb/>P, & F deor&longs;um moueantur; & VX erit tanquam vectis, in cuius <lb/>medio erit pondus appen&longs;um, & in VX duæ potentiæ æquales <lb/>&longs;extæ parti ponderis C. potentiæ enim in MO hoc e&longs;t funes PV <lb/>FX &longs;extam &longs;u&longs;tinent partem ponderis C. totus igitur orbiculus, <lb/>cuius centrum A &longs;ur&longs;um vnà cum trochlea mouebitur; non au­<lb/>tem circumuertetur. </s> </p> <p id="id.2.1.197.5.0.0.0" type="head"> <s id="id.2.1.197.5.1.1.0">PROPOSITIO XXV. </s> </p> <p id="id.2.1.197.6.0.0.0" type="main"> <s id="id.2.1.197.6.1.1.0">Si tribus duarum trochlearum orbiculis, <lb/>quarum altera binis in&longs;ignita rotulis à potentia <lb/>&longs;upernè detineatur; altera verò vnius tantùm <lb/>rotulæ infernè <expan abbr="cõ&longs;tituta">con&longs;tituta</expan>, ac ponderi alligata fue<lb/>rit, circumuoluatur funis; vtroq; eius extremo <lb/>alicuibi, non autem inferiori trochleæ religa­<lb/>to: dupla erit ponderis potentia. </s> </p> <pb xlink:href="036/01/210.jpg"/> <p id="id.2.1.197.8.0.0.0" type="main"> <s id="id.2.1.197.8.1.1.0">Sit pondus A trochleæ inferiori alligatum, <lb/>quæ orbiculum habeat, cuius centrum &longs;it B; tro<lb/>chlea verò &longs;uperior duos orbiculos habeat, <lb/>quorum centra &longs;int CD; &longs;itq; funis circa om<lb/>nes orbiculos reuolutus, qui in EF &longs;it religatus; <lb/>potentiaq; &longs;u&longs;tinens pondus &longs;it in G. </s> <s id="id.2.1.197.8.1.1.0.a">dico po<lb/>tentiam in G ponderis A duplam e&longs;&longs;e. </s> <s id="id.2.1.197.8.1.2.0">&longs;i enim <lb/><arrow.to.target n="note278"></arrow.to.target>in H k duæ e&longs;&longs;ent potentiæ pondus &longs;u&longs;tinen<lb/><arrow.to.target n="note279"></arrow.to.target>tes, e&longs;&longs;et vtraq; &longs;ubdupla ponderis A; &longs;ed po<lb/><arrow.to.target n="note280"></arrow.to.target>tentia in D dupla e&longs;t potentiæ in H, & poten<lb/>tia in C dupla potentiæ in K; quare duæ &longs;imul <lb/>potentiæ in CD vtriu&longs;q; &longs;imul potentiæ in H k <lb/>duplæ erunt. </s> <s id="id.2.1.197.8.1.3.0">&longs;ed potentiæ in H k ponderi A &longs;unt <lb/>æquales, & potentiæ in CD ip&longs;i potentiæ in G <lb/>&longs;unt etiam æquales; potentia igitur in G ponde­<lb/>ris A dupla erit. </s> <s id="id.2.1.197.8.1.4.0">quod oportebat demon&longs;trare. </s> </p> <p id="id.2.1.198.1.0.0.0" type="margin"> <s id="id.2.1.198.1.1.1.0"><margin.target id="note278"></margin.target>2. <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.198.1.1.2.0"><margin.target id="note279"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.198.1.1.3.0"><margin.target id="note280"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.199.1.0.0.0" type="main"> <s id="id.2.1.199.1.1.1.0">Si autem in G &longs;it potentia mouens pon­<lb/>dus, &longs;imiliter vt in præcedenti o&longs;tendetur &longs;pa<lb/>tium ponderis &longs;patii potentiæ duplum e&longs;&longs;e. <figure id="id.036.01.210.1.jpg" xlink:href="036/01/210/1.jpg"></figure></s> </p> <p id="id.2.1.199.2.0.0.0" type="main"> <s id="id.2.1.199.2.1.1.0">Hinc quoq; con&longs;iderandum e&longs;t vectem PQ <lb/>non moueri, quia vectis LM habet fulcimen <lb/>tum in L, potentia in medio, & pondus in M. </s> <s id="id.2.1.199.2.1.1.0.a"><lb/>vectis autem NO habet fulcimentum in O, <lb/>potentia in medio, & pondus in N. quare M, & N &longs;ur&longs;um mo<lb/>uebuntur. </s> <s id="id.2.1.199.2.1.2.0">in contrarias igitur partes orbiculi, quorum centra <lb/>&longs;unt CD mouentur. </s> <s id="id.2.1.199.2.1.3.0">idcirco vectis PQ in neutram partem mo<lb/>uebitur; eritq;, ac &longs;i in medio e&longs;&longs;et appen&longs;um pondus, & in PQ <lb/>duæ potentiæ æquales dimidio ponderis A. vtraq; enim potentia <lb/>in HK &longs;ubdupla e&longs;t ponderis A. totus igitur orbiculus, cuius <lb/>centrum B &longs;ur&longs;um mouebitur, &longs;ed non circumuertetur. </s> </p> <pb n="99" xlink:href="036/01/211.jpg"/> <p id="id.2.1.199.4.0.0.0" type="main"> <s id="id.2.1.199.4.1.1.0">Et &longs;i funis in F duobus aliis adhuc circumuol­<lb/>uatur orbiculis, quorum centra &longs;int HK, qui de­<lb/>inde religetur in L; erit proportio potentiæ in G <lb/>ad pondus A &longs;e&longs;quialtera. </s> </p> <p id="id.2.1.199.5.0.0.0" type="main"> <s id="id.2.1.199.5.1.1.0">Si enim in MNOP quatuor e&longs;&longs;ent poten<lb/>tiæ pondus &longs;u&longs;tinentes, vnaquæq; &longs;ubquadru<arrow.to.target n="note281"></arrow.to.target><lb/>pla e&longs;&longs;et ponderis A: &longs;ed cùm potentia in k <arrow.to.target n="note282"></arrow.to.target><lb/>&longs;it dupla potentiæ in N; erit potentia in k <lb/>ponderis A &longs;ubdupla. </s> <s id="id.2.1.199.5.1.2.0">& quoniam potentia <lb/>in D duabus in MO potentiis e&longs;t æqualis; erit <lb/>quoq; potentia in D ponderis A &longs;ubdupla. </s> <s id="id.2.1.199.5.1.3.0"><lb/>cùm autem adhuc potentia in C potentiæ in P <lb/>&longs;it dupla, erit &longs;imiliter <expan abbr="pot&etilde;tia">potentia</expan> in C ponderis A <lb/>&longs;ubdupla. </s> <s id="id.2.1.199.5.1.4.0">tres igitur potentiæ in CD k tribus <lb/>medietatibus ponderis A &longs;unt æquales. </s> <s id="id.2.1.199.5.1.5.0">quo­<lb/>niam autem potentia in G potentiis in CDK <lb/>e&longs;t æqualis, erit potentia in G tribus medie­<lb/>tatibus ponderis A æqualis. </s> <s id="id.2.1.199.5.1.6.0">Proportio igi­<lb/>tur potentiæ ad pondus &longs;e&longs;quialtera e&longs;t. </s> </p> <p id="id.2.1.200.1.0.0.0" type="margin"> <s id="id.2.1.200.1.1.1.0"><margin.target id="note281"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>huius<emph.end type="italics"/></s> <s id="id.2.1.200.1.1.2.0"><margin.target id="note282"></margin.target>15 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.201.1.0.0.0" type="main"> <s id="id.2.1.201.1.1.1.0">Si verò in G &longs;it potentia mouens, erit &longs;pa<lb/>tium ponderis &longs;patii potentiæ &longs;e&longs;quialterum. <figure id="id.036.01.211.1.jpg" xlink:href="036/01/211/1.jpg"></figure></s> </p> <p id="id.2.1.201.2.0.0.0" type="main"> <s id="id.2.1.201.2.1.1.0">Et &longs;i funis in L adhuc circa duos alios or<lb/>biculos reuoluatur, &longs;imiliter o&longs;tendetur pro­<lb/>portionem potentiæ ad pondus &longs;e&longs;quitertiam <lb/>e&longs;&longs;e. </s> <s id="id.2.1.201.2.1.2.0">& &longs;ic in infinitum omnes proportiones <lb/>potentiæ ad pondus &longs;uperparticulares inue­<lb/>niemus. </s> <s id="id.2.1.201.2.1.3.0">o&longs;tendemu&longs;q; potentiam pondus <lb/>&longs;u&longs;tinentem ad pondus ita e&longs;&longs;e, vt &longs;patium <lb/>ponderis moti ad &longs;patìum potentiæ pondus <lb/>mouentis. </s> </p> <pb xlink:href="036/01/212.jpg"/> <p id="id.2.1.201.4.0.0.0" type="main"> <s id="id.2.1.201.4.1.1.0">Motus vectium fiet hoc <lb/>modo, videlicet Q erit ful<lb/>cimentum vectis QR, po­<lb/>tentia in medio, pondus <lb/>in R; & vectis Z 9 fulci <lb/>mentum erit Z, pondus in <lb/>medio, potentiaq; in 9. &longs;i <lb/>militer X erit fulcimentum <lb/>vectis VX, potentia in me <lb/>dio, pondus in V. </s> <s id="id.2.1.201.4.1.1.0.a">& quo<lb/>niam V &longs;ur&longs;um mouetur, Y <lb/>quoq; &longs;ur&longs;um mouebitur; <lb/>& vectis YF fulcimentum <lb/>erit F: quare F, & Z in orbi <lb/>culis deor&longs;um mouebun­<lb/>tur. </s> <s id="id.2.1.201.4.1.2.0">& ob id vectis ST in <lb/>neutram mouebitur par­<lb/>tem; & ST erit tamquam <lb/>libra, cuius centrum D, & <lb/>pondera in ST æqualia <lb/>quartæ parti ponderis A. <lb/>vnu&longs;qui&longs;q; enim funis SZ <lb/>TF quartam &longs;u&longs;tinet par­<lb/>tem ponderis A. orbicu­<lb/>lus ergo, cuius centrum D, <lb/>&longs;ur&longs;um mouebitur; non au<lb/>tem circumuertetur. <figure id="id.036.01.212.1.jpg" xlink:href="036/01/212/1.jpg"></figure></s> </p> <pb n="100" xlink:href="036/01/213.jpg"/> <p id="id.2.1.201.6.0.0.0" type="main"> <s id="id.2.1.201.6.1.1.0">Hactenus proportiones ponderis ad potentiam multiplices, <lb/>& &longs;ubmultiplices; deinde &longs;uperparticulares, &longs;ub&longs;uperparticu­<lb/>lare&longs;qué declaratæ fuerunt: nunc autem reliquum e&longs;t, vt propor­<lb/>tiones inter pondus, & potentiam &longs;uperpartientes, & multi­<lb/>plices &longs;uperparticulares, multiplicesqué &longs;uperpartientes mani­<lb/>fe&longs;tentur. </s> </p> <p id="id.2.1.201.7.0.0.0" type="head"> <s id="id.2.1.201.7.1.1.0">PROPOSITIO XXVI. </s> </p> <p type="head"> <s id="id.2.1.201.7.3.1.0">PROBLEMA. </s> </p> <p id="id.2.1.201.8.0.0.0" type="main"> <s id="id.2.1.201.8.1.1.0">Si proportionem &longs;uperpartientem inuenire <lb/>volumus, quemadmodum &longs;i proportio, quam <lb/>habet pondus ad potentiam pondus &longs;u&longs;tinen­<lb/>tem fuerit &longs;uperbipartiens, &longs;icut quinque ad <lb/>tria. </s> </p> <pb xlink:href="036/01/214.jpg"/> <p id="id.2.1.201.10.0.0.0" type="main"> <s id="id.2.1.201.10.1.1.0"><arrow.to.target n="note283"></arrow.to.target>Exponatur potentia in A pondus B &longs;u&longs;ti<lb/>nens, proportionemq; habeat pondus B ad <lb/>potentiam in A, vt quinq; ad vnum; hoc e&longs;t, <lb/>&longs;it potentia in A &longs;ubquintupla ponderis B: de­<lb/>inde eodem fune circa alios orbiculos reuo­<lb/><arrow.to.target n="note284"></arrow.to.target>luto inueniatur potentia in C, quæ tripla &longs;it <lb/>potentiæ in A. </s> <s id="id.2.1.201.10.1.1.0.a">& quoniam pondus B ad po<lb/>tentiam in A e&longs;t, vt quinq; ad vnum; & <lb/>potentia in A ad potentiam in C e&longs;t, vt vnum <lb/>ad tria; erit pondus B ad potentiam in C, vt <lb/>quinq; ad tria; hoc e&longs;t &longs;uperbipartiens. </s> </p> <p id="id.2.1.202.1.0.0.0" type="margin"> <s id="id.2.1.202.1.1.1.0"><margin.target id="note283"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.202.1.1.2.0"><margin.target id="note284"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 17 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.203.1.0.0.0" type="main"> <s id="id.2.1.203.1.1.1.0">Et hoc modo omnes proportiones ponde<lb/>ris ad potentiam &longs;uperpartientes inuenientur; <lb/>vt &longs;i &longs;upertripartientem quis inuenire volue­<lb/>rit; eodem incedat ordine; fiat &longs;cilicet poten<lb/>tia in A &longs;u&longs;tinens pondus B &longs;ub&longs;eptupla ip­<lb/>&longs;ius ponderis B; deinde fiat potentia in C ip­<lb/>&longs;ius A quadrupla; erit pondus B ad poten­<lb/>tiam in C, vt &longs;eptem ad quatuor: vídelicet <lb/>&longs;upertripartiens. </s> </p> <p id="id.2.1.203.2.0.0.0" type="main"> <s id="id.2.1.203.2.1.1.0">Si verò in C &longs;it potentia mo­<lb/>uens pondus erit &longs;patium <expan abbr="pot&etilde;tiæ">potentiæ</expan> <lb/>&longs;patii ponderis &longs;uperbipartiens. <figure id="id.036.01.214.1.jpg" xlink:href="036/01/214/1.jpg"></figure></s> </p> <p id="id.2.1.203.3.0.0.0" type="main"> <s id="id.2.1.203.3.1.1.0"><arrow.to.target n="note285"></arrow.to.target>Spatium enim potentiæ in C tertia pars <lb/>e&longs;t &longs;patii potentiæ in A, ita videlicet &longs;e habent, <lb/>vt quinq; ad quindecim; & &longs;patium potentiæ <lb/><arrow.to.target n="note286"></arrow.to.target>in A quintuplum e&longs;t &longs;patii ponderis B, hoc <lb/>e&longs;t, vt quindecim ad tria; erit igitur &longs;patium <lb/>potentiæ in C ad &longs;patium ponderis B, vt <lb/>quinq; ad tria; videlicet &longs;uperbipartiens. </s> <s id="id.2.1.203.3.1.2.0">& &longs;emper o&longs;tendemus, ita <lb/>e&longs;&longs;e &longs;patium potentiæ mouentis ad &longs;patium ponderis; vt pondus <lb/>ad potentiam pondus &longs;u&longs;tinentem. </s> </p> <p id="id.2.1.204.1.0.0.0" type="margin"> <s id="id.2.1.204.1.1.1.0"><margin.target id="note285"></margin.target>17 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.204.1.1.2.0"><margin.target id="note286"></margin.target>14 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.205.1.0.0.0" type="main"> <s id="id.2.1.205.1.1.1.0">Similiq; pror&longs;us ratione proportionem potentiæ ad pondus &longs;u­<pb n="101" xlink:href="036/01/215.jpg"/>perpartientem inueniemus. </s> <s id="id.2.1.205.1.1.2.0">&longs;i enim C e&longs;&longs;et inferius, & in ip&longs;o <lb/>appen&longs;um e&longs;&longs;et pondus; B verò &longs;uperius, in quo e&longs;&longs;et potentia pon<lb/>dus in C &longs;u&longs;tinens, e&longs;&longs;et potentia in B &longs;uperbipartiens ponderis <lb/>in C appen&longs;i: cùm B ad A &longs;it, <expan abbr="vtquinq;">vt quinq;</expan> ad vnum; A verò ad <arrow.to.target n="note287"></arrow.to.target><lb/>C, vt vnum ad tria. <arrow.to.target n="note288"></arrow.to.target></s> </p> <p id="id.2.1.205.2.0.0.0" type="main"> <s id="id.2.1.205.2.1.1.0">Si autem multiplicem &longs;uperparticularem in­<lb/>uenire voluerimus; vt proportio, quam habet <lb/>pondus ad potentiam pondus &longs;u&longs;tinentem, &longs;it <lb/>duplex &longs;e&longs;quialtera, vt quinq; ad duo. </s> </p> <p id="id.2.1.206.1.0.0.0" type="margin"> <s id="id.2.1.206.1.1.1.0"><margin.target id="note287"></margin.target>18 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.206.1.1.2.0"><margin.target id="note288"></margin.target>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.207.1.0.0.0" type="main"> <s id="id.2.1.207.1.1.1.0">Eodem modo, quo &longs;uperpartientes inuenimus, has quo­<lb/>que omnes multiplices &longs;uperparticulares reperiemus. </s> <s id="id.2.1.207.1.1.2.0">vt fiat <arrow.to.target n="note289"></arrow.to.target><lb/>pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve<arrow.to.target n="note290"></arrow.to.target><lb/>ro in C ad potentiam in A, vt duo ad vnum; quod fiet, &longs;i fu­<lb/>nis &longs;it religatus in D, non autem trochleæ &longs;uperiori, vel in F: erit <lb/>pondus B ad potentiam in C, vt quinq; ad duo; hoc e&longs;t duplum <lb/>&longs;e&longs;quialterum. </s> </p> <p id="id.2.1.208.1.0.0.0" type="margin"> <s id="id.2.1.208.1.1.1.0"><margin.target id="note289"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.208.1.1.2.0"><margin.target id="note290"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 15, 16, <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.209.1.0.0.0" type="main"> <s id="id.2.1.209.1.1.1.0">Et è conuer&longs;o proportionem potentiæ ad pondus multiplicem <lb/>&longs;uperparticularem inueniemus; & vt in reliquis o&longs;tendetur, ita e&longs; <lb/>&longs;e &longs;patium potentiæ mouentis ad &longs;patium ponderis, vt pondus <lb/>ad potentiam pondus &longs;u&longs;tinentem. </s> </p> <p id="id.2.1.209.2.0.0.0" type="main"> <s id="id.2.1.209.2.1.1.0">Omnem quoq; multiplicem &longs;uperpartientem <lb/>eodem modo inueniemus; vt &longs;i proportio, quam <lb/>habet pondus ad potentiam, &longs;it duplex &longs;uperbi <lb/>partiens, vt octo ad tria. </s> </p> <p id="id.2.1.209.3.0.0.0" type="main"> <s id="id.2.1.209.3.1.1.0">Fiat potentia in A pondus B &longs;u&longs;tinens &longs;uboctupla ponderis B; <arrow.to.target n="note291"></arrow.to.target><lb/>& potentia in C potentiæ in A &longs;it tripla; erit pondus B ad po<lb/>tentiam in C, vt octo ad tria. </s> <s id="id.2.1.209.3.1.2.0">& è conuer&longs;o omnem potentiæ ad <pb xlink:href="036/01/216.jpg"/>pondus proportionem <expan abbr="multipticem">multiplicem</expan> &longs;uperpartientem in ueniemus. </s> <s id="id.2.1.209.3.1.3.0"><lb/>& vt in cæteris reperiemus ita e&longs;&longs;e pondus ad potentiam pondus <lb/>&longs;u&longs;tinentem, vt &longs;patium potentiæ mouentis ad &longs;patium pon­<lb/>deris. </s> </p> <p id="id.2.1.210.1.0.0.0" type="margin"> <s id="id.2.1.210.1.1.1.0"><margin.target id="note291"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius Ex<emph.end type="italics"/> 17 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.211.1.0.0.0" type="main"> <s id="id.2.1.211.1.1.1.0">Notandum autem e&longs;t, quòd cùm in præcedentibus <expan abbr="demo&longs;tratio">demonstratio</expan><lb/>nibus &longs;æpius dictum fuerit, potentiam pondus &longs;u&longs;tinentem ip&longs;ius <lb/>ponderis duplam e&longs;&longs;e, vel triplam, & huiu&longs;modi; vt in