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| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 29 Nov 2017 16:55:37 +0100 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <archimedes xmlns:xlink="http://www.w3.org/1999/xlink" > <info> <author>Monte, Guidobaldo del</author> <title>Mechanicorum Liber, old version (275 pages)</title> <date>1577</date> <place>Pisauri</place> <translator></translator> <lang>la</lang> <cvs_file>monte_mecha_036_la_1577.xml</cvs_file> <cvs_version>2635.10</cvs_version> <locator>036.xml</locator> <echodir>/permanent/archimedes/monte_mecha_036_la_1577</echodir> </info> <text> <front> <section> <pb xlink:href="036/01/001.jpg"></pb> <p id="id.2.1.1.1.0.0.0" type="head"> <s id="id.2.1.1.1.2.1.0">GVIDIVBALDI <lb></lb>E MARCHIONIBVS <lb></lb>MONTIS <lb></lb>MECHANICORVM <lb></lb>LIBER. </s> </p> <figure id="id.036.01.001.1.jpg" place="text" 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></lb>Apud Hieronymum Concordiam. </s> <lb></lb> <s id="id.2.1.1.1.8.1.0">M. D. LXXVII. </s> <lb></lb> <s id="id.2.1.1.1.10.1.0">Cum Licentia Superiorum. </s> </p> <pb xlink:href="036/01/002.jpg"></pb> <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></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"></pb> <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></lb>MARIAM II <lb></lb>VRBINATVM <lb></lb>AMPLISSIMVM DVCEM <lb></lb>GVIDIVBALDI <lb></lb>E MARCHIONIBVS <lb></lb>MONTIS </s> </p> <p id="N10084" 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></lb>CEPS) quæ ad conciliandas homi<lb></lb>nibus facultates, vtilitas nempè, & <lb></lb>nobilitas, plurimùm valere conſue<lb></lb>uerunt. </s> <s id="id.2.1.1.12.1.2.0">illæ ad exornandam mecha<lb></lb>nicam facultatem, & eam præ om<lb></lb>nibus alijs appetibilem reddendam conſpiraſſe <lb></lb>mihi videntur: nam ſi nobilitatem (quod pleriq; <lb></lb>modò faciunt) ortu ipſo metimur, occurret hinc <lb></lb>Geometria, illinc verò Phiſica; quorum gemina<lb></lb>to complexu nobiliſſima artium prodit mechani<lb></lb>ca. </s> <s id="id.2.1.1.12.1.3.0">ſi enim nobilitatem magis, tùm ſtratæ materiæ, <lb></lb>tùm argumentorum neceſſitati (quod Ariſtote<lb></lb>les fatetur aliquandò) relatam volumus, omnium <lb></lb>procul dubiò nobiliſſimam perſpiciemus. </s> <s id="id.2.1.1.12.1.4.0">quæ <pb xlink:href="036/01/004.jpg"></pb>quidem non ſolum geometriam (vt Pappus teſta<lb></lb>tur) abſoluit, & perficit; verùm etiam & phiſica<lb></lb>rum rerum imperium habet: quandoquidem <lb></lb>quodcunq; Fabris, Architectis, Baiulis, Agricolis, <lb></lb>Nautis, & quàm plurimis alijs (repugnantibus na<lb></lb>turæ legibus) opitulatur; id omne mechanicum <lb></lb>eſt imperium. </s> <s id="id.2.1.1.12.1.5.0">quippè quod aduerſus naturam <lb></lb>vel eiuſdem emulata leges exercet; ſumma id <lb></lb>certè admiratione dignum; veriſſimum tamen, <lb></lb>& à quocunque liberaliter admiſſum, qui pri<lb></lb>us ab Ariſtotele didicerit, omnia mechanica, <lb></lb>tùm problemata, tùm theoremata ad rotundam <lb></lb>machinam reduci, atq; ideo illo niti principio, <lb></lb><expan abbr="nõ">non</expan> minus ſenſui, quàm rationi noto. </s> <s id="id.2.1.1.12.1.6.0">Rotunda ma<lb></lb>china eſt mouentiſſima, & quò maior, eò mouen<lb></lb>tior. </s> <s id="id.2.1.1.12.1.7.0">Verùm huic nobilitati adnexa eſt ſumma re <lb></lb>rum ad vitam pertinentium vtilitas, quæ propte<lb></lb>rea omnes alias à diuerſis artibus propagatas an<lb></lb>tecellit; quòd aliæ facultates poſt mundi geneſim <lb></lb>longa temporis intercapedine ſuos explicarunt <lb></lb>vſus; iſta verò & in ipſis mundi primordijs ita fuit <lb></lb>hominibus neceſſaria, vt ea ſublata Sol de mun<lb></lb>do ſublatus videretur. </s> <s id="id.2.1.1.12.1.8.0">nam quacunq; neceſſita<lb></lb>te Adæ vita degeretur; & quamuis etiam caſis <lb></lb>contectis ſtramine, & anguſtis tugurijs, ac gurgu<lb></lb>ſtijs cœli defenderet iniurias; ſic & in corporis ve<lb></lb>ſtitu, licet ipſe nihil aliud ſpectaret, niſi vt imbres, <pb xlink:href="036/01/005.jpg"></pb>vt niues, vt ventos; vt Solem, vt frigus arceret; <lb></lb>quodcunque tamen id fuit, omne mechanicum <lb></lb>fuit. </s> <s id="id.2.1.1.12.1.9.0">neq; tamen huic facultati contingit, quod <lb></lb>ventis ſolet, qui cùm vndè oriuntur, ibi vehe<lb></lb>mentiſſimi ſint, ad longinqua tamen fracti, de<lb></lb>bilitatiquè perueniunt: ſed quod magnis flumini<lb></lb>bus crebriuſ accidit, quæ cùm in ipſo ortu parua <lb></lb>ſint, perpetuò tamen aucta, eò ampliori ferun<lb></lb>tur alueo, quò à fontibus ſuis longius receſſe<lb></lb>runt. </s> <s id="id.2.1.1.12.1.10.0">Nam & temporis progreſſu mechanica fa <lb></lb>cultas ſub iugo æquum arationis laborem di<lb></lb>ſpenſare, atque aratrum agris circumagere cæ<lb></lb>pit. </s> <s id="id.2.1.1.12.1.11.0">deinceps bigis, & quadrigis docuit comea<lb></lb>tus, merces, onera quælibet vehere, è finibus <lb></lb>noſtriſ ad finitimos populos exportare, & ex il<lb></lb>lis contra importare ad nos. </s> <s id="id.2.1.1.12.1.12.0">præterea cùm iam <lb></lb>res non tantùm neceſſitate, verùm etiam orna<lb></lb>tu, & commoditate metirentur, mechanicæ <lb></lb>fuit ſubtilitatis, quòd nauigia remo impellere<lb></lb>mus; quòd gubernaculo exiguo in extrema pup<lb></lb>pi collocato ingentes triremium moles inflecte<lb></lb>remus; quòd vnius ſæpè manu pro multis fabro<lb></lb>rum manibus modò pondera lapidum, & tra<lb></lb>bium Fabris, & Architectis ſubleuaremus; mo<lb></lb>dò tollenonis ſpecie aquas è puteis olitoribus e<lb></lb>xhauriremus. </s> <s id="id.2.1.1.12.1.13.0">hinc etiam è liquidorum prælis vi<lb></lb>na, olea, vnguenta expreſſa, & quicquid liquo<pb xlink:href="036/01/006.jpg"></pb>ris habent, perſoluere domino compulſa. </s> <s id="id.2.1.1.12.1.14.0">hinc <lb></lb>magnas <expan abbr="arborũ">arborum</expan>, & marmorum moles duobus in <lb></lb>contrarias partes <expan abbr="diſtrahẽtibus">diſtrahentibus</expan> vectibus diremp<lb></lb>ſimus; hinc militiæ in aggeribus extruendis, in <lb></lb>conſerenda manu, in opugnando, propugnan<lb></lb>doq; loca infinitæ ferè redundarunt vtilitates; <lb></lb>hinc demum Lignatores, Lapicidæ, Marmorarij <lb></lb>Vinitores, Olearij, Vnguentarij, Ferrarij, Auri<lb></lb>fices, Metallici, Chirurgi, Tonſores, Piſtores, Sar<lb></lb>tores, omnes deniq; opifices beneficiarij, tot, tan<lb></lb>taq; vitæ humanæ ſuppeditarunt commoda. </s> <s id="id.2.1.1.12.1.15.0">Eant <lb></lb>nunc noui logodedali quidam mechanicorum <lb></lb>contemptores, perfricent frontem, ſi quam ha<lb></lb>bent, & ignobilitatem, atquè inutilitatem falſò <lb></lb>criminari deſinant: quòd ſi & adhuc id minimè <lb></lb>velint, eos quæſo in inſcitia ſua relinquamus: <lb></lb>Ariſtotelemquè potius philoſophorum cory<lb></lb>phæum imitemur, cuius mechanici amoris ardo<lb></lb>rem acutiſſimæ illæ mechanicæ quæſtiones poſte <lb></lb>ris traditæ ſatis declarant: qua quidem laude <lb></lb>Platonem magnificè ſuperauit; qui (vt teſtatur <lb></lb>Plutarcus) Architam, & Eudoxum mechanicæ <lb></lb>vtilitatem impenſius colentes ab inſtituto deter<lb></lb>ruit; quòd nobiliſſimam philoſophorum poſſeſ<lb></lb>ſionem in vulgus indicarent, ac publicarent; & <lb></lb>velut arcana philoſophiæ myſteria proderent. </s> <s id="id.2.1.1.12.1.16.0"><lb></lb>res ſanè meo quidem iudicio proſus vituperan<pb xlink:href="036/01/007.jpg"></pb>da, niſi fortè velimus tam nobilis diſciplinæ con<lb></lb>templationem quidem ocioſam laudare; fructum <lb></lb>verò, & vſum, artiſq; finem improbare. </s> <s id="id.2.1.1.12.1.17.0">ſed præ <lb></lb>omnibus mathematicis vnus Archimedes ore <lb></lb>laudandus eſt pleniore, quem voluit Deus in me<lb></lb>chanicis velut ideam ſingularem eſſe, quam om<lb></lb>nes earum ſtudioſi ad imitandum ſibi propone<lb></lb>rent. </s> <s id="id.2.1.1.12.1.18.0">is enim Cœleſtem globum exiguo admo<lb></lb>dum, fragili què vitreo orbe concluſum ita efin<lb></lb>xit, ſimulatis aſtris viuum naturæ opus, ac iura <lb></lb>poli motibus certis adeò præ ſe ferentibus; vt <lb></lb>æmula naturæ manus tale de ſe encomium ſit <lb></lb>promerita: ſic manus naturam, vt natura ma<lb></lb>num ipſa immitata putetur. </s> <s id="id.2.1.1.12.1.19.0">is poliſpaſtu manu <lb></lb>leua, & ſola, quinquies millenum modiorum <lb></lb>pondus attraxit. </s> <s id="id.2.1.1.12.1.20.0">nauem in ſiccum litus eductam, <lb></lb>ac grauius oneratam ſolus machinis ſuis ad ſe <lb></lb>perindè pertraxit, ac ſi in mari remis, veliſuè <lb></lb>impulſa moueretur, <expan abbr="quã">quam</expan> & poſtea in litore (quod <lb></lb>omnes Siciliæ vires non potuerunt) in mare de<lb></lb>duxit. </s> <s id="id.2.1.1.12.1.21.0">ab iſto etiam ea extiterunt bellica tor<lb></lb>menta, quibus Syracuſæ aduerſus Marcellum <lb></lb>ita defenſæ ſunt, vt paſſim eorum machinator <lb></lb>Briareus, & centimanus à Romanis appellare<lb></lb>tur. </s> <s id="id.2.1.1.12.1.22.0">demum hac arte confiſus eò proceſſit au<lb></lb>daciæ, vt eam vocem naturæ legibus adeò re<lb></lb>pugnantem protulerit. </s> <s id="id.2.1.1.12.1.23.0">Da mihi, vbi ſiſtam, ter<pb xlink:href="036/01/008.jpg"></pb>ramq; mouebo. </s> <s id="id.2.1.1.12.1.24.0">quod tamen non modò nos <lb></lb>vecte tantùm fieri potuiſſe in præſenti libro doce<lb></lb>mus; verùm etiam, & omnis antiquitas (quod <lb></lb>multis fortaſſè mirabile videbitur) id penitus <lb></lb>credidiſſe mihi videtur; quæ Neptuno tri<lb></lb>dentem tanquam vectem attribuit; cuius ope <lb></lb>terræ concuſſor vbiq; nuncupatur à poetis. </s> <s id="id.2.1.1.12.1.25.0">ad <lb></lb>quod etiam aſpiciens celeberrimus noſter poeta <lb></lb>Neptunum inducit iſta machina ſyrtes, quò ma<lb></lb>gis apparerent Troianis, ſubleuantem. </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 ipſe tridenti <lb></lb>& vaſtas aperit ſyrtes.” </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, Cteſibius, <lb></lb>& Pappus, qui licet ad mechanicæ apicem, perin<lb></lb>de atq; Archimedes, euecti fortaſſè minimè ſint; <lb></lb>mechanicam tamen facultatem egregiè percal<lb></lb>luerunt; taleſq; fuerunt, & præſertim Pappus, vt <lb></lb>eum me ducem ſequentem nemo (vt opinor) cul<lb></lb>pauerit. </s> <s id="id.2.1.1.14.1.2.0">quod & propterea libentius feci, quòd <lb></lb>nè latum quidem vnguem ab Archimedeis prin<lb></lb>cipijs Pappus recedat. </s> <s id="id.2.1.1.14.1.3.0">ego enim in hac præſertim <lb></lb>facultate Archimedis veſtigijs hærere ſemper vo <lb></lb>lui: & licet eius lucubrationes ad <expan abbr="mechanicã">mechanicam</expan> per<pb xlink:href="036/01/009.jpg"></pb>tinentes multis ab hinc annis paſſim ſoleant do<lb></lb>ctis deſiderari: eruditiſſimus tamen libellus de æ<lb></lb>queponderantibus præ manibus <expan abbr="hominũ">hominum</expan> adhuc <lb></lb>verſatur, in quò tanquam in copioſiſſima pœnu <lb></lb>omnia ferè mechanica dogmata repoſita mihi vi<lb></lb>dentur; quem ſanè libellum, ſi ætatis noſtræ mathe<lb></lb>matici ſibi magis familiarem adhibuiſſent; reperiſ<lb></lb>ſent ſanè <expan abbr="ſentẽtias">ſententias</expan> multas, quas modó ipſi firmas, <lb></lb>& ratas eſſe docent; ſubtiliſſimè, atquè veriſ<lb></lb>ſimè conuulſas, & labefactatas. </s> <s id="id.2.1.1.14.1.4.0">ſed hoc vi<lb></lb>derint ipſi. </s> <s id="id.2.1.1.14.1.5.0">ego enim ad Pappum redeo, qui <lb></lb>ad vſum mathematicarum vberiorem, emulu<lb></lb>mentorumquè acceſſiones amplificandas peni<lb></lb>tus conuerſus, de quinque principibus machi<lb></lb>nis, Vecte nempè, Trochlea, Axe in peri<lb></lb>trochio, Cuneo, & Cochlea, multa egre<lb></lb>giè philoſophatus eſt; demonſtrauit què quicquid <lb></lb>in machinis, aut cogitari peritè, aut acutè <lb></lb>definiri, aut certò ſtatui poteſt, id omne quin<lb></lb>què illis infinita vi præditis machinis referen<lb></lb>dum eſſe. </s> <s id="id.2.1.1.14.1.6.0">atquè vtinam iniuria temporis ni<lb></lb>hil è tanti viri ſcriptis abraſiſſet: nec enim tam <lb></lb>denſa inſcitiæ caligo vniuerſum propè terra<lb></lb>rum orbem obtexiſſet, neque tanta mechani<lb></lb>cæ facultatis eſſet ignoratio conſecuta, vt ma<lb></lb>thematicarum proceres exiſtimarentur illi, qui <lb></lb>modò ineptiſſima quadam diſtinctione, diffi<pb xlink:href="036/01/010.jpg"></pb>cultates nonnullas, nec illas tamen ſatis ar<lb></lb>duas, & obſcuras è medio tollunt. </s> <s id="id.2.1.1.14.1.7.0">reperiun<lb></lb>tur enim aliqui, noſtraq; ætate emunctæ naris <lb></lb>mathematici, qui mechanicam, tùm mathe<lb></lb>maticè ſeorſum, tùm phiſicè conſiderari poſ<lb></lb>ſe affirmant; ac ſi aliquando, vel ſine demon<lb></lb>ſtrationibus geometricis, vel ſine vero motu <lb></lb>res mechanicæ conſiderari poſſint: qua ſanè di<lb></lb>ſtinctione (vt leuius cum illis agam) nihil aliud mi<lb></lb>hi comminiſci videntur, quàm vt dum ſe, tùm <lb></lb>phiſicos, tùm mathematicos proferant, vtra<lb></lb>que (quod aiunt) ſella excludantur. </s> <s id="id.2.1.1.14.1.8.0">nequè <lb></lb>enim amplius mechanica, ſi à machinis abſtra<lb></lb>hatur, & ſeiungatur, mechanica poteſt appel<lb></lb>lari. </s> <s id="id.2.1.1.14.1.9.0">Emicuit tamen inter iſtas tenebras (quam<lb></lb>uis alij quoquè nonnulli fuerint præclariſſimi) <lb></lb>Solis inſtar Federicus Commandinus, qui multis <lb></lb>doctiſſimis elucubrationibus amiſſum mathema<lb></lb>ticarum patrimonium non modò reſtaurauit, <lb></lb>verùm etiam auctiùs, & locupletiùs effecit. </s> <s id="id.2.1.1.14.1.10.0"><lb></lb>erat enim ſummus iſte vir omnibus adeò facul<lb></lb>tatibus mathematicis ornatus, vt in eo Archi<lb></lb>tas, Eudoxus, Heron, Euclides, Theon, Ari<lb></lb>ſtarcus, Diophantus, Theodoſius, Ptolemæus <lb></lb>Apollonius, Serenus, Pappus, quin & ip<lb></lb>ſemet Archimedes (ſiquidem ipſius in Archi<lb></lb>medem ſcripta Archimedis olent lucernam) re <pb xlink:href="036/01/011.jpg"></pb>uixiſſe viderentur. </s> <s id="id.2.1.1.14.1.11.0">& ecce repentè è tenebris (vt <lb></lb>confidimus) ac vinculis corporis in lucem, li<lb></lb>bertatem què productus mathematicas alieniſ<lb></lb>ſimo tempore optimo, & præſtantiſſimo patre <lb></lb>orbatas, nos verò ita conſternatos reliquit, vt e<lb></lb>ius deſiderium vix longo ſermone mitigare <lb></lb>poſſe videamur. </s> <s id="id.2.1.1.14.1.12.0">Ille tamen perpetuò in alia<lb></lb>rum mathematicarum explicationem verſans, <lb></lb>mechanicam facultatem, aut penitus præter<lb></lb>miſit, aut modicè attigit. </s> <s id="id.2.1.1.14.1.13.0">Quapropter in hoc <lb></lb>ſtudium ardentiùs ego incumbere cæpi, nec me <lb></lb>vnquam per omne mathematum genus vagan<lb></lb>tem ea ſolicitudo deſeruit; ecquid ex vno <lb></lb>quoquè decerpi, ac delibari poſſit; quo ad me<lb></lb>chanicam expoliendam, & exornandam acco<lb></lb>modatior eſſe poſſem. </s> <s id="id.2.1.1.14.1.14.0">Nunc verò cùm mihi <lb></lb>videar, noni ea quidem omnia, quæ ad mecha<lb></lb>nicam pertinent, perfeciſſe; ſed eò vſq; tamen <lb></lb>progreſſus, vt ijs, qui ex Pappo, ex Vitruuio, <lb></lb>& ex alijs didicerint, quid ſit Vectis, quid Tro<lb></lb>chlea, quid Axis in peritrochio, quid Cuneus, <lb></lb>quid Cochlea; quomodoq; vt pondera moueri <lb></lb>poſſint, aptari debeant; adhuc tamen acciden<lb></lb>tia permulta, quæ inter potentiam, & pondus <lb></lb>vectis virtute illis inſunt inſtrumentis, perdiſce<lb></lb>re cupiunt, opis aliquid adferre poſſim; putaui <lb></lb>tempus iam poſtulare, vt prodirem; & nauatæ <pb xlink:href="036/01/012.jpg"></pb>in hoc genere operæ ſpecimen aliquod darem. </s> <s id="id.2.1.1.14.1.15.0"><lb></lb>Verùm quò facilius totius operis ſubſtructio <lb></lb>ad faſtigium ſuum per duceretur, nonnulla quo<lb></lb>què de libra fuerunt pertractanda, & præſer<lb></lb>tim dum vnico pondere alterum ſolum ipſius <lb></lb>brachium penitus deprimitur: que in re mi<lb></lb>rum eſt quantas fecerint ruinas Iordanus (qui <lb></lb>inter recentiores maximæ fuit auctoritatis) & <lb></lb>alij; qui hanc rem ſibi diſcutiendam propoſue<lb></lb>runt. </s> <s id="id.2.1.1.14.1.16.0">opus ſanè arduum, & forſan viribus no<lb></lb>ſtris impar aggreſsi ſumus; in eo tamen digni, vt <lb></lb>noſtros conatus, & induſtriam ad præclara ten<lb></lb>dentem bonorum omnium perpetuus applau<lb></lb>ſus, approbatioq; comitetur; quòd ad ſtudium <lb></lb>tàm illuſtre, tam magnificum, tam laudabile <lb></lb>contulimus quicquid habuimus virium. </s> <s id="id.2.1.1.14.1.17.0">quod <lb></lb>ſanè qualecunq; ſit, tibi celeberrime PRINCEPS <lb></lb>nuncupandum cenſuimus; cuius ſanè conſilij, <lb></lb>atq; inſtituti noſtri rationes multas reddere in <lb></lb>promptu eſt: & primùm hæreditaria tibi in fa<lb></lb>miliam noſtram promerita, quibus nos ita de<lb></lb>uictos habes; vt facilè intelligamus ad fortunas <lb></lb>non modò noſtras, verùm & ad ſanguinem, & <lb></lb>vitam quoq; pro tua dignitate propendendam <lb></lb>paratiſſimos eſſe debere. </s> <s id="id.2.1.1.14.1.18.0">Præterea illud non <lb></lb>parui quoq; ponderis accedit, quòd à pueri<lb></lb>tia literarum omnium, ſed præcipuè mathe<pb xlink:href="036/01/013.jpg"></pb>maticarum deſiderio ita fueris incenſus, vt ni<lb></lb>ſi illis adeptis vitam tibi acerbam, atq; inſua<lb></lb>uem ſtatueres. </s> <s id="id.2.1.1.14.1.19.0">proinde in earum ſtudio infi<lb></lb>xus primam ætatis partem in illis percipiendis <lb></lb>exegiſti, eamquè ſæpius verè principe dignam <lb></lb>vocem protuliſti, te propterea mathematicis <lb></lb>præſertim delectari, quòd iſtæ maximè ex do<lb></lb>meſtico illo, & vmbratili vitæ genere in Solem <lb></lb>(quod dicitur) & puluerem prodire poſsint: cu<lb></lb>ius ſanè rei tuum flagrantiſsimum ab ineunte æta <lb></lb>te peritiæ militaris deſiderium, exploratum in<lb></lb>dicium poterat eſſe, niſi nimis emendicatæ men<lb></lb>tis eſſet ea proponere, quæ à te ſperari poſſent; <lb></lb>quando tu penitus adoleſcens, egregia multa fa<lb></lb>cinora proficere maturaſti. </s> <s id="id.2.1.1.14.1.20.0">Tu enim cùm iam <lb></lb>à ſanctiſſimo Pontifice Pio V ſaluberrimæ Prin<lb></lb>cipum Chriſtianorum coniunctionis fundamen<lb></lb>ta iacta eſſent, alacer admodum ad debellan<lb></lb>dos Chriſti hoſtes profectus, ſolidiſſimam, ac ve<lb></lb>riſſimam gloriam tibi comparaſti. </s> <s id="id.2.1.1.14.1.21.0">Tu quoties de <lb></lb>ſumma rerum deliberatum eſt, eas ſententias <lb></lb>dixiſti, quæ ſummam prudentiam cùm ſumma <lb></lb>animi excelſitate coniunctam indicarent. </s> <s id="id.2.1.1.14.1.22.0">ommit<lb></lb>tam interim pleraq; alia illis temporibus egre<lb></lb>giè, viriliter què à te geſta, ne tibi ipſi ea, quæ <lb></lb>omnibus ſunt manifeſta, palàm facere videar: <pb xlink:href="036/01/014.jpg"></pb>quæ cùm omnia magna, & præclara ſint; mul<lb></lb>tò tamen à te maiora, & præclara expectant <lb></lb>adhuc homines. </s> <s id="id.2.1.1.14.1.23.0">Vale interim præſtantiſſimum <lb></lb>orbis decus, & ſi quando aliquid otij nactus <lb></lb>fueris has meas vigiliolas aſpicere ne dedi<lb></lb>gneris. </s> </p> <pb n="1" xlink:href="036/01/015.jpg"></pb> <p id="id.2.1.1.15.0.0.0" type="head"> <s id="id.2.1.1.16.1.1.0">GVIDIVBALDI <lb></lb>E MARCHIONIBVS <lb></lb>MONTIS. </s> </p> <p id="N10397" type="head"> <s id="id.2.1.1.16.3.1.0">MECHANICORVM <lb></lb>LIBER. </s> </p> </section> </front> <body> <chap id="N1039F"> <p id="id.id.2.1.1.16.5.1.0.a" type="main"> <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 vniuſcu<lb></lb>iuſq; corporis eſt punctum quod<lb></lb>dam intra poſitum, à quo ſi gra<lb></lb>ue appenſum mente concipiatur, <lb></lb>dum fertur, quieſcit; & ſeruat eam, <lb></lb>quam in principio habebat poſi<lb></lb>tionem: neq; in ipſa 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></lb>octauo Mathematicarum collectionum libro tradidit. </s> <s id="id.2.1.1.18.1.2.0">Federicus <lb></lb>verò Commandinus in libro de centro grauitatis ſolidorum idem <lb></lb>centrum deſcribendo 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 vniuſcuiuſq; ſolidæ figu<lb></lb>ræ eſt punctum illud intra poſitum, circa quod <lb></lb>vndiq; partes æqualium momentorum conſi<lb></lb>ſtunt. </s> <s id="id.2.1.1.19.1.2.0">ſi enim per tale centrum ducatur planum <lb></lb>figuram quomodocunq; ſecans ſemper in par<lb></lb>tes æqueponderantes ipſam diuidet. </s> </p> <pb xlink:href="036/01/016.jpg"></pb> <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 id="N103E3" 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></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></lb>ciantur, tota ſimul 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 ſe æquè ſunt <lb></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 id="N10412" 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 eſt centrum gra<lb></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 ſemper in <lb></lb>eodem eſt ſitu reſpectu ſui 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></lb>ſum feruntur. </s> </p> </chap> <pb n="2" xlink:href="036/01/017.jpg"></pb> <chap id="N1043F"> <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 ſermo ha<lb></lb>beatur, vtres clarior eluceſcat, ſit <lb></lb>libra AB recta linea; CD verò <lb></lb>trutina, quæ ſecundum commu<lb></lb>nem conſuetudinem horizonti <lb></lb>ſemper eſt perpendicularis. </s> <s id="id.2.1.1.35.1.2.0">pun<lb></lb>ctum autem C immobile, circa quod vertitur li<lb></lb>bra, centrum libræ <lb></lb>vocetur. </s> <s id="id.2.1.1.35.1.3.0">itidemque <lb></lb>(quamuis tamen im<lb></lb>proprie) ſiue ſupra, <lb></lb>ſiue infra libram fue<lb></lb>rit conſtitutum. </s> <s id="id.2.1.1.35.1.4.0">CA <lb></lb>verò, & CB, tum di<lb></lb>ſtantiæ, tum libræ <lb></lb>brachia nuncupen<lb></lb>tur. </s> <s id="id.2.1.1.35.1.5.0">& ſi à centro li<lb></lb>bræ ſupra, vel infra <lb></lb><figure id="id.036.01.017.1.jpg" place="text" xlink:href="036/01/017/1.jpg"></figure><lb></lb>libram conſtituto ipſi AB perpendicularis duca<lb></lb>tur, hæc perpendiculum vocetur, quæ libram AB <lb></lb>ſubſtinebit; & quocunque modo moueatur libra, <lb></lb>ipſi ſemper perpendicularis exiſtet. </s> </p> <pb xlink:href="036/01/018.jpg"></pb> <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></lb>metro AB circulus deſcribatur AEBD, cuius <lb></lb>centrum C. </s> <s id="id.2.1.1.38.1.1.0.a">Dico punctum B infimum eſſe lo<lb></lb>cum circumferentiæ circuli AEBD; punctum <lb></lb>verò A ſublimiorem; & quælibet puncta, vt DE <lb></lb>æqualiter à puncto A diſtantia æqualiter eſſe <lb></lb>deorſum; quæ verò propius ſunt ipſi A eis, quæ <lb></lb>magis diſtant, ſublimiora eſſe. </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 vſq; ad mundi cen<lb></lb>trum, quod ſit F; deinde in circuli circum<lb></lb><arrow.to.target n="note1"></arrow.to.target>ferentia quoduis accipiatur punctum G; <lb></lb>connectanturq; FG FD FE. </s> <s id="id.2.1.1.39.1.2.0">Quoniam <lb></lb>n. BF minima eſt omnium, quæ à puncto <lb></lb>F ad circumferentiam AEBD ducun<lb></lb>tur; erit BF ipſa FG minor. </s> <s id="id.2.1.1.39.1.3.0">quare punctum <lb></lb>B propius erit puncto F, quàm G. </s> <s id="id.2.1.1.39.1.3.0.a">hacq; <lb></lb>ratione oſtendetur punctum B quouis alio <lb></lb>puncto circumferentiæ circuli AEDB <lb></lb>mundi centro propius eſſe. </s> <s id="id.2.1.1.39.1.4.0">erit igitur pun<lb></lb>ctum B circumferentiæ circuli AEBD <lb></lb>infimus locus. </s> <s id="id.2.1.1.39.1.5.0">Deinde quoniam AF per <lb></lb>centrum ducta maior eſt ipſa GF; erit <lb></lb>punctum A non <expan abbr="ſolũ">ſolum</expan> ipſo G, verum etiam <lb></lb>quouis alio puncto circumferentiæ circuli <lb></lb>AEBD ſublimius. </s> <s id="id.2.1.1.39.1.6.0">Præterea quoniam DF <lb></lb>FE ſunt æquales; puncta DE æqualiter <lb></lb><figure id="id.036.01.018.1.jpg" place="text" xlink:href="036/01/018/1.jpg"></figure><lb></lb>mundi centro diſtabunt. </s> <s id="id.2.1.1.39.1.7.0">& cum DF maior ſit FG; erit pun<lb></lb>ctum D ipſi A propius puncto G ſublimius. </s> <s id="id.2.1.1.39.1.8.0">quæ omnia demon<lb></lb>ſtrare 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"></emph>Tertil.<emph.end type="italics"></emph.end></s> </p> <pb n="3" xlink:href="036/01/019.jpg"></pb> <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 ſu<lb></lb>ſtineatur linea, nunquam manebit, niſi eadem li<lb></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></lb>uitatis B, quod à linea CE ſuſti<lb></lb>neatur. </s> <s id="id.2.1.3.3.1.2.0">Dico pondus nunquam <lb></lb>permanſurum, niſi CB horizonti <lb></lb>perpendicularis exiſtat. </s> <s id="id.2.1.3.3.1.3.0">ſit pun<lb></lb>ctum C immobile, quod vt pon<lb></lb>dus ſuſtineatur, neceſſe eſt. </s> <s id="id.2.1.3.3.1.4.0">& cum <lb></lb>punctum C ſit immobile, ſi pon<lb></lb>dus A mouebitur, punctum B cir<lb></lb>culi circumferentiam deſcribet, <lb></lb>cuius ſemidiameter erit CB. qua<lb></lb>re centro C, ſpatio verò BC, cir<lb></lb>culus deſcribatur BFDE. </s> <s id="id.2.1.3.3.1.4.0.a">ſitq; <lb></lb><figure id="id.036.01.019.1.jpg" place="text" xlink:href="036/01/019/1.jpg"></figure><lb></lb>primum BC horizonti perpendicularís, quæ vſq; ad D produca<lb></lb>tur; atq; punctum C ſit 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></lb>A ſecundum grauitatis centrum B deorſum mouetur; punctum <lb></lb>B deorſum in centrum mundi, quò naturaliter tendit, per re<lb></lb>ctam lineam BD mouebitur: totum ergo pondus A eius cen<lb></lb>tro grauitatis B ſuper rectam lineam BC graueſcet. </s> <s id="id.2.1.3.3.1.5.0">cum au<lb></lb>tem pondus à linea CB ſuſtineatur, linea CB totum ſuſti<lb></lb>nebit pondus A; ſuper quam deorſum moueri non poteſt, cum <lb></lb>ab ipſa prohibeatur: per definitionem igitur centri grauitatis pun<lb></lb>ctum B, ponduſq; A in hoc ſitu manebunt. </s> <s id="id.2.1.3.3.1.6.0">& quamquam B quo<lb></lb>cunq; alio puncto circuli ſit ſublimius, ab hoc tamen ſitu deorſum <lb></lb>per circuli circumferentiam nequaquam mouebitur non enim ver<lb></lb>ſus F magis, quàm verſus E inclinabitur, cum ex vtraq; parte æqua<lb></lb>lis ſit deſcenſus; neq; pondus A in vnam magis, quàm in alteram <lb></lb>partem propenſionem habeat: quod non accidit in quouis alio <lb></lb>puncto circumferentiæ circuli (præter D) ſit ponderis eiuſdem <pb xlink:href="036/01/020.jpg"></pb>centrum grauitatis, vt in F; cum ex <lb></lb>puncto F verſus D ſit deſcenſus, at <lb></lb>verò verſus B aſcenſus. </s> <s id="id.2.1.3.3.1.7.0">quare pun<lb></lb>ctum F deorſum mouebitur. </s> <s id="id.2.1.3.3.1.8.0">& quo<lb></lb>niam per rectam lineam in centrum <lb></lb>mundi moueri non poteſt, cum à <lb></lb>puncto C immobili propter lineam <lb></lb>CF prohibeatur; deorſum tamen <lb></lb>ſicuti eius natura poſtulat, ſemper <lb></lb>mouebitur. </s> <s id="id.2.1.3.3.1.9.0">& cum infimus locus ſit <lb></lb>D, per <expan abbr="circumferentiã">circumferentiam</expan> FD mouebi<lb></lb>tur, donec in D perueniat, in quo <lb></lb>ſitu manebit, <expan abbr="põduſq">ponduſq</expan>; immobile exi <lb></lb><figure id="id.036.01.020.1.jpg" place="text" xlink:href="036/01/020/1.jpg"></figure><lb></lb>ſtet. </s> <s id="id.2.1.3.3.1.10.0">tum quia deorſum amplius moueri non poteſt, cum ex pun<lb></lb>cto C ſit appenſum; tum etiam, quia in eius centro grauitatis ſuſti<lb></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></lb>ſimulq; horizonti perpendicularis. </s> <s id="id.2.1.3.3.1.12.0">pondus ergo nunquam mane<lb></lb>bit, donec linea CF horizonti perpendicularis non exiſtat. quod <lb></lb>oſtendere oportebat. </s> <s id="id.2.1.3.3.1.13.0">quod <lb></lb>oſtendere 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"></emph>Supp.<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></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 poteſt, pondus quocunq; modo <lb></lb>in dato puncto ſuſtineatur, nunquam manere; ni <lb></lb>ſi quando a centro grauitatis ponderis ad id pun<lb></lb>ctum ducta linea horizonti ſit 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 iiſdem poſitis, ſuſtineatur <lb></lb>pondus à lineis CG CH. </s> <s id="id.2.1.5.2.1.1.0.a">Dico <lb></lb>ſi ducta BC horizonti ſit perpen<lb></lb>dicularis, pondus A manere. </s> <s id="id.2.1.5.2.1.2.0">ſi verò <lb></lb>ducta CF non ſit horizonti per<lb></lb>pendicularis, punctum F deorſum <lb></lb>vſq; ad D moueri; in quo ſitu pon<lb></lb>dus manebit, ductaq; CD horizon<lb></lb>ti perpendicularis exiſtet. </s> <s id="id.2.1.5.2.1.3.0">quæ om<lb></lb>nia eadem ratione oſtendentur. <figure id="id.036.01.020.2.jpg" place="text" xlink:href="036/01/020/2.jpg"></figure></s> <pb n="4" xlink:href="036/01/021.jpg"></pb> <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 æquidiſtans, cuius centrum <lb></lb>ſit ſupra libram, æqualia in extremitatibus, æqua <lb></lb>literq; à perpendiculo diſtantia habens pondera, <lb></lb>ſi ab eiuſmodi moueatur ſitu, in eundem rurſus <lb></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></lb>nea horizonti æquidi<lb></lb>ſtans, cuius centrum C <lb></lb>ſit ſupra libram; ſitq; CD <lb></lb><expan abbr="perpendiculũ">perpendiculum</expan>, quod ho<lb></lb>rizonti perpendiculare <lb></lb>erit: atq; diſtantia DA ſit <lb></lb>diſtantiæ DB æqualis; <lb></lb>ſintq; in AB pondera æ<lb></lb>qualia, <expan abbr="quorũ">quorum</expan> grauitatis <lb></lb>centra ſint in AB <expan abbr="pũctis">punctis</expan>. </s> <s id="id.2.1.5.4.1.2.0"><lb></lb>Moueatur AB libra ab <lb></lb><figure id="id.036.01.021.1.jpg" place="text" xlink:href="036/01/021/1.jpg"></figure><lb></lb>hoc ſitu, putá in EF, deinde relinquatur. </s> <s id="id.2.1.5.4.1.3.0">dico libram EF in AB ho<lb></lb>rizonti æquidiſtantem redire, ibíq; manere. </s> <s id="id.2.1.5.4.1.4.0">Quoniam autem pun<lb></lb>ctum C eſt immobile, dum libra mouetur, punctum D circuli cir<lb></lb>cumferentiam deſcribet, cuius ſemidiameter erit CD. quare cen<lb></lb>tro C, ſpatio verò CD, circulus deſcribatur DGH. </s> <s id="id.2.1.5.4.1.4.0.a">Quoniam <lb></lb>enim CD ipſi libræ ſemper eſt perpendicularis, dum libra erit in <lb></lb>EF, linea CD erit in CG, ita vt CG ſit ipſi EF perpendicula<lb></lb>ris. </s> <s id="id.2.1.5.4.1.5.0">Cùm autem AB bifariam à puncto D diuidatur, & pondera <lb></lb>in AB ſint æqualia; erit magnitudinis ex ipſis AB compoſitæ cen<arrow.to.target n="note3"></arrow.to.target><lb></lb>trum grauitatis in medio, hoc eſt in D. & <expan abbr="quãdo">quando</expan> libra vná cum pon<lb></lb>deribus erit in EF; erit magnitudinis ex vtriſq; EF compoſitæ cen<lb></lb>trum grauitatis G. </s> <s id="id.2.1.5.4.1.5.0.a">& quoniam CG horizonti non eſt perpendi<lb></lb>cularis; <arrow.to.target n="note4"></arrow.to.target>magnitudo ex ponderibus EF compoſita in hoc ſitu mi<lb></lb>nimè perſiſtet, ſed deorſum <expan abbr="ſecũdùm">ſecundùm</expan> eius centrum grauitatis G per <lb></lb>circumferentiam GD mouebitur; donec CG horizonti fiat per<pb xlink:href="036/01/022.jpg"></pb>pendicularis, ſcilicet do<lb></lb>nec CG in CD redeat. </s> <s id="id.2.1.5.4.1.6.0"><lb></lb>Quando autem CG erit <lb></lb>in CD, linea EF, cùm <lb></lb>ipſi CG ſemper ad rectos <lb></lb>ſit angulos, erit in AB; in <lb></lb><arrow.to.target n="note5"></arrow.to.target>quo ſitu quoq; manebit. </s> <s id="id.2.1.5.4.1.7.0">li<lb></lb>bra ergo EF in AB hori<lb></lb>zonti <expan abbr="æquidiſtãtem">æquidiſtantem</expan> redi<lb></lb>bit, ibíq; manebit. </s> <s id="id.2.1.5.4.1.8.0">quod <lb></lb>demonſtrare 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"></emph>primi Archi<lb></lb>medis de <lb></lb>æqueponde<lb></lb>rantibus.<emph.end type="italics"></emph.end></s> <s id="id.2.1.6.1.1.2.0"><margin.target id="note4"></margin.target>1. <emph type="italics"></emph>Huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.6.1.1.3.0"><margin.target id="note5"></margin.target>1. <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.036.01.022.1.jpg" place="text" 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 æquidiſtans æqualia in extre<lb></lb>mitatibus, æqualiterq; à perpendiculo diſtan<lb></lb>tia habens pondera, centro infernè collocato, in <lb></lb>hoc ſitu manebit. </s> <s id="id.2.1.7.2.1.2.0">ſi verò inde moueatur, deor<lb></lb>ſum relicta, ſecundùm partem decliuiorem mo<lb></lb>uebitur. <figure id="id.036.01.022.2.jpg" place="text" 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></lb>nea horizonti æquidi<lb></lb>ſtans, cuius centrum C <lb></lb>ſit infra libram; perpen<lb></lb>diculumq; ſit CD, quod <lb></lb>horizonti perpendiculare <lb></lb>erit; & diſtantia AD ſit <lb></lb>diſtantiæ DB æqualis; <lb></lb>ſintq; in AB pondera <lb></lb>æqualia, quorum grauita<lb></lb>tis centra ſint in punctis <lb></lb>AB. </s> <s id="id.2.1.7.3.1.1.0.a">Dico primùm libram AB in hoc ſitu manere. </s> <s id="id.2.1.7.3.1.2.0">Quoniam <lb></lb>enim AB bifariam diuiditur à puncto D, & pondera in AB ſunt <lb></lb>æqualia; erit punctum D centrum grauitatis magnitudinis ex <pb n="5" xlink:href="036/01/023.jpg"></pb>vtriſq; AB ponderibus compoſitæ. </s> <s id="id.2.1.7.3.1.3.0">& CD libram ſuſtinens ho<lb></lb>rizonti <arrow.to.target n="note6"></arrow.to.target>eſt perpendicularis, libra ergo AB in hoc ſitu manebit. <arrow.to.target n="note7"></arrow.to.target><lb></lb>moueatur autem libra AB ab hoc ſitu, putà in EF, deinde relinqua<lb></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></lb>ipſi libræ ſemper eſt perpendicularis, dum libra erit in EF, erit <lb></lb>CD in CG ipſi EF perpendicularis. </s> <s id="id.2.1.7.3.1.6.0">& punctum G magnitudi<lb></lb>nis ex EF compoſitæ centrum grauitatis erit; quod dum moue<lb></lb>tur, circuli circumferentiam deſcribet DGH, cuius ſemidiameter <lb></lb>CD, & centrum C. </s> <s id="id.2.1.7.3.1.6.0.a">Quoniam autem CG horizonti non eſt per<lb></lb>pendicularis, magnitudo ex EF ponderibus compoſita in hoc ſi<lb></lb>tu minimè manebit; ſed ſecundùm eius grauitatis centrum G deor<lb></lb>ſum per circumferentiam GH mouebitur. </s> <s id="id.2.1.7.3.1.7.0">libra ergo EF ex par <lb></lb>te F deorſum mouebitur, quod demonſtrare 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"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.8.1.1.3.0"><margin.target id="note7"></margin.target>1. <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 æquidiſtans æqualia in ex<lb></lb>tremitatibus, æqualiterq; à centro in ipſa libra <lb></lb>collocato, diſtantia habens pondera; ſiue inde <lb></lb>moueatur, ſiue minus; vbicunq; relicta, manebit. <figure id="id.036.01.023.1.jpg" place="text" 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></lb>B horizonti æquidiſtans, <lb></lb>cuius centrum C in ea<lb></lb>dem ſit linea AB; diſtan<lb></lb>tia verò CA ſit diſtantiæ <lb></lb>CB æqualis: ſintq; pon<lb></lb>dera in AB æqualia, quo<lb></lb>rum centra grauitatis ſint <lb></lb>in <expan abbr="puntis">punctis</expan> AB. </s> <s id="id.2.1.9.3.1.1.0.a">Moueatur <lb></lb>libra, vt in DE, ibiquè <lb></lb>relinquatur. </s> <s id="id.2.1.9.3.1.2.0">Dico primùm libram DE non moueri, in eoquè ſitu <lb></lb>manere. </s> <s id="id.2.1.9.3.1.3.0">Quoniam enim pondera AB ſunt æqualia; erit magni<lb></lb>tudinis ex vtroq; pondere, videlicet A, & B compoſitæ centrum <lb></lb>grauitatis C. quare idem punctum C, & centrum libræ, & <expan abbr="centrũ">centrum</expan><lb></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"></pb>C, dum libra AB vnà <lb></lb>cum ponderibus in DE <lb></lb>mouetur, immobile re<lb></lb>manet, centrum quoq; <lb></lb>grauitatis, quod eſt idem <lb></lb>C, non mouebitur. </s> <s id="id.2.1.9.3.1.5.0">nec <lb></lb>igitur libra DE mouebi<lb></lb>tur, per definitionem <lb></lb>centri grauitatis, cum in <lb></lb>ipſo ſuſpendatur. </s> <s id="id.2.1.9.3.1.6.0">Idip<lb></lb><figure id="id.036.01.024.1.jpg" place="text" xlink:href="036/01/024/1.jpg"></figure><lb></lb>ſum quoq; contingit libra in AB horizonti æquidiſtante, vel in <lb></lb>quocunq; alio ſitu exiſtente. </s> <s id="id.2.1.9.3.1.7.0">Manebit ergo libra, vbi relinque<lb></lb>tur. </s> <s id="id.2.1.9.3.1.8.0">quod demonſtrare 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 ſunt, grauitatis tantùm magnitudi<lb></lb>num, quæ in extremitatibus libræ poſitæ ſunt æquales, abſq; lí<lb></lb>bræ grauitate conſiderauerimus; quoniam tamen adhuc libræ bra<lb></lb>chia ſunt æqualia, idcirco idem libræ, eius grauitate conſiderata, <lb></lb>vnà cum ponderibus, vel ſine ponderibus eueniet. </s> <s id="id.2.1.9.4.1.2.0">idem enim cen<lb></lb>trum grauitatis fine ponderibus libræ tantùm grauitatis centrum <lb></lb>erit. </s> <s id="id.2.1.9.4.1.3.0">Similiter ſi pondera in libræ extremitatibus appendantur, vt <lb></lb>fieri ſolet, idem eueniet; dummodo ex ſuſpenſionum punctis ad <lb></lb>centra grauitatum ponderum ductæ lineæ (quocunq; modo mo<lb></lb>ueatur libra) ſi protrahantur, in centrum mundi concurrant. </s> <s id="id.2.1.9.4.1.4.0">vbi <lb></lb>enim pondera hoc modo ſunt appenſa, ibi graueſcunt, ac ſi in iiſ<lb></lb>dem punctis centra grauitatum haberent. </s> <s id="id.2.1.9.4.1.5.0">præterea, quæ ſequun<lb></lb>tur, eodem prorſus modo conſiderare 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></lb>aliter ſentientibus dicta officere videntur; idcirco in hac parte ali<lb></lb><arrow.to.target n="note9"></arrow.to.target>quantulum immorari oportebit; & pro viribus, non ſolum pro<lb></lb>priam ſententiam, ſed Archimedem ipſum, qui in hac eadem eſſe <lb></lb><arrow.to.target n="note10"></arrow.to.target>ſententia videtur, defendere conabor. <pb n="6" xlink:href="036/01/025.jpg"></pb> <figure id="id.036.01.025.1.jpg" place="text" 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">Iiſdem poſitis, duca<lb></lb>tur FCG ipſi AB, & <lb></lb>horizonti perpendicula<lb></lb>ris; & centro C, ſpatio<lb></lb>què CA, circulus deſcri<lb></lb>batur ADFBEG. erunt <lb></lb>puncta ADBE in circu<lb></lb>li circumferentia; cum li<lb></lb>bræ brachia ſint æqualia. </s> <s id="id.2.1.9.6.1.2.0"><lb></lb>& quoniam in vnam con<lb></lb>ueniunt ſententiam, aſſe<lb></lb>rentes ſcilicet libram DE <lb></lb>neq; in FG moueri, ne<lb></lb>que in DE manere, ſed in AB horizonti æquidiſtantem rediré. </s> <s id="id.2.1.9.6.1.3.0"><lb></lb>hanc eorum ſententiam nullo modo conſiſtere poſſe oſtendam. </s> <s id="id.2.1.9.6.1.4.0"><lb></lb>Non enim, ſed ſi quod aiunt, euenerit, vel ideo erit, quia pondus <lb></lb>D pondere E grauius fuerit, vel ſi pondera ſunt æqualia, diſtantiæ, <lb></lb>quibus ſunt poſita, non erunt æquales, hoc eſt CD ipſi CE non erit <lb></lb>æqualis, ſed maior. </s> <s id="id.2.1.9.6.1.5.0">Quòd autem pondera in DE ſint æqualia, & <lb></lb>diſtantia CD ſit æqualis diſtantiæ CE: hæc ex ſuppoſitione pa<lb></lb>tent. </s> <s id="id.2.1.9.6.1.6.0">Sed quoniam dicunt pondus in D in eo ſitu pondere in E <lb></lb>grauius eſſe in altero ſitu deorſum: dum pondera ſunt in DE, pun<lb></lb>ctum C non erit amplius centrum grauitatis, nam non manent, ſi <lb></lb>ex C ſuſpendantur; ſed erit in linea CD, ex tertia primi Archi<lb></lb>medis de æqueponderantibus. </s> <s id="id.2.1.9.6.1.7.0">non autem erit in linea CE, cum pon<lb></lb>dus D grauius ſit pondere E. ſit igitur in H, in quo ſi ſuſpendan<lb></lb>tur, manebunt. </s> <s id="id.2.1.9.6.1.8.0">Quoniam autem centrum grauitatis ponderum <lb></lb>in AB connexorum eſt punctum C; ponderum verò in DE eſt <lb></lb>punctum H: dum igitur pondera AB mouentur in DE, centrum <lb></lb>grauitatis C verſus D mouebitur, & ad D propius accedet; quod <lb></lb>eſt impoſsibile: cum pondera eandem inter ſe ſe ſeruent diſtantiam. </s> <s id="id.2.1.9.6.1.9.0"><lb></lb>Vniuſcuiuſq; enim corporis centrum grauitatis in eodem ſemper <arrow.to.target n="note11"></arrow.to.target><lb></lb>eſt ſitu reſpectu ſui corporis. </s> <s id="id.2.1.9.6.1.10.0">& quamquam punctum C ſit duo<lb></lb>rum corporum AB centrum grauitatis, quia tamen inter ſe ſe ita à <lb></lb>libra connexa ſunt, vt ſemper eodem modo ſe ſe habeant; Ideo <lb></lb>punctum C ita eorum erit centrum grauitatis, ac ſi vna tantum <pb xlink:href="036/01/026.jpg"></pb><arrow.to.target n="note12"></arrow.to.target>eſſet magnitudo. </s> <s id="id.2.1.9.6.1.11.0">libra <lb></lb>enim vna cum ponderi<lb></lb>bus vnum tantum conti<lb></lb>nuum efficit, cuius cen<lb></lb>trum grauitatis erit ſem<lb></lb>per in medio. </s> <s id="id.2.1.9.6.1.12.0">non igitur <lb></lb>pondus in D pondere in <lb></lb>E eſt grauius. </s> <s id="id.2.1.9.6.1.13.0">Si autem <lb></lb>dicerent centrum graui<lb></lb>tatis non in linea CD, <lb></lb>ſed in CE eſſe debere; <lb></lb>idem eueniet abſurdum. <figure id="id.036.01.026.1.jpg" place="text" 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 ſi pondus D <lb></lb>deorſum mouebitur, pondus E ſurſum mouebit. </s> <s id="id.2.1.9.7.1.2.0">pondus igitur gra<lb></lb>uius, quàm ſit E, in eodemmet ſitu ponderi D æqueponderabit, & <lb></lb>grauia inæqualia æquali diſtantia poſita æqueponderabunt. </s> <s id="id.2.1.9.7.1.3.0">Adii<lb></lb>ciatur ergo ponderi E aliquod graue, ita vt ipſi D contraponde<lb></lb>ret, ſi ex C ſuſpendantur. </s> <s id="id.2.1.9.7.1.4.0">ſed cum ſupra oſtenſum ſit punctum C <lb></lb>centrum eſſe grauitatis æqualium ponderum in DE; ſi igitur pon<lb></lb><arrow.to.target n="note13"></arrow.to.target>dus E grauius fuerit pondere D, erit centrum grauitatis in linea <lb></lb>CE. </s> <s id="id.2.1.9.7.1.4.0.a">ſitq; hoc centrum K. </s> <s id="id.2.1.9.7.1.4.0.b">at per definitionem centri grauitatis, ſi <lb></lb>pondera ſuſpendantur ex K, manebunt. </s> <s id="id.2.1.9.7.1.5.0">ergo ſi ſuſpendantur ex <lb></lb>C, non manebunt, quod eſt contra hypoteſim: ſed pondus E deor<lb></lb>ſum mouebitur. </s> <s id="id.2.1.9.7.1.6.0">quòd ſi ex C quoque ſuſpenſa æqueponderarent; <lb></lb><arrow.to.target n="note14"></arrow.to.target>vnius magnitudinis duo eſſent centra grauitatis; quod eſt impoſsi<lb></lb>bile. </s> <s id="id.2.1.9.7.1.7.0">Non igitur pondus in E grauius eo, quod eſt in D, ipſi D æque<lb></lb>ponderabit, cum ex puncto C fiat ſuſpenſio. </s> <s id="id.2.1.9.7.1.8.0">Pondera ergo in DE <lb></lb>æqualia ex eorum grauitatis centro C ſuſpenſa, æqueponderabunt, <lb></lb>manebuntquè. </s> <s id="id.2.1.9.7.1.9.0">quod demonſtrare fuerat propoſitum. </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"></emph>Iordanus de Ponderibus. <emph.end type="italics"></emph.end></s> <s id="id.2.1.10.1.1.2.0"><margin.target id="note9"></margin.target><emph type="italics"></emph>Hyerommus Cardanus de ſubtilitate. <emph.end type="italics"></emph.end></s> <s id="id.2.1.10.1.1.3.0"><margin.target id="note10"></margin.target><emph type="italics"></emph>Nicolaus Tartalea de quæſitis, ac inuentionibus. <emph.end type="italics"></emph.end></s> <s id="id.2.1.10.1.1.4.0"><margin.target id="note11"></margin.target>2. <emph type="italics"></emph>Sup. huius. <emph.end type="italics"></emph.end></s> <s id="id.2.1.10.1.1.6.0"><margin.target id="note12"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>primi Archim de Aequep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.10.1.1.7.0"><margin.target id="note13"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 3. <emph type="italics"></emph>primi Archim de Aequep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.10.1.1.8.0"><margin.target id="note14"></margin.target>1. <emph type="italics"></emph>Suppoſ. huius.<emph.end type="italics"></emph.end></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 poſtremo inconuenienti occurrunt dicentes, im<lb></lb>poſsibile eſſe addere ipſi E pondus adeo minimum, quin adhuc ſi <lb></lb>ex C ſuſpendantur, pondus E ſemper deorſum verſus G moueatur. </s> <s id="id.2.1.11.1.1.2.0"><lb></lb>quod nos fieri poſſe ſuppoſuimus, atque fieri poſſe credebamus. </s> <s id="id.2.1.11.1.1.3.0">ex<lb></lb>ceſſum enim ponderis D ſupra pondus E, cum quantitatis ratio<lb></lb>nem habeat, non ſolum minimum eſſe, verum in infinitum diuidi <lb></lb>poſſe immaginabamur, quod quidem ipſi, non ſolum minimum, <pb n="7" xlink:href="036/01/027.jpg"></pb>ſed ne minimum quidem eſſe, cum reperiri non poſsit, hoc mo<lb></lb>do demonſtrare nituntur. <figure id="id.036.01.027.1.jpg" place="text" 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></lb>à punctiſquè DE hori<lb></lb>zonti <expan abbr="perpẽdiculares">perpendiculares</expan> du<lb></lb><expan abbr="cãtur">cantur</expan> DHEK, atq; alius <lb></lb>ſit circulus LDM, cu<lb></lb>ius <expan abbr="centrũ">centrum</expan> N, qui FDG <lb></lb>in puncto D contingat, <lb></lb>ipſiq; FDG ſit æqualis: <lb></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></lb>quoniam angulus KEC <lb></lb>angulo HDN eſt æqua <arrow.to.target n="note17"></arrow.to.target><lb></lb>lis, angulusq; CEG an<lb></lb>gulo NDM eſt etiam <lb></lb>æqualis; cum à ſemidiametris, æqualibusq; circumferentiis conti<lb></lb>neatur; erit reliquus mixtuſquè angulus KEG reliquo mixtoquè <lb></lb>HDM æqualis. </s> <s id="id.2.1.11.2.1.4.0">& quia ſupponunt, quò minor eſt angulus linea <lb></lb>horizonti perpendiculari, & circumferentia contentus, eò pondus <lb></lb>in eo ſitu grauius eſſe. </s> <s id="id.2.1.11.2.1.5.0">vt quò minor eſt angulus HD, & circumfe<lb></lb>rentia DG contentus angulo KEG, hoc eſt angulo HDM; ita ſe<lb></lb>cundum hanc proportionem pondus in D grauius eſſe pondere in <lb></lb>E. </s> <s id="id.2.1.11.2.1.5.0.a">Proportio autem anguli MDH ad angulum HDG minor eſt <lb></lb>qualibet proportione, quæ ſit inter maiorem, & minorem quanti<lb></lb>tatem: ergo proportio ponderum DE omnium proportionum mi<lb></lb>nima erit. </s> <s id="id.2.1.11.2.1.6.0">immo neq; erit ferè proportio, cum ſit omnium pro <lb></lb>portionum minima. </s> <s id="id.2.1.11.2.1.7.0">quòd autem proportio MDH ad HDG ſit <lb></lb>omnium minima, ex hac neceſsitate oſtendunt; quia MDH exce<lb></lb>dit HDG angulo curuilineo MDG, qui quidem angulus omnium <lb></lb>angulorum rectilineorum minimus exiſtit: ergo cum non poſsit da <lb></lb>ri angulus minor MDG, erit proportio MDH ad HDG <expan abbr="omniũ">omnium</expan><lb></lb>proportionum minima. </s> <s id="id.2.1.11.2.1.8.0">quæ ratio inutilis valde videtur eſſe; quia <lb></lb>quamquam angulus MDG ſit omnibus rectilineis angulis minor, <lb></lb>non idcirco ſequitur, abſolutè, ſimpliciterq; omnium eſſe <expan abbr="angulorũ">angulorum</expan><lb></lb>minimum: nam ducatur à puncto D linea DO ipſi NC perpendicu<lb></lb>laris, hæc vtraſq; tanget circumferentias LDM FDG in puncto <arrow.to.target n="note18"></arrow.to.target> <pb xlink:href="036/01/028.jpg"></pb>D. </s> <s id="N109F9">quia verò circumfe<lb></lb>rentiæ ſunt æquales, erit <lb></lb>angulus MDO mixtus <lb></lb>angulo ODG mixto <lb></lb>æqualis; alter ergo an<lb></lb>gulus, vt ODG minor <lb></lb>erit MDG, hoc eſt mi <lb></lb>nor minimo. </s> <s id="id.2.1.11.2.1.9.0">angulus <lb></lb>deinde OGH minor <lb></lb>erit angulo MDH; qua <lb></lb>re ODH ad angulum <lb></lb><arrow.to.target n="note19"></arrow.to.target>HDG minorem habe<lb></lb>bit <expan abbr="proportionẽ">proportionem</expan>, quàm <lb></lb><figure id="id.036.01.028.1.jpg" place="text" xlink:href="036/01/028/1.jpg"></figure><lb></lb>MDH ad eundem HDG. </s> <s id="N10A25">dabitur ergo quoquè proportio mi<lb></lb>nor minima, quam in infinitum adhuc minorem ita oſtende<lb></lb>mus. </s> <s id="id.2.1.11.2.1.10.0">Deſcribatur circulus DR, cuius centrum E, & ſemidiame<lb></lb><arrow.to.target n="note20"></arrow.to.target>ter ED. continget circumferentia DR circumferentiam DG in <lb></lb><arrow.to.target n="note21"></arrow.to.target>puncto D, lineamquè DO in puncto D; quare minor erit angu<lb></lb>lus RDG angulo ODG. ſimiliter & angulus RDH angulo <lb></lb>ODH. </s> <s id="id.2.1.11.2.1.10.0.a">minorem igitur proportionem habebit RDH ad HDG, <lb></lb>quàm ODH ad HDG. </s> <s id="id.2.1.11.2.1.10.0.b">Accipiatur deinde inter EC vtcun<lb></lb>que punctum P, ex quo in diſtantia PD alia deſcribatur circum<lb></lb>ferentia DQ, quæ circumferentiam DR, circumferentiamquè <lb></lb>DG in puncto D continget; & angulus QDH minor erit <lb></lb>angulo RDH: ergo QDH ad HDG minorem habebit propor<lb></lb>tionem, quàm RDH ad HDG. </s> <s id="N10A4E">eodemquè prorſus modo, ſi <lb></lb>inter PC aliud accipiatur punctum, & inter hoc &C aliud, & ſic <lb></lb>deinceps, infinitæ deſcribentur circumferentiæ inter DO, & cir<lb></lb>cumferentiam DG; ex quibus proportionem in infinitum ſemper <lb></lb>minorem inueniemus. </s> <s id="id.2.1.11.2.1.11.0">atque ideo proportionem ponderis in D <lb></lb>ad pondus in E non adeo minorem eſſe ſequitur, quin ad infini <lb></lb>tum ipſa ſemper minorem reperiri poſsit. </s> <s id="id.2.1.11.2.1.12.0">& quia angulus MDG <lb></lb>in infinitum diuidi poteſt; exceſſus quoque grauitatis D ſupra E <lb></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"></emph>Tartalea ſexta propoſitione octaui libri.<emph.end type="italics"></emph.end></s> <s id="id.2.1.12.1.1.2.0"><margin.target id="note16"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 12. <emph type="italics"></emph>tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.12.1.1.3.0"><margin.target id="note17"></margin.target>29. <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.12.1.1.4.0"><margin.target id="note18"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.12.1.1.5.0"><margin.target id="note19"></margin.target>8. <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.12.1.1.6.0"><margin.target id="note20"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 11. <emph type="italics"></emph>tertit.<emph.end type="italics"></emph.end></s> <s id="id.2.1.12.1.1.7.0"><margin.target id="note21"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 18. <emph type="italics"></emph>tertii.<emph.end type="italics"></emph.end></s> </p> <pb n="8" xlink:href="036/01/029.jpg"></pb> <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></lb>eſt, ipſos in demonſtratio<lb></lb>ne angulum KEG maiorem <lb></lb>eſſe angulo HDG, tanquam <lb></lb>notum accepiſſe. </s> <s id="id.2.1.13.1.2.2.0">quod eſt <lb></lb>quidem verum, ſi DHEK <lb></lb>inter ſe ſe ſint æquidiſtan<lb></lb>tes. </s> <s id="id.2.1.13.1.2.3.0">Quoniam autem (vt <lb></lb>ipſi quoque ſupponunt) li<lb></lb>neæ DHEK in centrum <lb></lb>mundi conueniunt; lineæ <lb></lb>DHEK æquidiſtantes nun<lb></lb>quam erunt, & angulus KEG <lb></lb>angulo HDG non ſolum <lb></lb>maior erit, ſed minor. </s> <s id="id.2.1.13.1.2.4.0">vt <lb></lb>exempli gratia, producatur <lb></lb>FG vſque ad centrum mun<lb></lb>di, quod ſit S; connectan<lb></lb>tur〈qué〉 DSES. </s> <s id="N10AF9">oſtenden<lb></lb>dum eſt angulum SEG mi<lb></lb>norem eſſe angulo SDG. </s> <s id="id.2.1.13.1.2.4.0.a">du<lb></lb><figure id="id.036.01.029.1.jpg" place="text" xlink:href="036/01/029/1.jpg"></figure><lb></lb>catur à puncto E linea ET circulum DGEF contingens, ab eo <lb></lb>dem〈qué〉 puncto ipſi DS æquidiſtans ducatur EV. </s> <s id="id.2.1.13.1.2.4.0.b">Quoniam igi<lb></lb>tur EVDS inter ſe ſe ſunt æquidiſtantes: ſimiliter ETDO æqui <lb></lb>diſtantes: erit angulus VET angulo SDO æqualis. </s> <s id="id.2.1.13.1.2.5.0">& angulus <lb></lb>TEG angulo ODM eſt æqualis; cum à lineis contingentibus, <lb></lb>circumferentiiſ〈qué〉 æqualibus contineatur: totus ergo angulus <lb></lb>VEG angulo SDM æqualis erit. </s> <s id="id.2.1.13.1.2.6.0">Auferatur ab angulo SDM <lb></lb>angulus curuilineus MDG; ab angulo autem VEG angulus au<lb></lb>feratur VES; & angulus VES rectilineus maior eſt curuilineo <lb></lb>MDG; erit reliquus angulus SEG minor angulo SDG. </s> <s id="id.2.1.13.1.2.6.0.a"><lb></lb>Quare ex ipſorum ſuppoſitionibus non ſolum pondus in D gra<lb></lb>uius erit pondere in E; verùm è conuerſo, pondus in E ipſo D <lb></lb>grauius exiſtet. </s> </p> <pb xlink:href="036/01/030.jpg"></pb> <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></lb>ferunt, quibus demon<lb></lb>ſtrare nituntur, libram <lb></lb>DE in AB horizon<lb></lb>ti æquidiſtantem ex <lb></lb>neceſsitate redire. </s> <s id="id.2.1.13.3.1.2.0"><expan abbr="Primùm">Pri<lb></lb>mum</expan> quidem oſten<lb></lb>dunt, idem pondus <lb></lb>grauius eſſe in A, <lb></lb>quàm in alio ſitu, quem <lb></lb>æqualitatis ſitum no<lb></lb>minant, cum linea <lb></lb>AB ſit horizonti æ<lb></lb><figure id="id.036.01.030.1.jpg" place="text" xlink:href="036/01/030/1.jpg"></figure><lb></lb>quidiſtans. </s> <s id="id.2.1.13.3.1.3.0">deinde quò propius eſt ipſi A, quouis alio remotiori <lb></lb>grauius eſſe. </s> <s id="id.2.1.13.3.1.4.0">Vt pondus in A grauius eſſe, quàm in D; & in D, <lb></lb>quàm in L. ſimiliter in A grauius, quam in N; & in N grauius, <lb></lb>quàm in M. </s> <s id="id.2.1.13.3.1.4.0.a">Vnum tantùm conſiderando pondus in altero libræ <lb></lb><arrow.to.target n="note22"></arrow.to.target>brachio ſurſum deorſumq; moto. </s> <s id="id.2.1.13.3.1.5.0">Quia (inquiunt) poſita trutina <lb></lb>in CF, pondus in A longius eſt à trutina, quàm in D: & in D <lb></lb>longius, quàm in L. </s> <s id="N10B77">ductis enim DO LP ipſi CF perpendicula<lb></lb><arrow.to.target n="note23"></arrow.to.target>ribus, linea AC maior eſt, quàm DO, & DO ipſa LP. </s> <s id="N10B7E">quod <lb></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></lb>dus velocius mouetur, ibi grauius eſt; velocius autem ex A, quàm <lb></lb>ab alio ſitu mouetur; ergo in A grauius eſt. </s> <s id="id.2.1.13.3.1.6.0">ſimili modo, quò <lb></lb>propius eſt ipſi A, velocius quoque mouetur; ergo in D gra<lb></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 cauſa, quam ex rectiori, & obli<lb></lb><arrow.to.target n="note26"></arrow.to.target>quiori motu deducunt, eſt; quò pondus in arcubus æqualibus re<lb></lb>ctius deſcendit, grauius eſſe videtur; cum pondus liberum, atq; <lb></lb><arrow.to.target n="note27"></arrow.to.target>ſolutum ſuaptè natura rectè moueatur; ſed in A rectius deſcen<lb></lb>dit; ergo in A grauius erit. </s> <s id="id.2.1.13.3.1.7.0">hocq; oſtendunt accipiendo arcum <lb></lb>AN arcui LD æqualem; à punctiſq; NL lineæ FG (quam <lb></lb>etiam directionis vocant) æquidiſtantes ducantur NRLQ, quæ <lb></lb>lineas AB DO ſecent in QR; & à puncto N ipſi FG perpen<lb></lb>dicularis ducatur NT. </s> <s id="id.2.1.13.3.1.7.0.a">rectèq; demonſtrant LQ ipſi PO æqua<lb></lb>lem eſſe, & NR ipſi CT; lineamq; NR ipſa LQ maiorem eſſe. </s> <s id="id.2.1.13.3.1.8.0"><lb></lb>Quoniam autem deſcenſu; ponderis ex A vſq; ad N per circum<pb n="9" xlink:href="036/01/031.jpg"></pb>ferentiam AN maiorem portionem lineæ FG pertranſit (quod <lb></lb>ipſi vocant capere de directo) quàm deſcenſus ex L in D per cir<lb></lb>cumferentiam LD; cùm deſcenſus AN lineam CT pertranſeat, <lb></lb>deſcenſus verò LD lineam PO; & CT maior eſt PO; rectior erit <lb></lb>deſcenſus AN, quám deſcenſus LD. </s> <s id="id.2.1.13.3.1.8.0.a">grauius ergo erit pondus <lb></lb>in A, quàm in L, & in quouis alio ſitu. </s> <s id="id.2.1.13.3.1.9.0">eodemq; prorſus <lb></lb>modo oſtendunt, quò propius eſt ipſi A, grauius eſſe. </s> <s id="id.2.1.13.3.1.10.0"><lb></lb>Vt ſint circumferentiæ LD DA inter ſe ſe æquales, & à puncto <lb></lb>D ipſi AB perpendicularis ducatur DR; erit DR ipſi CO æqua <arrow.to.target n="note28"></arrow.to.target><lb></lb>lis. </s> <s id="id.2.1.13.3.1.11.0">lineam deinde DR ipſa LQ maiorem eſſe demonſtrant. </s> <s id="id.2.1.13.3.1.12.0">di<lb></lb>cuntq; deſcenſum DA magis capere de directo deſcenſu LD, ma<lb></lb>ior enim eſt linea CO, quàm OP; quare pondus grauius erit <lb></lb>in D, quàm in L. quod ipſum euenit in punctis NM. </s> <s id="id.2.1.13.3.1.12.0.a">Suppo<lb></lb>ſitionem itaq;, qua libram DE in AB redire demonſtrant, vt <arrow.to.target n="note29"></arrow.to.target><lb></lb>notam, manifeſtamq; proferunt. </s> <s id="id.2.1.13.3.1.13.0">Nempè Secundùm ſitum pon<lb></lb>dus grauius eſſe, quanto in eodem ſitu minus obliquus eſt deſcen<lb></lb>ſus. </s> <s id="id.2.1.13.3.1.14.0">huiuſq; reditus cauſam eam eſſe dicunt; Quoniam ſcilicet <arrow.to.target n="note30"></arrow.to.target><lb></lb>deſcenſus ponderis in D rectior eſt deſcenſu ponderis in E, cùm <lb></lb>minus capiat de directo pondus in E deſcendendo, quàm pon<arrow.to.target n="note31"></arrow.to.target><lb></lb>dus in D ſim liter deſcendendo. </s> <s id="id.2.1.13.3.1.15.0">Vt ſi arcus EV ſit ipſi DA <lb></lb>æqualis, ducanturq; VH ET ipſi FG perpendiculares; maior <lb></lb>erit DR, quàm TH. </s> <s id="N10C0D">quare per ſuppoſitionem pondus in D ra<lb></lb>tione ſitus grauius erit pondere in E. </s> <s id="id.2.1.13.3.1.15.0.a">pondus ergo in D, cùm ſit <lb></lb>grauius, deorſum mouebitur; pondus verò in E ſurſum, donec li<lb></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"></emph>Cardanus primo de ſubtilitate. <emph.end type="italics"></emph.end></s> <s id="id.2.1.14.1.1.2.0"><margin.target id="note23"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 15. <emph type="italics"></emph>tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.14.1.1.3.0"><margin.target id="note24"></margin.target><emph type="italics"></emph>Cardanus. <emph.end type="italics"></emph.end></s> <s id="id.2.1.14.1.1.4.0"><margin.target id="note25"></margin.target><emph type="italics"></emph>Cardanus. <emph.end type="italics"></emph.end></s> <s id="id.2.1.14.1.1.5.0"><margin.target id="note26"></margin.target><emph type="italics"></emph>Iordanus propoſitio ne<emph.end type="italics"></emph.end> 4. </s> <s id="id.2.1.14.1.1.6.0"><margin.target id="note27"></margin.target><emph type="italics"></emph>Tartalea propoſitione<emph.end type="italics"></emph.end> 5. </s> <s id="id.2.1.14.1.1.7.0"><margin.target id="note28"></margin.target>34 <emph type="italics"></emph>Primi. <emph.end type="italics"></emph.end></s> <s id="id.2.1.14.1.1.8.0"><margin.target id="note29"></margin.target><emph type="italics"></emph>Iordanus ſuppoſitione<emph.end type="italics"></emph.end> 4. </s> <s id="id.2.1.14.1.1.9.0"><margin.target id="note30"></margin.target><emph type="italics"></emph>Iordanus propoſitio ne<emph.end type="italics"></emph.end> 3. </s> <s id="id.2.1.14.1.1.10.0"><margin.target id="note31"></margin.target><emph type="italics"></emph>Tartalea propoſitio ne<emph.end type="italics"></emph.end> 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 eſt, cùm trutina ſupra libram <arrow.to.target n="note32"></arrow.to.target><lb></lb>eſt in CF; linea CG eſt meta. </s> <s id="id.2.1.15.1.1.2.0">& quoniam angulus GCD ma<lb></lb>ior eſt angulo GCE, & maior à meta angulus grauius reddit <lb></lb>pondus; trutina igitur ſuperius exiſtente, grauius erit pondus in <lb></lb>D, quàm in E. </s> <s id="N10C95">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"></emph>Cardanus.<emph.end type="italics"></emph.end></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 oſtendere libram DE in AB re<lb></lb>dire; quæ meo quidem iuditio facile ſolui poſſunt. </s> </p> <pb xlink:href="036/01/032.jpg"></pb> <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></lb>tum attinet ad ratio<lb></lb>nes pondus in A gra<lb></lb>uius eſſe, quàm in a<lb></lb>lio ſitu oſtendentes, <lb></lb>quas ex longiori, & <lb></lb>propinquiori <expan abbr="diſtãtia">diſtantia</expan> à <lb></lb>linea FG, & ex velo<lb></lb>ciori, & rectiori mo <lb></lb>tu à puncto A dedu<lb></lb>cunt; primùm quidem <lb></lb>non demonſtrant, cur <lb></lb>pondus ex A velocius <lb></lb><figure id="id.036.01.032.1.jpg" place="text" xlink:href="036/01/032/1.jpg"></figure><lb></lb>moueatur, quàm ex alio ſitu. </s> <s id="id.2.1.17.3.1.2.0">nec quia CA eſt DO maior, <lb></lb>& DO ipſa LP, propterea ſequitur tanquam ex vera cauſa, pon<lb></lb>dus in A grauius eſſe, quàm in D; & in D, quàm in L. </s> <s id="id.2.1.17.3.1.2.0.a">neq; <lb></lb>enim intellectus quieſcit, niſi alia huius oſtendatur cauſa; cùm po<lb></lb>tius ſignum, quàm vera cauſa eſſe videatur. </s> <s id="id.2.1.17.3.1.3.0">id ipſum quoq; al<lb></lb>teri rationi contintingit, quam ex rectiori & obliquiori motu de<lb></lb>ducunt. </s> <s id="id.2.1.17.3.1.4.0">Præterea quæcunq; ex velociori, & rectiori motu per<lb></lb>ſuadent pondus in A grauius eſſe, quàm in D; non ideo de<lb></lb>monſtrant pondus in A, quatenus eſt in A, grauius eſſe pon<lb></lb>dere in D, quatenus eſt in D; ſed quatenus à punctis DA rece<lb></lb>dit. </s> <s id="id.2.1.17.3.1.5.0">Idcirco antequàm vlterius progrediar, oſtendam primùm <lb></lb>pondus, quò propius eſt ipſis FG, minus grauitare; tum qua<lb></lb>tenus in eo ſitu, in quo reperitur, manet: tum quatenus ab eo <lb></lb>recedit. </s> <s id="id.2.1.17.3.1.6.0">ſimulq; falſum eſſe, pondus in A grauius eſſe, quàm in <lb></lb>alio ſitu. </s> </p> <pb n="10" xlink:href="036/01/033.jpg"></pb> <p id="id.2.1.17.5.0.0.0" type="main"> <s id="id.2.1.17.5.1.1.0">Producatur FG vſq; ad mundi cen<lb></lb>trum, quod ſit S. </s> <s id="N10D12">& à puncto S circu<lb></lb>lum AFBG contingens ducatur. </s> <s id="id.2.1.17.5.1.2.0">neq; <lb></lb>enim linea à puncto S circulum con<lb></lb>tingere poteſt in A; nam ducta AS <lb></lb>triangulum ACS duos haberet angu<lb></lb>los rectos, nempè SAC ACS, quod <arrow.to.target n="note33"></arrow.to.target><lb></lb>eſt impoſsibile. </s> <s id="id.2.1.17.5.1.3.0">neq; ſupra punctum A <lb></lb>in circumferentia AF continget; cir<lb></lb>culum enim ſecaret. </s> <s id="id.2.1.17.5.1.4.0">tanget igitur in<lb></lb>fra, ſitq; SO. </s> <s id="N10D32">connectantur deinde SD <lb></lb>SL, quæ circumferentiam AOG in <lb></lb>punctis KH ſecent. </s> <s id="id.2.1.17.5.1.5.0">& Ck CH con<lb></lb>iungantur. </s> <s id="id.2.1.17.5.1.6.0">Et quoniam pondus, quanto <lb></lb>propius eſt ipſi F, magis quoque inni<lb></lb>titur centro; vt pondus in D magis ver<lb></lb>ſionis puncto C innititur tanquam <lb></lb>centro; hoc eſt in D magis ſupra li<lb></lb>neam CD grauitat, quàm ſi eſſet in A <lb></lb>ſupra lineam CA; & adhuc magis in <lb></lb>L ſupra lineam CL; Nam cùm tres <lb></lb>anguli cuiuſcunq; trianguli duobus re<lb></lb><figure id="id.036.01.033.1.jpg" place="text" xlink:href="036/01/033/1.jpg"></figure><lb></lb>ctis ſint æquales, & trianguli DCk æquicruris angulus DCk <lb></lb>minor ſit angulo LCH æquicruris trianguli LCH: erunt reli<lb></lb>qui ad baſim ſcilicet CDk CkD ſimul ſumpti reliquis CLH <lb></lb>CHL maiores. </s> <s id="id.2.1.17.5.1.7.0">& horum dimidii; hoc eſt angulus CDS angu<lb></lb>lo CLS maior erit. </s> <s id="id.2.1.17.5.1.8.0">cùm itaq; CLS ſit minor, linea CL ma<lb></lb>gis adhærebit motui naturali ponderis in L prorſus ſoluti. </s> <s id="id.2.1.17.5.1.9.0">hoc <lb></lb>eſt 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></lb>berum, atq; ſolutum in centrum mundi per LS moueretur, pon<lb></lb>dusq; in D per DS. </s> <s id="id.2.1.17.5.1.9.0.b">quoniam verò pondus in L totum ſuper LS <lb></lb>grauitat, in D verò ſuper DS: pondus in L magis ſupra lineam <lb></lb>CL grauitabit, quàm exiſtens in D ſupra lineam DC. </s> <s id="N10D7F">ergo <lb></lb>linea CL pondus magis ſuſtentabit, quàm linea CD. </s> <s id="id.2.1.17.5.1.9.0.c">Eodem<lb></lb>〈qué〉 modo, quò pondus propius fuerit ipſi F, magis ob hanc cau<lb></lb>ſam à linea CL ſuſtineri oſtendetur; ſemper enim angulus CLS <pb xlink:href="036/01/034.jpg"></pb>minor eſſet. </s> <s id="id.2.1.17.5.1.10.0">quod etiam patet; quia ſi <lb></lb>lineæ CL, & LS in vnam coinciderent <lb></lb>lineam, quod euenit in FCS; tunc linea <lb></lb>CF totum ſuſtineret pondus in F, im<lb></lb>mobilemq; redderet: neq; vllam pror<lb></lb>ſus grauitatem in circumferentia circu<lb></lb>li haberet. </s> <s id="id.2.1.17.5.1.11.0">Idem ergo pondus propter <lb></lb>ſituum diuerſitatem grauius, leuiuſq; erit. </s> <s id="id.2.1.17.5.1.12.0"><lb></lb>non autem quia ratione ſitus interdum <lb></lb>maiorem re vera acquirat grauitatem, <lb></lb>interdum verò amittat, cùm eiuſdem ſit <lb></lb>ſemper grauitatis, vbicunque reperiatur; <lb></lb>ſed quia magis, minuſuè in circumferen<lb></lb>tia grauitat, vt in D magis ſupra circum<lb></lb>ferentiam DA grauitat, quàm in L ſupra <lb></lb>circumferentiam LD. </s> <s id="id.2.1.17.5.1.12.0.a">hoc eſt, ſi pon<lb></lb>dus à circumferentiis, rectiſq; lineis ſu<lb></lb>ſtineatur; circumferentia AD magis ſu<lb></lb>ſtinebit pondus in D, quàm circumfe<lb></lb>rentia DL pondere exiſtente in <emph type="italics"></emph>L.