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Colored diff for /texts/archimedes/xml/Attic/monte_mecha_01_la_1577.xml between version 1.63 and 1.64

version 1.63, 2003/08/26 16:17:51

version 1.64, 2003/08/27 06:14:38

Line 47

<s id="id.2.1.1.12.1.2.0">illæ ad exornandam mecha<lb/>nicam facultatem, & eam præ om<lb/>nibus alijs appetibilem reddendam con&longs;pira&longs;&longs;e <lb/>mihi videntur: nam &longs;i nobilitatem (quod pleriq; <lb/>modò faciunt) ortu ip&longs;o metimur, occurret hinc <lb/>Geometria, illinc verò Phi&longs;ica; quorum gemina<lb/>to complexu nobili&longs;&longs;ima artium prodit mechani<lb/>ca. </s>

<s id="id.2.1.1.12.1.3.0">&longs;i enim nobilitatem magis, tùm &longs;tratæ materiæ, <lb/>tùm argumentorum nece&longs;&longs;itati (quod Ari&longs;tote<lb/>les fatetur aliquandò) relatam volumus, omnium <lb/>procul dubiò nobili&longs;&longs;imam per&longs;piciemus. </s>

<s id="id.2.1.1.12.1.4.0">quæ <pb xlink:href="036/01/004.jpg"/>quidem non &longs;olum geometriam (vt Pappus te&longs;ta<lb/>tur) ab&longs;oluit, & perficit; verùm etiam & phi&longs;ica<lb/>rum rerum imperium habet: quandoquidem <lb/>quodcunq; Fabris, Architectis, Baiulis, Agricolis, <lb/>Nautis, & quàm plurimis alijs (repugnantibus na<lb/>turæ legibus) opitulatur; id omne mechanicum <lb/>e&longs;t imperium. </s>

<s id="id.2.1.1.12.1.5.0">quippè quod aduer&longs;us naturam <lb/>vel eiu&longs;dem emulata leges exercet; &longs;umma id <lb/>certè admiratione dignum; veri&longs;&longs;imum tamen, <lb/>& à quocun&que; liberaliter admi&longs;&longs;um, qui pri<lb/>us ab Ari&longs;totele didicerit, omnia mechanica, <lb/>tùm problemata, tùm theoremata ad rotundam <lb/>machinam reduci, atq; ideo illo niti principio, <lb/><expan abbr="nõ">non</expan> minus &longs;en&longs;ui, quàm rationi noto. </s>

<s id="id.2.1.1.12.1.5.0">quippè quod aduer&longs;us naturam <lb/>vel eiu&longs;dem emulata leges exercet; &longs;umma id <lb/>certè admiratione dignum; veri&longs;&longs;imum tamen, <lb/>& à quocunque liberaliter admi&longs;&longs;um, qui pri<lb/>us ab Ari&longs;totele didicerit, omnia mechanica, <lb/>tùm problemata, tùm theoremata ad rotundam <lb/>machinam reduci, atq; ideo illo niti principio, <lb/><expan abbr="nõ">non</expan> minus &longs;en&longs;ui, quàm rationi noto. </s>

<s id="id.2.1.1.12.1.6.0">Rotunda ma<lb/>china e&longs;t mouenti&longs;&longs;ima, & quò maior, eò mouen<lb/>tior. </s>

<s id="id.2.1.1.12.1.7.0">Verùm huic nobilitati adnexa e&longs;t &longs;umma re <lb/>rum ad vitam pertinentium vtilitas, quæ propte<lb/>rea omnes alias à diuer&longs;is artibus propagatas an<lb/>tecellit; quòd aliæ facultates po&longs;t mundi gene&longs;im <lb/>longa temporis intercapedine &longs;uos explicarunt <lb/>v&longs;us; i&longs;ta verò & in ip&longs;is mundi primordijs ita fuit <lb/>hominibus nece&longs;&longs;aria, vt ea &longs;ublata Sol de mun<lb/>do &longs;ublatus videretur. </s>

<s id="id.2.1.1.12.1.8.0">nam quacunq; nece&longs;&longs;ita<lb/>te Adæ vita degeretur; & quamuis etiam ca&longs;is <lb/>contectis &longs;tramine, & angu&longs;tis tugurijs, ac gurgu<lb/>&longs;tijs c&oelig;li de fenderet iniurias; &longs;ic & in corporis ve<lb/>&longs;titu, licet ip&longs;e nihil aliud &longs;pectaret, ni&longs;i vt imbres, <pb xlink:href="036/01/005.jpg"/>vt niues, vt ventos; vt Solem, vt frigus arceret; <lb/>quodcun&que; tamen id fuit, omne mechanicum <lb/>fuit. </s>

<s id="id.2.1.1.12.1.9.0">neq; tamen huic facultati contingit, quod <lb/>ventis &longs;olet, qui cùm vndè oriuntur, ibi vehe<lb/>menti&longs;&longs;imi &longs;int, ad longinqua tamen fracti, de<lb/>bilitatiquè perueniunt: &longs;ed quod magnis flumini<lb/>bus crebriu&longs; accidit, quæ cùm in ip&longs;o ortu parua <lb/>&longs;int, perpetuò tamen aucta, eò ampliori ferun<lb/>tur alueo, quò à fontibus &longs;uis longius rece&longs;&longs;e<lb/>runt. </s>

<s id="id.2.1.1.12.1.10.0">Nam & temporis progre&longs;&longs;u mechanica fa <lb/>cultas &longs;ub iugo æquum arationis laborem di<lb/>&longs;pen&longs;are, at&que; aratrum agris circumagere cæ<lb/>pit. </s>

<s id="id.2.1.1.12.1.10.0">Nam & temporis progre&longs;&longs;u mechanica fa <lb/>cultas &longs;ub iugo æquum arationis laborem di<lb/>&longs;pen&longs;are, atque aratrum agris circumagere cæ<lb/>pit. </s>

<s id="id.2.1.1.12.1.11.0">deinceps bigis, & quadrigis docuit comea<lb/>tus, merces, onera quælibet vehere, è finibus <lb/>no&longs;tri&longs; ad finitimos populos exportare, & ex il<lb/>lis contra importare ad nos. </s>

<s id="id.2.1.1.12.1.12.0">præterea cùm iam <lb/>res non tantùm nece&longs;&longs;itate, verùm etiam orna<lb/>tu, & commoditate metirentur, mechanicæ <lb/>fuit &longs;ubtilitatis, quòd nauigia remo impellere<lb/>mus; quòd gubernaculo exiguo in extrema pup<lb/>pi collocato ingentes triremium moles inflecte<lb/>remus; quòd vnius &longs;æpè manu pro multis fabro<lb/>rum manibus modò pondera lapidum, & tra<lb/>bium Fabris, & Architectis &longs;ubleuaremus; mo<lb/>dò tollenonis &longs;pecie aquas è puteis olitoribus e<lb/>xhauriremus. </s>

<s id="id.2.1.1.12.1.13.0">hinc etiam è liquidorum prælis vi<lb/>na, olea, vnguenta expre&longs;&longs;a, & quicquid liquo<pb xlink:href="036/01/006.jpg"/>ris habent, per&longs;oluere domino compul&longs;a. </s>

<s id="id.2.1.1.12.1.14.0">hinc <lb/>magnas <expan abbr="arbor&utilde;">arborum</expan>, & marmorum moles duobus in <lb/>contrarias partes <expan abbr="di&longs;trah&etilde;tibus">di&longs;trahentibus</expan> vectibus diremp<lb/>&longs;imus; hinc militiæ in aggeribus extruendis, in <lb/>con&longs;erenda manu, in opugnando, propugnan<lb/>doq; loca infinitæ ferè redundarunt vtilitates; <lb/>hinc demum Lignatores, Lapicidæ, Marmorarij <lb/>Vinitores, Olearij, Vnguentarij, Ferrarij, Auri<lb/>fices, Metallici, Chirurgi, Ton&longs;ores, Pi&longs;tores, Sar<lb/>tores, omnes deniq; opifices beneficiarij, tot, tan<lb/>taq; vitæ humanæ &longs;uppeditarunt commoda. </s>

<s id="id.2.1.1.12.1.15.0">Eant <lb/>nunc noui logodedali quidam mechanicorum <lb/>contemptores, perfricent frontem, &longs;i quam ha<lb/>bent, & ignobilitatem, atquè inutilitatem fal&longs;ò <lb/>criminari de&longs;inant: quòd &longs;i & adhuc id minimè <lb/>velint, eos quæ&longs;o in in&longs;citia &longs;ua relinquamus: <lb/>Ari&longs;totelemquè potius philo&longs;ophorum cory<lb/>phæum imitemur, cuius mechanici amoris ardo<lb/>rem acuti&longs;&longs;imæ illæ mechanicæ quæ&longs;tiones po&longs;te <lb/>ris traditæ &longs;atis declarant: qua quidem laude <lb/>Platonem magnificè &longs;uperauit; qui (vt te&longs;tatur <lb/>Plutarcus) Architam, & Eudoxum mechanicæ <lb/>vtilitatem impen&longs;ius colentes ab in&longs;tituto deter<lb/>ruit; quòd nobili&longs;&longs;imam philo&longs;ophorum po&longs;&longs;e&longs;<lb/>&longs;ionem in vulgus indicarent, ac publicarent; & <lb/>velut arcana philo&longs;ophiæ my&longs;teria proderent. </s>

<s id="id.2.1.1.12.1.16.0"><lb/>res &longs;anè meo quidem iudicio pro&longs;us vituperan<pb xlink:href="036/01/007.jpg"/>da, ni&longs;i fortè velimus tam nobilis di&longs;ciplinæ con<lb/>templationem quidem ocio&longs;am laudare; fructum <lb/>verò, & v&longs;um, arti&longs;q; finem improbare. </s>

<s id="id.2.1.1.12.1.17.0">&longs;ed præ <lb/>omnibus mathematicis vnus Archimedes ore <lb/>laudandus e&longs;t pleniore, &que;m voluit Deus in me<lb/>chanicis velut ideam &longs;ingularem e&longs;&longs;e, quam om<lb/>nes earum &longs;tudio&longs;i ad imitandum &longs;ibi propone<lb/>rent. </s>

<s id="id.2.1.1.12.1.17.0">&longs;ed præ <lb/>omnibus mathematicis vnus Archimedes ore <lb/>laudandus e&longs;t pleniore, quem voluit Deus in me<lb/>chanicis velut ideam &longs;ingularem e&longs;&longs;e, quam om<lb/>nes earum &longs;tudio&longs;i ad imitandum &longs;ibi propone<lb/>rent. </s>

<s id="id.2.1.1.12.1.18.0">is enim C&oelig;le&longs;tem globum exiguo admo<lb/>dum, fragili què vitreo orbe conclu&longs;um ita efin<lb/>xit, &longs;imulatis a&longs;tris viuum naturæ opus, ac iura <lb/>poli motibus certis adeò præ &longs;e ferentibus; vt <lb/>æmula naturæ manus tale de &longs;e encomium &longs;it <lb/>promerita: &longs;ic manus naturam, vt natura ma<lb/>num ip&longs;a immitata putetur. </s>

<s id="id.2.1.1.12.1.19.0">is poli&longs;pa&longs;tu manu <lb/>leua, & &longs;ola, quinquies millenum modiorum <lb/>pondus attraxit. </s>

<s id="id.2.1.1.12.1.20.0">nauem in &longs;iccum litus eductam, <lb/>ac grauius oneratam &longs;olus machinis &longs;uis ad &longs;e <lb/>perindè pertraxit, ac &longs;i in mari remis, veli&longs;uè <lb/>impul&longs;a moueretur, <expan abbr="quã">quam</expan> & po&longs;tea in litore (quod <lb/>omnes Siciliæ vires non potuerunt) in mare de<lb/>duxit. </s>

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<s id="id.2.1.1.12.1.24.0">quod tamen non modò nos <lb/>vecte tantùm fieri potui&longs;&longs;e in præ&longs;enti libro doce<lb/>mus; verùm etiam, & omnis antiquitas (quod <lb/>multis forta&longs;&longs;è mirabile videbitur) id penitus <lb/>credidi&longs;&longs;e mihi videtur; quæ Neptuno tri<lb/>dentem tanquam vectem attribuit; cuius ope <lb/>terræ concu&longs;&longs;or vbiq; nuncupatur à poetis. </s>

<s id="id.2.1.1.12.1.25.0">ad <lb/>quod etiam a&longs;piciens celeberrimus no&longs;ter poeta <lb/>Neptunum inducit i&longs;ta machina &longs;yrtes, quò ma<lb/>gis apparerent Troianis, &longs;ubleuantem. </s>

<s id="id.2.1.1.13.1.1.0">“Leuat ip&longs;e tridenti <lb/>& va&longs;tas aperit &longs;yrtes.” </s>

<s id="id.2.1.1.14.1.1.0">Mechanici præterea fuerunt Heron, Cte&longs;ibius, <lb/>& Pappus, qui licet ad mechanicæ apicem, perin<lb/>de atq; Archimedes, euecti forta&longs;&longs;è minimè &longs;int; <lb/>mechanicam tamen facultatem egregiè percal<lb/>luerunt; tale&longs;q; fuerunt, & præ&longs;ertim Pappus, vt <lb/>eum me ducem &longs;e&que;ntem nemo (vt opinor) cul<lb/>pauerit. </s>

<s id="id.2.1.1.14.1.2.0">quod & propterea libentius feci, quòd <lb/>nè latum quidem vnguem ab Archimedeis prin<lb/>cipijs Pappus recedat. </s>

<s id="id.2.1.1.14.1.3.0">ego enim in hac præ&longs;ertim <lb/>facultate Archimedis ve&longs;tigijs hærere &longs;emper vo <lb/>lui: & licet eius lucubrationes ad <expan abbr="mechanicã">mechanicam</expan> per<pb xlink:href="036/01/009.jpg"/>tinentes multis ab hinc annis pa&longs;&longs;im &longs;oleant do<lb/>ctis de&longs;iderari: eruditi&longs;&longs;imus tamen libellus de æ<lb/>&que;ponderantibus præ manibus <expan abbr="homin&utilde;">hominum</expan> adhuc <lb/>ver&longs;atur, in quò tanquam in copio&longs;i&longs;&longs;ima p&oelig;nu <lb/>omnia ferè mechanica dogmata repo&longs;ita mihi vi<lb/>dentur; &que;m &longs;anè libellum, &longs;i ætatis no&longs;træ mathe<lb/>matici &longs;ibi magis familiarem adhibui&longs;&longs;ent; reperi&longs;<lb/>&longs;ent &longs;anè <expan abbr="&longs;ent&etilde;tias">&longs;ententias</expan> multas, quas modó ip&longs;i firmas, <lb/>& ratas e&longs;&longs;e docent; &longs;ubtili&longs;&longs;imè, atquè veri&longs;<lb/>&longs;imè conuul&longs;as, & labefactatas. </s>

<s id="id.2.1.1.14.1.3.0">ego enim in hac præ&longs;ertim <lb/>facultate Archimedis ve&longs;tigijs hærere &longs;emper vo <lb/>lui: & licet eius lucubrationes ad <expan abbr="mechanicã">mechanicam</expan> per<pb xlink:href="036/01/009.jpg"/>tinentes multis ab hinc annis pa&longs;&longs;im &longs;oleant do<lb/>ctis de&longs;iderari: eruditi&longs;&longs;imus tamen libellus de æ<lb/>queponderantibus præ manibus <expan abbr="homin&utilde;">hominum</expan> adhuc <lb/>ver&longs;atur, in quò tanquam in copio&longs;i&longs;&longs;ima p&oelig;nu <lb/>omnia ferè mechanica dogmata repo&longs;ita mihi vi<lb/>dentur; quem &longs;anè libellum, &longs;i ætatis no&longs;træ mathe<lb/>matici &longs;ibi magis familiarem adhibui&longs;&longs;ent; reperi&longs;<lb/>&longs;ent &longs;anè <expan abbr="&longs;ent&etilde;tias">&longs;ententias</expan> multas, quas modó ip&longs;i firmas, <lb/>& ratas e&longs;&longs;e docent; &longs;ubtili&longs;&longs;imè, atquè veri&longs;<lb/>&longs;imè conuul&longs;as, & labefactatas. </s>

<s id="id.2.1.1.14.1.4.0">&longs;ed hoc vi<lb/>derint ip&longs;i. </s>

<s id="id.2.1.1.14.1.5.0">ego enim ad Pappum redeo, qui <lb/>ad v&longs;um mathematicarum vberiorem, emulu<lb/>mentorumquè acce&longs;&longs;iones amplificandas peni<lb/>tus conuer&longs;us, de quin&que; principibus machi<lb/>nis, Vecte nempè, Trochlea, Axe in peri<lb/>trochio, Cuneo, & Cochlea, multa egre<lb/>giè philo&longs;ophatus e&longs;t; demon&longs;trauit què quicquid <lb/>in machinis, aut cogitari peritè, aut acutè <lb/>definiri, aut certò &longs;tatui pote&longs;t, id omne quin<lb/>què illis infinita vi præditis machinis referen<lb/>dum e&longs;&longs;e. </s>

<s id="id.2.1.1.14.1.5.0">ego enim ad Pappum redeo, qui <lb/>ad v&longs;um mathematicarum vberiorem, emulu<lb/>mentorumquè acce&longs;&longs;iones amplificandas peni<lb/>tus conuer&longs;us, de quinque principibus machi<lb/>nis, Vecte nempè, Trochlea, Axe in peri<lb/>trochio, Cuneo, & Cochlea, multa egre<lb/>giè philo&longs;ophatus e&longs;t; demon&longs;trauit què quicquid <lb/>in machinis, aut cogitari peritè, aut acutè <lb/>definiri, aut certò &longs;tatui pote&longs;t, id omne quin<lb/>què illis infinita vi præditis machinis referen<lb/>dum e&longs;&longs;e. </s>

<s id="id.2.1.1.14.1.6.0">atquè vtinam iniuria temporis ni<lb/>hil è tanti viri &longs;criptis abra&longs;i&longs;&longs;et: nec enim tam <lb/>den&longs;a in&longs;citiæ caligo vniuer&longs;um propè terra<lb/>rum orbem obtexi&longs;&longs;et, ne&que; tanta mechani<lb/>cæ facultatis e&longs;&longs;et ignoratio con&longs;ecuta, vt ma<lb/>thematicarum proceres exi&longs;timarentur illi, qui <lb/>modò inepti&longs;&longs;ima quadam di&longs;tinctione, diffi<pb xlink:href="036/01/010.jpg"/>cultates nonnullas, nec illas tamen &longs;atis ar<lb/>duas, & ob&longs;curas è medio tollunt. </s>

<s id="id.2.1.1.14.1.6.0">atquè vtinam iniuria temporis ni<lb/>hil è tanti viri &longs;criptis abra&longs;i&longs;&longs;et: nec enim tam <lb/>den&longs;a in&longs;citiæ caligo vniuer&longs;um propè terra<lb/>rum orbem obtexi&longs;&longs;et, neque tanta mechani<lb/>cæ facultatis e&longs;&longs;et ignoratio con&longs;ecuta, vt ma<lb/>thematicarum proceres exi&longs;timarentur illi, qui <lb/>modò inepti&longs;&longs;ima quadam di&longs;tinctione, diffi<pb xlink:href="036/01/010.jpg"/>cultates nonnullas, nec illas tamen &longs;atis ar<lb/>duas, & ob&longs;curas è medio tollunt. </s>

<s id="id.2.1.1.14.1.7.0">reperiun<lb/>tur enim aliqui, no&longs;traq; ætate emunctæ naris <lb/>mathematici, qui mechanicam, tùm mathe<lb/>maticè &longs;eor&longs;um, tùm phi&longs;icè con&longs;iderari po&longs;<lb/>&longs;e affirmant; ac &longs;i aliquando, vel &longs;ine demon<lb/>&longs;trationibus geometricis, vel &longs;ine vero motu <lb/>res mechanicæ con&longs;iderari po&longs;&longs;int: qua &longs;anè di<lb/>&longs;tinctione (vt leuius cum illis agam) nihil aliud mi<lb/>hi commini&longs;ci videntur, quàm vt dum &longs;e, tùm <lb/>phi&longs;icos, tùm mathematicos proferant, vtra<lb/>&que; (quod aiunt) &longs;ella excludantur. </s>

<s id="id.2.1.1.14.1.8.0">nequè <lb/>enim amplius mechanica, &longs;i à machinis ab&longs;tra<lb/>hatur, & &longs;eiungatur, mechanica pote&longs;t appel<lb/>lari. </s>

<s id="id.2.1.1.14.1.9.0">Emicuit tamen inter i&longs;tas tenebras (quam<lb/>uis alij quoquè nonnulli fuerint præclari&longs;&longs;imi) <lb/>Solis in&longs;tar Federicus Commandinus, qui multis <lb/>docti&longs;&longs;imis elucubrationibus ami&longs;&longs;um mathema<lb/>ticarum patrimonium non modò re&longs;taurauit, <lb/>verùm etiam auctiùs, & locupletiùs effecit. </s>

<s id="id.2.1.1.14.1.10.0"><lb/>erat enim &longs;ummus i&longs;te vir omnibus adeò facul<lb/>tatibus mathematicis ornatus, vt in eo Archi<lb/>tas, Eudoxus, Heron, Euclides, Theon, Ari<lb/>&longs;tarcus, Diophantus, Theodo&longs;ius, Ptolemæus <lb/>Apollonius, Serenus, Pappus, quin & ip<lb/>&longs;emet Archimedes (&longs;iquidem ip&longs;ius in Archi<lb/>medem &longs;cripta Archimedis olent lucernam) re <pb xlink:href="036/01/011.jpg"/>uixi&longs;&longs;e viderentur. </s>

Line 83

<s id="id.2.1.1.14.1.12.0">Ille tamen perpetuò in alia<lb/>rum mathematicarum explicationem ver&longs;ans, <lb/>mechanicam facultatem, aut penitus præter<lb/>mi&longs;it, aut modicè attigit. </s>

<s id="id.2.1.1.14.1.13.0">Quapropter in hoc <lb/>&longs;tudium ardentiùs ego incumbere cæpi, nec me <lb/>vnquam per omne mathematum genus vagan<lb/>tem ea &longs;olicitudo de&longs;eruit; ecquid ex vno <lb/>quoquè decerpi, ac delibari po&longs;&longs;it; quo ad me<lb/>chanicam expoliendam, & exornandam acco<lb/>modatior e&longs;&longs;e po&longs;&longs;em. </s>

<s id="id.2.1.1.14.1.14.0">Nunc verò cùm mihi <lb/>videar, noni ea quidem omnia, quæ ad mecha<lb/>nicam pertinent, perfeci&longs;&longs;e; &longs;ed eò v&longs;q; tamen <lb/>progre&longs;&longs;us, vt ijs, qui ex Pappo, ex Vitruuio, <lb/>& ex alijs didicerint, quid &longs;it Vectis, quid Tro<lb/>chlea, quid Axis in peritrochio, quid Cuneus, <lb/>quid Cochlea; quomodoq; vt pondera moueri <lb/>po&longs;&longs;int, aptari debeant; adhuc tamen acciden<lb/>tia permulta, quæ inter potentiam, & pondus <lb/>vectis virtute illis in&longs;unt in&longs;trumentis, perdi&longs;ce<lb/>re cupiunt, opis aliquid adferre po&longs;&longs;im; putaui <lb/>tempus iam po&longs;tulare, vt prodirem; & nauatæ <pb xlink:href="036/01/012.jpg"/>in hoc genere operæ &longs;pecimen aliquod darem. </s>

<s id="id.2.1.1.14.1.15.0"><lb/>Verùm quò facilius totius operis &longs;ub&longs;tructio <lb/>ad fa&longs;tigium &longs;uum per duceretur, nonnulla quo<lb/>què de libra fuerunt pertractanda, & præ&longs;er<lb/>tim dum vnico pondere alterum &longs;olum ip&longs;ius <lb/>brachium penitus deprimitur: &que; in re mi<lb/>rum e&longs;t quantas fecerint ruinas Iordanus (qui <lb/>inter recentiores maximæ fuit auctoritatis) & <lb/>alij; qui hanc rem &longs;ibi di&longs;cutiendam propo&longs;ue<lb/>runt. </s>

<s id="id.2.1.1.14.1.15.0"><lb/>Verùm quò facilius totius operis &longs;ub&longs;tructio <lb/>ad fa&longs;tigium &longs;uum per duceretur, nonnulla quo<lb/>què de libra fuerunt pertractanda, & præ&longs;er<lb/>tim dum vnico pondere alterum &longs;olum ip&longs;ius <lb/>brachium penitus deprimitur: que in re mi<lb/>rum e&longs;t quantas fecerint ruinas Iordanus (qui <lb/>inter recentiores maximæ fuit auctoritatis) & <lb/>alij; qui hanc rem &longs;ibi di&longs;cutiendam propo&longs;ue<lb/>runt. </s>

<s id="id.2.1.1.14.1.16.0">opus &longs;anè arduum, & for&longs;an viribus no<lb/>&longs;tris impar aggre&longs;si &longs;umus; in eo tamen digni, vt <lb/>no&longs;tros conatus, & indu&longs;triam ad præclara ten<lb/>dentem bonorum omnium perpetuus applau<lb/>&longs;us, approbatioq; comitetur; quòd ad &longs;tudium <lb/>tàm illu&longs;tre, tam magnificum, tam laudabile <lb/>contulimus quicquid habuimus virium. </s>

<s id="id.2.1.1.14.1.17.0">quod <lb/>&longs;anè qualecunq; &longs;it, tibi celeberrime PRINCEPS <lb/>nuncupandum cen&longs;uimus; cuius &longs;anè con&longs;ilij, <lb/>atq; in&longs;tituti no&longs;tri rationes multas reddere in <lb/>promptu e&longs;t: & primùm hæreditaria tibi in fa<lb/>miliam no&longs;tram promerita, quibus nos ita de<lb/>uictos habes; vt facilè intelligamus ad fortunas <lb/>non modò no&longs;tras, verùm & ad &longs;anguinem, & <lb/>vitam quoq; pro tua dignitate propendendam <lb/>parati&longs;&longs;imos e&longs;&longs;e debere. </s>

<s id="id.2.1.1.14.1.18.0">Præterea illud non <lb/>parui quoq; ponderis accedit, quòd à pueri<lb/>tia literarum omnium, &longs;ed præcipuè mathe<pb xlink:href="036/01/013.jpg"/>maticarum de&longs;iderio ita fueris incen&longs;us, vt ni<lb/>&longs;i illis adeptis vitam tibi acerbam, atq; in&longs;ua<lb/>uem &longs;tatueres. </s>

Line 100

<s id="id.2.1.1.18.1.1.0">Hanc centri grauitatis definitionem Pappus Alexandrinus in <lb/>octauo Mathematicarum collectionum libro tradidit. </s>

<s id="id.2.1.1.18.1.2.0">Federicus <lb/>verò Commandinus in libro de centro grauitatis &longs;olidorum idem <lb/>centrum de&longs;cribendo ita explicauit. </s>

<s id="id.2.1.1.19.1.1.0">Centrum grauitatis vniu&longs;cuiu&longs;q; &longs;olidæ figu<lb/>ræ e&longs;t punctum illud intra po&longs;itum, circa quod <lb/>vndiq; partes æqualium momentorum con&longs;i<lb/>&longs;tunt. </s>

<s id="id.2.1.1.19.1.2.0">&longs;i enim per tale centrum ducatur planum <lb/>figuram quomodocunq; &longs;ecans &longs;emper in par<lb/>tes æ&que;ponderantes ip&longs;am diuidet. </s><pb xlink:href="036/01/016.jpg"/>

<s id="id.2.1.1.19.1.2.0">&longs;i enim per tale centrum ducatur planum <lb/>figuram quomodocunq; &longs;ecans &longs;emper in par<lb/>tes æqueponderantes ip&longs;am diuidet. </s><pb xlink:href="036/01/016.jpg"/>

<s id="id.2.1.1.21.1.1.0">COMMVNES NOTIONES. </s>

<s id="id.2.1.1.22.1.1.0">Si ab æ&que;ponderantibus æ&que;ponderantia au<lb/>ferantur, reliqua æ&que;ponderabunt. </s>

<s id="id.2.1.1.22.1.1.0">Si ab æqueponderantibus æqueponderantia au<lb/>ferantur, reliqua æqueponderabunt. </s>

<s id="id.2.1.1.24.1.1.0">Si æ&que;ponderantibus æ&que;ponderantia adii<lb/>ciantur, tota &longs;imul æ&que;ponderabunt. </s>

<s id="id.2.1.1.24.1.1.0">Si æqueponderantibus æqueponderantia adii<lb/>ciantur, tota &longs;imul æqueponderabunt. </s>

<s id="id.2.1.1.26.1.1.0">Quæ eidem æ&que;ponderant, inter &longs;e æquè &longs;unt <lb/>grauia. </s>

<s id="id.2.1.1.26.1.1.0">Quæ eidem æqueponderant, inter &longs;e æquè &longs;unt <lb/>grauia. </s>

<s id="id.2.1.1.27.1.1.0">SVPPOSITIONES. </s>

<s id="id.2.1.1.28.1.1.0">Vnius corporis vnum tantùm e&longs;t centrum gra<lb/>uitatis. </s>

Line 122

<s id="id.2.1.1.34.1.1.0">DE LIBRA. </s>

<s id="id.2.1.1.35.1.1.0">Anteqvam de libra &longs;ermo ha<lb/>beatur, vtres clarior eluce&longs;cat, &longs;it <lb/>libra AB recta linea; CD verò <lb/>trutina, quæ &longs;ecundum commu<lb/>nem con&longs;uetudinem horizonti <lb/>&longs;emper e&longs;t perpendicularis. </s>

<s id="id.2.1.1.35.1.2.0">pun<lb/>ctum autem C immobile, circa quod vertitur li<lb/>bra, centrum libræ <lb/>vocetur. </s>

<s id="id.2.1.1.35.1.3.0">itidem&que; <lb/>(quamuis tamen im<lb/>proprie) &longs;iue &longs;upra, <lb/>&longs;iue infra libram fue<lb/>rit con&longs;titutum. </s>

<s id="id.2.1.1.35.1.3.0">itidemque <lb/>(quamuis tamen im<lb/>proprie) &longs;iue &longs;upra, <lb/>&longs;iue infra libram fue<lb/>rit con&longs;titutum. </s>

<s id="id.2.1.1.35.1.4.0">CA <lb/>verò, & CB, tum di<lb/>&longs;tantiæ, tum libræ <lb/>brachia nuncupen<lb/>tur. </s>

<s id="id.2.1.1.35.1.5.0">& &longs;i à centro li<lb/>bræ &longs;upra, vel infra <lb/><figure id="id.036.01.017.1.jpg" place="text" xlink:href="036/01/017/1.jpg"/><lb/>libram con&longs;tituto ip&longs;i AB perpendicularis duca<lb/>tur, hæc perpendiculum vocetur, quæ libram AB <lb/>&longs;ub&longs;tinebit; & quocun&que; modo moueatur libra, <lb/>ip&longs;i &longs;emper perpendicularis exi&longs;tet. </s><pb xlink:href="036/01/018.jpg"/>

<s id="id.2.1.1.35.1.5.0">& &longs;i à centro li<lb/>bræ &longs;upra, vel infra <lb/><figure id="id.036.01.017.1.jpg" place="text" xlink:href="036/01/017/1.jpg"/><lb/>libram con&longs;tituto ip&longs;i AB perpendicularis duca<lb/>tur, hæc perpendiculum vocetur, quæ libram AB <lb/>&longs;ub&longs;tinebit; & quocunque modo moueatur libra, <lb/>ip&longs;i &longs;emper perpendicularis exi&longs;tet. </s><pb xlink:href="036/01/018.jpg"/>

<s id="id.2.1.1.37.1.1.0">LEMMA. </s>

<s id="id.2.1.1.38.1.1.0">Sit linea AB horizonti perpendicularis, & dia <lb/>metro AB circulus de&longs;cribatur AEBD, cuius <lb/>centrum C. </s>

Line 176

<s id="id.2.1.5.4.1.6.0"><lb/>Quando autem CG erit <lb/>in CD, linea EF, cùm <lb/>ip&longs;i CG &longs;emper ad rectos <lb/>&longs;it angulos, erit in AB; in <lb/><arrow.to.target n="note5"/>quo &longs;itu quoq; manebit. </s>

<s id="id.2.1.5.4.1.7.0">li<lb/>bra ergo EF in AB hori<lb/>zonti <expan abbr="æquidi&longs;tãtem">æquidi&longs;tantem</expan> redi<lb/>bit, ibíq; manebit. </s>

<s id="id.2.1.5.4.1.8.0">quod <lb/>demon&longs;trare oportebat. </s>

<s id="id.2.1.6.1.1.1.0"><margin.target id="note3"/>4. <emph type="italics"/>primi Archi<lb/>medis de <lb/>æ&que;ponde<lb/>rantibus.<emph.end type="italics"/></s>

<s id="id.2.1.6.1.1.1.0"><margin.target id="note3"/>4. <emph type="italics"/>primi Archi<lb/>medis de <lb/>æqueponde<lb/>rantibus.<emph.end type="italics"/></s>

<s id="id.2.1.6.1.1.2.0"><margin.target id="note4"/>1. <emph type="italics"/>Huius<emph.end type="italics"/></s>

<s id="id.2.1.6.1.1.3.0"><margin.target id="note5"/>1. <emph type="italics"/>Huius.<emph.end type="italics"/></s>

Line 193

<s id="id.2.1.7.3.1.6.0">& punctum G magnitudi<lb/>nis ex EF compo&longs;itæ centrum grauitatis erit; quod dum moue<lb/>tur, circuli circumferentiam de&longs;cribet DGH, cuius &longs;emidiameter <lb/>CD, & centrum C. </s>

<s id="id.2.1.7.3.1.6.0.a">Quoniam autem CG horizonti non e&longs;t per<lb/>pendicularis, magnitudo ex EF ponderibus compo&longs;ita in hoc &longs;i<lb/>tu minimè manebit; &longs;ed &longs;ecundùm eius grauitatis centrum G deor<lb/>&longs;um per circumferentiam GH mouebitur. </s>

<s id="id.2.1.7.3.1.7.0">libra ergo EF ex par <lb/>te F deor&longs;um mouebitur, quod demon&longs;trare oportebat. </s>

<s id="id.2.1.8.1.1.1.0"><margin.target id="note6"/>4. <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.8.1.1.1.0"><margin.target id="note6"/>4. <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.8.1.1.3.0"><margin.target id="note7"/>1. <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.9.1.1.1.0">PROPOSITIO IIII. </s>

<s id="id.2.1.9.2.1.1.0">Libra horizonti æquidi&longs;tans æqualia in ex<lb/>tremitatibus, æqualiterq; à centro in ip&longs;a libra <lb/>collocato, di&longs;tantia habens pondera; &longs;iue inde <lb/>moueatur, &longs;iue minus; vbicunq; relicta, manebit. <figure id="id.036.01.023.1.jpg" place="text" xlink:href="036/01/023/1.jpg"/></s>

Line 204

<s id="id.2.1.9.3.1.4.0">Quoniam autem centrum libræ <pb xlink:href="036/01/024.jpg"/>C, dum libra AB vnà <lb/>cum ponderibus in DE <lb/>mouetur, immobile re<lb/>manet, centrum quoq; <lb/>grauitatis, quod e&longs;t idem <lb/>C, non mouebitur. </s>

<s id="id.2.1.9.3.1.5.0">nec <lb/>igitur libra DE mouebi<lb/>tur, per definitionem <lb/>centri grauitatis, cum in <lb/>ip&longs;o &longs;u&longs;pendatur. </s>

<s id="id.2.1.9.3.1.6.0">Idip<lb/><figure id="id.036.01.024.1.jpg" place="text" xlink:href="036/01/024/1.jpg"/><lb/>&longs;um quoq; contingit libra in AB horizonti æquidi&longs;tante, vel in <lb/>quocunq; alio &longs;itu exi&longs;tente. </s>

<s id="id.2.1.9.3.1.7.0">Manebit ergo libra, vbi relin&que;<lb/>tur. </s>

<s id="id.2.1.9.3.1.7.0">Manebit ergo libra, vbi relinque<lb/>tur. </s>

<s id="id.2.1.9.3.1.8.0">quod demon&longs;trare oportebat. </s>

<s id="id.2.1.9.4.1.1.0">Cum verò in iis, quæ dicta &longs;unt, grauitatis tantùm magnitudi<lb/>num, quæ in extremitatibus libræ po&longs;itæ &longs;unt æquales, ab&longs;q; lí<lb/>bræ grauitate con&longs;iderauerimus; quoniam tamen adhuc libræ bra<lb/>chia &longs;unt æqualia, idcirco idem libræ, eius grauitate con&longs;iderata, <lb/>vnà cum ponderibus, vel &longs;ine ponderibus eueniet. </s>

<s id="id.2.1.9.4.1.2.0">idem enim cen<lb/>trum grauitatis fine ponderibus libræ tantùm grauitatis centrum <lb/>erit. </s>

Line 214

<s id="id.2.1.9.5.1.1.0"><arrow.to.target n="note8"/>Quoniam autem huic determinationi vltimæ multa à nonnullis <lb/>aliter &longs;entientibus dicta officere videntur; idcirco in hac parte ali<lb/><arrow.to.target n="note9"/>quantulum immorari oportebit; & pro viribus, non &longs;olum pro<lb/>priam &longs;ententiam, &longs;ed Archimedem ip&longs;um, qui in hac eadem e&longs;&longs;e <lb/><arrow.to.target n="note10"/>&longs;ententia videtur, defendere conabor. <pb n="6" xlink:href="036/01/025.jpg"/>

<s id="id.2.1.9.6.1.1.0">Ii&longs;dem po&longs;itis, duca<lb/>tur FCG ip&longs;i AB, & <lb/>horizonti perpendicula<lb/>ris; & centro C, &longs;patio<lb/>què CA, circulus de&longs;cri<lb/>batur ADFBEG. erunt <lb/>puncta ADBE in circu<lb/>li circumferentia; cum li<lb/>bræ brachia &longs;int æqualia. </s>