decima­<lb/>quinta huius o&longs;ten&longs;um e&longs;t; quia tamen potentia non &longs;olum pon<lb/>dus, verùm etiam trochleam &longs;u&longs;tinet; idcirco maioris longè vir­<lb/>tutis, maiori&longs;q; ip&longs;i ponderi proportionis con&longs;tituenda videtur <lb/>ip&longs;a potentia. </s> <s id="id.2.1.211.1.1.2.0">quod quidem verum e&longs;t, &longs;i etiam trochleæ graui<lb/>tatem con&longs;iderare voluerimus. </s> <s id="id.2.1.211.1.1.3.0">&longs;ed quoniam inter potentiam, & <lb/>pondus proportionem quærimus: ideo hanc trochleæ grauitatem <lb/>ommi&longs;imus, quam &longs;iquis etiam con&longs;iderare voluerit, vim ip&longs;i po­<lb/>tentiæ æqualem trochleæ addere poterit. </s> <s id="id.2.1.211.1.1.4.0">Quod ip&longs;um etiam in <lb/>fune ob&longs;eruari poterit. </s> <s id="id.2.1.211.1.1.5.0">& &longs;icut hoc in decimaquinta con&longs;ideraui<lb/>mus, idem quoq; in reliquis aliis con&longs;iderare poterimus. </s> </p> <pb n="97" xlink:href="036/01/217.jpg"/> <p id="id.2.1.211.3.0.0.0" type="main"> <s id="id.2.1.211.3.1.1.0">Noui&longs;&longs;e etiam oportet, quòd &longs;icuti proportio <lb/>nes omnes inter potentiam, & pondus vnico <lb/>fune inuentæ fuerunt; ita etiam pluribus funi­<lb/>bus, trochlei&longs;qué eædem inueniri poterunt. </s> <s id="id.2.1.211.3.1.2.0">vt <lb/>&longs;i multiplicem &longs;uperparticularem proportionem <lb/>pluribus funibus inuenire voluerimus, veluti &longs;i <lb/>proportio, quam habet pondus ad potentiam <lb/>pondus &longs;u&longs;tinentem, fuerit duplex &longs;e&longs;quialtera, vt <lb/>quinq; ad duo; oportet hanc proportionem ex <lb/>pluribus componere. </s> <s id="id.2.1.211.3.1.3.0">vt (exempli gratia) ex pro­<lb/>portione &longs;e&longs;quiquarta, vt quinqué ad quatuor, <lb/>& ex dupla, vt quatuor ad duo. </s> <s id="id.2.1.211.3.1.4.0">exponatur igitur po<arrow.to.target n="note292"></arrow.to.target><lb/>tentia in A pondus B &longs;u&longs;tinens, ad quam pondus <lb/><expan abbr="proportion&etilde;">proportionem</expan> habeat &longs;e&longs;quiquartam, vt quinq; ad <lb/>quatuor: deinde alio fune inueniatur <expan abbr="pot&etilde;tia">potentia</expan> in C,<arrow.to.target n="note293"></arrow.to.target><lb/>cuius dupla &longs;it potentia in A. </s> <s id="id.2.1.211.3.1.4.0.a">& <expan abbr="quoniã">quoniam</expan> B ad A e&longs;t, <lb/>vt quinq; ad quatuor; & A ad C, vt quatuor ad <lb/>duo; erit pondus B ad potentiam in C, vt quin<lb/>que ad duo; hoc e&longs;t proportionem habebit du­<lb/>plicem &longs;e&longs;quialteram. <figure id="id.036.01.217.1.jpg" xlink:href="036/01/217/1.jpg"></figure></s> </p> <p id="id.2.1.211.4.0.0.0" type="main"> <s id="id.2.1.211.4.1.1.0">Et notandum e&longs;t hanc quoq; <expan abbr="proportion&etilde;">proportionem</expan> inue<lb/>niri po&longs;&longs;e, &longs;i proportionem quinq; ad duo ex pluri<lb/>bus componamus, vt quinq; ad quindecim & quin<lb/>decim ad viginti & viginti ad duo. </s> <s id="id.2.1.211.4.1.2.0">Et hoc modo <lb/>non &longs;olum omnem aliam proportionem inuenie<lb/>mus, &longs;ed quamcunq, multis, infinitisqué mo­<lb/>dis comperiemus. </s> <s id="id.2.1.211.4.1.3.0">omnis enim proportio ex infi­<lb/>nitis proportionibus componi pote&longs;t. </s> <s id="id.2.1.211.4.1.4.0">vt patet <lb/>in commentario Eutocii in quartam propo&longs;itio­<lb/>nem &longs;ecundi libri Archimedis de &longs;phera, & cy­<lb/>lindro. </s> </p> <p id="id.2.1.212.1.0.0.0" type="margin"> <s id="id.2.1.212.1.1.1.0"><margin.target id="note292"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 21 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.212.1.1.2.0"><margin.target id="note293"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 2 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <p id="id.2.1.213.1.0.0.0" type="main"> <s id="id.2.1.213.1.1.1.0">Po&longs;&longs;umus quoq; pluribus funibus, trochleis <lb/>verò inferioribus tantùm, vel &longs;uperioribus vti. </s> </p> <pb xlink:href="036/01/218.jpg"/> <p id="id.2.1.213.3.0.0.0" type="main"> <s id="id.2.1.213.3.1.1.0">Sit pondus A, cui alligata &longs;it trochlea <lb/>orbiculum habens, cuius centrum B; <lb/>religetur funis in C, qui circa orbiculum <lb/>reuoluatur, funi&longs;q; perueniat in D: erit <lb/><arrow.to.target n="note294"></arrow.to.target>potentia in D &longs;u&longs;tinens pondus A &longs;ub­<lb/>dupla ponderis A. </s> <s id="id.2.1.213.3.1.1.0.a">deinde funis in D <lb/>alteri trochleæ religetur, & circa huius <lb/>trochleæ orbiculum alius reuoluatur fu<lb/>nis, qui religetur in E, & perueniat in <lb/><arrow.to.target n="note295"></arrow.to.target>F; erit potentia in F &longs;ubdupla eius, <lb/>quod &longs;u&longs;tinet <expan abbr="pot&etilde;tia">potentia</expan> in D: e&longs;t enim ac &longs;i <lb/>D dimidium ponderis A &longs;u&longs;tineret &longs;i <lb/>ne trochlea; quare potentia in F &longs;ubqua­<lb/>drupla erit ponderis A. & &longs;i adhuc fu <lb/>nis in F alteri trochleæ religetur, & <lb/>per eius orbiculum circumuoluatur a­<lb/>lius funis, qui religetur in G, & per <lb/>ueniat in H; erit potentia in H &longs;ub <lb/>dupla potentiæ in F. </s> <s id="id.2.1.213.3.1.1.0.b">ergo potentia in <lb/>H &longs;uboctupla erit ponderis A. & &longs;ic <lb/>in infinitum &longs;emper &longs;ubduplam poten<lb/>tiam <expan abbr="præced&etilde;tis">præcedentis</expan> potentiæ inueniemus. <figure id="id.036.01.218.1.jpg" xlink:href="036/01/218/1.jpg"></figure></s> </p> <p id="id.2.1.213.4.0.0.0" type="main"> <s id="id.2.1.213.4.1.1.0">Et &longs;i in H &longs;it potentia mouens, erit <lb/>&longs;patium potentiæ &longs;patii ponderis octu<lb/><arrow.to.target n="note296"></arrow.to.target>plum. </s> <s id="id.2.1.213.4.1.2.0">&longs;patium enim D duplum e&longs;t &longs;pa<lb/>tii ponderis A, & &longs;patium F &longs;patii D <lb/>duplum; erit &longs;patium F &longs;patii ponde<lb/>ris A quadruplum. </s> <s id="id.2.1.213.4.1.3.0">&longs;imiliter quoniam <lb/>&longs;patium potentiæ in H <expan abbr="duplũ">duplum</expan> e&longs;t &longs;patii <lb/>F, erit &longs;patium potentiæ in H &longs;patii <lb/>ponderis A octuplum. </s> </p> <p id="id.2.1.214.1.0.0.0" type="margin"> <s id="id.2.1.214.1.1.1.0"><margin.target id="note294"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.214.1.1.2.0"><margin.target id="note295"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.214.1.1.3.0"><margin.target id="note296"></margin.target>11 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb n="103" xlink:href="036/01/219.jpg"/> <p id="id.2.1.215.1.0.0.0" type="main"> <s id="id.2.1.215.1.2.1.0">Sit deinde pondus A funi alliga­<lb/>tum, qui orbiculo trochleæ &longs;uperio<lb/>ris &longs;it circumuolutus, & religatus in <arrow.to.target n="note297"></arrow.to.target><lb/>B; &longs;itq; potentia in C &longs;u&longs;tinens pon<lb/>dus A: erit potentia in C ponderis A <lb/>dupla, deinde C alteri funi religetur, <lb/>qui per alterius trochleæ orbicu<lb/>lum circumuoluatur, & religetur <lb/>in D; erit potentia in E dupla po<arrow.to.target n="note298"></arrow.to.target><lb/>tentiæ in C. </s> <s id="id.2.1.215.1.2.1.0.a">Quare potentia in E <lb/>quadrupla erit ponderis A. </s> <s id="id.2.1.215.1.2.1.0.b">& &longs;i ad <lb/>huc E alteri funi religetur, qui etiam <lb/>circa orbiculum alterius trochleæ re<lb/>uoluatur, & religetur in F; erit poten<lb/>tia in G dupla potentiæ in E. </s> <s id="id.2.1.215.1.2.1.0.c">ergo <lb/>potentia in G octupla erit ponderis <lb/>A. & &longs;ic in infinitum &longs;emper præ <lb/>cedentis potentiæ potentiam du­<lb/>plam inueniemus. <figure id="id.036.01.219.1.jpg" xlink:href="036/01/219/1.jpg"></figure></s> </p> <p id="id.2.1.215.2.0.0.0" type="main"> <s id="id.2.1.215.2.1.1.0">Si autem in G &longs;it potentia mo­<lb/>uens, <arrow.to.target n="note299"></arrow.to.target>erit &longs;patium ponderis octu­<lb/>plum &longs;patii potentiæ in G. &longs;patium <lb/>enim ponderis A duplum e&longs;t &longs;patii <lb/>potentiæ in C, & C duplum e&longs;t &longs;patii <lb/>ip&longs;ius E; quare &longs;patium ponderis <lb/>A &longs;patii potentiæ in E quadruplum <lb/>erit. </s> <s id="id.2.1.215.2.1.2.0">&longs;imiliter quoniam &longs;patium E <lb/>duplum e&longs;t &longs;patii potentiæ in G; erit ergo &longs;patium ponderis A <lb/>octuplum &longs;patii potentiæ in G. </s> </p> <p id="id.2.1.216.1.0.0.0" type="margin"> <s id="id.2.1.216.1.1.1.0"><margin.target id="note297"></margin.target>15 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.216.1.1.2.0"><margin.target id="note298"></margin.target><emph type="italics"/>Ex e adem.<emph.end type="italics"/></s> <s id="id.2.1.216.1.1.3.0"><margin.target id="note299"></margin.target>16 <emph type="italics"/>Huius.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/220.jpg"/> <p id="id.2.1.217.1.0.0.0" type="head"> <s id="id.2.1.217.1.2.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.217.2.0.0.0" type="main"> <s id="id.2.1.217.2.1.1.0">Ex his manife&longs;tum e&longs;t maiorem &longs;emper ha­<lb/>bere proportionem &longs;patium potentiæ mouen­<lb/>tis ad &longs;patium ponderis moti, quàm pondus <lb/>ad eandem potentiam. </s> </p> <p id="id.2.1.217.3.0.0.0" type="main"> <s id="id.2.1.217.3.1.1.0">Hoc autem ex iis, quæ in corollario quartæ huius de vecte dicta <lb/>&longs;unt, patet. </s> </p> <p id="id.2.1.217.4.0.0.0" type="head"> <s id="id.2.1.217.4.1.1.0">PROPOSITIO XXVII. </s> </p> <p type="head"> <s id="id.2.1.217.4.3.1.0">PROBLEMA. </s> </p> <p id="id.2.1.217.5.0.0.0" type="main"> <s id="id.2.1.217.5.1.1.0">Datum pondus à data potentia trochleis <lb/>moueri. </s> </p> <p id="id.2.1.217.6.0.0.0" type="main"> <s id="id.2.1.217.6.1.1.0">Data potentia, vel e&longs;t maior, vel æqualis, vel minor dato <lb/>pondere. </s> </p> <pb n="104" xlink:href="036/01/221.jpg"/> <p id="id.2.1.217.8.0.0.0" type="main"> <s id="id.2.1.217.8.1.1.0">Et &longs;i e&longs;t maior, tunc poten­<lb/>tia, vel ab&longs;q; alio in&longs;trumento, <lb/>vel fune circa orbiculum trochleæ <lb/>&longs;ur&longs;um appen&longs;æ reuoluto datum <lb/>pondus mouebit. </s> <s id="id.2.1.217.8.1.2.0">Minor enim po<arrow.to.target n="note300"></arrow.to.target><lb/>tentia; quàm data, ponderi æque­<lb/>ponderat, data ergo mouebit. </s> <s id="id.2.1.217.8.1.3.0"><lb/>Quod idem fieri pote&longs;t iuxta om­<lb/>nes propo&longs;itiones, quibus poten­<lb/>tia pondus &longs;u&longs;tinens, vel æqualis, <lb/>vel minor pondere o&longs;ten&longs;a e&longs;t. <figure id="id.036.01.221.1.jpg" xlink:href="036/01/221/1.jpg"></figure></s> </p> <p id="id.2.1.217.9.0.0.0" type="main"> <s id="id.2.1.217.9.1.1.0">Si autem æqualis, <lb/>pondus mouebit fune <lb/>per orbiculum trochleæ <lb/>ponderi alligatæ circum <lb/>uoluto. </s> <s id="id.2.1.217.9.1.2.0">potentia enim <arrow.to.target n="note301"></arrow.to.target><lb/>&longs;u&longs;tinens pondus &longs;ubdu<lb/>pla e&longs;t ponderis, poten<lb/>tia igitur ponderi æqua <lb/>lis datum pondus mo­<lb/>uebit. </s> <s id="id.2.1.217.9.1.3.0">Quod etiam <expan abbr="&longs;e­cundùm">&longs;e­<lb/>cundum</expan> propo&longs;itiones, <lb/>quibus potentiam pon<lb/>dere minorem e&longs;&longs;e o­<lb/>&longs;ten&longs;um e&longs;t, fieri po­<lb/>te&longs;t. <figure id="id.036.01.221.2.jpg" xlink:href="036/01/221/2.jpg"></figure></s> </p> <pb xlink:href="036/01/222.jpg"/> <p id="id.2.1.217.11.0.0.0" type="main"> <s id="id.2.1.217.11.1.1.0">Si verò minor, &longs;it datum pondus <lb/>vt &longs;exaginta, potentia verò mouens <lb/><arrow.to.target n="note302"></arrow.to.target>data &longs;it tredecim. </s> <s id="id.2.1.217.11.1.2.0">inueniatur poten­<lb/>tia in A &longs;u&longs;tinens pondus B, quæ pon<lb/>deris B &longs;it &longs;ubquintupla. </s> <s id="id.2.1.217.11.1.3.0">& quoniam <lb/>potentia in A pondus &longs;u&longs;tinens e&longs;t <lb/>vt duodecim; maior igitur poten­<lb/>tia, quàm duodecim in A pondus <lb/>B mouebit. </s> <s id="id.2.1.217.11.1.4.0">Quare potentia vt tre­<lb/>decim in A pondus B mouebit. </s> <s id="id.2.1.217.11.1.5.0">quod. <lb/>facere oportebat. </s> </p> <p id="id.2.1.218.1.0.0.0" type="margin"> <s id="id.2.1.218.1.1.1.0"><margin.target id="note300"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 1 <emph type="italics"/>huius<emph.end type="italics"/></s> <s id="id.2.1.218.1.1.2.0"><margin.target id="note301"></margin.target>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.218.1.1.3.0"><margin.target id="note302"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius<emph.end type="italics"/></s> </p> <p id="id.2.1.219.1.0.0.0" type="main"> <s id="id.2.1.219.1.1.1.0"><expan abbr="Animaduertendũ">Animaduertendum</expan> quoq; e&longs;t in mo <lb/>uendis ponderibus, potentiam ali­<lb/>quando for&longs;itan melius mouere mo <lb/>uendo &longs;e deor&longs;um, quàm mouendo <lb/>&longs;e &longs;ur&longs;um. </s> <s id="id.2.1.219.1.1.2.0">vt circumuoluatur adhuc <lb/>funis per alium trochleæ &longs;uperioris <lb/>orbiculum, cuius centrum C, funi&longs;q; <lb/><arrow.to.target n="note303"></arrow.to.target>perueniat in D; erit <expan abbr="pot&etilde;tia">potentia</expan> in D <expan abbr="&longs;u&longs;ti­n&etilde;s">&longs;u&longs;ti­nens</expan> <expan abbr="põdus">pondus</expan> B &longs;imiliter duodecim, <expan abbr="qu&etilde;">quem</expan><lb/>admodum erat in A. </s> <s id="id.2.1.219.1.1.2.0.a">Ideo poten­<lb/>tia vt tredecim in D pondus B mo­<lb/>uebit. </s> <s id="id.2.1.219.1.1.3.0">& quia mouet &longs;e deor&longs;um, <lb/>forta&longs;&longs;e trahet facilius, quàm in A; <lb/>atq; tempus e&longs;t idem, &longs;icut etiam <lb/>erat in A. <figure id="id.036.01.222.1.jpg" xlink:href="036/01/222/1.jpg"></figure></s> </p> <pb n="105" xlink:href="036/01/223.jpg"/> <p id="id.2.1.219.3.0.0.0" type="head"> <s id="id.2.1.219.3.1.1.0">PROPOSITIO XXVIII. </s> </p> <p id="id.2.1.220.1.0.0.0" type="margin"> <s id="id.2.1.220.1.1.1.0"><margin.target id="note303"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 5 <emph type="italics"/>Huius<emph.end type="italics"/></s> </p> <p id="id.2.1.221.1.0.0.0" type="head"> <s id="id.2.1.221.1.2.1.0">PROBLEMA. </s> </p> <p id="id.2.1.221.2.0.0.0" type="main"> <s id="id.2.1.221.2.1.1.0">Propo&longs;itum &longs;it nobis efficere, potentiam pon<lb/>dus mouentem, & pondus per data &longs;patia &longs;ibi in <lb/>uicem longitudine commen&longs;urabilia moueri. </s> </p> <p id="id.2.1.221.3.0.0.0" type="main"> <s id="id.2.1.221.3.1.1.0">Sit datum &longs;patium potentiæ, vt tria, <arrow.to.target n="note304"></arrow.to.target><lb/>ponderis verò, vt quatuor. </s> <s id="id.2.1.221.3.1.2.0">inueniatur po<lb/>tentia in A pondus B &longs;u&longs;tinens, quæ pon<lb/>deris &longs;it &longs;e&longs;quitertia, vt quatuor ad trìa. </s> <s id="id.2.1.221.3.1.3.0">&longs;i <lb/>igitur in A &longs;it potentia mouens pondus; <arrow.to.target n="note305"></arrow.to.target><lb/>erit &longs;patium ponderis &longs;patii potentiæ &longs;e&longs;­<lb/>quitertium, vt quatuor ad tria. </s> <s id="id.2.1.221.3.1.4.0">quod face <lb/>re oportebat. <figure id="id.036.01.223.1.jpg" xlink:href="036/01/223/1.jpg"></figure></s> </p> <p id="id.2.1.221.4.0.0.0" type="main"> <s id="id.2.1.221.4.1.1.0">Hoc autem & ex iis, quæ dicta &longs;unt in <lb/>vige&longs;ima &longs;ecunda, & in vige&longs;imaquinta <lb/>huius efficere po&longs;&longs;umus &longs;olo fune. </s> <s id="id.2.1.221.4.1.2.0">Quòd &longs;i <lb/>pluribus funibus id efficere voluerimus, <lb/>non &longs;olum multis, &longs;ed infinitis modis hoc <lb/>efficere poterimus, vt &longs;upra dictum e&longs;t. <arrow.to.target n="note306"></arrow.to.target><lb/>Quare hoc affirmare po&longs;&longs;umus, quod qui­<lb/>dem mirum e&longs;&longs;e videtur: videlicet. </s> </p> <p id="id.2.1.222.1.0.0.0" type="margin"> <s id="id.2.1.222.1.1.1.0"><margin.target id="note304"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 22 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.222.1.1.2.0"><margin.target id="note305"></margin.target><emph type="italics"/>Ex eadem.