<emph.end type="italics"></emph.end> mi<lb></lb>nus enim coadiuuat CD, quàm CL. </s> <s id="id.2.1.17.5.1.12.0.b"><lb></lb>Præterea quando pondus eſt in L, ſi eſ<lb></lb><figure id="id.036.01.034.1.jpg" place="text" xlink:href="036/01/034/1.jpg"></figure><lb></lb>ſet omnino liberum, penituſq; ſolutum, deorſum per LS moueretur; <lb></lb>niſi à linea CL prohiberetur, quæ pondus in L vltra lineam LS per <lb></lb><expan abbr="circumferentiã">circumferentiam</expan> LD moueri cogit; ipſumq; quodammodo impellit, <lb></lb>impellendoq; pondus partim ſuſtentabit. </s> <s id="id.2.1.17.5.1.13.0">niſi enim ſuſtineret, ipſiq; <lb></lb>reniteretur, deorſum per lineam LS moueretur, non autem per <lb></lb>circumferentiam LD. </s> <s id="N10DE3">ſimiliter CD ponderi in D renititur, cùm <lb></lb>illud per circumferentiam DA moueri cogat. </s> <s id="id.2.1.17.5.1.14.0">eodemq; modo <lb></lb>exiſtente pondere in A, linea CA pondus vltra lineam AS per <lb></lb>circumferentiam AO moueri compellet. </s> <s id="id.2.1.17.5.1.15.0">eſt enim angulus CAS <lb></lb>acutus; cùm angulus ACS ſit rectus. </s> <s id="id.2.1.17.5.1.16.0">lineæ igitur CA CD ali<lb></lb>qua ex parte, non tamen ex æquo ponderi renituntur. </s> <s id="id.2.1.17.5.1.17.0">& quotieſ <lb></lb>cunque angulus in circumferentia circuli à lineis à centro <lb></lb>mundi S, & centro C prodeuntibus, fuerit acutus; idem eue<lb></lb>nire ſimiliter oſtendemus. </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"></pb>æqualis eſt angulo CDA, cùm à ſemidiametris, eademq; circumfe<lb></lb>rentia contineantur; & angulus C<emph type="italics"></emph>L<emph.end type="italics"></emph.end>S angulo CDS eſt minor; <lb></lb>erit reliquus <emph type="italics"></emph>S<emph.end type="italics"></emph.end>LD reliquo SDA maior. </s> <s id="id.2.1.17.5.1.19.0">quare circumferentia <lb></lb>DA, hoc eſt deſcenſus ponderis in D propior erit motui natu<lb></lb>rali ponderis in D ſoluti, lineæ ſcilicet DS, quàm circumferen<lb></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></lb>tur, quàm linea CL ponderi in L. </s> <s id="id.2.1.17.5.1.19.0.b">linea ideo CD minus ſuſtinet, <lb></lb>quàm CL; ponduſq; magis liberum erit in D, quàm in L: <lb></lb>cùm pondus naturaliter magis per DA moueatur, quàm per LD. <lb></lb></s> <s id="N10E2F">quare grauius erit in D, quàm in L. </s> <s id="N10E31">ſimiliter oſtendemus CA <lb></lb>minus ſuſtinere, quàm CD: ponduſq; magis in A, quàm in D li<lb></lb>berum, grauiuſq, eſſe. </s> <s id="id.2.1.17.5.1.20.0">Ex parte deinde inferiori ob eaſdem cauſas, <lb></lb>quò pondus propius fuerit ipſi G, magis detinebitur, vt in H ma<lb></lb>gis à linea CH, quàm in K à linea CK. </s> <s id="N10E3E">nam cùm angulus CHS <lb></lb>maior ſit angulo CkS, ad rectitudinem magis appropinquabunt <arrow.to.target n="note34"></arrow.to.target><lb></lb>ſe ſe lineæ CH HS, quàm Ck kS; atq; ob id pondus magis deti<lb></lb>nebitur à CH, quàm à Ck ſi enim CH HS in vnam conuenirent <lb></lb>lineam vt euenit pondere exiſtente in G; tunc linea CG totum ſu<lb></lb>ſtineret' pondus in G, ita vt immobilis perſiſteret. </s> <s id="id.2.1.17.5.1.21.0">quò igitur <lb></lb>minor erit angulus linea CH, & deſcenſu ponderis ſoluti, ſcilicet <lb></lb>HS contentus, eò minus quoq; eiuſmodi linea pondus detinebit. </s> <s id="id.2.1.17.5.1.22.0"><lb></lb>& vbi minus detinebitur, ibi magis liberum, grauiuſq; exiſtet. </s> <s id="id.2.1.17.5.1.23.0"><lb></lb>Præterea ſi pondus in k liberum eſſet, atq; ſolutum, per lineam <lb></lb>k S moueretur; à linea verò Ck prohibetur, quæ cogit pondus <lb></lb>citrà lineam k S per circumferentiam k H moueri. </s> <s id="id.2.1.17.5.1.24.0">ipſum enim <lb></lb>quodammodo retrahit, retrahendoq; ſuſtinet. </s> <s id="id.2.1.17.5.1.25.0">niſi enim ſuſtineret. </s> <s id="id.2.1.17.5.1.26.0"><lb></lb>pondus deorſum per rectam k S moueretur, non autem per cir<lb></lb>cumferentiam k H. </s> <s id="N10E6E">ſimiliter CH pondus retinet, cùm per circum<lb></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></lb>ior eſt angulo CKS, <expan abbr="dẽptis">demptis</expan> æqualibus angulis CHG CkH; erit <lb></lb>reliquus SHG reliquo SKH maior. </s> <s id="id.2.1.17.5.1.28.0">circumferentia igitur k H, hoc <lb></lb>eſt deſcenſus ponderis in k, propior erit motui naturali ponderis in <lb></lb>k ſoluti, hoc eſt lineæ k S, quàm circumferentia HG lineæ HS. </s> <s id="N10E8A">mi<lb></lb>nus idcirco detinet linea Ck, quàm CH: cùm pondus naturali<lb></lb>ter magis moueatur per k H, quàm per HG. </s> <s id="id.2.1.17.5.1.28.0.a">ſimili ratione oſten<lb></lb>detur, quò minor erit angulus SkH, lineam Ck minus ſuſtinere. </s> <s id="id.2.1.17.5.1.29.0"><pb xlink:href="036/01/036.jpg"></pb>exiſtente igitur pondere in O, quia angu<lb></lb>lus SOC non ſolum minor eſt angulo <lb></lb>CKS, verùm etiam omnium angulorum <lb></lb>à punctis CS prodeuntium, verticemq; <lb></lb>in circumferuntia OkG habentium mi<lb></lb>nimus; erit <expan abbr="anglus">angulus</expan> SOK, & angulo SkH, <lb></lb>& eiuſmodi omnium minimus. </s> <s id="id.2.1.17.5.1.30.0">ergo de<lb></lb>ſcenſus ponderis in O propior erit motui <lb></lb>naturali ipſius in O ſoluti, quàm in alio <lb></lb>ſitu circumferentiæ OkG. </s> <s id="N10EB4">lineaq; CO <lb></lb>minus pondus ſuſtinebit, quàm ſi pon<lb></lb>dus in quouis alio fuerit ſitu eiuſdem cir<lb></lb>cumferentiæ OG. </s> <s id="id.2.1.17.5.1.30.0.a">ſimiliter quoniam con<lb></lb>tingentiæ angulus SOk, & angulo SDA, <lb></lb>& SAO, ac quibuſcunq; ſimilibus eſt mi <lb></lb>nor; erit deſcenſus ponderis in O motui <lb></lb>naturali ipſius ponderis in O ſoluti pro<lb></lb>pior, quàm in alio ſitu circumferentiæ <lb></lb>ODF. </s> <s id="id.2.1.17.5.1.30.0.b">Præterea quoniam linea GO pon<lb></lb>dus in O dum deorſum mouetur, impelle<lb></lb>re non poteſt, ita vt vltra lineam OS mo<lb></lb>ueatur; cùm linea OS circulum non ſecet, <lb></lb><figure id="id.036.01.036.1.jpg" place="text" xlink:href="036/01/036/1.jpg"></figure><lb></lb>ſed contingat; anguluſq; SOC ſit rectus, & non acutus; pondus <lb></lb>in O nihil ſupra 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></lb>admodum in quouis alio puncto ſupra O accideret. </s> <s id="id.2.1.17.5.1.33.0">erit igitur pon<lb></lb>dus in O magis ob has cauſas liberum, atq; ſolutum in hoc ſitu, <lb></lb>quàm in quouis alio circumferentiæ FOG. </s> <s id="N10EED">ac idcirco in hoc <lb></lb>grauius erit, hoc eſt magis grauitabit, quàm in alio ſitu. </s> <s id="id.2.1.17.5.1.34.0">& quò <lb></lb>propius fuerit ipſi O remotiori grauius erit. </s> <s id="id.2.1.17.5.1.35.0">lineaq; CO horizonti <lb></lb>æquidiſtans erit. </s> <s id="id.2.1.17.5.1.36.0">non tamen puncti C horizonti (vt ipſi exiſti<lb></lb>mant) ſed ponderis in O conſtituti, cùm ex centro grauitatis <lb></lb>ponderis ſummendus ſit horizon. </s> <s id="id.2.1.17.5.1.37.0">quæ omnia demonſtrare opor<lb></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"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.18.1.1.2.0"><margin.target id="note34"></margin.target>21 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> </p> <pb n="12" xlink:href="036/01/037.jpg"></pb> <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 ipſo CO <lb></lb>fuerit maius, putá quantitate CD; erit <lb></lb>quoq; pondus in O grauius. </s> <s id="id.2.1.19.1.2.2.0">circulus de<lb></lb>ſcribatur OH, cuius centrum ſit D, ſe<arrow.to.target n="note35"></arrow.to.target><lb></lb>midiameterq; DO. </s> <s id="N10F36">tanget circulus OH <lb></lb>circulum FOG in puncto O, lineamq; <arrow.to.target n="note36"></arrow.to.target><lb></lb>OS, quæ ponderis in O rectus, natura<lb></lb>liſq; eſt deſcenſus, in eodem puncto con<lb></lb>tinget. </s> <s id="id.2.1.19.1.2.3.0">& quoniam angulus SOH mi<lb></lb>nor eſt angulo SOG, erit deſcenſus <lb></lb>ponderis in O per circumferentiam OH <lb></lb>motui naturali OS propior, quàm per <lb></lb>circumferentiam OG. </s> <s id="id.2.1.19.1.2.3.0.a">magis ergo li<lb></lb>berum, atq; ſolutum, ac per conſequens <lb></lb>grauius erit in O, centro libræ exiſten<lb></lb>te in D, quàm in C. </s> <s id="N10F57">ſimiliter oſten<lb></lb>detur, quò maius fuerit brachium DO, <lb></lb>pondus in O adhuc grauius eſſe. <figure id="id.036.01.037.1.jpg" place="text" xlink:href="036/01/037/1.jpg"></figure></s> </p> <pb xlink:href="036/01/038.jpg"></pb> <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></lb>cuius centrum ſit R, propius fuerit <lb></lb>mundi centro S; circulum〈qué〉 à pun<lb></lb>cto S ducatur contingens ST; punctum <lb></lb>T (vbi grauius eſt pondus) magis <lb></lb>à puncto A diſtabit, quàm punctum <lb></lb>O. ducantur enim à punctis OT ipſi <lb></lb>CS perpendiculares OMTN; conne<lb></lb>ctanturq; RT; ſitq; centrum R in li<lb></lb>nea CS; lineaq; ARB ipſi ACB æqui <lb></lb><arrow.to.target n="note37"></arrow.to.target>diſtans. </s> <s id="id.2.1.19.3.1.2.0">Quoniam igitur triangula COS <lb></lb>RTS ſunt rectangula; erit SC ad CO, <lb></lb>vt CO ad CM. </s> <s id="N10F89">ſimiliter SR ad RT, <lb></lb>vt RT ad RN. </s> <s id="N10F8D">cùm itaq; ſit RT ip<lb></lb><arrow.to.target n="note38"></arrow.to.target>ſi CO æqualis, & SC ipſa SR maior: <lb></lb>maiorem habebit proportionem SC <lb></lb>ad CO, quàm SR ad RT. </s> <s id="N10F98">quare ma<lb></lb>iorem quoq; proportionem habebit <lb></lb>CO ad CM, quàm RT ad RN. </s> <s id="id.2.1.19.3.1.2.0.a">mi<lb></lb><arrow.to.target n="note39"></arrow.to.target>nor ergo erit CM, quàm RN. </s> <s id="N10FA6">ſecetur <lb></lb>igitur RN in P, ita vt RP ſit ipſi <lb></lb><figure id="id.036.01.038.1.jpg" place="text" xlink:href="036/01/038/1.jpg"></figure><lb></lb>CM æqualis; & à puncto P ipſis MONT æquidiſtans ducatur <lb></lb>PQ, quæ circumferentiam AT ſecet in Q: deniq; connectatur <lb></lb>RQ. </s> <s id="N10FB6">quoniam enim duæ CO CM duabus RQRP ſunt æqua<lb></lb><arrow.to.target n="note40"></arrow.to.target>les, & angulus CMO angulo RPQ eſt æqualis; erit & angu<lb></lb>lus MCO angulo PRQ æqualis. </s> <s id="id.2.1.19.3.1.3.0">angulus autem MCA rectus <lb></lb><arrow.to.target n="note41"></arrow.to.target>recto PRA eſt æqualis; ergo reliquus OCA reliquo QRA <lb></lb>æqualis, & circumferentia OA circumferentiæ QA æqualis quo<lb></lb>que erit. </s> <s id="id.2.1.19.3.1.4.0">punctum idcirco T, quia magis à puncto A diſtat, <lb></lb>quàm Q; magis quoq; à puncto A diſtabit, quàm punctum O. <lb></lb></s> <s id="N10FD1">ſimiliter oſtendetur, quò propius fuerit circulus mundi centro, eun<lb></lb>dem magis diſtare. </s> <s id="id.2.1.19.3.1.5.0">atq; ita vt prius demonſtrabitur pondus in cir<lb></lb>cumferentia TAF centro R inniti, in circumferentia verò TG <lb></lb>à linea detineri; atq; in puncto T grauius eſſe. </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"></emph>Ex<emph.end type="italics"></emph.end> 11 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.20.1.1.2.0"><margin.target id="note36"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 18 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.20.1.1.3.0"><margin.target id="note37"></margin.target><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 8 <emph type="italics"></emph>ſexti<emph.end type="italics"></emph.end></s> <s id="id.2.1.20.1.1.4.0"><margin.target id="note38"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 8 <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> <s id="id.2.1.20.1.1.5.0"><margin.target id="note39"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 10 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.20.1.1.6.0"><margin.target id="note40"></margin.target>7 <emph type="italics"></emph>Sexti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.20.1.1.7.0"><margin.target id="note41"></margin.target>26 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> </p> <pb n="13" xlink:href="036/01/039.jpg"></pb> <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 eſſet <lb></lb>in centro mundi; tunc quò <lb></lb>pondus propius fuerit ipſi G, <lb></lb>grauius erit: & vbicunq; po<lb></lb>natur pondus præterquàm in <lb></lb>ipſo G, ſemper centro C inni<lb></lb>tetur, vt in K. </s> <s id="N11054">nam ducta <lb></lb>G k, efficiet hæc (ſecun<lb></lb>dùm quam fit ponderis natu<lb></lb>ralis motus) vná cum libræ <lb></lb>brachio k C angulum acu<lb></lb>tum. </s> <s id="id.2.1.21.1.2.2.0">æquicruris enim trian<lb></lb>guli CkG ad baſim anguli <lb></lb>ad k, & G ſunt ſemper acuti. </s> <s id="id.2.1.21.1.2.3.0"><lb></lb><figure id="id.036.01.039.1.jpg" place="text" xlink:href="036/01/039/1.jpg"></figure><lb></lb>Conferantur autem inuicem hæc duo, pondus videlicet in k, & <lb></lb>pondus in D: erit pondus in k grauius, quàm in D. </s> <s id="N11073">nam iuncta <lb></lb>DG, cùm tres anguli cuiuſcunque trianguli duobus ſint rectis <lb></lb>æquales, & trianguli CDG æquicruris angulus DCG maior ſit <lb></lb>angulo kCG æquicruris trianguli CkG: erunt reliqui ad baſim an<lb></lb>guli DGC GDC ſimul ſumpti reliquis KGCGkC ſimul ſumptis <lb></lb>minores. </s> <s id="id.2.1.21.1.2.4.0">horumq; dimidii; angulus ſcilicet CDG angulo CKG <lb></lb>minor erit. </s> <s id="id.2.1.21.1.2.5.0">quare cùm pondus in k ſolutum naturaliter per <lb></lb>KG moueatur, pondusq; in D per DG, tanquam per ſpatia, <lb></lb>quibus in centrum mundi feruntur; linea CD, hoc eſt libræ <lb></lb>brachium magis adhærebit motui naturali ponderis in D pror<lb></lb>ſus ſoluti, lineæ ſcilicet DG; quàm Ck motui ſecundùm kG <lb></lb>effecto. </s> <s id="id.2.1.21.1.2.6.0">magis igitur ſuſtinebit linea CD, quàm Ck. </s> <s id="id.2.1.21.1.2.7.0">ac pro<lb></lb>pterea pondus in k ex ſuperius dictis grauius erit, quàm in D. </s> <s id="id.2.1.21.1.2.7.0.a"><lb></lb>Præterea quoniam pondus in K ſi eſſet omnino liberum, prorſuſq; <lb></lb>ſolutum, deorſum per k G moueretur; niſi à linea C k prohibere<lb></lb>tur, quæ pondus vltra lineam KG per circumferentiam KH mo<lb></lb>ueri cogit; linea C k pondus partim ſuſtinebit, ipſiq; renitetur; <lb></lb>cùm illud per circumferentiam k H moueri compellat. </s> <s id="id.2.1.21.1.2.8.0">& <lb></lb>quoniam angulus CDG minor eſt angulo CkG, & angulus CDk <lb></lb>angulo CkH eſt æqualis; erit reliquus GDk reliquo G k H maior. </s> <s id="id.2.1.21.1.2.9.0"><lb></lb>circumferentia igitur k H motui naturali ponderis in k ſoluti, li<pb xlink:href="036/01/040.jpg"></pb>neæ ſcilicet KG propior erit, <lb></lb>quàm circumferentia Dk li<lb></lb>neæ DG. </s> <s id="N110B8">quare linea CD <lb></lb>ponderi in D magis renititur, <lb></lb>quàm linea C k ipſi ponde<lb></lb>ri in K. </s> <s id="id.2.1.21.1.2.9.0.a">ergo pondus in k <lb></lb>grauius erit, quàm in D. </s> <s id="id.2.1.21.1.2.9.0.b"><lb></lb>Similiter oſtendetur pondus, <lb></lb>quò fuerit ipſi F propius, vt <lb></lb>in L, minus grauitare: pro<lb></lb>pius verò ipſi G, vt in H, <lb></lb>grauius eſſe. <figure id="id.036.01.040.1.jpg" place="text" 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></lb>S eſſet inter puncta CG; <lb></lb>primùm quidem ſimili<lb></lb>ter oſtendetur pondus vbi<lb></lb>cunq; poſitum centro C <lb></lb>initi, vt in H. </s> <s id="N110E6">ductis enim <lb></lb>HG HS, angulus ad <lb></lb>baſim GHC æquicruris tri<lb></lb>anguli CHG eſt ſemper <lb></lb>acutus: quare & SHC ip<lb></lb>ſo minor erit quoq; ſem<lb></lb>per acutus. </s> <s id="id.2.1.21.2.1.2.0">ducatur au<lb></lb>tem à puncto S ipſi CS <lb></lb>perpendicularis Sk. </s> <s id="id.2.1.21.2.1.3.0">di<lb></lb><figure id="id.036.01.040.2.jpg" place="text" xlink:href="036/01/040/2.jpg"></figure><lb></lb>co pondus grauius eſſe in k, quàm in alio ſitu circumferentiæ FKG. <lb></lb>& quò propius fuerit ipſi F, vel G, minus grauitare. </s> <s id="id.2.1.21.2.1.4.0">Accipiantur <lb></lb>verſus F puncta DL, connectanturq; LC LS DC DS, produ<lb></lb>canturq; LS DS k SHS vſq; ad circuli circumferentiam in EM <lb></lb>NO; connectanturq; CE, CM, CN, CO. </s> <s id="id.2.1.21.2.1.4.0.a">Quoniam enim <lb></lb><arrow.to.target n="note42"></arrow.to.target>LE DM ſe inuicem ſecant in S; erit rectangulum LSE rectan<lb></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></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 eſt LS, quàm DS; & SM ipſa SE. </s> <s id="id.2.1.21.2.1.5.0.b"><pb n="14" xlink:href="036/01/041.jpg"></pb>ergo LS SE ſimul ſumptæ ipſis 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></lb>ratione kN minorem eſſe DM oſtendetur. </s> <s id="id.2.1.21.2.1.7.0">rurſus quoniam re<lb></lb>ctangulum OSH æquale eſt rectangulo kSN; ob eandem cauſam <lb></lb>HO maior erit kN. </s> <s id="N1113F">eodemq; prorſus modo kN omnibus a<lb></lb>liis per punctum S tranſeuntibus minorem eſſe demonſtrabitur. </s> <s id="id.2.1.21.2.1.8.0"><lb></lb>& quoniam æquicrurium triangulorum CLE DCM latera LC <lb></lb>CE lateribus DC CM ſunt æqualia; baſis verò LE maior eſt <lb></lb>DM: erit angulus LCE angulo DCM maior. </s> <s id="id.2.1.21.2.1.9.0">quare ad baſim <arrow.to.target n="note46"></arrow.to.target><lb></lb>anguli C<emph type="italics"></emph>L<emph.end type="italics"></emph.end>E CEL ſimul ſumpti angulis CDM CMD mi<lb></lb>nores erunt. </s> <s id="id.2.1.21.2.1.10.0">& horum dimidii, angulus ſcilicet CLS angulo CDS <lb></lb>minor erit. </s> <s id="id.2.1.21.2.1.11.0">ergo pondus in <emph type="italics"></emph>L<emph.end type="italics"></emph.end> magis ſupra lineam LC, quàm <lb></lb>in D ſupra DC grauitabit. </s> <s id="id.2.1.21.2.1.11.0.a">magis〈qué〉 centro innitetur in L, quàm <lb></lb>in D. </s> <s id="id.2.1.21.2.1.11.0.b">ſimiliter oſtendetur in D magis <expan abbr="cẽtro">centro</expan> C inniti, quàm in k. </s> <s id="id.2.1.21.2.1.12.0">ergo <lb></lb><expan abbr="ponds">pondus</expan> in k grauius erit, quàm in D; & in D, quàm in L. </s> <s id="N1117F">eademq; pror<lb></lb>ſus ratione quoniam kN minor eſt HO, erit angulus CKS an<lb></lb>gulo CHS maior. </s> <s id="id.2.1.21.2.1.13.0">quare pondus in H magis centro C innite<lb></lb>tur, quàm in k. </s> <s id="id.2.1.21.2.1.14.0">& hoc modo oſtendetur, vbicunq; in circum<lb></lb>ferentia FDG fuerit pondus, minus in K centro C inniti, quàm <lb></lb>in alio ſitu: & quò propius fuerit ipſi F, vel G, magis inniti. </s> <s id="id.2.1.21.2.1.15.0">dein<lb></lb>de quoniam angulus CkS maior eſt CDS, & CDk æqualis <lb></lb>eſt CkH: erit reliquus SkH reliquo SDk minor. </s> <s id="id.2.1.21.2.1.16.0">quare cir<lb></lb>cumferentia k H propior erit motui naturali recto ponderis in K <lb></lb>ſoluti, lineæ ſcilicet k S, quàm circumferentia D k motui DS. </s> <s id="id.2.1.21.2.1.16.0.a">& <lb></lb>ideo linea CD magis ipſi ponderi in D renititur, quàm CK <lb></lb>ponderi in k conſtituto. </s> <s id="id.2.1.21.2.1.17.0">hacq; ratione oſtendetur angulum <lb></lb>SHG maiorem eſſe SkH: & per conſequens lineam CH magis <lb></lb>ponderi in H reniti, quàm CK ponderi in K. </s> <s id="N111AD">ſimiliter demon<lb></lb>ſtrabitur lineam C<emph type="italics"></emph>L<emph.end type="italics"></emph.end> magis pondus ſuſtinere, quàm CD: ob <lb></lb>eaſdemq; cauſas oſtendetur pondus in K minus ſupra lineam Ck <lb></lb>grauitare, quàm in quouis alio ſitu fuerit circumferentiæ FDG. <lb></lb></s> <s id="N111BC">& quò propius fuerit ipſi F, vel G, minus grauitare. </s> <s id="id.2.1.21.2.1.18.0">grauius ergo <lb></lb>erit in k, quàm in alio ſitu: minuſq; graue erit, quò propius fue<lb></lb>rit ipſi F, vel G. <pb xlink:href="036/01/042.jpg"></pb></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"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.22.1.1.2.0"><margin.target id="note43"></margin.target>16 <emph type="italics"></emph>Sexti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.22.1.1.3.0"><margin.target id="note44"></margin.target>7 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.22.1.1.4.0"><margin.target id="note45"></margin.target>25 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.22.1.1.5.0"><margin.target id="note46"></margin.target>25 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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></lb>eſſet in centro mundi, <lb></lb>pondus vbicunque con<lb></lb>ſtitutum manere mani<lb></lb>feſtum eſt. </s> <s id="id.2.1.23.1.1.2.0">vt poſito pon<lb></lb>dere in D, linea CD to<lb></lb>tum ſuſtinebit pondus; <lb></lb>cùm ipſius ponderis in D <lb></lb>horizonti ſit perpendicu<lb></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></lb>nebit. <figure id="id.036.01.042.1.jpg" place="text" 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 demonſtratis, nullam de gra<lb></lb>uitate brachii libræ mentionem fecimus, idcirco ſi brachii quoq; <lb></lb>grauitatem conſiderare voluerimus, centrum grauitatis magnitu<lb></lb>dinis ex pondere, brachioq; compoſitæ inueniri poterit, circulo<lb></lb>rumq; circumferentiæ ſecundum diſtantiam à centro libræ ad <lb></lb>hoc ipſum grauitatis centrum deſcribentur, ac ſi in ipſo (vt re ue<lb></lb>ra eſt) pondus conſtitutum fuerit; omnia, ſicuti abſq; libræ bra<lb></lb>chii grauitate conſiderata inuenimus; hoc quoq; modo eius conſi<lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="15" xlink:href="036/01/043.jpg"></pb> <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, conſiderando li<lb></lb>bram, vt longè à mundi centro a<lb></lb>beſt, quemadmodum ipſi fecere, ſi<lb></lb>cuti etiam actu eſt, apparet falſitas <lb></lb>dicentium pondus in A grauius eſſe, <lb></lb>quàm in alio ſitu. </s> <s id="id.2.1.25.1.2.2.0">ſimulq; falſum eſſe, <lb></lb>quò pondus à linea FG magis diſtat <lb></lb><expan abbr="grauiuis">grauius</expan> eſſe. </s> <s id="id.2.1.25.1.2.3.0">nam punctum O pro<lb></lb>pius eſt ipſi FG, quàm punctum A. <lb></lb></s> <s id="N1126C">eſt enim linea à puncto O ipſi FG <arrow.to.target n="note48"></arrow.to.target><lb></lb>perpendicularis ipſa CA minor. </s> <s id="id.2.1.25.1.2.4.0">de<lb></lb>inde ex puncto A pondus velocius mo<lb></lb>ueri, quàm ab alio ſitu, eſt quoque <lb></lb>falſum. </s> <s id="id.2.1.25.1.2.5.0">ex puncto enim O pondus ve<lb></lb>locius mouebitur, quàm ex puncto <lb></lb>A; cùm in O ſit magis liberum, atq; <lb></lb>ſolutum, quàm in alio ſitu: deſcenſus <lb></lb>〈qué〉 ex puncto O propior ſit motui na<lb></lb>turali recto, quàm quilibet alius de<lb></lb>ſcenſus. <figure id="id.036.01.043.1.jpg" place="text" 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></lb>ctiori, & obliquiori <expan abbr="deſcẽſu">deſcen<lb></lb>ſu</expan> oſtendunt, pondus in <lb></lb>A <expan abbr="grauiur">grauior</expan> eſſe, quàm in <lb></lb>D; & in D, quàm in <lb></lb>L; primùm quidem fal<lb></lb>ſum exiſtimant, ſi pon<lb></lb>dus aliquod collocatum <lb></lb>fuerit in quocunq; ſitu <lb></lb>circunferentiæ, vt in D, <lb></lb>rectum eius deſcenſum <lb></lb>per rectam lineam DR <lb></lb>ipſi FG parallelam, tam <lb></lb>quàm ſecundùm mo<figure id="id.036.01.043.2.jpg" place="text" xlink:href="036/01/043/2.jpg"></figure> <pb xlink:href="036/01/044.jpg"></pb>tum naturalem fieri de<lb></lb>bere; ſicuti prius dictum <lb></lb>eſt. </s> <s id="id.2.1.25.2.1.2.0">In quocunq; enim <lb></lb>ſitu pondus aliquod con<lb></lb>ſtituatur, ſi naturalem <lb></lb>eius ad propium locum <lb></lb>motionem ſpectemus, <lb></lb>cùm rectá ad eum ſua<lb></lb>ptè natura moueatur, ſup<lb></lb>poſita totius vniuerſi figu<lb></lb>ra, eiuſmodi erit; vt <lb></lb>ſemper <expan abbr="ſpatiũ">ſpatium</expan>, per quod <lb></lb>naturaliter mouetur, ra<lb></lb>tionem habere videatur <lb></lb><figure id="id.036.01.044.1.jpg" place="text" xlink:href="036/01/044/1.jpg"></figure><lb></lb>lineæ à circumferentia ad centrum productæ. </s> <s id="id.2.1.25.2.1.3.0">non igitur natura<lb></lb>les deſcenſus recti cuiuslibet ſoluti ponderis per lineas fieri poſ<lb></lb>ſunt inter ſe ſe parallelas; cùm omnes in centrum mundi conue<lb></lb>niant. </s> <s id="id.2.1.25.2.1.4.0">ſupponunt deinde ponderis ex D in A per rectam lineam <lb></lb>verſus centrum mundi motum eiuſdem eſſe quantitatis, ac ſi fuiſ<lb></lb>ſet ex O in C: ita vt punctum A æqualiter à centro mundi ſit <lb></lb>diſtans, vt C. </s> <s id="N112FC">quod eſt etiam falſum; nam punctum A magis <lb></lb>à centro mundi diſtat, quàm C: maior enim eſt linea à cen<lb></lb><arrow.to.target n="note49"></arrow.to.target>tro mundi vſq; ad A, quàm à centro mundi vſq; ad C: cùm li<lb></lb>nea à centro mundi vſq; ad A rectum ſubtendat angulum à li<lb></lb>neis AC, & à puncto C ad centrum mundi contentum. </s> <s id="id.2.1.25.2.1.5.0">ex qui<lb></lb>bus non ſolum ſuppoſitio illa, qua libram DE in AB redire demon<lb></lb>ſtrant, verùm etiam omnes ferè ipſorum demonſtrationes ruunt. </s> <s id="id.2.1.25.2.1.6.0"><lb></lb>niſi fortaſſe dixerint, hæc omnia propter maximam à centro mun<lb></lb>di vſq; ad nos diſtantiam adeo inſenſibilia eſſe, vt propter inſen<lb></lb>ſibilitatem tanquam vera ſupponi poſsint: cùm omnes <expan abbr="quidẽ">quidem</expan> alii, qui <lb></lb>hæc tractauerunt, tanquam nota ſuppoſuerint. </s> <s id="id.2.1.25.2.1.7.0">præſertim quia <lb></lb>ſenſibilitas illa non efficit, quin deſcenſus ponderis ex L in D <lb></lb>(vt eorum verbis vtar) minus capiat de directo, quàm deſcen<lb></lb>ſus DA. </s> <s id="N11327">ſimiliter arcus DA magis de directo capiet, quàm cir<lb></lb>cumferentia EV. </s> <s id="N1132B">quocirca vera erit ſuppoſitio; aliæq; demon<lb></lb>ſtrationes in ſuo 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"></pb>dus in A grauius eſſe, quàm in alio ſitu; rectumq; ponderis de<lb></lb>ſcenſum per rectam lineam ipſi FG parallelam fieri debere; & <lb></lb>quælibet puncta in lineis horizonti æquidiſtantibus accepta æ<lb></lb>qualiter à centro mundi diſtare: non tamen propterea ſequetur, <lb></lb>veram eſſe demonſtrationem, qua inferunt pondus in A grauius <lb></lb>eſſe, quàm in alio ſitu, vt in L. </s> <s id="N11341">ſi enim verum eſſet, quò pon<lb></lb>dus hoc modo rectius deſcendit, ibi grauius eſſe; ſequeretur etiam, <lb></lb>quò idem pondus in æqualibus arcubus æqualiter rectè deſcende<lb></lb>ret, vt in iiſdem locis æqualem haberet grauitatem, quod fal<lb></lb>ſum eſſe ita demonſtratur. </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"></emph>Ex<emph.end type="italics"></emph.end> 15 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.26.1.1.2.0"><margin.target id="note49"></margin.target>18 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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 ſe ſe æquales; & conne<lb></lb>ctatur LM, quæ AB ſecet in X: erit LM ipſi FG æquidiſtans, <lb></lb>ipſiq; AB perpendicularis. </s> <s id="id.2.1.27.1.1.2.0">& XM ipſi XL æqualis erit. </s> <s id="id.2.1.27.1.1.3.0">ſi igi<arrow.to.target n="note50"></arrow.to.target><lb></lb>tur pondus ex L moueatur in A per circumferentiam LA, rectus <lb></lb>eius motus erit ſecundùm lineam LX. </s> <s id="id.2.1.27.1.1.3.0.a">ſi verò moueatur ex A <lb></lb>in M per circumferentiam AM, ſecundùm rectam eius motus <lb></lb>erit XM. </s> <s id="id.2.1.27.1.1.3.0.b">quare deſcenſus ex L in A æqualis erit deſcenſui ex A <lb></lb>in M; tum ob circumferentias æquales, tum propter rectas li<lb></lb>neas ipſi AB perpendiculares æquales. </s> <s id="id.2.1.27.1.1.4.0">ergo idem pondus in L <lb></lb>æquè graue erit, vt in A, quod eſt falſum. </s> <s id="id.2.1.27.1.1.5.0">cum longé grauius ſit <lb></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"></emph>Ex<emph.end type="italics"></emph.end> 3 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></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 ſecundùm ipſos de directo <lb></lb>capiant; dicent fortaſſe, quia tamen principium deſcenſus ex L <lb></lb>ſcilicet LD minus de directo capit, quàm principium deſcenſus <lb></lb>ex A, ſcilicet AN; pondus in A grauius erit, quàm in L. </s> <s id="id.2.1.29.1.1.1.0.a">nam <lb></lb>cùm circumferentia AN ſit ipſi LD (vt ſupra poſitum eſt) <lb></lb>æqualis, quæ ſecundùm ipſos de directo capit CT; LD verò <lb></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></lb></s> <s id="id.2.1.29.1.1.1.0.c">quod ſi verum eſſet, ſequeretur idem pondus in eodem ſitu diuer<lb></lb>ſo duntaxat modo conſideratum in habitudine ad eundem ſitum, <lb></lb>tum grauius, tum leuius eſſe. </s> <s id="id.2.1.29.1.1.2.0">quod eſt impoſsibile. </s> <s id="id.2.1.29.1.1.3.0">hoc eſt, ſi <lb></lb>deſcenſum conſideremus ponderis in L, quatenus ex L in A de<lb></lb>ſcendit, grauius erit, quàm ſi eiuſdem ponderis deſcenſum con<lb></lb>ſideremus ex L in D tantùm. </s> <s id="id.2.1.29.1.1.4.0">neq; enim negare poſſunt ex eiſ<lb></lb>demmet dictis, quin deſcenſus ponderis ex L in A de directo ca<lb></lb>piat LX, ſiue PC. </s> <s id="N113DB">deſcenſus verò AM, quin ſimiliter de directo <pb xlink:href="036/01/046.jpg"></pb>capiat XM: cùm ipſi <lb></lb>quoq; hoc modo acci<lb></lb>piant, atq; ita accipe<lb></lb>re ſit neceſſe. </s> <s id="id.2.1.29.1.1.5.0">ſi enim li<lb></lb>bram DE in AB redire <lb></lb>demonſtrare volunt, com<lb></lb>parando deſcenſus pon<lb></lb>deris in D cum deſcen<lb></lb>ſu ponderis in E, neceſſe <lb></lb>eſt, vt oſtendant rectum <lb></lb>deſcenſum OC corre<lb></lb>ſpondentem circumferen<lb></lb>tiæ DA maiorem eſſe re<lb></lb>cto deſcenſu TH circum<lb></lb><figure id="id.036.01.046.1.jpg" place="text" xlink:href="036/01/046/1.jpg"></figure><lb></lb>ferentiæ EV correſpondente. </s> <s id="id.2.1.29.1.1.6.0">ſi enim partem tantùm totius de<lb></lb>ſcenſus ex D in A acciperent, vt D k; oſtenderentq; magis cape<lb></lb>re de directo deſcenſum Dk, quàm æqualis portio deſcenſus ex <lb></lb>puncto E. </s> <s id="N1140F">ſequetur pondus in D ſecundùm ipſos grauius eſſe pon<lb></lb>dere in E; & vſq; ad k tantùm deorſum moueri: ita vt libra mo<lb></lb>ta ſit in kI. </s> <s id="N11415">ſimiliter ſi libram KI in AB redire demonſtrare vo<lb></lb>lunt accipiendo portionem deſcenſus ex k in A; hoc eſt k S; <lb></lb>oſtenderentq; k S magis de directo capere, quàm ex aduerſo æ<lb></lb>qualis deſcenſus ex puncto I: ſimili modo ſequetur pondus in k <lb></lb>grauius eſſe, quàm in I; & vſq; ad S tantùm moueri. </s> <s id="id.2.1.29.1.1.7.0">& ſi rurſus <lb></lb>oſtenderent portionem deſcenſus ex S in A, atq; ita deinceps, re<lb></lb>ctiorem eſſe æquali deſcenſu ponderis oppoſiti; ſemper ſequetur <lb></lb>libram SI ad AB propius accedere, nunquam tamen in AB per<lb></lb>uenire demonſtrabunt. </s> <s id="id.2.1.29.1.1.8.0">ſi igitur libram DE in AB redire demon<lb></lb>ſtrare volunt, neceſſe eſt, vt deſcenſum ponderis ex D in A de di<lb></lb>recro capere quantitatem lineæ ex puncto D ipſi AB ad rectos <lb></lb>angulos ductæ accipiant. </s> <s id="id.2.1.29.1.1.9.0">atq; ita, ſi æquales deſcenſus DA AN <lb></lb>inuicem comparemus, qui æqualiter de directo capient OC CT, <lb></lb>eueniet idem pondus in D æquè graue eſſe, vt in A. </s> <s id="N1143A">ſi verò por<lb></lb>tiones tantum ex D A accipiamus; grauius erit in A, quàm <lb></lb>in D. </s> <s id="N11440">ergo ex diuerſitate tantùm modi conſiderandi, idem pon<lb></lb>dus, & grauius, & leuius eſſe continget. </s> <s id="id.2.1.29.1.1.10.0">non autem ex ipſa na<pb n="17" xlink:href="036/01/047.jpg"></pb>tura rei. </s> <s id="id.2.1.29.1.1.11.0">Inſuper ipſorum ſuppoſitio non aſſerit, pondus ſecun<lb></lb>dùm ſitum grauius eſſe, quantò in eodem ſitu minus obliquum <lb></lb>eſt principium ipſius deſcenſus. </s> <s id="id.2.1.29.1.1.12.0">Suppoſitio igitur ſuperius alla<lb></lb>ta, hoc eſt, ſecundùm ſitum pondus grauius eſſe, quantò in eo <lb></lb>dem ſitu minus obliquus eſt deſcenſus; non ſolum ex his, quæ <lb></lb>diximus, vllo modo concedi poteſt; ſed quoniam huius oppoſi<lb></lb>tum oſtendere quoq; non eſt difficile: ſcilicet idem pondus in <lb></lb>æqualibus circumferentiis, quò minus obliquus eſt deſcenſus, ibi <lb></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></lb><expan abbr="cumferentræ">cumferentiae</expan> AL AM <lb></lb>inter ſe ſe æquales; ſitq; <lb></lb>punctum L propè F. </s> <s id="N11471">& <lb></lb>connectatur LM, quæ <lb></lb>ipſi AB perpendicularis <lb></lb>erit. </s> <s id="id.2.1.29.2.1.2.0">& LX ipſi XM <lb></lb>æqualis. </s> <s id="id.2.1.29.2.1.3.0">deinde propè <lb></lb>M inter MG quoduis <lb></lb>accipiatur punctum P. <lb></lb>fiatq; circumferentia PO <lb></lb>circumferentiæ AM æ<lb></lb>qualis. </s> <s id="id.2.1.29.2.1.4.0">erit punctum O <lb></lb><figure id="id.036.01.047.1.jpg" place="text" xlink:href="036/01/047/1.jpg"></figure><lb></lb>propè A. </s> <s id="N11496">connectanturq; CL, CO, CM, CP, OP. </s> <s id="N11498">& à <lb></lb>puncto P ipſi OC perpendicularis ducatur PN. </s> <s id="id.2.1.29.2.1.4.0.a">& quoniam cir<lb></lb>cumferentia AM circumferentiæ OP eſt æqualis: erit angu<lb></lb>lus <arrow.to.target n="note51"></arrow.to.target>ACM æqualis angulo OCP; & angulus CXM rectus re<lb></lb>cto CNP eſt æqualis: erit quoq; reliquus XMC trianguli MCX <arrow.to.target n="note52"></arrow.to.target><lb></lb>reliquo NPC trianguli PCN æqualis. </s> <s id="id.2.1.29.2.1.5.0">ſed & latus CM lateri <arrow.to.target n="note53"></arrow.to.target><lb></lb>CP eſt æquale: ergo triangulum MCX triangulo PCN æquale <lb></lb>erit. </s> <s id="id.2.1.29.2.1.6.0">latuſq; MX lateri NP æquale. </s> <s id="id.2.1.29.2.1.7.0">quare linea PN ipſi LX æqua<lb></lb>lis erit. </s> <s id="id.2.1.29.2.1.8.0">ducatur præterea à puncto O linea OT ipſi AC æqui<lb></lb>diſtans, quæ NP ſecet in V. </s> <s id="N114C5">atq; ipſi OT à puncto P perpendi<lb></lb>cularis ducatur, quæ quidem inter OV cadere non poteſt; nam <lb></lb>cùm angulus ONV ſit 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></lb>obtuſus erit. </s> <s id="id.2.1.29.2.1.10.0">non igitur linea à puncto P ipſi OT intra OV <pb xlink:href="036/01/048.jpg"></pb>perpendicularis cadet. </s> <s id="id.2.1.29.2.1.11.0"><lb></lb>duo enim anguli vnius <lb></lb>trianguli, vnus quidem <lb></lb>rectus, alter verò ob<lb></lb>tuſus eſſet. </s> <s id="id.2.1.29.2.1.12.0">quod eſt im<lb></lb>poſsibile. </s> <s id="id.2.1.29.2.1.13.0">cadet ergo in <lb></lb>linea OT in parte VT. </s> <s id="id.2.1.29.2.1.13.0.a">ſitq; PT. <lb></lb></s> <s id="N114EF">erit PT ſecun<lb></lb>dùm ipſos rectus circum<lb></lb>ferentiæ OP deſcenſus. </s> <s id="id.2.1.29.2.1.14.0"><lb></lb>Quoniam igitur angulus <lb></lb>ONV eſt rectus; erit <lb></lb><arrow.to.target n="note55"></arrow.to.target>linea OV ipſa ON ma<lb></lb>ior. </s> <s id="id.2.1.29.2.1.15.0">quare OT ipſa <lb></lb><figure id="id.036.01.048.1.jpg" place="text" xlink:href="036/01/048/1.jpg"></figure><lb></lb>quoq; ON maior exiſtet. </s> <s id="id.2.1.29.2.1.16.0">Cùm itaq; linèa OP angulos ſubten<lb></lb>dat rectos ONP OTP; erit quadratum ex OP quadratis ex <lb></lb><arrow.to.target n="note56"></arrow.to.target>ON NP ſimul ſumptis æquale. </s> <s id="id.2.1.29.2.1.17.0">ſimiliter quadratis ex OT TP <lb></lb>ſimul æquale. </s> <s id="id.2.1.29.2.1.18.0">quare quadrata ſimul ex ON NP quadratis ex <lb></lb>OT TP ſimul æqualia erunt. </s> <s id="id.2.1.29.2.1.19.0">quadratum autem ex OT maius <lb></lb>eſt quadrato ex ON; cum linea OT ſit ipſa ON maior. </s> <s id="id.2.1.29.2.1.20.0">ergo qua<lb></lb>dratum ex NP maius erit quadrato ex TP. </s> <s id="N1152B">ac propterea linea <lb></lb>TP minor erit linea PN, & linea LX. </s> <s id="N1152F">minus obliquus igitur eſt <lb></lb>deſcenſus arcus LA, quàm arcus OP. </s> <s id="id.2.1.29.2.1.20.0.a">ergo pondus in L, ex ip<lb></lb>ſorum dictis, grauius erit, quàm in O. quod ex iis, quæ ſupra di<lb></lb>ximus eſt manifeſtè falſum, cùm pondus in O grauius ſit, quàm <lb></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></lb>ligi poteſt, ſecundùm ſitum pondus grauius eſſe, quantò in eo<lb></lb>dem ſitu minus obliquus eſt deſcenſus. </s> <s id="id.2.1.29.2.1.21.0">Atq; hinc oritur omnis <lb></lb>fermé ipſorum error in hac re, atq; deceptio: nam quamuis per <lb></lb>accidens interdum ex falſis ſequatur verum, per ſe tamen ex fal<lb></lb>ſis falſum ſequitur, quemadmodum ex veris ſemper verum, nil <lb></lb>idcirco mirum, ſi dum falſa accipiunt; illiſq; tanquam veriſsi<lb></lb>mis innituntur; falſiſsima omninò colligunt, atq; concludunt. </s> <s id="id.2.1.29.2.1.22.0"><lb></lb>decipiuntur quinetiam, dùm libræ contemplationem mathemati<lb></lb>cè ſimpliciter aſſummunt; cùm eius conſideratio ſit prorſus me<lb></lb>chanica: nec vllo modo abſq; vero motu, ac ponderibus (en<pb n="18" xlink:href="036/01/049.jpg"></pb>tibus omninò naturalibus) de ipſa ſermo haberi poſsit: ſine qui<lb></lb>bus eorum, quæ libræ accidunt, veræ caulæ reperiri nullo mo <lb></lb>do poſsint. </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"></emph>Ex<emph.end type="italics"></emph.end> 27 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.30.1.1.2.0"><margin.target id="note52"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 32 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.30.1.1.3.0"><margin.target id="note53"></margin.target>26 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.30.1.1.4.0"><margin.target id="note54"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 13 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.30.1.1.5.0"><margin.target id="note55"></margin.target>19 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.30.1.1.6.0"><margin.target id="note56"></margin.target>47 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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 ſi adhuc ſup<lb></lb>poſitionem conceda<lb></lb>mus; à conſideratione <lb></lb>libræ longè recedunt; <lb></lb>dum eo pacto, vt libra <lb></lb>DE in AB redire de<lb></lb>beat, diſcurrunt. </s> <s id="id.2.1.31.1.1.2.0">ſemper <lb></lb>enim alterum pondus <lb></lb>ſeorſum accipiunt, putá <lb></lb>D, vel E; ac ſi modò <expan abbr="vnũ">vnum</expan><lb></lb>modò alterum in libra <lb></lb>conſtitutum eſſet, nec <lb></lb>vllo modo ambo con<lb></lb><figure id="id.036.01.049.1.jpg" place="text" xlink:href="036/01/049/1.jpg"></figure><lb></lb>nexa; cuius tamen oppoſitum omninò fieri oportet; neq; alterum <lb></lb>ſine altero rectè conſiderari poteſt; cùm de ipſis in libra conſti<lb></lb>tutis ſermo habeatur. </s> <s id="id.2.1.31.1.1.3.0">cùm enim dicunt, deſcenſum ponderis in <lb></lb>D minus obliquum eſſe deſcenſu ponderis in E; erit pondus in <lb></lb>D per ſuppoſitionem grauius pondere in E: quare cùm ſit graui<lb></lb>us, neceſſe eſt deorſum moueri, libramq; DE in AB redire: di<lb></lb>ſcurſus iſte nullius prorſus momenti eſt. </s> <s id="id.2.1.31.1.1.4.0">Primùm quidem ſem<lb></lb>per argumentantur, ac ſi pondera in DE deſcendere debeant, <lb></lb>vnius tantùm ſine alterius connexione conſiderando deſcenſum. </s> <s id="id.2.1.31.1.1.5.0"><lb></lb>poſtremò tamen ob ponderum deſcenſuum comparationem colli<lb></lb>gentes inferunt, pondus in D deorſum moueri, & pondus in E <lb></lb>ſurſum, vtraq; ſimul in libra inuicem connexa accipientes. </s> <s id="id.2.1.31.1.1.6.0">ve<lb></lb>rùm ex iiſdemmet, quibus vtuntur, principiis, ac demonſtratio<lb></lb>nibus, oppoſitum eius, quod defendere conantur, facillimè col<lb></lb>ligi poteſt. </s> <s id="id.2.1.31.1.1.7.0">Nam ſi comparetur deſcenſus ponderis in D cum a<lb></lb>ſcenſu ponderis in E, vt ductis EK DH ipſi AB perpendicula<lb></lb>ribus; cùm angulus DCH ſit æqualis angulo ECk; & angulus <arrow.to.target n="note57"></arrow.to.target><lb></lb>DHC rectus æqualis eſt recto E k C; & latus DC lateri CE æqua<lb></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"></pb>teri Ek æquale. </s> <s id="id.2.1.31.1.1.8.0">cùm <lb></lb>autem angulus DCA <lb></lb>ſit angulo ECB æqua<lb></lb>lis: erit quoq; circum<lb></lb>ferentia DA <expan abbr="cirferen">circumferen</expan><lb></lb>tiæ BE æqualis. </s> <s id="id.2.1.31.1.1.9.0">dum <lb></lb>itaq; pondus in D de<lb></lb>ſcendit per circumfe<lb></lb>rentiam DA, pondus <lb></lb>in E per circumferen<lb></lb>tiam EB ipſi DA æ<lb></lb>qualem aſcendit. </s> <s id="id.2.1.31.1.1.10.0">& de<lb></lb>ſcenſus <expan abbr="põderis">ponderis</expan> in D de <lb></lb>directo (more <expan abbr="ipſorũ">ipſorum</expan>) <lb></lb><figure id="id.036.01.050.1.jpg" place="text" xlink:href="036/01/050/1.jpg"></figure><lb></lb>capiet DH; aſcenſus verò ponderis in E de directo capiet Ek ip<lb></lb>ſi DH æqualem: erit itaq; deſcenſus ponderis in D aſcenſui pon<lb></lb>deris in E æqualis, & qualis erit propenſio vnius ad motum deor<lb></lb>sum, talis etiam erit reſiſtentia alterius ad motum ſurſum. </s> <s id="id.2.1.31.1.1.11.0">re<lb></lb>ſiſtentia ſcilicet violentiæ ponderis in E in aſcenſu naturali po<lb></lb>tentiæ ponderis in D in deſcenſu contrà nitendo apponitur; cùm <lb></lb>ſit ipſi æqualis. </s> <s id="id.2.1.31.1.1.12.0">quò enim pondus in D naturali potentia deor<lb></lb>ſum velocius deſcendit, eò tardius pondus in E violenter aſcendit. </s> <s id="id.2.1.31.1.1.13.0"><lb></lb>quare neutrum ipſorum alteri præponderabit, cùm ab æquali non <lb></lb>proueniat actio. </s> <s id="id.2.1.31.1.1.14.0">Non igitur pondus in D pondus in E ſurſum <lb></lb>mouebit. </s> <s id="id.2.1.31.1.1.15.0">ſi enim moueret; neceſſe eſſet, pondus in D maiorem <lb></lb>habere virtutem deſcendendo, quàm pondus in E aſcendendo; <lb></lb>ſed hæc ſunt æqualia: ergo pondera manebunt. </s> <s id="id.2.1.31.1.1.16.0">& grauitas pon<lb></lb>deris in D grauitati ponderis in E æqualis erit. </s> <s id="id.2.1.31.1.1.17.0">Præterea quoniam <lb></lb>ſupponunt, quò pondus à linea directionis FG magis diſtat, eò <lb></lb>grauius eſſe: Idcirco ductis quoq; à punctis DE ipſi FG perpen<lb></lb>dicularibus DO EI; ſimili modo demonſtrabitur, triangulum <lb></lb>CDO triangulo CEI æqualem eſſe: & lineam DO ipſi EI æqua<lb></lb>lem. </s> <s id="id.2.1.31.1.1.18.0">tam igitur diſtat à linea FG pondus in D, quàm pondus in <lb></lb>E. </s> <s id="N1168B">ex ipſorum igitur rationibus, atq; ſuppoſitionibus, pondera <lb></lb>in DE æquè grauia erunt. </s> <s id="id.2.1.31.1.1.19.0">Amplius quid prohibet, quin libram <lb></lb>DE ex neceſsitate in FG moueri ſimili ratione oſtendatur? </s> <s id="id.2.1.31.1.1.20.0">Pri<pb n="19" xlink:href="036/01/051.jpg"></pb>mùm quidem ex eorummet demonſtrationibus colligi poteſt, a<lb></lb>ſcenſum ponderis in E verſus B rectiorem eſſe aſcenſu ponderis <lb></lb>in D verſus F; hoc eſt minus capere de directo aſcenſum pon<lb></lb>deris in D in arcubus æqualibus aſcenſu ponderis in E. </s> <s id="id.2.1.31.1.1.20.0.a">ſuppona<lb></lb>tur ergo ſecundùm ſitum pondus leuius eſſe, quantò in eodem ſi<lb></lb>tu minus rectus eſt aſcenſus: quæ quidem ſuppoſitio, adeò ma<lb></lb>nifeſta eſſe videtur, veluti ipſorum altera. </s> <s id="id.2.1.31.1.1.21.0">Quoniam igitur aſcen<lb></lb>ſus ponderis in E rectior eſt aſcenſu ponderis in D; per ſuppoſi<lb></lb>tionem pondus in D leuius erit pondere in E. ergo pondus in <lb></lb>D ſurſum à pondere in E mouebitur, ita vt libra in FG perue<lb></lb>niat. </s> <s id="id.2.1.31.1.1.22.0">atq; ita demonſtrari poterit, libram DE in FG moueri.<lb></lb></s> <s id="id.2.1.31.1.1.23.0">quæ quidem demonſtratio inutilis eſt prorſus, eaſdemq; patitur <lb></lb>difficultates. </s> <s id="id.2.1.31.1.1.24.0">licet enim tanquàm verum admittatur pondus in E <lb></lb>aſcendendo grauius eſſe pondere in D ſimiliter aſcendendo, <lb></lb>non tamen ex hoc ſequitur, pondus in E deſcendendo grauius <lb></lb>eſſe pondere in D aſcendendo. </s> <s id="id.2.1.31.1.1.25.0">Neutra igitur harum demon<lb></lb>ſtrationum libram DE, vel in AB redire, vel in FG moue<lb></lb>ri, oſtendentium, vera eſt. </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"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.32.1.1.2.0"><margin.target id="note58"></margin.target>26 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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 ſi ipſorum ſuppoſitionem, eorumq; verborum vim <lb></lb>rectè perpendamus; alium certè habere ſenſum conſpiciemus. </s> <s id="id.2.1.33.1.1.2.0">nam <lb></lb>cùm ſemper ſpatium, per quod naturaliter pondus mouetur, à cen<lb></lb>tro grauitatis ipſius ponderis ad centrum mundi, inſtar rectæ li<lb></lb>neæ à centro grauitatis ad centrum mundi productæ, ſit ſumendum; <lb></lb>tantò huiusmodi ponderis deſcenſus, magis, minusuè obliquus <lb></lb>dicetur; quantò ſecundùm ſpatium inſtar prædictæ lineæ deſigna <lb></lb>tum, magis, aut minus (naturalem tamen locum petens, ſemperq; <lb></lb>magis ipſi appropinquans) mouebitur; ita vt tantò obliquior de<lb></lb>ſcenſus dicatur, quantò recedit ab eiuſmodi ſpatio: rectior verò, <lb></lb>quantò ad idem accedit. </s> <s id="id.2.1.33.1.1.3.0">& in hoc ſenſu ſuppoſitio illa nemini <lb></lb>difficultatem parere debet, adeò enim veritas eius conſpicua eſt; <lb></lb>rationiq; conſentanea: vt nulla proſus manifeſtatione egere vi<lb></lb>deatur. </s> </p> <pb xlink:href="036/01/052.jpg"></pb> <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 ſolutum in ſitu D <lb></lb>collocatum ad propium locum mo<lb></lb>ueri debeat; proculdubio poſito cen<lb></lb>tro mundi S, per lineam DS moue<lb></lb>bitur. </s> <s id="id.2.1.33.3.1.2.0">ſimiliter pondus in E ſolutum <lb></lb>per lineam ES mouebitur. </s> <s id="id.2.1.33.3.1.3.0">quare ſi <lb></lb>(vt rei veritas eſt) ponderis deſcen<lb></lb>ſus magis, minuſuè obliquus dicetur <lb></lb>ſecundùm receſſum, & acceſſum ad <lb></lb>ſpatia per lineas DSES deſignata, <lb></lb>iuxta naturales ipſorum ad propria lo <lb></lb>ca lationes; conſpicuum eſt, minus <lb></lb>obliquum eſſe deſcenſum ipſius E <lb></lb>per EG, quàm ipſius D per DA: <lb></lb>cùm angulum SEG angulo SDA <lb></lb>minorem eſſe ſupra oſtenſum ſit. </s> <s id="id.2.1.33.3.1.4.0">qua <lb></lb>re in E pondus magis grauitabit, <lb></lb>quàm in D. quod eſt penitus oppo<lb></lb>ſitum eius, quod ipſi oſtendere cona<lb></lb>ti ſunt. </s> <s id="id.2.1.33.3.1.5.0">Inſurgent autem fortaſſe <lb></lb>contrarios, ſi igitur (dicent) pondus <lb></lb>in E grauius eſt pondere in D, libra <lb></lb><figure id="id.036.01.052.1.jpg" place="text" xlink:href="036/01/052/1.jpg"></figure><lb></lb>DE in hoc ſitu minimè perſiſtet, quod <expan abbr="equidẽ">equidem</expan> tueri propoſuimus: <lb></lb>ſed in FG mouebitur. </s> <s id="id.2.1.33.3.1.6.0">quibus reſpondemus, plurimum referre, ſiue <lb></lb>conſideremus pondera, quatenus ſunt inuicem diſiuncta, ſiue quate <lb></lb>nus ſunt ſibi inuicem connexa. </s> <s id="id.2.1.33.3.1.7.0">alia eſt enim ratio ponderis in E ſine <lb></lb>connexione ponderis in D, alia verò eiuſdem alteri ponderi con<lb></lb>nexi; ita vt alterum ſine altero moueri non poſsit. </s> <s id="id.2.1.33.3.1.8.0">nam ponde<lb></lb>ris in E, quatenus eſt ſine alterius ponderis connexione, rectus <lb></lb>naturalis deſcenſus eſt per lineam ES; quatenus verò connexum <lb></lb>eſt ponderi in D, eius naturalis deſcenſus non erit amplius per <lb></lb>lineam ES, ſed per lineam ipſi CS parallelam. </s> <s id="id.2.1.33.3.1.9.0">magnitudo enim <lb></lb>ex ponderibus ED, & libra DE compoſita, cuius grauitatis cen<lb></lb>trum eſt C, ſi nullibi ſuſtineatur, deorſum eo modo, quo reperi<lb></lb>tur, ſecundùm grauitatis centrum per rectam à centro grauita<lb></lb>tis C ad centrum mundi S ductam naturaliter mouebitur, donec <pb n="20" xlink:href="036/01/053.jpg"></pb>centrum C in centrum S perueniat. </s> <s id="id.2.1.33.3.1.10.0">libra igitur DE vná cum pon<lb></lb>deribus eo modo, quo reperitur, deorſum mouebitur, ita vt pun<lb></lb>ctum C per lineam CS moueatur, donec C in S, libraq; DE in <lb></lb>Hk perueniat; habeatq; libra in Hk eandem, quam prius habe<lb></lb>bat poſitionem; hoc eſt Hk ſit ipſi DE æquidiſtans. </s> <s id="id.2.1.33.3.1.11.0">connectantur <lb></lb>igitur DH Ek. </s> <s id="id.2.1.33.3.1.12.0">manifeſtum eſt, dum libra DE in Hk mouetur pun<lb></lb>cta DE per lineas DH Ek moueri, quippe exiſtentibus inter ſe <arrow.to.target n="note59"></arrow.to.target><lb></lb>ſe, ipſiq; CS æqualibus, & æquidiſtantibus. </s> <s id="id.2.1.33.3.1.13.0">Quare pondera in <lb></lb>DE, quatenus ſunt ſibi inuicem connexa, ſi ipſorum naturalem mo <lb></lb>tum ſpectemus, non ſecundùm lineas DS ES, ſed ſecundùm <lb></lb>LDH MEk ipſi CS æquidiſtantes mouebuntur. </s> <s id="id.2.1.33.3.1.14.0">ponderis ve<lb></lb>rò in E liberi, ac ſoluti, naturalis propenſio erit per ES: ponderis <lb></lb>autem in D ſimiliter ſoluti erit per DS. ac propterea non eſt incon<lb></lb>ueniens idem pondus modò in E, modò in D, grauius eſſe in E, <lb></lb>quàm in D. </s> <s id="id.2.1.33.3.1.14.0.a">ſi verò pondera in ED ſibi inuicem connexa, quate<lb></lb>nusq; ſunt connexa conſiderauerimus; erit ponderis in E natura<lb></lb>lis propenſio per lineam MEK: grauitas enim alterius ponde<lb></lb>ris in D efficit, nè pondus in E per lineam ES grauitet, ſed per <lb></lb>Ek. </s> <s id="id.2.1.33.3.1.15.0">quod ipſum quoq; grauitas ponderis in E efficit, nè ſcilicet <lb></lb>pondus in D per rectam DS degrauet; ſed ſecundùm DH: vtra<lb></lb>que enim ſe impediunt, nè ad propria loca <expan abbr="permeent">permeant</expan>. </s> <s id="id.2.1.33.3.1.16.0">Cùm igi<lb></lb>tur naturalis deſcenſus rectus ponderum in DE ſit ſecundùm <lb></lb>LDH MEK: erit <expan abbr="ſimliter">similiter</expan> rectus eorum aſcenſus ſecundùm eaſ<lb></lb>dem lineas HDL KEM. </s> <s id="id.2.1.33.3.1.16.0.a">atq; aſcenſus ponderis in E magis, mi<lb></lb>nuſuè obliquus dicetur; quantò ſecundùm ſpatium magis, mi<lb></lb>nuſuè iuxta lineam Mk mouebitur. </s> <s id="id.2.1.33.3.1.17.0">hocq; prorſus modo iuxta li<lb></lb>neam LH ſummendus eſt, tùm deſcenſus, tùm aſcenſus ponde<lb></lb>ris in D. </s> <s id="N117DE">ſi itaq; pondus in E deorſum per EG moueretur; pon<lb></lb>dus in D ſurſum 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></lb>æqualis eſt angulo CDL, & angulus CEG angulo CDF æqua<lb></lb>lis; erit reliquus GEK reliquo LDF æqualis. </s> <s id="id.2.1.33.3.1.19.0">cùm autem ſup<lb></lb>poſitio illa, quæ ait, ſecundúm ſitum pondus grauius eſſe, quan<lb></lb>tò in eodem ſitu minus obliquus eſt deſcenſus; tanquam clara, <lb></lb>atq; conſpicua admittatur; proculdubio hæc quoq; accipienda <lb></lb>erit; nempè, ſecundúm ſitum pondus grauius eſſe, quantò in eo<lb></lb>dem ſitu minus obliquus eſt aſcenſus. </s> <s id="id.2.1.33.3.1.20.0">cùm non minus manifeſta, <pb xlink:href="036/01/054.jpg"></pb>rationiq; ſit conſentanea. </s> <s id="id.2.1.33.3.1.21.0">æqualis <lb></lb>igitur erit deſcenſus ponderis in E <lb></lb>aſcenſui ponderis in D. </s> <s id="N11807">eandem <lb></lb>enim obliquitatem habet deſcenſus <lb></lb>ponderis in E, quam habet aſcen<lb></lb>ſus ponderis in D; & qualis erit <lb></lb>propenſio vnius ad motum deorſum, <lb></lb>talis quoq; erit reſiſtentia alterius ad <lb></lb>motum ſurſum. </s> <s id="id.2.1.33.3.1.22.0"><expan abbr="nõ">non</expan> ergo pondus in E <lb></lb>pondus in D ſurſum mouebit. </s> <s id="id.2.1.33.3.1.23.0">neq; <lb></lb>pondus in D deorſum mouebitur, ita <lb></lb>vt ſurſum moueat pondus in E. </s> <s id="id.2.1.33.3.1.23.0.a">nam <lb></lb><expan abbr="cũ">cum</expan> angulus CEB ſit ipſi CDA æqua<lb></lb><arrow.to.target n="note61"></arrow.to.target>lis, & Angulus CEM ſit angulo <lb></lb>CDH æqualis; erit reliquus MEB <lb></lb>reliquo HDA æqualis. </s> <s id="id.2.1.33.3.1.24.0">deſcenſus <lb></lb>igitur ponderis in D aſcenſui ponde<lb></lb>ris in E æqualis erit. </s> <s id="id.2.1.33.3.1.25.0">non ergo pon<lb></lb>dus in D pondus in E ſurſum moue<lb></lb>bit. </s> <s id="id.2.1.33.3.1.26.0">ex quibus ſequitur pondera in <lb></lb>DE, quatenus ſunt ſibi inuicem con<lb></lb>nexa, æquè grauia eſſe. <figure id="id.036.01.054.1.jpg" place="text" 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></lb>bram ſimiliter DE in AB <lb></lb>redire oſtendens, cùm in<lb></lb>quiunt, exiſtente trutina in <lb></lb>CF meta eſt CG. </s> <s id="id.2.1.33.4.1.1.0.a">& quo<lb></lb>niam angulus DCG maior <lb></lb>eſt angulo ECG; pondus <lb></lb>in D grauius erit pondere <lb></lb>in E; ergo libra DE in AB <lb></lb>redibit: nihil meo iudicio <lb></lb>concludit. </s> <s id="id.2.1.33.4.1.2.0">figmentumq; <lb></lb>hoc de trutina, & meta po<lb></lb>tius omittendum, ac ſilen<figure id="id.036.01.054.2.jpg" place="text" xlink:href="036/01/054/2.jpg"></figure> <pb n="21" xlink:href="036/01/055.jpg"></pb>tio <expan abbr="prætereundũ">prætereundum</expan> eſſet, quàm <expan abbr="verbũ">verbum</expan> <expan abbr="vllũ">vllum</expan> in eius confutatione ſumen<lb></lb>dum; cùm ſit prorſus voluntarium. </s> <s id="id.2.1.33.4.1.3.0">neceſsitas enim cur pondus <lb></lb>in D ex maiore angulo ſit grauius; curq; maior angulus maioris <lb></lb>ſit cauſa grauitatis; nuſquam apparet. </s> <s id="id.2.1.33.4.1.4.0">ſi autem comparentur in<lb></lb>uicem anguli, cùm angulus GCD ſit æqualis angulo FCE; ſi angu<lb></lb>lus GCD eſt cauſa grauitatis; quare angulus FCE ſimiliter gra<lb></lb>uitatis non eſt cauſa? </s> <s id="id.2.1.33.4.1.5.0">Huius autem rei eam in medium rationem <lb></lb>afferre videntur, quoniam CG eſt meta, & CF trutina. </s> <s id="id.2.1.33.4.1.6.0">ſi (inquiunt) <lb></lb>CG eſſet trutina, & CF meta, tunc angulus FCE grauitatis eſſet <lb></lb>cauſa; non autem DCG ipſi æqualis. </s> <s id="id.2.1.33.4.1.7.0">quæ quidem ratio imma<lb></lb>ginaria prorſus, ac voluntaria eſſe videtur. </s> <s id="id.2.1.33.4.1.8.0">quid enim refert, ſiue tru<lb></lb>tina ſit in CF, ſiue in CG, cùm libra DE in eodem ſemper pun<lb></lb>cto C ſuſtineatur? </s> <s id="id.2.1.33.4.1.9.0">Vt autem eorum deceptio clarius appa<lb></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"></emph>Prmi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.34.1.1.2.0"><margin.target id="note60"></margin.target>29 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.34.1.1.3.0"><margin.target id="note61"></margin.target>29 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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></lb>ius medium C. </s> <s id="id.2.1.35.1.1.1.0.a">ſit deinde <lb></lb>tota FG trutina. </s> <s id="id.2.1.35.1.1.2.0">eaq; im<lb></lb>mobilis exiſtat; quæ libram <lb></lb>AB in puncto C ſuſtineat. </s> <s id="id.2.1.35.1.1.3.0"><lb></lb>moueaturq; libra in DE. </s> <s id="N118EA">& <lb></lb>quoniam trutina eſt, & ſu<lb></lb>pra, & infra libram, quis <lb></lb>nam angulus erit cauſa gra<lb></lb>uitatis, cùm libra DE in <lb></lb><figure id="id.036.01.055.1.jpg" place="text" xlink:href="036/01/055/1.jpg"></figure> <expan abbr="eodẽ"><lb></lb>eodem</expan> ſemper puncto ſuſtineatur? </s> <s id="id.2.1.35.1.1.4.0">dicent forſan, ſi trutina à potentia <lb></lb>in F ſuſtiteneatur, tunc CG erit tanquam meta, & angulus <lb></lb>DCG grauitatis erit cauſa. </s> <s id="id.2.1.35.1.1.5.0">ſi verò ſuſtineatur in G, tunc FCE <lb></lb>erit cauſa grauitatis, CF verò tanquam meta erit. </s> <s id="id.2.1.35.1.1.6.0">cuius quidem <lb></lb>rei nulla videtur eſſe cauſa, niſi immaginaria. </s> <s id="id.2.1.35.1.1.7.0">meta enim (quod <lb></lb>aiunt) nullam prorſus vim attractiuam, quandoq; ex maioris an<lb></lb>guli parte, quandoq; ex parte minoris habere videtur. </s> <s id="id.2.1.35.1.1.8.0">Verùm à dua<lb></lb>bus potentiis ſuſtineatur trutina, in F ſcilicet, & in G, quod præ ne<lb></lb>ceſsitate fieri poteſt, veluti ſi potentia in F ſit adeò debilis, vt ex ſe <lb></lb>ipſa medietatem tantùm ponderis ſuſtinere quæat: ſitq; potentia in <lb></lb>G ipſi potentiæ in F æqualis, vtræq; <expan abbr="autẽ">autem</expan> ſimul libram vná cum pon<lb></lb>deribus ſuſtineant. </s> <s id="id.2.1.35.1.1.9.0">tunc quis nam angulus erit cauſa grauitatis? </s> <s id="id.2.1.35.1.1.10.0">non <pb xlink:href="036/01/056.jpg"></pb>FCE, quia trutina eſt in <lb></lb>CF, & in F ſuſtinetur. </s> <s id="id.2.1.35.1.1.11.0">neq; <lb></lb>DCG, cùm trutina ſit in <figure id="id.036.01.056.1.jpg" place="text" xlink:href="036/01/056/1.jpg"></figure><lb></lb>CG, & in G quoq; ſuſti<lb></lb>neatur; non igitur anguli <lb></lb>grauitatis cauſa erunt. </s> <s id="id.2.1.35.1.1.12.0">ergo <lb></lb>neq; libra DE ab hoc ſitu <lb></lb>ob hanc cauſam mouebi<lb></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></lb>ſententiam dupliciter con<lb></lb>firmare videntur. </s> <s id="id.2.1.35.1.1.14.0">primùm quidem aſſerunt Ariſtotelem in quæſtio<lb></lb>nibus mechanicis has duas tantùm quæſtiones propoſuiſſe; eiuſq; <lb></lb>demonſtrationes, tum maiori, & minori angulo, tùm trutinæ poſi<lb></lb>tioni inniti. </s> <s id="id.2.1.35.1.1.15.0">Affirmant deinde experientiam hoc idem docere; <lb></lb>hoc eſt libram DE trutina exiſtente in CF, in AB horizonti <lb></lb>æquidiſtantem redire. </s> <s id="id.2.1.35.1.1.16.0">quando autem trutina eſt in CG, in FG <lb></lb>moueri. </s> <s id="id.2.1.35.1.1.17.0">Verùm neq; Ariſtoteles, neq; experientia huic eorum <lb></lb>opinioni fauent, quin potius aduerſantur. </s> <s id="id.2.1.35.1.1.18.0">quantùm enim atti<lb></lb>net ad experientiam decipiuntur, ipſa quidem experientia ma<lb></lb>nifeſtum eſt hoc accidere, quando libræ quoq; centrum, vel ſu<lb></lb>pra, vel infra libram fuerit collocatum: non autem trutina dun<lb></lb>taxat ſupra, vel infra exiſtente, 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"></emph>Cardanus.<emph.end type="italics"></emph.end></s> </p> <pb n="22" xlink:href="036/01/057.jpg"></pb> <p id="id.2.1.37.1.0.0.0" type="main"> <s id="id.2.1.37.1.2.1.0">Nam ſi libra AB habeat <lb></lb>centrum C ſupra libram; <lb></lb>ſitq; trutina CD infra li<lb></lb>bram; moueaturq; libra in <lb></lb>EF; tunc EF rurſus in AB <lb></lb>horizonti æquidiſtantem <arrow.to.target n="note63"></arrow.to.target><lb></lb>redibit. </s> <s id="id.2.1.37.1.2.2.0">ſimiliter ſi libra <lb></lb>centrum C habeat infra li<lb></lb>bram, ſitq; trutina CD ſu<lb></lb>pra libram, & moueatur <lb></lb>libra in EF; patet libram <arrow.to.target n="note64"></arrow.to.target><lb></lb>ex parte F deorſum moue <lb></lb>ri, trutina ſupra libram e<lb></lb>xiſtente. </s> <s id="id.2.1.37.1.2.3.0">& in quocunq; a<lb></lb>lio ſitu fuerit trutina, idem <lb></lb>ſemper eueniet. </s> <s id="id.2.1.37.1.2.4.0">non igitur <lb></lb>trutina, ſed centrum libræ <lb></lb>harum diuerſitatum cau<lb></lb>ſa erit. <figure id="id.036.01.057.1.jpg" place="text" 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 eſt <lb></lb>itaq; in hac parte difficulter materialem libram conſtitui poſſe, <lb></lb>quæ in vno tantùm puncto ſuſtineatur; quemadmodum mente <lb></lb>concipimus. </s> <s id="id.2.1.37.2.1.2.0">brachiaq; ab eiuſmodi centro adeò æqualia habeat, <lb></lb>non ſolum in longitudine, verùm etiam in latitudine, & profun<lb></lb>ditate, vt omnes partes hinc indé ad vnguem æqueponderent. </s> <s id="id.2.1.37.2.1.3.0"><lb></lb>hoc enim materia difficilimè patitur. </s> <s id="id.2.1.37.2.1.4.0">quocirca ſi centrum in ipſa <lb></lb>libra eſſe conſiderauerimus, ad ſenſum confugiendum non eſt: <lb></lb>cùm artificilia ad ſummum illud perfectionis gradum ab artifice <lb></lb>deduci minimè poſsint. </s> <s id="id.2.1.37.2.1.5.0">In aliis verò experientia quidem appa<lb></lb>rentia docere poterit; propterea quod, quamquam centrum libræ <lb></lb>ſit ſemper punctum, quando tamen ſupra libram fuerit, parùm re<lb></lb>fert, ſi libra in eo puncto adamuſſim minimè ſuſtineatur; quia cùm <lb></lb>ſit ſemper ſupra libram, idem ſemper eueniet. </s> <s id="id.2.1.37.2.1.6.0">ſimili quoq; modo <lb></lb>quando eſt infra libram: quod tamen non accidit centro in ipſa li<lb></lb>bra exiſtente. </s> <s id="id.2.1.37.2.1.7.0">ſi enim ad vnguem ſemper in illo medio non ſu<lb></lb>ſtineatur, diuerſitatem efficiet; cùm facillimum ſit, centrum il<pb xlink:href="036/01/058.jpg"></pb>lud, dùm libra mouetur, proprium mutare ſitum. </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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.38.1.1.2.0"><margin.target id="note64"></margin.target>3 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 Ariſtoteles duas tantùm quæſtiones propo<lb></lb>ſuerit, cur ſcilicet trutina ſuperius exiſtente, ſi libra non ſit <lb></lb>horizonti æquidiſtans in æquilibrium, hoc eſt horizonti æqui <lb></lb>diſtans redit: ſi autem trutina deorſum fuerit conſtituta, non <lb></lb>redit; ſed adhuc ſecundùm partem depreſſam mouetur: verum <lb></lb>quidem eſt. </s> <s id="id.2.1.39.1.1.2.0">non tamen eius demonſtrationes maiori, & mino <lb></lb>ri angulo, poſitioni〈qué〉 trutinæ (vt ipſi dicunt) innituntur. </s> <s id="id.2.1.39.1.1.3.0">In <lb></lb>hoc enim mentem philoſophi aſignantis rationem diuerſitatis <lb></lb>motuum libræ minimè attingunt. </s> <s id="id.2.1.39.1.1.4.0">tantùm enim abeſt philoſo<lb></lb>phum has diuerſitates in angulos referre, vt potius in cauſa eſſe <lb></lb>dicat magnitudinis alterius brachii libræ exceſſum à perpendiculo, <lb></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 ſuperius in <lb></lb>CF exiſtente, perpendicu<lb></lb>lum erit FCG, quod ſe<lb></lb>cundùm ipſum in centrum <lb></lb>mundi ſemper vergit; <lb></lb>quod quidem libram mo<lb></lb>tam in DE in partes di<lb></lb>uidit inæquales; & maior <lb></lb>pars eſt verſus D: id au<lb></lb>tem, quod plus eſt, deor<lb></lb>ſum fertur; ergo ex par<lb></lb>te D deorſum libra moue<lb></lb>bitur, donec in AB re<lb></lb>deat. </s> <s id="id.2.1.39.2.1.2.0">ſi verò trutina ſit <lb></lb><figure id="id.036.01.058.1.jpg" place="text" xlink:href="036/01/058/1.jpg"></figure><lb></lb>in CG deorſum, erit GCF perpendiculum, quod libram DE <lb></lb>in partes inæquales ſimiliter diuidit: maior autem pars erit verſus <lb></lb>E; quare ex parte E deorſum libra mouebitur. </s> <s id="id.2.1.39.2.1.3.0">quod vt rectè in<lb></lb>telligatur, cùm trutina eſt ſupra libram, libræ quoq; centrum ſu<lb></lb>pra libram eſſe intelligendum eſt; & ſi deorſum, centrum quoque <lb></lb>deorſum: vt infra patebit. </s> <s id="id.2.1.39.2.1.4.0">Aliter ipſa Ariſtotelis demonſtratio <lb></lb>nihil concluderet. </s> <s id="id.2.1.39.2.1.5.0">exiſtente enim centro in ipſa libra, vt in C; quo<lb></lb>cunq; modo moueatur libra, nunquam perpendiculum FG libram, <pb n="23" xlink:href="036/01/059.jpg"></pb>niſi in puncto C, & in partes diuidet æquales. </s> <s id="id.2.1.39.2.1.6.0">quare Ariſtotelis <lb></lb>ſententia ipſis non ſolum non fauet, verùm etiam maximè aduer<lb></lb>ſatur. </s> <s id="id.2.1.39.2.1.7.0">quòd non ſolum ex ſecunda, & tertia huius liquet; verùm <lb></lb>quia exiſtente centro ſupra libram pondus eleuatum maiorem <lb></lb>propter ſitum acquirit grauitatem. </s> <s id="id.2.1.39.2.1.8.0">ex quò contingit redditus li<lb></lb>bræ ad æqualem horizonti diſtantiam. </s> <s id="id.2.1.39.2.1.9.0">è contra verò, quando <lb></lb>centrum eſt infra libram. </s> <s id="id.2.1.39.2.1.10.0">Quæ omnia hoc modo oſtendentur; <lb></lb>ſupponendo ea, quæ ſupra declarata ſunt. </s> <s id="id.2.1.39.2.1.11.0">ſcilicet pondus ex quò <lb></lb>loco rectius deſcendit, grauius fieri. </s> <s id="id.2.1.39.2.1.12.0">& ex quo rectius aſcendit, gra<lb></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></lb>æquidiſtans, cuius centrum <lb></lb>C ſit ſupra libram, perpen<lb></lb>diculumq; ſit CD. ſintq; in <lb></lb>AB ponderum æqualium <lb></lb>centra grauitatis poſita: mo<lb></lb>taq; ſit libra in EF. </s> <s id="id.2.1.39.3.1.1.0.a">Dico <lb></lb>pondus in E maiorem ha<lb></lb>bere grauitatem, quàm pon<lb></lb>dus in F. </s> <s id="N11ACB">& ob id libram <lb></lb>EF in AB redire. </s> <s id="id.2.1.39.3.1.2.0">Produ<lb></lb>catur primùm CD vſq; ad <lb></lb>mundi <expan abbr="centrũ">centrum</expan>, quod ſit S. </s> <s id="id.2.1.39.3.1.2.0.a">de<lb></lb>inde AC CB EC CF HS <lb></lb><expan abbr="cõnectantur">connectantur</expan>, à punctiſq; EF <lb></lb>ipſi HS æquidiſtantes du<lb></lb>cantur Ek GFL. </s> <s id="id.2.1.39.3.1.2.0.b">Quoniam <lb></lb>igitur naturalis deſcenſus re<lb></lb>ctus totius magnitudinis, <lb></lb>libræ ſcilicet EF ſic conſti<lb></lb>tutæ vná cum ponderibus, <lb></lb>eſt <expan abbr="ſcundùm">secundum</expan> grauitatis cen<lb></lb>trum H per rectam HS; erit <lb></lb><figure id="id.036.01.059.1.jpg" place="text" xlink:href="036/01/059/1.jpg"></figure><lb></lb>quoq; ponderum in EF ita poſsitorum deſcenſus ſecundùm re<lb></lb>ctas Ek FL ipſi HS parallelas; ſicuti ſupra demonſtrauimus. </s> <s id="id.2.1.39.3.1.3.0"><pb xlink:href="036/01/060.jpg"></pb>Deſcenſus igitur, & aſcen<lb></lb>ſus ponderum in EF ma<lb></lb>gis, minuſuè obliquus di<lb></lb>cetur ſecundùm acceſſum, <lb></lb>& receſſum iuxta lineas Ek <lb></lb>FL deſignatum. </s> <s id="id.2.1.39.3.1.4.0"><expan abbr="Quoniã">Quoniam</expan> <expan abbr="autẽ">au<lb></lb>tem</expan> duo latera AD DC duo<lb></lb>bus lateribus BD DE ſunt <lb></lb>æqualia; anguliq; ad D ſunt <lb></lb><arrow.to.target n="note65"></arrow.to.target>recti; erit latus AC lateri <lb></lb>CB æquale. </s> <s id="id.2.1.39.3.1.5.0">& cùm pun<lb></lb>ctum C ſit immobile; dum <lb></lb>puncta AB mouentur, cir<lb></lb>culi circumferentiam deſcri<lb></lb>bent, cuius ſemidiameter <lb></lb>erit AC. </s> <s id="id.2.1.39.3.1.5.0.a">quare centro C, <lb></lb>circulus deſcribatur AEBF. <lb></lb></s> <s id="id.2.1.39.3.1.5.0.b">puncta AB EF in circuli <lb></lb>circumferentia erunt. </s> <s id="id.2.1.39.3.1.6.0">ſed <lb></lb>cùm EF ſit ipſi AB æqua <lb></lb><arrow.to.target n="note66"></arrow.to.target>lis; erit circumferentia <lb></lb>EAF circumferentiæ AFB <lb></lb>æqualis. </s> <s id="id.2.1.39.3.1.7.0">quare dempta <lb></lb><figure id="id.036.01.060.1.jpg" place="text" xlink:href="036/01/060/1.jpg"></figure><lb></lb>communi AF, erit circumferentia EA circumferentiæ FB æqua<lb></lb>lis. </s> <s id="id.2.1.39.3.1.8.0">Quoniam autem mixtus angulus CEA eſt æqualis mixto <lb></lb>CFB; & HFB ipſo CFB eſt maior; angulus verò HEA ipſo <lb></lb>CEA minor; erit angulus HFB angulo HEA maior. </s> <s id="id.2.1.39.3.1.9.0">à quibus <lb></lb><arrow.to.target n="note67"></arrow.to.target>ſi auferantur anguli HFG HEk æquales; erit angulus GFB an <lb></lb>gulo kEA maior. </s> <s id="id.2.1.39.3.1.10.0">ergo deſcenſus ponderis in E minus obliquus <lb></lb>erit aſcenſu ponderis in F. </s> <s id="N11B6C">& quamquam pondus in E deſcen<lb></lb>dendo, & pondus in F aſcendendo per circumferentias mouean<lb></lb>tur æquales; quia tamen pondus in E ex hoc loco rectius deſcen<lb></lb>dit, quàm pondus in F aſcendit: idcirco naturalis potentia pon<lb></lb>deris in E reſiſtentiam violentiæ ponderis F ſuperabit. </s> <s id="id.2.1.39.3.1.11.0">quare <lb></lb>maiorem grauitatem habebit pondus in E, quàm pondus in F. </s> <s id="id.2.1.39.3.1.11.0.a"><lb></lb>ergo pondus in E deorſum, pondus verò in F ſurſum mouebitur: <pb n="24" xlink:href="036/01/061.jpg"></pb>donec libra EF in AB redeat. </s> <s id="id.2.1.39.3.1.12.0">quod demonſtrare 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"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.40.1.1.2.0"><margin.target id="note66"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 28 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.40.1.1.3.0"><margin.target id="note67"></margin.target>29 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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 Ariſtotele poſita, hic manifeſta in <arrow.to.target n="note68"></arrow.to.target><lb></lb>tueri poteſt. </s> <s id="id.2.1.41.1.1.2.0">ſit enim punctum N vbi CS EF ſe inuicem ſecant. </s> <s id="id.2.1.41.1.1.3.0"><lb></lb>& quoniam HE eſt ipſi HF æqualis; erit NE maior NF. </s> <s id="N11BBF">li<lb></lb>nea ergo CS, quam perpendiculum vocat, libram EF in partes di<lb></lb>uidet inæquales. </s> <s id="id.2.1.41.1.1.4.0">cùm itaq; pars libræ NE ſit maior NF; atq; id, <lb></lb>quod plus eſt, neceſſe eſt, deorſum ferri: libra ergo EF ex parte E <lb></lb>deorſum 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"></emph>Ariſtotelis ratio.