<s id="id.2.1.9.6.1.2.0"><lb/>& quoniam in vnam con<lb/>ueniunt &longs;ententiam, a&longs;&longs;e<lb/>rentes &longs;cilicet libram DE <lb/>neq; in FG moueri, ne<lb/>&que; in DE manere, &longs;ed in AB horizonti æquidi&longs;tantem rediré. </s>

<s id="id.2.1.9.6.1.2.0"><lb/>& quoniam in vnam con<lb/>ueniunt &longs;ententiam, a&longs;&longs;e<lb/>rentes &longs;cilicet libram DE <lb/>neq; in FG moueri, ne<lb/>que in DE manere, &longs;ed in AB horizonti æquidi&longs;tantem rediré. </s>

<s id="id.2.1.9.6.1.3.0"><lb/>hanc eorum &longs;ententiam nullo modo con&longs;i&longs;tere po&longs;&longs;e o&longs;tendam. </s>

<s id="id.2.1.9.6.1.4.0"><lb/>Non enim, &longs;ed &longs;i quod aiunt, euenerit, vel ideo erit, quia pondus <lb/>D pondere E grauius fuerit, vel &longs;i pondera &longs;unt æqualia, di&longs;tantiæ, <lb/>quibus &longs;unt po&longs;ita, non erunt æquales, hoc e&longs;t CD ip&longs;i CE non erit <lb/>æqualis, &longs;ed maior. </s>

<s id="id.2.1.9.6.1.5.0">Quòd autem pondera in DE &longs;int æqualia, & <lb/>di&longs;tantia CD &longs;it æqualis di&longs;tantiæ CE: hæc ex &longs;uppo&longs;itione pa<lb/>tent. </s>

<s id="id.2.1.9.6.1.6.0">Sed quoniam dicunt pondus in D in eo &longs;itu pondere in E <lb/>grauius e&longs;&longs;e in altero &longs;itu deor&longs;um: dum pondera &longs;unt in DE, pun<lb/>ctum C non erit amplius centrum grauitatis, nam non manent, &longs;i <lb/>ex C &longs;u&longs;pendantur; &longs;ed erit in linea CD, ex tertia primi Archi<lb/>medis de æ&que;ponderantibus. </s>

<s id="id.2.1.9.6.1.7.0">non autem erit in linea CE, cum pon<lb/>dus D grauius &longs;it pondere E. &longs;it igitur in H, in quo &longs;i &longs;u&longs;pendan<lb/>tur, manebunt. </s>

<s id="id.2.1.9.6.1.8.0">Quoniam autem centrum grauitatis ponderum <lb/>in AB connexorum e&longs;t punctum C; ponderum verò in DE e&longs;t <lb/>punctum H: dum igitur pondera AB mouentur in DE, centrum <lb/>grauitatis C ver&longs;us D mouebitur, & ad D propius accedet; quod <lb/>e&longs;t impo&longs;sibile: cum pondera eandem inter &longs;e &longs;e &longs;eruent di&longs;tantiam. </s>

<s id="id.2.1.9.6.1.9.0"><lb/>Vniu&longs;cuiu&longs;q; enim corporis centrum grauitatis in eodem &longs;emper <arrow.to.target n="note11"/><lb/>e&longs;t &longs;itu re&longs;pectu &longs;ui corporis. </s>

Line 227

<s id="id.2.1.9.6.1.12.0">non igitur <lb/>pondus in D pondere in <lb/>E e&longs;t grauius. </s>

<s id="id.2.1.9.6.1.13.0">Si autem <lb/>dicerent centrum graui<lb/>tatis non in linea CD, <lb/>&longs;ed in CE e&longs;&longs;e debere; <lb/>idem eueniet ab&longs;urdum. <figure id="id.036.01.026.1.jpg" place="text" xlink:href="036/01/026/1.jpg"/></s>

<s id="id.2.1.9.7.1.1.0">Amplius &longs;i pondus D <lb/>deor&longs;um mouebitur, pondus E &longs;ur&longs;um mouebit. </s>

<s id="id.2.1.9.7.1.2.0">pondus igitur gra<lb/>uius, quàm &longs;it E, in eodemmet &longs;itu ponderi D æ&que;ponderabit, & <lb/>grauia inæqualia æquali di&longs;tantia po&longs;ita æ&que;ponderabunt. </s>

<s id="id.2.1.9.7.1.2.0">pondus igitur gra<lb/>uius, quàm &longs;it E, in eodemmet &longs;itu ponderi D æqueponderabit, & <lb/>grauia inæqualia æquali di&longs;tantia po&longs;ita æqueponderabunt. </s>

<s id="id.2.1.9.7.1.3.0">Adii<lb/>ciatur ergo ponderi E aliquod graue, ita vt ip&longs;i D contraponde<lb/>ret, &longs;i ex C &longs;u&longs;pendantur. </s>

<s id="id.2.1.9.7.1.4.0">&longs;ed cum &longs;upra o&longs;ten&longs;um &longs;it punctum C <lb/>centrum e&longs;&longs;e grauitatis æqualium ponderum in DE; &longs;i igitur pon<lb/><arrow.to.target n="note13"/>dus E grauius fuerit pondere D, erit centrum grauitatis in linea <lb/>CE. </s>

<s id="id.2.1.9.7.1.4.0.a">&longs;itq; hoc centrum K. </s>

<s id="id.2.1.9.7.1.4.0.b">at per definitionem centri grauitatis, &longs;i <lb/>pondera &longs;u&longs;pendantur ex K, manebunt. </s>

<s id="id.2.1.9.7.1.5.0">ergo &longs;i &longs;u&longs;pendantur ex <lb/>C, non manebunt, quod e&longs;t contra hypote&longs;im: &longs;ed pondus E deor<lb/>&longs;um mouebitur. </s>

<s id="id.2.1.9.7.1.6.0">quòd &longs;i ex C quo&que; &longs;u&longs;pen&longs;a æ&que;ponderarent; <lb/><arrow.to.target n="note14"/>vnius magnitudinis duo e&longs;&longs;ent centra grauitatis; quod e&longs;t impo&longs;si<lb/>bile. </s>

<s id="id.2.1.9.7.1.6.0">quòd &longs;i ex C quoque &longs;u&longs;pen&longs;a æqueponderarent; <lb/><arrow.to.target n="note14"/>vnius magnitudinis duo e&longs;&longs;ent centra grauitatis; quod e&longs;t impo&longs;si<lb/>bile. </s>

<s id="id.2.1.9.7.1.7.0">Non igitur pondus in E grauius eo, quod e&longs;t in D, ip&longs;i D æ&que;<lb/>ponderabit, cum ex puncto C fiat &longs;u&longs;pen&longs;io. </s>

<s id="id.2.1.9.7.1.7.0">Non igitur pondus in E grauius eo, quod e&longs;t in D, ip&longs;i D æque<lb/>ponderabit, cum ex puncto C fiat &longs;u&longs;pen&longs;io. </s>

<s id="id.2.1.9.7.1.8.0">Pondera ergo in DE <lb/>æqualia ex eorum grauitatis centro C &longs;u&longs;pen&longs;a, æ&que;ponderabunt, <lb/>manebuntquè. </s>

<s id="id.2.1.9.7.1.8.0">Pondera ergo in DE <lb/>æqualia ex eorum grauitatis centro C &longs;u&longs;pen&longs;a, æqueponderabunt, <lb/>manebuntquè. </s>

<s id="id.2.1.9.7.1.9.0">quod demon&longs;trare fuerat propo&longs;itum. </s>

<s id="id.2.1.10.1.1.1.0"><margin.target id="note8"/><emph type="italics"/>Iordanus de Ponderibus. <emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.2.0"><margin.target id="note9"/><emph type="italics"/>Hyerommus Cardanus de &longs;ubtilitate. <emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.3.0"><margin.target id="note10"/><emph type="italics"/>Nicolaus Tartalea de quæ&longs;itis, ac inuentionibus. <emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.4.0"><margin.target id="note11"/>2. <emph type="italics"/>Sup. huius. <emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.6.0"><margin.target id="note12"/><emph type="italics"/>Ex<emph.end type="italics"/> 4. <emph type="italics"/>primi Archim de Ae&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.6.0"><margin.target id="note12"/><emph type="italics"/>Ex<emph.end type="italics"/> 4. <emph type="italics"/>primi Archim de Aequep.<emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.7.0"><margin.target id="note13"/><emph type="italics"/>Ex<emph.end type="italics"/> 3. <emph type="italics"/>primi Archim de Ae&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.7.0"><margin.target id="note13"/><emph type="italics"/>Ex<emph.end type="italics"/> 3. <emph type="italics"/>primi Archim de Aequep.<emph.end type="italics"/></s>

<s id="id.2.1.10.1.1.8.0"><margin.target id="note14"/>1. <emph type="italics"/>Suppo&longs;. huius.<emph.end type="italics"/></s>

<s id="id.2.1.11.1.1.1.0"><arrow.to.target n="note15"/>Huic autem po&longs;tremo inconuenienti occurrunt dicentes, im<lb/>po&longs;sibile e&longs;&longs;e addere ip&longs;i E pondus adeo minimum, quin adhuc &longs;i <lb/>ex C &longs;u&longs;pendantur, pondus E &longs;emper deor&longs;um ver&longs;us G moueatur. </s>

<s id="id.2.1.11.1.1.2.0"><lb/>quod nos fieri po&longs;&longs;e &longs;uppo&longs;uimus, at&que; fieri po&longs;&longs;e credebamus. </s>

<s id="id.2.1.11.1.1.2.0"><lb/>quod nos fieri po&longs;&longs;e &longs;uppo&longs;uimus, atque fieri po&longs;&longs;e credebamus. </s>

<s id="id.2.1.11.1.1.3.0">ex<lb/>ce&longs;&longs;um enim ponderis D &longs;upra pondus E, cum quantitatis ratio<lb/>nem habeat, non &longs;olum minimum e&longs;&longs;e, verum in infinitum diuidi <lb/>po&longs;&longs;e immaginabamur, quod quidem ip&longs;i, non &longs;olum minimum, <pb n="7" xlink:href="036/01/027.jpg"/>&longs;ed ne minimum quidem e&longs;&longs;e, cum reperiri non po&longs;sit, hoc mo<lb/>do demon&longs;trare nituntur. <figure id="id.036.01.027.1.jpg" place="text" xlink:href="036/01/027/1.jpg"/></s>

<s id="id.2.1.11.2.1.1.0">Exponantur eadem. </s>

<s id="id.2.1.11.2.1.2.0"><lb/>à puncti&longs;què DE hori<lb/>zonti <expan abbr="perp&etilde;diculares">perpendiculares</expan> du<lb/><expan abbr="cãtur">cantur</expan> DHEK, atq; alius <lb/>&longs;it circulus LDM, cu<lb/>ius <expan abbr="centr&utilde;">centrum</expan> N, qui FDG <lb/>in puncto D contingat, <lb/>ip&longs;iq; FDG &longs;it æqualis: <lb/>erit NC recta linea. </s>

Line 262

<s id="N10A25">dabitur ergo quoquè proportio mi<lb/>nor minima, quam in infinitum adhuc minorem ita o&longs;tende<lb/>mus. </s>

<s id="id.2.1.11.2.1.10.0">De&longs;cribatur circulus DR, cuius centrum E, & &longs;emidiame<lb/><arrow.to.target n="note20"/>ter ED. continget circumferentia DR circumferentiam DG in <lb/><arrow.to.target n="note21"/>puncto D, lineamquè DO in puncto D; quare minor erit angu<lb/>lus RDG angulo ODG. &longs;imiliter & angulus RDH angulo <lb/>ODH. </s>

<s id="id.2.1.11.2.1.10.0.a">minorem igitur proportionem habebit RDH ad HDG, <lb/>quàm ODH ad HDG. </s>

<s id="id.2.1.11.2.1.10.0.b">Accipiatur deinde inter EC vtcun<lb/>&que; punctum P, ex quo in di&longs;tantia PD alia de&longs;cribatur circum<lb/>ferentia DQ, quæ circumferentiam DR, circumferentiamquè <lb/>DG in puncto D continget; & angulus QDH minor erit <lb/>angulo RDH: ergo QDH ad HDG minorem habebit propor<lb/>tionem, quàm RDH ad HDG. </s>

<s id="id.2.1.11.2.1.10.0.b">Accipiatur deinde inter EC vtcun<lb/>que punctum P, ex quo in di&longs;tantia PD alia de&longs;cribatur circum<lb/>ferentia DQ, quæ circumferentiam DR, circumferentiamquè <lb/>DG in puncto D continget; & angulus QDH minor erit <lb/>angulo RDH: ergo QDH ad HDG minorem habebit propor<lb/>tionem, quàm RDH ad HDG. </s>

<s id="N10A4E">eodemquè pror&longs;us modo, &longs;i <lb/>inter PC aliud accipiatur punctum, & inter hoc &C aliud, & &longs;ic <lb/>deinceps, infinitæ de&longs;cribentur circumferentiæ inter DO, & cir<lb/>cumferentiam DG; ex quibus proportionem in infinitum &longs;emper <lb/>minorem inueniemus. </s>

<s id="id.2.1.11.2.1.11.0">at&que; ideo proportionem ponderis in D <lb/>ad pondus in E non adeo minorem e&longs;&longs;e &longs;equitur, quin ad infini <lb/>tum ip&longs;a &longs;emper minorem reperiri po&longs;sit. </s>

<s id="id.2.1.11.2.1.11.0">atque ideo proportionem ponderis in D <lb/>ad pondus in E non adeo minorem e&longs;&longs;e &longs;equitur, quin ad infini <lb/>tum ip&longs;a &longs;emper minorem reperiri po&longs;sit. </s>

<s id="id.2.1.11.2.1.12.0">& quia angulus MDG <lb/>in infinitum diuidi pote&longs;t; exce&longs;&longs;us quo&que; grauitatis D &longs;upra E <lb/>diuidi ad infinitum poterit. </s>

<s id="id.2.1.11.2.1.12.0">& quia angulus MDG <lb/>in infinitum diuidi pote&longs;t; exce&longs;&longs;us quoque grauitatis D &longs;upra E <lb/>diuidi ad infinitum poterit. </s>

<s id="id.2.1.12.1.1.1.0"><margin.target id="note15"/><emph type="italics"/>Tartalea &longs;exta propo&longs;itione octaui libri.<emph.end type="italics"/></s>

<s id="id.2.1.12.1.1.2.0"><margin.target id="note16"/><emph type="italics"/>Ex<emph.end type="italics"/> 12. <emph type="italics"/>tertii.<emph.end type="italics"/></s>

<s id="id.2.1.12.1.1.3.0"><margin.target id="note17"/>29. <emph type="italics"/>Primi.<emph.end type="italics"/></s>

Line 274

<s id="id.2.1.12.1.1.6.0"><margin.target id="note20"/><emph type="italics"/>Ex<emph.end type="italics"/> 11. <emph type="italics"/>tertit.<emph.end type="italics"/></s>

<s id="id.2.1.12.1.1.7.0"><margin.target id="note21"/><emph type="italics"/>Ex<emph.end type="italics"/> 18. <emph type="italics"/>tertii.<emph.end type="italics"/></s><pb n="8" xlink:href="036/01/029.jpg"/>

<s id="id.2.1.13.1.2.1.0">Sed ne&que; prætereundum <lb/>e&longs;t, ip&longs;os in demon&longs;tratio<lb/>ne angulum KEG maiorem <lb/>e&longs;&longs;e angulo HDG, tanquam <lb/>notum accepi&longs;&longs;e. </s>

<s id="id.2.1.13.1.2.1.0">Sed neque prætereundum <lb/>e&longs;t, ip&longs;os in demon&longs;tratio<lb/>ne angulum KEG maiorem <lb/>e&longs;&longs;e angulo HDG, tanquam <lb/>notum accepi&longs;&longs;e. </s>

<s id="id.2.1.13.1.2.2.0">quod e&longs;t <lb/>quidem verum, &longs;i DHEK <lb/>inter &longs;e &longs;e &longs;int æquidi&longs;tan<lb/>tes. </s>

<s id="id.2.1.13.1.2.3.0">Quoniam autem (vt <lb/>ip&longs;i quo&que; &longs;upponunt) li<lb/>neæ DHEK in centrum <lb/>mundi conueniunt; lineæ <lb/>DHEK æquidi&longs;tantes nun<lb/>quam erunt, & angulus KEG <lb/>angulo HDG non &longs;olum <lb/>maior erit, &longs;ed minor. </s>

<s id="id.2.1.13.1.2.3.0">Quoniam autem (vt <lb/>ip&longs;i quoque &longs;upponunt) li<lb/>neæ DHEK in centrum <lb/>mundi conueniunt; lineæ <lb/>DHEK æquidi&longs;tantes nun<lb/>quam erunt, & angulus KEG <lb/>angulo HDG non &longs;olum <lb/>maior erit, &longs;ed minor. </s>

<s id="id.2.1.13.1.2.4.0">vt <lb/>exempli gratia, producatur <lb/>FG v&longs;&que; ad centrum mun<lb/>di, quod &longs;it S; connectan<lb/>tur&queacute; DSES. </s>

<s id="id.2.1.13.1.2.4.0">vt <lb/>exempli gratia, producatur <lb/>FG v&longs;que ad centrum mun<lb/>di, quod &longs;it S; connectan<lb/>tur&queacute; DSES. </s>

<s id="N10AF9">o&longs;tenden<lb/>dum e&longs;t angulum SEG mi<lb/>norem e&longs;&longs;e angulo SDG. </s>

<s id="id.2.1.13.1.2.4.0.a">du<lb/><figure id="id.036.01.029.1.jpg" place="text" xlink:href="036/01/029/1.jpg"/><lb/>catur à puncto E linea ET circulum DGEF contingens, ab eo <lb/>dem&queacute; puncto ip&longs;i DS æquidi&longs;tans ducatur EV. </s>

<s id="id.2.1.13.1.2.4.0.b">Quoniam igi<lb/>tur EVDS inter &longs;e &longs;e &longs;unt æquidi&longs;tantes: &longs;imiliter ETDO æqui <lb/>di&longs;tantes: erit angulus VET angulo SDO æqualis. </s>

Line 286

<s id="id.2.1.13.1.2.6.0.a"><lb/>Quare ex ip&longs;orum &longs;uppo&longs;itionibus non &longs;olum pondus in D gra<lb/>uius erit pondere in E; verùm è conuer&longs;o, pondus in E ip&longs;o D <lb/>grauius exi&longs;tet. </s><pb xlink:href="036/01/030.jpg"/>

<s id="id.2.1.13.3.1.1.0">Rationes tamen af<lb/>ferunt, quibus demon<lb/>&longs;trare nituntur, libram <lb/>DE in AB horizon<lb/>ti æquidi&longs;tantem ex <lb/>nece&longs;sitate redire. </s>

<s id="id.2.1.13.3.1.2.0"><expan abbr="Primùm">Pri<lb/>mum</expan> quidem o&longs;ten<lb/>dunt, idem pondus <lb/>grauius e&longs;&longs;e in A, <lb/>quàm in alio &longs;itu, &que;m <lb/>æqualitatis &longs;itum no<lb/>minant, cum linea <lb/>AB &longs;it horizonti æ<lb/><figure id="id.036.01.030.1.jpg" place="text" xlink:href="036/01/030/1.jpg"/><lb/>quidi&longs;tans. </s>

<s id="id.2.1.13.3.1.2.0"><expan abbr="Primùm">Pri<lb/>mum</expan> quidem o&longs;ten<lb/>dunt, idem pondus <lb/>grauius e&longs;&longs;e in A, <lb/>quàm in alio &longs;itu, quem <lb/>æqualitatis &longs;itum no<lb/>minant, cum linea <lb/>AB &longs;it horizonti æ<lb/><figure id="id.036.01.030.1.jpg" place="text" xlink:href="036/01/030/1.jpg"/><lb/>quidi&longs;tans. </s>

<s id="id.2.1.13.3.1.3.0">deinde quò propius e&longs;t ip&longs;i A, quouis alio remotiori <lb/>grauius e&longs;&longs;e. </s>

<s id="id.2.1.13.3.1.4.0">Vt pondus in A grauius e&longs;&longs;e, quàm in D; & in D, <lb/>quàm in L. &longs;imiliter in A grauius, quam in N; & in N grauius, <lb/>quàm in M. </s>

<s id="id.2.1.13.3.1.4.0.a">Vnum tantùm con&longs;iderando pondus in altero libræ <lb/><arrow.to.target n="note22"/>brachio &longs;ur&longs;um deor&longs;umq; moto. </s>

Line 294

<s id="N10B77">ductis enim DO LP ip&longs;i CF perpendicula<lb/><arrow.to.target n="note23"/>ribus, linea AC maior e&longs;t, quàm DO, & DO ip&longs;a LP. </s>

<s id="N10B7E">quod <lb/><arrow.to.target n="note24"/>idem euenit in punctis NM. </s>

<s id="id.2.1.13.3.1.5.0.a">deinde ex quo loco (aiunt) pon<lb/>dus velocius mouetur, ibi grauius e&longs;t; velocius autem ex A, quàm <lb/>ab alio &longs;itu mouetur; ergo in A grauius e&longs;t. </s>

<s id="id.2.1.13.3.1.6.0">&longs;imili modo, quò <lb/>propius e&longs;t ip&longs;i A, velocius quo&que; mouetur; ergo in D gra<lb/><arrow.to.target n="note25"/>uius erit, quàm in L. </s>

<s id="id.2.1.13.3.1.6.0">&longs;imili modo, quò <lb/>propius e&longs;t ip&longs;i A, velocius quoque mouetur; ergo in D gra<lb/><arrow.to.target n="note25"/>uius erit, quàm in L. </s>

<s id="id.2.1.13.3.1.6.0.a">Altera deinde cau&longs;a, quam ex rectiori, & obli<lb/><arrow.to.target n="note26"/>quiori motu deducunt, e&longs;t; quò pondus in arcubus æqualibus re<lb/>ctius de&longs;cendit, grauius e&longs;&longs;e videtur; cum pondus liberum, atq; <lb/><arrow.to.target n="note27"/>&longs;olutum &longs;uaptè natura rectè moueatur; &longs;ed in A rectius de&longs;cen<lb/>dit; ergo in A grauius erit. </s>

<s id="id.2.1.13.3.1.7.0">hocq; o&longs;tendunt accipiendo arcum <lb/>AN arcui LD æqualem; à puncti&longs;q; NL lineæ FG (quam <lb/>etiam directionis vocant) æquidi&longs;tantes ducantur NRLQ, quæ <lb/>lineas AB DO &longs;ecent in QR; & à puncto N ip&longs;i FG perpen<lb/>dicularis ducatur NT. </s>

<s id="id.2.1.13.3.1.7.0.a">rectèq; demon&longs;trant LQ ip&longs;i PO æqua<lb/>lem e&longs;&longs;e, & NR ip&longs;i CT; lineamq; NR ip&longs;a LQ maiorem e&longs;&longs;e. </s>

Line 341

<s id="id.2.1.17.5.1.4.0">tanget igitur in<lb/>fra, &longs;itq; SO. </s>

<s id="N10D32">connectantur deinde SD <lb/>SL, quæ circumferentiam AOG in <lb/>punctis KH &longs;ecent. </s>

<s id="id.2.1.17.5.1.5.0">& Ck CH con<lb/>iungantur. </s>

<s id="id.2.1.17.5.1.6.0">Et quoniam pondus, quanto <lb/>propius e&longs;t ip&longs;i F, magis quo&que; inni<lb/>titur centro; vt pondus in D magis ver<lb/>&longs;ionis puncto C innititur tanquam <lb/>centro; hoc e&longs;t in D magis &longs;upra li<lb/>neam CD grauitat, quàm &longs;i e&longs;&longs;et in A <lb/>&longs;upra lineam CA; & adhuc magis in <lb/>L &longs;upra lineam CL; Nam cùm tres <lb/>anguli cuiu&longs;cunq; trianguli duobus re<lb/><figure id="id.036.01.033.1.jpg" place="text" xlink:href="036/01/033/1.jpg"/><lb/>ctis &longs;int æquales, & trianguli DCk æquicruris angulus DCk <lb/>minor &longs;it angulo LCH æquicruris trianguli LCH: erunt reli<lb/>qui ad ba&longs;im &longs;cilicet CDk CkD &longs;imul &longs;umpti reliquis CLH <lb/>CHL maiores. </s>

<s id="id.2.1.17.5.1.6.0">Et quoniam pondus, quanto <lb/>propius e&longs;t ip&longs;i F, magis quoque inni<lb/>titur centro; vt pondus in D magis ver<lb/>&longs;ionis puncto C innititur tanquam <lb/>centro; hoc e&longs;t in D magis &longs;upra li<lb/>neam CD grauitat, quàm &longs;i e&longs;&longs;et in A <lb/>&longs;upra lineam CA; & adhuc magis in <lb/>L &longs;upra lineam CL; Nam cùm tres <lb/>anguli cuiu&longs;cunq; trianguli duobus re<lb/><figure id="id.036.01.033.1.jpg" place="text" xlink:href="036/01/033/1.jpg"/><lb/>ctis &longs;int æquales, & trianguli DCk æquicruris angulus DCk <lb/>minor &longs;it angulo LCH æquicruris trianguli LCH: erunt reli<lb/>qui ad ba&longs;im &longs;cilicet CDk CkD &longs;imul &longs;umpti reliquis CLH <lb/>CHL maiores. </s>

<s id="id.2.1.17.5.1.7.0">& horum dimidii; hoc e&longs;t angulus CDS angu<lb/>lo CLS maior erit. </s>

<s id="id.2.1.17.5.1.8.0">cùm itaq; CLS &longs;it minor, linea CL ma<lb/>gis adhærebit motui naturali ponderis in L pror&longs;us &longs;oluti. </s>

<s id="id.2.1.17.5.1.9.0">hoc <lb/>e&longs;t lineæ LS, quàm CD motui DS. </s>

Line 351

<s id="id.2.1.17.5.1.9.0.c">Eodem<lb/>&queacute; modo, quò pondus propius fuerit ip&longs;i F, magis ob hanc cau<lb/>&longs;am à linea CL &longs;u&longs;tineri o&longs;tendetur; &longs;emper enim angulus CLS <pb xlink:href="036/01/034.jpg"/>minor e&longs;&longs;et. </s>

<s id="id.2.1.17.5.1.10.0">quod etiam patet; quia &longs;i <lb/>lineæ CL, & LS in vnam coinciderent <lb/>lineam, quod euenit in FCS; tunc linea <lb/>CF totum &longs;u&longs;tineret pondus in F, im<lb/>mobilemq; redderet: neq; vllam pror<lb/>&longs;us grauitatem in circumferentia circu<lb/>li haberet. </s>

<s id="id.2.1.17.5.1.11.0">Idem ergo pondus propter <lb/>&longs;ituum diuer&longs;itatem grauius, leuiu&longs;q; erit. </s>

<s id="id.2.1.17.5.1.12.0"><lb/>non autem quia ratione &longs;itus interdum <lb/>maiorem re vera acquirat grauitatem, <lb/>interdum verò amittat, cùm eiu&longs;dem &longs;it <lb/>&longs;emper grauitatis, vbicun&que; reperiatur; <lb/>&longs;ed quia magis, minu&longs;uè in circumferen<lb/>tia grauitat, vt in D magis &longs;upra circum<lb/>ferentiam DA grauitat, quàm in L &longs;upra <lb/>circumferentiam LD. </s>

<s id="id.2.1.17.5.1.12.0"><lb/>non autem quia ratione &longs;itus interdum <lb/>maiorem re vera acquirat grauitatem, <lb/>interdum verò amittat, cùm eiu&longs;dem &longs;it <lb/>&longs;emper grauitatis, vbicunque reperiatur; <lb/>&longs;ed quia magis, minu&longs;uè in circumferen<lb/>tia grauitat, vt in D magis &longs;upra circum<lb/>ferentiam DA grauitat, quàm in L &longs;upra <lb/>circumferentiam LD. </s>

<s id="id.2.1.17.5.1.12.0.a">hoc e&longs;t, &longs;i pon<lb/>dus à circumferentiis, recti&longs;q; lineis &longs;u<lb/>&longs;tineatur; circumferentia AD magis &longs;u<lb/>&longs;tinebit pondus in D, quàm circumfe<lb/>rentia DL pondere exi&longs;tente in <emph type="italics"/>L.<emph.end type="italics"/> mi<lb/>nus enim coadiuuat CD, quàm CL. </s>

<s id="id.2.1.17.5.1.12.0.b"><lb/>Præterea quando pondus e&longs;t in L, &longs;i e&longs;<lb/><figure id="id.036.01.034.1.jpg" place="text" xlink:href="036/01/034/1.jpg"/><lb/>&longs;et omnino liberum, penitu&longs;q; &longs;olutum, deor&longs;um per LS moueretur; <lb/>ni&longs;i à linea CL prohiberetur, quæ pondus in L vltra lineam LS per <lb/><expan abbr="circumferentiã">circumferentiam</expan> LD moueri cogit; ip&longs;umq; quodammodo impellit, <lb/>impellendoq; pondus partim &longs;u&longs;tentabit. </s>

<s id="id.2.1.17.5.1.13.0">ni&longs;i enim &longs;u&longs;tineret, ip&longs;iq; <lb/>reniteretur, deor&longs;um per lineam LS moueretur, non autem per <lb/>circumferentiam LD. </s>

Line 359

<s id="id.2.1.17.5.1.14.0">eodemq; modo <lb/>exi&longs;tente pondere in A, linea CA pondus vltra lineam AS per <lb/>circumferentiam AO moueri compellet. </s>

<s id="id.2.1.17.5.1.15.0">e&longs;t enim angulus CAS <lb/>acutus; cùm angulus ACS &longs;it rectus. </s>

<s id="id.2.1.17.5.1.16.0">lineæ igitur CA CD ali<lb/>qua ex parte, non tamen ex æquo ponderi renituntur. </s>

<s id="id.2.1.17.5.1.17.0">& quotie&longs; <lb/>cun&que; angulus in circumferentia circuli à lineis à centro <lb/>mundi S, & centro C prodeuntibus, fuerit acutus; idem eue<lb/>nire &longs;imiliter o&longs;tendemus. </s>

<s id="id.2.1.17.5.1.17.0">& quotie&longs; <lb/>cunque angulus in circumferentia circuli à lineis à centro <lb/>mundi S, & centro C prodeuntibus, fuerit acutus; idem eue<lb/>nire &longs;imiliter o&longs;tendemus. </s>

<s id="id.2.1.17.5.1.18.0">Quoniam autem mixtus angulus CLD <pb n="11" xlink:href="036/01/035.jpg"/>æqualis e&longs;t angulo CDA, cùm à &longs;emidiametris, eademq; circumfe<lb/>rentia contineantur; & angulus C<emph type="italics"/>L<emph.end type="italics"/>S angulo CDS e&longs;t minor; <lb/>erit reliquus <emph type="italics"/>S<emph.end type="italics"/>LD reliquo SDA maior. </s>

<s id="id.2.1.17.5.1.19.0">quare circumferentia <lb/>DA, hoc e&longs;t de&longs;cen&longs;us ponderis in D propior erit motui natu<lb/>rali ponderis in D &longs;oluti, lineæ &longs;cilicet DS, quàm circumferen<lb/>tia LD lineæ LS. </s>

<s id="id.2.1.17.5.1.19.0.a">minus igitur linea CD ponderi in D reniti<lb/>tur, quàm linea CL ponderi in L. </s>

Line 385

<s id="id.2.1.17.5.1.30.0.a">&longs;imiliter quoniam con<lb/>tingentiæ angulus SOk, & angulo SDA, <lb/>& SAO, ac quibu&longs;cunq; &longs;imilibus e&longs;t mi <lb/>nor; erit de&longs;cen&longs;us ponderis in O motui <lb/>naturali ip&longs;ius ponderis in O &longs;oluti pro<lb/>pior, quàm in alio &longs;itu circumferentiæ <lb/>ODF. </s>

<s id="id.2.1.17.5.1.30.0.b">Præterea quoniam linea GO pon<lb/>dus in O dum deor&longs;um mouetur, impelle<lb/>re non pote&longs;t, ita vt vltra lineam OS mo<lb/>ueatur; cùm linea OS circulum non &longs;ecet, <lb/><figure id="id.036.01.036.1.jpg" place="text" xlink:href="036/01/036/1.jpg"/><lb/>&longs;ed contingat; angulu&longs;q; SOC &longs;it rectus, & non acutus; pondus <lb/>in O nihil &longs;upra lineam CO grauitabit. </s>

<s id="id.2.1.17.5.1.31.0">neq; centro innitetur. </s>

<s id="id.2.1.17.5.1.32.0">&que;m <lb/>admodum in quouis alio puncto &longs;upra O accideret. </s>

<s id="id.2.1.17.5.1.32.0">quem <lb/>admodum in quouis alio puncto &longs;upra O accideret. </s>

<s id="id.2.1.17.5.1.33.0">erit igitur pon<lb/>dus in O magis ob has cau&longs;as liberum, atq; &longs;olutum in hoc &longs;itu, <lb/>quàm in quouis alio circumferentiæ FOG. </s>

<s id="N10EED">ac idcirco in hoc <lb/>grauius erit, hoc e&longs;t magis grauitabit, quàm in alio &longs;itu. </s>

<s id="id.2.1.17.5.1.34.0">& quò <lb/>propius fuerit ip&longs;i O remotiori grauius erit. </s>

Line 399

<s id="id.2.1.19.1.2.2.0">circulus de<lb/>&longs;cribatur OH, cuius centrum &longs;it D, &longs;e<arrow.to.target n="note35"/><lb/>midiameterq; DO. </s>

<s id="N10F36">tanget circulus OH <lb/>circulum FOG in puncto O, lineamq; <arrow.to.target n="note36"/><lb/>OS, quæ ponderis in O rectus, natura<lb/>li&longs;q; e&longs;t de&longs;cen&longs;us, in eodem puncto con<lb/>tinget. </s>

<s id="id.2.1.19.1.2.3.0">& quoniam angulus SOH mi<lb/>nor e&longs;t angulo SOG, erit de&longs;cen&longs;us <lb/>ponderis in O per circumferentiam OH <lb/>motui naturali OS propior, quàm per <lb/>circumferentiam OG. </s>

<s id="id.2.1.19.1.2.3.0.a">magis ergo li<lb/>berum, atq; &longs;olutum, ac per con&longs;e&que;ns <lb/>grauius erit in O, centro libræ exi&longs;ten<lb/>te in D, quàm in C. </s>

<s id="id.2.1.19.1.2.3.0.a">magis ergo li<lb/>berum, atq; &longs;olutum, ac per con&longs;equens <lb/>grauius erit in O, centro libræ exi&longs;ten<lb/>te in D, quàm in C. </s>

<s id="N10F57">&longs;imiliter o&longs;ten<lb/>detur, quò maius fuerit brachium DO, <lb/>pondus in O adhuc grauius e&longs;&longs;e. <figure id="id.036.01.037.1.jpg" place="text" xlink:href="036/01/037/1.jpg"/></s><pb xlink:href="036/01/038.jpg"/>

<s id="id.2.1.19.3.1.1.0">Si verò idem circulus AFBG, <lb/>cuius centrum &longs;it R, propius fuerit <lb/>mundi centro S; circulum&queacute; à pun<lb/>cto S ducatur contingens ST; punctum <lb/>T (vbi grauius e&longs;t pondus) magis <lb/>à puncto A di&longs;tabit, quàm punctum <lb/>O. ducantur enim à punctis OT ip&longs;i <lb/>CS perpendiculares OMTN; conne<lb/>ctanturq; RT; &longs;itq; centrum R in li<lb/>nea CS; lineaq; ARB ip&longs;i ACB æqui <lb/><arrow.to.target n="note37"/>di&longs;tans. </s>

Line 410

<s id="N10FA6">&longs;ecetur <lb/>igitur RN in P, ita vt RP &longs;it ip&longs;i <lb/><figure id="id.036.01.038.1.jpg" place="text" xlink:href="036/01/038/1.jpg"/><lb/>CM æqualis; & à puncto P ip&longs;is MONT æquidi&longs;tans ducatur <lb/>PQ, quæ circumferentiam AT &longs;ecet in Q: deniq; connectatur <lb/>RQ. </s>

<s id="N10FB6">quoniam enim duæ CO CM duabus RQRP &longs;unt æqua<lb/><arrow.to.target n="note40"/>les, & angulus CMO angulo RPQ e&longs;t æqualis; erit & angu<lb/>lus MCO angulo PRQ æqualis. </s>

<s id="id.2.1.19.3.1.3.0">angulus autem MCA rectus <lb/><arrow.to.target n="note41"/>recto PRA e&longs;t æqualis; ergo reliquus OCA reliquo QRA <lb/>æqualis, & circumferentia OA circumferentiæ QA æqualis quo<lb/>&que; erit. </s>

<s id="id.2.1.19.3.1.4.0">punctum idcirco T, quia magis à puncto A di&longs;tat, <lb/>quàm Q; magis quoq; à puncto A di&longs;tabit, quàm punctum O. <lb/></s>

<s id="N10FD1">&longs;imiliter o&longs;tendetur, quò propius fuerit circulus mundi centro, eun<lb/>dem magis di&longs;tare. </s>

<s id="id.2.1.19.3.1.5.0">atq; ita vt prius demon&longs;trabitur pondus in cir<lb/>cumferentia TAF centro R inniti, in circumferentia verò TG <lb/>à linea detineri; atq; in puncto T grauius e&longs;&longs;e. </s>

Line 426

<s id="N11054">nam ducta <lb/>G k, efficiet hæc (&longs;ecun<lb/>dùm quam fit ponderis natu<lb/>ralis motus) vná cum libræ <lb/>brachio k C angulum acu<lb/>tum. </s>

<s id="id.2.1.21.1.2.2.0">æquicruris enim trian<lb/>guli CkG ad ba&longs;im anguli <lb/>ad k, & G &longs;unt &longs;emper acuti. </s>