<emph.end type="italics"/></s> <s id="id.2.1.222.1.1.3.0"><margin.target id="note306"></margin.target><emph type="italics"/>In<emph.end type="italics"/> 26 <emph type="italics"/>huius.<emph.end type="italics"/></s> </p> <pb xlink:href="036/01/224.jpg"/> <p id="id.2.1.223.1.0.0.0" type="head"> <s id="id.2.1.223.1.2.1.0">COROLLARIVM. I. </s> </p> <p id="id.2.1.223.2.0.0.0" type="main"> <s id="id.2.1.223.2.1.1.0">Ex his manife&longs;tum e&longs;&longs;e, Quamlibet datam in <lb/>numeris proportionem inter pondus, & poten<lb/>tiam; & inter &longs;patium ponderis moti, & &longs;patium <lb/>potentiæ motæ; infinitis modis trochleis inueni­<lb/>ri po&longs;&longs;e. </s> </p> <p id="id.2.1.223.3.0.0.0" type="head"> <s id="id.2.1.223.3.1.1.0">COROLLARIVM II. </s> </p> <p id="id.2.1.223.4.0.0.0" type="main"> <s id="id.2.1.223.4.1.1.0">Ex dictis etiam manife&longs;tum e&longs;t, quò pondus <lb/>facilius mouetur, eò quoq; tempus maius e&longs;&longs;e; <lb/>quò verò difficilius, eò minus e&longs;&longs;e. & è con­<lb/>uer&longs;o. </s> </p> </chap> <pb n="106" xlink:href="036/01/225.jpg"/> <chap> <p id="id.2.1.223.5.0.0.0" type="head"> <s id="id.2.1.223.6.1.1.0">DE AXE IN <lb/>PERITROCHIO. </s> </p> <figure id="id.036.01.225.1.jpg" xlink:href="036/01/225/1.jpg"> </figure> <p id="id.2.1.223.7.0.0.0" type="main"> <s id="id.2.1.223.7.1.1.0">Fabricam, & <expan abbr="cõ&longs;tructionem">con&longs;tructionem</expan> hu­<lb/>ius in&longs;trumenti Pappus in octauo <lb/>mathematicarum collectionum <lb/>libro docet; axemq; vocat AB, <lb/>tympanum verò CD circa idem <lb/>centrum; & &longs;cytalas in foramini­<lb/>bus tympani EF GH & c. </s> <s id="id.2.1.223.7.1.2.0">ita vt potentia, <pb xlink:href="036/01/226.jpg"/><figure id="id.036.01.226.1.jpg" xlink:href="036/01/226/1.jpg"></figure><lb/>quæ &longs;emper in &longs;cytalis e&longs;t, vt in F, dum circum­<lb/>uertit tympanum, & axem, &longs;ur&longs;um moueat pon­<lb/>dus K axi appen&longs;um fune LM circa axem reuo<lb/>luto. </s> <s id="id.2.1.223.7.1.3.0">Nobis igitur re&longs;tat, vt o&longs;tendamus, cur ma­<lb/>gna pondera ab exigua virtute, quouè etiam mo <lb/>do hoc in&longs;trumento moueantur; temporis quin <lb/>etiam, &longs;patiiq; mouentis inuicem potentiæ, ac <lb/>moti ponderis rationem aperiamus; huiu&longs;modi­<lb/>que in&longs;trumenti v&longs;um ad vectem reducamus. </s> </p> <pb n="107" xlink:href="036/01/227.jpg"/> <p id="id.2.1.223.8.0.0.0" type="head"> <s id="id.2.1.223.9.1.1.0">PROPOSITIO I. </s> </p> <p id="id.2.1.223.10.0.0.0" type="main"> <s id="id.2.1.223.10.1.1.0">Potentia pondus &longs;u&longs;tinens axe in peritrochio <lb/>ad pondus eandem habet proportionem, quam <lb/>&longs;emidiameter axis ad &longs;emidiametrum tympani <lb/>vná cum &longs;cytala. <figure id="id.036.01.227.1.jpg" xlink:href="036/01/227/1.jpg"></figure></s> </p> <p id="id.2.1.223.11.0.0.0" type="main"> <s id="id.2.1.223.11.1.1.0">Sit diameter axis AB, cuius centrum C; &longs;it diameter tympani <lb/>DCE circa idem centrum; &longs;intq; AB DE in eadem recta linea; <lb/>&longs;int deinde &longs;cytalæ in foraminibus tympani DF GH & c.inter &longs;e &longs;e <lb/>æquales, atq; æquè di&longs;tantes; &longs;itq; FE horizonti æquidi&longs;tans; <pb xlink:href="036/01/228.jpg"/><figure id="id.036.01.228.1.jpg" xlink:href="036/01/228/1.jpg"></figure><lb/>pondus autem K in fune BL circa axem volubili &longs;it appen&longs;um. </s> <s id="id.2.1.223.11.1.2.0">& <lb/>potentia in F &longs;u&longs;tineat pondus K. </s> <s id="id.2.1.223.11.1.2.0.a">Dico potentiam in F ad pondus <lb/>k ita &longs;e habere, vt CB ad CF. fiat vt CF ad CB, ita pondus <lb/>k ad aliud M, quod appendatur in F. </s> <s id="id.2.1.223.11.1.2.0.b">& quoniam pondera M k <lb/>appen&longs;a &longs;unt in FB; erit FB tanquam vectis, &longs;iue libra; quia ve <lb/>rò Ce&longs;t punctum immobile, circa quod axis, tympanusq; reuol­<lb/>uuntur; erit C fulcimentum vectis FB; vellibræ centrum. </s> <s id="id.2.1.223.11.1.3.0">cùm <lb/><arrow.to.target n="note307"></arrow.to.target>autem it a &longs;it CF ad CB, vt k ad M, pondera k M æqueponde­<lb/>rabunt. </s> <s id="id.2.1.223.11.1.4.0">Potentia igitur in F &longs;u&longs;tinens pondus k, ne deor&longs;um ver­<lb/>gat, ponderi K æqueponderabit; ip&longs;iq; M æqualis erit. </s> <s id="id.2.1.223.11.1.5.0">idem enim <lb/>præ&longs;tat potentia, quod pondus M. </s> <s id="id.2.1.223.11.1.5.0.a">pondus igitur K ad poten<lb/><arrow.to.target n="note308"></arrow.to.target>tiam in F erit, vt CF ad CB; & conuertendo, potentia ad <lb/>pondus erit, vt CB ad CF, hoc e&longs;t, &longs;emidiameter axis ad &longs;emi<pb n="108" xlink:href="036/01/229.jpg"/>diametrum tympani vnà cum &longs;cytala DF. </s> <s id="id.2.1.223.11.1.5.0.b">Similiter etiam o&longs;ten­<lb/>detur, &longs;i potentia pondus &longs;u&longs;tinens fuerit in q. tunc enim &longs;u&longs;ti­<lb/>neret vecte CQ; & ad pondus eam haberet proportionem, quam <arrow.to.target n="note309"></arrow.to.target><lb/>habet CB ad Cq. Videlicet &longs;emidiameter axis ad &longs;emidiame­<lb/>trum tympani vná cum &longs;cytala Eq. quod demon&longs;trare opor­<lb/>tebat. </s> </p> <p id="id.2.1.224.1.0.0.0" type="margin"> <s id="id.2.1.224.1.1.1.0"><margin.target id="note307"></margin.target>6. <emph type="italics"/>Primi Archim. de æquepon.<emph.end type="italics"/></s> <s id="id.2.1.224.1.1.3.0"><margin.target id="note308"></margin.target><emph type="italics"/>Cor.<emph.end type="italics"/> 4. <emph type="italics"/>quinti.<emph.end type="italics"/></s> <s id="id.2.1.224.1.1.4.0"><margin.target id="note309"></margin.target>2 <emph type="italics"/><expan abbr="Huuius">Huius</expan>. de vecte.<emph.end type="italics"/></s> </p> <p id="id.2.1.225.1.0.0.0" type="head"> <s id="id.2.1.225.1.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.225.2.0.0.0" type="main"> <s id="id.2.1.225.2.1.1.0">Manife&longs;tum e&longs;t potentiam &longs;emper minorem <lb/>e&longs;&longs;e pondere. </s> </p> <p id="id.2.1.225.3.0.0.0" type="main"> <s id="id.2.1.225.3.1.1.0">Semidiameter enim axis &longs;emper &longs;emidiametro tympani mi­<lb/>nor e&longs;t. </s> <s id="id.2.1.225.3.1.2.0">& potentia eò minor e&longs;t pondere, quò &longs;emidiameter axis <lb/>minor e&longs;t &longs;emidiametro tympani vná cum &longs;cytala. </s> <s id="id.2.1.225.3.1.3.0">quare quò lon<lb/>gior e&longs;t CF, vel CQ; & quò breuior e&longs;t CB, minor adhuc &longs;em<lb/>per potentia in F, vel in Q pondus k &longs;u&longs;tinebit. </s> <s id="id.2.1.225.3.1.4.0">quò enim minor <lb/>e&longs;t CB, eò minorem habebit proportionem &longs;emidiameter axis <lb/>ad &longs;emidiametrum tympani vná cum &longs;cytala. </s> </p> <p id="id.2.1.225.4.0.0.0" type="main"> <s id="id.2.1.225.4.1.1.0">Hoc autem loco con&longs;iderandum occurrit, quòd &longs;i in alia &longs;cyta­<lb/>la appendatur pondus, vt in T, &longs;u&longs;tinens pondus k; it a nempè, vt <lb/>pondus in T appen&longs;um, pondusq; k circa axem con&longs;titutum <lb/>maneant; erit pondus in T grauius pondere M in F appen&longs;o. </s> <s id="id.2.1.225.4.1.2.0"><lb/>iungatur enim TB, & à puncto C horizonti perpendicularis du­<lb/>catur CI, quæ lineam TB &longs;ecet in I; tandemq; connectatur <lb/>TC, quæ æqualis erit CF. </s> <s id="id.2.1.225.4.1.2.0.a">Quoniam autem pondera appen&longs;a <lb/>&longs;unt in TB, perindè &longs;e &longs;e habebunt, ac &longs;i in punctis TB ip&longs;orum <lb/>centra grauitatum haberent; vt antea dictum e&longs;t. </s> <s id="id.2.1.225.4.1.3.0">& quia ma­<lb/>nent, erit punctum I (ex prima huius de libra) amborum &longs;imul <lb/>grauitatis centrum; cùm &longs;it CI horizonti perpendicularis. </s> <s id="id.2.1.225.4.1.4.0">&longs;ed <lb/>quoniam angulus BCI e&longs;t rectus, erit BIC acutus, lineaq; BI <arrow.to.target n="note310"></arrow.to.target><lb/>ip&longs;a BC maior erit. </s> <s id="id.2.1.225.4.1.5.0">quare angulus CIT erit obtu&longs;us; atq; <arrow.to.target n="note311"></arrow.to.target><lb/>ideo linea CT ip&longs;a TI maior erit. </s> <s id="id.2.1.225.4.1.6.0">Cùm autem CT maior &longs;it <lb/>TI, & IB maior BC; maiorem habebit proportionem TC ad <lb/>CB, quàm TI ad IB; & conuertendo, minorem habebit pro­<pb xlink:href="036/01/230.jpg"/><figure id="id.036.01.230.1.jpg" xlink:href="036/01/230/1.jpg"></figure><lb/>portionem BC ad CT, hoc e&longs;t ad CF, quàm BI ad IT; vt ex <lb/>vige&longs;ima &longs;exta quinti elementorum (iuxta Commandini editio­<lb/>nem) patet. </s> <s id="id.2.1.225.4.1.7.0">Quoniam verò punctum I e&longs;t ponderum in TB <lb/><arrow.to.target n="note312"></arrow.to.target>exi&longs;tentium centrum grauitatis; erit pondus in T ad pondus in B, <lb/>vt BI ad IT. </s> <s id="id.2.1.225.4.1.7.0.a">pondus verò in F ad idem pondus in B e&longs;t, vt BC <lb/>ad CF; maiorem igitur proportionem habebit pondus in T ad <lb/>pondus in B, quàm pondus in F ad idem pondus in B. </s> <s id="id.2.1.225.4.1.7.0.b">ergo <lb/><arrow.to.target n="note313"></arrow.to.target>grauius erit pondus in T, quàm pondus in F. </s> </p> <p id="id.2.1.226.1.0.0.0" type="margin"> <s id="id.2.1.226.1.1.1.0"><margin.target id="note310"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 19 <emph type="italics"/>primi.<emph.end type="italics"/></s> <s id="id.2.1.226.1.1.2.0"><margin.target id="note311"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 13 <emph type="italics"/>primi.<emph.end type="italics"/></s> <s id="id.2.1.226.1.1.3.0"><margin.target id="note312"></margin.target>6. <emph type="italics"/>Primi Archim. de æquepon.<emph.end type="italics"/></s> <s id="id.2.1.226.1.1.5.0"><margin.target id="note313"></margin.target>10. <emph type="italics"/>Quinti.<emph.end type="italics"/></s> </p> <p id="id.2.1.227.1.0.0.0" type="main"> <s id="id.2.1.227.1.1.1.0">Si verò loco ponderis in T animata potentia &longs;u&longs;tinens pon­<lb/>dus k con&longs;tituatur; quæ ita degrauet &longs;e, ac &longs;i in centrum mundi <lb/>tendere vellet; quemadmodum &longs;uapte natura efficit pondus in T <lb/>appen&longs;um; erit hæc eadem ponderi in T appen&longs;o æqualis; alio­<lb/>quin non &longs;u&longs;tineret; quæ quidem ip&longs;a potentia in F collocata ma<pb n="109" xlink:href="036/01/231.jpg"/>ior erit. </s> <s id="id.2.1.227.1.1.2.0">&longs;icuti enim &longs;e &longs;e habet pondus in T ad pondus in F, ita <lb/>& potentia in T ad potentiam in F; cùm potentiæ &longs;int ponderi­<lb/>bus æquales. </s> <s id="id.2.1.227.1.1.3.0">verùm &longs;i vnaquæq; potentia &longs;eor&longs;um &longs;umpta, tàm <lb/>in T, quàm in F &longs;u&longs;tinens pondus <expan abbr="&longs;ecundũ">&longs;ecundum</expan> <expan abbr="circũferentiam">circunferentiam</expan> THFN <lb/>moueri &longs;e vellet, veluti apprehen&longs;a manu &longs;cytala; tunc eademmet <lb/>potentia, vel in F, vel in T con&longs;tituta idem pondus k &longs;u&longs;tinere po<lb/>terit; cùm &longs;emper in cuiu&longs;cunq; extremitate &longs;cytalæ ponatur, ab <lb/>eodem centro C æquidi&longs;tans fuerit, ac &longs;ecundum eandem circum<lb/>ferentiam ab eodem centro æqualiter &longs;emper di&longs;tantem perpen&longs;io<lb/>nem habeat. </s> <s id="id.2.1.227.1.1.4.0">neq; enim (&longs;icuti pondus) proprio nutu magis in <lb/>centrum ferri exoptat, quam circulariter moueri; cùm vtrunq;, &longs;eu <lb/>quemlibet alium motum nullo pror&longs;us re&longs;piciat di&longs;crimine. </s> <s id="id.2.1.227.1.1.5.0">pro­<lb/>pterea non eodem modo res &longs;e &longs;e habet, &longs;iue pondera, &longs;iue anímatæ <lb/>potentiæ ii&longs;dem locis eodem munere abeundo fuerint con&longs;titutæ. </s> </p> <p id="id.2.1.227.2.0.0.0" type="main"> <s id="id.2.1.227.2.1.1.0">Potentia autem mouet pondus vecte FB, videlicet dum po<lb/>tentia in F circumuertit tympanum, circumuertit etiam axem; & <lb/>FB fit tamquam vectis, cuius fulcimentum C, potentia mouens <lb/>in F, & <expan abbr="podus">pondus</expan> in B appen&longs;um. </s> <s id="id.2.1.227.2.1.2.0">& dum punctum F peruenit in N; <lb/>punctum H erit in F, & punctum B erit in O; ita vt ducta NO <lb/>tran&longs;eat per C; eodemq; tempore pondus k motum erit in P, ita <lb/>vt OBP &longs;it æqualis ip&longs;i BL, cùm &longs;it idem funis. </s> </p> <p id="id.2.1.227.3.0.0.0" type="main"> <s id="id.2.1.227.3.1.1.0">Deinde ex quarta huius de vecte facilè eliciemus &longs;patium po­<lb/>tentiæ mouentis ad &longs;patium ponderis moti ita e&longs;&longs;e, vt &longs;emidiame<lb/>ter tympani cùm &longs;cytala ad &longs;emidiametrum axis, hoc e&longs;t, vt CF <lb/>ad CB, cùm circumferentia FN ad BO, &longs;it vt CF ad CB. </s> <s id="id.2.1.227.3.1.1.0.a">& quo<arrow.to.target n="note314"></arrow.to.target><lb/>niam BL, e&longs;t æqualis OBP, dempta communi BP, erit OB ip<lb/>&longs;i PL æqualis. </s> <s id="id.2.1.227.3.1.2.0">quare FN &longs;patium potentiæ ad PL &longs;patium pon­<lb/>deris erit, vt CF ad CB, videlicet &longs;emidiameter tympani cùm <lb/>&longs;cytala ad &longs;emidiametrum axis. </s> <s id="id.2.1.227.3.1.3.0">Quod idem o&longs;tendetur, poten­<lb/>tia vel in Q, vel in qualibet alia &longs;cytala exi&longs;tente, vt in S. cùm <lb/>enim &longs;cytalæ &longs;int &longs;ibi inuicem æquales, atq; æqualiter di&longs;tantes; <lb/>vbicunq; &longs;it potentia æquali mota velocitate &longs;emper æquali tem­<lb/>pore æquale &longs;patium pertran&longs;ibit, hoc e&longs;t ex Q in R, vel ex Sin T <lb/>eodem tempore mouebitur, quò ex F in N. </s> <s id="id.2.1.227.3.1.3.0.a">&longs;ed quò tempore po<lb/>tentia ex F in N mouetur, eodemmet pror&longs;us pondus k ex L in <lb/>P quoq; mouetur; vbicunq; igitur &longs;it potentia, erit &longs;patium poten­<pb xlink:href="036/01/232.jpg"/><figure id="id.036.01.232.1.jpg" xlink:href="036/01/232/1.jpg"></figure><lb/>tiæ ad &longs;patium ponderis moti, vt CF ad CB, hoc e&longs;t &longs;emidia­<lb/>meter tympani cum &longs;cytala, ad &longs;emidiametrum axis. </s> </p> <p id="id.2.1.228.1.0.0.0" type="margin"> <s id="id.2.1.228.1.1.1.0"><margin.target id="note314"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 4 <emph type="italics"/>huius de vecte.<emph.end type="italics"/></s> </p> <p id="id.2.1.229.1.0.0.0" type="head"> <s id="id.2.1.229.1.1.1.0">COROLLARIVM. I. </s> </p> <p id="id.2.1.229.2.0.0.0" type="main"> <s id="id.2.1.229.2.1.1.0">Ex his manife&longs;tum e&longs;t, ita e&longs;&longs;e pondus ad po­<lb/>tentiam pondus &longs;u&longs;tinentem, vt &longs;patium poten­<lb/>tiæ mouentis ad &longs;patium ponderis moti. </s> </p> <pb n="110" xlink:href="036/01/233.jpg"/> <p id="id.2.1.229.3.0.0.0" type="head"> <s id="id.2.1.229.4.1.1.0">COROLLARIVM II. </s> </p> <p id="id.2.1.229.5.0.0.0" type="main"> <s id="id.2.1.229.5.1.1.0">Manife&longs;tum e&longs;t etiam, maiorem &longs;emper ha­<lb/>bere proportionem &longs;patium potentiæ mouentis <lb/>ad &longs;patium ponderis moti, quàm pondus ad ean<lb/>dem potentiam. </s> </p> <p id="id.2.1.229.6.0.0.0" type="main"> <s id="id.2.1.229.6.1.1.0">Præterea quò circulus FHN circa &longs;cytalas e&longs;t maior, eò quoq; <lb/>in pondere mouendo maius &longs;umetur tempus; dummodo potentia <lb/>æquali moueatur velocitate. </s> <s id="id.2.1.229.6.1.2.0">tempu&longs;q; eò maius erit, quò diame<lb/>ter vnius diametro alterius e&longs;t maior. </s> <s id="id.2.1.229.6.1.3.0">circulorum enim circumfe-<arrow.to.target n="note315"></arrow.to.target><lb/>rentiæ ita &longs;e habent, vt diametri. </s> <s id="id.2.1.229.6.1.4.0">Cùm vero ex trige&longs;ima &longs;exta <lb/>quarti libri Pappi Mathematicarum collectionum, duorum inæ <lb/>qualium circulorum æquales circumferentias inuenire po&longs;simus; <lb/>ideo tempus quoq; portionum circulorum inæqualium hoc modo <lb/>inueniemus. </s> <s id="id.2.1.229.6.1.5.0">è conuer&longs;o autem, quò maior erit axis circumferen<lb/>tia citius pondus &longs;ur&longs;um mouebitur. </s> <s id="id.2.1.229.6.1.6.0">maior enim pars funis BL <lb/>in vna circumuer&longs;ione completa circa circulum ABO reuoluitur, <lb/>quàm &longs;i minor e&longs;&longs;et; cùm funis circumuolutus &longs;it circumferen­<lb/>tiæ circuli æqualis, circa quem reuoluitur. </s> </p> <p id="id.2.1.230.1.0.0.0" type="margin"> <s id="id.2.1.230.1.1.1.0"><margin.target id="note315"></margin.target>23 <emph type="italics"/>Octaui libri Pappi.