<emph.end type="italics"></emph.end></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></lb>ctenus dicta ſunt inferre li<lb></lb>cet, libram EF velocius ab <lb></lb>eo ſitu in AB moueri; vndè <lb></lb>linea EF in directum pro<lb></lb>tracta in centrum mundi <lb></lb>perueniat. </s> <s id="id.2.1.43.1.1.2.0">vt ſit EFS recta <lb></lb>linea. </s> <s id="id.2.1.43.1.1.3.0">& quoniam CD <lb></lb>CH, ſunt inter ſe ſe æqua<lb></lb>les. </s> <s id="id.2.1.43.1.1.4.0">ſi igitur centro C, ſpa<lb></lb>tioq; CD, circulus deſcri<lb></lb>batur DHM; erunt pun<lb></lb>cta DH in circuli circum<lb></lb>ferentia. </s> <s id="id.2.1.43.1.1.5.0">Quoniam au<lb></lb>tem CH ipſi EF eſt per<lb></lb>pendicularis; continget li<lb></lb>nea EHS circulum DHM <lb></lb>in puncto H. </s> <s id="id.2.1.43.1.1.5.0.a">pondus igi<lb></lb>tur in H (ſicuti ſupra de<lb></lb>monſtrauimus) grauius <lb></lb><figure id="id.036.01.061.1.jpg" place="text" xlink:href="036/01/061/1.jpg"></figure><lb></lb>erit, quàm in alio ſitu circuli DHM. </s> <s id="id.2.1.43.1.1.5.0.b">ergo magnitudo ex EF <lb></lb>ponderibus, & libra EF compoſita, cuius centrum grauitatis eſt <lb></lb>in H, in hoc ſitu magis grauitabit, quàm in quocunq; alio ſitu <pb xlink:href="036/01/062.jpg"></pb>circuli fuerit punctum H. <lb></lb></s> <s id="N11C27">ab hoc igitur ſitu velo<lb></lb>cius, quàm à quocunq; <lb></lb>alio mouebitur. </s> <s id="id.2.1.43.1.1.6.0">& ſi H <lb></lb>propius fuerit ipſi D mi <lb></lb>nus grauitabit, minuſq; <lb></lb>ab eo ſitu mouebitur. </s> <s id="id.2.1.43.1.1.7.0"><lb></lb>ſemper enim deſcenſus <lb></lb>obliquior eſt, & minus re<lb></lb>ctus. </s> <s id="id.2.1.43.1.1.8.0">libra ergo EF velo<lb></lb>cius ab hoc ſitu mouebi<lb></lb>tur, quàm ab alio ſitu. </s> <s id="id.2.1.43.1.1.9.0">& <lb></lb>ſi propius ad AB acce<lb></lb>det, inde minus mouebi<lb></lb>tur. </s> <s id="id.2.1.43.1.1.10.0">Deinde quò longius <lb></lb>punctum H à puncto C <lb></lb>diſtabit, velocius moue<lb></lb>bitur; quod <expan abbr="nõ">non</expan> <expan abbr="ſolũ">ſolum</expan> ex Ari<lb></lb>ſtotele in principio quæſt<lb></lb>io num mechanicarum, & <lb></lb><figure id="id.036.01.062.1.jpg" place="text" xlink:href="036/01/062/1.jpg"></figure><lb></lb>ex ſuperius dictis patet; verùm etiam ex iis, quæ infra in ſexta <lb></lb>propoſitione dicemus, manifeſtum erit. </s> <s id="id.2.1.43.1.1.11.0">libra igitur EF, quò ma<lb></lb>gis ab eius centro diſtabit, adhuc velocius mouebitur. </s> </p> <pb n="25" xlink:href="036/01/063.jpg"></pb> <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></lb>cuius centrum C ſit infra li<lb></lb>bram; ſintq; in AB pon<lb></lb>dera æqualia; libraq; ſit <lb></lb>mota in EF. </s> <s id="id.2.1.43.3.1.1.0.a">Dico maio<lb></lb>rem habere grauitatem <lb></lb>pondus in F, quàm pondus <lb></lb>in E. </s> <s id="id.2.1.43.3.1.1.0.b">atq; ideo libram EF <lb></lb>deorſum ex parte F moue<lb></lb>ri. </s> <s id="id.2.1.43.3.1.2.0">Producatur DC ex <lb></lb>vtraq; parte vſq; ad mun<lb></lb>di centrum S, & vſq; ad <lb></lb>O, lineaq; HS ducatur, <lb></lb>cui à punctis EF æquidi<lb></lb>ſtantes ducantur GEk FL; <lb></lb>connectanturq; CE CF: <lb></lb>atq; centro C, ſpatioq; CE <lb></lb>circulus deſcribatur AEO <lb></lb>BF. </s> <s id="id.2.1.43.3.1.2.0.a">ſimiliter demonſtra<lb></lb>bitur puncta ABEF in <lb></lb>circuli circumferentia eſſe; <lb></lb>deſcenſumq; libræ EF vná <lb></lb>cum ponderibus rectum ſe<lb></lb>cundùm lineam HS fieri; <lb></lb>ponderumq; in EF ſecun<lb></lb><figure id="id.036.01.063.1.jpg" place="text" xlink:href="036/01/063/1.jpg"></figure>dùm<lb></lb> lineas GK FL ipſi HS æquidiſtantes. </s> <s id="id.2.1.43.3.1.3.0">Quoniam autem an<lb></lb>gulus CFP æqualis eſt angulo CEO: erit angulus HFP angulo <lb></lb>HEO maior. </s> <s id="id.2.1.43.3.1.4.0">angulus verò HFL æqualis eſt angulo HEG. </s> <s id="id.2.1.43.3.1.4.0.a">à <arrow.to.target n="note69"></arrow.to.target><lb></lb>quibus igitur ſi demantur anguli HFP HEO, erit angulus <lb></lb>LFP angulo GEO minor. </s> <s id="id.2.1.43.3.1.5.0">quare deſcenſus ponderis in F rectior <lb></lb>erit aſcenſu ponderis in E. </s> <s id="id.2.1.43.3.1.5.0.a">ergo naturalis potentia ponderis in <lb></lb>F reſiſtentiam violentiæ ponderis in E ſuperabit. </s> <s id="id.2.1.43.3.1.6.0">& ideo ma<lb></lb>iorem habebit grauitatem pondus in F, quàm pondus in E. </s> <s id="id.2.1.43.3.1.6.0.a"><lb></lb>Pondus igitur in F deorſum, pondus verò in E ſurſum mo<lb></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"></emph>Primi.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.45.1.0.0.0" type="main"> <s id="id.2.1.45.1.1.1.0">Ariſtotelis quoq; ratio hic perſpicua erit. </s> <s id="id.2.1.45.1.1.2.0">ſit enim punctum <arrow.to.target n="note70"></arrow.to.target> <pb xlink:href="036/01/064.jpg"></pb>N vbi CO EF ſe inuicem <lb></lb>ſecant; erit NF maior <lb></lb>NE. </s> <s id="id.2.1.45.1.1.2.0.a">& quoniam CO per<lb></lb>pendiculum (ſecundùm <lb></lb>ipſum) libram EF in par<lb></lb>tes inæquales diuidit, & <lb></lb>maior pars eſt verſus F, hoc <lb></lb>eſt NF; libra EF ex par<lb></lb>te F deorſum mouebitur: <lb></lb>cùm id, quod plus eſt, deor<lb></lb>ſum 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"></emph>Ariſtotelis ratio.<emph.end type="italics"></emph.end></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></lb>quoq; eliciemus libram EF <lb></lb>centrum habens infra li<lb></lb>bram, quò magis à ſitu <lb></lb>AB diſtabit, velocius mo <lb></lb>ueri. </s> <s id="id.2.1.47.1.1.2.0">centrum enim graui<lb></lb>tatis H, quò magis á pun<lb></lb>cto D diſtat, eò volecius <lb></lb>pondus ex EF ponderibus, <lb></lb>libraq; EF compoſitum <lb></lb>mouebitur, donec angulus <lb></lb>CHS rectus euadat. </s> <s id="id.2.1.47.1.1.3.0">ad<lb></lb>huc inſuper velocius moue<lb></lb>bitur, quò libram à centro <lb></lb>C magis diſtabit. <figure id="id.036.01.064.1.jpg" place="text" 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 ipſorum quinetiam rationibus, ac falſis ſupoſitionibus iam <lb></lb>declaratos libræ effectus, ac motus deducere, ac manifeſtare libet; <lb></lb>vt quanta ſit veritatis efficacia appareat, quippè ex falſis etiam <lb></lb>eluceſcere contendit. </s> </p> <pb n="26" xlink:href="036/01/065.jpg"></pb> <p id="id.2.1.47.4.0.0.0" type="main"> <s id="id.2.1.47.4.1.1.0">Exponantur eadem, ſci <lb></lb>licet ſit circulus AEBF; <lb></lb>libra〈qué〉 AB, cuius cen<lb></lb>trum C ſit ſupra libram, <lb></lb>moueatur in EF. </s> <s id="id.2.1.47.4.1.1.0.a">dico <lb></lb>pondus in E maiorem ibi <lb></lb>habere grauitatem, quàm <lb></lb>pondus in F; libramq; EF <lb></lb>in AB redire. </s> <s id="id.2.1.47.4.1.2.0">Ducantur <lb></lb>à punctis EF ipſi AB <lb></lb>perpendiculares EL FM, <lb></lb>quæ inter ſe æquidiſtan<lb></lb>tes <arrow.to.target n="note71"></arrow.to.target> <figure id="id.036.01.065.1.jpg" place="text" xlink:href="036/01/065/1.jpg"></figure>erunt; ſitq; punctum N, vbi AB EF ſe inuicem ſecant. </s> <s id="id.2.1.47.4.1.3.0"><lb></lb>Quoniam igitur angulus FNM eſt æqualis angulo ENL, & an<lb></lb>gulus <arrow.to.target n="note72"></arrow.to.target>F MN rectus recto ELN æqualis, ac reliquus NFM reli<lb></lb>quo <arrow.to.target n="note73"></arrow.to.target>NEL eſt etiam æqualis; erit triangulum NLE triangu<lb></lb>lo NMF ſimile. </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></lb>mutando vt EN ad NF, ita EL ad FM. </s> <s id="id.2.1.47.4.1.4.0.a">ſed cùm ſit HE ipſi <arrow.to.target n="note75"></arrow.to.target><lb></lb>HF æqualis, erit EN maior NF; quare & EL maior erit FM. </s> <s id="id.2.1.47.4.1.4.0.b"><lb></lb>& quoniam dum pondus in E per <expan abbr="circumferentiiam">circumferentiam</expan> EA deſcendit, <lb></lb>pondus in F per circumferentiam FB ipſi circumferentiæ EA <lb></lb>æqualem aſcendit; deſcenſuſq; ponderis in E de directo (vt ip<lb></lb>ſi dicunt) capit EL: aſcenſus verò ponderis in F de directo ca<lb></lb>pit FM; minus de directo capiet aſcenſus ponderis in F, quàm <lb></lb>deſcenſus ponderis in E. </s> <s id="id.2.1.47.4.1.4.0.c">maiorem igitur grauitatem habebit pon<lb></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"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.48.1.1.2.0"><margin.target id="note72"></margin.target>15 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.48.1.1.3.0"><margin.target id="note73"></margin.target>29 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.48.1.1.4.0"><margin.target id="note74"></margin.target>4 <emph type="italics"></emph>Sexti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.48.1.1.5.0"><margin.target id="note75"></margin.target>16 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></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></lb>puncto S ſecet. </s> <s id="id.2.1.49.1.1.2.0">& quoniam (vt aiunt) quò magis pondus à li<lb></lb>nea directionis OP diſtat, eò fit grauius; idcirco hoc quoq; me <lb></lb>dio pondus in E maiorem habere <expan abbr="grauitauitatem">grauitatem</expan> pondere in F o<lb></lb>ſtendetur. </s> <s id="id.2.1.49.1.1.3.0">Ducantur à punctis EF ipſi OP perpendiculares EQ <lb></lb>FR. </s> <s id="id.2.1.49.1.1.3.0.a">ſimili ratione oſtendetur, triangulum QES triangulo RFS <lb></lb>ſimile eſſe; lineamq; EQ ipſa RF maiorem eſſe. </s> <s id="id.2.1.49.1.1.4.0">pondus itaq; <lb></lb>in E magis à linea OP diſtabit, quàm pondus in F; ac propterea <lb></lb>pondus in E maiorem habebit grauitatem pondere in F. </s> <s id="id.2.1.49.1.1.4.0.a">ex quibus <lb></lb>reditus libræ EF in AB manifeſtus apparet. </s> </p> <pb xlink:href="036/01/066.jpg"></pb> <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></lb>ſit infra libram, tunc pon<lb></lb>dus depreſſum maiorem <lb></lb>habere grauitatem eleuato <lb></lb>iiſdem mediis oſtendetur. </s> <s id="id.2.1.49.3.1.2.0"><lb></lb>ducantur à punctis EF ip<lb></lb>ſi AB perpendiculares EL <lb></lb>FM. </s> <s id="N11E30">ſimiliter demonſtra<lb></lb>bitur EL maiorem eſſe <lb></lb>FM; & ob id deſcenſus <lb></lb>ponderis in F minus de di <lb></lb>recto capiet, quàm aſcen<lb></lb><figure id="id.036.01.066.1.jpg" place="text" xlink:href="036/01/066/1.jpg"></figure><lb></lb>ſus ponderis in E: quocirca reſiſtentia violentiæ ponderis in E ſu<lb></lb>perabit naturalem propenſionem ponderis in F. </s> <s id="N11E44">ergo pondus in E <lb></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; ipſiq; à punctis <lb></lb>EF perpendiculares ducantur EQ FR. </s> <s id="N11E50">eodem prorſus modo <lb></lb>oſtendetur, lineam EQ maiorem eſſe FR. </s> <s id="N11E54">pondus ideò in E ma<lb></lb>gis à linea directionis OP diſtabit, quàm pondus in F. </s> <s id="N11E58">maio<lb></lb>rem igitur grauitatem habebit pondus in E, quàm pondus in F. <lb></lb></s> <s id="N11E5D">ex quibus ſequitur, libram EF ex parte E deorſum 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">Ariſtoteles itaq; has duas tantùm quæſtiones propoſuit, ter<lb></lb>tiamq; reliquit; ſcilicet cùm centrum libræ in ipſa eſt libra: hanc <lb></lb>autem ommiſsit, vt notam, quemadmodum res valde notas præ<lb></lb>termittere ſolet. </s> <s id="id.2.1.49.5.1.2.0">nam cui dubium, ſi pondus in eius centro gra<lb></lb>uitatis ſuſtineatur, quin maneat? </s> <s id="id.2.1.49.5.1.3.0">Ea verò, quæ ex ipſius ſenten<lb></lb>tia attulimus, aliquis reprehendere poſſet, nos integram eius ſenten<lb></lb>tiam minimè protuliſſe <expan abbr="affimans">affirmans</expan>. </s> <s id="id.2.1.49.5.1.4.0">nam cùm in ſecunda parte ſe<lb></lb>cundæ quæſtionis proponit, cur libra, trutina deorſum conſtituta, <lb></lb>quando deorſum lato pondere quiſpiam id amouet, non aſcen<lb></lb>dit, ſed manet? </s> <s id="id.2.1.49.5.1.5.0">non aſſerit adhuc libram deorſum moueri; ſed <lb></lb>manere. </s> <s id="id.2.1.49.5.1.6.0">quod in vltima quoq; concluſione colligiſſe videtur. </s> <s id="id.2.1.49.5.1.7.0">Ve <lb></lb>rùm hoc non ſolum nobis non repugnat, ſed ſi rectè intelligitur, <lb></lb>maximè ſuffragatur. </s> </p> <pb n="27" xlink:href="036/01/067.jpg"></pb> <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></lb>horizonti æquidiſtans, <lb></lb>cuius centrum E ſit <lb></lb>infra libram. </s> <s id="id.2.1.49.7.1.2.0">quia ve <lb></lb>rò Ariſtoteles libram, <lb></lb>ſicuti actu eſt, conſide<lb></lb>rat; ideò neceſſe eſt <lb></lb>trutinam, vel aliquid <lb></lb>aliud infra centrum E <lb></lb>collocare, vt EF <lb></lb>(quod quidem truti<lb></lb>na erit) ita vt centrum <lb></lb>E ſuſtineat. </s> <s id="id.2.1.49.7.1.3.0">ſitq; per<lb></lb><figure id="id.036.01.067.1.jpg" place="text" xlink:href="036/01/067/1.jpg"></figure><lb></lb>pendiculum ECD. </s> <s id="N11EC3">& vt libra AB ab hoc moueatur ſitu; dicit <lb></lb>Ariſtoteles, ponatur pondus in B, quod cùm ſit graue, libram ex <lb></lb>parte B deorſum mouebit; putá in G. </s> <s id="N11EC9">ita vt propter impedimen<lb></lb>tum deorſum amplius moueri non poterit. </s> <s id="id.2.1.49.7.1.4.0">non enim dicit Ari<lb></lb>ſtoteles, moueatur libra ex parte B deorſum, quouſq; libuerit; dein <lb></lb>de relinquatur, vt nos diximus: ſed præcipit, vt in ipſo B po<lb></lb>natur pondus, quod ex ipſius natura deorſum ſemper mouebi<lb></lb>tur; donec libra trutinæ, ſiue alicui alii adhæreat. </s> <s id="id.2.1.49.7.1.5.0">& quando B erit <lb></lb>in G, erit libra in GH; in quo ſitu, ablato pondere, manebit: <lb></lb>cùm maior pars libræ à perpendiculo ſit verſus G, quæ eſt DG, <lb></lb>quàm DH. </s> <s id="id.2.1.49.7.1.5.0.a">nec deorſum amplius mouebitur; nam libra, vel <lb></lb>trutinæ, vel alteri cuipiam, quod centrum libræ ſuſtineat, incum<lb></lb>bet. </s> <s id="id.2.1.49.7.1.6.0">ſi enim huic non adhæreret, libra ex parte G deorſum ex <lb></lb>ipſius ſententia moueretur; cùm id, quod plus eſt, ſcilicet DG, <lb></lb>deorſum ferri ſit neceſſe. </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, ſi paruum imponatur pon<lb></lb>dus in B, mouebitur quidem libra deorſum, non autem vſq; ad <lb></lb>G. </s> <s id="N11EF9">in quò ſitu ſecundùm Ariſtotelem, ablato pondere, mane<lb></lb>re deberet. </s> <s id="id.2.1.49.8.1.2.0">quod experimento patet; cùm in vna tantùm libræ <lb></lb>extremitate, impoſito onere, hocq; vel maiore, vel minore, libra <lb></lb>plus, minuſuè inclinetur. </s> <s id="id.2.1.49.8.1.3.0">Quod eſt quidem veriſſimum, centro ſupra <lb></lb>libram, non autem infra, neq; in ipſa libra collocato. </s> <s id="id.2.1.49.8.1.4.0">Vt exempli <lb></lb>gratia. </s> </p> <pb xlink:href="036/01/068.jpg"></pb> <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></lb>quidiſtans AB, cuius cen<lb></lb>trum C ſit ſupra libram, <lb></lb>perpendiculumq; CD ho<lb></lb>rizonti perpendiculare, <lb></lb>quod ex parte D produca<lb></lb>tur in H. </s> <s id="id.2.1.49.10.1.1.0.a">Quoniam enim <lb></lb>conſiderata libræ grauita<lb></lb>te, erit punctum D libræ <lb></lb>centrum grauitatis. </s> <s id="id.2.1.49.10.1.2.0">ſi ergo <lb></lb>in B paruum imponatur <lb></lb>pondus, cuius centrum <lb></lb><figure id="id.036.01.068.1.jpg" place="text" xlink:href="036/01/068/1.jpg"></figure><lb></lb>grauitatis ſit in puncto B; magnitudinis ex libra AB, & pondere <lb></lb>in B compoſitæ non erit amplius centrum grauitatis D; ſed erit in <lb></lb><arrow.to.target n="note76"></arrow.to.target>linea DB, vt in E: ita vt DE ad EB ſit, vt pondus in B ad gra<lb></lb>uitatem libræ AB. </s> <s id="N11F44">Connectatur CE. </s> <s id="id.2.1.49.10.1.2.0.a">Quoniam autem pun<lb></lb>ctum C eſt immobile, dum libra mouetur, punctum E circuli cir<lb></lb>cumferentiam EFG deſcribet, cuius ſemidiameter CE, & cen<lb></lb>trum C. </s> <s id="N11F4F">quia verò CD horizonti eſt perpendicularis, linea CE <lb></lb>horizonti perpendicularis nequaquam erit. </s> <s id="id.2.1.49.10.1.3.0">quare magnitudo ex <lb></lb>AB, & pondere in B compoſita minimè in hoc ſitu manebit; ſed <lb></lb><arrow.to.target n="note77"></arrow.to.target>deorſum ſecundùm eius grauitatis centrum E per circumferen<lb></lb>tiam EFG mouebitur; donec CE horizonti perpendicularis eua<lb></lb>dat; hoc eſt, donec CE in CDF perueniat. </s> <s id="id.2.1.49.10.1.4.0">atq; tunc libra AB <lb></lb>mota erit in kL, in quo ſitu libra vná cum pondere manebit. </s> <s id="id.2.1.49.10.1.5.0">nec <lb></lb>deorſum amplius mouebitur. </s> <s id="id.2.1.49.10.1.6.0">Si verò in B ponatur pondus graui<lb></lb>us; centrum grauitatis totius magnitudinis erit ipſi B propius, vt in <lb></lb>M. </s> <s id="N11F72">& tunc libra deorſum, donec iuncta CM in linea CDH per <lb></lb>ueniat, mouebitur. </s> <s id="id.2.1.49.10.1.7.0">Ex maiore igitur, & minore pondere in B po<lb></lb>ſito, libra plus, minuſuè inclinabitur. </s> <s id="id.2.1.49.10.1.8.0">ex quo ſequitur pondus B <lb></lb>quarta circuli parte minorem ſemper circumferentiam deſcribe<lb></lb>re, cùm angulus FCE ſit ſemper acutus. </s> <s id="id.2.1.49.10.1.9.0">nunquam enim punctum <lb></lb>B vſq; ad lineam CH perueniet, cùm centrum grauitatis ponde<lb></lb>ris, & libræ ſimul ſemper inter DB exiſtat. </s> <s id="id.2.1.49.10.1.10.0">quò tamen pondus <lb></lb>in B grauius fuerit, maiorem quoq; circumferentiam deſcribet. </s> <s id="id.2.1.49.10.1.11.0"><lb></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"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.50.1.1.3.0"><margin.target id="note77"></margin.target>1. <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="28" xlink:href="036/01/069.jpg"></pb> <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></lb>centrum C in ipſa libra, atq; <lb></lb>in eius medio: erit C libræ <lb></lb>centrum quoq; grauitatis; <lb></lb>à quo ipſi AB, horizontiq; <lb></lb>perpendicularis ducatur FC <lb></lb>G. </s> <s id="N11FBF">ponatur deinde in B <lb></lb>quoduis pondus; erit totius <lb></lb>magnitudinis centrum gra<lb></lb>uitatis putá in E; ita vt CE <lb></lb><figure id="id.036.01.069.1.jpg" place="text" xlink:href="036/01/069/1.jpg"></figure><lb></lb>ad EB ſit, vt pondus in B ad libræ grauitatem. </s> <s id="id.2.1.51.1.2.2.0">& quoniam CE <lb></lb>non eſt horizonti perpendicularis, libra AB, atq; pondus in B <lb></lb>in hoc ſitu nunquam manebunt; ſed deorſum ex parte B mouebun<lb></lb>tur, donec CE horizonti fiat perpendicularis. </s> <s id="id.2.1.51.1.2.3.0">hoc eſt donec li<lb></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></lb>circuli quartam ſemper deſcribere. </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></lb>fra libram AB. </s> <s id="N11FEA">ſitq; DCE <lb></lb>perpendiculum. </s> <s id="id.2.1.51.2.1.2.0">ſimiliter <lb></lb>poſito in B pondere, cen<lb></lb>trum grauitatis magnitudi<lb></lb>nis ex AB libra, & ponde<lb></lb>re in B compoſitæ in linea <lb></lb>DB erit; vt in F; ita vt DF <lb></lb>ad FB ſit, vt pondus in B <lb></lb><figure id="id.036.01.069.2.jpg" place="text" xlink:href="036/01/069/2.jpg"></figure><lb></lb>ad libræ pondus. </s> <s id="id.2.1.51.2.1.3.0">Iungatur CF. </s> <s id="N12008">& quoniam CD horizonti eſt <lb></lb>perpendicularis; linea CF horizonti nequaquam perpendicula<lb></lb>ris exiſtet. </s> <s id="id.2.1.51.2.1.4.0">quare magnitudo ex AB libra, ac pondere in B com<lb></lb>poſita in hoc ſitu nunquam perſiſtet; ſed deorſum, niſi aliquid <lb></lb>impediat, mouebitur; donec CF in DCE perueniat: in quo ſitu <lb></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></lb>punctum A in H, libraq; GH non amplius centrum infra, ſed ſu<lb></lb>pra ipſam habebit. </s> <s id="id.2.1.51.2.1.6.0">quod idem ſemper eueniet; quamuis mini<lb></lb>mum imponatur pondus in B. </s> <s id="N12023">ergo priuſquam B perueniat ad <lb></lb>G; neceſſe eſt libram, ſiue trutinæ deorſum poſitæ, vel alicui <pb xlink:href="036/01/070.jpg"></pb>alteri, quod centrum C ſu<lb></lb>ſtineat, occurrere; ibiq; ad<lb></lb>hærere. </s> <s id="id.2.1.51.2.1.7.0">ex hoc ſequitur, pon<lb></lb>dus in B vltra lineam Dk <lb></lb>ſemper moueri; ac circuli <lb></lb>quarta maiorem ſemper cir<lb></lb><expan abbr="cumferẽtiam">cumferentiam</expan> deſcribere: eſt <lb></lb>enim angulus FCE ſemper <lb></lb>obtuſus, cùm angulus DCF <lb></lb>ſemper ſit acutus. </s> <s id="id.2.1.51.2.1.8.0">quò au<lb></lb><figure id="id.036.01.070.1.jpg" place="text" xlink:href="036/01/070/1.jpg"></figure><lb></lb>tem pondus in B fuerit leuius, maiorem tamen adhuc circumfe<lb></lb>rentiam deſcribet. </s> <s id="id.2.1.51.2.1.9.0">nam quò pondus in G leuius fuerit, eò ma<lb></lb>gis pondus in G eleuabitur; libraq; GH ad ſitum horizonti æqui<lb></lb>diſtantem propius accedet. </s> <s id="id.2.1.51.2.1.10.0">quæ omnia ex iis, quæ ſupra dixi<lb></lb>mus, manifeſta ſunt. </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 demonſtratis. </s> <s id="id.2.1.51.3.1.2.0">Manifeſtum eſt, centrum libræ cauſam eſſe <lb></lb>diuerſitatis effectuum in libra. </s> <s id="id.2.1.51.3.1.3.0">atq; patet omnes Archimedis de <lb></lb>æqueponderantibus propoſitiones ad hoc pertinentes in omni ſitu <lb></lb>veras eſſe. </s> <s id="id.2.1.51.3.1.4.0">hoc eſt ſiue libra ſit horizonti æquidiſtans, ſiue non: <lb></lb>dummodo centrum libræ in ipſa ſit libra; quemadmodum ipſe <lb></lb>conſiderat. </s> <s id="id.2.1.51.3.1.5.0">& quamquam libra brachia habeat inæqualia, idem eue<lb></lb>niet; eodemq; proſus modo oſtendetur, centrum libræ diuerſimo<lb></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></lb>zonti æquidiſtans; & in AB <lb></lb>ſint pondera inæqualia, quo <lb></lb>rum grauitatis centrum ſit <lb></lb>C: ſuſpendaturq; libra in <lb></lb>eodem puncto C. </s> <s id="N1208C">& mo<lb></lb>ueatur libra in DE. </s> <s id="id.2.1.51.4.1.1.0.a">mani<lb></lb><arrow.to.target n="note78"></arrow.to.target>feſtum eſt libram non ſo<lb></lb>lum in DE, ſed in quouis <lb></lb>alio ſitu manere. <figure id="id.036.01.070.2.jpg" place="text" xlink:href="036/01/070/2.jpg"></figure></s> </p> <pb n="29" xlink:href="036/01/071.jpg"></pb> <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></lb>AB ſupra C in F; ſitq; <lb></lb>FC ipſi AB, & horizonti <lb></lb>perpendicularis: & ſi mo<lb></lb>ueatur libra in DE, linea <lb></lb>CF mota erit in FG; quæ <lb></lb>cùm non ſit horizonti per<lb></lb>pendicularis, libra DE <arrow.to.target n="note79"></arrow.to.target><lb></lb>deorſum ex parte D moue<lb></lb>bitur, donec FG in FC <lb></lb>redeat: atq; tunc libra DE <lb></lb>in AB erit, in quò ſitu <lb></lb>quoq; manebit. <figure id="id.036.01.071.1.jpg" place="text" 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 ſi centrum libræ F <lb></lb>ſit infra libram; ſitq; mota <lb></lb>libra in DE; primùm qui<lb></lb>dem manifeſtum eſt li<lb></lb>bram in AB manere; in <arrow.to.target n="note80"></arrow.to.target><lb></lb>DE verò deorſum ex par <lb></lb>te E moueri: cùm linea <lb></lb>FG non ſit horizonti per<lb></lb>pendicularis. <figure id="id.036.01.071.2.jpg" place="text" 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 ſi libra ſit <lb></lb>arcuata, vel libræ brachia angulum <lb></lb>conſtituant; centrumq; diuerſimo<lb></lb>dè collocetur (quamquam hæc pro<lb></lb>priè non ſit libra) varios tamen <lb></lb>huius quoq; effectus oſtendere pote<lb></lb>rimus. </s> <s id="id.2.1.51.8.1.2.0">Vt ſit libra ACB, cuius <lb></lb>centrum, circa quod vertitur, ſit C. <lb></lb></s> <s id="N12101">ductaq; AB, ſit arcus ſiue angulus <lb></lb><figure id="id.036.01.071.3.jpg" place="text" xlink:href="036/01/071/3.jpg"></figure><lb></lb>ACB ſupra lineam AB; & in AB grauitatis centra ponderum <lb></lb>ponantur, quæ in hoc ſitu maneant. </s> <s id="id.2.1.51.8.1.3.0">moueatur deinde libra ab <pb xlink:href="036/01/072.jpg"></pb>hoc ſitu, putá in ECF. </s> <s id="id.2.1.51.8.1.3.0.a">Dico li<lb></lb>bram ECF in ACB redire. </s> <s id="id.2.1.51.8.1.4.0">to<lb></lb>tius magnitudinis centrum grauita<lb></lb>tis inueniatur D. </s> <s id="N12120">& CD iunga<lb></lb>tur. </s> <s id="id.2.1.51.8.1.5.0">Quoniam enim pondera AB <lb></lb><arrow.to.target n="note81"></arrow.to.target>manent, linea CD horizonti per<lb></lb>pendicularis erit. </s> <s id="id.2.1.51.8.1.6.0">quando igitur <lb></lb>libra erit in ECF, linea CD erit <lb></lb>putá in CG; quæ cùm non ſit ho<lb></lb><figure id="id.036.01.072.1.jpg" place="text" xlink:href="036/01/072/1.jpg"></figure><lb></lb>rizonti perpendicularis; libra ECF in ACB redibit. </s> <s id="id.2.1.51.8.1.7.0">quod idem <lb></lb>eueniet, ſi centrum C ſupra libram conſtituatur, 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"></emph>Per def. <expan abbr="cẽtri">centri</expan> grauitatis. <emph.end type="italics"></emph.end></s> <s id="id.2.1.52.1.1.2.0"><margin.target id="note79"></margin.target>1 <emph type="italics"></emph>Huius. <emph.end type="italics"></emph.end></s> <s id="id.2.1.52.1.1.3.0"><margin.target id="note80"></margin.target>1. <emph type="italics"></emph>Huius. <emph.end type="italics"></emph.end></s> <s id="id.2.1.52.1.1.4.0"><margin.target id="note81"></margin.target>1 <emph type="italics"></emph>Huius. <emph.end type="italics"></emph.end></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, ſiue angulus <lb></lb>ACB, ſit infra lineam AB; eo <lb></lb>dem modo libram ECF, cuius <lb></lb>centrum, ſiue ſit in C, ſiue in H, <lb></lb>deorſum ex parte F moueri o<lb></lb>ſtendemus. <figure id="id.036.01.072.2.jpg" place="text" xlink:href="036/01/072/2.jpg"></figure> <figure id="id.036.01.072.3.jpg" place="text" 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 ſupra lineam AB; ac libræ centrum <lb></lb>ſit H; lineaq; CH libram ſuſtineat; & moueatur libra in EKF: <lb></lb>libra EkF in ACB redibit. </s> </p> <pb n="30" xlink:href="036/01/073.jpg"></pb> <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 ſit D, quocunq; modo moueatur libra; <lb></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 ſit infra lineam AB; tunc libra EkF <lb></lb>deorſum 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; prorſus ratione, ſi an<lb></lb>gulus ACB ſit infra lineam AB; <lb></lb>ſitq; libræ centrum H; ſuſtineaturq; <lb></lb>libra linea CH; ſi libra ab hoc mo<lb></lb>ueatur ſitu, deorſum ex parte pon<lb></lb>deris inferioris mouebitur. </s> <s id="id.2.1.53.6.1.2.0">& ſi cen<lb></lb>trum libræ ſit D; vbi relinquetur, <lb></lb>manebit. </s> <s id="id.2.1.53.6.1.3.0">ſi verò ſit in K; ſi ab eiuſ <lb></lb><figure id="id.036.01.073.1.jpg" place="text" xlink:href="036/01/073/1.jpg"></figure><lb></lb>modi moueatur ſitu, in eundem proſus redibit. </s> <s id="id.2.1.53.6.1.4.0">quæ omnia ex iis, <lb></lb>quæ in principio diximus, ſunt manifeſta. </s> <s id="id.2.1.53.6.1.5.0">ſimiliter ſi centrum li<lb></lb>bræ, vel in altero brachiorum, vel intra, vel extra vtcunq; po<lb></lb>natur; eadem inueniemus. </s> </p> <pb xlink:href="036/01/074.jpg"></pb> <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 appenſa, ſi libra inter <lb></lb>hæc ita diuidatur, vt partes ponderibus per<lb></lb>mutatim reſpondeant; tàm in punctis appenſis <lb></lb>ponderabunt, quàm ſi vtraq; ex diuiſionis pun<lb></lb>cto ſuſpendantur. <figure id="id.036.01.074.1.jpg" place="text" 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; ſintq; duo pondera EF ex pun<lb></lb>ctis BG ſuſpenſa: diuidaturq; BG in H, ita vt BH ad HG <lb></lb>eandem habeat proportionem, quam pondus E ad pondus F. </s> <s id="id.2.1.53.10.1.1.0.a"><lb></lb>Dico pondera EF tàm in BG ponderare, quàm ſi vtraq; ex pun<lb></lb>cto H ſuſpendantur. </s> <s id="id.2.1.53.10.1.2.0">fiat AC ipſi CH æqualis. </s> <s id="id.2.1.53.10.1.3.0">& vt AC ad <lb></lb>CG, ita fiat pondus E ad pondus L. </s> <s id="N1220A">ſimiliter vt AC ad CB, <lb></lb>ita fiat pondus F ad pondus M. </s> <s id="N1220E">ponderaq; LM ex puncto A ſu<lb></lb>ſpendantur. </s> <s id="id.2.1.53.10.1.4.0">Quoniam enim AC eſt æqualis CH, erit BC ad <lb></lb>CH vt pondus M ad pondus F. </s> <s id="id.2.1.53.10.1.4.0.a">& quoniam maior eſt BC, <lb></lb>quàm CH; erit & pondus M ipſo F maius. </s> <s id="id.2.1.53.10.1.5.0">diuidatur igitur pon<lb></lb>dus M in duas partes QR, ſitq; pars Q ipſi F æqualis; erit BC <lb></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></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></lb>niam CH eſt æqualis ipſi CA, erit HC ad CG, vt pondus <lb></lb>E ad pondus L: maior autem eſt HC, quàm CG; erit & pon<pb n="31" xlink:href="036/01/075.jpg"></pb>dus E pondere L maius. </s> <s id="id.2.1.53.10.1.6.0">diuidatur itaq; pondus E in duas partes <lb></lb>NO ita, vt pars O ſit ipſi L æqualis, erit HC ad CG, vt to<lb></lb>tum NO ad O; & diuidendo, vt HG ad GC, ita N ad O: <arrow.to.target n="note84"></arrow.to.target><lb></lb>conuertendoq; vt CG ad GH, ita O ad N. </s> <s id="N12243">& iterum com<lb></lb>ponendo, vt CH ad HG, ita ON ad N. </s> <s id="N12247">vt autem GH <arrow.to.target n="note85"></arrow.to.target><lb></lb>ad HB, ita eſt F ad ON. </s> <s id="N1224E">quare ex æquali, vt CH ad HB, ita F <arrow.to.target n="note86"></arrow.to.target><lb></lb>ad N. ſed vt CH ad HB ita eſt Q ad R: erit igitur Q ad R, vt <arrow.to.target n="note87"></arrow.to.target><lb></lb>F ad N; & permutando, vt Q ad F, ita R ad N. </s> <s id="N1225A">eſt autem pars <arrow.to.target n="note88"></arrow.to.target><lb></lb>Q ipſi F æqualis; quare & pars R ipſi N æqualis erit. </s> <s id="id.2.1.53.10.1.7.0">Itaq; cùm <lb></lb>pondus L ſit ipſi O æquale, & pondus F ipſi Q etiam æquale, atq; <lb></lb>pars R ipſi N æqualis; erunt pondera LM ipſis EF ponderibus <lb></lb>æqualia. </s> <s id="id.2.1.53.10.1.8.0">& quoniam eſt, vt AC ad CG, ita pondus E ad pon<lb></lb>dus L; pondera EL æqueponderabunt. </s> <s id="id.2.1.53.10.1.9.0">ſimiliter quoniam eſt, vt <arrow.to.target n="note89"></arrow.to.target><lb></lb>AC ad CB, ita <expan abbr="pundus">pondus</expan> F ad pondus M; pondera quoq; FM <lb></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></lb>appenſis æqueponderabunt. </s> <s id="id.2.1.53.10.1.11.0">cùm autem diſtantia CA æqualis ſit <lb></lb>diſtantiæ CH; ſi igitur vtraq; pondera EF in H appendantur, <lb></lb>pondera LM ipſis EF ponderibus in H appenſis æquepondera<lb></lb>bunt. </s> <s id="id.2.1.53.10.1.12.0">ſed LM ipſis EF in GB quoq; æqueponderant: æquè <arrow.to.target n="note91"></arrow.to.target><lb></lb>igitur grauia erunt pondera EF in GB, vt in H appenſa. </s> <s id="id.2.1.53.10.1.13.0">tàm igi<lb></lb>tur ponderabunt in BG, quàm in H appenſa. <figure id="id.036.01.075.1.jpg" place="text" 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 appenſa; ſitq; C libræ centrum; <lb></lb>& diuidatur CB in H, ita vt CH ad HB ſit, vt pondus in F ad <lb></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></lb>fiat CA ipſi CH æqualis, & vt CA ad CB, ita fiat pondus F ad <lb></lb>aliud D, quod appendatur in A. </s> <s id="id.2.1.53.11.1.1.0.c">Quoniam enim CH eſt æqua<pb xlink:href="036/01/076.jpg"></pb> <figure id="id.036.01.076.1.jpg" place="text" xlink:href="036/01/076/1.jpg"></figure><lb></lb>lis CA, erit CH ad CB, vt F ad D; & maior quidem eſt CB, <lb></lb>quàm CH; idcirco D pondere F maius erit. </s> <s id="id.2.1.53.11.1.2.0">Diuidatur ergo D <lb></lb>in duas partes Gk, ſitq; G ipſi F æqualis; erit vt BC ad CH, <lb></lb>vt Gk ad G; & diuidendo, vt BH ad HC, ita K ad G; & conuer<lb></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 eſt <lb></lb><arrow.to.target n="note93"></arrow.to.target>F ad E. </s> <s id="N122D6">vt igitur G ad k, ita eſt F ad E; & permutando vt G <lb></lb><arrow.to.target n="note94"></arrow.to.target>ad F, ita k ad E. </s> <s id="N122DD">ſunt autem GF æqualia; erunt & kE inter ſe <lb></lb>ſe æqualia. </s> <s id="id.2.1.53.11.1.4.0">cùm itaq; pars G ſit ipſi F æqualis, & K ipſi E; erit <lb></lb>totum C k ipſis EF ponderibus æquale. </s> <s id="id.2.1.53.11.1.5.0">& quoniam AC eſt ip<lb></lb>ſi CH æqualis; ſi igitur pondera EF ex puncto H ſuſpendantur, <lb></lb>pondus D ipſis EF in H appenſis æqueponderabit. </s> <s id="id.2.1.53.11.1.6.0">ſed & ipſis <lb></lb>æqueponderat in CB, hoc eſt F in B, & E in C; cùm ſit vt AC <lb></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 ſuſpen<lb></lb>ſum non efficit, vt libra in alterutram moueatur partem. </s> <s id="id.2.1.53.11.1.8.0">tàm igi<lb></lb>tur grauia erunt pondera EF in CB, quàm in H appenſa. <pb n="32" xlink:href="036/01/077.jpg"></pb> <figure id="id.036.01.077.1.jpg" place="text" 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 ſuſpenſa ſint pondera <lb></lb>EF; ſitq; centrum libræ C intra pondera; diuidaturq; AB in <lb></lb>D, ita vt AD ad DB ſit, vt pondus F ad pondus E. </s> <s id="id.2.1.53.12.1.1.0.a">Dico pon<lb></lb>dera EF tàm in AB ponderare, quám ſi vtraq; ex puncto D ſuſpen<lb></lb>dantur. </s> <s id="id.2.1.53.12.1.2.0">fiat CG æqualis ipſi CD; & vt DC ad CA, ita fiat <lb></lb>pondus E ad aliud H; quod appendatur in D. </s> <s id="id.2.1.53.12.1.2.0.a">vt autem GC ad <lb></lb>CB, ita fiat pondus F ad aliud K; appendaturq; k in G. </s> <s id="id.2.1.53.12.1.2.0.b"><expan abbr="Quoniã">Quoniam</expan> enim <lb></lb>eſt, vt BC ad CG, hoc eſt ad CD, ita pondus k ad F; erit K ma <lb></lb>ius pondere F. </s> <s id="N12329">quare diuidatur pondus k in L, & MN; fiatq; <lb></lb>pars L ipſi F æqualis; erit vt BC ad CD, vt totum LMN ad <lb></lb>L; & diuidendo, vt BD ad DC, ita pars MN ad partem L. </s> <s id="N1232F">vt <arrow.to.target n="note95"></arrow.to.target><lb></lb>igitur BD ad DC, ita pars MN ad F. </s> <s id="N12336">vt autem AD ad DB, <lb></lb>ita F ad E: quare ex æquali, vt AD ad DC, ita MN ad E. </s> <s id="N1233A">cùm <arrow.to.target n="note96"></arrow.to.target><lb></lb>verò AD ſit ipſa CD maior; erit & pars MN pondere E <lb></lb>maior: diuidatur ergo MN in duas partes MN, ſitq; M æqua <lb></lb>lis ipſi E. </s> <s id="N12345">erit vt AD ad DC, vt NM ad M; & diuidendo, vt <arrow.to.target n="note97"></arrow.to.target><lb></lb>AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M <lb></lb>ad N. </s> <s id="N1234E">vt autem DC ad CA, ita eſt E ad H; erit igitur M ad N <arrow.to.target n="note98"></arrow.to.target><lb></lb>vt E ad H; & permutando, vt M ad E, ita N ad H. </s> <s id="N12355">ſed ME <arrow.to.target n="note99"></arrow.to.target><lb></lb>ſunt inter ſe æqualia, erunt NH inter ſeſe quoq; æqualia. </s> <s id="id.2.1.53.12.1.3.0">& quo<lb></lb>niam ita eſt AC ad CD, vt H ad E: pondera HE æqueponde<lb></lb>rabunt. <arrow.to.target n="note100"></arrow.to.target></s> <s id="id.2.1.53.12.1.4.0">ſimiliter quoniam eſt vt GC ad CB, ita F ad k, ponde<pb xlink:href="036/01/078.jpg"></pb> <figure id="id.036.01.078.1.jpg" place="text" xlink:href="036/01/078/1.jpg"></figure><lb></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></lb>bra AB, cuius centrum C, æqueponderabunt. </s> <s id="id.2.1.53.12.1.6.0">cùm autem GC <lb></lb>ipſi CD ſit æqualis, & pondus H ſit ipſi N æquale; pondera NH <lb></lb>æqueponderabunt. </s> <s id="id.2.1.53.12.1.7.0">& quoniam omnia æqueponderant, demptis <lb></lb><arrow.to.target n="note102"></arrow.to.target>HN ponderibus, quæ æqueponderant, reliqua æqueponderabunt; <lb></lb>hoc eſt pondera EF & pondus LM ex centro libræ C ſuſpenſa. </s> <s id="id.2.1.53.12.1.8.0"><lb></lb>quia verò pars L ipſi F eſt æqualis, & pars M ipſi E æqualis; erit <lb></lb>totum LM ipſis FE ponderibus ſimul ſumptis æquale. </s> <s id="id.2.1.53.12.1.9.0">& cùm <lb></lb>ſit CG ipſi CD æqualis, ſi igitur pondera EF ex puncto D ſuſpen<lb></lb>dantur, pondera EF in D appenſa ipſi LM æqueponderabunt. </s> <s id="id.2.1.53.12.1.10.0">quare <lb></lb>LM tàm ipſis EF in AB appenſis æqueponderat, quàm in pun<lb></lb>cto D appenſis. </s> <s id="id.2.1.53.12.1.11.0">libra enim ſemper eodem modo manet. </s> <s id="id.2.1.53.12.1.12.0">Ponde<lb></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></lb><expan abbr="demonſtre">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"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.2.0"><margin.target id="note83"></margin.target><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.3.0"><margin.target id="note84"></margin.target>17 <emph type="italics"></emph>Quinti. </s> <s id="id.2.1.54.1.1.4.0">Cor.<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.5.0"><margin.target id="note85"></margin.target>18 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.6.0"><margin.target id="note86"></margin.target>23 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.7.0"><margin.target id="note87"></margin.target>11 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.8.0"><margin.target id="note88"></margin.target>16 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.9.0"><margin.target id="note89"></margin.target>6 <emph type="italics"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.11.0"><margin.target id="note90"></margin.target>2 <emph type="italics"></emph>Com. not. huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.14.0"><margin.target id="note91"></margin.target>3 <emph type="italics"></emph>Com. not. huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.16.0"><margin.target id="note92"></margin.target>17 <emph type="italics"></emph>Quinti. </s> <s id="id.2.1.54.1.1.17.0">Cor.<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.18.0"><margin.target id="note93"></margin.target>11 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.19.0"><margin.target id="note94"></margin.target>16 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.20.0"><margin.target id="note95"></margin.target>17 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.21.0"><margin.target id="note96"></margin.target>23 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.22.0"><margin.target id="note97"></margin.target>17 <emph type="italics"></emph>Quinti. </s> <s id="id.2.1.54.1.1.23.0">Cor.<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>quinti<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.24.0"><margin.target id="note98"></margin.target>11 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.25.0"><margin.target id="note99"></margin.target>16 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.26.0"><margin.target id="note100"></margin.target>6 <emph type="italics"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.28.0"><margin.target id="note101"></margin.target>2 <emph type="italics"></emph>Com.not. huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.30.0"><margin.target id="note102"></margin.target>1 <emph type="italics"></emph>Com.not. huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.54.1.1.32.0"><margin.target id="note103"></margin.target>3 <emph type="italics"></emph>Com.not. huius.<emph.end type="italics"></emph.end></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></lb>gis) aliter oſtendemus. <pb n="33" xlink:href="036/01/079.jpg"></pb> <figure id="id.036.01.079.1.jpg" place="text" 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; ſintq; vt in primo caſu duo pon<lb></lb>dera EF ex punctis BG ſuſpenſa: ſitq; GH ad HB, vt pondus <lb></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></lb>ſi vtraq; ex diuiſionis puncto H ſuſpendantur. </s> <s id="id.2.1.55.2.1.2.0">Conſtruantur ea <lb></lb>dem, hoc eſt fiat AC ipſi CH æqualis, & ex puncto A duo ap<lb></lb>pendantur pondera LM, ita vt pondus E ad pondus L, ſit vt <lb></lb>CA ad CG; vt autem CB ad CA, ita ſit pondus M ad pondus <lb></lb>F. </s> <s id="id.2.1.55.2.1.2.0.a">pondera LM ipſis EF in GB appenſis (vt ſupra dictum eſt) <lb></lb>æqueponderabunt. </s> <s id="id.2.1.55.2.1.3.0">Sint deinde puncta NO centra grauitatis pon<lb></lb>derum EF; connectanturq; GN BO; iungaturq; NO, quæ tan<lb></lb>quam libra erit; quæ etiam efficiat lineas GN BO inter ſe ſe æqui<lb></lb>diſtantes eſſe; à punctoq; H horizonti perpendicularis ducatur <lb></lb>HP, quæ NO ſecet in P, atq; ipſis GN BO ſit æquidiſtans. <lb></lb></s> <s id="id.2.1.55.2.1.3.0.a">deniq; connectatur GO, quæ HP ſecet in R. </s> <s id="id.2.1.55.2.1.4.0">Quoniam igitur <lb></lb>HR eſt lateri BO trianguli GBO æquidiſtans; erit GH ad HB, <lb></lb>vt GR ad RO. </s> <s id="N124F8">ſimiliter quoniam RP eſt lateri GN trianguli <arrow.to.target n="note104"></arrow.to.target><lb></lb>OGN æquidiſtans; erit GR ad RO, vt NP ad PO. </s> <s id="N124FF">quare <lb></lb>vt GH ad HB, ita eſt NP ad PO. </s> <s id="N12503">vt autem GH ad HB, ita <arrow.to.target n="note105"></arrow.to.target><lb></lb>eſt pondus F ad pondus E; vt igitur NP ad PO, ita eſt pondus <lb></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></lb>tudinis ex vtriſq; EF ponderibus compoſitæ. </s> <s id="id.2.1.55.2.1.5.0">Intelligantur itaq; <arrow.to.target n="note106"></arrow.to.target><lb></lb>pondera EF ita eſſe à libra NO connexa, ac ſi vna tantùm eſſet <lb></lb>magnitudo ex vtriſq; EF compoſita, in punctiſq; BG appenſa. </s> <s id="id.2.1.55.2.1.6.0">ſi <lb></lb>igitur ponderum ſuſpenſiones BG ſoluantur, manebunt pondera <arrow.to.target n="note107"></arrow.to.target><lb></lb>EF ex HP ſuſpenſa; ſicuti in GB prius manebant. </s> <s id="id.2.1.55.2.1.7.0">pondera verò EF <lb></lb>in GB appenſa ipſis LM ponderibus æqueponderant, & pondera <pb xlink:href="036/01/080.jpg"></pb> <figure id="id.036.01.080.1.jpg" place="text" xlink:href="036/01/080/1.jpg"></figure><lb></lb>EF ex puncto H ſuſpenſa, eandem habent conſtitutionem ad li<lb></lb>bram AB, quam in BG appenſa: eadem ergo pondera EF ex <lb></lb>H ſuſpenſa eiſdem ponderibus LM æqueponderabunt. </s> <s id="id.2.1.55.2.1.8.0">æquè igi<lb></lb>tur ſunt grauia pondera EF in GB, vt in H appenſa. <figure id="id.036.01.080.2.jpg" place="text" 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 demonſtrabitur, pondera EF in quibuſcunq; aliis pun<lb></lb>ctis appenſa tàm <expan abbr="põderare">ponderare</expan>, quàm ſi vtraq; ex diuiſionis puncto H ſu<lb></lb>ſpendantur. </s> <s id="id.2.1.55.3.1.2.0">ſi enim (vt ſupra docuimus) in libra pondera inue<lb></lb>niantur, quibus pondera EF æqueponderent; eadem pondera EF <lb></lb>ex H ſuſpenſa eiſdem inuentis ponderibus æqueponderabunt; cùm <lb></lb>punctum P ſit ſemper eorum centrum grauitatis; & HP horizon <lb></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"></emph>Sexti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.56.1.1.2.0"><margin.target id="note105"></margin.target>11 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.56.1.1.3.0"><margin.target id="note106"></margin.target>6 <emph type="italics"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.56.1.1.5.0"><margin.target id="note107"></margin.target>1 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="34" xlink:href="036/01/081.jpg"></pb> <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 appenſa eam in gra<lb></lb>uitate proportionem habent; quam diſtantiæ, ex <lb></lb>quibus appenduntur. <figure id="id.036.01.081.1.jpg" place="text" 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 ſuſpenſa ex puncto A; & ſecetur AC vtcunq; <lb></lb>in D: ex punctis autem DC appendantur æqualia pondera EF. <lb></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></lb>bere, quam habet diſtantia CA ad diſtantiam AD. </s> <s id="id.2.1.57.3.1.1.0.b">fiat enim vt <lb></lb>CA ad AD, ita pondus F ad aliud pondus, quod ſit G. </s> <s id="id.2.1.57.3.1.1.0.c">Dico pri<lb></lb>múm pondera GF ex puncto C ſuſpenſa tantùm ponderare, quan<lb></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></lb>ex H appendantur vtraq; pondera EF. </s> <s id="N125BE">ponderabunt EF ſimul <lb></lb>ſumpta in eo ſitu, quantùm ponderant in DC. ponatur BA <arrow.to.target n="note108"></arrow.to.target><lb></lb>æqualis AH, ſeceturq; BA in K, ita vt ſit KA æqualis AD: <lb></lb>deinde ex puncto B appendatur pondus L duplum ponderis F, <lb></lb>hoc eſt æquale duobus ponderibus EF, quod quidem æqueponde<lb></lb>rabit ponderibus EF in H appenſis, hoc eſt appenſis in DC. </s> <s id="id.2.1.57.3.1.1.0.e"><expan abbr="Quoniã">Quoniam</expan><lb></lb>igitur, vt CA ad AD, ita eſt pondus F ad pondus G; erit compo<lb></lb>nendo vt CA AD ad AD, hoc eſt vt Ck ad AD, ita ponde<lb></lb>ra <arrow.to.target n="note109"></arrow.to.target>FG ad pondus G. </s> <s id="N125DC">ſed cùm ſit, vt CA ad AD, ita F pon<lb></lb>dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <arrow.to.target n="note110"></arrow.to.target><lb></lb>G ad pondus F; & conſequentium dupla, vt DA ad duplam ipſius <lb></lb>AC, ita pondus G ad duplum ponderis F, hoc eſt ad pondus <lb></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"></pb> <figure id="id.036.01.082.1.jpg" place="text" xlink:href="036/01/082/1.jpg"></figure><lb></lb><arrow.to.target n="note111"></arrow.to.target>AD ad <expan abbr="duplã">duplam</expan> ipſius AC, ita pondus G ad pondus L; ergo ex æquali, <lb></lb>vt Ck ad <expan abbr="duplã">duplam</expan> ipſius AC, ita pondera FG ad pondus L. </s> <s id="N12603">ſed vt Ck <lb></lb>ad duplam AC, ita dimidia CK, videlicet AH, hoc eſt BA, ad <lb></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></lb>re ex ſexta eiuſdem primi Archimedis, duo pondera FG ex pun<lb></lb>cto C ſuſpenſa tantùm ponderabunt, quantùm pondus L ex B; <lb></lb>hoc eſt quantùm pondera EF ex punctis DC ſuſpenſa. </s> <s id="id.2.1.57.3.1.2.0">Itaq; quo<lb></lb>niam pondera FG tantùm ponderant, quantum pondera EF; ſu<lb></lb>blato communi pondere F, tàm ponderabit pondus G in C ap<lb></lb>penſum, quàm pondus E in D. </s> <s id="id.2.1.57.3.1.2.0.a">ac propterea pondus F ad pon<lb></lb><arrow.to.target n="note112"></arrow.to.target>dus E eam in grauitate proportionem habet, quam habet ad pon<lb></lb>dus G. </s> <s id="N12628">ſed pondus F ad G erat, vt CA ad AD: ergo & F pon<lb></lb>dus ad pondus E eam in grauitate proportionem habebit, quam ha<lb></lb>bet CA ad AD. </s> <s id="N1262E">quod demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.58.1.1.2.0"><margin.target id="note109"></margin.target>18 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.58.1.1.3.0"><margin.target id="note110"></margin.target><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.58.1.1.4.0"><margin.target id="note111"></margin.target>22 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.58.1.1.5.0"><margin.target id="note112"></margin.target>7 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></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></lb>BAC pondera EF <lb></lb>æqualia ex punctis <lb></lb>BC ſuſpendantur; ſi<lb></lb>militer dico pondus <lb></lb>E ad pondus F eam <lb></lb><figure id="id.036.01.082.2.jpg" place="text" xlink:href="036/01/082/2.jpg"></figure><lb></lb>in grauitate proportionem habere, quàm habet diſtantia CA ad di<lb></lb>ſtantiam AB. </s> <s id="id.2.1.59.1.1.1.0.a">fiat AD ipſi AB æqualis, & ex puncto D ſuſpen<lb></lb>datur pondus G æquale ponderi F; quod etiam ipſi E erit æquale. </s> <s id="id.2.1.59.1.1.2.0"><lb></lb>& quoniam AD eſt æqualis ipſi AB; pondera FG æqueponde<lb></lb>rabunt, eandemq; habebunt grauitatem. </s> <s id="id.2.1.59.1.1.3.0">cùm autem grauitas pon<lb></lb>deris E ad grauitatem ponderis G ſit, vt CA ad AD; erit graui<lb></lb>tas ponderis E ad grauitatem ponderis F, vt CA ad AD, hoc eſt <lb></lb>CA ad AB. quod erat quoq; oſtendendum. </s> </p> <pb n="35" xlink:href="036/01/083.jpg"></pb> <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></lb>ius centrum A; in pun<lb></lb>ctis verò BC pondera <lb></lb>appendantur æqualia G <lb></lb>F: ſitq; primùm cen<lb></lb>trum A vtcunque inter <lb></lb>BC. </s> <s id="id.2.1.59.4.1.1.0.a">Dico pondus F ad <lb></lb>pondus G eam in graui<lb></lb><figure id="id.036.01.083.1.jpg" place="text" xlink:href="036/01/083/1.jpg"></figure><lb></lb>tate proportionem habere, quam habet diſtantia CA ad diſtan<lb></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></lb>pendatur in B: pondera HF ex A æqueponderabunt. </s> <s id="id.2.1.59.4.1.2.0">ſed cùm <arrow.to.target n="note113"></arrow.to.target><lb></lb>pondera FG ſint æqualia, habebit pondus H ad pondus G ean<lb></lb>dem proportionem, quam habet ad F. </s> <s id="N126D2">vt igitur CA ad AB, ita <arrow.to.target n="note114"></arrow.to.target><lb></lb>eſt H ad G. </s> <s id="N126D9">vt autem H ad G, ita eſt grauitas ipſius H ad graui<lb></lb>tatem ipſius G; cùm in eodem puncto B ſint appenſa. </s> <s id="id.2.1.59.4.1.3.0">quare vt CA <lb></lb>ad AB, ita grauitas ponderis H ad grauitatem ponderis G. </s> <s id="N126E2">cùm au<lb></lb>tem grauitas ponderis F in C appenſi ſit æqualis grauitati ponderis <lb></lb>H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA <lb></lb>ad AB, videlicet vt diſtantia ad diſtantiam. </s> <s id="id.2.1.59.4.1.4.0">quod demonſtrare <lb></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"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.60.1.1.3.0"><margin.target id="note114"></margin.target>7 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></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></lb>AC ſecetur vtcunq; <lb></lb>in D, & in DC ap<lb></lb>pendantur pondera <lb></lb>æqualia EF. </s> <s id="id.2.1.61.1.1.1.0.a">Dico <lb></lb>ſimiliter ita eſſe gra<lb></lb><figure id="id.036.01.083.2.jpg" place="text" xlink:href="036/01/083/2.jpg"></figure><lb></lb>uitatem ponderis F ad grauitatem ponderis E, vt diſtantia CA ad <lb></lb>diſtantiam AD. </s> <s id="id.2.1.61.1.1.1.0.b">fiat AB æqualis ipſi AD, & in B appendatur <lb></lb>pondus G æquale ponderi E, & ponderi F. </s> <s id="id.2.1.61.1.1.1.0.c">Quoniam enim AB eſt <lb></lb>æqualis AD; pondera GE æqueponderabunt. </s> <s id="id.2.1.61.1.1.2.0">ſed cùm grauitas <lb></lb>ponderis F ad grauitatem ponderis G ſit, vt CA ad AB, & graui<lb></lb>tas ponderis E ſit æqualis grauitati ponderis G; erit grauitas pon<lb></lb>deris F ad grauitatem ponderis E, vt CA ad AB, hoc eſt vt CA <lb></lb>ad AD. </s> <s id="N12738">quod demonſtrare oportebat. </s> </p> <pb xlink:href="036/01/084.jpg"></pb> <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 manifeſtum eſt, quò pondus à centro <lb></lb>libræ magis diſtat, eò grauius eſſe; & per conſe<lb></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 ſtateræ quoq; ratio facilè oſten<lb></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"></emph>Stateræ ratio.<emph.end type="italics"></emph.end></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 ſtate<lb></lb>ræ ſcapus AB, cu<lb></lb>ius trutina ſit in <lb></lb>C; ſitq; ſtateræ <lb></lb>appendiculum E. <lb></lb></s> <s id="N12773">appendatur in A <lb></lb>pondus D, quod <lb></lb>æqueponderet ap<lb></lb>pendiculo E in F <lb></lb><figure id="id.036.01.084.1.jpg" place="text" xlink:href="036/01/084/1.jpg"></figure><lb></lb>appenſo. </s> <s id="id.2.1.63.1.1.2.0">aliud quoq; appendatur pondus G in A, quod etiam <lb></lb>appendiculo E in B appenſo æqueponderet. </s> <s id="id.2.1.63.1.1.3.0">Dico grauitatem <lb></lb>ponderis D ad grauitatem ponderis G ita eſſe, vt CF ad CB. </s> <s id="id.2.1.63.1.1.3.0.a"><lb></lb>Quoniam enim grauitas ponderis D eſt æqualis grauitati ponde<lb></lb>ris E in F appenſi, & grauitas ponderis G eſt æqualis grauitati pon<lb></lb>deris E in B; erit grauitas ponderis D ad grauitatem ponderis E in <lb></lb>F, vt grauitas ponderis G ad grauitatem ponderis E in B: & permu<lb></lb><arrow.to.target n="note116"></arrow.to.target>tando, vt grauitas ponderis D ad grauitatem ponderis G, ita graui<lb></lb>tas ipſius E in F, ad grauitatem ipſius E in B; grauitas autem pon<lb></lb><arrow.to.target n="note117"></arrow.to.target>deris E in F ad grauitatem ponderis E in B eſt, vt CF ad CB; vt <lb></lb>igitur grauitas ponderis D ad grauitatem ponderis G, ita eſt CF <lb></lb>ad CB. </s> <s id="id.2.1.63.1.1.3.0.b">ſi ergo pars ſcapi CB in partes diuidatur æquales, ſolo <lb></lb>pondere E, & propius, & longius à puncto C poſito; ponderum <lb></lb>grauitates, quæ ex puncto A ſuſpenduntur inter ſe ſe notæ erunt. </s> <s id="id.2.1.63.1.1.4.0"><pb n="36" xlink:href="036/01/085.jpg"></pb>Vt ſi diſtantia CB tripla ſit diſtantiæ CF, erit quoq; grauitas ip<lb></lb>ſius G grauitatis ipſius D tripla, quod demonſtrare 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"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.64.1.1.2.0"><margin.target id="note117"></margin.target>6 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 ſtatera vti poſſumus, vt <lb></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 ſcapus AB, cuius tru<lb></lb>tina ſit in C; ſitq; ſtateræ ap<lb></lb>pendiculum E, quod appen<lb></lb>datur in A; ſint〈qué〉 pon<lb></lb>dera DG inæqualia, quorum <lb></lb>inter ſe ſe grauitatum propor<lb></lb>tiones quærimus: appenda<lb></lb>tur pondus D in B, ita vt ipſi <lb></lb><figure id="id.036.01.085.1.jpg" place="text" xlink:href="036/01/085/1.jpg"></figure><lb></lb>E æqueponderet. </s> <s id="id.2.1.65.2.1.2.0">ſimiliter pondus G appendatur in F, quod ei<lb></lb>dem ponderi E æqueponderet. </s> <s id="id.2.1.65.2.1.3.0">dico D ad G ita eſſe, vt CF ad <lb></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></lb>vt CA ad CB. </s> <s id="N12801">cùm autem pondera quoque GE æquepon<lb></lb>derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua <lb></lb>li pondus D ad pondus G ita erit, vt CF ad CB. </s> <s id="N12807">quod oſtende<arrow.to.target n="note119"></arrow.to.target><lb></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"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.66.1.1.3.0"><margin.target id="note119"></margin.target>23 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/086.jpg"></pb> <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 id="N1282E" 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></lb>vbicunque appenſis, centrum libræ inuenire, <lb></lb>ex quo ſi ſuſpendatur libra, data pondera ma<lb></lb>neant. <figure id="id.036.01.086.1.jpg" place="text" 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, ſintq; data quotcunque pondera CDEFG. <lb></lb></s> <s id="id.2.1.67.3.1.1.0.a">accipiantur in libra vtcunque puncta AHkLB, ex quibus <lb></lb>data pondera <expan abbr="ſpuſpendantur">suspendantur</expan>. </s> <s id="id.2.1.67.3.1.2.0">Centrum libræ inuenire oportet, <lb></lb>ex quo ſi fiat ſuſpenſio, data pondera maneant. </s> <s id="id.2.1.67.3.1.3.0">Diuidatur <pb n="37" xlink:href="036/01/087.jpg"></pb> <figure id="id.036.01.087.1.jpg" place="text" xlink:href="036/01/087/1.jpg"></figure><lb></lb>AH in M, ita vt HM ad MA, ſit vt grauitas ponderis <lb></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></lb>vt LN ad NB, ſit vt grauitas ponderis G ad grauitatem pon<lb></lb>deris F. </s> <s id="N12870">diuidaturq; MN in O, ita vt MO ad ON ſit, vt <lb></lb>grauitas ponderum FG ad grauitatem ponderum CD. </s> <s id="id.2.1.67.3.1.3.0.b">tandem<lb></lb>què diuidatur kO in P, ita vt kP ad PO, ſit vt grauitas pon<lb></lb>derum CDFG ad grauitatem ponderis E. </s> <s id="id.2.1.67.3.1.3.0.c">Quoniam igitur pon<lb></lb>dera CDFG tàm ponderant in O, quàm CD in M, & FG in N; <arrow.to.target n="note120"></arrow.to.target><lb></lb>æqueponderabunt pondera CD in M, & FG in N, & pondus E <lb></lb>in K, ſi ex puncto P ſuſpendantur. </s> <s id="id.2.1.67.3.1.4.0">cùm verò pondera CD tan<lb></lb>tùm ponderent in M, quantùm in AH, & FG in N, quantùm <lb></lb>in LB; pondera CDFG ex AHLB punctis ſuſpenſa, & pon<lb></lb>dus E ex k, ſi ex P ſuſpendantur, æqueponderabunt, atq; mane<lb></lb>bunt. </s> <s id="id.2.1.67.3.1.5.0">Inuentum eſt ergo centrum libræ P, ex quo data pondera <lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/088.jpg"></pb> <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 manifeſtum eſt, ſi ponderum CDEFG <lb></lb>centra grauitatis eſſent in AHKLB punctis; eſ<lb></lb>ſet punctum P magnitudinis ex omnibus CD <lb></lb>EFG ponderibus compoſitæ centrum graui<lb></lb>tatis. <figure id="id.036.01.088.1.jpg" place="text" 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></lb>ra, ſi ex puncto P ſuſpendantur, maneant. </s> </p> </chap> <pb n="38" xlink:href="036/01/089.jpg"></pb> <chap id="N128CF"> <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 id="N128D6" 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></lb>BCD; ſitq; A maior B, & C ma<lb></lb>ior D. </s> <s id="id.2.1.69.6.1.1.0.a">Dico A ad D maiorem <lb></lb>habere proportionem; quàm <lb></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></lb>portionem, quàm B ad C; & A ad D maio<lb></lb>rem <arrow.to.target n="note121"></arrow.to.target>quoq; habet proportionem, quam habet <lb></lb>ad C: A igitur ad D maiorem habebit, quam B <lb></lb>ad C. quod demonſtrare 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"></emph>Quinti.<emph.end type="italics"></emph.end></s> </p> <figure id="id.036.01.089.1.jpg" place="text" 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 ſuſtinens pondus vecti appenſum; <lb></lb>eandem ad ipſum pondus proportionem habe<lb></lb>bit, quam vectis diſtantia inter fulcimentum, ac <lb></lb>ponderis ſuſpenſionem ad diſtantiam à fulcimen<lb></lb>to ad potentiam interiectam. <pb xlink:href="036/01/090.jpg"></pb> <figure id="id.036.01.090.1.jpg" place="text" 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; ſitq; pondus D ex A ſu<lb></lb>ſpenſum AH, ita vt AH ſit ſemper horizonti perpendicularis: <lb></lb>ſitq; potentia ſuſtinens pondus in B. </s> <s id="id.2.1.71.3.1.1.0.a">Dico potentiam in B ad pon<lb></lb>dus D ita eſſe, 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></lb><arrow.to.target n="note122"></arrow.to.target>ad aliud pondus E, quippè quod ſi in B appendatur; ipſi D æque <lb></lb>ponderabit, exiſtente C amborum grauitatis centro. </s> <s id="id.2.1.71.3.1.2.0">quare poten<lb></lb>tia æqualis ipſi E ibidem conſtituta ipſi D æqueponderabit, vecte <lb></lb>AB, eius fulcimento in C collocato, hoc eſt prohibebit, ne pon<lb></lb>dus D deorſum vergat, quemadmodum prohibet pondus E. </s> <s id="id.2.1.71.3.1.2.0.a">Po<lb></lb><arrow.to.target n="note123"></arrow.to.target>tentia verò in B ad pondus D eandem habet proportionem, quam <lb></lb>pondus E ad idem pondus D: ergo potentia in B ad pondus D <lb></lb>erit, vt CA ad CB; hoc eſt vectis diſtantia à fulcimento ad pon<lb></lb>deris ſuſpendium ad diſtantiam à fulcimento ad potentiam. </s> <s id="id.2.1.71.3.1.3.0">quod <lb></lb>demonſtrare 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"></emph>Primi Archim. de æquep.<emph.end type="italics"></emph.end></s> <s id="id.2.1.72.1.1.3.0"><margin.target id="note123"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 7 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></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 oſtendi poteſt, fulcimentum quò <lb></lb>ponderi fuerit propius, minorem ad idem pon<lb></lb>dus ſuſtinendum 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">Iiſdem poſi<lb></lb>tis, ſit fulcimen <lb></lb>tum in F ipſi A <lb></lb>propius, quàm <lb></lb>C; fiatq; vt BF <lb></lb>ad FA, ita pon<lb></lb>dus D ad aliud <lb></lb><figure id="id.036.01.090.2.jpg" place="text" xlink:href="036/01/090/2.jpg"></figure><lb></lb>G, quod ſi appendatur in B, pondera DG ex fulcimento E <lb></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 eſt BC, & CA <lb></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"></pb>& ideo maior quoq; erit proportio ponderis D ad pondus G, <lb></lb>quàm idem D ad E: pondus igitur G minus erit pondere E. cùm <arrow.to.target n="note126"></arrow.to.target><lb></lb>autem potentia in B ipſi G æqualis ponderi D æqueponderet, mi<lb></lb>nor potentia, quàm ea, quæ ponderi E eſt æqualis, pondus D ſu<lb></lb>ſtinebit; exiſtente vecte AB, eius verò fulcimento vbi F, quàm ſi <lb></lb>fuerit vbi C. ſimiliter quoq; oſtendetur, quò propius erit fulci<lb></lb>mentum ponderi D, adhuc ſemper minorem requiri potentiam <lb></lb>ad ſuſtinendum 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"></emph>Ex eadem Sexta.<emph.end type="italics"></emph.end></s> <s id="id.2.1.74.1.1.2.0"><margin.target id="note125"></margin.target><emph type="italics"></emph>Lemma.<emph.end type="italics"></emph.end></s> <s id="id.2.1.74.1.1.3.0"><margin.target id="note126"></margin.target>10 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></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, exiſtente AF ipſa <lb></lb>FB minore, minorem quoq; requiri potentiam <lb></lb>in ipſo B pondere D ſuſtinendo. </s> <s id="id.2.1.75.2.1.2.0">æquali verò <lb></lb>æqualem. </s> <s id="N129F8">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 poſsumus. </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></lb>fulcimentum ſit B, & <lb></lb>pondus C vtcunq; in <lb></lb>D inter AB appen<lb></lb>ſum; ſitq; potentia in <lb></lb>A ſuſtinens pondus C. </s> <s id="id.2.1.75.5.1.1.0.a"><lb></lb>Dico vt BD ad BA, <lb></lb><figure id="id.036.01.091.1.jpg" place="text" xlink:href="036/01/091/1.jpg"></figure><lb></lb>ita eſſe potentiam in A ad pondus C. </s> <s id="N12A22">appendatur in A pondus <lb></lb>E æquale ipſi C; & vt AB ad BD, ita fiat pondus E ad aliud F. <lb></lb></s> <s id="N12A23">& quoniam pondera CE ſunt inter ſe ſe æqualia, erit pondus C <lb></lb>ad pondus F, vt AB ad BD. </s> <s id="N12A2A">appendatur quoq; pondus F in A. <lb></lb></s> <s id="N12A2D">& quoniam pondus E ad pondus F eſt, vt grauitas ipſius E ad gra<lb></lb>uitatem <arrow.to.target n="note127"></arrow.to.target>ipſius F; & pondus E ad F eſt, vt AB ad BD; vt igitur <lb></lb>grauitas ponderis E ad grauitatem ponderis F, ita eſt AB ab BD. <lb></lb></s> <s id="N12A38">vt autem AB ad BD, ita eſt grauitas ponderis E ad grauitatem <arrow.to.target n="note128"></arrow.to.target> <pb xlink:href="036/01/092.jpg"></pb>ponderis C: quare gra<lb></lb>uitas ponderis E ad <lb></lb>grauitatem ponderis <lb></lb>F ita erit, vt grauitas <lb></lb>ponderis E ad gra<lb></lb>uitatem ponderis C. </s> <s id="id.2.1.75.5.1.1.0.b"><lb></lb>Pondera igitur CF <lb></lb><figure id="id.036.01.092.1.jpg" place="text" xlink:href="036/01/092/1.jpg"></figure><lb></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 ſuſtinens <lb></lb>pondus F; erit potentia in A æqualis ipſi ponderi F. </s> <s id="id.2.1.75.5.1.2.0.a">& quoniam <lb></lb>pondus F in A appenſum æquè graue eſt, vt pondus C in D ap<lb></lb>penſum; eandem proportionem habebit potentia in A ad grauita<lb></lb><arrow.to.target n="note130"></arrow.to.target>tem ponderis F in A appenſi, quam habet ad grauitatem ponde<lb></lb>ris C in D appenſi. </s> <s id="id.2.1.75.5.1.3.0">Potentia verò in A ipſi F æqualis ſuſtinet <lb></lb>pondus F, ergo potentia in A pondus quoq; C ſuſtinebit. </s> <s id="id.2.1.75.5.1.4.0">Itaq; <lb></lb>cùm potentia in A ſit æqualis ponderi F, & pondus C ad pon<lb></lb>dus F ſit, vt AB ad BD; erit pondus C ad potentiam in A, vt <lb></lb><arrow.to.target n="note131"></arrow.to.target>AB ad BD. </s> <s id="id.2.1.75.5.1.4.0.a">& è conuerſo, vt BD ad BA, ita potentia in A ad <lb></lb>pondus C. </s> <s id="id.2.1.75.5.1.4.0.b">potentia ergo ad pondus ita erit, vt diſtantia fulci<lb></lb>mento, ac ponderis ſuſpenſioni intercepta ad diſtantiam à fulci <lb></lb>mento ad potentiam. </s> <s id="id.2.1.75.5.1.5.0">quod oportebat demonſtrare. </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"></emph>In ſexta huius de libra Ex<emph.end type="italics"></emph.end> 11 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.76.1.1.2.0"><margin.target id="note128"></margin.target>6 <emph type="italics"></emph>Huius. de libra.