<s id="id.2.1.21.1.2.3.0"><lb/><figure id="id.036.01.039.1.jpg" place="text" xlink:href="036/01/039/1.jpg"/><lb/>Conferantur autem inuicem hæc duo, pondus videlicet in k, & <lb/>pondus in D: erit pondus in k grauius, quàm in D. </s>

<s id="N11073">nam iuncta <lb/>DG, cùm tres anguli cuiu&longs;cun&que; trianguli duobus &longs;int rectis <lb/>æquales, & trianguli CDG æquicruris angulus DCG maior &longs;it <lb/>angulo kCG æquicruris trianguli CkG: erunt reliqui ad ba&longs;im an<lb/>guli DGC GDC &longs;imul &longs;umpti reliquis KGCGkC &longs;imul &longs;umptis <lb/>minores. </s>

<s id="N11073">nam iuncta <lb/>DG, cùm tres anguli cuiu&longs;cunque trianguli duobus &longs;int rectis <lb/>æquales, & trianguli CDG æquicruris angulus DCG maior &longs;it <lb/>angulo kCG æquicruris trianguli CkG: erunt reliqui ad ba&longs;im an<lb/>guli DGC GDC &longs;imul &longs;umpti reliquis KGCGkC &longs;imul &longs;umptis <lb/>minores. </s>

<s id="id.2.1.21.1.2.4.0">horumq; dimidii; angulus &longs;cilicet CDG angulo CKG <lb/>minor erit. </s>

<s id="id.2.1.21.1.2.5.0">quare cùm pondus in k &longs;olutum naturaliter per <lb/>KG moueatur, pondusq; in D per DG, tanquam per &longs;patia, <lb/>quibus in centrum mundi feruntur; linea CD, hoc e&longs;t libræ <lb/>brachium magis adhærebit motui naturali ponderis in D pror<lb/>&longs;us &longs;oluti, lineæ &longs;cilicet DG; quàm Ck motui &longs;ecundùm kG <lb/>effecto. </s>

<s id="id.2.1.21.1.2.6.0">magis igitur &longs;u&longs;tinebit linea CD, quàm Ck. </s>

Line 462

<s id="id.2.1.21.2.1.15.0">dein<lb/>de quoniam angulus CkS maior e&longs;t CDS, & CDk æqualis <lb/>e&longs;t CkH: erit reliquus SkH reliquo SDk minor. </s>

<s id="id.2.1.21.2.1.16.0">quare cir<lb/>cumferentia k H propior erit motui naturali recto ponderis in K <lb/>&longs;oluti, lineæ &longs;cilicet k S, quàm circumferentia D k motui DS. </s>

<s id="id.2.1.21.2.1.16.0.a">& <lb/>ideo linea CD magis ip&longs;i ponderi in D renititur, quàm CK <lb/>ponderi in k con&longs;tituto. </s>

<s id="id.2.1.21.2.1.17.0">hacq; ratione o&longs;tendetur angulum <lb/>SHG maiorem e&longs;&longs;e SkH: & per con&longs;e&que;ns lineam CH magis <lb/>ponderi in H reniti, quàm CK ponderi in K. </s>

<s id="id.2.1.21.2.1.17.0">hacq; ratione o&longs;tendetur angulum <lb/>SHG maiorem e&longs;&longs;e SkH: & per con&longs;equens lineam CH magis <lb/>ponderi in H reniti, quàm CK ponderi in K. </s>

<s id="N111AD">&longs;imiliter demon<lb/>&longs;trabitur lineam C<emph type="italics"/>L<emph.end type="italics"/> magis pondus &longs;u&longs;tinere, quàm CD: ob <lb/>ea&longs;demq; cau&longs;as o&longs;tendetur pondus in K minus &longs;upra lineam Ck <lb/>grauitare, quàm in quouis alio &longs;itu fuerit circumferentiæ FDG. <lb/></s>

<s id="N111BC">& quò propius fuerit ip&longs;i F, vel G, minus grauitare. </s>

<s id="id.2.1.21.2.1.18.0">grauius ergo <lb/>erit in k, quàm in alio &longs;itu: minu&longs;q; graue erit, quò propius fue<lb/>rit ip&longs;i F, vel G. <pb xlink:href="036/01/042.jpg"/></s>

Line 471

<s id="id.2.1.22.1.1.3.0"><margin.target id="note44"/>7 <emph type="italics"/>Tertii.<emph.end type="italics"/></s>

<s id="id.2.1.22.1.1.4.0"><margin.target id="note45"/>25 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.22.1.1.5.0"><margin.target id="note46"/>25 <emph type="italics"/>Primi.<emph.end type="italics"/></s>

<s id="id.2.1.23.1.1.1.0">Si deniq; centrum C <lb/>e&longs;&longs;et in centro mundi, <lb/>pondus vbicun&que; con<lb/>&longs;titutum manere mani<lb/>fe&longs;tum e&longs;t. </s>

<s id="id.2.1.23.1.1.1.0">Si deniq; centrum C <lb/>e&longs;&longs;et in centro mundi, <lb/>pondus vbicunque con<lb/>&longs;titutum manere mani<lb/>fe&longs;tum e&longs;t. </s>

<s id="id.2.1.23.1.1.2.0">vt po&longs;ito pon<lb/>dere in D, linea CD to<lb/>tum &longs;u&longs;tinebit pondus; <lb/>cùm ip&longs;ius ponderis in D <lb/>horizonti &longs;it perpendicu<lb/><arrow.to.target n="note47"/>laris. </s>

<s id="id.2.1.23.1.1.3.0">pondus ergo ma <lb/>nebit. <figure id="id.036.01.042.1.jpg" place="text" xlink:href="036/01/042/1.jpg"/></s>

<s id="id.2.1.23.2.1.1.0">Quoniam autem in his hactenus demon&longs;tratis, nullam de gra<lb/>uitate brachii libræ mentionem fecimus, idcirco &longs;i brachii quoq; <lb/>grauitatem con&longs;iderare voluerimus, centrum grauitatis magnitu<lb/>dinis ex pondere, brachioq; compo&longs;itæ inueniri poterit, circulo<lb/>rumq; circumferentiæ &longs;ecundum di&longs;tantiam à centro libræ ad <lb/>hoc ip&longs;um grauitatis centrum de&longs;cribentur, ac &longs;i in ip&longs;o (vt re ue<lb/>ra e&longs;t) pondus con&longs;titutum fuerit; omnia, &longs;icuti ab&longs;q; libræ bra<lb/>chii grauitate con&longs;iderata inuenimus; hoc quoq; modo eius con&longs;i<lb/>derata grauitate reperiemus. </s>

<s id="id.2.1.24.1.1.1.0"><margin.target id="note47"/>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s><pb n="15" xlink:href="036/01/043.jpg"/>

<s id="id.2.1.25.1.2.1.0">Ex dictis igitur, con&longs;iderando li<lb/>bram, vt longè à mundi centro a<lb/>be&longs;t, &que;madmodum ip&longs;i fecere, &longs;i<lb/>cuti etiam actu e&longs;t, apparet fal&longs;itas <lb/>dicentium pondus in A grauius e&longs;&longs;e, <lb/>quàm in alio &longs;itu. </s>

<s id="id.2.1.25.1.2.1.0">Ex dictis igitur, con&longs;iderando li<lb/>bram, vt longè à mundi centro a<lb/>be&longs;t, quemadmodum ip&longs;i fecere, &longs;i<lb/>cuti etiam actu e&longs;t, apparet fal&longs;itas <lb/>dicentium pondus in A grauius e&longs;&longs;e, <lb/>quàm in alio &longs;itu. </s>

<s id="id.2.1.25.1.2.2.0">&longs;imulq; fal&longs;um e&longs;&longs;e, <lb/>quò pondus à linea FG magis di&longs;tat <lb/><expan abbr="grauiuis">grauius</expan> e&longs;&longs;e. </s>

<s id="id.2.1.25.1.2.3.0">nam punctum O pro<lb/>pius e&longs;t ip&longs;i FG, quàm punctum A. <lb/></s>

<s id="N1126C">e&longs;t enim linea à puncto O ip&longs;i FG <arrow.to.target n="note48"/><lb/>perpendicularis ip&longs;a CA minor. </s>

<s id="id.2.1.25.1.2.4.0">de<lb/>inde ex puncto A pondus velocius mo<lb/>ueri, quàm ab alio &longs;itu, e&longs;t quo&que; <lb/>fal&longs;um. </s>

<s id="id.2.1.25.1.2.4.0">de<lb/>inde ex puncto A pondus velocius mo<lb/>ueri, quàm ab alio &longs;itu, e&longs;t quoque <lb/>fal&longs;um. </s>

<s id="id.2.1.25.1.2.5.0">ex puncto enim O pondus ve<lb/>locius mouebitur, quàm ex puncto <lb/>A; cùm in O &longs;it magis liberum, atq; <lb/>&longs;olutum, quàm in alio &longs;itu: de&longs;cen&longs;us <lb/>&queacute; ex puncto O propior &longs;it motui na<lb/>turali recto, quàm quilibet alius de<lb/>&longs;cen&longs;us. <figure id="id.036.01.043.1.jpg" place="text" xlink:href="036/01/043/1.jpg"/></s>

<s id="id.2.1.25.2.1.1.0">Præterea cùm ex re<lb/>ctiori, & obliquiori <expan abbr="de&longs;c&etilde;&longs;u">de&longs;cen<lb/>&longs;u</expan> o&longs;tendunt, pondus in <lb/>A <expan abbr="grauiur">grauior</expan> e&longs;&longs;e, quàm in <lb/>D; & in D, quàm in <lb/>L; primùm quidem fal<lb/>&longs;um exi&longs;timant, &longs;i pon<lb/>dus aliquod collocatum <lb/>fuerit in quocunq; &longs;itu <lb/>circunferentiæ, vt in D, <lb/>rectum eius de&longs;cen&longs;um <lb/>per rectam lineam DR <lb/>ip&longs;i FG parallelam, tam <lb/>quàm &longs;ecundùm mo<figure id="id.036.01.043.2.jpg" place="text" xlink:href="036/01/043/2.jpg"/>

<pb xlink:href="036/01/044.jpg"/>tum naturalem fieri de<lb/>bere; &longs;icuti prius dictum <lb/>e&longs;t. </s>

Line 494

<s id="id.2.1.25.2.1.7.0">præ&longs;ertim quia <lb/>&longs;en&longs;ibilitas illa non efficit, quin de&longs;cen&longs;us ponderis ex L in D <lb/>(vt eorum verbis vtar) minus capiat de directo, quàm de&longs;cen<lb/>&longs;us DA. </s>

<s id="N11327">&longs;imiliter arcus DA magis de directo capiet, quàm cir<lb/>cumferentia EV. </s>

<s id="N1132B">quocirca vera erit &longs;uppo&longs;itio; aliæq; demon<lb/>&longs;trationes in &longs;uo robore permanebunt. </s>

<s id="id.2.1.25.2.1.8.0">Concedamus etiam pon<pb n="16" xlink:href="036/01/045.jpg"/>dus in A grauius e&longs;&longs;e, quàm in alio &longs;itu; rectumq; ponderis de<lb/>&longs;cen&longs;um per rectam lineam ip&longs;i FG parallelam fieri debere; & <lb/>quælibet puncta in lineis horizonti æquidi&longs;tantibus accepta æ<lb/>qualiter à centro mundi di&longs;tare: non tamen propterea &longs;e&que;tur, <lb/>veram e&longs;&longs;e demon&longs;trationem, qua inferunt pondus in A grauius <lb/>e&longs;&longs;e, quàm in alio &longs;itu, vt in L. </s>

<s id="id.2.1.25.2.1.8.0">Concedamus etiam pon<pb n="16" xlink:href="036/01/045.jpg"/>dus in A grauius e&longs;&longs;e, quàm in alio &longs;itu; rectumq; ponderis de<lb/>&longs;cen&longs;um per rectam lineam ip&longs;i FG parallelam fieri debere; & <lb/>quælibet puncta in lineis horizonti æquidi&longs;tantibus accepta æ<lb/>qualiter à centro mundi di&longs;tare: non tamen propterea &longs;equetur, <lb/>veram e&longs;&longs;e demon&longs;trationem, qua inferunt pondus in A grauius <lb/>e&longs;&longs;e, quàm in alio &longs;itu, vt in L. </s>

<s id="N11341">&longs;i enim verum e&longs;&longs;et, quò pon<lb/>dus hoc modo rectius de&longs;cendit, ibi grauius e&longs;&longs;e; &longs;e&que;retur etiam, <lb/>quò idem pondus in æqualibus arcubus æqualiter rectè de&longs;cende<lb/>ret, vt in ii&longs;dem locis æqualem haberet grauitatem, quod fal<lb/>&longs;um e&longs;&longs;e ita demon&longs;tratur. </s>

<s id="N11341">&longs;i enim verum e&longs;&longs;et, quò pon<lb/>dus hoc modo rectius de&longs;cendit, ibi grauius e&longs;&longs;e; &longs;equeretur etiam, <lb/>quò idem pondus in æqualibus arcubus æqualiter rectè de&longs;cende<lb/>ret, vt in ii&longs;dem locis æqualem haberet grauitatem, quod fal<lb/>&longs;um e&longs;&longs;e ita demon&longs;tratur. </s>

<s id="id.2.1.26.1.1.1.0"><margin.target id="note48"/><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>Tertii.<emph.end type="italics"/></s>

<s id="id.2.1.26.1.1.2.0"><margin.target id="note49"/>18 <emph type="italics"/>Primi.<emph.end type="italics"/></s>

<s id="id.2.1.27.1.1.1.0">Sint circumferentiæ AL AM inter &longs;e &longs;e æquales; & conne<lb/>ctatur LM, quæ AB &longs;ecet in X: erit LM ip&longs;i FG æquidi&longs;tans, <lb/>ip&longs;iq; AB perpendicularis. </s>

Line 509

<s id="id.2.1.29.1.1.1.0">Quamuis autem AMLA æqualiter &longs;ecundùm ip&longs;os de directo <lb/>capiant; dicent forta&longs;&longs;e, quia tamen principium de&longs;cen&longs;us ex L <lb/>&longs;cilicet LD minus de directo capit, quàm principium de&longs;cen&longs;us <lb/>ex A, &longs;cilicet AN; pondus in A grauius erit, quàm in L. </s>

<s id="id.2.1.29.1.1.1.0.a">nam <lb/>cùm circumferentia AN &longs;it ip&longs;i LD (vt &longs;upra po&longs;itum e&longs;t) <lb/>æqualis, quæ &longs;ecundùm ip&longs;os de directo capit CT; LD verò <lb/>de directo capit PO. </s>

<s id="id.2.1.29.1.1.1.0.b">ideo pondus grauius erit in A, quàm in L. <lb/></s>

<s id="id.2.1.29.1.1.1.0.c">quod &longs;i verum e&longs;&longs;et, &longs;e&que;retur idem pondus in eodem &longs;itu diuer<lb/>&longs;o duntaxat modo con&longs;ideratum in habitudine ad eundem &longs;itum, <lb/>tum grauius, tum leuius e&longs;&longs;e. </s>

<s id="id.2.1.29.1.1.1.0.c">quod &longs;i verum e&longs;&longs;et, &longs;equeretur idem pondus in eodem &longs;itu diuer<lb/>&longs;o duntaxat modo con&longs;ideratum in habitudine ad eundem &longs;itum, <lb/>tum grauius, tum leuius e&longs;&longs;e. </s>

<s id="id.2.1.29.1.1.2.0">quod e&longs;t impo&longs;sibile. </s>

<s id="id.2.1.29.1.1.3.0">hoc e&longs;t, &longs;i <lb/>de&longs;cen&longs;um con&longs;ideremus ponderis in L, quatenus ex L in A de<lb/>&longs;cendit, grauius erit, quàm &longs;i eiu&longs;dem ponderis de&longs;cen&longs;um con<lb/>&longs;ideremus ex L in D tantùm. </s>

<s id="id.2.1.29.1.1.4.0">neq; enim negare po&longs;&longs;unt ex ei&longs;<lb/>demmet dictis, quin de&longs;cen&longs;us ponderis ex L in A de directo ca<lb/>piat LX, &longs;iue PC. </s>

<s id="N113DB">de&longs;cen&longs;us verò AM, quin &longs;imiliter de directo <pb xlink:href="036/01/046.jpg"/>capiat XM: cùm ip&longs;i <lb/>quoq; hoc modo acci<lb/>piant, atq; ita accipe<lb/>re &longs;it nece&longs;&longs;e. </s>

<s id="id.2.1.29.1.1.5.0">&longs;i enim li<lb/>bram DE in AB redire <lb/>demon&longs;trare volunt, com<lb/>parando de&longs;cen&longs;us pon<lb/>deris in D cum de&longs;cen<lb/>&longs;u ponderis in E, nece&longs;&longs;e <lb/>e&longs;t, vt o&longs;tendant rectum <lb/>de&longs;cen&longs;um OC corre<lb/>&longs;pondentem circumferen<lb/>tiæ DA maiorem e&longs;&longs;e re<lb/>cto de&longs;cen&longs;u TH circum<lb/><figure id="id.036.01.046.1.jpg" place="text" xlink:href="036/01/046/1.jpg"/><lb/>ferentiæ EV corre&longs;pondente. </s>

<s id="id.2.1.29.1.1.6.0">&longs;i enim partem tantùm totius de<lb/>&longs;cen&longs;us ex D in A acciperent, vt D k; o&longs;tenderentq; magis cape<lb/>re de directo de&longs;cen&longs;um Dk, quàm æqualis portio de&longs;cen&longs;us ex <lb/>puncto E. </s>

<s id="N1140F">&longs;e&que;tur pondus in D &longs;ecundùm ip&longs;os grauius e&longs;&longs;e pon<lb/>dere in E; & v&longs;q; ad k tantùm deor&longs;um moueri: ita vt libra mo<lb/>ta &longs;it in kI. </s>

<s id="N1140F">&longs;equetur pondus in D &longs;ecundùm ip&longs;os grauius e&longs;&longs;e pon<lb/>dere in E; & v&longs;q; ad k tantùm deor&longs;um moueri: ita vt libra mo<lb/>ta &longs;it in kI. </s>

<s id="N11415">&longs;imiliter &longs;i libram KI in AB redire demon&longs;trare vo<lb/>lunt accipiendo portionem de&longs;cen&longs;us ex k in A; hoc e&longs;t k S; <lb/>o&longs;tenderentq; k S magis de directo capere, quàm ex aduer&longs;o æ<lb/>qualis de&longs;cen&longs;us ex puncto I: &longs;imili modo &longs;e&que;tur pondus in k <lb/>grauius e&longs;&longs;e, quàm in I; & v&longs;q; ad S tantùm moueri. </s>

<s id="N11415">&longs;imiliter &longs;i libram KI in AB redire demon&longs;trare vo<lb/>lunt accipiendo portionem de&longs;cen&longs;us ex k in A; hoc e&longs;t k S; <lb/>o&longs;tenderentq; k S magis de directo capere, quàm ex aduer&longs;o æ<lb/>qualis de&longs;cen&longs;us ex puncto I: &longs;imili modo &longs;equetur pondus in k <lb/>grauius e&longs;&longs;e, quàm in I; & v&longs;q; ad S tantùm moueri. </s>

<s id="id.2.1.29.1.1.7.0">& &longs;i rur&longs;us <lb/>o&longs;tenderent portionem de&longs;cen&longs;us ex S in A, atq; ita deinceps, re<lb/>ctiorem e&longs;&longs;e æquali de&longs;cen&longs;u ponderis oppo&longs;iti; &longs;emper &longs;e&que;tur <lb/>libram SI ad AB propius accedere, nunquam tamen in AB per<lb/>uenire demon&longs;trabunt. </s>

<s id="id.2.1.29.1.1.7.0">& &longs;i rur&longs;us <lb/>o&longs;tenderent portionem de&longs;cen&longs;us ex S in A, atq; ita deinceps, re<lb/>ctiorem e&longs;&longs;e æquali de&longs;cen&longs;u ponderis oppo&longs;iti; &longs;emper &longs;equetur <lb/>libram SI ad AB propius accedere, nunquam tamen in AB per<lb/>uenire demon&longs;trabunt. </s>

<s id="id.2.1.29.1.1.8.0">&longs;i igitur libram DE in AB redire demon<lb/>&longs;trare volunt, nece&longs;&longs;e e&longs;t, vt de&longs;cen&longs;um ponderis ex D in A de di<lb/>recro capere quantitatem lineæ ex puncto D ip&longs;i AB ad rectos <lb/>angulos ductæ accipiant. </s>

<s id="id.2.1.29.1.1.9.0">atq; ita, &longs;i æquales de&longs;cen&longs;us DA AN <lb/>inuicem comparemus, qui æqualiter de directo capient OC CT, <lb/>eueniet idem pondus in D æquè graue e&longs;&longs;e, vt in A. </s>

<s id="N1143A">&longs;i verò por<lb/>tiones tantum ex D A accipiamus; grauius erit in A, quàm <lb/>in D. </s>

Line 557

<s id="N1152F">minus obliquus igitur e&longs;t <lb/>de&longs;cen&longs;us arcus LA, quàm arcus OP. </s>

<s id="id.2.1.29.2.1.20.0.a">ergo pondus in L, ex ip<lb/>&longs;orum dictis, grauius erit, quàm in O. quod ex iis, quæ &longs;upra di<lb/>ximus e&longs;t manife&longs;tè fal&longs;um, cùm pondus in O grauius &longs;it, quàm <lb/>in L. </s>

<s id="id.2.1.29.2.1.20.0.b">non igitur ex rectiori, & obliquiori motu ita accepto col<lb/>ligi pote&longs;t, &longs;ecundùm &longs;itum pondus grauius e&longs;&longs;e, quantò in eo<lb/>dem &longs;itu minus obliquus e&longs;t de&longs;cen&longs;us. </s>

<s id="id.2.1.29.2.1.21.0">Atq; hinc oritur omnis <lb/>fermé ip&longs;orum error in hac re, atq; deceptio: nam quamuis per <lb/>accidens interdum ex fal&longs;is &longs;equatur verum, per &longs;e tamen ex fal<lb/>&longs;is fal&longs;um &longs;equitur, &que;madmodum ex veris &longs;emper verum, nil <lb/>idcirco mirum, &longs;i dum fal&longs;a accipiunt; illi&longs;q; tanquam veri&longs;si<lb/>mis innituntur; fal&longs;i&longs;sima omninò colligunt, atq; concludunt. </s>

<s id="id.2.1.29.2.1.21.0">Atq; hinc oritur omnis <lb/>fermé ip&longs;orum error in hac re, atq; deceptio: nam quamuis per <lb/>accidens interdum ex fal&longs;is &longs;equatur verum, per &longs;e tamen ex fal<lb/>&longs;is fal&longs;um &longs;equitur, quemadmodum ex veris &longs;emper verum, nil <lb/>idcirco mirum, &longs;i dum fal&longs;a accipiunt; illi&longs;q; tanquam veri&longs;si<lb/>mis innituntur; fal&longs;i&longs;sima omninò colligunt, atq; concludunt. </s>

<s id="id.2.1.29.2.1.22.0"><lb/>decipiuntur quinetiam, dùm libræ contemplationem mathemati<lb/>cè &longs;impliciter a&longs;&longs;ummunt; cùm eius con&longs;ideratio &longs;it pror&longs;us me<lb/>chanica: nec vllo modo ab&longs;q; vero motu, ac ponderibus (en<pb n="18" xlink:href="036/01/049.jpg"/>tibus omninò naturalibus) de ip&longs;a &longs;ermo haberi po&longs;sit: &longs;ine qui<lb/>bus eorum, quæ libræ accidunt, veræ caulæ reperiri nullo mo <lb/>do po&longs;sint. </s>

<s id="id.2.1.30.1.1.1.0"><margin.target id="note51"/><emph type="italics"/>Ex<emph.end type="italics"/> 27 <emph type="italics"/>Tertii.<emph.end type="italics"/></s>

<s id="id.2.1.30.1.1.2.0"><margin.target id="note52"/><emph type="italics"/>Ex<emph.end type="italics"/> 32 <emph type="italics"/>primi.<emph.end type="italics"/></s>

Line 614

<s id="id.2.1.33.3.1.13.0">Quare pondera in <lb/>DE, quatenus &longs;unt &longs;ibi inuicem connexa, &longs;i ip&longs;orum naturalem mo <lb/>tum &longs;pectemus, non &longs;ecundùm lineas DS ES, &longs;ed &longs;ecundùm <lb/>LDH MEk ip&longs;i CS æquidi&longs;tantes mouebuntur. </s>

<s id="id.2.1.33.3.1.14.0">ponderis ve<lb/>rò in E liberi, ac &longs;oluti, naturalis propen&longs;io erit per ES: ponderis <lb/>autem in D &longs;imiliter &longs;oluti erit per DS. ac propterea non e&longs;t incon<lb/>ueniens idem pondus modò in E, modò in D, grauius e&longs;&longs;e in E, <lb/>quàm in D. </s>

<s id="id.2.1.33.3.1.14.0.a">&longs;i verò pondera in ED &longs;ibi inuicem connexa, quate<lb/>nusq; &longs;unt connexa con&longs;iderauerimus; erit ponderis in E natura<lb/>lis propen&longs;io per lineam MEK: grauitas enim alterius ponde<lb/>ris in D efficit, nè pondus in E per lineam ES grauitet, &longs;ed per <lb/>Ek. </s>

<s id="id.2.1.33.3.1.15.0">quod ip&longs;um quoq; grauitas ponderis in E efficit, nè &longs;cilicet <lb/>pondus in D per rectam DS degrauet; &longs;ed &longs;ecundùm DH: vtra<lb/>&que; enim &longs;e impediunt, nè ad propria loca <expan abbr="permeent">permeant</expan>. </s>

<s id="id.2.1.33.3.1.15.0">quod ip&longs;um quoq; grauitas ponderis in E efficit, nè &longs;cilicet <lb/>pondus in D per rectam DS degrauet; &longs;ed &longs;ecundùm DH: vtra<lb/>que enim &longs;e impediunt, nè ad propria loca <expan abbr="permeent">permeant</expan>. </s>

<s id="id.2.1.33.3.1.16.0">Cùm igi<lb/>tur naturalis de&longs;cen&longs;us rectus ponderum in DE &longs;it &longs;ecundùm <lb/>LDH MEK: erit <expan abbr="&longs;imliter">similiter</expan> rectus eorum a&longs;cen&longs;us &longs;ecundùm ea&longs;<lb/>dem lineas HDL KEM. </s>

<s id="id.2.1.33.3.1.16.0.a">atq; a&longs;cen&longs;us ponderis in E magis, mi<lb/>nu&longs;uè obliquus dicetur; quantò &longs;ecundùm &longs;patium magis, mi<lb/>nu&longs;uè iuxta lineam Mk mouebitur. </s>

<s id="id.2.1.33.3.1.17.0">hocq; pror&longs;us modo iuxta li<lb/>neam LH &longs;ummendus e&longs;t, tùm de&longs;cen&longs;us, tùm a&longs;cen&longs;us ponde<lb/>ris in D. </s>

Line 672

<s id="id.2.1.37.1.2.2.0">&longs;imiliter &longs;i libra <lb/>centrum C habeat infra li<lb/>bram, &longs;itq; trutina CD &longs;u<lb/>pra libram, & moueatur <lb/>libra in EF; patet libram <arrow.to.target n="note64"/><lb/>ex parte F deor&longs;um moue <lb/>ri, trutina &longs;upra libram e<lb/>xi&longs;tente. </s>

<s id="id.2.1.37.1.2.3.0">& in quocunq; a<lb/>lio &longs;itu fuerit trutina, idem <lb/>&longs;emper eueniet. </s>

<s id="id.2.1.37.1.2.4.0">non igitur <lb/>trutina, &longs;ed centrum libræ <lb/>harum diuer&longs;itatum cau<lb/>&longs;a erit. <figure id="id.036.01.057.1.jpg" place="text" xlink:href="036/01/057/1.jpg"/></s>

<s id="id.2.1.37.2.1.1.0">Animaduertendum e&longs;t <lb/>itaq; in hac parte difficulter materialem libram con&longs;titui po&longs;&longs;e, <lb/>quæ in vno tantùm puncto &longs;u&longs;tineatur; &que;madmodum mente <lb/>concipimus. </s>

<s id="id.2.1.37.2.1.2.0">brachiaq; ab eiu&longs;modi centro adeò æqualia habeat, <lb/>non &longs;olum in longitudine, verùm etiam in latitudine, & profun<lb/>ditate, vt omnes partes hinc indé ad vnguem æ&que;ponderent. </s>

<s id="id.2.1.37.2.1.3.0"><lb/>hoc enim materia difficilimè patitur. </s>

<s id="id.2.1.37.2.1.4.0">quocirca &longs;i centrum in ip&longs;a <lb/>libra e&longs;&longs;e con&longs;iderauerimus, ad &longs;en&longs;um confugiendum non e&longs;t: <lb/>cùm artificilia ad &longs;ummum illud perfectionis gradum ab artifice <lb/>deduci minimè po&longs;sint. </s>

<s id="id.2.1.37.2.1.5.0">In aliis verò experientia quidem appa<lb/>rentia docere poterit; propterea quod, quamquam centrum libræ <lb/>&longs;it &longs;emper punctum, quando tamen &longs;upra libram fuerit, parùm re<lb/>fert, &longs;i libra in eo puncto adamu&longs;&longs;im minimè &longs;u&longs;tineatur; quia cùm <lb/>&longs;it &longs;emper &longs;upra libram, idem &longs;emper eueniet. </s>

Line 687

<s id="id.2.1.39.1.1.4.0">tantùm enim abe&longs;t philo&longs;o<lb/>phum has diuer&longs;itates in angulos referre, vt potius in cau&longs;a e&longs;&longs;e <lb/>dicat magnitudinis alterius brachii libræ exce&longs;&longs;um à perpendiculo, <lb/>modò ex vna, modò ex altera parte contingentem. </s>

<s id="id.2.1.39.2.1.1.0">Vt trutina &longs;uperius in <lb/>CF exi&longs;tente, perpendicu<lb/>lum erit FCG, quod &longs;e<lb/>cundùm ip&longs;um in centrum <lb/>mundi &longs;emper vergit; <lb/>quod quidem libram mo<lb/>tam in DE in partes di<lb/>uidit inæquales; & maior <lb/>pars e&longs;t ver&longs;us D: id au<lb/>tem, quod plus e&longs;t, deor<lb/>&longs;um fertur; ergo ex par<lb/>te D deor&longs;um libra moue<lb/>bitur, donec in AB re<lb/>deat. </s>

<s id="id.2.1.39.2.1.2.0">&longs;i verò trutina &longs;it <lb/><figure id="id.036.01.058.1.jpg" place="text" xlink:href="036/01/058/1.jpg"/><lb/>in CG deor&longs;um, erit GCF perpendiculum, quod libram DE <lb/>in partes inæquales &longs;imiliter diuidit: maior autem pars erit ver&longs;us <lb/>E; quare ex parte E deor&longs;um libra mouebitur. </s>

<s id="id.2.1.39.2.1.3.0">quod vt rectè in<lb/>telligatur, cùm trutina e&longs;t &longs;upra libram, libræ quoq; centrum &longs;u<lb/>pra libram e&longs;&longs;e intelligendum e&longs;t; & &longs;i deor&longs;um, centrum quo&que; <lb/>deor&longs;um: vt infra patebit. </s>

<s id="id.2.1.39.2.1.4.0">Aliter ip&longs;a Ari&longs;totelis demon&longs;tratio <lb/>nihil concluderet. </s>

<s id="id.2.1.39.2.1.5.0">exi&longs;tente enim centro in ip&longs;a libra, vt in C; quo<lb/>cunq; modo moueatur libra, nunquam perpendiculum FG libram, <pb n="23" xlink:href="036/01/059.jpg"/>ni&longs;i in puncto C, & in partes diuidet æquales. </s>

<s id="id.2.1.39.2.1.6.0">quare Ari&longs;totelis <lb/>&longs;ententia ip&longs;is non &longs;olum non fauet, verùm etiam maximè aduer<lb/>&longs;atur. </s>

<s id="id.2.1.39.2.1.7.0">quòd non &longs;olum ex &longs;ecunda, & tertia huius li&que;t; verùm <lb/>quia exi&longs;tente centro &longs;upra libram pondus eleuatum maiorem <lb/>propter &longs;itum acquirit grauitatem. </s>

<s id="id.2.1.39.2.1.7.0">quòd non &longs;olum ex &longs;ecunda, & tertia huius liquet; verùm <lb/>quia exi&longs;tente centro &longs;upra libram pondus eleuatum maiorem <lb/>propter &longs;itum acquirit grauitatem. </s>

<s id="id.2.1.39.2.1.8.0">ex quò contingit redditus li<lb/>bræ ad æqualem horizonti di&longs;tantiam. </s>

<s id="id.2.1.39.2.1.9.0">è contra verò, quando <lb/>centrum e&longs;t infra libram. </s>

<s id="id.2.1.39.2.1.10.0">Quæ omnia hoc modo o&longs;tendentur; <lb/>&longs;upponendo ea, quæ &longs;upra declarata &longs;unt. </s>

Line 796

<s id="N11E54">pondus ideò in E ma<lb/>gis à linea directionis OP di&longs;tabit, quàm pondus in F. </s>

<s id="N11E58">maio<lb/>rem igitur grauitatem habebit pondus in E, quàm pondus in F. <lb/></s>

<s id="N11E5D">ex quibus &longs;equitur, libram EF ex parte E deor&longs;um moueri. </s>

<s id="id.2.1.49.5.1.1.0">Ari&longs;toteles itaq; has duas tantùm quæ&longs;tiones propo&longs;uit, ter<lb/>tiamq; reliquit; &longs;cilicet cùm centrum libræ in ip&longs;a e&longs;t libra: hanc <lb/>autem ommi&longs;sit, vt notam, &que;madmodum res valde notas præ<lb/>termittere &longs;olet. </s>

<s id="id.2.1.49.5.1.1.0">Ari&longs;toteles itaq; has duas tantùm quæ&longs;tiones propo&longs;uit, ter<lb/>tiamq; reliquit; &longs;cilicet cùm centrum libræ in ip&longs;a e&longs;t libra: hanc <lb/>autem ommi&longs;sit, vt notam, quemadmodum res valde notas præ<lb/>termittere &longs;olet. </s>

<s id="id.2.1.49.5.1.2.0">nam cui dubium, &longs;i pondus in eius centro gra<lb/>uitatis &longs;u&longs;tineatur, quin maneat? </s>

<s id="id.2.1.49.5.1.3.0">Ea verò, quæ ex ip&longs;ius &longs;enten<lb/>tia attulimus, aliquis reprehendere po&longs;&longs;et, nos integram eius &longs;enten<lb/>tiam minimè protuli&longs;&longs;e <expan abbr="affimans">affirmans</expan>. </s>

<s id="id.2.1.49.5.1.4.0">nam cùm in &longs;ecunda parte &longs;e<lb/>cundæ quæ&longs;tionis proponit, cur libra, trutina deor&longs;um con&longs;tituta, <lb/>quando deor&longs;um lato pondere qui&longs;piam id amouet, non a&longs;cen<lb/>dit, &longs;ed manet? </s>

Line 835

<s id="id.2.1.49.10.1.9.0">nunquam enim punctum <lb/>B v&longs;q; ad lineam CH perueniet, cùm centrum grauitatis ponde<lb/>ris, & libræ &longs;imul &longs;emper inter DB exi&longs;tat. </s>

<s id="id.2.1.49.10.1.10.0">quò tamen pondus <lb/>in B grauius fuerit, maiorem quoq; circumferentiam de&longs;cribet. </s>

<s id="id.2.1.49.10.1.11.0"><lb/>eò enim magis punctum B ad lineam CH accedet. </s>

<s id="id.2.1.50.1.1.1.0"><margin.target id="note76"/>6 <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.50.1.1.1.0"><margin.target id="note76"/>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.50.1.1.3.0"><margin.target id="note77"/>1. <emph type="italics"/>Huius.<emph.end type="italics"/></s><pb n="28" xlink:href="036/01/069.jpg"/>

<s id="id.2.1.51.1.2.1.0">Habeat autem libra AB <lb/>centrum C in ip&longs;a libra, atq; <lb/>in eius medio: erit C libræ <lb/>centrum quoq; grauitatis; <lb/>à quo ip&longs;i AB, horizontiq; <lb/>perpendicularis ducatur FC <lb/>G. </s>

Line 858

<s id="id.2.1.51.2.1.10.0">quæ omnia ex iis, quæ &longs;upra dixi<lb/>mus, manife&longs;ta &longs;unt. </s>

<s id="id.2.1.51.3.1.1.0">His demon&longs;tratis. </s>

<s id="id.2.1.51.3.1.2.0">Manife&longs;tum e&longs;t, centrum libræ cau&longs;am e&longs;&longs;e <lb/>diuer&longs;itatis effectuum in libra. </s>

<s id="id.2.1.51.3.1.3.0">atq; patet omnes Archimedis de <lb/>æ&que;ponderantibus propo&longs;itiones ad hoc pertinentes in omni &longs;itu <lb/>veras e&longs;&longs;e. </s>

<s id="id.2.1.51.3.1.3.0">atq; patet omnes Archimedis de <lb/>æqueponderantibus propo&longs;itiones ad hoc pertinentes in omni &longs;itu <lb/>veras e&longs;&longs;e. </s>

<s id="id.2.1.51.3.1.4.0">hoc e&longs;t &longs;iue libra &longs;it horizonti æquidi&longs;tans, &longs;iue non: <lb/>dummodo centrum libræ in ip&longs;a &longs;it libra; &que;madmodum ip&longs;e <lb/>con&longs;iderat. </s>

<s id="id.2.1.51.3.1.5.0">& quamquam libra brachia habeat inæqualia, idem eue<lb/>niet; eodemq; pro&longs;us modo o&longs;tendetur, centrum libræ diuer&longs;imo<lb/>dè collocatum varios producere effectus. </s>