<emph.end type="italics"/></s> </p> <p id="id.2.1.231.1.0.0.0" type="head"> <s id="id.2.1.231.1.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.231.2.0.0.0" type="main"> <s id="id.2.1.231.2.1.1.0">Ex his manife&longs;tum e&longs;t, quò facilius pondus mo<lb/>uetur, tempus quoq; eò maius e&longs;&longs;e; & quò dif­<lb/>ficilius, eò tempus minus e&longs;&longs;e. </s> <s id="id.2.1.231.2.1.2.0">& è conuer&longs;o. </s> </p> <pb xlink:href="036/01/234.jpg"/> <p id="id.2.1.231.4.0.0.0" type="head"> <s id="id.2.1.231.4.1.1.0">PROPOSITIO II. </s> </p> <p type="head"> <s id="id.2.1.231.4.3.1.0">PROBLEMA. </s> </p> <p id="id.2.1.231.5.0.0.0" type="main"> <s id="id.2.1.231.5.1.1.0">Datum pondus à data potentia axe in peritro­<lb/>chio moueri. </s> </p> <p id="id.2.1.231.6.0.0.0" type="main"> <s id="id.2.1.231.6.1.1.0">Sit datum pondus &longs;exagin<lb/>ta; potentia verò vt decem. </s> <s id="id.2.1.231.6.1.2.0"><lb/>exponatur quædam recta li­<lb/>nea AB, quæ diuidatur in C, <lb/>ita vt AC ad CB eandem <lb/><figure id="id.036.01.234.1.jpg" xlink:href="036/01/234/1.jpg"></figure><lb/>habeat proportionem, quam &longs;exaginta ad decem. </s> <s id="id.2.1.231.6.1.3.0">& &longs;i CB axis <lb/>&longs;emidiameter e&longs;&longs;et, & CA &longs;emidiameter tympani cùm &longs;cytalis; <lb/><arrow.to.target n="note316"></arrow.to.target>patet potentiam vt decem in A ponderi &longs;exaginta in B æquepon<lb/>derare. </s> <s id="id.2.1.231.6.1.4.0">Accipiatur autem inter BC quoduis punctum D; fiatq; <lb/>BD &longs;emidiameter axis, & DA &longs;emidiameter tympani cùm &longs;cy­<lb/>talis; ponaturq; pondus &longs;exaginta in B fune circa axem, & potentia <lb/><arrow.to.target n="note317"></arrow.to.target><emph type="italics"/>in A. </s> <s id="id.2.1.231.6.1.4.0.a">Quoniam enim AD ad DB maiorem habet proportio­<lb/>nem, quam AC ad CB; maiorem habebit proportionem AD ad <lb/>DB, quam pondus &longs;exaginta in B appen&longs;um ad potentiam vt decem<emph.end type="italics"/> <lb/><arrow.to.target n="note318"></arrow.to.target>in A. </s> <s id="id.2.1.231.6.1.4.0.b">Quare potentia in A pondus &longs;exaginta axe in peritro­<lb/>chio mouebit, cuius axis &longs;emidiameter e&longs;t BD, & DA &longs;emidia<lb/>meter tympani cùm &longs;cytalis. </s> <s id="id.2.1.231.6.1.5.0">quod erat faciendum. </s> </p> <p id="id.2.1.232.1.0.0.0" type="margin"> <s id="id.2.1.232.1.1.1.0"><margin.target id="note316"></margin.target><emph type="italics"/>Per præcedentem.<emph.end type="italics"/></s> <s id="id.2.1.232.1.1.2.0"><margin.target id="note317"></margin.target><emph type="italics"/>Lemma in primi huius de vecte.<emph.end type="italics"/></s> <s id="id.2.1.232.1.1.3.0"><margin.target id="note318"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 11 <emph type="italics"/>huius de vecte.<emph.end type="italics"/></s> </p> <pb n="111" xlink:href="036/01/235.jpg"/> <p id="id.2.1.233.1.0.0.0" type="head"> <s id="id.2.1.233.1.2.1.0">ALITER. </s> </p> <p type="head"> <s id="id.2.1.233.1.4.1.0">Organicè verò melius erit hoc pacto. </s> </p> <p id="id.2.1.233.2.0.0.0" type="main"> <s id="id.2.1.233.2.1.1.0">Exponatur axis, cuius <lb/>diameter &longs;it BD, & cen­<lb/>trum C, quem quidem <lb/>axem maiorem, vel mino <lb/>rem con&longs;tituemus, veluti <lb/><figure id="id.036.01.235.1.jpg" xlink:href="036/01/235/1.jpg"></figure><lb/>magnitudo, ponderi&longs;q; grauitas po&longs;tulat. </s> <s id="id.2.1.233.2.1.2.0">producatur deinde BD <lb/>v&longs;q; ad A: fiatq; BC ad CA, vt decem ad &longs;exaginta. </s> <s id="id.2.1.233.2.1.3.0">& &longs;i CA tym<lb/>pani cùm &longs;cytalis &longs;emidiameter e&longs;&longs;et, potentia decem in A ponde<lb/>ri &longs;exaginta in B æqueponderaret. </s> <s id="id.2.1.233.2.1.4.0">producatur verò BA ex parte <lb/>A, & in hac producta linea quoduis accipiatur punctum E; fiatq; <lb/>CE &longs;emidiameter tympani cùm &longs;cytalis; ponaturq; potentia vt <lb/>decem in E; habebit EC ad CB maiorem proportionem, quàm <lb/>pondus &longs;exaginta in B ad potentiam vt decem in E. </s> <s id="id.2.1.233.2.1.4.0.a">potentia igi­<lb/>tur vt decem in E mouebit pondus &longs;exaginta in B appen&longs;um fune <lb/>circa axem, cuius &longs;emidiameter e&longs;t CB, & CE &longs;emidiameter tym<lb/>pani cùm &longs;cytalis. </s> <s id="id.2.1.233.2.1.5.0">quod facere oportebat. </s> </p> <pb xlink:href="036/01/236.jpg"/> <p id="id.2.1.233.4.0.0.0" type="main"> <s id="id.2.1.233.4.1.1.0">Sub hoc facultatis genere &longs;unt ergatæ, &longs;uccu­<lb/>læ, terebræ, tympana cum &longs;uis axibus, &longs;iue dentata, <lb/>&longs;iue non; & &longs;imilia. </s> </p> <p id="id.2.1.233.5.0.0.0" type="main"> <s id="id.2.1.233.5.1.1.0">Terebra verò habet etiam ne&longs;cioquid cochleæ; dum enim mo­<lb/>uet pondus, &longs;cilicet dum perforat, ex &longs;ua ferè natura &longs;emper vlte­<lb/>rius progreditur<emph type="italics"/>:<emph.end type="italics"/> habet enim ferè helices tamquam circa conum <lb/>de&longs;criptas. </s> <s id="id.2.1.233.5.1.2.0">quoniam autem verticem habet acutum, ad cunei quoq; <lb/>rationem commodè referri poterit. <figure id="id.036.01.236.1.jpg" xlink:href="036/01/236/1.jpg"></figure></s> </p> </chap> <pb n="112" xlink:href="036/01/237.jpg"/> <chap> <p id="id.2.1.233.5.0.0.0.a" type="head"> <s id="id.2.1.233.5.3.1.0">DE CVNEO. </s> </p> <p id="id.2.1.233.6.0.0.0" type="main"> <s id="id.2.1.233.6.1.1.0">Aristoteles in quæ&longs;tioni­<lb/>bus Mechanicis quæ&longs;tione deci­<lb/>ma&longs;eptima a&longs;&longs;erit, cuneum &longs;cin­<lb/>dendo ponderi duorum vicem <lb/>pror&longs;us gerere vectium &longs;ibi inui­<lb/>cem contrariorum hoc <expan abbr="niodo">modo</expan>. </s> </p> <p id="id.2.1.233.7.0.0.0" type="main"> <s id="id.2.1.233.7.1.1.0">Sit cuneus ABC, cu<lb/>ius vertex B, & &longs;it AB <lb/>æqualis BC; quod au<lb/>tem &longs;cindendum e&longs;t, <lb/>&longs;it DEFG; &longs;itq; pars <lb/>cunei HB k intra DE <lb/>FG, & HB æqualis <lb/>&longs;it ip&longs;i Bk. </s> <s id="id.2.1.233.7.1.2.0">percutiatur <lb/>(vt fieri &longs;olet) cuneus <lb/>in AC, dum cuneus in <lb/>AC percutitur, AB fit <lb/>vectis, cuius fulcimen <lb/>tum e&longs;t H, & pondus in <lb/>B. </s> <s id="id.2.1.233.7.1.2.0.a">eodemq; modo CB <lb/>fit vectis, cuius fulci­<lb/><figure id="id.036.01.237.1.jpg" xlink:href="036/01/237/1.jpg"></figure><lb/>mentum e&longs;t K, & pondus &longs;imiliter in B. </s> <s id="id.2.1.233.7.1.2.0.b">&longs;ed dum percutitur cu­<lb/>neus, maiori adhuc ip&longs;ius portione ip&longs;um DEFG ingreditur, <lb/>quàm prius e&longs;&longs;et: &longs;it autem portio hæc MBL; &longs;itq; M B ip&longs;i BL <lb/>æqualis. </s> <s id="id.2.1.233.7.1.3.0">& cùm MB BI. &longs;int ip&longs;is HB BK maiores; erit ML maior <pb xlink:href="036/01/238.jpg"/>Hk. </s> <s id="id.2.1.233.7.1.4.0">dum igitur ML <lb/>erit in &longs;itu Hk; opor­<lb/><expan abbr="ter">tet</expan>, vt fiat maior &longs;ci&longs;sio; <lb/>& D moueatur ver&longs;us <lb/>O, G autem ver&longs;us N: <lb/>& quò maior pars cu<lb/>nei intra DEFG ingre<lb/>dietur, eò maior fiet <lb/>&longs;ci&longs;sio; & DG ma­<lb/>gis adhuc impellentur <lb/>ver&longs;us ON. </s> <s id="id.2.1.233.7.1.4.0.a">pars igi<lb/>tur KG eius, quod &longs;cin<lb/>ditur, mouebitur à ve­<lb/>cte AB, cuius fulcimen<lb/>tum e&longs;t H, & pondus <lb/><figure id="id.036.01.238.1.jpg" xlink:href="036/01/238/1.jpg"></figure><lb/>in B; ita vt punctum B ip&longs;ius vectis AB impellat partem KG. <lb/>& pars HD mouebitur à vecte CB, cuius fulcimentum e&longs;t k; ita <lb/>vt B vecte CB partem HD impellat. </s> </p> <p id="id.2.1.233.8.0.0.0" type="main"> <s id="id.2.1.233.8.1.1.0">Cùm autem tria &longs;int vectium genera, vt &longs;upra <lb/>o&longs;ten&longs;um e&longs;t; idcirco conuenientius erit forta&longs;&longs;è <lb/>cuneum hoc modo con&longs;iderare. </s> </p> <p id="id.2.1.233.9.0.0.0" type="main"> <s id="id.2.1.233.9.1.1.0">Ii&longs;dem po&longs;itis, intelligatur vectis AB, cuius fulcimentum B, & <lb/>pondus in H, vt in &longs;ecunda huius de vecte diximus. </s> <s id="id.2.1.233.9.1.2.0">&longs;imiliter ve­<lb/>ctis CB, cuius fulcimentum B, & pondus in K; ita vt pars HD <lb/>moueatur à vecte AB, cuius fulcimentum e&longs;t B, & pondus in H; <lb/>ita vt punctum H ip&longs;ius vectis AB impellat partem HD. &longs;imi <lb/>li quoq; modo pars KG moueatur à vecte CB, cuius fulcimentum <lb/>e&longs;t B, & pondus in k, it aut k ip&longs;ius uectis CB partem k G mo­<lb/>ueat. </s> <s id="id.2.1.233.9.1.3.0">quod quidem for&longs;itan rationi magis con&longs;entaneum erit. </s> </p> <pb n="113" xlink:href="036/01/239.jpg"/> <p id="id.2.1.233.11.0.0.0" type="main"> <s id="id.2.1.233.11.1.1.0">Sit enim cuneus ABC; <lb/>&longs;intq; duo pondera &longs;epa­<lb/>rat a DEFG, & HIkL, <lb/>intra quæ &longs;it pars cunei <lb/>DBH, cuius uertex B <lb/>medium inter utrumq; &longs;i <lb/>tum obtineat. </s> <s id="id.2.1.233.11.1.2.0">percutia­<lb/>tur autem cuneus, ita ut <lb/>magis adhuc intra pon­<lb/>dera propellatur, &longs;icuti <lb/>prius dictum e&longs;t; ponde­<lb/><figure id="id.036.01.239.1.jpg" xlink:href="036/01/239/1.jpg"></figure><lb/>ra enim &longs;unt, ac &longs;i unum tantùm continuum e&longs;&longs;et GFkL, quod <lb/>&longs;cindendum e&longs;&longs;et: eodem enim modo pars DG, dum cuneus <lb/>ulterius impellitur, mouebitur uer&longs;us M; & pars HL uer&longs;us N. </s> <s id="id.2.1.233.11.1.2.0.a"><lb/>Moueatur itaq; pars DG uer&longs;us M, & pars HL uer&longs;us N, B uerò <lb/>dum ulterius progreditur, &longs;emper medium inter utrunq; pondus <lb/>remaneat. </s> <s id="id.2.1.233.11.1.3.0">dum autem DG à cuneo mouetur uer&longs;us M; patet B <lb/>non mouere partem DG uer&longs;us M uecte CB, cuius fulcimentum <lb/>H; <expan abbr="punctũ">punctum</expan> enim B non tangit pondus; &longs;ed DG mouebitur à pun­<lb/>cto uectis D uecte AB, cuius fulcimentum B; punctum enim D tan<lb/>git pondus, & in&longs;trumenta mouent per contactum. </s> <s id="id.2.1.233.11.1.4.0">Similiter <lb/>HL mouebitur ab H uecte CB, cuius fulcimentum B; & uterq; <lb/>uectis utriq; re&longs;i&longs;tit in B, ita ut B potius fulcimenti uice fungatur, <lb/>quàm mouendi ponderis. </s> <s id="id.2.1.233.11.1.5.0">quod ip&longs;um hoc quoq; modo manife­<lb/>&longs;tum erit. </s> </p> <pb xlink:href="036/01/240.jpg"/> <p id="id.2.1.233.13.0.0.0" type="main"> <s id="id.2.1.233.13.1.1.0">Sit, quod &longs;cindendum e&longs;t A <lb/>BCD <expan abbr="parallelogrammũ">parallelogrammum</expan> rectan­<lb/>gulum; &longs;intq; duo vectes æqua­<lb/>les EF GF, & partes vectium <lb/>HF KF &longs;int intra ABCD; &longs;itq; <lb/>HF æqualis Fk, & HA æqua<lb/>lis KB. </s> <s id="id.2.1.233.13.1.1.0.a">Oporteat verò vecti­<lb/>bus EF GF &longs;cindere ABCD <lb/>ab&longs;q; percu&longs;sione, videlicet &longs;int <lb/>potentiæ mouentes in EG æqua<lb/>les. </s> <s id="id.2.1.233.13.1.2.0">vt autem &longs;cindatur ABCD, <lb/>oportet partem HA moueri uer<lb/><figure id="id.036.01.240.1.jpg" xlink:href="036/01/240/1.jpg"></figure><lb/>&longs;us M. & kB ver&longs;us N; &longs;ed dum vectes mouentur, putá alter in <lb/>M, alter verò in N; nece&longs;&longs;e e&longs;t, vt punctum F immobile rema <lb/>neat; in illo enim fit vectium occur&longs;us. </s> <s id="id.2.1.233.13.1.3.0">quare F erit fulcimen­<lb/>tum vtriu&longs;q; vectis, & FG mouebit partem kB, cuius fulcimen <lb/>tum erit F, & potentia mouens in G; & pondus in k. </s> <s id="id.2.1.233.13.1.4.0">&longs;imi­<lb/>liter pars HA mouebitur à vecte EF, cuius fulcimentum F, po<lb/>tentia in E, & pondus in H. </s> </p> <p id="id.2.1.233.14.0.0.0" type="main"> <s id="id.2.1.233.14.1.1.0">Si autem k H e&longs;&longs;ent fulcimenta immobilia, & pondera in F; <lb/>dum vectis FG conatur mouere pondus in F, tunc ei re&longs;i&longs;tit ve­<lb/>ctis EF, qui etiam conatur mouere pondus in F ad partem op<lb/>po&longs;itam; &longs;ed quoniam potentiæ &longs;unt æquales, & cætera æqualia; <lb/>ergo in F non fiet motus: æquale enim non mouet æquale. </s> <s id="id.2.1.233.14.1.2.0">patet <lb/>igitur in F maximam fieri vectium &longs;ibi inuicem occurrentium re&longs;i<lb/>&longs;tentiam, ita ut F &longs;it quoddam immobile. </s> <s id="id.2.1.233.14.1.3.0">Quare con&longs;iderando <lb/>cuneum, <expan abbr="vtmouet">vt mouet</expan> vectibus &longs;ibi inuicem aduer&longs;is, for&longs;itan eis po<lb/>tius utitur hoc &longs;ecundo modo, quàm primo. </s> </p> <p id="id.2.1.233.15.0.0.0" type="main"> <s id="id.2.1.233.15.1.1.0">Quoniam autem totus cuneus &longs;cindendo mo<lb/>uetur, po&longs;&longs;umus idcirco eundem alio quoq; mo<lb/>do con&longs;iderare; videlicet dum ingreditur id, <pb n="114" xlink:href="036/01/241.jpg"/>quod &longs;cinditur, nihil aliud e&longs;&longs;e, ni&longs;i pondus &longs;u<lb/>pra planum horizonti inclinatum mouere. <figure id="id.036.01.241.1.jpg" xlink:href="036/01/241/1.jpg"></figure></s> </p> <p id="id.2.1.233.16.0.0.0" type="main"> <s id="id.2.1.233.16.1.1.0">Sit planum horizonti æquidi&longs;tans tran&longs;iens per AB; &longs;it cuneus <lb/>CDB, & CD æqualis ip&longs;i DB; & latus cunei DB &longs;it &longs;emper in <lb/>&longs;ubiecto plano. </s> <s id="id.2.1.233.16.1.2.0">&longs;it deinde pondus AEFG immobile in A; &longs;itq; <lb/>pars cunei EDH &longs;ub AEFG. </s> <s id="id.2.1.233.16.1.2.0.a">Quoniam enim dum percutitur cu<lb/>neus in CB, maior pars cunei ingreditur &longs;ub AEFG, quàm &longs;it <lb/>EDH; &longs;it hæc pars IDH. </s> <s id="id.2.1.233.16.1.2.0.b">& quoniam latus cunei DB &longs;emper <lb/>e&longs;t in &longs;ubiecto plano per AB ducto horizonti parallelo, tunc quan<lb/>do pars cunei kDI erit &longs;ub AEFG; erit punctum k in H, & I <lb/>&longs;ub E. </s> <s id="id.2.1.233.16.1.2.0.c">&longs;ed Ik maior e&longs;t HE; punctum igitur E &longs;ur&longs;um motum <lb/>erit. </s> <s id="id.2.1.233.16.1.3.0">& dum cuneus &longs;ub AEFG ingreditur, punctum E &longs;ur&longs;um <lb/>&longs;uper latus cunei EI mouebitur, eodemq; modo &longs;i cuneus vlterius <lb/>progredietur, &longs;emper punctum E &longs;uper latus cunei DC mouebitur: <lb/>punctum igitur E ponderis &longs;uper planum DC mouebitur horizonti <lb/>inclinatum, cuius inclinatio e&longs;t angulus BDC. quod demon­<lb/>&longs;trare oportebat. <pb xlink:href="036/01/242.jpg"/><figure id="id.036.01.242.1.jpg" xlink:href="036/01/242/1.jpg"></figure></s> </p> <p id="id.2.1.233.17.0.0.0" type="main"> <s id="id.2.1.233.17.1.1.0">In hoc exemplo, con&longs;iderando cuneum in&longs;tar vectis mouen­<lb/>tem, manife&longs;tum e&longs;t, cuneum BCD pondus AEFG vecte CD <lb/>mouere; ita vt D &longs;it fulcimentum, & pondus in E. non autem ve<lb/>cte BD, cuius fulcimentum H, & pondus in D. </s> </p> <p id="id.2.1.233.18.0.0.0" type="main"> <s id="id.2.1.233.18.1.1.0">Vt autem res clarior reddatur, alio vtamur <lb/>exemplo. </s> </p> <p id="id.2.1.233.19.0.0.0" type="main"> <s id="id.2.1.233.19.1.1.0">Sit planum hori­<lb/>zonti æquidi&longs;tans <lb/>tran&longs;iens per AB; &longs;it <lb/>cuneus CAB, cuius <lb/>latus AB &longs;it &longs;emper <lb/>in &longs;ubiecto plano; <expan abbr="&longs;it­qué">&longs;it­<lb/>que</expan> pondus AEFG, <lb/>quod nullum alium <lb/>habeat motum, ni&longs;i <lb/><figure id="id.036.01.242.2.jpg" xlink:href="036/01/242/2.jpg"></figure><lb/>&longs;ur&longs;um, & deor&longs;um ad rectos angulos horizonti; ita vt ducta IGk <lb/>&longs;ubiecto plano, ip&longs;iqué AB perpendicularis, punctum G &longs;it &longs;em<lb/>per in linea IGk. </s> <s id="id.2.1.233.19.1.2.0">& quoniam dum cuneus percutitur in CB, to<lb/>tus &longs;uper