<emph.end type="italics"></emph.end></s> <s id="id.2.1.76.1.1.4.0"><margin.target id="note129"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 9 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.76.1.1.5.0"><margin.target id="note130"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 7 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.76.1.1.6.0"><margin.target id="note131"></margin.target><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></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" place="text" 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 ſit B, & pondus E ex puncto <lb></lb>C ſuſpenſum; ſitq; vis in A ſuſtinens pondus E. </s> <s id="id.2.1.77.2.1.1.0.a">Dico vt BC ad BA, <lb></lb>ita eſſe potentiam in A ad pondus E. </s> <s id="id.2.1.77.2.1.1.0.b">Producatur AB in C, & <lb></lb>fiat BD æqualis BC; & ex puncto D appendatur pondus F æqua <lb></lb>le ponderi E; itemq; ex puncto A ſuſpendatur pondus G ita, vt <lb></lb>pondus F ad pondus G eandem habeat proportionem, quam AB <pb n="40" xlink:href="036/01/093.jpg"></pb>ad BA. </s> <s id="N12AF6">pondera FG æqueponderabunt. </s> <s id="id.2.1.77.2.1.2.0">cùm autem ſit CB æqua <lb></lb>lis BD, pondera quoq; FE æqualia æqueponderabunt. </s> <s id="id.2.1.77.2.1.3.0">pondera <lb></lb>verò FEG in libra, ſeu vecte DBA appenſa, cuius fulcimentum <lb></lb>eſt B, non æqueponderabunt; ſed ex parte A deorſum tendent. </s> <s id="id.2.1.77.2.1.4.0">po<lb></lb>natur itaq; in A tanta vis, vt pondera FEG æqueponderent; erit <lb></lb>potentia in A æqualis ponderi G. </s> <s id="N12B0B">pondera enim FE <expan abbr="æqueponderãt">æqueponderant</expan>, <lb></lb>& vis in A nihil aliud efficere debet, niſi ſuſtinere <expan abbr="põdus">pondus</expan> G, ne deſcen<lb></lb>dat. </s> <s id="id.2.1.77.2.1.5.0">& quoniam pondera FEG, & potentia in A æqueponderant, <lb></lb>demptis igitur FG ponderibus, quæ æqueponderant, reliqua æque <lb></lb>ponderabunt; ſcilicet potentia in A ponderi E, hoc eſt potentia <lb></lb>in A pondus E ſuſtinebit, ita vt vectis AB maneat, vt prius erat. </s> <s id="id.2.1.77.2.1.6.0"><lb></lb>Cùm autem potentia in A ſit æqualis ponderi G, & pondus E pon<lb></lb>deri F æquale; habebit potentia in A ad pondus E eandem pro<lb></lb>portionem, quam habet BD, hoc eſt BC ad BA. </s> <s id="N12B2A">quod demon<lb></lb>ſtrare 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) manifeſtum eſſe po<lb></lb>teſt, ſi ponatur pondus E propius fulcimento B, <lb></lb>vt in H; minorem potentiam in A ſuſtinere poſ<lb></lb>ſe ipſum 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></lb>BA. </s> <s id="N12B4B">& quò propius pondus erit fulcimento, adhuc ſemper mino <lb></lb>rem poſſe potentiam ſuſtinere pondus E ſimiliter oſtendetur. </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"></emph>Quinti.<emph.end type="italics"></emph.end></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 ſemper mino <lb></lb>rem eſſe 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, ſemper BC <lb></lb>minor erit BA. </s> </p> <pb xlink:href="036/01/094.jpg"></pb> <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 poteſt, ſi duæ fuerint poten<lb></lb>tiæ, vna in A, altera in B, & vtraq; ſuſtentet <lb></lb>pondus E; potentiam in A ad potentiam in B eſ<lb></lb>ſe, 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></lb>tur officio duorum <expan abbr="vectiũ">vectium</expan>; <lb></lb>& AB ſunt tanquam duo <lb></lb>fulcimenta, hoc eſt quan<lb></lb>do AB eſt vectis, & poten<lb></lb>tia ſuſtinens in A; erit eius <lb></lb><figure id="id.036.01.094.1.jpg" place="text" xlink:href="036/01/094/1.jpg"></figure><lb></lb>fulcimentum B. </s> <s id="id.2.1.79.7.1.1.0.a">Quando verò BA eſt vectis, & potentia in B; <lb></lb>erit A fulcimentum: & pondus ſemper ex puncto C remanet ſu<lb></lb>ſpenſum. </s> <s id="id.2.1.79.7.1.2.0">& quoniam potentia in A ad pondus E eſt, vt BC ad <lb></lb>BA; vt autem pondus E ad potentiam, quæ eſt in B, ita eſt <lb></lb><arrow.to.target n="note133"></arrow.to.target>BA ad AC; erit ex æquali, potentia in A ad potentiam in B, vt <lb></lb>BC ad CA. </s> <s id="N12BB6">& hoc modo facilè etiam proportionem, quæ in <lb></lb>Quæſtionibus Mechanicis quæſtione vigeſima nona ab Ariſtotele <lb></lb>ponitur, nouiſſe 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"></emph>Quinti.<emph.end type="italics"></emph.end></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">Eſt etiam manifeſtum, vtraſq; potentias in A, <lb></lb>& B ſimul ſumptas æquales eſſe 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 eſt, vt BA ad BC; & idem <lb></lb>pondus E ad potentiam in B eſt, vt BA ad AC; quare pondus <lb></lb>E ad vtraſq; potentias in A, & B ſimul ſumptas eſt, vt AB ad BC <lb></lb>CA ſimul, hoc eſt ad BA. </s> <s id="N12BE3">pondus igitur E vtriſq; potentiis ſimul <lb></lb>ſumptis æquale erit. </s> </p> <pb n="41" xlink:href="036/01/095.jpg"></pb> <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 poſsumus. </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></lb>cuius fulcimentum <lb></lb>B; ſitq; ex puncto <lb></lb>A pondus C appen<lb></lb>ſum; ſitq; potentia <lb></lb>in D vtcunq; inter <lb></lb>AB ſuſtinens pon<lb></lb>dus C. </s> <s id="id.2.1.81.7.1.1.0.a">Dico vt AB <lb></lb><figure id="id.036.01.095.1.jpg" place="text" xlink:href="036/01/095/1.jpg"></figure><lb></lb>ad BD, ita eſſe potentiam in D ad pondus C. </s> <s id="id.2.1.81.7.1.1.0.b">Appendatur ex <lb></lb>puncto D pondus E æquale ipſi C; & vt BD ad BA, ita fiat pon<lb></lb>dus E ad aliud F. </s> <s id="N12C1D">& cùm pondera CE ſint inter ſe ſe æqualia; erit <lb></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></lb>F quoq; in D. </s> <s id="id.2.1.81.7.1.1.0.d">& quoniam pondus E ad ipſum F eſt, vt grauitas <lb></lb>ponderis E ad grauitatem ponderis F; & pondus E ad pondus F <arrow.to.target n="note134"></arrow.to.target><lb></lb>eſt, vt BD ad BA: vt igitur grauitas ponderis E ad grauitatem <lb></lb>ponderis F, ita eſt BD ad BA. </s> <s id="N12C32">vt autem BD ad BA, ita eſt gra<arrow.to.target n="note135"></arrow.to.target><lb></lb>uitas ponderis E ad grauitatem ponderis C; quare grauitas ponde<lb></lb>ris E ad grauitatem ponderis F eandem habet proportionem, <lb></lb>quam habet ad grauitatem ponderis C. </s> <s id="N12C33">pondera ergo CF eandem <arrow.to.target n="note136"></arrow.to.target><lb></lb>habent grauitatem. </s> <s id="id.2.1.81.7.1.2.0">ſit igitur potentia in D ſuſtinens pondus F, <lb></lb>erit potentia in D ipſi ponderi F æqualis. </s> <s id="id.2.1.81.7.1.3.0">& quoniam pondus F <lb></lb>in D æquè graue eſt, vt pondus C in A; habebit potentia in D <lb></lb>eandem proportionem ad grauitatem ponderis F, quam habet ad <arrow.to.target n="note137"></arrow.to.target><lb></lb>grauitatem ponderis C. </s> <s id="id.2.1.81.7.1.3.0.a">ſed potentia in D pondus F ſuſtinet; po<lb></lb>tentia igitur in D pondus quoq; C ſuſtinebit: & pondus C ad po<lb></lb>tentiam in D ita erit, vt pondus C ad pondus F; & C ad F eſt, vt <lb></lb>BD ad BA; erit igitur pondus C ad potentiam in D, vt BD ad <lb></lb>BA: & conuertendo, vt AB ad BD, ita potentia in D ad pondus <lb></lb>C. </s> <s id="id.2.1.81.7.1.3.0.b">potentia ergo ad pondus eſt, vt diſtantia à fulcimento ad pon<lb></lb>deris ſuſpendium ad diſtantiam à fulcimento ad potentiam. </s> <s id="id.2.1.81.7.1.4.0">quod <lb></lb>demonſtrare 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"></emph>In ſexta huius de libra.<emph.end type="italics"></emph.end></s> <s id="id.2.1.82.1.1.2.0"><margin.target id="note135"></margin.target>6 <emph type="italics"></emph>Huius de libra.<emph.end type="italics"></emph.end></s> <s id="id.2.1.82.1.1.3.0"><margin.target id="note136"></margin.target>9 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.82.1.1.4.0"><margin.target id="note137"></margin.target>7 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/096.jpg"></pb> <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" place="text" 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 ſit pon<lb></lb>dus C ſuſpenſum; ſitq; potentia in D ſuſtinens pondus C. </s> <s id="id.2.1.83.2.1.1.0.a">Dico <lb></lb>vt AB ad BD, ita eſſe potentiam in D ad pondus C. </s> <s id="id.2.1.83.2.1.1.0.b">Produca<lb></lb>tur AB in E, fiatq; BE æqualis ipſi BA; & ex puncto E appen<lb></lb>datur pondus F æquale ponderi C; & vt BD ad BE, ita fiat pon<lb></lb>dus F ad aliud G, quod ex puncto D ſuſpendatur. </s> <s id="id.2.1.83.2.1.2.0">pondera FG <lb></lb>æqueponderabunt. </s> <s id="id.2.1.83.2.1.3.0">& quoniam AB eſt æqualis BE, & pondera <lb></lb>FC æqualia; ſimiliter pondera FC æqueponderabunt. </s> <s id="id.2.1.83.2.1.4.0">Pondera <lb></lb>verò FGC ſuſpenſa in vecte EBA, cuius fulcimentum eſt B, non <lb></lb>æqueponderabunt; ſed ex parte A deorſum tendent. </s> <s id="id.2.1.83.2.1.5.0">Ponatur igi<lb></lb>tur in D tanta vis, vt pondera FGC æqueponderent; erit po<lb></lb>tentia in D æqualis ponderi G: pondera enim FC æqueponde<lb></lb>rant, & potentia in D nil aliud efficere debet, niſi ſuſtinere pon<lb></lb>dus G ne deſcendat. </s> <s id="id.2.1.83.2.1.6.0">& quoniam pondera FGC, & potentia in <lb></lb>D æqueponderant, demptis igitur FG ponderibus, quæ æquepon<lb></lb>derant; reliqua æqueponderabunt, ſcilicet potentia in D ponderi C. <lb></lb></s> <s id="N12CDC">hoc eſt potentia in D pondus C ſuſtinebit, ita vt vectis AB ma<lb></lb>neat, vt prius. </s> <s id="id.2.1.83.2.1.7.0">& cùm potentia in D ſit æqualis ponderi G, & pon<lb></lb>dus C æquale ponderi F; habebit potentia in D ad pondus C ean<lb></lb>dem proportionem, quam EB, hoc eſt AB ad BD. </s> <s id="id.2.1.83.2.1.7.0.a">quod de<lb></lb>monſtrare 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, ſi coftituatur pon<lb></lb>dus fulcimento B propius, vt in H; à minori po<lb></lb>tentia pondus ipſum ſubſtineri debere. </s> </p> <pb n="42" xlink:href="036/01/097.jpg"></pb> <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></lb>BD. </s> <s id="id.2.1.83.6.1.1.0.a">& quò propius erit fulcimento, adhuc ſemper minorem re<lb></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"></emph>Quinti.<emph.end type="italics"></emph.end></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">Manifeſtum quoq; eſt, potentiam in D ſemper <lb></lb>maiorem eſſe 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 ſumatur quoduis punctum D, ſemper AB <lb></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 eſt haſce, quas attulimus demonſtrationes <lb></lb>non ſolum vectibus horizonti æquidiſtantibus, verùm etiam ve<lb></lb>ctibus horizonti inclinatis ad hæc omnia oſtendenda commodè <lb></lb>aptari poſſe. </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 appenſum mo<lb></lb>ueat; erit ſpatium potentiæ motæ ad ſpatium <lb></lb>moti ponderis, vt diſtantia à fulcimento ad po<lb></lb>tentiam ad diſtantiam ab eodem ad ponderis ſu<lb></lb>ſpenſionem. </s> </p> <pb xlink:href="036/01/098.jpg"></pb> <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></lb>cimentum C; & ex puncto B <lb></lb>ſit pondus D ſuſpenſum; ſitq; <lb></lb>potentia in A mouens pon<lb></lb>dus D vecte AB. </s> <s id="id.2.1.85.8.1.1.0.a">Dico ſpa<lb></lb>tium potentiæ in A ad ſpa<lb></lb>tium ponderis ita eſſe, vt CA <lb></lb>ad CB. </s> <s id="id.2.1.85.8.1.1.0.b">Moueatur vectis AB, <lb></lb>& vt pondus D ſurſum mo<lb></lb>ueatur, oportet B ſurſum mo <lb></lb>ueri, A verò deorſum. </s> <s id="id.2.1.85.8.1.2.0">& quo<lb></lb>niam C eſt punctum immobi<lb></lb>le; idcirco dum A, & B mo<lb></lb>uentur, <expan abbr="circulorũ">circulorum</expan> circumferen<lb></lb>tias deſcribent. </s> <s id="id.2.1.85.8.1.3.0">Moueatur igi<lb></lb>tur AB in EF; erunt AE <lb></lb><figure id="id.036.01.098.1.jpg" place="text" xlink:href="036/01/098/1.jpg"></figure><lb></lb>BF circulorum circumferentiæ, quorum ſemidiametri ſunt CA <lb></lb>CB. </s> <s id="N12D91">tota compleatur circumferentia AGE, & tota BHF; ſintq; <lb></lb>KH puncta, vbi AB, & EF circulum BHF ſecant. </s> <s id="id.2.1.85.8.1.4.0">Quoniam e<lb></lb><arrow.to.target n="note139"></arrow.to.target>nim angulus BCF eſt æqualis angulo HCk; erit circumferentia <lb></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></lb>kH ſint ſub eodem angulo ACE, & circumferentia AE ad to<lb></lb>tam circumferentiam AGE ſit, vt angulus ACE ad quatuor re<lb></lb>ctos; vt autem idem angulus HCk ad quatuor rectos, ita quoq; <lb></lb>eſt circumferentia HK ad totam circumferentiam HBK; erit cir<lb></lb>cumferentia AE ad totam circumferentiam AGE, vt circumfe<lb></lb><arrow.to.target n="note141"></arrow.to.target>rentia kH ad totam kFH. </s> <s id="id.2.1.85.8.1.5.0.a">& permutando, vt circumferentia <lb></lb>AE ad circumferentiam kH, hoc eſt BF, ita tota circumferen<lb></lb>tia AGE ad totam circumferentiam BHF. </s> <s id="id.2.1.85.8.1.5.0.b">tota verò circumfe<lb></lb>rentia AGE ita ſe habet ad totam BHF, vt diameter circuli AEG <lb></lb><arrow.to.target n="note142"></arrow.to.target>ad diametrum circuli BHF. </s> <s id="id.2.1.85.8.1.5.0.c">Vt igitur circumferentia AE ad cir<lb></lb><arrow.to.target n="note143"></arrow.to.target>cumferentiam BF, ita diameter circuli AGE ad diametrum cir<lb></lb>culi BHF: vt autem diameter ad diametrum, ita ſemidiameter <lb></lb>ad ſemidiametrum, hoc eſt CA ad CB: quare vt circumferen<lb></lb>tia AE ad circumferentiam BF, ita CA ad CF. </s> <s id="N12DD0">circumferentia <lb></lb>verò AE ſpatium eſt potentiæ motæ, & circumferentia BF eſt <pb n="43" xlink:href="036/01/099.jpg"></pb>æqualis ſpatio ponderis D moti. </s> <s id="id.2.1.85.8.1.6.0">ſpatium enim motus ponderis <lb></lb>D ſemper æquale eſt ſpatio motus puncti B, cùm in B ſit appen<lb></lb>ſum: ſpatium ergo potentiæ motæ ad ſpatium moti ponderis eſt, <lb></lb>vt CA ad CB; hoc eſt vt diſtantia à fulcimento ad potentiam <lb></lb>ad diſtantiam ab eodem ad ponderis ſuſpenſionem. </s> <s id="id.2.1.85.8.1.7.0">quod demon<lb></lb>ſtrare 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"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.86.1.1.2.0"><margin.target id="note140"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 26 <emph type="italics"></emph>tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.86.1.1.3.0"><margin.target id="note141"></margin.target>16 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.86.1.1.4.0"><margin.target id="note142"></margin.target>23 <emph type="italics"></emph>Octaui Pappi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.86.1.1.5.0"><margin.target id="note143"></margin.target>11 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></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></lb>ius fulcimentum B; potentia<lb></lb>〈qué〉 mouens in A; & pondus <lb></lb>in C. </s> <s id="id.2.1.87.1.1.1.0.a">dico ſpatium potentiæ <lb></lb>translatæ ad ſpatium transla<lb></lb>ti ponderis ita eſſe, vt BA ad <lb></lb>BC. </s> <s id="id.2.1.87.1.1.1.0.b">Moueatur vectis, & vt <lb></lb>pondus sursum attollatur, ne<lb></lb>ceſſe eſt puncta C A ſurſum <lb></lb>moueri. </s> <s id="id.2.1.87.1.1.2.0">Moueatur igitur A <lb></lb>ſurſum vſq; ad D; ſitq; ve<lb></lb>ctis motus BD. </s> <s id="id.2.1.87.1.1.2.0.a">eodemq; <lb></lb>modo (vt prius dictum eſt) <lb></lb>oſtendemus puncta CA cir<lb></lb>culorum circumferentias de<lb></lb><figure id="id.036.01.099.1.jpg" place="text" xlink:href="036/01/099/1.jpg"></figure><lb></lb>ſcribere, <expan abbr="quorũ">quorum</expan> ſemidiametri ſunt BA BC. </s> <s id="id.2.1.87.1.1.2.0.b">ſimiliterq; oſtendemus <lb></lb>ita eſſe AD ad CE, vt ſemidiameter AB ad ſemidiametrum 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, ſi potentia eſſet in C, & pondus in A, <lb></lb>oſtendetur ita eſſe CE ad AD, vt BC ad BA; hoc eſt diſtan<lb></lb>tia à fulcimento ad potentiam ad diſtantiam ab eodem ad ponde<lb></lb>ris ſuſpenſionem. </s> <s id="id.2.1.87.2.1.2.0">quod oportebat demonſtrare. </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 manifeſtum eſt maiorem habere pro<lb></lb>portionem ſpatium potentiæ mouentis ad ſpa<lb></lb>tium ponderis moti, quàm pondus ad eandem <lb></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 ſpatium ponderis eandem habet, <pb xlink:href="036/01/100.jpg"></pb>quam pondus ad potentiam pondus ſuſtinentem; potentia ve<lb></lb>rò ſuſtinens minor eſt potentia mouente, quare minorem habebit <lb></lb><arrow.to.target n="note144"></arrow.to.target>proportionem pondus ad potentiam ipſum mouentem, quàm ad <lb></lb>potentiam ipſum ſuſtinentem. </s> <s id="id.2.1.87.5.1.2.0">ſpatium igitur potentiæ mouentis <lb></lb>ad ſpatium ponderis maiorem habebit proportionem, quàm pon<lb></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"></emph>Quinti.<emph.end type="italics"></emph.end></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 ſuſti<lb></lb>nens ad ipſum pondus eandem habebit propor<lb></lb>tionem, quam diſtantia à fulcimento ad punctum, <lb></lb>vbi à centro grauitatis ponderis horizonti ducta <lb></lb>perpendicularis vectem ſecat, intercepta, ad <lb></lb>diſtantiam 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></lb>horizonti æqui<lb></lb>diſtans, cuius ful<lb></lb>cimentum N; ſit <lb></lb>deinde pondus <lb></lb>AC, cuius cen<lb></lb>trum grauitatis <lb></lb>ſit D, quod pri<lb></lb>mùm ſit infra ve<lb></lb>ctem; pondus ve<lb></lb>rò ſit ex punctis <lb></lb>AO ſuſpenſum; <lb></lb><figure id="id.036.01.100.1.jpg" place="text" xlink:href="036/01/100/1.jpg"></figure><lb></lb>& à puncto D horizonti, & ipſi AB perpendicularis ducatur DE. </s> <s id="id.2.1.89.3.1.1.0.a"><lb></lb>ſi verò alii ſint quoq; vectes AF AG, quorum fulcimenta ſint <lb></lb>HK; ponduſq; AC in vecte AG ex punctis AQ ſit appenſum; <lb></lb>in vecte autem AF in punctis AP: lineaq; DE producta ſecet <lb></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 ſuſtinen<lb></lb>tem ad ipſum pondus eam habere proportionem, quam habet kL <pb n="44" xlink:href="036/01/101.jpg"></pb>ad kF; & potentiam in B ad pondus eam habere, quam NE ad <lb></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></lb>Quoniam enim DL horizonti eſt perpendicularis, pondus AC <lb></lb>vbicunq; in linea DL fuerit appenſum, eodem modo, quo reperi<lb></lb>tur, manebit. </s> <s id="id.2.1.89.3.1.2.0">quare in vecte AB ſi ſuſpenſiones, quæ ſunt ad AO <lb></lb>ſoluantur, pondus AC in E appenſum eodem modo manebit, ſi<lb></lb>cuti nunc manet; hoc eſt ſublato puncto A, & linea QO, codem <lb></lb>modo pondus in E appenſum manebit, vt ab ipſis AO pun<lb></lb>ctis ſuſtinebatur; ex commentario Federici Commandini in ſextam <lb></lb>Archimedis <expan abbr="propoſionẽ">propoſitionem</expan> de quadratura parabolæ, & ex prima huius <lb></lb>de libra. </s> <s id="id.2.1.89.3.1.3.0">Itaq; quoniam pondus AC eandem ad libram habet conſti<lb></lb>tutionem, ſiue in AO ſuſtineatur, ſiue ex puncto E ſit appenſum; <lb></lb>eadem potentia in B idem pondus AC, ſiue in E, ſiue in AO <lb></lb>ſuſpenſum ſuſtinebit. </s> <s id="id.2.1.89.3.1.4.0">potentia verò in B ſuſtinens pondus AC <lb></lb>in E appenſum ad ipſum pondus ita ſe habet, vt NE ad NB; po<lb></lb>tentia <arrow.to.target n="note145"></arrow.to.target>igitur in B ſuſtinens pondus AC ex punctis AO ſuſpen<lb></lb>ſum ad ipſum pondus ita erit, vt NE ad NB. </s> <s id="id.2.1.89.3.1.4.0.a">Non aliter oſten <lb></lb>detur pondus AC ex puncto L ſuſpenſum manere, ſicuti à pun<lb></lb>ctis AP ſuſtinetur; potentiamq; in F ad ipſum pondus ita eſſe, vt kL <lb></lb>ad KF. </s> <s id="id.2.1.89.3.1.4.0.b">In vecte verò AG pondus AC in M appenſum ita mane <lb></lb>re, vt à punctis AQ ſuſtinetur; potentiamq; in G ad pondus <lb></lb>AC ita eſſe, vt HM ad HG; hoc eſt vt diſtantia à fulcimento <lb></lb>ad punctum, vbi à centro grauitatis ponderis horizonti ducta <lb></lb>perpendicularis vectem ſecat, ad diſtantiam à fulcimento ad poten<lb></lb>tiam. </s> <s id="id.2.1.89.3.1.5.0">quod demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></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 eſſent vectium fulcimenta, potentiæq; eſſent <lb></lb>in KNH pondus ſuſtinentes, ſimili modo oſtendetur ita eſſe po<lb></lb>tentiam in H ad pondus, vt GM ad GH; & potentiam in N ad <lb></lb>pondus, vt BE ad BN; ac potentiam in k ad pondus, vt FL <lb></lb>ad Fk. </s> </p> <pb xlink:href="036/01/102.jpg"></pb> <p id="id.2.1.91.3.0.0.0" type="main"> <s id="id.2.1.91.3.1.1.0">Et ſi vectes AB <lb></lb>AF AG habeant <lb></lb>fulcimenta in A, <lb></lb>& pondus ſit NO; <lb></lb>deinde ab eius <lb></lb>centro grauitatis <lb></lb>D ducatur ipſi A <lb></lb>B, & horizonti <lb></lb><expan abbr="perpẽdicularis">perpendicularis</expan> D <lb></lb>MEL; ſintq; po<lb></lb>tentiæ in FBG: <lb></lb>ſimiliter oſtende<lb></lb>tur ita eſſe poten<lb></lb><figure id="id.036.01.102.1.jpg" place="text" xlink:href="036/01/102/1.jpg"></figure><lb></lb>tiam in G pondus NO ſuſtinentem ad ipſum pondus, vt AM <lb></lb>ad AG; ac potentiam in B, vt AE ad AB; & potentiam in F, <lb></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></lb>vectis AB ho<lb></lb>rizonti æqui<lb></lb>diſtans, cuius <lb></lb>fulcimentum <lb></lb>D; & ſit BE <lb></lb>pondus, cuius <lb></lb>centrum <expan abbr="gaui">graui</expan><lb></lb>tatis ſit F ſu<lb></lb>pra vectem: à <lb></lb>punctoq; F ho<lb></lb>rizonti, & ipſi <lb></lb>AB ducatur <lb></lb><figure id="id.036.01.102.2.jpg" place="text" xlink:href="036/01/102/2.jpg"></figure><lb></lb>FH; ponduſq; à puncto B, & PQ ſuſtineatur. </s> <s id="id.2.1.91.4.1.2.0">Sint deinde alii ve<lb></lb>ctes BL BM, quorum fulcimenta ſint NO; lineaq; FH producta ſe<lb></lb>cet BM in k, & BL in G; pondus autem in vecte BL in pun<lb></lb>ctis BP ſuſtineatur; in vecte autem BM à puncto B, & PR. </s> <s id="id.2.1.91.4.1.2.0.a">Di<lb></lb>co potentiam in L pondus BE vecte BL ſuſtinentem ad ipſum <lb></lb>pondus eam habere proportionem, quam NG ad NL; & po<pb n="45" xlink:href="036/01/103.jpg"></pb>tentiam in A ad pondus eam habere, quam DH ad DA; poten<lb></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></lb>nim à centro grauitatis F ducta eſt kF horizonti perpendicularis, <lb></lb>ex quocunq; puncto lineæ kF ſuſtineatur pondus, manebit; vt <arrow.to.target n="note146"></arrow.to.target><lb></lb>nunc ſe habet. </s> <s id="id.2.1.91.4.1.3.0">ſi igitur ſuſtineatur in H, manebit vt prius; ſcili<lb></lb>cet ſublato puncto B, & PQ, quæ pondus ſuſtinent, pondus BE <lb></lb>manebit, ſicuti ab ipſis ſuſtinebatur. </s> <s id="id.2.1.91.4.1.4.0">quare in vecte AB graueſcet <lb></lb>in H, & ad vectem eandem habebit conſtitutionem, quam prius; <lb></lb>idcirco erit, ac ſi in H eſſet appenſum. </s> <s id="id.2.1.91.4.1.5.0">eadem igitur potentia ìdem <lb></lb>pondus BE, ſiue in H, ſiue in B, & Q ſuffultum, ſuſtinebit. </s> <s id="id.2.1.91.4.1.6.0">Potentia ve<arrow.to.target n="note147"></arrow.to.target><lb></lb>rò in A ſuſtinens pondus BE vecte AB in H appenſum ad ipſum <lb></lb>pondus eandem habet proportionem, quam DH ad DA; eadem <lb></lb>ergo potentia in A ſuſtinens pondus BE in punctis BQ ſuſtenta <lb></lb>tum ad ipſum pondus erit, vt DH ad DA. </s> <s id="id.2.1.91.4.1.6.0.a">Similiter oſtende<lb></lb>tur pondus BE ſi in G ſuſtineatur, manere; ſicuti à punctis BP <lb></lb>ſuſtinebatur: & in puncto k, vt à punctis BR. </s> <s id="N12FFF">quare potentia in <lb></lb>L ſuſtinens pondus BE ad ipſum pondus ita erit, vt NG ad NL. <lb></lb></s> <s id="N13004">potentia verò in M ad pondus, vt OK ad OM; hoc eſt vt diſtan<lb></lb>tia à fulcimento ad punctum, vbi à centro grauitatis ponderis ho<lb></lb>rizonti ducta perpendicularis vectem ſecat, ad diſtantiam à fulci<lb></lb>mento ad potentiam. </s> <s id="id.2.1.91.4.1.7.0">quod demonſtrare 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"></emph>Huius de libra.<emph.end type="italics"></emph.end></s> <s id="id.2.1.92.1.1.2.0"><margin.target id="note147"></margin.target>1 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 eſſent fulcimenta, & potentiæ in NDO; ſimi <lb></lb>liter oſtendetur ita eſſe potentiam in N ad pondus, vt LG ad L <lb></lb>N; & potentiam in D, vt AH ad AD; & potentiam in O, vt <lb></lb>Mk ad MO. <pb xlink:href="036/01/104.jpg"></pb></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 ſi vectes BA <lb></lb>BL BM habeant <lb></lb>fulcimenta in B, & <lb></lb>pondus ſupra <expan abbr="vectẽ">vectem</expan><lb></lb>ſit NO; & ab eius <lb></lb>centro grauitatis F <lb></lb>ducatur ipſi AB, & <lb></lb>horizonti perpendi<lb></lb>cularis FDEG; ſint <lb></lb>〈qué〉 potentiæ in L <lb></lb>AM; ſimiliter o<lb></lb>ſtendetur ita eſſe po<lb></lb>tentiam in L pon<lb></lb><figure id="id.036.01.104.1.jpg" place="text" xlink:href="036/01/104/1.jpg"></figure><lb></lb>dus ſuſtinentem ad ipſum pondus, vt BD ad BL; & potentiam <lb></lb>in A ad pondus, vt BE ad BA, atq; potentiam in M, vt BG <lb></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></lb>vectis AB ho<lb></lb>rizonti æqui<lb></lb>diſtans, cuius <lb></lb>fulcimentum <lb></lb>C, & pondus <lb></lb>DE habeat <expan abbr="cẽ">cen</expan><lb></lb>trum grauita<lb></lb>tis F in ipſo <lb></lb>vecte AB; <lb></lb>ſintq; deniq; <lb></lb>alii vectes G <lb></lb>H kL, quo<lb></lb><figure id="id.036.01.104.2.jpg" place="text" xlink:href="036/01/104/2.jpg"></figure><lb></lb>rum fulcimenta ſint MN; pondusq; in vecte GH ſuſtineatur à <lb></lb>punctis GO; in vecte autem AB à punctis AP; & in uecte KL <lb></lb>à punctis KQ; & centrum grauitatis F ſit quoq; in utroq; uecte <lb></lb>GH kL; ſintq; potentiæ in HBL. </s> <s id="id.2.1.93.3.1.1.0.a">Dico potentiam in H ad <lb></lb>pondus ita eſſe, ut NF ad NH; & potentiam in B ad pondus, ut <lb></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></lb>niam enim F centrum eſt grauitatis ponderis DE, ſi igitur in F <pb n="46" xlink:href="036/01/105.jpg"></pb>ſuſtineatur, pondus DE manebit ſicut prius, per definitionem cen<lb></lb>tri grauitatis; eritq; ac ſi in F eſſet appenſum; atq; in vecte eodem <lb></lb>modo manebit, ſiue à punctis AP, ſiue à puncto F ſuſtineatur. </s> <s id="id.2.1.93.3.1.2.0"><lb></lb>quod idem in vectibus GH kL eueniet; ſcilicet pondus eodem mo <lb></lb>do manere, ſiue in F, ſiue in GO, vel in kQ ſuſtineatur. </s> <s id="id.2.1.93.3.1.3.0">eadem <lb></lb>igitur potentia in B idem pondus DE, vel in F, vel in AP appenſum <lb></lb>ſuſtinebit: & quando appenſum eſt in F ad ipſum pon<lb></lb>dus eſt, vt CF ad CB, ergo potentia ſuſtinens pondus DE in <lb></lb>AP appenſum ad ipſum pondus erit, vt CF ad CB. </s> <s id="N130B7">eodemq; mo <lb></lb>do potentia in H ad pondus in GO appenſum ita erit, vt NF ad <lb></lb>NH. </s> <s id="N130BD">potentiaq; in L ad pondus in kQ appenſum erit, vt MF <lb></lb>ad ML. </s> <s id="N130C1">quod oſtendere 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 eſſent fulcimenta, & potentiæ eſſent in NCM; ſi<lb></lb>militer oſtendetur potentiam in N ad pondus ita eſſe, vt HF ad <lb></lb>HN; & potentiam in C, vt BF ad BC, & potentiam in M, vt <lb></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 ſi vectes BA <lb></lb>BC BD <expan abbr="habeãt">habeant</expan> ful<lb></lb>cimenta in B, ſintq; <lb></lb>pondera in EF GH <lb></lb>kL, ita vt eorum <lb></lb>centra MNO gra<lb></lb>uitatis ſint in vecti<lb></lb>bus; ſintq; poten<lb></lb>tiæ in CAD: ſimi <lb></lb>liter oſtendetur po<lb></lb>tentiam in C ad <lb></lb>pondus EF ita eſſe, <lb></lb><figure id="id.036.01.105.1.jpg" place="text" xlink:href="036/01/105/1.jpg"></figure><lb></lb>vt BM ad BC, & potentiam in A ad pondus GH, vt BN ad <lb></lb>BA, potentiamq; in D ad pondus KL, vt BO ad BD. </s> </p> <pb xlink:href="036/01/106.jpg"></pb> <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 ſit rectos <lb></lb>AD, quæ ex parte A producatur vtcunq; vſq; <lb></lb>ad C; connectaturq; CB, quæ ex parte B quoq; <lb></lb>producatur vſq; ad E. ducantur deinde à pun<lb></lb>cto B vtcunq; inter AB BE lineæ BF BG ipſi <lb></lb>AB æquales; à punctiſq; FG ipſis perpendicula<lb></lb>res ducantur FH GK, quæ & inter ſe ſe, & ipſi <lb></lb>AD conſtituantur æ<lb></lb>quales, ac ſi BA AD <lb></lb>motæ ſint in BF FH, <lb></lb>& in BG GK; con<lb></lb>nectanturq; CH CK, <lb></lb>quæ lineas BF BG <lb></lb>in punctis MN ſe<lb></lb>cent. </s> <s id="id.2.1.93.8.1.2.0">Dico BN mi<lb></lb>norem eſſe BM, & <lb></lb>BM ipſa BA. <lb></lb><figure id="id.036.01.106.1.jpg" place="text" 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></lb>BK. </s> <s id="N13139">& quoniam duæ lineæ <lb></lb>DA AB duabus HF FB <lb></lb>ſunt æquales, & angulus <lb></lb>DAB rectus recto HFB eſt <lb></lb><arrow.to.target n="note148"></arrow.to.target>etiam æqualis; erunt reliqui <lb></lb>anguli reliquis angulis æqua<lb></lb>les, & HB ipſi DB æqualis. </s> <s id="id.2.1.93.9.1.2.0"><lb></lb>ſimiliter oſtendetur triangu<lb></lb>lum BkG triangulo BHF æqualem eſſe. </s> <s id="id.2.1.93.9.1.3.0">quare centro B, inter<pb n="47" xlink:href="036/01/107.jpg"></pb>uallo quidem vna ipſarum circulus deſcribatur DH kE, qui li<lb></lb>neas CH CK ſecet in punctis OP; connectanturq; OB PB. </s> <s id="id.2.1.93.9.1.3.0.a"><lb></lb>Quoniam igitur punctum k propius eſt ipſi E, quàm H; erit linea <arrow.to.target n="note149"></arrow.to.target><lb></lb>Ck maior ipſa CH, & CP ipſa CO minor: ergo PK ipſa OH <lb></lb>maior erit. </s> <s id="id.2.1.93.9.1.4.0">Quoniam autem triangulum BkP æquicrure latera <lb></lb>Bk BP lateribus BH BO trianguli BHO æquicruris æqualia ha<lb></lb>bet, baſim verò KP baſi HO maiorem, erit angulus kBP an<lb></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 baſim anguli, hoc eſt kPB <lb></lb>PkB ſimul ſumpti, qui inter ſe ſunt æquales, reliquis ad baſim an<lb></lb>gulis, nempè OHB HOB, qui etiam inter ſe ſunt æquales, mino<lb></lb>res <arrow.to.target n="note151"></arrow.to.target>erunt: cùm omnes anguli cuiuſcunq; trianguli duobus ſint rectis <lb></lb>æquales. </s> <s id="id.2.1.93.9.1.6.0">quare & horum dimidii, ſcilicet NkB minor MHB. </s> <s id="id.2.1.93.9.1.6.0.a"><lb></lb>Cùm autem angulus BkG æqualis ſit angulo BHF, erit NkG <lb></lb>ipſo MHF maior. </s> <s id="id.2.1.93.9.1.7.0">ſi igitur à puncto k conſtituatur angulus GKQ <lb></lb>ipſi FHM æqualis, fiet triangulum GkQ triangulo FHM æqua <lb></lb>le; nam duo anguli ad FH vnius duobus ad Gk alterius ſunt <lb></lb>æquales, & latus FH lateri Gk eſt æquale, erit GQ ipſi FM æ<lb></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 ipſa FM. </s> <s id="id.2.1.93.9.1.8.0.a">Cùm itaq; BG ipſi BF ſit æqua <lb></lb>lis, erit BN minor ipſa BM. </s> <s id="id.2.1.93.9.1.8.0.b">Quòd autem BM ſit ipſa BA minor, <lb></lb>eſt manifeſtum; cùm BM ipſa BF, quæ ipſi BA eſt æqualis, ſit <lb></lb>minor. </s> <s id="id.2.1.93.9.1.9.0">quod demonſtrare 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"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.94.1.1.2.0"><margin.target id="note149"></margin.target>8 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.94.1.1.3.0"><margin.target id="note150"></margin.target>25 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.94.1.1.4.0"><margin.target id="note151"></margin.target>5 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.94.1.1.5.0"><margin.target id="note152"></margin.target>26 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.95.1.0.0.0" type="main"> <s id="id.2.1.95.1.1.1.0">Inſuper ſi intra BG BE alia vtcunq; ducatur linea ipſi BG æ<lb></lb>qualis; fiatq; operatio, quemadmodum ſupra dictum eſt; ſimili<lb></lb>ter oſtendetur lineam BR minorem eſſe BN. </s> <s id="id.2.1.95.1.1.1.0.a">& quò propius fue<lb></lb>rit ipſi BE, adhuc minorem ſemper eſſe. </s> </p> <pb xlink:href="036/01/108.jpg"></pb> <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 ſint <lb></lb>deorſum inter BC BA conſtituta; connectan<lb></lb>turq; HC KC, quæ lineas BF BG ex parte <lb></lb>FG productas in punctis MN ſecent erit BN <lb></lb>maior BM, & BM ipſa 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></lb>Ck vſq; ad circumferentiam <lb></lb>in OP, Connectanturq; BO <lb></lb>BP; ſimili modo oſtende<lb></lb>tur lineam Pk maiorem eſ <lb></lb>ſe OH, angulumq; PkB mi<lb></lb>norem eſſe angulo OHB. </s> <s id="id.2.1.95.4.1.1.0.a">& <lb></lb>quoniam angulus BHF eſt <lb></lb>æqualis angulo BkG; erit to<lb></lb>tus PKG angulus angulo <lb></lb>OHF minor: quare reliquus <lb></lb>GKN reliquo FHM maior <lb></lb>erit. </s> <s id="id.2.1.95.4.1.2.0">ſi it aq; conſtituatur angu<lb></lb>lus GkQ ipſi FHM æqua <lb></lb>lis, linea KQ ipſam GN ita <lb></lb>ſecabit, vt GQ ipſi FM æqua <lb></lb>lis euadat: quare maior. </s> <s id="id.2.1.95.4.1.3.0">erit <lb></lb>GN, quàm FM; quibus ſi <lb></lb>æquales adiiciantur BF BG, <lb></lb>erit BN ipſa BM maior. </s> <s id="id.2.1.95.4.1.4.0">& <lb></lb>cùm BM ſit ipſa FB maior, <lb></lb>erit quoq; ipſa BA maior. </s> <s id="id.2.1.95.4.1.5.0">ſi <lb></lb>militer oſtendetur, quò pro <lb></lb>pius fuerit BG ipſi BC, li<lb></lb>neam BN ſemper maiorem <lb></lb>eſſe. <figure id="id.036.01.108.1.jpg" place="text" xlink:href="036/01/108/1.jpg"></figure></s> </p> <pb n="48" xlink:href="036/01/109.jpg"></pb> <p id="id.2.1.95.5.0.0.0" type="head"> <s id="id.2.1.95.4.3.1.0">PROPOSITIO VII. </s> </p> <p id="id.2.1.95.5.0.0.0.a" type="main"> <s id="id.2.1.95.5.1.1.0">Sit recta linea AB, cuì perpendicularis exi<lb></lb>ſtat AD, quæ ex parte D producatur vtcunq; vſq; <lb></lb>ad C; connectaturq; CB, quæ producatur e<lb></lb>tiam vſq; ad E; & inter AB BE lineæ ſimiliter <lb></lb>vtcunq; ducantur BF BG ipſi AB æquales; à <lb></lb>punctisq; FG lineæ FH GK ipſi AB æquales, <lb></lb>ipſis verò BF BG <expan abbr="perpẽdiculares">per<lb></lb>pendiculares</expan> ducantur; <lb></lb>ac ſi BA AD motæ <lb></lb>ſint in BF FH BG <lb></lb>GK: Connectanturq; <lb></lb>CH CK, quæ lineas <lb></lb>BF BG productas ſe<lb></lb>cent in punctis MN. </s> <s id="id.2.1.95.5.1.1.0.a"><lb></lb>Dico BN maiorem eſ <lb></lb>ſe BM, & BM ipſa BA. <lb></lb><figure id="id.036.01.109.1.jpg" place="text" 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></lb>& centro B, interuallo quidem <lb></lb>BD, circulus deſcribatur. </s> <s id="id.2.1.95.6.1.2.0">ſimi <lb></lb>liter vt in præcedenti demon<lb></lb>ſtrabimus puncta kHDOP in <lb></lb>circuli circumferentia eſſe, trian<lb></lb>gulaq; ABD FBH GBk in<lb></lb>ter ſe ſe æqualia eſſe, atq; lineam <lb></lb>Pk maiorem OH, angulumq; <lb></lb>PKB minorem eſſe angulo O <lb></lb>HB. </s> <s id="id.2.1.95.6.1.2.0.a">Quoniam igitur angulus BHF æqualis eſt angulo BkG, <pb xlink:href="036/01/110.jpg"></pb>erit totus angulus PkG angu<lb></lb>lo OHF minor: quare reliquus <lb></lb>GkN reliquo FHM maior <lb></lb>erit. </s> <s id="id.2.1.95.6.1.3.0">ſi igitur fiat angulus GK <lb></lb>Q ipſi FHM æqualis, erit trian<lb></lb>gulum GKQ triangulo FHM <lb></lb>æquale, & latus GQ lateri FM <lb></lb>æquale; ergo maior erit GN ip<lb></lb>ſa FM; ac propterea BN ma<lb></lb>ior erit BM. </s> <s id="id.2.1.95.6.1.3.0.a">BM autem ma<lb></lb>ior erit BA; nam BM maior eſt <lb></lb>ipſa BF. </s> <s id="N132BF">quod demonſtrare <lb></lb>oportebat. <figure id="id.036.01.110.1.jpg" place="text" 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; prorſus modo, quo <lb></lb>propius fuerit BG ipſi BE, li<lb></lb>neam BN ſemper maiorem eſſe <lb></lb>oſtendetur. </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 deorſum in<lb></lb>ter AB BC conſtituantur, ducanturq; CHO <lb></lb>CKP, quæ lineas BF BG ſecent in punctis M <lb></lb>N; erit linea BN minor ipſa BM, & BM <lb></lb>ipſa BA. </s> </p> <pb n="49" xlink:href="036/01/111.jpg"></pb> <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></lb>ſimiliter oſtendetur angulum <lb></lb>PKB minorem eſſe OHB. </s> <s id="id.2.1.95.10.1.1.0.a">& <lb></lb>quoniam angulus FHB æqua<lb></lb>lis eſt angulo GkB; erit angu<lb></lb>lus GkN angulo FHM ma<lb></lb>ior: quare & linea GN ma<lb></lb>ior erit ipſa FM. </s> <s id="N132FD">ideoq; linea <lb></lb><expan abbr="nea"></expan> BN minor erit linea BM. </s> <s id="id.2.1.95.10.1.1.0.b"><lb></lb>Cùm autem maior ſit BF ipſa <lb></lb>BM; erit BM ipſa BA minor. </s> <s id="id.2.1.95.10.1.2.0">Si<lb></lb>miliq; modo oſtendetur, quò <lb></lb>propius fuerit BG ipſi BC, li<lb></lb>neam BN ſemper minorem <lb></lb>eſſe. </s> </p> <figure id="id.036.01.111.1.jpg" place="text" 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 ſuſtinens centrum grauitatis <lb></lb>ſupra vectem horizonti æquidiſtantem habens, <lb></lb>quò magis pondus ab hoc ſitu vecte eleuabitur; <lb></lb>minori ſemper, vt ſuſtineatur, egebit potentia: <lb></lb>ſi verò deprimetur, maiori. <pb xlink:href="036/01/112.jpg"></pb> <figure id="id.036.01.112.1.jpg" place="text" 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 æquidiſtans, cuius fulcimentum C; <lb></lb>pondus autem BD, eiuſdem verò grauitatis centrum ſit ſupra ve<lb></lb>ctem vbi H: ſitq; potentia ſuſtinens in A. </s> <s id="id.2.1.95.12.1.1.0.a">moueatur deinde ve<lb></lb>ctis AB in EF, ſitq; pondus motum in FG. </s> <s id="id.2.1.95.12.1.1.0.b">Dico primùm mino <lb></lb>rem <expan abbr="potentiã">potentiam</expan> in E ſuſtinere pondus FG vecte EF, quàm <expan abbr="potẽtia">potentia</expan> in <lb></lb>A pondus BD vecte AB. </s> <s id="id.2.1.95.12.1.1.0.c">ſit k centrum grauitatis ponderis FG; <lb></lb>deinde tùm ex H, tùm ex K ducantur HL kM ipſorum horizon<lb></lb>tibus perpendiculares, quæ in <expan abbr="centrũ">centrum</expan> mundi conuenient; ſitq; HL ip<lb></lb>ſi quoq; AB perpendicularis. </s> <s id="id.2.1.95.12.1.2.0">ducatur deinde kN ipſi EF perpen<lb></lb>dicularis, quæ ipſi HL æqualis erit, & CN ipſi CL æqualis. </s> <s id="id.2.1.95.12.1.3.0">Quo<lb></lb><arrow.to.target n="note153"></arrow.to.target>niam enim HL horizonti eſt perpendicularis, potentia in A ſu<lb></lb>ſtinens pondus BD ad ipſum pondus eam habebit proportionem, <lb></lb>quam CL ad CA. </s> <s id="id.2.1.95.12.1.3.0.a">rurſus quoniam kM horizonti eſt perpendicu<lb></lb>laris, potentia in E pondus FG ſuſtinens ita erit ad pondus, vt <lb></lb>CM ad CE. </s> <s id="id.2.1.95.12.1.3.0.b">Cùm autem CN NK ipſis CL LH ſint æquales, <lb></lb><arrow.to.target n="note154"></arrow.to.target>angulosq; rectos contineant; erit CM minor ipſa CL; ergo CM <lb></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"></pb>CA ipſi CE eſt æqualis, minorem igitur proportionem habebit <lb></lb>CM ad CE. quàm CL ad CA: & cùm pondera BD FG ſint <lb></lb>æqualia, eſt enim idem pondus; ergo minor erit proportio po<lb></lb>tentiæ in E pondus FG ſuſtinentis ad ipſum pondus, quàm po<lb></lb>tentiæ in A pondus BD ſuſtinentis ad ipſum pondus. </s> <s id="id.2.1.95.12.1.4.0">Quare <arrow.to.target n="note156"></arrow.to.target><lb></lb>minor potentia in E ſuſtinebit pondus FG, quàm potentia in A <lb></lb>pondus BD. </s> <s id="N1339E">& quò pondus magis eleuabitur; ſemper oſtendetur <lb></lb>minorem adhuc potentiam pondus ſuſtinere; cùm linea PC mi <arrow.to.target n="note157"></arrow.to.target><lb></lb>nor ſit linea CM. </s> <s id="id.2.1.95.12.1.4.0.a">ſit deinde vectis in QR, & pondus in QS, <lb></lb>cuius <expan abbr="centrũ">centrum</expan> grauitatis ſit O. </s> <s id="id.2.1.95.12.1.4.0.b">dico maiorem requiri potentiam in R <lb></lb>ad <expan abbr="ſuſtinendũ">ſuſtinendum</expan> pondus QS, quàm in A ad pondus BD. </s> <s id="N133B9">ducatur à cen<lb></lb>tro grauitatis O linea OT horizonti perpendicularis. </s> <s id="id.2.1.95.12.1.5.0">& quo<lb></lb>niam HL OT, ſi ex parte L, atq; T producantur, in centrum <lb></lb>mundi conuenient; erit CT maior CL: eſt autem CA ipſi CR <arrow.to.target n="note158"></arrow.to.target><lb></lb>æqualis, habebit ergo TC ad CR maiorem proportionem, quàm <lb></lb>LC ad CA. </s> <s id="id.2.1.95.12.1.5.0.a">Maior igitur erit potentia in R ſuſtinens pondus <arrow.to.target n="note159"></arrow.to.target><lb></lb>QS, quàm in A ſuſtinens BD. </s> <s id="N133D3">ſimiliter oſtendetur; quò vectis <lb></lb>RQ magis à vecte AB diſtabit deorſum vergens, ſemper maio<lb></lb>rem potentiam requiri ad ſuſtinendum pondus: diſtantia enim CV <arrow.to.target n="note160"></arrow.to.target><lb></lb>longior eſt CT. </s> <s id="id.2.1.95.12.1.5.0.b">Quò igitur pondus à ſitu horizonti æquidiſtan<lb></lb>te magis eleuabitur à minori ſemper potentia pondus ſuſtinebitur; <lb></lb>quò verò magis deprimetur, maiori, vt ſuſtineatur, egebit potentia. <lb></lb></s> <s id="id.2.1.95.12.1.6.0"><lb></lb>quod demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.96.1.1.2.0"><margin.target id="note154"></margin.target>6 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.96.1.1.3.0"><margin.target id="note155"></margin.target>8 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.96.1.1.4.0"><margin.target id="note156"></margin.target>10 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.96.1.1.5.0"><margin.target id="note157"></margin.target>6 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.96.1.1.6.0"><margin.target id="note158"></margin.target>6 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.96.1.1.7.0"><margin.target id="note159"></margin.target>8 <emph type="italics"></emph>Quinti. </s> <s id="id.2.1.96.1.1.8.0">Ex<emph.end type="italics"></emph.end> 10 <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.96.1.1.9.0"><margin.target id="note160"></margin.target>6 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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></lb>tiam in E ita eſſe, 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 eſt LC ad CA, vt potentia in A ad pondus; vt au<lb></lb>tem CA, hoc eſt CE ad CM, ita eſt pondus ad potentiam in E; <lb></lb>quare ex æquali potentia in A ad potentiam in E ita erit, vt CL <arrow.to.target n="note161"></arrow.to.target><lb></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"></emph>Quinti.<emph.end type="italics"></emph.end></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 ſolum oſtendetur, potentiam in A ad po<lb></lb>tentiam in R ita eſſe, vt CL ad CT; ſed & potentiam quoq; in E <lb></lb>ad potentiam in R ita eſſe, vt CM ad CT. </s> <s id="N13474">& ita in reliquis. <pb xlink:href="036/01/114.jpg"></pb> <figure id="id.036.01.114.1.jpg" place="text" 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 æquidiſtans, cuius fulcimen<lb></lb>tum B; & centrum grauitatis H ponderis CD ſit ſupra vectem; <lb></lb>moueaturq; vectis in BE, ponduſq; in FG. </s> <s id="id.2.1.99.2.1.1.0.a">dico minorem po<lb></lb>tentiam in E ſuſtinere pondus FG vecte EB, quàm potentia in <lb></lb>A pondus CD vecte AB. </s> <s id="id.2.1.99.2.1.1.0.b">ſit k centrum grauitatis ponderis FG, <lb></lb>& à centris grauitatum Hk ipſorum horizontibus perpendicu<lb></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 ſupra demonſtratis) <lb></lb><arrow.to.target n="note163"></arrow.to.target>BM minor eſt BL, & BE ipſi BA æqualis; minorem habebit <lb></lb><arrow.to.target n="note164"></arrow.to.target>proportionem BM ad BE, quàm BL ad BA. </s> <s id="N134A6">ſed vt BM ad <lb></lb>BE, ita potentia in E ſuſtinens pondus FG ad ipſum pondus; & <lb></lb>vt BL ad BA, ita potentia in A ad pondus CD; minorem <lb></lb>habebit proportionem potentia in E ad pondus FG, quàm poten<lb></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></lb>tia in A. </s> <s id="N134B8">ſimiliter oſtendetur, quò magis pondus eleuabitur, ſem<lb></lb>per minorem potentiam pondus ſuſtinere. </s> <s id="id.2.1.99.2.1.2.0">Sit autem vectis in <lb></lb>BO, & pondus in PQ, cuius centrum grauitatis ſit R. </s> <s id="id.2.1.99.2.1.2.0.a">dico maio<lb></lb>rem potentiam in O requiri ad ſuſtinendum pondus PQ vecte BO, <lb></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></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 eſt BL, habebit BS ad <lb></lb>BO maiorem proportionem, quàm BL ad BA; quare maior erit <lb></lb>potentia in O ſuſtinens pondus PQ, quàm potentia in A ſuſti<lb></lb>nens pondus CD. </s> <s id="id.2.1.99.2.1.2.0.d">& hoc modo oſtendetur' quò vectis BO ma<lb></lb>gis à vecte AB deorſum tendens diſtabit, ſemper maiorem ponderi <pb n="51" xlink:href="036/01/115.jpg"></pb>ſuſtinendo 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.100.1.1.2.0"><margin.target id="note163"></margin.target>8 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.100.1.1.3.0"><margin.target id="note164"></margin.target>5 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.100.1.1.4.0"><margin.target id="note165"></margin.target>10 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.100.1.1.5.0"><margin.target id="note166"></margin.target>6 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 ſupra patet pontentiam in A ad potentiam in E eſ <lb></lb>ſe, vt BL ad BM; potentiamq; in A ad potentiam in O, vt BL <lb></lb>ad BS. </s> <s id="id.2.1.101.1.1.1.0.a">atque potentiam in E ad potentiam in O, vt BM <lb></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 ſi in B alia intelligatur potentia, ita vt duæ ſint poten<lb></lb>tiæ pondus ſuſtinentes; minor erit potentia in B ſuſtinens pon<lb></lb>dus PQ vecte BO, quàm pondus CD vecte BA aduerſo au<lb></lb>tem maior requiritur potentia in B ad ſuſtinendum pondus FG ve<lb></lb>cte BE, quàm pondus CD vecte AB. </s> <s id="N1352F">ducta enim kN ipſi EB <lb></lb>perpendicularis, erit EN ipſi AL æqualis: quare EM ipſa LA <lb></lb>maior erit. </s> <s id="id.2.1.101.2.1.2.0">ergo maiorem habebit proportionem EM ad E<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, <arrow.to.target n="note167"></arrow.to.target><lb></lb>quàm LA ad A<emph type="italics"></emph>B<emph.end type="italics"></emph.end>; & LA ad A<emph type="italics"></emph>B<emph.end type="italics"></emph.end> maiorem, quàm SO ad O<emph type="italics"></emph>B<emph.end type="italics"></emph.end>; <arrow.to.target n="note168"></arrow.to.target><lb></lb>quæ ſunt 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"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.102.1.1.2.0"><margin.target id="note168"></margin.target>5 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 oſtendetur potentiam in <emph type="italics"></emph>B<emph.end type="italics"></emph.end> pondus vecte A<emph type="italics"></emph>B<emph.end type="italics"></emph.end> ſuſti<lb></lb>nentem ad potentiam in eodem puncto <emph type="italics"></emph>B<emph.end type="italics"></emph.end> vecte E<emph type="italics"></emph>B<emph.end type="italics"></emph.end> ſuſtinentem <lb></lb>eſſe, vt LA ad EM; ad potentiam autem in B pondus vecte O<emph type="italics"></emph>B<emph.end type="italics"></emph.end><lb></lb>ſuſtinentem ita eſſe, vt AL ad OS. </s> <s id="N1359A">quæ verò vectibus E<emph type="italics"></emph>B<emph.end type="italics"></emph.end> OB <lb></lb>ſuſtinent inter ſe ſe eſſe, 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 ſuperius dicta ſunt, demonſtrabimus po<lb></lb>tentiam in <emph type="italics"></emph>B<emph.end type="italics"></emph.end> ad potentiam in E eam habere proportionem, quam <arrow.to.target n="note169"></arrow.to.target><lb></lb>EM ad M<emph type="italics"></emph>B<emph.end type="italics"></emph.end>; & potentiam in <emph type="italics"></emph>B<emph.end type="italics"></emph.end> ad potentiam in A ita eſſe, vt AL ad <arrow.to.target n="note170"></arrow.to.target><lb></lb>L<emph type="italics"></emph>B<emph.end type="italics"></emph.end>, potentiamq; in <emph type="italics"></emph>B<emph.end type="italics"></emph.end> ad potentiam in O, vt OS ad S<emph type="italics"></emph>B.<emph.end type="italics"></emph.end></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"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.104.1.1.2.0"><margin.target id="note170"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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"></emph>B<emph.end type="italics"></emph.end><lb></lb>horizonti æquidiſtans, <lb></lb>cuius fulcimentum <emph type="italics"></emph>B<emph.end type="italics"></emph.end>, <lb></lb>grauitatiſq; centrum H <lb></lb>ponderis AC ſit ſupra <lb></lb>vectem: moueaturq; ve<lb></lb>ctis in <emph type="italics"></emph>B<emph.end type="italics"></emph.end>E, ac pondus <lb></lb>in EF, potentiaq; in G. <lb></lb></s> <s id="N13616">ſimiliter vt ſupra oſten<lb></lb>detur potentiam in G <lb></lb>pondus EF <expan abbr="ſuiſtinen">sustinen</expan><lb></lb><figure id="id.036.01.115.1.jpg" place="text" xlink:href="036/01/115/1.jpg"></figure><lb></lb>tem minorem eſſe potentia in D pondus AC ſuſtinente. </s> <s id="id.2.1.105.1.1.2.0">cùm <pb xlink:href="036/01/116.jpg"></pb>enim minor ſit BM ipſa <lb></lb>BL, minorem habebit <lb></lb>proportionem MB ad <lb></lb>BG, quàm LB ad BD. <lb></lb></s> <s id="N13636">atq; hoc modo oſten<lb></lb>detur, quò pondus ve<lb></lb>cte magis eleuabitur, mi<lb></lb>norem ſemper. </s> <s id="N1363E">ad pon<lb></lb>dus ſuſtinendum requi<lb></lb>ri potentiam. </s> <s id="id.2.1.105.1.1.4.0">Simili<lb></lb>ter ſi moueatur vectis <lb></lb>in BO, potentiaq; ſu<lb></lb><figure id="id.036.01.116.1.jpg" place="text" xlink:href="036/01/116/1.jpg"></figure><lb></lb>ſtinens in N, oſtendetur potentiam in N maiorem eſſe potentia in <lb></lb>D. </s> <s id="N13655">maiorem enim habet proportionem SB ad BN, quàm LB <lb></lb>ad BD. </s> <s id="N13659">oſtendetur etiam, quò magis pondus deprimetur; ma<lb></lb>iorem ſemper (vt ſuſtineatur) requiri potentiam. </s> <s id="id.2.1.105.1.1.5.0">quod demon<lb></lb>ſtrare 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 ſe ſe ita eſſe, vt <lb></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 manifeſtum eſt; ſi potentia vecte ſur<lb></lb>ſum moueat pondus, cuius centrum grauitatis <lb></lb>ſit ſupra vectem, quò magis pondus eleuabitur; <lb></lb>ſemper minorem potentiam requiri vt pondus <lb></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 ſuſtinens eſt ſemper minor, erit <lb></lb>quoq; potentia ipſum mouens ſemper minor. <pb n="52" xlink:href="036/01/117.jpg"></pb> <figure id="id.036.01.117.1.jpg" place="text" 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 demonſtrabitur, ſi centrum grauitatis eiuſdem pon<lb></lb>deris, ſiue propinquius, ſiue remotius fuerit à vecte AB horizon<lb></lb>ti æquidiſtante, eandem potentiam in A pondus nihilominus <lb></lb>ſuſtinere: vt ſi centrum grauitatis H ponderis BD longius abſit <lb></lb>à vecte BA, quàm centrum grauitatis N ponderis PV, dum<lb></lb>modo ducta à puncto H perpendicularis HL horizonti, vectiq; <lb></lb>AB tranſeat per N; ſitq; pondus PV ponderi BD æquale; <lb></lb>erit tùm pondus BD, tùm pondus PV, ac ſi ambo in L eſ<lb></lb>ſent appenſa; atque ſunt æqualia, cùm loco vnius ponderis ac<lb></lb>cipiantur, eadem igitur potentia in A ſuſtinens pondus BD, <lb></lb>pondus quoq; PV ſuſtinebit. </s> <s id="id.2.1.105.6.1.2.0">Vecte autem EF, quò centrum <lb></lb>grauitatis longius fuerit à vecte, eò facilius potentia idem pon<lb></lb>dus ſuſtinebit: vt ſi centrum grauitatis k ponderis FG longius <lb></lb>ſit à vecte EF, quàm centrum grauitatis X ponderis YZ; ita ta<lb></lb>men vt ducta à puncto k vecti FE perpendicularis tranſeat per <lb></lb>X; ſitq; pondus FG ponderi YZ æquale; & à punctis kX ip<lb></lb>ſorum horizontibus perpendiculares ducantur KM X9; erit C9 <lb></lb>maior CM; ac propterea pondus FG in vecte erit, ac ſi in M eſ <lb></lb>ſet appenſum, & pondus YZ, ac ſi in 9 eſſet appenſum. </s> <s id="id.2.1.105.6.1.3.0">quo<pb xlink:href="036/01/118.jpg"></pb> <figure id="id.036.01.118.1.jpg" place="text" xlink:href="036/01/118/1.jpg"></figure><lb></lb><arrow.to.target n="note171"></arrow.to.target>niam autem maiorem habet proportionem C9 ad CE, quàm <lb></lb>CM ad CE, maior potentia in E ſuſtinebit pondus YZ, quàm <lb></lb>FG. </s> <s id="id.2.1.105.6.1.3.0.a">In vecte autem QR è conuerſo demonſtrabitur, ſcilicet <lb></lb>quò centrum grauitatis eiuſdem ponderis ſit longius à vecte, eò <lb></lb>maiorem eſſe potentiam pondus ſuſtinentem. </s> <s id="id.2.1.105.6.1.4.0">maior enim eſt <lb></lb>CT, quàm CI; & ob id maiorem habebit proportionem CT <lb></lb>ad CR, quàm CI ad CR. </s> <s id="id.2.1.105.6.1.4.0.a">Similiter demonſtrabitur, ſi pondus <lb></lb>intra potentiam, & fulcimentum fuerit collocatum; vel poten<lb></lb>tia intra fulcimentum, & pondus. </s> <s id="id.2.1.105.6.1.5.0">Quod idem etiam potentiæ <lb></lb>eueniet mouenti. </s> <s id="id.2.1.105.6.1.6.0">vbi enim minor potentia ſuſtinet pondus, ibi <lb></lb>minor potentia mouebit; & vbi maior in ſuſtinendo, ibi maior <lb></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"></emph>Quinti.<emph.end type="italics"></emph.end></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 ſuſtinens infra vectem ho<lb></lb>rizonti æquidiſtantem ipſius centrum grauitatis <pb n="53" xlink:href="036/01/119.jpg"></pb>habens, quò magis ab hoc ſitu vecte pondus ele<lb></lb>uabitur maiori ſemper potentia, vt ſuſtineatur, <lb></lb>egebit. </s> <s id="id.2.1.107.2.1.2.0">ſi verò deprimetur, minori. <figure id="id.036.01.119.1.jpg" place="text" 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 æquidiſtans, cuius fulcimentum C; <lb></lb>ſitq; pondus AD, cuius centrum grauitatis L ſit infra vectem; <lb></lb>ſitq; potentia in B ſuſtinens pondus AD: moueatur deinde ve<lb></lb>ctis in FG, & pondus in FH. </s> <s id="id.2.1.107.3.1.1.0.a">Dico primum maiorem requiri <lb></lb>potentiam in G ad ſuſtinendum pondus FH vecte FG, quàm <lb></lb>ſit potentia in B pondere exiſtente AD vecte autem AB. </s> <s id="id.2.1.107.3.1.1.0.b">ſit M <lb></lb>grauitatis centrum ponderis FH, & à punctis LM ipſorum ho<lb></lb>rizontibus perpendiculares ducantur Lk MN: ipſi verò FG per<lb></lb>pendicularis ducatur MS, quæ æqualis erit LK, & CK ipſi CS <lb></lb>erit etiam æqualis. </s> <s id="id.2.1.107.3.1.2.0">Quoniam igitur CN maior eſt Ck, habe<lb></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></lb>tentia uerò in B ad pondus AD eandem habet, quam kC ad CB: <arrow.to.target n="note174"></arrow.to.target><lb></lb>& vt potentia in G ad pondus FH, ita eſt NC ad CG; ergo <lb></lb>maiorem habebit proportionem potentia in G ad pondus FH, <lb></lb>quàm potentia in B ad pondus AD. </s> <s id="id.2.1.107.3.1.2.0.a">maior igitur eſt potentia <arrow.to.target n="note175"></arrow.to.target><lb></lb>in G ipſa potentia in B. </s> <s id="N1375E">ſi verò vectis ſit in OP, & pondus in <lb></lb>OQ; erit potentia in B maior, quàm in P. </s> <s id="N13762">eodem enim mo<lb></lb>do oſtendetur CR minorem eſſe Ck, & CR ad CP minorem <arrow.to.target n="note176"></arrow.to.target><pb xlink:href="036/01/120.jpg"></pb><figure id="id.036.01.120.1.jpg" place="text" xlink:href="036/01/120/1.jpg"></figure><lb></lb>habere proportionem, quàm Ck ad CB; & ob id potentiam in <lb></lb>B maiorem eſſe potentia in P. </s> <s id="N13775">& hoc modo oſtendetur, quò ma<lb></lb>gis à ſitu AB pondus eleuabitur, ſemper maiorem potentiam ad <lb></lb>pondus ſuſtinendum requiri. </s> <s id="id.2.1.107.3.1.3.0">è contra verò ſi deprimetur. </s> <s id="id.2.1.107.3.1.4.0">quod <lb></lb>demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.108.1.1.2.0"><margin.target id="note173"></margin.target>8 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.108.1.1.3.0"><margin.target id="note174"></margin.target>5 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.108.1.1.4.0"><margin.target id="note175"></margin.target>10 <emph type="italics"></emph>Quinti<emph.end type="italics"></emph.end></s> <s id="id.2.1.108.1.1.5.0"><margin.target id="note176"></margin.target>7 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 poteſt potentias in PBG inter ſe ſe ita <lb></lb>eſſe, vt CR ad Ck; & vt Ck ad CN; atq; vt CN ad CR. <lb></lb><figure id="id.036.01.120.2.jpg" place="text" 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 æquidiſtans, cuius fulcimentum <lb></lb>B; ponduſq; CD habeat centrum grauitatis O infra vectem; ſitq; <lb></lb>potentia in A ſuſtinens 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"></pb>BE BF, ponduſq; transferatur in GH kL. </s> <s id="id.2.1.109.2.1.1.0.b">Dico maiorem re<lb></lb>quiri potentiam in E, vt pondus ſuſtineatur, quàm in A; & ma<lb></lb>iorem in A, quàm in F. </s> <s id="N137DF">ducantur à centris grauitatum horizon<lb></lb>tibus perpendiculares NM OP QR, quæ ex parte NOQ <lb></lb>protractæ in centrum mundi conuenient. </s> <s id="id.2.1.109.2.1.2.0">ſimiliter vt ſupra oſten<lb></lb>detur BM <expan abbr="maiorẽ">maiorem</expan> eſſe BP, & <emph type="italics"></emph>B<emph.end type="italics"></emph.end>P maiorem BR; & BM ad BE ma<lb></lb>iorem <arrow.to.target n="note177"></arrow.to.target>habere proportionem, <expan abbr="qaàm">quàm</expan> BP ad BA; & BP ad BA ma<lb></lb>iorem, quàm BR ad BF: & propter hoc potentiam in E maio<lb></lb>rem eſſe potentia in A; & potentiam in A maiorem potentia in <lb></lb>F. </s> <s id="N13804">& quò vectis magis à ſitu AB eleuabitur, ſemper oſtendetur, <lb></lb>maiorem requiri potentiam ponderi ſuſtinendo. </s> <s id="N13808">ſi verò depri<lb></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"></emph>Huius.<emph.end type="italics"></emph.end></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 ſe ſe ita eſſe, vt BM ad <lb></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">Inſuper ſi in B altera ſit potentia, ita vt duæ ſint potentiæ pondus <lb></lb>ſuſtinentes, maiore opus eſt potentia in B pondus kL ſuſtinente <lb></lb>vecte BF, quàm pondus CD vecte AB. </s> <s id="id.2.1.111.2.1.1.0.a">& adhuc maiore vecte <lb></lb>AB, quàm vecte BE. </s> <s id="id.2.1.111.2.1.1.0.b">maiorem enim habet proportionem RF <lb></lb>ad FB, quàm PA ad AB; & PA ad AB maiorem habet, quàm <lb></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; oſtendetur potentias in B pondus vectibus ſuſtinen<lb></lb>tes inter ſe ſe ita eſſe, vt EM ad AP; & ut <lb></lb>AP ad FR; atque ut <lb></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></lb>RB; & potentia in B ad potentiam in A, ut PA ad PB, & po<lb></lb>tentia <arrow.to.target n="note179"></arrow.to.target>in <emph type="italics"></emph>B<emph.end type="italics"></emph.end> ad potentiam in E, ut EM ad M<emph type="italics"></emph>B.<emph.end type="italics"></emph.end></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"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.112.1.1.2.0"><margin.target id="note179"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/122.jpg"></pb> <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></lb>AB horizonti æqui<lb></lb>diſtans, cuius fulci<lb></lb>mentum B; & pon<lb></lb>dus AC, cuius cen<lb></lb>trum grauitatis ſit in<lb></lb>fra vectem: ſitq; po<lb></lb>tentia in D pondus <lb></lb><expan abbr="ſuſtinẽs">ſuſtinens</expan>; moueaturq; <lb></lb>vectis in BE BF, & <lb></lb>potentia in GH: ſi<lb></lb>militer oſtendetur po<lb></lb><figure id="id.036.01.122.1.jpg" place="text" xlink:href="036/01/122/1.jpg"></figure><lb></lb>tentiam in G maiorem eſſe debere potentia in D; & potentiam in <lb></lb>D maiorem potentia in H. </s> <s id="id.2.1.113.1.2.1.0.a">maiorem enim proportionem habet <lb></lb>KB ad BG, quàm BL ad BD; & BL ad BD maiorem, quàm <lb></lb>MB ad BH. </s> <s id="id.2.1.113.1.2.1.0.b">& hoc modo oſtendetur, quò vectis magis à ſitu <lb></lb>AB eleuabitur, adhuc ſemper maiorem eſſe debere potentiam pon<lb></lb>dus ſuſtinentem. </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></lb>demonſtrare 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 ſe ſe ita. erunt, vt BK <lb></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, ſi potentia vecte ſurſum <lb></lb>moueat pondus, cuius centrum grauitatis ſit in<lb></lb>fra vectem; quò magis pondus eleuabitur, ſem<lb></lb>per maiorem requiri potentiam, vt pondus mo<lb></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 ſi potentia pondus ſuſtinens ſemper eſt maior: erit quoq; <lb></lb>potentia mouens ſemper maior. <pb n="55" xlink:href="036/01/123.jpg"></pb> <figure id="id.036.01.123.1.jpg" place="text" 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, ſi centrum grauitatis eiuſdem pon<lb></lb>deris, ſiue propius, ſiue remotius fuerit à vecte AB horizonti æ<lb></lb>quidiſtante; eandem potentiam in B pondus ſuſtinere. </s> <s id="id.2.1.113.6.1.2.0">vt ſi cen<lb></lb>trum grauitatis L ponderis AD ſit remotius à vecte BA, quàm <lb></lb>centrum grauitatis N ponderis PV; dummodo ducta à puncto L <lb></lb>perpendicularis LK horizonti, vectiq; AB tranſeat per N: ſimili<lb></lb>ter vt in præcedenti oſtendetur, eandem potentiam in B, & pondus <lb></lb>AD, & pondus PV ſuſtinere. </s> <s id="id.2.1.113.6.1.3.0">In vecte auté EF, quò <expan abbr="centrũ">centrum</expan> grauitatis <lb></lb>longius aberit à vecte, eò maiori opus erit potentia ponderi ſuſti<lb></lb>nendo. </s> <s id="id.2.1.113.6.1.4.0">vt centrum grauitatis M ponderis FH remotius ſit à ue<lb></lb>cte EF, quàm S centrum grauitatis ponderis XZ; ducantur à pun<lb></lb>ctis MS horizontibus perpendiculares MI SG; erit CI maior <lb></lb>CG: ac propterea maior eſſe debet potentia in E pondus FH ſu<lb></lb>ſtinens, quàm pondus XZ. </s> <s id="id.2.1.113.6.1.4.0.a">Contra uerò in uecte OR oſtende<lb></lb>tur, quò ſcilicet centrum grauitatis eiuſdem ponderis longius ab <lb></lb>ſit à uecte, à minori potentia pondus ſuſtineri. </s> <s id="id.2.1.113.6.1.5.0">minor enim eſt <lb></lb>CY, quàm CT. </s> <s id="id.2.1.113.6.1.5.0.a">Simili quoq; modo demonſtrabitur, ſi pondus <lb></lb>ſit intra potentiam, & fulcimentum; uel potentia intra fulci<lb></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"></pb>vbi enim minor potentia ſuſtinet pondus, ibi minor potentia mo<lb></lb>uebit. </s> <s id="id.2.1.113.6.1.7.0">& vbi maior potentia in ſuſtinendo; ibi quoq; maior in mo<lb></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 ſuſtinens in ipſo vecte cen<lb></lb>trum grauitatis habens, quomodocunq; vecte <lb></lb>transferatur pondus; eadem ſemper, vt ſuſtinea<lb></lb>tur, potentia opus erit. <figure id="id.036.01.124.1.jpg" place="text" 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 æquidiſtàns, cuius fulcimentum C. <lb></lb></s> <s id="N1394E">E verò centrum grauitatis ponderis in ipſo ſit vecte. </s> <s id="id.2.1.113.9.1.2.0">Moueatur <lb></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></lb>dem potentiam in kBG idemmet ſemper ſuſtinere pondus. </s> <s id="id.2.1.113.9.1.3.0"><lb></lb>Quoniam enim pondus in uecte AB perinde ſe habet, ac ſi eſſet <lb></lb><arrow.to.target n="note180"></arrow.to.target>appenſum in E; & in uecte GF, ac ſi eſſet appenſum in L; & in <lb></lb>uecte Hk. </s> <s id="id.2.1.113.9.1.4.0">ac ſi in M eſſet appenſum; diſtantiæ uerò CL CE <lb></lb>CM ſunt inter ſe ſe æquales; nec non CK CB CG inter ſe æ<lb></lb>quales; erit potentia in B ad pondus, ut CE ad CB; atque poten<pb n="56" xlink:href="036/01/125.jpg"></pb>tia in k ad pondus, ut CM ad Ck; & potentia in G ad pondus, <lb></lb>vt CL ad CG. </s> <s id="id.2.1.113.9.1.4.0.a">eadem igitur potentia in k<emph type="italics"></emph>B<emph.end type="italics"></emph.end>G idem translatum <lb></lb>pondus ſuſtinebit. </s> <s id="id.2.1.113.9.1.5.0">quod demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></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 oſtendetur, ſi pondus eſſet intra potentiam, & fulci<lb></lb>mentum; vel potentia inter fulcimentum, & pondus. </s> <s id="id.2.1.115.1.1.2.0">quod idem <lb></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 diſtantia inter fulcimentum, & poten<lb></lb>tiam ad diſtantiam fulcimento, punctoq;, vbi <lb></lb>à centro grauitatis ponderis horizonti ducta <lb></lb>perpendicularis vectem ſecat, interiectam ma<lb></lb>iorem habuerit proportionem, quàm pondus <lb></lb>ad potentiam; pondus vtiq; à potentia moue<lb></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></lb>punctoq; A ſuſpenda<lb></lb>tur pondus C; hoc eſt <lb></lb>punctum A ſemper ſit <lb></lb>punctum, vbi perpen<lb></lb>dicularis à grauitatis <lb></lb>centro ponderis du<lb></lb>cta vectem ſecat; ſitq; <lb></lb><figure id="id.036.01.125.1.jpg" place="text" xlink:href="036/01/125/1.jpg"></figure><lb></lb>potentia in B, ac fulcimentum ſit D; & DB ad DA maiorem <lb></lb>habeat proportionem, quàm pondus C ad potentiam in B. </s> <s id="id.2.1.115.4.1.1.0.a">Di<lb></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></lb>pondus E ad potentiam in B; atq; pondus E quoq; appendatur <lb></lb>in A: patet potentiam in B æqueponderare ipſi E; hoc eſt pon<lb></lb>dus <arrow.to.target n="note181"></arrow.to.target>E ſuſtinere. </s> <s id="id.2.1.115.4.1.3.0">& quoniam BD ad DA maiorem habet pro<lb></lb>portionem, quàm C ad potentiam in B; & vt BD ad DA, ita <pb xlink:href="036/01/126.jpg"></pb>eſt pondus E ad po<lb></lb>tentiam: igitur E ad <lb></lb>potentiam maiorem <lb></lb>habebit proportio<lb></lb>nem, quàm pondus <lb></lb>C ad eandem poten<lb></lb><arrow.to.target n="note182"></arrow.to.target>tiam. </s> <s id="id.2.1.115.4.1.4.0">quare pondus <lb></lb>E maius erit ponde<lb></lb><figure id="id.036.01.126.1.jpg" place="text" xlink:href="036/01/126/1.jpg"></figure><lb></lb>re C. </s> <s id="N13A05">& cùm potentia ipſa E æqueponderet, potentia igitur ipſi <lb></lb>C non æqueponderabit, ſed ſua ui deorſum verget. </s> <s id="id.2.1.115.4.1.5.0">pondus igitur <lb></lb>C à potentia in B mouebitur vecte AB, cuius fulcimentum <lb></lb>eſt 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.116.1.1.2.0"><margin.target id="note182"></margin.target>10 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></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 ſit vectis AB, & <lb></lb>fulcimentum A, ponduſq; C <lb></lb>in D appenſum, & potentia <lb></lb>in B; & BA ad AD maio<lb></lb>rem habeat proportionem, <lb></lb>quàm pondus C ad poten<lb></lb>tiam in B. </s> <s id="id.2.1.117.1.1.1.0.a">dico pondus C à <lb></lb><figure id="id.036.01.126.2.jpg" place="text" xlink:href="036/01/126/2.jpg"></figure><lb></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></lb><arrow.to.target n="note183"></arrow.to.target>tiam in B: & ſi E appendatur in D, potentia in B pondus E ſuſti<lb></lb>nebit. </s> <s id="id.2.1.117.1.1.3.0">ſed cùm BA ad AD maiorem habeat proportionem, <lb></lb>quàm pondus C ad potentiam in B; & vt BA ad AD, ita eſt <lb></lb>pondus E ad potentiam in B: pondus igitur E ad potentiam, <lb></lb>quæ eſt in B, maiorem habebit proportionem, quàm pondus C <lb></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></lb></s> <s id="N13A60">potentia verò in B ſuſtinet pondus E; ergo potentia in B pondus <lb></lb>C minus pondere E in D appenſum mouebit vecte AB, cuius fulci <lb></lb>mentum eſt 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.118.1.1.2.0"><margin.target id="note184"></margin.target>10 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> </p> <pb n="57" xlink:href="036/01/127.jpg"></pb> <p id="id.2.1.119.1.0.0.0" type="main"> <s id="id.2.1.119.1.2.1.0">Sit rurſus vectis <lb></lb>AB, cuius fulcimen <lb></lb><expan abbr="tũ">tum</expan> A; & pondus C in <lb></lb>B ſit appenſum; ſitq; <lb></lb>potentia in D: & <lb></lb>DA ad AB maio<lb></lb>rem habeat propor<lb></lb>tionem, quàm pon<lb></lb><figure id="id.036.01.127.1.jpg" place="text" xlink:href="036/01/127/1.jpg"></figure><lb></lb>dus C ad potentiam, quæ eſt in D. </s> <s id="id.2.1.119.1.2.1.0.a">dico pondus C à potentia <lb></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></lb>D; & ſit pondus E ex puncto B ſuſpenſum: potentia in D pondus <lb></lb>E ſuſtinebit. </s> <s id="id.2.1.119.1.2.3.0">ſed DA ad AB maiorem habet proportionem, <lb></lb>quàm C ad potentiam in D; & vt DA ad AB, ita eſt pondus E <lb></lb>ad potentiam in D; pondus igitur E ad potentiam, quæ eſt in D, <lb></lb>maiorem habebit proportionem, quàm pondus C ad eandem po<lb></lb>tentiam. </s> <s id="id.2.1.119.1.2.4.0">quare pondus E maius eſt pondere C. </s> <s id="N13ABA">& cùm poten<lb></lb>tia in D pondus E ſuſtineat, potentia igitur in D pondus C in B <lb></lb>appenſum vecte AB, cuius fulcimentum eſt A, mouebit. </s> <s id="id.2.1.119.1.2.5.0">quod <lb></lb>demonſtrare 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></lb>pondus C in A ap<lb></lb>penſum & poten<lb></lb>tia in B; ſit〈qué〉 fulci<lb></lb>mentum D: & DB <lb></lb><figure id="id.036.01.127.2.jpg" place="text" xlink:href="036/01/127/2.jpg"></figure><lb></lb>ad DA maiorem habeat proportionem, quàm pondus C ad po<lb></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></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></lb>tet enim BE ad EA minorem habere proportionem, quàm DB <lb></lb>ad DA, & ideo BE minor erit BD. </s> <s id="id.2.1.119.3.1.2.0.b">& quoniam potentia in B ſu<arrow.to.target n="note185"></arrow.to.target><lb></lb>ſtinet pondus C in A appenſum uecte AB, cuius <expan abbr="fulcimentũ">fulcimentum</expan> E; minor <lb></lb>igitur potentia in B, quàm data, idem pondus ſuſtinebit fulcimen<lb></lb>to D. </s> <s id="N13B02">data ergo potentia in B pondus C mouebit uecte AB, cuius <lb></lb>fulcimentum eſt D. <pb xlink:href="036/01/128.jpg"></pb></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"></emph>Huius.