<s id="id.2.1.51.4.1.1.0">Sit enim libra AB hori<lb/>zonti æquidi&longs;tans; & in AB <lb/>&longs;int pondera inæqualia, quo <lb/>rum grauitatis centrum &longs;it <lb/>C: &longs;u&longs;pendaturq; libra in <lb/>eodem puncto C. </s>

<s id="N1208C">& mo<lb/>ueatur libra in DE. </s>

Line 885

<s id="id.2.1.53.2.1.1.0">Sit autem angulus ACB &longs;upra lineam AB; ac libræ centrum <lb/>&longs;it H; lineaq; CH libram &longs;u&longs;tineat; & moueatur libra in EKF: <lb/>libra EkF in ACB redibit. </s><pb n="30" xlink:href="036/01/073.jpg"/>

<s id="id.2.1.53.4.1.1.0">Si verò centrum libræ &longs;it D, quocunq; modo moueatur libra; <lb/>vbi relin&que;tur, manebit. </s>

<s id="id.2.1.53.4.1.1.0">Si verò centrum libræ &longs;it D, quocunq; modo moueatur libra; <lb/>vbi relinquetur, manebit. </s>

<s id="id.2.1.53.5.1.1.0">Si deinde punctum H &longs;it infra lineam AB; tunc libra EkF <lb/>deor&longs;um ex parte F mouebitur. </s>

<s id="id.2.1.53.6.1.1.0">Similiq; pror&longs;us ratione, &longs;i an<lb/>gulus ACB &longs;it infra lineam AB; <lb/>&longs;itq; libræ centrum H; &longs;u&longs;tineaturq; <lb/>libra linea CH; &longs;i libra ab hoc mo<lb/>ueatur &longs;itu, deor&longs;um ex parte pon<lb/>deris inferioris mouebitur. </s>

<s id="id.2.1.53.6.1.2.0">& &longs;i cen<lb/>trum libræ &longs;it D; vbi relin&que;tur, <lb/>manebit. </s>

<s id="id.2.1.53.6.1.2.0">& &longs;i cen<lb/>trum libræ &longs;it D; vbi relinquetur, <lb/>manebit. </s>

<s id="id.2.1.53.6.1.3.0">&longs;i verò &longs;it in K; &longs;i ab eiu&longs; <lb/><figure id="id.036.01.073.1.jpg" place="text" xlink:href="036/01/073/1.jpg"/><lb/>modi moueatur &longs;itu, in eundem pro&longs;us redibit. </s>

<s id="id.2.1.53.6.1.4.0">quæ omnia ex iis, <lb/>quæ in principio diximus, &longs;unt manife&longs;ta. </s>

<s id="id.2.1.53.6.1.5.0">&longs;imiliter &longs;i centrum li<lb/>bræ, vel in altero brachiorum, vel intra, vel extra vtcunq; po<lb/>natur; eadem inueniemus. </s><pb xlink:href="036/01/074.jpg"/>

Line 911

<s id="N1224E">quare ex æquali, vt CH ad HB, ita F <arrow.to.target n="note86"/><lb/>ad N. &longs;ed vt CH ad HB ita e&longs;t Q ad R: erit igitur Q ad R, vt <arrow.to.target n="note87"/><lb/>F ad N; & permutando, vt Q ad F, ita R ad N. </s>

<s id="N1225A">e&longs;t autem pars <arrow.to.target n="note88"/><lb/>Q ip&longs;i F æqualis; quare & pars R ip&longs;i N æqualis erit. </s>

<s id="id.2.1.53.10.1.7.0">Itaq; cùm <lb/>pondus L &longs;it ip&longs;i O æquale, & pondus F ip&longs;i Q etiam æquale, atq; <lb/>pars R ip&longs;i N æqualis; erunt pondera LM ip&longs;is EF ponderibus <lb/>æqualia. </s>

<s id="id.2.1.53.10.1.8.0">& quoniam e&longs;t, vt AC ad CG, ita pondus E ad pon<lb/>dus L; pondera EL æ&que;ponderabunt. </s>

<s id="id.2.1.53.10.1.8.0">& quoniam e&longs;t, vt AC ad CG, ita pondus E ad pon<lb/>dus L; pondera EL æqueponderabunt. </s>

<s id="id.2.1.53.10.1.9.0">&longs;imiliter quoniam e&longs;t, vt <arrow.to.target n="note89"/><lb/>AC ad CB, ita <expan abbr="pundus">pondus</expan> F ad pondus M; pondera quoq; FM <lb/>æ&que;ponderabunt. </s>

<s id="id.2.1.53.10.1.10.0">Pondera igitur LM ponderibus EF in BG <arrow.to.target n="note90"/><lb/>appen&longs;is æ&que;ponderabunt. </s>

<s id="id.2.1.53.10.1.10.0">Pondera igitur LM ponderibus EF in BG <arrow.to.target n="note90"/><lb/>appen&longs;is æqueponderabunt. </s>

<s id="id.2.1.53.10.1.11.0">cùm autem di&longs;tantia CA æqualis &longs;it <lb/>di&longs;tantiæ CH; &longs;i igitur vtraq; pondera EF in H appendantur, <lb/>pondera LM ip&longs;is EF ponderibus in H appen&longs;is æ&que;pondera<lb/>bunt. </s>

<s id="id.2.1.53.10.1.12.0">&longs;ed LM ip&longs;is EF in GB quoq; æ&que;ponderant: æquè <arrow.to.target n="note91"/><lb/>igitur grauia erunt pondera EF in GB, vt in H appen&longs;a. </s>

<s id="id.2.1.53.10.1.12.0">&longs;ed LM ip&longs;is EF in GB quoq; æqueponderant: æquè <arrow.to.target n="note91"/><lb/>igitur grauia erunt pondera EF in GB, vt in H appen&longs;a. </s>

<s id="id.2.1.53.10.1.13.0">tàm igi<lb/>tur ponderabunt in BG, quàm in H appen&longs;a. <figure id="id.036.01.075.1.jpg" place="text" xlink:href="036/01/075/1.jpg"/></s>

<s id="id.2.1.53.11.1.1.0">Sint autem pondera EF in CB appen&longs;a; &longs;itq; C libræ centrum; <lb/>& diuidatur CB in H, ita vt CH ad HB &longs;it, vt pondus in F ad <lb/>E. </s>

<s id="id.2.1.53.11.1.1.0.a">Dico pondera EF tàm in CB ponderare, quàm in puncto H. </s>

Line 927

<s id="N122D6">vt igitur G ad k, ita e&longs;t F ad E; & permutando vt G <lb/><arrow.to.target n="note94"/>ad F, ita k ad E. </s>

<s id="N122DD">&longs;unt autem GF æqualia; erunt & kE inter &longs;e <lb/>&longs;e æqualia. </s>

<s id="id.2.1.53.11.1.4.0">cùm itaq; pars G &longs;it ip&longs;i F æqualis, & K ip&longs;i E; erit <lb/>totum C k ip&longs;is EF ponderibus æquale. </s>

<s id="id.2.1.53.11.1.5.0">& quoniam AC e&longs;t ip<lb/>&longs;i CH æqualis; &longs;i igitur pondera EF ex puncto H &longs;u&longs;pendantur, <lb/>pondus D ip&longs;is EF in H appen&longs;is æ&que;ponderabit. </s>

<s id="id.2.1.53.11.1.6.0">&longs;ed & ip&longs;is <lb/>æ&que;ponderat in CB, hoc e&longs;t F in B, & E in C; cùm &longs;it vt AC <lb/>ad CB, ita F ad. D. </s>

<s id="id.2.1.53.11.1.6.0">&longs;ed & ip&longs;is <lb/>æqueponderat in CB, hoc e&longs;t F in B, & E in C; cùm &longs;it vt AC <lb/>ad CB, ita F ad. D. </s>

<s id="id.2.1.53.11.1.7.0">pondus enim E ex centro libræ C &longs;u&longs;pen<lb/>&longs;um non efficit, vt libra in alterutram moueatur partem. </s>

<s id="id.2.1.53.11.1.8.0">tàm igi<lb/>tur grauia erunt pondera EF in CB, quàm in H appen&longs;a. <pb n="32" xlink:href="036/01/077.jpg"/>

Line 944

<s id="N12345">erit vt AD ad DC, vt NM ad M; & diuidendo, vt <arrow.to.target n="note97"/><lb/>AC ad CD, ita N ad M: conuertendoq; vt DC ad CA, ita M <lb/>ad N. </s>

<s id="N1234E">vt autem DC ad CA, ita e&longs;t E ad H; erit igitur M ad N <arrow.to.target n="note98"/><lb/>vt E ad H; & permutando, vt M ad E, ita N ad H. </s>

<s id="N12355">&longs;ed ME <arrow.to.target n="note99"/><lb/>&longs;unt inter &longs;e æqualia, erunt NH inter &longs;e&longs;e quoq; æqualia. </s>

<s id="id.2.1.53.12.1.3.0">& quo<lb/>niam ita e&longs;t AC ad CD, vt H ad E: pondera HE æ&que;ponde<lb/>rabunt. <arrow.to.target n="note100"/></s>

<s id="id.2.1.53.12.1.3.0">& quo<lb/>niam ita e&longs;t AC ad CD, vt H ad E: pondera HE æqueponde<lb/>rabunt. <arrow.to.target n="note100"/></s>

<s id="id.2.1.53.12.1.4.0">&longs;imiliter quoniam e&longs;t vt GC ad CB, ita F ad k, ponde<pb xlink:href="036/01/078.jpg"/>

<figure id="id.036.01.078.1.jpg" place="text" xlink:href="036/01/078/1.jpg"/><lb/><arrow.to.target n="note101"/>ra etiam kF æ&que;ponderabunt. </s>

<figure id="id.036.01.078.1.jpg" place="text" xlink:href="036/01/078/1.jpg"/><lb/><arrow.to.target n="note101"/>ra etiam kF æqueponderabunt. </s>

<s id="id.2.1.53.12.1.5.0">pondera igitur Ek HF in li<lb/>bra AB, cuius centrum C, æ&que;ponderabunt. </s>

<s id="id.2.1.53.12.1.5.0">pondera igitur Ek HF in li<lb/>bra AB, cuius centrum C, æqueponderabunt. </s>

<s id="id.2.1.53.12.1.6.0">cùm autem GC <lb/>ip&longs;i CD &longs;it æqualis, & pondus H &longs;it ip&longs;i N æquale; pondera NH <lb/>æ&que;ponderabunt. </s>

<s id="id.2.1.53.12.1.6.0">cùm autem GC <lb/>ip&longs;i CD &longs;it æqualis, & pondus H &longs;it ip&longs;i N æquale; pondera NH <lb/>æqueponderabunt. </s>

<s id="id.2.1.53.12.1.7.0">& quoniam omnia æ&que;ponderant, demptis <lb/><arrow.to.target n="note102"/>HN ponderibus, quæ æ&que;ponderant, reliqua æ&que;ponderabunt; <lb/>hoc e&longs;t pondera EF & pondus LM ex centro libræ C &longs;u&longs;pen&longs;a. </s>

<s id="id.2.1.53.12.1.7.0">& quoniam omnia æqueponderant, demptis <lb/><arrow.to.target n="note102"/>HN ponderibus, quæ æqueponderant, reliqua æqueponderabunt; <lb/>hoc e&longs;t pondera EF & pondus LM ex centro libræ C &longs;u&longs;pen&longs;a. </s>

<s id="id.2.1.53.12.1.8.0"><lb/>quia verò pars L ip&longs;i F e&longs;t æqualis, & pars M ip&longs;i E æqualis; erit <lb/>totum LM ip&longs;is FE ponderibus &longs;imul &longs;umptis æquale. </s>

<s id="id.2.1.53.12.1.9.0">& cùm <lb/>&longs;it CG ip&longs;i CD æqualis, &longs;i igitur pondera EF ex puncto D &longs;u&longs;pen<lb/>dantur, pondera EF in D appen&longs;a ip&longs;i LM æ&que;ponderabunt. </s>

<s id="id.2.1.53.12.1.10.0">quare <lb/>LM tàm ip&longs;is EF in AB appen&longs;is æ&que;ponderat, quàm in pun<lb/>cto D appen&longs;is. </s>

<s id="id.2.1.53.12.1.10.0">quare <lb/>LM tàm ip&longs;is EF in AB appen&longs;is æqueponderat, quàm in pun<lb/>cto D appen&longs;is. </s>

<s id="id.2.1.53.12.1.11.0">libra enim &longs;emper eodem modo manet. </s>

<s id="id.2.1.53.12.1.12.0">Ponde<lb/><arrow.to.target n="note103"/>ra ergo EF tàm in AB ponderabunt, quàm in puncto D. </s>

<s id="id.2.1.53.12.1.9.0.a">quod <lb/><expan abbr="demon&longs;tre">demonstrare</expan> oportebat. </s>

Line 964

<s id="id.2.1.54.1.1.6.0"><margin.target id="note86"/>23 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.7.0"><margin.target id="note87"/>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.8.0"><margin.target id="note88"/>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.9.0"><margin.target id="note89"/>6 <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.9.0"><margin.target id="note89"/>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.11.0"><margin.target id="note90"/>2 <emph type="italics"/>Com. not. huius.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.14.0"><margin.target id="note91"/>3 <emph type="italics"/>Com. not. huius.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.16.0"><margin.target id="note92"/>17 <emph type="italics"/>Quinti. </s>

Line 977

<s id="id.2.1.54.1.1.23.0">Cor.<emph.end type="italics"/> 4 <emph type="italics"/>quinti<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.24.0"><margin.target id="note98"/>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.25.0"><margin.target id="note99"/>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.26.0"><margin.target id="note100"/>6 <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.26.0"><margin.target id="note100"/>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.28.0"><margin.target id="note101"/>2 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.30.0"><margin.target id="note102"/>1 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/></s>

<s id="id.2.1.54.1.1.32.0"><margin.target id="note103"/>3 <emph type="italics"/>Com.not. huius.<emph.end type="italics"/></s>

Line 986

<s id="id.2.1.55.2.1.1.0">Sit libra AB, cuius centrum C; &longs;intq; vt in primo ca&longs;u duo pon<lb/>dera EF ex punctis BG &longs;u&longs;pen&longs;a: &longs;itq; GH ad HB, vt pondus <lb/>F ad pondus E. </s>

<s id="id.2.1.55.2.1.1.0.a">Dico pondera EF tàm in GB ponderare, quàm <lb/>&longs;i vtraq; ex diui&longs;ionis puncto H &longs;u&longs;pendantur. </s>

<s id="id.2.1.55.2.1.2.0">Con&longs;truantur ea <lb/>dem, hoc e&longs;t fiat AC ip&longs;i CH æqualis, & ex puncto A duo ap<lb/>pendantur pondera LM, ita vt pondus E ad pondus L, &longs;it vt <lb/>CA ad CG; vt autem CB ad CA, ita &longs;it pondus M ad pondus <lb/>F. </s>

<s id="id.2.1.55.2.1.2.0.a">pondera LM ip&longs;is EF in GB appen&longs;is (vt &longs;upra dictum e&longs;t) <lb/>æ&que;ponderabunt. </s>

<s id="id.2.1.55.2.1.2.0.a">pondera LM ip&longs;is EF in GB appen&longs;is (vt &longs;upra dictum e&longs;t) <lb/>æqueponderabunt. </s>

<s id="id.2.1.55.2.1.3.0">Sint deinde puncta NO centra grauitatis pon<lb/>derum EF; connectanturq; GN BO; iungaturq; NO, quæ tan<lb/>quam libra erit; quæ etiam efficiat lineas GN BO inter &longs;e &longs;e æqui<lb/>di&longs;tantes e&longs;&longs;e; à punctoq; H horizonti perpendicularis ducatur <lb/>HP, quæ NO &longs;ecet in P, atq; ip&longs;is GN BO &longs;it æquidi&longs;tans. <lb/></s>

<s id="id.2.1.55.2.1.3.0.a">deniq; connectatur GO, quæ HP &longs;ecet in R. </s>

<s id="id.2.1.55.2.1.4.0">Quoniam igitur <lb/>HR e&longs;t lateri BO trianguli GBO æquidi&longs;tans; erit GH ad HB, <lb/>vt GR ad RO. </s>

Line 996

<s id="id.2.1.55.2.1.4.0.a">punctum ergo P centrum erit grauitatis magni<lb/>tudinis ex vtri&longs;q; EF ponderibus compo&longs;itæ. </s>

<s id="id.2.1.55.2.1.5.0">Intelligantur itaq; <arrow.to.target n="note106"/><lb/>pondera EF ita e&longs;&longs;e à libra NO connexa, ac &longs;i vna tantùm e&longs;&longs;et <lb/>magnitudo ex vtri&longs;q; EF compo&longs;ita, in puncti&longs;q; BG appen&longs;a. </s>

<s id="id.2.1.55.2.1.6.0">&longs;i <lb/>igitur ponderum &longs;u&longs;pen&longs;iones BG &longs;oluantur, manebunt pondera <arrow.to.target n="note107"/><lb/>EF ex HP &longs;u&longs;pen&longs;a; &longs;icuti in GB prius manebant. </s>

<s id="id.2.1.55.2.1.7.0">pondera verò EF <lb/>in GB appen&longs;a ip&longs;is LM ponderibus æ&que;ponderant, & pondera <pb xlink:href="036/01/080.jpg"/>

<s id="id.2.1.55.2.1.7.0">pondera verò EF <lb/>in GB appen&longs;a ip&longs;is LM ponderibus æqueponderant, & pondera <pb xlink:href="036/01/080.jpg"/>

<figure id="id.036.01.080.1.jpg" place="text" xlink:href="036/01/080/1.jpg"/><lb/>EF ex puncto H &longs;u&longs;pen&longs;a, eandem habent con&longs;titutionem ad li<lb/>bram AB, quam in BG appen&longs;a: eadem ergo pondera EF ex <lb/>H &longs;u&longs;pen&longs;a ei&longs;dem ponderibus LM æ&que;ponderabunt. </s>

<s id="id.2.1.55.2.1.8.0">æquè igi<lb/>tur &longs;unt grauia pondera EF in GB, vt in H appen&longs;a. <figure id="id.036.01.080.2.jpg" place="text" xlink:href="036/01/080/2.jpg"/></s>

<s id="id.2.1.55.3.1.1.0">Similiter demon&longs;trabitur, pondera EF in quibu&longs;cunq; aliis pun<lb/>ctis appen&longs;a tàm <expan abbr="põderare">ponderare</expan>, quàm &longs;i vtraq; ex diui&longs;ionis puncto H &longs;u<lb/>&longs;pendantur. </s>

<s id="id.2.1.55.3.1.2.0">&longs;i enim (vt &longs;upra docuimus) in libra pondera inue<lb/>niantur, quibus pondera EF æ&que;ponderent; eadem pondera EF <lb/>ex H &longs;u&longs;pen&longs;a ei&longs;dem inuentis ponderibus æ&que;ponderabunt; cùm <lb/>punctum P &longs;it &longs;emper eorum centrum grauitatis; & HP horizon <lb/>ri perpendicularis. </s>

<s id="id.2.1.55.3.1.2.0">&longs;i enim (vt &longs;upra docuimus) in libra pondera inue<lb/>niantur, quibus pondera EF æqueponderent; eadem pondera EF <lb/>ex H &longs;u&longs;pen&longs;a ei&longs;dem inuentis ponderibus æqueponderabunt; cùm <lb/>punctum P &longs;it &longs;emper eorum centrum grauitatis; & HP horizon <lb/>ri perpendicularis. </s>

<s id="id.2.1.56.1.1.1.0"><margin.target id="note104"/>2 <emph type="italics"/>Sexti.<emph.end type="italics"/></s>

<s id="id.2.1.56.1.1.2.0"><margin.target id="note105"/>11 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.56.1.1.3.0"><margin.target id="note106"/>6 <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.56.1.1.3.0"><margin.target id="note106"/>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.56.1.1.5.0"><margin.target id="note107"/>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s><pb n="34" xlink:href="036/01/081.jpg"/>

<s id="id.2.1.57.1.2.1.0">PROPOSITIO. VI. </s>

Line 1013

<s id="id.2.1.57.3.1.1.0.b">fiat enim vt <lb/>CA ad AD, ita pondus F ad aliud pondus, quod &longs;it G. </s>

<s id="id.2.1.57.3.1.1.0.c">Dico pri<lb/>múm pondera GF ex puncto C &longs;u&longs;pen&longs;a tantùm ponderare, quan<lb/>tùm pondera EF ex punctis DC. </s>

<s id="id.2.1.57.3.1.1.0.d">Secetur DC bifariam in H, & <lb/>ex H appendantur vtraq; pondera EF. </s>

<s id="N125BE">ponderabunt EF &longs;imul <lb/>&longs;umpta in eo &longs;itu, quantùm ponderant in DC. ponatur BA <arrow.to.target n="note108"/><lb/>æqualis AH, &longs;eceturq; BA in K, ita vt &longs;it KA æqualis AD: <lb/>deinde ex puncto B appendatur pondus L duplum ponderis F, <lb/>hoc e&longs;t æquale duobus ponderibus EF, quod quidem æ&que;ponde<lb/>rabit ponderibus EF in H appen&longs;is, hoc e&longs;t appen&longs;is in DC. </s>

<s id="N125BE">ponderabunt EF &longs;imul <lb/>&longs;umpta in eo &longs;itu, quantùm ponderant in DC. ponatur BA <arrow.to.target n="note108"/><lb/>æqualis AH, &longs;eceturq; BA in K, ita vt &longs;it KA æqualis AD: <lb/>deinde ex puncto B appendatur pondus L duplum ponderis F, <lb/>hoc e&longs;t æquale duobus ponderibus EF, quod quidem æqueponde<lb/>rabit ponderibus EF in H appen&longs;is, hoc e&longs;t appen&longs;is in DC. </s>

<s id="id.2.1.57.3.1.1.0.e"><expan abbr="Quoniã">Quoniam</expan><lb/>igitur, vt CA ad AD, ita e&longs;t pondus F ad pondus G; erit compo<lb/>nendo vt CA AD ad AD, hoc e&longs;t vt Ck ad AD, ita ponde<lb/>ra <arrow.to.target n="note109"/>FG ad pondus G. </s>

<s id="N125DC">&longs;ed cùm &longs;it, vt CA ad AD, ita F pon<lb/>dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <arrow.to.target n="note110"/><lb/>G ad pondus F; & con&longs;e&que;ntium dupla, vt DA ad duplam ip&longs;ius <lb/>AC, ita pondus G ad duplum ponderis F, hoc e&longs;t ad pondus <lb/>L. </s>

<s id="N125DC">&longs;ed cùm &longs;it, vt CA ad AD, ita F pon<lb/>dus ad pondus G; erit conuertendo, vt DA ad AC, ita pondus <arrow.to.target n="note110"/><lb/>G ad pondus F; & con&longs;equentium dupla, vt DA ad duplam ip&longs;ius <lb/>AC, ita pondus G ad duplum ponderis F, hoc e&longs;t ad pondus <lb/>L. </s>

<s id="id.2.1.57.3.1.1.0.f">Quare vt Ck ad DA, ita pondera EF ad pondus G; & vt <pb xlink:href="036/01/082.jpg"/>

<figure id="id.036.01.082.1.jpg" place="text" xlink:href="036/01/082/1.jpg"/><lb/><arrow.to.target n="note111"/>AD ad <expan abbr="duplã">duplam</expan> ip&longs;ius AC, ita pondus G ad pondus L; ergo ex æquali, <lb/>vt Ck ad <expan abbr="duplã">duplam</expan> ip&longs;ius AC, ita pondera FG ad pondus L. </s>

<s id="N12603">&longs;ed vt Ck <lb/>ad duplam AC, ita dimidia CK, videlicet AH, hoc e&longs;t BA, ad <lb/>AC. </s>

Line 1032

<s id="id.2.1.58.1.1.5.0"><margin.target id="note112"/>7 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.59.1.1.1.0">Si verò in libra <lb/>BAC pondera EF <lb/>æqualia ex punctis <lb/>BC &longs;u&longs;pendantur; &longs;i<lb/>militer dico pondus <lb/>E ad pondus F eam <lb/><figure id="id.036.01.082.2.jpg" place="text" xlink:href="036/01/082/2.jpg"/><lb/>in grauitate proportionem habere, quàm habet di&longs;tantia CA ad di<lb/>&longs;tantiam AB. </s>

<s id="id.2.1.59.1.1.1.0.a">fiat AD ip&longs;i AB æqualis, & ex puncto D &longs;u&longs;pen<lb/>datur pondus G æquale ponderi F; quod etiam ip&longs;i E erit æquale. </s>

<s id="id.2.1.59.1.1.2.0"><lb/>& quoniam AD e&longs;t æqualis ip&longs;i AB; pondera FG æ&que;ponde<lb/>rabunt, eandemq; habebunt grauitatem. </s>

<s id="id.2.1.59.1.1.2.0"><lb/>& quoniam AD e&longs;t æqualis ip&longs;i AB; pondera FG æqueponde<lb/>rabunt, eandemq; habebunt grauitatem. </s>

<s id="id.2.1.59.1.1.3.0">cùm autem grauitas pon<lb/>deris E ad grauitatem ponderis G &longs;it, vt CA ad AD; erit graui<lb/>tas ponderis E ad grauitatem ponderis F, vt CA ad AD, hoc e&longs;t <lb/>CA ad AB. quod erat quoq; o&longs;tendendum. </s><pb n="35" xlink:href="036/01/083.jpg"/>

<s id="id.2.1.59.3.1.1.0">ALITER. </s>

<s id="id.2.1.59.4.1.1.0">Sit libra BAC, cu<lb/>ius centrum A; in pun<lb/>ctis verò BC pondera <lb/>appendantur æqualia G <lb/>F: &longs;itq; primùm cen<lb/>trum A vtcun&que; inter <lb/>BC. </s>

<s id="id.2.1.59.4.1.1.0">Sit libra BAC, cu<lb/>ius centrum A; in pun<lb/>ctis verò BC pondera <lb/>appendantur æqualia G <lb/>F: &longs;itq; primùm cen<lb/>trum A vtcunque inter <lb/>BC. </s>

<s id="id.2.1.59.4.1.1.0.a">Dico pondus F ad <lb/>pondus G eam in graui<lb/><figure id="id.036.01.083.1.jpg" place="text" xlink:href="036/01/083/1.jpg"/><lb/>tate proportionem habere, quam habet di&longs;tantia CA ad di&longs;tan<lb/>tiam AB. </s>

<s id="id.2.1.59.4.1.1.0.b">fiat vt BA ad AC, ita pondus F ad aliud H, quod ap<lb/>pendatur in B: pondera HF ex A æ&que;ponderabunt. </s>

<s id="id.2.1.59.4.1.1.0.b">fiat vt BA ad AC, ita pondus F ad aliud H, quod ap<lb/>pendatur in B: pondera HF ex A æqueponderabunt. </s>

<s id="id.2.1.59.4.1.2.0">&longs;ed cùm <arrow.to.target n="note113"/><lb/>pondera FG &longs;int æqualia, habebit pondus H ad pondus G ean<lb/>dem proportionem, quam habet ad F. </s>

<s id="N126D2">vt igitur CA ad AB, ita <arrow.to.target n="note114"/><lb/>e&longs;t H ad G. </s>

<s id="N126D9">vt autem H ad G, ita e&longs;t grauitas ip&longs;ius H ad graui<lb/>tatem ip&longs;ius G; cùm in eodem puncto B &longs;int appen&longs;a. </s>

<s id="id.2.1.59.4.1.3.0">quare vt CA <lb/>ad AB, ita grauitas ponderis H ad grauitatem ponderis G. </s>

<s id="N126E2">cùm au<lb/>tem grauitas ponderis F in C appen&longs;i &longs;it æqualis grauitati ponderis <lb/>H in B; erit grauitas ponderis F ad grauitatem ponderis G, vt CA <lb/>ad AB, videlicet vt di&longs;tantia ad di&longs;tantiam. </s>

<s id="id.2.1.59.4.1.4.0">quod demon&longs;trare <lb/>oportebat. </s>

<s id="id.2.1.60.1.1.1.0"><margin.target id="note113"/>6 <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.60.1.1.1.0"><margin.target id="note113"/>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.60.1.1.3.0"><margin.target id="note114"/>7 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.61.1.1.1.0">Si verò libra B <lb/>AC &longs;ecetur vtcunq; <lb/>in D, & in DC ap<lb/>pendantur pondera <lb/>æqualia EF. </s>

<s id="id.2.1.61.1.1.1.0.a">Dico <lb/>&longs;imiliter ita e&longs;&longs;e gra<lb/><figure id="id.036.01.083.2.jpg" place="text" xlink:href="036/01/083/2.jpg"/><lb/>uitatem ponderis F ad grauitatem ponderis E, vt di&longs;tantia CA ad <lb/>di&longs;tantiam AD. </s>

<s id="id.2.1.61.1.1.1.0.b">fiat AB æqualis ip&longs;i AD, & in B appendatur <lb/>pondus G æquale ponderi E, & ponderi F. </s>

<s id="id.2.1.61.1.1.1.0.c">Quoniam enim AB e&longs;t <lb/>æqualis AD; pondera GE æ&que;ponderabunt. </s>

<s id="id.2.1.61.1.1.1.0.c">Quoniam enim AB e&longs;t <lb/>æqualis AD; pondera GE æqueponderabunt. </s>

<s id="id.2.1.61.1.1.2.0">&longs;ed cùm grauitas <lb/>ponderis F ad grauitatem ponderis G &longs;it, vt CA ad AB, & graui<lb/>tas ponderis E &longs;it æqualis grauitati ponderis G; erit grauitas pon<lb/>deris F ad grauitatem ponderis E, vt CA ad AB, hoc e&longs;t vt CA <lb/>ad AD. </s>

<s id="N12738">quod demon&longs;trare oportebat. </s><pb xlink:href="036/01/084.jpg"/>

<s id="id.2.1.61.3.1.1.0">COROLLARIVM. </s>

<s id="id.2.1.61.4.1.1.0">Ex hoc manife&longs;tum e&longs;t, quò pondus à centro <lb/>libræ magis di&longs;tat, eò grauius e&longs;&longs;e; & per con&longs;e<lb/>&que;ns velocius moueri. </s>

<s id="id.2.1.61.4.1.1.0">Ex hoc manife&longs;tum e&longs;t, quò pondus à centro <lb/>libræ magis di&longs;tat, eò grauius e&longs;&longs;e; & per con&longs;e<lb/>quens velocius moueri. </s>

<s id="id.2.1.61.5.1.1.0"><arrow.to.target n="note115"/>Hinc præterea &longs;tateræ quoq; ratio facilè o&longs;ten<lb/>detur. </s>

<s id="id.2.1.62.1.1.1.0"><margin.target id="note115"/><emph type="italics"/>Stateræ ratio.<emph.end type="italics"/></s>

<s id="id.2.1.63.1.1.1.0">Sit enim &longs;tate<lb/>ræ &longs;capus AB, cu<lb/>ius trutina &longs;it in <lb/>C; &longs;itq; &longs;tateræ <lb/>appendiculum E. <lb/></s>

<s id="N12773">appendatur in A <lb/>pondus D, quod <lb/>æ&que;ponderet ap<lb/>pendiculo E in F <lb/><figure id="id.036.01.084.1.jpg" place="text" xlink:href="036/01/084/1.jpg"/><lb/>appen&longs;o. </s>

<s id="N12773">appendatur in A <lb/>pondus D, quod <lb/>æqueponderet ap<lb/>pendiculo E in F <lb/><figure id="id.036.01.084.1.jpg" place="text" xlink:href="036/01/084/1.jpg"/><lb/>appen&longs;o. </s>

<s id="id.2.1.63.1.1.2.0">aliud quoq; appendatur pondus G in A, quod etiam <lb/>appendiculo E in B appen&longs;o æ&que;ponderet. </s>

<s id="id.2.1.63.1.1.2.0">aliud quoq; appendatur pondus G in A, quod etiam <lb/>appendiculo E in B appen&longs;o æqueponderet. </s>

<s id="id.2.1.63.1.1.3.0">Dico grauitatem <lb/>ponderis D ad grauitatem ponderis G ita e&longs;&longs;e, vt CF ad CB. </s>

<s id="id.2.1.63.1.1.3.0.a"><lb/>Quoniam enim grauitas ponderis D e&longs;t æqualis grauitati ponde<lb/>ris E in F appen&longs;i, & grauitas ponderis G e&longs;t æqualis grauitati pon<lb/>deris E in B; erit grauitas ponderis D ad grauitatem ponderis E in <lb/>F, vt grauitas ponderis G ad grauitatem ponderis E in B: & permu<lb/><arrow.to.target n="note116"/>tando, vt grauitas ponderis D ad grauitatem ponderis G, ita graui<lb/>tas ip&longs;ius E in F, ad grauitatem ip&longs;ius E in B; grauitas autem pon<lb/><arrow.to.target n="note117"/>deris E in F ad grauitatem ponderis E in B e&longs;t, vt CF ad CB; vt <lb/>igitur grauitas ponderis D ad grauitatem ponderis G, ita e&longs;t CF <lb/>ad CB. </s>

<s id="id.2.1.63.1.1.3.0.b">&longs;i ergo pars &longs;capi CB in partes diuidatur æquales, &longs;olo <lb/>pondere E, & propius, & longius à puncto C po&longs;ito; ponderum <lb/>grauitates, quæ ex puncto A &longs;u&longs;penduntur inter &longs;e &longs;e notæ erunt. </s>

Line 1068

<s id="id.2.1.64.1.1.1.0"><margin.target id="note116"/>16 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.64.1.1.2.0"><margin.target id="note117"/>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.65.1.1.1.0">Alio quoq; modo &longs;tatera vti po&longs;&longs;umus, vt <lb/>ponderum grauitates notæ reddantur. </s>

<s id="id.2.1.65.2.1.1.0">Sit &longs;capus AB, cuius tru<lb/>tina &longs;it in C; &longs;itq; &longs;tateræ ap<lb/>pendiculum E, quod appen<lb/>datur in A; &longs;int&queacute; pon<lb/>dera DG inæqualia, quorum <lb/>inter &longs;e &longs;e grauitatum propor<lb/>tiones quærimus: appenda<lb/>tur pondus D in B, ita vt ip&longs;i <lb/><figure id="id.036.01.085.1.jpg" place="text" xlink:href="036/01/085/1.jpg"/><lb/>E æ&que;ponderet. </s>

<s id="id.2.1.65.2.1.2.0">&longs;imiliter pondus G appendatur in F, quod ei<lb/>dem ponderi E æ&que;ponderet. </s>

<s id="id.2.1.65.2.1.2.0">&longs;imiliter pondus G appendatur in F, quod ei<lb/>dem ponderi E æqueponderet. </s>

<s id="id.2.1.65.2.1.3.0">dico D ad G ita e&longs;&longs;e, vt CF ad <lb/>CB. </s>

<s id="id.2.1.65.2.1.3.0.a">Quoniam enim pondera DE æ&que;ponderant, erit D ad E, <arrow.to.target n="note118"/><lb/>vt CA ad 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"/><lb/>vt CA ad CB. </s>

<s id="N12801">cùm autem pondera quo&que; GE æ&que;pon<lb/>derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua <lb/>li pondus D ad pondus G ita erit, vt CF ad CB. </s>

<s id="N12801">cùm autem pondera quoque GE æquepon<lb/>derent, erit pondus E ad pondus G, vt FC ad CA; quare ex æqua <lb/>li pondus D ad pondus G ita erit, vt CF ad CB. </s>

<s id="N12807">quod o&longs;tende<arrow.to.target n="note119"/><lb/>re quoq; oportebat. </s>

<s id="id.2.1.66.1.1.1.0"><margin.target id="note118"/>6 <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.66.1.1.1.0"><margin.target id="note118"/>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.66.1.1.3.0"><margin.target id="note119"/>23 <emph type="italics"/>Quinti.<emph.end type="italics"/></s><pb xlink:href="036/01/086.jpg"/>

<s id="id.2.1.67.1.2.1.0">PROPOSITIO VII. </s>

<s id="id.2.1.67.1.4.1.0">PROBLEMA. </s>

<s id="id.2.1.67.2.1.1.0">Quotcun&que; datis in libra ponderibus <lb/>vbicun&que; appen&longs;is, centrum libræ inuenire, <lb/>ex quo &longs;i &longs;u&longs;pendatur libra, data pondera ma<lb/>neant. <figure id="id.036.01.086.1.jpg" place="text" xlink:href="036/01/086/1.jpg"/></s>

<s id="id.2.1.67.2.1.1.0">Quotcunque datis in libra ponderibus <lb/>vbicunque appen&longs;is, centrum libræ inuenire, <lb/>ex quo &longs;i &longs;u&longs;pendatur libra, data pondera ma<lb/>neant. <figure id="id.036.01.086.1.jpg" place="text" xlink:href="036/01/086/1.jpg"/></s>

<s id="id.2.1.67.3.1.1.0">Sit libra AB, &longs;intq; data quotcun&que; pondera CDEFG. <lb/></s>

<s id="id.2.1.67.3.1.1.0">Sit libra AB, &longs;intq; data quotcunque pondera CDEFG. <lb/></s>

<s id="id.2.1.67.3.1.1.0.a">accipiantur in libra vtcun&que; puncta AHkLB, ex quibus <lb/>data pondera <expan abbr="&longs;pu&longs;pendantur">suspendantur</expan>. </s>

<s id="id.2.1.67.3.1.1.0.a">accipiantur in libra vtcunque puncta AHkLB, ex quibus <lb/>data pondera <expan abbr="&longs;pu&longs;pendantur">suspendantur</expan>. </s>

<s id="id.2.1.67.3.1.2.0">Centrum libræ inuenire oportet, <lb/>ex quo &longs;i fiat &longs;u&longs;pen&longs;io, data pondera maneant. </s>