AB vlterius progreditur; pondus AEFG eleuabitur ex <pb n="115" xlink:href="036/01/243.jpg"/>iis, quæ &longs;upra diximus. </s> <s id="id.2.1.233.19.1.3.0">Moueatur cuneus ita, vt E tandem per­<lb/>ueniat in C, & po&longs;itio cunei ABC &longs;it MNO, & po&longs;itio pon­<lb/>deris AEFG &longs;it PMQI, & G &longs;it in I. </s> <s id="id.2.1.233.19.1.3.0.a">Quoniam itaq; dum cu<lb/>neus &longs;uper lineam BO mouetur, pondus AEFG &longs;ur&longs;um moue­<lb/>tur à linea AC. & dum cuneus ABC vlterius progreditur, &longs;em<lb/>per pondus AEFG magis à latere cunei AC eleuatur: pondus igi<lb/>tur AEFG &longs;uper planum cunei AC mouebitur; quod quidem <lb/>nihil aliud e&longs;t, ni&longs;i planum horizonti inclinatum, cuius inclinatio <lb/>e&longs;t angulus BAC. </s> </p> <p id="id.2.1.233.20.0.0.0" type="main"> <s id="id.2.1.233.20.1.1.0">Hic motus facilè ad libram, vectemq; reducitur. </s> <s id="id.2.1.233.20.1.2.0">quod enim <lb/>&longs;uper planum horizonti inclinatum mouetur ex nona Pappi octa­<lb/>ui libri Mathematicarum collectionum reducitur ad libram. </s> <s id="id.2.1.233.20.1.3.0">ea­<lb/>dem enim e&longs;t ratio, &longs;iue manente cuneo, vt pondus &longs;uper cunei <lb/>latus moueatur; &longs;iue eodem etiam moto, pondus adhuc &longs;uper ip<lb/>&longs;ius latus moueatur; tamquam &longs;uper planum horizonti incli­<lb/>natum. </s> </p> <p id="id.2.1.233.21.0.0.0" type="main"> <s id="id.2.1.233.21.1.1.0">Ea verò, quæ &longs;cinduntur, quomodo tam­<lb/>quam &longs;uper plana horizonti inclinata mouean­<lb/>tur, o&longs;tendamus. </s> </p> <p id="id.2.1.233.22.0.0.0" type="main"> <s id="id.2.1.233.22.1.1.0">Sit cuneus ABC, <lb/>& AB ip&longs;i BC æqua­<lb/>lis. </s> <s id="id.2.1.233.22.1.2.0">Diuidatur AC <lb/>bifariam in D, conne­<lb/>ctaturq; BD. </s> <s id="id.2.1.233.22.1.2.0.a">&longs;it dein­<lb/>de linea EF, per quam <lb/>tran&longs;eat planum hori<lb/>zonti æquidi&longs;tans; &longs;itq; <lb/>BD in eadem linea EF; <lb/>& dum cuneus percuti<lb/>tur, dumq; mouetur ver<lb/><figure id="id.036.01.243.1.jpg" xlink:href="036/01/243/1.jpg"></figure><lb/>&longs;us E, &longs;emper BD &longs;it in linea EF. quod verò &longs;cindendum e&longs;t <lb/>&longs;it GHLM, intra quod &longs;it pars cunei kBI. manife&longs;tum e&longs;t, <pb xlink:href="036/01/244.jpg"/>dum cuneus uer&longs;us E <lb/>mouetur, partem kG <lb/>ver&longs;us N moueri; & par<lb/>tem HI uer&longs;us O. per<lb/>cutiatur cuneus, ita vt <lb/>AC &longs;it in linea NO; <lb/>tunc k erit in A, & I in <lb/>C: & k ex &longs;uperius di<lb/>ctis motum erit &longs;uper <lb/>kA, & I &longs;uper IC. <lb/>quare dum cuneus mo<lb/><figure id="id.036.01.244.1.jpg" xlink:href="036/01/244/1.jpg"></figure><lb/>uetur, pars KG &longs;uper BA latus cunei mouebitur, & pars IH &longs;uper <lb/>latus BC. </s> <s id="id.2.1.233.22.1.2.0.b">pars igitur kG &longs;uper planum mouetur horizonti incli­<lb/>natum, cuius inclinatio e&longs;t angulus FBA. &longs;imiliter IH moue­<lb/>tur &longs;uper planum BC in angulo FBC. </s> <s id="id.2.1.233.22.1.2.0.c">Partes ergo eius, quod <lb/>&longs;cinditur &longs;uper plana horizonti inclinata mouebuntur. </s> <s id="id.2.1.233.22.1.3.0">& quam­<lb/>quam planum BC &longs;it &longs;ub horizonte; pars tamen IH &longs;uper IC mo<lb/>uetur, tamquam &longs;i BC e&longs;&longs;et &longs;upra <expan abbr="horizont&etilde;">horizontem</expan> in angulo DBC. partes <lb/>enim eius quod <expan abbr="&longs;inditur">scinditur</expan>, eodem tempore, ab eadem potentia mo­<lb/>uentur; eadem ergo erit ratio motus partis IH, ac partis KG. &longs;i­<lb/>militer eadem e&longs;t ratio, &longs;iue EF &longs;it horizonti æquidi&longs;tans, &longs;iue <lb/>horizonti perpendicularis, vel alio modo. </s> <s id="id.2.1.233.22.1.4.0">nece&longs;&longs;e e&longs;t enim poten<lb/>tiam cuneum mouentem eandem e&longs;&longs;e, cùm cætera eadem rema <lb/>neant. </s> <s id="id.2.1.233.22.1.5.0">eadem igitur erit ratio. </s> </p> <p id="id.2.1.233.23.0.0.0" type="main"> <s id="id.2.1.233.23.1.1.0">Po&longs;t hæc con&longs;iderandum e&longs;t, quæ nam &longs;int ea, quæ efficiunt, <lb/>vt aliquod facilius moueatur, &longs;iue &longs;cindatur. </s> <s id="id.2.1.233.23.1.2.0">quæ quidem duo <lb/>&longs;unt. </s> </p> <p id="id.2.1.233.24.0.0.0" type="main"> <s id="id.2.1.233.24.1.1.0">Primum, quod efficit, vt aliquod facilè &longs;cin<lb/>datur, quod etiam ad e&longs;&longs;entiam cunei magis per­<lb/>tinet, e&longs;t angulus ad verticem cunei; quò enim <lb/>minor e&longs;t angulus, eò facilius mouet, ac &longs;cindit. </s> </p> <pb n="116" xlink:href="036/01/245.jpg"/> <p id="id.2.1.233.26.0.0.0" type="main"> <s id="id.2.1.233.26.1.1.0">Sint duo cunei ABC DEF, & angulus <lb/>ABC ad verticem minor &longs;it angulo DEF. </s> <s id="id.2.1.233.26.1.1.0.a"><lb/>dico aliquod facilius moueri, &longs;iue &longs;cindi à cu<lb/>neo ABC, quàm à DEF. diuidantur AC <lb/>DF bifariam in G H punctis; connectan­<lb/>turq; BG, & EH. </s> <s id="id.2.1.233.26.1.1.0.b">Quoniam enim partes <lb/>eius, quod &longs;cinditur à cuneo ABC, &longs;u­<lb/>per planum horizonti inclinatum mouen­<lb/>tur, cuius inclinatio e&longs;t GBA: quæ <expan abbr="ve­rò">ve­<lb/>ro</expan> à cuneo DEF, &longs;uper planum horizonti <lb/>inclinatum mouentur, cuius inclinatio e&longs;t <lb/><figure id="id.036.01.245.1.jpg" xlink:href="036/01/245/1.jpg"></figure><lb/>HED; & angulus GBA minor e&longs;t angulo HED; cùm <lb/>CBA minor &longs;it DEF: & ex nona Pappi octaui libri mathe<lb/>maticarum collectionum, quod mouetur &longs;uper planum AB faci­<lb/>lius mouebitur, & à minore potentia, quàm &longs;uper ED; Quod <lb/>ergo &longs;cinditur à cuneo ABC facilius, & à minore potentia &longs;cin<lb/>detur, quàm à cuneo DEF. &longs;imiliter o&longs;tendetur, quò magis an­<lb/>gulus ad verticem cunei erit acutus, eò facilius aliquod moueri, <lb/>ac &longs;cindi. </s> <s id="id.2.1.233.26.1.2.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.233.27.0.0.0" type="main"> <s id="id.2.1.233.27.1.1.0">Po&longs;&longs;umus etiam hoc alia ratione o&longs;tendere <lb/>con&longs;iderando cuneum, vt vectibus &longs;ibi inuicem <lb/>aduer&longs;is mouet, &longs;icuti &longs;ecundo modo dictum e&longs;t. </s> <s id="id.2.1.233.27.1.2.0"><lb/>hoc autem prius o&longs;tendere oportet. </s> </p> <pb xlink:href="036/01/246.jpg"/> <p id="id.2.1.233.29.0.0.0" type="main"> <s id="id.2.1.233.29.1.1.0">Sit vectis AB, cuius fulcimentum <lb/>&longs;it B immobile; quod autem mouen­<lb/>dum e&longs;t, &longs;it CDEF rectangulum ita <lb/>accommodatum, vt deor&longs;um ex par <lb/>te FE moueri non po&longs;sit; & punctum <lb/>E&longs;it immobile, & tamquam centrum; <lb/>ita vt punctum D moueatur per cir­<lb/>cumferentiam circuli DH, cuius cen­<lb/>trum &longs;it E. & C per circumferentiam <lb/>CL, ita vt iuncta CE &longs;it eius &longs;emi<lb/>diameter. </s> <s id="id.2.1.233.29.1.2.0">tangat in&longs;uper CDEF ve<lb/><figure id="id.036.01.246.1.jpg" xlink:href="036/01/246/1.jpg"></figure><lb/>ctem AB in C, atq; vectis AB moueat pondus CDEF, & po<lb/>tentia mouens &longs;it in A, fulcimentum B, & pondus in C. </s> <s id="id.2.1.233.29.1.2.0.a">&longs;it <lb/>deinde alius vectis MCN, qui etiam moueat CDEF, cuius ful<lb/>cimentum immobile &longs;it N; potentia mouens in M, & pondus <lb/>&longs;imiliter in C; &longs;itq; CN æqualis ip&longs;i CB, & CM ip&longs;i CA; al<lb/>ternatimq; moueatur pondus CDEF vectibus AB MN. </s> <s id="id.2.1.233.29.1.2.0.b">dico <lb/>CDEF facilius ab eadem potentia moueri vecte AB, quàm ve<lb/>cte MN. </s> </p> <p id="id.2.1.233.30.0.0.0" type="main"> <s id="id.2.1.233.30.1.1.0">Fiat centrum B, & interuallo BC circumferentia de&longs;cribatur <lb/>CO. &longs;imiliter centro N, interuallo quidem NC, circumferen<lb/>tia de&longs;cribatur CP. </s> <s id="id.2.1.233.30.1.1.0.a">Quoniam enim dum vectis AB mouet CD <lb/>EF, punctum vetis C mouetur &longs;uper circumferentiam CO; cùm <lb/>&longs;it B fulcimentum, & centrum immobile. </s> <s id="id.2.1.233.30.1.2.0">&longs;imiliter dum vectis <lb/>MN mouet CDEF, punctum C mouetur per circumferentiam <lb/>CP; dum igitur vectis AB mouet CDEF, conatur mouere pun<lb/>ctum C ponderis &longs;uper circumferentiam CO; quod quidem effi<lb/>cere non pote&longs;t: quia C mouetur &longs;uper circumferentiam CL. qua <lb/>re in motu vectis AB &longs;ecundùm partem ip&longs;i re&longs;pondentem, ac mo <lb/>tu ponderis &longs;ecundum C facto, contingit repugnantia quædam; <lb/>in diuer&longs;as enim partes mouentur. </s> <s id="id.2.1.233.30.1.3.0">&longs;imiliter dum vectis MN mo<lb/>uet CDEF, conatur mouere C &longs;uper circumferentiam CP; at­<lb/>que ideo in hoc etiam vtroq; motu &longs;imilis oritur repugnantia. </s> <s id="id.2.1.233.30.1.4.0"><lb/>quoniam autem circumferentia CO propior e&longs;t circumferentiæ <lb/>CL, quam &longs;it CP; hoc e&longs;t propior e&longs;t motui, quem facit pun­<lb/>ctum C ponderis; ideo minor erit repugnantia inter motum vectis <pb n="117" xlink:href="036/01/247.jpg"/>AB, & motum C ponderis, quàm inter motum vectis MN, & <lb/>motum eiu&longs;dem C. quod etiam patet, &longs;i intelligatur CF hori­<lb/>zonti perpendicularis, tunc enim circumferentia CP magis ten<lb/>dit deor&longs;um, quàm CO; & CL tendit &longs;ur&longs;um. </s> <s id="id.2.1.233.30.1.5.0">& ideo minor fit re <lb/>pugnantia inter vectem AB, & motum C, quàm inter <expan abbr="vect&etilde;">vectem</expan> MN, & <lb/>motum C. &longs;ed vbi minor repugnantia ibi maior facilitas. </s> <s id="id.2.1.233.30.1.6.0">ergo faci<lb/>lius mouebitur CD EF vecte AB, quàm vecte MN. quod demon<lb/>&longs;trare oportebat. </s> </p> <p id="id.2.1.233.31.0.0.0" type="head"> <s id="id.2.1.233.31.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.233.32.0.0.0" type="main"> <s id="id.2.1.233.32.1.1.0">Ex hoc manife&longs;tum e&longs;t, quò minor e&longs;t an­<lb/>gulus à linea CF, vel CE, vel CD contentus; <lb/>hoc e&longs;t, quò minor e&longs;t angulus BCF, vel BCE, <lb/>vel etiam BCD, eò facilius pondus moueri. </s> <s id="id.2.1.233.32.1.2.0"><lb/>quod quidem eodem modo o&longs;tendetur. </s> </p> <p id="id.2.1.233.33.0.0.0" type="main"> <s id="id.2.1.233.33.1.1.0">Quod autem propo&longs;itum e&longs;t, &longs;ic demon­<lb/>&longs;trabimus. </s> </p> <p id="id.2.1.233.34.0.0.0" type="main"> <s id="id.2.1.233.34.1.1.0">Sint cunei ABC DE <lb/>F, & angulus ABC mi­<lb/>nor &longs;it angulo DEF, & <lb/>AB BC DE EF &longs;int in <lb/>ter &longs;e &longs;e æquales. </s> <s id="id.2.1.233.34.1.2.0">Sint de­<lb/>inde quatuor pondera æ­<lb/>qualia GH IL NO QR <lb/>rectangula; &longs;intq; LM <lb/>kH in eadem recta linea: <lb/><figure id="id.036.01.247.1.jpg" xlink:href="036/01/247/1.jpg"></figure><lb/>&longs;imiliter RS PO in recta linea; erunt GK IM parallelæ, & NP <arrow.to.target n="note319"></arrow.to.target><lb/>QS parallelæ. </s> <s id="id.2.1.233.34.1.3.0">&longs;it IBG pars cunei intra pondera GH IL; & cu<lb/>nei pars QEN intra pondera NO QR; &longs;intqué IB BG QE <lb/>EN inter &longs;e &longs;e æquales. </s> <s id="id.2.1.233.34.1.4.0">dico pondera GH IL facilius ab eadem <pb xlink:href="036/01/248.jpg"/>potentia moueri cuneo <lb/>ABC, quàm pondera <lb/>NO QR cuneo DEF. </s> </p> <p id="id.2.1.234.1.0.0.0" type="margin"> <s id="id.2.1.234.1.1.1.0"><margin.target id="note319"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 28 <emph type="italics"/>primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.235.1.0.0.0" type="main"> <s id="id.2.1.235.1.1.1.0">Diuidantur AC DF <lb/>bifariam in TV, iungan<lb/>turq; TBVE, erunt an­<lb/>guli ad T, & V recti. </s> <s id="id.2.1.235.1.1.2.0">con<lb/>nectatur IG, quæ &longs;ecet <lb/>BT in X. </s> <s id="id.2.1.235.1.1.2.0.a">Quoniam e­<lb/><figure id="id.036.01.248.1.jpg" xlink:href="036/01/248/1.jpg"></figure><lb/>nim IB e&longs;t æqualis BG, & BA æqualis BC; erit IA ip&longs;i GC <lb/><arrow.to.target n="note320"></arrow.to.target>æqualis. </s> <s id="id.2.1.235.1.1.3.0">quare vt BI ad IA, ita e&longs;t BG ad GC. </s> <s id="id.2.1.235.1.1.3.0.a">parallela igitur <lb/><arrow.to.target n="note321"></arrow.to.target>e&longs;t IG ip&longs;i AC. ac propterea anguli ad X &longs;unt recti: &longs;ed & an<lb/><arrow.to.target n="note322"></arrow.to.target>guli XG k XIM &longs;unt recti, rectangulum enim e&longs;t GM; quare <lb/>TB æquidi&longs;tans e&longs;t ip&longs;is Gk IM. angulus igitur TBC æqua­<lb/>lis e&longs;t angulo BGK, & TBA ip&longs;i BIM æqualis. </s> <s id="id.2.1.235.1.1.4.0">&longs;imiliter demon<lb/>&longs;trabimus angulum VEF æqualem e&longs;&longs;e ENP, & VED æqualem <lb/>EQS. cùm autem angulus ABC minor &longs;it angulo DEF; erit <lb/>& angulus TBC minor VEN. quare & BGk minor ENP. <lb/>&longs;imili modo BIM minor EQS. </s> <s id="id.2.1.235.1.1.4.0.a">quoniam autem cuneus ABC <lb/>duobus mouet vectibus AB BC, quorum fulcimenta &longs;unt in B; <lb/>& pondera in GI: &longs;imiliter cuneus DEF duobus vectibus mouet <lb/>DE EF, quorum fulcimenta &longs;unt in E; & pondera in N Q: per <lb/>præcedentem pondera GH IL facilius vectibus AB BC mo­<lb/>uebuntur, quàm pondera NO QR vectibus DE EF. </s> <s id="id.2.1.235.1.1.4.0.b">ponde­<lb/>ra ergo GH IL facilius cuneo ABC mouebuntur, quàm ponde­<lb/>ra NO QR cuneo DEF. </s> <s id="id.2.1.235.1.1.4.0.c">& quia eadem e&longs;t ratio in mouendo, <lb/>atq; in &longs;cindendo; facilius idcirco aliquod cuneo ABC &longs;cindetur <lb/>quàm cuneo DEF. &longs;imiliterq; o&longs;tendetur, quò minor e&longs;t angu<lb/>lus ad verticem cunei, eò facilius aliquod moueri, vel &longs;cindi. </s> <s id="id.2.1.235.1.1.5.0">quod <lb/>demon&longs;trare oportebat. </s> </p> <p id="id.2.1.236.1.0.0.0" type="margin"> <s id="id.2.1.236.1.1.1.0"><margin.target id="note320"></margin.target>2 <emph type="italics"/>Sexti.<emph.end type="italics"/></s> <s id="id.2.1.236.1.1.2.0"><margin.target id="note321"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 29 <emph type="italics"/>primi.