<emph.end type="italics"></emph.end></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></lb>mentum A, & pondus C in D <lb></lb>appenſum, ſitq; potentia in B; & <lb></lb>AB ad AD maiorem habeat pro<lb></lb>portionem, quàm pondus C ad <lb></lb>potentiam in B. </s> <s id="id.2.1.121.1.1.1.0.a">dico pondus C <lb></lb><figure id="id.036.01.128.1.jpg" place="text" xlink:href="036/01/128/1.jpg"></figure><lb></lb>à 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></lb><arrow.to.target n="note186"></arrow.to.target>tiam; erit ſimiliter punctum E inter BD. </s> <s id="N13B39">neceſſe eſt enim AE <lb></lb>maiorem eſſe AD. </s> <s id="N13B3D">& ſi pondus C eſſet in E appenſum, potentia <lb></lb><arrow.to.target n="note187"></arrow.to.target>in B illud ſuſtineret. </s> <s id="id.2.1.121.1.1.3.0">minor autem potentia in B, quàm data, ſuſti<lb></lb><arrow.to.target n="note188"></arrow.to.target>net pondus C in D appenſum; data ergo potentia in B pondus C in <lb></lb><arrow.to.target n="note189"></arrow.to.target>D appenſum vecte AB, cuius fulcimentum eſt 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"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.122.1.1.2.0"><margin.target id="note187"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.122.1.1.3.0"><margin.target id="note188"></margin.target>1 <emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.122.1.1.4.0"><margin.target id="note189"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 rurſus vectis AB, cu<lb></lb>ius fulcimentum A, & pon<lb></lb>dus C in B ſit appenſum; <lb></lb>ſitq; potentia in D; & DA <lb></lb>ad AB maiorem habeat <lb></lb><figure id="id.036.01.128.2.jpg" place="text" xlink:href="036/01/128/2.jpg"></figure><lb></lb>proportionem, quàm pondus C ad potentiam in D. </s> <s id="id.2.1.123.1.1.1.0.a">dico pon<lb></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></lb><arrow.to.target n="note190"></arrow.to.target>ita DA ad AE; erit AE maior AB; cùm maior ſit proportio <lb></lb>DA ad AB, quàm DA ad AE. </s> <s id="N13BA1">& ſi pondus C appendatur in <lb></lb><arrow.to.target n="note191"></arrow.to.target>E, patet potentiam in D ſuſtinere pondus C in E appenſum. </s> <s id="id.2.1.123.1.1.3.0">mi<lb></lb><arrow.to.target n="note192"></arrow.to.target>nor autem potentia, quàm data, ſuſtinet idem pondus C in B; <lb></lb><arrow.to.target n="note193"></arrow.to.target>data igitur potentia in D pondus C in B appenſum mouebit ve<lb></lb>cte AB, cuius fulcimentum eſt A. </s> <s id="id.2.1.123.1.1.3.0.a">quod oportebat demon<lb></lb>ſtrare. </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"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.124.1.1.2.0"><margin.target id="note191"></margin.target>3 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.124.1.1.3.0"><margin.target id="note192"></margin.target>1 <emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.124.1.1.4.0"><margin.target id="note193"></margin.target>3 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 id="N13BEA" 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></lb>moueri. <pb n="58" xlink:href="036/01/129.jpg"></pb> <figure id="id.036.01.129.1.jpg" place="text" 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 ſit vt decem; <lb></lb>ſitq; datus vectis BC. </s> <s id="id.2.1.125.3.1.1.0.a">oportet potentiam, quæ eſt decem pondus <lb></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></lb>ad DB eandem habeat proportionem, quàm habet centum ad <lb></lb>decem, hoc eſt decem ad vnum; etenim ſi D fieret fulcimentum, <lb></lb>conſtat potentiam vt decem in C æqueponderare ponderi A in B <arrow.to.target n="note194"></arrow.to.target><lb></lb>appenſo: hoc eſt pondus A ſuſtinere. </s> <s id="id.2.1.125.3.1.3.0">accipiatur inter BD quod <lb></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></lb>eſt proportio CE ad EB, quàm CD ad DB; maiorem habebit <lb></lb>proportionem CE ad EB, quàm pondus A ad potentiam decem <lb></lb>in C: potentia igitur decem in C pondus A centum in B appen<lb></lb>ſum vecte BC, cuius fulcimentum ſit 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 ſit vectis <lb></lb>BC, & fulcimen<lb></lb>tum B. </s> <s id="N13C3B">diuidatur CB <lb></lb>in D, ita vt CB ad <lb></lb>BD eandem habeat <lb></lb>proportionem, <expan abbr="quã">quam</expan><lb></lb><figure id="id.036.01.129.2.jpg" place="text" xlink:href="036/01/129/2.jpg"></figure><lb></lb> habet centum ad decem: & ſi pondus A in D ſuſpendatur, & po<lb></lb>tentia in C, potentia vt decem in C pondus A in D appenſum ſu<arrow.to.target n="note197"></arrow.to.target><lb></lb>ſtinebit. </s> <s id="id.2.1.125.4.1.2.0">accipiatur inter DB quoduis punctum E, ponaturq; pon<lb></lb>dus A in E; & cùm ſit maior proportio CB ad BE, quàm <arrow.to.target n="note198"></arrow.to.target><lb></lb>BC ad BD; maiorem habebit proportionem CB ad BE, quàm <lb></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></lb>in C pondus A centum in E appenſum mouebit vecte BC, cu<lb></lb>ius fulcimentum eſt B. </s> <s id="N13C6B">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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.126.1.1.2.0"><margin.target id="note195"></margin.target><emph type="italics"></emph>Lemma huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.126.1.1.3.0"><margin.target id="note196"></margin.target>11 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.126.1.1.4.0"><margin.target id="note197"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.126.1.1.5.0"><margin.target id="note198"></margin.target>8 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.126.1.1.6.0"><margin.target id="note199"></margin.target>11 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/130.jpg"></pb> <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></lb>teſt exiſtente vecte BC, cuius <lb></lb>fulcimentum ſit B, & pondus <lb></lb>A centum in C appenſum: po<lb></lb>natur enim potentia ſuſtinens <lb></lb>pondus A vtcunq; inter BC, <lb></lb><arrow.to.target n="note200"></arrow.to.target>vt in D, ſemper potentia ma<lb></lb><arrow.to.target n="note201"></arrow.to.target>ior erit pondere A. </s> <s id="N13CC8">quare opor<lb></lb><figure id="id.036.01.130.1.jpg" place="text" xlink:href="036/01/130/1.jpg"></figure><lb></lb>tet datam potentiam maiorem eſſe pondere A. </s> <s id="N13CD2">ſit igitur poten<lb></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></lb>ad BD ſit, vt centum quinquaginta ad centum; hoc eſt tria ad duo: <lb></lb><arrow.to.target n="note202"></arrow.to.target>& ſi ponatur potentia in D, patet potentiam in D ſuſtinere pon<lb></lb>dus A in C <expan abbr="appepſum">appensum</expan>. </s> <s id="id.2.1.127.1.2.3.0">accipiatur itaq; inter DC quoduis pun<lb></lb><arrow.to.target n="note203"></arrow.to.target>ctum E, ponaturq; potentia mouens in E; & cùm maior ſit pro<lb></lb>portio EB ad BC, quàm DB ad BC; habebit EB ad BC maio<lb></lb>rem proportionem, quàm pondus A ad potentiam in E. </s> <s id="id.2.1.127.1.2.3.0.a">poten<lb></lb><arrow.to.target n="note204"></arrow.to.target>tia igitur vt centum quinquaginta in E pondus A centum in C <lb></lb>appenſum vecte BC, cuius fulcimentum eſt B, mouebit. </s> <s id="id.2.1.127.1.2.4.0">quod <lb></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"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.128.1.1.2.0"><margin.target id="note201"></margin.target>3 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.128.1.1.3.0"><margin.target id="note202"></margin.target>3 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.128.1.1.4.0"><margin.target id="note203"></margin.target>8 <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.128.1.1.5.0"><margin.target id="note204"></margin.target>11 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 manifeſtum eſt ſi data potentia ſit dato <lb></lb>pondere maior; hoc fieri poſſe, ſiue ita exiſten<lb></lb>te vecte, vt eius fulcimentum ſit inter pondus, <lb></lb>& potentiam; ſiue pondus inter fulcimentum, <lb></lb>& potentiam habente; ſiue demum potentia in<lb></lb>ter pondus, & fulcimentum conſtituta. </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></lb>dato pondere fuerit; palam quoq; eſt id ipſum <lb></lb>dumtaxat aſſe qui poſſe vecte ita exiſtente, vt eius <lb></lb>fulcimentum ſit inter pondus, & pontentiam; <pb n="59" xlink:href="036/01/131.jpg"></pb>vel pondus intra fulcimentum, & potentiam <lb></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 id="N13D65" 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 vbicun<lb></lb>què appenſis, cuius fulcimentum ſit quoq; da<lb></lb>tum, potentiam inuenire, quæ in dato puncto <lb></lb>data pondera ſuſtineat. <figure id="id.036.01.131.1.jpg" place="text" 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></lb>vbicunq; in punctis DGH appenſa: collocandaq; ſit potentia in <lb></lb>puncto E. </s> <s id="N13D85">potentiam inuenire oportet, quæ in E data pondera <lb></lb>ABC vecte DE ſuſtineat. </s> <s id="id.2.1.129.6.1.2.0">diuidatur DG in k, ita vt Dk ad KG <lb></lb>ſit, vt pondus B ad pondus A; deinde diuidatur kH in L, ita vt kL <lb></lb>ad LH, ſit vt pondus C ad pondera BA; atq; vt FE ad FL, ita <lb></lb>fiant pondera ABC ſimul ad potentiam, quæ ponatur in E. </s> <s id="id.2.1.129.6.1.2.0.a">di<lb></lb>co potentiam in E data pondera ABC in DGH appenſa vecte <lb></lb>DE, cuius fulcimentum eſt F, ſuſtinere. </s> <s id="id.2.1.129.6.1.3.0">Quoniam enim ſi ponde<lb></lb>ra ABC ſimul eſſent in L appenſa, potentia in E data pondera <arrow.to.target n="note205"></arrow.to.target><lb></lb>in L appenſa ſuſtineret; pondera verò ABC tàm in L ponderant, <arrow.to.target n="note206"></arrow.to.target> <expan abbr="quàm"><lb></lb>quam</expan> ſi C in H, & BA ſimul in K eſſent appenſa; & AB in k tàm <pb xlink:href="036/01/132.jpg"></pb> <figure id="id.036.01.132.1.jpg" place="text" xlink:href="036/01/132/1.jpg"></figure><lb></lb>ponderant, quàm ſi A in D, & B in G appenſa eſſent; ergo po<lb></lb>tentia in E data pondera ABC in DGH appenſa vecte DE, cu<lb></lb>ius fulcimentum eſt F, ſuſtinebit. </s> <s id="id.2.1.129.6.1.4.0">Si autem potentia in quouis <lb></lb>alio puncto vectis DE (præterquàm in F) conſtituenda eſſet, <lb></lb>vt in k; fiat vt Fk ad FL, ita pondera ABC ad potentiam: ſi<lb></lb><arrow.to.target n="note207"></arrow.to.target>militer demonſtrabimus potentiam in k pondera ABC in pun<lb></lb>ctis DGH appenſa ſuſtinere. </s> <s id="id.2.1.129.6.1.5.0">quod facere oportebat. <figure id="id.036.01.132.2.jpg" place="text" 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, ſi pondera ABC ſint in vecte <lb></lb>DE quomodocunq; poſita; oporteatq; potentiam inuenire, quæ <lb></lb>in E data pondera ſuſtinere debeat: ducantur à centris grauita<lb></lb>tum ponderum ABC horizontibus perpendiculares, quæ ve<lb></lb>ctem DE in DGH punctis ſecent; cæteraq; eodem modo fiant: <lb></lb>Manifeſtum eſt, potentiam in E, vel in K data pondera ſuſtinere. </s> <s id="id.2.1.129.7.1.2.0"><lb></lb>idem enim eſt, ac ſi pondera in DGH eſſent appenſa. </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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.130.1.1.2.0"><margin.target id="note206"></margin.target>5 <emph type="italics"></emph>Huius. de libra.<emph.end type="italics"></emph.end></s> <s id="id.2.1.130.1.1.4.0"><margin.target id="note207"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="60" xlink:href="036/01/133.jpg"></pb> <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 id="N13E0E" 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></lb>cunq; & quomodocunq; poſita à data potentia <lb></lb>moueri. <figure id="id.036.01.133.1.jpg" place="text" 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, & ſint data pondera vt in præcedenti co<lb></lb>rollario; ſitq; A vt centum, B vt quinquaginta, C vt triginta; <lb></lb>dataq; potentia ſit vt triginta. </s> <s id="id.2.1.131.3.1.2.0">exponantur eadem, inueniaturq; <lb></lb>punctum L; deinde diuidatur LE in F, ita vt FE ad FL ſit, vt <lb></lb>centum octoginta ad triginta, hoc eſt ſex ad vnum: & ſi F fieret <lb></lb>fulcimentum, potentia vt triginta in E ſuſtineret pondera ABC. </s> <s id="id.2.1.131.3.1.2.0.a"><arrow.to.target n="note208"></arrow.to.target><lb></lb>accipiatur igitur inter LF quoduis punctum M, fiatq; M fulci<lb></lb>mentum: manifeſtum eſt potentiam in E vt triginta pondera <arrow.to.target n="note209"></arrow.to.target><lb></lb>ABC vt centum octoginta vecte DE mouere. </s> <s id="id.2.1.131.3.1.3.0">quod facere <lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.132.1.1.2.0"><margin.target id="note209"></margin.target>11 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 aſſequi minimè poterimus, ſi in extremita<lb></lb>te vectis fulcimentum eſſet, vt in D; quia proportio DE, ad DL <lb></lb>hoc eſt proportio ponderum ABC ad potentiam, quæ pondera <lb></lb>ſuſtinere debeat, ſemper eſt data. </s> <s id="id.2.1.133.1.1.2.0">quod multo quoq; minus fieri <lb></lb>poſſet, ſi ponenda eſſet potentia inter DL. </s> </p> <pb xlink:href="036/01/134.jpg"></pb> <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 id="N13E79" 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></lb>vectis quoq; grauitatem habet, cuius nulla ha<lb></lb>ctenus mentio facta eſt: idcirco primùm quo<lb></lb>modo inueniatur potentia, quæ in dato puncto <lb></lb>datum vectem, cuius fulcimentum ſit quoq; da<lb></lb>tum, ſuſtineat, oſtendamus. <figure id="id.036.01.134.1.jpg" place="text" 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 ſit datum C; ſitq; <lb></lb>punctum D, in quo collocanda ſit potentia, quæ vectem AB ſu<lb></lb>ſtinere debeat, ita vt immobilis perſiſtat. </s> <s id="id.2.1.133.5.1.2.0">ducatur à puncto C <lb></lb>linea CE horizonti perpendicularis, quæ vectem AB in duas di<lb></lb>uidat partes AE EF, ſitq; partis AE centrum grauitatis G, & <lb></lb>partis EF centrum grauitatis H; à punctis〈qué〉 GH horizon<lb></lb>tibus perpendiculares ducantur Gk HL, quæ lineam AF <lb></lb>in punctis KL ſecent. </s> <s id="id.2.1.133.5.1.3.0">quoniam enim vectis AB à linea CE in duas <lb></lb>diuiditur partes AE EF; ideo vectis AB nihil aliud erit, niſi <lb></lb>duo pondera AE EF in vecte, ſiue libra AF poſita; cuius ſu<lb></lb>ſpenſio, ſiue fulcimentum eſt C. quare pondera AE EF ita erunt <lb></lb>poſita, ac ſi in kL eſſent appenſa. </s> <s id="id.2.1.133.5.1.4.0">diuidatur ergo kL in M, <lb></lb>ita vt kM ad ML, ſit vt grauitas partis EF ad grauitatem par<lb></lb>tis AE; & vt CA ad CM, ita fiat grauitas totius vectis AB ad <lb></lb>potentiam, quæ ſi collocetur in D (dummodo DA horizonti <pb n="61" xlink:href="036/01/135.jpg"></pb>perpendicularis exiſtat) vecti æqueponderabit; hoc eſt vectem <arrow.to.target n="note210"></arrow.to.target><lb></lb>AB deorſum premendo ſuſtinebit. </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"></emph>Huius.<emph.end type="italics"></emph.end></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 eſſet. </s> <s id="id.2.1.135.1.1.2.0">fiat vt CF ad CM <lb></lb>ita pondus AB ad potentiam. </s> <s id="id.2.1.135.1.1.3.0">ſimili modo oſtendetur poten<lb></lb>tiam in B vectem AB ſuſtinere. </s> <s id="id.2.1.135.1.1.4.0">ſimiliterq; demonſtrabitur in quo<lb></lb>cunq; alio ſitu (præterquàm in e) ponenda fuerit potentia, vt in <lb></lb>N. </s> <s id="N13EEF">fiat enim vt CO ad CM, ita AB ad potentiam; quæ ſi pona<lb></lb>tur in N, vectem AB ſuſtinebit. </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 appenſum, <lb></lb>ſiue poſitum; vt iisdem poſitis ſit pondus P in <lb></lb>A appenſum; potentiaq; ſit ponenda in B, ita <lb></lb>vt vectem AB vnà cum pondere P ſuſtineat. <figure id="id.036.01.135.1.jpg" place="text" 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 ſit, ut grauitas ue<lb></lb>ctis AB ad grauitatem ponderis P; deinde ut CF ad CQ, ita fat <lb></lb>grauitas AB, & P ſimul ad potentiam, quæ ponatur in B: patet <lb></lb>potentiam in B uectem AB unà cum pondere P ſuſtinere. </s> <s id="id.2.1.135.3.1.2.0">Si ue<arrow.to.target n="note211"></arrow.to.target><lb></lb>rò eſſet CA ad CM, vt AB ad P; eſſet punctum C eorum centrum <arrow.to.target n="note212"></arrow.to.target><lb></lb>grauitatis, & ideo vectis AB vná cum pondere P abſq; potentia in <arrow.to.target n="note213"></arrow.to.target><lb></lb>B manebit. </s> <s id="id.2.1.135.3.1.3.0">ſed ſi ponderum grauitatis centrum eſſet inter CF, vt <lb></lb>in O; fiat vt CF ad CO, ita AB&P ſimul ad potentiam, quæ <lb></lb>in B, & vectem AB, & pondus P ſuſtinebit. <pb xlink:href="036/01/136.jpg"></pb> <figure id="id.036.01.136.1.jpg" place="text" 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 oſtendetur, ſi plura eſſent pondera in vecte AB ubi<lb></lb>cunq;, & quomodocunq; poſita. </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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.136.1.1.2.0"><margin.target id="note212"></margin.target><emph type="italics"></emph>Ex ſexta<emph.end type="italics"></emph.end></s> <s id="id.2.1.136.1.1.3.0"><margin.target id="note213"></margin.target>1 <emph type="italics"></emph>Arch. de æquep.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.137.1.0.0.0" type="main"> <s id="id.2.1.137.1.1.1.0">Inſuper ex his non ſolum, ut in decimaquarta huius docuimus, <lb></lb>quomodo ſcilicet data pondera ubicunq; in uecte poſita data poten<lb></lb>tia dato uecte mouere poſſumus, eodem modo grauitate uectis <lb></lb>conſiderata idem facere poterimus; uerùm etiam accidentia reli<lb></lb>qua, quæ ſupra abſq; uectis grauitatis conſideratione demonſtra<lb></lb>ta ſunt; ſimili modo uectis grauitate conſiderata vná cum ponde<lb></lb>ribus, uel ſine ponderibus oſtendentur. </s> </p> </chap> <pb n="62" xlink:href="036/01/137.jpg"></pb> <chap id="N13F6F"> <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 inſtrumento pon<lb></lb>dus multipliciter moueri poteſt; <lb></lb>quia verò in omnibus eſt eadem <lb></lb>ratio: ideo (vt res euidentior ap<lb></lb>pareat) in iis, quæ dicenda ſunt, <lb></lb>intelligatur pondus ſurſum ad re<lb></lb>ctos horizontis plano angulos hoc modo ſem<lb></lb>per moueri. </s> </p> <pb xlink:href="036/01/138.jpg"></pb> <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 ipſi ho<lb></lb>rizontis plano ſurſum ad rectos <lb></lb>angulos ſit attollendum; & vt <lb></lb>fieri ſolet, trochlea duos habens <lb></lb>orbiculos, quorum axiculi ſint <lb></lb>in BC, ſupernè appendatur; <lb></lb>trochlea verò duos ſimiliter ha<lb></lb>bens orbiculos, quorum axicu<lb></lb>li ſint in DE, ponderi alligetur: <lb></lb>ac per omnes vtriuſq; trochleæ <lb></lb>orbiculos circunducatur ducta<lb></lb>rius funis, quem in altero eius ex <lb></lb>tremo, putá in F, oportet eſſe <lb></lb>religatum. </s> <s id="id.2.1.137.6.1.2.0">potentia autem mo<lb></lb>uens ponatur in G, quæ dum <lb></lb>deſcendit, pondus A ſurſum ex <lb></lb>aduerſo attolletur; quemadmo<lb></lb>dum Pappus in octauo libro Ma<lb></lb>thematicarum collectionum aſ<lb></lb>ſerit; nec non Vitruuius in deci <lb></lb>mo de Architectura, & alii. <figure id="id.036.01.138.1.jpg" place="text" 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 inſtrumen<lb></lb>tum reducatur ad vectem; cur magnum pondus <lb></lb>ab exigua virtute, & quomodo, quantoq; in tem<lb></lb>pore moueatur; cur funis in vno capite debeat <lb></lb>eſſe religatus; quodq; ſuperioris, inferiorisq́ue <lb></lb>trochleæ fuerit officium; & quomodo omnis in <pb n="63" xlink:href="036/01/139.jpg"></pb>numeris data proportio inter potentiam, & pon<lb></lb>dus inueniri poſsit; 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></lb>punctis AC circulum ACE contingant, cuius <lb></lb>centrum F: & FA FC connectantur. </s> <s id="id.2.1.137.9.1.2.0">Dico <lb></lb>AFC rectam lineam eſſe. </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 ipſis AB CD æquidiſtans. </s> <s id="id.2.1.137.10.1.2.0"><lb></lb>& quoniam AB, & FE ſunt parallelæ, & <lb></lb>angulus BAF eſt rectus; erit & AFE re<lb></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></lb>nea igitur <arrow.to.target n="note214"></arrow.to.target>AFC recta eſt. </s> <s id="id.2.1.137.10.1.5.0">quod erat de<lb></lb>monſtrandum. <arrow.to.target n="note215"></arrow.to.target><arrow.to.target n="note216"></arrow.to.target><lb></lb></s> </p> <figure id="id.036.01.139.1.jpg" place="text" 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"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.138.1.1.2.0"><margin.target id="note215"></margin.target>29 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.138.1.1.3.0"><margin.target id="note216"></margin.target>14 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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 ſupernè appenſæ orbiculo <lb></lb>circunducatur, alterumq; eius extremum pon<lb></lb>deri alligetur, altero interim à potentia pondus <lb></lb>ſuſtinente apprehenſo: erit potentia ponderi <lb></lb>æqualis. </s> </p> <pb xlink:href="036/01/140.jpg"></pb> <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></lb>cui alligatus ſit fu<lb></lb>nis in B; trochleaq; <lb></lb>habens orbiculum C <lb></lb>EF, cuius centrum <lb></lb>D, ſurſum appenda<lb></lb>tur; ſitq; D quoq; <lb></lb>centrum axiculi; & <lb></lb>circa orbiculum uo<lb></lb>luatur funis BC EF <lb></lb>G; ſitq; potentia <lb></lb>in G ſuſtinens pon<lb></lb>dus A. </s> <s id="id.2.1.139.4.1.1.0.a">dico poten<lb></lb>tiam in G ponderi A <lb></lb>æqualem eſſe. </s> <s id="id.2.1.139.4.1.2.0">Sit FG <lb></lb>æquidiſtans CB. </s> <s id="id.2.1.139.4.1.2.0.a"><lb></lb>Quoniam igitur pon<lb></lb><arrow.to.target n="note217"></arrow.to.target>dus A manet; erit <lb></lb><figure id="id.036.01.140.1.jpg" place="text" xlink:href="036/01/140/1.jpg"></figure><lb></lb>CB horizonti plano perpendicularis: quare FG eidem plano per<lb></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></lb>in horizontis <expan abbr="planũ">planum</expan> ad rectos angulos deſcendunt; tangent BC FG <lb></lb><expan abbr="orbiculũ">orbiculum</expan> CEF in punctis CF. </s> <s id="N140A0"><expan abbr="orbiculũ">orbiculum</expan> enim ſecare <expan abbr="nõ">non</expan> poſſunt. </s> <s id="id.2.1.139.4.1.4.0">con<lb></lb>nectantur DC DF; erit CF recta linea, & anguli DCB DFG recti. </s> <s id="id.2.1.139.4.1.5.0"><lb></lb><arrow.to.target n="note219"></arrow.to.target> <expan abbr="Quoniã">Quoniam</expan> <expan abbr="autẽ">autem</expan> BC tùm horizonti, tùm ipſi CF eſt perpendicularis; <lb></lb>erit linea CF horizonti æquidiſtans. </s> <s id="id.2.1.139.4.1.6.0">cùm verò <expan abbr="põdus">pondus</expan> appenſum ſit <lb></lb><arrow.to.target n="note220"></arrow.to.target>in BC, & potentia ſit in G; quod idem eſt, ac ſi eſſet in F; erit <lb></lb>CF tanquam libra, ſiue vectis, cuius centrum, ſiue fulcimentum eſt <lb></lb>D; nam in axiculo <expan abbr="orbuculus">orbiculus</expan> ſuſtinetur; atq; punctum D, cùm ſit <lb></lb>centrum axiculi, & orbiculi, etiam vtriſque circumuolutis <lb></lb>immobile remanet. </s> <s id="id.2.1.139.4.1.7.0">Itaq; cùm diſtantia DC ſit æqualis diſtantiæ <lb></lb>DF, potentiaq; in F ponderi A in C appenſo æqueponderet, cùm <lb></lb><arrow.to.target n="note221"></arrow.to.target>pondus ſuſtineat, ne deorſum vergat; erit potentia in F, ſiue in G <lb></lb>(nam idem eſt) conſtituta ponderi A æqualis. </s> <s id="id.2.1.139.4.1.8.0">Idem enim effi<lb></lb>cit potentia in G, ac ſi in G aliud eſſet appenſum pondus æquale <lb></lb>ponderi A; quæ pondera in CF appenſa æquæponderabunt. </s> <s id="id.2.1.139.4.1.9.0">Præ<lb></lb>terea, cùm in neutram fiat motus partem, idem erit vnico exi<pb n="64" xlink:href="036/01/141.jpg"></pb>ſtente fune BC EFG hoc modo orbiculo circumuoluto, ac ſi duo <lb></lb>eſſent funes BC FG alligati in vecte, ſiue 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"></emph>Huius. de libra.<emph.end type="italics"></emph.end></s> <s id="id.2.1.140.1.1.3.0"><margin.target id="note218"></margin.target>8 <emph type="italics"></emph>Vndecimi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.140.1.1.4.0"><margin.target id="note219"></margin.target>18 <emph type="italics"></emph>Tertii.<emph.end type="italics"></emph.end></s> <s id="id.2.1.140.1.1.5.0"><margin.target id="note220"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 28 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.140.1.1.6.0"><margin.target id="note221"></margin.target>1 <emph type="italics"></emph>Primi. Archim. de æquepond.<emph.end type="italics"></emph.end></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 manifeſtum eſſe poteſt, idem pon<lb></lb>dus ab eadem potentia abſq; ullo huius tro<lb></lb>chleæ auxilio nihilominus ſuſtineri poſſe. </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></lb>ponderi A, cui alligatus ſit funis <lb></lb>kL; ſitq; potentia in L ſuſtinens <lb></lb>pondus H. </s> <s id="N1414E">cùm autem pondus <lb></lb>abſq; vllo adminiculo ſuſtinere <lb></lb>volentes tanta vi opus ſit, quanta <lb></lb>ponderi eſt æqualis; erit potentia <lb></lb>in L ponderi H æqualis; pondus <lb></lb>verò H ipſi ponderi A eſt æquale, <lb></lb>cui potentia in G eſt æqualis; erit <lb></lb>igitur potentia in G potentiæ in L <lb></lb>æqualis. </s> <s id="id.2.1.141.3.1.2.0">quod idem eſt, ac ſi <expan abbr="eadẽ">eadem</expan><lb></lb>potentia idem pondus ſuſtineret. <figure id="id.036.01.141.1.jpg" place="text" 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 ſi potentiæ in G, & <lb></lb>in L inuicem fuerint æquales, ſeor<lb></lb>ſum autem ponderibus minores; <lb></lb>patet potentias ponderibus ſuſti<lb></lb>nendis non ſufficere. </s> <s id="id.2.1.141.4.1.2.0">ſi verò maiores, manifeſtum eſt pondera à <lb></lb>pontentiis moueri. </s> <s id="id.2.1.141.4.1.3.0">& ſic in eadem eſſe proportione potentiam in <lb></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 demonſtratione aſſumptum fuit axiculum cir<lb></lb>cumuerti, qui vt plurimum immobilis manet; idcirco immobili <lb></lb>quoq; manente axiculo idem oſtendatur. </s> </p> <pb xlink:href="036/01/142.jpg"></pb> <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></lb>ius centrum D; ſitq; axiculus GHk, <lb></lb>cuius idem ſit centrum D. </s> <s id="id.2.1.141.7.1.1.0.a">Ducatur <lb></lb>CG DkF diameter horizonti æ<lb></lb>quidiſtans. </s> <s id="id.2.1.141.7.1.2.0">& quoniam dum orbi<lb></lb>culus circumuertitur, circumferen<lb></lb>tia circuli CEF ſemper eſt æquidi<lb></lb>ſtans circumferentiæ axiculi GHk; <lb></lb>circa enim axiculum circumuerti<lb></lb>tur; & circulorum æquidiſtantes cir<lb></lb>cumferentiæ idem habent centrum; <lb></lb>erit punctum D ſemper & orbiculi, <lb></lb><figure id="id.036.01.142.1.jpg" place="text" xlink:href="036/01/142/1.jpg"></figure><lb></lb>& axiculi centrum. </s> <s id="id.2.1.141.7.1.3.0">Itaq; cùm DC ſit æqualis DF, & DG ipſi <lb></lb>Dk; erit GC ipſi kF æqualis. </s> <s id="id.2.1.141.7.1.4.0">ſi igitur in vecte, ſiue libra CF <lb></lb>pondera appendantur æqualia, æqueponderabunt. </s> <s id="id.2.1.141.7.1.5.0">diſtantia enim <lb></lb>CG æqualis eſt diſtantiæ kF; axiculuſq; GHK immobilis gerit <lb></lb>vicem centri, ſiue fulcimenti. </s> <s id="id.2.1.141.7.1.6.0">immobili igitur manente axicu<lb></lb>lo, ſi ponatur in F potentia ſuſtinens pondus in C appenſum; erit <lb></lb>potentia in F ipſi ponderi æqualis. </s> <s id="id.2.1.141.7.1.7.0">quod erat oſtendendum. </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 prorſus ſit, ſiue axiculus circumuertatur, ſiue mi<lb></lb>nus; liceat propterea in iis, quæ dicenda ſunt, loco axiculi cen<lb></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></lb>circumducatur, altero eius extremo alicubi reli<lb></lb>gato, altero uerò à potentia pondus ſuſtinente <lb></lb>apprehenſo; erit potentia ponderis ſubdupla. </s> </p> <pb n="65" xlink:href="036/01/143.jpg"></pb> <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; ſit BCD <lb></lb>orbiculus trochleæ pon<lb></lb>deri A alligate, cuius cen<lb></lb>trum E; funis deinde FB <lb></lb>CDG circa orbiculum <lb></lb>voluatur, qui religetur in <lb></lb>F; ſitq; potentia in G ſu<lb></lb>ſtinens pondus A. </s> <s id="id.2.1.141.12.1.1.0.a">dico <lb></lb>potentiam in G ſubdu<lb></lb>plam eſſe ponderis A. </s> <s id="id.2.1.141.12.1.1.0.b">ſint <lb></lb>funes FB GD puncti E <lb></lb>horizonti perpendicula<lb></lb>res, qui inter ſe ſe æqui<lb></lb>diſtantes <arrow.to.target n="note222"></arrow.to.target>erunt; tangantq; <lb></lb>funes FB GD circulum <lb></lb>BCD in BD punctis. </s> <s id="id.2.1.141.12.1.2.0"><lb></lb>connectatur BD; erit BD <lb></lb>per centrum E ducta, <arrow.to.target n="note223"></arrow.to.target><lb></lb><figure id="id.036.01.143.1.jpg" place="text" xlink:href="036/01/143/1.jpg"></figure><lb></lb>ipſiuſ〈qué〉 centri horizonti æquidiſtans. </s> <s id="id.2.1.141.12.1.3.0">Cùm autem potén<lb></lb>tia in G trochlea pondus A ſuſtinere debeat, funem ex altero ex<lb></lb>tremo religatum eſſe oportet, puta in F; ita vt F æqualiter ſaltem <lb></lb>potentiæ in G reſiſtat, alioquin potentia in G nullatenus pondus <lb></lb>ſuſtinere poſſet. </s> <s id="id.2.1.141.12.1.4.0">Et quoniam potentia fune ſuſtinet orbiculum, <lb></lb>qui reliquam trochleæ partem, cui appenſum eſt pondus, ſuſtinet <lb></lb>axiculo; grauitabit hæc trochleæ pars in axiculo, hoc eſt in centro <lb></lb>E. </s> <s id="N1424A">quare pondus A in eodem quoq; centro E ponderabit, ac ſi <lb></lb>in E eſſet appenſum. </s> <s id="id.2.1.141.12.1.5.0">poſita igitur potentia, quæ in G, vbi D <lb></lb>(idem enim prorſus eſt) erit BD tanquam vectis, cuius fulci<lb></lb>mentum erit B, pondus in E appenſum, & potentia in D. </s> <s id="N14255">con<lb></lb>uenienter enim fulcimenti rationem ipſum B ſubire poteſt, exi<lb></lb>ſtente fune FB immobili. </s> <s id="id.2.1.141.12.1.6.0">cæterum hoc poſterius magis eluceſcet. </s> <s id="id.2.1.141.12.1.7.0"><lb></lb>Quoniam autem potentia ad pondus eandem habet proportio<lb></lb>nem, <arrow.to.target n="note224"></arrow.to.target>quàm BE ad BD; & BE in ſubdupla eſt proportione <lb></lb>ad BD: potentia igitur in G ponderis A ſubdupla erit. </s> <s id="id.2.1.141.12.1.8.0">quod de<lb></lb>monſtrare 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"></emph>Vndecimi<emph.end type="italics"></emph.end></s> <s id="id.2.1.142.1.1.2.0"><margin.target id="note223"></margin.target><emph type="italics"></emph>Ex præcedenti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.142.1.1.3.0"><margin.target id="note224"></margin.target>2 <emph type="italics"></emph>Huius de vecte.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/144.jpg"></pb> <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 ſe ha<lb></lb>bet vnico exiſtente fune <lb></lb>FBC DG ipſi orbiculo <lb></lb>circumducto, ac ſi duo eſ<lb></lb>ſent funes BF GD ve<lb></lb>cti BD alligati, cuius ful<lb></lb>cimentum erit B, pon<lb></lb>dus in E appenſum, & <lb></lb>potentia ſuſtinens in D, <lb></lb>vel quod idem eſt in G. </s> </p> <figure id="id.036.01.144.1.jpg" place="text" 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; manifeſtum eſt, pondus hoc mo<lb></lb>do à minori in ſubdupla proportione potentia <lb></lb>ſuſtineri, quam ſine vllo huiuſmodi trochleæ <lb></lb>auxilio. </s> </p> <pb n="66" xlink:href="036/01/145.jpg"></pb> <p id="id.2.1.143.4.0.0.0" type="main"> <s id="id.2.1.143.4.1.1.0">Veluti ſit pondus H ponderi A <lb></lb>æquale, cui religatus ſit funis kL; <lb></lb>potentiaq; in L ſuſtineat pondus H; <lb></lb>erit potentia in L ſeorſum ponderi <lb></lb>H, & ponderi A æqualis; ſed poten<lb></lb>tia in G ſubdupla eſt ponderis A, <lb></lb>quare potentia in G ſubdupla erit po<lb></lb>tentiæ, quæ eſt in L. </s> <s id="N142D9">& hoc modo in <lb></lb>huiuſcemodi reliquis omnibus pro <lb></lb>portio inueniri poterit. </s> </p> <figure id="id.036.01.145.1.jpg" place="text" 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">Manifeſtum eſt etiam; ſi duæ fuerint poten<lb></lb>tiæ vna in G, altera in F, pondus A ſuſtinentes; <lb></lb>vtraſq; ſimul ponderi A æquales eſſe: & vnam <lb></lb>quamque ſuſtinere 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 ſecundæ huius in <lb></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 innoteſcit, cur ſcilicet fu<lb></lb>nis ex altero religatus eſſe debeat extremo. </s> </p> <pb xlink:href="036/01/146.jpg"></pb> <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 vtriſq; duarum trochlearum ſingulis or<lb></lb>biculis, quarum altera ſupernè, altera verò in<lb></lb>fernè conſtituta, ponderiq; alligata fuerit, cir<lb></lb>cunducatur funis; altero eius extremo alicubi <lb></lb>religato, altero verò à potentia pondus ſuſti<lb></lb>nente detento; erit potentia ponderis ſub du<lb></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; ſit BCD orbiculus trochleæ pon<lb></lb>deri A alligatæ, cuius centrum K; EFG verò <lb></lb>ſit trochleæ ſurſum appenſæ, cuius centrum H. <lb></lb></s> <s id="N14332">deinde LBC DME FGN funis circa orbicu<lb></lb>los ducatur, qui religetur in L; ſitq; potentia in <lb></lb>N ſuſtinens pondus A. </s> <s id="id.2.1.143.12.1.1.0.a">dico potentiam in N <lb></lb>ſubduplam eſſe ponderis A. </s> <s id="N1433D">ſi enim potentia ſu<lb></lb>ſtinens pondus A vbi M collocata foret, eſſet <lb></lb>vtiq; potentia in M ſubdupla ponderis A. </s> <s id="N14343">po<lb></lb><arrow.to.target n="note225"></arrow.to.target>tentiæ verò in M æqualis eſt vis in N. </s> <s id="N1434A">eſt e<lb></lb><arrow.to.target n="note226"></arrow.to.target>nim ac ſi potentia in M dimidium ponderis <lb></lb>A ſine trochlea ſuſtineret, cui æqueponderat <lb></lb>pondus in N ponderis A dimidio æquale. </s> <s id="id.2.1.143.12.1.2.0"><lb></lb>quare vis in N æqualis dimidio ponderis A <lb></lb>ipſum A ſuſtinebit. </s> <s id="id.2.1.143.12.1.3.0">Potentia igitur in N ſuſti<lb></lb>nens pondus A ſubdupla eſt ipſius A. </s> <s id="N14360">quod <lb></lb>demonſtrare oportebat. <figure id="id.036.01.146.1.jpg" place="text" xlink:href="036/01/146/1.jpg"></figure></s> </p> <pb n="67" xlink:href="036/01/147.jpg"></pb> <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 ſecunda figura ſit fu<lb></lb>nis BC DEF GHkL orbiculis cir<lb></lb>cum uolutus, & religatus in B; poten<lb></lb>tiaq; in L pondus A ſuſtineat: erit <lb></lb>potentia in L ſimiliter ponderis ſubdu<lb></lb>pla. </s> <s id="id.2.1.143.14.1.2.0">orbiculus enim trochleæ ſupe<lb></lb>rioris, ipſa〈qué〉 trochlea penitus ſunt <lb></lb>inutiles: & idem eſt, ac ſi funis reli<lb></lb>gatus eſſet in F, & potentia in L ſu<lb></lb>ſtineret pondus ſola trochlea ponderi <lb></lb>alligata, quæ potentia ponderis A oſten<lb></lb>ſa eſt ſubdupla. </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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.144.1.1.2.0"><margin.target id="note226"></margin.target>1 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.036.01.147.1.jpg" place="text" 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 ſequitur, ſi duæ ſint potentiæ in BL; <lb></lb>vtraſq; inter ſe ſe æquales eſſe. </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 ſeorſum eſt ipſius A ſubdupla. </s> </p> <pb xlink:href="036/01/148.jpg"></pb> <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 ſit A; qui <lb></lb>bifariam diuidatur in D: ſitq; pondus C in D <lb></lb>appenſum; duæq; ſint potentiæ æquales in BD <lb></lb>pondus C ſuſtinentes. </s> <s id="id.2.1.145.6.1.2.0">Dico unamquamq; poten<lb></lb>tiam in BD ponderis C ſubtriplam eſſe. </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></lb>potentia eſt in D colloca<lb></lb>ta, & pondus C in eodem <lb></lb>puncto D eſt appenſum; <lb></lb>potentia in D partem <lb></lb>ponderis C ſuſtinebit ip<lb></lb>ſi potentiæ D æqualem. </s> <s id="id.2.1.145.7.1.2.0"><lb></lb><figure id="id.036.01.148.1.jpg" place="text" xlink:href="036/01/148/1.jpg"></figure><lb></lb>quare potentia in B partem ſuſtinebit reliquam, quæ pars dupla erit <lb></lb>ipſius potentiæ in B; cùm pondus ad potentiam eandem habeat <lb></lb>proportionem, quam AB ad AD: & potentiæ in BD ſunt æqua<lb></lb>les; ergo potentia in B duplam ſuſtinebit partem eius, quam ſuſti<lb></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></lb>rum vna ſit reliquæ dupla; quod fiet, ſi in tres partes æquales EFG <lb></lb>diuiſerimus: tunc enim FG dupla erit ipſius E. </s> <s id="id.2.1.145.7.1.2.0.b">Itaq; potentia <lb></lb>in D partem E ſuſtinebit, & potentiam in B reliquas FG. </s> <s id="N14406">vtreq; <lb></lb>igitur inter ſe ſe æquales potentiæ in BD ſimul totum ſuſtinebunt <lb></lb>pondus C. </s> <s id="id.2.1.145.7.1.2.0.c">& quoniam potentia in D partem E ſuſtinet, quæ ter<lb></lb>tia eſt pars ponderis C, ipſiq; eſt æqualis; erit potentia in D ſub <lb></lb>tripla ponderis C. </s> <s id="N14413">& cùm potentia in B ſuſtineat partes FG, qua <lb></lb>rum potentia in B eſt ſubdupla; erit in B potentia vni partium FG, <lb></lb>putà G æqualis. </s> <s id="id.2.1.145.7.1.3.0">G verò tertia eſt pars ponderis C; potentia <lb></lb>igitur in B ſubtripla erit ponderis C. </s> <s id="id.2.1.145.7.1.3.0.a">Vnaquæq; ergo potentia in <lb></lb>BD ſubtripla eſt ponderis C. </s> <s id="N14423">quod demonſtrare oportebat. <pb n="68" xlink:href="036/01/149.jpg"></pb> <figure id="id.036.01.149.1.jpg" place="text" 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 ſi duo eſſent vectes AB EF bifariam in GD diuiſi, quorum <lb></lb>fulcimenta eſſent AF, & pondus C in DG vtriq; vecti appen<lb></lb>ſum, ita tamen vt in vtroq; æqualiter ponderet; duæq; eſſent <lb></lb>æquales potentiæ in BG: eadem prorſus ratione oſtendetur, <lb></lb>vnamquamq; potentiam in B, & G ponderis C ſubtriplam <lb></lb>eſſe. </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 vtriſq; duarum <expan abbr="trochlearũ">trochlearum</expan> ſingulis orbiculis, <lb></lb>quarum altera ſupernè, altera verò infernè conſti<lb></lb>tuta, ponderiq; alligata fuerit, circumducatur fu<lb></lb>nis; altero eius extremo inferiori trochleæ reli<lb></lb>gato, altero verò à potentia pondus ſuſtinente <lb></lb>detento: erit potentia ponderis ſubtripla. </s> </p> <pb xlink:href="036/01/150.jpg"></pb> <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; ſit BCD orbiculus tro<lb></lb>chleæ ponderi A alligate, cuius centrum <lb></lb>E; & FGH trochleæ ſurſum appenſæ, cu<lb></lb>ius centrum k; & LFGHBCDM funis <lb></lb>orbiculis circumducatur, qui religetur in L <lb></lb>trochleæ inferiori; ſitq; potentia in M ſu<lb></lb>ſtinens pondus A. </s> <s id="id.2.1.145.12.1.1.0.a">dico potentiam in M <lb></lb>ſubtriplam eſſe ponderis A. </s> <s id="id.2.1.145.12.1.1.0.b">ducantur FH <lb></lb>BD per centra kE horizonti æquidiſtan<lb></lb>tes, ſicut in præcedentibus dictum eſt. </s> <s id="N14479">Quo<lb></lb>niam enim funis FL trochleam ſuſtinet in<lb></lb>feriorem, quæ ſuſtinet orbiculum in eius <lb></lb>centro E; erit funis in L vt potentia ſuſti<lb></lb>nens orbiculum, ac ſi in ipſo E centro eſſet; <lb></lb>potentia verò in M eſt, ac ſi eſſet in D; <lb></lb>efficietur igitur DB tanquam vectis, cuius <lb></lb><arrow.to.target n="note227"></arrow.to.target>fulcimentum erit B; pondus verò A (vt ſu<lb></lb>pra oſtenſum eſt) ex E ſuſpenſum à dua<lb></lb>bus potentiis altera in D, altera in E ſuſten<lb></lb>tatum. </s> <s id="id.2.1.145.12.1.2.0">Cùm autem in pondere ſuſtinendo <lb></lb>vectes FH BD immobiles maneant, ſi in <lb></lb>funibus FL HB appendantur pondera, e<lb></lb><arrow.to.target n="note228"></arrow.to.target>runt hæc ipſa æqualia; cùm vectis FH ha<lb></lb>beat fulcimentum in medio; alioquin ex al<lb></lb>tera parte deorſum fieret motus, quod <expan abbr="tamẽ">tamen</expan><lb></lb>non contingit. </s> <s id="id.2.1.145.12.1.3.0">tam igitur ſuſtinet funis FL, <lb></lb>quàm HB. </s> <s id="N144AC">deinde quoniam ex medio ve<lb></lb><figure id="id.036.01.150.1.jpg" place="text" xlink:href="036/01/150/1.jpg"></figure><lb></lb>cte BD pondus ſuſpenditur, idcirco ſi duæ fuerint potentiæ in BD <lb></lb><arrow.to.target n="note229"></arrow.to.target>pondus ſuſtinentes, 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"></pb>FL ipſe quoq; pondus ſuſtineat, cùm potentiæ in E <expan abbr="vicẽ">vicem</expan> gerat; quia <lb></lb>tamen ex eodemmet puncto ſuſtinet, vbi appenſum eſt pondus, non <lb></lb>efficiet propterea, quin potentiæ in BD ſint inter ſe ſe æquales; <lb></lb>opitulatur enim tàm vni, quàm alteri. </s> <s id="id.2.1.145.12.1.5.0">potentiæ verò in BD eæ<lb></lb>dem ſunt, ac ſi eſſent in HM; quare tàm ſuſtinebit funis MD, <lb></lb>quàm HB. </s> <s id="id.2.1.145.12.1.5.0.a">ita verò ſuſtinet HB, atq; FL; funis igitur MD ita <lb></lb>ſuſtinebit, ſicut FL, hoc eſt, ac ſi in D, & L appenſa eſſent pon<lb></lb>dera æqualia. </s> <s id="id.2.1.145.12.1.6.0">Cùm itaq; æqualia pondera à potentiis ſuſtinean<lb></lb>tur æqualibus, potentiæ in ML æquales erunt; quarum eadem pror<lb></lb>ſus eſt ratio, ac ſi eſſent ambæ in DE. </s> <s id="id.2.1.145.12.1.6.0.a">Itaq; cùm pondus A in <lb></lb>medio vectis BD ſit appenſum, duæq; potentiæ ſint æquales in <lb></lb>DE pondus ſuſtinentes; erit B fulcimentum, ac vnaquæq; potentia, <arrow.to.target n="note230"></arrow.to.target><lb></lb>ſiue in DE, ſiue in ML ſubtripla ponderis A. </s> <s id="N144EE">ergo potentia in M <lb></lb>ſuſtinens pondus ſubtripla erit ponderis A. </s> <s id="N144F2">quod oſtendere o<lb></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"></emph>In<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>Huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.146.1.1.2.0"><margin.target id="note228"></margin.target>1 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.146.1.1.3.0"><margin.target id="note229"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 3 <emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>Huius vecte.<emph.end type="italics"></emph.end></s> <s id="id.2.1.146.1.1.4.0"><margin.target id="note230"></margin.target>4 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 manifeſtum eſt, vnumquemq; funem <lb></lb>MD FL HB tertiam ſuſtinere partem pon<lb></lb>deris A. <pb xlink:href="036/01/152.jpg"></pb></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, ſi funis ex M per a<lb></lb>lium adhuc deferatur orbiculum ſu<lb></lb>periorem in trochlea ſurſum ſimi<lb></lb>liter appenſa conſtitutum, cuius <lb></lb>centrum N; ita vt perueniat in O; <lb></lb>ibiq; à potentia detineatur; erit po<lb></lb>tentia in O ſuſtinens pondus A iti <lb></lb>dem ſubtripla ipſius ponderis. </s> <s id="id.2.1.147.3.1.2.0">fu<lb></lb>nis enim MD tantùm ponderis ſu<lb></lb>ſtinet, ac ſi in D appenſum eſſet <lb></lb>pondus æquale tertiæ parti ponde<lb></lb><arrow.to.target n="note231"></arrow.to.target>ris A, cui æquiualet potentia in <lb></lb>O ipſi æqualis, hoc eſt ſubtripla <lb></lb>ponderis A. </s> <s id="id.2.1.147.3.1.2.0.a">Potentia igitur in O <lb></lb>ſubtripla eſt ponderis A. <lb></lb><figure id="id.036.01.152.1.jpg" place="text" 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 ſæpius repetatur, no<lb></lb>uiſſe oportet potentiam in O ſem<lb></lb>per æqualem eſſe ei, quæ eſt in M; <lb></lb>hoc eſt ſi potentia in M eſſet ſub <lb></lb>quadrupla, ſubquintupla, vel huiuſ <lb></lb>modi aliter ipſius ponderis; poten<lb></lb>tia quoq; in O erit itidem ſubqua<lb></lb>drupla, ſubquintupla, atq; ita dein<lb></lb>ceps eiuſdemmet ponderis, quem<lb></lb>madmodum ſe habet potentia <lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="70" xlink:href="036/01/153.jpg"></pb> <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 diuiſi in <lb></lb>EF, quorum fulcimenta ſint. </s> <s id="id.2.1.149.2.1.2.0">in BD; ſitq; pon<lb></lb>dus G in EF vtriq; vecti appenſum, ita ut ex <lb></lb>vtroq; æqualiter ponderet; duæq; ſint potentiæ <lb></lb>in AC æquales pondus ſuſtinentes. </s> <s id="id.2.1.149.2.1.3.0">Dico unam <lb></lb>quamq; potentiam in AC ſubquadruplam eſ<lb></lb>ſe 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></lb>AC totum ſuſtineant pon<lb></lb>dus G, potentiaq; in A ad <lb></lb>partem ponderis, quod ſuſti<lb></lb>net, ſit vt BE ad BA; po<lb></lb>tentia <arrow.to.target n="note232"></arrow.to.target>verò in C ad partem <lb></lb>ipſius G, quod ſuſtinet, ita <lb></lb>ſit vt DF ad DC; & vt BE <lb></lb>ad BA, ita eſt DF ad DC; <lb></lb><figure id="id.036.01.153.1.jpg" place="text" xlink:href="036/01/153/1.jpg"></figure><lb></lb>erit potentia in A ad partem ponderis, quod ſuſtinet, vt poten<lb></lb>tia in C ad ipſius ponderis, quod ſuſtinet, partem; & potentiæ <lb></lb>in AC ſunt æquales; æquales igitur erunt partes ponderis G, <lb></lb>quæ à potentiis ſuſtinentur. </s> <s id="id.2.1.149.3.1.2.0">quare vnaquæq; potentia in A C di<lb></lb>midium ſuſtinebit ponderis G. </s> <s id="id.2.1.149.3.1.2.0.a">Potentia verò in A ſubdupla eſt pon<lb></lb>deris, quod ſuſtinet: ergo potentia in A dimidio dimidii, hoc <lb></lb>eſt quartæ portioni ponderis G æqualis erit; ideoq; ſubquadrupla <lb></lb>erit ponderis G. </s> <s id="id.2.1.149.3.1.2.0.b">neq; aliter demonſtrabitur potentiam in C ſub<lb></lb>quadruplam eſſe eiuſdem ponderis G. </s> <s id="N145F9">quod demonſtrare opor<lb></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"></emph>Huius. de vecte.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/154.jpg"></pb> <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 ſint vectes <lb></lb>AB CD EF bifariam di<lb></lb>uiſi in GHk, quorum fulci <lb></lb>menta ſint BDF; & pondus <lb></lb>L eodem modo in GHK <lb></lb>appenſum; ſintq; tres poten<lb></lb>tiæ in ACE æquales pondus <lb></lb>ſuſtinentes; ſimiliter oſten<lb></lb>detur vnamquamque po<lb></lb>tentiam ſubſexcuplam eſſe <lb></lb>ponderis L. </s> <s id="N14627">atq; hoc ordi<lb></lb>ne ſi quatuor eſſent vectes, <lb></lb>& quatuor potentiæ; erit vnaquæq; potentia ſuboctupla ponderis. </s> <lb></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" place="text" 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></lb>altera ſupernè vnico duntaxat, altera verò infer<lb></lb>nè duobus autem inſignita orbiculis, ponderiq; <lb></lb>alligata conſtituta fuerit, funis circumponatur; al<lb></lb>tero eius extremo alicubi religato, altero verò à <lb></lb>potentia pondus ſuſtinente retento; erit potentia <lb></lb>ponderis ſubquadrupla. </s> </p> <pb n="71" xlink:href="036/01/155.jpg"></pb> <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; ſint tres orbiculi, quorum <lb></lb>centra BCD; orbiculuſq;, cuius centrum D, <lb></lb>ſit trochleæ ſurſum appenſæ; quorum verò <lb></lb>ſunt centra BC, ſint trochleæ ponderi A alli<lb></lb>gatæ; funiſq; EFGHkLNOP per omnes <lb></lb>circumducatur orbiculos, qui religetur in E; <lb></lb>ſitq; vis in P ſuſtinens pondus A. </s> <s id="id.2.1.151.4.1.1.0.a">dico po<lb></lb>tentiam in P ſubquadruplam eſſe ponderis <lb></lb>A. </s> <s id="id.2.1.151.4.1.1.0.b">ducantur kL GF ON per rotularum <lb></lb>centra, & horizonti æquidiſtantes, quæ (ex <lb></lb>iis, quæ dicta ſunt) tanquam vectes erunt. </s> <s id="id.2.1.151.4.1.2.0"><lb></lb>& quoniam propter vectem, ſiue libram kL, <lb></lb>cuius fulcimentum, ſiue centrum eſt in me <lb></lb>dio, tàm ſuſtinet funis kG, quàm LN, cùm <arrow.to.target n="note233"></arrow.to.target><lb></lb>in neutram partem fiat motus. </s> <s id="id.2.1.151.4.1.3.0">nec non <lb></lb>propter vectem GF, è cuius medio veluti ſu<lb></lb>ſpenſum dependet onus; ſi duæ eſſent in GF <lb></lb>potentiæ, ſeu in HE (eſt enim par vtriuſq; <lb></lb>ſitus ratio, vt iam ſepius dictum eſt) eſſent <arrow.to.target n="note234"></arrow.to.target><lb></lb>vtiq; huiuſmodi potentiæ inuicem æquales. </s> <s id="id.2.1.151.4.1.4.0"><lb></lb>quare ita ſuſtinet funis HG, vt EF. </s> <s id="N14696">ſimiliter <lb></lb>oſten detur funem PO tàm ſuſtinere, quàm <lb></lb>LN: quare funes PO kG EF LN æqua<lb></lb>liter ſuſtinent. </s> <s id="id.2.1.151.4.1.5.0">æqualiter igitur funis PO ſu<lb></lb>ſtinet, vt kG. </s> <s id="N146A3">ſi ergo duæ intelligantur eſ <lb></lb><figure id="id.036.01.155.1.jpg" place="text" xlink:href="036/01/155/1.jpg"></figure><lb></lb>ſe potentiæ in OG, ſeu in PH, quod idem eſt, pondus nihilomi<lb></lb>nus ſuſtinentes, quemadmodum funes ſuſtinent, æquales vtiq; eſ<lb></lb>ſent; & GF ON duorum vectium vires gerent; quorum fulci <lb></lb>menta erunt FN, & pondus A in BC medio vectium appenſum. </s> <s id="id.2.1.151.4.1.6.0"><lb></lb>& quoniam omnes funes æqualiter ſuſtinent, tàm ſuſtinebunt <lb></lb>duo PO LN, quàm duo KGEF; tàm igitur ſuſtinebit vectis <lb></lb>ON, quàm vectis GF. </s> <s id="N146BB">quare in vtroq; vecte ON GF æquali <lb></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 ſubquadru<arrow.to.target n="note235"></arrow.to.target><lb></lb>pla ponderis A. </s> <s id="N146CB">& cùm funis KG potentiæ loco ſumatur, quippè <lb></lb>qui haud ſecus ſuſtinet, quàm PO; erit potentia in P ſuſtinens pon<lb></lb>dus A ipſius ponderis ſubquadrupla. </s> <s id="id.2.1.151.4.1.8.0">quod demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.152.1.1.2.0"><margin.target id="note234"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end>2 <emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.152.1.1.3.0"><margin.target id="note235"></margin.target>6 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/156.jpg"></pb> <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 manifeſtum eſt vnumquemq; funem EF <lb></lb>GK LN OP quartam ſuſtinere partem pon<lb></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></lb>non minus eo, cuius centrum eſt B, ſuſtinere. </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 iiſdem poſitis, ſi duæ eſſent poten<lb></lb>tiæ æquales pondus A ſuſtinentes, vna in O <lb></lb><arrow.to.target n="note236"></arrow.to.target>altera in C; eſſet vnaquæq; dictarum poten<lb></lb>tiarum ponderis A ſubtripla. </s> <s id="id.2.1.153.6.1.2.0">ſed quoniam <lb></lb>vectis GF, cuius fulcimentum eſt F bifariam <lb></lb>diuiſus eſt in C; ſi igitur ponatur in G poten<lb></lb>tia idem pondus ſuſtinens, vt potentia in C; <lb></lb>erit potentia in G ſubdupla potentiæ, quæ eſ <lb></lb>ſet in C; nam ſi potentia in C ſe ipſa pon<lb></lb>dus in C appenſum ſuſtineret, eſſet vtiq; ip<lb></lb>ſi ponderi æqualis; & idem pondus, ſi à po<lb></lb><arrow.to.target n="note237"></arrow.to.target>tentia in G ſuſtineretur, eſſet ipſius poten<lb></lb>tiæ in G duplum; potentia veró in C ſubtri<lb></lb>pla eſſet ponderis A; ergo potentia in G <lb></lb>ſubſexcupla eſſet ponderis A. </s> <s id="id.2.1.153.6.1.2.0.a">Cùm itaq; <lb></lb>potentia in O ſubtripla ſit ponderis A, & <lb></lb>potentia in G ſubſexcupla; erunt vtræq; ſi<lb></lb>mul potentiæ in OG ipſius ponderis A ſub <lb></lb>duplæ. </s> <s id="id.2.1.153.6.1.3.0">tertia enim pars cum ſexta dimi<lb></lb>dium efficit. </s> <s id="id.2.1.153.6.1.4.0">quoniam autem potentiæ in <lb></lb>OG, ſiue in PH (vt prius dictum eſt) <lb></lb>ſunt inter ſe æquales, ac vtræq; ſimul ſubdu<lb></lb>plæ ſunt ponderis A. erit vnaquæq; poten<lb></lb><figure id="id.036.01.156.1.jpg" place="text" xlink:href="036/01/156/1.jpg"></figure> <pb n="72" xlink:href="036/01/157.jpg"></pb>tia in P H ipſius A ſubquadrupla. </s> <s id="id.2.1.153.6.1.5.0">Potentia igitur in P ſuſtinens pon<lb></lb>dus A ipſius ponderis A ſubquadrupla erit. </s> <s id="id.2.1.153.6.1.6.0">quod erat oſten<lb></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"></emph>Ex<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>Huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.154.1.1.2.0"><margin.target id="note237"></margin.target>2 <emph type="italics"></emph>Huius. de vecte.<emph.end type="italics"></emph.end></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></lb>& ſecundùm quatuor adhuc <lb></lb>circumuoluatur orbiculos, per <lb></lb>ueniatq; ad P. </s> <s id="N147A8">ſimiliter oſten<lb></lb>detur potentiam in P ſubqua<lb></lb>druplam eſſe ponderis A. <lb></lb></s> <s id="N147AF">idem enim eſt, ac ſi funis re<lb></lb>ligatus eſſet in L, potentiaq; <lb></lb>ſuſtineret pondus fune tribus <lb></lb>tantùm orbiculis circumdu<lb></lb>cto, quorum centra eſſent B <lb></lb>CQ. </s> <s id="N147BB">orbiculus enim cuius <lb></lb>centrum D eſt pœnitus inu<lb></lb>tilis. <figure id="id.036.01.157.1.jpg" place="text" xlink:href="036/01/157/1.jpg"></figure></s> <pb xlink:href="036/01/158.jpg"></pb> <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 diuiſi in EF, <lb></lb>quorum fulcimenta ſint AC, & pondus G in <lb></lb>punctis EF vtriq; vecti ſit appenſum, ita vt ex <lb></lb>vtroq; æqualiter ponderet; treſq; ſint potentiæ <lb></lb>æquales in BDE pondus G ſuſtinentes. </s> <s id="id.2.1.155.2.1.2.0">Dico <lb></lb>vnamquamq; ſeorſum ex dictis potentiis ſub<lb></lb>quintuplam eſſe 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></lb>appenſum eſt in EF, & tres <lb></lb>ſunt potentiæ in EBD æqua<lb></lb>les; ideo potentia in E partem <lb></lb>tantùm ponderis G ſuſtinebit <lb></lb>ipſi potentiæ in E æqualem; <lb></lb>potentiæ verò in BD partem <lb></lb>ſuſtinebunt reliquam; & pars, <lb></lb><arrow.to.target n="note238"></arrow.to.target>quam ſuſtinet B, erit ipſius <lb></lb>dupla; pars autem, quam ſu<lb></lb><figure id="id.036.01.158.1.jpg" place="text" xlink:href="036/01/158/1.jpg"></figure><lb></lb>ſtinet D, erit ſimiliter ipſius D dupla; propter proportionem <lb></lb>BA ad AE, & DC ad CF. </s> <s id="id.2.1.155.3.1.1.0.a">Cùm itaq; potentiæ in BD ſint æqua<lb></lb><arrow.to.target n="note239"></arrow.to.target>les, erunt (ex iis, quæ ſupra dictum eſt) partes ponderis G, quæ <lb></lb>à potentiis BD ſuſtinentur, inter ſe ſe æquales; & vnaquæq; du<lb></lb>pla eius partis, quæ à potentia in E ſuſtinetur. </s> <s id="id.2.1.155.3.1.2.0">diuidatur er<lb></lb>go pondus G in tres partes, quarum duæ ſint inter ſe ſe æquales, <lb></lb>nec non vnaquæq; ſeorſum alterius tertiæ partis dupla. </s> <s id="id.2.1.155.3.1.3.0">quod <lb></lb>fiet, ſi in quinq; partes æquales HKLMN diuidatur; pars <lb></lb>enim compoſita ex duabus partibus kL dupla eſt partis H; pars <lb></lb>quoq; MN eiuſdem partis H eſt ſimiliter dupla. </s> <s id="id.2.1.155.3.1.4.0">quare & pars <lb></lb>kL parti MN erit æqualis. </s> <s id="id.2.1.155.3.1.5.0">Suſtineat autem potentia in E par<lb></lb>tem H; & potentia in B partes KL; potentia verò in D partes <pb n="73" xlink:href="036/01/159.jpg"></pb>MN: tres igitur potentiæ æquales in BDE totum ſuſtinebunt pon<lb></lb>dus G; & vnaquæq; potentia in BD duplum ſuſtinebit eius, quod <lb></lb>ſuſtinet potentia in E. </s> <s id="id.2.1.155.3.1.5.0.a">Cùm itaq; potentia in E partem H ſuſti<lb></lb>neat, quæ quinta eſt pars ponderis G, ipſiq; ſit æqualis; erit po<lb></lb>tentia in E ſubquintupla ponderis G. </s> <s id="id.2.1.155.3.1.5.0.b">& quoniam potentia in B <lb></lb>partes kL ſuſtinet, quæ quidem duplæ ſunt potentiæ B, & partis H; <lb></lb>erit quoq; potentia in B ipſi H æqualis: quare ſubquintupla erit <lb></lb>ponderis G. </s> <s id="id.2.1.155.3.1.5.0.c">Non aliter oſtendetur potentiam in D ſubquintu<lb></lb>plam eſſe ponderis G. </s> <s id="N1484A">vnaquæq; igitur potentia in BDE ſubquin<lb></lb>tupla eſt ponderis G. </s> <s id="N1484E">quod demonſtrare 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"></emph>Huius. de vecte.