<s id="id.2.1.67.3.1.3.0">Diuidatur <pb n="37" xlink:href="036/01/087.jpg"/>

<figure id="id.036.01.087.1.jpg" place="text" xlink:href="036/01/087/1.jpg"/><lb/>AH in M, ita vt HM ad MA, &longs;it vt grauitas ponderis <lb/>C ad grauitatem ponderis D. </s>

<s id="id.2.1.67.3.1.3.0.a">deinde diuidatur BL in N, ita <lb/>vt LN ad NB, &longs;it vt grauitas ponderis G ad grauitatem pon<lb/>deris F. </s>

<s id="N12870">diuidaturq; MN in O, ita vt MO ad ON &longs;it, vt <lb/>grauitas ponderum FG ad grauitatem ponderum CD. </s>

<s id="id.2.1.67.3.1.3.0.b">tandem<lb/>què diuidatur kO in P, ita vt kP ad PO, &longs;it vt grauitas pon<lb/>derum CDFG ad grauitatem ponderis E. </s>

<s id="id.2.1.67.3.1.3.0.c">Quoniam igitur pon<lb/>dera CDFG tàm ponderant in O, quàm CD in M, & FG in N; <arrow.to.target n="note120"/><lb/>æ&que;ponderabunt pondera CD in M, & FG in N, & pondus E <lb/>in K, &longs;i ex puncto P &longs;u&longs;pendantur. </s>

<s id="id.2.1.67.3.1.3.0.c">Quoniam igitur pon<lb/>dera CDFG tàm ponderant in O, quàm CD in M, & FG in N; <arrow.to.target n="note120"/><lb/>æqueponderabunt pondera CD in M, & FG in N, & pondus E <lb/>in K, &longs;i ex puncto P &longs;u&longs;pendantur. </s>

<s id="id.2.1.67.3.1.4.0">cùm verò pondera CD tan<lb/>tùm ponderent in M, quantùm in AH, & FG in N, quantùm <lb/>in LB; pondera CDFG ex AHLB punctis &longs;u&longs;pen&longs;a, & pon<lb/>dus E ex k, &longs;i ex P &longs;u&longs;pendantur, æ&que;ponderabunt, atq; mane<lb/>bunt. </s>

<s id="id.2.1.67.3.1.5.0">Inuentum e&longs;t ergo centrum libræ P, ex quo data pondera <lb/>manent. </s>

<s id="id.2.1.67.3.1.6.0">quod facere oportebat. </s>

<s id="id.2.1.68.1.1.1.0"><margin.target id="note120"/>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s><pb xlink:href="036/01/088.jpg"/>

Line 1113

<s id="id.2.1.71.3.1.1.0">Sit vectis AB, cuius fulcimentum C; &longs;itq; pondus D ex A &longs;u<lb/>&longs;pen&longs;um AH, ita vt AH &longs;it &longs;emper horizonti perpendicularis: <lb/>&longs;itq; potentia &longs;u&longs;tinens pondus in B. </s>

<s id="id.2.1.71.3.1.1.0.a">Dico potentiam in B ad pon<lb/>dus D ita e&longs;&longs;e, vt CA ad CB. </s>

<s id="id.2.1.71.3.1.1.0.b">fiat vt BC ad CA, ita pondus D <lb/><arrow.to.target n="note122"/>ad aliud pondus E, quippè quod &longs;i in B appendatur; ip&longs;i D æ&que; <lb/>ponderabit, exi&longs;tente C amborum grauitatis centro. </s>

<s id="id.2.1.71.3.1.1.0.b">fiat vt BC ad CA, ita pondus D <lb/><arrow.to.target n="note122"/>ad aliud pondus E, quippè quod &longs;i in B appendatur; ip&longs;i D æque <lb/>ponderabit, exi&longs;tente C amborum grauitatis centro. </s>

<s id="id.2.1.71.3.1.2.0">quare poten<lb/>tia æqualis ip&longs;i E ibidem con&longs;tituta ip&longs;i D æ&que;ponderabit, vecte <lb/>AB, eius fulcimento in C collocato, hoc e&longs;t prohibebit, ne pon<lb/>dus D deor&longs;um vergat, &que;madmodum prohibet pondus E. </s>

<s id="id.2.1.71.3.1.2.0">quare poten<lb/>tia æqualis ip&longs;i E ibidem con&longs;tituta ip&longs;i D æqueponderabit, vecte <lb/>AB, eius fulcimento in C collocato, hoc e&longs;t prohibebit, ne pon<lb/>dus D deor&longs;um vergat, quemadmodum prohibet pondus E. </s>

<s id="id.2.1.71.3.1.2.0.a">Po<lb/><arrow.to.target n="note123"/>tentia verò in B ad pondus D eandem habet proportionem, quam <lb/>pondus E ad idem pondus D: ergo potentia in B ad pondus D <lb/>erit, vt CA ad CB; hoc e&longs;t vectis di&longs;tantia à fulcimento ad pon<lb/>deris &longs;u&longs;pendium ad di&longs;tantiam à fulcimento ad potentiam. </s>

<s id="id.2.1.71.3.1.3.0">quod <lb/>demon&longs;trare oportebat. </s>

<s id="id.2.1.72.1.1.1.0"><margin.target id="note122"/>6 <emph type="italics"/>Primi Archim. de æ&que;p.<emph.end type="italics"/></s>

<s id="id.2.1.72.1.1.1.0"><margin.target id="note122"/>6 <emph type="italics"/>Primi Archim. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.72.1.1.3.0"><margin.target id="note123"/><emph type="italics"/>Ex<emph.end type="italics"/> 7 <emph type="italics"/>quinti.<emph.end type="italics"/></s>

<s id="id.2.1.73.1.1.1.0">Hinc facilè o&longs;tendi pote&longs;t, fulcimentum quò <lb/>ponderi fuerit propius, minorem ad idem pon<lb/>dus &longs;u&longs;tinendum requiri potentiam. </s>

<s id="id.2.1.73.2.1.1.0">Ii&longs;dem po&longs;i<lb/>tis, &longs;it fulcimen <lb/>tum in F ip&longs;i A <lb/>propius, quàm <lb/>C; fiatq; vt BF <lb/>ad FA, ita pon<lb/>dus D ad aliud <lb/><figure id="id.036.01.090.2.jpg" place="text" xlink:href="036/01/090/2.jpg"/><lb/>G, quod &longs;i appendatur in B, pondera DG ex fulcimento E <lb/><arrow.to.target n="note124"/>æ&que;ponderabunt. </s>

<s id="id.2.1.73.2.1.2.0">quoniam autem BF maior e&longs;t BC, & CA <lb/><arrow.to.target n="note125"/>maior AC; maior erit proportio BF ad FA, quàm BC ad CA: <pb n="39" xlink:href="036/01/091.jpg"/>& ideo maior quoq; erit proportio ponderis D ad pondus G, <lb/>quàm idem D ad E: pondus igitur G minus erit pondere E. cùm <arrow.to.target n="note126"/><lb/>autem potentia in B ip&longs;i G æqualis ponderi D æ&que;ponderet, mi<lb/>nor potentia, quàm ea, quæ ponderi E e&longs;t æqualis, pondus D &longs;u<lb/>&longs;tinebit; exi&longs;tente vecte AB, eius verò fulcimento vbi F, quàm &longs;i <lb/>fuerit vbi C. &longs;imiliter quoq; o&longs;tendetur, quò propius erit fulci<lb/>mentum ponderi D, adhuc &longs;emper minorem requiri potentiam <lb/>ad &longs;u&longs;tinendum pondus D. </s>

<s id="id.2.1.73.2.1.2.0">quoniam autem BF maior e&longs;t BC, & CA <lb/><arrow.to.target n="note125"/>maior AC; maior erit proportio BF ad FA, quàm BC ad CA: <pb n="39" xlink:href="036/01/091.jpg"/>& ideo maior quoq; erit proportio ponderis D ad pondus G, <lb/>quàm idem D ad E: pondus igitur G minus erit pondere E. cùm <arrow.to.target n="note126"/><lb/>autem potentia in B ip&longs;i G æqualis ponderi D æqueponderet, mi<lb/>nor potentia, quàm ea, quæ ponderi E e&longs;t æqualis, pondus D &longs;u<lb/>&longs;tinebit; exi&longs;tente vecte AB, eius verò fulcimento vbi F, quàm &longs;i <lb/>fuerit vbi C. &longs;imiliter quoq; o&longs;tendetur, quò propius erit fulci<lb/>mentum ponderi D, adhuc &longs;emper minorem requiri potentiam <lb/>ad &longs;u&longs;tinendum pondus D. </s>

<s id="id.2.1.74.1.1.1.0"><margin.target id="note124"/><emph type="italics"/>Ex eadem Sexta.<emph.end type="italics"/></s>

<s id="id.2.1.74.1.1.2.0"><margin.target id="note125"/><emph type="italics"/>Lemma.<emph.end type="italics"/></s>

<s id="id.2.1.74.1.1.3.0"><margin.target id="note126"/>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

Line 1158

<s id="id.2.1.77.2.1.1.0">Sit vectis AB, cuius fulcimentum &longs;it B, & pondus E ex puncto <lb/>C &longs;u&longs;pen&longs;um; &longs;itq; vis in A &longs;u&longs;tinens pondus E. </s>

<s id="id.2.1.77.2.1.1.0.a">Dico vt BC ad BA, <lb/>ita e&longs;&longs;e potentiam in A ad pondus E. </s>

<s id="id.2.1.77.2.1.1.0.b">Producatur AB in C, & <lb/>fiat BD æqualis BC; & ex puncto D appendatur pondus F æqua <lb/>le ponderi E; itemq; ex puncto A &longs;u&longs;pendatur pondus G ita, vt <lb/>pondus F ad pondus G eandem habeat proportionem, quam AB <pb n="40" xlink:href="036/01/093.jpg"/>ad BA. </s>

<s id="N12AF6">pondera FG æ&que;ponderabunt. </s>

<s id="N12AF6">pondera FG æqueponderabunt. </s>

<s id="id.2.1.77.2.1.2.0">cùm autem &longs;it CB æqua <lb/>lis BD, pondera quoq; FE æqualia æ&que;ponderabunt. </s>

<s id="id.2.1.77.2.1.2.0">cùm autem &longs;it CB æqua <lb/>lis BD, pondera quoq; FE æqualia æqueponderabunt. </s>

<s id="id.2.1.77.2.1.3.0">pondera <lb/>verò FEG in libra, &longs;eu vecte DBA appen&longs;a, cuius fulcimentum <lb/>e&longs;t B, non æ&que;ponderabunt; &longs;ed ex parte A deor&longs;um tendent. </s>

<s id="id.2.1.77.2.1.3.0">pondera <lb/>verò FEG in libra, &longs;eu vecte DBA appen&longs;a, cuius fulcimentum <lb/>e&longs;t B, non æqueponderabunt; &longs;ed ex parte A deor&longs;um tendent. </s>

<s id="id.2.1.77.2.1.4.0">po<lb/>natur itaq; in A tanta vis, vt pondera FEG æ&que;ponderent; erit <lb/>potentia in A æqualis ponderi G. </s>

<s id="id.2.1.77.2.1.4.0">po<lb/>natur itaq; in A tanta vis, vt pondera FEG æqueponderent; erit <lb/>potentia in A æqualis ponderi G. </s>

<s id="N12B0B">pondera enim FE <expan abbr="æ&que;ponderãt">æ&que;ponderant</expan>, <lb/>& vis in A nihil aliud efficere debet, ni&longs;i &longs;u&longs;tinere <expan abbr="põdus">pondus</expan> G, ne de&longs;cen<lb/>dat. </s>

<s id="N12B0B">pondera enim FE <expan abbr="æqueponderãt">æqueponderant</expan>, <lb/>& vis in A nihil aliud efficere debet, ni&longs;i &longs;u&longs;tinere <expan abbr="põdus">pondus</expan> G, ne de&longs;cen<lb/>dat. </s>

<s id="id.2.1.77.2.1.5.0">& quoniam pondera FEG, & potentia in A æ&que;ponderant, <lb/>demptis igitur FG ponderibus, quæ æ&que;ponderant, reliqua æ&que; <lb/>ponderabunt; &longs;cilicet potentia in A ponderi E, hoc e&longs;t potentia <lb/>in A pondus E &longs;u&longs;tinebit, ita vt vectis AB maneat, vt prius erat. </s>

<s id="id.2.1.77.2.1.5.0">& quoniam pondera FEG, & potentia in A æqueponderant, <lb/>demptis igitur FG ponderibus, quæ æqueponderant, reliqua æque <lb/>ponderabunt; &longs;cilicet potentia in A ponderi E, hoc e&longs;t potentia <lb/>in A pondus E &longs;u&longs;tinebit, ita vt vectis AB maneat, vt prius erat. </s>

<s id="id.2.1.77.2.1.6.0"><lb/>Cùm autem potentia in A &longs;it æqualis ponderi G, & pondus E pon<lb/>deri F æquale; habebit potentia in A ad pondus E eandem pro<lb/>portionem, quam habet BD, hoc e&longs;t BC ad BA. </s>

<s id="N12B2A">quod demon<lb/>&longs;trare oportebat. </s>

<s id="id.2.1.77.3.1.1.0">COROLLARIVM I. </s>

Line 1213

<s id="id.2.1.83.2.1.1.0">Sit vectis AB, cuius fulcimentum B; & ex puncto A &longs;it pon<lb/>dus C &longs;u&longs;pen&longs;um; &longs;itq; potentia in D &longs;u&longs;tinens pondus C. </s>

<s id="id.2.1.83.2.1.1.0.a">Dico <lb/>vt AB ad BD, ita e&longs;&longs;e potentiam in D ad pondus C. </s>

<s id="id.2.1.83.2.1.1.0.b">Produca<lb/>tur AB in E, fiatq; BE æqualis ip&longs;i BA; & ex puncto E appen<lb/>datur pondus F æquale ponderi C; & vt BD ad BE, ita fiat pon<lb/>dus F ad aliud G, quod ex puncto D &longs;u&longs;pendatur. </s>

<s id="id.2.1.83.2.1.2.0">pondera FG <lb/>æ&que;ponderabunt. </s>

<s id="id.2.1.83.2.1.2.0">pondera FG <lb/>æqueponderabunt. </s>

<s id="id.2.1.83.2.1.3.0">& quoniam AB e&longs;t æqualis BE, & pondera <lb/>FC æqualia; &longs;imiliter pondera FC æ&que;ponderabunt. </s>

<s id="id.2.1.83.2.1.3.0">& quoniam AB e&longs;t æqualis BE, & pondera <lb/>FC æqualia; &longs;imiliter pondera FC æqueponderabunt. </s>

<s id="id.2.1.83.2.1.4.0">Pondera <lb/>verò FGC &longs;u&longs;pen&longs;a in vecte EBA, cuius fulcimentum e&longs;t B, non <lb/>æ&que;ponderabunt; &longs;ed ex parte A deor&longs;um tendent. </s>

<s id="id.2.1.83.2.1.4.0">Pondera <lb/>verò FGC &longs;u&longs;pen&longs;a in vecte EBA, cuius fulcimentum e&longs;t B, non <lb/>æqueponderabunt; &longs;ed ex parte A deor&longs;um tendent. </s>

<s id="id.2.1.83.2.1.5.0">Ponatur igi<lb/>tur in D tanta vis, vt pondera FGC æ&que;ponderent; erit po<lb/>tentia in D æqualis ponderi G: pondera enim FC æ&que;ponde<lb/>rant, & potentia in D nil aliud efficere debet, ni&longs;i &longs;u&longs;tinere pon<lb/>dus G ne de&longs;cendat. </s>

<s id="id.2.1.83.2.1.5.0">Ponatur igi<lb/>tur in D tanta vis, vt pondera FGC æqueponderent; erit po<lb/>tentia in D æqualis ponderi G: pondera enim FC æqueponde<lb/>rant, & potentia in D nil aliud efficere debet, ni&longs;i &longs;u&longs;tinere pon<lb/>dus G ne de&longs;cendat. </s>

<s id="id.2.1.83.2.1.6.0">& quoniam pondera FGC, & potentia in <lb/>D æ&que;ponderant, demptis igitur FG ponderibus, quæ æ&que;pon<lb/>derant; reliqua æ&que;ponderabunt, &longs;cilicet potentia in D ponderi C. <lb/></s>

<s id="id.2.1.83.2.1.6.0">& quoniam pondera FGC, & potentia in <lb/>D æqueponderant, demptis igitur FG ponderibus, quæ æquepon<lb/>derant; reliqua æqueponderabunt, &longs;cilicet potentia in D ponderi C. <lb/></s>

<s id="N12CDC">hoc e&longs;t potentia in D pondus C &longs;u&longs;tinebit, ita vt vectis AB ma<lb/>neat, vt prius. </s>

<s id="id.2.1.83.2.1.7.0">& cùm potentia in D &longs;it æqualis ponderi G, & pon<lb/>dus C æquale ponderi F; habebit potentia in D ad pondus C ean<lb/>dem proportionem, quam EB, hoc e&longs;t AB ad BD. </s>

<s id="id.2.1.83.2.1.7.0.a">quod de<lb/>mon&longs;trare oportebat. </s>

Line 1332

<s id="id.2.1.94.1.1.3.0"><margin.target id="note150"/>25 <emph type="italics"/>Primi.<emph.end type="italics"/></s>

<s id="id.2.1.94.1.1.4.0"><margin.target id="note151"/>5 <emph type="italics"/>Primi.<emph.end type="italics"/></s>

<s id="id.2.1.94.1.1.5.0"><margin.target id="note152"/>26 <emph type="italics"/>Primi.<emph.end type="italics"/></s>

<s id="id.2.1.95.1.1.1.0">In&longs;uper &longs;i intra BG BE alia vtcunq; ducatur linea ip&longs;i BG æ<lb/>qualis; fiatq; operatio, &que;madmodum &longs;upra dictum e&longs;t; &longs;imili<lb/>ter o&longs;tendetur lineam BR minorem e&longs;&longs;e BN. </s>

<s id="id.2.1.95.1.1.1.0">In&longs;uper &longs;i intra BG BE alia vtcunq; ducatur linea ip&longs;i BG æ<lb/>qualis; fiatq; operatio, quemadmodum &longs;upra dictum e&longs;t; &longs;imili<lb/>ter o&longs;tendetur lineam BR minorem e&longs;&longs;e BN. </s>

<s id="id.2.1.95.1.1.1.0.a">& quò propius fue<lb/>rit ip&longs;i BE, adhuc minorem &longs;emper e&longs;&longs;e. </s><pb xlink:href="036/01/108.jpg"/>

<s id="id.2.1.95.3.1.1.0">Si verò æqualia triangula BFH BGK &longs;int <lb/>deor&longs;um inter BC BA con&longs;tituta; connectan<lb/>turq; HC KC, quæ lineas BF BG ex parte <lb/>FG productas in punctis MN &longs;ecent erit BN <lb/>maior BM, & BM ip&longs;a BA. </s>

Line 1417

<s id="id.2.1.100.1.1.4.0"><margin.target id="note165"/>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.100.1.1.5.0"><margin.target id="note166"/>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.101.1.1.1.0">Hinc quoq; vt &longs;upra patet pontentiam in A ad potentiam in E e&longs; <lb/>&longs;e, vt BL ad BM; potentiamq; in A ad potentiam in O, vt BL <lb/>ad BS. </s>

<s id="id.2.1.101.1.1.1.0.a">at&que; potentiam in E ad potentiam in O, vt BM <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/>ad BS. </s>

<s id="id.2.1.101.2.1.1.0">Præterea &longs;i in B alia intelligatur potentia, ita vt duæ &longs;int poten<lb/>tiæ pondus &longs;u&longs;tinentes; minor erit potentia in B &longs;u&longs;tinens pon<lb/>dus PQ vecte BO, quàm pondus CD vecte BA aduer&longs;o au<lb/>tem maior requiritur potentia in B ad &longs;u&longs;tinendum pondus FG ve<lb/>cte BE, quàm pondus CD vecte AB. </s>

<s id="N1352F">ducta enim kN ip&longs;i EB <lb/>perpendicularis, erit EN ip&longs;i AL æqualis: quare EM ip&longs;a LA <lb/>maior erit. </s>

<s id="id.2.1.101.2.1.2.0">ergo maiorem habebit proportionem EM ad E<emph type="italics"/>B<emph.end type="italics"/>, <arrow.to.target n="note167"/><lb/>quàm LA ad A<emph type="italics"/>B<emph.end type="italics"/>; & LA ad A<emph type="italics"/>B<emph.end type="italics"/> maiorem, quàm SO ad O<emph type="italics"/>B<emph.end type="italics"/>; <arrow.to.target n="note168"/><lb/>quæ &longs;unt proportiones potentiæ ad pondus. </s>

Line 1437

<s id="N13655">maiorem enim habet proportionem SB ad BN, quàm LB <lb/>ad BD. </s>

<s id="N13659">o&longs;tendetur etiam, quò magis pondus deprimetur; ma<lb/>iorem &longs;emper (vt &longs;u&longs;tineatur) requiri potentiam. </s>

<s id="id.2.1.105.1.1.5.0">quod demon<lb/>&longs;trare oportebat. </s>

<s id="id.2.1.105.2.1.1.0">Hinc quoq; li&que;t potentias in GDN inter &longs;e &longs;e ita e&longs;&longs;e, vt <lb/>BM ad BL, atq; vt BL ad BS, deniq; vt BM ad BS. </s>

<s id="id.2.1.105.2.1.1.0">Hinc quoq; liquet potentias in GDN inter &longs;e &longs;e ita e&longs;&longs;e, vt <lb/>BM ad BL, atq; vt BL ad BS, deniq; vt BM ad BS. </s>

<s id="id.2.1.105.3.1.1.0">COROLLARIVM. </s>

<s id="id.2.1.105.4.1.1.0">Ex his manife&longs;tum e&longs;t; &longs;i potentia vecte &longs;ur<lb/>&longs;um moueat pondus, cuius centrum grauitatis <lb/>&longs;it &longs;upra vectem, quò magis pondus eleuabitur; <lb/>&longs;emper minorem potentiam requiri vt pondus <lb/>moueatur. </s>

<s id="id.2.1.105.5.1.1.0">Vbi enim potentia pondus &longs;u&longs;tinens e&longs;t &longs;emper minor, erit <lb/>quoq; potentia ip&longs;um mouens &longs;emper minor. <pb n="52" xlink:href="036/01/117.jpg"/>

<s id="id.2.1.105.6.1.1.0">Ex iis etiam demon&longs;trabitur, &longs;i centrum grauitatis eiu&longs;dem pon<lb/>deris, &longs;iue propinquius, &longs;iue remotius fuerit à vecte AB horizon<lb/>ti æquidi&longs;tante, eandem potentiam in A pondus nihilominus <lb/>&longs;u&longs;tinere: vt &longs;i centrum grauitatis H ponderis BD longius ab&longs;it <lb/>à vecte BA, quàm centrum grauitatis N ponderis PV, dum<lb/>modo ducta à puncto H perpendicularis HL horizonti, vectiq; <lb/>AB tran&longs;eat per N; &longs;itq; pondus PV ponderi BD æquale; <lb/>erit tùm pondus BD, tùm pondus PV, ac &longs;i ambo in L e&longs;<lb/>&longs;ent appen&longs;a; at&que; &longs;unt æqualia, cùm loco vnius ponderis ac<lb/>cipiantur, eadem igitur potentia in A &longs;u&longs;tinens pondus BD, <lb/>pondus quoq; PV &longs;u&longs;tinebit. </s>

<s id="id.2.1.105.6.1.1.0">Ex iis etiam demon&longs;trabitur, &longs;i centrum grauitatis eiu&longs;dem pon<lb/>deris, &longs;iue propinquius, &longs;iue remotius fuerit à vecte AB horizon<lb/>ti æquidi&longs;tante, eandem potentiam in A pondus nihilominus <lb/>&longs;u&longs;tinere: vt &longs;i centrum grauitatis H ponderis BD longius ab&longs;it <lb/>à vecte BA, quàm centrum grauitatis N ponderis PV, dum<lb/>modo ducta à puncto H perpendicularis HL horizonti, vectiq; <lb/>AB tran&longs;eat per N; &longs;itq; pondus PV ponderi BD æquale; <lb/>erit tùm pondus BD, tùm pondus PV, ac &longs;i ambo in L e&longs;<lb/>&longs;ent appen&longs;a; atque &longs;unt æqualia, cùm loco vnius ponderis ac<lb/>cipiantur, eadem igitur potentia in A &longs;u&longs;tinens pondus BD, <lb/>pondus quoq; PV &longs;u&longs;tinebit. </s>

<s id="id.2.1.105.6.1.2.0">Vecte autem EF, quò centrum <lb/>grauitatis longius fuerit à vecte, eò facilius potentia idem pon<lb/>dus &longs;u&longs;tinebit: vt &longs;i centrum grauitatis k ponderis FG longius <lb/>&longs;it à vecte EF, quàm centrum grauitatis X ponderis YZ; ita ta<lb/>men vt ducta à puncto k vecti FE perpendicularis tran&longs;eat per <lb/>X; &longs;itq; pondus FG ponderi YZ æquale; & à punctis kX ip<lb/>&longs;orum horizontibus perpendiculares ducantur KM X9; erit C9 <lb/>maior CM; ac propterea pondus FG in vecte erit, ac &longs;i in M e&longs; <lb/>&longs;et appen&longs;um, & pondus YZ, ac &longs;i in 9 e&longs;&longs;et appen&longs;um. </s>

<figure id="id.036.01.118.1.jpg" place="text" xlink:href="036/01/118/1.jpg"/><lb/><arrow.to.target n="note171"/>niam autem maiorem habet proportionem C9 ad CE, quàm <lb/>CM ad CE, maior potentia in E &longs;u&longs;tinebit pondus YZ, quàm <lb/>FG. </s>

Line 1483

<s id="id.2.1.111.2.1.1.0">In&longs;uper &longs;i in B altera &longs;it potentia, ita vt duæ &longs;int potentiæ pondus <lb/>&longs;u&longs;tinentes, maiore opus e&longs;t potentia in B pondus kL &longs;u&longs;tinente <lb/>vecte BF, quàm pondus CD vecte AB. </s>

<s id="id.2.1.111.2.1.1.0.a">& adhuc maiore vecte <lb/>AB, quàm vecte BE. </s>

<s id="id.2.1.111.2.1.1.0.b">maiorem enim habet proportionem RF <lb/>ad FB, quàm PA ad AB; & PA ad AB maiorem habet, quàm <lb/>EM ad EB. </s>

<s id="id.2.1.111.3.1.1.0">Similiterq; o&longs;tendetur potentias in B pondus vectibus &longs;u&longs;tinen<lb/>tes inter &longs;e &longs;e ita e&longs;&longs;e, vt EM ad AP; & ut <lb/>AP ad FR; at&que; ut <lb/>EM ad FR. </s>

<s id="id.2.1.111.3.1.1.0">Similiterq; o&longs;tendetur potentias in B pondus vectibus &longs;u&longs;tinen<lb/>tes inter &longs;e &longs;e ita e&longs;&longs;e, vt EM ad AP; & ut <lb/>AP ad FR; atque ut <lb/>EM ad FR. </s>

<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"/><lb/>RB; & potentia in B ad potentiam in A, ut PA ad PB, & po<lb/>tentia <arrow.to.target n="note179"/>in <emph type="italics"/>B<emph.end type="italics"/> ad potentiam in E, ut EM ad M<emph type="italics"/>B.<emph.end type="italics"/></s>

<s id="id.2.1.112.1.1.2.0"><margin.target id="note179"/>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s><pb xlink:href="036/01/122.jpg"/>

Line 1514

<s id="id.2.1.113.9.1.2.0">Moueatur <lb/>deinde uectis in FG, Hk; & centrum grauitatis in LM. </s>

<s id="id.2.1.113.9.1.2.0.a">dico ean<lb/>dem potentiam in kBG idemmet &longs;emper &longs;u&longs;tinere pondus. </s>

<s id="id.2.1.113.9.1.3.0"><lb/>Quoniam enim pondus in uecte AB perinde &longs;e habet, ac &longs;i e&longs;&longs;et <lb/><arrow.to.target n="note180"/>appen&longs;um in E; & in uecte GF, ac &longs;i e&longs;&longs;et appen&longs;um in L; & in <lb/>uecte Hk. </s>

<s id="id.2.1.113.9.1.4.0">ac &longs;i in M e&longs;&longs;et appen&longs;um; di&longs;tantiæ uerò CL CE <lb/>CM &longs;unt inter &longs;e &longs;e æquales; nec non CK CB CG inter &longs;e æ<lb/>quales; erit potentia in B ad pondus, ut CE ad CB; at&que; poten<pb n="56" xlink:href="036/01/125.jpg"/>tia in k ad pondus, ut CM ad Ck; & potentia in G ad pondus, <lb/>vt CL ad CG. </s>

<s id="id.2.1.113.9.1.4.0">ac &longs;i in M e&longs;&longs;et appen&longs;um; di&longs;tantiæ uerò CL CE <lb/>CM &longs;unt inter &longs;e &longs;e æquales; nec non CK CB CG inter &longs;e æ<lb/>quales; erit potentia in B ad pondus, ut CE ad CB; atque poten<pb n="56" xlink:href="036/01/125.jpg"/>tia in k ad pondus, ut CM ad Ck; & potentia in G ad pondus, <lb/>vt CL ad CG. </s>

<s id="id.2.1.113.9.1.4.0.a">eadem igitur potentia in k<emph type="italics"/>B<emph.end type="italics"/>G idem translatum <lb/>pondus &longs;u&longs;tinebit. </s>

<s id="id.2.1.113.9.1.5.0">quod demon&longs;trare oportebat. </s>

<s id="id.2.1.114.1.1.1.0"><margin.target id="note180"/>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

Line 1524

<s id="id.2.1.115.3.1.1.0">Si vectis di&longs;tantia inter fulcimentum, & poten<lb/>tiam ad di&longs;tantiam fulcimento, punctoq;, vbi <lb/>à centro grauitatis ponderis horizonti ducta <lb/>perpendicularis vectem &longs;ecat, interiectam ma<lb/>iorem habuerit proportionem, quàm pondus <lb/>ad potentiam; pondus vtiq; à potentia moue<lb/>bitur. </s>

<s id="id.2.1.115.4.1.1.0">Sit véctis AB, ex <lb/>punctoq; A &longs;u&longs;penda<lb/>tur pondus C; hoc e&longs;t <lb/>punctum A &longs;emper &longs;it <lb/>punctum, vbi perpen<lb/>dicularis à grauitatis <lb/>centro ponderis du<lb/>cta vectem &longs;ecat; &longs;itq; <lb/><figure id="id.036.01.125.1.jpg" place="text" xlink:href="036/01/125/1.jpg"/><lb/>potentia in B, ac fulcimentum &longs;it D; & DB ad DA maiorem <lb/>habeat proportionem, quàm pondus C ad potentiam in B. </s>

<s id="id.2.1.115.4.1.1.0.a">Di<lb/>co pondus Cà potentia in B moueri. </s>

<s id="id.2.1.115.4.1.2.0">fiat vt BD ad DA, ita <lb/>pondus E ad potentiam in B; atq; pondus E quoq; appendatur <lb/>in A: patet potentiam in B æ&que;ponderare ip&longs;i E; hoc e&longs;t pon<lb/>dus <arrow.to.target n="note181"/>E &longs;u&longs;tinere. </s>

<s id="id.2.1.115.4.1.2.0">fiat vt BD ad DA, ita <lb/>pondus E ad potentiam in B; atq; pondus E quoq; appendatur <lb/>in A: patet potentiam in B æqueponderare ip&longs;i E; hoc e&longs;t pon<lb/>dus <arrow.to.target n="note181"/>E &longs;u&longs;tinere. </s>

<s id="id.2.1.115.4.1.3.0">& quoniam BD ad DA maiorem habet pro<lb/>portionem, quàm C ad potentiam in B; & vt BD ad DA, ita <pb xlink:href="036/01/126.jpg"/>e&longs;t pondus E ad po<lb/>tentiam: igitur E ad <lb/>potentiam maiorem <lb/>habebit proportio<lb/>nem, quàm pondus <lb/>C ad eandem poten<lb/><arrow.to.target n="note182"/>tiam. </s>

<s id="id.2.1.115.4.1.4.0">quare pondus <lb/>E maius erit ponde<lb/><figure id="id.036.01.126.1.jpg" place="text" xlink:href="036/01/126/1.jpg"/><lb/>re C. </s>

<s id="N13A05">& cùm potentia ip&longs;a E æ&que;ponderet, potentia igitur ip&longs;i <lb/>C non æ&que;ponderabit, &longs;ed &longs;ua ui deor&longs;um verget. </s>

<s id="N13A05">& cùm potentia ip&longs;a E æqueponderet, potentia igitur ip&longs;i <lb/>C non æqueponderabit, &longs;ed &longs;ua ui deor&longs;um verget. </s>

<s id="id.2.1.115.4.1.5.0">pondus igitur <lb/>C à potentia in B mouebitur vecte AB, cuius fulcimentum <lb/>e&longs;t D. </s>

<s id="id.2.1.116.1.1.1.0"><margin.target id="note181"/>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.116.1.1.2.0"><margin.target id="note182"/>10 <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

Line 1581

<s id="id.2.1.125.3.1.1.0">Sit pondus A vt centum, potentia verò mouens &longs;it vt decem; <lb/>&longs;itq; datus vectis BC. </s>

<s id="id.2.1.125.3.1.1.0.a">oportet potentiam, quæ e&longs;t decem pondus <lb/>A centum vecte BC mouere. </s>

<s id="id.2.1.125.3.1.2.0">Diuidatur BC in D, ita vt CD <lb/>ad DB eandem habeat proportionem, quàm habet centum ad <lb/>decem, hoc e&longs;t decem ad vnum; etenim &longs;i D fieret fulcimentum, <lb/>con&longs;tat potentiam vt decem in C æ&que;ponderare ponderi A in B <arrow.to.target n="note194"/><lb/>appen&longs;o: hoc e&longs;t pondus A &longs;u&longs;tinere. </s>

<s id="id.2.1.125.3.1.2.0">Diuidatur BC in D, ita vt CD <lb/>ad DB eandem habeat proportionem, quàm habet centum ad <lb/>decem, hoc e&longs;t decem ad vnum; etenim &longs;i D fieret fulcimentum, <lb/>con&longs;tat potentiam vt decem in C æqueponderare ponderi A in B <arrow.to.target n="note194"/><lb/>appen&longs;o: hoc e&longs;t pondus A &longs;u&longs;tinere. </s>

<s id="id.2.1.125.3.1.3.0">accipiatur inter BD quod <lb/>uis punctum E, & fiat E fulcimentum. </s>

<s id="id.2.1.125.3.1.4.0">Quoniam enim maior <arrow.to.target n="note195"/><lb/>e&longs;t proportio CE ad EB, quàm CD ad DB; maiorem habebit <lb/>proportionem CE ad EB, quàm pondus A ad potentiam decem <lb/>in C: potentia igitur decem in C pondus A centum in B appen<lb/>&longs;um vecte BC, cuius fulcimentum &longs;it E, mouebit. <arrow.to.target n="note196"/></s>

<s id="id.2.1.125.4.1.1.0">Si verò &longs;it vectis <lb/>BC, & fulcimen<lb/>tum B. </s>

Line 1647

<s id="id.2.1.133.5.1.1.0">Sit datus vectis AB, cuius fulcimentum &longs;it datum C; &longs;itq; <lb/>punctum D, in quo collocanda &longs;it potentia, quæ vectem AB &longs;u<lb/>&longs;tinere debeat, ita vt immobilis per&longs;i&longs;tat. </s>

<s id="id.2.1.133.5.1.2.0">ducatur à puncto C <lb/>linea CE horizonti perpendicularis, quæ vectem AB in duas di<lb/>uidat partes AE EF, &longs;itq; partis AE centrum grauitatis G, & <lb/>partis EF centrum grauitatis H; à punctis&queacute; GH horizon<lb/>tibus perpendiculares ducantur Gk HL, quæ lineam AF <lb/>in punctis KL &longs;ecent. </s>

<s id="id.2.1.133.5.1.3.0">quoniam enim vectis AB à linea CE in duas <lb/>diuiditur partes AE EF; ideo vectis AB nihil aliud erit, ni&longs;i <lb/>duo pondera AE EF in vecte, &longs;iue libra AF po&longs;ita; cuius &longs;u<lb/>&longs;pen&longs;io, &longs;iue fulcimentum e&longs;t C. quare pondera AE EF ita erunt <lb/>po&longs;ita, ac &longs;i in kL e&longs;&longs;ent appen&longs;a. </s>

<s id="id.2.1.133.5.1.4.0">diuidatur ergo kL in M, <lb/>ita vt kM ad ML, &longs;it vt grauitas partis EF ad grauitatem par<lb/>tis AE; & vt CA ad CM, ita fiat grauitas totius vectis AB ad <lb/>potentiam, quæ &longs;i collocetur in D (dummodo DA horizonti <pb n="61" xlink:href="036/01/135.jpg"/>perpendicularis exi&longs;tat) vecti æ&que;ponderabit; hoc e&longs;t vectem <arrow.to.target n="note210"/><lb/>AB deor&longs;um premendo &longs;u&longs;tinebit. </s>

<s id="id.2.1.133.5.1.4.0">diuidatur ergo kL in M, <lb/>ita vt kM ad ML, &longs;it vt grauitas partis EF ad grauitatem par<lb/>tis AE; & vt CA ad CM, ita fiat grauitas totius vectis AB ad <lb/>potentiam, quæ &longs;i collocetur in D (dummodo DA horizonti <pb n="61" xlink:href="036/01/135.jpg"/>perpendicularis exi&longs;tat) vecti æqueponderabit; hoc e&longs;t vectem <arrow.to.target n="note210"/><lb/>AB deor&longs;um premendo &longs;u&longs;tinebit. </s>

<s id="id.2.1.133.5.1.5.0">quod inuenire oportebat. </s>

<s id="id.2.1.134.1.1.1.0"><margin.target id="note210"/>13 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.135.1.1.1.0">Si verò potentia in puncto B ponenda e&longs;&longs;et. </s>