<emph.end type="italics"/></s> <s id="id.2.1.236.1.1.3.0"><margin.target id="note322"></margin.target>28 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.237.1.0.0.0" type="main"> <s id="id.2.1.237.1.1.1.0">Præterea quæ mouentur à cuneo DEF, per maiora mouentur <lb/>&longs;patia; quàm ea, quæ à cuneo ABC. nam vt DF &longs;it intra QN, <lb/>& AC &longs;it intra IG; nece&longs;&longs;e e&longs;t, vt QN per &longs;patia moueantur <lb/>maiora; &longs;cilicet vnum dextror&longs;um, alter &longs;ini&longs;tror&longs;um, quàm IG; <lb/>cùm DF maior &longs;it AC; dummodo totus cuneus intra pondera in­<pb n="118" xlink:href="036/01/249.jpg"/>grediatur. </s> <s id="id.2.1.237.1.1.2.0">à potentia verò facilius eodem tempore mouetur ali­<lb/>quod per minus &longs;patium, quàm per maius; dummodo cætera, qui­<lb/>bus fit motus, &longs;int æqualia: &longs;i ergo eodem tempore AC DF in <lb/>IG QN <expan abbr="perueniãt">perueniant</expan>, cùm AI CG DQ FN &longs;int inter &longs;e &longs;e æqua<lb/>les; facilius à potentia mouebuntur GI cuneo ABC, quàm QN <lb/>cuneo DEF. quare facilius pondera GH IL à potentia mouebun<lb/>tur cuneo ABC, quàm pondera NO QR cuneo DEF. <expan abbr="&longs;imiliter­qué">&longs;imiliter­<lb/>que</expan> o&longs;tendetur, quò angulus ad verticem cunei minor e&longs;&longs;et, eò fa<lb/>cilius pondera moueri, vel &longs;cindi. </s> </p> <p id="id.2.1.237.2.0.0.0" type="main"> <s id="id.2.1.237.2.1.1.0">Secundum, quod efficit, vt aliquod facilius <lb/>&longs;cindatur, e&longs;t percu&longs;sio; qua cuneus mouetur, & <lb/>mouet; hoc e&longs;t percutitur, ac &longs;cindit. <figure id="id.036.01.249.1.jpg" xlink:href="036/01/249/1.jpg"></figure></s> </p> <p id="id.2.1.237.3.0.0.0" type="main"> <s id="id.2.1.237.3.1.1.0">Sit cuneus A, quod &longs;cinditur B, quod <lb/>percutit C; quod quidem, vel ex &longs;e ip&longs;o, <lb/>vel à regente, atq; ip&longs;um mouente poten<lb/>tia percutit, atq; mouet. </s> <s id="id.2.1.237.3.1.2.0">&longs;i quidem ex <lb/>&longs;e ip&longs;o, Primùm quò grauius erit, eò <lb/>maior fiet percu&longs;sio. </s> <s id="id.2.1.237.3.1.3.0">quinetiam, quò <lb/>longior fuerit di&longs;tantia inter AC, maior <lb/>itidem fiet percu&longs;sio. </s> <s id="id.2.1.237.3.1.4.0">graue enim vnum­<lb/>quodq; dum mouetur; grauitatis ma­<lb/>gis a&longs;&longs;umit motum, quàm quie&longs;cens: & <lb/>adhuc magis quo longius mouetur. <figure id="fig209" place="text" xlink:href=""></figure></s> </p> <pb xlink:href="036/01/250.jpg"/> <p id="id.2.1.237.5.0.0.0" type="main"> <s id="id.2.1.237.5.1.1.0">Si verò C ab aliqua moueatur po<lb/>tentia, vt &longs;i per manubrium DE mo<lb/>ueatur; primùm quò grauius erit C, <lb/>deinde quò longius erit DE, eò ma­<lb/>ior fiet percu&longs;sio. </s> <s id="id.2.1.237.5.1.2.0">&longs;i enim ponatur po<lb/>tentia mouens in E, erit C magis di <lb/>&longs;tans à centro & ideo citius mouebi<lb/>tur. </s> <s id="id.2.1.237.5.1.3.0">vt in quæ&longs;tionibus Mechanicis <lb/>latè mon&longs;trat Ari&longs;toteles; nec non <lb/>ex iis, quæ in tractatu de libra di­<lb/>cta fuere, patere pote&longs;t, quò magis <lb/><figure id="id.036.01.250.1.jpg" xlink:href="036/01/250/1.jpg"></figure><lb/>pondus Cà centro di&longs;tat, eò grauius reddi. </s> <s id="id.2.1.237.5.1.4.0">quod ip&longs;um etiam va<lb/>lidiori pellet impul&longs;u virtute in E potentiore exi&longs;tente. </s> </p> <p id="id.2.1.237.6.0.0.0" type="main"> <s id="id.2.1.237.6.1.1.0">Hoc verò &longs;ecundùm e&longs;t, quod efficit, vt hoc in&longs;trumento ma­<lb/>gna moueantur, &longs;cindanturq; pondera. </s> <s id="id.2.1.237.6.1.2.0">percu&longs;sio enim vis e&longs;t ua<lb/>lidi&longs;sima, vt ex decimanona <expan abbr="quæ&longs;tionū">quæ&longs;tionum</expan> Mechanicarum Ari&longs;totelis <lb/>patet. </s> <s id="id.2.1.237.6.1.3.0">&longs;i enim &longs;upra cuneum maximum imponatur onus; tunc cu­<lb/>neus nihil ferè efficiet, præ&longs;ertim ictus comparatione. </s> <s id="id.2.1.237.6.1.4.0">quod &longs;i ad <lb/>huc ip&longs;i cuneo vectem, vel cochleam, vel quoduis aliud huiu&longs;mo<lb/>di aptetur in&longs;trumentum ad cuneum ponderi intimius propellen­<lb/>dum, nullius ferè momenti præ ictu continget effectus. </s> <s id="id.2.1.237.6.1.5.0">cuius qui­<pb n="119" xlink:href="036/01/251.jpg"/>dem rei indicio e&longs;&longs;e pote&longs;t, &longs;i fuerit <lb/>corpus A <expan abbr="lapideũ">lapideum</expan>, ex quo aliquam eius <lb/>partem detrahere qui&longs;piam voluerit, pu<lb/>tá partem anguli B; tunc malleo ferreo <lb/>ab&longs;q; alio in&longs;trumento percutiendo in B, <lb/>facilè aliquam anguli B partem franget. </s> <s id="id.2.1.237.6.1.6.0"><lb/>quod quidem nullo alio in&longs;trumento <lb/>percu&longs;sionis munere carente, ni&longs;i maxi<lb/>ma cùm difficultate efficere poterit; &longs;iue <lb/><figure id="id.036.01.251.1.jpg" xlink:href="036/01/251/1.jpg"></figure><lb/>fuerit vectis, &longs;iue cochlea, &longs;iue quoduis aliud huiu&longs;modi. </s> <s id="id.2.1.237.6.1.7.0">quare <lb/>percu&longs;sio in cau&longs;a e&longs;t, quo magna &longs;cindantur pondera. </s> <s id="id.2.1.237.6.1.8.0">cùm autem <lb/>&longs;ola percu&longs;sio tantam vim habeat, &longs;i ei aliquod adiiciamus in&longs;tru<lb/>mentum ad mouendum, &longs;cindendumq; accomodatum, admiran<lb/>da profectò videbimus. </s> <s id="id.2.1.237.6.1.9.0">In&longs;trumentum huiu&longs; <lb/>modi cuneus e&longs;t, in quo duo (quantum ad ip­<lb/>&longs;ius formam attinet) con&longs;ideranda occurrunt. </s> <s id="id.2.1.237.6.1.10.0"><lb/>Alterum e&longs;t, cuneum ad &longs;u&longs;cipiendam, &longs;u&longs;tinen<lb/>damq; percu&longs;sionem apti&longs;simum e&longs;&longs;e; alterum <lb/>e&longs;t quòd propter eius in altera parte &longs;ubtilita­<lb/>tem facilè intra corpora ingreditur, vt manife<lb/>&longs;tè patet. </s> <s id="id.2.1.237.6.1.11.0">Cuneus ergo cum percu&longs;sione ip&longs;ius <lb/>efficit, vt in mouendis, &longs;cindendi&longs;q; ponderi­<lb/>bus ferè miracula cernamus. <figure id="id.036.01.251.2.jpg" xlink:href="036/01/251/2.jpg"></figure></s> </p> <pb xlink:href="036/01/252.jpg"/> <p id="id.2.1.237.8.0.0.0" type="main"> <s id="id.2.1.237.8.1.1.0">Ad huiu&longs;modi facultatis in&longs;trumentum, ea <lb/>quoquè omnia commodè referri po&longs;&longs;unt, quæ <lb/>percu&longs;sione, &longs;iue impul&longs;u incidunt, diuidunt, <lb/>perforant, huiu&longs;modiq; alia obeunt munera. </s> <s id="id.2.1.237.8.1.2.0">vt <lb/>en&longs;es, gladii, mucrones, &longs;ecures, & &longs;imilia. </s> <s id="id.2.1.237.8.1.3.0">&longs;erra <lb/>quoq; ad hoc reducetur; dentes enim percu­<lb/>tiunt, cuneiq; in&longs;tar exi&longs;tunt. </s> </p> </chap> <pb n="120" xlink:href="036/01/253.jpg"/> <chap> <p id="id.2.1.237.9.0.0.0" type="head"> <s id="id.2.1.237.10.1.1.0">DE COCHLEA. </s> </p> <p id="id.2.1.237.11.0.0.0" type="main"> <s id="id.2.1.237.11.1.1.0">Pappvs in eodem octauo libro <lb/>multa pertractans de cochlea, do<lb/>cet quomodo conficienda &longs;it; & <lb/>quomodo magna huiu&longs;modi in­<lb/>&longs;trumento moueantur pondera; <lb/>nec non alia theoremata ad eius <lb/>cognitionem valdè vtilia. </s> <s id="id.2.1.237.11.1.2.0">Quoniam autem in­<lb/>ter cætera pollicetur, &longs;e o&longs;tendere velle, co­<lb/>chleam nihil aliud e&longs;&longs;e præter a&longs;&longs;umptum cu­<lb/>neum percu&longs;sionis expertem vecte motionem <lb/>facientem; hoc autem in ip&longs;o de&longs;ideratur; pro­<lb/>pterea idip&longs;um o&longs;tendere conabimur, nec non <lb/>eiu&longs;dem cochleæ ad vectem, libramq; reductio­<lb/>nem; vt ip&longs;ius tandem completa habeatur co­<lb/>gnitio. <pb xlink:href="036/01/254.jpg"/><figure id="id.036.01.254.1.jpg" xlink:href="036/01/254/1.jpg"></figure></s> </p> <p id="id.2.1.237.12.0.0.0" type="main"> <s id="id.2.1.237.12.1.1.0">Sit cuneus ABC, qui circa cylindrum DE circumuoluatur:&longs;itq; <lb/>IGH cuneus circa cylindrum reuolutus, cuius vertex &longs;it I. </s> <s id="id.2.1.237.12.1.1.0.a">&longs;it de­<lb/>inde cylindrus cum circumpo&longs;ito cuneo ita accomodatus, vt ab&longs;q; <lb/>vllo <expan abbr="impedim&etilde;to">impedimento</expan> manubrio kF eius axi annexo circumuerti po&longs;sit. </s> <s id="id.2.1.237.12.1.2.0"><lb/>&longs;itq; LMNO, quod &longs;cindendum e&longs;t; quod etiam ex parte MN <lb/>&longs;it immobile: vt in iis, quæ &longs;cinduntur, fieri &longs;olet: & &longs;it vertex <lb/>I intra RS. circumuertatur kF, & perueniat ad kP; dum autem kF <lb/>circumuertitur, circumuertitur etiam totus cylindrus DE, & cu­<lb/>neus IGH: quare dum KF erit in kP, vertex I non erit amplius <lb/>intra RS, &longs;ed cunei pars alia, vt TV: &longs;ed TV maior e&longs;t, quàm <lb/>RS; &longs;emper enim pars cunei, quæ magis à vertice di&longs;tat, maior <lb/>e&longs;t ea, quæ ip&longs;i e&longs;t propinquior: vt igitur TV &longs;it intra RS, opor­<lb/>tet, vt R cedat, moueaturq; ver&longs;us X, & S ver&longs;us Z, vt faciunt <lb/>ea, quæ &longs;cinduntur. </s> <s id="id.2.1.237.12.1.3.0">totum ergo LMNO &longs;cindetur. </s> <s id="id.2.1.237.12.1.4.0">&longs;imiliter <lb/>què demon&longs;trabimus, dum manubrium kP erit in kQ, tunc GH <lb/>e&longs;&longs;e intra RS: & vt GH &longs;it intra RS, nece&longs;&longs;e e&longs;t, vt R &longs;it in X, <lb/>& S in Z; ita vt <emph type="italics"/>X<emph.end type="italics"/>Z &longs;it æqualis GH; &longs;emperq; LMNO amplius <lb/>&longs;cindetur. </s> <s id="id.2.1.237.12.1.5.0">&longs;ic igitur patet, dum kF circumuertitur, &longs;emper R moue <lb/>ri ver&longs;us X, atq; S ver&longs;us Z: & R &longs;emper &longs;uper ITG moueri, S au<lb/>tem &longs;uper IVH, hoc e&longs;t &longs;uper latera cunei circa cylindrum circum <lb/>uoluti. </s> </p> <pb n="121" xlink:href="036/01/255.jpg"/> <p id="id.2.1.237.13.0.0.0" type="head"> <s id="id.2.1.237.14.1.1.0">PROPOSITIO I. </s> </p> <p id="id.2.1.237.15.0.0.0" type="main"> <s id="id.2.1.237.15.1.1.0">Cuneus hoc modo circa cylindrum accommo­<lb/>datus, nihil e&longs;t aliud; ni&longs;i cochlea duas habens he<lb/>lices in vnico puncto inuicem coniunctas. <figure id="id.036.01.255.1.jpg" xlink:href="036/01/255/1.jpg"></figure></s> </p> <p id="id.2.1.237.16.0.0.0" type="main"> <s id="id.2.1.237.16.1.1.0">Sit cuneus ABC; & AB <lb/>ip&longs;i BC æqualis. </s> <s id="id.2.1.237.16.1.2.0">diuidatur <lb/>AC bifariam in D, iunga<lb/>turq; BD; erit BD ip&longs;i AC <lb/>perpendicularis; & AD <lb/>ip&longs;i DC æqualis, triangu­<lb/>lumq; ABD triangulo C <lb/>BD æquale. </s> <s id="id.2.1.237.16.1.3.0">fiant deinde <lb/>triangula rectangula EFG <lb/>HIk non &longs;olum inter &longs;e, <lb/>verùm etiam vtriq; ADB <lb/>& CDB æqualia. </s> <s id="id.2.1.237.16.1.4.0">&longs;itq; cy<lb/>lindrus LMNO, cuius perimeter &longs;it æqualis vtriq; FG kI. & <lb/>LMNO &longs;it parallelogrammum per axem. </s> <s id="id.2.1.237.16.1.5.0">fiatq; MP æqualis <lb/>FE; & PN æqualis HI. ponaturq; HI in NP, circumuolua­<lb/>turq; triangulum HIk circa cylindrum; & &longs;ecundùm kH helix <lb/>de&longs;cribatur NQP, vt Pappus quoq; docet in octauo libro propo<lb/>&longs;itione vige&longs;ima quarta. </s> <s id="id.2.1.237.16.1.6.0">&longs;imiliter ponatur EF in MP, circum­<lb/>uoluaturq; triangulum EFG circa cylindrum; de&longs;cribaturq; per <lb/>EG helix PRM. cùm itaq; PMPN &longs;int æquales EFHI, erit <lb/>MN æqualis ip&longs;i AC, & cùm helices PRM PQN &longs;int æquales <lb/>lineis EGHk; helices igitur ip&longs;is ABBC æquales erunt. </s> <s id="id.2.1.237.16.1.7.0">cu­<lb/>neus ergo ABC totus circumuolutus erit circa cylindrum LMNO. <pb xlink:href="036/01/256.jpg"/>incidantur deinde helices, <lb/>vt docet Pappus &longs;ecundùm <lb/>latitudinem cunei; & hoc <lb/>modo cuneus vná cum cy<lb/>lindro nihil aliud erit, <lb/>quàm cochlea duas habens <lb/>helices PRMPQN cir<lb/>ca cylindrum LN in vnico <lb/>puncto P inuicem coniun<lb/>ctas. </s> <s id="id.2.1.237.16.1.8.0">quod demon&longs;trare o­<lb/>portebat. </s> </p> <figure id="id.036.01.256.1.jpg" xlink:href="036/01/256/1.jpg"> </figure> <p id="id.2.1.237.16.3.1.0" type="head"> <s id="id.2.1.237.16.5.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.237.17.0.0.0" type="main"> <s id="id.2.1.237.17.1.1.0">Hinc manife&longs;tum e&longs;&longs;e pote&longs;t, quomodo heli­<lb/>ces in ip&longs;a cochlea de&longs;cribi po&longs;sint. </s> </p> <p id="id.2.1.237.18.0.0.0" type="main"> <s id="id.2.1.237.18.1.1.0">Quomodo autem pondera &longs;uper helices co­<lb/>chleæ moueantur, o&longs;tendamus. <figure id="id.036.01.256.2.jpg" xlink:href="036/01/256/2.jpg"></figure></s> </p> <p id="id.2.1.237.19.0.0.0" type="main"> <s id="id.2.1.237.19.1.1.0">Sit (veluti prius) cuneus IGH circa cylindrum DE reuolutus, <lb/>cuius vertex &longs;it I. apteturq; cylindrus ita, vt liberè vna cum &longs;uo <lb/>axe circumuertatur. </s> <s id="id.2.1.237.19.1.2.0">&longs;intq; duo pondera MN cuiu&longs;cunq; figuræ <lb/>voluerimus, ita tamen aptata, vt moueri non po&longs;sint, ni&longs;i &longs;uper <pb n="122" xlink:href="036/01/257.jpg"/>rectam lineam LO, quæ axi cylindri &longs;it æquidi&longs;tans. </s> <s id="id.2.1.237.19.1.3.0">&longs;intq; MN <lb/>iuxta cunei verticem I. Circumuertatur KF, & perueniat ad kP: <lb/>dum autem kF erit in kP, tunc TV erit intra pondera MN; &longs;i­<lb/>cut &longs;upra diximus. </s> <s id="id.2.1.237.19.1.4.0">M igitur ver&longs;us L mouebitur, & N ver&longs;us O. <lb/>&longs;imiliter o&longs;tendetur, dum kP erit in KQ, tunc GH e&longs;&longs;e intra pon­<lb/>dera MN; & M erit in X, & N in Z; ita vt XZ &longs;it æqualis GH. <lb/>quare dum kF circumuertitur, &longs;emper pondus N mouetur ver&longs;us <lb/>O, & &longs;uper helicem IRS; M verò &longs;uper aliam helicem. <figure id="id.036.01.257.1.jpg" xlink:href="036/01/257/1.jpg"></figure></s> </p> <p id="id.2.1.237.20.0.0.0" type="main"> <s id="id.2.1.237.20.1.1.0">Similiter &longs;i cochlea plures habeat hæ­<lb/>lices, vt in &longs;ecunda figura, pondus A, <lb/>dum cochlea circumuertitur, &longs;emper &longs;u­<lb/>per helices BCDEFG mouebitur; <lb/>dummodo pondus A aptetur ita vt mo­<lb/>ueri non po&longs;sit, ni&longs;i &longs;uper rectam HI ip&longs;i <lb/>cylindro æquidi&longs;tantem. </s> <s id="id.2.1.237.20.1.2.0">eodem enim <lb/>modo, quo &longs;uper primam mouetur heli<lb/>cem, mouetur etiam &longs;upra &longs;ecundam, <lb/>& tertiam, & cætera. </s> <s id="id.2.1.237.20.1.3.0">quotcunq; enim <lb/>fuerint helices, nihil aliud &longs;unt, quàm <lb/>latus cunei circa idem cylindrum iterum <lb/>atq; iterum circumuolutum. </s> <s id="id.2.1.237.20.1.4.0">& &longs;iue co­<lb/>chlea fuerit horizonti perpendicularis, <lb/>&longs;iue horizonti æquidi&longs;tans, vel alio mo­<lb/>do collocata, nihil refert: &longs;emper enim <lb/>eadem erit ratio. <pb xlink:href="036/01/258.jpg"/><figure id="id.036.01.258.1.jpg" xlink:href="036/01/258/1.jpg"></figure></s> </p> <p id="id.2.1.237.21.0.0.0" type="main"> <s id="id.2.1.237.21.1.1.0">Si verò (vt in tertia figura) &longs;upra cochleam imponatur aliquod, <lb/>vt B, quod quidem tylum vocant, ita accommodatum, vt inferio <lb/>ri parte helices habeat concauas ip&longs;i cochleæ appo&longs;itè admodum <lb/>congruentes; per&longs;picuum &longs;atis e&longs;&longs;e poterit, ip&longs;um B, dum <expan abbr="coclhea">cochlea</expan> <lb/>circumuertitur, &longs;uper helices cochleæ eo pror&longs;us modo moueri; <lb/>quo pondus iuxta primam <expan abbr="figurã">figuram</expan> mouebatur: dummodo tylum ap­<lb/>tetur, vt docet Pappus in octauo libro; ita &longs;cilicet vt tantùm <expan abbr="an­tè">an­<lb/>te</expan>, retrouè axi cylindri æquidi&longs;tans moueatur. <figure id="id.036.01.258.2.jpg" xlink:href="036/01/258/2.jpg"></figure></s> </p> <p id="id.2.1.237.22.0.0.0" type="main"> <s id="id.2.1.237.22.1.1.0">Et &longs;i loco tyli, quod helices habet concauas in parte inferiori, con<lb/>&longs;tituatur, vt in quarta figura, cylindrus concauus vt D, & in eius <lb/>concaua &longs;uperficie de&longs;cribantur helices, incidanturq; ita, vt aptè <pb n="123" xlink:href="036/01/259.jpg"/>cùm cochlea congruant (eodem enim modo de&longs;cribentur helices <lb/>in &longs;uperficie concauia cylindri, &longs;icuti fit in conuexa) &longs;i deinde co­<lb/>chlea in &longs;uis polis firmetur, &longs;cilicet in &longs;uo axe, circumuertaturq;; <lb/>patet D ad motum circumuer&longs;ionis cochleæ quemmadmodum ty<lb/>lum moueri. </s> <s id="id.2.1.237.22.1.2.0">nec non &longs;i D in EF firmetur, ita vt immobilis ma <lb/>neat, dum circumuertitur cochlea; &longs;uper helices cylindri D, ad <lb/>motum &longs;uæ circumuer&longs;ionis dextror&longs;um, vel &longs;ini&longs;tror&longs;um factæ; <lb/>tùm in anteriorem, tùm in po&longs;teriorem partem mouebitur. </s> <s id="id.2.1.237.22.1.3.0">cylin­<lb/>drus autem D hoc modo <expan abbr="accõmodatus">accommodatus</expan> vulgò mater, &longs;iue cochleæ <lb/>fæmina nuncupatur. <figure id="id.036.01.259.1.jpg" xlink:href="036/01/259/1.jpg"></figure></s> </p> <p id="id.2.1.237.23.0.0.0" type="main"> <s id="id.2.1.237.23.1.1.0">Si autem cochleæ (vt in quinta figura) tympanum C dentibus <lb/>obliquis dentatum apponatur, vt docet Pappus in eodem octauo li­<lb/>bro; vel etiam rectis; ita tamen con&longs;tructis, vt facilè cum cochlea <lb/>conueniant: &longs;imiliter manife&longs;tum e&longs;t ad motum cochleæ circumuer<lb/>ti etiam tympanum C. </s> <s id="id.2.1.237.23.1.1.0.a">eodemq; modo tympani dentes &longs;uper he<lb/>lices cochleæ moueri. </s> <s id="id.2.1.237.23.1.2.0">& hæc dicitur cochlea infinita, quia & co<lb/>chlea, & tympanum dum circumuertuntur, &longs;emper eodem modo <lb/>&longs;e &longs;e habent. </s> </p> <pb xlink:href="036/01/260.jpg"/> <p id="id.2.1.237.25.0.0.0" type="main"> <s id="id.2.1.237.25.1.1.0">Hæc diximus, vt manife&longs;tum &longs;it cochleam in mouendo pondere <lb/>cunei munere ab&longs;q; percu&longs;sione fungi. </s> <s id="id.2.1.237.25.1.2.0">Illud