<emph.end type="italics"></emph.end></s> <s id="id.2.1.156.1.1.3.0"><margin.target id="note239"></margin.target><emph type="italics"></emph>In<emph.end type="italics"></emph.end> 6 <emph type="italics"></emph>Huius<emph.end type="italics"></emph.end></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 ſint tres vectes AB <lb></lb>CD EF bifariam diuiſi in <lb></lb>GHk, quorum fulcimenta <lb></lb>ſint ACE; & pondus L eo<lb></lb>dem modo in GHk ſit ap<lb></lb>penſum; quatuorq; ſint po<lb></lb>tentiæ æquales in BDFG <lb></lb>pondus L ſuſtinentes; ſimili <lb></lb>modo oſtendetur vnam<lb></lb>quamq; potentiam in BD <lb></lb>FG ſubſeptuplam eſſe ponde<lb></lb>ris L. </s> <s id="N14888">& ſi quatuor eſſent vectes, & quinq; potentiæ æquales pon<lb></lb>dus ſuſtinentes; eodem quoq; modo oſtendetur vnamquamq; <lb></lb>potentiam ſubnonuplam eſſe 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" place="text" 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></lb>culis, quarum altera ſupernè, altera vero in<lb></lb>fernè, ponderiq; alligata, diſpoſita fuerit, cir<lb></lb>cumducatur funis; altero eius extremo inferiori <pb xlink:href="036/01/160.jpg"></pb>trochleæ religato, altero verò à potentia pon<lb></lb>dus ſuſtinente retento: erit potentia ponderis <lb></lb>ſubquintupla. </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 ſit trochlea duos <lb></lb>habens orbiculos, quorum centra ſint BC; <lb></lb>ſitq; trochlea ſurſum appenſa duos alios ha<lb></lb>bens orbiculos, quorum centra ſint DE; funiſq; <lb></lb>per omnes circumducatur orbiculos, qui tro<lb></lb>chleæ inferiori religetur in F; ſit〈qué〉 poten<lb></lb>tia in G ſuſtinens pondus A. </s> <s id="id.2.1.157.3.1.1.0.a">dico poten<lb></lb>tiam in G ſubquintuplam eſſe ponderis A. <lb></lb></s> <s id="N148C8">ducantur Hk LM per centra BC horizon<lb></lb>ti æquidiſtantes, quas eodem modo, quo ſu<lb></lb>pra dictum eſt, eſſe tanquam vectes oſtende<lb></lb>mus, quorum fulcimenta kM, & pondus A <lb></lb>ex medio vtriuſq; vectis BC ſuſpenſum, & tres <lb></lb>potentiæ in LHC pondus ſuſtinentes, quas <lb></lb>ſimili modo æquales eſſe demonſtrabimus; fu<lb></lb>nes enim idem efficiunt, ac ſi eſſent potentiæ. </s> <s id="id.2.1.157.3.1.2.0"><lb></lb>& quoniam pondus æqualiter ex vtroq; ve<lb></lb>cte HK LM ponderat, quod quidem oſten<lb></lb>detur quoque, vt in præcedentibus demon<lb></lb><arrow.to.target n="note240"></arrow.to.target>ſtratum eſt: erit vnaquæq; potentia, tùm in <lb></lb>L, ſeu in G, quod idem eſt; tùm in H, atq; <lb></lb>in C, hoc eſt in F, ſubquintupla ponderis A. </s> <s id="id.2.1.157.3.1.2.0.a"><lb></lb>Potentia ergo in G ſuſtinens pondus A ipſius <lb></lb>A ſubquintupla erit. </s> <s id="id.2.1.157.3.1.3.0">quod oſtendere opor<lb></lb>tebat. <figure id="id.036.01.160.1.jpg" place="text" xlink:href="036/01/160/1.jpg"></figure></s> </p> <pb n="74" xlink:href="036/01/161.jpg"></pb> <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></lb>feratur circa alium orbiculum, <lb></lb>cuius centrum N, qui religetur <lb></lb>in O; ſimiliter duplici medio <lb></lb>(vt in ſeptima huius) demon<lb></lb>ſtrabitur potentiam in G pon<lb></lb>dus A ſuſtinentem ſubſexcu<arrow.to.target n="note241"></arrow.to.target><lb></lb>plam eſſe ponderis A. </s> <s id="id.2.1.157.5.1.1.0.a">Primùm <lb></lb>quidem ex tribus vectibus LM <lb></lb>Hk FP, quorum fulcimenta <lb></lb>ſunt MkP, & pondus in me <lb></lb>dio vectium appenſum; & tres <lb></lb>potentiæ in LHF æquales pon<lb></lb>dus ſuſtinéres. </s> <s id="id.2.1.157.5.1.2.0">deinde ex poten<arrow.to.target n="note242"></arrow.to.target><lb></lb>tiis in LHN, quarum vnaquæq; <lb></lb>ſubquintupla eſſet ponderis A. <lb></lb></s> <s id="N1492E">eſſent enim ambæ ſimul poten<lb></lb>tiæ in LH ſubduplæ ſexquialte<lb></lb>ræ ipſius ponderis, <expan abbr="potẽtia">potentia</expan> verò <lb></lb>in F ſubdecupla eſſet, cùm ſit ip<lb></lb>ſius N ſubdupla: ſed duæ quin <lb></lb>tæ cùm decima dimidium ef<lb></lb>ficiunt, quòd ſi per terna diui <lb></lb>datur, ſexta pars ponderis re<lb></lb>ſpondebit vnicuiq; potentiæ in <lb></lb>LHF. </s> <s id="N14946">ex quibus patet poten<lb></lb>tiam in G ſubſexcuplam eſſe <lb></lb>ponderis A. </s> <s id="N1494C">ſimiliterq; demon<lb></lb>ſtrabitur vnumquemque orbi<lb></lb>culum æqualem ſuſtinere por<lb></lb>tionem. <figure id="id.036.01.161.1.jpg" place="text" xlink:href="036/01/161/1.jpg"></figure></s> </p> <pb xlink:href="036/01/162.jpg"></pb> <p id="id.2.1.157.7.0.0.0" type="main"> <s id="id.2.1.157.7.1.1.0">Quòd ſi, vt in tertia figura <lb></lb>funis in O protrahatur; per <lb></lb>aliumq; circumducatur orbi<lb></lb>culum, cuius centrum Q; qui <lb></lb>deinde in R trochleæ relige<lb></lb>tur inferiori; erit potentia in <lb></lb><arrow.to.target n="note243"></arrow.to.target>G ponderis ſubſeptupla. </s> <s id="id.2.1.157.7.1.2.0">atq; <lb></lb>ita in infinitum procedendo <lb></lb>proportio potentiæ ad pon<lb></lb>dus quotcunq; ſubmulti<lb></lb>plex inueniri poterit. </s> <s id="id.2.1.157.7.1.3.0">dein<lb></lb>de ſemper oſtendetur vt in <lb></lb>præcedentibus; ſi potentia <lb></lb>pondus ſuſtinens fuerit, vel <lb></lb>ſubquadrupla, vel ſubquitu<lb></lb>pla, vel quouis alio modo ſe <lb></lb>habebit ad pondus; ſimiliter <lb></lb>vnumquemque funem, vel <lb></lb>quartam, vel quintam, vel <lb></lb>quamuis aliam partem ſuſti<lb></lb>nere ponderis, quemadmo<lb></lb>dum potentia ipſa; funes e<lb></lb>nim idem efficiunt, ac ſi tot <lb></lb>eſſent potentiæ: orbiculi ve<lb></lb>rò, ac ſi tot eſſent 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.158.1.1.2.0"><margin.target id="note241"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 6 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.158.1.1.3.0"><margin.target id="note242"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 8 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.158.1.1.4.0"><margin.target id="note243"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 8 <emph type="italics"></emph>Huius<emph.end type="italics"></emph.end></s> </p> <figure id="id.036.01.162.1.jpg" place="text" 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 manifeſtum eſt orbiculos trochleæ, cui <lb></lb>eſt alligatum pondus, efficere, vt pondus mino<pb n="75" xlink:href="036/01/163.jpg"></pb>re ſuſtineatur potentia, quàm ſit ipſum pondus; <lb></lb>quod quidem trochleæ ſuperioris orbiculi non <lb></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">Nouiſſe tamen oportet, quòd (vt fieri ſolet) inferioris tro<lb></lb>chleæ orbiculus, cuius centrum N, minor eſſe debet eo, cuius cen<lb></lb>trum C; hic autem minor adhuc eo, cuius centrum B; ac deniq; <lb></lb>ſi plures fuerint orbiculi in trochlea inferiori ponderi alligata, ſem<lb></lb>per cæteris maior eſſe debet, qui annexo ponderi eſt propinquior. </s> <s id="id.2.1.159.3.1.2.0"><lb></lb>oppoſito autem modo diſponendi ſunt in trochlea ſuperiori. </s> <s id="id.2.1.159.3.1.3.0">quod <lb></lb>fieri conſueuit, ne funes inuicem complicentur; nam quantùm <lb></lb>ad orbiculos attinet, ſiue magni fuerint, ſiue parui, nihil refert; <lb></lb>cùm ſemper idem ſequatur. </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 eſt, quod etiam ex dictis facilè patet, ſi <lb></lb>funis, ſiue religetur in R trochleæ inferiori, ſiue in S, maximam <lb></lb>indè oriri differentiam inter potentiam, & pondus: nam ſi relige<lb></lb>tur in S, erit potentia in G ponderis ſubſexcupla. </s> <s id="id.2.1.159.4.1.2.0">ſi verò in R, <lb></lb>ſubſeptupla. </s> <s id="id.2.1.159.4.1.3.0">quod trochleæ ſuperiori non contingit, quia ſiue <lb></lb>religetur funis (vt in præcedenti figura) in T, ſiue in O; ſem<lb></lb>per potentia in G ſubſexcupla erit ipſius 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">Poſt hæc conſiderandum eſt, quonam modo vis moueat pon<lb></lb>dus; necnon potentiæ mouentis, ponderiſq; moti ſpatium, atque <lb></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 ſurſum appenſæ <lb></lb>fuerit circumuolutus, cuius altero extremo ſit al<lb></lb>ligatum pondus; alteri autem mouens collocata <lb></lb>ſit potentia: mouebit hæc vecte horizonti ſem<lb></lb>per æquidiſtante. </s> </p> <pb xlink:href="036/01/164.jpg"></pb> <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, ſit orbiculus trochleæ ſur<lb></lb>ſum appenſæ' cuius centrum K; ſit deinde <lb></lb>funis HBCDEF aligatus ponderi A in H, <lb></lb>orbiculoq; circumductus; ſitq; trochlea ita in <lb></lb>L appenſa, & nullum alium habeat motum <lb></lb>præter liberam orbiculi circa axem verſionem; <lb></lb>ſitq; potentia in F mouens pondus A. </s> <s id="id.2.1.159.9.1.1.0.a">Dico <lb></lb>potentiam in F ſemper mouere pondus A <lb></lb>vecte horizonti æquidiſtante. </s> <s id="id.2.1.159.9.1.2.0">ducatur BKE <lb></lb>horizonti æquidiſtans; ſintq; BE puncta, vbi <lb></lb>funes BH, & EF circulum tangunt; erit BkE <lb></lb><arrow.to.target n="note244"></arrow.to.target>vectis, cuius fulcimentum eſt in eius medio <lb></lb>k. </s> <s id="id.2.1.159.9.1.3.0">ſicut ſupra oſtenſum eſt. </s> <s id="id.2.1.159.9.1.4.0">dum itaq; vis <lb></lb>in F deorſum tendit verſus M, vectis EB <lb></lb>mouebitur, cùm totus orbiculus moueatur, <lb></lb><figure id="id.036.01.164.1.jpg" place="text" xlink:href="036/01/164/1.jpg"></figure><lb></lb>hoc eſt circumuertatur. </s> <s id="id.2.1.159.9.1.5.0">dum igitur F eſt in M, ſit punctum E ve<lb></lb>ctis vſq; ad I motum; B autem vſq; ad C, ita vt vectis ſit in <lb></lb>CI. </s> <s id="id.2.1.159.9.1.5.0.a">fiat deinde NM æqualis ipſi FE: & quando punctum E <lb></lb>erit in I, <expan abbr="tnnc">tunc</expan> funis punctum, quod erat in E, erit in N: quod au<lb></lb>tem erat in B erit in C; ita vt ducta CI per centrum K tranſeat. </s> <s id="id.2.1.159.9.1.6.0"><lb></lb>dum autem B eſt in C, ſit punctum H in G; eritq; BH ipſi <lb></lb>CBG æqualis; cùm ſit idem funis. </s> <s id="id.2.1.159.9.1.7.0">& quoniam dum EF tendit <lb></lb>in NM, adhuc ſemper remanet EFM horizonti perpendicularis, <lb></lb>circulumq; tangens in puncto E; ita vt ducta à puncto E per cen<lb></lb>trum k, ſit ſemper horizonti æquidiſtans. </s> <s id="id.2.1.159.9.1.8.0">quod idem euenit funi <lb></lb>BG, & puncto B. </s> <s id="N14AA4">dum igitur circulus, ſiue orbiculus circumuer<lb></lb>titur, ſemper mouetur vectis EB, ſemperq; adhuc remanet alius <lb></lb>vectis in EB. </s> <s id="id.2.1.159.9.1.8.0.a">ſiquidem ex ipſius rotulæ natura, in qua ſemper <lb></lb>dum mouetur, remanet diameter ex B in E (quæ vectis vicem ge<lb></lb>rit) euenit, vt recedente vna, ſemper altera ſuccedat; eiuſmodi <lb></lb>durante circumductione: atq; ita fit, vt potentia ſemper moueat <lb></lb>pondus vecte EB horizonti æquidiſtante. </s> <s id="id.2.1.159.9.1.9.0">quod demonſtrare opor<lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="76" xlink:href="036/01/165.jpg"></pb> <p id="id.2.1.161.1.0.0.0" type="main"> <s id="id.2.1.161.1.2.1.0">Iiſdem poſitis, ſpatium potentiæ pondus <lb></lb>mouentis eſt æquale ſpatio eiuſdem ponderis <lb></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 oſtenſum eſt, dum F eſt in M, pondus A, hoc <lb></lb>eſt punctum H eſſe in G; & cùm funis HBCDEF ſit æqualis <lb></lb>GBCDENFM, eſt enim idem funis; dempto igitur communi <lb></lb>GBCDENF, erit HG ipſi FM æqualis. </s> <s id="id.2.1.161.2.1.2.0">ſimiliterq; oſtende<lb></lb>tur, deſcenſum F ſemper æqualem eſſe aſcenſui H. </s> <s id="N14AE6">ergo ſpatium <lb></lb>potentiæ æquale eſt ſpatio ponderis. </s> <s id="id.2.1.161.2.1.3.0">quod erat demonſtran<lb></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></lb>ſpatium in æquali tempore mouet, tàm fune <lb></lb>hoc modo orbiculo trochleæ ſurſum appenſæ <lb></lb>circumuoluto, quàm ſine trochlea: dummo<lb></lb>do ipſius potentiæ lationes in velocitate ſint æ<lb></lb>quales. </s> </p> <pb xlink:href="036/01/166.jpg"></pb> <p id="id.2.1.161.5.0.0.0" type="main"> <s id="id.2.1.161.5.1.1.0">Iiſdem poſitis ſit aliud pondus P <lb></lb>æquale ponderi A, cui alligatus ſit <lb></lb>funis TQ <expan abbr="horizõti">horizonti</expan> <expan abbr="perpẽdicularis">perpendicularis</expan>; <lb></lb>et ſit TQ ipſi HB æqualis; moueat<lb></lb>〈qué〉 <expan abbr="potẽtia">potentia</expan> in Q <expan abbr="põdus">pondus</expan> P ſurſum <lb></lb>ad rectos angulos horizonti, quem <lb></lb>admodum mouetur pondus A. </s> <s id="id.2.1.161.5.1.1.0.a">di<lb></lb>co per æquale ſpatium in eodem <lb></lb>tempore potentiam in Q pondus <lb></lb>P, & potentiam in F pondus A <lb></lb>mouere. </s> <s id="id.2.1.161.5.1.2.0">quod idem eſt, ac ſi eſſet <lb></lb>idem pondus in æquali tempore <lb></lb>motum; ſicut propoſuimus. </s> <s id="id.2.1.161.5.1.3.0">Pro<lb></lb>ducatur EF in S, & TQ in R; <lb></lb>fiantq; QR FS non ſolum inter <lb></lb>ſe ſe, verùm etiam ipſi BH æqua<lb></lb>les. </s> <s id="id.2.1.161.5.1.4.0">Cùm autem TQ QR ſint <lb></lb>ipſis HB FS æquales, & vis in Q <lb></lb>moueat pondus P per rectam T <lb></lb>QR; vis autem in F moueat A <lb></lb>per rectam HB, & velocitates <lb></lb><figure id="id.036.01.166.1.jpg" place="text" xlink:href="036/01/166/1.jpg"></figure><lb></lb>motuum vtriuſq; potentiæ ſint æquales; tunc in eodem tempore <lb></lb>potentia in Q erit in R, & potentia in F erit in S; cùm ſpatia ſint <lb></lb>æqualia. </s> <s id="id.2.1.161.5.1.5.0">ſed dum potentia in Q eſt in R, pondus P, hoc eſt <lb></lb>punctum T erit in Q; cùm TQ ſit ipſi QR æqualis. </s> <s id="id.2.1.161.5.1.6.0">& dum po<lb></lb>tentia in F eſt in S, pondus A, hoc eſt punctum H erit in B; ſed <lb></lb>ſpatium TQ æquale eſt ſpatio HB, potentiæ ergo in FQ æquali <lb></lb>ter motæ pondera PA æqualia per æqualia ſpatia in eodem tempo<lb></lb>re mouebunt. </s> <s id="id.2.1.161.5.1.7.0">quod erat demonſtrandum </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></lb>fuerit circumuolutus, qui in altero eius extre<pb n="77" xlink:href="036/01/167.jpg"></pb>mo alicubi religetur, altero autem à potentia <lb></lb>mouente pondus appræhenſo; vecte ſemper ho<lb></lb>rizonti æquiſtante 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></lb>CED trochleæ ponderi A alli<lb></lb>gatæ ex kH; ſitq; KH ad rectos <lb></lb>angulos horizonti, ita vt pon<lb></lb>dus ſemper trochleæ motum, ſi<lb></lb>ue ſurſum, ſiue deorſum factum <lb></lb>ſequatur; ſitq; orbiculi centrum <lb></lb>K; & funis orbiculo circumuo<lb></lb>lutus ſit BCDEF, qui relige<lb></lb>tur in B, ita vt in B immobilis <lb></lb>maneat; & ſit potentia in F mo<lb></lb>uens pondus A. </s> <s id="id.2.1.161.8.1.2.0.a">dico potentiam <lb></lb>in F ſemper mouere <expan abbr="põdus">pondus</expan> A ve<lb></lb>cte horizonti æquidiſtante. </s> <s id="id.2.1.161.8.1.3.0">ſint <lb></lb>BC EF inter ſe ſe, ipſiq; kH æ<lb></lb>quidiſtantes, & eiuſdem kH ho<lb></lb>rizonti perpendiculares, tangen<lb></lb>teſq; <expan abbr="circulũ">circulum</expan> CED in EC <expan abbr="pũctis">punctis</expan>; <lb></lb>et connectatur EC, quæ per cen<arrow.to.target n="note245"></arrow.to.target><lb></lb>trum k tranſibit, horizontiq; <lb></lb>æquidiſtans erit; ſicuti prius di<lb></lb>ctum eſt. </s> <s id="id.2.1.161.8.1.4.0">Quoniam enim or<lb></lb>biculus CED circa eius cen<lb></lb>trum K vertitur; ideo dum vis <lb></lb>in F trahit ſurſum punctum E, <lb></lb>deberet punctum C deſcende<lb></lb>re, ac trahere deorſum B; ſed fu<lb></lb><figure id="id.036.01.167.1.jpg" place="text" xlink:href="036/01/167/1.jpg"></figure><lb></lb>nis in B eſt immobilis, & BC <expan abbr="deſcedere">descendere</expan> non poteſt; quare dum <lb></lb>potentia in F trahit ſurſum E, totus orbiculus ſurſum mouebitur; <lb></lb>ac per conſequens tota trochlea, & pondus; & EkC erit tanquam <arrow.to.target n="note246"></arrow.to.target><lb></lb>vectis, cuius fulcimentum erit C; eſt enim punctum C propter BC <lb></lb>ferè immobile, potentia verò mouens vectem eſt in F fune EF, <pb xlink:href="036/01/168.jpg"></pb>& pondus in k appenſum. </s> <s id="id.2.1.161.8.1.5.0"><lb></lb>quòd ſi punctum C omnino fue<lb></lb>rit immobile, moueaturq; ve<lb></lb>ctis EC in NC; & diuidatur <lb></lb>NC bifariam in L: erunt CL <lb></lb>LN ipſis Ck KE æquales. </s> <s id="id.2.1.161.8.1.6.0"><lb></lb>quare ſi vectis EC eſſet in CN, <lb></lb>punctum k eſſet in L; & ſi du<lb></lb>catur LM horizonti perpendi<lb></lb>cularis, quæ ſit etiam æqualis <lb></lb>kH; eſſet pondus A, hoc eſt <lb></lb>punctum H in M. </s> <s id="id.2.1.161.8.1.6.0.a">ſed quoniam <lb></lb>potentia in F dum tendit ſur<lb></lb>ſum mouendo orbiculum, ſem<lb></lb>per mouetur ſuper rectam EFG, <lb></lb>quæ ſemper eſt quoq; æquidi<lb></lb>ſtans BC; neceſſe erit orbicu<lb></lb>lum trochleæ ſemper inter li<lb></lb>neas EG BC eſſe: & centrum <lb></lb>k, cum ſit in medio, ſuper <lb></lb>rectam lineam HkT ſemper <lb></lb>moueri. </s> <s id="id.2.1.161.8.1.7.0">Itaq; ducatur per L li<lb></lb>nea PTLQ horizonti, & EC <lb></lb>æquidiſtans, quæ ſecet Hk pro<lb></lb>ductam in T; & centro T, ſpa<lb></lb>tio verò TQ, circulus deſcriba<lb></lb><figure id="id.036.01.168.1.jpg" place="text" xlink:href="036/01/168/1.jpg"></figure><lb></lb>tur QRPS, qui æqualis erit circulo CED; & puncta PQ tangent fu<lb></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 eſt PECQ, & <lb></lb>PT TQ ipſis EK kC ſunt æquales. </s> <s id="id.2.1.161.8.1.9.0">deinde per T ducatur R <lb></lb>TS diameter circuli PQS æquidiſtans ipſi NC; fiat〈qué〉 TO æqua <lb></lb>lis kH. </s> <s id="id.2.1.161.8.1.9.0.a">dum autem centrum k motum erit vſq; ad lineam PQ, <lb></lb>tunc centrum k erit in T. </s> <s id="N14C4A">oſtenſum eſt enim centrum orbiculi ſu<lb></lb>per rectam HT ſemper moueri. </s> <s id="id.2.1.161.8.1.10.0">idcirco vt centrum k ſit in li<lb></lb>nea PQ ipſi EC æquidiſtante, neceſſe eſt vt ſit in T. </s> <s id="N14C53">& vt vectis <lb></lb>EC eleuetur in angulo ECN, neceſſe eſt, vt ſit in RS, non au<lb></lb><arrow.to.target n="note248"></arrow.to.target>tem in CN: angulus enim RSE angulo NCE eſt æqualis, & ſic <pb n="78" xlink:href="036/01/169.jpg"></pb>fulcimentum C non eſt penitus immobile. </s> <s id="id.2.1.161.8.1.11.0">cùm totus orbiculus ſur<lb></lb>ſum moueatur, toruſq; mutet totum locum; habet tamen C ratio <lb></lb>nem fulcimenti, quia minus mouetur C, quàm k, & E: punctum <lb></lb>enim E mouetur vſq; ad R, & K vſq; ad T, punctum verò C vſq; <lb></lb>ad S tantùm. </s> <s id="id.2.1.161.8.1.12.0">quare dum centrum K eſt in T, poſitio orbiculi erit <lb></lb>QR PS: & pondus A. hoc eſt punctum H erit in O; cùm TO <lb></lb>ſit æqualis kH; poſitio verò EC, ſcilicet vectis moti, erit RS, po<lb></lb>tentiaq; in F mota erit ſurſum per rectam EFG. </s> <s id="id.2.1.161.8.1.12.0.a">eodem autem <lb></lb>tempore, quo k erit in T, ſit potentia in G: dum autem vectis EC <lb></lb>hoc modo mouetur, adhuc ſemper remanent GP BQ inter ſe ſe æ<lb></lb>quidiſtantes, atq; horizonti perpendiculares, ita vt vbi orbiculum <lb></lb>tangunt, vt in punctis PQ; ſemper linea PQ erit diameter orbi<lb></lb>culi, & tanquam vectis horizonti æquidiſtans. </s> <s id="id.2.1.161.8.1.13.0">dum igitur orbi<lb></lb>culus mouetur, & circumuertitur, ſemper etiam mouetur vectis <lb></lb>EC, & ſemper remanet alius vectis in orbiculo horizonti æquiſtans, <lb></lb>vt PQ; ita vt potentia in F ſemper moueat pondus vecte hori<lb></lb>zonti æquidiſtante, cuius fulcimentum erit ſemper in linea CB; & <lb></lb>pondus in medio vectis appenſum; potentiaq; in linea EG. </s> <s id="id.2.1.161.8.1.13.0.a">quod <lb></lb>erat oſtendendum. </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"></emph>Ex<emph.end type="italics"></emph.end> 1 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.162.1.1.2.0"><margin.target id="note246"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.162.1.1.3.0"><margin.target id="note247"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 34 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.162.1.1.4.0"><margin.target id="note248"></margin.target>29 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.163.1.0.0.0" type="main"> <s id="id.2.1.163.1.1.1.0">Iiſdem poſitis, ſpatium potentiæ pondus <lb></lb>mouentis duplum eſt ſpatii eiuſdem ponderis <lb></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 oſtenſum ſit, dum k eſt in T, pondus A, hoc eſt <lb></lb>punctum H eſſe in O, & in eodem etiam tempore potentiam in <lb></lb>F eſſe in G: & quoniam funis BCDEF eſt æqualis funi BQS <lb></lb>PG; funis enim eſt idem; & funis circa ſemicirculum CDE eſt <lb></lb>æqualis funi circa ſemicirculum QSP; demptis igitur communi<lb></lb>bus BQ, & FP; erit reliquus FG ipſis CQ, & EP ſimul ſumptis <lb></lb>æqualis. </s> <s id="id.2.1.163.2.1.2.0">ſed EP ipſi TK eſt æqualis, & CQ ipſi quoq; Tk æqualis, <lb></lb>ſunt enim Pk TC parallelogramma rectangula; quare lineæ EP <lb></lb>CQ ſimul ipſius Tk duplæ erunt. </s> <s id="id.2.1.163.2.1.3.0">funis igitur FC ipſius TK du<lb></lb>plus erit. </s> <s id="id.2.1.163.2.1.4.0">& quoniam kH eſt æqualis TO, dempto communi kO, <lb></lb>erit kT ipſi HO æqualis; quare funis FG ipſius HO duplus erit; <pb xlink:href="036/01/170.jpg"></pb>hoc eſt ſpatium potentiæ ſpatii ponderis duplum. </s> <s id="id.2.1.163.2.1.5.0">quod erat <lb></lb>demonſtrandum. </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></lb>pore per dimidium ſpatium mouebit fune circa <lb></lb>orbiculum trochleæ ponderi alligatæ reuoluto, <lb></lb>quàm ſine trochlea; dummodo ipſius potentiæ <lb></lb>velocitates motuum ſint 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 (iiſdem poſi<lb></lb>tis) aliud pondus V æqua <lb></lb>le ponderi A, cui alligatus <lb></lb>ſit funis 9X; ſitq; poten<lb></lb>tia in X mouens pondus <lb></lb>V. </s> <s id="id.2.1.163.4.1.1.0.a">dico ſi vtriuſq; poten<lb></lb>tiæ motuum velocitates <lb></lb>ſint æquales, in eodem <lb></lb>tempore potentiam in F <lb></lb>mouere pondus A per di<lb></lb>midium ſpatium eius, per <lb></lb>quod à potentia in X mo<lb></lb>uetur pondus V; quod <lb></lb>idem eſt, ac ſi eſſet idem <lb></lb>pondus in æquali tempo<lb></lb>re motum. </s> <s id="id.2.1.163.4.1.2.0">Moueat po<lb></lb>tentia in X pondus V, po<lb></lb>tentiaq; perueniat in Y; <lb></lb>ſitq; XY æqualis ipſi FG; <lb></lb>& fiat YZ æqualis X9, ita <lb></lb>vt quando potentia in X <lb></lb>erit in Y, ſit pondus V, <lb></lb>hoc eſt punctum 9 in Z. </s> <s id="id.2.1.163.4.1.2.0.a"><lb></lb>ſed 9 Z eſt æqualis FG, <lb></lb><figure id="id.036.01.170.1.jpg" place="text" xlink:href="036/01/170/1.jpg"></figure> <pb n="79" xlink:href="036/01/171.jpg"></pb>cùm ſit æqualis XY; ergo 9 Z ipſius HO dupla erit. </s> <s id="id.2.1.163.4.1.3.0">Itaq; dum poten<lb></lb>tiæ erunt in GY, pondera AV erunt in OZ. </s> <s id="N14D5D">in eodem autem <lb></lb>tempore erunt potentiæ in GY, ipſarum enim velocitates mo<lb></lb>tuum ſunt æquales; quare vis in F pondus A in eodem tempore <lb></lb>mouebit per dimidium ſpatium eius, per quod mouetur à poten<lb></lb>tia in X pondus V: & pondera ſunt æqualia; Potentia ergo idem <lb></lb>pondus in æquali tempore per dimidium ſpatium mouebit fune, <lb></lb>trochleaq; hoc modo ponderi alligata, quàm ſine trochlea; dum <lb></lb>modo potentiæ motuum velocitates ſint æquales. </s> <s id="id.2.1.163.4.1.4.0">quod erat de<lb></lb>monſtrandum. </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></lb>tero eius extremo alicubi religato, altero au<lb></lb>tem à potentia pondus mouente detento; poten<lb></lb>tia vectibus horizonti ſemper æquidiſtantibus <lb></lb>mouebit. </s> </p> <pb xlink:href="036/01/172.jpg"></pb> <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, ſit orbiculus CED tro<lb></lb>chleæ ponderi alligatæ ex kS ad rectos an<lb></lb>gulos horizonti; ita vt pondus ſemper eius <lb></lb>motum ſurſum, ac deorſum factum ſequa<lb></lb>tur. </s> <s id="id.2.1.163.8.1.2.0">ſit deinde orbiculus circa centrum L <lb></lb>trochleæ ſurſum appenſæ ſitq; funis circa <lb></lb>orbiculos reuolutus BCDEHMNO, <lb></lb>qui religatus ſit in B; ſitq; vis in O mouens <lb></lb>pondus A mouendo ſe deorſum per OP. </s> <s id="id.2.1.163.8.1.2.0.a"><lb></lb>dico potentiam in O ſemper mouere pon<lb></lb>dus A vectibus horizonti ſemper æquidi<lb></lb>ſtantibus. </s> <s id="id.2.1.163.8.1.4.0">ducatur NH per centrum L ho<lb></lb><arrow.to.target n="note249"></arrow.to.target>rizonti æquidiſtans, quæ erit vectis orbi<lb></lb>culi, cuius centrum eſt L. </s> <s id="N14DB4">ducatur deinde <lb></lb>EC per centrum k ſimiliter horizonti æqui <lb></lb><arrow.to.target n="note250"></arrow.to.target>diſtans, quæ etiam erit vectis orbiculi, cu<lb></lb>ius centrum eſt k. </s> <s id="id.2.1.163.8.1.5.0">Moueatur potentia in <lb></lb>O deorſum, quæ dum deorſum mouetur, ve<lb></lb>ctem NH mouebit; & dum vectis moue<lb></lb><arrow.to.target n="note251"></arrow.to.target>tur, N deorſum mouebitur, H verò ſur<lb></lb>ſum, vti ſupra dictum eſt. </s> <s id="id.2.1.163.8.1.6.0">dum autem H <lb></lb>mouetur ſurſum, mouet etiam ſurſum E; & <lb></lb>vectem EC, cuius fulcimentum eſt C, ſed <lb></lb>fulcimentum C non poteſt mouere deor<lb></lb>ſum B; ideo orbiculus, cuius centrum K, ſur<lb></lb><figure id="id.036.01.172.1.jpg" place="text" xlink:href="036/01/172/1.jpg"></figure><lb></lb>ſum mouebitur, & per conſequens trochlea, & pondus A; vt in <lb></lb>præcedenti dictum eſt. </s> <s id="id.2.1.163.8.1.7.0">& quoniam ob eandem cauſam in præce<lb></lb>dentibus aſsignatam in HN, & EC ſemper remanent vectes hori<lb></lb>zonti æquidiſtantes; potentia ergo mouens pondus A ſemper <lb></lb>eum mouebit vectibus horizonti æquidiſtantibus. </s> <s id="id.2.1.163.8.1.8.0">quod erat o<lb></lb>ſtendendum. </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"></emph>Et<emph.end type="italics"></emph.end> 10 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.164.1.1.2.0"><margin.target id="note250"></margin.target>11 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.164.1.1.3.0"><margin.target id="note251"></margin.target>10 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 ſi funis circa plures ſit reuolutus orbiculos; ſimiliter oſtende<lb></lb>tur, potentiam mouere pondus vectibus horizonti ſemper æqui<lb></lb>diſtantibus: & vectes orbiculorum trochleæ ſuperioris ſemper <lb></lb>eſſe, vt HN, quorum fulcimenta erunt ſemper in medio: vectes au<lb></lb>tem orbiculorum trochleæ inferioris ſemper exiſtere, vt EC; quo<pb n="80" xlink:href="036/01/173.jpg"></pb>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">Iiſdem poſitis, ſpatium potentiæ duplum eſt <lb></lb>ſpatii 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 vſq; ad centrum R; & orbiculus ſit FTG. <lb></lb></s> <s id="N14E39">deinde per centrum R ducatur GF ipſi EC æquidiſtans: tangent <lb></lb>funes EH CB orbiculum in GF punctis. </s> <s id="id.2.1.165.3.1.2.0">fiat deniq; RQ æqua <lb></lb>lis KS. </s> <s id="N14E42">dum igitur k erit in R; pondus A, ſcilicet punctum S erit <lb></lb>in q. </s> <s id="N14E46">& dum centrum orbiculi eſt in R, ſit potentia in O mota <lb></lb>in P. </s> <s id="id.2.1.165.3.1.2.0.a">& quoniam funis BCDEHMNO eſt æqualis funi BFT <lb></lb>GHMNP; eſt enim idem funis; & FTG æqualis eſt CDE; dem<lb></lb>ptis igitur communibus BF, & GHMNO, erit reliquus OP ip<lb></lb>ſis FCEG ſimul ſumptis æqualis: & per conſequens duplus kR, <lb></lb>& QS & cùm OP ſit ſpatium potentiæ motæ, & SQ ſpatium pon<lb></lb>deris moti; erit ſpatium potentiæ duplum ſpatii ponderis. </s> <s id="id.2.1.165.3.1.3.0">quod <lb></lb>erat oſtendendum. </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></lb>tempore per dimidium ſpatium mouebit fune <lb></lb>circa duos orbiculos reuoluto, quorum vnus <lb></lb>ſit trochleæ ſuperioris, alter verò ſit trochleæ <lb></lb>ponderi alligatæ; quàm ſine trochleis: dummo<lb></lb>do ipſius potentiæ lationes ſint æqualiter ve<lb></lb>loces. </s> </p> <pb xlink:href="036/01/174.jpg"></pb> <p id="id.2.1.165.6.0.0.0" type="main"> <s id="id.2.1.165.6.1.1.0">Iiſdem namq; poſitis, ſit pon<lb></lb>dus V æquale ipſi A, cui alliga<lb></lb>tus ſit funis X9; ſitq; <expan abbr="potẽtia">potentia</expan> in X <lb></lb>mouens <expan abbr="põdus">pondus</expan> V; quæ dum pon<lb></lb>dus mouet, perueniat in Y: fiant <lb></lb>〈qué〉 XY Z9 ipſi OP æquales; <lb></lb>erit Z9 dupla QS. </s> <s id="N14E8B">& ſi vtriuſ<lb></lb>que potentiæ velocitates mo<lb></lb>tuum ſint æquales; patet pon<lb></lb>dus V duplum pertranſire ſpa<lb></lb>tium in eodem tempore eìus, <lb></lb>quod pertranſit pondus A. </s> <s id="id.2.1.165.6.1.1.0.a">in eo<lb></lb>dem enim tempore potentia in <lb></lb>X peruenit ad Y, & potentia in <lb></lb>O ad P; ponderaq; ſimiliter in <lb></lb>Z Q. </s> <s id="N14EA2">quod erat demonſtran<lb></lb>dum. </s> </p> <figure id="id.036.01.174.1.jpg" place="text" 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 ſingulos duarum trochlearum <lb></lb>orbiculos, quarum altera ſupernè, altera verò <lb></lb>infernè, ponderiq; alligata fuerit, reuoluto; <lb></lb>altero etiam eius extremo inferiori trochleæ re<pb n="81" xlink:href="036/01/175.jpg"></pb>ligata, altero autem à mouente potentia deten<lb></lb>to: erit decurſum trahentis potentiæ ſpatium, mo<lb></lb>ti ponderis ſpatii 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; ſit BCD orbiculus tro<lb></lb>chleæ ponderi A ex EQ ſuſpenſo alligatæ; <lb></lb>ſitq; orbiculi centrum E; ſit deinde FGH <lb></lb>orbiculus trochleæ ſurſum appenſæ, cuius <lb></lb>centrum k; ſitq; funis LFGHDCBM <lb></lb>circa omnes reuolutus orbiculos, tro<lb></lb>chleæq; inferiori in L religatus: ſitq; in <lb></lb>M potentia mouens. </s> <s id="id.2.1.165.8.1.2.0">dico ſpatium de<lb></lb>curſum à potentia in M, dum mouet pon<lb></lb>dus, triplum eſſe ſpatii moti ponderis A. </s> <s id="id.2.1.165.8.1.2.0.a"><lb></lb>Moueatur potentia in M vſq; ad N; & <lb></lb>centrum E ſit motum vſq; ad O; & L vſ<lb></lb>que ad P; atq; pondus A, hoc eſt pun<lb></lb>ctum Q vſq; ad R; orbiculuſq; motus, ſit <lb></lb>TSV. </s> <s id="N14EED">ducantur per EO lineæ ST BD <lb></lb>horizonti æquidiſtantes, quæ inter ſe ſe <lb></lb>quoq; æquidiſtantes erunt. </s> <s id="id.2.1.165.8.1.3.0">quoniam au<lb></lb>tem dum E eſt in O, punctum Q eſt in <lb></lb>R; erit EQ æqualis OR, & EO ipſi QR <lb></lb>æqualis; ſimiliter LQ æqualis erit PR, <lb></lb>& L P ipſi QR æqualis. </s> <s id="id.2.1.165.8.1.4.0">tres igitur QR <lb></lb>EO LP inter ſe ſe æquales erunt; quibus <lb></lb>etiam ſunt æquales BS DT. </s> <s id="id.2.1.165.8.1.4.0.a">& quoniam fu<lb></lb>nis LFGHDCBM æqualis eſt funi PF <lb></lb>GHTVSN, cùm ſit idem funis, & qui <lb></lb>circa ſemicirculum TVS eſt æqualis funi <lb></lb>circa ſemicirculum BCD; demptis igi<lb></lb>tur communibus PFGHT' & SM; erit <lb></lb>reliquus MN tribus BS LP DT ſimul <lb></lb>ſumptis æqualis. </s> <s id="id.2.1.165.8.1.5.0">BS verò LP DT ſimul <lb></lb>tripli ſunt EO, & ex conſequenti QR. <lb></lb><figure id="id.036.01.175.1.jpg" place="text" xlink:href="036/01/175/1.jpg"></figure></s> <pb xlink:href="036/01/176.jpg"></pb> <s id="id.2.1.165.8.1.5.0.a">ſpatium igitur MN translatæ potentiæ ſpatii QR ponderis mo<lb></lb>ti triplum erit. </s> <s id="id.2.1.165.8.1.6.0">quod erat demonſtrandum. </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 manifeſtum eſt, eadem enim po<lb></lb>tentia in æquali tempore ſpatio ſecundùm triplum ampliori ſine <lb></lb>huiuſmodi trochleis idem pondus mouebit, quàm cum eiſdem <lb></lb>hoc modo accomodatis. </s> <s id="id.2.1.165.9.1.2.0">ſpatium ponderis ſine trochleis moti <lb></lb>æquale eſt ſpatio potentiæ. </s> <s id="id.2.1.165.9.1.3.0">& hoc modo in omnibus inueniemus <lb></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></lb>los, quarum altera ſupernè vnico dumtaxat, al <lb></lb>tera verò inſernè, duobus autem inſignita or<lb></lb>biculis, ponderiq́ue alligata fuerit, reuoluto; <lb></lb>altero eius eſtremo alicubi religato, altero autem <lb></lb>à potentia pondus mouente detento: erit decur<lb></lb>ſum trahentis potentiæ ſpatium moti ponderis <lb></lb>ſpatii quadruplum. </s> </p> <pb n="82" xlink:href="036/01/177.jpg"></pb> <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, ſint duo orbiculi, <expan abbr="quorũ">quorum</expan> <expan abbr="cẽtra;">cen<lb></lb>tra</expan> k I trochleæ ponderi alligatæ k <foreign lang="grc">α</foreign>; ita vt <lb></lb>pondus motum trochleæ ſurſum, & deorſum <lb></lb>ſemper ſequatur: ſit deinde orbiculus, cuius cen<lb></lb>trum L, trochleæ ſurſum appenſæ in <foreign lang="el">d</foreign>; ſitq; <lb></lb>funis circa omnes orbiculos circumuolutus BC<lb></lb>DEFGHZMNO, religatuſq; in B; ſitq; po<lb></lb>tentia in O mouens pondus A. </s> <s id="id.2.1.165.13.1.1.0.a">dico ſpatium, <lb></lb>quod mouendo pertranſit potentia in O, qua<lb></lb>druplum eſſe ſpatii moti ponderis A. </s> <s id="id.2.1.165.13.1.1.0.b">mouean<lb></lb>tur orbiculi trochleæ ponderi alligatæ; & dum <lb></lb>centrum k eſt in R, centrum I ſit in S, & pon<lb></lb>dus A, hoc eſt punctum <foreign lang="grc">α</foreign>in <foreign lang="grc">β</foreign>: erunt IS kR <lb></lb> <foreign lang="grc">αβ</foreign>inter ſe ſe æquales, itemq; k I ipſi RS e<lb></lb>rit æqualis. </s> <s id="id.2.1.165.13.1.2.0">orbiculi enim inter ſe ſe eandem <lb></lb>ſemper ſeruant diſtantiam; & k <foreign lang="grc">α</foreign>ipſi R <foreign lang="grc">β</foreign>æ<lb></lb>qualis erit. </s> <s id="id.2.1.165.13.1.3.0">ducantur per orbiculorum centra <lb></lb>lineæ FH QT EC VX NZ horizonti æqui<lb></lb>diſtantes, quæ tangent funes in FHQTEC <lb></lb>VX NZ punctis, & inter ſe ſe quoq; æquidi<lb></lb>ſtantes erunt: & EQ CT VN XZ non ſo<lb></lb>lum inter ſe ſe, ſed etiam ipſis IS KR <foreign lang="grc">αβ</foreign>æqua<lb></lb>les erunt. </s> <s id="id.2.1.165.13.1.4.0">& dum centra kI ſunt in RS, po<lb></lb>tentia in O ſit mota in P. </s> <s id="id.2.1.165.13.1.4.0.a">& quoniam funis <lb></lb>BCDEFGHZMNO eſt æqualis funi BT9 <lb></lb>QFGHXYVP, eſt enim <expan abbr="idẽ">idem</expan> funis, & funes cir<lb></lb><figure id="id.036.01.177.1.jpg" place="text" xlink:href="036/01/177/1.jpg"></figure><lb></lb>ca T9Q XYV ſemicirculos ſunt æquales funibus, qui ſunt circa <lb></lb>CDE ZMN; Demptis igitur communibus BT, QF GHX, <lb></lb>& VO; erit OP æqualis ipſis VN XZ CT QE ſimul ſumptis. </s> <s id="id.2.1.165.13.1.5.0"><lb></lb>quatuor verò VN ZX CT QE ſunt inter ſe ſe æquales, & ſimul <lb></lb>quadruplæ kR, & <foreign lang="grc">αβ</foreign>; quare OP quadrupla erit ipſius <foreign lang="grc">αβ</foreign>. </s> <s id="id.2.1.165.13.1.6.0">ſpa<lb></lb>tium igitur potentiæ quadruplum eſt ſpatii ponderis. </s> <s id="id.2.1.165.13.1.7.0">quod erat <lb></lb>oſtendendum. </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 ſi funis in P circa alium adhuc reuoluatur orbiculum verſus <lb></lb><foreign lang="el">d</foreign>, potentia〈qué〉 mouendo ſe deorſum moueat ſurſum pondus; ſimi <lb></lb>liter oſtendetur ſpatium potentiæ quadruplum eſſe ſpatii ponderis. </s> </p> <pb xlink:href="036/01/178.jpg"></pb> <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></lb>teri orbiculo, qui deinde trochleæ in<lb></lb><arrow.to.target n="note252"></arrow.to.target>feriori religetur; erit potentia in O <lb></lb>ſuſtinens pondus A ſubquintupla pon<lb></lb>deris. </s> <s id="id.2.1.165.16.1.2.0">& ſi in O ſit potentia mouens <lb></lb>pondus A; ſimiliter demonſtrabitur <lb></lb>ſpatium potentiæ in O quintuplum eſ<lb></lb>ſe ſpatii ponderis A. <lb></lb><figure id="id.036.01.178.1.jpg" place="text" 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 ſi funis ita circa orbiculos apte<lb></lb>tur, vt potentia in O ſuſtinens pon<lb></lb>dus ſit ponderis ſubſextupla; & loco <lb></lb>potentiæ ſuſtinentis ponatur in O po<lb></lb>tentia mouens pondus: eodem modo <lb></lb>oſtendetur ſpatium potentiæ ſextu<lb></lb>plum eſſe ſpatii ponderis moti. </s> <s id="id.2.1.165.17.1.2.0">& ſic <lb></lb>procedendo in infinitum proportiones <lb></lb>ſpatii potentiæ ad ſpatium ponderis <lb></lb>moti quotcunq; multiplices inuenien<lb></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"></emph>Huius.<emph.end type="italics"></emph.end></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 manifeſtum eſt ita ſe habere pondus <lb></lb>ad potentiam ipſum ſuſtinentem, ſicuti ſpatium <lb></lb>potentiæ mouentis ad ſpatium 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 ſi pondus A quintuplum ſit potentiæ in O pondus A ſuſti<lb></lb>nentis; erit & ſpatium OP potentiæ pondus mouentis quintuplum <lb></lb>ſpatii <foreign lang="grc">αβ</foreign>ponderis moti. </s> </p> <pb n="83" xlink:href="036/01/179.jpg"></pb> <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 ſunt, orbiculos <lb></lb>trochleæ, quæ ponderi eſt alligata, efficere; vt à <lb></lb>moto pondere minus, quàm à trahente poten<lb></lb>tia deſcribatur ſpatium; maioriq; tempore datum <lb></lb>æquale ſpatium deſcribi, quàm ſine illis. </s> <s id="id.2.1.167.6.1.2.0">quod <lb></lb>quidem orbiculi trochleæ ſuperioris non effi<lb></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 oſtenſa ponderis ad potentiam proportione, iam ex <lb></lb>aduerſo potentiæ ad pondus proportio multiplex oſtendatur. </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 ſurſum <lb></lb>detentæ fuerit circumuolutus; altero eius extre<lb></lb>mo alicubi religato, alteri verò pondere appen<lb></lb>ſo; dupla erit ponderis potentia. </s> </p> <pb xlink:href="036/01/180.jpg"></pb> <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></lb>centrum A; & ſit pondus B alligatum fu<lb></lb>ni CDEFG, qui circa orbiculum ſit re<lb></lb>uolutus, ac tandem religatus in G: ſitq; <lb></lb>potentia in H ſuſtinens pondus. </s> <s id="id.2.1.167.11.1.2.0">dico po<lb></lb>tentiam in H duplam eſſe ponderis B. </s> <s id="N150B6">du<lb></lb>catur DF per <expan abbr="centrũ">centrum</expan> A horizonti æquidi<lb></lb>ſtans. </s> <s id="id.2.1.167.11.1.3.0"><expan abbr="quoniã">quoniam</expan> igitur potentia in H ſuſtinet <lb></lb><expan abbr="trochleã">trochleam</expan>, quæ ſuſtinet <expan abbr="orbiculũ">orbiculum</expan> in eius <expan abbr="cẽtro">centro</expan><lb></lb>A, qui pondus ſuſtinet; erit potentia ſuſti<lb></lb>nens <expan abbr="orbiculũ">orbiculum</expan>, ac ſi in A <expan abbr="cõſtituta">conſtituta</expan> eſſet; ipſa <lb></lb>ergo in A exiſtente, pondere verò in D <lb></lb>appenſo, funiq; CD religato; erit DF <lb></lb>tanquam vectis, cuius fulcimentum erit <lb></lb>F, pondus in D, & potentia in A. </s> <s id="id.2.1.167.11.1.3.0.a">po<lb></lb><arrow.to.target n="note253"></arrow.to.target>tentia verò ad pondus eſt, vt DF ad <lb></lb>ad FA, & DF dupla eſt ipſius FA; Po<lb></lb><figure id="id.036.01.180.1.jpg" place="text" xlink:href="036/01/180/1.jpg"></figure><lb></lb>tentia igitur in A, ſiue in H, quod idem eſt, ponderis B dupla erit. </s> <lb></lb> <s id="id.2.1.167.11.1.4.0">quod demonſtrare 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"></emph>Huius. de vecte.<emph.end type="italics"></emph.end></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 conſiderandum occurrit, cùm hæc omnia maneant, <lb></lb>idem eſſe vnico exiſtente fune CD EFG hoc modo orbiculo <expan abbr="cicum">circum</expan><lb></lb>uoluto, ac ſi duo eſſent funes CD FG in vecte ſiue libra DF al<lb></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">Iiſdem poſitis, ſi in G appenſum eſſet pondus k æquale pon<lb></lb>deri B, pondera B k æqueponderabunt in libra DF, cuius centrum <lb></lb>A. </s> <s id="id.2.1.169.3.1.1.0.a">potentia verò in H ſuſtinens pondera Bk eſt ipſis ſimul ſum<lb></lb>ptis æqualis, & pondera BK ipſius B ſunt dupla; potentia ergo in <lb></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></lb>liud efficit, niſi quòd pondus B ſuſtinet, ne deſcendat; quod idem <lb></lb>efficit pondus k in G appenſum: potentia igitur in H ſuſtinens <lb></lb>pondus B, fune religato in G, dupla eſt ponderis B. </s> <s id="N15138">quod de<lb></lb>monſtrare oportebat. </s> </p> <pb n="84" xlink:href="036/01/181.jpg"></pb> <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">Iiſdem poſitis ſi in H ſit potentia mouens pon<lb></lb>dus, mouebit hæc eadem vecte horizonti ſem<lb></lb>per æquidiſtante. </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 (ſicut in ſuperioribus dictum <lb></lb>eſt) oſtendetur. </s> <s id="id.2.1.169.7.1.2.0">moueatur enim orbiculus <lb></lb>ſurſum, poſitionemq; habeat MNO, cuius <lb></lb>centrum L: & per L ducatur MLO ipſi DF, <lb></lb>& horizonti æquidiſtans. </s> <s id="id.2.1.169.7.1.3.0">& quoniam funes <lb></lb>tangunt circulum MON in punctis MO; <lb></lb>ideo cùm potentia in A, ſeu in H, quod <lb></lb>idem eſt, moueat pondus B in D appenſum <lb></lb>vecte DF, cuius fulcimentum eſt F; ſemper <lb></lb>adhuc remanebit alius vectis, vt MO hori<lb></lb>zonti æquidiſtans, ita vt ſemper potentia <lb></lb>moueat pondus vecte horizonti æquidiſtan<lb></lb>te, cuius fulcimentum eſt ſemper in linea <lb></lb>OG, & pondus in MC, potentiaq; in cen<lb></lb>tro orbiculi. <figure id="id.036.01.181.1.jpg" place="text" 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">Iiſdem poſitis, ſpatium ponderis moti duplum <lb></lb>eſt ſpatii potentiæ mouentis. </s> </p> <pb xlink:href="036/01/182.jpg"></pb> <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></lb>vſq; ad centrum L; & pondus B, <lb></lb>hoc eſt punctum C, in eodem tem<lb></lb>pore ſit motum in P; & potentia in <lb></lb>H vſq; ad K; erit AH ipſi LK æqua <lb></lb>lis, & AL ipſi Hk. </s> <s id="id.2.1.169.10.1.2.0">& quoniam fu<lb></lb>nis CDEFG eſt æqualis funi PM <lb></lb>NOG, idem enim eſt funis, & fu <lb></lb>nis circa ſemicirculum MNO æ<lb></lb>qualis eſt funi circa ſemicirculum <lb></lb>DEF; demptis igitur communi<lb></lb>bus DP FG, erit PC æqualis <lb></lb>DM FO ſimul ſumptis, qui funes <lb></lb>ſunt dupli ipſius AL, & conſequen<lb></lb>ter ipſius Hk. </s> <s id="id.2.1.169.10.1.3.0">ſpatium ergo pon<lb></lb>deris moti CP duplum eſt ſpatii <lb></lb>Hk potentiæ. </s> <s id="id.2.1.169.10.1.4.0">quod oportebat de<lb></lb>monſtrare. </s> </p> <figure id="id.036.01.182.1.jpg" place="text" 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 manifeſtum eſt, idem pondus trahi <lb></lb>ab eadem potentia in æquali tempore per du<lb></lb>plum ſpatium trochlea hoc modo accommoda<lb></lb>ta, quàm ſine trochlea; dummodo ipſius poten<lb></lb>tiæ lationes in velocitate ſint 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 ſine trochlea æquale eſt ſpatio <lb></lb>potentiæ. </s> </p> <pb n="85" xlink:href="036/01/183.jpg"></pb> <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></lb>orbiculum, cuius centrum k; ſitq; huiuſmo<lb></lb>di orbiculi trochlea deorſum affixa, quæ nul<lb></lb>lum alium habeat motum, niſi liberam orbi <lb></lb>culi circa axem reuolutionem; funiſq; relige<lb></lb>tur in M; erit potentia in H ſuſtinens pondus <lb></lb>B ſimiliter ipſius ponderis dupla. </s> <s id="id.2.1.169.14.1.2.0">quod qui <lb></lb>dem manifeſtum eſt, cùm idem prorſus ſit, <lb></lb>ſiue funis ſit religatus in M, ſiue in G. </s> <s id="N151F7">orbicu<lb></lb>lus enim, cuius centrum k, nihil efficit; penituſ<lb></lb>〈qué〉 inutilis eſt. <figure id="id.036.01.183.1.jpg" place="text" 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 ſit potentia in M ſuſtinens pon<lb></lb>dus B, & trochlea ſuperior ſit ſurſum appen<lb></lb>ſa; 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 ſuſtinens <arrow.to.target n="note254"></arrow.to.target><lb></lb>pondus B æqualis eſt ponderi B, & ipſi po<lb></lb>tentiæ in G æqualis eſt potentia in L; eſt <lb></lb>enim GL vectis, cuius fulcimentum eſt k; <lb></lb>& diſtantia Gk diſtantiæ kL eſt æqualis; <lb></lb>erit igitur potentia in L, ſiue (quod idem eſt) <lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.171.1.0.0.0" type="main"> <s id="id.2.1.171.1.1.1.0">Huiuſmodi autem motus fit vectibus DF LG, quorum fulci<lb></lb>menta ſunt kA, & pondus in D, & potentia in F. </s> <s id="N15236">ſed in vecte <lb></lb>LG potentia eſt in L, pondus verò, ac ſi eſſet 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 ſit potentia mouens pondus, transferaturq; po<lb></lb>tentia in N, pondus autem motum fuerit vſq; ad O; erit MN <lb></lb>ſpatium potentiæ æquale ſpatio CO ponderis. </s> <s id="id.2.1.171.2.1.2.0">Cùm enim funis <lb></lb>MLGFDC æqualis ſit funi NLGFDO.</s> <s id="N15249"> eſt enim idem funis; <lb></lb>dempto communi MLGFDO; erit ſpatium MN potentiæ æ<lb></lb>quale ſpatio 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 ſi funis in M circa plures reuoluatur orbiculos, ſemper erit <lb></lb>potentia altero eius extremo pondus ſuſtinens æqualis ipſi ponderi. </s> <s id="id.2.1.171.3.1.2.0"><lb></lb>ſpatiaq; ponderis, atq; potentiæ mouentis ſemper oſtendentur <lb></lb>æqualia. </s> </p> <pb xlink:href="036/01/184.jpg"></pb> <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 vtriſq; duarum trochlearum ſingulis orbicu<lb></lb>lis, quarum vna ſupernè à potentia ſuſtineatur, <lb></lb>altera verò infernè, ibiq; affixa, conſtituta fue<lb></lb>rit, funis circumducatur; altero eius extremo ſu<lb></lb>periori trochleæ religato, alteri verò pondere <lb></lb>appenſo; 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></lb>chleæ infernè affixæ; & ſit funis BCD <lb></lb>EFG non ſolum huic orbiculo circumuo<lb></lb>lutus, verùm etiam orbiculo trochleæ ſu<lb></lb>perioris, cuius centrum k; ſitq; funis in <lb></lb>B ſuperiori trochleæ religatus; & in G ſit ap<lb></lb>penſum pondus H; potentiaq; in L ſuſti<lb></lb>neat pondus H. </s> <s id="id.2.1.171.7.1.1.0.a">dico potentiam in L tri<lb></lb>plam eſſe ponderis H. </s> <s id="id.2.1.171.7.1.1.0.b">ſi enim duæ eſſent <lb></lb>potentiæ pondus H <expan abbr="ſuſtidentes">sustinentes</expan>, vna in <lb></lb>K, altera in B, erunt vtræq; ſimul triplæ <lb></lb><arrow.to.target n="note255"></arrow.to.target>ponderis H potentia enim in k dupla eſt <lb></lb>ponderis H, & potentia in B ipſi ponderi <lb></lb>æqualis. </s> <s id="id.2.1.171.7.1.2.0">& quoniam ſola potentia in L <lb></lb>vtriſq; ſcilicet potentiæ in KB eſt æqua<lb></lb>lis. </s> <s id="id.2.1.171.7.1.3.0">ſuſtinet enim potentia in L; tùm po<lb></lb>tentiam in K, tùm potentiam in B; idem<lb></lb>〈qué〉 efficit potentia in L, ac ſi duæ eſſent <lb></lb>potentiæ, vna in k, altera in B: Tri<lb></lb>pla igitur erit potentia in L ponderis H. <lb></lb></s> <s id="N152B6">quod demonſtrare oportebat. <figure id="id.036.01.184.1.jpg" place="text" xlink:href="036/01/184/1.jpg"></figure></s> </p> <pb n="86" xlink:href="036/01/185.jpg"></pb> <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 ſit potentia mouens pondus. </s> <s id="id.2.1.171.9.1.2.0">di<lb></lb>co ſpatium ponderis moti triplum eſſe ſpatii po<lb></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"></emph>Huius. </s> <s id="id.2.1.172.1.1.2.0">In præcedenti.<emph.end type="italics"></emph.end></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></lb>biculi K vſq; ad M; cuius <lb></lb>quidem motus ſpatium <lb></lb>motæ potentiæ ſpatio eſt <arrow.to.target n="note256"></arrow.to.target><lb></lb>æquale, ſicuti ſupra dictum <lb></lb>eſt: & quando k erit in M, <lb></lb>B erit in N; & NB æqualis <lb></lb>erit M k; & dum k eſt in M, <lb></lb>ſit pondus H, hoc eſt pun<lb></lb>ctum G motum in O; & per <lb></lb>MK ducantur EF PQ ho<lb></lb>rizonti æquidiſtantes; erit <lb></lb>vnaquæq; EP BN FQ ip<lb></lb>ſi KM æqualis. </s> <s id="id.2.1.173.1.1.2.0">& quoniam <lb></lb>funis BCDEFG æqualis <lb></lb>eſt funi NCDPQO; <lb></lb>idem enim eſt funis; & fu<lb></lb>nis circa ſemicirculum ER <lb></lb>F æqualis eſt funi circa ſe<lb></lb>micirculum PSQ: dem<lb></lb>ptis igitur communibus <lb></lb>BCDE, & FO, erit OG <lb></lb>tribus QF NB PE ſimul <lb></lb>ſumptis æqualis. </s> <s id="id.2.1.173.1.1.3.0">ſed QF <lb></lb>NB PE ſimul triplæ ſunt <lb></lb>Mk, hoc eſt ſpatii poten<lb></lb>tiæ motæ; ſpatium ergo <lb></lb>GO ponderis H moti tri<lb></lb><figure id="id.036.01.185.1.jpg" place="text" xlink:href="036/01/185/1.jpg"></figure><lb></lb>plum eſt ſpatii potentiæ motæ. </s> <s id="id.2.1.173.1.1.4.0">quod oſtendere 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"></emph>In præcedenti.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/186.jpg"></pb> <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 vtriuſq; duarum trochlearum binis orbicu<lb></lb>lis, quarum altera ſupernè à potentia ſuſtineatur, <lb></lb>altera verò infernè, ibiq; annexa, collocata fue<lb></lb>rit, funis circumnectatur; altero eius extremo <lb></lb>alicubi, non autem ſuperiori trochleæ religato, <lb></lb>alteri verò pondere appenſo; quadrupla erit <lb></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></lb>quorum centra AB; ſit 〈qué〉 trochlea ſuperior <lb></lb>duos ſimiliter habens orbiculos, quorum cen<lb></lb>tra CD; funiſq; EFGHKLMNOP ſit cir<lb></lb>ca omnes orbiculos reuolutus, qui ſit religatus <lb></lb>in E; & in P appendatur pondus Q; ſitq; po<lb></lb>tentia in R. </s> <s id="id.2.1.175.3.1.1.0.a">dico potentiam in R quadruplam <lb></lb>eſſe ponderis q. </s> <s id="N1536C">Cùm enim ſi duæ intelligan<lb></lb>tur potentiæ, vna in k, altera in D, potentia <lb></lb><arrow.to.target n="note257"></arrow.to.target>in k ſuſtinens pondus Q fune k LMNOP æ<lb></lb>qualis erit ponderi; erunt duæ ſimul potentiæ, <lb></lb>vna in D, altera in k, pondus Q ſuſtinentes, <lb></lb>triplæ eiuſdem ponderis. </s> <s id="id.2.1.175.3.1.2.0">Potentia verò in C <lb></lb>dupla eſt potentiæ in k, & per conſequens pon<lb></lb>deris Q; idem enim eſt, ac ſi in k appenſum eſ<lb></lb><arrow.to.target n="note258"></arrow.to.target>ſet pondus æquale ponderi Q, cuius dupla eſt <lb></lb>potentia in C; duæ igitur potentiæ in DC qua<lb></lb>druplæ ſunt ponderis q. </s> <s id="N1538B">& cùm potentia in R <lb></lb>orbiculis ſuſtineat pondus Q, erit <expan abbr="potẽtia">potentia</expan> in R, <lb></lb>ac ſi duæ eſſent potentiæ, vna in D, altera in C, <lb></lb>& vtræq; ſimul pondus Q ſuſtinerent. </s> <s id="id.2.1.175.3.1.3.0">ergo po<lb></lb>tentia in R quadrupla eſt ponderis q. </s> <s id="N1539C">quod <lb></lb>oportebat demonſtrare. <figure id="id.036.01.186.1.jpg" place="text" xlink:href="036/01/186/1.jpg"></figure></s> <pb n="87" xlink:href="036/01/187.jpg"></pb> <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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.176.1.1.2.0"><margin.target id="note258"></margin.target>15 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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, ſi funis fuerit religatus in G, & <lb></lb>circa orbiculos, quorum centra ſunt BCD reuo<lb></lb>lutus; potentiam in R pondus ſuſtinentem ſimili<lb></lb>ter ponderis Q quadruplam eſſe. </s> <s id="id.2.1.177.1.1.2.0">orbiculus enim, <lb></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 ſit potentia mouens pondus. </s> <s id="id.2.1.177.2.1.2.0">dico <lb></lb>ſpatium ponderis moti quadruplum eſſe ſpatii <lb></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 vſq; ad <lb></lb>ST; erunt ex ſuperius dictis CS DT ſpatio <lb></lb>potentiæ æqualia; & per CSDT ducantur Hk <lb></lb>VX NO YZ horizonti æquidiſtantes; & <expan abbr="dũ">dum</expan><lb></lb>centra CD ſunt in ST, ſit pondus Q, hoc eſt <lb></lb>punctum P motum in 9. </s> <s id="N153F4">& quoniam funis EF <lb></lb>GHKLMNOP æqualis eſt funi EFGVX <lb></lb>LMYZ 9; cùm ſit idem funis: & funes circa <lb></lb>ſemicirculos NIO H <foreign lang="grc">α</foreign>k ſunt æquales funi<lb></lb>bus, qui ſunt circa ſemicirculos Y<foreign lang="el">d</foreign>Z V<foreign lang="grc">β</foreign>X; <lb></lb>demptis igitur communibus EFGH kLMN <lb></lb>& O9; erit P9 ipſis NY ZO VH <emph type="italics"></emph>X<emph.end type="italics"></emph.end>k ſi<lb></lb>mul ſumptis æqualis. </s> <s id="id.2.1.177.3.1.2.0">quatuor autem NY ZO <lb></lb>VH Xk ſimul quadrupli ſunt DT, hoc eſt <lb></lb>ſpatii potentiæ; ſpatium igitur P9 ponderis <lb></lb>quadruplum eſt ſpatii potentiæ quod demon<lb></lb>ſtrandum fuerat. <figure id="id.036.01.187.1.jpg" place="text" xlink:href="036/01/187/1.jpg"></figure></s> </p> <pb xlink:href="036/01/188.jpg"></pb> <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 ſit re<lb></lb>ligatus in E trochleæ ſu<lb></lb>periori, & potentia in R <lb></lb>ſuſtineat pondus Q; e<lb></lb>rit potentia in R ponde<lb></lb>ris Q quintupla. </s> <s id="id.2.1.177.5.1.2.0">& ſi in <lb></lb>R ſit potentia mouens <lb></lb>pondus; erit ſpatium pon<lb></lb>deris moti quintuplum <lb></lb>ſpatii potentiæ. </s> <s id="id.2.1.177.5.1.3.0">quæ om<lb></lb>nia ſimili modo oſten<lb></lb>dentur, ſicut in præce<lb></lb>dentibus demonſtra<lb></lb>tum eſt. <figure id="id.036.01.188.1.jpg" place="text" xlink:href="036/01/188/1.jpg"></figure></s> </p> <pb n="88" xlink:href="036/01/189.jpg"></pb> <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 ſubſtineat pon<lb></lb>dus Q trochlea tres orbiculos habente, <lb></lb>quorum centra ſint ABC; & ſit alia tro<lb></lb>chlea infernè affixa duos, vel tres orbicu<lb></lb>los habens, quorum centra DEF; ſitq; <lb></lb>funis circa omnes orbiculos reuolutus, ſi<lb></lb>ue in G, ſiue in H religatus; ſimiliter <lb></lb>oſtendetur potentiam in R ſexcuplam <lb></lb>eſſe ponderis q. </s> <s id="N1546A">Et ſi in R ſit potentia <lb></lb>mouens pondus, oſtendetur ſpatium pon<lb></lb>deris moti ſexcuplum eſſe ſpatii poten<lb></lb>tiæ. <figure id="id.036.01.189.1.jpg" place="text" 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 ſi funis ſit religatus in K trochleæ <lb></lb>ſuperiori, & in R ſit potentia pondus <lb></lb>ſuſtinens; ſimili modo oſtendetur poten<lb></lb>tiam in R ſeptuplam eſſe 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 ſi in R ſit potentia mouens, oſten <lb></lb>detur ſpatium ponderis Q ſeptuplum eſſe <lb></lb>ſpatii potentiæ. </s> <s id="id.2.1.177.9.1.2.0">atq; ita in infinitum <lb></lb>omnis potentiæ ad pondus multiplex <lb></lb>proportio inueniri poterit. </s> <s id="id.2.1.177.9.1.3.0">ſemperq; o<lb></lb>ſtendetur, ita eſſe pondus ad potentiam <lb></lb>ipſum ſuſtinentem, ſicuti ſpatium poten<lb></lb>tiæ pondus mouentis ad ſpatium ponde<lb></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 ipſorum orbiculorum <lb></lb>motus in his fit hoc modo, videlicet vectes <lb></lb>orbiculorum trochleæ ſuperioris mouen<lb></lb>tur, vti dictum eſt in decima ſexta huius; <lb></lb>hoc eſt habent fulcimentum in extremita<lb></lb>te, potentiam in medio, pondus in altera extremitate appenſum. </s> <s id="id.2.1.177.10.1.2.0">ve<lb></lb>ctes verò trochleæ inferioris habent fulcimentum in medio, pon<lb></lb>dus, & potentiam in extremitatibus. </s> </p> <pb xlink:href="036/01/190.jpg"></pb> <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">Manifeſtum eſt in his, orbiculos trochleæ ſu<lb></lb>perioris efficere, vt pondus moueatur maiori <lb></lb>potentia, quàm ſit ipſum pondus, & per maius <lb></lb>ſpatium potentiæ ſpatio, & per æquale tempo<lb></lb>re minori; quod quidem orbiculi trochleæ in<lb></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></lb>tionem inuenire poſſumus. </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 vtriuſq; duarum trochlearum ſingulis orbi <lb></lb>culis, quarum altera ſupernè appenſa, altera ve<lb></lb>rò infernè à ſuſtinente potentia <expan abbr="rententa">retenta</expan> fuerit, <lb></lb>funis circumuoluatur; altero eius extremo alicu<lb></lb>bi religato, alteri autem pondere appenſo; du<lb></lb>pla erit ponderis potentia. </s> </p> <pb n="89" xlink:href="036/01/191.jpg"></pb> <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 ſupernè appenſæ, cu<lb></lb>ius centrum ſit A; & BCD ſit trochleæ infe<lb></lb>rioris; ſit deinde funis EBC DFGHL reli<lb></lb>gatus in E; & in L ſit appenſum pondus M; <lb></lb>ſitq; potentia in N ſuſtinens pondus M. </s> <s id="id.2.1.177.18.1.1.0.a"><lb></lb>dico potentiam in N duplam eſſe ponderis <lb></lb>M. </s> <s id="id.2.1.177.18.1.1.0.b">Cùm enim ſupra oſtenſum ſit potentiam <lb></lb>in L, quæ pondus, exempli gratia, O ſuſti<lb></lb>neat <arrow.to.target n="note259"></arrow.to.target>in N appenſum, ſubduplam eſſe eiuſdem <lb></lb>ponderis; potentia igitur in N ponderi O æ<lb></lb>qualis pondus M potentiæ in L æquale ſuſti<lb></lb>nebit; ponderiſq; M dupla erit. </s> <s id="id.2.1.177.18.1.2.0">quod demon<lb></lb>ſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <figure id="id.036.01.191.1.jpg" place="text" 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">Iiſdem poſitis. </s> <s id="id.2.1.179.2.1.2.0">Quoniam potentia in F, <arrow.to.target n="note260"></arrow.to.target><lb></lb>ſeu in D, quod idem eſt, æqualis eſt ponde<lb></lb>ri M; & BD eſt vectis, cuius fulcimentum <lb></lb>eſt B, & potentia in N eſt, ac ſi eſſet in me<lb></lb>dio vectis, & pondus æquale ipſi M, ac ſi eſ<lb></lb>ſet in D propter funem FD; quod idem <lb></lb>eſt, ac ſi BCD eſſet orbiculus trochleæ ſupe<lb></lb>rioris, pondusq; appenſum eſſet in fune DF, <lb></lb>ſicut in decimaquinta, & decimaſexta dictum eſt; ergo potentia in <lb></lb>N dupla eſt ponderis M. </s> <s id="N15555">quod erat oſtendendum. </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"></emph>Huius.<emph.end type="italics"></emph.end></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 ſit potentia mouens pondus M, erit ſpatium <lb></lb>ponderis M duplum ſpatii potentiæ in N. </s> <s id="N1556C">quod ex duodecima <lb></lb>huius manifeſtum eſt; ſpatium enim puncti L deorſum ten<lb></lb>dentis duplum eſt ſpatii N ſurſum; erit igitur è conuerſo ſpatium <lb></lb>potentiæ in N deorſum tendentis dimidium <expan abbr="ſaptii">spatii</expan> ponderis M ſur<lb></lb>ſum 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, ſeptima huius, &c. </s> <s id="id.2.1.181.2.1.2.0">colligi poſſunt <lb></lb>ponderis O rationes quotcunq; multiplices ipſius potentiæ in L, <lb></lb><expan abbr="eodẽ">eodem</expan> quoq; modo oſtendi poterunt potentiæ in N pondus ſuſtinen<lb></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"></pb>decimaquarta rationes oſten<lb></lb>dentur quotcunq; multiplices <lb></lb>ſpatii ponderis M ad ſpatium <lb></lb>potentiæ mouentis in N conſti<lb></lb>tutæ. <figure id="id.036.01.192.1.jpg" place="text" 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 decimaſe<lb></lb>ptima decimaoctaua huius mul<lb></lb>tiplex inueniri proportio, quam <lb></lb>habet potentia pondus ſuſti<lb></lb>nens ad ipſum pondus; ſicut <lb></lb>proportio potentiæ in N ad pon<lb></lb>dus M ex decimaquinta, & deci<lb></lb>maſexta oſtendebatur: inuenie<lb></lb>turq; ita eſſe pondus ad poten<lb></lb>tiam pondus ſuſtinentem, vt ſpa<lb></lb>tium potentiæ mouentis ad ſpa<lb></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></lb>hoc modo, videlicet vectes or<lb></lb>biculorum trochleæ inferioris <lb></lb>mouentur, vt vectis BD, quæ <lb></lb>mouetur, ac ſi B eſſet fulcimen <lb></lb>tum, & pondus in D, & poten<lb></lb>tia in medio. </s> <s id="id.2.1.181.4.1.2.0">Vectes verò or<lb></lb>biculorum trochleæ ſuperioris mouentur, vt FH, cuius fulcimen <lb></lb>tum eſt 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 manifeſtum eſt, orbiculos trochleæ <lb></lb>inferioris in his efficere, vt pondus maiori po<pb n="90" xlink:href="036/01/193.jpg"></pb>tentia moueatur, quàm ſit ipſum pondus, & <lb></lb>per maius ſpatium ſpatio potentiæ, & minori <lb></lb>tempore per æquale. </s> <s id="id.2.1.181.6.1.2.0">quod quidem orbiculi ſu<lb></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 ſuperparticu<lb></lb>lares accedendum eſt. </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 vtriuſq; duarum trochlearum ſingulis or<lb></lb>biculis, quarum altera ſupernè à potentia ſuſti<lb></lb>neatur, altera verò infernè, ponderiq; alligata, <lb></lb><expan abbr="cõſtituta">conſtituta</expan> fuerit, funis reuoluatur; altero eius extre<lb></lb>mo alicuibi, altero verò inferiori trochleæ reli<lb></lb>gato; pondus potentiæ ſeſquialterum erit. </s> </p> <pb xlink:href="036/01/194.jpg"></pb> <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></lb>trochleæ ſuperioris, & <lb></lb>DEF trochleæ inferio<lb></lb>ris ponderi G alligatæ; <lb></lb>ſitq; funis HABCDE <lb></lb>Fk circa orbiculos re<lb></lb>uolutus, qui ſit religatus <lb></lb>in K, & in H trochleæ <lb></lb>inferiori; ſitq; potentia <lb></lb>in L ſuſtinens pondus <lb></lb>G. </s> <s id="id.2.1.181.11.1.1.0.a">dico pondus poten<lb></lb>tiæ ſeſquialterum eſſe. </s> <s id="id.2.1.181.11.1.2.0"><lb></lb><arrow.to.target n="note261"></arrow.to.target>Quoniam enim vterque <lb></lb>funis CD AH tertiam <lb></lb>ſuſtinet partem ponde<lb></lb>ris G, erit vnaquæq; po<lb></lb>tentia in DH ſubtripla <lb></lb>ponderis G; quibus ſi<lb></lb>mul aſſumptis eſt æqua<lb></lb><figure id="id.036.01.194.1.jpg" place="text" xlink:href="036/01/194/1.jpg"></figure><lb></lb><arrow.to.target n="note262"></arrow.to.target>lis potentia in L: potentia enim in L dupla eſt potentiæ in D, & <lb></lb>eius, quæ eſt in H. </s> <s id="N15654">quare potentia in L ſubſeſquialtera eſt ponde<lb></lb>ris G. </s> <s id="id.2.1.181.11.1.2.0.a">pondus ergo G ad pontentiam in L eſt, vt tria ad duo; <lb></lb>hoc eſt ſeſquialterum. </s> <s id="id.2.1.181.11.1.3.0">quod demonſtrare 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"></emph>Cor.<emph.end type="italics"></emph.end> 5 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.182.1.1.2.0"><margin.target id="note262"></margin.target><emph type="italics"></emph>Ex.<emph.end type="italics"></emph.end> 15 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <pb n="91" xlink:href="036/01/195.jpg"></pb> <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 ſit potentia mouens pondus. </s> <s id="id.2.1.183.1.2.2.0"><lb></lb>Dico ſpatium potentiæ ſpatii ponderis ſeſquial<lb></lb>terum eſſe. </s> </p> <p id="id.2.1.183.2.0.0.0" type="main"> <s id="id.2.1.183.2.1.1.0">Iiſdem poſitis, perueniat orbi<lb></lb>culus ABC vſq; ad MNO, & <lb></lb>DEF ad PQR; & H in S; & <lb></lb>pondus G vſq; ad T. </s> <s id="id.2.1.183.2.1.1.0.a">Et quoniam <lb></lb>funis HABCDEFK eſt æqualis <lb></lb>funi SMNOPQRk, cùm ſit <lb></lb>idem funis; & funes circa ſemicir<lb></lb>culos ABC MNO ſunt inter ſe <lb></lb>ſe æquales; qui verò ſunt circa <lb></lb>DEF PQR ſimiliter inter ſe æ<lb></lb>quales; Demptis igitur AS CP <lb></lb>RK communibus, erunt duo CO <lb></lb>MA tribus DP HS FR æqua<lb></lb>les. </s> <s id="id.2.1.183.2.1.2.0">ſed vterq; CO AM ſeorſum <lb></lb>eſt æqualis ſpatio potentiæ motæ. </s> <s id="id.2.1.183.2.1.3.0"><lb></lb>quare duo CO MA, ſimul ſpatii <lb></lb>potentiæ dupli erunt: treſq; DP <lb></lb>HS FR ſimul ſimili modo ſpatii <lb></lb>ponderis moti tripli erunt. </s> <s id="id.2.1.183.2.1.4.0">dimidia <lb></lb>verò pars, hoc eſt ſpatium poten<lb></lb>tiæ motæ ad tertiam, ad ſpatium <lb></lb>ſcilicet ponderis moti ita ſe habet, <lb></lb>vt duplum dimidii ad duplum ter<lb></lb>tii; hoc eſt, vt totum ad duas ter<lb></lb><figure id="id.036.01.195.1.jpg" place="text" xlink:href="036/01/195/1.jpg"></figure><lb></lb>tias, quod eſt vt tria ad duo. </s> <s id="id.2.1.183.2.1.5.0">ſpatium ergo potentiæ in L ſpa<lb></lb>tii ponderis G moti ſeſquialterum eſt. </s> <s id="id.2.1.183.2.1.6.0">quod oſtendere opor<lb></lb>tebat. </s> </p> <pb xlink:href="036/01/196.jpg"></pb> <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></lb>rum altera vnius tantùm orbiculi ſupernè à po<lb></lb>tentia ſuſtineatur, altera verò duorum infernè, <lb></lb>ponderiq; alligata, collocata fuerit, funis cir<lb></lb>cumuoluatur; altero eius extremo alicubi, altero <lb></lb>autem ſuperiori trochleæ religato: pondus poten<lb></lb>tiæ ſeſquitertium 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></lb>tum, quæ duos habeat orbiculos, quorum <lb></lb>centra ſint BC; ſuperiorq; trochlea orbicu<lb></lb>lum habeat, cuius centrum D; & ſit funis <lb></lb>EFGHkLMN circa omnes orbiculos re<lb></lb>uolutus, qui religatus ſit in N, & in E tro<lb></lb>chleæ ſuperiori; ſit〈qué〉 potentia in O <lb></lb>ſuſtinens pondus A. </s> <s id="id.2.1.183.6.1.1.0.a">dico pondus po<lb></lb><arrow.to.target n="note263"></arrow.to.target>tentiæ ſeſquitertium eſſe. </s> <s id="id.2.1.183.6.1.2.0">Quoniam enim <lb></lb>vnuſquiſq; funis NM HG EF KL quar<lb></lb>tam ſuſtinent partem ponderis A, & omnes <lb></lb>ſimul totum ſuſtinent pondus; tres HG <lb></lb>EF kL ſimul tres ſuſtinebunt partes pon<lb></lb>deris A. </s> <s id="N15726">quare pondus A ad hos omnes <lb></lb>ſimul erit, vt quatuor ad tria: & cùm po<lb></lb>tentia in O idem efficiat, quod HG EF kL <lb></lb>ſimul efficiunt; omnes enim ſuſtinet; erit po<lb></lb>tentia in O tribus ſimul HG EF kL æ<lb></lb>qualis; & ob id pondus A ad potentiam <lb></lb>in O erit, vt quatuor ad tria; hoc eſt ſeſqui<lb></lb>tertium. </s> <s id="id.2.1.183.6.1.3.0">quod demonſtrare oportebat. <figure id="id.036.01.196.1.jpg" place="text" xlink:href="036/01/196/1.jpg"></figure></s> </p> <pb n="92" xlink:href="036/01/197.jpg"></pb> <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 ſit potentia mouens pondus A. </s> <s id="id.2.1.183.8.1.1.0.a"><lb></lb>Dico ſpatium potentiæ in O decurſum ſpatii pon<lb></lb>deris A moti ſeſquitertium eſſe. </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"></emph>Cor.<emph.end type="italics"></emph.end> 1 <emph type="italics"></emph>ſeptimebuius.