Line 1663

<s id="id.2.1.135.4.1.1.0">Similiter o&longs;tendetur, &longs;i plura e&longs;&longs;ent pondera in vecte AB ubi<lb/>cunq;, & quomodocunq; po&longs;ita. </s>

<s id="id.2.1.136.1.1.1.0"><margin.target id="note211"/>13 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.136.1.1.2.0"><margin.target id="note212"/><emph type="italics"/>Ex &longs;exta<emph.end type="italics"/></s>

<s id="id.2.1.136.1.1.3.0"><margin.target id="note213"/>1 <emph type="italics"/>Arch. de æquep.<emph.end type="italics"/></s>

<s id="id.2.1.137.1.1.1.0">In&longs;uper ex his non &longs;olum, ut in decimaquarta huius docuimus, <lb/>quomodo &longs;cilicet data pondera ubicunq; in uecte po&longs;ita data poten<lb/>tia dato uecte mouere po&longs;&longs;umus, eodem modo grauitate uectis <lb/>con&longs;iderata idem facere poterimus; uerùm etiam accidentia reli<lb/>qua, quæ &longs;upra ab&longs;q; uectis grauitatis con&longs;ideratione demon&longs;tra<lb/>ta &longs;unt; &longs;imili modo uectis grauitate con&longs;iderata vná cum ponde<lb/>ribus, uel &longs;ine ponderibus o&longs;tendentur. </s></chap>

Line 1671

<s id="id.2.1.137.3.1.1.0">DE TROCHLEA. </s>

<s id="id.2.1.137.4.1.1.0">Trochleae in&longs;trumento pon<lb/>dus multipliciter moueri pote&longs;t; <lb/>quia verò in omnibus e&longs;t eadem <lb/>ratio: ideo (vt res euidentior ap<lb/>pareat) in iis, quæ dicenda &longs;unt, <lb/>intelligatur pondus &longs;ur&longs;um ad re<lb/>ctos horizontis plano angulos hoc modo &longs;em<lb/>per moueri. </s><pb xlink:href="036/01/138.jpg"/>

<s id="id.2.1.137.6.1.1.0">Sit pondus A, quod ip&longs;i ho<lb/>rizontis plano &longs;ur&longs;um ad rectos <lb/>angulos &longs;it attollendum; & vt <lb/>fieri &longs;olet, trochlea duos habens <lb/>orbiculos, quorum axiculi &longs;int <lb/>in BC, &longs;upernè appendatur; <lb/>trochlea verò duos &longs;imiliter ha<lb/>bens orbiculos, quorum axicu<lb/>li &longs;int in DE, ponderi alligetur: <lb/>ac per omnes vtriu&longs;q; trochleæ <lb/>orbiculos circunducatur ducta<lb/>rius funis, &que;m in altero eius ex <lb/>tremo, putá in F, oportet e&longs;&longs;e <lb/>religatum. </s>

<s id="id.2.1.137.6.1.1.0">Sit pondus A, quod ip&longs;i ho<lb/>rizontis plano &longs;ur&longs;um ad rectos <lb/>angulos &longs;it attollendum; & vt <lb/>fieri &longs;olet, trochlea duos habens <lb/>orbiculos, quorum axiculi &longs;int <lb/>in BC, &longs;upernè appendatur; <lb/>trochlea verò duos &longs;imiliter ha<lb/>bens orbiculos, quorum axicu<lb/>li &longs;int in DE, ponderi alligetur: <lb/>ac per omnes vtriu&longs;q; trochleæ <lb/>orbiculos circunducatur ducta<lb/>rius funis, quem in altero eius ex <lb/>tremo, putá in F, oportet e&longs;&longs;e <lb/>religatum. </s>

<s id="id.2.1.137.6.1.2.0">potentia autem mo<lb/>uens ponatur in G, quæ dum <lb/>de&longs;cendit, pondus A &longs;ur&longs;um ex <lb/>aduer&longs;o attolletur; &que;madmo<lb/>dum Pappus in octauo libro Ma<lb/>thematicarum collectionum a&longs;<lb/>&longs;erit; nec non Vitruuius in deci <lb/>mo de Architectura, & alii. <figure id="id.036.01.138.1.jpg" place="text" xlink:href="036/01/138/1.jpg"/></s>

<s id="id.2.1.137.6.1.2.0">potentia autem mo<lb/>uens ponatur in G, quæ dum <lb/>de&longs;cendit, pondus A &longs;ur&longs;um ex <lb/>aduer&longs;o attolletur; quemadmo<lb/>dum Pappus in octauo libro Ma<lb/>thematicarum collectionum a&longs;<lb/>&longs;erit; nec non Vitruuius in deci <lb/>mo de Architectura, & alii. <figure id="id.036.01.138.1.jpg" place="text" xlink:href="036/01/138/1.jpg"/></s>

<s id="id.2.1.137.7.1.1.0">Quomodo autem hoc trochleæ in&longs;trumen<lb/>tum reducatur ad vectem; cur magnum pondus <lb/>ab exigua virtute, & quomodo, quantoq; in tem<lb/>pore moueatur; cur funis in vno capite debeat <lb/>e&longs;&longs;e religatus; quodq; &longs;uperioris, inferioris&qacute;ue <lb/>trochleæ fuerit officium; & quomodo omnis in <pb n="63" xlink:href="036/01/139.jpg"/>numeris data proportio inter potentiam, & pon<lb/>dus inueniri po&longs;sit; dicamus. </s>

<s id="id.2.1.137.8.1.1.0">LEMMA. </s>

<s id="id.2.1.137.9.1.1.0">Sint rectæ lineæ AB CD parallelæ, quæ in <lb/>punctis AC circulum ACE contingant, cuius <lb/>centrum F: & FA FC connectantur. </s>

Line 1700

<expan abbr="Quoniã">Quoniam</expan>

<expan abbr="aut&etilde;">autem</expan> BC tùm horizonti, tùm ip&longs;i CF e&longs;t perpendicularis; <lb/>erit linea CF horizonti æquidi&longs;tans. </s>

<s id="id.2.1.139.4.1.6.0">cùm verò <expan abbr="põdus">pondus</expan> appen&longs;um &longs;it <lb/><arrow.to.target n="note220"/>in BC, & potentia &longs;it in G; quod idem e&longs;t, ac &longs;i e&longs;&longs;et in F; erit <lb/>CF tanquam libra, &longs;iue vectis, cuius centrum, &longs;iue fulcimentum e&longs;t <lb/>D; nam in axiculo <expan abbr="orbuculus">orbiculus</expan> &longs;u&longs;tinetur; atq; punctum D, cùm &longs;it <lb/>centrum axiculi, & orbiculi, etiam vtri&longs;&que; circumuolutis <lb/>immobile remanet. </s>

<s id="id.2.1.139.4.1.7.0">Itaq; cùm di&longs;tantia DC &longs;it æqualis di&longs;tantiæ <lb/>DF, potentiaq; in F ponderi A in C appen&longs;o æ&que;ponderet, cùm <lb/><arrow.to.target n="note221"/>pondus &longs;u&longs;tineat, ne deor&longs;um vergat; erit potentia in F, &longs;iue in G <lb/>(nam idem e&longs;t) con&longs;tituta ponderi A æqualis. </s>

<s id="id.2.1.139.4.1.7.0">Itaq; cùm di&longs;tantia DC &longs;it æqualis di&longs;tantiæ <lb/>DF, potentiaq; in F ponderi A in C appen&longs;o æqueponderet, cùm <lb/><arrow.to.target n="note221"/>pondus &longs;u&longs;tineat, ne deor&longs;um vergat; erit potentia in F, &longs;iue in G <lb/>(nam idem e&longs;t) con&longs;tituta ponderi A æqualis. </s>

<s id="id.2.1.139.4.1.8.0">Idem enim effi<lb/>cit potentia in G, ac &longs;i in G aliud e&longs;&longs;et appen&longs;um pondus æquale <lb/>ponderi A; quæ pondera in CF appen&longs;a æquæponderabunt. </s>

<s id="id.2.1.139.4.1.9.0">Præ<lb/>terea, cùm in neutram fiat motus partem, idem erit vnico exi<pb n="64" xlink:href="036/01/141.jpg"/>&longs;tente fune BC EFG hoc modo orbiculo circumuoluto, ac &longs;i duo <lb/>e&longs;&longs;ent funes BC FG alligati in vecte, &longs;iue libra CF. </s>

<s id="id.2.1.140.1.1.1.0"><margin.target id="note217"/>1 <emph type="italics"/>Huius. de libra.<emph.end type="italics"/></s>

<s id="id.2.1.140.1.1.3.0"><margin.target id="note218"/>8 <emph type="italics"/>Vndecimi.<emph.end type="italics"/></s>

<s id="id.2.1.140.1.1.4.0"><margin.target id="note219"/>18 <emph type="italics"/>Tertii.<emph.end type="italics"/></s>

<s id="id.2.1.140.1.1.5.0"><margin.target id="note220"/><emph type="italics"/>Ex<emph.end type="italics"/> 28 <emph type="italics"/>Primi.<emph.end type="italics"/></s>

<s id="id.2.1.140.1.1.6.0"><margin.target id="note221"/>1 <emph type="italics"/>Primi. Archim. de æ&que;pond.<emph.end type="italics"/></s>

<s id="id.2.1.140.1.1.6.0"><margin.target id="note221"/>1 <emph type="italics"/>Primi. Archim. de æquepond.<emph.end type="italics"/></s>

<s id="id.2.1.141.1.1.1.0">COROLLARIVM. </s>

<s id="id.2.1.141.2.1.1.0">Ex hoc manife&longs;tum e&longs;&longs;e pote&longs;t, idem pon<lb/>dus ab eadem potentia ab&longs;q; ullo huius tro<lb/>chleæ auxilio nihilominus &longs;u&longs;tineri po&longs;&longs;e. </s>

<s id="id.2.1.141.3.1.1.0">Sit enim pondus H æquale <lb/>ponderi A, cui alligatus &longs;it funis <lb/>kL; &longs;itq; potentia in L &longs;u&longs;tinens <lb/>pondus H. </s>

Line 1723

<s id="id.2.1.141.7.1.1.0.a">Ducatur <lb/>CG DkF diameter horizonti æ<lb/>quidi&longs;tans. </s>

<s id="id.2.1.141.7.1.2.0">& quoniam dum orbi<lb/>culus circumuertitur, circumferen<lb/>tia circuli CEF &longs;emper e&longs;t æquidi<lb/>&longs;tans circumferentiæ axiculi GHk; <lb/>circa enim axiculum circumuerti<lb/>tur; & circulorum æquidi&longs;tantes cir<lb/>cumferentiæ idem habent centrum; <lb/>erit punctum D &longs;emper & orbiculi, <lb/><figure id="id.036.01.142.1.jpg" place="text" xlink:href="036/01/142/1.jpg"/><lb/>& axiculi centrum. </s>

<s id="id.2.1.141.7.1.3.0">Itaq; cùm DC &longs;it æqualis DF, & DG ip&longs;i <lb/>Dk; erit GC ip&longs;i kF æqualis. </s>

<s id="id.2.1.141.7.1.4.0">&longs;i igitur in vecte, &longs;iue libra CF <lb/>pondera appendantur æqualia, æ&que;ponderabunt. </s>

<s id="id.2.1.141.7.1.4.0">&longs;i igitur in vecte, &longs;iue libra CF <lb/>pondera appendantur æqualia, æqueponderabunt. </s>

<s id="id.2.1.141.7.1.5.0">di&longs;tantia enim <lb/>CG æqualis e&longs;t di&longs;tantiæ kF; axiculu&longs;q; GHK immobilis gerit <lb/>vicem centri, &longs;iue fulcimenti. </s>

<s id="id.2.1.141.7.1.6.0">immobili igitur manente axicu<lb/>lo, &longs;i ponatur in F potentia &longs;u&longs;tinens pondus in C appen&longs;um; erit <lb/>potentia in F ip&longs;i ponderi æqualis. </s>

<s id="id.2.1.141.7.1.7.0">quod erat o&longs;tendendum. </s>

Line 1758

<s id="id.2.1.143.4.4.1.0">COROLLARIVM. II. </s>

<s id="id.2.1.143.5.1.1.0">Manife&longs;tum e&longs;t etiam; &longs;i duæ fuerint poten<lb/>tiæ vna in G, altera in F, pondus A &longs;u&longs;tinentes; <lb/>vtra&longs;q; &longs;imul ponderi A æquales e&longs;&longs;e: & vnam <lb/>quam&que; &longs;u&longs;tinere dimidium ponderis A. </s>

<s id="id.2.1.143.5.1.1.0">Manife&longs;tum e&longs;t etiam; &longs;i duæ fuerint poten<lb/>tiæ vna in G, altera in F, pondus A &longs;u&longs;tinentes; <lb/>vtra&longs;q; &longs;imul ponderi A æquales e&longs;&longs;e: & vnam <lb/>quamque &longs;u&longs;tinere dimidium ponderis A. </s>

<s id="id.2.1.143.6.1.1.0">Hoc autem ex tertio, & quarto corollario &longs;ecundæ huius in <lb/>tractatu de vecte patet. </s>

<s id="id.2.1.143.7.1.1.0">COROLLARIVM III. </s>

<s id="id.2.1.143.8.1.1.0">Illud quoq; præterea innote&longs;cit, cur &longs;cilicet fu<lb/>nis ex altero religatus e&longs;&longs;e debeat extremo. </s><pb xlink:href="036/01/146.jpg"/>

Line 1770

<s id="id.2.1.143.12.1.1.0.a">dico potentiam in N <lb/>&longs;ubduplam e&longs;&longs;e ponderis A. </s>

<s id="N1433D">&longs;i enim potentia &longs;u<lb/>&longs;tinens pondus A vbi M collocata foret, e&longs;&longs;et <lb/>vtiq; potentia in M &longs;ubdupla ponderis A. </s>

<s id="N14343">po<lb/><arrow.to.target n="note225"/>tentiæ verò in M æqualis e&longs;t vis in N. </s>

<s id="N1434A">e&longs;t e<lb/><arrow.to.target n="note226"/>nim ac &longs;i potentia in M dimidium ponderis <lb/>A &longs;ine trochlea &longs;u&longs;tineret, cui æ&que;ponderat <lb/>pondus in N ponderis A dimidio æquale. </s>

<s id="N1434A">e&longs;t e<lb/><arrow.to.target n="note226"/>nim ac &longs;i potentia in M dimidium ponderis <lb/>A &longs;ine trochlea &longs;u&longs;tineret, cui æqueponderat <lb/>pondus in N ponderis A dimidio æquale. </s>

<s id="id.2.1.143.12.1.2.0"><lb/>quare vis in N æqualis dimidio ponderis A <lb/>ip&longs;um A &longs;u&longs;tinebit. </s>

<s id="id.2.1.143.12.1.3.0">Potentia igitur in N &longs;u&longs;ti<lb/>nens pondus A &longs;ubdupla e&longs;t ip&longs;ius A. </s>

<s id="N14360">quod <lb/>demon&longs;trare oportebat. <figure id="id.036.01.146.1.jpg" place="text" xlink:href="036/01/146/1.jpg"/></s><pb n="67" xlink:href="036/01/147.jpg"/>

Line 1822

<s id="id.2.1.146.1.1.3.0"><margin.target id="note229"/><emph type="italics"/>Ex<emph.end type="italics"/> 3 <emph type="italics"/>Cor.<emph.end type="italics"/> 2 <emph type="italics"/>Huius vecte.<emph.end type="italics"/></s>

<s id="id.2.1.146.1.1.4.0"><margin.target id="note230"/>4 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.147.1.1.1.0">COROLLARIVM. </s>

<s id="id.2.1.147.2.1.1.0">Ex hoc manife&longs;tum e&longs;t, vnum&que;mq; funem <lb/>MD FL HB tertiam &longs;u&longs;tinere partem pon<lb/>deris A. <pb xlink:href="036/01/152.jpg"/></s>

<s id="id.2.1.147.2.1.1.0">Ex hoc manife&longs;tum e&longs;t, vnumquemq; funem <lb/>MD FL HB tertiam &longs;u&longs;tinere partem pon<lb/>deris A. <pb xlink:href="036/01/152.jpg"/></s>

<s id="id.2.1.147.3.1.1.0">Præterea, &longs;i funis ex M per a<lb/>lium adhuc deferatur orbiculum &longs;u<lb/>periorem in trochlea &longs;ur&longs;um &longs;imi<lb/>liter appen&longs;a con&longs;titutum, cuius <lb/>centrum N; ita vt perueniat in O; <lb/>ibiq; à potentia detineatur; erit po<lb/>tentia in O &longs;u&longs;tinens pondus A iti <lb/>dem &longs;ubtripla ip&longs;ius ponderis. </s>

<s id="id.2.1.147.3.1.2.0">fu<lb/>nis enim MD tantùm ponderis &longs;u<lb/>&longs;tinet, ac &longs;i in D appen&longs;um e&longs;&longs;et <lb/>pondus æquale tertiæ parti ponde<lb/><arrow.to.target n="note231"/>ris A, cui æquiualet potentia in <lb/>O ip&longs;i æqualis, hoc e&longs;t &longs;ubtripla <lb/>ponderis A. </s>

<s id="id.2.1.147.3.1.2.0.a">Potentia igitur in O <lb/>&longs;ubtripla e&longs;t ponderis A. <lb/><figure id="id.036.01.152.1.jpg" place="text" xlink:href="036/01/152/1.jpg"/></s>

<s id="id.2.1.147.4.1.1.0">Et ne idem &longs;æpius repetatur, no<lb/>ui&longs;&longs;e oportet potentiam in O &longs;em<lb/>per æqualem e&longs;&longs;e ei, quæ e&longs;t in M; <lb/>hoc e&longs;t &longs;i potentia in M e&longs;&longs;et &longs;ub <lb/>quadrupla, &longs;ubquintupla, vel huiu&longs; <lb/>modi aliter ip&longs;ius ponderis; poten<lb/>tia quoq; in O erit itidem &longs;ubqua<lb/>drupla, &longs;ubquintupla, atq; ita dein<lb/>ceps eiu&longs;demmet ponderis, &que;m<lb/>madmodum &longs;e habet potentia <lb/>in M. </s>

<s id="id.2.1.147.4.1.1.0">Et ne idem &longs;æpius repetatur, no<lb/>ui&longs;&longs;e oportet potentiam in O &longs;em<lb/>per æqualem e&longs;&longs;e ei, quæ e&longs;t in M; <lb/>hoc e&longs;t &longs;i potentia in M e&longs;&longs;et &longs;ub <lb/>quadrupla, &longs;ubquintupla, vel huiu&longs; <lb/>modi aliter ip&longs;ius ponderis; poten<lb/>tia quoq; in O erit itidem &longs;ubqua<lb/>drupla, &longs;ubquintupla, atq; ita dein<lb/>ceps eiu&longs;demmet ponderis, quem<lb/>madmodum &longs;e habet potentia <lb/>in M. </s>

<s id="id.2.1.148.1.1.1.0"><margin.target id="note231"/>1 <emph type="italics"/>Huius.<emph.end type="italics"/></s><pb n="70" xlink:href="036/01/153.jpg"/>

<s id="id.2.1.149.1.2.1.0">PROPOSITIO VI. </s>

Line 1840

<s id="N145F9">quod demon&longs;trare opor<lb/>tebat. </s>

<s id="id.2.1.150.1.1.1.0"><margin.target id="note232"/>2 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/></s><pb xlink:href="036/01/154.jpg"/>

<s id="id.2.1.151.1.2.1.0">Si verò tres &longs;int vectes <lb/>AB CD EF bifariam di<lb/>ui&longs;i in GHk, quorum fulci <lb/>menta &longs;int BDF; & pondus <lb/>L eodem modo in GHK <lb/>appen&longs;um; &longs;intq; tres poten<lb/>tiæ in ACE æquales pondus <lb/>&longs;u&longs;tinentes; &longs;imiliter o&longs;ten<lb/>detur vnamquam&que; po<lb/>tentiam &longs;ub&longs;excuplam e&longs;&longs;e <lb/>ponderis L. </s>

<s id="id.2.1.151.1.2.1.0">Si verò tres &longs;int vectes <lb/>AB CD EF bifariam di<lb/>ui&longs;i in GHk, quorum fulci <lb/>menta &longs;int BDF; & pondus <lb/>L eodem modo in GHK <lb/>appen&longs;um; &longs;intq; tres poten<lb/>tiæ in ACE æquales pondus <lb/>&longs;u&longs;tinentes; &longs;imiliter o&longs;ten<lb/>detur vnamquamque po<lb/>tentiam &longs;ub&longs;excuplam e&longs;&longs;e <lb/>ponderis L. </s>

<s id="N14627">atq; hoc ordi<lb/>ne &longs;i quatuor e&longs;&longs;ent vectes, <lb/>& quatuor potentiæ; erit vnaquæq; potentia &longs;uboctupla ponderis. </s><lb/>

<s id="id.2.1.151.1.2.2.0">atq; ita deinceps in infinitum. </s>

Line 1856

<s id="id.2.1.151.4.1.4.0"><lb/>quare ita &longs;u&longs;tinet funis HG, vt EF. </s>

<s id="N14696">&longs;imiliter <lb/>o&longs;ten detur funem PO tàm &longs;u&longs;tinere, quàm <lb/>LN: quare funes PO kG EF LN æqua<lb/>liter &longs;u&longs;tinent. </s>

<s id="id.2.1.151.4.1.5.0">æqualiter igitur funis PO &longs;u<lb/>&longs;tinet, vt kG. </s>

<s id="N146A3">&longs;i ergo duæ intelligantur e&longs; <lb/><figure id="id.036.01.155.1.jpg" place="text" xlink:href="036/01/155/1.jpg"/><lb/>&longs;e potentiæ in OG, &longs;eu in PH, quod idem e&longs;t, pondus nihilomi<lb/>nus &longs;u&longs;tinentes, &que;madmodum funes &longs;u&longs;tinent, æquales vtiq; e&longs;<lb/>&longs;ent; & GF ON duorum vectium vires gerent; quorum fulci <lb/>menta erunt FN, & pondus A in BC medio vectium appen&longs;um. </s>

<s id="N146A3">&longs;i ergo duæ intelligantur e&longs; <lb/><figure id="id.036.01.155.1.jpg" place="text" xlink:href="036/01/155/1.jpg"/><lb/>&longs;e potentiæ in OG, &longs;eu in PH, quod idem e&longs;t, pondus nihilomi<lb/>nus &longs;u&longs;tinentes, quemadmodum funes &longs;u&longs;tinent, æquales vtiq; e&longs;<lb/>&longs;ent; & GF ON duorum vectium vires gerent; quorum fulci <lb/>menta erunt FN, & pondus A in BC medio vectium appen&longs;um. </s>

<s id="id.2.1.151.4.1.6.0"><lb/>& quoniam omnes funes æqualiter &longs;u&longs;tinent, tàm &longs;u&longs;tinebunt <lb/>duo PO LN, quàm duo KGEF; tàm igitur &longs;u&longs;tinebit vectis <lb/>ON, quàm vectis GF. </s>

<s id="N146BB">quare in vtroq; vecte ON GF æquali <lb/>ter pondus <expan abbr="põderabit">ponderabit</expan>. </s>

<s id="id.2.1.151.4.1.7.0">erit ergo vnaquæq; potentia in PH &longs;ubquadru<arrow.to.target n="note235"/><lb/>pla ponderis A. </s>

Line 1867

<s id="id.2.1.152.1.1.3.0"><margin.target id="note235"/>6 <emph type="italics"/>Huius.<emph.end type="italics"/></s><pb xlink:href="036/01/156.jpg"/>

<s id="id.2.1.153.1.2.1.0">COROLLARIVM I. </s>

<s id="id.2.1.153.2.1.1.0">Hinc manife&longs;tum e&longs;t vnum&que;mq; funem EF <lb/>GK LN OP quartam &longs;u&longs;tinere partem pon<lb/>deris A. </s>

<s id="id.2.1.153.2.1.1.0">Hinc manife&longs;tum e&longs;t vnumquemq; funem EF <lb/>GK LN OP quartam &longs;u&longs;tinere partem pon<lb/>deris A. </s>

<s id="id.2.1.153.3.1.1.0">COROLLARIVM II. </s>

<s id="id.2.1.153.4.1.1.0">Patet etiam orbiculum, cuius centrum C, <lb/>non minus eo, cuius centrum e&longs;t B, &longs;u&longs;tinere. </s>

<s id="id.2.1.153.5.1.1.0">ALITER. </s>

Line 1912

<s id="id.2.1.157.3.1.1.0">Sit pondus A, cui alligata &longs;it trochlea duos <lb/>habens orbiculos, quorum centra &longs;int BC; <lb/>&longs;itq; trochlea &longs;ur&longs;um appen&longs;a duos alios ha<lb/>bens orbiculos, quorum centra &longs;int DE; funi&longs;q; <lb/>per omnes circumducatur orbiculos, qui tro<lb/>chleæ inferiori religetur in F; &longs;it&queacute; poten<lb/>tia in G &longs;u&longs;tinens pondus A. </s>

<s id="id.2.1.157.3.1.1.0.a">dico poten<lb/>tiam in G &longs;ubquintuplam e&longs;&longs;e ponderis A. <lb/></s>

<s id="N148C8">ducantur Hk LM per centra BC horizon<lb/>ti æquidi&longs;tantes, quas eodem modo, quo &longs;u<lb/>pra dictum e&longs;t, e&longs;&longs;e tanquam vectes o&longs;tende<lb/>mus, quorum fulcimenta kM, & pondus A <lb/>ex medio vtriu&longs;q; vectis BC &longs;u&longs;pen&longs;um, & tres <lb/>potentiæ in LHC pondus &longs;u&longs;tinentes, quas <lb/>&longs;imili modo æquales e&longs;&longs;e demon&longs;trabimus; fu<lb/>nes enim idem efficiunt, ac &longs;i e&longs;&longs;ent potentiæ. </s>

<s id="id.2.1.157.3.1.2.0"><lb/>& quoniam pondus æqualiter ex vtroq; ve<lb/>cte HK LM ponderat, quod quidem o&longs;ten<lb/>detur quo&que;, vt in præcedentibus demon<lb/><arrow.to.target n="note240"/>&longs;tratum e&longs;t: erit vnaquæq; potentia, tùm in <lb/>L, &longs;eu in G, quod idem e&longs;t; tùm in H, atq; <lb/>in C, hoc e&longs;t in F, &longs;ubquintupla ponderis A. </s>

<s id="id.2.1.157.3.1.2.0"><lb/>& quoniam pondus æqualiter ex vtroq; ve<lb/>cte HK LM ponderat, quod quidem o&longs;ten<lb/>detur quoque, vt in præcedentibus demon<lb/><arrow.to.target n="note240"/>&longs;tratum e&longs;t: erit vnaquæq; potentia, tùm in <lb/>L, &longs;eu in G, quod idem e&longs;t; tùm in H, atq; <lb/>in C, hoc e&longs;t in F, &longs;ubquintupla ponderis A. </s>

<s id="id.2.1.157.3.1.2.0.a"><lb/>Potentia ergo in G &longs;u&longs;tinens pondus A ip&longs;ius <lb/>A &longs;ubquintupla erit. </s>

<s id="id.2.1.157.3.1.3.0">quod o&longs;tendere opor<lb/>tebat. <figure id="id.036.01.160.1.jpg" place="text" xlink:href="036/01/160/1.jpg"/></s><pb n="74" xlink:href="036/01/161.jpg"/>

Line 1921

<s id="id.2.1.157.5.1.2.0">deinde ex poten<arrow.to.target n="note242"/><lb/>tiis in LHN, quarum vnaquæq; <lb/>&longs;ubquintupla e&longs;&longs;et ponderis A. <lb/></s>

<s id="N1492E">e&longs;&longs;ent enim ambæ &longs;imul poten<lb/>tiæ in LH &longs;ubduplæ &longs;exquialte<lb/>ræ ip&longs;ius ponderis, <expan abbr="pot&etilde;tia">potentia</expan> verò <lb/>in F &longs;ubdecupla e&longs;&longs;et, cùm &longs;it ip<lb/>&longs;ius N &longs;ubdupla: &longs;ed duæ quin <lb/>tæ cùm decima dimidium ef<lb/>ficiunt, quòd &longs;i per terna diui <lb/>datur, &longs;exta pars ponderis re<lb/>&longs;pondebit vnicuiq; potentiæ in <lb/>LHF. </s>

<s id="N14946">ex quibus patet poten<lb/>tiam in G &longs;ub&longs;excuplam e&longs;&longs;e <lb/>ponderis A. </s>

<s id="N1494C">&longs;imiliterq; demon<lb/>&longs;trabitur vnum&que;m&que; orbi<lb/>culum æqualem &longs;u&longs;tinere por<lb/>tionem. <figure id="id.036.01.161.1.jpg" place="text" xlink:href="036/01/161/1.jpg"/></s><pb xlink:href="036/01/162.jpg"/>

<s id="N1494C">&longs;imiliterq; demon<lb/>&longs;trabitur vnumquemque orbi<lb/>culum æqualem &longs;u&longs;tinere por<lb/>tionem. <figure id="id.036.01.161.1.jpg" place="text" xlink:href="036/01/161/1.jpg"/></s><pb xlink:href="036/01/162.jpg"/>

<s id="id.2.1.157.7.1.1.0">Quòd &longs;i, vt in tertia figura <lb/>funis in O protrahatur; per <lb/>aliumq; circumducatur orbi<lb/>culum, cuius centrum Q; qui <lb/>deinde in R trochleæ relige<lb/>tur inferiori; erit potentia in <lb/><arrow.to.target n="note243"/>G ponderis &longs;ub&longs;eptupla. </s>

<s id="id.2.1.157.7.1.2.0">atq; <lb/>ita in infinitum procedendo <lb/>proportio potentiæ ad pon<lb/>dus quotcunq; &longs;ubmulti<lb/>plex inueniri poterit. </s>

<s id="id.2.1.157.7.1.3.0">dein<lb/>de &longs;emper o&longs;tendetur vt in <lb/>præcedentibus; &longs;i potentia <lb/>pondus &longs;u&longs;tinens fuerit, vel <lb/>&longs;ubquadrupla, vel &longs;ubquitu<lb/>pla, vel quouis alio modo &longs;e <lb/>habebit ad pondus; &longs;imiliter <lb/>vnum&que;m&que; funem, vel <lb/>quartam, vel quintam, vel <lb/>quamuis aliam partem &longs;u&longs;ti<lb/>nere ponderis, &que;madmo<lb/>dum potentia ip&longs;a; funes e<lb/>nim idem efficiunt, ac &longs;i tot <lb/>e&longs;&longs;ent potentiæ: orbiculi ve<lb/>rò, ac &longs;i tot e&longs;&longs;ent vectes. </s>

<s id="id.2.1.157.7.1.3.0">dein<lb/>de &longs;emper o&longs;tendetur vt in <lb/>præcedentibus; &longs;i potentia <lb/>pondus &longs;u&longs;tinens fuerit, vel <lb/>&longs;ubquadrupla, vel &longs;ubquitu<lb/>pla, vel quouis alio modo &longs;e <lb/>habebit ad pondus; &longs;imiliter <lb/>vnumquemque funem, vel <lb/>quartam, vel quintam, vel <lb/>quamuis aliam partem &longs;u&longs;ti<lb/>nere ponderis, quemadmo<lb/>dum potentia ip&longs;a; funes e<lb/>nim idem efficiunt, ac &longs;i tot <lb/>e&longs;&longs;ent potentiæ: orbiculi ve<lb/>rò, ac &longs;i tot e&longs;&longs;ent vectes. </s>

<s id="id.2.1.158.1.1.1.0"><margin.target id="note240"/>8 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.158.1.1.2.0"><margin.target id="note241"/><emph type="italics"/>Ex<emph.end type="italics"/> 6 <emph type="italics"/>huius<emph.end type="italics"/></s>

<s id="id.2.1.158.1.1.3.0"><margin.target id="note242"/><emph type="italics"/>Ex<emph.end type="italics"/> 8 <emph type="italics"/>huius<emph.end type="italics"/></s>

Line 1940

<s id="id.2.1.159.4.1.1.0">Præterea notandum e&longs;t, quod etiam ex dictis facilè patet, &longs;i <lb/>funis, &longs;iue religetur in R trochleæ inferiori, &longs;iue in S, maximam <lb/>indè oriri differentiam inter potentiam, & pondus: nam &longs;i relige<lb/>tur in S, erit potentia in G ponderis &longs;ub&longs;excupla. </s>

<s id="id.2.1.159.4.1.2.0">&longs;i verò in R, <lb/>&longs;ub&longs;eptupla. </s>

<s id="id.2.1.159.4.1.3.0">quod trochleæ &longs;uperiori non contingit, quia &longs;iue <lb/>religetur funis (vt in præcedenti figura) in T, &longs;iue in O; &longs;em<lb/>per potentia in G &longs;ub&longs;excupla erit ip&longs;ius ponderis. </s>

<s id="id.2.1.159.5.1.1.0">Po&longs;t hæc con&longs;iderandum e&longs;t, quonam modo vis moueat pon<lb/>dus; necnon potentiæ mouentis, ponderi&longs;q; moti &longs;patium, at&que; <lb/>tempus. </s>

<s id="id.2.1.159.5.1.1.0">Po&longs;t hæc con&longs;iderandum e&longs;t, quonam modo vis moueat pon<lb/>dus; necnon potentiæ mouentis, ponderi&longs;q; moti &longs;patium, atque <lb/>tempus. </s>

<s id="id.2.1.159.6.1.1.0">PROPOSITIO X. </s>

<s id="id.2.1.159.7.1.1.0">Si funis orbiculo trochleæ &longs;ur&longs;um appen&longs;æ <lb/>fuerit circumuolutus, cuius altero extremo &longs;it al<lb/>ligatum pondus; alteri autem mouens collocata <lb/>&longs;it potentia: mouebit hæc vecte horizonti &longs;em<lb/>per æquidi&longs;tante. </s><pb xlink:href="036/01/164.jpg"/>

Line 1967

<s id="id.2.1.161.3.1.1.0">Præterea potentia idem pondus per æquale <lb/>&longs;patium in æquali tempore mouet, tàm fune <lb/>hoc modo orbiculo trochleæ &longs;ur&longs;um appen&longs;æ <lb/>circumuoluto, quàm &longs;ine trochlea: dummo<lb/>do ip&longs;ius potentiæ lationes in velocitate &longs;int æ<lb/>quales. </s><pb xlink:href="036/01/166.jpg"/>

<s id="id.2.1.161.5.1.1.0">Ii&longs;dem po&longs;itis &longs;it aliud pondus P <lb/>æquale ponderi A, cui alligatus &longs;it <lb/>funis TQ <expan abbr="horizõti">horizonti</expan>

<expan abbr="perp&etilde;dicularis">perpendicularis</expan>; <lb/>et &longs;it TQ ip&longs;i HB æqualis; moueat<lb/>&queacute; <expan abbr="pot&etilde;tia">potentia</expan> in Q <expan abbr="põdus">pondus</expan> P &longs;ur&longs;um <lb/>ad rectos angulos horizonti, &que;m <lb/>admodum mouetur pondus A. </s>

<s id="id.2.1.161.5.1.1.0.a">di<lb/>co per æquale &longs;patium in eodem <lb/>tempore potentiam in Q pondus <lb/>P, & potentiam in F pondus A <lb/>mouere. </s>

<s id="id.2.1.161.5.1.2.0">quod idem e&longs;t, ac &longs;i e&longs;&longs;et <lb/>idem pondus in æquali tempore <lb/>motum; &longs;icut propo&longs;uimus. </s>

<s id="id.2.1.161.5.1.3.0">Pro<lb/>ducatur EF in S, & TQ in R; <lb/>fiantq; QR FS non &longs;olum inter <lb/>&longs;e &longs;e, verùm etiam ip&longs;i BH æqua<lb/>les. </s>

Line 1981

<s id="id.2.1.161.8.1.2.0"><lb/>CED trochleæ ponderi A alli<lb/>gatæ ex kH; &longs;itq; KH ad rectos <lb/>angulos horizonti, ita vt pon<lb/>dus &longs;emper trochleæ motum, &longs;i<lb/>ue &longs;ur&longs;um, &longs;iue deor&longs;um factum <lb/>&longs;equatur; &longs;itq; orbiculi centrum <lb/>K; & funis orbiculo circumuo<lb/>lutus &longs;it BCDEF, qui relige<lb/>tur in B, ita vt in B immobilis <lb/>maneat; & &longs;it potentia in F mo<lb/>uens pondus A. </s>

<s id="id.2.1.161.8.1.2.0.a">dico potentiam <lb/>in F &longs;emper mouere <expan abbr="põdus">pondus</expan> A ve<lb/>cte horizonti æquidi&longs;tante. </s>

<s id="id.2.1.161.8.1.3.0">&longs;int <lb/>BC EF inter &longs;e &longs;e, ip&longs;iq; kH æ<lb/>quidi&longs;tantes, & eiu&longs;dem kH ho<lb/>rizonti perpendiculares, tangen<lb/>te&longs;q; <expan abbr="circul&utilde;">circulum</expan> CED in EC <expan abbr="p&utilde;ctis">punctis</expan>; <lb/>et connectatur EC, quæ per cen<arrow.to.target n="note245"/><lb/>trum k tran&longs;ibit, horizontiq; <lb/>æquidi&longs;tans erit; &longs;icuti prius di<lb/>ctum e&longs;t. </s>