enim remouet à loco, <lb/>vbi erat; quemadmodum cuneus remouet ea, quæ mouet, ac &longs;cindit. </s> <s id="id.2.1.237.25.1.3.0"><lb/>omnia enim hæc à cochlea mouentur, &longs;icuti pondus A in &longs;ecun­<lb/>da figura, & M in prima. </s> </p> <p id="id.2.1.237.26.0.0.0" type="main"> <s id="id.2.1.237.26.1.1.0">Quoniam autem duplici ratione mouentem cuneum con&longs;iderari <lb/>po&longs;&longs;e o&longs;tendimus, videlicet vt mouet vectibus, vel vt e&longs;t planum <lb/>horizonti inclinatum, dupliciter quoq; cochleam con&longs;iderabimus; <lb/><figure id="id.036.01.260.1.jpg" xlink:href="036/01/260/1.jpg"></figure><lb/>& primùm vt vectibus mouet, vt in prima figura circumuertatur <lb/>kF, & perueniat in KP; tunc, &longs;icut dictum e&longs;t, TV erit intra pon­<lb/>dera MN. & &longs;icut con&longs;ideramus vectes in cuneo, eodem quoq; <lb/>modo eos con&longs;iderare po&longs;&longs;umus in cochlea hoc pacto. </s> <s id="id.2.1.237.26.1.2.0">erit &longs;cilicet <lb/>IVH vectis, cuius fulcimentum I, & pondus in V. &longs;imiliter ITG ve<lb/>ctis, cuius fulcimentum I, & pondus in T. </s> <s id="id.2.1.237.26.1.2.0.a">potentiæ verò mo­<lb/>uentes GH e&longs;&longs;e deberent; &longs;ed &longs;icuti in cuneo potentia mouens <lb/>e&longs;t percu&longs;sio, quæ mouet cuneum; idcirco erit, ubi potentia mo­<lb/>uet cochleam; &longs;cilicet in P manubrio kP. cochlea enim &longs;ine per­<lb/>cu&longs;sione mouetur. </s> <s id="id.2.1.237.26.1.3.0">Hæc autem con&longs;ideratio propter vectes infle­<lb/>xos impropria for&longs;itan e&longs;&longs;e videbitur; Quocirca &longs;i id, quod moue<lb/>tur à cochlea, &longs;upra planum horizonti inclinatum moueri intelli<lb/>gatur; erit quidem huiu&longs;modi con&longs;ideratio (cùm ip&longs;i quoq; cuneo <lb/>conueniat) figuræ ip&longs;ius cochleæ magis conformis. </s> </p> <pb n="124" xlink:href="036/01/261.jpg"/> <p id="id.2.1.237.27.0.0.0" type="head"> <s id="id.2.1.237.28.1.1.0">PROPOSITIO II. </s> </p> <p id="id.2.1.237.29.0.0.0" type="main"> <s id="id.2.1.237.29.1.1.0">Si fuerit cochlea AB helices habens æquales <lb/>CDEFG. </s> <s id="id.2.1.237.29.1.1.0.a">Dico has nihil aliud e&longs;&longs;e præter pla<lb/>num horizonti inclinatum circa cylindrum re­<lb/>uolutum. <figure id="id.036.01.261.1.jpg" xlink:href="036/01/261/1.jpg"></figure></s> </p> <p id="id.2.1.237.30.0.0.0" type="main"> <s id="id.2.1.237.30.1.1.0">Sit cochlea AB horizonti perpendicularis duas habens helices <lb/>CDEFG. exponatur HI æqualis GC, quæ bifariam diui­<lb/>datur in k; erunt Hk kI non &longs;olum inter &longs;e &longs;e, verùm etiam <lb/>ip&longs;is GE EC æquales, & ip&longs;i HI ad rectos angulos ducatur LI; <lb/>& per LI intelligatur planum horizonti æquidi&longs;tans; &longs;itq; LI du<lb/>pla perimetro cylindri AB, quæ bifariam diuidatur in M; erunt <lb/>IM ML cylindri perimetro æquales. </s> <s id="id.2.1.237.30.1.2.0">connectatur HL, & à pun<lb/>cto M ducatur MN ip&longs;i HI æquidi&longs;tans, coniungaturq; KN. quo<lb/>niam enim &longs;imilia &longs;unt inter &longs;e &longs;e triangula HILNML, cùm <arrow.to.target n="note323"></arrow.to.target><pb xlink:href="036/01/262.jpg"/><figure id="id.036.01.262.1.jpg" xlink:href="036/01/262/1.jpg"></figure><lb/>NM &longs;it æquidi&longs;tans HI; erit LI ad IH, vt LM ad MN: & <lb/>permutando vt IL ad LM; ita HI ad NM. </s> <s id="id.2.1.237.30.1.2.0.a">&longs;ed IL dupla e&longs;t ip&longs;ius <lb/>LM; ergo & HI dupla erit MN. &longs;ed e&longs;t etiam dupla ip&longs;ius kI, <lb/>quare kI NM inter &longs;e æquales erunt. </s> <s id="id.2.1.237.30.1.3.0">& quoniam anguli ad MI <lb/>&longs;unt recti; erit kM parallelogrammum rectangulum, & kN æqua <lb/>lis erit IM. quare KN perimetro cylindri AB æqualis erit. </s> <s id="id.2.1.237.30.1.4.0">pona<lb/>tur itaq; HI in GC, erit Hk in GE. circumuoluatur deinde trian<lb/>gulum HkN circa cylindrum AB, de&longs;cribet HN helicen GFE; <lb/>cùm NK perimetro cylindri &longs;it æqualis; & punctum N erit in E; <lb/>& MN in CE. & quia ML æqualis e&longs;t perimetro cylindri; cir­<lb/>cumuoluatur rur&longs;us triangulum NML circa cylindrum AB, NL <lb/>de&longs;cribet helicen EDC. quare tota LH duas de&longs;cribet helices <lb/>CDEFG. patet igitur has helices cochleæ nihil aliud e&longs;&longs;e, ni­<lb/>&longs;i planum horizonti inclinatum; cuius inclinatio e&longs;t angulus HLI <lb/>circa cylindrum circumuolutum, &longs;upra quod pondus mouetur. </s> <lb/> <s id="id.2.1.237.30.1.5.0"><lb/>quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.238.1.0.0.0" type="margin"> <s id="id.2.1.238.1.1.1.0"><margin.target id="note323"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 4. <emph type="italics"/>&longs;exti.<emph.end type="italics"/></s> </p> <p id="id.2.1.239.1.0.0.0" type="main"> <s id="id.2.1.239.1.1.1.0">Quomodo autem hoc ad libram reducatur <expan abbr="mnnife&longs;tum">manifestum</expan> e&longs;t ex <lb/>nona octaui libri eiu&longs;dem Pappi. </s> </p> <pb n="125" xlink:href="036/01/263.jpg"/> <p id="id.2.1.239.3.0.0.0" type="main"> <s id="id.2.1.239.3.1.1.0">Po&longs;tquam vidimus quomodo pondera huiu&longs;modi moueantur <lb/>in&longs;trumento; nunc con&longs;iderandum e&longs;t, quæ nam &longs;int ea, quæ effi<lb/>ciunt, vt pondera facilè moueantur: hæc autem duo &longs;unt. </s> </p> <p id="id.2.1.239.4.0.0.0" type="main"> <s id="id.2.1.239.4.1.1.0">Primùm quidem, quod efficit, vt facilè pon­<lb/>dus moueatur, quod etiam ad e&longs;&longs;entiam cochleæ <lb/>magis pertinere videtur; e&longs;t helix circa co­<lb/>chleam. </s> <s id="id.2.1.239.4.1.2.0">vt &longs;i circa datam cochleam AB duæ <lb/>&longs;int helices inæquales CDA EFG, &longs;itq; AC mi <lb/>nor EG. </s> <s id="id.2.1.239.4.1.2.0.a">Dico idem pondus facilius &longs;uper heli<lb/>cen CDA moueri, quàm &longs;uper EFG. <lb/><figure id="id.036.01.263.1.jpg" xlink:href="036/01/263/1.jpg"></figure></s> </p> <p id="id.2.1.239.5.0.0.0" type="main"> <s id="id.2.1.239.5.1.1.0">Compleatur cuneus <lb/>ADCHI, hoc e&longs;t de­<lb/>&longs;cribatur helix CHI <lb/>æqualis CDA, & ver­<lb/>tex cunei &longs;it C. &longs;imili <lb/>ter compleatur cuneus <lb/>GFEKL, cuius ver­<lb/>tex E. exponatur de­<lb/>inde recta linea MN, <lb/>quæ &longs;it ip&longs;i AC æqua­<lb/>lis, cui ad rectos angu<lb/>los ducatur NP, quæ &longs;it <lb/>æqualis perimetro cy­<lb/>lindri AB: & conne­<lb/>ctatur <arrow.to.target n="note324"></arrow.to.target>PM; erit PM, <lb/>per ea, quæ dicta &longs;unt, <lb/>ip&longs;i CDA æqualis. </s> <s id="id.2.1.239.5.1.2.0"><lb/>producatur deinde M <lb/>N in O, fiatq; ON æ­<lb/>qualis MN, coniunga<lb/>turq; OP; erit OPM cuneus cuneo ADCHI æqualis. </s> <s id="id.2.1.239.5.1.3.0">&longs;imili-<arrow.to.target n="note325"></arrow.to.target><pb xlink:href="036/01/264.jpg"/>terq; exponatur cu­<lb/>neus STQ æqualis cu<lb/>neo GFEkL; erit TR <lb/>ip&longs;i PN, & perime­<lb/>tro cylindri æqualis; & <lb/>QR æqualis GE. <lb/>cùm autem GE ma­<lb/>ior &longs;it AC; erit & RQ <lb/>maior MN. &longs;ecetur <lb/>RQ in V; fiatq; RV <lb/>ip&longs;i MN æqualis, & <lb/>coniungatur TV; erit <lb/>triangulum TVR tri­<lb/>angulo MPN æquale: <lb/>duæ enim TR RV <lb/>duabus PN NM &longs;unt <lb/>æquales, & anguli, <lb/>quos continent, &longs;unt <lb/>æquales, nempe recti; <lb/><arrow.to.target n="note326"></arrow.to.target>angulus igitur RTV <lb/><figure id="id.036.01.264.1.jpg" xlink:href="036/01/264/1.jpg"></figure><lb/>angulo NPM æqualis erit. </s> <s id="id.2.1.239.5.1.4.0">quare angulus MPN minor e&longs;t angu­<lb/>lo QTR; & horum dupli, angulus &longs;cilicet MPO minor angulo <lb/>QTS. </s> <s id="id.2.1.239.5.1.4.0.a">quoniam autem cuneus, qui angulum ad verticem mino <lb/>rem habet, facilius mouet, ac &longs;cindit, quàm qui habet maiorem; <lb/>cuneus ergo MPO facilius mouebit, quàm QTS. </s> <s id="id.2.1.239.5.1.4.0.b">facilius igitur <lb/>pondus à cuneo ADCHI mouebitur, quàm à cuneo GFEkL. </s> <s id="id.2.1.239.5.1.4.0.c"><lb/>pondus ergo &longs;uper helicen CDA facilius mouebitur, quàm &longs;uper <lb/>EFG. eodemq; modo o&longs;tendetur, quò minor erit AC, eò faci­<lb/>lius pondus moueri. </s> <s id="id.2.1.239.5.1.5.0">quod demon&longs;trare oportebat. </s> </p> <p id="id.2.1.240.1.0.0.0" type="margin"> <s id="id.2.1.240.1.1.1.0"><margin.target id="note324"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.240.1.1.2.0"><margin.target id="note325"></margin.target>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s> <s id="id.2.1.240.1.1.3.0"><margin.target id="note326"></margin.target>4 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <pb n="126" xlink:href="036/01/265.jpg"/> <figure id="id.036.01.265.1.jpg" xlink:href="036/01/265/1.jpg"> </figure> <p id="id.2.1.241.1.2.1.0" type="head"> <s id="id.2.1.241.1.4.1.0">ALITER. </s> </p> <p id="id.2.1.241.2.0.0.0" type="main"> <s id="id.2.1.241.2.1.1.0">Sit data cochlea AB duas habens helices æquales CDEFG; &longs;it <lb/>deinde alius cylindrus <foreign lang="greek">ab</foreign>ip&longs;i AB æqualis, in quo &longs;ummatur OP ip<lb/>&longs;i CG æqualis; diuidaturq; OP in tres partes æquales OR RT <lb/>TP, & tres de&longs;cribantur helices OQRSTVP; erit vnaquæq; OR RT <lb/>TP minor CE, & EG: tertia enim pars minor e&longs;t dimidia. </s> <s id="id.2.1.241.2.1.2.0">dico <lb/>idem pondus facilius &longs;uper helices OQRSTVP moueri, quàm &longs;u<lb/>per CDEFG. exponatur HIL triangulum orthogonium, ita vt <lb/>HI &longs;it ip&longs;i CG æqualis, & IL duplo perimetri cylindri AB æqua<lb/>lis, & per <emph type="italics"/>L<emph.end type="italics"/>I intelligatur planum horizonti æqui&longs;tans; erit H<emph type="italics"/>L<emph.end type="italics"/> <lb/>æqualis CDEFG; & H<emph type="italics"/>L<emph.end type="italics"/>I inclinationis angulus erit. </s> <s id="id.2.1.241.2.1.3.0">exponatur <arrow.to.target n="note327"></arrow.to.target><lb/>&longs;imiliter <emph type="italics"/>X<emph.end type="italics"/>YZ triangulum orthogonium, ita vt XZ ip&longs;i OP &longs;it æ­<lb/>qualis, quæ etiam æqualis erit CG, & HI; &longs;itq; ZY cylindri pe­<lb/>rimetro tripla, erit XY æqualis OQRSTVP. diuidatur ZY in <pb xlink:href="036/01/266.jpg"/><figure id="id.036.01.266.1.jpg" xlink:href="036/01/266/1.jpg"></figure><lb/>tres partes æquales in <foreign lang="greek">g</foreign><35>; erit vnàquæq; Z <foreign lang="greek">g g</foreign><35> <35> Y perimetro cy<lb/>lindri <foreign lang="greek">ab</foreign>æqualis, quæ <expan abbr="etiã">etiam</expan> perimetro cylindri AB æquales erunt; & <lb/>per con&longs;equens ip&longs;is IM, & ML. connectatur X<35>. </s> <s id="id.2.1.241.2.1.4.0">& quoniam <lb/>duæ HI IL duabus XZ Z<35> &longs;unt æquales, & angulus HIL re­<lb/>ctus æqualis e&longs;t angulo XZ<35> recto; erit triangulum HIL trian­<lb/>gulo XZ<35> æquale; & angulus HLI angulo X<35>Z æqualis; & <lb/><arrow.to.target n="note328"></arrow.to.target>X<35> ip&longs;i HL æqualis. </s> <s id="id.2.1.241.2.1.5.0">&longs;ed quoniam angulus X<35>Z maior e&longs;t angu<lb/>lo <emph type="italics"/>X<emph.end type="italics"/>YZ; erit angulus HLI angulo <emph type="italics"/>X<emph.end type="italics"/>YZ maior. </s> <s id="id.2.1.241.2.1.6.0">ac propterea <expan abbr="planũ">planum</expan> <lb/>HL magis horizonti inclinat, quàm XY. quare <expan abbr="id&etilde;">idem</expan> <expan abbr="põdus">pondus</expan> à minore <lb/>potentia &longs;uper <expan abbr="planũ">planum</expan> XY, quàm &longs;uper <expan abbr="planũ">planum</expan> HL mouebitur; vt faci <lb/>lè elicitur ex <expan abbr="ead&etilde;">eadem</expan> nona Pappi. </s> <s id="id.2.1.241.2.1.7.0">cùm <expan abbr="aut&etilde;">autem</expan> helices OQRSTVP nihil <lb/>aliud &longs;int, quàm <expan abbr="planũ">planum</expan> XY horizonti <expan abbr="inclinatũ">inclinatum</expan> in angulo XYZ cir<lb/>ca cylindrum <foreign lang="greek">ab</foreign>circumuolutum; & helices CDEFG nihil &longs;unt <lb/>aliud, quàm planum HL horizonti inclinatum in angulo HLI cir<lb/>ca cylindrum AB circumuolutum; facilius ergo pondus &longs;uper he­<pb n="127" xlink:href="036/01/267.jpg"/>lices OQRSTVP mouebitur, quàm &longs;uper helices CDEFG. </s> </p> <p id="id.2.1.242.1.0.0.0" type="margin"> <s id="id.2.1.242.1.1.1.0"><margin.target id="note327"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 2 <emph type="italics"/>huius.<emph.end type="italics"/></s> <s id="id.2.1.242.1.1.2.0"><margin.target id="note328"></margin.target>21 <emph type="italics"/>Primi.<emph.end type="italics"/></s> </p> <p id="id.2.1.243.1.0.0.0" type="main"> <s id="id.2.1.243.1.1.1.0">Si autem OP diuidatur in quatuor partes æquales, <expan abbr="de&longs;cribantur­què">de&longs;cribantur­<lb/>que</expan> circa <foreign lang="greek">ab</foreign>quatuor helices; adhuc facilius pondus mouebitur &longs;u­<lb/>per has quatuor, quàm &longs;uper tres OQRSTVP. & quò plures <lb/>erunt helices, eò facilius pondus mouebitur. </s> <s id="id.2.1.243.1.1.2.0">quod demon&longs;trare <lb/>oportebat. </s> </p> <p id="id.2.1.243.2.0.0.0" type="main"> <s id="id.2.1.243.2.1.1.0">Tempus verò huius motus facilè patet, helices enim CDEFG <lb/>&longs;unt æquales HL; helices verò OQRSTVP &longs;unt æquales <lb/>XY: &longs;ed XY maior e&longs;t HL; ideo fiat Y<foreign lang="greek">e</foreign> ip&longs;i HL æqualis: &longs;i igi<arrow.to.target n="note329"></arrow.to.target><lb/>tur duo pondera &longs;uper lineas LHY<emph type="italics"/>X<emph.end type="italics"/> moueantur, & veloci­<lb/>tates motuum &longs;int æquales, citius pertran&longs;ibit quod mouetur &longs;uper <lb/>LH, quàm quod &longs;uper Y<emph type="italics"/>X<emph.end type="italics"/> mouetur. </s> <s id="id.2.1.243.2.1.2.0">in eodem enim tempore erunt <lb/>in H<foreign lang="greek">e</foreign>. </s> <s id="id.2.1.243.2.1.3.0">quare tempus eius, quod mouetur &longs;uper helices OQRS <lb/>TVP, maius erit eo, quod e&longs;t men&longs;ura eius, quod mouetur &longs;uper C <lb/>DEFG. & quò plures erunt helices, eò maius erit tempus. </s> <s id="id.2.1.243.2.1.4.0">cùm au<lb/>tem datæ &longs;int lineæ HI<emph type="italics"/>XZ<emph.end type="italics"/>, & IL<emph type="italics"/>Z<emph.end type="italics"/>Y: datæ enim &longs;unt cochleæ AB <lb/><foreign lang="greek">ab</foreign>; & anguli ad IZ recti dati; erit HL data. </s> <s id="id.2.1.243.2.1.5.0">&longs;imiliter & <emph type="italics"/>X<emph.end type="italics"/>Y data <arrow.to.target n="note330"></arrow.to.target><lb/>erit. </s> <s id="id.2.1.243.2.1.6.0">quare & harum proportio data erit. </s> <s id="id.2.1.243.2.1.7.0">temporum igitur propor<arrow.to.target n="note331"></arrow.to.target><lb/>tio eorum, quæ &longs;uper helices mouentur data erit. </s> </p> <p id="id.2.1.244.1.0.0.0" type="margin"> <s id="id.2.1.244.1.1.1.0"><margin.target id="note329"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 18 <emph type="italics"/>Primi.<emph.end type="italics"/></s> <s id="id.2.1.244.1.1.2.0"><margin.target id="note330"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 48 <emph type="italics"/>primi.<emph.end type="italics"/></s> <s id="id.2.1.244.1.1.3.0"><margin.target id="note331"></margin.target>1 <emph type="italics"/>Datorum & Ex &longs;exta primi Ioannis de Monte rego de triangulis.