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.185.1.0.0.0" type="main"> <s id="id.2.1.185.1.1.1.0">Iiſdem poſitis, ſit centrum B motum <lb></lb>in P; &C vſq; ad Q; & D in R; & E in <lb></lb>S eodem tempore: & per centra ducantur <lb></lb>ML 9Z FG TV Hk XY horizonti, <lb></lb>& inter ſe ſe æquidiſtantes. </s> <s id="id.2.1.185.1.1.2.0">Similiter, vt in <lb></lb>præcedente oſtendetur tres <emph type="italics"></emph>X<emph.end type="italics"></emph.end>H SE Yk <lb></lb>quatuor TG VF ZL 9M æquales eſſe. </s> <s id="id.2.1.185.1.1.3.0">& <lb></lb>quoniam tres XH SE Yk ſimul triplæ <lb></lb>ſunt ſpatii potentiæ, quatuor verò TG VF <lb></lb>ZL 9M ſimul quadruplæ ſunt ſpatii pon<lb></lb>deris moti; erit ſpatium potentiæ ad ſpa<lb></lb>tium ponderis, vt tertia pars ad quartam. </s> <s id="id.2.1.185.1.1.4.0"><lb></lb>ſed tertia pars ad quartam eſt, vt tres ter<lb></lb>tiæ ad tres quartas, hoc eſt, vt totum ad <lb></lb>tres quartas; quod eſt, vt quatuor ad tria. </s> <s id="id.2.1.185.1.1.5.0"><lb></lb>ſpatium ergo potentiæ ſpatii ponderis mo<lb></lb>ti ſeſquitertium eſt. </s> <s id="id.2.1.185.1.1.6.0">quod erat demon<lb></lb>ſtrandum. <figure id="id.036.01.197.1.jpg" place="text" 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></lb>uatur orbiculum, qui deinde trochleæ in <lb></lb>feriori religetur; ſimiliter oſtendetur pro <lb></lb>portionem ponderis ad <expan abbr="potentiã">potentiam</expan> in O pon<lb></lb>dus ſuſtinentem ſeſquiquartam eſſe. </s> <s id="id.2.1.185.2.1.2.0">quòd <lb></lb>ſi in O ſit potentia mouens pondus, oſten <lb></lb>detur ſpatium potentiæ ſpatii ponderis ſeſ<lb></lb>quiquartum eſſe. </s> <s id="id.2.1.185.2.1.3.0">& ſic in infinitum proce<lb></lb>dendo quamcunq; ſuperparticularem pro <lb></lb>portionem ponderis ad potentiam inuenie<lb></lb>mus; ſemperq; reperiemus, ita eſſe pondus <lb></lb>ad potentiam pondus ſuſtinentem, vt ſpa<lb></lb>tium potentiæ mouentis ad ſpatium ponde<lb></lb>ris moti. </s> </p> <pb xlink:href="036/01/198.jpg"></pb> <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></lb>do, videlicet vectis ML fulci<lb></lb>mentum eſt M, cùm funis ſit re <lb></lb>ligatus in N, & pondus in me<lb></lb>dio, & potentia in L. </s> <s id="N157DB">ve<lb></lb>rò punctum L tendit ſurſum, quod <lb></lb>à fune KL mouetur, idcirco K ſur<lb></lb>ſum mouebitur, & vectis HK ful<lb></lb>cimentum erit H, pondus ac ſi eſ<lb></lb>ſent in k, & potentia in medio; <lb></lb>vectis autem FG fulcimentum <lb></lb>erit G, pondus in medio; & poten<lb></lb>tia in F. </s> <s id="id.2.1.185.4.1.1.0.a">punctum enim F ſurſum <lb></lb>mouetur à fune EF. </s> <s id="id.2.1.185.4.1.1.0.b">Præterea <lb></lb>G in orbiculo deorſum tendit, <lb></lb>quia H quoque in eius orbiculo <lb></lb>deorſum mouetur. <figure id="id.036.01.198.1.jpg" place="text" xlink:href="036/01/198/1.jpg"></figure></s> <pb n="93" xlink:href="036/01/199.jpg"></pb> <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 vtriſque duarum trochlearum ſingulis <lb></lb>orbiculis, quarum altera ſupernè à potentia <lb></lb>ſuſtineatur, altera verò infernè, ponderiq; alli<lb></lb>gata, collocata fuerit, circumducatur funis; al<lb></lb>tero eius extremo alicubi, altero autem ſuperio<lb></lb>ri trochleæ religato. </s> <s id="id.2.1.185.5.1.2.0">erit potentia ponderis ſeſ<lb></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></lb>ligatæ; & EFG trochleæ ſuperioris, cuius <lb></lb>centrum H; ſit deinde funis k ABCEFGL <lb></lb>circa orbiculos reuolutus, & religatus in L, & <lb></lb>in k trochleæ ſuperiori; ſitq; potentia in M <lb></lb>ſuſtinens pondus D. </s> <s id="id.2.1.185.6.1.1.0.a">dico potentiam ponde<lb></lb>ris ſeſquialteram eſſe. </s> <s id="id.2.1.185.6.1.2.0">Quoniam enim poten<arrow.to.target n="note264"></arrow.to.target><lb></lb>tia in E ſuſtinens pondus D ſubdupla eſt pon<arrow.to.target n="note265"></arrow.to.target><lb></lb>deris D, potentiæ verò in E dupla eſt poten<arrow.to.target n="note266"></arrow.to.target><lb></lb>tia in H; erit potentia in H ponderi D æqua <arrow.to.target n="note267"></arrow.to.target><lb></lb>lis; & cùm potentia in K ſubdupla ſit ponde<lb></lb>ris D; erunt vtræq; ſimul potentiæ in H k ſeſ<lb></lb>quialteræ ponderis D. </s> <s id="id.2.1.185.6.1.2.0.a">Itaq; cùm potentia in <lb></lb>M duabus potentiis in Hk ſimul ſumptis ſit <lb></lb>æqualis, quemadmodum in ſuperioribus o<lb></lb>ſtenſum eſt; erit potentia in M ſeſquialtera <lb></lb>ponderis D. </s> <s id="N15857">quod oportebat demonſtrare. </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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.186.1.1.2.0"><margin.target id="note265"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 15 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.186.1.1.3.0"><margin.target id="note266"></margin.target>2 <emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.186.1.1.4.0"><margin.target id="note267"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 ſit potentia mouens pondus, <lb></lb>ſimiliter vt in præcedentibus oſtendetur, ſpa<lb></lb>tium ponderis ſpatii potentiæ ſeſquialterum <lb></lb>eſſe. <figure id="id.036.01.199.1.jpg" place="text" xlink:href="036/01/199/1.jpg"></figure></s> </p> <pb xlink:href="036/01/200.jpg"></pb> <p id="id.2.1.187.3.0.0.0" type="main"> <s id="id.2.1.187.3.1.1.0">Et ſi funis in K per alium circumuoluatur <lb></lb>orbiculum, cuius centrum ſit N; qui dein<lb></lb>de trochleæ inferiori religetur in O; & po<lb></lb>tentia in M ſuſtineat pondus D. </s> <s id="id.2.1.187.3.1.1.0.a">dico pro<lb></lb>portionem potentiæ ad pondus ſeſquiter<lb></lb>tiam eſſe. <figure id="id.036.01.200.1.jpg" place="text" 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 ſuſtinens <lb></lb><arrow.to.target n="note268"></arrow.to.target>pondus D fune ECB AKPO ſubtripla eſt <lb></lb><arrow.to.target n="note269"></arrow.to.target>ipſius D, ipſius autem E dupla eſt potentia <lb></lb>in H; erit potentia in H ſubſeſquialtera pon<lb></lb>deris D. </s> <s id="id.2.1.187.4.1.1.0.a">ſimili quoq; modo quoniam po<lb></lb>tentia in O, quæ eſt, ac ſi eſſet in centro or<lb></lb><arrow.to.target n="note270"></arrow.to.target>biculi ABC, ſubtripla eſt ponderis D; ip<lb></lb>ſius autem O dupla eſt potentia in N; erit <lb></lb>quoq; potentia in N ſubſeſquialtera ponde<lb></lb>ris D. </s> <s id="N158D6">quare duæ ſimul potentiæ in HN pon<lb></lb>dus D ſuperant tertia parte, ſe ſe habentq; ad <lb></lb>D in ratione ſeſquitertia: & cùm potentia <lb></lb>in M duabus ſit potentiis in HN ſimul ſum<lb></lb>ptis æqualis, ſuperabit itidem potentia in <lb></lb>M pondus D tertia parte. </s> <s id="id.2.1.187.4.1.2.0">ergo proportio <lb></lb>potentiæ in M ad pondus D ſeſquitertia <lb></lb>eſt. </s> <s id="id.2.1.187.4.1.3.0">quod demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.188.1.1.2.0"><margin.target id="note269"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 15 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.188.1.1.3.0"><margin.target id="note270"></margin.target>3, 15,<emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 ſit potentia mouens pon<lb></lb>dus, ſimili modo oſtendetur ſpatium ponderis D ſpatii potentiæ in <lb></lb>M ſeſquitertium eſſe. </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 ſi funis in O per alium circumuoluatur orbiculum, qui tro<lb></lb>chleæ ſuperiori deinde religetur; eodem modo demonſtrabimus <lb></lb>proportionem potentiæ in M pondus ſuſtinentis ad pondus ſeſ<lb></lb>quiquartam eſſe. </s> <s id="id.2.1.189.2.1.2.0">& ſi in M ſit potentia mouens, ſimiliter oſten<lb></lb>detur ſpatium ponderis ſpatii potentiæ ſeſquiquartum eſſe. </s> <s id="id.2.1.189.2.1.3.0">pro<lb></lb>cedendoq; hoc modo in infinitum quamcunq; proportionem <lb></lb>potentiæ ad pondus ſuperparticularem inueniemus; ſemper〈qué〉 <pb n="94" xlink:href="036/01/201.jpg"></pb>oſtendemus potentiam pondus ſuſtinentem ita eſſe ad pondus, <lb></lb>vt ſpatium ponderis ad ſpatium 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 eſt, ac ſi G eſſet fulcimentum, cùm <lb></lb>funis ſit religatus in L; pondus ac ſi in E eſſet appenſum, & po<lb></lb>tentia in medio. </s> <s id="id.2.1.189.3.1.2.0">Vectis verò CA fulcimentum eſt A pondus in <lb></lb>medio, & potentia in C. </s> <s id="N1594A">& K fulcimentum eſt vectis Pk, pon<lb></lb>dus in P, & potentia in medio. </s> <s id="id.2.1.189.3.1.3.0">quæ omnia ſicut in præceden<lb></lb>ti oſtendentur. </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 vtriſq; duarum trochlearum ſingulis or<lb></lb>biculis, quarum altera ſupernè à potentia ſuſti<lb></lb>neatur, altera verò infernè, ponderiq; alligata, <lb></lb><expan abbr="cõſtituta">conſtituta</expan> fuerit, circumferatur funis; vtroq; eius <lb></lb>extremo alicuibi, non autem trochleis religato; <lb></lb>æqualis erit ponderi potentia. </s> </p> <pb xlink:href="036/01/202.jpg"></pb> <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 ſuperioris <lb></lb>ABC, cuius centrum D; & EFG <lb></lb>trochleæ ponderi H alligatæ, cu<lb></lb>ius centrum k; & ſit funis LEF <lb></lb>GABCM circa orbiculos reuo<lb></lb>lutus, religatuſq; in LM; ſitq; <lb></lb>potentia in N ſuſtinens pondus <lb></lb>H. </s> <s id="id.2.1.189.7.1.1.0.a">dico potentiam in N æqua<lb></lb>lem eſſe ponderi H. </s> <s id="id.2.1.189.7.1.1.0.b">Accipiatur <lb></lb>quoduis punctum O in AG. </s> <s id="id.2.1.189.7.1.1.0.c">& <lb></lb>quoniam ſi in O eſſet potentia ſu<lb></lb><arrow.to.target n="note271"></arrow.to.target>ſtinens pondus H, ſubdupla eſſet <lb></lb><arrow.to.target n="note272"></arrow.to.target>ponderis H, & potentiæ in O <lb></lb>dupla eſt ea, quæ eſt in D, ſiue <lb></lb>(quod idem eſt) in N; erit po<lb></lb>tentia in N ponderi H æqualis. </s> <lb></lb> <s id="id.2.1.189.7.1.2.0">quod demonſtrare oportebat. <figure id="id.036.01.202.1.jpg" place="text" 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 ſi in N ſit potentia mouens pondus. </s> <s id="id.2.1.189.8.1.2.0">Dico <lb></lb>ſpatium potentiæ in N æqualem eſſe ſpatio pon<lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.190.1.1.2.0"><margin.target id="note272"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 15 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></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 ſpatium puncti O moti, duplum eſt, tùm ſpatii <lb></lb><arrow.to.target n="note273"></arrow.to.target>ponderis H moti, tùm ſpatii potentiæ in N motæ; erit ſpatium <lb></lb><arrow.to.target n="note274"></arrow.to.target>potentiæ in N ſpatio 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.192.1.1.2.0"><margin.target id="note274"></margin.target>16 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="95" xlink:href="036/01/203.jpg"></pb> <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">Iiſdem poſitis, transfera<lb></lb>tur centrum orbiculi ABC <lb></lb>vſq; ad P; orbiculuſq; poſi<lb></lb>tionem habeat QRS; dein<lb></lb>de eodem tempore orbiculus <lb></lb>EFG ſit in TVX, cuius cen<lb></lb>trum ſit Y; & pondus perue<lb></lb>nerit in Z. ducantur per or<lb></lb>biculorum centra lineæ GE <lb></lb>TX AC QS horizonti æqui <lb></lb>diſtantes. </s> <s id="id.2.1.193.2.1.2.0">& ſicut in aliis <lb></lb>demonſtratum fuit, duo fu<lb></lb>nes AQ CS duobus XG <lb></lb>TE æquales erunt; ſed AQ <lb></lb>CS ſimul dupli ſunt ſpatii po<lb></lb>tentiæ motæ; & duo XG TE <lb></lb>ſimul ſunt ſimiliter dupli ſpa<lb></lb>tii ponderis; erit igitur <expan abbr="ſpatiũ">ſpatium</expan><lb></lb>potentiæ ſpatio ponderis æ<lb></lb>quale. </s> <s id="id.2.1.193.2.1.3.0">quod demonſtrare o<lb></lb>portebat. <figure id="id.036.01.203.1.jpg" place="text" xlink:href="036/01/203/1.jpg"></figure></s> </p> <pb xlink:href="036/01/204.jpg"></pb> <p id="id.2.1.193.4.0.0.0" type="main"> <s id="id.2.1.193.4.1.1.0">Quod etiam ſi vtraq; trochlea duos <lb></lb>habuerit orbiculos, quorum centra <lb></lb>ſint ABCD, funiſq; per omnes cir<lb></lb>cumuoluatur, qui in LM religetur; <lb></lb>ſimiliter oſtendetur potentiam in N <lb></lb>æqualem eſſe ponderi H. </s> <s id="N15A54">vnaquæq; <lb></lb>enim potentia in EF ſuſtinens pon<lb></lb>dus ſubquadrupla eſt ponderis; & po<lb></lb>tentiæ in CD duplæ ſunt earum, <lb></lb>quæ ſunt in EF; erit vnaquæq; po<lb></lb>tentia in CD ſubdupla ponderis H. <lb></lb></s> <s id="N15A61">quare potentiæ in CD ſimul ſumptæ <lb></lb>ponderi H erunt æquales. </s> <s id="id.2.1.193.4.1.2.0">& quo<lb></lb>niam potentia in N duabus in CD <lb></lb>pontentiis eſt æqualis; erit potentia <lb></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 ſi in N ſit potentia mouens, ſi <lb></lb>mili modo oſtendetur, ſpatium po<lb></lb>tentiæ æquale eſſe ſpatio 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></lb>quatuor, vel quotcunq; habeat orbi<lb></lb>culos; ſemper oſtendetur <expan abbr="potẽtiam">potentiam</expan> in <lb></lb>N æqualem eſſe ponderi H; & ſpa<lb></lb>tium potentiæ pondus mouentis æ<lb></lb>quale eſſe ſpatio ponderis moti. <figure id="id.036.01.204.1.jpg" place="text" 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 ſe habent; orbiculorum qui <lb></lb>dem trochleæ ſuperioris, veluti AC in præcedenti figura fulcimen <lb></lb>tum eſt C, pondus verò in A appenſum, & potentia in D medio. </s> <s id="id.2.1.193.7.1.2.0"><lb></lb>vectes autem orbiculorum trochleæ inferioris ita mouentur, vt ip<lb></lb>ſius GE fulcimentum ſit E, pondus in medio appenſum, & po<lb></lb>tentia in G. </s> </p> <pb n="96" xlink:href="036/01/205.jpg"></pb> <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></lb>rum altera vnius dumtaxat orbiculi ſupernè à <lb></lb>potentia ſuſtineatur, altera verò duorum infer<lb></lb>nè, ponderiq; alligata fuerit conſtituta, cir<lb></lb>cundetur funis; vtroq; eius extremo alicubi, ſed <lb></lb>non ſuperiori trochleæ religato: duplum erit <lb></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></lb>trochleæ ponderi C alligatæ; D ve<lb></lb>rò ſit centrum orbiculi trochleæ ſu<lb></lb>perioris; ſit deinde funis per om<lb></lb>nes orbiculos circumuolutus, reli<lb></lb>gatuſq; in EF; & ſit potentia in <lb></lb>G ſuſtinens pondus C. </s> <s id="id.2.1.193.11.1.1.0.a">dico pon<lb></lb>dus C duplum eſſe potentiæ in G. </s> <s id="id.2.1.193.11.1.1.0.b"><lb></lb>Quoniam enim ſi in H k duæ eſ<lb></lb>ſent potentiæ pondus ſuſtinentes <lb></lb>duobus funibus orbiculis trochleæ <lb></lb>inferioris tantùm circumuolutis, eſ<lb></lb>ſet vtiq; vtraq; potentia in k H ſub <arrow.to.target n="note275"></arrow.to.target><lb></lb>quadrupla ponderis C; ſed poten<lb></lb>tia in G æqualis eſt potentiis in Hk <arrow.to.target n="note276"></arrow.to.target><lb></lb>ſimul ſumptis; vniuſcuiuſq; enim <lb></lb>potentiæ in H, & k dupla eſt: erit <lb></lb>potentia in G ſubdupla ponderis <lb></lb>C. </s> <s id="N15AF4">pondus ergo potentiæ duplum <lb></lb>erit. </s> <s id="id.2.1.193.11.1.2.0">quod demonſtrare opor<lb></lb>tebat. <figure id="id.036.01.205.1.jpg" place="text" xlink:href="036/01/205/1.jpg"></figure></s> </p> <pb xlink:href="036/01/206.jpg"></pb> <p id="id.2.1.193.13.0.0.0" type="main"> <s id="id.2.1.193.13.1.1.0">Et ſi in G ſit potentia mouens pondus. </s> <s id="id.2.1.193.13.1.2.0">Dico <lb></lb>ſpatium potentiæ duplum eſſe ſpatii 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"></emph>Ex<emph.end type="italics"></emph.end> 7 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.194.1.1.2.0"><margin.target id="note276"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 15 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.195.1.0.0.0" type="main"> <s id="id.2.1.195.1.1.1.0">Iiſdem poſitis, ſint <lb></lb>moti orbiculi, ſimiliter <lb></lb>demonſtrabitur ambos <lb></lb>illos LM NO æquales <lb></lb>eſſe quatuor PQ RS <lb></lb>TV XY. </s> <s id="N15B41">ſed LM NO <lb></lb>ſimul dupli ſunt ſpatii po<lb></lb>tentiæ in G motæ; & <lb></lb>quatuor PQ RS TV <lb></lb>XY ſimul quadrupli ſunt <lb></lb>ſpatii ponderis moti. </s> <s id="N15B4D">ſpa<lb></lb>tium igitur potentiæ ad <lb></lb>ſpatium ponderis eſt tan<lb></lb>quam ſubduplum ad ſub <lb></lb>quadruplum. </s> <s id="id.2.1.195.1.1.2.0">erit ergo <lb></lb>potentiæ ſpatium pon<lb></lb>deris ſpatii duplum. <figure id="id.036.01.206.1.jpg" place="text" xlink:href="036/01/206/1.jpg"></figure></s> </p> <pb n="97" xlink:href="036/01/207.jpg"></pb> <p id="id.2.1.195.3.0.0.0" type="main"> <s id="id.2.1.195.3.1.1.0">Hinc autem conſiderandum <lb></lb>eſt quomodo fiat motus; quia, <lb></lb>cùm funis ſit religatur in F, vectis <lb></lb>NO in prima figura habebit ful<lb></lb>cimentum O, pondus in medio, <lb></lb>& potentia in N. </s> <s id="N15B77">ſimiliter quo<lb></lb>niam funis eſt religatus in E, ve<lb></lb>ctis PQ habebit <expan abbr="fulcimentũ">fulcimentum</expan> P, & <lb></lb>pondus in medio, & potentia in <lb></lb>q. </s> <s id="N15B78">idcirco partes orbiculorum <lb></lb>in N, & Q ſurſum mouebuntur; <lb></lb>orbiculi ergo non in eandem, ſed <lb></lb>in contrarias mouebuntur partes, <lb></lb>videlicet vnus <expan abbr="dextroſum">dextrorsum</expan>, alter ſi<lb></lb>niſtrorſum. </s> <s id="id.2.1.195.3.1.2.0">& quoniam potentiæ <lb></lb>in NQ eædem ſunt, quæ ſunt in <lb></lb>LM; potentiæ igitur in LM æ<lb></lb>quales ſurſum mouebuntur. </s> <s id="id.2.1.195.3.1.3.0">ve<lb></lb>ctis igitur LM in neutram moue<lb></lb>bitur partem. </s> <s id="id.2.1.195.3.1.4.0">quare neq; orbicu<lb></lb>lus circumuertetur. </s> <s id="id.2.1.195.3.1.5.0">Itaq; LM <lb></lb>erit tanquam libra, cuius centrum <lb></lb>D, pondera〈qué〉 appenſa in LM <lb></lb>æqualia quartæ parti ponderis C; <lb></lb>vnuſquiſq; enim funis LN MQ <lb></lb>quartam ſuſtinet partem ponderis C. </s> <s id="N15BB5">mouebitur ergo totus orbi <lb></lb>culus, cuius centrum D, ſurſum; ſed non circumuertetur. <figure id="id.036.01.207.1.jpg" place="text" xlink:href="036/01/207/1.jpg"></figure></s> </p> <pb xlink:href="036/01/208.jpg"></pb> <p id="id.2.1.195.5.0.0.0" type="main"> <s id="id.2.1.195.5.1.1.0">Et ſi funis in F circa alios duos <lb></lb>voluatur orbiculos, quorum cen<lb></lb>tra ſint HK, qui deinde religetur <lb></lb>in L; erit proportio ponderis ad <lb></lb>potentiam ſeſquialtera. </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 eſſent potentiæ <lb></lb><arrow.to.target n="note277"></arrow.to.target>in MNOI, eſſet vnaquæq; ſubſeſ<lb></lb>cupla ponderis C, quare quatuor <lb></lb>ſimul potentiæ in MNOI qua<lb></lb>tuor ſextæ erunt ponderis C. </s> <s id="N15BE0">& <lb></lb>quoniam duæ ſimul potentiæ in <lb></lb>HD quatuor potentiis in MNOI <lb></lb>ſunt æquales; & potentia in G æ<lb></lb>qualis eſt potentiis in DH: erit <lb></lb>potentia in G quatuor ſimul po<lb></lb>tentiis in MNOI æqualis; & ob <lb></lb>id quatuor ſextæ erit ponderis C. </s> <s id="id.2.1.195.6.1.1.0.a"><lb></lb>proportio igitur ponderis C ad po<lb></lb>tentiam in G ſeſquialtera eſt. </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"></emph>Ex<emph.end type="italics"></emph.end> 9 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></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 ſi in G ſit potentia mouens, <lb></lb>ſimili modo oſtendetur ſpatium <lb></lb>potentiæ ſpatii ponderis ſeſquialte<lb></lb>rum eſſe. <figure id="id.036.01.208.1.jpg" place="text" 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 ſi funis in L adhuc circa duos <lb></lb>alios orbiculos reuoluatur ſimi<lb></lb>liter oſtendetur proportionem <lb></lb>ponderis ad potentiam ſeſqui<lb></lb>tertiam eſſe. </s> <s id="id.2.1.197.2.1.2.0">quòd ſi in G ſit <lb></lb>potentia mouens, oſtende<lb></lb>tur ſpatium potentiæ ſpatii ponde<lb></lb>ris ſeſquitertium eſſe, atq; ita dein<lb></lb>ceps in infinitum procedendo, <lb></lb>quamcunq; proportionem ponderis ad potentiam ſuperparticula<lb></lb>rem inueniemus ſemperq; reperiemus ita eſſe pondus ad poten<lb></lb>tiam pondus ſuſtinentem, vt ſpatium potentiæ mouentis ad ſpa<lb></lb>tium ponderis à potentia moti. </s> </p> <pb n="98" xlink:href="036/01/209.jpg"></pb> <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 ſit religatus <lb></lb>in E, habet fulcimentum in Y, pondus in B medio appenſum, & <lb></lb>potentia in Z. </s> <s id="N15C48">& vectis PQ habet fulcimentum in P potentia in <lb></lb>medio, & pondus in q. </s> <s id="N15C4C">oportet enim orbiculos, quorum cen<lb></lb>tra ſunt BD in eandem partem moueri, videlicet vt QZ ſur<lb></lb>ſum moueantur. </s> <s id="id.2.1.197.4.1.2.0">& quoniam funis religatus eſt in L, erit T fulci <lb></lb>mentum vectis ST, qui pondus habet in medio, & potentia in <lb></lb>S. </s> <s id="N15C59">& quia S mouetur ſurſum, neceſſe eſt etiam R ſurſum moue <lb></lb>ri; & ideo F erit fulcimentum vectis FR, & pondus erit in R, <lb></lb>& potentia in medio. </s> <s id="id.2.1.197.4.1.3.0">orbiculi igitur, quorum centra ſunt H k, <lb></lb>in contrariam mouentur partem eorum, quorum centra ſunt BD: <lb></lb>quare partes <expan abbr="orbiculorũ">orbiculorum</expan> PF in orbiculis deorſum <expan abbr="tendẽt">tendent</expan>; videlicet <lb></lb>verſus XV. </s> <s id="id.2.1.197.4.1.3.0.a">vectis igitur VX in neutram partem mouebitur, cùm <lb></lb>P, & F deorſum moueantur; & VX erit tanquam vectis, in cuius <lb></lb>medio erit pondus appenſum, & in VX duæ potentiæ æquales <lb></lb>ſextæ parti ponderis C. </s> <s id="N15C79">potentiæ enim in MO hoc eſt funes PV <lb></lb>FX ſextam ſuſtinent partem ponderis C. </s> <s id="N15C7D">totus igitur orbiculus, <lb></lb>cuius centrum A ſurſum vnà cum trochlea mouebitur; non au<lb></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></lb>quarum altera binis inſignita rotulis à potentia <lb></lb>ſupernè detineatur; altera verò vnius tantùm <lb></lb>rotulæ infernè <expan abbr="cõſtituta">conſtituta</expan>, ac ponderi alligata fue<lb></lb>rit, circumuoluatur funis; vtroq; eius extremo <lb></lb>alicuibi, non autem inferiori trochleæ religa<lb></lb>to: dupla erit ponderis potentia. </s> </p> <pb xlink:href="036/01/210.jpg"></pb> <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></lb>quæ orbiculum habeat, cuius centrum ſit B; tro<lb></lb>chlea verò ſuperior duos orbiculos habeat, <lb></lb>quorum centra ſint CD; ſitq; funis circa om<lb></lb>nes orbiculos reuolutus, qui in EF ſit religatus; <lb></lb>potentiaq; ſuſtinens pondus ſit in G. </s> <s id="id.2.1.197.8.1.1.0.a">dico po<lb></lb>tentiam in G ponderis A duplam eſſe. </s> <s id="id.2.1.197.8.1.2.0">ſi enim <lb></lb><arrow.to.target n="note278"></arrow.to.target>in H k duæ eſſent potentiæ pondus ſuſtinen<lb></lb><arrow.to.target n="note279"></arrow.to.target>tes, eſſet vtraq; ſubdupla ponderis A; ſed po<lb></lb><arrow.to.target n="note280"></arrow.to.target>tentia in D dupla eſt potentiæ in H, & poten<lb></lb>tia in C dupla potentiæ in K; quare duæ ſimul <lb></lb>potentiæ in CD vtriuſq; ſimul potentiæ in H k <lb></lb>duplæ erunt. </s> <s id="id.2.1.197.8.1.3.0">ſed potentiæ in H k ponderi A ſunt <lb></lb>æquales, & potentiæ in CD ipſi potentiæ in G <lb></lb>ſunt etiam æquales; potentia igitur in G ponde<lb></lb>ris A dupla erit. </s> <s id="id.2.1.197.8.1.4.0">quod oportebat demonſtrare. </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"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.198.1.1.2.0"><margin.target id="note279"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.198.1.1.3.0"><margin.target id="note280"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 15 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></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 ſit potentia mouens pon<lb></lb>dus, ſimiliter vt in præcedenti oſtendetur ſpa<lb></lb>tium ponderis ſpatii potentiæ duplum eſſe. <figure id="id.036.01.210.1.jpg" place="text" 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; conſiderandum eſt vectem PQ <lb></lb>non moueri, quia vectis LM habet fulcimen<lb></lb>tum in L, potentia in medio, & pondus in M. </s> <s id="id.2.1.199.2.1.1.0.a"><lb></lb>vectis autem NO habet fulcimentum in O, <lb></lb>potentia in medio, & pondus in N. </s> <s id="N15D20">quare M, & N ſurſum mo<lb></lb>uebuntur. </s> <s id="id.2.1.199.2.1.2.0">in contrarias igitur partes orbiculi, quorum centra <lb></lb>ſunt CD mouentur. </s> <s id="id.2.1.199.2.1.3.0">idcirco vectis PQ in neutram partem mo<lb></lb>uebitur; eritq;, ac ſi in medio eſſet appenſum pondus, & in PQ <lb></lb>duæ potentiæ æquales dimidio ponderis A. </s> <s id="N15D30">vtraq; enim potentia <lb></lb>in HK ſubdupla eſt ponderis A. </s> <s id="N15D34">totus igitur orbiculus, cuius <lb></lb>centrum B ſurſum mouebitur, ſed non circumuertetur. </s> </p> <pb n="99" xlink:href="036/01/211.jpg"></pb> <p id="id.2.1.199.4.0.0.0" type="main"> <s id="id.2.1.199.4.1.1.0">Et ſi funis in F duobus aliis adhuc circumuol<lb></lb>uatur orbiculis, quorum centra ſint HK, qui de<lb></lb>inde religetur in L; erit proportio potentiæ in G <lb></lb>ad pondus A ſeſquialtera. </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 eſſent poten<lb></lb>tiæ pondus ſuſtinentes, vnaquæq; ſubquadru<arrow.to.target n="note281"></arrow.to.target><lb></lb>pla eſſet ponderis A: ſed cùm potentia in k <arrow.to.target n="note282"></arrow.to.target><lb></lb>ſit dupla potentiæ in N; erit potentia in k <lb></lb>ponderis A ſubdupla. </s> <s id="id.2.1.199.5.1.2.0">& quoniam potentia <lb></lb>in D duabus in MO potentiis eſt æqualis; erit <lb></lb>quoq; potentia in D ponderis A ſubdupla. </s> <s id="id.2.1.199.5.1.3.0"><lb></lb>cùm autem adhuc potentia in C potentiæ in P <lb></lb>ſit dupla, erit ſimiliter <expan abbr="potẽtia">potentia</expan> in C ponderis A <lb></lb>ſubdupla. </s> <s id="id.2.1.199.5.1.4.0">tres igitur potentiæ in CD k tribus <lb></lb>medietatibus ponderis A ſunt æquales. </s> <s id="id.2.1.199.5.1.5.0">quo<lb></lb>niam autem potentia in G potentiis in CDK <lb></lb>eſt æqualis, erit potentia in G tribus medie<lb></lb>tatibus ponderis A æqualis. </s> <s id="id.2.1.199.5.1.6.0">Proportio igi<lb></lb>tur potentiæ ad pondus ſeſquialtera eſt. </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"></emph>Ex<emph.end type="italics"></emph.end> 7 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.200.1.1.2.0"><margin.target id="note282"></margin.target>15 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 ſit potentia mouens, erit ſpa<lb></lb>tium ponderis ſpatii potentiæ ſeſquialterum. <figure id="id.036.01.211.1.jpg" place="text" 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 ſi funis in L adhuc circa duos alios or<lb></lb>biculos reuoluatur, ſimiliter oſtendetur pro<lb></lb>portionem potentiæ ad pondus ſeſquitertiam <lb></lb>eſſe. </s> <s id="id.2.1.201.2.1.2.0">& ſic in infinitum omnes proportiones <lb></lb>potentiæ ad pondus ſuperparticulares inue<lb></lb>niemus. </s> <s id="id.2.1.201.2.1.3.0">oſtendemuſq; potentiam pondus <lb></lb>ſuſtinentem ad pondus ita eſſe, vt ſpatium <lb></lb>ponderis moti ad ſpatìum potentiæ pondus <lb></lb>mouentis. </s> </p> <pb xlink:href="036/01/212.jpg"></pb> <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></lb>modo, videlicet Q erit ful<lb></lb>cimentum vectis QR, po<lb></lb>tentia in medio, pondus <lb></lb>in R; & vectis Z 9 fulci <lb></lb>mentum erit Z, pondus in <lb></lb>medio, potentiaq; in 9. </s> <s id="N15DDC">ſi<lb></lb>militer X erit fulcimentum <lb></lb>vectis VX, potentia in me <lb></lb>dio, pondus in V. </s> <s id="id.2.1.201.4.1.1.0.a">& quo<lb></lb>niam V ſurſum mouetur, Y <lb></lb>quoq; ſurſum mouebitur; <lb></lb>& vectis YF fulcimentum <lb></lb>erit F: quare F, & Z in orbi <lb></lb>culis deorſum mouebun<lb></lb>tur. </s> <s id="id.2.1.201.4.1.2.0">& ob id vectis ST in <lb></lb>neutram mouebitur par<lb></lb>tem; & ST erit tamquam <lb></lb>libra, cuius centrum D, & <lb></lb>pondera in ST æqualia <lb></lb>quartæ parti ponderis A. <lb></lb></s> <s id="N15E01">vnuſquiſq; enim funis SZ <lb></lb>TF quartam ſuſtinet par<lb></lb>tem ponderis A. </s> <s id="N15E07">orbicu<lb></lb>lus ergo, cuius centrum D, <lb></lb>ſurſum mouebitur; non au<lb></lb>tem circumuertetur. <figure id="id.036.01.212.1.jpg" place="text" xlink:href="036/01/212/1.jpg"></figure></s> </p> <pb n="100" xlink:href="036/01/213.jpg"></pb> <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></lb>& ſubmultiplices; deinde ſuperparticulares, ſubſuperparticu<lb></lb>lareſ〈qué〉 declaratæ fuerunt: nunc autem reliquum eſt, vt propor<lb></lb>tiones inter pondus, & potentiam ſuperpartientes, & multi<lb></lb>plices ſuperparticulares, multiplices〈qué〉 ſuperpartientes mani<lb></lb>feſtentur. </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 id="N15E2E" 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 ſuperpartientem inuenire <lb></lb>volumus, quemadmodum ſi proportio, quam <lb></lb>habet pondus ad potentiam pondus ſuſtinen<lb></lb>tem fuerit ſuperbipartiens, ſicut quinque ad <lb></lb>tria. </s> </p> <pb xlink:href="036/01/214.jpg"></pb> <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 ſuſti<lb></lb>nens, proportionemq; habeat pondus B ad <lb></lb>potentiam in A, vt quinq; ad vnum; hoc eſt, <lb></lb>ſit potentia in A ſubquintupla ponderis B: de<lb></lb>inde eodem fune circa alios orbiculos reuo<lb></lb><arrow.to.target n="note284"></arrow.to.target>luto inueniatur potentia in C, quæ tripla ſit <lb></lb>potentiæ in A. </s> <s id="id.2.1.201.10.1.1.0.a">& quoniam pondus B ad po<lb></lb>tentiam in A eſt, vt quinq; ad vnum; & <lb></lb>potentia in A ad potentiam in C eſt, vt vnum <lb></lb>ad tria; erit pondus B ad potentiam in C, vt <lb></lb>quinq; ad tria; hoc eſt ſuperbipartiens. </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"></emph>Ex<emph.end type="italics"></emph.end> 9 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.202.1.1.2.0"><margin.target id="note284"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 17 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></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></lb>ris ad potentiam ſuperpartientes inuenientur; <lb></lb>vt ſi ſupertripartientem quis inuenire volue<lb></lb>rit; eodem incedat ordine; fiat ſcilicet poten<lb></lb>tia in A ſuſtinens pondus B ſubſeptupla ip<lb></lb>ſius ponderis B; deinde fiat potentia in C ip<lb></lb>ſius A quadrupla; erit pondus B ad poten<lb></lb>tiam in C, vt ſeptem ad quatuor: vídelicet <lb></lb>ſupertripartiens. </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 ſit potentia mo<lb></lb>uens pondus erit ſpatium <expan abbr="potẽtiæ">potentiæ</expan><lb></lb>ſpatii ponderis ſuperbipartiens. <figure id="id.036.01.214.1.jpg" place="text" 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></lb>eſt ſpatii potentiæ in A, ita videlicet ſe habent, <lb></lb>vt quinq; ad quindecim; & ſpatium potentiæ <lb></lb><arrow.to.target n="note286"></arrow.to.target>in A quintuplum eſt ſpatii ponderis B, hoc <lb></lb>eſt, vt quindecim ad tria; erit igitur ſpatium <lb></lb>potentiæ in C ad ſpatium ponderis B, vt <lb></lb>quinq; ad tria; videlicet ſuperbipartiens. </s> <s id="id.2.1.203.3.1.2.0">& ſemper oſtendemus, ita <lb></lb>eſſe ſpatium potentiæ mouentis ad ſpatium ponderis; vt pondus <lb></lb>ad potentiam pondus ſuſtinentem. </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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.204.1.1.2.0"><margin.target id="note286"></margin.target>14 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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; prorſus ratione proportionem potentiæ ad pondus ſu<pb n="101" xlink:href="036/01/215.jpg"></pb>perpartientem inueniemus. </s> <s id="id.2.1.205.1.1.2.0">ſi enim C eſſet inferius, & in ipſo <lb></lb>appenſum eſſet pondus; B verò ſuperius, in quo eſſet potentia pon<lb></lb>dus in C ſuſtinens, eſſet potentia in B ſuperbipartiens ponderis <lb></lb>in C appenſi: cùm B ad A ſit, <expan abbr="vtquinq;">vt quinq;</expan> ad vnum; A verò ad <arrow.to.target n="note287"></arrow.to.target><lb></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 ſuperparticularem in<lb></lb>uenire voluerimus; vt proportio, quam habet <lb></lb>pondus ad potentiam pondus ſuſtinentem, ſit <lb></lb>duplex ſeſquialtera, 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.206.1.1.2.0"><margin.target id="note288"></margin.target>5 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 ſuperpartientes inuenimus, has quo<lb></lb>que omnes multiplices ſuperparticulares reperiemus. </s> <s id="id.2.1.207.1.1.2.0">vt fiat <arrow.to.target n="note289"></arrow.to.target><lb></lb>pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve<arrow.to.target n="note290"></arrow.to.target><lb></lb>ro in C ad potentiam in A, vt duo ad vnum; quod fiet, ſi fu<lb></lb>nis ſit religatus in D, non autem trochleæ ſuperiori, vel in F: erit <lb></lb>pondus B ad potentiam in C, vt quinq; ad duo; hoc eſt duplum <lb></lb>ſeſquialterum. </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"></emph>Ex<emph.end type="italics"></emph.end> 9 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.208.1.1.2.0"><margin.target id="note290"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 15, 16, <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></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 conuerſo proportionem potentiæ ad pondus multiplicem <lb></lb>ſuperparticularem inueniemus; & vt in reliquis oſtendetur, ita eſ<lb></lb>ſe ſpatium potentiæ mouentis ad ſpatium ponderis, vt pondus <lb></lb>ad potentiam pondus ſuſtinentem. </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 ſuperpartientem <lb></lb>eodem modo inueniemus; vt ſi proportio, quam <lb></lb>habet pondus ad potentiam, ſit duplex ſuperbi <lb></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 ſuſtinens ſuboctupla ponderis B; <arrow.to.target n="note291"></arrow.to.target><lb></lb>& potentia in C potentiæ in A ſit tripla; erit pondus B ad po<lb></lb>tentiam in C, vt octo ad tria. </s> <s id="id.2.1.209.3.1.2.0">& è conuerſo omnem potentiæ ad <pb xlink:href="036/01/216.jpg"></pb>pondus proportionem <expan abbr="multipticem">multiplicem</expan> ſuperpartientem in ueniemus. </s> <s id="id.2.1.209.3.1.3.0"><lb></lb>& vt in cæteris reperiemus ita eſſe pondus ad potentiam pondus <lb></lb>ſuſtinentem, vt ſpatium potentiæ mouentis ad ſpatium pon<lb></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"></emph>Ex<emph.end type="italics"></emph.end> 9 <emph type="italics"></emph>huius Ex<emph.end type="italics"></emph.end> 17 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></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 eſt, quòd cùm in præcedentibus <expan abbr="demoſtratio">demonstratio</expan><lb></lb>nibus ſæpius dictum fuerit, potentiam pondus ſuſtinentem ipſius <lb></lb>ponderis duplam eſſe, vel triplam, & huiuſmodi; vt in decima<lb></lb>quinta huius oſtenſum eſt; quia tamen potentia non ſolum pon<lb></lb>dus, verùm etiam trochleam ſuſtinet; idcirco maioris longè vir<lb></lb>tutis, maioriſq; ipſi ponderi proportionis conſtituenda videtur <lb></lb>ipſa potentia. </s> <s id="id.2.1.211.1.1.2.0">quod quidem verum eſt, ſi etiam trochleæ graui<lb></lb>tatem conſiderare voluerimus. </s> <s id="id.2.1.211.1.1.3.0">ſed quoniam inter potentiam, & <lb></lb>pondus proportionem quærimus: ideo hanc trochleæ grauitatem <lb></lb>ommiſimus, quam ſiquis etiam conſiderare voluerit, vim ipſi po<lb></lb>tentiæ æqualem trochleæ addere poterit. </s> <s id="id.2.1.211.1.1.4.0">Quod ipſum etiam in <lb></lb>fune obſeruari poterit. </s> <s id="id.2.1.211.1.1.5.0">& ſicut hoc in decimaquinta conſideraui<lb></lb>mus, idem quoq; in reliquis aliis conſiderare poterimus. </s> </p> <pb n="97" xlink:href="036/01/217.jpg"></pb> <p id="id.2.1.211.3.0.0.0" type="main"> <s id="id.2.1.211.3.1.1.0">Nouiſſe etiam oportet, quòd ſicuti proportio <lb></lb>nes omnes inter potentiam, & pondus vnico <lb></lb>fune inuentæ fuerunt; ita etiam pluribus funi<lb></lb>bus, trochleiſ〈qué〉 eædem inueniri poterunt. </s> <s id="id.2.1.211.3.1.2.0">vt <lb></lb>ſi multiplicem ſuperparticularem proportionem <lb></lb>pluribus funibus inuenire voluerimus, veluti ſi <lb></lb>proportio, quam habet pondus ad potentiam <lb></lb>pondus ſuſtinentem, fuerit duplex ſeſquialtera, vt <lb></lb>quinq; ad duo; oportet hanc proportionem ex <lb></lb>pluribus componere. </s> <s id="id.2.1.211.3.1.3.0">vt (exempli gratia) ex pro<lb></lb>portione ſeſquiquarta, vt quin〈qué〉 ad quatuor, <lb></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></lb>tentia in A pondus B ſuſtinens, ad quam pondus <lb></lb><expan abbr="proportionẽ">proportionem</expan> habeat ſeſquiquartam, vt quinq; ad <lb></lb>quatuor: deinde alio fune inueniatur <expan abbr="potẽtia">potentia</expan> in C,<arrow.to.target n="note293"></arrow.to.target><lb></lb>cuius dupla ſit potentia in A. </s> <s id="id.2.1.211.3.1.4.0.a">& <expan abbr="quoniã">quoniam</expan> B ad A eſt, <lb></lb>vt quinq; ad quatuor; & A ad C, vt quatuor ad <lb></lb>duo; erit pondus B ad potentiam in C, vt quin<lb></lb>que ad duo; hoc eſt proportionem habebit du<lb></lb>plicem ſeſquialteram. <figure id="id.036.01.217.1.jpg" place="text" 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 eſt hanc quoq; <expan abbr="proportionẽ">proportionem</expan> inue<lb></lb>niri poſſe, ſi proportionem quinq; ad duo ex pluri<lb></lb>bus componamus, vt quinq; ad quindecim & quin<lb></lb>decim ad viginti & viginti ad duo. </s> <s id="id.2.1.211.4.1.2.0">Et hoc modo <lb></lb>non ſolum omnem aliam proportionem inuenie<lb></lb>mus, ſed quamcunq, multis, infinitis〈qué〉 mo<lb></lb>dis comperiemus. </s> <s id="id.2.1.211.4.1.3.0">omnis enim proportio ex infi<lb></lb>nitis proportionibus componi poteſt. </s> <s id="id.2.1.211.4.1.4.0">vt patet <lb></lb>in commentario Eutocii in quartam propoſitio<lb></lb>nem ſecundi libri Archimedis de ſphera, & cy<lb></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"></emph>Ex<emph.end type="italics"></emph.end> 21 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.212.1.1.2.0"><margin.target id="note293"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <p id="id.2.1.213.1.0.0.0" type="main"> <s id="id.2.1.213.1.1.1.0">Poſſumus quoq; pluribus funibus, trochleis <lb></lb>verò inferioribus tantùm, vel ſuperioribus vti. </s> </p> <pb xlink:href="036/01/218.jpg"></pb> <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 ſit trochlea <lb></lb>orbiculum habens, cuius centrum B; <lb></lb>religetur funis in C, qui circa orbiculum <lb></lb>reuoluatur, funiſq; perueniat in D: erit <lb></lb><arrow.to.target n="note294"></arrow.to.target>potentia in D ſuſtinens pondus A ſub<lb></lb>dupla ponderis A. </s> <s id="id.2.1.213.3.1.1.0.a">deinde funis in D <lb></lb>alteri trochleæ religetur, & circa huius <lb></lb>trochleæ orbiculum alius reuoluatur fu<lb></lb>nis, qui religetur in E, & perueniat in <lb></lb><arrow.to.target n="note295"></arrow.to.target>F; erit potentia in F ſubdupla eius, <lb></lb>quod ſuſtinet <expan abbr="potẽtia">potentia</expan> in D: eſt enim ac ſi <lb></lb>D dimidium ponderis A ſuſtineret ſi <lb></lb>ne trochlea; quare potentia in F ſubqua<lb></lb>drupla erit ponderis A. </s> <s id="N160B4">& ſi adhuc fu <lb></lb>nis in F alteri trochleæ religetur, & <lb></lb>per eius orbiculum circumuoluatur a<lb></lb>lius funis, qui religetur in G, & per <lb></lb>ueniat in H; erit potentia in H ſub <lb></lb>dupla potentiæ in F. </s> <s id="id.2.1.213.3.1.1.0.b">ergo potentia in <lb></lb>H ſuboctupla erit ponderis A. </s> <s id="N160C5">& ſic <lb></lb>in infinitum ſemper ſubduplam poten<lb></lb>tiam <expan abbr="præcedẽtis">præcedentis</expan> potentiæ inueniemus. <figure id="id.036.01.218.1.jpg" place="text" 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 ſi in H ſit potentia mouens, erit <lb></lb>ſpatium potentiæ ſpatii ponderis octu<lb></lb><arrow.to.target n="note296"></arrow.to.target>plum. </s> <s id="id.2.1.213.4.1.2.0">ſpatium enim D duplum eſt ſpa<lb></lb>tii ponderis A, & ſpatium F ſpatii D <lb></lb>duplum; erit ſpatium F ſpatii ponde<lb></lb>ris A quadruplum. </s> <s id="id.2.1.213.4.1.3.0">ſimiliter quoniam <lb></lb>ſpatium potentiæ in H <expan abbr="duplũ">duplum</expan> eſt ſpatii <lb></lb>F, erit ſpatium potentiæ in H ſpatii <lb></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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.214.1.1.2.0"><margin.target id="note295"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.214.1.1.3.0"><margin.target id="note296"></margin.target>11 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb n="103" xlink:href="036/01/219.jpg"></pb> <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></lb>tum, qui orbiculo trochleæ ſuperio<lb></lb>ris ſit circumuolutus, & religatus in <arrow.to.target n="note297"></arrow.to.target><lb></lb>B; ſitq; potentia in C ſuſtinens pon<lb></lb>dus A: erit potentia in C ponderis A <lb></lb>dupla, deinde C alteri funi religetur, <lb></lb>qui per alterius trochleæ orbicu<lb></lb>lum circumuoluatur, & religetur <lb></lb>in D; erit potentia in E dupla po<arrow.to.target n="note298"></arrow.to.target><lb></lb>tentiæ in C. </s> <s id="id.2.1.215.1.2.1.0.a">Quare potentia in E <lb></lb>quadrupla erit ponderis A. </s> <s id="id.2.1.215.1.2.1.0.b">& ſi ad <lb></lb>huc E alteri funi religetur, qui etiam <lb></lb>circa orbiculum alterius trochleæ re<lb></lb>uoluatur, & religetur in F; erit poten<lb></lb>tia in G dupla potentiæ in E. </s> <s id="id.2.1.215.1.2.1.0.c">ergo <lb></lb>potentia in G octupla erit ponderis <lb></lb>A. </s> <s id="N16151">& ſic in infinitum ſemper præ <lb></lb>cedentis potentiæ potentiam du<lb></lb>plam inueniemus. <figure id="id.036.01.219.1.jpg" place="text" 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 ſit potentia mo<lb></lb>uens, <arrow.to.target n="note299"></arrow.to.target>erit ſpatium ponderis octu<lb></lb>plum ſpatii potentiæ in G. ſpatium <lb></lb>enim ponderis A duplum eſt ſpatii <lb></lb>potentiæ in C, & C duplum eſt ſpatii <lb></lb>ipſius E; quare ſpatium ponderis <lb></lb>A ſpatii potentiæ in E quadruplum <lb></lb>erit. </s> <s id="id.2.1.215.2.1.2.0">ſimiliter quoniam ſpatium E <lb></lb>duplum eſt ſpatii potentiæ in G; erit ergo ſpatium ponderis A <lb></lb>octuplum ſpatii 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.216.1.1.2.0"><margin.target id="note298"></margin.target><emph type="italics"></emph>Ex e adem.<emph.end type="italics"></emph.end></s> <s id="id.2.1.216.1.1.3.0"><margin.target id="note299"></margin.target>16 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/220.jpg"></pb> <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 manifeſtum eſt maiorem ſemper ha<lb></lb>bere proportionem ſpatium potentiæ mouen<lb></lb>tis ad ſpatium ponderis moti, quàm pondus <lb></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></lb>ſunt, 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 id="N161BE" 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></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 eſt maior, vel æqualis, vel minor dato <lb></lb>pondere. </s> </p> <pb n="104" xlink:href="036/01/221.jpg"></pb> <p id="id.2.1.217.8.0.0.0" type="main"> <s id="id.2.1.217.8.1.1.0">Et ſi eſt maior, tunc poten<lb></lb>tia, vel abſq; alio inſtrumento, <lb></lb>vel fune circa orbiculum trochleæ <lb></lb>ſurſum appenſæ reuoluto datum <lb></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></lb>tentia; quàm data, ponderi æque<lb></lb>ponderat, data ergo mouebit. </s> <s id="id.2.1.217.8.1.3.0"><lb></lb>Quod idem fieri poteſt iuxta om<lb></lb>nes propoſitiones, quibus poten<lb></lb>tia pondus ſuſtinens, vel æqualis, <lb></lb>vel minor pondere oſtenſa eſt. <figure id="id.036.01.221.1.jpg" place="text" 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></lb>pondus mouebit fune <lb></lb>per orbiculum trochleæ <lb></lb>ponderi alligatæ circum<lb></lb>uoluto. </s> <s id="id.2.1.217.9.1.2.0">potentia enim <arrow.to.target n="note301"></arrow.to.target><lb></lb>ſuſtinens pondus ſubdu<lb></lb>pla eſt ponderis, poten<lb></lb>tia igitur ponderi æqua<lb></lb>lis datum pondus mo<lb></lb>uebit. </s> <s id="id.2.1.217.9.1.3.0">Quod etiam <expan abbr="ſecundùm">ſe<lb></lb>cundum</expan> propoſitiones, <lb></lb>quibus potentiam pon<lb></lb>dere minorem eſſe o<lb></lb>ſtenſum eſt, fieri po<lb></lb>teſt. <figure id="id.036.01.221.2.jpg" place="text" xlink:href="036/01/221/2.jpg"></figure></s> </p> <pb xlink:href="036/01/222.jpg"></pb> <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, ſit datum pondus <lb></lb>vt ſexaginta, potentia verò mouens <lb></lb><arrow.to.target n="note302"></arrow.to.target>data ſit tredecim. </s> <s id="id.2.1.217.11.1.2.0">inueniatur poten<lb></lb>tia in A ſuſtinens pondus B, quæ pon<lb></lb>deris B ſit ſubquintupla. </s> <s id="id.2.1.217.11.1.3.0">& quoniam <lb></lb>potentia in A pondus ſuſtinens eſt <lb></lb>vt duodecim; maior igitur poten<lb></lb>tia, quàm duodecim in A pondus <lb></lb>B mouebit. </s> <s id="id.2.1.217.11.1.4.0">Quare potentia vt tre<lb></lb>decim in A pondus B mouebit. </s> <s id="id.2.1.217.11.1.5.0">quod <lb></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"></emph>Ex<emph.end type="italics"></emph.end> 1 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></s> <s id="id.2.1.218.1.1.2.0"><margin.target id="note301"></margin.target>2 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.218.1.1.3.0"><margin.target id="note302"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 9 <emph type="italics"></emph>huius<emph.end type="italics"></emph.end></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; eſt in mo <lb></lb>uendis ponderibus, potentiam ali<lb></lb>quando forſitan melius mouere mo<lb></lb>uendo ſe deorſum, quàm mouendo <lb></lb>ſe ſurſum. </s> <s id="id.2.1.219.1.1.2.0">vt circumuoluatur adhuc <lb></lb>funis per alium trochleæ ſuperioris <lb></lb>orbiculum, cuius centrum C, funiſq; <lb></lb><arrow.to.target n="note303"></arrow.to.target>perueniat in D; erit <expan abbr="potẽtia">potentia</expan> in D <expan abbr="ſuſtinẽs">ſuſtinens</expan> <expan abbr="põdus">pondus</expan> B ſimiliter duodecim, <expan abbr="〈quẽ〉">quem</expan><lb></lb>admodum erat in A. </s> <s id="id.2.1.219.1.1.2.0.a">Ideo poten<lb></lb>tia vt tredecim in D pondus B mo<lb></lb>uebit. </s> <s id="id.2.1.219.1.1.3.0">& quia mouet ſe deorſum, <lb></lb>fortaſſe trahet facilius, quàm in A; <lb></lb>atq; tempus eſt idem, ſicut etiam <lb></lb>erat in A. <figure id="id.036.01.222.1.jpg" place="text" xlink:href="036/01/222/1.jpg"></figure></s> </p> <pb n="105" xlink:href="036/01/223.jpg"></pb> <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"></emph>Ex<emph.end type="italics"></emph.end> 5 <emph type="italics"></emph>Huius<emph.end type="italics"></emph.end></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">Propoſitum ſit nobis efficere, potentiam pon<lb></lb>dus mouentem, & pondus per data ſpatia ſibi in <lb></lb>uicem longitudine commenſurabilia 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 ſpatium potentiæ, vt tria, <arrow.to.target n="note304"></arrow.to.target><lb></lb>ponderis verò, vt quatuor. </s> <s id="id.2.1.221.3.1.2.0">inueniatur po<lb></lb>tentia in A pondus B ſuſtinens, quæ pon<lb></lb>deris ſit ſeſquitertia, vt quatuor ad trìa. </s> <s id="id.2.1.221.3.1.3.0">ſi <lb></lb>igitur in A ſit potentia mouens pondus; <arrow.to.target n="note305"></arrow.to.target><lb></lb>erit ſpatium ponderis ſpatii potentiæ ſeſ<lb></lb>quitertium, vt quatuor ad tria. </s> <s id="id.2.1.221.3.1.4.0">quod face<lb></lb>re oportebat. <figure id="id.036.01.223.1.jpg" place="text" 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 ſunt in <lb></lb>vigeſima ſecunda, & in vigeſimaquinta <lb></lb>huius efficere poſſumus ſolo fune. </s> <s id="id.2.1.221.4.1.2.0">Quòd ſi <lb></lb>pluribus funibus id efficere voluerimus, <lb></lb>non ſolum multis, ſed infinitis modis hoc <lb></lb>efficere poterimus, vt ſupra dictum eſt. </s> <s id="N16332"><arrow.to.target n="note306"></arrow.to.target><lb></lb>Quare hoc affirmare poſſumus, quod qui<lb></lb>dem mirum eſſe 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"></emph>Ex<emph.end type="italics"></emph.end> 22 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.222.1.1.2.0"><margin.target id="note305"></margin.target><emph type="italics"></emph>Ex eadem.<emph.end type="italics"></emph.end></s> <s id="id.2.1.222.1.1.3.0"><margin.target id="note306"></margin.target><emph type="italics"></emph>In<emph.end type="italics"></emph.end> 26 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> </p> <pb xlink:href="036/01/224.jpg"></pb> <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 manifeſtum eſſe, Quamlibet datam in <lb></lb>numeris proportionem inter pondus, & poten<lb></lb>tiam; & inter ſpatium ponderis moti, & ſpatium <lb></lb>potentiæ motæ; infinitis modis trochleis inueni<lb></lb>ri poſſe. </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 manifeſtum eſt, quò pondus <lb></lb>facilius mouetur, eò quoq; tempus maius eſſe; <lb></lb>quò verò difficilius, eò minus eſſe. & è con<lb></lb>uerſo. </s> </p> </chap> <pb n="106" xlink:href="036/01/225.jpg"></pb> <chap id="N16391"> <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></lb>PERITROCHIO. </s> </p> <figure id="id.036.01.225.1.jpg" place="text" 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="cõſtructionem">conſtructionem</expan> hu<lb></lb>ius inſtrumenti Pappus in octauo <lb></lb>mathematicarum collectionum <lb></lb>libro docet; axemq; vocat AB, <lb></lb>tympanum verò CD circa idem <lb></lb>centrum; & ſcytalas in foramini<lb></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"></pb> <figure id="id.036.01.226.1.jpg" place="text" xlink:href="036/01/226/1.jpg"></figure><lb></lb>quæ ſemper in ſcytalis eſt, vt in F, dum circum<lb></lb>uertit tympanum, & axem, ſurſum moueat pon<lb></lb>dus K axi appenſum fune LM circa axem reuo<lb></lb>luto. </s> <s id="id.2.1.223.7.1.3.0">Nobis igitur reſtat, vt oſtendamus, cur ma<lb></lb>gna pondera ab exigua virtute, quouè etiam mo <lb></lb>do hoc inſtrumento moueantur; temporis quin <lb></lb>etiam, ſpatiiq; mouentis inuicem potentiæ, ac <lb></lb>moti ponderis rationem aperiamus; huiuſmodi<lb></lb>que inſtrumenti vſum ad vectem reducamus. </s> </p> <pb n="107" xlink:href="036/01/227.jpg"></pb> <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 ſuſtinens axe in peritrochio <lb></lb>ad pondus eandem habet proportionem, quam <lb></lb>ſemidiameter axis ad ſemidiametrum tympani <lb></lb>vná cum ſcytala. <figure id="id.036.01.227.1.jpg" place="text" 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; ſit diameter tympani <lb></lb>DCE circa idem centrum; ſintq; AB DE in eadem recta linea; <lb></lb>ſint deinde ſcytalæ in foraminibus tympani DF GH & c inter ſe ſe <lb></lb>æquales, atq; æquè diſtantes; ſitq; FE horizonti æquidiſtans; <pb xlink:href="036/01/228.jpg"></pb> <figure id="id.036.01.228.1.jpg" place="text" xlink:href="036/01/228/1.jpg"></figure><lb></lb>pondus autem K in fune BL circa axem volubili ſit appenſum. </s> <s id="id.2.1.223.11.1.2.0">& <lb></lb>potentia in F ſuſtineat pondus K. </s> <s id="id.2.1.223.11.1.2.0.a">Dico potentiam in F ad pondus <lb></lb>k ita ſe habere, vt CB ad CF. </s> <s id="N16410">fiat vt CF ad CB, ita pondus <lb></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></lb>appenſa ſunt in FB; erit FB tanquam vectis, ſiue libra; quia ve<lb></lb>rò C eſt punctum immobile, circa quod axis, tympanusq; reuol<lb></lb>uuntur; erit C fulcimentum vectis FB; vellibræ centrum. </s> <s id="id.2.1.223.11.1.3.0">cùm <lb></lb><arrow.to.target n="note307"></arrow.to.target>autem it a ſit CF ad CB, vt k ad M, pondera k M æqueponde<lb></lb>rabunt. </s> <s id="id.2.1.223.11.1.4.0">Potentia igitur in F ſuſtinens pondus k, ne deorſum ver<lb></lb>gat, ponderi K æqueponderabit; ipſiq; M æqualis erit. </s> <s id="id.2.1.223.11.1.5.0">idem enim <lb></lb>præſtat potentia, quod pondus M. </s> <s id="id.2.1.223.11.1.5.0.a">pondus igitur K ad poten<lb></lb><arrow.to.target n="note308"></arrow.to.target>tiam in F erit, vt CF ad CB; & conuertendo, potentia ad <lb></lb>pondus erit, vt CB ad CF, hoc eſt, ſemidiameter axis ad ſemi<pb n="108" xlink:href="036/01/229.jpg"></pb>diametrum tympani vnà cum ſcytala DF. </s> <s id="id.2.1.223.11.1.5.0.b">Similiter etiam oſten<lb></lb>detur, ſi potentia pondus ſuſtinens fuerit in q. </s> <s id="N16445">tunc enim ſuſti<lb></lb>neret vecte CQ; & ad pondus eam haberet proportionem, quam <arrow.to.target n="note309"></arrow.to.target><lb></lb>habet CB ad Cq. </s> <s id="N1644E">Videlicet ſemidiameter axis ad ſemidiame<lb></lb>trum tympani vná cum ſcytala Eq. </s> <s id="N16452">quod demonſtrare opor<lb></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"></emph>Primi Archim. de æquepon.<emph.end type="italics"></emph.end></s> <s id="id.2.1.224.1.1.3.0"><margin.target id="note308"></margin.target><emph type="italics"></emph>Cor.<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>quinti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.224.1.1.4.0"><margin.target id="note309"></margin.target>2 <emph type="italics"></emph> <expan abbr="Huuius">Huius</expan>. de vecte.<emph.end type="italics"></emph.end></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">Manifeſtum eſt potentiam ſemper minorem <lb></lb>eſſe 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 ſemper ſemidiametro tympani mi<lb></lb>nor eſt. </s> <s id="id.2.1.225.3.1.2.0">& potentia eò minor eſt pondere, quò ſemidiameter axis <lb></lb>minor eſt ſemidiametro tympani vná cum ſcytala. </s> <s id="id.2.1.225.3.1.3.0">quare quò lon<lb></lb>gior eſt CF, vel CQ; & quò breuior eſt CB, minor adhuc ſem<lb></lb>per potentia in F, vel in Q pondus k ſuſtinebit. </s> <s id="id.2.1.225.3.1.4.0">quò enim minor <lb></lb>eſt CB, eò minorem habebit proportionem ſemidiameter axis <lb></lb>ad ſemidiametrum tympani vná cum ſcytala. </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 conſiderandum occurrit, quòd ſi in alia ſcyta<lb></lb>la appendatur pondus, vt in T, ſuſtinens pondus k; it a nempè, vt <lb></lb>pondus in T appenſum, pondusq; k circa axem conſtitutum <lb></lb>maneant; erit pondus in T grauius pondere M in F appenſo. </s> <s id="id.2.1.225.4.1.2.0"><lb></lb>iungatur enim TB, & à puncto C horizonti perpendicularis du<lb></lb>catur CI, quæ lineam TB ſecet in I; tandemq; connectatur <lb></lb>TC, quæ æqualis erit CF. </s> <s id="id.2.1.225.4.1.2.0.a">Quoniam autem pondera appenſa <lb></lb>ſunt in TB, perindè ſe ſe habebunt, ac ſi in punctis TB ipſorum <lb></lb>centra grauitatum haberent; vt antea dictum eſt. </s> <s id="id.2.1.225.4.1.3.0">& quia ma<lb></lb>nent, erit punctum I (ex prima huius de libra) amborum ſimul <lb></lb>grauitatis centrum; cùm ſit CI horizonti perpendicularis. </s> <s id="id.2.1.225.4.1.4.0">ſed <lb></lb>quoniam angulus BCI eſt rectus, erit BIC acutus, lineaq; BI <arrow.to.target n="note310"></arrow.to.target><lb></lb>ipſa BC maior erit. </s> <s id="id.2.1.225.4.1.5.0">quare angulus CIT erit obtuſus; atq; <arrow.to.target n="note311"></arrow.to.target><lb></lb>ideo linea CT ipſa TI maior erit. </s> <s id="id.2.1.225.4.1.6.0">Cùm autem CT maior ſit <lb></lb>TI, & IB maior BC; maiorem habebit proportionem TC ad <lb></lb>CB, quàm TI ad IB; & conuertendo, minorem habebit pro<pb xlink:href="036/01/230.jpg"></pb> <figure id="id.036.01.230.1.jpg" place="text" xlink:href="036/01/230/1.jpg"></figure><lb></lb>portionem BC ad CT, hoc eſt ad CF, quàm BI ad IT; vt ex <lb></lb>vigeſima ſexta quinti elementorum (iuxta Commandini editio<lb></lb>nem) patet. </s> <s id="id.2.1.225.4.1.7.0">Quoniam verò punctum I eſt ponderum in TB <lb></lb><arrow.to.target n="note312"></arrow.to.target>exiſtentium centrum grauitatis; erit pondus in T ad pondus in B, <lb></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 eſt, vt BC <lb></lb>ad CF; maiorem igitur proportionem habebit pondus in T ad <lb></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></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"></emph>Ex<emph.end type="italics"></emph.end> 19 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.226.1.1.2.0"><margin.target id="note311"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 13 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.226.1.1.3.0"><margin.target id="note312"></margin.target>6. <emph type="italics"></emph>Primi Archim. de æquepon.<emph.end type="italics"></emph.end></s> <s id="id.2.1.226.1.1.5.0"><margin.target id="note313"></margin.target>10. <emph type="italics"></emph>Quinti.<emph.end type="italics"></emph.end></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 ſuſtinens pon<lb></lb>dus k conſtituatur; quæ ita degrauet ſe, ac ſi in centrum mundi <lb></lb>tendere vellet; quemadmodum ſuapte natura efficit pondus in T <lb></lb>appenſum; erit hæc eadem ponderi in T appenſo æqualis; alio<lb></lb>quin non ſuſtineret; quæ quidem ipſa potentia in F collocata ma<pb n="109" xlink:href="036/01/231.jpg"></pb>ior erit. </s> <s id="id.2.1.227.1.1.2.0">ſicuti enim ſe ſe habet pondus in T ad pondus in F, ita <lb></lb>& potentia in T ad potentiam in F; cùm potentiæ ſint ponderi<lb></lb>bus æquales. </s> <s id="id.2.1.227.1.1.3.0">verùm ſi vnaquæq; potentia ſeorſum ſumpta, tàm <lb></lb>in T, quàm in F ſuſtinens pondus <expan abbr="ſecundũ">ſecundum</expan> <expan abbr="circũferentiam">circunferentiam</expan> THFN <lb></lb>moueri ſe vellet, veluti apprehenſa manu ſcytala; tunc eademmet <lb></lb>potentia, vel in F, vel in T conſtituta idem pondus k ſuſtinere po<lb></lb>terit; cùm ſemper in cuiuſcunq; extremitate ſcytalæ ponatur, ab <lb></lb>eodem centro C æquidiſtans fuerit, ac ſecundum eandem circum<lb></lb>ferentiam ab eodem centro æqualiter ſemper diſtantem perpenſio<lb></lb>nem habeat. </s> <s id="id.2.1.227.1.1.4.0">neq; enim (ſicuti pondus) proprio nutu magis in <lb></lb>centrum ferri exoptat, quam circulariter moueri; cùm vtrunq;, ſeu <lb></lb>quemlibet alium motum nullo prorſus reſpiciat diſcrimine. </s> <s id="id.2.1.227.1.1.5.0">pro<lb></lb>pterea non eodem modo res ſe ſe habet, ſiue pondera, ſiue anímatæ <lb></lb>potentiæ iiſdem locis eodem munere abeundo fuerint conſtitutæ. </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></lb>tentia in F circumuertit tympanum, circumuertit etiam axem; & <lb></lb>FB fit tamquam vectis, cuius fulcimentum C, potentia mouens <lb></lb>in F, & <expan abbr="podus">pondus</expan> in B appenſum. </s> <s id="id.2.1.227.2.1.2.0">& dum punctum F peruenit in N; <lb></lb>punctum H erit in F, & punctum B erit in O; ita vt ducta NO <lb></lb>tranſeat per C; eodemq; tempore pondus k motum erit in P, ita <lb></lb>vt OBP ſit æqualis ipſi BL, cùm ſit 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 ſpatium po<lb></lb>tentiæ mouentis ad ſpatium ponderis moti ita eſſe, vt ſemidiame<lb></lb>ter tympani cùm ſcytala ad ſemidiametrum axis, hoc eſt, vt CF <lb></lb>ad CB, cùm circumferentia FN ad BO, ſit 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></lb>niam BL, eſt æqualis OBP, dempta communi BP, erit OB ip<lb></lb>ſi PL æqualis. </s> <s id="id.2.1.227.3.1.2.0">quare FN ſpatium potentiæ ad PL ſpatium pon<lb></lb>deris erit, vt CF ad CB, videlicet ſemidiameter tympani cùm <lb></lb>ſcytala ad ſemidiametrum axis. </s> <s id="id.2.1.227.3.1.3.0">Quod idem oſtendetur, poten<lb></lb>tia vel in Q, vel in qualibet alia ſcytala exiſtente, vt in S. </s> <s id="N165BA">cùm <lb></lb>enim ſcytalæ ſint ſibi inuicem æquales, atq; æqualiter diſtantes; <lb></lb>vbicunq; ſit potentia æquali mota velocitate ſemper æquali tem<lb></lb>pore æquale ſpatium pertranſibit, hoc eſt ex Q in R, vel ex Sin T <lb></lb>eodem tempore mouebitur, quò ex F in N. </s> <s id="id.2.1.227.3.1.3.0.a">ſed quò tempore po<lb></lb>tentia ex F in N mouetur, eodemmet prorſus pondus k ex L in <lb></lb>P quoq; mouetur; vbicunq; igitur ſit potentia, erit ſpatium poten<pb xlink:href="036/01/232.jpg"></pb> <figure id="id.036.01.232.1.jpg" place="text" xlink:href="036/01/232/1.jpg"></figure><lb></lb>tiæ ad ſpatium ponderis moti, vt CF ad CB, hoc eſt ſemidia<lb></lb>meter tympani cum ſcytala, ad ſemidiametrum 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"></emph>Ex<emph.end type="italics"></emph.end> 4 <emph type="italics"></emph>huius de vecte.<emph.end type="italics"></emph.end></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 manifeſtum eſt, ita eſſe pondus ad po<lb></lb>tentiam pondus ſuſtinentem, vt ſpatium poten<lb></lb>tiæ mouentis ad ſpatium ponderis moti. </s> </p> <pb n="110" xlink:href="036/01/233.jpg"></pb> <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">Manifeſtum eſt etiam, maiorem ſemper ha<lb></lb>bere proportionem ſpatium potentiæ mouentis <lb></lb>ad ſpatium ponderis moti, quàm pondus ad ean<lb></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 ſcytalas eſt maior, eò quoq; <lb></lb>in pondere mouendo maius ſumetur tempus; dummodo potentia <lb></lb>æquali moueatur velocitate. </s> <s id="id.2.1.229.6.1.2.0">tempuſq; eò maius erit, quò diame<lb></lb>ter vnius diametro alterius eſt maior. </s> <s id="id.2.1.229.6.1.3.0">circulorum enim circumfe<arrow.to.target n="note315"></arrow.to.target><lb></lb>rentiæ ita ſe habent, vt diametri. </s> <s id="id.2.1.229.6.1.4.0">Cùm vero ex trigeſima ſexta <lb></lb>quarti libri Pappi Mathematicarum collectionum, duorum inæ<lb></lb>qualium circulorum æquales circumferentias inuenire poſsimus; <lb></lb>ideo tempus quoq; portionum circulorum inæqualium hoc modo <lb></lb>inueniemus. </s> <s id="id.2.1.229.6.1.5.0">è conuerſo autem, quò maior erit axis circumferen<lb></lb>tia citius pondus ſurſum mouebitur. </s> <s id="id.2.1.229.6.1.6.0">maior enim pars funis BL <lb></lb>in vna circumuerſione completa circa circulum ABO reuoluitur, <lb></lb>quàm ſi minor eſſet; cùm funis circumuolutus ſit circumferen<lb></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"></emph>Octaui libri Pappi.<emph.end type="italics"></emph.end></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 manifeſtum eſt, quò facilius pondus mo<lb></lb>uetur, tempus quoq; eò maius eſſe; & quò dif<lb></lb>ficilius, eò tempus minus eſſe. </s> <s id="id.2.1.231.2.1.2.0">& è conuerſo. </s> </p> <pb xlink:href="036/01/234.jpg"></pb> <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 id="N16668" 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></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 ſexagin<lb></lb>ta; potentia verò vt decem. </s> <s id="id.2.1.231.6.1.2.0"><lb></lb>exponatur quædam recta li<lb></lb>nea AB, quæ diuidatur in C, <lb></lb>ita vt AC ad CB eandem <lb></lb><figure id="id.036.01.234.1.jpg" place="text" xlink:href="036/01/234/1.jpg"></figure><lb></lb>habeat proportionem, quam ſexaginta ad decem. </s> <s id="id.2.1.231.6.1.3.0">& ſi CB axis <lb></lb>ſemidiameter eſſet, & CA ſemidiameter tympani cùm ſcytalis; <lb></lb><arrow.to.target n="note316"></arrow.to.target>patet potentiam vt decem in A ponderi ſexaginta in B æquepon<lb></lb>derare. </s> <s id="id.2.1.231.6.1.4.0">Accipiatur autem inter BC quoduis punctum D; fiatq; <lb></lb>BD ſemidiameter axis, & DA ſemidiameter tympani cùm ſcy<lb></lb>talis; ponaturq; pondus ſexaginta in B fune circa axem, & potentia <lb></lb><arrow.to.target n="note317"></arrow.to.target><emph type="italics"></emph>in A. </s> <s id="id.2.1.231.6.1.4.0.a">Quoniam enim AD ad DB maiorem habet proportio<lb></lb>nem, quam AC ad CB; maiorem habebit proportionem AD ad <lb></lb>DB, quam pondus ſexaginta in B appenſum ad potentiam vt decem<emph.end type="italics"></emph.end><lb></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 ſexaginta axe in peritro<lb></lb>chio mouebit, cuius axis ſemidiameter eſt BD, & DA ſemidia<lb></lb>meter tympani cùm ſcytalis. </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"></emph>Per præcedentem.<emph.end type="italics"></emph.end></s> <s id="id.2.1.232.1.1.2.0"><margin.target id="note317"></margin.target><emph type="italics"></emph>Lemma in primi huius de vecte.<emph.end type="italics"></emph.end></s> <s id="id.2.1.232.1.1.3.0"><margin.target id="note318"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 11 <emph type="italics"></emph>huius de vecte.