<s id="id.2.1.161.8.1.4.0">Quoniam enim or<lb/>biculus CED circa eius cen<lb/>trum K vertitur; ideo dum vis <lb/>in F trahit &longs;ur&longs;um punctum E, <lb/>deberet punctum C de&longs;cende<lb/>re, ac trahere deor&longs;um B; &longs;ed fu<lb/><figure id="id.036.01.167.1.jpg" place="text" xlink:href="036/01/167/1.jpg"/><lb/>nis in B e&longs;t immobilis, & BC <expan abbr="de&longs;cedere">descendere</expan> non pote&longs;t; quare dum <lb/>potentia in F trahit &longs;ur&longs;um E, totus orbiculus &longs;ur&longs;um mouebitur; <lb/>ac per con&longs;e&que;ns tota trochlea, & pondus; & EkC erit tanquam <arrow.to.target n="note246"/><lb/>vectis, cuius fulcimentum erit C; e&longs;t enim punctum C propter BC <lb/>ferè immobile, potentia verò mouens vectem e&longs;t in F fune EF, <pb xlink:href="036/01/168.jpg"/>& pondus in k appen&longs;um. </s>

<s id="id.2.1.161.8.1.4.0">Quoniam enim or<lb/>biculus CED circa eius cen<lb/>trum K vertitur; ideo dum vis <lb/>in F trahit &longs;ur&longs;um punctum E, <lb/>deberet punctum C de&longs;cende<lb/>re, ac trahere deor&longs;um B; &longs;ed fu<lb/><figure id="id.036.01.167.1.jpg" place="text" xlink:href="036/01/167/1.jpg"/><lb/>nis in B e&longs;t immobilis, & BC <expan abbr="de&longs;cedere">descendere</expan> non pote&longs;t; quare dum <lb/>potentia in F trahit &longs;ur&longs;um E, totus orbiculus &longs;ur&longs;um mouebitur; <lb/>ac per con&longs;equens tota trochlea, & pondus; & EkC erit tanquam <arrow.to.target n="note246"/><lb/>vectis, cuius fulcimentum erit C; e&longs;t enim punctum C propter BC <lb/>ferè immobile, potentia verò mouens vectem e&longs;t in F fune EF, <pb xlink:href="036/01/168.jpg"/>& pondus in k appen&longs;um. </s>

<s id="id.2.1.161.8.1.5.0"><lb/>quòd &longs;i punctum C omnino fue<lb/>rit immobile, moueaturq; ve<lb/>ctis EC in NC; & diuidatur <lb/>NC bifariam in L: erunt CL <lb/>LN ip&longs;is Ck KE æquales. </s>

<s id="id.2.1.161.8.1.6.0"><lb/>quare &longs;i vectis EC e&longs;&longs;et in CN, <lb/>punctum k e&longs;&longs;et in L; & &longs;i du<lb/>catur LM horizonti perpendi<lb/>cularis, quæ &longs;it etiam æqualis <lb/>kH; e&longs;&longs;et pondus A, hoc e&longs;t <lb/>punctum H in M. </s>

<s id="id.2.1.161.8.1.6.0.a">&longs;ed quoniam <lb/>potentia in F dum tendit &longs;ur<lb/>&longs;um mouendo orbiculum, &longs;em<lb/>per mouetur &longs;uper rectam EFG, <lb/>quæ &longs;emper e&longs;t quoq; æquidi<lb/>&longs;tans BC; nece&longs;&longs;e erit orbicu<lb/>lum trochleæ &longs;emper inter li<lb/>neas EG BC e&longs;&longs;e: & centrum <lb/>k, cum &longs;it in medio, &longs;uper <lb/>rectam lineam HkT &longs;emper <lb/>moueri. </s>

Line 2025

<s id="id.2.1.163.8.1.4.0">ducatur NH per centrum L ho<lb/><arrow.to.target n="note249"/>rizonti æquidi&longs;tans, quæ erit vectis orbi<lb/>culi, cuius centrum e&longs;t L. </s>

<s id="N14DB4">ducatur deinde <lb/>EC per centrum k &longs;imiliter horizonti æqui <lb/><arrow.to.target n="note250"/>di&longs;tans, quæ etiam erit vectis orbiculi, cu<lb/>ius centrum e&longs;t k. </s>

<s id="id.2.1.163.8.1.5.0">Moueatur potentia in <lb/>O deor&longs;um, quæ dum deor&longs;um mouetur, ve<lb/>ctem NH mouebit; & dum vectis moue<lb/><arrow.to.target n="note251"/>tur, N deor&longs;um mouebitur, H verò &longs;ur<lb/>&longs;um, vti &longs;upra dictum e&longs;t. </s>

<s id="id.2.1.163.8.1.6.0">dum autem H <lb/>mouetur &longs;ur&longs;um, mouet etiam &longs;ur&longs;um E; & <lb/>vectem EC, cuius fulcimentum e&longs;t C, &longs;ed <lb/>fulcimentum C non pote&longs;t mouere deor<lb/>&longs;um B; ideo orbiculus, cuius centrum K, &longs;ur<lb/><figure id="id.036.01.172.1.jpg" place="text" xlink:href="036/01/172/1.jpg"/><lb/>&longs;um mouebitur, & per con&longs;e&que;ns trochlea, & pondus A; vt in <lb/>præcedenti dictum e&longs;t. </s>

<s id="id.2.1.163.8.1.6.0">dum autem H <lb/>mouetur &longs;ur&longs;um, mouet etiam &longs;ur&longs;um E; & <lb/>vectem EC, cuius fulcimentum e&longs;t C, &longs;ed <lb/>fulcimentum C non pote&longs;t mouere deor<lb/>&longs;um B; ideo orbiculus, cuius centrum K, &longs;ur<lb/><figure id="id.036.01.172.1.jpg" place="text" xlink:href="036/01/172/1.jpg"/><lb/>&longs;um mouebitur, & per con&longs;equens trochlea, & pondus A; vt in <lb/>præcedenti dictum e&longs;t. </s>

<s id="id.2.1.163.8.1.7.0">& quoniam ob eandem cau&longs;am in præce<lb/>dentibus a&longs;signatam in HN, & EC &longs;emper remanent vectes hori<lb/>zonti æquidi&longs;tantes; potentia ergo mouens pondus A &longs;emper <lb/>eum mouebit vectibus horizonti æquidi&longs;tantibus. </s>

<s id="id.2.1.163.8.1.8.0">quod erat o<lb/>&longs;tendendum. </s>

<s id="id.2.1.164.1.1.1.0"><margin.target id="note249"/>1, <emph type="italics"/>Et<emph.end type="italics"/> 10 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

Line 2038

<s id="id.2.1.165.3.1.2.0">fiat deniq; RQ æqua <lb/>lis KS. </s>

<s id="N14E42">dum igitur k erit in R; pondus A, &longs;cilicet punctum S erit <lb/>in q. </s>

<s id="N14E46">& dum centrum orbiculi e&longs;t in R, &longs;it potentia in O mota <lb/>in P. </s>

<s id="id.2.1.165.3.1.2.0.a">& quoniam funis BCDEHMNO e&longs;t æqualis funi BFT <lb/>GHMNP; e&longs;t enim idem funis; & FTG æqualis e&longs;t CDE; dem<lb/>ptis igitur communibus BF, & GHMNO, erit reliquus OP ip<lb/>&longs;is FCEG &longs;imul &longs;umptis æqualis: & per con&longs;e&que;ns duplus kR, <lb/>& QS & cùm OP &longs;it &longs;patium potentiæ motæ, & SQ &longs;patium pon<lb/>deris moti; erit &longs;patium potentiæ duplum &longs;patii ponderis. </s>

<s id="id.2.1.165.3.1.2.0.a">& quoniam funis BCDEHMNO e&longs;t æqualis funi BFT <lb/>GHMNP; e&longs;t enim idem funis; & FTG æqualis e&longs;t CDE; dem<lb/>ptis igitur communibus BF, & GHMNO, erit reliquus OP ip<lb/>&longs;is FCEG &longs;imul &longs;umptis æqualis: & per con&longs;equens duplus kR, <lb/>& QS & cùm OP &longs;it &longs;patium potentiæ motæ, & SQ &longs;patium pon<lb/>deris moti; erit &longs;patium potentiæ duplum &longs;patii ponderis. </s>

<s id="id.2.1.165.3.1.3.0">quod <lb/>erat o&longs;tendendum. </s>

<s id="id.2.1.165.4.1.1.0">Præterea potentia idem pondus in æquali <lb/>tempore per dimidium &longs;patium mouebit fune <lb/>circa duos orbiculos reuoluto, quorum vnus <lb/>&longs;it trochleæ &longs;uperioris, alter verò &longs;it trochleæ <lb/>ponderi alligatæ; quàm &longs;ine trochleis: dummo<lb/>do ip&longs;ius potentiæ lationes &longs;int æqualiter ve<lb/>loces. </s><pb xlink:href="036/01/174.jpg"/>

<s id="id.2.1.165.6.1.1.0">Ii&longs;dem namq; po&longs;itis, &longs;it pon<lb/>dus V æquale ip&longs;i A, cui alliga<lb/>tus &longs;it funis X9; &longs;itq; <expan abbr="pot&etilde;tia">potentia</expan> in X <lb/>mouens <expan abbr="põdus">pondus</expan> V; quæ dum pon<lb/>dus mouet, perueniat in Y: fiant <lb/>&queacute; XY Z9 ip&longs;i OP æquales; <lb/>erit Z9 dupla QS. </s>

<s id="N14E8B">& &longs;i vtriu&longs;<lb/>&que; potentiæ velocitates mo<lb/>tuum &longs;int æquales; patet pon<lb/>dus V duplum pertran&longs;ire &longs;pa<lb/>tium in eodem tempore eìus, <lb/>quod pertran&longs;it pondus A. </s>

<s id="N14E8B">& &longs;i vtriu&longs;<lb/>que potentiæ velocitates mo<lb/>tuum &longs;int æquales; patet pon<lb/>dus V duplum pertran&longs;ire &longs;pa<lb/>tium in eodem tempore eìus, <lb/>quod pertran&longs;it pondus A. </s>

<s id="id.2.1.165.6.1.1.0.a">in eo<lb/>dem enim tempore potentia in <lb/>X peruenit ad Y, & potentia in <lb/>O ad P; ponderaq; &longs;imiliter in <lb/>Z Q. </s>

<s id="N14EA2">quod erat demon&longs;tran<lb/>dum. </s>

Line 2052

<s id="id.2.1.165.7.1.1.0">Fune circa &longs;ingulos duarum trochlearum <lb/>orbiculos, quarum altera &longs;upernè, altera verò <lb/>infernè, ponderiq; alligata fuerit, reuoluto; <lb/>altero etiam eius extremo inferiori trochleæ re<pb n="81" xlink:href="036/01/175.jpg"/>ligata, altero autem à mouente potentia deten<lb/>to: erit decur&longs;um trahentis potentiæ &longs;patium, mo<lb/>ti ponderis &longs;patii triplum. </s>

<s id="id.2.1.165.8.1.1.0">Sit pondus A; &longs;it BCD orbiculus tro<lb/>chleæ ponderi A ex EQ &longs;u&longs;pen&longs;o alligatæ; <lb/>&longs;itq; orbiculi centrum E; &longs;it deinde FGH <lb/>orbiculus trochleæ &longs;ur&longs;um appen&longs;æ, cuius <lb/>centrum k; &longs;itq; funis LFGHDCBM <lb/>circa omnes reuolutus orbiculos, tro<lb/>chleæq; inferiori in L religatus: &longs;itq; in <lb/>M potentia mouens. </s>

<s id="id.2.1.165.8.1.2.0">dico &longs;patium de<lb/>cur&longs;um à potentia in M, dum mouet pon<lb/>dus, triplum e&longs;&longs;e &longs;patii moti ponderis A. </s>

<s id="id.2.1.165.8.1.2.0.a"><lb/>Moueatur potentia in M v&longs;q; ad N; & <lb/>centrum E &longs;it motum v&longs;q; ad O; & L v&longs;<lb/>&que; ad P; atq; pondus A, hoc e&longs;t pun<lb/>ctum Q v&longs;q; ad R; orbiculu&longs;q; motus, &longs;it <lb/>TSV. </s>

<s id="id.2.1.165.8.1.2.0.a"><lb/>Moueatur potentia in M v&longs;q; ad N; & <lb/>centrum E &longs;it motum v&longs;q; ad O; & L v&longs;<lb/>que ad P; atq; pondus A, hoc e&longs;t pun<lb/>ctum Q v&longs;q; ad R; orbiculu&longs;q; motus, &longs;it <lb/>TSV. </s>

<s id="N14EED">ducantur per EO lineæ ST BD <lb/>horizonti æquidi&longs;tantes, quæ inter &longs;e &longs;e <lb/>quoq; æquidi&longs;tantes erunt. </s>

<s id="id.2.1.165.8.1.3.0">quoniam au<lb/>tem dum E e&longs;t in O, punctum Q e&longs;t in <lb/>R; erit EQ æqualis OR, & EO ip&longs;i QR <lb/>æqualis; &longs;imiliter LQ æqualis erit PR, <lb/>& L P ip&longs;i QR æqualis. </s>

<s id="id.2.1.165.8.1.4.0">tres igitur QR <lb/>EO LP inter &longs;e &longs;e æquales erunt; quibus <lb/>etiam &longs;unt æquales BS DT. </s>

<s id="id.2.1.165.8.1.4.0.a">& quoniam fu<lb/>nis LFGHDCBM æqualis e&longs;t funi PF <lb/>GHTVSN, cùm &longs;it idem funis, & qui <lb/>circa &longs;emicirculum TVS e&longs;t æqualis funi <lb/>circa &longs;emicirculum BCD; demptis igi<lb/>tur communibus PFGHT' & SM; erit <lb/>reliquus MN tribus BS LP DT &longs;imul <lb/>&longs;umptis æqualis. </s>

<s id="id.2.1.165.8.1.5.0">BS verò LP DT &longs;imul <lb/>tripli &longs;unt EO, & ex con&longs;e&que;nti QR. <lb/><figure id="id.036.01.175.1.jpg" place="text" xlink:href="036/01/175/1.jpg"/></s>

<s id="id.2.1.165.8.1.5.0">BS verò LP DT &longs;imul <lb/>tripli &longs;unt EO, & ex con&longs;equenti QR. <lb/><figure id="id.036.01.175.1.jpg" place="text" xlink:href="036/01/175/1.jpg"/></s>

<s id="id.2.1.165.8.1.5.0.a">&longs;patium igitur MN translatæ potentiæ &longs;patii QR ponderis mo<lb/>ti triplum erit. </s>

<s id="id.2.1.165.8.1.6.0">quod erat demon&longs;trandum. </s>

Line 2106

<s id="id.2.1.168.1.1.1.0"><margin.target id="note253"/>3 <emph type="italics"/>Huius. de vecte.<emph.end type="italics"/></s>

<s id="id.2.1.169.1.1.1.0">Præterea con&longs;iderandum occurrit, cùm hæc omnia maneant, <lb/>idem e&longs;&longs;e vnico exi&longs;tente fune CD EFG hoc modo orbiculo <expan abbr="cicum">circum</expan><lb/>uoluto, ac &longs;i duo e&longs;&longs;ent funes CD FG in vecte &longs;iue libra DF al<lb/>ligati. </s>

<s id="id.2.1.169.2.1.1.0">ALITER. </s>

<s id="id.2.1.169.3.1.1.0">Ii&longs;dem po&longs;itis, &longs;i in G appen&longs;um e&longs;&longs;et pondus k æquale pon<lb/>deri B, pondera B k æ&que;ponderabunt in libra DF, cuius centrum <lb/>A. </s>

<s id="id.2.1.169.3.1.1.0">Ii&longs;dem po&longs;itis, &longs;i in G appen&longs;um e&longs;&longs;et pondus k æquale pon<lb/>deri B, pondera B k æqueponderabunt in libra DF, cuius centrum <lb/>A. </s>

<s id="id.2.1.169.3.1.1.0.a">potentia verò in H &longs;u&longs;tinens pondera Bk e&longs;t ip&longs;is &longs;imul &longs;um<lb/>ptis æqualis, & pondera BK ip&longs;ius B &longs;unt dupla; potentia ergo in <lb/>H ponderis B dupla erit. </s>

<s id="id.2.1.169.3.1.2.0">& quoniam funis religatus in G nihil a<lb/>liud efficit, ni&longs;i quòd pondus B &longs;u&longs;tinet, ne de&longs;cendat; quod idem <lb/>efficit pondus k in G appen&longs;um: potentia igitur in H &longs;u&longs;tinens <lb/>pondus B, fune religato in G, dupla e&longs;t ponderis B. </s>

<s id="N15138">quod de<lb/>mon&longs;trare oportebat. </s><pb n="84" xlink:href="036/01/181.jpg"/>

Line 2119

<s id="id.2.1.169.8.1.1.0">Ii&longs;dem po&longs;itis, &longs;patium ponderis moti duplum <lb/>e&longs;t &longs;patii potentiæ mouentis. </s><pb xlink:href="036/01/182.jpg"/>

<s id="id.2.1.169.10.1.1.0">Sit motus orbiculus à centro A <lb/>v&longs;q; ad centrum L; & pondus B, <lb/>hoc e&longs;t punctum C, in eodem tem<lb/>pore &longs;it motum in P; & potentia in <lb/>H v&longs;q; ad K; erit AH ip&longs;i LK æqua <lb/>lis, & AL ip&longs;i Hk. </s>

<s id="id.2.1.169.10.1.2.0">& quoniam fu<lb/>nis CDEFG e&longs;t æqualis funi PM <lb/>NOG, idem enim e&longs;t funis, & fu <lb/>nis circa &longs;emicirculum MNO æ<lb/>qualis e&longs;t funi circa &longs;emicirculum <lb/>DEF; demptis igitur communi<lb/>bus DP FG, erit PC æqualis <lb/>DM FO &longs;imul &longs;umptis, qui funes <lb/>&longs;unt dupli ip&longs;ius AL, & con&longs;e&que;n<lb/>ter ip&longs;ius Hk. </s>

<s id="id.2.1.169.10.1.3.0">&longs;patium ergo pon<lb/>deris moti CP duplum e&longs;t &longs;patii <lb/>Hk potentiæ. </s>

<s id="id.2.1.169.10.1.4.0">quod oportebat de<lb/>mon&longs;trare. </s>

Line 2166

<s id="id.2.1.175.3.1.1.0">Sit trochlea inferior, duos habens orbiculos, <lb/>quorum centra AB; &longs;it &queacute; trochlea &longs;uperior <lb/>duos &longs;imiliter habens orbiculos, quorum cen<lb/>tra CD; funi&longs;q; EFGHKLMNOP &longs;it cir<lb/>ca omnes orbiculos reuolutus, qui &longs;it religatus <lb/>in E; & in P appendatur pondus Q; &longs;itq; po<lb/>tentia in R. </s>

<s id="id.2.1.175.3.1.1.0.a">dico potentiam in R quadruplam <lb/>e&longs;&longs;e ponderis q. </s>

<s id="N1536C">Cùm enim &longs;i duæ intelligan<lb/>tur potentiæ, vna in k, altera in D, potentia <lb/><arrow.to.target n="note257"/>in k &longs;u&longs;tinens pondus Q fune k LMNOP æ<lb/>qualis erit ponderi; erunt duæ &longs;imul potentiæ, <lb/>vna in D, altera in k, pondus Q &longs;u&longs;tinentes, <lb/>triplæ eiu&longs;dem ponderis. </s>

<s id="id.2.1.175.3.1.2.0">Potentia verò in C <lb/>dupla e&longs;t potentiæ in k, & per con&longs;e&que;ns pon<lb/>deris Q; idem enim e&longs;t, ac &longs;i in k appen&longs;um e&longs;<lb/><arrow.to.target n="note258"/>&longs;et pondus æquale ponderi Q, cuius dupla e&longs;t <lb/>potentia in C; duæ igitur potentiæ in DC qua<lb/>druplæ &longs;unt ponderis q. </s>

<s id="id.2.1.175.3.1.2.0">Potentia verò in C <lb/>dupla e&longs;t potentiæ in k, & per con&longs;equens pon<lb/>deris Q; idem enim e&longs;t, ac &longs;i in k appen&longs;um e&longs;<lb/><arrow.to.target n="note258"/>&longs;et pondus æquale ponderi Q, cuius dupla e&longs;t <lb/>potentia in C; duæ igitur potentiæ in DC qua<lb/>druplæ &longs;unt ponderis q. </s>

<s id="N1538B">& cùm potentia in R <lb/>orbiculis &longs;u&longs;tineat pondus Q, erit <expan abbr="pot&etilde;tia">potentia</expan> in R, <lb/>ac &longs;i duæ e&longs;&longs;ent potentiæ, vna in D, altera in C, <lb/>& vtræq; &longs;imul pondus Q &longs;u&longs;tinerent. </s>

<s id="id.2.1.175.3.1.3.0">ergo po<lb/>tentia in R quadrupla e&longs;t ponderis q. </s>

<s id="N1539C">quod <lb/>oportebat demon&longs;trare. <figure id="id.036.01.186.1.jpg" place="text" xlink:href="036/01/186/1.jpg"/></s>

Line 2230

<s id="id.2.1.181.11.1.1.0">Sit ABC orbiculus <lb/>trochleæ &longs;uperioris, & <lb/>DEF trochleæ inferio<lb/>ris ponderi G alligatæ; <lb/>&longs;itq; funis HABCDE <lb/>Fk circa orbiculos re<lb/>uolutus, qui &longs;it religatus <lb/>in K, & in H trochleæ <lb/>inferiori; &longs;itq; potentia <lb/>in L &longs;u&longs;tinens pondus <lb/>G. </s>

<s id="id.2.1.181.11.1.1.0.a">dico pondus poten<lb/>tiæ &longs;e&longs;quialterum e&longs;&longs;e. </s>

<s id="id.2.1.181.11.1.2.0"><lb/><arrow.to.target n="note261"/>Quoniam enim vter&que; <lb/>funis CD AH tertiam <lb/>&longs;u&longs;tinet partem ponde<lb/>ris G, erit vnaquæq; po<lb/>tentia in DH &longs;ubtripla <lb/>ponderis G; quibus &longs;i<lb/>mul a&longs;&longs;umptis e&longs;t æqua<lb/><figure id="id.036.01.194.1.jpg" place="text" xlink:href="036/01/194/1.jpg"/><lb/><arrow.to.target n="note262"/>lis potentia in L: potentia enim in L dupla e&longs;t potentiæ in D, & <lb/>eius, quæ e&longs;t in H. </s>

<s id="id.2.1.181.11.1.2.0"><lb/><arrow.to.target n="note261"/>Quoniam enim vterque <lb/>funis CD AH tertiam <lb/>&longs;u&longs;tinet partem ponde<lb/>ris G, erit vnaquæq; po<lb/>tentia in DH &longs;ubtripla <lb/>ponderis G; quibus &longs;i<lb/>mul a&longs;&longs;umptis e&longs;t æqua<lb/><figure id="id.036.01.194.1.jpg" place="text" xlink:href="036/01/194/1.jpg"/><lb/><arrow.to.target n="note262"/>lis potentia in L: potentia enim in L dupla e&longs;t potentiæ in D, & <lb/>eius, quæ e&longs;t in H. </s>

<s id="N15654">quare potentia in L &longs;ub&longs;e&longs;quialtera e&longs;t ponde<lb/>ris G. </s>

<s id="id.2.1.181.11.1.2.0.a">pondus ergo G ad pontentiam in L e&longs;t, vt tria ad duo; <lb/>hoc e&longs;t &longs;e&longs;quialterum. </s>

<s id="id.2.1.181.11.1.3.0">quod demon&longs;trare oportebat. </s>

Line 2271

<s id="id.2.1.185.4.1.1.0">Motus verò vectium fit hoc mo <lb/>do, videlicet vectis ML fulci<lb/>mentum e&longs;t M, cùm funis &longs;it re <lb/>ligatus in N, & pondus in me<lb/>dio, & potentia in L. </s>

<s id="N157DB">ve<lb/>rò punctum L tendit &longs;ur&longs;um, quod <lb/>à fune KL mouetur, idcirco K &longs;ur<lb/>&longs;um mouebitur, & vectis HK ful<lb/>cimentum erit H, pondus ac &longs;i e&longs;<lb/>&longs;ent in k, & potentia in medio; <lb/>vectis autem FG fulcimentum <lb/>erit G, pondus in medio; & poten<lb/>tia in F. </s>

<s id="id.2.1.185.4.1.1.0.a">punctum enim F &longs;ur&longs;um <lb/>mouetur à fune EF. </s>

<s id="id.2.1.185.4.1.1.0.b">Præterea <lb/>G in orbiculo deor&longs;um tendit, <lb/>quia H quo&que; in eius orbiculo <lb/>deor&longs;um mouetur. <figure id="id.036.01.198.1.jpg" place="text" xlink:href="036/01/198/1.jpg"/></s>

<s id="id.2.1.185.4.1.1.0.b">Præterea <lb/>G in orbiculo deor&longs;um tendit, <lb/>quia H quoque in eius orbiculo <lb/>deor&longs;um mouetur. <figure id="id.036.01.198.1.jpg" place="text" xlink:href="036/01/198/1.jpg"/></s>

<s id="id.2.1.185.4.3.1.0">PROPOSITIO XXII. </s>

<s id="id.2.1.185.5.1.1.0">Si vtri&longs;&que; duarum trochlearum &longs;ingulis <lb/>orbiculis, quarum altera &longs;upernè à potentia <lb/>&longs;u&longs;tineatur, altera verò infernè, ponderiq; alli<lb/>gata, collocata fuerit, circumducatur funis; al<lb/>tero eius extremo alicubi, altero autem &longs;uperio<lb/>ri trochleæ religato. </s>

<s id="id.2.1.185.5.1.1.0">Si vtri&longs;que duarum trochlearum &longs;ingulis <lb/>orbiculis, quarum altera &longs;upernè à potentia <lb/>&longs;u&longs;tineatur, altera verò infernè, ponderiq; alli<lb/>gata, collocata fuerit, circumducatur funis; al<lb/>tero eius extremo alicubi, altero autem &longs;uperio<lb/>ri trochleæ religato. </s>

<s id="id.2.1.185.5.1.2.0">erit potentia ponderis &longs;e&longs;<lb/>quialtera. </s>

<s id="id.2.1.185.6.1.1.0">Sit orbiculus ABC trochleæ ponderi D al <lb/>ligatæ; & EFG trochleæ &longs;uperioris, cuius <lb/>centrum H; &longs;it deinde funis k ABCEFGL <lb/>circa orbiculos reuolutus, & religatus in L, & <lb/>in k trochleæ &longs;uperiori; &longs;itq; potentia in M <lb/>&longs;u&longs;tinens pondus D. </s>

<s id="id.2.1.185.6.1.1.0.a">dico potentiam ponde<lb/>ris &longs;e&longs;quialteram e&longs;&longs;e. </s>

<s id="id.2.1.185.6.1.2.0">Quoniam enim poten<arrow.to.target n="note264"/><lb/>tia in E &longs;u&longs;tinens pondus D &longs;ubdupla e&longs;t pon<arrow.to.target n="note265"/><lb/>deris D, potentiæ verò in E dupla e&longs;t poten<arrow.to.target n="note266"/><lb/>tia in H; erit potentia in H ponderi D æqua <arrow.to.target n="note267"/><lb/>lis; & cùm potentia in K &longs;ubdupla &longs;it ponde<lb/>ris D; erunt vtræq; &longs;imul potentiæ in H k &longs;e&longs;<lb/>quialteræ ponderis D. </s>

<s id="id.2.1.185.6.1.2.0.a">Itaq; cùm potentia in <lb/>M duabus potentiis in Hk &longs;imul &longs;umptis &longs;it <lb/>æqualis, &que;madmodum in &longs;uperioribus o<lb/>&longs;ten&longs;um e&longs;t; erit potentia in M &longs;e&longs;quialtera <lb/>ponderis D. </s>

<s id="id.2.1.185.6.1.2.0.a">Itaq; cùm potentia in <lb/>M duabus potentiis in Hk &longs;imul &longs;umptis &longs;it <lb/>æqualis, quemadmodum in &longs;uperioribus o<lb/>&longs;ten&longs;um e&longs;t; erit potentia in M &longs;e&longs;quialtera <lb/>ponderis D. </s>

<s id="N15857">quod oportebat demon&longs;trare. </s>

<s id="id.2.1.186.1.1.1.0"><margin.target id="note264"/>2 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.186.1.1.2.0"><margin.target id="note265"/><emph type="italics"/>Ex<emph.end type="italics"/> 15 <emph type="italics"/>huius.<emph.end type="italics"/></s>

Line 2423

<s id="id.2.1.201.6.1.1.0">Hactenus proportiones ponderis ad potentiam multiplices, <lb/>& &longs;ubmultiplices; deinde &longs;uperparticulares, &longs;ub&longs;uperparticu<lb/>lare&longs;&queacute; declaratæ fuerunt: nunc autem reliquum e&longs;t, vt propor<lb/>tiones inter pondus, & potentiam &longs;uperpartientes, & multi<lb/>plices &longs;uperparticulares, multiplices&queacute; &longs;uperpartientes mani<lb/>fe&longs;tentur. </s>

<s id="id.2.1.201.7.1.1.0">PROPOSITIO XXVI. </s>

<s id="id.2.1.201.7.3.1.0">PROBLEMA. </s>

<s id="id.2.1.201.8.1.1.0">Si proportionem &longs;uperpartientem inuenire <lb/>volumus, &que;madmodum &longs;i proportio, quam <lb/>habet pondus ad potentiam pondus &longs;u&longs;tinen<lb/>tem fuerit &longs;uperbipartiens, &longs;icut quin&que; ad <lb/>tria. </s><pb xlink:href="036/01/214.jpg"/>

<s id="id.2.1.201.8.1.1.0">Si proportionem &longs;uperpartientem inuenire <lb/>volumus, quemadmodum &longs;i proportio, quam <lb/>habet pondus ad potentiam pondus &longs;u&longs;tinen<lb/>tem fuerit &longs;uperbipartiens, &longs;icut quinque ad <lb/>tria. </s><pb xlink:href="036/01/214.jpg"/>

<s id="id.2.1.201.10.1.1.0"><arrow.to.target n="note283"/>Exponatur potentia in A pondus B &longs;u&longs;ti<lb/>nens, proportionemq; habeat pondus B ad <lb/>potentiam in A, vt quinq; ad vnum; hoc e&longs;t, <lb/>&longs;it potentia in A &longs;ubquintupla ponderis B: de<lb/>inde eodem fune circa alios orbiculos reuo<lb/><arrow.to.target n="note284"/>luto inueniatur potentia in C, quæ tripla &longs;it <lb/>potentiæ in A. </s>

<s id="id.2.1.201.10.1.1.0.a">& quoniam pondus B ad po<lb/>tentiam in A e&longs;t, vt quinq; ad vnum; & <lb/>potentia in A ad potentiam in C e&longs;t, vt vnum <lb/>ad tria; erit pondus B ad potentiam in C, vt <lb/>quinq; ad tria; hoc e&longs;t &longs;uperbipartiens. </s>

Line 2440

<s id="id.2.1.205.2.1.1.0">Si autem multiplicem &longs;uperparticularem in<lb/>uenire voluerimus; vt proportio, quam habet <lb/>pondus ad potentiam pondus &longs;u&longs;tinentem, &longs;it <lb/>duplex &longs;e&longs;quialtera, vt quinq; ad duo. </s>

<s id="id.2.1.206.1.1.1.0"><margin.target id="note287"/>18 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.206.1.1.2.0"><margin.target id="note288"/>5 <emph type="italics"/>Huius.<emph.end type="italics"/></s>

<s id="id.2.1.207.1.1.1.0">Eodem modo, quo &longs;uperpartientes inuenimus, has quo<lb/>&que; omnes multiplices &longs;uperparticulares reperiemus. </s>

<s id="id.2.1.207.1.1.1.0">Eodem modo, quo &longs;uperpartientes inuenimus, has quo<lb/>que omnes multiplices &longs;uperparticulares reperiemus. </s>

<s id="id.2.1.207.1.1.2.0">vt fiat <arrow.to.target n="note289"/><lb/>pondus B ad potentiam in A, vt quinq; ad vnum; potentia ve<arrow.to.target n="note290"/><lb/>ro in C ad potentiam in A, vt duo ad vnum; quod fiet, &longs;i fu<lb/>nis &longs;it religatus in D, non autem trochleæ &longs;uperiori, vel in F: erit <lb/>pondus B ad potentiam in C, vt quinq; ad duo; hoc e&longs;t duplum <lb/>&longs;e&longs;quialterum. </s>

<s id="id.2.1.208.1.1.1.0"><margin.target id="note289"/><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius.<emph.end type="italics"/></s>

<s id="id.2.1.208.1.1.2.0"><margin.target id="note290"/><emph type="italics"/>Ex<emph.end type="italics"/> 15, 16, <emph type="italics"/>Huius.<emph.end type="italics"/></s>

Line 2460

<s id="id.2.1.211.3.1.2.0">vt <lb/>&longs;i multiplicem &longs;uperparticularem proportionem <lb/>pluribus funibus inuenire voluerimus, veluti &longs;i <lb/>proportio, quam habet pondus ad potentiam <lb/>pondus &longs;u&longs;tinentem, fuerit duplex &longs;e&longs;quialtera, vt <lb/>quinq; ad duo; oportet hanc proportionem ex <lb/>pluribus componere. </s>

<s id="id.2.1.211.3.1.3.0">vt (exempli gratia) ex pro<lb/>portione &longs;e&longs;quiquarta, vt quin&queacute; ad quatuor, <lb/>& ex dupla, vt quatuor ad duo. </s>

<s id="id.2.1.211.3.1.4.0">exponatur igitur po<arrow.to.target n="note292"/><lb/>tentia in A pondus B &longs;u&longs;tinens, ad quam pondus <lb/><expan abbr="proportion&etilde;">proportionem</expan> habeat &longs;e&longs;quiquartam, vt quinq; ad <lb/>quatuor: deinde alio fune inueniatur <expan abbr="pot&etilde;tia">potentia</expan> in C,<arrow.to.target n="note293"/><lb/>cuius dupla &longs;it potentia in A. </s>

<s id="id.2.1.211.3.1.4.0.a">& <expan abbr="quoniã">quoniam</expan> B ad A e&longs;t, <lb/>vt quinq; ad quatuor; & A ad C, vt quatuor ad <lb/>duo; erit pondus B ad potentiam in C, vt quin<lb/>&que; ad duo; hoc e&longs;t proportionem habebit du<lb/>plicem &longs;e&longs;quialteram. <figure id="id.036.01.217.1.jpg" place="text" xlink:href="036/01/217/1.jpg"/></s>

<s id="id.2.1.211.3.1.4.0.a">& <expan abbr="quoniã">quoniam</expan> B ad A e&longs;t, <lb/>vt quinq; ad quatuor; & A ad C, vt quatuor ad <lb/>duo; erit pondus B ad potentiam in C, vt quin<lb/>que ad duo; hoc e&longs;t proportionem habebit du<lb/>plicem &longs;e&longs;quialteram. <figure id="id.036.01.217.1.jpg" place="text" xlink:href="036/01/217/1.jpg"/></s>

<s id="id.2.1.211.4.1.1.0">Et notandum e&longs;t hanc quoq; <expan abbr="proportion&etilde;">proportionem</expan> inue<lb/>niri po&longs;&longs;e, &longs;i proportionem quinq; ad duo ex pluri<lb/>bus componamus, vt quinq; ad quindecim & quin<lb/>decim ad viginti & viginti ad duo. </s>

<s id="id.2.1.211.4.1.2.0">Et hoc modo <lb/>non &longs;olum omnem aliam proportionem inuenie<lb/>mus, &longs;ed quamcunq, multis, infinitis&queacute; mo<lb/>dis comperiemus. </s>

<s id="id.2.1.211.4.1.3.0">omnis enim proportio ex infi<lb/>nitis proportionibus componi pote&longs;t. </s>

Line 2501

<s id="id.2.1.217.6.1.1.0">Data potentia, vel e&longs;t maior, vel æqualis, vel minor dato <lb/>pondere. </s><pb n="104" xlink:href="036/01/221.jpg"/>

<s id="id.2.1.217.8.1.1.0">Et &longs;i e&longs;t maior, tunc poten<lb/>tia, vel ab&longs;q; alio in&longs;trumento, <lb/>vel fune circa orbiculum trochleæ <lb/>&longs;ur&longs;um appen&longs;æ reuoluto datum <lb/>pondus mouebit. </s>

<s id="id.2.1.217.8.1.2.0">Minor enim po<arrow.to.target n="note300"/><lb/>tentia; quàm data, ponderi æ&que;<lb/>ponderat, data ergo mouebit. </s>

<s id="id.2.1.217.8.1.2.0">Minor enim po<arrow.to.target n="note300"/><lb/>tentia; quàm data, ponderi æque<lb/>ponderat, data ergo mouebit. </s>

<s id="id.2.1.217.8.1.3.0"><lb/>Quod idem fieri pote&longs;t iuxta om<lb/>nes propo&longs;itiones, quibus poten<lb/>tia pondus &longs;u&longs;tinens, vel æqualis, <lb/>vel minor pondere o&longs;ten&longs;a e&longs;t. <figure id="id.036.01.221.1.jpg" place="text" xlink:href="036/01/221/1.jpg"/></s>

<s id="id.2.1.217.9.1.1.0">Si autem æqualis, <lb/>pondus mouebit fune <lb/>per orbiculum trochleæ <lb/>ponderi alligatæ circum<lb/>uoluto. </s>

<s id="id.2.1.217.9.1.2.0">potentia enim <arrow.to.target n="note301"/><lb/>&longs;u&longs;tinens pondus &longs;ubdu<lb/>pla e&longs;t ponderis, poten<lb/>tia igitur ponderi æqua<lb/>lis datum pondus mo<lb/>uebit. </s>

Line 2517

<s id="id.2.1.218.1.1.3.0"><margin.target id="note302"/><emph type="italics"/>Ex<emph.end type="italics"/> 9 <emph type="italics"/>huius<emph.end type="italics"/></s>

<s id="id.2.1.219.1.1.1.0"><expan abbr="Animaduertend&utilde;">Animaduertendum</expan> quoq; e&longs;t in mo <lb/>uendis ponderibus, potentiam ali<lb/>quando for&longs;itan melius mouere mo<lb/>uendo &longs;e deor&longs;um, quàm mouendo <lb/>&longs;e &longs;ur&longs;um. </s>