<emph.end type="italics"/></s> </p> <p id="id.2.1.245.1.0.0.0" type="main"> <s id="id.2.1.245.1.1.1.0">Alterum, quod efficit, vt pondera facilè mo­<lb/>ueantur, &longs;unt &longs;cytalæ, aut manubria, quibus co­<lb/>chlea circumuertitur. <pb xlink:href="036/01/268.jpg"/><figure id="id.036.01.268.1.jpg" xlink:href="036/01/268/1.jpg"></figure></s> </p> <p id="id.2.1.245.2.0.0.0" type="main"> <s id="id.2.1.245.2.1.1.0">Sit cochlea habens helices ABCD, quæ etiam &longs;cytalas ha­<lb/>beat EFGH foraminibus cochleæ impo&longs;itas. </s> <s id="id.2.1.245.2.1.2.0">&longs;it infra helices <lb/>cylindrus MN, in quo non &longs;int inci&longs;æ helices; & circa cylindrum <lb/>funis circumuoluatur trahens pondus O, quod ad motum &longs;cytala <lb/>rum EFGH moueatur, ac &longs;i ergatæ in&longs;trumento traheretur. </s> <s id="id.2.1.245.2.1.3.0">du<lb/>catur (per ea quæ prius dicta &longs;unt de axe in peritrochio) Lk &longs;cy<lb/>talæ æqualis, axiq; cylindri perpendicularis, eumq; &longs;ecans in I: <lb/>patet quò longior &longs;it LI, & quò breuior &longs;it Ik, pondus O facilius <lb/>moueri. </s> <s id="id.2.1.245.2.1.4.0">e&longs;t autem animaduertendum, quòd dum cochlea mouet <lb/>pondus, &longs;i mente concipiatur, quòd loco trahendi pondus O fune, <lb/>pondus &longs;uper helices ABCD moueat; pondus quoq; in k, quod <lb/>&longs;it R, &longs;uper helices etiam facilius mouebit. </s> <s id="id.2.1.245.2.1.5.0">e&longs;t enim LK vectis, cuius <lb/><arrow.to.target n="note332"></arrow.to.target>fulcimentum e&longs;t I: cùm circa axem cochlea circumuertatur; po­<lb/><arrow.to.target n="note333"></arrow.to.target>tentia mouens in L; & pondus in k. </s> <s id="id.2.1.245.2.1.6.0">facilius enim mouetur pon<lb/>dus vecte Lk, quàm &longs;ine vecte; quia LI &longs;emper maior e&longs;t Ik. </s> <s id="id.2.1.245.2.1.7.0"><pb n="128" xlink:href="036/01/269.jpg"/>Intelligatur itaq; manente cochlea pondus R moueri à potentia <lb/>in L vecte Lk &longs;uper helicen Ck: vel quod idem e&longs;t, &longs;icut etiam <lb/>&longs;upra diximus, &longs;i pondus R aptetur ita, vt moueri non po&longs;sit, ni <lb/>&longs;i &longs;uper rectam PQ axi cylindri æquidi&longs;tantem; circumuertaturq; <lb/>cochlea, potentia exi&longs;tente in L; mouebitur pondus R &longs;uper he­<lb/>licen CD eodem modo, ac &longs;i à vecte Lk moueretur. </s> <s id="id.2.1.245.2.1.8.0">idem enim <lb/>e&longs;t, &longs;iue pondus manente cochlea &longs;uper helicen moueatur; &longs;iue he<lb/>lix circumuertatur, ita vt pondus &longs;uper ip&longs;am moueatur. </s> <s id="id.2.1.245.2.1.9.0">cùm <lb/>ab eadem potentia in L moueatur. </s> <s id="id.2.1.245.2.1.10.0">&longs;imiliter o&longs;tendetur, quò lon<lb/>gior &longs;it LI, adhuc pondus facilius &longs;emper moueri. </s> <s id="id.2.1.245.2.1.12.0">à minori enim <arrow.to.target n="note334"></arrow.to.target><lb/>potentia moueretur. </s> <s id="id.2.1.245.2.1.13.0">quod erat propo&longs;itum. </s> </p> <p id="id.2.1.246.1.0.0.0" type="margin"> <s id="id.2.1.246.1.1.1.0"><margin.target id="note332"></margin.target>2 <emph type="italics"/>Cor.<emph.end type="italics"/></s> <s id="id.2.1.246.1.1.2.0"><margin.target id="note333"></margin.target>1 <emph type="italics"/>huius de vecte.<emph.end type="italics"/></s> <s id="id.2.1.246.1.1.3.0"><margin.target id="note334"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 1 <emph type="italics"/>huius de vecte.<emph.end type="italics"/></s> </p> <p id="id.2.1.247.1.0.0.0" type="main"> <s id="id.2.1.247.1.1.1.0">Tempus quoq; huius motus manife&longs;tum e&longs;t, quò enim longior <lb/>e&longs;t LI, eò tempus maius erit: dummodo potentiæ motuum &longs;int <lb/>in velocitate æquales; &longs;icuti dictum e&longs;t de axe in peritrochio. </s> </p> <p id="id.2.1.247.2.0.0.0" type="head"> <s id="id.2.1.247.2.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.247.3.0.0.0" type="main"> <s id="id.2.1.247.3.1.1.0">Ex his manife&longs;tum e&longs;t. </s> <s id="id.2.1.247.3.1.2.0">quò plures &longs;unt heli­<lb/>ces; & quò longiores &longs;unt &longs;cytalæ, &longs;iue manu­<lb/>bria, pondus ip&longs;um facilius quidem, tardius au<lb/>tem moueri. </s> </p> <p id="id.2.1.247.4.0.0.0" type="main"> <s id="id.2.1.247.4.1.1.0">Virtus deniq; mouentis, atq; in &longs;cytalis con­<lb/>&longs;titutæ potentiæ, hinc manife&longs;ta fiet. <pb xlink:href="036/01/270.jpg"/><figure id="id.036.01.270.1.jpg" xlink:href="036/01/270/1.jpg"></figure></s> </p> <p id="id.2.1.247.5.0.0.0" type="main"> <s id="id.2.1.247.5.1.1.0">Sit datum A centum; &longs;it planum horizonti inclinatum CD in <lb/>angulo DCE. inueniatur ex eadem nona Pappi quanta vi pondus <lb/>A &longs;uper CD mouetur; quæ &longs;it decem. </s> <s id="id.2.1.247.5.1.2.0">exponatur cochlea LM <lb/>helices habens GHIK &c. in angulo ECD; per ea, quæ dicta <lb/>&longs;unt, potentia decem pondus A &longs;uper helices GHIk mouebit. </s> <s id="id.2.1.247.5.1.4.0">&longs;i <lb/>autem hac cochlea volumus pondus A mouere, & potentia mo­<lb/>uens &longs;it vt duo. </s> <s id="id.2.1.247.5.1.5.0">ducatur NP axi cochleæ perpendicularis, axem <lb/>&longs;ecans in O; fiatq; PO ad ON, vt vnum ad quinq; hoc e&longs;t duo ad <lb/><arrow.to.target n="note335"></arrow.to.target>decem. </s> <s id="id.2.1.247.5.1.6.0">Quoniam enim potentia mouens pondus A in P, ide&longs;t <lb/>&longs;uper helices e&longs;t vt decem, cui potentiæ re&longs;i&longs;tit, & æqualis e&longs;t po<lb/>tentia in N vt duo; e&longs;t enim NP vectis, cuius fulcimentum e&longs;t <lb/>O. potentia ergo vt duo in N pondus A &longs;uper helices cochleæ <lb/>mouebit. </s> <s id="id.2.1.247.5.1.7.0">efficiantur igitur &longs;cytalæ, &longs;iue manubria, quæ v&longs;q; ad N <pb n="129" xlink:href="036/01/271.jpg"/>perueniant; manife&longs;tum e&longs;t, potentiam vt duo in his pondus cen­<lb/>tum cochlea <emph type="italics"/>L<emph.end type="italics"/>M mouere. </s> </p> <p id="id.2.1.248.1.0.0.0" type="margin"> <s id="id.2.1.248.1.1.1.0"><margin.target id="note335"></margin.target><emph type="italics"/>Ex<emph.end type="italics"/> 1 <emph type="italics"/>huius de vecte.<emph.end type="italics"/></s> </p> <p id="id.2.1.249.1.0.0.0" type="main"> <s id="id.2.1.249.1.1.1.0">Si igitur &longs;it cochlea QR helices habens in angulo DCE, & cir­<lb/>ca ip&longs;am &longs;it eius mater S, quæ &longs;i pependerit centum, adiiciatur ST <lb/>manubrium quoddam, &longs;iue &longs;cytala; ita vt T in eadem proportio­<lb/>ne di&longs;tet ab axe cylindri, vt NOP; patet potentiam vt duo in T <lb/>mouere S &longs;uper helices cochleæ. </s> <s id="id.2.1.249.1.1.2.0">nihil enim aliud e&longs;t S, ni&longs;i pon­<lb/>dus &longs;uper helices cochleæ motum. </s> <s id="id.2.1.249.1.1.3.0">&longs;imiliter &longs;i S &longs;it immobilis, cir­<lb/>cumuertaturq; cochlea manubrio, &longs;iue &longs;cytala QX in eadem pro­<lb/>portione con&longs;ecta; fueritq; cochlea centum pondo (quòd qui­<lb/>dem, vel ex &longs;e ip&longs;a, vel cum pondere V cochleæ appen&longs;o, vel cum <lb/>pondere Y cochleæ &longs;uper impo&longs;ito centum pependerit) manife­<lb/>&longs;tum e&longs;t potentiam vt duo in X mouere cochleam QR &longs;uper he<lb/>lices intra matricem cochleæ inci&longs;as. </s> <s id="id.2.1.249.1.1.4.0">atq; ita in aliis, quæ cochleæ <lb/>in&longs;trumento mouentur; proportionem potentiæ ad pondus inue­<lb/>niemus. </s> </p> <p id="id.2.1.249.2.0.0.0" type="head"> <s id="id.2.1.249.2.1.1.0">COROLLARIVM. </s> </p> <p id="id.2.1.249.3.0.0.0" type="main"> <s id="id.2.1.249.3.1.1.0">Ex hoc manife&longs;tum e&longs;t, quomodo datum pon<lb/>dus à data potentia cochlea moueatur. <pb xlink:href="036/01/272.jpg"/><figure id="id.036.01.272.1.jpg" xlink:href="036/01/272/1.jpg"></figure></s> </p> <p id="id.2.1.249.4.0.0.0" type="main"> <s id="id.2.1.249.4.1.1.0">Illud quoq; præterea hoc loco ob&longs;eruandum occurrit; quò plu­<lb/>res erunt matricis cochleæ helices, eò minus in pondere mouen­<lb/>do cochleam pati. </s> <s id="id.2.1.249.4.1.2.0">&longs;i enim matrix vnicam duntaxat helicen po&longs;&longs;e<lb/>derit, tunc pondus vt centrum à &longs;ola cochleæ &longs;u&longs;tinebitur helice; <lb/>&longs;i verò plures, in plures quoque, ac totidem cochleæ heli­<lb/>ces ponderis grauitas di&longs;tribuetur; vt &longs;i quatuor contineat helices, <lb/>tunc quatuor vici&longs;sim cochleæ helices vniuer&longs;o ponderi &longs;u&longs;tinendo <lb/>incumbent; &longs;iquidem vnaquæquè quartam totius ponderis portio­<lb/>nem &longs;u&longs;tentabit. </s> <s id="id.2.1.249.4.1.3.0">quòd &longs;i adhuc plures contineat helices, ponderis <lb/>quoq; totius in plures, atque ideo minores portiones fiet di&longs;tri­<lb/>butio. </s> </p> <pb n="130" xlink:href="036/01/273.jpg"/> <p id="id.2.1.249.6.0.0.0" type="main"> <s id="id.2.1.249.6.1.1.0">O&longs;ten&longs;um e&longs;t igitur pondus à cochlea moueri <lb/>tamquam à cuneo percu&longs;sionis experte: loco e­<lb/>nim percu&longs;sionis mouet vecte, hoc e&longs;t &longs;cytala, &longs;i­<lb/>ue manubrio. </s> </p> <p id="id.2.1.249.7.0.0.0" type="main"> <s id="id.2.1.249.7.1.1.0">His demon&longs;tratis liquet, quomodo <expan abbr="datũ">datum</expan> pon­<lb/>dus à data potentia moueri po&longs;sit. </s> <s id="id.2.1.249.7.1.2.0">quòd &longs;i vecte <lb/>hoc a&longs;&longs;equi volumus; po&longs;&longs;umus & dato vecte da <lb/>tum pondus data potentia mouere. </s> <s id="id.2.1.249.7.1.3.0">quod quidem <lb/>in nullis ex aliis fieri po&longs;&longs;e ab&longs;olutè contingit: &longs;iue <lb/>&longs;it cochlea, &longs;iue axis in peritrochio, &longs;iue trochlea. </s> <s id="id.2.1.249.7.1.4.0"><lb/>non enim datis trochleis, neq; dato axe in peri­<lb/>trochio, neq; data cochlea, datum pondus à data <lb/>potentia moueri pote&longs;t, cùm potentia in his &longs;em­<lb/>per &longs;it determinata: &longs;i igitur <expan abbr="pot&etilde;tia">potentia</expan>, quæ pondus <lb/>mouere debeat, hac minor &longs;it data, nunquam pon<lb/>dus mouebit. </s> <s id="id.2.1.249.7.1.5.0">po&longs;&longs;umus tamen dato axe, & tympa­<lb/>no ab&longs;q; &longs;cytalis datum pondus data <expan abbr="pot&etilde;tia">potentia</expan> mo­<lb/>uere; cùm &longs;cytalas con&longs;truere po&longs;simus, ita vt &longs;e<lb/>midiameter tympani dati vná cum longitudine <lb/>&longs;cytalæ ad axis &longs;emidiametrum <expan abbr="datã">datam</expan> habeat pro­<lb/>portionem. </s> <s id="id.2.1.249.7.1.6.0">quod idem cochleæ contingere po<lb/>te&longs;t, &longs;cilicet datum pondus data cochlea &longs;ine ma<lb/>nubrio, vel &longs;cytala, data potentia mouere. </s> <s id="id.2.1.249.7.1.7.0">co­<lb/>gnita enim potentia, quæ pondus &longs;uper helices <lb/>moueat, po&longs;&longs;umus manubrium, &longs;iue &longs;cytalam ita <pb xlink:href="036/01/274.jpg"/>con&longs;truere, vt data potentia in &longs;cytala eandem <lb/>vim habeat, quam potentia pondus &longs;uper helices <lb/>mouens cùm autem hoc datis trochleis nullo mo <lb/>do fieri po&longs;sit. </s> <s id="id.2.1.249.7.1.8.0">datum tamen pondus data poten­<lb/>tia trochleis infinitis modis mouere po&longs;&longs;umus. </s> <s id="id.2.1.249.7.1.9.0"><lb/>datum verò pondus data potentia cunei in&longs;tru­<lb/>mento mouere, hoc minimè fieri po&longs;&longs;e clarum e&longs; <lb/>&longs;e videtur; non enim data potentia datum pon­<lb/>dus &longs;uper planum horizonti inclinatum mouere <lb/>pote&longs;t, neq; datum pondus à data potentia moue<lb/>bitur vectibus &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> aduer&longs;is, quemmadmo­<lb/>dum in cuneo in&longs;unt; cùm in vectibus cunei pro­<lb/>pria, veraq; vectis proportio &longs;eruari non po&longs;sit. </s> <s id="id.2.1.249.7.1.10.0"><lb/>vectium enim fulcimenta non &longs;unt immobilia, <lb/>cùm totus cuneus moueatur. </s> </p> <p id="id.2.1.249.8.0.0.0" type="main"> <s id="id.2.1.249.8.1.1.0">Poterit deinde quis &longs;truere machinas, atq; eas <lb/>ex pluribus componere; vt ex trochleis, & &longs;uc­<lb/>culis, vel ergatis, pluribu&longs;uè dentatis tympanis, <lb/>uel quocunq; alio modo; & ex ijs, quæ diximus; fa<lb/>cilè inter pondus, & potentiam proportionem <lb/>inuenire. </s> </p> <p id="id.2.1.249.9.0.0.0" type="head"> <s id="id.2.1.249.9.1.1.0">FINIS. </s> </p> <pb xlink:href="036/01/275.jpg"/> </chap> </body> <back> <section> <p id="id.2.1.249.11.0.0.0" type="head"> <s id="id.2.1.249.11.1.1.0">Locorum aliquot, quæ inter imprimendum deprauata <lb/>&longs;unt, emendatior lectio.</s> </p> <p id="id.2.1.249.12.0.0.0" type="main"> <s id="id.2.1.249.12.1.1.0"><emph type="italics"/>Pagina<emph.end type="italics"/> 2, <emph type="italics"/>b, ver&longs;u<emph.end type="italics"/> 19, <emph type="italics"/>AEBD<emph.end type="italics"/> ¶ 5, <emph type="italics"/>a<emph.end type="italics"/>, 6, <emph type="italics"/>ip&longs;i<emph.end type="italics"/> ¶ 7, <emph type="italics"/>b<emph.end type="italics"/>, 9, <emph type="italics"/>ODH<emph.end type="italics"/> ¶ 9, <emph type="italics"/>b<emph.end type="italics"/>, 19, <emph type="italics"/><expan abbr="cõtingit">contingit</expan><emph.end type="italics"/> <expan abbr="¶"><lb/>¶</expan>15, <emph type="italics"/>a<emph.end type="italics"/>, 24, <emph type="italics"/>grauius<emph.end type="italics"/> ¶ 16, <emph type="italics"/>b<emph.end type="italics"/>, 30, <emph type="italics"/>recto<emph.end type="italics"/> ¶ 21, <emph type="italics"/>a<emph.end type="italics"/>, 26, <emph type="italics"/>&longs;u&longs;tineatur<emph.end type="italics"/> ¶ 23, <emph type="italics"/>b<emph.end type="italics"/>, 8, <emph type="italics"/>BD DC<emph.end type="italics"/> ¶ 31, <emph type="italics"/>b<emph.end type="italics"/>, <lb/>9, <emph type="italics"/>totum GK<emph.end type="italics"/> ¶ 34, <emph type="italics"/>a<emph.end type="italics"/>, 24, <emph type="italics"/>pondera FG<emph.end type="italics"/> ¶ 38, <emph type="italics"/>b<emph.end type="italics"/>, 27, <emph type="italics"/>maior AF<emph.end type="italics"/> ¶ 39, <emph type="italics"/>b<emph.end type="italics"/>, 24 <emph type="italics"/>AB in D<emph.end type="italics"/> ¶ 40, <lb/><emph type="italics"/>a<emph.end type="italics"/>, 1, <emph type="italics"/>ad BD<emph.end type="italics"/> ¶ 44, <emph type="italics"/>b<emph.end type="italics"/>, 24, <emph type="italics"/>graui<emph.end type="italics"/> ¶ 48, <emph type="italics"/>a<emph.end type="italics"/>, 7, <emph type="italics"/>ip&longs;i AD<emph.end type="italics"/> ¶ 50, <emph type="italics"/>b<emph.end type="italics"/>, 12 <emph type="italics"/>pondus<emph.end type="italics"/> ¶ 54, <emph type="italics"/>a<emph.end type="italics"/>, 7, <emph type="italics"/>quàm<emph.end type="italics"/> ¶ 61, <lb/><emph type="italics"/>a<emph.end type="italics"/>, 6, <emph type="italics"/>præterquam in E<emph.end type="italics"/> ¶ 65, <emph type="italics"/>a<emph.end type="italics"/>, 33, <emph type="italics"/>quam<emph.end type="italics"/> ¶ 81, <emph type="italics"/>a<emph.end type="italics"/>, 1, <emph type="italics"/>ligato<emph.end type="italics"/> ¶ 85, <emph type="italics"/>b<emph.end type="italics"/>, 22, <emph type="italics"/>vtriq;<emph.end type="italics"/> ¶ 97, <emph type="italics"/>a<emph.end type="italics"/>, 14, <lb/><emph type="italics"/>dextror&longs;um<emph.end type="italics"/> ¶ 98, <emph type="italics"/>b<emph.end type="italics"/>, 20, <emph type="italics"/>Hic<emph.end type="italics"/> ¶ 110, <emph type="italics"/>b, in po&longs;till.</s> <s id="id.2.1.249.12.1.2.0">Lemma in <expan abbr="primã">primam</expan><emph.end type="italics"/> ¶ 122, <emph type="italics"/>a<emph.end type="italics"/>, 8, <emph type="italics"/>&<emph.end type="italics"/> 17, <emph type="italics"/>helicen<emph.end type="italics"/> <expan abbr="¶"><lb/>¶</expan>123, <emph type="italics"/>b<emph.end type="italics"/>, 15, <emph type="italics"/>ventes in GH<emph.end type="italics"/> ¶ 124, <emph type="italics"/>b<emph.end type="italics"/>, 17, <emph type="italics"/>manife&longs;tum<emph.end type="italics"/> ¶ 127, <emph type="italics"/>a, in po&longs;til.</s> <s id="id.2.1.249.12.1.3.0">Monteregio<emph.end type="italics"/> <expan abbr="¶"><lb/>¶</expan>127, <emph type="italics"/>b, in po&longs;til.</s> <s id="id.2.1.249.12.1.4.0">ex Cor.<emph.end type="italics"/></s> <lb/> <s id="id.2.1.249.12.3.1.0">REGISTRVM. </s> <lb/> <s id="id.2.1.249.12.5.1.0"><12><12><12> ABCDEFGHIKLMNOPQRSTVX <lb/>YZ, Aa Bb Cc Dd Ee Ff Gg Hh Ii Kk. </s> <lb/> <s id="id.2.1.249.12.7.1.0">Omnes duerni. </s> <lb/> <s id="id.2.1.249.12.9.1.0">PISAVRI </s> <lb/> <s id="id.2.1.249.12.11.1.0">Apud Hieronymum Concordiam. </s> <lb/> <s id="id.2.1.249.12.13.1.0">M. D. LXXVII. </s> </p> </section> </back> </text> </archimedes>