<emph.end type="italics"></emph.end></s> </p> <pb n="111" xlink:href="036/01/235.jpg"></pb> <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 id="N166ED" 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></lb>diameter ſit BD, & cen<lb></lb>trum C, quem quidem <lb></lb>axem maiorem, vel mino<lb></lb>rem conſtituemus, veluti <lb></lb><figure id="id.036.01.235.1.jpg" place="text" xlink:href="036/01/235/1.jpg"></figure><lb></lb>magnitudo, ponderiſq; grauitas poſtulat. </s> <s id="id.2.1.233.2.1.2.0">producatur deinde BD <lb></lb>vſq; ad A: fiatq; BC ad CA, vt decem ad ſexaginta. </s> <s id="id.2.1.233.2.1.3.0">& ſi CA tym<lb></lb>pani cùm ſcytalis ſemidiameter eſſet, potentia decem in A ponde<lb></lb>ri ſexaginta in B æqueponderaret. </s> <s id="id.2.1.233.2.1.4.0">producatur verò BA ex parte <lb></lb>A, & in hac producta linea quoduis accipiatur punctum E; fiatq; <lb></lb>CE ſemidiameter tympani cùm ſcytalis; ponaturq; potentia vt <lb></lb>decem in E; habebit EC ad CB maiorem proportionem, quàm <lb></lb>pondus ſexaginta in B ad potentiam vt decem in E. </s> <s id="id.2.1.233.2.1.4.0.a">potentia igi<lb></lb>tur vt decem in E mouebit pondus ſexaginta in B appenſum fune <lb></lb>circa axem, cuius ſemidiameter eſt CB, & CE ſemidiameter tym<lb></lb>pani cùm ſcytalis. </s> <s id="id.2.1.233.2.1.5.0">quod facere oportebat. </s> </p> <pb xlink:href="036/01/236.jpg"></pb> <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 ſunt ergatæ, ſuccu<lb></lb>læ, terebræ, tympana cum ſuis axibus, ſiue dentata, <lb></lb>ſiue non; & ſimilia. </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 neſcioquid cochleæ; dum enim mo<lb></lb>uet pondus, ſcilicet dum perforat, ex ſua ferè natura ſemper vlte<lb></lb>rius progreditur<emph type="italics"></emph>:<emph.end type="italics"></emph.end> habet enim ferè helices tamquam circa conum <lb></lb>deſcriptas. </s> <s id="id.2.1.233.5.1.2.0">quoniam autem verticem habet acutum, ad cunei quoq; <lb></lb>rationem commodè referri poterit. <figure id="id.036.01.236.1.jpg" place="text" xlink:href="036/01/236/1.jpg"></figure></s> </p> </chap> <pb n="112" xlink:href="036/01/237.jpg"></pb> <chap id="N16758"> <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 quæſtioni<lb></lb>bus Mechanicis quæſtione deci<lb></lb>maſeptima aſſerit, cuneum ſcin<lb></lb>dendo ponderi duorum vicem <lb></lb>prorſus gerere vectium ſibi inui<lb></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></lb>ius vertex B, & ſit AB <lb></lb>æqualis BC; quod au<lb></lb>tem ſcindendum eſt, <lb></lb>ſit DEFG; ſitq; pars <lb></lb>cunei HB k intra DE <lb></lb>FG, & HB æqualis <lb></lb>ſit ipſi Bk. </s> <s id="id.2.1.233.7.1.2.0">percutiatur <lb></lb>(vt fieri ſolet) cuneus <lb></lb>in AC, dum cuneus in <lb></lb>AC percutitur, AB fit <lb></lb>vectis, cuius fulcimen <lb></lb>tum eſt H, & pondus in <lb></lb>B. </s> <s id="id.2.1.233.7.1.2.0.a">eodemq; modo CB <lb></lb>fit vectis, cuius fulci<lb></lb><figure id="id.036.01.237.1.jpg" place="text" xlink:href="036/01/237/1.jpg"></figure><lb></lb>mentum eſt K, & pondus ſimiliter in B. </s> <s id="id.2.1.233.7.1.2.0.b">ſed dum percutitur cu<lb></lb>neus, maiori adhuc ipſius portione ipſum DEFG ingreditur, <lb></lb>quàm prius eſſet: ſit autem portio hæc MBL; ſitq; M B ipſi BL <lb></lb>æqualis. </s> <s id="id.2.1.233.7.1.3.0">& cùm MB BI ſint ipſis HB BK maiores; erit ML maior <pb xlink:href="036/01/238.jpg"></pb>Hk. </s> <s id="id.2.1.233.7.1.4.0">dum igitur ML <lb></lb>erit in ſitu Hk; opor<lb></lb><expan abbr="ter">tet</expan>, vt fiat maior ſciſsio; <lb></lb>& D moueatur verſus <lb></lb>O, G autem verſus N: <lb></lb>& quò maior pars cu<lb></lb>nei intra DEFG ingre<lb></lb>dietur, eò maior fiet <lb></lb>ſciſsio; & DG ma<lb></lb>gis adhuc impellentur <lb></lb>verſus ON. </s> <s id="id.2.1.233.7.1.4.0.a">pars igi<lb></lb>tur KG eius, quod ſcin<lb></lb>ditur, mouebitur à ve<lb></lb>cte AB, cuius fulcimen<lb></lb>tum eſt H, & pondus <lb></lb><figure id="id.036.01.238.1.jpg" place="text" xlink:href="036/01/238/1.jpg"></figure><lb></lb>in B; ita vt punctum B ipſius vectis AB impellat partem KG. <lb></lb></s> <s id="N167E1">& pars HD mouebitur à vecte CB, cuius fulcimentum eſt k; ita <lb></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 ſint vectium genera, vt ſupra <lb></lb>oſtenſum eſt; idcirco conuenientius erit fortaſſè <lb></lb>cuneum hoc modo conſiderare. </s> </p> <p id="id.2.1.233.9.0.0.0" type="main"> <s id="id.2.1.233.9.1.1.0">Iiſdem poſitis, intelligatur vectis AB, cuius fulcimentum B, & <lb></lb>pondus in H, vt in ſecunda huius de vecte diximus. </s> <s id="id.2.1.233.9.1.2.0">ſimiliter ve<lb></lb>ctis CB, cuius fulcimentum B, & pondus in K; ita vt pars HD <lb></lb>moueatur à vecte AB, cuius fulcimentum eſt B, & pondus in H; <lb></lb>ita vt punctum H ipſius vectis AB impellat partem HD. </s> <s id="N16800">ſimi<lb></lb>li quoq; modo pars KG moueatur à vecte CB, cuius fulcimentum <lb></lb>eſt B, & pondus in k, it aut k ipſius uectis CB partem k G mo<lb></lb>ueat. </s> <s id="id.2.1.233.9.1.3.0">quod quidem forſitan rationi magis conſentaneum erit. </s> </p> <pb n="113" xlink:href="036/01/239.jpg"></pb> <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></lb>ſintq; duo pondera ſepa<lb></lb>rat a DEFG, & HIkL, <lb></lb>intra quæ ſit pars cunei <lb></lb>DBH, cuius uertex B <lb></lb>medium inter utrumq; ſi <lb></lb>tum obtineat. </s> <s id="id.2.1.233.11.1.2.0">percutia<lb></lb>tur autem cuneus, ita ut <lb></lb>magis adhuc intra pon<lb></lb>dera propellatur, ſicuti <lb></lb>prius dictum eſt; ponde<lb></lb><figure id="id.036.01.239.1.jpg" place="text" xlink:href="036/01/239/1.jpg"></figure><lb></lb>ra enim ſunt, ac ſi unum tantùm continuum eſſet GFkL, quod <lb></lb>ſcindendum eſſet: eodem enim modo pars DG, dum cuneus <lb></lb>ulterius impellitur, mouebitur uerſus M; & pars HL uerſus N. </s> <s id="id.2.1.233.11.1.2.0.a"><lb></lb>Moueatur itaq; pars DG uerſus M, & pars HL uerſus N, B uerò <lb></lb>dum ulterius progreditur, ſemper medium inter utrunq; pondus <lb></lb>remaneat. </s> <s id="id.2.1.233.11.1.3.0">dum autem DG à cuneo mouetur uerſus M; patet B <lb></lb>non mouere partem DG uerſus M uecte CB, cuius fulcimentum <lb></lb>H; <expan abbr="punctũ">punctum</expan> enim B non tangit pondus; ſed DG mouebitur à pun<lb></lb>cto uectis D uecte AB, cuius fulcimentum B; punctum enim D tan<lb></lb>git pondus, & inſtrumenta mouent per contactum. </s> <s id="id.2.1.233.11.1.4.0">Similiter <lb></lb>HL mouebitur ab H uecte CB, cuius fulcimentum B; & uterq; <lb></lb>uectis utriq; reſiſtit in B, ita ut B potius fulcimenti uice fungatur, <lb></lb>quàm mouendi ponderis. </s> <s id="id.2.1.233.11.1.5.0">quod ipſum hoc quoq; modo manife<lb></lb>ſtum erit. </s> </p> <pb xlink:href="036/01/240.jpg"></pb> <p id="id.2.1.233.13.0.0.0" type="main"> <s id="id.2.1.233.13.1.1.0">Sit, quod ſcindendum eſt A <lb></lb>BCD <expan abbr="parallelogrammũ">parallelogrammum</expan> rectan<lb></lb>gulum; ſintq; duo vectes æqua<lb></lb>les EF GF, & partes vectium <lb></lb>HF KF ſint intra ABCD; ſitq; <lb></lb>HF æqualis Fk, & HA æqua<lb></lb>lis KB. </s> <s id="id.2.1.233.13.1.1.0.a">Oporteat verò vecti<lb></lb>bus EF GF ſcindere ABCD <lb></lb>abſq; percuſsione, videlicet ſint <lb></lb>potentiæ mouentes in EG æqua<lb></lb>les. </s> <s id="id.2.1.233.13.1.2.0">vt autem ſcindatur ABCD, <lb></lb>oportet partem HA moueri uer<lb></lb><figure id="id.036.01.240.1.jpg" place="text" xlink:href="036/01/240/1.jpg"></figure><lb></lb>ſus M. & kB verſus N; ſed dum vectes mouentur, putá alter in <lb></lb>M, alter verò in N; neceſſe eſt, vt punctum F immobile rema <lb></lb>neat; in illo enim fit vectium occurſus. </s> <s id="id.2.1.233.13.1.3.0">quare F erit fulcimen<lb></lb>tum vtriuſq; vectis, & FG mouebit partem kB, cuius fulcimen <lb></lb>tum erit F, & potentia mouens in G; & pondus in k. </s> <s id="id.2.1.233.13.1.4.0">ſimi<lb></lb>liter pars HA mouebitur à vecte EF, cuius fulcimentum F, po<lb></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 eſſent fulcimenta immobilia, & pondera in F; <lb></lb>dum vectis FG conatur mouere pondus in F, tunc ei reſiſtit ve<lb></lb>ctis EF, qui etiam conatur mouere pondus in F ad partem op<lb></lb>poſitam; ſed quoniam potentiæ ſunt æquales, & cætera æqualia; <lb></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></lb>igitur in F maximam fieri vectium ſibi inuicem occurrentium reſi<lb></lb>ſtentiam, ita ut F ſit quoddam immobile. </s> <s id="id.2.1.233.14.1.3.0">Quare conſiderando <lb></lb>cuneum, <expan abbr="vtmouet">vt mouet</expan> vectibus ſibi inuicem aduerſis, forſitan eis po<lb></lb>tius utitur hoc ſecundo 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 ſcindendo mo<lb></lb>uetur, poſſumus idcirco eundem alio quoq; mo<lb></lb>do conſiderare; videlicet dum ingreditur id, <pb n="114" xlink:href="036/01/241.jpg"></pb>quod ſcinditur, nihil aliud eſſe, niſi pondus ſu<lb></lb>pra planum horizonti inclinatum mouere. <figure id="id.036.01.241.1.jpg" place="text" 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 æquidiſtans tranſiens per AB; ſit cuneus <lb></lb>CDB, & CD æqualis ipſi DB; & latus cunei DB ſit ſemper in <lb></lb>ſubiecto plano. </s> <s id="id.2.1.233.16.1.2.0">ſit deinde pondus AEFG immobile in A; ſitq; <lb></lb>pars cunei EDH ſub AEFG. </s> <s id="id.2.1.233.16.1.2.0.a">Quoniam enim dum percutitur cu<lb></lb>neus in CB, maior pars cunei ingreditur ſub AEFG, quàm ſit <lb></lb>EDH; ſit hæc pars IDH. </s> <s id="id.2.1.233.16.1.2.0.b">& quoniam latus cunei DB ſemper <lb></lb>eſt in ſubiecto plano per AB ducto horizonti parallelo, tunc quan<lb></lb>do pars cunei kDI erit ſub AEFG; erit punctum k in H, & I <lb></lb>ſub E. </s> <s id="id.2.1.233.16.1.2.0.c">ſed Ik maior eſt HE; punctum igitur E ſurſum motum <lb></lb>erit. </s> <s id="id.2.1.233.16.1.3.0">& dum cuneus ſub AEFG ingreditur, punctum E ſurſum <lb></lb>ſuper latus cunei EI mouebitur, eodemq; modo ſi cuneus vlterius <lb></lb>progredietur, ſemper punctum E ſuper latus cunei DC mouebitur: <lb></lb>punctum igitur E ponderis ſuper planum DC mouebitur horizonti <lb></lb>inclinatum, cuius inclinatio eſt angulus BDC. </s> <s id="N16905">quod demon<lb></lb>ſtrare oportebat. <pb xlink:href="036/01/242.jpg"></pb> <figure id="id.036.01.242.1.jpg" place="text" 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, conſiderando cuneum inſtar vectis mouen<lb></lb>tem, manifeſtum eſt, cuneum BCD pondus AEFG vecte CD <lb></lb>mouere; ita vt D ſit fulcimentum, & pondus in E. </s> <s id="N1691B">non autem ve<lb></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></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></lb>zonti æquidiſtans <lb></lb>tranſiens per AB; ſit <lb></lb>cuneus CAB, cuius <lb></lb>latus AB ſit ſemper <lb></lb>in ſubiecto plano; ſit<lb></lb>〈qué〉 pondus AEFG, <lb></lb>quod nullum alium <lb></lb>habeat motum, niſi <lb></lb><figure id="id.036.01.242.2.jpg" place="text" xlink:href="036/01/242/2.jpg"></figure><lb></lb>ſurſum, & deorſum ad rectos angulos horizonti; ita vt ducta IGk <lb></lb>ſubiecto plano, ipſi〈qué〉 AB perpendicularis, punctum G ſit ſem<lb></lb>per in linea IGk. </s> <s id="id.2.1.233.19.1.2.0">& quoniam dum cuneus percutitur in CB, to<lb></lb>tus ſuper AB vlterius progreditur; pondus AEFG eleuabitur ex <pb n="115" xlink:href="036/01/243.jpg"></pb>iis, quæ ſupra diximus. </s> <s id="id.2.1.233.19.1.3.0">Moueatur cuneus ita, vt E tandem per<lb></lb>ueniat in C, & poſitio cunei ABC ſit MNO, & poſitio pon<lb></lb>deris AEFG ſit PMQI, & G ſit in I. </s> <s id="id.2.1.233.19.1.3.0.a">Quoniam itaq; dum cu<lb></lb>neus ſuper lineam BO mouetur, pondus AEFG ſurſum moue<lb></lb>tur à linea AC. </s> <s id="id.2.1.233.19.1.3.0.b">& dum cuneus ABC vlterius progreditur, ſem<lb></lb>per pondus AEFG magis à latere cunei AC eleuatur: pondus igi<lb></lb>tur AEFG ſuper planum cunei AC mouebitur; quod quidem <lb></lb>nihil aliud eſt, niſi planum horizonti inclinatum, cuius inclinatio <lb></lb>eſt 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></lb>ſuper planum horizonti inclinatum mouetur ex nona Pappi octa<lb></lb>ui libri Mathematicarum collectionum reducitur ad libram. </s> <s id="id.2.1.233.20.1.3.0">ea<lb></lb>dem enim eſt ratio, ſiue manente cuneo, vt pondus ſuper cunei <lb></lb>latus moueatur; ſiue eodem etiam moto, pondus adhuc ſuper ip<lb></lb>ſius latus moueatur; tamquam ſuper planum horizonti incli<lb></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 ſcinduntur, quomodo tam<lb></lb>quam ſuper plana horizonti inclinata mouean<lb></lb>tur, oſtendamus. </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></lb>& AB ipſi BC æqua<lb></lb>lis. </s> <s id="id.2.1.233.22.1.2.0">Diuidatur AC <lb></lb>bifariam in D, conne<lb></lb>ctaturq; BD. </s> <s id="id.2.1.233.22.1.2.0.a">ſit dein<lb></lb>de linea EF, per quam <lb></lb>tranſeat planum hori<lb></lb>zonti æquidiſtans; ſitq; <lb></lb>BD in eadem linea EF; <lb></lb>& dum cuneus percuti<lb></lb>tur, dumq; mouetur ver<lb></lb><figure id="id.036.01.243.1.jpg" place="text" xlink:href="036/01/243/1.jpg"></figure><lb></lb>ſus E, ſemper BD ſit in linea EF. </s> <s id="N169B3">quod verò ſcindendum eſt <lb></lb>ſit GHLM, intra quod ſit pars cunei kBI. </s> <s id="N169B7">manifeſtum eſt, <pb xlink:href="036/01/244.jpg"></pb>dum cuneus uerſus E <lb></lb>mouetur, partem kG <lb></lb>verſus N moueri; & par<lb></lb>tem HI uerſus O. </s> <s id="N169C3">per<lb></lb>cutiatur cuneus, ita vt <lb></lb>AC ſit in linea NO; <lb></lb>tunc k erit in A, & I in <lb></lb>C: & k ex ſuperius di<lb></lb>ctis motum erit ſuper <lb></lb>kA, & I ſuper IC. <lb></lb></s> <s id="N169D2">quare dum cuneus mo<lb></lb><figure id="id.036.01.244.1.jpg" place="text" xlink:href="036/01/244/1.jpg"></figure><lb></lb>uetur, pars KG ſuper BA latus cunei mouebitur, & pars IH ſuper <lb></lb>latus BC. </s> <s id="id.2.1.233.22.1.2.0.b">pars igitur kG ſuper planum mouetur horizonti incli<lb></lb>natum, cuius inclinatio eſt angulus FBA. </s> <s id="id.2.1.233.22.1.2.0.c">ſimiliter IH moue<lb></lb>tur ſuper planum BC in angulo FBC. </s> <s id="id.2.1.233.22.1.2.0.d">Partes ergo eius, quod <lb></lb>ſcinditur ſuper plana horizonti inclinata mouebuntur. </s> <s id="id.2.1.233.22.1.3.0">& quam<lb></lb>quam planum BC ſit ſub horizonte; pars tamen IH ſuper IC mo<lb></lb>uetur, tamquam ſi BC eſſet ſupra <expan abbr="horizontẽ">horizontem</expan> in angulo DBC. </s> <s id="N169F5">partes <lb></lb>enim eius quod <expan abbr="ſinditur">scinditur</expan>, eodem tempore, ab eadem potentia mo<lb></lb>uentur; eadem ergo erit ratio motus partis IH, ac partis KG. </s> <s id="N169F6">ſi<lb></lb>militer eadem eſt ratio, ſiue EF ſit horizonti æquidiſtans, ſiue <lb></lb>horizonti perpendicularis, vel alio modo. </s> <s id="id.2.1.233.22.1.4.0">neceſſe eſt enim poten<lb></lb>tiam cuneum mouentem eandem eſſe, cùm cætera eadem rema <lb></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">Poſt hæc conſiderandum eſt, quæ nam ſint ea, quæ efficiunt, <lb></lb>vt aliquod facilius moueatur, ſiue ſcindatur. </s> <s id="id.2.1.233.23.1.2.0">quæ quidem duo <lb></lb>ſunt. </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 ſcin<lb></lb>datur, quod etiam ad eſſentiam cunei magis per<lb></lb>tinet, eſt angulus ad verticem cunei; quò enim <lb></lb>minor eſt angulus, eò facilius mouet, ac ſcindit. </s> </p> <pb n="116" xlink:href="036/01/245.jpg"></pb> <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></lb>ABC ad verticem minor ſit angulo DEF. </s> <s id="id.2.1.233.26.1.1.0.a"><lb></lb>dico aliquod facilius moueri, ſiue ſcindi à cu<lb></lb>neo ABC, quàm à DEF. </s> <s id="N16A38">diuidantur AC <lb></lb>DF bifariam in G H punctis; connectan<lb></lb>turq; BG, & EH. </s> <s id="id.2.1.233.26.1.1.0.b">Quoniam enim partes <lb></lb>eius, quod ſcinditur à cuneo ABC, ſu<lb></lb>per planum horizonti inclinatum mouen<lb></lb>tur, cuius inclinatio eſt GBA: quæ ve<lb></lb>rò à cuneo DEF, ſuper planum horizonti <lb></lb>inclinatum mouentur, cuius inclinatio eſt <lb></lb><figure id="id.036.01.245.1.jpg" place="text" xlink:href="036/01/245/1.jpg"></figure><lb></lb>HED; & angulus GBA minor eſt angulo HED; cùm <lb></lb>CBA minor ſit DEF: & ex nona Pappi octaui libri mathe<lb></lb>maticarum collectionum, quod mouetur ſuper planum AB faci<lb></lb>lius mouebitur, & à minore potentia, quàm ſuper ED; Quod <lb></lb>ergo ſcinditur à cuneo ABC facilius, & à minore potentia ſcin<lb></lb>detur, quàm à cuneo DEF. ſimiliter oſtendetur, quò magis an<lb></lb>gulus ad verticem cunei erit acutus, eò facilius aliquod moueri, <lb></lb>ac ſcindi. </s> <s id="id.2.1.233.26.1.2.0">quod demonſtrare 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">Poſſumus etiam hoc alia ratione oſtendere <lb></lb>conſiderando cuneum, vt vectibus ſibi inuicem <lb></lb>aduerſis mouet, ſicuti ſecundo modo dictum eſt. </s> <s id="id.2.1.233.27.1.2.0"><lb></lb>hoc autem prius oſtendere oportet. </s> </p> <pb xlink:href="036/01/246.jpg"></pb> <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></lb>ſit B immobile; quod autem mouen<lb></lb>dum eſt, ſit CDEF rectangulum ita <lb></lb>accommodatum, vt deorſum ex par <lb></lb>te FE moueri non poſsit; & punctum <lb></lb>E ſit immobile, & tamquam centrum; <lb></lb>ita vt punctum D moueatur per cir<lb></lb>cumferentiam circuli DH, cuius cen<lb></lb>trum ſit E. </s> <s id="N16A8B">& C per circumferentiam <lb></lb>CL, ita vt iuncta CE ſit eius ſemi<lb></lb>diameter. </s> <s id="id.2.1.233.29.1.2.0">tangat inſuper CDEF ve<lb></lb><figure id="id.036.01.246.1.jpg" place="text" xlink:href="036/01/246/1.jpg"></figure><lb></lb>ctem AB in C, atq; vectis AB moueat pondus CDEF, & po<lb></lb>tentia mouens ſit in A, fulcimentum B, & pondus in C. </s> <s id="id.2.1.233.29.1.2.0.a">ſit <lb></lb>deinde alius vectis MCN, qui etiam moueat CDEF, cuius ful<lb></lb>cimentum immobile ſit N; potentia mouens in M, & pondus <lb></lb>ſimiliter in C; ſitq; CN æqualis ipſi CB, & CM ipſi CA; al<lb></lb>ternatimq; moueatur pondus CDEF vectibus AB MN. </s> <s id="id.2.1.233.29.1.2.0.b">dico <lb></lb>CDEF facilius ab eadem potentia moueri vecte AB, quàm ve<lb></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 deſcribatur <lb></lb>CO. </s> <s id="N16AB8">ſimiliter centro N, interuallo quidem NC, circumferen<lb></lb>tia deſcribatur CP. </s> <s id="id.2.1.233.30.1.1.0.a">Quoniam enim dum vectis AB mouet CD <lb></lb>EF, punctum vetis C mouetur ſuper circumferentiam CO; cùm <lb></lb>ſit B fulcimentum, & centrum immobile. </s> <s id="id.2.1.233.30.1.2.0">ſimiliter dum vectis <lb></lb>MN mouet CDEF, punctum C mouetur per circumferentiam <lb></lb>CP; dum igitur vectis AB mouet CDEF, conatur mouere pun<lb></lb>ctum C ponderis ſuper circumferentiam CO; quod quidem effi<lb></lb>cere non poteſt: quia C mouetur ſuper circumferentiam CL. </s> <s id="N16ACE">qua<lb></lb>re in motu vectis AB ſecundùm partem ipſi reſpondentem, ac mo<lb></lb>tu ponderis ſecundum C facto, contingit repugnantia quædam; <lb></lb>in diuerſas enim partes mouentur. </s> <s id="id.2.1.233.30.1.3.0">ſimiliter dum vectis MN mo<lb></lb>uet CDEF, conatur mouere C ſuper circumferentiam CP; at<lb></lb>que ideo in hoc etiam vtroq; motu ſimilis oritur repugnantia. </s> <s id="id.2.1.233.30.1.4.0"><lb></lb>quoniam autem circumferentia CO propior eſt circumferentiæ <lb></lb>CL, quam ſit CP; hoc eſt propior eſt motui, quem facit pun<lb></lb>ctum C ponderis; ideo minor erit repugnantia inter motum vectis <pb n="117" xlink:href="036/01/247.jpg"></pb>AB, & motum C ponderis, quàm inter motum vectis MN, & <lb></lb>motum eiuſdem C. quod etiam patet, ſi intelligatur CF hori<lb></lb>zonti perpendicularis, tunc enim circumferentia CP magis ten<lb></lb>dit deorſum, quàm CO; & CL tendit ſurſum. </s> <s id="id.2.1.233.30.1.5.0">& ideo minor fit re <lb></lb>pugnantia inter vectem AB, & motum C, quàm inter <expan abbr="vectẽ">vectem</expan> MN, & <lb></lb>motum C. </s> <s id="N16AFB">ſed vbi minor repugnantia ibi maior facilitas. </s> <s id="id.2.1.233.30.1.6.0">ergo faci<lb></lb>lius mouebitur CD EF vecte AB, quàm vecte MN. </s> <s id="N16B02">quod demon<lb></lb>ſtrare 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 manifeſtum eſt, quò minor eſt an<lb></lb>gulus à linea CF, vel CE, vel CD contentus; <lb></lb>hoc eſt, quò minor eſt angulus BCF, vel BCE, <lb></lb>vel etiam BCD, eò facilius pondus moueri. </s> <s id="id.2.1.233.32.1.2.0"><lb></lb>quod quidem eodem modo oſtendetur. </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 propoſitum eſt, ſic demon<lb></lb>ſtrabimus. </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></lb>F, & angulus ABC mi<lb></lb>nor ſit angulo DEF, & <lb></lb>AB BC DE EF ſint in <lb></lb>ter ſe ſe æquales. </s> <s id="id.2.1.233.34.1.2.0">Sint de<lb></lb>inde quatuor pondera æ<lb></lb>qualia GH IL NO QR <lb></lb>rectangula; ſintq; LM <lb></lb>kH in eadem recta linea: <lb></lb><figure id="id.036.01.247.1.jpg" place="text" xlink:href="036/01/247/1.jpg"></figure><lb></lb>ſimiliter RS PO in recta linea; erunt GK IM parallelæ, & NP <arrow.to.target n="note319"></arrow.to.target><lb></lb>QS parallelæ. </s> <s id="id.2.1.233.34.1.3.0">ſit IBG pars cunei intra pondera GH IL; & cu<lb></lb>nei pars QEN intra pondera NO QR; ſint〈qué〉 IB BG QE <lb></lb>EN inter ſe ſe 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"></pb>potentia moueri cuneo <lb></lb>ABC, quàm pondera <lb></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"></emph>Ex<emph.end type="italics"></emph.end> 28 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></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></lb>bifariam in TV, iungan<lb></lb>turq; TBVE, erunt an<lb></lb>guli ad T, & V recti. </s> <s id="id.2.1.235.1.1.2.0">con<lb></lb>nectatur IG, quæ ſecet <lb></lb>BT in X. </s> <s id="id.2.1.235.1.1.2.0.a">Quoniam e<lb></lb><figure id="id.036.01.248.1.jpg" place="text" xlink:href="036/01/248/1.jpg"></figure><lb></lb>nim IB eſt æqualis BG, & BA æqualis BC; erit IA ipſi GC <lb></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 eſt BG ad GC. </s> <s id="id.2.1.235.1.1.3.0.a">parallela igitur <lb></lb><arrow.to.target n="note321"></arrow.to.target>eſt IG ipſi AC. </s> <s id="N16B9C">ac propterea anguli ad X ſunt recti: ſed & an<lb></lb><arrow.to.target n="note322"></arrow.to.target>guli XG k XIM ſunt recti, rectangulum enim eſt GM; quare <lb></lb>TB æquidiſtans eſt ipſis Gk IM. </s> <s id="N16BA5">angulus igitur TBC æqua<lb></lb>lis eſt angulo BGK, & TBA ipſi BIM æqualis. </s> <s id="id.2.1.235.1.1.4.0">ſimiliter demon<lb></lb>ſtrabimus angulum VEF æqualem eſſe ENP, & VED æqualem <lb></lb>EQS. </s> <s id="N16BB0">cùm autem angulus ABC minor ſit angulo DEF; erit <lb></lb>& angulus TBC minor VEN. </s> <s id="N16BB4">quare & BGk minor ENP. <lb></lb></s> <s id="N16BB7">ſimili modo BIM minor EQS. </s> <s id="id.2.1.235.1.1.4.0.a">quoniam autem cuneus ABC <lb></lb>duobus mouet vectibus AB BC, quorum fulcimenta ſunt in B; <lb></lb>& pondera in GI: ſimiliter cuneus DEF duobus vectibus mouet <lb></lb>DE EF, quorum fulcimenta ſunt in E; & pondera in N Q: per <lb></lb>præcedentem pondera GH IL facilius vectibus AB BC mo<lb></lb>uebuntur, quàm pondera NO QR vectibus DE EF. </s> <s id="id.2.1.235.1.1.4.0.b">ponde<lb></lb>ra ergo GH IL facilius cuneo ABC mouebuntur, quàm ponde<lb></lb>ra NO QR cuneo DEF. </s> <s id="id.2.1.235.1.1.4.0.c">& quia eadem eſt ratio in mouendo, <lb></lb>atq; in ſcindendo; facilius idcirco aliquod cuneo ABC ſcindetur <lb></lb>quàm cuneo DEF. </s> <s id="N16BD4">ſimiliterq; oſtendetur, quò minor eſt angu<lb></lb>lus ad verticem cunei, eò facilius aliquod moueri, vel ſcindi. </s> <s id="id.2.1.235.1.1.5.0">quod <lb></lb>demonſtrare 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"></emph>Sexti.<emph.end type="italics"></emph.end></s> <s id="id.2.1.236.1.1.2.0"><margin.target id="note321"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 29 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.236.1.1.3.0"><margin.target id="note322"></margin.target>28 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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></lb>ſpatia; quàm ea, quæ à cuneo ABC. </s> <s id="id.2.1.237.1.1.1.0.a">nam vt DF ſit intra QN, <lb></lb>& AC ſit intra IG; neceſſe eſt, vt QN per ſpatia moueantur <lb></lb>maiora; ſcilicet vnum dextrorſum, alter ſiniſtrorſum, quàm IG; <lb></lb>cùm DF maior ſit AC; dummodo totus cuneus intra pondera in<pb n="118" xlink:href="036/01/249.jpg"></pb>grediatur. </s> <s id="id.2.1.237.1.1.2.0">à potentia verò facilius eodem tempore mouetur ali<lb></lb>quod per minus ſpatium, quàm per maius; dummodo cætera, qui<lb></lb>bus fit motus, ſint æqualia: ſi ergo eodem tempore AC DF in <lb></lb>IG QN <expan abbr="perueniãt">perueniant</expan>, cùm AI CG DQ FN ſint inter ſe ſe æqua<lb></lb>les; facilius à potentia mouebuntur GI cuneo ABC, quàm QN <lb></lb>cuneo DEF. </s> <s id="N16C27">quare facilius pondera GH IL à potentia mouebun<lb></lb>tur cuneo ABC, quàm pondera NO QR cuneo DEF. </s> <s id="N16C2B">ſimiliter<lb></lb>〈qué〉 oſtendetur, quò angulus ad verticem cunei minor eſſet, eò fa<lb></lb>cilius pondera moueri, vel ſcindi. </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></lb>ſcindatur, eſt percuſsio; qua cuneus mouetur, & <lb></lb>mouet; hoc eſt percutitur, ac ſcindit. <figure id="id.036.01.249.1.jpg" place="text" 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 ſcinditur B, quod <lb></lb>percutit C; quod quidem, vel ex ſe ipſo, <lb></lb>vel à regente, atq; ipſum mouente poten<lb></lb>tia percutit, atq; mouet. </s> <s id="id.2.1.237.3.1.2.0">ſi quidem ex <lb></lb>ſe ipſo, Primùm quò grauius erit, eò <lb></lb>maior fiet percuſsio. </s> <s id="id.2.1.237.3.1.3.0">quinetiam, quò <lb></lb>longior fuerit diſtantia inter AC, maior <lb></lb>itidem fiet percuſsio. </s> <s id="id.2.1.237.3.1.4.0">graue enim vnum<lb></lb>quodq; dum mouetur; grauitatis ma<lb></lb>gis aſſumit motum, quàm quieſcens: & <lb></lb>adhuc magis quo longius mouetur. <figure id="fig209" place="text" xlink:href=""></figure></s> </p> <pb xlink:href="036/01/250.jpg"></pb> <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></lb>tentia, vt ſi per manubrium DE mo<lb></lb>ueatur; primùm quò grauius erit C, <lb></lb>deinde quò longius erit DE, eò ma<lb></lb>ior fiet percuſsio. </s> <s id="id.2.1.237.5.1.2.0">ſi enim ponatur po<lb></lb>tentia mouens in E, erit C magis di <lb></lb>ſtans à centro & ideo citius mouebi<lb></lb>tur. </s> <s id="id.2.1.237.5.1.3.0">vt in quæſtionibus Mechanicis <lb></lb>latè monſtrat Ariſtoteles; nec non <lb></lb>ex iis, quæ in tractatu de libra di<lb></lb>cta fuere, patere poteſt, quò magis <lb></lb><figure id="id.036.01.250.1.jpg" place="text" xlink:href="036/01/250/1.jpg"></figure><lb></lb>pondus C à centro diſtat, eò grauius reddi. </s> <s id="id.2.1.237.5.1.4.0">quod ipſum etiam va<lb></lb>lidiori pellet impulſu virtute in E potentiore exiſtente. </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 ſecundùm eſt, quod efficit, vt hoc inſtrumento ma<lb></lb>gna moueantur, ſcindanturq; pondera. </s> <s id="id.2.1.237.6.1.2.0">percuſsio enim vis eſt ua<lb></lb>lidiſsima, vt ex decimanona <expan abbr="quæſtionũ">quæſtionum</expan> Mechanicarum Ariſtotelis <lb></lb>patet. </s> <s id="id.2.1.237.6.1.3.0">ſi enim ſupra cuneum maximum imponatur onus; tunc cu<lb></lb>neus nihil ferè efficiet, præſertim ictus comparatione. </s> <s id="id.2.1.237.6.1.4.0">quod ſi ad <lb></lb>huc ipſi cuneo vectem, vel cochleam, vel quoduis aliud huiuſmo<lb></lb>di aptetur inſtrumentum ad cuneum ponderi intimius propellen<lb></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"></pb>dem rei indicio eſſe poteſt, ſi fuerit <lb></lb>corpus A <expan abbr="lapideũ">lapideum</expan>, ex quo aliquam eius <lb></lb>partem detrahere quiſpiam voluerit, pu<lb></lb>tá partem anguli B; tunc malleo ferreo <lb></lb>abſq; alio inſtrumento percutiendo in B, <lb></lb>facilè aliquam anguli B partem franget. </s> <s id="id.2.1.237.6.1.6.0"><lb></lb>quod quidem nullo alio inſtrumento <lb></lb>percuſsionis munere carente, niſi maxi<lb></lb>ma cùm difficultate efficere poterit; ſiue <lb></lb><figure id="id.036.01.251.1.jpg" place="text" xlink:href="036/01/251/1.jpg"></figure><lb></lb>fuerit vectis, ſiue cochlea, ſiue quoduis aliud huiuſmodi. </s> <s id="id.2.1.237.6.1.7.0">quare <lb></lb>percuſsio in cauſa eſt, quo magna ſcindantur pondera. </s> <s id="id.2.1.237.6.1.8.0">cùm autem <lb></lb>ſola percuſsio tantam vim habeat, ſi ei aliquod adiiciamus inſtru<lb></lb>mentum ad mouendum, ſcindendumq; accomodatum, admiran<lb></lb>da profectò videbimus. </s> <s id="id.2.1.237.6.1.9.0">Inſtrumentum huiuſ <lb></lb>modi cuneus eſt, in quo duo (quantum ad ip<lb></lb>ſius formam attinet) conſideranda occurrunt. </s> <s id="id.2.1.237.6.1.10.0"><lb></lb>Alterum eſt, cuneum ad ſuſcipiendam, ſuſtinen<lb></lb>damq; percuſsionem aptiſsimum eſſe; alterum <lb></lb>eſt quòd propter eius in altera parte ſubtilita<lb></lb>tem facilè intra corpora ingreditur, vt manife<lb></lb>ſtè patet. </s> <s id="id.2.1.237.6.1.11.0">Cuneus ergo cum percuſsione ipſius <lb></lb>efficit, vt in mouendis, ſcindendiſq; ponderi<lb></lb>bus ferè miracula cernamus. <figure id="id.036.01.251.2.jpg" place="text" xlink:href="036/01/251/2.jpg"></figure></s> </p> <pb xlink:href="036/01/252.jpg"></pb> <p id="id.2.1.237.8.0.0.0" type="main"> <s id="id.2.1.237.8.1.1.0">Ad huiuſmodi facultatis inſtrumentum, ea <lb></lb>quoquè omnia commodè referri poſſunt, quæ <lb></lb>percuſsione, ſiue impulſu incidunt, diuidunt, <lb></lb>perforant, huiuſmodiq; alia obeunt munera. </s> <s id="id.2.1.237.8.1.2.0">vt <lb></lb>enſes, gladii, mucrones, ſecures, & ſimilia. </s> <s id="id.2.1.237.8.1.3.0">ſerra <lb></lb>quoq; ad hoc reducetur; dentes enim percu<lb></lb>tiunt, cuneiq; inſtar exiſtunt. </s> </p> </chap> <pb n="120" xlink:href="036/01/253.jpg"></pb> <chap id="N16D2B"> <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></lb>multa pertractans de cochlea, do<lb></lb>cet quomodo conficienda ſit; & <lb></lb>quomodo magna huiuſmodi in<lb></lb>ſtrumento moueantur pondera; <lb></lb>nec non alia theoremata ad eius <lb></lb>cognitionem valdè vtilia. </s> <s id="id.2.1.237.11.1.2.0">Quoniam autem in<lb></lb>ter cætera pollicetur, ſe oſtendere velle, co<lb></lb>chleam nihil aliud eſſe præter aſſumptum cu<lb></lb>neum percuſsionis expertem vecte motionem <lb></lb>facientem; hoc autem in ipſo deſideratur; pro<lb></lb>pterea idipſum oſtendere conabimur, nec non <lb></lb>eiuſdem cochleæ ad vectem, libramq; reductio<lb></lb>nem; vt ipſius tandem completa habeatur co<lb></lb>gnitio. <pb xlink:href="036/01/254.jpg"></pb> <figure id="id.036.01.254.1.jpg" place="text" 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: ſitq; <lb></lb>IGH cuneus circa cylindrum reuolutus, cuius vertex ſit I. </s> <s id="id.2.1.237.12.1.1.0.a">ſit de<lb></lb>inde cylindrus cum circumpoſito cuneo ita accomodatus, vt abſq; <lb></lb>vllo <expan abbr="impedimẽto">impedimento</expan> manubrio kF eius axi annexo circumuerti poſsit. </s> <s id="id.2.1.237.12.1.2.0"><lb></lb>ſitq; LMNO, quod ſcindendum eſt; quod etiam ex parte MN <lb></lb>ſit immobile: vt in iis, quæ ſcinduntur, fieri ſolet: & ſit vertex <lb></lb>I intra RS. </s> <s id="N16D7A">circumuertatur kF, & perueniat ad kP; dum autem kF <lb></lb>circumuertitur, circumuertitur etiam totus cylindrus DE, & cu<lb></lb>neus IGH: quare dum KF erit in kP, vertex I non erit amplius <lb></lb>intra RS, ſed cunei pars alia, vt TV: ſed TV maior eſt, quàm <lb></lb>RS; ſemper enim pars cunei, quæ magis à vertice diſtat, maior <lb></lb>eſt ea, quæ ipſi eſt propinquior: vt igitur TV ſit intra RS, opor<lb></lb>tet, vt R cedat, moueaturq; verſus X, & S verſus Z, vt faciunt <lb></lb>ea, quæ ſcinduntur. </s> <s id="id.2.1.237.12.1.3.0">totum ergo LMNO ſcindetur. </s> <s id="id.2.1.237.12.1.4.0">ſimiliter <lb></lb>què demonſtrabimus, dum manubrium kP erit in kQ, tunc GH <lb></lb>eſſe intra RS: & vt GH ſit intra RS, neceſſe eſt, vt R ſit in X, <lb></lb>& S in Z; ita vt <emph type="italics"></emph>X<emph.end type="italics"></emph.end>Z ſit æqualis GH; ſemperq; LMNO amplius <lb></lb>ſcindetur. </s> <s id="id.2.1.237.12.1.5.0">ſic igitur patet, dum kF circumuertitur, ſemper R moue<lb></lb>ri verſus X, atq; S verſus Z: & R ſemper ſuper ITG moueri, S au<lb></lb>tem ſuper IVH, hoc eſt ſuper latera cunei circa cylindrum circum <lb></lb>uoluti. </s> </p> <pb n="121" xlink:href="036/01/255.jpg"></pb> <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></lb>datus, nihil eſt aliud; niſi cochlea duas habens he<lb></lb>lices in vnico puncto inuicem coniunctas. <figure id="id.036.01.255.1.jpg" place="text" 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></lb>ipſi BC æqualis. </s> <s id="id.2.1.237.16.1.2.0">diuidatur <lb></lb>AC bifariam in D, iunga<lb></lb>turq; BD; erit BD ipſi AC <lb></lb>perpendicularis; & AD <lb></lb>ipſi DC æqualis, triangu<lb></lb>lumq; ABD triangulo C <lb></lb>BD æquale. </s> <s id="id.2.1.237.16.1.3.0">fiant deinde <lb></lb>triangula rectangula EFG <lb></lb>HIk non ſolum inter ſe, <lb></lb>verùm etiam vtriq; ADB <lb></lb>& CDB æqualia. </s> <s id="id.2.1.237.16.1.4.0">ſitq; cy<lb></lb>lindrus LMNO, cuius perimeter ſit æqualis vtriq; FG kI. </s> <s id="id.2.1.237.16.1.4.0.a">& <lb></lb>LMNO ſit parallelogrammum per axem. </s> <s id="id.2.1.237.16.1.5.0">fiatq; MP æqualis <lb></lb>FE; & PN æqualis HI. </s> <s id="N16DEE">ponaturq; HI in NP, circumuolua<lb></lb>turq; triangulum HIk circa cylindrum; & ſecundùm kH helix <lb></lb>deſcribatur NQP, vt Pappus quoq; docet in octauo libro propo<lb></lb>ſitione vigeſima quarta. </s> <s id="id.2.1.237.16.1.6.0">ſimiliter ponatur EF in MP, circum<lb></lb>uoluaturq; triangulum EFG circa cylindrum; deſcribaturq; per <lb></lb>EG helix PRM. </s> <s id="N16DFD">cùm itaq; PMPN ſint æquales EFHI, erit <lb></lb>MN æqualis ipſi AC, & cùm helices PRM PQN ſint æquales <lb></lb>lineis EGHk; helices igitur ipſis ABBC æquales erunt. </s> <s id="id.2.1.237.16.1.7.0">cu<lb></lb>neus ergo ABC totus circumuolutus erit circa cylindrum LMNO. </s> <s id="N16E08"><pb xlink:href="036/01/256.jpg"></pb>incidantur deinde helices, <lb></lb>vt docet Pappus ſecundùm <lb></lb>latitudinem cunei; & hoc <lb></lb>modo cuneus vná cum cy<lb></lb>lindro nihil aliud erit, <lb></lb>quàm cochlea duas habens <lb></lb>helices PRMPQN cir<lb></lb>ca cylindrum LN in vnico <lb></lb>puncto P inuicem coniun<lb></lb>ctas. </s> <s id="id.2.1.237.16.1.8.0">quod demonſtrare o<lb></lb>portebat. </s> </p> <figure id="id.036.01.256.1.jpg" place="text" 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 manifeſtum eſſe poteſt, quomodo heli<lb></lb>ces in ipſa cochlea deſcribi poſsint. </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 ſuper helices co<lb></lb>chleæ moueantur, oſtendamus. <figure id="id.036.01.256.2.jpg" place="text" 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></lb>cuius vertex ſit I. </s> <s id="N16E4C">apteturq; cylindrus ita, vt liberè vna cum ſuo <lb></lb>axe circumuertatur. </s> <s id="id.2.1.237.19.1.2.0">ſintq; duo pondera MN cuiuſcunq; figuræ <lb></lb>voluerimus, ita tamen aptata, vt moueri non poſsint, niſi ſuper <pb n="122" xlink:href="036/01/257.jpg"></pb>rectam lineam LO, quæ axi cylindri ſit æquidiſtans. </s> <s id="id.2.1.237.19.1.3.0">ſintq; MN <lb></lb>iuxta cunei verticem I. </s> <s id="N16E5F">Circumuertatur KF, & perueniat ad kP: <lb></lb>dum autem kF erit in kP, tunc TV erit intra pondera MN; ſi<lb></lb>cut ſupra diximus. </s> <s id="id.2.1.237.19.1.4.0">M igitur verſus L mouebitur, & N verſus O. <lb></lb></s> <s id="N16E69">ſimiliter oſtendetur, dum kP erit in KQ, tunc GH eſſe intra pon<lb></lb>dera MN; & M erit in X, & N in Z; ita vt XZ ſit æqualis GH. <lb></lb></s> <s id="N16E6E">quare dum kF circumuertitur, ſemper pondus N mouetur verſus <lb></lb>O, & ſuper helicem IRS; M verò ſuper aliam helicem. <figure id="id.036.01.257.1.jpg" place="text" 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 ſi cochlea plures habeat hæ<lb></lb>lices, vt in ſecunda figura, pondus A, <lb></lb>dum cochlea circumuertitur, ſemper ſu<lb></lb>per helices BCDEFG mouebitur; <lb></lb>dummodo pondus A aptetur ita vt mo<lb></lb>ueri non poſsit, niſi ſuper rectam HI ipſi <lb></lb>cylindro æquidiſtantem. </s> <s id="id.2.1.237.20.1.2.0">eodem enim <lb></lb>modo, quo ſuper primam mouetur heli<lb></lb>cem, mouetur etiam ſupra ſecundam, <lb></lb>& tertiam, & cætera. </s> <s id="id.2.1.237.20.1.3.0">quotcunq; enim <lb></lb>fuerint helices, nihil aliud ſunt, quàm <lb></lb>latus cunei circa idem cylindrum iterum <lb></lb>atq; iterum circumuolutum. </s> <s id="id.2.1.237.20.1.4.0">& ſiue co<lb></lb>chlea fuerit horizonti perpendicularis, <lb></lb>ſiue horizonti æquidiſtans, vel alio mo<lb></lb>do collocata, nihil refert: ſemper enim <lb></lb>eadem erit ratio. <pb xlink:href="036/01/258.jpg"></pb> <figure id="id.036.01.258.1.jpg" place="text" 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) ſupra cochleam imponatur aliquod, <lb></lb>vt B, quod quidem tylum vocant, ita accommodatum, vt inferio <lb></lb>ri parte helices habeat concauas ipſi cochleæ appoſitè admodum <lb></lb>congruentes; perſpicuum ſatis eſſe poterit, ipſum B, dum <expan abbr="coclhea">cochlea</expan><lb></lb>circumuertitur, ſuper helices cochleæ eo prorſus modo moueri; <lb></lb>quo pondus iuxta primam <expan abbr="figurã">figuram</expan> mouebatur: dummodo tylum ap<lb></lb>tetur, vt docet Pappus in octauo libro; ita ſcilicet vt tantùm an<lb></lb>tè, retrouè axi cylindri æquidiſtans moueatur. <figure id="id.036.01.258.2.jpg" place="text" 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 ſi loco tyli, quod helices habet concauas in parte inferiori, con<lb></lb>ſtituatur, vt in quarta figura, cylindrus concauus vt D, & in eius <lb></lb>concaua ſuperficie deſcribantur helices, incidanturq; ita, vt aptè <pb n="123" xlink:href="036/01/259.jpg"></pb>cùm cochlea congruant (eodem enim modo deſcribentur helices <lb></lb>in ſuperficie concauia cylindri, ſicuti fit in conuexa) ſi deinde co<lb></lb>chlea in ſuis polis firmetur, ſcilicet in ſuo axe, circumuertaturq;; <lb></lb>patet D ad motum circumuerſionis cochleæ quemmadmodum ty<lb></lb>lum moueri. </s> <s id="id.2.1.237.22.1.2.0">nec non ſi D in EF firmetur, ita vt immobilis ma <lb></lb>neat, dum circumuertitur cochlea; ſuper helices cylindri D, ad <lb></lb>motum ſuæ circumuerſionis dextrorſum, vel ſiniſtrorſum factæ; <lb></lb>tùm in anteriorem, tùm in poſteriorem partem mouebitur. </s> <s id="id.2.1.237.22.1.3.0">cylin<lb></lb>drus autem D hoc modo <expan abbr="accõmodatus">accommodatus</expan> vulgò mater, ſiue cochleæ <lb></lb>fæmina nuncupatur. <figure id="id.036.01.259.1.jpg" place="text" 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></lb>obliquis dentatum apponatur, vt docet Pappus in eodem octauo li<lb></lb>bro; vel etiam rectis; ita tamen conſtructis, vt facilè cum cochlea <lb></lb>conueniant: ſimiliter manifeſtum eſt ad motum cochleæ circumuer<lb></lb>ti etiam tympanum C. </s> <s id="id.2.1.237.23.1.1.0.a">eodemq; modo tympani dentes ſuper he<lb></lb>lices cochleæ moueri. </s> <s id="id.2.1.237.23.1.2.0">& hæc dicitur cochlea infinita, quia & co<lb></lb>chlea, & tympanum dum circumuertuntur, ſemper eodem modo <lb></lb>ſe ſe habent. </s> </p> <pb xlink:href="036/01/260.jpg"></pb> <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 manifeſtum ſit cochleam in mouendo pondere <lb></lb>cunei munere abſq; percuſsione fungi. </s> <s id="id.2.1.237.25.1.2.0">Illud enim remouet à loco, <lb></lb>vbi erat; quemadmodum cuneus remouet ea, quæ mouet, ac ſcindit. </s> <s id="id.2.1.237.25.1.3.0"><lb></lb>omnia enim hæc à cochlea mouentur, ſicuti pondus A in ſecun<lb></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 conſiderari <lb></lb>poſſe oſtendimus, videlicet vt mouet vectibus, vel vt eſt planum <lb></lb>horizonti inclinatum, dupliciter quoq; cochleam conſiderabimus; <lb></lb><figure id="id.036.01.260.1.jpg" place="text" xlink:href="036/01/260/1.jpg"></figure><lb></lb>& primùm vt vectibus mouet, vt in prima figura circumuertatur <lb></lb>kF, & perueniat in KP; tunc, ſicut dictum eſt, TV erit intra pon<lb></lb>dera MN. </s> <s id="N16F44">& ſicut conſideramus vectes in cuneo, eodem quoq; <lb></lb>modo eos conſiderare poſſumus in cochlea hoc pacto. </s> <s id="id.2.1.237.26.1.2.0">erit ſcilicet <lb></lb>IVH vectis, cuius fulcimentum I, & pondus in V. </s> <s id="id.2.1.237.26.1.2.0.a">ſimiliter ITG ve<lb></lb>ctis, cuius fulcimentum I, & pondus in T. </s> <s id="id.2.1.237.26.1.2.0.b">potentiæ verò mo<lb></lb>uentes GH eſſe deberent; ſed ſicuti in cuneo potentia mouens <lb></lb>eſt percuſsio, quæ mouet cuneum; idcirco erit, ubi potentia mo<lb></lb>uet cochleam; ſcilicet in P manubrio kP. </s> <s id="N16F58">cochlea enim ſine per<lb></lb>cuſsione mouetur. </s> <s id="id.2.1.237.26.1.3.0">Hæc autem conſideratio propter vectes infle<lb></lb>xos impropria forſitan eſſe videbitur; Quocirca ſi id, quod moue<lb></lb>tur à cochlea, ſupra planum horizonti inclinatum moueri intelli<lb></lb>gatur; erit quidem huiuſmodi conſideratio (cùm ipſi quoq; cuneo <lb></lb>conueniat) figuræ ipſius cochleæ magis conformis. </s> </p> <pb n="124" xlink:href="036/01/261.jpg"></pb> <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></lb>CDEFG. </s> <s id="id.2.1.237.29.1.1.0.a">Dico has nihil aliud eſſe præter pla<lb></lb>num horizonti inclinatum circa cylindrum re<lb></lb>uolutum. <figure id="id.036.01.261.1.jpg" place="text" 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></lb>CDEFG. </s> <s id="id.2.1.237.30.1.1.0.a">exponatur HI æqualis GC, quæ bifariam diui<lb></lb>datur in k; erunt Hk kI non ſolum inter ſe ſe, verùm etiam <lb></lb>ipſis GE EC æquales, & ipſi HI ad rectos angulos ducatur LI; <lb></lb>& per LI intelligatur planum horizonti æquidiſtans; ſitq; LI du<lb></lb>pla perimetro cylindri AB, quæ bifariam diuidatur in M; erunt <lb></lb>IM ML cylindri perimetro æquales. </s> <s id="id.2.1.237.30.1.2.0">connectatur HL, & à pun<lb></lb>cto M ducatur MN ipſi HI æquidiſtans, coniungaturq; KN. </s> <s id="id.2.1.237.30.1.2.0.a">quo<lb></lb>niam enim ſimilia ſunt inter ſe ſe triangula HILNML, cùm <arrow.to.target n="note323"></arrow.to.target> <pb xlink:href="036/01/262.jpg"></pb> <figure id="id.036.01.262.1.jpg" place="text" xlink:href="036/01/262/1.jpg"></figure><lb></lb>NM ſit æquidiſtans HI; erit LI ad IH, vt LM ad MN: & <lb></lb>permutando vt IL ad LM; ita HI ad NM. </s> <s id="id.2.1.237.30.1.2.0.b">ſed IL dupla eſt ipſius <lb></lb>LM; ergo & HI dupla erit MN. </s> <s id="N16FB2">ſed eſt etiam dupla ipſius kI, <lb></lb>quare kI NM inter ſe æquales erunt. </s> <s id="id.2.1.237.30.1.3.0">& quoniam anguli ad MI <lb></lb>ſunt recti; erit kM parallelogrammum rectangulum, & kN æqua <lb></lb>lis erit IM. </s> <s id="id.2.1.237.30.1.3.0.a">quare KN perimetro cylindri AB æqualis erit. </s> <s id="id.2.1.237.30.1.4.0">pona<lb></lb>tur itaq; HI in GC, erit Hk in GE. </s> <s id="id.2.1.237.30.1.4.0.a">circumuoluatur deinde trian<lb></lb>gulum HkN circa cylindrum AB, deſcribet HN helicen GFE; <lb></lb>cùm NK perimetro cylindri ſit æqualis; & punctum N erit in E; <lb></lb>& MN in CE. </s> <s id="N16FC8">& quia ML æqualis eſt perimetro cylindri; cir<lb></lb>cumuoluatur rurſus triangulum NML circa cylindrum AB, NL <lb></lb>deſcribet helicen EDC. </s> <s id="N16FCE">quare tota LH duas deſcribet helices <lb></lb>CDEFG. </s> <s id="N16FD2">patet igitur has helices cochleæ nihil aliud eſſe, ni<lb></lb>ſi planum horizonti inclinatum; cuius inclinatio eſt angulus HLI <lb></lb>circa cylindrum circumuolutum, ſupra quod pondus mouetur. <lb></lb></s> <s id="id.2.1.237.30.1.5.0"><lb></lb>quod demonſtrare 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"></emph>Ex<emph.end type="italics"></emph.end> 4. <emph type="italics"></emph>ſexti.<emph.end type="italics"></emph.end></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="mnnifeſtum">manifestum</expan> eſt ex <lb></lb>nona octaui libri eiuſdem Pappi. </s> </p> <pb n="125" xlink:href="036/01/263.jpg"></pb> <p id="id.2.1.239.3.0.0.0" type="main"> <s id="id.2.1.239.3.1.1.0">Poſtquam vidimus quomodo pondera huiuſmodi moueantur <lb></lb>inſtrumento; nunc conſiderandum eſt, quæ nam ſint ea, quæ effi<lb></lb>ciunt, vt pondera facilè moueantur: hæc autem duo ſunt. </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></lb>dus moueatur, quod etiam ad eſſentiam cochleæ <lb></lb>magis pertinere videtur; eſt helix circa co<lb></lb>chleam. </s> <s id="id.2.1.239.4.1.2.0">vt ſi circa datam cochleam AB duæ <lb></lb>ſint helices inæquales CDA EFG, ſitq; AC mi<lb></lb>nor EG. </s> <s id="id.2.1.239.4.1.2.0.a">Dico idem pondus facilius ſuper heli<lb></lb>cen CDA moueri, quàm ſuper EFG. <lb></lb><figure id="id.036.01.263.1.jpg" place="text" 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></lb>ADCHI, hoc eſt de<lb></lb>ſcribatur helix CHI <lb></lb>æqualis CDA, & ver<lb></lb>tex cunei ſit C. </s> <s id="N17035">ſimili<lb></lb>ter compleatur cuneus <lb></lb>GFEKL, cuius ver<lb></lb>tex E. </s> <s id="N1703D">exponatur de<lb></lb>inde recta linea MN, <lb></lb>quæ ſit ipſi AC æqua<lb></lb>lis, cui ad rectos angu<lb></lb>los ducatur NP, quæ ſit <lb></lb>æqualis perimetro cy<lb></lb>lindri AB: & conne<lb></lb>ctatur <arrow.to.target n="note324"></arrow.to.target>PM; erit PM, <lb></lb>per ea, quæ dicta ſunt, <lb></lb>ipſi CDA æqualis. </s> <s id="id.2.1.239.5.1.2.0"><lb></lb>producatur deinde M <lb></lb>N in O, fiatq; ON æ<lb></lb>qualis MN, coniunga<lb></lb>turq; OP; erit OPM cuneus cuneo ADCHI æqualis. </s> <s id="id.2.1.239.5.1.3.0">ſimili<arrow.to.target n="note325"></arrow.to.target> <pb xlink:href="036/01/264.jpg"></pb>terq; exponatur cu<lb></lb>neus STQ æqualis cu<lb></lb>neo GFEkL; erit TR <lb></lb>ipſi PN, & perime<lb></lb>tro cylindri æqualis; & <lb></lb>QR æqualis GE. <lb></lb></s> <s id="N17074">cùm autem GE ma<lb></lb>ior ſit AC; erit & RQ <lb></lb>maior MN. </s> <s id="N1707A">ſecetur <lb></lb>RQ in V; fiatq; RV <lb></lb>ipſi MN æqualis, & <lb></lb>coniungatur TV; erit <lb></lb>triangulum TVR tri<lb></lb>angulo MPN æquale: <lb></lb>duæ enim TR RV <lb></lb>duabus PN NM ſunt <lb></lb>æquales, & anguli, <lb></lb>quos continent, ſunt <lb></lb>æquales, nempe recti; <lb></lb><arrow.to.target n="note326"></arrow.to.target>angulus igitur RTV <lb></lb><figure id="id.036.01.264.1.jpg" place="text" xlink:href="036/01/264/1.jpg"></figure><lb></lb>angulo NPM æqualis erit. </s> <s id="id.2.1.239.5.1.4.0">quare angulus MPN minor eſt angu<lb></lb>lo QTR; & horum dupli, angulus ſcilicet MPO minor angulo <lb></lb>QTS. </s> <s id="id.2.1.239.5.1.4.0.a">quoniam autem cuneus, qui angulum ad verticem mino <lb></lb>rem habet, facilius mouet, ac ſcindit, quàm qui habet maiorem; <lb></lb>cuneus ergo MPO facilius mouebit, quàm QTS. </s> <s id="id.2.1.239.5.1.4.0.b">facilius igitur <lb></lb>pondus à cuneo ADCHI mouebitur, quàm à cuneo GFEkL. </s> <s id="id.2.1.239.5.1.4.0.c"><lb></lb>pondus ergo ſuper helicen CDA facilius mouebitur, quàm ſuper <lb></lb>EFG. </s> <s id="N170B6">eodemq; modo oſtendetur, quò minor erit AC, eò faci<lb></lb>lius pondus moueri. </s> <s id="id.2.1.239.5.1.5.0">quod demonſtrare 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"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.240.1.1.2.0"><margin.target id="note325"></margin.target>1 <emph type="italics"></emph>Huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.240.1.1.3.0"><margin.target id="note326"></margin.target>4 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> </p> <pb n="126" xlink:href="036/01/265.jpg"></pb> <figure id="id.036.01.265.1.jpg" place="text" 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; ſit <lb></lb>deinde alius cylindrus <foreign lang="grc">αβ</foreign>ipſi AB æqualis, in quo ſummatur OP ip<lb></lb>ſi CG æqualis; diuidaturq; OP in tres partes æquales OR RT <lb></lb>TP, & tres deſcribantur helices OQRSTVP; erit vnaquæq; OR RT <lb></lb>TP minor CE, & EG: tertia enim pars minor eſt dimidia. </s> <s id="id.2.1.241.2.1.2.0">dico <lb></lb>idem pondus facilius ſuper helices OQRSTVP moueri, quàm ſu<lb></lb>per CDEFG. </s> <s id="id.2.1.241.2.1.2.0.a">exponatur HIL triangulum orthogonium, ita vt <lb></lb>HI ſit ipſi CG æqualis, & IL duplo perimetri cylindri AB æqua<lb></lb>lis, & per <emph type="italics"></emph>L<emph.end type="italics"></emph.end>I intelligatur planum horizonti æquiſtans; erit H<emph type="italics"></emph>L<emph.end type="italics"></emph.end><lb></lb>æqualis CDEFG; & H<emph type="italics"></emph>L<emph.end type="italics"></emph.end>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></lb>ſimiliter <emph type="italics"></emph>X<emph.end type="italics"></emph.end>YZ triangulum orthogonium, ita vt XZ ipſi OP ſit æ<lb></lb>qualis, quæ etiam æqualis erit CG, & HI; ſitq; ZY cylindri pe<lb></lb>rimetro tripla, erit XY æqualis OQRSTVP. </s> <s id="N1712F">diuidatur ZY in <pb xlink:href="036/01/266.jpg"></pb> <figure id="id.036.01.266.1.jpg" place="text" xlink:href="036/01/266/1.jpg"></figure><lb></lb>tres partes æquales in <foreign lang="grc">γ</foreign><foreign lang="el">d</foreign>; erit vnàquæq; Z <foreign lang="grc">γ γ</foreign><foreign lang="el">d</foreign> <foreign lang="el">d</foreign> Y perimetro cy<lb></lb>lindri <foreign lang="grc">αβ</foreign>æqualis, quæ <expan abbr="etiã">etiam</expan> perimetro cylindri AB æquales erunt; & <lb></lb>per conſequens ipſis IM, & ML. </s> <s id="N1714F">connectatur X<foreign lang="el">d</foreign>. </s> <s id="id.2.1.241.2.1.4.0">& quoniam <lb></lb>duæ HI IL duabus XZ Z<foreign lang="el">d</foreign> ſunt æquales, & angulus HIL re<lb></lb>ctus æqualis eſt angulo XZ<foreign lang="el">d</foreign> recto; erit triangulum HIL trian<lb></lb>gulo XZ<foreign lang="el">d</foreign> æquale; & angulus HLI angulo X<foreign lang="el">d</foreign>Z æqualis; & <lb></lb><arrow.to.target n="note328"></arrow.to.target>X<foreign lang="el">d</foreign> ipſi HL æqualis. </s> <s id="id.2.1.241.2.1.5.0">ſed quoniam angulus X<foreign lang="el">d</foreign>Z maior eſt angu<lb></lb>lo <emph type="italics"></emph>X<emph.end type="italics"></emph.end>YZ; erit angulus HLI angulo <emph type="italics"></emph>X<emph.end type="italics"></emph.end>YZ maior. </s> <s id="id.2.1.241.2.1.6.0">ac propterea <expan abbr="planũ">planum</expan><lb></lb>HL magis horizonti inclinat, quàm XY. </s> <s id="N17178">quare <expan abbr="idẽ">idem</expan> <expan abbr="põdus">pondus</expan> à minore <lb></lb>potentia ſuper <expan abbr="planũ">planum</expan> XY, quàm ſuper <expan abbr="planũ">planum</expan> HL mouebitur; vt faci<lb></lb>lè elicitur ex <expan abbr="eadẽ">eadem</expan> nona Pappi. </s> <s id="id.2.1.241.2.1.7.0">cùm <expan abbr="autẽ">autem</expan> helices OQRSTVP nihil <lb></lb>aliud ſint, quàm <expan abbr="planũ">planum</expan> XY horizonti <expan abbr="inclinatũ">inclinatum</expan> in angulo XYZ cir<lb></lb>ca cylindrum <foreign lang="grc">αβ</foreign>circumuolutum; & helices CDEFG nihil ſunt <lb></lb>aliud, quàm planum HL horizonti inclinatum in angulo HLI cir<lb></lb>ca cylindrum AB circumuolutum; facilius ergo pondus ſuper he<pb n="127" xlink:href="036/01/267.jpg"></pb>lices OQRSTVP mouebitur, quàm ſuper 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"></emph>Ex<emph.end type="italics"></emph.end> 2 <emph type="italics"></emph>huius.<emph.end type="italics"></emph.end></s> <s id="id.2.1.242.1.1.2.0"><margin.target id="note328"></margin.target>21 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></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, deſcribantur<lb></lb>què circa <foreign lang="grc">αβ</foreign>quatuor helices; adhuc facilius pondus mouebitur ſu<lb></lb>per has quatuor, quàm ſuper tres OQRSTVP. </s> <s id="N171DB">& quò plures <lb></lb>erunt helices, eò facilius pondus mouebitur. </s> <s id="id.2.1.243.1.1.2.0">quod demonſtrare <lb></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></lb>ſunt æquales HL; helices verò OQRSTVP ſunt æquales <lb></lb>XY: ſed XY maior eſt HL; ideo fiat Y<foreign lang="grc">ε</foreign> ipſi HL æqualis: ſi igi<arrow.to.target n="note329"></arrow.to.target><lb></lb>tur duo pondera ſuper lineas LHY<emph type="italics"></emph>X<emph.end type="italics"></emph.end> moueantur, & veloci<lb></lb>tates motuum ſint æquales, citius pertranſibit quod mouetur ſuper <lb></lb>LH, quàm quod ſuper Y<emph type="italics"></emph>X<emph.end type="italics"></emph.end> mouetur. </s> <s id="id.2.1.243.2.1.2.0">in eodem enim tempore erunt <lb></lb>in H<foreign lang="grc">ε</foreign>. </s> <s id="id.2.1.243.2.1.3.0">quare tempus eius, quod mouetur ſuper helices OQRS <lb></lb>TVP, maius erit eo, quod eſt menſura eius, quod mouetur ſuper C <lb></lb>DEFG. </s> <s id="N17217">& quò plures erunt helices, eò maius erit tempus. </s> <s id="id.2.1.243.2.1.4.0">cùm au<lb></lb>tem datæ ſint lineæ HI<emph type="italics"></emph>XZ<emph.end type="italics"></emph.end>, & IL<emph type="italics"></emph>Z<emph.end type="italics"></emph.end>Y: datæ enim ſunt cochleæ AB <lb></lb> <foreign lang="grc">αβ</foreign>; & anguli ad IZ recti dati; erit HL data. </s> <s id="id.2.1.243.2.1.5.0">ſimiliter & <emph type="italics"></emph>X<emph.end type="italics"></emph.end>Y data <arrow.to.target n="note330"></arrow.to.target><lb></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></lb>tio eorum, quæ ſuper 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"></emph>Ex<emph.end type="italics"></emph.end> 18 <emph type="italics"></emph>Primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.244.1.1.2.0"><margin.target id="note330"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 48 <emph type="italics"></emph>primi.<emph.end type="italics"></emph.end></s> <s id="id.2.1.244.1.1.3.0"><margin.target id="note331"></margin.target>1 <emph type="italics"></emph>Datorum & Ex ſexta primi Ioannis de Monte rego de triangulis.<emph.end type="italics"></emph.end></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></lb>ueantur, ſunt ſcytalæ, aut manubria, quibus co<lb></lb>chlea circumuertitur. <pb xlink:href="036/01/268.jpg"></pb> <figure id="id.036.01.268.1.jpg" place="text" 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 ſcytalas ha<lb></lb>beat EFGH foraminibus cochleæ impoſitas. </s> <s id="id.2.1.245.2.1.2.0">ſit infra helices <lb></lb>cylindrus MN, in quo non ſint inciſæ helices; & circa cylindrum <lb></lb>funis circumuoluatur trahens pondus O, quod ad motum ſcytala <lb></lb>rum EFGH moueatur, ac ſi ergatæ inſtrumento traheretur. </s> <s id="id.2.1.245.2.1.3.0">du<lb></lb>catur (per ea quæ prius dicta ſunt de axe in peritrochio) Lk ſcy<lb></lb>talæ æqualis, axiq; cylindri perpendicularis, eumq; ſecans in I: <lb></lb>patet quò longior ſit LI, & quò breuior ſit Ik, pondus O facilius <lb></lb>moueri. </s> <s id="id.2.1.245.2.1.4.0">eſt autem animaduertendum, quòd dum cochlea mouet <lb></lb>pondus, ſi mente concipiatur, quòd loco trahendi pondus O fune, <lb></lb>pondus ſuper helices ABCD moueat; pondus quoq; in k, quod <lb></lb>ſit R, ſuper helices etiam facilius mouebit. </s> <s id="id.2.1.245.2.1.5.0">eſt enim LK vectis, cuius <lb></lb><arrow.to.target n="note332"></arrow.to.target>fulcimentum eſt I: cùm circa axem cochlea circumuertatur; po<lb></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></lb>dus vecte Lk, quàm ſine vecte; quia LI ſemper maior eſt Ik. </s> <s id="id.2.1.245.2.1.7.0"><pb n="128" xlink:href="036/01/269.jpg"></pb>Intelligatur itaq; manente cochlea pondus R moueri à potentia <lb></lb>in L vecte Lk ſuper helicen Ck: vel quod idem eſt, ſicut etiam <lb></lb>ſupra diximus, ſi pondus R aptetur ita, vt moueri non poſsit, ni <lb></lb>ſi ſuper rectam PQ axi cylindri æquidiſtantem; circumuertaturq; <lb></lb>cochlea, potentia exiſtente in L; mouebitur pondus R ſuper he<lb></lb>licen CD eodem modo, ac ſi à vecte Lk moueretur. </s> <s id="id.2.1.245.2.1.8.0">idem enim <lb></lb>eſt, ſiue pondus manente cochlea ſuper helicen moueatur; ſiue he<lb></lb>lix circumuertatur, ita vt pondus ſuper ipſam moueatur. </s> <s id="id.2.1.245.2.1.9.0">cùm <lb></lb>ab eadem potentia in L moueatur. </s> <s id="id.2.1.245.2.1.10.0">ſimiliter oſtendetur, quò lon<lb></lb>gior ſit LI, adhuc pondus facilius ſemper moueri. </s> <s id="id.2.1.245.2.1.12.0">à minori enim <arrow.to.target n="note334"></arrow.to.target><lb></lb>potentia moueretur. </s> <s id="id.2.1.245.2.1.13.0">quod erat propoſitum. </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"></emph>Cor.<emph.end type="italics"></emph.end></s> <s id="id.2.1.246.1.1.2.0"><margin.target id="note333"></margin.target>1 <emph type="italics"></emph>huius de vecte.<emph.end type="italics"></emph.end></s> <s id="id.2.1.246.1.1.3.0"><margin.target id="note334"></margin.target><emph type="italics"></emph>Ex<emph.end type="italics"></emph.end> 1 <emph type="italics"></emph>huius de vecte.<emph.end type="italics"></emph.end></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 manifeſtum eſt, quò enim longior <lb></lb>eſt LI, eò tempus maius erit: dummodo potentiæ motuum ſint <lb></lb>in velocitate æquales; ſicuti dictum eſt 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 manifeſtum eſt. </s> <s id="id.2.1.247.3.1.2.0">quò plures ſunt heli<lb></lb>ces; & quò longiores ſunt ſcytalæ, ſiue manu<lb></lb>bria, pondus ipſum facilius quidem, tardius au<lb></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 ſcytalis con<lb></lb>ſtitutæ potentiæ, hinc manifeſta fiet. <pb xlink:href="036/01/270.jpg"></pb> <figure id="id.036.01.270.1.jpg" place="text" 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; ſit planum horizonti inclinatum CD in <lb></lb>angulo DCE. </s> <s id="N17346">inueniatur ex eadem nona Pappi quanta vi pondus <lb></lb>A ſuper CD mouetur; quæ ſit decem. </s> <s id="id.2.1.247.5.1.2.0">exponatur cochlea LM <lb></lb>helices habens GHIK &c. in angulo ECD; per ea, quæ dicta <lb></lb>ſunt, potentia decem pondus A ſuper helices GHIk mouebit. </s> <s id="id.2.1.247.5.1.4.0">ſi <lb></lb>autem hac cochlea volumus pondus A mouere, & potentia mo<lb></lb>uens ſit vt duo. </s> <s id="id.2.1.247.5.1.5.0">ducatur NP axi cochleæ perpendicularis, axem <lb></lb>ſecans in O; fiatq; PO ad ON, vt vnum ad quinq; hoc eſt duo ad <lb></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, ideſt <lb></lb>ſuper helices eſt vt decem, cui potentiæ reſiſtit, & æqualis eſt po<lb></lb>tentia in N vt duo; eſt enim NP vectis, cuius fulcimentum eſt <lb></lb>O. </s> <s id="N1736D">potentia ergo vt duo in N pondus A ſuper helices cochleæ <lb></lb>mouebit. </s> <s id="id.2.1.247.5.1.7.0">efficiantur igitur ſcytalæ, ſiue manubria, quæ vſq; ad N <pb n="129" xlink:href="036/01/271.jpg"></pb>perueniant; manifeſtum eſt, potentiam vt duo in his pondus cen<lb></lb>tum cochlea <emph type="italics"></emph>L<emph.end type="italics"></emph.end>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"></emph>Ex<emph.end type="italics"></emph.end> 1 <emph type="italics"></emph>huius de vecte.<emph.end type="italics"></emph.end></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 ſit cochlea QR helices habens in angulo DCE, & cir<lb></lb>ca ipſam ſit eius mater S, quæ ſi pependerit centum, adiiciatur ST <lb></lb>manubrium quoddam, ſiue ſcytala; ita vt T in eadem proportio<lb></lb>ne diſtet ab axe cylindri, vt NOP; patet potentiam vt duo in T <lb></lb>mouere S ſuper helices cochleæ. </s> <s id="id.2.1.249.1.1.2.0">nihil enim aliud eſt S, niſi pon<lb></lb>dus ſuper helices cochleæ motum. </s> <s id="id.2.1.249.1.1.3.0">ſimiliter ſi S ſit immobilis, cir<lb></lb>cumuertaturq; cochlea manubrio, ſiue ſcytala QX in eadem pro<lb></lb>portione conſecta; fueritq; cochlea centum pondo (quòd qui<lb></lb>dem, vel ex ſe ipſa, vel cum pondere V cochleæ appenſo, vel cum <lb></lb>pondere Y cochleæ ſuper impoſito centum pependerit) manife<lb></lb>ſtum eſt potentiam vt duo in X mouere cochleam QR ſuper he<lb></lb>lices intra matricem cochleæ inciſas. </s> <s id="id.2.1.249.1.1.4.0">atq; ita in aliis, quæ cochleæ <lb></lb>inſtrumento mouentur; proportionem potentiæ ad pondus inue<lb></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 manifeſtum eſt, quomodo datum pon<lb></lb>dus à data potentia cochlea moueatur. <pb xlink:href="036/01/272.jpg"></pb> <figure id="id.036.01.272.1.jpg" place="text" 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 obſeruandum occurrit; quò plu<lb></lb>res erunt matricis cochleæ helices, eò minus in pondere mouen<lb></lb>do cochleam pati. </s> <s id="id.2.1.249.4.1.2.0">ſi enim matrix vnicam duntaxat helicen poſſe<lb></lb>derit, tunc pondus vt centrum à ſola cochleæ ſuſtinebitur helice; <lb></lb>ſi verò plures, in plures quoque, ac totidem cochleæ heli<lb></lb>ces ponderis grauitas diſtribuetur; vt ſi quatuor contineat helices, <lb></lb>tunc quatuor viciſsim cochleæ helices vniuerſo ponderi ſuſtinendo <lb></lb>incumbent; ſiquidem vnaquæquè quartam totius ponderis portio<lb></lb>nem ſuſtentabit. </s> <s id="id.2.1.249.4.1.3.0">quòd ſi adhuc plures contineat helices, ponderis <lb></lb>quoq; totius in plures, atque ideo minores portiones fiet diſtri<lb></lb>butio. </s> </p> <pb n="130" xlink:href="036/01/273.jpg"></pb> <p id="id.2.1.249.6.0.0.0" type="main"> <s id="id.2.1.249.6.1.1.0">Oſtenſum eſt igitur pondus à cochlea moueri <lb></lb>tamquam à cuneo percuſsionis experte: loco e<lb></lb>nim percuſsionis mouet vecte, hoc eſt ſcytala, ſi<lb></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 demonſtratis liquet, quomodo <expan abbr="datũ">datum</expan> pon<lb></lb>dus à data potentia moueri poſsit. </s> <s id="id.2.1.249.7.1.2.0">quòd ſi vecte <lb></lb>hoc aſſequi volumus; poſſumus & dato vecte da <lb></lb>tum pondus data potentia mouere. </s> <s id="id.2.1.249.7.1.3.0">quod quidem <lb></lb>in nullis ex aliis fieri poſſe abſolutè contingit: ſiue <lb></lb>ſit cochlea, ſiue axis in peritrochio, ſiue trochlea. </s> <s id="id.2.1.249.7.1.4.0"><lb></lb>non enim datis trochleis, neq; dato axe in peri<lb></lb>trochio, neq; data cochlea, datum pondus à data <lb></lb>potentia moueri poteſt, cùm potentia in his ſem<lb></lb>per ſit determinata: ſi igitur <expan abbr="potẽtia">potentia</expan>, quæ pondus <lb></lb>mouere debeat, hac minor ſit data, nunquam pon<lb></lb>dus mouebit. </s> <s id="id.2.1.249.7.1.5.0">poſſumus tamen dato axe, & tympa<lb></lb>no abſq; ſcytalis datum pondus data <expan abbr="potẽtia">potentia</expan> mo<lb></lb>uere; cùm ſcytalas conſtruere poſsimus, ita vt ſe<lb></lb>midiameter tympani dati vná cum longitudine <lb></lb>ſcytalæ ad axis ſemidiametrum <expan abbr="datã">datam</expan> habeat pro<lb></lb>portionem. </s> <s id="id.2.1.249.7.1.6.0">quod idem cochleæ contingere po<lb></lb>teſt, ſcilicet datum pondus data cochlea ſine ma<lb></lb>nubrio, vel ſcytala, data potentia mouere. </s> <s id="id.2.1.249.7.1.7.0">co<lb></lb>gnita enim potentia, quæ pondus ſuper helices <lb></lb>moueat, poſſumus manubrium, ſiue ſcytalam ita <pb xlink:href="036/01/274.jpg"></pb>conſtruere, vt data potentia in ſcytala eandem <lb></lb>vim habeat, quam potentia pondus ſuper helices <lb></lb>mouens cùm autem hoc datis trochleis nullo mo <lb></lb>do fieri poſsit. </s> <s id="id.2.1.249.7.1.8.0">datum tamen pondus data poten<lb></lb>tia trochleis infinitis modis mouere poſſumus. </s> <s id="id.2.1.249.7.1.9.0"><lb></lb>datum verò pondus data potentia cunei inſtru<lb></lb>mento mouere, hoc minimè fieri poſſe clarum eſ<lb></lb>ſe videtur; non enim data potentia datum pon<lb></lb>dus ſuper planum horizonti inclinatum mouere <lb></lb>poteſt, neq; datum pondus à data potentia moue<lb></lb>bitur vectibus ſibi <expan abbr="inuicẽ">inuicem</expan> aduerſis, quemmadmo<lb></lb>dum in cuneo inſunt; cùm in vectibus cunei pro<lb></lb>pria, veraq; vectis proportio ſeruari non poſsit. </s> <s id="id.2.1.249.7.1.10.0"><lb></lb>vectium enim fulcimenta non ſunt immobilia, <lb></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 ſtruere machinas, atq; eas <lb></lb>ex pluribus componere; vt ex trochleis, & ſuc<lb></lb>culis, vel ergatis, pluribuſuè dentatis tympanis, <lb></lb>uel quocunq; alio modo; & ex ijs, quæ diximus; fa<lb></lb>cilè inter pondus, & potentiam proportionem <lb></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"></pb> </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></lb>ſunt, 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"></emph>Pagina<emph.end type="italics"></emph.end> 2, <emph type="italics"></emph>b, verſu<emph.end type="italics"></emph.end> 19, <emph type="italics"></emph>AEBD<emph.end type="italics"></emph.end> ¶ 5, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 6, <emph type="italics"></emph>ipſi<emph.end type="italics"></emph.end> ¶ 7, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 9, <emph type="italics"></emph>ODH<emph.end type="italics"></emph.end> ¶ 9, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 19, <emph type="italics"></emph> <expan abbr="cõtingit">contingit</expan><emph.end type="italics"></emph.end><lb></lb>¶ 15, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 24, <emph type="italics"></emph>grauius<emph.end type="italics"></emph.end> ¶ 16, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 30, <emph type="italics"></emph>recto<emph.end type="italics"></emph.end> ¶ 21, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 26, <emph type="italics"></emph>ſuſtineatur<emph.end type="italics"></emph.end> ¶ 23, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 8, <emph type="italics"></emph>BD DC<emph.end type="italics"></emph.end> ¶ 31, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, <lb></lb>9, <emph type="italics"></emph>totum GK<emph.end type="italics"></emph.end> ¶ 34, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 24, <emph type="italics"></emph>pondera FG<emph.end type="italics"></emph.end> ¶ 38, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 27, <emph type="italics"></emph>maior AF<emph.end type="italics"></emph.end> ¶ 39, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 24 <emph type="italics"></emph>AB in D<emph.end type="italics"></emph.end> ¶ 40, <lb></lb><emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 1, <emph type="italics"></emph>ad BD<emph.end type="italics"></emph.end> ¶ 44, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 24, <emph type="italics"></emph>graui<emph.end type="italics"></emph.end> ¶ 48, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 7, <emph type="italics"></emph>ipſi AD<emph.end type="italics"></emph.end> ¶ 50, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 12 <emph type="italics"></emph>pondus<emph.end type="italics"></emph.end> ¶ 54, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 7, <emph type="italics"></emph>quàm<emph.end type="italics"></emph.end> ¶ 61, <lb></lb><emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 6, <emph type="italics"></emph>præterquam in E<emph.end type="italics"></emph.end> ¶ 65, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 33, <emph type="italics"></emph>quam<emph.end type="italics"></emph.end> ¶ 81, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 1, <emph type="italics"></emph>ligato<emph.end type="italics"></emph.end> ¶ 85, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 22, <emph type="italics"></emph>vtriq;<emph.end type="italics"></emph.end> ¶ 97, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 14, <lb></lb><emph type="italics"></emph>dextrorſum<emph.end type="italics"></emph.end> ¶ 98, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 20, <emph type="italics"></emph>Hic<emph.end type="italics"></emph.end> ¶ 110, <emph type="italics"></emph>b, in poſtill.</s> <s id="id.2.1.249.12.1.2.0">Lemma in <expan abbr="primã">primam</expan><emph.end type="italics"></emph.end> ¶ 122, <emph type="italics"></emph>a<emph.end type="italics"></emph.end>, 8, <emph type="italics"></emph>&<emph.end type="italics"></emph.end> 17, <emph type="italics"></emph>helicen<emph.end type="italics"></emph.end><lb></lb>¶ 123, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 15, <emph type="italics"></emph>ventes in GH<emph.end type="italics"></emph.end> ¶ 124, <emph type="italics"></emph>b<emph.end type="italics"></emph.end>, 17, <emph type="italics"></emph>manifeſtum<emph.end type="italics"></emph.end> ¶ 127, <emph type="italics"></emph>a, in poſtil.</s> <s id="id.2.1.249.12.1.3.0">Monteregio<emph.end type="italics"></emph.end><lb></lb>¶ 127, <emph type="italics"></emph>b, in poſtil.</s> <s id="id.2.1.249.12.1.4.0">ex Cor.<emph.end type="italics"></emph.end></s> <lb></lb> <s id="id.2.1.249.12.3.1.0">REGISTRVM. </s> <lb></lb> <s id="id.2.1.249.12.5.1.0"><12><12><12> ABCDEFGHIKLMNOPQRSTVX <lb></lb>YZ, Aa Bb Cc Dd Ee Ff Gg Hh Ii Kk. </s> <lb></lb> <s id="id.2.1.249.12.7.1.0">Omnes duerni. </s> <lb></lb> <s id="id.2.1.249.12.9.1.0">PISAVRI </s> <lb></lb> <s id="id.2.1.249.12.11.1.0">Apud Hieronymum Concordiam. </s> <lb></lb> <s id="id.2.1.249.12.13.1.0">M. D. LXXVII. </s> </p> </section> </back> </text> </archimedes>