<s id="id.2.1.219.1.1.2.0">vt circumuoluatur adhuc <lb/>funis per alium trochleæ &longs;uperioris <lb/>orbiculum, cuius centrum C, funi&longs;q; <lb/><arrow.to.target n="note303"/>perueniat in D; erit <expan abbr="pot&etilde;tia">potentia</expan> in D <expan abbr="&longs;u&longs;tin&etilde;s">&longs;u&longs;tinens</expan>

<expan abbr="põdus">pondus</expan> B &longs;imiliter duodecim, <expan abbr="&quetilde;">&que;m</expan><lb/>admodum erat in A. </s>

<expan abbr="põdus">pondus</expan> B &longs;imiliter duodecim, <expan abbr="&quetilde;">quem</expan><lb/>admodum erat in A. </s>

<s id="id.2.1.219.1.1.2.0.a">Ideo poten<lb/>tia vt tredecim in D pondus B mo<lb/>uebit. </s>

<s id="id.2.1.219.1.1.3.0">& quia mouet &longs;e deor&longs;um, <lb/>forta&longs;&longs;e trahet facilius, quàm in A; <lb/>atq; tempus e&longs;t idem, &longs;icut etiam <lb/>erat in A. <figure id="id.036.01.222.1.jpg" place="text" xlink:href="036/01/222/1.jpg"/></s><pb n="105" xlink:href="036/01/223.jpg"/>

Line 2549

<s id="id.2.1.223.7.1.1.0">Fabricam, & <expan abbr="cõ&longs;tructionem">con&longs;tructionem</expan> hu<lb/>ius in&longs;trumenti Pappus in octauo <lb/>mathematicarum collectionum <lb/>libro docet; axemq; vocat AB, <lb/>tympanum verò CD circa idem <lb/>centrum; & &longs;cytalas in foramini<lb/>bus tympani EF GH & c. </s>

<s id="id.2.1.223.7.1.2.0">ita vt potentia, <pb xlink:href="036/01/226.jpg"/>

<figure id="id.036.01.226.1.jpg" place="text" xlink:href="036/01/226/1.jpg"/><lb/>quæ &longs;emper in &longs;cytalis e&longs;t, vt in F, dum circum<lb/>uertit tympanum, & axem, &longs;ur&longs;um moueat pon<lb/>dus K axi appen&longs;um fune LM circa axem reuo<lb/>luto. </s>

<s id="id.2.1.223.7.1.3.0">Nobis igitur re&longs;tat, vt o&longs;tendamus, cur ma<lb/>gna pondera ab exigua virtute, quouè etiam mo <lb/>do hoc in&longs;trumento moueantur; temporis quin <lb/>etiam, &longs;patiiq; mouentis inuicem potentiæ, ac <lb/>moti ponderis rationem aperiamus; huiu&longs;modi<lb/>&que; in&longs;trumenti v&longs;um ad vectem reducamus. </s><pb n="107" xlink:href="036/01/227.jpg"/>

<s id="id.2.1.223.7.1.3.0">Nobis igitur re&longs;tat, vt o&longs;tendamus, cur ma<lb/>gna pondera ab exigua virtute, quouè etiam mo <lb/>do hoc in&longs;trumento moueantur; temporis quin <lb/>etiam, &longs;patiiq; mouentis inuicem potentiæ, ac <lb/>moti ponderis rationem aperiamus; huiu&longs;modi<lb/>que in&longs;trumenti v&longs;um ad vectem reducamus. </s><pb n="107" xlink:href="036/01/227.jpg"/>

<s id="id.2.1.223.9.1.1.0">PROPOSITIO I. </s>

<s id="id.2.1.223.10.1.1.0">Potentia pondus &longs;u&longs;tinens axe in peritrochio <lb/>ad pondus eandem habet proportionem, quam <lb/>&longs;emidiameter axis ad &longs;emidiametrum tympani <lb/>vná cum &longs;cytala. <figure id="id.036.01.227.1.jpg" place="text" xlink:href="036/01/227/1.jpg"/></s>

Line 2559

<s id="id.2.1.223.11.1.2.0.a">Dico potentiam in F ad pondus <lb/>k ita &longs;e habere, vt CB ad CF. </s>

<s id="N16410">fiat vt CF ad CB, ita pondus <lb/>k ad aliud M, quod appendatur in F. </s>

<s id="id.2.1.223.11.1.2.0.b">& quoniam pondera M k <lb/>appen&longs;a &longs;unt in FB; erit FB tanquam vectis, &longs;iue libra; quia ve<lb/>rò C e&longs;t punctum immobile, circa quod axis, tympanusq; reuol<lb/>uuntur; erit C fulcimentum vectis FB; vellibræ centrum. </s>

<s id="id.2.1.223.11.1.3.0">cùm <lb/><arrow.to.target n="note307"/>autem it a &longs;it CF ad CB, vt k ad M, pondera k M æ&que;ponde<lb/>rabunt. </s>

<s id="id.2.1.223.11.1.3.0">cùm <lb/><arrow.to.target n="note307"/>autem it a &longs;it CF ad CB, vt k ad M, pondera k M æqueponde<lb/>rabunt. </s>

<s id="id.2.1.223.11.1.4.0">Potentia igitur in F &longs;u&longs;tinens pondus k, ne deor&longs;um ver<lb/>gat, ponderi K æ&que;ponderabit; ip&longs;iq; M æqualis erit. </s>

<s id="id.2.1.223.11.1.4.0">Potentia igitur in F &longs;u&longs;tinens pondus k, ne deor&longs;um ver<lb/>gat, ponderi K æqueponderabit; ip&longs;iq; M æqualis erit. </s>

<s id="id.2.1.223.11.1.5.0">idem enim <lb/>præ&longs;tat potentia, quod pondus M. </s>

<s id="id.2.1.223.11.1.5.0.a">pondus igitur K ad poten<lb/><arrow.to.target n="note308"/>tiam in F erit, vt CF ad CB; & conuertendo, potentia ad <lb/>pondus erit, vt CB ad CF, hoc e&longs;t, &longs;emidiameter axis ad &longs;emi<pb n="108" xlink:href="036/01/229.jpg"/>diametrum tympani vnà cum &longs;cytala DF. </s>

<s id="id.2.1.223.11.1.5.0.b">Similiter etiam o&longs;ten<lb/>detur, &longs;i potentia pondus &longs;u&longs;tinens fuerit in q. </s>

<s id="N16445">tunc enim &longs;u&longs;ti<lb/>neret vecte CQ; & ad pondus eam haberet proportionem, quam <arrow.to.target n="note309"/><lb/>habet CB ad Cq. </s>

<s id="N1644E">Videlicet &longs;emidiameter axis ad &longs;emidiame<lb/>trum tympani vná cum &longs;cytala Eq. </s>

<s id="N16452">quod demon&longs;trare opor<lb/>tebat. </s>

<s id="id.2.1.224.1.1.1.0"><margin.target id="note307"/>6. <emph type="italics"/>Primi Archim. de æ&que;pon.<emph.end type="italics"/></s>

<s id="id.2.1.224.1.1.1.0"><margin.target id="note307"/>6. <emph type="italics"/>Primi Archim. de æquepon.<emph.end type="italics"/></s>

<s id="id.2.1.224.1.1.3.0"><margin.target id="note308"/><emph type="italics"/>Cor.<emph.end type="italics"/> 4. <emph type="italics"/>quinti.<emph.end type="italics"/></s>

<expan abbr="Huuius">Huius</expan>. de vecte.<emph.end type="italics"/></s>

Line 2590

<s id="id.2.1.225.4.1.7.0.b">ergo <lb/><arrow.to.target n="note313"/>grauius erit pondus in T, quàm pondus in F. </s>

<s id="id.2.1.226.1.1.1.0"><margin.target id="note310"/><emph type="italics"/>Ex<emph.end type="italics"/> 19 <emph type="italics"/>primi.<emph.end type="italics"/></s>

<s id="id.2.1.226.1.1.2.0"><margin.target id="note311"/><emph type="italics"/>Ex<emph.end type="italics"/> 13 <emph type="italics"/>primi.<emph.end type="italics"/></s>

<s id="id.2.1.226.1.1.3.0"><margin.target id="note312"/>6. <emph type="italics"/>Primi Archim. de æ&que;pon.<emph.end type="italics"/></s>

<s id="id.2.1.226.1.1.3.0"><margin.target id="note312"/>6. <emph type="italics"/>Primi Archim. de æquepon.<emph.end type="italics"/></s>

<s id="id.2.1.226.1.1.5.0"><margin.target id="note313"/>10. <emph type="italics"/>Quinti.<emph.end type="italics"/></s>

<s id="id.2.1.227.1.1.1.0">Si verò loco ponderis in T animata potentia &longs;u&longs;tinens pon<lb/>dus k con&longs;tituatur; quæ ita degrauet &longs;e, ac &longs;i in centrum mundi <lb/>tendere vellet; &que;madmodum &longs;uapte natura efficit pondus in T <lb/>appen&longs;um; erit hæc eadem ponderi in T appen&longs;o æqualis; alio<lb/>quin non &longs;u&longs;tineret; quæ quidem ip&longs;a potentia in F collocata ma<pb n="109" xlink:href="036/01/231.jpg"/>ior erit. </s>

<s id="id.2.1.227.1.1.1.0">Si verò loco ponderis in T animata potentia &longs;u&longs;tinens pon<lb/>dus k con&longs;tituatur; quæ ita degrauet &longs;e, ac &longs;i in centrum mundi <lb/>tendere vellet; quemadmodum &longs;uapte natura efficit pondus in T <lb/>appen&longs;um; erit hæc eadem ponderi in T appen&longs;o æqualis; alio<lb/>quin non &longs;u&longs;tineret; quæ quidem ip&longs;a potentia in F collocata ma<pb n="109" xlink:href="036/01/231.jpg"/>ior erit. </s>

<s id="id.2.1.227.1.1.2.0">&longs;icuti enim &longs;e &longs;e habet pondus in T ad pondus in F, ita <lb/>& potentia in T ad potentiam in F; cùm potentiæ &longs;int ponderi<lb/>bus æquales. </s>

<s id="id.2.1.227.1.1.3.0">verùm &longs;i vnaquæq; potentia &longs;eor&longs;um &longs;umpta, tàm <lb/>in T, quàm in F &longs;u&longs;tinens pondus <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan>

<expan abbr="circ&utilde;ferentiam">circunferentiam</expan> THFN <lb/>moueri &longs;e vellet, veluti apprehen&longs;a manu &longs;cytala; tunc eademmet <lb/>potentia, vel in F, vel in T con&longs;tituta idem pondus k &longs;u&longs;tinere po<lb/>terit; cùm &longs;emper in cuiu&longs;cunq; extremitate &longs;cytalæ ponatur, ab <lb/>eodem centro C æquidi&longs;tans fuerit, ac &longs;ecundum eandem circum<lb/>ferentiam ab eodem centro æqualiter &longs;emper di&longs;tantem perpen&longs;io<lb/>nem habeat. </s>

<s id="id.2.1.227.1.1.4.0">neq; enim (&longs;icuti pondus) proprio nutu magis in <lb/>centrum ferri exoptat, quam circulariter moueri; cùm vtrunq;, &longs;eu <lb/>&que;mlibet alium motum nullo pror&longs;us re&longs;piciat di&longs;crimine. </s>

<s id="id.2.1.227.1.1.4.0">neq; enim (&longs;icuti pondus) proprio nutu magis in <lb/>centrum ferri exoptat, quam circulariter moueri; cùm vtrunq;, &longs;eu <lb/>quemlibet alium motum nullo pror&longs;us re&longs;piciat di&longs;crimine. </s>

<s id="id.2.1.227.1.1.5.0">pro<lb/>pterea non eodem modo res &longs;e &longs;e habet, &longs;iue pondera, &longs;iue anímatæ <lb/>potentiæ ii&longs;dem locis eodem munere abeundo fuerint con&longs;titutæ. </s>

<s id="id.2.1.227.2.1.1.0">Potentia autem mouet pondus vecte FB, videlicet dum po<lb/>tentia in F circumuertit tympanum, circumuertit etiam axem; & <lb/>FB fit tamquam vectis, cuius fulcimentum C, potentia mouens <lb/>in F, & <expan abbr="podus">pondus</expan> in B appen&longs;um. </s>

<s id="id.2.1.227.2.1.2.0">& dum punctum F peruenit in N; <lb/>punctum H erit in F, & punctum B erit in O; ita vt ducta NO <lb/>tran&longs;eat per C; eodemq; tempore pondus k motum erit in P, ita <lb/>vt OBP &longs;it æqualis ip&longs;i BL, cùm &longs;it idem funis. </s>

Line 2618

<s id="id.2.1.229.6.1.3.0">circulorum enim circumfe<arrow.to.target n="note315"/><lb/>rentiæ ita &longs;e habent, vt diametri. </s>

<s id="id.2.1.229.6.1.4.0">Cùm vero ex trige&longs;ima &longs;exta <lb/>quarti libri Pappi Mathematicarum collectionum, duorum inæ<lb/>qualium circulorum æquales circumferentias inuenire po&longs;simus; <lb/>ideo tempus quoq; portionum circulorum inæqualium hoc modo <lb/>inueniemus. </s>

<s id="id.2.1.229.6.1.5.0">è conuer&longs;o autem, quò maior erit axis circumferen<lb/>tia citius pondus &longs;ur&longs;um mouebitur. </s>

<s id="id.2.1.229.6.1.6.0">maior enim pars funis BL <lb/>in vna circumuer&longs;ione completa circa circulum ABO reuoluitur, <lb/>quàm &longs;i minor e&longs;&longs;et; cùm funis circumuolutus &longs;it circumferen<lb/>tiæ circuli æqualis, circa &que;m reuoluitur. </s>

<s id="id.2.1.229.6.1.6.0">maior enim pars funis BL <lb/>in vna circumuer&longs;ione completa circa circulum ABO reuoluitur, <lb/>quàm &longs;i minor e&longs;&longs;et; cùm funis circumuolutus &longs;it circumferen<lb/>tiæ circuli æqualis, circa quem reuoluitur. </s>

<s id="id.2.1.230.1.1.1.0"><margin.target id="note315"/>23 <emph type="italics"/>Octaui libri Pappi.<emph.end type="italics"/></s>

<s id="id.2.1.231.1.1.1.0">COROLLARIVM. </s>

<s id="id.2.1.231.2.1.1.0">Ex his manife&longs;tum e&longs;t, quò facilius pondus mo<lb/>uetur, tempus quoq; eò maius e&longs;&longs;e; & quò dif<lb/>ficilius, eò tempus minus e&longs;&longs;e. </s>

Line 2629

<s id="id.2.1.231.5.1.1.0">Datum pondus à data potentia axe in peritro<lb/>chio moueri. </s>

<s id="id.2.1.231.6.1.1.0">Sit datum pondus &longs;exagin<lb/>ta; potentia verò vt decem. </s>

<s id="id.2.1.231.6.1.2.0"><lb/>exponatur quædam recta li<lb/>nea AB, quæ diuidatur in C, <lb/>ita vt AC ad CB eandem <lb/><figure id="id.036.01.234.1.jpg" place="text" xlink:href="036/01/234/1.jpg"/><lb/>habeat proportionem, quam &longs;exaginta ad decem. </s>

<s id="id.2.1.231.6.1.3.0">& &longs;i CB axis <lb/>&longs;emidiameter e&longs;&longs;et, & CA &longs;emidiameter tympani cùm &longs;cytalis; <lb/><arrow.to.target n="note316"/>patet potentiam vt decem in A ponderi &longs;exaginta in B æ&que;pon<lb/>derare. </s>

<s id="id.2.1.231.6.1.4.0">Accipiatur autem inter BC quoduis punctum D; fiatq; <lb/>BD &longs;emidiameter axis, & DA &longs;emidiameter tympani cùm &longs;cy<lb/>talis; ponaturq; pondus &longs;exaginta in B fune circa axem, & potentia <lb/><arrow.to.target n="note317"/><emph type="italics"/>in A. </s>

<s id="id.2.1.231.6.1.4.0.a">Quoniam enim AD ad DB maiorem habet proportio<lb/>nem, quam AC ad CB; maiorem habebit proportionem AD ad <lb/>DB, quam pondus &longs;exaginta in B appen&longs;um ad potentiam vt decem<emph.end type="italics"/><lb/><arrow.to.target n="note318"/>in A. </s>

<s id="id.2.1.231.6.1.4.0.b">Quare potentia in A pondus &longs;exaginta axe in peritro<lb/>chio mouebit, cuius axis &longs;emidiameter e&longs;t BD, & DA &longs;emidia<lb/>meter tympani cùm &longs;cytalis. </s>

Line 2640

<s id="id.2.1.233.1.2.1.0">ALITER. </s>

<s id="id.2.1.233.1.4.1.0">Organicè verò melius erit hoc pacto. </s>

<s id="id.2.1.233.2.1.1.0">Exponatur axis, cuius <lb/>diameter &longs;it BD, & cen<lb/>trum C, &que;m quidem <lb/>axem maiorem, vel mino<lb/>rem con&longs;tituemus, veluti <lb/><figure id="id.036.01.235.1.jpg" place="text" xlink:href="036/01/235/1.jpg"/><lb/>magnitudo, ponderi&longs;q; grauitas po&longs;tulat. </s>

<s id="id.2.1.233.2.1.1.0">Exponatur axis, cuius <lb/>diameter &longs;it BD, & cen<lb/>trum C, quem quidem <lb/>axem maiorem, vel mino<lb/>rem con&longs;tituemus, veluti <lb/><figure id="id.036.01.235.1.jpg" place="text" xlink:href="036/01/235/1.jpg"/><lb/>magnitudo, ponderi&longs;q; grauitas po&longs;tulat. </s>

<s id="id.2.1.233.2.1.2.0">producatur deinde BD <lb/>v&longs;q; ad A: fiatq; BC ad CA, vt decem ad &longs;exaginta. </s>

<s id="id.2.1.233.2.1.3.0">& &longs;i CA tym<lb/>pani cùm &longs;cytalis &longs;emidiameter e&longs;&longs;et, potentia decem in A ponde<lb/>ri &longs;exaginta in B æ&que;ponderaret. </s>

<s id="id.2.1.233.2.1.3.0">& &longs;i CA tym<lb/>pani cùm &longs;cytalis &longs;emidiameter e&longs;&longs;et, potentia decem in A ponde<lb/>ri &longs;exaginta in B æqueponderaret. </s>

<s id="id.2.1.233.2.1.4.0">producatur verò BA ex parte <lb/>A, & in hac producta linea quoduis accipiatur punctum E; fiatq; <lb/>CE &longs;emidiameter tympani cùm &longs;cytalis; ponaturq; potentia vt <lb/>decem in E; habebit EC ad CB maiorem proportionem, quàm <lb/>pondus &longs;exaginta in B ad potentiam vt decem in E. </s>

<s id="id.2.1.233.2.1.4.0.a">potentia igi<lb/>tur vt decem in E mouebit pondus &longs;exaginta in B appen&longs;um fune <lb/>circa axem, cuius &longs;emidiameter e&longs;t CB, & CE &longs;emidiameter tym<lb/>pani cùm &longs;cytalis. </s>

<s id="id.2.1.233.2.1.5.0">quod facere oportebat. </s><pb xlink:href="036/01/236.jpg"/>

Line 2742

<s id="id.2.1.233.30.1.1.0.a">Quoniam enim dum vectis AB mouet CD <lb/>EF, punctum vetis C mouetur &longs;uper circumferentiam CO; cùm <lb/>&longs;it B fulcimentum, & centrum immobile. </s>

<s id="id.2.1.233.30.1.2.0">&longs;imiliter dum vectis <lb/>MN mouet CDEF, punctum C mouetur per circumferentiam <lb/>CP; dum igitur vectis AB mouet CDEF, conatur mouere pun<lb/>ctum C ponderis &longs;uper circumferentiam CO; quod quidem effi<lb/>cere non pote&longs;t: quia C mouetur &longs;uper circumferentiam CL. </s>

<s id="N16ACE">qua<lb/>re in motu vectis AB &longs;ecundùm partem ip&longs;i re&longs;pondentem, ac mo<lb/>tu ponderis &longs;ecundum C facto, contingit repugnantia quædam; <lb/>in diuer&longs;as enim partes mouentur. </s>

<s id="id.2.1.233.30.1.3.0">&longs;imiliter dum vectis MN mo<lb/>uet CDEF, conatur mouere C &longs;uper circumferentiam CP; at<lb/>&que; ideo in hoc etiam vtroq; motu &longs;imilis oritur repugnantia. </s>

<s id="id.2.1.233.30.1.3.0">&longs;imiliter dum vectis MN mo<lb/>uet CDEF, conatur mouere C &longs;uper circumferentiam CP; at<lb/>que ideo in hoc etiam vtroq; motu &longs;imilis oritur repugnantia. </s>

<s id="id.2.1.233.30.1.4.0"><lb/>quoniam autem circumferentia CO propior e&longs;t circumferentiæ <lb/>CL, quam &longs;it CP; hoc e&longs;t propior e&longs;t motui, &que;m facit pun<lb/>ctum C ponderis; ideo minor erit repugnantia inter motum vectis <pb n="117" xlink:href="036/01/247.jpg"/>AB, & motum C ponderis, quàm inter motum vectis MN, & <lb/>motum eiu&longs;dem C. quod etiam patet, &longs;i intelligatur CF hori<lb/>zonti perpendicularis, tunc enim circumferentia CP magis ten<lb/>dit deor&longs;um, quàm CO; & CL tendit &longs;ur&longs;um. </s>

<s id="id.2.1.233.30.1.4.0"><lb/>quoniam autem circumferentia CO propior e&longs;t circumferentiæ <lb/>CL, quam &longs;it CP; hoc e&longs;t propior e&longs;t motui, quem facit pun<lb/>ctum C ponderis; ideo minor erit repugnantia inter motum vectis <pb n="117" xlink:href="036/01/247.jpg"/>AB, & motum C ponderis, quàm inter motum vectis MN, & <lb/>motum eiu&longs;dem C. quod etiam patet, &longs;i intelligatur CF hori<lb/>zonti perpendicularis, tunc enim circumferentia CP magis ten<lb/>dit deor&longs;um, quàm CO; & CL tendit &longs;ur&longs;um. </s>

<s id="id.2.1.233.30.1.5.0">& ideo minor fit re <lb/>pugnantia inter vectem AB, & motum C, quàm inter <expan abbr="vect&etilde;">vectem</expan> MN, & <lb/>motum C. </s>

<s id="N16AFB">&longs;ed vbi minor repugnantia ibi maior facilitas. </s>

<s id="id.2.1.233.30.1.6.0">ergo faci<lb/>lius mouebitur CD EF vecte AB, quàm vecte MN. </s>

Line 2854

<s id="id.2.1.237.20.1.4.0">& &longs;iue co<lb/>chlea fuerit horizonti perpendicularis, <lb/>&longs;iue horizonti æquidi&longs;tans, vel alio mo<lb/>do collocata, nihil refert: &longs;emper enim <lb/>eadem erit ratio. <pb xlink:href="036/01/258.jpg"/>

<s id="id.2.1.237.21.1.1.0">Si verò (vt in tertia figura) &longs;upra cochleam imponatur aliquod, <lb/>vt B, quod quidem tylum vocant, ita accommodatum, vt inferio <lb/>ri parte helices habeat concauas ip&longs;i cochleæ appo&longs;itè admodum <lb/>congruentes; per&longs;picuum &longs;atis e&longs;&longs;e poterit, ip&longs;um B, dum <expan abbr="coclhea">cochlea</expan><lb/>circumuertitur, &longs;uper helices cochleæ eo pror&longs;us modo moueri; <lb/>quo pondus iuxta primam <expan abbr="figurã">figuram</expan> mouebatur: dummodo tylum ap<lb/>tetur, vt docet Pappus in octauo libro; ita &longs;cilicet vt tantùm an<lb/>tè, retrouè axi cylindri æquidi&longs;tans moueatur. <figure id="id.036.01.258.2.jpg" place="text" xlink:href="036/01/258/2.jpg"/></s>

<s id="id.2.1.237.22.1.1.0">Et &longs;i loco tyli, quod helices habet concauas in parte inferiori, con<lb/>&longs;tituatur, vt in quarta figura, cylindrus concauus vt D, & in eius <lb/>concaua &longs;uperficie de&longs;cribantur helices, incidanturq; ita, vt aptè <pb n="123" xlink:href="036/01/259.jpg"/>cùm cochlea congruant (eodem enim modo de&longs;cribentur helices <lb/>in &longs;uperficie concauia cylindri, &longs;icuti fit in conuexa) &longs;i deinde co<lb/>chlea in &longs;uis polis firmetur, &longs;cilicet in &longs;uo axe, circumuertaturq;; <lb/>patet D ad motum circumuer&longs;ionis cochleæ &que;mmadmodum ty<lb/>lum moueri. </s>

<s id="id.2.1.237.22.1.2.0">nec non &longs;i D in EF firmetur, ita vt immobilis ma <lb/>neat, dum circumuertitur cochlea; &longs;uper helices cylindri D, ad <lb/>motum &longs;uæ circumuer&longs;ionis dextror&longs;um, vel &longs;ini&longs;tror&longs;um factæ; <lb/>tùm in anteriorem, tùm in po&longs;teriorem partem mouebitur. </s>

<s id="id.2.1.237.22.1.3.0">cylin<lb/>drus autem D hoc modo <expan abbr="accõmodatus">accommodatus</expan> vulgò mater, &longs;iue cochleæ <lb/>fæmina nuncupatur. <figure id="id.036.01.259.1.jpg" place="text" xlink:href="036/01/259/1.jpg"/></s>

<s id="id.2.1.237.23.1.1.0">Si autem cochleæ (vt in quinta figura) tympanum C dentibus <lb/>obliquis dentatum apponatur, vt docet Pappus in eodem octauo li<lb/>bro; vel etiam rectis; ita tamen con&longs;tructis, vt facilè cum cochlea <lb/>conueniant: &longs;imiliter manife&longs;tum e&longs;t ad motum cochleæ circumuer<lb/>ti etiam tympanum C. </s>

Line 2862

<s id="id.2.1.237.23.1.2.0">& hæc dicitur cochlea infinita, quia & co<lb/>chlea, & tympanum dum circumuertuntur, &longs;emper eodem modo <lb/>&longs;e &longs;e habent. </s><pb xlink:href="036/01/260.jpg"/>

<s id="id.2.1.237.25.1.1.0">Hæc diximus, vt manife&longs;tum &longs;it cochleam in mouendo pondere <lb/>cunei munere ab&longs;q; percu&longs;sione fungi. </s>

<s id="id.2.1.237.25.1.2.0">Illud enim remouet à loco, <lb/>vbi erat; &que;madmodum cuneus remouet ea, quæ mouet, ac &longs;cindit. </s>

<s id="id.2.1.237.25.1.2.0">Illud enim remouet à loco, <lb/>vbi erat; quemadmodum cuneus remouet ea, quæ mouet, ac &longs;cindit. </s>

<s id="id.2.1.237.25.1.3.0"><lb/>omnia enim hæc à cochlea mouentur, &longs;icuti pondus A in &longs;ecun<lb/>da figura, & M in prima. </s>

<s id="id.2.1.237.26.1.1.0">Quoniam autem duplici ratione mouentem cuneum con&longs;iderari <lb/>po&longs;&longs;e o&longs;tendimus, videlicet vt mouet vectibus, vel vt e&longs;t planum <lb/>horizonti inclinatum, dupliciter quoq; cochleam con&longs;iderabimus; <lb/><figure id="id.036.01.260.1.jpg" place="text" xlink:href="036/01/260/1.jpg"/><lb/>& primùm vt vectibus mouet, vt in prima figura circumuertatur <lb/>kF, & perueniat in KP; tunc, &longs;icut dictum e&longs;t, TV erit intra pon<lb/>dera MN. </s>

<s id="N16F44">& &longs;icut con&longs;ideramus vectes in cuneo, eodem quoq; <lb/>modo eos con&longs;iderare po&longs;&longs;umus in cochlea hoc pacto. </s>

Line 2923

<s id="id.2.1.241.2.1.2.0.a">exponatur HIL triangulum orthogonium, ita vt <lb/>HI &longs;it ip&longs;i CG æqualis, & IL duplo perimetri cylindri AB æqua<lb/>lis, & per <emph type="italics"/>L<emph.end type="italics"/>I intelligatur planum horizonti æqui&longs;tans; erit H<emph type="italics"/>L<emph.end type="italics"/><lb/>æqualis CDEFG; & H<emph type="italics"/>L<emph.end type="italics"/>I inclinationis angulus erit. </s>

<s id="id.2.1.241.2.1.3.0">exponatur <arrow.to.target n="note327"/><lb/>&longs;imiliter <emph type="italics"/>X<emph.end type="italics"/>YZ triangulum orthogonium, ita vt XZ ip&longs;i OP &longs;it æ<lb/>qualis, quæ etiam æqualis erit CG, & HI; &longs;itq; ZY cylindri pe<lb/>rimetro tripla, erit XY æqualis OQRSTVP. </s>

<s id="N1712F">diuidatur ZY in <pb xlink:href="036/01/266.jpg"/>

<figure id="id.036.01.266.1.jpg" place="text" xlink:href="036/01/266/1.jpg"/><lb/>tres partes æquales in <foreign lang="greek">g</foreign><foreign lang="el">d</foreign>; erit vnàquæq; Z <foreign lang="greek">g g</foreign><foreign lang="el">d</foreign> <foreign lang="el">d</foreign> Y perimetro cy<lb/>lindri <foreign lang="greek">ab</foreign>æqualis, quæ <expan abbr="etiã">etiam</expan> perimetro cylindri AB æquales erunt; & <lb/>per con&longs;e&que;ns ip&longs;is 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/>duæ HI IL duabus XZ Z<foreign lang="el">d</foreign> &longs;unt æquales, & angulus HIL re<lb/>ctus æqualis e&longs;t angulo XZ<foreign lang="el">d</foreign> recto; erit triangulum HIL trian<lb/>gulo XZ<foreign lang="el">d</foreign> æquale; & angulus HLI angulo X<foreign lang="el">d</foreign>Z æqualis; & <lb/><arrow.to.target n="note328"/>X<foreign lang="el">d</foreign> ip&longs;i HL æqualis. </s>

<s id="id.2.1.241.2.1.5.0">&longs;ed quoniam angulus X<foreign lang="el">d</foreign>Z maior e&longs;t angu<lb/>lo <emph type="italics"/>X<emph.end type="italics"/>YZ; erit angulus HLI angulo <emph type="italics"/>X<emph.end type="italics"/>YZ maior. </s>

Line 2988

<s id="id.2.1.249.3.1.1.0">Ex hoc manife&longs;tum e&longs;t, quomodo datum pon<lb/>dus à data potentia cochlea moueatur. <pb xlink:href="036/01/272.jpg"/>

<s id="id.2.1.249.4.1.1.0">Illud quoq; præterea hoc loco ob&longs;eruandum occurrit; quò plu<lb/>res erunt matricis cochleæ helices, eò minus in pondere mouen<lb/>do cochleam pati. </s>

<s id="id.2.1.249.4.1.2.0">&longs;i enim matrix vnicam duntaxat helicen po&longs;&longs;e<lb/>derit, tunc pondus vt centrum à &longs;ola cochleæ &longs;u&longs;tinebitur helice; <lb/>&longs;i verò plures, in plures quo&que;, ac totidem cochleæ heli<lb/>ces ponderis grauitas di&longs;tribuetur; vt &longs;i quatuor contineat helices, <lb/>tunc quatuor vici&longs;sim cochleæ helices vniuer&longs;o ponderi &longs;u&longs;tinendo <lb/>incumbent; &longs;iquidem vnaquæquè quartam totius ponderis portio<lb/>nem &longs;u&longs;tentabit. </s>

<s id="id.2.1.249.4.1.2.0">&longs;i enim matrix vnicam duntaxat helicen po&longs;&longs;e<lb/>derit, tunc pondus vt centrum à &longs;ola cochleæ &longs;u&longs;tinebitur helice; <lb/>&longs;i verò plures, in plures quoque, ac totidem cochleæ heli<lb/>ces ponderis grauitas di&longs;tribuetur; vt &longs;i quatuor contineat helices, <lb/>tunc quatuor vici&longs;sim cochleæ helices vniuer&longs;o ponderi &longs;u&longs;tinendo <lb/>incumbent; &longs;iquidem vnaquæquè quartam totius ponderis portio<lb/>nem &longs;u&longs;tentabit. </s>

<s id="id.2.1.249.4.1.3.0">quòd &longs;i adhuc plures contineat helices, ponderis <lb/>quoq; totius in plures, at&que; ideo minores portiones fiet di&longs;tri<lb/>butio. </s><pb n="130" xlink:href="036/01/273.jpg"/>

<s id="id.2.1.249.4.1.3.0">quòd &longs;i adhuc plures contineat helices, ponderis <lb/>quoq; totius in plures, atque ideo minores portiones fiet di&longs;tri<lb/>butio. </s><pb n="130" xlink:href="036/01/273.jpg"/>

<s id="id.2.1.249.6.1.1.0">O&longs;ten&longs;um e&longs;t igitur pondus à cochlea moueri <lb/>tamquam à cuneo percu&longs;sionis experte: loco e<lb/>nim percu&longs;sionis mouet vecte, hoc e&longs;t &longs;cytala, &longs;i<lb/>ue manubrio. </s>

<s id="id.2.1.249.7.1.1.0">His demon&longs;tratis li&que;t, quomodo <expan abbr="dat&utilde;">datum</expan> pon<lb/>dus à data potentia moueri po&longs;sit. </s>

<s id="id.2.1.249.7.1.1.0">His demon&longs;tratis liquet, quomodo <expan abbr="dat&utilde;">datum</expan> pon<lb/>dus à data potentia moueri po&longs;sit. </s>

<s id="id.2.1.249.7.1.2.0">quòd &longs;i vecte <lb/>hoc a&longs;&longs;equi volumus; po&longs;&longs;umus & dato vecte da <lb/>tum pondus data potentia mouere. </s>

<s id="id.2.1.249.7.1.3.0">quod quidem <lb/>in nullis ex aliis fieri po&longs;&longs;e ab&longs;olutè contingit: &longs;iue <lb/>&longs;it cochlea, &longs;iue axis in peritrochio, &longs;iue trochlea. </s>

<s id="id.2.1.249.7.1.4.0"><lb/>non enim datis trochleis, neq; dato axe in peri<lb/>trochio, neq; data cochlea, datum pondus à data <lb/>potentia moueri pote&longs;t, cùm potentia in his &longs;em<lb/>per &longs;it determinata: &longs;i igitur <expan abbr="pot&etilde;tia">potentia</expan>, quæ pondus <lb/>mouere debeat, hac minor &longs;it data, nunquam pon<lb/>dus mouebit. </s>

Line 3000

<s id="id.2.1.249.7.1.6.0">quod idem cochleæ contingere po<lb/>te&longs;t, &longs;cilicet datum pondus data cochlea &longs;ine ma<lb/>nubrio, vel &longs;cytala, data potentia mouere. </s>

<s id="id.2.1.249.7.1.7.0">co<lb/>gnita enim potentia, quæ pondus &longs;uper helices <lb/>moueat, po&longs;&longs;umus manubrium, &longs;iue &longs;cytalam ita <pb xlink:href="036/01/274.jpg"/>con&longs;truere, vt data potentia in &longs;cytala eandem <lb/>vim habeat, quam potentia pondus &longs;uper helices <lb/>mouens cùm autem hoc datis trochleis nullo mo <lb/>do fieri po&longs;sit. </s>

<s id="id.2.1.249.7.1.8.0">datum tamen pondus data poten<lb/>tia trochleis infinitis modis mouere po&longs;&longs;umus. </s>

<s id="id.2.1.249.7.1.9.0"><lb/>datum verò pondus data potentia cunei in&longs;tru<lb/>mento mouere, hoc minimè fieri po&longs;&longs;e clarum e&longs;<lb/>&longs;e videtur; non enim data potentia datum pon<lb/>dus &longs;uper planum horizonti inclinatum mouere <lb/>pote&longs;t, neq; datum pondus à data potentia moue<lb/>bitur vectibus &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> aduer&longs;is, &que;mmadmo<lb/>dum in cuneo in&longs;unt; cùm in vectibus cunei pro<lb/>pria, veraq; vectis proportio &longs;eruari non po&longs;sit. </s>

<s id="id.2.1.249.7.1.9.0"><lb/>datum verò pondus data potentia cunei in&longs;tru<lb/>mento mouere, hoc minimè fieri po&longs;&longs;e clarum e&longs;<lb/>&longs;e videtur; non enim data potentia datum pon<lb/>dus &longs;uper planum horizonti inclinatum mouere <lb/>pote&longs;t, neq; datum pondus à data potentia moue<lb/>bitur vectibus &longs;ibi <expan abbr="inuic&etilde;">inuicem</expan> aduer&longs;is, quemmadmo<lb/>dum in cuneo in&longs;unt; cùm in vectibus cunei pro<lb/>pria, veraq; vectis proportio &longs;eruari non po&longs;sit. </s>

<s id="id.2.1.249.7.1.10.0"><lb/>vectium enim fulcimenta non &longs;unt immobilia, <lb/>cùm totus cuneus moueatur. </s>

<s id="id.2.1.249.8.1.1.0">Poterit deinde quis &longs;truere machinas, atq; eas <lb/>ex pluribus componere; vt ex trochleis, & &longs;uc<lb/>culis, vel ergatis, pluribu&longs;uè dentatis tympanis, <lb/>uel quocunq; alio modo; & ex ijs, quæ diximus; fa<lb/>cilè inter pondus, & potentiam proportionem <lb/>inuenire. </s>

<s id="id.2.1.249.9.1.1.0">FINIS. </s><pb xlink:href="036/01/275.jpg"/>

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